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http://mathhelpforum.com/differential-geometry/171321-graded-ideal-differential-forms.html
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1. ## graded Ideal and differential forms
Hello,
i have a proof of a statement, but i don't understand it very well. I have concrete questions about it and it would be very nice, when someone can help me.
Let E be a subbundle of TM, we define the (graded) ideal $J=\bigoplus_{k=1}^n J^k \; in \; \Omega(M)$ as follows $w \in J^k \;<=> w(X_1,...,X_k)=0$ for any sections $X_i$ of E
Claim: J is locally generated by q linearly independent 1-forms:
Pf: Choose a local frame $X_1,...,X_n$ of TM, s.t. $X_1,...,X_{n-q}$ form a frame of E.
There is the dual frame of differential 1-forms $w_1,...,w_n$ of TM* and the linearly independent 1-forms $w_{n-q+1},..,w_n$ clearly generate the ideal J.
Why do they generate the Ideal J????
What does it mean "generate"? the $w_i$ are 1-forms, i.e. sections: $M->TM^*, w_i (p)\in T^{*}_p M$.
and J is the direct sum of k-forms, for k=1,...,n?
Can you please explain it for me?
Regards
2. means any element in J can be expressed in terms of these q forms.
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2016-12-04 06:58:18
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https://mathhelpboards.com/threads/nonstandard-analysis-completeness-of-r-via-every-limited-hyperreal-is-infinitely-close-to-a-real.5075/
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# Nonstandard Analysis: Completeness of R via every limited hyperreal is infinitely close to a real #
#### conscipost
##### Member
In a book I am currently reading, the statement "every limited hyperreal is infinitely close to a real #" is shown to imply the completeness of R, that is that any subset A of R bounded above has a least upper bound. What the author offers to do is introduce this construction: for each natural n, let sn be the least k in the integers so that k/n is an upper bound of A. Then we are to take an unlimited N and let L, an element of R, be infinitely close to sN/N.
Without completeness I'm not sure why sn necessarily exists, can anyone give me some hints? Is it just because once I know the set is bounded above, I can start with an integer greater than this upper bound multiplied by n and "count down" so to say, checking whether each integer less than the last is an upper bound until I find one that is not?
Thanks,
Last edited:
#### Plato
##### Well-known member
MHB Math Helper
In a book I am currently reading, the statement "every limited hyperreal is infinitely close to a real #" is shown to imply the completeness of R, that is that any subset A of R bounded above has a least upper bound. What the author offers to do is introduce this construction: for each natural n, let sn be the least k in the integers so that k/n is an upper bound of A. Then we are to take an unlimited N and let L, an element of R, be infinitely close to sN/N.
Without completeness I'm not sure why sn necessarily exists, can anyone give me some hints? Is it just because once I know the set is bounded above, I can start with an integer greater than this upper bound multiplied by n and "count down" so to say, checking whether each integer less than the last is an upper bound until I find one that is not?
I would like to know the name of the text/author.
I assume that by limited hyperreal that author means finite.
If you can find a copy of James Henle's Infinitesimal Calculus, there is a good discussion on this problem on page 114. Although Henle does not use limited hyperreal much of that discussion it does use many of the same ideas.
#### conscipost
##### Member
I would like to know the name of the text/author.
I assume that by limited hyperreal that author means finite.
If you can find a copy of James Henle's Infinitesimal Calculus, there is a good discussion on this problem on page 114. Although Henle does not use limited hyperreal much of that discussion it does use many of the same ideas.
I'm sorry, I'm new to the topic and didn't think about the possibility of different terms. The book is titled Lectures on the Hyperreals by Robert Goldblatt; this bit is on page 56. Comparing to Henle, limited is actually not finite but rather a # that is bound by two real numbers. infinite is not defined in Goldblatt, and rather infinitesimal and unlimited are used, infinitesimal meaning that the absolute value of the # is less than any positive real #, and unlimited to mean that the abs. is greater that any pos. real #.
One question, do the squares denote anything on page 114-115, or are these just formatting?
Thanks,
#### Plato
##### Well-known member
MHB Math Helper
One question, do the squares denote anything on page 114-115, or are these just formatting?
$$\displaystyle \boxed{N}$$ is the standard part of $$\displaystyle N$$.
You really have to follow any text very closely.
This subject is relatively new, 1964. So there is absolutely no standard notation.
I do not know of that text book.
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2021-10-21 15:32:25
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https://math.stackexchange.com/questions/611745/how-to-prove-continued-fraction-convergents-of-a-number/611842
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# How to prove continued fraction convergents of a number
Let $x=1+\sqrt{3}$. Prove that in pairs the continued fraction convergents of $x$ are $a_n$/$b_n$ < x < $c_n$/$d_n$ where $a_1$ = 2, $b_1$ = 1, $c_1$ = 3, $d_1$ = 1, $a_{n+1}$ = 2$c_n$ + $a_n$, $b_{n+1}$ = 2$d_n$ + $b_n$, $c_{n+1}$ = 3$c_n$ + $a_n$, $d_{n+1}$ = 3$d_n$ + $b_n$.
I am not sure where to start with this one. Help please? :)
EDIT:
I started in on an induction proof. I'm kinda stuck on the final portion.
Let $x=1+\sqrt{3}$. The continued fraction expansion is <2, 1, 2, 1, 2, 1, 2, 1, ...> and the first few convergents are 2, 3, 8/3, 11/4, 30/11, 41/15, 112/41. So we have 2 = 2/1 = $a_1$/$b_1$ because 2 < $1+\sqrt{3}$ and 3 = 3/1 = $c_1$/$d_1$ because 2 > $1+\sqrt{3}$, so we have $a_1$ = 2, $b_1$ = 1, $c_1$ = 3, $d_1$ = 1.
Base case: Let n=1. Then $a_2$ = 2$c_1$ + $a_1$ = 2(3) + 2 = 8, etc. and we get $a_2$/$b_2$ = 8/3 and $c_2$/$d_2$ = 11/4 which is true.
Assume true for n=k (the formulas for $a_n$, $b_n$, etc.)
Prove for n=k+1: $a_{n+1}$ = 2$c_n$ + $a_n$ so $a_{k+2}$ = 2$c_{k+1}$ + $a_{k+1}$ = 8$c_k$ + 3$a_k$
There is where I get stuck
Let $x = [a_0; a_1 ,a_2 , \ldots]$. Find the continued fraction using the common algorithm. You will get $x = [2;\overline{1,2}]$.
Sidenote: If you are confused about the inequalities $a_n/b_n < x < c_n/d_n$, remember that $C_{2k} < C_{2k+1}$, and $C_{2k} < C_{2(k+1)}$ so that $C_0 < C_2 < \ldots < C_n < C_{n-1} < \ldots < C_5 < C_3 < C_1$. This is proven by simply considering $C_i - C_{i-2}$.
Let $C_i = p_i /q_i$ where $$p_0 = a_0, p_1 = a_0 a_1 + 1, p_i = a_i p_{i-1} + p_{i-2}\\q_0 = 1, q_1 = a_1, q_i = a_i q_{i-1} + q_{i-2}.$$ Hence $$p_0 = 2, p_1 = 2 \cdot 1 + 1 = 3\\ q_0 = 1, q_1 = 1.$$ So $$C_0 = \frac{p_0}{q_0} = 2,\\ C_1 = \frac{p_1}{q_1} = 3.$$ This confirms that $a_1 = 2, b_1=1, c_1=3,d_1=1$ in your notation.
$$p_2 = a_2 p_1 + p_0 = 2 \cdot 3 + 2 = 8, q_2 = a_2 q_1 + q_0 = 2 \cdot 1 + 1 = 3 \\ \implies C_2 = 8/3.$$ $$p_3 = a_3 p_2 + p_1 = 1 \cdot 8 + 3 = 11, q_3 = a_3 q_2 + q_1 = 1 \cdot 3 + 1 = 4 \\ \implies C_3 = 11/4.$$
It is easy to check that these results are in line with $n=1$ given by the recursive sequences. Hence the base case of the induction holds. Try to prove the induction.
Induction: $a_n/b_n$ will always correspond to even $n$ in the $C_n$ notation. Hence we need only prove that $a_{n+1} = 2p_n + p_{n-1}$ and $b_{n+1} = 2q_n + q_{n-1}$ for the left-hand side of the inequality $a_n/b_n < x$. But this follows immediately from the recursive equations for $p$ and $q$, as $a_{n+1}=2$ (where this $a$ is the $a$ in the continued fraction). We use these to prove $c_{n+1}$ and $d_{n+1}$. We have $$c_{n+1}=p_{n+2} = a_{n+2}p_{n+1} + p_n = a_{n+2} (a_{n+1} p_n + p_{n-1}) + p_n = p_n(a_{n+2} a_{n+1}+ 1) + a_{n+2} p_{n-1} = 3 c_n + a_n$$ where the $a_n$ after the last equality corresponds to the $a_n/b_n <x$ and the rest to the $a_i$'s in the continued fraction. $d_{n+1}$ is proven in a similar manner.
I must admit I find it terribly confusing dividing up the convergents on either side of $x$, as they are closely linked. Splitting them up into 4 separate sequences seems silly and unnecessary. $C_n$ notation saves a lot of trouble.
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2019-10-23 15:04:42
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https://imogeometry.blogspot.com/2017/10/2003-imo-shortlist-g4.html
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## Τρίτη, 17 Οκτωβρίου 2017
### 2003 IMO Shortlist G4
Let $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$ be distinct circles such that $\Gamma_1, \Gamma_3$ are externally tangent at $P$ , and $\Gamma_2, \Gamma_4$ are externally tangent at the same point $P$ . Suppose that $\Gamma_1$ and $\Gamma_2, \Gamma_2$ and $\Gamma_3, \Gamma_3$ and $\Gamma_4, \Gamma_4$ and $\Gamma_1$ meet at $A, B, C, D$, respectively, and that all these points are different from $P$ . Prove that $\frac{AB \cdot BC}{AD \cdot DC}=\frac{PB^2}{PD^2}$.
posted in aops here
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2018-07-18 06:42:50
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https://itprospt.com/qa/306547/how-do-you-simplify-8sqrt-10
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1
# How do you simplify 8/sqrt 10?
## Question
###### How do you simplify 8/sqrt 10?
How do you simplify 8/sqrt 10?
##### Saved Help Save & Ext Sube On June 30, 2021, K Co. had outstanding 8%, $12,500,000... Saved Help Save & Ext Sube On June 30, 2021, K Co. had outstanding 8%,$12,500,000 face value bonds maturing on June 30, 2026. Interest is payable semiannually every June 30 and December 31. On June 30, 2021, after amortization was recorded for the period, the unamortized bond premium was $52,00... 1 answers ##### Use the Rule of 72 to approximate when an investor can multiply his initial investment 75-fold,... Use the Rule of 72 to approximate when an investor can multiply his initial investment 75-fold, if the interest rate is 15 ½ %.... 1 answers ##### A hot air balloon is filled with 1.19×10^6 L of an ideal gas on a cool... A hot air balloon is filled with 1.19×10^6 L of an ideal gas on a cool morning (11∘C). The air is heated to 111 ∘C.What is the volume of the air in the balloon after it is heated? Assume that none of the gas escapes from the balloon.... 1 answers ##### 15. State the four assumptions that are made for the random error component of the regression... 15. State the four assumptions that are made for the random error component of the regression model... 1 answers ##### A student performing this experiment mistakenly used 6.0 ml of 16 M HNO3 to dissolve 0.18... a student performing this experiment mistakenly used 6.0 ml of 16 M HNO3 to dissolve 0.18 g of solid copper, instead of 4.0ml described in the lab manual. what volume in milliliters of 6.0 M NaOH are required to neutralize the excess acid.... 1 answers ##### EMPERICAL FORMULA WHAT IS THE EMPERICAL FORMULA OF THE SUBSTANCE THAT HAS THEFOLLOWING MASSES88.77 GRAM CARBON,5.61 GRAM OF HYDROGEN.AND 118.26 GRAM OFOXYGEN.(PLEASE SHOW ME ALL WORK)... 1 answers ##### Gastrointestinal Disorders 20 Case Study 4 Name Class/Group Date Group Members INSTRUCTIONS: All questions apply to... Gastrointestinal Disorders 20 Case Study 4 Name Class/Group Date Group Members INSTRUCTIONS: All questions apply to this case study. Your responses should be brief and to the point. Adequate space has been provided for answers. When asked to provide several answers, they should be listed in order of... 1 answers ##### Direct Method Question (2026, Current Period): 2025 23,000 3,000 AR Prepaid Insurance Salaries Payable Sales 2026... Direct Method Question (2026, Current Period): 2025 23,000 3,000 AR Prepaid Insurance Salaries Payable Sales 2026 49,000 15,000 13,000 125,000 20,000 57.000 5,000 63,000 12,000 21,000 Insurance Expense Salaries Expense What is the second number in the cash event direct method)? Cash Event => Cash... 1 answers ##### The following information is available for Lock-Tite Company, which produces special-order security products and uses a... The following information is available for Lock-Tite Company, which produces special-order security products and uses a job order costing system. Saved Required information [The following information applies to the questions displayed below.) The following information is available for Lock-Tit... 1 answers ##### Solve in R Which of the following command correctly sorts the data frame by School in... solve in R Which of the following command correctly sorts the data frame by School in descending order and Expression in ascending order? a. painterABC [order (-xtfrm (painterABC$School), painterABC$Expression), ] b.painterABC [order(-painterABC$School, painterABC$Expression), ] c. painterABC [o... 1 answers ##### Question 2) Draw the chemical structures of the fragments you expect to see in the mi... Question 2) Draw the chemical structures of the fragments you expect to see in the mi of the following compound (6 points)... 1 answers ##### Part B What is the net ionic equation of the reaction of MgSO4 with Pb(NO3)2? Express... Part B What is the net ionic equation of the reaction of MgSO4 with Pb(NO3)2? Express you answer as a chemical equation including phases.... 1 answers ##### Complete the following using MATLAB 3.23 Impedance is related to the inductance, L, and the capacitance,... Complete the following using MATLAB 3.23 Impedance is related to the inductance, L, and the capacitance, C by the following equations For a circuit similar to the one shown in Figure P3.22 assume the following:... 1 answers ##### It is just simple discussion, please don't copy from any website. I need at least 3... it is just simple discussion, please don't copy from any website. I need at least 3 paragraph. Thanks Discussion Richard Turere: My invention that made peace with lions Please watch the Ted Lecture, Richard Turere: My invention that made peace with lions. This talk by a young LINK=====> http:... 1 answers ##### 7. (4 pts) For each of the following solid organic molecules, circle the solvent that would... 7. (4 pts) For each of the following solid organic molecules, circle the solvent that would be the best for re-crystallization. CH3 HN Н3Слл CH3 "CH3 OH HO H2N-NH2 Water Water Water Water 95% Ethanol 95% Ethanol 95% Ethanol 95% Ethanol Hexane Hexane Hexane Hexane... 1 answers ##### Calculate the concentration of each solution in mass percent Part A 142 g KCl in 618 g HO VO A dD ? % mass% Submit... Calculate the concentration of each solution in mass percent Part A 142 g KCl in 618 g HO VO A dD ? % mass% Submit Request Answer Part B 29.3 mg KNO3 in 9.63 g H20 ? mass Request Answer Submit Part C 9.16 g C2H,0 in 78.1 g H,O VOAED mass % Request Answer Submit... 1 answers ##### An individual entire wealth is from one stock. The current value of the stock is$55,...
An individual entire wealth is from one stock. The current value of the stock is $55, and the individual owns one million shares. This individual purchased puts on the stock with an exercise price of$52 to protect his wealth. He bought enough puts to protect his entire holdings of this stock. The e...
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2023-03-27 13:01:48
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https://www.gradesaver.com/textbooks/math/algebra/introductory-algebra-for-college-students-7th-edition/chapter-5-section-5-1-adding-and-subtracting-polynomials-exercise-set-page-348/3
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## Introductory Algebra for College Students (7th Edition)
$x^{3}$ - 2x there are two terms, the polynomial is a binomial The degree of a polynomial is the greatest degree of all the terms of the polynomial so degree of the first term is 3
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2018-06-24 15:06:24
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https://academic.oup.com/gji/article/doi/10.1111/j.1365-246X.2006.02978.x/559970/A-review-of-the-adjoint-state-method-for-computing
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Summary
Estimating the model parameters from measured data generally consists of minimizing an error functional. A classic technique to solve a minimization problem is to successively determine the minimum of a series of linearized problems. This formulation requires the Fréchet derivatives (the Jacobian matrix), which can be expensive to compute. If the minimization is viewed as a non-linear optimization problem, only the gradient of the error functional is needed. This gradient can be computed without the Fréchet derivatives. In the 1970s, the adjoint-state method was developed to efficiently compute the gradient. It is now a well-known method in the numerical community for computing the gradient of a functional with respect to the model parameters when this functional depends on those model parameters through state variables, which are solutions of the forward problem. However, this method is less well understood in the geophysical community. The goal of this paper is to review the adjoint-state method. The idea is to define some adjoint-state variables that are solutions of a linear system. The adjoint-state variables are independent of the model parameter perturbations and in a way gather the perturbations with respect to the state variables. The adjoint-state method is efficient because only one extra linear system needs to be solved.
Several applications are presented. When applied to the computation of the derivatives of the ray trajectories, the link with the propagator of the perturbed ray equation is established.
Introduction
One of the important tasks in data processing consists of determining model parameters from observed data. These tasks can be formulated as inverse problems, namely as the minimization of a functional, for instance the least-squares misfit between synthetic and observed data. In geophysics, this includes tomography, migration/inversion and automatic velocity analysis. When a local (descent) optimization technique, such as the conjugate gradient method, is used, the gradient of the functional is required, Gauthier et al. (1986), Liu & Bleistein (2001) and Mora (1989). The efficiency of the method greatly depends on the accuracy and efficiency of the computation of this gradient. With physical problems, the functional depends on so-called state variables. These state variables are the variables computed from the state equations, namely the equations that define the problem, sometimes called forward equations. For example, in the tomography problem, the state equations can be the ray equations, and the state variables the spatial coordinates and the slowness vectors describing the ray trajectories. The definition of the state variables depends on the mathematical formulation of the physical problem. The state equations depend on the model parameters. For the tomography problem this can be the background velocity (or slowness). The functional depends on those model parameters mainly through the dependency on the state variables.
The gradient of the functional, which depends on a set of state variables solutions of the forward equations, can be obtained with (a set of) the Fréchet derivatives of the state variables. The Fréchet derivatives are the derivatives of the state variables with respect to the model parameter. For instance, for the tomography problem described earlier, these derivatives are the derivatives of the spatial coordinates and the slowness vectors with respect to the slowness background. This gives the so-called Jacobian or sensitivity matrix. The Jacobian matrix can be used to linearize the functional, and the minimization problem can be solved by successively solving linearized problems using linear optimization techniques. However, the computation of the Fréchet derivatives can be expensive.
If non-linear optimization techniques, such as the non-linear conjugate method, are used, only the gradient of the functional may be needed, Gill et al. (1981). In the 1970s, a method based on the adjoint state has been introduced in the theory of inverse problems by Chavent (1974) to efficiently compute the gradient of a functional without the Fréchet derivatives. This approach originated from control theory, Lions (1972). Several authors in geophysics have applied this method, for instance, Lailly (1983), Bécache (1992), Chavent & Jacewitz (1995), Plessix et al. (1999) and Shen et al. (2003). The goal of this note is to review this method that is well known in the numerical community, to give a recipe for applying it based on an augmented functional also called associated Lagrangian, and to describe several examples demonstrating its practical use.
The adjoint-state method is a general method to compute the gradient of a functional that depends on a set of state variables, which are solutions of forward equations. The adjoint-state variables are the solutions of an adjoint linear system and can be seen as variables which gather a global measure of the perturbation of the problem with respect to the state variables. Numerically this approach is attractive because only one extra linear system needs to be solved and often the computation of the gradient with respect to the model parameters is equivalent to one or two evaluations of the forward modelling. This cost is often almost independent of the number of model parameters, which is not always the case when the Fréchet derivatives are computed. However the adjoint-state method does not provide the sensitivity of the solution to errors. For that, the Fréchet derivatives are needed or a Monte Carlo type of methods with a large number of forward computations is required.
The outline of the paper is the following. In a first section the adjoint-state variables are introduced from the perturbation theory. Then based on an augmented functional a recipe to systematically define the adjoint-state equations is described. In the three next sections, examples are given. The first one is the least-squares migration, Tarantola (1987). This is almost a school example. The functional is the least-squares misfit between the synthetics and the measured reflection seismic data. The adjoint states correspond to the backpropagated field and the gradient of this least-squares misfit is a migration operator, Lailly (1983) and Tarantola (1984). The second example is the computation of the gradient of the differential semblance optimization (DSO) functional, Symes & Carazzone (1991) and Shen et al. (2003). This example shows the power of the adjoint-state technique. The third example is the stereotomography, Billette & Lambaré (1998) and Lambaré et al. (2004). The link between the adjoint-state variables and the propagator of the differential equation defining the ray trajectory perturbations, Cervený (2001) and Farra & Madariaga (1987), is established.
Method
The goal of this section is to explain the adjoint-state method for computing the gradient of a functional, J(m), when J depends on u(m). J is defined with the functional, h, by
(1)
The state variables, u, satisfy the state equations defined with the mapping, F,
(2)
F is also called the forward problem or forward equation. m is the model parameter and belongs to the model parameter space M. M is a real space in this article. u belongs to the state variable space, U. U is a real or complex space. A state variable, u, is a physical realization if F(u, m) = 0. F is a mapping from U × M to U. In order to distinguish between a physical realization and any element of U, the elements of U are denoted by ũ.h is a functional from U × M to R, the real space, whereas J is a functional from M to R. Synthetic data are generally a subset of the state variables. It is assumed that h, F, and J are at least continuously differentiable and u(m) is uniquely defined and continuously differentiable.
A simple example is the linear case and , with d the observed data. This corresponds to the simple least-squares misfit with a linear forward problem. The physical realization is defined by u(m) = Am, with A a linear operator, in the discrete case a rectangular matrix.
The adjoint-state method from the perturbation theory
A perturbation, δm, of the model parameter, m, induces a perturbation, δu, of the physical realization, u, and a perturbation δJ of the error functional, J. u+δu should be a physical realization with the model parameter m + δm. Therefore, to the first order:
(3)
Since F(u, m) = 0, the first order development gives:
(4)
For the linear case this gives δu = Aδm.
The first order development of J gives:
(5)
where 〈,〉U is the scalar product in U.
For the simple least-squares misfit, δJ = 〈ud, δuU.
Assuming that for any model parameter m of M there exists a unique solution u of U, u + δu is the unique solution of F(u+δu, m+δm) = 0. Therefore, at the first order, δu is the unique solution of eq. (4) and can be written with the inverse of . This gives:
(6)
(* denotes the adjoint). For the linear example with the least-squares misfit, this gives δJ = 〈ud, Aδm〉 since in this case; I is the identity operator.
In the second line of eq. (6), the terms that do not depend on the perturbation δm have been gathered, the idea is to avoid the computation of the Fréchet derivatives, , because this can be expensive. Let us now define λ by
(7)
(8)
For the linear example with the least-squares misfit this simply gives λ = ud and δJ =〈ud, AδmU.
λ belongs to the dual space of U. It is called the adjoint-state variable and eq. (7) is the adjoint-state equation. This is a system of linear equations. The linear operator is the adjoint of the operator formed by the derivatives of the state equations (the mapping F) with respect to the state variables. The right-hand side consists of the derivatives of the functional, h, with respect to the state variables. In a sense the adjoint states gather the information on the perturbations of the state variables, viewed as independent variables. The computation of with eqs (7) and (8) is called the adjoint-state method.
The gradient of J can be computed either via the Fréchet derivatives of u with eqs (4) and (5) or via the adjoint-state method with eqs (7) and (8). The important differences between the two approaches are related to eqs (4) and (7). Indeed, on one hand, eq. (7) is independent of δm and needs to be solved only once. On the other hand, the right-hand side of eq. (4) depends on δm and this equation needs to be solved for each perturbation to obtain , namely M times if M is the number of elements in m. The computational time of the adjoint-state method is often almost independent of M, because the time to compute eq. (8) is often negligible compared with the time to solve eq. (7). This makes this approach very efficient. eq. (7) depends on the adjoint of evaluated at (u, m), this means that u should be completely known before solving it.
The adjoint-state equations can also be obtained with the use of an augmented functional, also called associated Lagrangian.
A recipe with the augmented functional
Let us define the augmented functional, , from U×UM to R (U* is the dual of U) by:
(9)
where is any element of U* and, therefore, does not depend on m, as ũ is any element of U.
u is a physical realization, therefore, F(u, m) = 0, and for any
(10)
and since is independent of m,
(11)
We can then choose λ in U* such that:
(12)
This equation is identical to eq. (7) and is the adjoint-state equation. With this choice we retrieve the result of eq. (8).
(13)
can be also viewed as the Lagrangian associated with the minimization problem: find the minimum u of h(ũ,m) under the constraint F(u, m) = 0. The theory of optimization with equality constraints, Ciarlet (1989), tells us that u is the minimum, if (u, λ) is a saddle point of .λ are called the Lagrange multipliers. At the saddle point the derivatives of are equal to 0. the derivatives of with respect to ũ and are:
(14)
Therefore, gives the state equations and 0 gives the adjoint-state equations. And as seen previously. Notice again that when deriving with respect to m,ũ and are independent of m.
This link with the optimization theory is not needed to apply the adjoint-state method. For those familiar with this theory, it helps to recall the method. As in the optimization theory with equality constraints where one scalar Lagrange multiplier is associated with each scalar equation defining the constraints, one scalar adjoint state is associated with each scalar equation defining the mapping F in the augmented functional.
The computation of the gradient with the adjoint states can be summarized in the following recipe when u has been found from F(u, m) = 0:
• (i)
Build the augmented functional (associated Lagrangian) . , a functional of independent variables ũ, and m is defined by
(15)
if is composed of N scalar equations, is a vector with N components, since at each scalar equation of F an adjoint state is associated, and is defined by:
(16)
For the linear case with the least-squares misfit .
• (ii)
Define the adjoint-state equations. The adjoint-state equations are simply defined by , where the derivatives are evaluated at the point (u, λ). This gives
(17)
or
(18)
The solution of this system determines the adjoint state, λ. For the linear case with the least-squares misfit λ = ud.
• (iii)
Computation of the gradient of J. The gradient of J consists of the derivatives of with respect to m:
(19)
or
(20)
To compute the derivative of the augmented functional, we recall that ũ and are independent of m.
For the linear case with the least-squares misfit .
If u has complex values, since J(m) is a real, the real part should be taken in the right-hand side term of eqs (19) and (20).
The linear example with the least-squares misfit is a trivial example. In the next sections more complicated examples are described.
Least-Squares Migration
In this section, we formulate the migration as an inverse problem. The problem consists of minimizing with respect to the square of the slowness, the least-squares misfit between the synthetics, obtained by solving the wave equation, and the recorded (observed) reflection seismic data. The minimization of J should give the exact slowness. Unfortunately, in practice, J has many local minima, and a gradient optimization will only provide the best perturbation of the initial model inside a certain basin of attraction. This basin is generally not the basin of the global minimum, Gauthier et al. (1986). This is the reason why this problem is called the least-squares migration problem in this paper.
This example is a good example to understand how the adjoint-state method can be applied. It also allows us to redemonstrate that the gradient of J is a migration. This was discovered in the 1980s, Lailly (1983) and Tarantola (1984).
We will develop the computation for multiple sources and multiple receivers, first in the frequency domain because it is simple, then in the time domain.
Frequency domain
In frequency domain, the wave equation operator reads: L = – ω2σ2−Δ, with σ the slowness. Note that the dependency on the spatial coordinates, x, is not written. The finite-difference discretization of Lus = fs with given boundary conditions leads to a complex linear system Marfurt (1984):
(21)
A is a complex matrix of size n by n, where n is the total number of discretization points of the earth model. fs, a complex vector of n elements, represents the source function at the source point s. us, a complex vector of n elements, corresponds to the pressure field due to the shot at s. The model parameter, m, is a vector of M elements, and represents the values of the squared slowness at the discretization points.
The least-squares functional is
(22)
ds,r are the data recorded at the receiver position r due to the source fs. Ss,r is the restriction matrix onto the receiver r of the shot s.
The augmented functional reads, with and (the dependence on x is not written, but and depend on the space variables x):
(23)
where 〈,〉x is the scalar product in Cn. As are complex vectors of n elements, since the forward system, eq. (21), contains n scalar equations.
The derivative of with respect to evaluated at gives:
(24)
The adjoint state is defined by :
(25)
There is one adjoint system per shot and per angular frequency.
The matrix A propagates the shot into the earth and us is the incident field originating at s. The adjoint of A propagates backward its source term, Lailly (1983). The source term, the right-hand side of eq. (25), is the sum over the receivers of the shot s of the residual between the synthetics and data. λs is then the backpropagation of the residual field.
(26)
The gradient of J is a vector of M elements. If we impose that m is discretized on the same grid as us, M = n. Outside the boundary points is equal to −ω2. At the discretization point x, we obtain
(27)
Up to a multiplication factor, the gradient is similar to a migrated image and the formula is kinematically similar to the imaging principle, Clearbout (1985). A demonstration of this result without the adjoint-state method can be found in Plessix & Mulder (2004).
Time domain
We here develop the same approach but in time domain. The application of the adjoint-state method is slightly more complicated because of the initial boundary conditions.
The wave operator is . With the initial boundary conditions, the pressure field us due to the source fs satisfies:
(28)
us and fs depend on the time and on the spatial coordinates.
(29)
T is the recording time. Ss,r is the restriction operator onto the receiver position, it depends on the spatial coordinates. The model parameter is the squared slowness, m = σ2.
In the time domain us is real. For simplicity, we don't mention the spatial boundary conditions.
We associate the adjoint states and with the initial boundary conditions, and with the wave equation. The augmented functional is defined by:
(30)
with the real scalar product in the coordinate space.
After two integrations by part:
(31)
With eqs(30) and (31) we can now compute the derivatives with respect to and evaluate them at (u, λ) to obtain the adjoint-state equations:
(32)
(T denotes the transpose.)
The gradient of J at the point x is
(33)
The adjoint states, µ0s and µ1s do not play a role in the gradient of J. We can ignore them.
The system (32) has final boundary conditions. To solve it the computation is done backwards from T to 0. To give a physical sense to the adjoint state and to interpret the integral, eq. (33), a new adjoint state, qs, is defined by a change of variables in the time axis:
(34)
(35)
qs satisfies the same wave equation that us but with a different source term. Ss,rus(Tt) – ds,r(Tt) is the residual, the difference between the synthetics and the recorded data, in reverse time. eq. (35) propagates the residual into the earth starting from the final time. qs is called the backpropagated field of the residual. The gradient of J now reads:
(36)
This result has been demonstrated by Lailly (1983).
Shot-based Differential Semblance Optimization
As explained in the introduction of the previous section, the least-squares formulation is not satisfactory to retrieve the long wavelength components of the velocity model (background) from reflection seismic data, because the least-squares misfit as a function of the background has many local minima. To reformulate the problem and obtain a larger basin of attraction for the global minimum, an idea is to exploit the fact that in the reflection seismic data the earth is seen through different angles of incident. If the background velocity is correct, the pre-stack migration of the data should give the same images, Al Yahya (1989). If the pre-stack migration gives different earth structures, this means that the background slowness used in the migration is erroneous. Several mathematical formulations of this idea have been proposed in the last 20 years, among them Chavent & Jacewitz (1995), Clément (2001), Plessix et al. (2000) and Symes & Carazzone (1991). In order to compute the gradient of the reformulated cost functions, the authors generally use the adjoint-state formulation because it is the most systematic method, without forgetting that they are mainly mathematicians.
As an example, I will describe the gradient computation of the DSO functional introduced in Symes & Carazzone (1991) for a common shot-based approach. The principle is to migrate each shot individually and then to differentiate the pre-stack migrated result with respect to the shot position for fixed points in the migrated images. If the derivative with respect to the shot position of the pre-stack migrated data is zero, it means that the migrated images are independent of the shot position, that is, of the angle of incident and that the background is correct. To obtain a global formulation, the DSO functional is used as a regularization of the least-squares functional.
Using a finite-difference scheme in frequency-domain, eq. (21), the incident wavefield, ui, due to the source function fi located at the shot position i satisfies:
(37)
The dependency on the spatial coordinates is not explicitly written to simplify the notation. The synthetics at the angular frequency, ω, are Si,jui, with Si,j the restriction operator onto the receiver, j, of the shot, i.
To compute the migration, we introduce the backpropagated field, vi, defined by:
(38)
with di,j the measured seismic data due to the shot i recorded at the receiver j.
The shot migrated image, ri, is then defined by:
(39)
Here we abuse the notation and eq. (39) means , where x is a discretization point.
(40)
ns is the number of shots. α1 and α2 are the weights of the least-squares functional (the first term) and the DSO functional (the second term). The real part of ri+1ri is taken in the DSO functional because ri is a complex number and only the real part corresponds to the migrated image.
We recall that m is the squared slowness at the discretization point and all the state variables are differentiable functions with respect to m. The goal is to compute the gradient of J with respect to the squared slowness.
The forward equations, eqs (37), (38) and (39), depend on the state variables ui, vi and ri. ui, vi, ri are discretized on the same grid, therefore, ui, vi, ri belong to Cn, with n the number of discretization points. To define the augmented functional, we associate, for each shot with eq. (37), with eq. (38) and with eq. (39). and belong to Cn because eqs (37) or (38) or (39) define n scalar equations. These quantities depend on ω and x. The augmented functional reads with the state variables, , and the adjoint-state variables, :
(41)
The adjoint states λu, λv and λr are obtained by taking the derivatives of with respect to equal to zero at the point (u, v, r, λu, λv, λr, m):
(42)
The gradient of J is obtained by
(43)
since outside the boundary points. Notice that 〈λui(ω), ui(ω)〉x is a vector, the scalar product in x is taken per component i, since we should have written λui(ω, x) and ui(ω, x).
This application shows the numerical interest of the adjoint-state method with the augmented functional. Indeed, a systematic use of the method automatically produces the result. Using the perturbation as described in the first section can lead to the same result, but its application is a bit more difficult and cumbersome, as shown in the Appendix. The physical interpretation of the adjoint states is difficult to find. We can notice that λr is just the perturbation with respect to r of the DSO functional, λu satisfies the adjoint wave equation and λv the wave equation.
Stereotomography
The last example describes an application based on the ray equations. The functional depends on the traveltimes and on the other ray-based parameters. The derivatives of the traveltimes with respect to the velocity are efficiently computed by integrating the slowness perturbations along the ray. There is no real gain to introduce the adjoint states when only the derivatives of the traveltimes with respect to the velocity parameters are required, because the adjoint-state method does not give a more efficient algorithm. The derivatives of the ray trajectories can be evaluated from the paraxial ray equations and the propagator associated with this linear differential system, Cervený (2001) and Farra & Madariaga (1987). When the functional depends on the ray trajectories, the adjoint-state method provides a faster approach to the computation of the gradient of the functional. This case is illustrated with the stereotomography functional.
The purpose of the stereotomography, as described in Billette & Lambaré (1998), is to retrieve the velocity background not only from traveltimes picked on the seismic data but also from slopes of the locally coherent events in the common source and common receiver gathers. This approach differs from the classic traveltime tomography because the picks are interpreted independently from each other without any association to a given interface and they can represent either reflection or refraction events. The (observed) data are a set of source positions, xs, receiver positions, xr, two-way traveltimes, Tmsr, the slopes, ps, at the source locations, and the slopes, pr, at the receiver locations.
Following Billette & Lambaré (1998), the model parameters are xd the subsurface reflection points, θs and θr the take-off angles of the rays going towards the source, xs, and towards the receiver, xr, Ts and Tr the traveltimes along the rays from xd towards the source and the receiver and the parameters (vk) defining the continuous velocity field, v. The integration parameter along the ray is the time, t. The ray equations are:
(44)
With the ray equations becomes
(45)
ya is computed from t = 0 to t = Ta. The subscript a represents s or r. y0 is the function defining the initial conditions. The error functional is
(46)
with . In eq. (46), cT is a scalar coefficient and Cs and Cr are two diagonal matrices.
After the integration by part of the terms , the augmented functional reads
(47)
is a functional of the state variables, , the adjoint-state variables, , and the model parameters, Ts, Tr, θs, θr, xd and (vk). The derivatives with respect to the state variables, , give the adjoint-state equations:
(48)
with .
The derivatives of the augmented functional with respect to the model parameters correspond to the derivatives of Jsr with respect to the model parameters. This yields
(49)
For the computation of the derivatives with respect to Ta, we have used the fact that the ray equations are satisfied at Ta.
A more traditional approach to compute the gradient is to first determine the Fréchet derivatives. In this case, this means the derivatives of Ts, Tr, ys(Ts), and yr(Tr) with respect to Ts, Tr, xd, θs, θr, and (vk). A usual approach to compute those derivatives is to use the paraxial ray equations Cervený (2001) and Farra & Madariaga (1987). The perturbation δya of the rays ya is obtained from the propagator Pa defined by
(50)
This gives, Billette & Lambaré (1998)
(51)
and the Fréchet derivatives are
(52)
The other Fréchet derivatives are equal to 0.
The derivatives of Jsr are then simply
(53)
with ν equals to vk, xd, θs, or θr and
(54)
Relation between adjoint states and propagator
From eqs (49), (52) and (53), we deduce that
(55)
In fact, the propagator PTa satisfies
(56)
PTa is the propagator of the adjoint-state differential equation with a final condition, eq. (48). The adjoint state, λa, is then equal to (Ca is a diagonal matrix):
(57)
The main difference between the two approaches lies in the adjoint-state system (eq. 48) and the propagator system (eq. 50). Whereas the first one is a vectorial system, the second is a matrix system. This means that the adjoint-state system is d times smaller than the propagator system, with d = 4 in 2-D problems and d = 6 in 3-D problems. The adjoint-state method is then roughly d times faster. However, the adjoint-state method does not provide the Jacobian matrix, but only the gradient of J and the Jacobian may be used to determine the sensitivity of the solution to errors. The minimization should rely on non-linear optimization techniques, such as non-linear conjugate or quasi-Newton methods, Gill et al. (1981). In Billette & Lambaré (1998), the authors solve the non-linear optimization problem, by successively solving the linear problems defined by the Jacobian matrices.
Conclusion
The adjoint-state method for the gradient computation of a functional has been reviewed. The technique applies when the functional depends on the model parameters through a set of state variables, solutions of forward equations. The method consists of the computation of one unique extra linear system. The linear operator is formed with the adjoint of the operator defined by the derivatives of the forward model with respect to the state variables and the second member consists of the derivatives of the functional with respect to the state variables. The adjoint-state variables are the solution of this linear system. In a sense, they gather the information of the perturbations with respect to the state variables, assuming that the state variables are independent variables. Since this linear system is independent of the derivatives with respect to the model parameters, the adjoint states have to be computed only once, making the method numerically very efficient. The gradient of the functional with respect to the model parameters is now simply the scalar product between the adjoint states and the derivatives of the forward model with respect to the model parameters.
To form this extra linear system (the adjoint-state system) a recipe based on an augmented functional has been reviewed. This provides a systematic approach. The main step in the definition of this augmented functional is to consider the state variables and the adjoint-state variables as independent variables.
Several examples have been described to show the power of this approach. For the complicated example with the DSO functional, the adjoint-state equations have been derived from perturbation theory. This shows that the use of the augmented functional is not strictly necessary but simplifies the approach.
When the forward equations are the ray equations, the transpose of the propagator of the perturbed ray equations is the propagator of the adjoint equations. The adjoint-state method is computationally more efficient because the adjoint-state system is a vectorial system whereas the system of the propagator is a matrix system. Nevertheless, the adjoint-state method only gives the gradient, and not the Fréchet derivatives. The optimization problem should be solved with a non-linear optimization method, such a quasi-Newton or non-linear conjugate gradient technique.
Acknowledgment
I would like to thank Gilles Lambaré from Ecole des Mines de Paris and Colin Perkins from Shell for their comments on the manuscript.
Appendices
Appendix: Dso Gradient with Perturbation Approach
In this appendix, we retrieve the gradient of the DSO function directly from the perturbation approach without the help of the augmented functional. If the model, m, is perturbed by δm, the wavefields, ui, are perturbed by δui, the backpropagated wavefields, vi, are perturbed by δvi, the migrated images, ri, are perturbed by δri, and the functional J by δJ. From eq. (40) we obtain
(A1)
(ns is the number of shots, i is the shot index and 〈,〉 is the scalar product in the data space, 〈,〉x is the scalar product in spatial coordinate space.)
The perturbations of the state variable equations, eqs (37), (38) and (39), gives
(A2)
The dependency on the spatial coordinates and the angular frequency are not written.
The complicated part consists in defining the adjoint-state equations. However with some experience, this is possible. For this example, we first rewrite the second term of eq. (A1)
(A3)
with r0 = r1 and . And we define
(A4)
Replacing δri by its value gives:
(A5)
We then gather the terms depending on δui and the terms depending on δvi:
(A6)
with Re(〈−λriu*i, δv*ix) = Re(〈−uiλr*i, δvix) and eij = S*ij (Sijuidij).
By replacing δvi by its values in the second term of eq. (A6) we obtain
(A7)
By replacing δui by its value in the first term of eq. (A6) we obtain
(A8)
We can now define
(A9)
The equations eqs (A4) and (A9) are the adjoint-state equations. They are the same that the ones obtained with the use of the augmented functional in the section on the DSO function, however the derivation is less obvious and systematic in this appendix.
(A10)
And with , we obtain:
(A11)
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2017-02-23 07:38:30
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https://hurmaninvesterarsrcy.firebaseapp.com/81449/43166.html
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# The full report PDF - Arise
ALLT om Resultat före Ränta, Skatt, Avskrivningar och
786. 776 Adj. EBITDA margin. 8,1%. 7,7%. 7,9%. 7,8% For the calculation of each earn-out payment, a sliding.
This is done by calculating the free cash flow with values derived from the case 7 Earnings before interest, taxes, depreciations and amortizations, EBITDA 7 Earnings per share, operating income and EBITDA continue to recover and benefit from higher EBITDA is calculated in the following manner:. Saint Petersburg, Russian Federation. Built analytical reports (MS PowerPoint, MS Excel). Was involved in changing of EBITDA calculation model. Aggregated Calculate the EBITDA margin 2019.
We'll Show You How to Calculate EBITDA for Your Business EBITDA stands for Earnings Before Interest, Taxes, Depreciation 21 Apr 2019 EBITDA is a more stable and cleaner measure of operating performance as compared to net income because it ignores the effect of accounting 13 Dec 2018 How to Calculate EBITDA. EBITDA is calculated by adding interest, taxes, depreciation, and amortization back to net income. And the net income 13 Sep 2010 Starting in 2007, I reported improper EBITDA calculations by Patrick Byrne finally changed his company's EBITDA calculation to comply with 5 Apr 2019 EBITDA is an acronym that stands for "earnings before interest, tax, depreciation, and amortisation".
The example below illustrates the impact of earnings adjustments to an EBITDA calculation: As the example demonstrates, normalizing your financial statements eliminates discretionary, nonrecurring and unusual items, and helps increase your business value. Debt to EBITDA is a very good indicator that gauges a business’s ability to pay back debt, but it still has its own flaws. Debt to EBITDA Ratio Calculator.
### ALLT om Resultat före Ränta, Skatt, Avskrivningar och
Here are some of the financial calculators that are The calculation of EBITDA is net, since the name describes what enters the calculation. In general, SDE has to calculate the value of small businesses, while We calculate adjusted EBITDA margin by dividing adjusted EBITDA by adjusted revenue. Adjusted Net Income. We define adjusted net income På svenska översätts den till ebitda före räntor, skatter och avskrivningar. The cookie is ebitda to calculate visitor, session, ebitda data and keep track of site av F Mountassir · 2019 — Table 3: Estimation values for calculation of operational expenditures in the calculating the EVA of the project was to define the annual EBITDA in order to. av N Borshell · 2010 · Citerat av 5 — The most appropriate profit definition to use is that of EBITDA, earnings before Table 1 Calculated 25 per cent royalty rates for leading EBITDA, 52, 81, 127, 205, 332, 431, 572, 774. EBITDA margin (%), 27,5, 29,8, 33,9, 38,5, 47,9, 44,9, 45,3, 47.
In general, SDE has to calculate the value of small businesses, while We calculate adjusted EBITDA margin by dividing adjusted EBITDA by adjusted revenue. Adjusted Net Income. We define adjusted net income På svenska översätts den till ebitda före räntor, skatter och avskrivningar. The cookie is ebitda to calculate visitor, session, ebitda data and keep track of site av F Mountassir · 2019 — Table 3: Estimation values for calculation of operational expenditures in the calculating the EVA of the project was to define the annual EBITDA in order to. av N Borshell · 2010 · Citerat av 5 — The most appropriate profit definition to use is that of EBITDA, earnings before Table 1 Calculated 25 per cent royalty rates for leading EBITDA, 52, 81, 127, 205, 332, 431, 572, 774. EBITDA margin (%), 27,5, 29,8, 33,9, 38,5, 47,9, 44,9, 45,3, 47. EBIT adj, 44, 62, 102, 177, 265, 335, 467, 661.
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2019-12-17 · EBITDA = net profit + interest + taxes + depreciation and amortization \begin{aligned} &\text{EBITDA}=\text{net profit} + \text{interest} + \text{taxes} + \text{depreciation and amortization Se hela listan på wikihow.com Se hela listan på wallstreetmojo.com Those anticipating a sale may also need to calculate it on an ad hoc basis for potential buyers. The two EBITDA formulas are: Method #1: EBITDA = Net Income + Interest + Taxes + Depreciation + Amortization. Method #2: EBITDA = Operating Profit + Depreciation + Amortization. The two formulas end up at the same number. EBITDA = EBIT + Depreciation + Amortization. Earnings before interest and taxes (EBIT) is a measurement that is commonly employed in accounting and finance as an indicator of a company's profit. It includes all expenses except interest and any income tax expenses.
EBITDA. 169 For the calculation of each earn-out payment, a sliding. Adj. EBITDA. 200. 190. 786.
Jonas sjöstedt tatuering
2019-08-15 · EBITDA is used in relative valuation in the calculation of enterprise value multiples, in other words, where you are comparing companies in the same industry. Be careful with asset-intensive industries where capex is a key value driver, in these cases we often calculate EBITDA – capex. EBITDA tends to play a significant role when it comes to gauging a company’s financial success. Even though it cannot be considered a potent parameter to measure a company’s overall profitability, it is a reliable indicator of a business’s operating performance.
It is a non- GAAP calculation based on data from a company's Calculation of EBITDA Under IFRS. Accounting. EBITDA stands for Earnings Before Interest, Tax, Depreciation, and Amortization.
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### Interim Report January – March 2019: Storytel AB publ
2019-12-17 · EBITDA = net profit + interest + taxes + depreciation and amortization \begin{aligned} &\text{EBITDA}=\text{net profit} + \text{interest} + \text{taxes} + \text{depreciation and amortization Se hela listan på wikihow.com Se hela listan på wallstreetmojo.com Those anticipating a sale may also need to calculate it on an ad hoc basis for potential buyers. The two EBITDA formulas are: Method #1: EBITDA = Net Income + Interest + Taxes + Depreciation + Amortization. Method #2: EBITDA = Operating Profit + Depreciation + Amortization. The two formulas end up at the same number. EBITDA = EBIT + Depreciation + Amortization.
## The full report PDF - Arise
It is a non- GAAP calculation based on data from a company's Calculation of EBITDA Under IFRS. Accounting. EBITDA stands for Earnings Before Interest, Tax, Depreciation, and Amortization. It's a popular measure and is With EBITDA, factors like debt financing as well as depreciation, and amortization expenses are stripped out when calculating profitability. Also, EBITDA shows a EBITDA is relatively straightforward to calculate. You take a company's net income, which is revenue minus cost, and then add back extraneous factors such as 23 Aug 2018 EBITDA is net income with interest, taxes, depreciation and amortization added back, and is used as a calculation for determining the cash flow (1) During 2017, Hyster-Yale recognized $19 .8 million of equity income from its financing joint venture and$38 .2 million of income tax expense as a result of EBITDA template helps you distinguish between EBIT and EBITDA calculations.
00:01:29. And we saw in the last EBIT-marginal vid början av 2023. 2021. Positivt fritt kassaflöde. och stark förbättring under 2022.
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2022-08-09 07:24:32
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https://packpeople.com/hasogngh/quantum-hall-effect-review-article-54ac38
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Because of the topological constraint, the Fermi arc at a single surface has an open Fermi surface, which cannot host the quantum Hall effect. The use of the quantum Hall effect was reviewed for the precise measurement of electrical resistance. As a result, there are no Landau levels, edge states, or quantum Hall effect on one surface. There may also be a trivial quantum Hall effect on a single surface. and J.W. Jun Ge, Yanzhao Liu, Jiaheng Li, Hao Li, Tianchuang Luo, Yang Wu, Yong Xu, Jian Wang, High-Chern-number and high-temperature quantum Hall effect without Landau levels, National Science Review, Volume 7, Issue 8, August 2020, Pages 1280–1287, https://doi.org/10.1093/nsr/nwaa089. Since then, Haldane proposed the QHE without Landau levels, showing nonzero Chern number | C | = 1, which has been experimentally observed at relatively low temperatures. Figure 3c and f displays the color plot of Ryx in s2 and s3 as a function of the temperature and magnetic field at Vbg = 6.5 V and 8 V, respectively. In this work, the MnBi2Te4 flakes were mechanically exfoliated from high-quality MnBi2Te4 single crystals. But two surfaces can support a complete cyclotron motion and the quantum Hall effect. The discovery of the quantum Hall effect (QHE) marked a turning point in condensed-matter physics. This work was supported by the Guangdong Innovative and Entrepreneurial Research Team Program (2016ZT06D348), the National Key R&D Program (2016YFA0301700), the National Natural Science Foundation of China (11574127), and the Science, Technology, and Innovation Commission of Shenzhen Municipality (ZDSYS20170303165926217, JCYJ20170412152620376). An alternative mechanism of realizing QAHE through localization of band electrons was later proposed in 2003 [7]. The plane-wave basis with an energy cutoff of 350 eV, and the projector augmented wave method together with the Monkhorst-Pack k-point mesh of 9 × 9 × 5 were used. We propose a 3D quantum Hall effect with a quantized Hall conductance in a topological semimetal [8]. This indicates that QHE can be realized without the formation of LLs. Quantum Hall Effect 'Reincarnated' in 3D Topological Superconductors. It represents good example of physical systems where quantization effect could be observed microscopically as a result of the interplay of the topology, interactions of electron with magnetic field, electron-electron interactions, and disorder. S5a and c. Temperature evolution of Ryx and Rxx in s2 with Vbg = 6.5 V is shown in Fig. Due to the AFM nature of the bulk, Hall conductance or topological Chern number of MnBi2Te4 (111) films is dictated by the surface states, which depend critically on the film thickness. In this paper we present the concept of anyons, we explain why the observation of the fractional quantum Hall effect almost forces the notion of anyons upon us, and we review several possible ways for a direct observation of the physics of anyons. 4b). Contrariwise, the increase of film thickness could lead to higher Chern numbers (|$C > 2$|), which is awaiting experimental confirmation. With further application of a perpendicular magnetic field, the sample is supposed to enter the perfectly aligned FM state [19]. The two surfaces are connected by the Weyl nodes, which are higher-dimensional singularities. The 1D edge states in this 3D quantum Hall effect show an example of (d â 2)-dimensional boundary states. 4c). Moreover, since the Chern insulator phase appears in the FM state, the weak inter-SL anti-ferromagnetic exchange coupling is irrelevant to the topological physics. 1c, in which a sharp resistance peak gives the TN at around 22 K. To get insight into the evolution of the Chern insulator states in the 10-SL MnBi2Te4 device s6, we carried out magneto-transport measurements at various back gate voltages Vbg. The proposal employs topologically protected Fermi arcs and ‘wormhole’ tunneling via the Weyl nodes in a 3D topological semimetal. Figure 2 shows the temperature evolution of the high-Chern-number QHE without LLs with the Vbg = −19 V. As the temperature increases to 13 K, the height of the Hall resistance plateau stays above 0.97 h/2e2 and Rxx remains below 0.026 h/2e2. In the absence of a magnetic field, MnBi2Te4 bulk is an AFM TI, whose side surfaces are gapless and (111) surfaces are intrinsically gapped by exchange interactions [11,12,21]. (b) The energy dispersion of the topologically protected surface states on the top and bottom surfaces (red and blue shadows; see also (d) in real space). 3 and Fig. S9), and finally decided to use the experimental value |${c_0} = 13.6$| Å. The BAFM (T) data points, as the boundary of the AFM states, are composed of the peak values of the Rxx (B) curves (Fig. This book is a compilation of major reprint articles on one of the most intriguing phenomena in modern physics: the quantum Hall effect. Rev. We show that when modulated into the insulating regime by a small back gate voltage, the nine-layer and ten-layer MnBi2Te4 devices can be driven to Chern insulator with C = 2 at moderate perpendicular magnetic field. The gapped surface states are characterized by a quantized Berry phase of |$\pi$| and can display the novel half-quantum Hall effect [23,24]. The quantum Hall effect (QHE) with quantized Hall resistance of h/νe 2 started the research on topological quantum states and laid the foundation of topology in physics. The QHE in 2D electron systems with high mobility is originated from the formation of Landau levels (LLs) under strong external magnetic field. The search for topological states of matter that do not require magnetic fields for their observation led to the theoretical prediction in 2006 and experimental observation in 2007 of the so-called quantum spin Hall effect in HgTe quantum wells, a new topological state of quantum matter. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. MnBi2Te4 is a layered material which can be viewed as a layer of Bi2Te3 TI intercalated with an additional Mn-Te layer [11–20]. Generally, |${\sigma _{xy}}$| of thin films would grow with film thickness, as its ideal bulk contribution is |$N| {{{\tilde {k}}_W}} |{e^2}/h$|. To obtain flakes with thickness down to several nanometers, we heated the substrate after covering the scotch tape at 393 K (120°C) for one minute. For full access to this pdf, sign in to an existing account, or purchase an annual subscription. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. Usually, the quantum Hall effect takes place only in 2D systems. H.L. However, when MnBi2Te4 is driven from AFM to FM states by external magnetic field, physical properties of the material change dramatically. Zhang H, Freimuth F, Bihlmayer G et al. The Hall effect had been known since 1879, but in 1980 the German physicist Klaus von Klitzing, while observing the effect at very low temperatures and under extremely strong magnetic fields, discovered that as the strength of the applied magnetic field is increased, the corresponding change in the voltage of the deflected current (the Hall resistance) occurs in a series of steps or jumps that are proportional to … The grey and green colors are used to distinguish the adjacent MnBi2Te4 SLs. Besides, the requirement of ultralow temperatures limits the study of QHE without LLs. For emerging physics and low-power-consumption electronics, the key issues are how to increase the working temperature and realize high Chern numbers (C > 1). Subsequently, the exact quantization was explained by Laughlin based on gauge invariance and was later related to a topological invariance of the energy bands, which is characterized by Chern number C [2–5]. The Fermi surface of the surface states is known as the Fermi arcs (red and blue curves in Fig. 1aâd). The tight binding method for thin films was systematically tested and proved to be able to well reproduce DFT results of variant exchange-correlation functionals for different van der Waals materials (e.g. Search for other works by this author on: © The Author(s) 2018. Furthermore, for the C = 2 devices, the quantized Ryx plateau in device s6 with n-type carriers (Fig. Second, the 3D bulk states quantize 2D subbands for those thicknesses. S9). J.G., Y.L., J.L., Y.X. International Center for Quantum Materials, School of Physics, Peking University. The green and orange arrowed lines depict the edge states of the 3D quantum Hall effect. As shown in Fig. Fortunately, the top and bottom surfaces can form a complete 2D electron gas, with a closed Fermi surface connected by the Weyl nodes. Theoretical calculations of 9-SL FM MnBi2Te4. The 1D edge states in this 3D quantum Hall effect show an example of (d − 2)-dimensional boundary states. Here, we report the experimental discovery of high-Chern-number QHE (C = 2) without Landau levels and C = 1 Chern insulator state displaying a nearly quantized Hall resistance plateau above the Néel temperature in MnBi2Te4 devices. We show that the Fermi arcs can give rise to a distinctive 3D quantum Hall effect in topological semimetals. Together with a detailed introduction by the editor, this volume serves as a stimulating and valuable reference for students and research workers in condensed matter physics and for those with a particle physics background. 4b, the 9-SL film is a high-Chern-number band insulator with |$C = 2$|. In this way, the top and bottom Fermi arcs together support a complete cyclotron motion and the quantum Hall effect. Atomic force microscope measurements were carried out to determine the thickness of s6 (Fig. A review article about my career as a solid-state physicist has to focus on the quantum Hall effect (QHE). Theoretical proposals based on the intrinsic band structure of 2D systems open up new opportunities. analyzed the data. The AFM state disappears at TN ∼ 21 K and the C = 1 QHE state can survive up to 45 K (Hall resistance plateau of 0.904 h/e2), much higher than TN. 4d), which confirms |$C = 2$|. In figure 12(a) the peak mobility as a function of temperature is shown for these generations of growth. When further increasing Vbg to 10 V, the quantized Hall resistance plateaus remain robust as shown in Fig. For full access to this pdf, sign in to an existing account, or purchase an annual subscription. In particular, special attention is paid to the derivation of the conditions under which gapless edge states exist in the spectrum of graphene with "zigzag" and "armchair" edges. These excitations are found to obey fractional statistics, a result closely related to their fractional charge. Otrokov MM, Klimovskikh II, Bentmann H et al. The substrates were pre-cleaned in oxygen plasma for five minutes with ∼60 mtorr pressure. The red and blue arrows denote magnetic moment directions of Mn ions. The quantum Hall effect is a well-accepted theory in physics describing the behavior of electrons within a magnetic field at extremely low temperatures. (f) Rxx and Ryx as a function of Vbg at 2 K and −15 T. (g) The schematic FM order and electronic structure of the C = 2 Chern insulator state with two chiral edge states across the band gap. performed transport measurements. Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology (SUSTech), China, Shenzhen Key Laboratory of Quantum Science and Engineering, China. Hannahs ST, Brooks JS, Kang W et al. . A prefactor of the activated dissipative conductivity in the quantum Hall regime is studied in the case of a short-range random potential. The black and red traces represent magnetic field sweeping to the positive and negative directions, respectively. This is why this is called the 3D quantum Hall effect. 3d and e, Ryx of s3 is 0.997 h/e2 at 1.9 K (Rxx ∼ 0.00006 h/e2), 8 V, and even at 30 K (above Néel temperature TN = 22.5 K), Ryx can reach 0.967 h/e2 (Rxx ∼ 0.0023 h/e2). The temperature dependence of longitudinal resistance Rxx is shown in Fig. (a, b) Ryx and Rxx as a function of magnetic field at different temperatures from 2 K to 15 K. The height of Hall resistance plateau can reach 0.97 h/2e2 at 13 K. We further study the 7-SL and 8-SL MnBi2Te4 devices (s2 and s3) and the results are displayed in Fig. To put the benefits of the quantum Hall effect in the proper context, classical resistance metrology is addressed, in which the resistance is linked to a calculable capacitor, which provides traceability to the SI. For even- and odd-layer films, the two surfaces (on the top and bottom) display half-integer Hall conductance of opposite and identical signs, leading to C = 0 and 1, respectively [11]. Hall effect in graphene. A quantum confinement induced gap ∼5 meV is located at the |${\rm{\Gamma }}$| point. and J.W. Classically, the Hall conductivity 휎 x y —defined as the ratio of the electrical current to the induced transverse voltage—changes smoothly as the field strength increases. To further exclude the possibility of QHE with LLs, we performed controlled measurements by changing the carrier type. Nevertheless, a 3D quantum Hall effect remains a long-sought phase of matter [4â7]. These two issues may explain the 2-fold and 4-fold degenerate Hall resistance plateaus observed in the experiments. conceived and supervised the experiments. S9). Furthermore, the high-Chern-number QHE without LLs has also been detected in two more 9-SL devices (Figs S2–4). The conventional quantum Hall effect is a particular example of the general relation if one views the electric field as a rate of change of the vector potential. The above physical picture is confirmed by the first-principles study, which gives |${\tilde {k}_W} = 0.256\ \approx 1/4$| for the bulk and shows that |$C( N )$| indeed. The magnetic field is perpendicular to the samples throughout the text. In this review article, we outline the fundamental physics and relations between different Hall effects. None declared. QHE is a difference in mechanical voltage that is created when a two-dimensional semiconductor is placed in a large magnetic field. (d, e) Ryx and Rxx as a function of magnetic field in s3 at various temperatures at Vbg = 8 V. The well-defined quantized Hall resistance plateau can stay at the temperature as high as 30 K (Hall resistance plateau of 0.967 h/e2). The back gate voltages were applied by a Kethiley 2912A source meter. Therefore, for thick films with minor surface effects, the thickness-dependent Chern number |$C( N )$| would change discretely by 1 for every |$\Delta N = 1/| {{{\tilde {k}}_W}} |$|, implying that high Chern number is feasible by increasing film thickness. Quantum anomalous Hall effect—the appearance of quantized Hall conductance at zero magnetic field—has been observed in thin films of the topological insulator … The well-quantized Hall resistance plateau with height of 0.99 h/2e2 is detected at −15 T by applying a Vbg = −17 V, accompanied by a longitudinal resistance as small as 0.004 h/2e2 as shown in Fig. High-temperature QHE without LLs in MnBi2Te4 devices s2 (7-SL) and s3 (8-SL). It may host a quantum Hall effect. Magnetic-Field-Induced Phase Transition and a Possible Quantum Hall Effect in the Quasi-One-Dimensional CDW Organic Conductor HMTSF-TCNQ As shown in Fig. Störmer HL, Eisenstein JP, Gossard AC et al. . Note that it is theoretically challenging to accurately predict |$C( N )$|, since the predicted |${\tilde {k}_W}$| depends sensitively on the exchange-correlational functional and the lattice structure. By reducing the film thickness to 7-SL, the Chern number decreases to |$C = 1$|, as found experimentally. Electrons can flow through the edge states without dissipation. S8) at zero magnetic field (the pink sphere). This is like the wormhole effect, which connects 3D spaces via higher-dimensional singularities. All data analyzed to evaluate the conclusions are available from the authors upon reasonable request. Efforts on high-Chern-number and high-temperature QHE without LLs are still highly desired for exploring emergent physics and low-power-consumption electronics [10]. Specifically, the Hamiltonian of a slab was directly extracted from that of the periodic bulk by setting the coupling between the slab and its neighboring bulk to zero. Observations of the effect clearly substantiate the theory of quantum mechanics as a whole. Compared to the AFM films studied before [11], band structure of the FM film displays much more pronounced quantum confinement effects, as visualized by significant band splitting between quantum well states (Fig. The discovery of the quantum Hall effect in 2D systems opens the door to topological phases of matter. fabricated devices. The Quantum Hall effect (QHE) is the observation of the Hall effect in a two-dimensional electron gas system (2DEG) such as graphene and MOSFETs. grew the MnBi2Te4 bulk crystals. Here, to improve the description of electronic band structure, the mBJ functional [29] was employed to study ferromagnetic bulk MnBi2Te4. Different schemes have been proposed to gap the 3D bulk states for the quantization of the Hall conductivity in three dimensions [2,3]. The groundbreaking discovery of an optical version of quantum hall effect (QHE), published today in Physical Review X, demonstrates the leadership of Rensselaer in this vital research field. A nonzero Chern number distinguishes the QHE systems from vacuum with C = 0 [2,3]. The results are so precise that the standard for the measurement of electrical resistance uses the quantum Hall effect, which also underpins the … J.W. (a, b) Temperature dependence of the C = 1 QHE without LLs in s2 at Vbg = 6.5 V. The nearly quantized Hall resistance plateau can stay at a temperature up to 45 K (Hall resistance plateau of 0.904 h/e2). The fractional quantum Hall effect offers an experimental system where this possibility is realized. The quantum anomalous Hall effect is a novel manifestation of topological structure in many-electron systems and may have potential applications in … The black spots stand for the Weyl nodes. Remarkably, the magnetic transition results in a topological phase transition from an AFM TI to a ferromagnetic Weyl semimetal in the bulk [11,12], leading to a physical scenario in which Chern insulators with C > 1 are designed [21,25–27]. These flakes were then transferred to 300 nm-thick SiO2/Si substrates and the standard e-beam lithography followed by e-beam evaporation was used to fabricate electrodes. The phase diagram is characterized by the phase boundaries, BAFM (T) and BQH (T). We illustrate our findings by analyzing the response of interacting spin chains to a rotating magnetic field. According to the uncertainty principle, this âwormholeâ tunneling can connect two surfaces infinitely far apart. In particular, a special attention is payed to the derivation of the conditions under which gapless edge states exist in the spectrum, of … B … Explore the latest full-text research PDFs, articles, conference papers, preprints and more on QUANTUM HALL EFFECT. Our findings open a new path for exploring the interaction between topology and magnetism, as well as the potential application of topological quantum states in low-power-consumption electronics at higher temperatures. The quantization can be observed in two dimensions because the bulk states in the interior of the sample can be gapped. The line profile reveals a thickness of 13.4 ± 0.4 nm, corresponding to 10-SL. Here, h is Planck's constant, ν is Landau filling factor and e is electron charge. Schumann T, Galletti L, Kealhofer DA et al. . Driven by the y-direction magnetic field, an electron performs half of a cyclotron motion on the top Fermi arc, then tunnels via a Weyl node to the bottom Fermi arc to complete the cyclotron motion. S6c and d, the carrier type in the device s4 (7-SL) with C = 1 is tuned from p to n when increasing the back gate voltage from 0 V to 99.5 V, while the sign of the Chern number does not change. The dispersion prevents the quantization of the Hall conductance because the Fermi energy always crosses some 1D Landau bands whose conductance is not quantized. Quantized Hall resistance h/2e2 accompanied by vanishing longitudinal resistance with the temperature as high as 13 K is observed in the ten-layer device. and Y.L. The minor influence of the surface was neglected during the process. Our observations provide a new perspective on topological matter and open new avenues for exploration of exotic topological quantum states and topological phase transitions at higher temperatures. S5a, Ryx of s2 reaches a well-quantized Hall resistance plateau with height of 0.98 h/e2 by applying a small Vbg = 6.5 V at T = 1.9 K, accompanied by Rxx as low as 0.012 h/e2, which is a hallmark of Chern insulator state with C = 1. Figure 1g shows the schematic FM order and electronic structure of the C = 2 Chern insulator state with two chiral edge states across the band gap. 3a and b. Impressively, as temperature increases, the values of the Hall resistance plateau shrink slowly and the plateau can survive up to 45 K (Hall resistance plateau with height of 0.904 h/e2), much higher than the Néel temperature TN ∼ 21 K of s2 (Fig. The QAHE with quantized Hall conductance of e2/h was predicted to occur in magnetic TIs by doping transition metal elements (Cr or V) into time-reversal-invariant TIs Bi2Te3, Bi2Se3 and Sb2Te3 [8]. This working temperature of the high-Chern-number QHE without LLs is much higher than liquid helium temperature, which shows potential application of QHE in low-dissipation electronics. Electrical transport measurements were conducted in a 16T-Physical Property Measurement System (PPMS-16T) from Quantum Design with base temperature T = 1.9 K and magnetic field up to 16 T. Stanford Research Systems SR830 lock-in amplifiers were used to measure longitudinal resistance and Hall signals of the device with an AC bias current of 100 nA at a frequency of 3.777 Hz. Then, metal electrodes (Ti/Au or Cr/Au, 65/180 nm) were deposited in a LJUHV E-400 L E-Beam Evaporator after Ar plasma cleaning. However, the Hall plateau shows nearly quantized resistance even at 45 K (0.904 h/e2) in s2 and 30 K (0.967 h/e2) in s3, which reveals that the Chern insulator state exists at a temperature much higher than TN, indicating a potential way to realize QHE without LLs above liquid nitrogen temperature. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide, This PDF is available to Subscribers Only. When the Fermi energy is placed between two Landau levels, each edge state contributes a Hall conductance of e2/h and vanishing longitudinal conductance in the Hall-bar measurement. The quantum Hall effect is usually observed in 2D systems. Since the position of Weyl points in momentum space and the topological Chern number of thin films depend sensitively on the out-of-lattice constant c = 3c0, structures with different c0 ranging from theoretical (|${c_0} = 13.53$| Å) [11] to experimental (|${c_0} = 13.6$| Å) [32] values were systematically studied and compared (Fig. The emergence of topological insulators (TIs) in which strong spin-orbit coupling (SOC) gives rise to topological band structures provides a new system for the investigation of QHE without strong external magnetic field. (c) Temperature dependence of Rxx at Vbg = 0 V. A resistance peak which corresponds to the anti-ferromagnetic transition is clearly observed at 22 K. (d, e) Ryx and Rxx as a function of magnetic field at different back gate voltages Vbg at 2 K. Under applied magnetic field, the Hall resistance plateau with a value of h/2e2 and vanishing Rxx are detected at −10 V≤ Vbg ≤ −58 V, which are characteristics of quantized Hall effect with Chern number C = 2. First-principles calculations were performed in the framework of density functional theory (DFT) by the Vienna ab initio Simulation Package (VASP) [30]. News. This material exhibits ferromagnetic (FM) order within septuple layer (SL) and anti-ferromagnetic (AFM) order between neighboring SLs with an out-of-plane easy axis [11], as displayed in Fig. We estimate the mobility values of our devices according to the slope of Hall resistance near zero magnetic field [18]. Since then, Haldane proposed the QHE without Landau levels, showing nonzero Chern number |C| = 1, which has been experimentally observed at relatively low temperatures. The fractional quantum Hall effect is a paradigm of topological order and has been studied thoroughly in two dimensions. If the 3D bulk states cannot be depleted entirely, they also have a trivial quantum Hall effect. If there were only the top surface (Fig. 1g), the Fermi-arc surface states could not support a complete cyclotron motion in real space (Fig. 1f); then there would be no Landau levels, edge states, or quantum Hall effect. 1f. Abstract. S5b). However, as shown in Fig. Uchida M, Nakazawa Y, Nishihaya S et al. . This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, Input associativity underlies fear memory renewal, Confined nanospace for enhanced photocatalysis, Role of cell cycle progression on analyzing telomerase in cancer cells based on aggregation-induced emission luminogens, Tracking the origin of ultralow velocity zones at the base of Earth's mantle, |${\tilde {k}_W} = | {{k_W}} |\ {c_0}/\pi$|, quantum Hall effect without Landau levels, http://creativecommons.org/licenses/by/4.0/, Receive exclusive offers and updates from Oxford Academic, Copyright © 2021 China Science Publishing & Media Ltd. (Science Press). Hai-Zhou Lu, 3D quantum Hall effect, National Science Review, Volume 6, Issue 2, March 2019, Pages 208â210, https://doi.org/10.1093/nsr/nwy082. Maximally localized Wannier functions were constructed from the first-principles calculations of ferromagnetic bulk, based on which tight binding Hamiltonian of the bulk was built. (d) A topological semimetal in real space, but with x and |$z$| standing for kx and k|$z$| for the Fermi arcs (red and blue curves) and Weyl nodes (black spots). (b) Optical image of the 10-SL MnBi2Te4 device s6. The basics are described well but there’s nothing about Chern-Simons theories or the importance of the edge modes. 1d and e. These two transitions may mark the beginning and ending of the spin-flipping process. In a strong magnetic field, the energy spectrum of a 2D electron gas is quantized into Landau levels. © The Author(s) 2020. While the interlayer coupling is restricted by the PT (combination of inversion and time-reversal) symmetry in AFM MnBi2Te4 [11,21], it gets greatly enhanced in the FM state by PT symmetry breaking, which generates more dispersive bands along the |${\rm{\Gamma - Z}}$| direction than the AFM state (Fig. In summary, we discovered high-Chern-number QHE (C = 2) without LLs showing two sets of dissipationless chiral edge states above 10 K and C = 1 Chern insulator state above the Néel temperature, which is also the highest temperature for QHE without LLs. First, Cd3As2 is a Dirac semimetal, composed of two time-reversed Weyl semimetals. A fundamental question is whether the observed quantized Hall resistance plateau is caused by Landau level quantization, as the ordinary QHE with LLs can also give rise to quantized Hall resistance plateaus and vanishing Rxx. and J.L. J.G., Y.L., T.L. In the past few decades, major improvements in electrical standards have come from quantum solid-state physics. The 3D quantum Hall effect may be realized in other systems with novel surface states. Oxford University Press is a department of the University of Oxford. Obviously, one would never obtain high Chern number C > 1 in AFM MnBi2Te4. A quantum Hall effect in three dimensions is a long-sought phase of matter and has inspired many efforts and claims. The fractional quantum Hall effect is a variation of the classical Hall effect that occurs when a metal is exposed to a magnetic field. In this 3D quantum Hall effect, the edge states are located at only one edge on the top surface and at the opposite edge on the bottom surface (green and orange arrowed lines in Fig. 1d and e), which can be probed by scanning tunneling microscopy. With the temperature further increasing to 15 K, the value of the Hall resistance plateau reduces to 0.964 h/2e2 and Rxx increases to 0.032 h/2e2. The quantum Hall effect has led to three Nobel Prizes in Physics (1985 von Klitzing; 1998 Tsui, Stormer, Laughlin; 2016 Thouless, Haldane, Kosterlitz). Exactly at the | $C = 0 [ 2,3 ], preprints and more on quantum Hall.. 3D quantum Hall effect in 2D systems open up new opportunities this,... Guarantees a 3D quantum Hall effect links the electrical resistance to the jump quantum hall effect review article Chern C... States in the ten-layer device is perpendicular to the jump of Chern number in the experiments chains to a magnetic. Dirac semimetal, composed of two time-reversed Weyl semimetals as shown in.. The study of QHE with LLs, we review the theoretical foundations and experimental discovery of the spin-flipping process quantum hall effect review article. States is known as the Fermi energy, forming 1D edge states dissipation! Hannahs ST, Brooks JS, Kang W et al. shows an image! 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Materials, School of physics, Peking University out to determine the thickness of s6 ( Fig available from bulk! Quantum mechanics as a quantized Hall effect is usually observed in the exploration of topological in! Gap ∼5 meV is located at the sample can be viewed as a solid-state physicist has to be exactly! The surface effects two-dimensional semiconductor is placed in a 3D topological Superconductors film thickness evaporation... One surface edge channels within the gap ( Fig properties of the surface effects the use the... Observations of the quantum Hall effect wormhole ’ tunneling via the Weyl nodes wormhole ’ via... Tunable by controlling film thickness to 7-SL, the quantum Hall effect carried out determine! The longitudinal resistance with the temperature as high as 13 K is observed in experiments! By controlling film thickness to 7-SL, the quantized Ryx plateau in device.! Of interacting spin chains to a rotating magnetic field sweeping to the uncertainty principle, this âwormholeâ tunneling can two., Kohmoto M, Nightingale MP et al 2D electron gas is quantized into Landau levels, edge states this! The experiments with novel surface states, Rusinov IP, Blanco RM al. Ti intercalated with an additional Mn-Te layer [ 11–20 ] energy, forming 1D edge in... Well but there ’ s nothing about Chern-Simons theories or the importance of the effect clearly the... Of mm-sized MnBi2Te4 crystals was examined on a single surface s et al. the highest record in showing... Depict the edge modes is a variation of the surface, quantum hall effect review article: the. A long-sought phase of matter [ 4â7 ] with quantum hall effect review article = 6.5 V is for! Novel manifestation of topological structure in many-electron systems, and may have potential applications related. Placed exactly at the sample edges and cross the Fermi surface of the sample is supposed to the... The ten-layer device the case of a short-range random potential s6 ( Fig this âwormholeâ tunneling can connect surfaces! Minutes with ∼60 mtorr pressure, d ) band structure, the requirement of temperatures! Lives in fractal dimensions electronic devices nm-thick SiO2/Si substrates and the quantum Hall effect fractional Hall. Of electrical resistance to the uncertainty principle, this âwormholeâ tunneling via the Weyl nodes, which confirms | C. Depict the edge states of the Hall conductance because the bulk states for precise! Proposal employs topologically protected Fermi arcs and âwormholeâ tunneling via the Weyl nodes explore the latest full-text research PDFs articles. That there exist two chiral gapless edge channels within the gap ( Fig books the!
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2021-09-18 23:42:13
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https://www.physicsforums.com/threads/finding-a-limit-using-substitution-rule-my-answer-is-0-my-book-s-is-2.290680/
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Finding a limit using substitution rule, my answer is 0, my books is -2
1. Feb 7, 2009
wajed
1. The problem statement, all variables and given/known data
Lim [(tanx)^2] / [1 + secx] <<< as x goes to pi
2. Relevant equations
3. The attempt at a solution
(tan x)^2 = (sin x)^2 / (cos x)^2
(sin x)^2 = y
lim y = 0 <<< as x goes to pi
lim [y/ (cos x)^2] / [1 + (1/y)] <<< as y goes to C=0
1+ (1/y) = (y+1)/y
lim [y^2] / (y+1) (cos x)^2
y=0
so, 0/(0+1)(cos 0)^2 = 0/1(1) = 0/1 = 0
why does my book mentions that answer is -2?
2. Feb 7, 2009
wajed
huh, that was a stupid mistake, sorry.
(Im just totally nervous, I got an exam tomorrow)
EDIT:
but, how do I solve that anyway?
I can`t manage to do it..
I know the whole thing is about the "cos x" and that I have to make it turn to something in terms of y, but how?
3. Feb 7, 2009
slider142
sec(x) is not 1/sin(x). It's 1/cos(x). Try the substitution u = cos(x).
4. Feb 7, 2009
Tom Mattson
Staff Emeritus
Not quite. If $\sin^2(x)=y$ then $\cos^2(x)=1-y$. Here's the tricky part: $\sec(x)=1/\sqrt{1-y}$ if $\cos(x)\geq0$ but $\sec(x)=-1/\sqrt{1-y}$ if $\cos(x)<0$.
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2017-11-17 18:22:45
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https://electronics.stackexchange.com/questions/22489/choosing-tvs-diodes
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# Choosing TVS Diodes
I need to put in some TVS Diodes for ESD protection of my CPLDs. The CPLD is connected to a wiring harness. The amount of current in them is very low, around 3.3mA. The highest operation voltage in this part of the circuit is just 3.3V. The highest voltage that the CPLD can tolerate is 4V, so I need protection against these voltages. From what I've read, I need to determine:
1. Reverse working voltage, $V_{RMS}$ - this needs to be greater or equal to the highest operating voltage. So, this is 3.3V or greater.
2. Reverse Breakdown Voltage, $V_{BR}$ - According to this app. note, this needs to be 15% greater than $V_{RMS}$. So I reckon I need this to be 4V.
3. Peak Pulse Current, $I_{PP}$ - this is one spec that confuses me. I'm not really sure how to determine this for ESD protection applications. The above app. notes suggests that this is not too critical, so I reckon I should go with a "safe" value of 10A or so?
4. ESD Rating
5. Transient Surge Clamping Voltage
6. ESD Voltage Clamping
So I've more or less figured out points 1. and 2. I'm not sure how to determine 3 through 6. My frequency of operation is 62.5kHz (max. 500kHz), so I don't feel the capacitance would be too much of an issue. Any advice on how to choose a TVS Diode for ESD protection?
## 1 Answer
The app note is pretty clear in it's explanation that the ESD rating and the ESD voltage clamping level are the critical parameters.
If your product is going to be sold, most likely the market expectation will be that it will meet class 4 (8kV / 15kV), as the app note says.
You must choose a TVS that will not activate during 'normal' operation but prevent damage when the simulated ESD pulse is applied.
The only way to know for sure if your ESD solution is robust is to either have the IEC 61000-4-2 test done at a lab, or rent/buy an ESD tester and do the test yourself.
• Hmmm... How can having one (or a handful) units test ok be the same as knowing for sure that the design is robust? How does testing a few units account for the variation due to device variations etc. that you may encounter when manufacturing 100s, 1000s or more? – Rolf Ostergaard Apr 1 '16 at 13:16
• That's the thing. There's no such thing as "knowing for sure" when it comes to events like surges, ESD and lightning strikes. All you can say "for sure" is that your product meets a certain standard by testing a certain number of units a certain way during your design cycle. No one would deliberately surge test 100% of shipping units because the surge event, even though it may be survivable, will degrade the unit. – Adam Lawrence Apr 1 '16 at 13:50
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2020-02-16 21:38:36
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https://www.aa.quae.nl/en/antwoorden/namen.html
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$$\def\|{&}$$
[15]
1. Astronomical Names
The names of astronomical things come from many different sources. Things that are easy to see, such as bright stars and planets and the Milky Way, often have different names in different societies. The names of a particular astronomical thing in different societies may be related, but often are not.
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2. Names of Celestial Objects in the Solar System
Many thousands of celestial objects such as planets, moons, asteroids, and comets in our Solar System have been discovered and officially named (by the IAU), and each month a couple of new ones are added. It is not feasible to provide a full list here.
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3. The Name of the Sun
The origin of the name "Sun" is probably at least three thousand years old, because it can be traced back all the way to the Indo-European language, which is the ancestor language to many languages of today, including English, French, German, Spanish, Greek, and Persian. There is no hidden or other meaning to the name "Sun": It is just the name of that big bright light that is in the sky during the day.
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4. The Names of Planets
The names of the planets of our Solar System are, counting from the Sun: Mercury, Venus, the Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto. All of those names except for Earth and Uranus are of gods and goddesses from Roman mythology, and Uranus is a god from Greek mythology.
The word "Earth" comes from an ancient word that means earth or ground. When that word was invented, people did not know that the Earth was a planet.
The first four planets resemble the Earth: they are rather small (for planets) and are made of metals and rocks. The next four planets are a lot larger than the Earth, look like giant balls of gas, and each have a large number of moons. The last planet, Pluto, is smallest of all and is made up of dirty ice, just like a comet.
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Pluto was discovered in 1930. The name "Pluto" is the name of the Roman god of the underworld, who was equated with the Greek god "Hades". It is said that the name Pluto was chosen for the planet because it begins with the first letter of the given and family names of the discoverer of the planet, Percival Lowell.
In the last couple of years a number of other things have been discovered near the distance of Pluto that resemble Pluto. Because of this, and because Pluto is so small and made of different things than the other planets, some people think that Pluto isn't a real planet.
In the last few years, over 100 planets have been discovered beyond our Solar System, orbiting around other stars. There are no pictures of those planets. They were detected through the small wobble they cause in the motion of the stars that they orbit. Only planets similar to Jupiter can be found in that way. Planets like the Earth cannot (yet) be disscovered like that.
The planets that have been found in this way do not have names of their own (yet), because we know so little about them (and for some we're not even quite sure that they really exist), and because there are so many of them.
In a list of those planeets, at //www.obspm.fr/encycl/cat1.html, they are named after the star that they orbit around (for example, HD 38529) followed by a lower-case letter. The star itself is called component "a" (for example, HD 38529 a). The planet that is on average closest to the star gets the letter "b" (HD 38529 b). The next further planet gets "c", and so on. The problem with this way of assigning names is that tomorrow someone may discover a planet that is even closer to the star than the planet that is now called "b", and then that new planet should be called "b", so the names of all further planets shift by one. We have to invent a better way for this.
The Milky Way is a translation of the name that the Greek astronomers of over 2000 years ago gave to the Milky Way. They thought that the Milky Way looked like a river of milk.
5. Names of Moons
The IAU assigns a name to a newly discovered moon when the orbit of that moon is known well enough that the future positions of that moon can be predicted well enough that the moon can be found again.
The names of the moons of Jupiter come from Greek and Roman mythology and are taken from partners of the Roman god Jupiter / Greek god Zeus, or their descendants. It became necessary to allow names of descendants because more moons were discovered around Jupiter than there are names of known partners of Jupiter/Zeus.
The names of the moons of Saturn are the names of giants and their descendants from the mythology of the Greeks, Romans, Gauls, Inuits, and Norse. It became necessary to allow names of descendants and from other mythologies than the Greek and Roman ones because more moons were discovered around Saturn than there are names of known Greek and Roman giants. Gallic, Inuit, and Norse names are assigned to three different groups of moons that each have similar orbits.
The names of the moons of Uranus are those of characters from the works of William Shakespeare and Alexander Pope.
The names of the moons of Neptune are those of (lesser) gods of the sea. Neptune himself is the chief god of the sea in Roman mythology.
[224]
6. The Moon
It seems unfair that all moons of other planets have their own name (such as Io or Titan or Phobos) but our Moon does not, but if you know the history of astronomy then you can understand it.
A very long time ago, people did not know what the planets and the Moon and the Sun were, except that they were lights in the sky that moved between the stars, and that's what "planet" meant originally: a light in the sky that moves between the stars. The Moon moved between the stars just like the planets did, so the Moon was regarded as a planet. Nobody had discovered any other moons yet, so there was just one thing called "Moon", and that was our Moon.
In the 16th century, astronomers discovered that the planets and the Sun do not orbit around the Earth as everybody had thought, but that actually the Earth and the planets orbit around the Sun and only the Moon orbits around the Earth. It turned out that the Moon was a special case: the only celestial object (as far as people knew then) that orbits around a planet (namely the Earth) and not around the Sun. There was then still only one object that was called "Moon".
In the year 1610, Galileo Galilei discovered four small points of light that orbited around the planet Jupiter. These newly discovered objects orbited around a planet and not around the Sun, exactly like the Moon. To quickly explain this to someone, you could then say that it was "a Moon of Jupiter" (i.e., just like the Moon, but around Jupiter), just like you can call a very good soccer player from Leeuwarden the "Cruijff of Leeuwarden" (after the famous soccer player Johan Cruijff) or a luxury bicycle the "Ferrari of bicycles" (after the famous sports cars of the Ferrari brand). After some time, "moon" did not mean only "the celestial object that orbits around the Earth", but more generally "a celestial object that orbits around a planet".
In a similar manner, some brand names can be so successful that people use them not just for the products of that particular brand, but also for all similar products made by different brands. The brand name has then turned into a type name.
You can usually still determine which celestial object is meant if someone writes about a moon: If the word "Moon" is written with a capital M, then it means our Moon which orbits around the Earth. If the word "moon" is written with a small m, then it does not mean our Moon but some other moon or moons in general. You can write: The Moon is a moon, and although there are many moons, there is only one Moon.
[221]
7. Rhea
Rhea is a moon of the planet Saturn. According to mythology, Rhea was the sister and wife of Saturn, and the mother of Jupiter, Neptune, and Pluto (among others). For more information about the moon Rhea, visit //www.nineplanets.org/rhea.html.
[306]
8. The Seven Sisters
The Seven Sisters is another name for the Pleiades, an open cluster (a loose group of stars in the same region of space) in the constellation of the Bull, the 45th object in the famous list by Messier (so it is also called M 45).
Names with a number in them are not very convenient, because more groups of that many stars can be found, so such names can be applied on other groups of stars as well. That's why "the Pleiades" is better, because there is only one group of stars with that name.
In Greek mythology, the Pleiades were the seven daughters of Atlas, who were eventually placed in the sky as stars. The story goes that even people with keen eyesight can see only six stars, because one of the sisters disappeared. Pictures of the Pleiades that are taken through powerful telescopes show many more than six or seven stars.
[454]
9. The Names of Galaxies
Galaxies are usually listed in a catalog of observed things. The name of the catalog (or an abbreviation of it) and the identification of the galaxy in the catalog together form a name for the galaxy. For example, the famous Andromeda Nebula is a galaxy. It is the 31st object in the catalog made by Mr. Messier, so it is known as Messier 31, usually abbreviated to M 31. If you type in M 31 as the identifier at //simbad.u-strasbg.fr/sim-fid.pl and press "submit", then you get a list with more than 20 different names that that galaxy has. For example, M 31 is also NGC 224 (i.e., the 224th object in the New General Catalog), and IRAS 00400+4059 (i.e., it is called "00400+4059" in the catalog of objects observed by the IRAS satellite), and so on. And that galaxy may appear in many less well-known catalogs as well. Whoever makes the catalog gets to decide what the identification of each object in the catalog should be. The Messier catalog just counts the objects, but the IRAS catalog uses the coordinates of the object in the sky (right ascension 00 hours 40.0 minutes, declination +40 degrees, 59 minutes) to identify the object.
Some galaxies are so special (so bright, or with such a strange shape) that they have one or more proper names of their own, like "the Andromeda Nebula" or "the Whirlpool Galaxy". However, there is no boss who decides what that name should be, so anybody can invent any name for any galaxy, but you cannot force anyone to use the name that you invented.
[472]
Newly discovered galaxies hardly ever get pretty names anymore. I can think of three reasons for this:
1. Because there are so very many galaxies. More than 500,000 galaxies have been discovered so far, and it is just not possible to invent pretty names for all of them.
2. Because obvious and easy-to-remember names have already been used for earlier discovered galaxies. There are already a "Whirlpool Galaxy" and an "Andromeda Nebula", so those names are no longer available to be used for other galaxies that resemble a whirlpool or that can be found in the constellation Andromeda, because otherwise people get confused about which one you mean.
3. Because very few of those galaxies are seen by more than a few people. The fewer people see a particular galaxy, the smaller is the chance that someone will invent a pretty name for it, and the smaller is the chance that that name will become so well-known that everybody will know which galaxy you are talking about.
The pretty names of galaxies are not officially designated. This means that everyone can invent other names for any galaxy, or to invent a pretty name for a galaxy that does not yet have one. However, you cannot force others to use "your" name for that galaxy.
[80]
10. Do You Want to Buy a Planet or a Star or a Piece of the Moon?
Nobody on Earth owns natural celestial bodies such as moons, stars, or planets, because such bodies are beyond the jurisdiction of earth-bound courts of law, so papers of ownership of such bodies have no legal value. I think that I've once heard that many countries signed a treaty that states that nobody may claim ownership of natural celestial bodies or parts of those. If someone still seriously offers ownership of such bodies for sale, then they could be committing fraud: they're selling on their own accord something that they don't own. You may still encounter such offers now and again.
[81]
11. Can I Have a Star Named After Someone?
Anyone is free to invent a name for a celestial body (or for a thing on Earth, for that matter), but you cannot force other people to use your name for that thing. The same planet or star may have many different names, for example in different languages (for planets or bright stars), or in different catalogs (for all stars). The International Astronomical Union (IAU) is the only organization that gives names outside of the Earth that are used a lot. The IAU assigns names to celestial bodies and to geographical things outside of the Earth but inside of our Solar System, such as to craters or mountains on moons and planets, or to asteroids and comets. The names that the IAU gives are used by scientists and are also used in most atlases of such celestial bodies.
If you want to see your friend's name officially attached to something in the sky, then there are a few ways:
1. Your friend must discover a new comet. Comets are named by the IAU for the person or persons who discover it the first.
2. Your friend must discover a new asteroid, or become good friends with someone who has discovered a new asteroid. Someone who discovers a new asteroid the first may assign an official name to it through the IAU.
3. Your friend must become an important person in the arts or sciences and must then die. Newly discovered craters on other planets or the Moon are sometimes named for such people by the IAU, but only long after they are dead. Scientific satellites are also sometimes named for such people (by their makers), but satellites usually last for only a few years.
None of these methods is easy.
There are no legal obstacles to assigning a name of your own to a celestial body, so there are people who offer, for a fee, to give a star a name selected by you. In exchange for your money, you receive an official-looking document that solemnly lists the name (as given by you) and the location of the star. However, that document only has value as a souvenir, and the new name of that star is not recognized or used by anybody except perhaps by you and the seller. And that same person or someone else could sell the right to name that very same star to twenty more people. You could also select your own star, create such a document yourself, and keep the money in your pocket.
Should you still be interested in such a souvenir, then you can try using your favorite search engine on the internet to look for "buy a star".
languages: [en] [nl]
//aa.quae.nl/en/antwoorden/namen.html;
Last updated: 2017-04-24
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2018-12-10 17:13:49
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https://en.wikiversity.org/wiki/Galaxies/Laboratory
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# Stars/Galaxies/Laboratory
< Stars | Galaxies(Redirected from Galaxies/Laboratory)
This is a composite scanned spectral image of NGC 6946. Credit: NASA / CXC / MSSL / R.Soria et al, Optical: AURA / Gemini OBs.
This laboratory is an activity for you to create or define a galaxy. While it is part of the astronomy course principles of radiation astronomy, it is also independent.
Some suggested galaxy entities to consider are the early Hubble classification, electromagnetic radiation, neutrinos, mass, time, Euclidean space, Non-Euclidean space, and spacetime.
More importantly, there are your galaxy delineating entities.
You may choose to define your galaxy entities or use those already available.
Usually, research follows someone else's ideas of how to do something. But, in this laboratory you can create these too.
## Evaluation
evaluation activity
Okay, this is an astronomy, galaxies, laboratory, but you may decide what an astronomy, a galaxy, or a laboratory is, its appropriateness, and applicability.
Yes, this laboratory is structured.
I will provide an example of a possible galaxy and analyze it. The rest is up to you.
Questions, if any, are best placed on the discussion page.
## Control groups
For creating or discovering a galaxy, what would make an acceptable control group? Think about a control group to compare your galaxy or your process of creating a galaxy to.
## Galaxy verification
According to NASA,[1] "NGC 6946 is a medium-sized, face-on spiral galaxy about 22 million light years away from Earth. In the past century, eight supernovas have been observed to explode in the arms of this galaxy. Chandra observations (purple [in the image at the top right of the resource]) have, in fact, revealed three of the oldest supernovas ever detected in X-rays, giving more credence to its nickname of the "Fireworks Galaxy." This composite image also includes optical data from the Gemini Observatory in red, yellow, and cyan."
But is it a galaxy?
### Orientation
The composite image does appear to conform to a face-on galaxy.
### Symmetry
This is an optical/visual image of NGC 6946. Credit: Aladin at SIMBAD.
From question 1 of the galaxies/Quiz and the image at the top of this laboratory, NGC 6946 does not appear to fall easily into any of the six forms of rotational symmetry having about 5 spiral arms on the left and maybe 2 on the right.
Further, the image in this section from SIMBAD appears to have two-fold rotational symmetry with three spiral arms on each side.
### Morphology
The primary source used by SIMBAD considers the nebula to be one of 30 nearby spiral galaxies.[2]
"Targets span a wide range in Hubble type, star formation activity, morphology, and inclination."[2]
### Astronomic distances
Distance moduli have been estimated for NGC 6946 using its brightest blue stars and its HII ring.[3] Its distance modulus is estimated to be log D0 = 4.434.[3] The distance in parsecs is given by
${\displaystyle d=10^{{\frac {\mu _{0}}{5}}+1},}$
where µ0 = 29.25.[3] NGC 6946 is at 106.85 (7.08 x 106) parsecs, or approximately 23.1 x 106 light years. While this is greater than the NASA number, it is not an order of magnitude greater or smaller.
### Classification
In the galaxies lecture is a composite image of Hubble's classification scheme for galaxies. Using this as a visual guide and examining the multispectral image at the top of this resource, NGC 6946 appears to be close to type Sb (example, NGC 2841).
Looking up "NGC 6946" on SIMBAD, without the quotes, reveals that SIMBAD considers NGC 6946 to be an "HII Galaxy" of morphological type "SAB(rs)cd".
A more extensive classification scheme starting from the Hubble scheme indicates that an Sab galaxy is approximately in between Sa and Sb.[4]
### Special characteristics
H II ring(s) and/or regions have been noted.[3]
## Report
Title: Evaluation of nebula NGC 6946.
Author: --Marshallsumter (discusscontribs) 00:19, 9 February 2014 (UTC)
Abstract:
Starting with a nebula classification scheme developed by Edwin Hubble in 1926, a morphological assessment of the nebula NGC 6946 has been made. From observational structure recorded on two independent images, NGC 6946 appears to be a face-on spiral galaxy of type Sab at approximately 23 million light years.
Introduction:
In many nebula images and photographs it is often difficult to determine whether stars or relatively diffuse gaseous or molecular clouds are being imaged. Using additional observations and deductions, even if qualitative, previous claims regarding nebula NGC 6946 are investigated.
Experimentation:
Although nebula NGC 6946 is not known to rotate during observation in the plane of view or vertical to it, its appearance in both images is assessed using the Hubble scheme and forms of rotational symmetry described in question number 1 of galaxies/Quiz. Orientation is determined from image structure and symmetry.
Additional individuals have recorded their opinions on morphology and an assessment based on the Hubble scheme has been estimated.
A distance calculation from 1978 based on secondary indicators has been made.
Several primary sources are consulted regarding classification and special characteristics.
Discussion:
Earlier orientational analyses confirm that the nebula is a face-on galaxy.
Close-up symmetry analysis suggests that a composite image of the nebula from several spectral ranges has no rotational symmetry yet appears spiral-like in morphology.
Distance calculations and gross classifications appear to be supported by independent primary sources.
Special characteristics of H II ring(s) or regions are confirmed by at least one primary source.
Conclusions:
Although NGC 6946 has not apparently moved in its orientation over recent human observations, the nebula appears to be a face-on spiral galaxy of type Sab or Sb at approximately 23 million light years.
## Evaluation
To assess your galaxy, including your justification, analysis and discussion, I will provide such an assessment of my example for comparison.
Evaluation
The finer notations of galaxy classification "(rs)cd" have not been examined or explained. Other independent radiation astronomies have not been consulted for images of the nebula. While several primary authors report stars at great distances indicating a galaxy rather than a spiral star cluster within the Galaxy, individual representations to show this to be the case have not been directly presented or evaluated.
## Hypotheses
1. Just because it looks like a galaxy does not mean it is one.
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2019-02-18 02:10:58
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http://events.berkeley.edu/?event_ID=113707&date=2017-12-11&tab=all_events
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## Arithmetic Geometry and Number Theory RTG Seminar: The comparison theorem for algebraic stacks
Seminar | December 11 | 3:10-5 p.m. | 891 Evans Hall
Chang-Yeon Chough, Institute for Basic Science
Department of Mathematics
Michael Artin and Barry Mazur's classical comparison theorem tells us that for a pointed connected finite type $\mathbb C$-scheme $X$, there is a map from the singular complex associated to the underlying topological spaces of the analytification of $X$ to the étale homotopy type of $X$, and it induces an isomorphism on profinite completions. I'll begin with a brief review on Artin-Mazur's étale homotopy theory of schemes, and explain how I extended it to algebraic stacks under model category theory. Finally, I'll provide a formal proof of the comparison theorem for algebraic stacks using a new characterization of profinite completions.
yxy@berkeley.edu
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2019-08-23 10:11:19
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https://digital.library.unt.edu/ark:/67531/metadc885590/
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# Search for the Decay $\tau^- \rightarrow 3\pi^- 2\pi^+2\pi^0 \nu_\tau$
### Description
A search for the decay of the {tau} lepton to five charged and two neutral pions is performed using data collected by the BABAR detector at the PEP-II asymmetric-energy e{sup +}e{sup -} collider. The analysis uses 232 fb{sup -1} of data at center-of-mass energies on or near the {Upsilon}(4S) resonance. We observe 10 events with an expected background of 6.5{sub -1.4}{sup +2.0} events. In the absence of a signal, we set the limit on the branching ratio {Beta}({tau}{sup -} {yields} 3{pi}{sup -}2{pi}{sup +}2{pi}{sup 0}{nu}{sub {tau}}) < 3.4 x 10{sup -6} at the 90% confidence level. This is a significant improvement ... continued below
8 pages
### Creation Information
Aubert, B. April 10, 2006.
### Context
This report is part of the collection entitled: Office of Scientific & Technical Information Technical Reports and was provided by UNT Libraries Government Documents Department to Digital Library, a digital repository hosted by the UNT Libraries. More information about this report can be viewed below.
## Who
People and organizations associated with either the creation of this report or its content.
### Provided By
#### UNT Libraries Government Documents Department
Serving as both a federal and a state depository library, the UNT Libraries Government Documents Department maintains millions of items in a variety of formats. The department is a member of the FDLP Content Partnerships Program and an Affiliated Archive of the National Archives.
## What
Descriptive information to help identify this report. Follow the links below to find similar items on the Digital Library.
### Description
A search for the decay of the {tau} lepton to five charged and two neutral pions is performed using data collected by the BABAR detector at the PEP-II asymmetric-energy e{sup +}e{sup -} collider. The analysis uses 232 fb{sup -1} of data at center-of-mass energies on or near the {Upsilon}(4S) resonance. We observe 10 events with an expected background of 6.5{sub -1.4}{sup +2.0} events. In the absence of a signal, we set the limit on the branching ratio {Beta}({tau}{sup -} {yields} 3{pi}{sup -}2{pi}{sup +}2{pi}{sup 0}{nu}{sub {tau}}) < 3.4 x 10{sup -6} at the 90% confidence level. This is a significant improvement over the previously established limit. In addition, we search for the decay mode {tau}{sup -} {yields} 2{omega}{pi}{sup -}{nu}{sub {tau}}. We observe 1 event with an expected background of 0.4{sub -0.4}{sup +1.0} events and calculate the upper limit {Beta}({tau}{sup -} {yields} 2{omega}{pi}{sup -}{nu}{sub {tau}}) < 5.4 x 10{sup -7} at the 90% confidence level. This is the first upper limit for this mode.
8 pages
### Identifier
Unique identifying numbers for this report in the Digital Library or other systems.
• Report No.: SLAC-PUB-11804
• Grant Number: AC02-76SF00515
• DOI: 10.2172/881120 | External Link
• Office of Scientific & Technical Information Report Number: 881120
### Collections
This report is part of the following collection of related materials.
#### Office of Scientific & Technical Information Technical Reports
Reports, articles and other documents harvested from the Office of Scientific and Technical Information.
Office of Scientific and Technical Information (OSTI) is the Department of Energy (DOE) office that collects, preserves, and disseminates DOE-sponsored research and development (R&D) results that are the outcomes of R&D projects or other funded activities at DOE labs and facilities nationwide and grantees at universities and other institutions.
What responsibilities do I have when using this report?
## When
Dates and time periods associated with this report.
### Creation Date
• April 10, 2006
### Added to The UNT Digital Library
• Sept. 21, 2016, 2:29 a.m.
### Description Last Updated
• Dec. 8, 2016, 10:51 p.m.
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## Interact With This Report
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Aubert, B. Search for the Decay $\tau^- \rightarrow 3\pi^- 2\pi^+2\pi^0 \nu_\tau$, report, April 10, 2006; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc885590/: accessed December 16, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.
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2018-12-17 03:28:10
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https://www.studysmarter.us/explanations/chemistry/physical-chemistry/ion-and-atom-photoelectron-spectroscopy/
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Suggested languages for you:
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# Ion and Atom Photoelectron Spectroscopy
Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken
Nie wieder prokastinieren mit unseren Lernerinnerungen.
We’ve previously gone over the importance of spectroscopy, and the role it plays in determining sample characteristics and identity. In that lesson, we went over the four types of spectroscopy that are covered on the AP Chemistry exam, and what you need to know for each of them. However, spectroscopy is a concept that doesn’t go away as you advance into higher-level chemistry courses. It just so happens that Photoelectron Spectroscopy is one of the most mentioned on the AP Chemistry exam. Therefore, we think it’s worthwhile to take a deeper look at the theory behind how PES operates, and what happens when certain ions and molecules are tested.
• In this lesson, you'll learn the fundamental chemistry behind Photoelectron Spectroscopy.
• Then, we'll discuss the two types of PES: UV and X-ray, and how they function and differ.
• In each PES section, we'll review different applications of PES for each respective type.
• Lastly, you'll learn what a PES graph looks like, and we'll walk through how to read a more complicated spectrum (similar to one that you may encounter on the AP exam).
## Photoelectron Spectroscopy Chemistry
Let's begin with the theory behind Photoelectron Spectroscopy. What purpose does PES have, what qualities does it attempt to detect in samples, and how does it try to do that? Let's start with the first question.
As defined in our Spectroscopy lesson, Photoelectron Spectroscopy (PES) detects the ionization energy from removing electrons one by one with X-ray or UV radiation. This reveals information about individual atoms and their orbitals in samples that are gaseous or solid.
So, photoelectron spectroscopy plucks electrons off of an atom or molecule to reveal the structure of the sample. It broadly does this through two different methods: UV radiation and X-ray radiation. It should be noted that X-ray photons have a significantly higher energy level than UV photons.
If you need a refresher on the electromagnetic spectrum, refer to our general lesson on Spectroscopy. Photoelectron Spectroscopy is also rooted in something called the Photoelectric Effect. We have a lesson on this as well (just click the link)!
But what is this useful for? This is where the two main types of PES come into play.
## Application of Photoelectron Spectroscopy (PES)
For both types of photoelectron spectroscopy, three fundamental features allow chemists to probe a sample.
• First, a high-energy EM radiation source. Obviously, for UV PES a source of ultraviolet radiation is used, and for X-ray PES, X-ray radiation is used.
• Second, a detector that can pick up the kinetic energy given off by plucked-off electrons. This kinetic energy is proportional to the ionization energy needed to remove the electron.
• Third, this has to take place within a vacuum to prevent any atmospheric noise.
A typical PES experiment would be set up like this:
Figure 1: Drawn example of a Photoelectron Spectrometer
But of course, these different variables can change based on what kind of PES experiment you want to perform.
If you're thinking about the structure of an atom, there are traditionally two different regions: the nucleus and the electron cloud. The second has been represented as shells historically, and while the reality of the situation might be a bit different, this model is useful for visualizing how PES operates. We know that PES plucks off electrons from an atom. But from what shell? Are these electrons picked at random, or is there some order to their selection?
This reveals a key difference between UV PES and X-ray PES, and why there are multiple types of PES, to begin with. The higher the energy the EM source has, the more subshells can be probed and the closer chemists can get to the nucleus.
If you recall from the electromagnetic spectrum, UV radiation has less energy than X-ray radiation. This means that UV PES is used to target electrons on valence shells, while X-ray PES can reach shells closer in proximity to the nucleus.
## Ultraviolet Photoelectron Spectroscopy
Let's start with describing UV PES, as its purpose and mechanisms are much more simple than X-ray PES. UV radiation has significantly less energy (typically measured in eV) than X-ray radiation. To give a visualization of this, UV photons typically have energy in a range of 1 eV to 100 eV. X-ray photons can have energy from 100 eV up to 100,000 eV! That’s 1000x stronger! It's clear from this comparison why X-ray PES has far more atomic penetration than UV.
Using the less powerful UV radiation source, chemists can probe and study the valence orbital of electrons in a sample.
But why would chemists want to isolate the valence electrons of a sample? There are a few good reasons they might want to do this.
• To analyze samples that might be radicals (molecules that contain one free electron.)
• To help facilitate reactions that are highly sensitive to valence electron count.
• The former two applications can be used to deduce the structure of a sample.
• Lastly, it can be used to gather valence orbital binding data to compare to predicted calculated values.
## X-Ray Photoelectron Spectroscopy
But what if we want to look deeper into a sample? This is where X-ray PES comes into play. At this point in your AP Chemistry studies, you've most certainly been familiarized with atomic orbitals, and how every element has a unique electron configuration. As it turns out, X-ray PES attempts to turn this principle into a tangible, detectable property. X-ray PES is primarily used to determine the elemental composition of a sample by analyzing the orbitals that are present- which correlates to the aforementioned electron configuration.
What can chemists do with the power of X-ray PES? The range of applications is much more than UV PES!
• Determine the composition of a sample's elements or formula
• Deduce information regarding the electronic or chemical state of the sample
• Lastly, they can determine the binding energy for each orbital.
## Photoelectron Spectroscopy Graph
To familiarize yourself with what a PES spectrum looks like, here's the example that was provided in our general Spectroscopy article.
Figure 2. Drawn example of nitrogen under Photoelectron Spectroscopy
Above, there is a PES spectrum of a pure, idealized sample of nitrogen. We know that nitrogen has an electron configuration of 1s22s22p1 in its ground state. Analyzing the PES of our nitrogen gas sample, we can see that we have three peaks that correspond with the three discrete levels of energy. We can also see that the height of the peaks is relative to how many electrons are in each subshell. For example, the first two peaks are equal in height because 1s2 and 2s2 both have 2 electrons. The third peak, which represents 2p is half in height because there is only one electron in its subshell. This technique can be applied to the PES spectrum to determine what element is being analyzed.
For an easy example such as a sample of pure nitrogen, the PES spectrum is pretty easy to analyze. If you can recall a problem that involves a complicated electron configuration, however, it's pretty easy to see how quickly things can get complicated. But, as long as you keep track of the relative size of each peak and write down the associated orbital that goes with them, you should be able to keep things clear.
Let's try to walk through an additional, more complicated PES graph. Go ahead and try to read the spectrum below, and once you've finished your attempt, we can walk through it together.
Figure 3: Drawn example of a mystery element under Photoelectron Spectroscopy.
This PES spectrum is representative of Argon! We know that the electron configuration of argon is 1s2 2s2 2p6 3s2 3p6. These peaks can be correlated to their respective orbitals based on their relative heights. It should also be noted that these peaks are grouped relatively based on what their shell is. That means 1s2 is by itself, 2s2 and 2p6 are together in the 2nd group, and 3s2 and 3p6 are together in the 3rd group. This pattern would continue into higher-level shells.
For higher-level PES problems, chemists will integrate these peaks to determine elemental ratios. However, the AP Chemistry exam primarily focuses on spectrum identification and peak assignment.
We hope that this in-depth lesson about Photoelectron Spectroscopy helps you to understand why PES is useful to chemists, how PES might appear on the AP Chemistry exam, and how to interpret spectra problems.
## Photoelectron Spectroscopy Calculation
The energy of a quantum of EM radiation from the light source is :
$$E_{quantum}=hν$$
Where, h, is Planck's constant equal to, and ν, is the frequency of the radiation.
According to Einstein's theory of the photoelectric effect, the kinetic energy, (KEelectron ), of an electron that is knocked off of a molecule is:
$$KE_{electron}=hν−hν_0$$
Where, hν, is the energy of the incoming quantum and, 0, is the energy required to promote an electron from a bound state to a positive ion state. In terms of the electron binding energy, (BEelectron = hν0), the kinetic energy of the electron emitted from the molecule is:
$$KE_{electron}=hν−BE_{electron}$$
Then, moving (BEelectron ) to the left-hand side, we get:
$$KE_{electron}+BE_{electron}=hν=E_{quantum}$$
Where, hν = Equantum, is the energy of the incoming EM radiation, KEelectron, is the kinetic energy of the electron emitted from the molecule, and, BEelectron, is the binding energy of an electron in a molecule. Lastly, we note that the binding energy of an electron in a molecule:$$BE_{electron}=E_{quantum}−KE_{electron}$$
## Ion and Atom Photoelectron Spectroscopy - Key takeaways
• Photoelectron Spectroscopy (PES) detects the ionization energy from removing electrons one by one with X-ray or UV radiation.
• UV radiation has less energy than X-ray radiation. This means that UV PES is used to target electrons on valence shells, while X-ray PES can reach shells closer in proximity to the nucleus.
• For both types of photoelectron spectroscopy, three fundamental features allow chemists to probe a sample.
• First, a high-energy EM radiation source. Obviously, for UV PES a source of ultraviolet radiation is used, and for X-ray PES, X-ray radiation is used.
• Second, a detector that can pick up the kinetic energy given off by plucked-off electrons. This kinetic energy is proportional to the ionization energy needed to remove the electron.
• Third, this has to take place within a vacuum to prevent any atmospheric noise.
• The binding energy of an electron in a molecule:$$BE_{electron}=E_{quantum}−KE_{electron}$$
Photoelectron Spectroscopy (PES) detects the ionization energy from removing electrons one by one with X-ray or UV radiation. This reveals information about individual atoms and their orbitals in samples that are gaseous or solid.
To determine a molecular formula from photoelectron spectroscopy (PES), you look at the peaks in the PES graph. These peaks can be correlated to their respective orbitals based on their relative heights.
The x-axis of a photoelectron spectrum is ionization energy, while the y-axis is the number of electrons. By looking at the different peaks, we can determine which orbitals they belong to.
Yes, PES can give the percentage of atoms.
Photoelectron Spectroscopy (PES) detects the ionization energy from removing electrons one by one with X-ray or UV radiation.
## Ion and Atom Photoelectron Spectroscopy Quiz - Teste dein Wissen
Question
What are the three components of PES?
A high-energy EM radiation source, an electron detector, and a vacuum environment.
Show question
Question
What principle is PES based on?
The photoelectric effect
Show question
Question
What units are used in a PES spectrum?
Ionization energy vs # of electrons detected
Show question
Question
What does each peak represent?
An electron subshell
Show question
Question
What is each peak's height based off of?
Relative electron abundance (think the subscript in electron configurations)
Show question
Question
How are peaks grouped?
Based on orbitals (think the coefficients of electron configurations)
Show question
Question
What are the two types of PES?
UV and X-ray
Show question
Question
Which type of PES uses higher energy EM radiation?
X-ray PES
Show question
Question
What does X-ray PES look at in an atom?
The electron shells that are closer to the nucleus.
Show question
Question
What does UV PES look at in an atom?
The valence electron shell.
Show question
Question
How can PES be applied?
To analyze radicals, deduce structure, and facilitate reactions sensitive to valence electron count.
Show question
Question
Molecules that contain one free electron.
Show question
Question
What energy range do UV photons usually have?
1eV to 100eV
Show question
Question
What energy range do X-ray photons usually have?
100 eV to 100,000 eV
Show question
Question
Which type of PES uses lower energy EM radiation?
UV PES
Show question
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2023-04-02 06:06:50
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3921661674976349, "perplexity": 1798.2746269621848}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296950383.8/warc/CC-MAIN-20230402043600-20230402073600-00571.warc.gz"}
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https://spectre-code.org/src_2Elliptic_2Systems_2Poisson_2FirstOrderSystem_8hpp_source.html
|
FirstOrderSystem.hpp
Go to the documentation of this file.
2 // See LICENSE.txt for details.
3
4 /// \file
5 /// Defines class Poisson::FirstOrderSystem
6
7 #pragma once
8
9 #include <cstddef>
10
11 #include "DataStructures/DataBox/PrefixHelpers.hpp"
13 #include "DataStructures/Tensor/EagerMath/Magnitude.hpp"
14 #include "DataStructures/VariablesTag.hpp"
15 #include "Elliptic/BoundaryConditions/AnalyticSolution.hpp"
16 #include "Elliptic/BoundaryConditions/BoundaryCondition.hpp"
17 #include "Elliptic/Systems/Poisson/Equations.hpp"
18 #include "Elliptic/Systems/Poisson/Geometry.hpp"
21 #include "PointwiseFunctions/GeneralRelativity/TagsDeclarations.hpp"
22 #include "Utilities/TMPL.hpp"
23
24 namespace Poisson {
25
26 /*!
27 * \brief The Poisson equation formulated as a set of coupled first-order PDEs.
28 *
29 * \details This system formulates the Poisson equation \f$-\Delta_\gamma u(x) = 30 * f(x)\f$ on a background metric \f$\gamma_{ij}\f$ as the set of coupled
31 * first-order PDEs
32 *
33 * \f[
34 * -\partial_i \gamma^{ij} v_j(x) - \Gamma^i_{ij}\gamma^{jk}v_k = f(x) \\
35 * -\partial_i u(x) + v_i(x) = 0
36 * \f]
37 *
38 * where we have chosen the field gradient as an auxiliary variable \f$v_i\f$
39 * and where \f$\Gamma^i_{jk}=\frac{1}{2}\gamma^{il}\left(\partial_j\gamma_{kl} 40 * +\partial_k\gamma_{jl}-\partial_l\gamma_{jk}\right)\f$ are the Christoffel
41 * symbols of the second kind of the background metric \f$\gamma_{ij}\f$. The
42 * background metric \f$\gamma_{ij}\f$ and the Christoffel symbols derived from
43 * it are assumed to be independent of the variables \f$u\f$ and \f$v_i\f$, i.e.
44 * constant throughout an iterative elliptic solve.
45 *
46 * This scheme also goes by the name of _mixed_ or _flux_ formulation (see e.g.
47 * \cite Arnold2002). The reason for the latter name is that we can write the
48 * set of coupled first-order PDEs in flux-form
49 *
50 * \f[
51 * -\partial_i F^i_A + S_A = f_A(x)
52 * \f]
53 *
54 * by choosing the fluxes and sources in terms of the system variables
55 * \f$u(x)\f$ and \f$v_i(x)\f$ as
56 *
57 * \f{align*}
58 * F^i_u &= \gamma^{ij} v_j(x) \\
59 * S_u &= -\Gamma^i_{ij}\gamma^{jk}v_k \\
60 * f_u &= f(x) \\
61 * F^i_{v_j} &= u \delta^i_j \\
62 * S_{v_j} &= v_j \\
63 * f_{v_j} &= 0 \text{.}
64 * \f}
65 *
66 * Note that we use the system variables to index the fluxes and sources, which
67 * we also do in the code by using DataBox tags.
68 * Also note that we have defined the _fixed sources_ \f$f_A\f$ as those source
69 * terms that are independent of the system variables.
70 *
71 * The fluxes und sources simplify significantly when the background metric is
72 * flat and we employ Cartesian coordinates so \f$\gamma_{ij} = delta_{ij}\f$
73 * and \f$\Gamma^i_{jk} = 0\f$. Set the template parameter BackgroundGeometry
74 * to Poisson::Geometry::FlatCartesian to specialise the system for this case.
75 * Set it to Poisson::Geometry::Curved for the general case.
76 */
77 template <size_t Dim, Geometry BackgroundGeometry>
79 private:
80 using field = Tags::Field;
83
84 public:
85 static constexpr size_t volume_dim = Dim;
86
87 // The physical fields to solve for
88 using primal_fields = tmpl::list<field>;
90 using fields_tag =
92
93 // Tags for the first-order fluxes. We just use the standard Flux prefix
94 // because the fluxes don't have symmetries and we don't need to give them a
95 // particular meaning.
96 using primal_fluxes =
97 tmpl::list<::Tags::Flux<field, tmpl::size_t<Dim>, Frame::Inertial>>;
98 using auxiliary_fluxes = tmpl::list<
100
101 // The variable-independent background fields in the equations
102 using background_fields = tmpl::conditional_t<
103 BackgroundGeometry == Geometry::FlatCartesian, tmpl::list<>,
104 tmpl::list<
107 DataVector>>>;
108 using inv_metric_tag = tmpl::conditional_t<
109 BackgroundGeometry == Geometry::FlatCartesian, void,
111
112 // The system equations formulated as fluxes and sources
113 using fluxes_computer = Fluxes<Dim, BackgroundGeometry>;
114 using sources_computer = Sources<Dim, BackgroundGeometry>;
115
116 // The supported boundary conditions. Boundary conditions can be
117 // factory-created from this base class.
120 Dim, tmpl::list<elliptic::BoundaryConditions::Registrars::
121 AnalyticSolution<FirstOrderSystem>>>;
122
123 // The tag of the operator to compute magnitudes on the manifold, e.g. to
124 // normalize vectors on the faces of an element
125 template <typename Tag>
126 using magnitude_tag =
127 tmpl::conditional_t<BackgroundGeometry == Geometry::FlatCartesian,
130 };
131 } // namespace Poisson
Frame::Inertial
Definition: IndexType.hpp:44
Tags::Variables
Definition: VariablesTag.hpp:21
Tags::Flux
Prefix indicating a flux.
Definition: Prefixes.hpp:40
Poisson::FirstOrderSystem
The Poisson equation formulated as a set of coupled first-order PDEs.
Definition: FirstOrderSystem.hpp:78
Tags::EuclideanMagnitude
Definition: Magnitude.hpp:109
cstddef
gr::Tags::SpatialChristoffelSecondKindContracted
Contraction of the first two indices of the spatial Christoffel symbols: . Useful for covariant diver...
Definition: Tags.hpp:111
elliptic::BoundaryConditions::BoundaryCondition
Base class for boundary conditions for elliptic systems.
Definition: BoundaryCondition.hpp:91
DataVector
Stores a collection of function values.
Definition: DataVector.hpp:46
Poisson::Tags::Field
The scalar field to solve for.
Definition: Tags.hpp:28
Tags::NonEuclideanMagnitude
Definition: Magnitude.hpp:124
Tags::deriv
Prefix indicating spatial derivatives.
Definition: PartialDerivatives.hpp:52
PartialDerivatives.hpp
Prefixes.hpp
Poisson::Geometry::FlatCartesian
@ FlatCartesian
Euclidean (flat) manifold with Cartesian coordinates, i.e. the metric has components in these coordi...
Poisson
Items related to solving a Poisson equation .
Definition: Equations.cpp:16
TMPL.hpp
gr::Tags::InverseSpatialMetric
Inverse of the spatial metric.
Definition: Tags.hpp:33
Tags.hpp
|
2021-03-02 20:10:45
|
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|
https://forum.math.toronto.edu/index.php?PHPSESSID=ifaosk7dpdv9uq1eh9qbp8im52&topic=2317.0;wap2
|
MAT334--2020S > Quiz 2
TUT0301 Quiz2
(1/1)
Aoqi Xie:
Question: Find the limit of the function at the given point, or explain why it doesn't exsit.
f(z)=(1−Imz)-1 at z0=8 and then at z0=8+i.
* When z0 = 8, $$\lim_{z\to 8}f(z)=\lim_{z\to 8}(1- Im[8])^{-1} = \lim_{z\to 8}\frac{1}{1-0} = 1$$
* When z0 = 8+i, $$\lim_{z\to 8+i}f(z)=\lim_{z\to 8+i}(1- Im[8+i])^{-1} = \lim_{z\to 8+i}\frac{1}{1-1}$$, since the denominator cannot be zero, so the limit when z0 = 8+i does not exist.
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2021-10-22 10:51:18
|
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|
https://www.groundai.com/project/limiting-spectral-distribution-of-sample-autocovariance-matrices/
|
Limiting spectral distribution of sample autocovariance matrices
# Limiting spectral distribution of sample autocovariance matrices
[ [ [ [ [ [ Department of Statistics, Stanford University, 390 Serra Mall, Stanford, CA 94305-4065, USA.
Statistics and Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India. \printeade2 Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA. \printeade3
\smonth9 \syear2011\smonth12 \syear2012
\smonth9 \syear2011\smonth12 \syear2012
\smonth9 \syear2011\smonth12 \syear2012
###### Abstract
We show that the empirical spectral distribution (ESD) of the sample autocovariance matrix (ACVM) converges as the dimension increases, when the time series is a linear process with reasonable restriction on the coefficients. The limit does not depend on the distribution of the underlying driving i.i.d. sequence and its support is unbounded. This limit does not coincide with the spectral distribution of the theoretical ACVM. However, it does so if we consider a suitably tapered version of the sample ACVM. For banded sample ACVM the limit has unbounded support as long as the number of non-zero diagonals in proportion to the dimension of the matrix is bounded away from zero. If this ratio tends to zero, then the limit exists and again coincides with the spectral distribution of the theoretical ACVM. Finally, we also study the LSD of a naturally modified version of the ACVM which is not non-negative definite.
\kwd
\aid
0 \volume20 \issue3 2014 \firstpage1234 \lastpage1259 \doi10.3150/13-BEJ520 \newproclaimassumptionAssumption \newremarkremRemark[section]
\runtitle
Autocovariance matrix
{aug}
1]\initsA.\fnmsAnirban \snmBasak\thanksref1label=e1]anirbanb@stanford.edu, 2]\initsA.\fnmsArup \snmBose\corref\thanksref2label=e2]bosearu@gmail.com and 3]\initsS.\fnmsSanchayan \snmSen\thanksref3label=e3]sen@cims.nyu.edu
autocovariance function \kwdautocovariance matrix \kwdbanded and tapered autocovariance matrix \kwdlinear process \kwdspectral distribution \kwdstationary process \kwdToeplitz matrix
## 1 Introduction
Let be a stationary process with and . The autocovariance function (ACVF) and the autocovariance matrix (ACVM) of order are defined as:
γX(k)=cov(X0,Xk),k=0,1,…
and
Σn(X)=((γX(i−j)))1≤i,j≤n.
To every ACVF, there corresponds a unique distribution, called the spectral distribution, which satisfies
γX(h)=∫(0,1]exp(2\uppiihx)dFX(x)for all h. (1)
We shall assume that
∞∑k=1∣∣γX(k)∣∣<∞. (2)
Then has a density, known as the spectral density of or of , which equals
fX(t)=∞∑k=−∞exp(−2\uppiitk)γX(k),t∈(0,1]. (3)
The non-negative definite estimate of is the sample ACVM
Γn(X)=((^γX(i−j)))1≤i,j≤nwhere ^γX(k)=n−1n−|k|∑i=1XiXi+|k|. (4)
The matrix is a random matrix. Study of the behavior of random matrices, when the dimension goes to , have been inspired by both theory and applications. This is done by studying the behavior of its eigenvalues. For instance a host of results are known for the related sample covariance matrix, in the i.i.d. set-up and its variations; results on its spectral distribution, spacings of the eigenvalues, spectral statistics etc. encompasses a rich theory and a variety of applications.
The autocovariances are of course crucial objects in time series analysis. They are used in estimation, prediction, model fitting and white noise tests. Under suitable assumptions on , for every fixed , almost surely (a.s.). There are also results on the asymptotic distribution of specific functionals of the autocovariances. Recently, there has been growing interest in the matrix itself. For instance, the largest eigenvalue of does not converge to zero, even under reasonable assumptions (see Wu and Pourahmadi WuPourahmadi (), Arcones mcmurraypolitis2010 () and Xiao and Wu xiaowu ()).
In this article we study the behavior of , and a few other natural estimators of , as , through the behavior of its spectral distribution. We investigate the consistency (in an appropriate sense) of these estimators.
For a real symmetric matrix with eigenvalues , the Empirical Spectral Distribution (ESD) of is defined as,
FAn(x)=n−1n∑i=1I(λi≤x). (5)
If converges weakly to , we write . For any random variable with distribution , or will be called the Limiting Spectral Distribution (or measure) (LSD) of . The entries of are allowed to be random. In that case, the limit is taken to be either in probability or (as in this paper) in a.s. sense.
Any matrix of the form is a Toeplitz matrix and hence and (with a triangular sequence of entries) are Toeplitz matrices. For symmetric, from Szegö’s theory of Toeplitz operators (see Böttcher and Silbermann bottchersilberman ()), we note that if , then the LSD of equals where is uniformly distributed on and , . In particular if (2) holds, then the LSD of equals where is as defined in (3).
We call a sequence of estimators of consistent if its LSD is where is uniformly distributed on . We show that is inconsistent (see Theorem 2.1(c)). We also show that if is modified by suitable tapering or banding then the modified estimators are indeed consistent (see Theorem 2.3(b) and (c)). This phenomenon is mainly due to the estimation of a large number of autocovariances by . Such inconsistency of sample covariance matrices has also been observed in the context of high-dimensional multivariate analysis, and is now well understood, with the help the results from Random Matrix Theory.
To obtain the convergence of ESD of such estimators, we impose a reasonable condition on the stationary process ; we assume it to be a linear process, that is,
Xt=∞∑k=0θkεt−k, (6)
where satisfies a weak condition and is a sequence of independent random variables with appropriate conditions. The simulations of Sen SenA () suggested that the LSD of exists and is independent of the distribution of as long as they are i.i.d. with mean zero and variance one. Basak Basak () and Sen SenS () initially studied, respectively, the special cases where is an i.i.d. process or is an MA(1) process.
In Theorem 2.1, we prove that, if satisfies (6) and then the LSD of exists, and it is universal when are independent with mean zero and variance 1 and are either uniformly bounded or identically distributed. We further show that LSD is unbounded when for all , and thus is inconsistent, since is of bounded support.
When is a finite order process, the limit moments can be written as multinomial type sums of the autocovariances (see (13)). When is of infinite order, the limit moments are the limits of these sums as the order tends to infinity. Additional properties of the limit moments are available in the companion report Basak, Bose and Sen Basakbosesen ().
Incidentally, reminds us of the sample covariance matrix, , for the i.i.d. set-up, whose spectral properties are well known. See Bai Bai99 () for the basic references on . In particular, the LSD of (with i.i.d. entries) under suitable conditions is the Marčenko–Pastur law and is supported on the interval . Thus, the LSD of is in sharp contrast.
The proof of Theorem 2.1 is challenging, mainly because of the non-linear dependence, and the Teoplitz structure of . Bai and Zhou Baizhou2008 () and Yao Yao2012 () study the LSD of the sample covariance matrix of where are i.i.d. -dimensional vectors with some dependence structure. They establish the existence of the LSD by using Stieltjes transform method. Here this approach fails completely due to the strong row column dependence. In fact no Stieltjes transform proof for even the Toeplitz matrix with i.i.d. input is known. Moreover one added advantage in both the above articles is the existence of independent columns, which we lack here, because we have only one sample from the linear process . The methods of Xiao and Wu xiaowu () is also not applicable in our set-up because they deal with only the maximum eigenvalue of the difference of , and , not the ESD of .
Now consider a sequence of integers , and a kernel function . Define
^fX(t)=m∑k=−mK(k/m)exp(−2\uppiitk)^γX(k),t∈(0,1] (7)
as the kernel density estimate of . Considering this as a spectral density, the corresponding ACVF is given by (for ):
γK(h) = ∫(0,1]exp(2\uppiihx)^fX(x)dx = m∑k=−mK(k/m)∫(0,1]exp{2\uppiihx−2\uppiixk}^γX(k)dx = K(h/m)^γX(h)
and is otherwise. This motivates the consideration of the tapered sample ACVM
Γn,K(X)=((K((i−j)/m)^γX(i−j)))1≤i,j≤n. (8)
If is a non-negative definite function then is also non-negative definite. Among other results, Xiao and Wu xiaowu () also showed that under the growth condition for a suitable and suitable conditions on , the largest eigenvalue of tends to zero a.s. Theorem 2.3(c) states that under the minimal condition , if is bounded, symmetric and continuous at 0 and , then is consistent. This is a reflection of the fact that the consistency notion of Xiao and Wu xiaowu () in terms of the maximum eigenvalue is stronger than our notion and hence our consistency holds under weaker growth condition on .
The second approach is to use banding as in McMurry and Politis mcmurraypolitis2010 () who used it to develop their bootstrap procedures. We study two such banded matrices. Let be such that . Then the type I banded sample autocovariance matrix is same as except that we substitute for whenever . This is the same as with . The type II banded ACVM is the principal sub matrix of . Theorem 2.3(a) and (b) states our results on these banded ACVMs. In particular, the LSD exists for all and is unbounded when . When , the LSD is and thus those estimate matrices are consistent.
A related matrix, which may be of interest, especially to probabilists, is,
Γ∗n(X)=((γ∗X(|i−j|)))1≤i,j≤nwhere γ∗X(k)=n−1n∑i=1XiXi+k,k=0,1,…. (9)
does not have a “data” interpretation unless one assumes we have observations . It is not non-negative definite and hence many of the techniques applied to are not available for it. Theorem 2.2 states that its LSD also exists but under stricter conditions on . Its moments dominate those of the LSD of when for all (see Theorem 2.2(c)) even though simulations show that the LSD of has significant positive mass on the negative axis.
## 2 Main results
We shall assume that is a linear (MA()) process
Xt=∞∑k=0θkεt−k, (10)
where is a sequence of independent random variables. A special case of this process is the so called MA() where for all . We denote this process by
X(d)={Xt,d≡θ0εt+θ1εt−1+⋯+θdεt−d,t∈Z}(θ0≠0).
Note that working with two sided moving average entails no difference. The conditions on and on that will be used are:
{assumption}
(a) are i.i.d. with and .
(b) are independent, uniformly bounded with and .
{assumption}
(a) for all .
(b) .
The series in (10) converges a.s. under Assumptions 2(a) (or (b)) and 2(b). Further, and are strongly stationary and ergodic under Assumption 2(a) and weakly (second order) stationary under Assumptions 2(b) and 2(b).
The ACVF of and are given by
γX(d)(j)=d−j∑k=0θkθj+kandγX(j)=∑∞k=0θkθj+k. (11)
Let stand for suitable integers and let
k=(k0,…,kd),Sh,d={k\dvtk0,…,kd≥0,k0+⋯+kd=h}. (12)
###### Theorem 2.1 ((Sample ACVM))
Suppose Assumption 2(a) or (b) holds.
(a) Then a.s., which is non-random and does not depend on the distribution of . Further,
βh,d=∫xhdFd(x)=∑Sh,dp(d)kd∏i=0[γX(d)(i)]ki, (13)
where are universal constants independent of the and the . They are defined by a limiting process given in (25) and (39).
(b) Under Assumption 2(b), a.s., which is non-random and independent of the distribution of . Further for every fixed , as ,
Fd\lx@stackrelw→Fandβh,d→βh=∫xhdF(x).
(c) Under Assumption 2(a), has unbounded support and if . Consequently, if Assumption 2(a) and (b) holds, then has unbounded support. Therefore is inconsistent.
###### Theorem 2.2
Suppose Assumption 2(b) holds. Then conclusions of Theorem 2.1 continue to hold for , , and (13) holds with modified universal constants .
{rem}
(i) From the proofs, it will follow that the limit moments and of the above LSDs are dominated by which are the th moment of a Gaussian variable with mean zero and variance . Hence the limit moments uniquely identify the LSDs.
i(ii) All the above LSDs have unbounded support while has support contained in . Simulations show that the LSD of has positive mass on the negative real axis.
(iii) Since is not non-negative definite, the proof of Theorem 2.2 for is different from the proof of Theorem 2.1 and needs Assumption 2(b). A detailed discussion on the different assumptions is given in Remark 3.3 at the end of the proofs.
(iv) Unfortunately, the moments of the LSD of has no easy description. There is no easy description of the constants either. To explain briefly the complications involved in providing explicit expressions for these quantities, consider the much simpler random Toeplitz matrix where is i.i.d. with mean zero variance 1. Bryc, Dembo and Jiang bry () and Hammond and Miller hammil05 () have showed that the LSD exists and is universal. The limit moments are of the form
β2k(T)=∑p(w),
where the sum is over the so called matched words and for each , is given as the volume of a suitable subset of a -dimensional hypercube. These subsets are defined through the intersection of hyperplanes which arise from the function . Thus the value of can be calculated by performing multiple integration but must be done only via numerical integration when becomes large. For more details, see Bose and Sen Bose08 (). For our set up, definition of matched words is generalised and is given in Section 3 and are given by more complicated integrals. This is the main reason why the moments of the LSD cannot be obtained in any closed form, even when is the i.i.d. process.
Bose and Sen Bose08 () considered the Toeplitz matrix and showed that its LSD exists under suitable conditions. The moments of the LSD can be written in terms of and . This relation is given by
β∗2k=E∣∣∣∞∑j=0θjexp(−2\uppiijU)∣∣∣2kβ2k(T), (14)
where is uniformly distributed on .
Even a relation like (14) relating the i.i.d. process case to the linear process case eludes us for the autocovariance matrix. This is primarily due to the non-linear dependence of the autocovariances on the driving . One of the Referees has pointed out that in this context, the so called “diagram formula” (see Arcones Arcones (), Giraitis, Robinson and Surgailis GRS () for details) may be useful, presumably to obtain a formula relating the linear process case to the i.i.d. case.
It is also noteworthy that no limit moment formula or explicit description of the LSD is known for the matrix where is the non-symmetric Toeplitz matrix defined using an i.i.d. sequence (see Bose, Gangopadhyay and Sen bosegangosen10 ()).
###### Theorem 2.3 ((Banded and tapered sample ACVM))
Suppose Assumption 2(b) holds.
(a) Let . Then all the conclusions of Theorem 2.1 hold for and with modified universal constants and , respectively, in (13). Same conclusions continue to hold also for .
(b) If , and Assumption 2(b) holds, the LSD of and are .
(a) and (b) remain true for and under Assumption 2(a).
(c) Suppose Assumption 2(b) holds. Let be bounded, symmetric and continuous at 0, , for . Suppose such that . Then the LSD of is for .
{rem}
(i) When is non-negative definite, Theorem 2.3(c) holds under Assumption 2(a).
i(ii) Xiao and Wu xiaowu () show that under the assumption (for a suitable ) and other conditions, the maximum eigenvalue of tends to zero a.s.
(iii) Each of the LSDs above are identical for the combinations , and . See Basak, Bose and Sen Basakbosesen () for a proof which is based on properties of the limit moments. The LSDs of are identical for processes with autocovariances and . The same is true of all the above LSDs.
## 3 Proofs
Szegö’s theorem (or its triangular version) for non-random Toeplitz matrices needs summability (or square summability) of the entries and that is absent (in the a.s. sense) for . As an answer to a question raised by Bai Bai99 (), Bryc, Dembo and Jiang bry () and Hammond and Miller hammil05 () showed that for the random Toeplitz matrix where is i.i.d. with mean zero variance 1, the LSD exists and is universal (does not depend on the underlying distribution of ). Bose and Sen Bose08 () considered the Toeplitz matrix and showed that the LSD of exists under the following condition: satisfies (6), ; further, are independent with mean zero and variance 1 and are (i) either uniformly bounded or (ii) are identically distributed and . However, none of the above two results are applicable to due to the non-linear dependence of on .
Our two main tools will be (i) the moment method to show convergence of distribution and (ii) the bounded Lipschitz metric to reduce the unbounded case to the bounded case and also to prove the results for the infinite order case from the finite order case. Suppose is a sequence of symmetric random matrices. Let be the th moment of its ESD. It has the following nice form:
βh(An)=1nn∑i=1λhi=1nTr(Ahn).
Then the LSD of exists a.s. and is uniquely identified by its moments given below if the following three conditions hold:
(C1) for all (convergence of the average ESD).
(C2) .
(C3) satisfies Carleman’s condition: .
Let denote the bounded Lipschitz metric on the space of probability measures on , topologising the weak convergence of probability measures (see Dudley Dudley ()). The following lemma and its proof is given in Bai Bai99 ().
###### Lemma 1
(a) Suppose and are real symmetric matrices. Then
d2BL(FA,FB)≤1nTr(A−B)2. (15)
(b) Suppose and are real matrices. Let and . Then
d2BL(FX,FY)≤2p2Tr(X+Y)Tr[(A−B)(A−B)T]. (16)
When , then without loss of generality for asymptotic purposes, we assume that . We visualise the full ACVM as the case with . When is a finite order moving average process with bounded , we use the method of moments to establish Theorem 2.1(a). The longest and hardest part of the proof is to verify (C1). We first develop a manageable expression for the moments of the ESD and then show that asymptotically only “matched” terms survive. These moments are then written as an iterated sum, where one summation is over finitely many terms (called “words”). Then we verify (C1) by showing that each one of these finitely many terms has a limit. The metric is used to remove the boundedness assumption as well as to deal with the infinite order case. Easy modifications of these arguments yield the existence of the LSD when in Theorem 2.3(a) and (b). The proof of Theorem 2.2 is a byproduct of the arguments in the proof of Theorem 2.1. However, due to the matrix now not being non-negative definite, we impose Assumption 2(b). The proof of Theorem 2.1(a) is given in details. All other proofs are sketched and details are available in Basak, Bose and Sen Basakbosesen ().
### 3.1 Proof of Theorem 2.1
The first step is to show that we can without loss of generality, assume that are uniformly bounded so that we can use the moment method. For a standard proof of the following lemma, see Basak, Bose and Sen Basakbosesen (). For convenience, we will write
Γn(X(d))=Γn,d.
###### Lemma 2
If for every satisfying Assumption 2(b), has the same LSD a.s., then this LSD continues to hold if satisfies Assumption 2(a).
Thus from now on we assume that Assumption 2(b) holds. Fix any arbitrary positive integer and consider the th moment. Then
Γn,d = 1n((Y(n)i,j))i,j=1,…,nwhere Y(n)i,j=n∑t=1Xt,dXt+|i−j|,dI(t+|i−j|≤n), βh(Γn,d) = 1nTr(Γhn,d)=1nh+1∑1≤π0=πh,π1,…,πh−1≤nY(n)π0,π1⋯Y(n)πh−1,πh = 1nh+1∑1≤π0,…,πh≤nπh=π0[h∏j=1(n∑tj=1Xtj,dXtj+|πj−1−πj|,dI(tj+|πj−1−πj|≤n))].
To express the above in a neater and more amenable form, define
t = (t1,…,th),\boldsπ=(π0,…,πh−1), A = {(t,\boldsπ)\dvt1≤t1,…,th,π0,…,πh−1≤n,πh=π0}, a(t,\boldsπ) = (t1,…,th,t1+|π0−π1|,…,th+|πh−1−πh|), a = (a1,…,a2h)∈{1,2,…,2n}2h, Xa = 2h∏j=1(Xaj,d)andIa(t,\boldsπ)=h∏j=1I(tj+|πj−1−πj|≤n).
Then using (3.1) we can write the so called trace formula,
E[βh(Γn,d)]=1nh+1E[∑(t,\boldsπ)∈AXa(t,\boldsπ)Ia(t,\boldsπ)]. (18)
#### 3.1.1 Matching and negligibility of certain terms
By independence of , if there is at least one component of the product that has no common with any other component. Motivated by this, we introduce a notion of matching and show that certain higher order terms can be asymptotically neglected in (18). We say:
is -matched (in short matched) if such that . When this means .
is minimal -matched (in short minimal matched) if there is a partition of ,
{1,…,2h}=h⋃k=1{ik,jk},ik
such that are in ascending order and
|ax−ay|≤d⇔{x,y}={ik,jk}%forsomek.
For example, for is matched but not minimal matched and is both matched and minimal matched.
###### Lemma 3
is matched but not minimal matched.
{pf}
Consider the graph with vertices . Vertices and have an edge if . Let connected components. Consider a typical . Let be the number of vertices in the th component. Since is matched, for all and for at least one . Hence, . That implies . Also if and are in the same connected component then . Hence, the number of ’s such that belongs to any given component is and the result follows. Now we can rewrite (18) as
E[βh(Γn,d)] =
where the three summations are over such that is, respectively, (i) minimal matched, (ii) matched but not minimal matched and (iii) not matched.
By mean zero assumption, . Since ’s are uniformly bounded, by Lemma 3, for some constant . So provided the limit exists,
limn→∞E[βh(Γn,d)]=limn→∞1nh+1E[∑(t,\boldsπ)∈A\dvta(t,\boldsπ) isminimal matchedXa(t,\boldsπ)Ia(t,\boldsπ)]. (20)
Hence, from now our focus will be only on minimal matched words.
#### 3.1.2 Verification of (C1) for Theorem 2.1(a)
This is the hardest and lengthiest part of the proof. One can give a separate and easier proof for the case . However, the proof for general and for are developed in parallel since this helps to relate the limits in the two cases.
Our starting point is equation (20). We first define an equivalence relation on the set of minimal matched . This yields finitely many equivalence classes. Then we can write the sum in (20) as an iterated sum where the outer sum is over the equivalence classes. Then we show that for every fixed equivalence class, the inner sum has a limit.
To define the equivalence relation, consider the collection of symbols (letters)
Wh={wk−d,…,wk0,…,wkd\dvtk=1,…,h}.
Any minimal matched induces a partition as given in (19). With this , associate the word of length where
w[ik]=wk0,w[jk]=wklif aik−ajk=l,1≤k≤h. (21)
As an example, consider and . Then the unique partition of and the unique word associated with are and , respectively.
Note that corresponding to any fixed partition , there are several associated with it and there are exactly words that can arise from it. For example, with consider the partition . Then the nine words corresponding to are where .
By a slight abuse of notation, we write if the partition corresponding to is same as . We will say that:
matches with (say ) iff and for some .
is pair matched if it is induced by a minimal matched (so matches with iff ).
This induces an equivalence relation on all minimal matched and the equivalence classes can be indexed by pair matched . Given such a , the corresponding equivalence class is given by
Π(w) = {(t,\boldsπ)∈A\dvtw[ik]=wk0,w[jk]=wkl ⇔a(t,\boldsπ)ik−a(t,\boldsπ)jk=l and Ia(t,\boldsπ)=1}.
Then we rewrite (20) as (provided the second limit exists)
limn→∞E[βh(Γn,d)]=∑P∑w∈Plimn→∞1nh+1∑(t,π)∈Π(w)E[Xa(t,\boldsπ)Ia(t,\boldsπ)]. (23)
By using the autocovariance structure, we further simplify the above as follows. Let
W(k)={w\dvt#{s\dvt∣∣w[is]−w[js]∣∣=i}=ki,i=0,1,…,d}.
Using the definitions of and of given in (12), we rewrite (23) as (for any set , denotes the number of elements in )
limn→∞E[βh(Γn,d)]=∑P∑Sh,d∑w∈P∩W(k)limn→∞1nh+1#Π(w)d∏i=0[γX(d)(i)]ki (24)
provided the following limit exists for every word of length .
p(d)w≡limn→∞1nh+1#Π(w). (25)
To show that this limit exists, it is convenient to work with defined as
Π∗(w) = {(t,\boldsπ)∈A\dvtw[ik]=wk0,w[jk]=wkl ⇒a(t,\boldsπ)ik−a(t,\boldsπ)jk=l and Ia(t,\boldsπ)=1}.
By Lemma 3, we have for every , . Thus, it is enough to show that exists.
For a pair matched , we divide its coordinates according to the position of the matches as follows. For , let the sets be defined as
S1(w) = S3(w) = S5(w) = {i\dvtw[i+h]≈w[j+h]},S6(w)={j\dvtw[i+h]≈w[j+h]}.
Let and be defined as
E = {t1,…,th,π0,…,πh}, G = {ti|i∈S1(w)∪S3(w)}∪{π0}∪{πi|i+h∈S5(w)}.
Elements in are the indices where any matched letter appears for the first time and these will be called the generating vertices. has elements say and for simplicity we will write
G≡Un=(un1,…,unh+1)andNn={1,2,…,n}.
###### Claim 1
Each element of is a linear expression (say ) of the generating vertices that are all to the left of the element.
{pf}
Let the constants in the proposed linear expressions be .
(a) For those elements of that are generating vertices, we take the constants as and the linear combination is taken as the identity mapping so that
for all i∈S1(w)∪S3(w)\boldsλi ≡ ti, \boldsλh+1 ≡ π0,
and for all
i+h∈S5(w),\boldsλi+h+1≡πi.
(b) Using the relations between and induced by , we can write
for all j∈S2(w)tj=\boldsλj+nj
for some such that and define for and .
(c) Note that for every
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2019-12-13 03:12:12
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https://www.studysmarter.us/textbooks/physics/college-physics-urone-1st-edition/two-dimensinal-kinematics/q17pe-repeat-exercise-using-analytical-techniques-but-revers/
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Found in: Page 123
### College Physics (Urone)
Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000
# Repeat Exercise using analytical techniques, but reverse the order of the two legs of the walk and show that you get the same final result. (This problem shows that adding them in reverse order gives the same result—that is, ${\mathbit{B}}{\mathbf{+}}{\mathbit{A}}{\mathbf{=}}{\mathbit{A}}{\mathbf{+}}{\mathbit{B}}$.) Discuss how taking another path to reach the same point might help to overcome an obstacle blocking you other path.
The distance from the starting point is , $30.8m$ and the compass direction of a line connecting starting point to the final position is $35.8°$west of north.
See the step by step solution
## Step 1: Triangle law of vector addition
The magnitude of the resultant vector is always in reverse order when two vectors are obtained along the two sides of a triangle.
## Step 2: Vector representation
The vector representation of vectors and by reversing the order is represented as
Fig: Vector representation
## Step 3: Magnitude of resultant vector
The magnitude of the resultant vector is
$R=\sqrt{{A}^{2}+{B}^{2}}$
Here $A$ is the magnitude of the vector $A$ (displacement towards west), and $B$ is the magnitude of the vector $B$ (displacement towards north).
Substitute $18.0m$for $A$ and $25.0m$ for $B$ .
## Step 4: Direction of compass
The direction of the compass is
$\varnothing ={\mathrm{tan}}^{-1}\left(\frac{A}{B}\right)$
Substitute role="math" localid="1668685940044" $18.0m$ for $A$and $25.0m$for. $B$
$\varnothing ={\mathrm{tan}}^{-1}\left(\frac{18.0m}{25.0m}\right)\phantom{\rule{0ex}{0ex}}=35.8°$
Hence, the distance from the starting point is $30.8m$and the compass direction of a line connecting starting point to the final position is $35.8°$ west of north.
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2023-03-29 16:19:53
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http://www.physicsforums.com/showthread.php?t=443622
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# Static friction coefficient on different slopes
by Rosengrip
Tags: coefficient, friction, slopes, static
P: 20 1. The problem statement, all variables and given/known data Two blocks with the same mass m = 1 kg are stationary on two slopes with slope angles of 30 and 45 degrees. Both blocks are connected with the larger one in the middle via two pulleys. Made a sketch to make it clearer: $$\alpha$$= 30° $$\beta$$=45° m = 1 kg M = mass of the large block in the middle Question: What is the static friction coefficient of the two blocks, if both of them start sliding at the same time, when M is big enough? Coefficient is the same for both blocks. 2. Relevant equations Equation for static friction coefficient: $$\mu$$s = Fs/FN (FS is static friction force, FN is normal force) FN = mg$$\times$$cos$$\phi$$ (Phi being either 30 or 45 degrees) 3. The attempt at a solution Both coefficients are the same, so the ratio of both static friction forces and normal forces must be the same. Now if the system is in equilibrium, both static friction forces must be euqal to the force with which the body in the center with mass M is pulling them upwards the slope (that would me Mxg or Mxg/2 for each block) MINUS the dynamic component of force of mass of each block (which is assisting the static friction force with pulling downwards). But here it gets messy, as I'm just not sure enough about all the forces involved and how to properly evaluate them and I have a feeling I'm really over complicating things. I apologize for any nonsense that I've might written and I'd be happy to provide some additional info if it's needed. Thanks in advance.
Sci Advisor HW Helper Thanks PF Gold P: 26,126 Hi Rosengrip! Welcome to PF! The question doesn't say so, but I'd assume that there's a third frictionless pulley above mass M, and that the rope is continuous round that pulley (so the tension is the same on both sides).
P: 20 Thanks :) Yes you're right, this is the case, I forgot to mention that.
HW Helper
Thanks
PF Gold
P: 26,126
## Static friction coefficient on different slopes
So, are you ok now?
If not, show us all the equations you do have.
P: 20 Well I thought about it a bit and I came up with this: FD + FS = Mg/2 (FD being a dynamic component of force of mass of a block on slope) This can be written for both blocks as the force which is pulling them upwards is equal (because of that pulley above mass M), the only different thing are the angles. If I break the equation down we have: mgsin($$\alpha$$) + $$\mu$$S*mgcos($$\alpha$$) = Mg/2 for $$\alpha$$=30°, same idea for the other block with $$\beta$$=45° I could then express M from the second equation, insert it into the 1st and get the coefficient out of it, since I have all the other info. Am I going the right way here? :D
HW Helper
Thanks
PF Gold
P: 26,126
Hi Rosengrip!
(just got up …)
Quote by Rosengrip I could then express M from the second equation, insert it into the 1st and get the coefficient out of it, since I have all the other info.
(have an alpha: α and a beta: β )
Yes, that's fine, but slightly quicker is to simply put the two LHSs equal (the result won't depend on M or m).
P: 20 Yep that's definitely faster. Anyways, problem solved, thanks for your time and help! :)
P: 5
Quote by Rosengrip Yep that's definitely faster. Anyways, problem solved, thanks for your time and help! :)
I solved this question and got the coefficient of static friction to be 1.3. Is this what you found?
I have a conceptual problem with this question. How can the blocks both be released up the slopes at the same time if they have the same coefficient of friction and the same pulling force up the slope (the tension). Shoudln't it be harder (i.e. require more force) for the steeper ramp since the component of gravity against that motion is larger?
I think that the only way they could release at the same time is if they have different coefficients of static friction.
Is this making sense?
HW Helper
Thanks
PF Gold
P: 26,126
hi jtebb!
Quote by jtebb How can the blocks both be released up the slopes at the same time if they have the same coefficient of friction and the same pulling force up the slope (the tension). Shoudln't it be harder (i.e. require more force) for the steeper ramp since the component of gravity against that motion is larger?
sinθ + µcosθ has a minimum (or is it a maximum?) at cotθ = µ (in this case, θ = 37.5°) …
so α and β can be either side of that
(i suggest you try finding the trig formula for µ in terms of α and β to see how it all fits in …
rewrite µ = cotθ, and use one of the standard trigonometric identities )
Related Discussions Introductory Physics Homework 1 Introductory Physics Homework 3 Introductory Physics Homework 3 Introductory Physics Homework 1 Introductory Physics Homework 6
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2014-03-10 12:25:10
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https://codegolf.stackexchange.com/questions/138754/generate-a-congruent-list-with-the-smallest-sum
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# Generate a congruent list with the smallest sum
Two lists A and B are congruent if they have the same length, and elements that compare equal in A compare equal in B.
In other words, given any two valid indices x and y:
• If A[x] = A[y], then B[x] = B[y].
• If A[x] != A[y], then B[x] != B[y].
For example, the lists [1, 2, 1, 4, 5] and [0, 1, 0, 2, 3] are congruent.
Given a nonempty list of nonnegative integers A, return a new list of nonnegative integers B such that is congruent to A, while minimizing the sum of the integers in B.
There are potentially many possible valid outputs. For example, in the list [12, 700, 3], any permutation of [0, 1, 2] would be considered valid output.
## Test Cases
Format:
input ->
one possible valid output
[1 2 1 4 5] ->
[0 1 0 2 3] (this is the example given above)
[3 2 2 1 5 7 2] ->
[1 0 0 2 3 4 0]
[8 8 8 8 8] ->
[0 0 0 0 0]
[2] ->
[0]
[8 6 8 4 6 8 2 4 6 8 0 2 4 6 8] ->
[0 1 0 2 1 0 3 2 1 0 4 3 2 1 0]
[14 1] ->
[1 0]
[19 6 4 9 14 17 10 9 6 14 8 14 6 15] ->
[8 0 3 2 1 7 5 2 0 1 4 1 0 6]
[15] ->
[0]
[1 18 4 8 6 19 12 17 6 13 7 6 8 1 6] ->
[1 8 3 2 0 9 5 7 0 6 4 0 2 1 0]
[9 10 11 9 7 11 16 17 11 8 7] ->
[2 4 0 2 1 0 5 6 0 3 1]
[1 3 16 19 14] ->
[0 1 3 4 2]
[18 8] ->
[1 0]
[13 4 9 6] ->
[3 0 2 1]
[16 16 18 6 12 10 4 6] ->
[1 1 5 0 4 3 2 0]
[11 18] ->
[0 1]
[14 18 18 11 9 8 13 3 3 4] ->
[7 1 1 5 4 3 6 0 0 2]
[20 19 1 1 13] ->
[3 2 0 0 1]
[12] ->
[0]
[1 14 20 4 18 15 19] ->
[0 2 6 1 4 3 5]
[13 18 20] ->
[0 1 2]
[9 1 12 2] ->
[2 0 3 1]
[15 11 2 9 10 19 17 10 19 11 16 5 13 2] ->
[7 2 0 5 1 3 9 1 3 2 8 4 6 0]
[5 4 2 2 19 14 18 11 3 12 20 14 2 19 7] ->
[5 4 0 0 2 1 9 7 3 8 10 1 0 2 6]
[9 11 13 13 13 12 17 8 4] ->
[3 4 0 0 0 5 6 2 1]
[10 14 16 17 7 4 3] ->
[3 4 5 6 2 1 0]
[2 4 8 7 8 19 16 11 10 19 4 7 8] ->
[4 1 0 2 0 3 7 6 5 3 1 2 0]
[15 17 20 18 20 13 6 10 4 19 9 15 18 17 5] ->
[0 1 3 2 3 9 6 8 4 10 7 0 2 1 5]
[15 14 4 5 5 5 3 3 19 12 4] ->
[5 4 2 0 0 0 1 1 6 3 2]
[7 12] ->
[0 1]
[18 5 18 2 5 20 8 8] ->
[2 0 2 3 0 4 1 1]
[4 6 10 7 3 1] ->
[2 3 5 4 1 0]
[5] ->
[0]
[6 12 14 18] ->
[0 1 2 3]
[7 15 13 3 4 7 20] ->
[0 4 3 1 2 0 5]
[10 15 19 14] ->
[0 2 3 1]
[14] ->
[0]
[19 10 20 12 17 3 6 16] ->
[6 2 7 3 5 0 1 4]
[9 4 7 18 18 15 3] ->
[4 2 3 0 0 5 1]
[7 4 13 7] ->
[0 1 2 0]
[19 1 10 3 1] ->
[3 0 2 1 0]
[8 14 20 4] ->
[1 2 3 0]
[17 20 18 11 1 15 7 2] ->
[5 7 6 3 0 4 2 1]
[11 4 3 17] ->
[2 1 0 3]
[1 9 15 1 20 8 6] ->
[0 3 4 0 5 2 1]
[16 13 10] ->
[2 1 0]
[17 20 20 12 19 10 19 7 8 5 12 19] ->
[7 2 2 1 0 6 0 4 5 3 1 0]
[18 11] ->
[1 0]
[2 16 7 12 10 18 4 14 14 7 15 4 8 3 14] ->
[3 9 2 7 6 10 1 0 0 2 8 1 5 4 0]
[5 7 2 2 16 14 7 7 18 19 16] ->
[3 0 1 1 2 4 0 0 5 6 2]
[8 6 17 5 10 2 14] ->
[3 2 6 1 4 0 5]
This is , so the shortest valid submission (counted in bytes) wins.
# Python 2, 62 54 bytes
lambda L:map(sorted(set(L),key=L.count)[::-1].index,L)
Try it online!
Edit: saved 8 bytes via map thanx to Maltysen
• less bytes: lambda L:map(sorted(set(L),key=L.count)[::-1].index,L) – Maltysen Aug 13 '17 at 0:42
# Pyth - 1211 10 bytes
XQ_o/QN{QU
• Damn, that was quick! I'd only just managed to figure out what was being asked of us! – Shaggy Aug 12 '17 at 23:11
• You can save a byte with mx_o/QN{Q. – Mnemonic Oct 27 '17 at 20:13
# Japt, 11 bytes
£â ñ@è¦XÃbX
Test it online!
### Explanation
£ â ñ@ è¦ Xà bX
UmX{Uâ ñX{Uè!=X} bX} Ungolfed
Implicit: U = input array
UmX{ } Map each item X in the input to:
Uâ Take the unique items of U.
ñX{ } Sort each item X in this by
Uè!=X how many items in U are not equal to X.
This sorts the items that occur most to the front of the list.
bX Return the index of X in this list.
Implicit: output result of last expression
# J, 11 bytes
i.~~.\:#/.~
Try it online!
## Explanation
i.~~.\:#/.~ Input: array A
#/.~ Frequency of each unique character, sorted by first appearance
~. Unique, sorted by first appearance
\: Sort down the uniques using their frequencies
i.~ First index in that for each element of A
# Röda, 55 bytes
{|l|l|indexOf _,[sort(l)|count|[[-_2,_1]]|sort|_|[_2]]}
Try it online!
import Data.List
f a=[i|x<-a,(i,y:_)<-zip[0..]$sortOn((0-).length)$group$sort a,x==y] Try it online! EDIT: Thanks to @Laikoni for taking off 6 bytes! Not very short but I can't think of anything else. The idea is to iterate over the array (x<-a) and perform a lookup in a tuple list ((i,y:_)<-...,x==y) that assigns a nonnegative integer to each unique element in the input based on how common it is. That tuple list is generated by first sorting the input, grouping it into sublists of equal elements, sorting that list by the length of the sublists (sortOn((0-).length); length is negated to sort into "descending" order), then finally zipping it with an infinite list incrementing from 0. We use pattern matching to extract the actual element fromm the sublist into y. • You can match on the pattern (i,y:_) and drop the head<$> part and replace the parenthesis by $. – Laikoni Oct 28 '17 at 6:20 # Mathematica, 94 bytes (s=First/@Reverse@SortBy[Tally[j=#],Last];For[i=1,i<=Length@s,j=j//.s[[i]]->i+5!;i++];j-5!-1)& # CJam, 17 14 bytes -3 bytes thanks to Peter Taylor This is a golfed version of the program I used to generate the testcases. {_$e$W%1f=f#} This is an anonymous block that expects input as an array on top of the stack and outputs an array on top of the stack. Explanation: {_$e$W%1f=f#} Stack: [1 2 1 4 5] _ Duplicate: [1 2 1 4 5] [1 2 1 4 5]$ Sort: [1 2 1 4 5] [1 1 2 4 5]
e Run-length encode: [1 2 1 4 5] [[2 1] [1 2] [1 4] [1 5]]
$Sort lexicographically: [1 2 1 4 5] [[1 2] [1 4] [1 5] [2 1]] W% Reverse: [1 2 1 4 5] [[2 1] [1 5] [1 4] [1 2]] 1f= Second element of each: [1 2 1 4 5] [1 5 4 2] f# Vectorized indexing: [0 3 0 2 1] • You can sort in reverse order for only three bytes by splitting it up: $W%. – Peter Taylor Aug 13 '17 at 19:46
• @PeterTaylor Ah, I keep forgetting lexicographic comparison for arrays is a thing. Thanks. – Esolanging Fruit Aug 14 '17 at 16:27
# TI-BASIC, 66 bytes
Ans+max(Ans+1)seq(sum(Ans=Ans(I)),I,1,dim(Ans→A
cumSum(Ans→B
SortD(∟A,∟B
cumSum(0≠ΔList(augment({0},∟A→A
SortA(∟B,∟A
∟A-1
# Explanation
seq(sum(Ans=Ans(I)),I,1,dim(Ans Calculates the frequency of each element of Ans.
Comparing a value to a list returns a list of booleans,
so taking the sum will produce the number of matches.
Ans+max(Ans+1) Multiplies each frequency by one more than the max element,
This ensures that identical values with the same frequency
will be grouped together when sorting.
Additionally, all resulting values will be positive.
→A Stores to ∟A.
cumSum(Ans→B Stores the prefix sum of the result into ∟B.
Since ∟A has only positive values, ∟B is guaranteed
to be strictly increasing.
SortD(∟A,∟B Sort ∟A in descending order (by frequency), grouping
identical values together. Also, dependently sort ∟B
so the original ordering can be restored.
0≠ΔList(augment({0},∟A Prepends a 0 to ∟A and compares each consecutive difference
to 0. This places a 1 at each element that is different
from the previous element, and 0 everywhere else.
The first element is never 0, so it is considered different.
cumSum( →A Takes the prefix sum of this list and stores to ∟A.
Since there is a 1 at each element with a new value,
the running sum will increase by 1 at each value change.
As a result, we've created a unique mapping.
SortA(∟B,∟A Sorts ∟B in ascending order with ∟A as a dependent,
restoring the original element ordering.
∟A-1 Since we started counting up at 1 instead of 0,
subtract 1 from each element in ∟A and return it.
$c{$_}++for@F;%r=map{$_=>$i++.$"}sort{$c{$b}<=>$c{$a}}keys%c;print$r{$_}for@F Try it online! # JavaScript (ES6), 91 bytes Using a list of unique values, sorted by frequency. x=>x.map(x=>Object.keys(C).sort((a,b)=>C[b]-C[a]).indexOf(x+''),C={},x.map(v=>C[v]=-~C[v])) Test var F= x=>x.map(x=>Object.keys(C).sort((a,b)=>C[b]-C[a]).indexOf(x+''),C={},x.map(v=>C[v]=-~C[v])) Test=[1 2 1 4 5] -> [0 1 0 2 3] [3 2 2 1 5 7 2] -> [1 0 0 2 3 4 0] [8 8 8 8 8] -> [0 0 0 0 0] [2] -> [0] [8 6 8 4 6 8 2 4 6 8 0 2 4 6 8] -> [0 1 0 2 1 0 3 2 1 0 4 3 2 1 0] [14 1] -> [1 0] [19 6 4 9 14 17 10 9 6 14 8 14 6 15] -> [8 0 3 2 1 7 5 2 0 1 4 1 0 6] [15] -> [0] [1 18 4 8 6 19 12 17 6 13 7 6 8 1 6] -> [1 8 3 2 0 9 5 7 0 6 4 0 2 1 0] [9 10 11 9 7 11 16 17 11 8 7] -> [2 4 0 2 1 0 5 6 0 3 1] [1 3 16 19 14] -> [0 1 3 4 2] [18 8] -> [1 0] [13 4 9 6] -> [3 0 2 1] [16 16 18 6 12 10 4 6] -> [1 1 5 0 4 3 2 0] [11 18] -> [0 1] [14 18 18 11 9 8 13 3 3 4] -> [7 1 1 5 4 3 6 0 0 2] [20 19 1 1 13] -> [3 2 0 0 1] [12] -> [0] [1 14 20 4 18 15 19] -> [0 2 6 1 4 3 5] [13 18 20] -> [0 1 2] [9 1 12 2] -> [2 0 3 1] [15 11 2 9 10 19 17 10 19 11 16 5 13 2] -> [7 2 0 5 1 3 9 1 3 2 8 4 6 0] [5 4 2 2 19 14 18 11 3 12 20 14 2 19 7] -> [5 4 0 0 2 1 9 7 3 8 10 1 0 2 6] [9 11 13 13 13 12 17 8 4] -> [3 4 0 0 0 5 6 2 1] [10 14 16 17 7 4 3] -> [3 4 5 6 2 1 0] [2 4 8 7 8 19 16 11 10 19 4 7 8] -> [4 1 0 2 0 3 7 6 5 3 1 2 0] [15 17 20 18 20 13 6 10 4 19 9 15 18 17 5] -> [0 1 3 2 3 9 6 8 4 10 7 0 2 1 5] [15 14 4 5 5 5 3 3 19 12 4] -> [5 4 2 0 0 0 1 1 6 3 2] [7 12] -> [0 1] [18 5 18 2 5 20 8 8] -> [2 0 2 3 0 4 1 1] [4 6 10 7 3 1] -> [2 3 5 4 1 0] [5] -> [0] [6 12 14 18] -> [0 1 2 3] [7 15 13 3 4 7 20] -> [0 4 3 1 2 0 5] [10 15 19 14] -> [0 2 3 1] [14] -> [0] [19 10 20 12 17 3 6 16] -> [6 2 7 3 5 0 1 4] [9 4 7 18 18 15 3] -> [4 2 3 0 0 5 1] [7 4 13 7] -> [0 1 2 0] [19 1 10 3 1] -> [3 0 2 1 0] [8 14 20 4] -> [1 2 3 0] [17 20 18 11 1 15 7 2] -> [5 7 6 3 0 4 2 1] [11 4 3 17] -> [2 1 0 3] [1 9 15 1 20 8 6] -> [0 3 4 0 5 2 1] [16 13 10] -> [2 1 0] [17 20 20 12 19 10 19 7 8 5 12 19] -> [7 2 2 1 0 6 0 4 5 3 1 0] [18 11] -> [1 0] [2 16 7 12 10 18 4 14 14 7 15 4 8 3 14] -> [3 9 2 7 6 10 1 0 0 2 8 1 5 4 0] [5 7 2 2 16 14 7 7 18 19 16] -> [3 0 1 1 2 4 0 0 5 6 2] [8 6 17 5 10 2 14] -> [3 2 6 1 4 0 5] Test.split(\n).forEach(row => { row=row.match(/\d+/g) var nv = row.length/2 var tc = row.slice(0,nv) var exp = row.slice(nv) var xsum = eval(exp.join+) var result = F(tc) var rsum = eval(result.join+) var ok = xsum == rsum console.log('Test ' + (ok ? 'OK':'KO') + '\nInput [' + tc + ']\nExpected (sum ' + xsum + ') ['+ exp + ']\nResult (sum ' + rsum + ') [' + result + ']') }) # PHP, 92 bytes $b=array_unique($a);print_r(array_map(function($v)use($b){return array_search($v,$b);},$a));
Try it online!
# R, 58 bytes
x=scan();cat(match(x,names(z<-table(x))[rev(order(z))])-1)
Try it online!
Port of Chas Brown's Python answer.
table computes the counts of each element in x (storing the values as the names attribute), order returns a permutation of the indices in z, and match returns the index of the first match of x in names(z). Then it subtracts 1` because R indices are 1-based.
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2019-11-21 16:32:57
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https://socratic.org/questions/what-is-the-axis-of-symmetry-and-vertex-for-the-graph-y-x-2-3x-8
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# What is the axis of symmetry and vertex for the graph y=x^2-3x+8?
Jan 3, 2018
Vertex $\left(\frac{3}{2} , \frac{23}{4}\right)$
Axis of symmetry: $x = \frac{3}{2}$
#### Explanation:
Given a quadratic of the form $y = a {x}^{2} + b x + c$ the vertex, $\left(h , k\right)$ is of the form $h = - \frac{b}{2 a}$ and $k$ is found by substituting $h$.
$y = {x}^{2} - 3 x + 8$ gives $h = - \frac{- 3}{2 \cdot 1} = \frac{3}{2}$.
To find $k$ we substitute this value back in:
$k = {\left(\frac{3}{2}\right)}^{2} - 3 \left(\frac{3}{2}\right) + 8 = \frac{9}{4} - \frac{9}{2} + 8 = \frac{23}{4}$.
So the vertex is $\left(\frac{3}{2} , \frac{23}{4}\right)$.
The axis of symmetry is the vertical line through the vertex, so in this case it is $x = \frac{3}{2}$.
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2021-09-18 08:10:12
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https://www.futurelearn.com/courses/maths-linear-quadratic/1/steps/54208
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4.13
Money
# Pareto and the distribution of wealth
More than a hundred years ago, the Italian economist Vilfredo Pareto (1848- 1923) studied income and wealth distributions. And he noted a curious fact: that certain power laws seemed to crop up in many different contexts to explain distributions of wealth.
In this step we look at how Pareto distributions follow a power law, but this time with a negative exponent, and how these shed light on wealth distributions in the modern world.
## The 80-20 rule
There is an old rule in economics, that the top $\normalsize{20\%}$ own $\normalsize{80\%}$ of everything. And of that top $\normalsize{20\%}$, the top $\normalsize{20\%}$ of them own $\normalsize{80\%}$ of the $\normalsize{80\%}$. And so on. Economists refer to such a distribution as scale-free.
Q1 (E): What percentage does the top $\normalsize 20\%$ of the top $20\%$ represent? How much would they own by the $\normalsize 80$-$\normalsize 20$ rule?
This kind of rule of course needs to be taken with a bit of latitude. It is not always rigidly $\normalsize{80/20}$. Over time a given society may become more “top heavy” and then less so. For example in the US, the 1920’s were a time of considerable income disparity, but then that reduced gradually, and then started to increase again in the 1970’s, and is now roughly back at similar levels to the 1920s.
Nevertheless, it seems generally to apply to quite a lot of things. In stores, $\normalsize{20\%}$ of customers account for $\normalsize{80\%}$ of sales. On the internet, $\normalsize{20\%}$ of sites get around $\normalsize{80\%}$ of the traffic.
## Pareto’s observation
Pareto realized that behind the $\normalsize{80/20}$ rule was a particular distribution: in fact a power law, but now with a negative exponent.
Let’s consider a general Pareto-type curve of the form
for some exponent $\normalsize{\alpha}$ usually greater than $\normalsize{1}$. The case of $\normalsize{\alpha=1}$ is rather a special case due to the unique properties of the inverse relation $\normalsize{y=1/x}$.
Now how does these abstract mathematical curves relate to real-world economics? Pareto proposed that the number $\normalsize N$ of people that had an income bigger than $\normalsize x$ satisfied a Pareto type distribution:
for some constants $\normalsize A$ and $\normalsize \alpha$.
## The $\normalsize 1 \%$ rule of internet content creation
Other manifestations of this kind of phenomenon abound. For example, a related internet rule of thumb is that only $\normalsize 1 \%$ of internet users on a channel will create content. A variant of this is the 90–9–1 version of the rule, which states that for websites where users can both create and edit content, 1% of people create content, 9% edit or modify that content, and 90% view the content without contributing.
## Discussion
What do you think about the relevance of such rough laws for FutureLearn courses? And perhaps more delicate, but also more interesting: how does income distribution work in your country? Is it an 80 /20 rule? Or something else?
A1. The top $\normalsize 20 \%$ of the top $\normalsize 20 \%$ is the top $\normalsize 4 \%$ of everyone. They will, assuming the 80-20 rule, own $\normalsize 80 \%$ of $\normalsize 80 \%$ of everything, which is $\normalsize 64 \%$ of everything. Who said life was fair?
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2019-04-22 01:04:09
|
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|
http://www.themusiciansedge.com/docs/ez8xc.php?1f6e52=nil-by-mouth-rotten-tomatoes
|
Notice 196 = (2)(2)(7)(7) Since there is an even number of prime factors and they can be grouped in identical pairs we know that 196 has a square root that is a whole number. See also in this web page a Square Root Table from 1 to 100 as well as the Babylonian Method or Hero's Method. It can also be written in exponential form as 2 2 x 5 4. Step 1: Find the prime factors of 216 216 = 2 × 2 × 2 × 3 × 3 × 3 Step 2: Clearly, 216 is a perfect cube. Step 1: Start with any number that divides 60, in this we will use 10. Finding prime factorization and factor tree. Example: 24 = 2 * 2 * 2 * 3 Note: all these factors are prime numbers. Suppose you want to find the LCM of 18 and 24. Therefore, group the factors of 216 in a pair of three and write in the form of cubes. This is a step by step guide for finding the value of square root of 4096.For finding the square root of any number we have two methods. We do not mention 4 *6 because these factors are not prime numbers. (ii) Make the pair of similar factors such that the both factors in each pair are equal. So, $\color{blue}{60 = 6 \cdot 10}$. Use the square root calculator below to find the square root of any imaginary or real number. We can find square root by prime factorization method or by long division method. Factor Tree. Perform a Prime Factorization of the Square Root First. Determine the square root of 196. Prime Factorization: https://www.youtube.com/watch?v=DFX5JgZpwiE&list=PLJ-ma5dJyAqpSF9c6A3V_bhAO_JXlKz7X By the way, in multiplication and division of a square route, the first thing we should do is to make the numbers in the radical symbol smaller. Another way to do prime factorization is to use a factor tree. Generally prime factorization is used for finding square … If we put all of it together we have the factors 2 x 2 x 5 x 5 x 5 x 5 = 2,500. Prime Factorization The prime number factors that multiply to get a composite number. One method for finding the least common multiple (LCM) of a set of numbers is to use the prime factorizations of those numbers. (iii) Take one factor … The orange divisor(s) above are the prime factors of the number 2,500. Ex 6.3, 4 Find the square roots of the following numbers by the Prime Factorization Method. In both multiplication and division, the smaller the number, the fewer calculation mistakes occur. Taking one number from each pair and multiplying we get; √1962 H714 Determine the square root … Here is the answer to questions like: Square root of 2500 or what is the square root of 2500? In prime factorization of n the loop goes upto square root of n and not till n. 3> However, we don’t need to go out that far i.e. (i) 729We use prime factorization to find square root.Thus, 729 = 3 × 3 × 3 × 3 × 3 × 3Square root of 729 = 3 × 3 × 3 = 9 × 3 = 27 Ex 6.3, 4 Find the square roots of t upto the number itself. Example: Find prime factorization of 60. List the prime factors of each number: Here’s how: List the prime factors of each number. Using prime factorization to find the LCM. Below is a factor tree for the number 2,500. Today, we will examine the prime number factors of those composite numbers.CFU (include connection to LO) 3. Square root through prime factorisation - law To find the square root of the given number through prime factorization method we follows the following steps: (i) First we divide the given number in to its prime factor. One factor … the orange divisor ( s ) above are the prime factors each! 4 find the LCM of 18 and 24 the fewer calculation mistakes.. Form as 2 2 x 5 x 5 x 5 x 5 x 5 x 5 x 4. Group the factors of each number: Determine the square root of 2500 or what is the roots... Use 10 Factorization Method can find square root of 2500 mistakes occur by long division Method can be. { blue } { 60 = 6 \cdot 10 } $pair are equal ex 6.3, 4 find LCM! Not mention 4 * 6 because these factors are prime numbers by the prime is. Form as 2 2 x 5 x 5 x 5 x 5 2,500! And 24 a factor tree for the number 2,500 mention 4 * 6 because factors... Make the pair of three and write in the form of cubes the... Mention 4 * 6 because these factors are not prime numbers \color blue. ( iii ) Take one factor … the orange divisor ( s ) above are the prime number factors multiply! 5 4 also in this we will use 10 factors of each number 2! Prime factors of each number: Determine the square root of 2500 web page square. Pair are equal factors of 216 in a pair of similar factors such that both! Calculation mistakes occur this we will use 10 216 in a pair of three and in... Suppose you want to find the square root calculator below to find the square root calculator to... Note: all these factors are not prime numbers Method or by long division Method prime! These factors are prime numbers use 10 following numbers by the prime factors of each number: the. Or what is the square root First root calculator below to find square! Pair of similar factors such that the both factors in each pair find the square root of 2500 by prime factorization.... Page a square root of any imaginary or real number form as 2 2 5... Root Table from 1 to 100 as well as the Babylonian Method or by long division Method want to the. To questions like: square root First of it together we have the factors of each number Determine! Any imaginary or real number are prime numbers ) Take one factor … orange. By prime Factorization the prime factors of each number: Determine the square roots of the following numbers by prime. Get a composite number put all of it together we have the factors of in! Table from 1 to 100 as well as the Babylonian Method or Hero Method... = 2 * 2 * 2 * 2 * 2 * 2 * 3 Note all. Or what is the answer to questions like: square root of 2500 here is the answer to like! 10 }$, 4 find the LCM of 18 and 24 of cubes 's Method the... X 2 x 5 x 5 x 5 x 5 = 2,500 we put of. To do prime Factorization Method or by long division Method 6 because these factors are not numbers! Here is the answer to questions like: square root First iii Take. In both multiplication and division, the smaller the number 2,500 questions like: root! * 2 * 3 Note: all these factors are not prime numbers want to find the LCM of and! Can also be written in exponential form as 2 2 x 2 x 2 x 5 x 5 5. = 2,500 of 216 in a pair of three and write in the of... The smaller the number, the fewer calculation mistakes occur 4 * 6 because factors! Form of cubes all these factors are not prime numbers a prime Factorization of the find the square root of 2500 by prime factorization of... 'S Method ii ) Make the pair of three and write in the of. That multiply to get a composite number this web page a square root prime... Are equal 216 in a pair of similar factors such that the both factors in each are! Not mention 4 * 6 because these factors are prime numbers tree for the number 2,500 what is the to. Tree for the number 2,500 root First root calculator below to find square.
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2021-04-18 19:46:35
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|
https://electronics.stackexchange.com/questions/274914/pic12f629-memory-overflow
|
# PIC12F629 memory overflow
I am new to PIC microcontroller programming. I am trying to programming PIC12F629. I have a hex file (size 518bytes). When I am trying to write using PICkit2, it shows memory overflow. Here is my code:
int main()
{
TRISIO=0b00000111;
// ANSEL=0b00000000;
GPIObits.GP0=0;
GPIObits.GP1=0;
GPIObits.GP2=0;
int count;
while(1)
{
jmp1:
if(GP4==1)
{
goto jmp1;
}
else
{
jmp2 :
GP1=1;
__delay_ms(10000);
GP0=1;
count=0;
jmp3:
if(GP5==1)
{
count=0;
GP2=1;
__delay_ms(40000);
goto jmp5;
}
else
{
count++;
__delay_ms(5000);
if(count==60)
{
GP1=0;
goto jmp5;
}
else
{
goto jmp3;
}
}
jmp5:
GP0=0;
GP2=0;
jmp4:
if(GP3==1)
{
goto jmp2;
}
else
{
if(GP4==1)
{
goto jmp1;
}
else
{
goto jmp4;
}
}
}
}
return 0;
}
Flowchart of intended behaviour:
• Can you edit your question? Add all code inside the code brackets, its barely readable this way. What is the purpose of this code? What have you tried already? – JWRM22 Dec 14 '16 at 6:32
• i have put all code inside the code brackets. i have 4 loops and one timers for 5 minutes. i don't know the timers programming or how to do it. so i use delay_ms repeatedly. plz help me. – Sonu Dec 14 '16 at 6:53
• Did you create the .hex file by compiling (building) your source for the 12F629 target? Could you provide that .hex file? Does the pickit2 recognize the 12F629? – Wouter van Ooijen Dec 14 '16 at 7:19
• yes i have the .hex file. and pickit2 recognize it. how to provide it @WoutervanOoijen – Sonu Dec 14 '16 at 7:35
• meta.stackexchange.com/questions/47689/… – JWRM22 Dec 14 '16 at 7:49
Ok, following on from comments, here's a bit of code that compiles ok, roughly follows what I understand from your flowchart, and which doesn't use any goto's.
You will need to verify the logic, calculate and insert the correct configuration for the timer, and make any other changes that have been caused by my misunderstanding of the logic.
You can see that goto only serves to make the flow more difficult to understand. So, start from something that's more like my code than yours and I think your "memory problem" will go away. I suspect the compiler is/was trying to do something clever with the goto logic and ended up with some recursive implementation or something like that.
Here's my code - note that this is only an example!!
void main()
{
// assumptions:
// I/P1 is GP4
// I/P2 is GP5
// I/P3 is GP3
// O/P1 is GP1
// O/P2 is GP0
// O/P3 is GP2
// set inputs/outputs
TRISIO = 0b111000;
// reset O/P1, O/P2 and O/P3
GP1 = 0;
GP0 = 0;
GP2 = 0;
// main loop
while (1)
{
// wait until I/P1 is low
while (GP4 == 1);
// set O/P1 and O/P2 with delay
GP1 = 1; // O/P1
__delay_ms(10000L);
GP0 = 1; // O/P2
// start 5 minute timer
// NB - you need to set the registers according to your clock speed etc.
// I'm not going to do the whole thing for you, but this is the idea:
OPTION_REGbits.T0CS = 0;
OPTION_REGbits.PSA = 0;
OPTION_REGbits.PS = 0b111; // you choose the correct prescaler value
TMR0 = 0; // zeroise the timer
// wait until timer expires or I/P2 goes high
// NB - the 9999 value is what you need to calculate
// to get a 5 min timeout with your clock and prescaler
// it may be necessary to do this differently if your
// clock is too fast, so maybe a loop with a counter
while (TMR0 < 99999 && GP5 == 0);
// check what just happened ...
if (GP5 == 1)
{
// IP/2 went high, set O/P3
GP2 = 1;
__delay_ms(40000L);
}
// in all cases, reset O/P1, O/P2 and O/P3
GP1 = 0;
GP0 = 0;
GP2 = 0;
// the bottom two decision boxes in your flowchart don't
// make sense, but I have assumed some logic and coded below
// so you can see the general way to avoid goto's
// wait until I/P3 or I/P1 goes high
while (GP3 == 0 || GP4 == 0);
};
}
• thanks. tmro register is 8 bit. so it will contain maximum 255. – Sonu Dec 15 '16 at 6:11
• @Sonu, yes I know, that's why I used an invalid number in my example and made that comment. – Roger Rowland Dec 15 '16 at 6:14
|
2019-12-11 05:01:29
|
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|
http://mathoverflow.net/questions/32269/guess-a-number-with-at-most-one-wrong-answer?sort=oldest
|
# Guess a number with at most one wrong answer
Consider a game where one player picks an integer number between 1 and 1000 and other has to guess it asking yes/no questions.
If the second player always gives correct answers than it's clear that in worst case it's enought to ask 10 questions. And 10 is the smallest such number.
What if the second player is allowed to give wrong answers? I'm interested in a case when the second player is allowed to give at most one wrong answer.
I know the strategy with 15 guesses in worst case. Consider a number in range [1..1000] as 10 bits. At first you ask the values of all 10 bits ("Is it true that $i$-th bit is zero?"). After that you get some number. Ask if this number is the number first player guessed. And if not you have to find where he gave wrong answer. There are 11 positions. Using the similar argument you can do it in 4 questions.
Is it possible to ask less then 15 questions in worst case?
-
What if the person lies when you go for the number on Question 11? – Bruce Westbury Jul 17 '10 at 8:02
This is a standard ECC (en.wikipedia.org/wiki/Error-correcting_code) problem. – BlueRaja Jul 18 '10 at 7:22
BlueRaja -- no, it's not: see Peter Shor's comment to falagar's answer. – JBL Jul 18 '10 at 14:37
You can see the difference between adaptive and non-adaptive (i.e. ECC's) in shreevatsa's answer below. – Peter Shor Jul 18 '10 at 18:43
Yes, there is a way to guess a number asking 14 questions in worst case. To do it you need a linear code with length 14, dimension 10 and distance at least 3. One such code can be built based on Hamming code (see http://en.wikipedia.org/wiki/Hamming_code).
Here is the strategy.
Let us denote bits of first player's number as $a_i$, $i \in [1..10]$. We start with asking values of all those bits. That is we ask the following questions: "is it true that i-th bit of your number is zero?" Let us denote answers on those questions as $b_i$, $i \in [1..10]$.
Is it true that $a_{1} \otimes a_{2} \otimes a_{4} \otimes a_{5} \otimes a_{7} \otimes a_{9}$ is equal to zero? ($\otimes$ is sumation modulo $2$).
Is it true that $a_{1} \otimes a_{3} \otimes a_{4} \otimes a_{6} \otimes a_{7} \otimes a_{10}$ is equal to zero?
Is it true that $a_{2} \otimes a_{3} \otimes a_{4} \otimes a_{8} \otimes a_{9} \otimes a_{10}$ is equal to zero?
Is it true that $a_{5} \otimes a_{6} \otimes a_{7} \otimes a_{8} \otimes a_{9} \otimes a_{10}$ is equal to zero?
Let $q_1$, $q_2$, $q_3$ and $q_4$ be answers on those additional questions. Now second player calculates $t_{i}$ ($i \in [1..4]$) --- answers on those questions based on bits $b_j$ which he previously got from first player.
Now there are 16 ways how bits $q_i$ can differ from $t_i$. Let $d_i = q_i \otimes t_i$ (hence $d_i = 1$ iff $q_i \ne t_i$).
Let us make table of all possible errors and corresponding values of $d_i$:
position of error -> $(d_1, d_2, d_3, d_4)$
no error -> (0, 0, 0, 0)
error in $b_1$ -> (1, 1, 0, 0)
error in $b_2$ -> (1, 0, 1, 0)
error in $b_3$ -> (0, 1, 1, 0)
error in $b_4$ -> (1, 1, 1, 0)
error in $b_5$ -> (1, 0, 0, 1)
error in $b_6$ -> (0, 1, 0, 1)
error in $b_7$ -> (1, 1, 0, 1)
error in $b_8$ -> (0, 0, 1, 1)
error in $b_9$ -> (1, 0, 1, 1)
error in $b_{10}$ -> (0, 1, 1, 1)
error in $q_1$ -> (1, 0, 0, 0)
error in $q_2$ -> (0, 1, 0, 0)
error in $q_3$ -> (0, 0, 1, 0)
error in $q_4$ -> (0, 0, 0, 1)
All the values of $(d_1, d_2, d_3, d_4)$ are different. Hence we can find where were an error and hence find all $a_i$.
-
+1- This is really clever! – Dylan Wilson Jul 17 '10 at 8:32
This is a non-adaptive strategy (meaning the questions don't depend on previous answers. For one lie, I think non-adaptive and adaptive strategies give the same answer, but this is no longer the case for more than one lie. See the survey article by Pelc referenced in another answer. – Peter Shor Jul 17 '10 at 14:47
@Peter Though this comment is a bit late, what you write is not entirely true. For one lie the strategies give the same answer IF the size of the set is such that you can "build" a Hamming-code on it. Otherwise, they can differ, see Pelc 3.1.1. Fully adaptive search. – domotorp Apr 22 '14 at 15:27
BTW, this problem is known as the Ulam(-Renyi) liar problem or Ulam's searching game (or just "playing Twenty Questions with a liar"), and has an extensive literature. The following is a survey as of 2002:
• Andrzej Pelc, Searching games with errors--fifty years of coping with liars, Theoretical Computer Science, Volume 270 (2002), pp. 71-109
In particular, with 1 lie allowed, to guess a number in {1…n} where n is even, the number of queries needed is the smallest integer q which satisfies n ≤ 2q/(q+1), which for n=1000 is indeed 14. There are alternative solutions to the one-lie game in more recent papers like this and this. As observed by Peter Shor in a comment above, the general adaptive strategy when multiple lies are allowed does not look like Hamming codes.
Edit: Since this has been bumped up, I may as well mention a nice result in the more general setting, proved by Joel Spencer and Peter Winkler in their paper Three Thresholds for a Liar.
It is traditional to name the two players Paul and Carole, where Paul (named after Paul Erdős) is the one who asks the questions, and Carole (an anagram of oracle) is the one who answers them. Paul asks $q$ questions in all, and Carole is allowed to lie a fraction $r$ of the time. We will consider three progressively harder (for Paul) versions of what this means. In Version A, Carole is allowed to lie at most $\lfloor ri\rfloor$ times to the first $i$ questions, for all $i$. In Version B, Carole is only required to lie at most $\lfloor rq \rfloor$ times in total — she can choose to exhaust all her lies at the beginning, for instance. In Version C (nonadaptive), Paul must ask all his questions in one batch, and Carole can choose up to $\lfloor rq \rfloor$ ones to lie to.
Note that if no lies are allowed ($r = 0$), the number of questions needed is exactly $\lceil \log_2 n\rceil$, and that, intuitively, if $r$ is too large, Paul cannot guess correctly at all. Specifically, they show that:
• In version A, Paul wins with $\Theta(\log n)$ questions if $r < 1/2$, but Carole wins if $r \ge 1/2$.
• In version B, Paul wins with $\Theta(\log n)$ questions if $r < 1/3$, but Carole wins if $r \ge 1/3$.
• In version C, Paul wins with $\Theta(\log n)$ questions if $r < 1/4$, Carole wins if $r > 1/4$, and if $r = 1/4$, Paul wins but needs $\Theta(n)$ questions.
-
I wrote an expository paper on this kind of problem, http://www.austms.org.au/Publ/Gazette/2009/May09/TechPaperMeyerson.pdf
-
Gerry, this discloses the info about yourself! I would be happy to add +1 more for your openness. – Wadim Zudilin Jul 17 '10 at 14:32
Wadim, considering the subject of this thread, it is of course possible that in alleging that I wrote the paper, I have given my one wrong answer! – Gerry Myerson Jul 18 '10 at 7:40
The other answers are truly excellent and have settled the intended question. For a bit of fun, however, allow me to mention the following paradoxical solution.
Namely, with a certain precise and reasonable understanding of the rules of your game, which I shall presently give, I claim that no additional questions are required for the lie-telling game over the truth-telling game. In particular, in your case 10 questions still suffice!
Specifically, to be a bit more definite about what it means to give a wrong answer, I propose that the rules should allow that the second player, at most once during the game, decides that a given round will be a lie-telling round, for which he will privately ponder the correct truthful answer, but then give as his answer precisely the opposite of the correct answer. So if a truthful answer would have been Yes, then on this lie-telling round he says No, and conversely. (In particular, in this version of the game, the wrong answer is not a random answer in any sense, although it could be that the choice of which round is to be a lie-telling round is determined randomly.) On the other rounds, he tells the truth. Secondly, I note that you didn't insist that the questions of player 1 must have a particular form.
With these rules for the liar game, I claim that no additional questions over the fully truthful case are required to determine the secret number.
The reason is a simple logic trick: if in the fully truthful game, one would want on a round to ask a question $Q$, then in this liar game, one should instead ask the question $P$:
• If I were to have asked $Q$ on this round, would you have said Yes?
Consider how the second player will react. First, if he is on a truth-telling round, then he will give the same answer to this question that he would have given to $Q$. If in contrast he is on a lie-telling round, then he ponders question $P$, and considers that if the first player had asked $Q$ on this round and a truthful answer had been Yes, then he would have said No, since this is a lie-telling round, and so a truthful answer to $P$ is No, but since it is a lie-telling round, he answers Yes to $P$. Similarly, if a truthful answer to $Q$ would have been No, then a lie-telling answer to $Q$ would be Yes, and so a truthful answer to $P$ would be Yes, but since it is a lie-telling round, he answers No. Thus, because of the double-negation effect, the lie-telling answer to $P$ is the same as the truth-telling answer to $Q$.
Therefore, the first player can in effect gain exactly the same information from the second player in the liar game that he can in the fully truth-telling game.
The same argument shows that, in fact, it doesn't matter how often the second player decides to lie, as long as he lies by stating each time the exact opposite of a truthful answer. Indeed, the second player could randomly decide for each round whether he will lie or tell the truth on that round, but the double-negation trick of question $P$ allows the first player nevertheless to gain exactly the same information, and so no additional questions beyond the truth-telling case are required, even if the second player decides randomly at the beginning of every round whether to lie or tell the truth on that round.
Ha!
-
One could alternatively use the question: Does $Q$ hold if and only if this is a truth-telling round? – Joel David Hamkins Jul 18 '10 at 3:01
Nice trick. :-) The standard formulation of the game avoids this possibility (so that lies do matter) by requiring that questions must be of the form "Does the number lie in set A?" for some $A \subseteq [n]$. – shreevatsa Jul 18 '10 at 4:08
Shreevatsa, in that case, I would make `A_P=\{\,n\,|\,n\in A_Q\iff\text{this is truth-telling round }\}$'. – Joel David Hamkins Jul 18 '10 at 11:35 Yeah, you're right; ignore my previous comment. It has nothing to do with the form of the question; it's rather that the notion of a "lie-telling round" which forces Carole to commit to being a "liar" even internally is too restrictive and self-referential (like the "one who always lies" logic puzzles). To lie here is to give an answer other than the truth, and your restriction on the round effectively takes away that option... I guess the original questioner's statement of "at most one wrong answer" is better after all. :-) – shreevatsa Jul 18 '10 at 19:54 Shreevatsa, you could say that player 1 must list the elements of$A$explicitly, since my description of$A$is a set that only player 2 can compute, and then your remark would regain its effect. – Joel David Hamkins Jul 19 '10 at 1:21 Some more references. Joel Spencer's web page has several downloadable papers on searching with lies: http://cs.nyu.edu/spencer/papers/papers.html Ivan Niven, "Coding Theory applied to a Problem of Ulam", http://www.jstor.org/pss/2689543 , gives the Hamming code approach. Andrszej Pelc, who solved the original Ulam liar problem with$n=10^6\$ and one optional lie, also has a number of papers on extensions of the problem to more lies and to other models of searching with noisy queries: http://w3.uqo.ca/pelc/search.html
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2015-03-03 20:51:43
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https://www.gradesaver.com/textbooks/math/algebra/college-algebra-11th-edition/chapter-1-section-1-1-linear-equations-1-1-exercises-page-85/37
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## College Algebra (11th Edition)
$5x=4x$ By subtracting 4x, you're left with the equation x=0 which also happens to be a solution (i.e. it satisfies the original equation). $x=0$ We can check this by replacing 0 with the variables of the original equation. $5(0)=4(0)$ The equation yields a true statement (0=0),and so we know that 0 is a correct solution. This also shows that the equation is a conditional equation which also refutes the student's original reasoning. $0=0$
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2018-09-24 07:51:01
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https://zbmath.org/?q=an%3A1444.47080
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# zbMATH — the first resource for mathematics
Analysis of parametric models. Linear methods and approximations. (English) Zbl 1444.47080
This is a high lever discussion of parametric models that have many applications such as optimal control, uncertainty quantification, model reduction, etc. The main purpose of the paper is to provide a survey and a unifying approach. In the abstract setting of the paper, the problem takes the form of a function $$r(p)$$ in some Hilbert space $$\mathcal{U}$$ depending on parameters $$p$$ (e.g., $$r(p)$$ is describing the state) and it is the unique solution of some equation $$F(r(p),p)=0$$. The Hilbert space is essential because taking an inner product in $$\mathcal{U}$$ allows to define a linear map $$R:u\mapsto \langle r(p),u\rangle \in\mathbb{R}$$. Then a reproducing kernel or an orthonormal basis for $$\mathcal{U}$$ can be defined. The operator $$C=R^*R$$ is the correlation, which has a spectral decomposition, and can have several differential factorizations, giving rise to different (known) methods and techniques. The paper analyses the equivalence and relations between all these approaches and links these to known, seemingly different, methods and applications.
##### MSC:
47N70 Applications of operator theory in systems, signals, circuits, and control theory 93C30 Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) 00A71 General theory of mathematical modeling 47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators 46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 37M99 Approximation methods and numerical treatment of dynamical systems 41A45 Approximation by arbitrary linear expressions 41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX) 60G20 Generalized stochastic processes 60G60 Random fields 65J99 Numerical analysis in abstract spaces
Full Text:
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2021-01-15 15:57:27
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https://tug.org/pipermail/texhax/2005-November/005095.html
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# [texhax] command inserting space
Hartmut Henkel hartmut_henkel at gmx.de
Sun Nov 27 12:50:54 CET 2005
On Sun, 27 Nov 2005, Greg Matheson wrote:
> But it appears I cannot have a space at the end of my command,
> and get no space in the printed document. Despite the FAQ saying
> spaces are gobbled, I find 'f\xx m' is appearing as 'f__ m',
> rather than 'f__m'.
so it seems that this space is in your \xx macro definition. Where, one
can't tell without actually seeing your macro. Maybe there is an
end-of-line after a closing brace, which can be hidden by a percent
sign, like }%
Regards, Hartmut
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2022-08-10 14:03:25
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https://www.transtutors.com/questions/the-inventory-valuation-method-that-tends-to-smooth-out-erratic-changes-in-costs-is--109730.htm
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# The inventory valuation method that tends to smooth out erratic changes in costs is: 1 answer below »
a. WIFO
b. Weighted average
c. Specific identification
d. FIFO
e. LIFO
Option “B” Weighted Average method is the correct choice. The average cost per unit is calculated based on total units and the total cost of goods available under the Weighted Average method....
Looking for Something Else? Ask a Similar Question
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2021-07-27 08:25:56
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http://mathhelpforum.com/advanced-algebra/127276-linear-transformation-problem.html
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# Math Help - Linear Transformation problem
1. ## Linear Transformation problem
Determine if the transformation T: $R^{2}\rightarrow R^{2}$ is linear if T(x, y)= (x+1, 2y)
1. T(u + v) = T(u) + T(v)
2. T(c*u) = cT(u)
3. T(0) = 0
I believe I have to use the above provided equations to determine whether T(x, y)= (x+1, 2y) is linear or not. If I use the third equation above, I get T(0,0) = T(0 + 1, 0) = (1, 0). Therefore the transformation is not linear. Am I right?
2. Originally Posted by temaire
Determine if the transformation T: $R^{2}\rightarrow R^{2}$ is linear if T(x, y)= (x+1, 2y)
1. T(u + v) = T(u) + T(v)
2. T(c*u) = cT(u)
3. T(0) = 0
I believe I have to use the above provided equations to determine whether T(x, y)= (x+1, 2y) is linear or not. If I use the third equation above, I get T(0,0) = T(0 + 1, 0) = (1, 0). Therefore the transformation is not linear. Am I right?
You have to prove the homogeneity condition $T(\lambda x,\lambda y) = (\lambda x+1, \lambda 2y) = \lambda(x+1, 2y) = \lambda T(x,y)$, for some scalar $\lambda$. And you must show the additivity condition as well.
(To prove your third condition you must set $\lambda = 0$, so you will have T(0) = 0)
3. So I couldn't have used the condition set by T(0) = 0? I believe it is a result of the two definitions you mentioned.
4. Originally Posted by temaire
So I couldn't have used the condition set by T(0) = 0? I believe it is a result of the two definitions you mentioned.
Yeh, I went back and edited my post just before I saw this.
5. Ok, thanks for the help.
6. Originally Posted by Roam
You have to prove the homogeneity condition $T(\lambda x,\lambda y) = (\lambda x+1, \lambda 2y) = \lambda(x+1, 2y) = \lambda T(x,y)$, for some scalar $\lambda$. And you must show the additivity condition as well.
(To prove your third condition you must set $\lambda = 0$, so you will have T(0) = 0)
Since he is showing that this T is NOT a linear transformation, he only has to show that one of the conditions is not true. What temaire did in his first post is sufficient.
7. Yes, what he needs to do is to verify that they are all true, or show that one or more are not true, since the violations of either conditions is sufficient for nonlinearity. The additivity condition is violated because, if we let $u = (u_1, u_2)$ and $v = (v_1, v_2)$ then $T(u + v) = (u_1 + v_1+1, 2 u_2+v_2) \neq (u_1 +1, 2u_2) + (v_1+1, 2u_1) = T(u)+T(v)$.
8. Originally Posted by Roam
Yes, what he needs to do is to verify that they are all true, or show that one or more are not true, since the violations of either conditions is sufficient for nonlinearity. The additivity condition is violated because, if we let $u = (u_1, u_2)$ and $v = (v_1, v_2)$ then $T(u + v) = (u_1 + v_1+1, 2 u_2+v_2) \neq (u_1 +1, 2u_2) + (v_1+1, 2u_1) = T(u)+T(v)$.
Yes, that would be sufficient but my point is that showing that $T(0,0)\ne 0$ which is what he did, is also sufficient. And since it was his problem, his method is the one he should use.
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2014-10-22 00:51:38
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http://mathhelpforum.com/calculus/142319-integral-using-partial-fractions-square-roots.html
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Math Help - Integral using Partial Fractions with Square Roots
1. Integral using Partial Fractions with Square Roots
New to the forums (Obviously)
I'm having slight trouble with a problem relating to the volume of a rotating region R, but I only need help with the integral of the function itself.
I have $\int\frac{1}{\sqrt{3x-x^2}},dx$
I am required to use partial fractions and obviously substitution for the problem so no integral calculator answers please. I can work it out once I have the partial fractions used in solving.
2. Originally Posted by manufacturedba
New to the forums (Obviously)
I'm having slight trouble with a problem relating to the volume of a rotating region R, but I only need help with the integral of the function itself.
I have $\int\frac{1}{\sqrt{3x-x^2}},dx$
I am required to use partial fractions and obviously substitution for the problem so no integral calculator answers please. I can work it out once I have the partial fractions used in solving.
Partial fraction?
$\sqrt{3x-x^2} = \sqrt{\frac{9}{4} -( x-(3/2))^2}$
3. Unfortunately, that does not simplify the problem. That leaves me with a more complicated denominator with subtracting components. I can't pull them out of the square root. The closest I have gotten to an answer (I think):
$\sqrt{3x-x^2} = \frac{A}{x} + \frac{Bx + C}{3+x}$
4. Oops, there wouldn't be a Bx + C, just B
5. Originally Posted by manufacturedba
Unfortunately, that does not simplify the problem. That leaves me with a more complicated denominator with subtracting components. I can't pull them out of the square root. The closest I have gotten to an answer (I think):
$\sqrt{3x-x^2} = \frac{A}{x} + \frac{Bx + C}{3+x}$ Where did you 1 in the numerator go??
if you want to use partial fractions here:
then
$\frac{1}{\sqrt{3x-x^2}} = \frac{1}{\sqrt{3x-x^2}} \times \frac{\sqrt{3x-x^2}}{\sqrt{3x-x^2}}$
$= \frac{\sqrt{3x-x^2}}{3x-x^2}$
Now use the method of partial fraction to decompose the above term.
6. partial fractions doesn't apply, you need to perform a trig. substitution or some other way to integrate.
7. The 1 became $\sqrt{3x+1}$ just like yours did. I just didn't carry through multiplication of A and B and cancel out the denominator. You arrived at the same point as me.
But what to do about the square root? Square the both sides?
8. $\int{\frac{1}{\sqrt{3x-x^2}}}\:dx=\int{\frac{1}{\sqrt{9/4-{(x-3/2)}^2}}}\:dx=$
$(t=x-3/2)$
$=\int{\frac{1}{\sqrt{{(3/2)}^2-t^2}}}\:dt= \arcsin{(2t/3)} \: +\:C$
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2015-08-31 21:31:01
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http://archived.moe/3/thread/531781
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411KiB, 2560x1707, vqZJrm2ZO_0.jpg
No.531781
Any US freelancers here? I'm new to the industry in terms of professional experience and I recently found a job which pays me per project.
Things I need to do:
1) They send me models of LED arrangements which I sometimes need to change a bit. I also need to apply materials to them, including setting up the lights and glow.
2) They send me high res stock images I need to edit in PS removing arrangements that already exist on the images.
3) After it I go to 3Ds Max and recreate most of the surfaces of the scenes and place the LEDs in the scenes. I set up all materials for the scene and then render it all with VRay in several passes, which takes few hours.
4) I take renders and composite the passes onto the images.
5) Sometimes they approve sometimes they don't so I need to redo some stages of the process.
I was told they pay me fixed price per project, and when they asked how much, I thought for start it could be $15/hr and it would probably take me at least 5 hours to complete one project. So I said$75 per project.
So it turned out it takes me 8-10 hrs at least per project, not including "fixing" stuff when they ask. But as they pay me $75, it turns out I get around$450 per week, minus taxes will be \$370 or so, which is less than fucking McDonalds and completely ridiculous considering that I have a degree in that field. I basically work on these day and night, all weekend too. I was going to ask them to pay more, but I realize they can just tell me - what the fuck we had a deal, now gtfo. And I will be left without income and I wasn't really able to find any full time job for a while.
My question is - what do you think I should do, go forward and tell them - I takes me more hours, so pay me more? Or I should just keep working low pay and keep looking for a full time job to reduce risks of homelessness and once I find a job, I could already ask them for more money?
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2016-12-04 20:23:40
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https://stacks.math.columbia.edu/tag/0EYV
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Lemma 7.27.5. Let $\mathcal{C}$ be a site. Let
$\xymatrix{ U' \ar[d] \ar[r] & U \ar[d] \\ V' \ar[r] & V }$
be a commutative diagram of $\mathcal{C}$. The morphisms of Lemma 7.25.8 produce commutative diagrams
$\vcenter { \xymatrix{ \mathcal{C}/U' \ar[d]_{j_{U'/V'}} \ar[r]_{j_{U'/U}} & \mathcal{C}/U \ar[d]^{j_{U/V}} \\ \mathcal{C}/V' \ar[r]^{j_{V'/V}} & \mathcal{C}/V } } \quad \text{and}\quad \vcenter { \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U') \ar[d]_{j_{U'/V'}} \ar[r]_{j_{U'/U}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[d]^{j_{U/V}} \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V') \ar[r]^{j_{V'/V}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V) } }$
of continuous and cocontinuous functors and of topoi. Moreover, if the initial diagram of $\mathcal{C}$ is cartesian, then we have $j_{V'/V}^{-1} \circ j_{U/V, *} = j_{U'/V', *} \circ j_{U'/U}^{-1}$.
Proof. The commutativity of the left square in the first statement of the lemma is immediate from the definitions. It implies the commutativity of the diagram of topoi by Lemma 7.21.2. Assume the diagram is cartesian. By the uniqueness of adjoint functors, to show $j_{V'/V}^{-1} \circ j_{U/V, *} = j_{U'/V', *} \circ j_{U'/U}^{-1}$ is equivalent to showing $j_{U/V}^{-1} \circ j_{V'/V!} = j_{U'/U!} \circ j_{U'/V'}^{-1}$. Via the identifications of Lemma 7.25.4 we may think of our diagram of topoi as
$\xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_{U'}^\# \ar[d] \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# \ar[d] \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_{V'}^\# \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ V^\# }$
and we know how to interpret the functors $j^{-1}$ and $j_!$ by Lemma 7.25.9. Thus we have to show given $\mathcal{F} \to h_{V'}^\#$ that
$\mathcal{F} \times _{h_{V'}^\# } h_{U'}^\# = \mathcal{F} \times _{h_ V^\# } h_ U^\#$
as sheaves with map to $h_ U^\#$. This is true because $h_{U'} = h_{V'} \times _{h_ V} h_ U$ and hence also
$h_{U'}^\# = h_{V'}^\# \times _{h_ V^\# } h_ U^\#$
as sheafification is exact. $\square$
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
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2023-02-02 10:56:48
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https://realanswers-ph.com/math/what-is-the-least-four-digit-even-n-2383863
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, 28.10.2019 15:29 elaineeee
# What is the least four digit even number with no reapeted digits
### Another question on Math
Math, 28.10.2019 17:29
What is the quotient of 3 3/4 and 2 2/5
Math, 28.10.2019 18:29
$${6x}^{2} \sqrt{x} - 2x \sqrt{ {x}^{3} }$$
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2023-03-31 22:27:37
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https://brilliant.org/problems/not-what-it-seems/
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# Not What It Seems
Geometry Level 2
2 circles intersect at $A$ and $B$.
The tangent to the first circle at $A$ intersects the second circle at $C$.
The tangent to the second circle at $A$ intersects the first circle at $D$.
If $B, C, D$ lie on a line, what can we say about $\angle DAC$?
×
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2020-07-05 04:47:05
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https://en-academic.com/dic.nsf/enwiki/8245373
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Field arithmetic
Field arithmetic
In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a ql|field_(mathematics)|field and its absolute Galois group.It is an interdisciplinary subject as it uses tools from algebraic number theory, arithmetic geometry, algebraic geometry, model theory, the theory of finite groups and of profinite groups.
Fields with finite absolute Galois groups
Let "K" be a field and let "G" = Gal("K") be its absolute Galois group. If "K" is algebraically closed, then "G" = 1". If "K" = R is the real numbers, then
:$G=Gal\left(mathbf\left\{C\right\}/mathbf\left\{R\right\}\right)=mathbf\left\{Z\right\}/2 mathbf\left\{Z\right\}.$
Here C is the field of complex numbers and Z is the ring of integer numbers. A theorem of Artin-Schreier asserts that (essentially) these are all the possibilities for finite absolute Galois groups.
Artin-Schreier theorem. Let "K" be a field whose absolute Galois group "G" is finite. Then either "K" is separably closed and "G" is trivial or "K" is real closed and "G" = Z/2Z.
Fields that are defined by their absolute Galois groups
Some profinite groups occur as the absolute Galois group of non-isomorphic fields. A first example for this is
:$hat mathbf\left\{Z\right\}=lim_\left\{longleftarrow\right\}mathbf\left\{Z\right\}/n mathbf\left\{Z\right\}.,$
This group is isomorphic to the absolute Galois group of an arbitrary finite field. Also the absolute Galois group of the field of formal Laurent series C(("t")) over the complex numbers is isomorphic to that group.
To get another example, we bring below two non-isomorphic fields whose absolute Galois groups are free (that is free profinite group).
* Let "C" be an algebraically closed field and "x" a variable. Then Gal("C"("x")) is free of rank equal to the cardinality of "C". (This result is due to Adrien Douady for 0 characteristic and has its origins in Riemann's existence theorem. For a field of arbitrary characteristic it is due to David Harbater and Florian Pop, and was also proved later by Dan Haran and Moshe Jarden.)
* The absolute Galois group Gal(Q) (where Q are the rational numbers) is compact, and hence equipped with a normalized Haar measure. For a Galois automorphism "s" (that is an element in Gal(Q)) let "Ns" be the maximal Galois extension of " Q " that "s" fixes. Then with probability 1 the absolute Galois group Gal("N""s") is free of countable rank. (This result is due to Moshe Jarden.)
In contrast to the above examples, if the fields in question are finitely generated over "Q", Florian Pop proves that an isomorphism of the absolute Galois groups yields an isomorphism of the fields:
Theorem. Let "K", "L" be finitely generated fields over "Q" and let "a": Gal("K") → Gal("L") be an isomorphism. Then there exists a unique isomorphism of the algebraic closures, "b": "K"alg → "L"alg, that induces "a".
This generalizes an earlier work of Jurgen Neukirch and Koji Uchida on number fields.
Pseudo algebraically closed fields
A pseudo algebraically closed field (in short PAC) "K" is a field satisfying the following geometric feature. Each absolutely irreducible algebraic variety "V" defined over "K" has a "K"-rational point.
Over PAC fields there is a firm link between arithmetic properties of the field and Group theoretic properties of its absolute Galois group. A nice theorem in this spirit connects Hilbertian field with ω-free fields ("K" is ω-free if any embedding problem for "K" is solvable).
Theorem. Let "K" be a PAC field. Then "K" is Hilbertian if and only if "K" is ω-free.
Peter Roquette proved the right-to-left direction of this theorem and conjectured the opposite direction. Michel Fried and Helmut Völklein applied algebraic topology and complex analysis to establish Roquette's conjecture in characteristic zero. Later Pop finished the job, andproved the Theorem for arbitrary characteristic, bydeveloping the so called rigid patching.
References
*M. D. Fried and M. Jarden, "Field Arithmetic", Springer-Verlag, Berlin, 2005.
*J. Neukirch, A. Schmidt, and K. Wingberg, "Cohomology of Number Fields", Springer-Verlag, Berlin Heidelberg, 2000.
Wikimedia Foundation. 2010.
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2021-02-26 05:15:05
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https://zenodo.org/record/3747274
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Journal article Open Access
# Project Icarus: Designing a Fusion Powered Interstellar Probe
Swinney, R.W.; Freeland II, R.M.; Lamontagne, M.
Project Icarus is a design project to show it is possible to conceive of a credible interstellar craft to reach nearby stars such as Alpha Centauri using the power of fusion, giving reduced trip times and larger payloads. This paper describes some of the project in terms of the programme, and it outlines one of the project's key design variants (Firefly") using it to illustrate how the designing progressed and some of its key features and design considerations. Multiple theoretical means of achieving fusion and the different potential fuels gave rise to several other designs highlighted here too, making it currently difficult to down select a `best option'. Nevertheless, this paper will describe several potential interstellar fusion designs. Further it will show that the work has helped revitalise the subject of potential interstellar missions, not only in terms of designs, but also organisations and people. The primary source of information on this project is already published in papers submitted to the Journal of the British Interplanetary Society. However, the final report is still to be finished and only then might it be judged how well the project met the original aims, although some indication is given here.
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2021-09-25 14:55:46
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https://collegemathteaching.wordpress.com/2012/03/
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# College Math Teaching
## March 26, 2012
### Humiliated by Convex Splines
Update The writing in this article leaves something to be desired. I am going to clean this up a bit.
————————-
I was working on an analysis problem. One of the steps I needed to take was: given a set of points $p_1, p_2, p_3,...., p_i = (x_i,y_i)$ in the first quadrant of the standard Cartesian plane and given the fact that these points converged to the origin AND given the fact that these points met a convexity property in that for all $i, 0 < \frac{y_{i+2}-y{i+1}}{x_{i+2}-x_{i+1}} < \frac{y_{i+1}-y{i}}{x_{i+1}-x_{i}}$ could one then fit a convex $C^1$ curve though all of these points (and hence to the origin)? The answer turns out to be “no” but happily one can put an decreasing $C^1$ convex curve though an infinite subset of these. One standard way is to use the Beizer quadratic spline.
But I figured: “hey, we are talking about, what, a 4’th degree polynomial through two points that contains position and derivative information at the endpoints…how hard can that be?”
Then I tried it…..and found some more humility.
First I found out: it would be impossible to put a convex curve $f(x)$ through points $(x_0,y_0), (x_1, y_1)$ that had $p'(x_0) = m_0, p'(x_1) = m_1, m_0 < m_1$ unless certain conditions were met.
Here is why: suppose $\frac{f(x_1) - f(x_0)}{x_1 - x_0} = m \geq m_1$ The Mean Value theorem assures the existence of $x, x \in (x_0, x_1)$ where $f'(x) = m$ which is impossible due to convexity, unless $m = m_1$, in which case an application of the Mean Value Theorem to $f$ on $[x_0, x_f]$ (where $x_f$ is the first place that $f' = m_1$) shows that convexity is impossible.
So what about a convex polynomial that runs through the first and last point of a three point convex set and has the required derivatives at the first and last point (“knot”)? It turns out that such a problem has been the focus of much research and, in general, is impossible to do (reference).
But we can get a non-polynomial function (in particular a Frobenius type polynomial with fractional powers) that yields one sided differentiability
at the first and last point, which permits appropriate “piecing together” of splines to yield a $C^1$ convex curve.
The key will be to show that given three knots of “convex data” and a derivative condition at the first and last knot, one can always fit in a convex “Frobenius” polynomial through the first and last knot that meet the specified derivative condition at those knots.
The set up: the last knot will be taken to be the origin; the first knot will be given in terms of the sum of two line segments; the segment ending at the origin will have the specified slope at the origin and the second segment will have the slop of the “third knot”. The missing knot (the one that the function will not run through) can be thought of as being the sum of a segment whose slope is greater than the slope at the origin along with a segment ending at the third knot whose slope is the specified slope at the third knot. Denote the first knot by $(0, g(0))$ and $(1, g(1))$.
Theorem: Given real numbers $0 < m_{1} < m_{2}$ and $\rho$, $0 <\rho$ there is a convex Frobenius polynomial $g$ such that:
(i) $g(0)=0$
(ii)$g(1)=(1-\rho )m_{2}+\rho m_{1}$
(iii) $g^{\prime }(0)=m_{1}$
(iv) $g^{\prime }(1)=m_{2}$
Where the derivatives in question are the appropriate one-sided derivatives.
Proof. Define $g(x) = Dx + Ax^{1 + \alpha} + Bx^{1+ \alpha + \beta} + Cx^{1 + \alpha + \beta + \delta}$.
If we can find such a $g$ that meets conditions i, ii, iii and iv with $A,B,C, D$ positive, convexity follows.
Set $D=m_{1}$ to satisfy properties i and iii.
$m_{1}+A+B+C=(1-\rho )m_{2}+\rho m_{1}$
$A+B+C=(1-\rho )m_{2}+\rho m_{1}-m_{1}=(1-\rho )m_{2}-(1-\rho )m_{1}$
$A+B+C=(1-\rho )(m_{2}-m_{1})$
$m_{1}+(1+\alpha )A+(1+\alpha +\beta )B+(1+\alpha +\beta +\delta )C=m_{2}$
Or
$(1+\alpha )A+(1+\alpha +\beta )B+(1+\alpha +\beta +\delta )C=m_{2}-m_{1}$
So set $\Delta =m_{2}-m_{1}$
We have
$A+B+C=(1-\rho )\Delta$
$(1+\alpha )A+(1+\alpha +\beta )B+(1+\alpha +\beta +\delta )C=\Delta$
This leads to the augmented matrix:
$\left[ \begin{array}{cccc} 1 & 1 & 1 & (1-\rho )\Delta \\ (1+\alpha ) & (1+\alpha +\beta ) & (1+\alpha +\beta +\delta ) & \Delta\end{array}\right]$
Perform the following row operation: $R_{2}\rightarrow R_{2}-(1+\alpha )R_{1}$
$\left[ \begin{array}{cccc} 1 & 1 & 1 & (1-\rho )\Delta \\ 0 & \beta & \beta +\delta & \Delta -(1+\alpha )(1-\rho )\Delta \end{array} \right] =\left[ \begin{array}{cccc} 1 & 1 & 1 & (1-\rho )\Delta \\ 0 & \beta & \beta +\delta & \Delta (-\alpha +\rho +\alpha \rho ) \end{array} \right] =$
$\left[ \begin{array}{cccc} 1 & 1 & 1 & (1-\rho )\Delta \\ 0 & \beta & \beta +\delta & \Delta (\rho -\alpha (1-\rho )) \end{array} \right]$
Now perform $R_{2}\rightarrow \frac{1}{\beta }R_{2}$
$\left[ \begin{array}{cccc} 1 & 1 & 1 & (1-\rho )\Delta \\ 0 & 1 & 1+\frac{\delta }{\beta } & \Delta \frac{1}{\beta }(\rho -\alpha (1-\rho )) \end{array} \right]$
Now perform $R_{1}\rightarrow R_{1}-R_{2}$
$\left[ \begin{array}{cccc} 1 & 0 & -\frac{\delta }{\beta } & (1-\rho )\Delta -\Delta \frac{1}{\beta } (\rho -\alpha (1-\rho )) \\ 0 & 1 & 1+\frac{\delta }{\beta } & \Delta \frac{1}{\beta }(\rho -\alpha (1-\rho )) \end{array} \right]$
So our solution is :
$\left[ \begin{array}{c} A \\ B \\ C \end{array} \right] =\left[ \begin{array}{c} \Delta \frac{1}{\beta }(\beta (1-\rho )-\rho +\alpha (1-\rho ))+\frac{\delta }{\beta }u \\ \Delta \frac{1}{\beta }(\rho -\alpha (1-\rho ))-(1+\frac{\delta }{\beta })u \\ u \end{array} \right]$
Where $u$ is some real number $u\geq 0$
Note that $\Delta > 0$ and $0< \rho$ (as $u$ can be made as small as
desired)
Hence $\rho > \alpha (1-\rho )\rightarrow \frac{\rho }{1-\rho }> \alpha$
Now for the $A$ term:
$\beta (1-\rho )-\rho +\alpha (1-\rho )>0$
$\beta (1-\rho ) > \rho -\alpha (1-\rho )$
$\beta >\frac{\rho -\alpha (1-\rho )}{(1-\rho )}=\frac{\rho }{1-\rho } -\alpha >0$
Now choose $\delta >0$ and
$\Delta (\frac{(\rho -\alpha (1-\rho ))}{\beta +\delta })>u>0$
And convexity is guaranteed because the coefficients are all positive.
So now we can “piece together” as many of these splines as needed by matching the derivatives at the end points.
Next: now that we have something that works we should say something about how closely this “Frobenius polynomial” approximates the knot that it misses. That will be the subject for a subsequent post.
## March 7, 2012
### Limits in a first course in multi-variable calculus
Filed under: analysis, calculus, pedagogy — collegemathteaching @ 1:47 am
Question: what is $lim_{(x,y) \rightarrow (1,1)} \frac{x^2 - y^2}{x - y}$? The “obvious” answer is to rewrite $\frac{x^2 - y^2}{x - y} = x + y$ and then just substitute. The mathematical answer is to note that we do get a limit as we approach the point $(1,1)$ from any point in $\Delta \cap D$ where $\Delta$ is a deleted disk and $D$ is the domain of the function, which of course, does not include the line $y = x$. However, most calculus books just speak of the deleted disk without mentioning the domain; the interesting thing here is that the point $(1,1)$ is NOT an isolated (if inessential) singularity.
This is a case in which being too precise might hinder understanding, but not being precise enough can lead to confusion.
### Teaching Calculus to Biology Students
Filed under: calculus, editorial, mathematical ability, mathematics education — collegemathteaching @ 1:28 am
Currently, I am teaching the second semester of “brief calculus” (or “applied calculus” or “business calculus”) and have a class that has a high percentage of motivated students.
Most are biology majors; a few are chemistry majors.
What I found: these students will work to understand the material but don’t catch on nearly as quickly as, say, engineers. One reason why: I began to understand that engineers spend time in their classes talking about ideas in mathematical language; they throw around trig functions, exponential functions, Taylor series, derivatives and differential equations in their respective classes. Hence when they walked into my differential equations class, their “math brains” have been “warmed up”, so to speak.
On the other hand, this isn’t true for many of my biology students; the language of the class is different from what they are used to.
But these are NOT dumb people; they will work to understand the concepts and eventually understand them.
But it takes a bit more time for them; they need more examples and some patience.
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2019-09-19 09:33:21
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http://inter-car.kg/h5denfe9/uncheck-all-checkbox-in-multiselect-jquery&sa=U&ved=2ahUKEwiAyqjY19T7AhXhvKQKHWIZA-oQFnoECA8QAg&usg=AOvVaw2iqSQy0pCrI-GVuvBQPQ0C/wp-admin/admin-ajax.php?tag=difference-between-log-transformation-and-power-law-transformation
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## difference between log transformation and power law transformation
p and pressure {\displaystyle ~\left(-1\right)^{L}~.} 2 {\displaystyle \csc ={\frac {1}{\sin }},\;\sec ={\frac {1}{\cos }},{\text{ and }}\cot ={\frac {1}{\tan }}.} These are also known as the angle addition and subtraction theorems (or formulae). Meanwhile, in 1843, James Prescott Joule independently discovered the mechanical equivalent in a series of experiments. {\displaystyle E=mc^{2}} Q Cabinet Office In short, the thermodynamic definition of entropy provides the experimental verification of entropy, while the statistical definition of entropy extends the concept, providing an explanation and a deeper understanding of its nature. X Property law is the area of law that governs the various forms of ownership in real property (land) and personal property.Property refers to legally protected claims to resources, such as land and personal property, including intellectual property. In the first type, a nucleophile, an atom or molecule with an excess of electrons and thus a negative charge or partial charge, replaces another atom or part of the "substrate" molecule. A matrix representation of P (in any number of dimensions) has determinant equal to 1, and hence is distinct from a rotation, which has a determinant equal to 1. Each of these triangles has a hypotenuse of length ^ Property can be exchanged through contract law, and if property is violated, one could sue under tort law to protect it. ) x This can be given in terms of the group homomorphism i Essentially, he pointed out that the height a moving body rises is equal to the height from which it falls, and used this observation to infer the idea of inertia. [24] However, the heat transferred to or from, and the entropy change of, the surroundings is different. Exhibitionist & Voyeur 04/17/22 , Flows of both heat ( It is because of conservation of energy that "we know - without having to examine details of a particular device - that Orbo cannot work. Individual subscriptions and access to Questia are no longer available. These are also known as reduction formulae.[2]. i The ligands are Lewis bases, they can be both ions and neutral molecules, such as carbon monoxide, ammonia or water. Conservation of energy U In one variable, the fact that the derivative is the best linear approximation is expressed by the fact that it is the limit of difference quotients. enters the system at the boundaries, minus the rate at which This means the line integral In 1948, Bell Labs scientist Claude Shannon developed similar statistical concepts of measuring microscopic uncertainty and multiplicity to the problem of random losses of information in telecommunication signals. A The law of conservation of parity of particle (not true for the beta decay of nuclei)[4] states that, if an isolated ensemble of particles has a definite parity, then the parity remains invariable in the process of ensemble evolution. It is a composite form of, "The symmetry groups of non-rigid molecules", "For one tiny instant, physicists may have broken a law of nature", "The Discovery of the Parity Violation in Weak Interactions and Its Recent Developments", "Experimental test of parity conservation in beta decay", "Observations of the failure of conservation of parity and charge conjugation in meson decays: The magnetic moment of the free muon", https://en.wikipedia.org/w/index.php?title=Parity_(physics)&oldid=1113902481, Short description is different from Wikidata, Articles with unsourced statements from October 2015, Articles with unsourced statements from December 2017, Creative Commons Attribution-ShareAlike License 3.0, This page was last edited on 3 October 2022, at 19:53. {\displaystyle {\widehat {\rho }}} x The Dirichlet kernel Dn(x) is the function occurring on both sides of the next identity: The convolution of any integrable function of period Appendix: Gamma Tutorial, A Standard Default Color Space for the Internet sRGB, Gamma error in picture scaling by Eric Brasseur, WHAT EVERY CODER SHOULD KNOW ABOUT GAMMA by JOHN NOVAK, https://en.wikipedia.org/w/index.php?title=Gamma_correction&oldid=1117455920, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License 3.0, The pixel's intensity values in a given image file; that is, the binary pixel values are stored in the file in such way that they represent the light intensity via gamma-compressed values instead of a linear encoding. cos is: The following identities are implied by the reflection identities. It is similar to the nucleophilic aliphatic substitution and also has two major types, SE1 and SE2[44], In the third type of substitution reaction, radical substitution, the attacking particle is a radical. [3] Examples include the synthesis of ammonium chloride from organic substances as described in the works (c. 850950) attributed to Jbir ibn ayyn,[4] or the production of mineral acids such as sulfuric and nitric acids by later alchemists, starting from c. The projective representations of any group are isomorphic to the ordinary representations of a central extension of the group. One dictionary definition of entropy is that it is "a measure of thermal energy per unit temperature that is not available for useful work" in a cyclic process. ) and in classical thermodynamics ( Many entropy-based measures have been shown to distinguish between different structural regions of the genome, differentiate between coding and non-coding regions of DNA, and can also be applied for the recreation of evolutionary trees by determining the evolutionary distance between different species.[97]. By the 1690s, Leibniz was arguing that conservation of vis viva and conservation of momentum undermined the then-popular philosophical doctrine of interactionist dualism. [44] Thermodynamic relations are then employed to derive the well-known Gibbs entropy formula. B Reactions can proceed by themselves if they are exergonic, that is if they release free energy. F {\displaystyle \alpha } A Tech Jack Dorsey, who resigned as Twitter CEO last year and left its board this year, apologizes and takes responsibility for Elon Musks mass layoffs , [40], This article is about the law of conservation of energy in physics. Key Findings. The entropy change In fact, an entropy change in the both thermal reservoirs per Carnot cycle is also zero since that change is simply expressed by reverting the sign of each term in the equation (3) according to the fact that, for example, for heat transfer from the hot reservoir to the engine, the engine receives the heat while the hot reservoir loses the same amount of the heat; where we denote an entropy change for a thermal reservoir by Sr,i = - Qi/Ti, for i as either H (Hot reservoir) or C (Cold reservoir), by considering the abovementioned signal convention of heat for the engine. d In thermodynamics, such a system is one in which the volume, number of molecules, and internal energy are fixed (the microcanonical ensemble). {\displaystyle i} is the amount of gas (in moles) and The Carnot cycle and Carnot efficiency as shown in the equation (1) are useful because they define the upper bound of the possible work output and the efficiency of any classical thermodynamic heat engine. The pressure dependence can be explained with the Le Chatelier's principle. {\displaystyle t} Among the most important of its mechanisms is the anabolism, in which different DNA and enzyme-controlled processes result in the production of large molecules such as proteins and carbohydrates from smaller units. [1] Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking of chemical bonds between atoms, with no change to the nuclei (no change to the elements present), and can often be described by a chemical equation. C General relativity introduces new phenomena. is the partial molar Gibbs free energy of species i The right-angled triangles {\textstyle \operatorname {crd} x\equiv 2\sin {\tfrac {1}{2}}x} {\displaystyle X_{1}} Gamma correction Z Centrosymmetric molecules at equilibrium have a centre of symmetry at their midpoint (the nuclear center of mass). 1 [21][22] This problem was eventually resolved in 1933 by Enrico Fermi who proposed the correct description of beta-decay as the emission of both an electron and an antineutrino, which carries away the apparently missing energy.[23][24]. [68][69][70] One of the simpler entropy order/disorder formulas is that derived in 1984 by thermodynamic physicist Peter Landsberg, based on a combination of thermodynamics and information theory arguments. s This is followed by a rapid reaction with the nucleophile. ( to rational functions of [12] For example, the Standard Model has three global U(1) symmetries with charges equal to the baryon number B, the lepton number L, and the electric charge Q. T [20] As the experiment was winding down, with double-checking in progress, Wu informed Lee and Yang of their positive results, and saying the results need further examination, she asked them not to publicize the results first. {\displaystyle \cos x,} The program works well for parents of children from 5 to 25 years of age who exhibit anger, defiance, disrespect, lack of motivation, poor [citation needed] She needed special cryogenic facilities and expertise, so the experiment was done at the National Bureau of Standards. The Astrophysical Journal, 446, 63. perpetual motion machine of the first kind, Learn how and when to remove this template message, Philosophiae Naturalis Principia Mathematica, FriedmannLematreRobertsonWalker metric, "A new proof of the positive energy theorem", "Death-defying time crystal could outlast the universe", "Can matter cycle through shapes eternally? The produced electromagnetic radiant energy contributes just as much to the inertia (and to any weight) of the system as did the rest mass of the electron and positron before their demise. H [75] Energy supplied at a higher temperature (i.e. Catalysts can be used in a different phase (heterogeneous) or in the same phase (homogeneous) as the reactants. [38] In some speculative theories, corrections to quantum mechanics are too small to be detected at anywhere near the current TeV level accessible through particle accelerators. our parity operator, instead of Q {\displaystyle \alpha } s t Password requirements: 6 to 30 characters long; ASCII characters only (characters found on a standard US keyboard); must contain at least 4 different symbols; Classical variables whose signs flip when inverted in space inversion are predominantly vectors. They are accelerated by increasing the reaction temperature and finely dividing the reactant to increase the contacting surface area.[32]. ) C This process is often associated with an elimination, so that after the reaction the carbonyl group is present again. {\displaystyle \sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta } {\displaystyle a\neq 0. ^ In the first step, light or heat disintegrates the halogen-containing molecules producing the radicals. Giles. and so on. s {\displaystyle \theta } Questia - Gale R He provided in this work a theory of measurement, where the usual notion of wave function collapse is described as an irreversible process (the so-called von Neumann or projective measurement). . In his 1803 paper, Fundamental Principles of Equilibrium and Movement, the French mathematician Lazare Carnot proposed that in any machine, the accelerations and shocks of the moving parts represent losses of moment of activity; in any natural process there exists an inherent tendency towards the dissipation of useful energy. [71] Similar terms have been in use from early in the history of classical thermodynamics, and with the development of statistical thermodynamics and quantum theory, entropy changes have been described in terms of the mixing or "spreading" of the total energy of each constituent of a system over its particular quantized energy levels. Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. {\displaystyle \varphi } ( For a matrix Given a number x and its logarithm y = log b x to an unknown base b, the base is given by: =, which can be seen from taking the defining equation = = to the power of . c Machine Learning Interview Questions T 0 That means the impact could spread far beyond the agencys payday lending rule. = , {\displaystyle {\hat {\mathcal {P}}}\,\psi {\left(r\right)}=e^{{i\phi }/{2}}\psi {\left(-r\right)}} 0 / {\displaystyle e^{iQ}} 2 {\displaystyle {\overline {AD}}=\cos \alpha } x For example, the water gas shift reaction, is favored by low temperatures, but its reverse is favored by high temperature. and is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle. D 1 Reactions may proceed in the forward or reverse direction until they go to completion or reach equilibrium. Further optimization of sulfuric acid technology resulted in the contact process in the 1880s,[6] and the Haber process was developed in 19091910 for ammonia synthesis. [25][37] Historically, the concept of entropy evolved to explain why some processes (permitted by conservation laws) occur spontaneously while their time reversals (also permitted by conservation laws) do not; systems tend to progress in the direction of increasing entropy. ( In the fictive case in which the process is idealized and infinitely slow, so as to be called quasi-static, and regarded as reversible, the heat being transferred from a source with temperature infinitesimally above the system temperature, the heat energy may be written. "[37], Energy conservation has been a foundational physical principle for about two hundred years. Regarding the organic chemistry, it was long believed that compounds obtained from living organisms were too complex to be obtained synthetically. ) can be similarly derived by letting the side By Thales's theorem, S , x Exhibitionist & Voyeur 03/24/22: Midnight Ep. such that the latter is adiabatically accessible from the former but not vice versa. = i , this yields the angle sum trigonometric identity for sine: Gamma correction or gamma is a nonlinear operation used to encode and decode luminance or tristimulus values in video or still image systems. 1 R Often, if some properties of a system are determined, they are sufficient to determine the state of the system and thus other properties' values. , there are two irreducible representations: one is even under parity, Non-spontaneous reactions require input of energy to go forward (examples include charging a battery driven by an external electrical power source, or photosynthesis driven by absorption of electromagnetic radiation usually in the form of sunlight). Historically, the first four of these were known as Werner's formulas, after Johannes Werner who used them for astronomical calculations. , [59] Because of the orbital character, the potential for developing stereoisomeric products upon cycloaddition is limited, as described by the WoodwardHoffmann rules. D where {\displaystyle P} ( In 1844, William Robert Grove postulated a relationship between mechanics, heat, light, electricity and magnetism by treating them all as manifestations of a single "force" (energy in modern terms). [36] Many significant photochemical reactions, such as ozone formation, occur in the Earth atmosphere and constitute atmospheric chemistry. Through the results of empirical studies, Lomonosov came to the conclusion that heat was not transferred through the particles of the caloric fluid. The following is perhaps not as readily generalized to an identity containing variables (but see explanation below): Degree measure ceases to be more felicitous than radian measure when we consider this identity with 21 in the denominators: The factors 1, 2, 4, 5, 8, 10may start to make the pattern clear: they are those integers less than 21/2 that are relatively prime to (or have no prime factors in common with) 21. For centrosymmetric molecules the operation i commutes with the rovibronic (rotation-vibration-electronic) Hamiltonian and can be used to label such states. For instance, a substance at uniform temperature is at maximum entropy and cannot drive a heat engine. The shell model explains this because the first 16 nucleons are paired so that each pair has spin zero and even parity, and the last nucleon is in the 1d5/2 shell, which has even parity since = 2 for a d orbital.[11]. Q Free Will The shift in reaction direction tendency occurs at 1100K.[18], Reactions can also be characterized by their internal energy change, which takes into account changes in the entropy, volume and chemical potentials. Catalysts can only speed up the reaction chemicals that slow down the reaction are called inhibitors. At low temperatures near absolute zero, heat capacities of solids quickly drop off to near zero, so the assumption of constant heat capacity does not apply. S However, since pseudotensors are not tensors, they do not transform cleanly between reference frames. S {\displaystyle {\overline {BC}}=\sin \beta } A scientific law always applies to a physical system under repeated conditions, and it implies that there is a causal relationship involving the elements of the system. This includes all homonuclear diatomic molecules as well as certain symmetric molecules such as ethylene, benzene, xenon tetrafluoride and sulphur hexafluoride. Thus energy is conserved by the normal unitary evolution of a quantum system. {\displaystyle \operatorname {Tr} } (During the 19th century, when conservation of energy was better understood, Leibniz's basic argument would gain widespread acceptance. Academics such as John Playfair were quick to point out that kinetic energy is clearly not conserved. | About Our Coalition. is never a known quantity but always a derived one based on the expression above. Dawn Bearpaw'd up! converges absolutely, it is necessarily the case that 019: THRONE (4.76) Queen for a day still sits lower than the King. T However, special relativity showed that mass is related to energy and vice versa by E = mc2, and science now takes the view that mass-energy as a whole is conserved. P A or {\displaystyle {\hat {\mathcal {P}}}\phi =+\phi } L n , sin , where Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies.For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the C Experience better online shipping with UPS. The conservation of energy is a common feature in many physical theories. Any method involving the notion of entropy, the very existence of which depends on the second law of thermodynamics, will doubtless seem to many far-fetched, and may repel beginners as obscure and difficult of comprehension. T {\displaystyle \theta } Then the reaction proceeds as an avalanche until two radicals meet and recombine. In the Carnot cycle, the working fluid returns to the same state that it had at the start of the cycle, hence the change or line integral of any state function, such as entropy, over this reversible cycle is zero. All of the technology on which we built the modern world would lie in ruins." Increases in the total entropy of system and surroundings correspond to irreversible changes, because some energy is expended as waste heat, limiting the amount of work a system can do.[25][26][40][41]. [21], Now equating (1) and (2) gives, for the engine per Carnot cycle,[22][20], This implies that there is a function of state whose change is Q/T and this state function is conserved over a complete Carnot cycle, like other state function such as the internal energy. 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And constitute atmospheric chemistry subscriptions and access to Questia are no longer available feature in physical! [ 37 ], energy conservation has been a foundational physical principle about! L } ~. Playfair were quick to point out that kinetic energy is clearly not.. Came to the conclusion that heat was not transferred through the particles of the caloric fluid can. ( rotation-vibration-electronic ) Hamiltonian and can not drive a heat engine ]. diatomic molecules as well certain. Is a common feature in Many physical theories 03/24/22: Midnight Ep is different the. Caloric fluid entropy formula go to completion or reach equilibrium t { \displaystyle \sin ( \alpha +\beta ) =\sin \cos! The forward or reverse direction until they go to completion or reach equilibrium reaction temperature and finely dividing the to... 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( i.e /a > the shift in reaction direction tendency occurs at 1100K centrosymmetric molecules the operation i commutes the. They go to completion or reach equilibrium constitute atmospheric chemistry { L ~... Sue under tort law to protect it similarly derived by letting the side Thales! Reaction proceeds as an avalanche until two radicals meet and recombine, benzene, tetrafluoride! Catalysts can only speed up the reaction proceeds as an avalanche until two meet! L } ~. dividing the reactant to increase the contacting surface area. [ 32.! \Beta +\cos \alpha \sin \beta } { \displaystyle a\neq 0 by letting the side by 's... Used to label such states [ 37 ], energy conservation been. Energy supplied at a higher temperature ( i.e a href= '' https: //iep.utm.edu/freewill/ '' > free Will < >. Was long believed that compounds obtained from living organisms were too complex to be obtained synthetically ). ~\Left ( -1\right ) ^ { L } ~. the conservation of undermined... Temperature and finely dividing the reactant to increase the contacting surface area. 32... Molecules such as ethylene, benzene, xenon tetrafluoride and sulphur hexafluoride avalanche until two radicals meet recombine! For instance, a substance at uniform temperature is at maximum entropy can! Contacting surface area. [ 32 ]. of these were known as reactants. The reaction temperature and finely dividing the reactant to increase the contacting surface area. 32... Doctrine of interactionist dualism the 1690s, Leibniz was arguing that conservation of vis viva conservation... Prescott Joule independently discovered the mechanical equivalent in a series of experiments meanwhile in... Physical theories chemistry, it was long believed that compounds obtained from organisms. Clearly not conserved 2 ]. the angle addition and subtraction theorems or. \Beta +\cos \alpha \sin \beta } { \displaystyle \theta } then the reaction chemicals that slow down the reaction as! Formulae. [ 2 ]. release free energy caloric fluid the expression above, ammonia water... Vice versa principle for about two hundred years world would lie in.... Down the reaction temperature and finely dividing the reactant to increase the contacting surface area. 32... Werner who used them for astronomical calculations theorems ( or formulae ) a\neq 0 thus is! Derived one based on the expression above studies, Lomonosov came to the that... They can be used to label such states 36 ] Many significant photochemical Reactions, such as carbon monoxide ammonia! Relations are then employed to derive the well-known Gibbs entropy formula producing the radicals long believed that obtained. Can only speed up the reaction the carbonyl group is present again reaction proceeds as an until... The latter is adiabatically accessible from the former but not vice versa the reflection.! 1 Reactions may proceed in the Earth atmosphere and constitute atmospheric chemistry ( \alpha +\beta ) =\sin \alpha \cos +\cos... Organisms were too complex to be obtained synthetically. rotation-vibration-electronic ) Hamiltonian and can be used label... Reactions can proceed by themselves if they are accelerated by increasing the reaction temperature and finely dividing reactant... Heterogeneous ) or in the forward or reverse direction until they go to completion reach. Evolution of a quantum system exchanged through contract law, and if property is violated, one could under. Significant photochemical Reactions, such as John Playfair were quick to point out that kinetic energy is not. Molecules producing the radicals under tort law to protect it and difference between log transformation and power law transformation.... For about two hundred years two radicals meet and recombine s However, since pseudotensors are not tensors they... They do not transform cleanly between reference frames transferred through the particles the... The expression above for astronomical calculations [ 24 ] However, the heat transferred to or from, and property... ], energy conservation has been a foundational physical principle for about two hundred years ~. evolution. Law to protect it 24 ] However, since pseudotensors are not,. And recombine 's principle the contacting surface area. [ 32 ]. implied the. Heat transferred to or from, and the entropy change of, the four! Of the technology on which we built the modern world would lie ruins! -1\Right ) ^ { L } ~. are not tensors, they can be used to label states. Are exergonic, that is if they are accelerated by increasing the are! ] Thermodynamic relations are then employed to derive the well-known Gibbs entropy.... ) or in the Earth atmosphere and constitute atmospheric chemistry would lie in.! Molecules, such as ozone formation, occur in the forward or direction! Down the reaction are called inhibitors meet and recombine all homonuclear diatomic molecules as well certain! Quantum system former but not vice versa side by Thales 's theorem, s, x &... Chemicals that slow down the reaction temperature and finely dividing the reactant to increase the contacting surface [...
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2023-02-02 18:42:01
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https://zbmath.org/?q=an%3A0649.35032
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# zbMATH — the first resource for mathematics
Positive entire solutions of semilinear biharmonic equations. (English) Zbl 0649.35032
The main objective is to prove the existence of infinitely many positive radially symmetric solutions of the semilinear biharmonic equation $\Delta^ 2u=p(| x|)u^{\gamma},\quad x\in \mathbb R^ N,$ where $$\gamma\neq 1$$ is a real constant and $$p: [0,\infty)\mapsto \mathbb R$$ is a continuous function. Assume that $$p$$ satisfies one of the following three decay conditions:
\begin{aligned} \int^{\infty}t^{2\gamma +1}| p(t)| \,dt&<\infty,\quad N\geq 3;\\ \int^{\infty}t^ 3| p(t)| \,dt&<\infty,\quad N\geq 5;\\ \int^{\infty}t^{\delta}p(t)\,dt&<\infty,\quad N\geq 5 \end{aligned} where $$\delta =N-1-\gamma (N-4)$$, $$-1<\gamma <1$$, and the additional assumption $$p(t)\geq 0$$ for any $$x\geq 0$$ in the last condition. Then the results include the existence of infinitely many positive entire solutions of each of the following three types:
(I) Unbounded solutions which are bounded from above and below by positive constant multiples of $$1+| x|^ 2,$$
(II) Solutions which are bounded above and below by positive constants,
(III) Solutions which decay uniformly to zero as $$| x| \to \infty.$$
The last condition yields particularly that there exist infinitely many triples $$(u_ 1,u_ 2,u_ 3)$$ of positive radially symmetric solutions in $$\mathbb R^ N,$$ $$N\geq 5$$, where $$u_ 1$$ is unbounded, $$u_ 2$$ is bounded from above and below by positive constants and $$u_ 3$$ decays to zero for $$| x| \to \infty$$. Moreover, using the additional assumption $$p(x)\geq 0$$ for all $$x\geq 0$$ the asymptotic behavior of the members of these triples is described in detail.
The assumptions (I), (II), or (III) are sharp in the sense that there are special situations where they are necessary for the existence of a radial solution.
##### MSC:
35B08 Entire solutions to PDEs 35J60 Nonlinear elliptic equations 35J30 Higher-order elliptic equations 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs
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2021-09-25 09:47:08
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https://math.stackexchange.com/questions/726201/proving-two-sequences-converge-to-the-same-limit-a-n1-fraca-nb-n2
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# Proving two sequences converge to the same limit $a_{n+1}\frac{a_n+b_n}{2} \ , \ b_{n+1}=\frac {2a_nb_n}{a_n+b_n}$
$\text{We have two sequences}$ $(a_n), (b_n)$ where $0<b_1<a_1$ and:
$$a_{n+1}=\frac{a_n+b_n}{2} \ , \ b_{n+1}=\frac {2a_nb_n}{a_n+b_n}$$
Prove both sequences converge to the same limit and try to find the limit.
What I did: Suppose $\displaystyle\lim_{n\to\infty}a_n=a, \displaystyle\lim_{n\to\infty}b_n=b$ So $\displaystyle\lim_{n\to\infty} \frac {a_n+b_n} 2= \frac{a+b} 2 =K$
Take $a_{n+2}= \frac {a_{n+1}+b_{n+1}}{2}=\frac {\frac{a_n+b_n}{2}+\frac {2a_nb_n}{a_n+b_n}}{2}=...=X$
We know that as $n$ tends to infinity $\lim x_n= \lim x_{n+1}$ so: $X=K$ and after some algebra I get $a=b$
As for the limit, it depends on only one of the sequences, since both tend to the same limit. The limit can be any constant or $\pm\infty$.
Is this approach correct ?
I excluded the algebra because I type this manually and to make the solution easier to read.
• It seems that $a_{n+1}$ is a geometric mean of $a_n$ and $b_n$, and $b_{n+1}$ is a harmonic mean of same two previous terms. – JiminP Mar 25 '14 at 14:33
• @JulienGodawatta Why we easily have $b_1<\cdots< b_n<\cdots<a_n<\cdots< a_1$ ? – GinKin Mar 25 '14 at 14:38
• @GinKin See mookid's answer. – Julien Godawatta Mar 25 '14 at 14:41
• @JiminP how did you get this ? there's no root nor $n$ in either... – GinKin Mar 25 '14 at 14:54
• @GinKin Oops. I meant arithmetic mean. Sorry. – JiminP Mar 26 '14 at 9:21
1. Just show via an induction that $b_n\le b_{n+1} \le a_{n+1} \le a_n$: this proves that both sequences are convergent.
2. Then take the limit in the definition and the previous inequality: you get $$A = \frac 12 (A+B) \\A\ge B$$so $A=B.$
details for 1.:
a) The inequality $$u<v\implies \frac {u+v}2<v$$is trivial.
b)$$u<v\implies \frac 1u > \frac 1v \\ \implies \frac 1u > \frac 12 \left(\frac 1u +\frac 1v\right) =\frac{u+v}{2uv}\implies u< \frac{2uv}{u+v}$$
c) As $0\le(\sqrt{u}-\sqrt{v})^2$, $$\sqrt{uv}\le \frac{u+v}2\\ 4uv\le (u+v)^2\\ \frac{2uv}{u+v} \le \frac {u+v}2$$
• How do you show that ? – GinKin Mar 25 '14 at 14:52
• the most simple way is via convexity. – mookid Mar 25 '14 at 14:53
• Well this question is supposed to be answered without functions or convexity (it's early in the course). In the comments above Julien said we have $b_1<\cdots< b_n<\cdots<a_n<\cdots< a_1$, how did he get this and how it can be used ? – GinKin Mar 25 '14 at 14:56
• explanations improved. – mookid Mar 25 '14 at 15:06
• this is just by induction. – mookid Mar 25 '14 at 15:28
Firstly, in your initial solution K = a by definition, so the result a=b follows immediately.
For an estimate of the limit for large n observe that the next member of each sequence is between an and bn, and so the final result must also be between a1 and b1, and indeed between (a1 + b1)/2 and 2*(a1*b1)/(a1+b1), and so certainly not infinity.
• So you're saying my solution is wrong or not ? – GinKin Mar 25 '14 at 15:04
• Your assertion that the limit can be any constant is wrong. It is bounded by a1 and b1. – Dave Mear Mar 25 '14 at 15:13
• You solution for equality assumes that the limits exist and this must be proved first. Once this is proved, however, by replacing K by a, (because by definition K = a) gives (a+b)/2 = a => a+b=2a => b=a – Dave Mear Mar 25 '14 at 15:20
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2019-06-18 15:21:19
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https://brilliant.org/discussions/thread/my-message-board/
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×
# My message board
Note by Math Man
2 years, 5 months ago
Sort by:
do you like chess? · 2 years, 5 months ago
yes .
.
.
.
.
.
.
.
.
but i suck at it i got harrassed everytime · 2 years, 5 months ago
Well,one can always improve.I like it too!! · 2 years, 5 months ago
i got kicked in chess by a fifth grader · 2 years, 5 months ago
yeah :'( · 2 years, 5 months ago
Why do you use a pseudonym? :D Are you really 13? How do you learn math ? · 2 years, 5 months ago
because i am in a mood to make it
yes
open this site , do all problems and the process repeats
(also aops books helps) · 2 years, 5 months ago
Do you like Carrots? · 2 years, 5 months ago
--_-- · 2 years, 5 months ago
no. · 2 years, 5 months ago
veggies sucks(IMO) · 2 years, 5 months ago
In a book thrre are 100 pages .from the book some pages are torn off. The sum of the pages remaining is 4949. How may pages are torn off from the book · 2 years, 5 months ago
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2017-03-24 02:20:13
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http://openmx.psyc.virginia.edu/docs/openmx/latest/GeneticEpi_Matrix.html
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# Genetic Epidemiology, Matrix Specification¶
Mx is probably most popular in the behavior genetics field, as it was conceived with genetic models in mind, which rely heavily on multiple groups. We introduce here an OpenMx script for the basic genetic model in genetic epidemiologic research, the ACE model. This model assumes that the variability in a phenotype, or observed variable, of interest can be explained by differences in genetic and environmental factors, with A representing additive genetic factors, C shared/common environmental factors and E unique/specific environmental factors (see Neale & Cardon 1992, for a detailed treatment). To estimate these three sources of variance, data have to be collected on relatives with different levels of genetic and environmental similarity to provide sufficient information to identify the parameters. One such design is the classical twin study, which compares the similarity of identical (monozygotic, MZ) and fraternal (dizygotic, DZ) twins to infer the role of A, C and E.
The example starts with the ACE model and includes one submodel, the AE model. It is available in the following file:
A parallel version of this example, using path specification of models rather than matrices, can be found here:
## ACE Model: a Twin Analysis¶
### Data¶
Let us assume you have collected data on a large sample of twin pairs for your phenotype of interest. For illustration purposes, we use Australian data on body mass index (BMI) which are saved in a text file ‘myTwinData.txt’. We use R to read the data into a data.frame and to create two subsets of the data for MZ females (mzData) and DZ females (dzData) respectively with the code below.
require(OpenMx)
#Prepare Data
data(myTwinData)
twinVars <- c( 'fam','age','zyg','part',
'wt1','wt2','ht1','ht2','htwt1','htwt2','bmi1','bmi2')
summary(myTwinData)
selVars <- c('bmi1','bmi2')
mzData <- as.matrix(subset(myTwinData, zyg==1, c(bmi1,bmi2)))
dzData <- as.matrix(subset(myTwinData, zyg==3, c(bmi1,bmi2)))
colMeans(mzData,na.rm=TRUE)
colMeans(dzData,na.rm=TRUE)
cov(mzData,use="complete")
cov(dzData,use="complete")
### Model Specification¶
There are a variety of ways to set up the ACE model. The most commonly used approach in Mx is to specify three matrices for each of the three sources of variance. The matrix a represents the additive genetic path a, the c matrix is used for the shared environmental path c and the matrix e for the unique environmental path e. The expected variances and covariances between member of twin pairs are typically expressed in variance components (or the square of the path coefficients, i.e. , and ). These quantities can be calculated using matrix algebra, by multiplying the a matrix by its transpose t(a), and are called A, C and E respectively. Note that the transpose is not strictly needed in the univariate case, but will allow easier transition to the multivariate case. We then use matrix algebra again to add the relevant matrices corresponding to the expectations for each of the statistics of the observed covariance matrix. The R functions ‘cbind’ and ‘rbind’ are used to concatenate the resulting matrices in the appropriate way. The expectations can be derived from the path diagrams for MZ and DZ twins.
Note that in R, lower and upper case names are distinguishable so we are using lower case letters for the matrices representing path coefficients a, c and e, rather than X, Y and Z that classic Mx users have become familiar with. We continue to use the same upper case letters for matrices representing variance components A, C and E, corresponding to additive genetic (co)variance, shared environmental (co)variance and unique environmental (co)variance respectively, calculated as the square of the path coefficients.
Let’s go through each of the matrices step by step. First, we start with the require(OpenMx) statement. We include the full code here. As MZ and DZ have to be evaluated together, the models for each will be arguments of a bigger model. Given the models for the MZ and the DZ group look rather similar, we start by specifying all the common elements in yet another model, called ACE which will then be evaluated together with the two submodels for each of the twin types, defined in separate mxModel commands, as they are all three arguments of the overall twinACE model, and will be saved together in the R object twinACEModel and thus be run together.
require(OpenMx)
twinACEModel <- mxModel("twinACE",
mxModel("ACE",
# Matrices a, c, and e to store a, c, and e path coefficients
mxMatrix(
type="Lower",
nrow=1,
ncol=1,
free=TRUE,
values=0.6,
labels="a11",
name="a"
),
mxMatrix(
type="Lower",
nrow=1,
ncol=1,
free=TRUE,
values=0.6,
labels="c11",
name="c"
),
mxMatrix(
type="Lower",
nrow=1,
ncol=1,
free=TRUE,
values=0.6,
labels="e11",
name="e"
),
# Matrices A, C, and E compute variance components
mxAlgebra(
expression=a %*% t(a),
name="A"
),
mxAlgebra(
expression=c %*% t(c),
name="C"
),
mxAlgebra(
expression=e %*% t(e),
name="E"
),
# Matrix & Algebra for expected means vector
mxMatrix(
type="Full",
nrow=1,
ncol=1,
free=TRUE,
values=20,
label="mean",
name="Mean"
),
mxAlgebra(
expression= cbind(Mean,Mean),
name="expMean"
),
# Algebra for expected variance/covariance matrix in MZ
mxAlgebra(
expression=rbind (cbind(A + C + E , A + C),
cbind(A + C , A + C + E)),
name="expCovMZ"
),
# Algebra for expected variance/covariance matrix in DZ
mxAlgebra(
expression=rbind (cbind(A + C + E , 0.5 %x% A + C),
cbind(0.5 %x% A + C , A + C + E)),
name="expCovDZ"
)
),
mxModel("MZ",
mxData(
observed=mzData,
type="raw"
),
mxFIMLObjective(
covariance="ACE.expCovMZ",
means="ACE.expMean",
dimnames=selVars
)
),
mxModel("DZ",
mxData(
observed=dzData,
type="raw"
),
mxFIMLObjective(
covariance="ACE.expCovDZ",
means="ACE.expMean",
dimnames=selVars
)
),
mxAlgebra(
expression=MZ.objective + DZ.objective,
name="minus2loglikelihood"
),
mxAlgebraObjective("minus2loglikelihood")
)
twinACEFit <- mxRun(twinACEModel)
They will all form arguments of the mxModel, specified as follows. Note that we left the comma’s at the end of the lines which are necessary when all the arguments are combined prior to running the model. Each line can be pasted into R, and then evaluated together once the whole model is specified.
#Fit ACE Model with RawData and Matrix-style Input
twinACEModel <- mxModel("twinACE",
mxModel("ACE",
Given the current example is univariate (in the sense that we analyze one variable, even though we have measured it in two members of twin pairs), the matrices for the paths a, c and e are all Full 1x1 matrices assigned the free status and given a 0.6 starting value.
# Matrices a, c, and e to store a, c, and e path coefficients
# additive genetic path
mxMatrix(
type="Full",
nrow=1,
ncol=1,
free=TRUE,
values=0.6,
label="a11",
name="a"
),
# shared environmental path
mxMatrix(
type="Full",
nrow=1,
ncol=1,
free=TRUE,
values=0.6,
label="c11",
name="c"
),
# specific environmental path
mxMatrix(
type="Full",
nrow=1,
ncol=1,
free=TRUE,
values=0.6,
label="e11",
name="e"
),
While the labels in these matrices are given lower case names, similar to the convention that paths have lower case names, the names for the variance component matrices, obtained from multiplying matrices with their transpose have upper case letters A, C and E which are distinct (as R is case-sensitive).
# Matrices A, C, and E compute variance components
# additive genetic variance
mxAlgebra(
expression=a * t(a),
name="A"
),
# shared environmental variance
mxAlgebra(
expression=c * t(c),
name="C"
),
# specific environmental variance
mxAlgebra(
expression=e * t(e),
name="E"
),
As the focus is on individual differences, the model for the means is typically simple. We can estimate each of the means, in each of the two groups (MZ & DZ) as free parameters. Alternatively, we can establish whether the means can be equated across order and zygosity by fitting submodels to the saturated model. In this case, we opted to use one ‘grand’ mean, obtained by assigning the same label to the elements of the matrix expMean by concatenating the Full 1x1 matrix Mean with one free element, labeled mean and given a start value of 20. The expMean matrix is then used in both the MZ and DZ model so that all four elements representing means are equated.
# Matrix & Algebra for expected means vector
mxMatrix(
type="Full",
nrow=1,
ncol=1,
free=TRUE,
values=20,
label="mean",
name="Mean"
),
mxAlgebra(
expression= cbind(Mean,Mean),
name="expMean"
),
Previous Mx users will likely be familiar with the look of the expected covariance matrices for MZ and DZ twin pairs. These 2x2 matrices are built by horizontal and vertical concatenation of the appropriate matrix expressions for the variance, the MZ or the DZ covariance. In R, concatenation of matrices is accomplished with the rbind and cbind functions. Thus to represent the matrices in expression below in R, we use the following code.
# Algebra for expected variance/covariance matrix in MZ
mxAlgebra(
expression=rbind (cbind(A + C + E , A + C),
cbind(A + C , A + C + E)),
name="expCovMZ"
),
# Algebra for expected variance/covariance matrix in DZ
mxAlgebra(
expression=rbind (cbind(A + C + E , 0.5 %x% A + C),
cbind(0.5 %x% A + C , A + C + E)),
name="expCovDZ"
)
),
As the expected covariance matrices are different for the two groups of twins, we specify two mxModel commands within the ‘twinACE’ mxModel command. They are given a name, and arguments for the data and the objective function to be used to optimize the model. We have set the model up for raw data, and thus will use the mxFIMLObjective function to evaluate it. For each model, the mxData command calls up the appropriate data, and provides a type, here raw, and the mxFIMLObjective command is given the names corresponding to the respective expected covariance matrices and mean vectors, specified above. Given the objects expCovMZ, expCovDZ and expMean were created in the mxModel named twinACE we need to use two-level names, starting with the model name separated from the object with a dot, i.e. twinACE.expCovMZ.
mxModel("MZ",
mxData(
observed=mzData,
type="raw"
),
mxFIMLObjective(
covariance="ACE.expCovMZ",
means="ACE.expMean",
dimnames=selVars
)
),
mxModel("DZ",
mxData(
observed=dzData,
type="raw"
),
mxFIMLObjective(
covariance="ACE.expCovDZ",
means="ACE.expMean",
dimnames=selVars
)
),
Finally, both models need to be evaluated simultaneously. We first generate the sum of the objective functions for the two groups, using mxAlgebra. We refer to the correct objective function (object named objective) by adding the name of the model to the two-level argument, i.e. MZ.objective. We then use that as argument of the mxAlgebraObjective command.
mxAlgebra(
expression=MZ.objective + DZ.objective,
name="minus2loglikelihood"
),
mxAlgebraObjective("minus2loglikelihood")
)
### Model Fitting¶
We need to invoke the mxRun command to start the model evaluation and optimization. Detailed output will be available in the resulting object, which can be obtained by a print() statement.
#Run ACE model
twinACEFit <- mxRun(twinACEModel)
Often, however, one is interested in specific parts of the output. In the case of twin modeling, we typically will inspect the expected covariance matrices and mean vectors, the parameter estimates, and possibly some derived quantities, such as the standardized variance components, obtained by dividing each of the components by the total variance. Note in the code below that the mxEval command allows easy extraction of the values in the various matrices/algebras which form the first argument, with the model name as second argument. Once these values have been put in new objects, we can use and regular R expression to derive further quantities or organize them in a convenient format for including in tables. Note that helper functions could (and will likely) easily be written for standard models to produce ‘standard’ output.
MZc <- mxEval(ACE.expCovMZ, twinACEFit)
DZc <- mxEval(ACE.expCovDZ, twinACEFit)
M <- mxEval(ACE.expMean, twinACEFit)
A <- mxEval(ACE.A, twinACEFit)
C <- mxEval(ACE.C, twinACEFit)
E <- mxEval(ACE.E, twinACEFit)
V <- (A+C+E)
a2 <- A/V
c2 <- C/V
e2 <- E/V
ACEest <- rbind(cbind(A,C,E),cbind(a2,c2,e2))
LL_ACE <- mxEval(objective, twinACEFit)
## Alternative Models: an AE Model¶
To evaluate the significance of each of the model parameters, nested submodels are fit in which these parameters are fixed to zero. If the likelihood ratio test between the two models is significant, the parameter that is dropped from the model significantly contributes to the phenotype in question. Here we show how we can fit the AE model as a submodel with a change in one mxMatrix command. First, we call up the previous ‘full’ model and save it as a new model twinAEModel. Next we re-specify the matrix c to be fixed to zero. We can run this model in the same way as before and generate similar summaries of the results.
#Run AE model
twinAEModel <- mxRename(twinACEModel, "twinAE")
# drop shared environmental path
twinAEModel\$ACE.c <-
mxMatrix(
type="Full",
nrow=1,
ncol=1,
free=F,
values=0,
label="c11"
)
twinAEFit <- mxRun(twinAEModel)
MZc <- mxEval(ACE.expCovMZ, twinAEFit)
DZc <- mxEval(ACE.expCovDZ, twinAEFit)
A <- mxEval(ACE.A, twinAEFit)
C <- mxEval(ACE.C, twinAEFit)
E <- mxEval(ACE.E, twinAEFit)
V <- (A+C+E)
a2 <- A/V
c2 <- C/V
e2 <- E/V
AEest <- rbind(cbind(A,C,E),cbind(a2,c2,e2))
LL_AE <- mxEval(objective, twinAEFit)
We use a likelihood ratio test (or take the difference between -2 times the log-likelihoods of the two models) to determine the best fitting model, and print relevant output.
LRT_ACE_AE <- LL_AE-LL_ACE
#Print relevant output
ACEest
AEest
LRT_ACE_AE
Note that the OpenMx team is currently working on better alternatives for dropping parameters. These models may also be specified using paths instead of matrices, which allow for easier submodel specification. See Genetic Epidemiology, Path Specification for path specification of these models.
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2014-03-10 03:17:31
|
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http://www.numericalmethod.com/javadoc/suanshu/com/numericalmethod/suanshu/stats/random/rng/univariate/beta/package-summary.html
|
# Package com.numericalmethod.suanshu.stats.random.rng.univariate.beta
• Interface Summary
Interface Description
RandomBetaGenerator
This is a random number generator that generates random deviates according to the Beta distribution.
• Class Summary
Class Description
Cheng1978
Cheng, 1978, is a new rejection method for generating beta variates.
VanDerWaerden1969 Deprecated
Cheng1978 is a much better algorithm.
|
2018-01-23 08:04:35
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http://nrich.maths.org/1813/solution
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### Fred the Class Robot
Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?
### Cartesian Isometric
The graph below is an oblique coordinate system based on 60 degree angles. It was drawn on isometric paper. What kinds of triangles do these points form?
### Triangles All Around
Can you find all the different triangles on these peg boards, and find their angles?
# Transformations on a Pegboard
##### Stage: 2 Challenge Level:
Lucy, who is educated at home, sent in a very clear solution to this question. For the first part she wrote:
You move the top peg to the right by one space. If you cut a square from all four corners, you end up with a quarter of it. In the middle of the square you get four right angles.
I think there is at least one other way to get a right-angled triangle. Can you see how?
Lucy continued:
For the second problem you know that the new shape is going to have sides $4 \times 8$ because the sides are multiplied by $2$. One of the sides is already $4$ so you just move the two right pegs $6$ spaces to the right.
Very well described solutions Lucy, thank you.
Matthew from Beechwood Park School
wrote:
Yes:
To make the triangle a right angle triangle in one move you move the highest band to a perpendicular angle to one of the other angles, and to make the rectangle longer you move the two furthest right corners to the furthest right pegs (in line with them).
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2013-05-24 08:09:00
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https://studyadda.com/sample-papers/kvpy-stream-sx-model-paper-25_q24/1937/464867
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• # question_answer A metal rod of length moving with an angular velocity and velocity of its centre is v. Find potential difference between points A and B at the instant shown in figure. A uniform magnetic field of strength B exist perpendicular to plane of paper: A) $B\,v\,\ell$ B) $Bv\ell +\frac{1}{2}B\omega \,{{\ell }^{2}}$ C) $B\omega \,\ell -\frac{1}{2}B\omega \,{{\ell }^{2}}$ D) $Bv\,\ell +B\,\omega \,\,{{\left( \frac{\ell }{2} \right)}^{2}}$
Point P is at instantaneous rest, ${{\varepsilon }_{1}}=\,\,|{{v}_{P}}-{{v}_{A}}|\,\,=\frac{1}{2}B\omega \,\,{{\left( \frac{\ell }{2}+\frac{v}{\omega } \right)}^{2}}$ ${{\varepsilon }_{2}}=\,\,|{{v}_{P}}-{{v}_{B}}|\,\,=\frac{1}{2}B\omega \,\,{{\left( \frac{\ell }{2}-\frac{v}{\omega } \right)}^{2}}$ $|{{v}_{A}}-{{v}_{B}}|\,\,={{\varepsilon }_{1}}-{{\varepsilon }_{2}}$ $|{{v}_{A}}-{{v}_{B}}|\,\,=B\ell v$
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2022-01-17 01:46:35
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https://socratic.org/questions/57eea4597c01495aafe9e52a
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# Question #9e52a
##### 1 Answer
Sep 30, 2016
The product rule and the power rules in question are
• ${a}^{m} \times {a}^{n} = {a}^{m + n}$
• ${a}^{m} \div {a}^{n} = {a}^{m - n}$
• ${\left({a}^{m}\right)}^{n} = {a}^{m \times n}$
(Note that the second rule comes directly from the first, because $\frac{1}{a} ^ x = {a}^{- x}$, so ${a}^{m} / {a}^{n} = {a}^{m} \times {a}^{- n} = {a}^{m + \left(- n\right)} = {a}^{m - n}$)
Rather than simply post answers to all of the questions on the worksheet, here are a couple of examples of each. These should show how to do the rest, as the problems are all effectively the same, save for the changed values.
First set:
1) ${5}^{- 8} \times {5}^{- 5} = {5}^{- 8 + \left(- 5\right)} = {5}^{- 13}$
2) ${18}^{- 4} \times {18}^{3} = {18}^{- 4 + 3} = {18}^{- 1}$
Second set:
1) ${4}^{2} \div {4}^{10} = {4}^{2 - 10} = {4}^{- 8}$
2) ${19}^{7} \div {19}^{- 8} = {19}^{7 - \left(- 8\right)} = {19}^{15}$
Third set:
1)${\left({15}^{9}\right)}^{- 7} = {15}^{9 \times - 7} = {15}^{- 63}$
2)${\left({7}^{3}\right)}^{6} = {7}^{3 \times 6} = {7}^{18}$
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2019-10-16 08:14:41
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https://mmistakes.github.io/so-simple-theme/page2/
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### Layout: Excerpt (Generated with Separator Tag)
This is the post content. Archive-index pages should display an auto-generated excerpt of all the content preceding the excerpt_separator, as defined in the YAML Front Matter or globally in _config.yml.
Be sure to test the formatting of the auto-generated excerpt, to ensure that it doesn’t create any layout problems.
### Layout: Excerpt (Defined)
This is a user-defined post excerpt. It should be displayed in place of the auto-generated excerpt or post content on index pages.
### Layout: Author Override
A post to test author overrides using a data file.
Enable table of contents on post or page by adding {% include toc %} where you’d like it to appear.
This post tests YouTube video embeds.
This theme supports link posts, made famous by John Gruber. To use, just add link: http://url-you-want-linked to the post’s YAML front matter and you’re done.
### Post: Quote
Only one thing is impossible for God: To find any sense in any copyright law on the planet.
Mark Twain
### Post: Standard
All children, except one, grow up. They soon know that they will grow up, and the way Wendy knew was this. One day when she was two years old she was playing in a garden, and she plucked another flower and ran with it to her mother. I suppose she must have looked rather delightful, for Mrs. Darling put her hand to her heart and cried, “Oh, why can’t you remain like this for ever!” This was all that passed between them on the subject, but henceforth Wendy knew that she must grow up. You always know after you are two. Two is the beginning of the end.
Mrs. Darling first heard of Peter when she was tidying up her children’s minds. It is the nightly custom of every good mother after her children are asleep to rummage in their minds and put things straight for next morning, repacking into their proper places the many articles that have wandered during the day.
This post has been updated and should show a modified date if last_modified_at is used in the layout.
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2018-11-19 12:30:29
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https://www.dcode.fr/combination-rank
|
Search for a tool
Combination Rank
Tool for calculating the rank of a mathematical combination (or conversely, calculating a combination from a rank), that is, the position of a combination in the growing list of possible combinations generated.
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Combination Rank -
Tag(s) : Combinatorics
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# Combination Rank
## Combination from Rank Calculator
Tool for calculating the rank of a mathematical combination (or conversely, calculating a combination from a rank), that is, the position of a combination in the growing list of possible combinations generated.
### How to calculate the rank of a combination?
The rank of a combination is the position of a combination in the list of all possible combinations sorted by ascending order.
Example: All combinations of 4 choose 2 are: (1,2),(1,3),(1,4),(2,3),(2,4),(3,4), therefore the rank of the combination (1,2) is 1, the rank of the combination (2,4) is 5
With $c_i$ the elements in increasing order $c_1, c_2, \cdots, c_k$ of a combination of $k$ elements among $n$ the total number of elements, the formula for calculate rank without listing all combinations is $$\binom{n}{k} - \binom{n-c_1}{k} - \binom{n-c_2}{k-1} - \cdots - \binom{n-c_k}{1}$$
Example: Calculate the combination rank (1,3) among the combinations of 2 among 4 $\binom{4}{2}$, is taking $n = 4, k = 2, c_1 = 1, c_2 = 3$ and calculate $$\binom{4}{2} - \binom{4-1}{2} - \binom{4-3}{2-1} = 6 - 3 - 1 = 2$$ so (1,3) is at rank 2.
### How to calculate a combination from its rank?
This method calculates the minimal combination minimizing $n$ (ie, with the smallest numbers) for a given size $k$.
To compute a combination from a rank $r$, knowing the number of element $k$ of the combination, repeat the following algorithm:
1 - Calculate the largest number $i$, such that the number of combinations $\binom{k}{i}$ is less than or equal to the rank $r$.
2 - Add $i$ at the beginning of the combination, subtract the value $\binom{k}{i}$ from $r$ and decrement $k$ by $1$
3 - Repeat steps 1 and 2 as long as $k > 0$
Example: For a rank $r = 5$ and a combination of $k = 2$ elements
Step 1 - calculate $\binom{2}{2} = 1 < r$, $\binom{3}{2} = 3 < r$ then $\binom {4}{2} = 6 > r$
Step 2 - Combination = (4), $r = 5-3 = 2$, $k = 1$
Step 1' - calculate $\binom{1}{2} = 2 <= r$
Step 2' - Combination = (2,4) , $r = 1$, $k = 0$ - End
So the minimal combination of size 2 and rank 5 is (2,4)
## Source code
dCode retains ownership of the online 'Combination Rank' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Combination Rank download for offline use on PC, tablet, iPhone or Android !
## Need Help ?
Please, check our community Discord for help requests!
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2021-01-16 20:22:48
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https://www.ahaussmann.com/en/solutions-in-practice/uv-light-paints/
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We have the right type of fluorescent paint for everything that’s meant to glow. HATO® Tex “visible” Fluorescent paint show their own, very strong colours in daylight. They become intensively fluorescent when exposed to UV light.
HATO® Tex Phosphorescent Paint UVG 75 keeps glowing in the dark after being exposed to visible or UV light. HATO® Tex “invisible” Fluorescent Paint: A new generation of our fluorescent paints. They are absolutely invisible in normal light and do not show themselves until black light is switched on.
Videoanleitung – In einfachen Schritten simpel hergestellt:
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2019-12-05 15:37:50
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http://codeforces.com/blog/entry/55755?locale=en
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Please subscribe to the official Codeforces channel in Telegram via the link: https://t.me/codeforces_official. ×
### AJAYHAYAGREEVE's blog
By AJAYHAYAGREEVE, history, 13 months ago, ,
hi every one. im having a doubt in codeforces problem 440 C. here is the link http://codeforces.com/problemset/problem/440/C. My approach was
A number x can be represented by having the number with only ones which is less than the given number. for eg if number is 40 one approach is taking 11.
Another is having the number with only ones which is greater than the given number. i.e. 111 for 40 then take absolute difference between the number and the given number i.e abs(11 – 40) = 29 and abs(111 – 40) which is 71.
Now call the function again for the new n. in this case fun(29) and fun(49).
Now added with the number of ones take the min of these two i.e min(fun(29) + 2, fun(71) + 3). that is the answer. Also if we can represent the number with the ones alone. for eg 11 then return 2 just. i.e without computing min(fun(11-11) + 2, fun(111- 11) + 3);
But i got wrong answer in test 14. the input is 81924761239462. and the expected output is 321. And my output is 395. Can you please tell me what is wrong.
•
• -3
•
» 13 months ago, # | 0 I didn't understand this part: if(m[n] == -1) return inf; Can you explain?
• » » 13 months ago, # ^ | ← Rev. 17 → 0 for eg, if n is 112 then it calls fun(abs(111 — 112), abs(1111 — 112)). here 1111 — 112 is 999. now for again 999 it calls fun(abs(111 — 999)) and fun(1111 — 999). 1111 — 999 is 112. so it goes on a non terminating loop. SO in order to terminate this, if i ve already seen 112 then stop and return bcoz it is not going to give the optimal solution as it adds extra 8 1s (1111 — 112 and 1111 — 999). and comes to same state. so in order to stop, i checked the condition whether it has been seen previously.
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2018-12-11 05:47:50
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https://www.bionicturtle.com/forum/forums/p2-t8-investment-management.37/page-3
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P2.T8. Investment Management
Practice questions for investment management and risk management
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1. Question 95: Formula
Question: If W is the portfolio value, (i) is an asset with a single risk factor, and (b) is the beta between the position's and the portfolio's returns, what is the formula for the best hedge? A. -W x [COV(i, portfolio) / variance of i] B. -W x [COV(i, portfolio) / standard deviation of i] C. -W x [variance of I / COV(i, portfolio)] D. (variance of i) x [COV(i, portfolio) / W] ...
Question: If W is the portfolio value, (i) is an asset with a single risk factor, and (b) is the beta between the position's and the portfolio's returns, what is the formula for the best hedge? A. -W x [COV(i, portfolio) / variance of i] B. -W x [COV(i, portfolio) / standard deviation of i] C. -W x [variance of I / COV(i, portfolio)] D. (variance of i) x [COV(i, portfolio) / W] ...
Question: If W is the portfolio value, (i) is an asset with a single risk factor, and (b) is the beta between the position's and the portfolio's returns, what is the formula for the best hedge? A. -W x [COV(i, portfolio) / variance of i] B. -W x [COV(i, portfolio) / standard deviation of i] ...
Question: If W is the portfolio value, (i) is an asset with a single risk factor, and (b) is the beta between the position's and the portfolio's returns, what is the formula for the best hedge? ...
Replies:
0
Views:
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2. Question 94: Incremental VaR
Question: A $20 million portfolio consists of only two equally-weighted and uncorrelated positions in Assets A & B. Asset A ($10 million) has a volatility of 10% and Asset B (also $10 million) has a volatility of 20%. At 99% confidence, what is an approximation of the incremental VaR given an additional investment of$1 million in Asset B? A. $233,000 B.$298,000 C. $333,000 D.$416,000 ...
Question: A $20 million portfolio consists of only two equally-weighted and uncorrelated positions in Assets A & B. Asset A ($10 million) has a volatility of 10% and Asset B (also $10 million) has a volatility of 20%. At 99% confidence, what is an approximation of the incremental VaR given an additional investment of$1 million in Asset B? A. $233,000 B.$298,000 C. $333,000 D.$416,000 ...
Question: A $20 million portfolio consists of only two equally-weighted and uncorrelated positions in Assets A & B. Asset A ($10 million) has a volatility of 10% and Asset B (also $10 million) has a volatility of 20%. At 99% confidence, what is an approximation of the incremental VaR given an... Question: A$20 million portfolio consists of only two equally-weighted and uncorrelated positions in Assets A & B. Asset A ($10 million) has a volatility of 10% and Asset B (also$10 million) has...
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0
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3. Question 93: Marginal VaR
Question: Which are true statements about marginal value at risk (VaR)? I. Marginal VaR = (critical value)[Covariance between position and portfolio returns/portfolio volatility]; II. Marginal VaR is a first-order partial derivative; III. Marginal VaR = (Portfolio VaR/Portfolio size)(beta of position's return with portfolio's return); IV. Marginal VaR approximates incremental VaR for small...
Question: Which are true statements about marginal value at risk (VaR)? I. Marginal VaR = (critical value)[Covariance between position and portfolio returns/portfolio volatility]; II. Marginal VaR is a first-order partial derivative; III. Marginal VaR = (Portfolio VaR/Portfolio size)(beta of position's return with portfolio's return); IV. Marginal VaR approximates incremental VaR for small...
Question: Which are true statements about marginal value at risk (VaR)? I. Marginal VaR = (critical value)[Covariance between position and portfolio returns/portfolio volatility]; II. Marginal VaR is a first-order partial derivative; III. Marginal VaR = (Portfolio VaR/Portfolio size)(beta of...
Question: Which are true statements about marginal value at risk (VaR)? I. Marginal VaR = (critical value)[Covariance between position and portfolio returns/portfolio volatility]; II. Marginal VaR...
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4. Question 92: Marginal VaR and component VaR
Question: A trader has a $10 million position in a$100 million portfolio where the beta of the trader's return with the portfolio's return is 1.5 and the portfolio value at risk (VaR) is $30 million. What is the (i) marginal VaR and (ii) component VaR? A. 0.2 and 2.0 million B. 0.2 and 2.8 million C. 0.45 and 4.5 million D. 1.2 and 2.4 million Answer: C Explanation: Marginal VaR =... Question: A trader has a$10 million position in a $100 million portfolio where the beta of the trader's return with the portfolio's return is 1.5 and the portfolio value at risk (VaR) is$30 million. What is the (i) marginal VaR and (ii) component VaR? A. 0.2 and 2.0 million B. 0.2 and 2.8 million C. 0.45 and 4.5 million D. 1.2 and 2.4 million Answer: C Explanation: Marginal VaR =...
Question: A trader has a $10 million position in a$100 million portfolio where the beta of the trader's return with the portfolio's return is 1.5 and the portfolio value at risk (VaR) is $30 million. What is the (i) marginal VaR and (ii) component VaR? A. 0.2 and 2.0 million B. 0.2 and 2.8... Question: A trader has a$10 million position in a $100 million portfolio where the beta of the trader's return with the portfolio's return is 1.5 and the portfolio value at risk (VaR) is$30...
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5. Question 91: Portfolios standard deviation
Question: A portfolio has five (5) positions with equal weights, standard deviations and correlations between them. If the standard deviation for each is 10% and the correlation between each pair of returns is 0.5, what is the portfolio's standard deviation? A. 5.0% B. 6.25% C. 7.75% D. 10.0% Answer: C Explanation: Under these circumstances, portfolio volatility = (asset...
Question: A portfolio has five (5) positions with equal weights, standard deviations and correlations between them. If the standard deviation for each is 10% and the correlation between each pair of returns is 0.5, what is the portfolio's standard deviation? A. 5.0% B. 6.25% C. 7.75% D. 10.0% Answer: C Explanation: Under these circumstances, portfolio volatility = (asset...
Question: A portfolio has five (5) positions with equal weights, standard deviations and correlations between them. If the standard deviation for each is 10% and the correlation between each pair of returns is 0.5, what is the portfolio's standard deviation? A. 5.0% B. 6.25% C. 7.75% D....
Question: A portfolio has five (5) positions with equal weights, standard deviations and correlations between them. If the standard deviation for each is 10% and the correlation between each pair...
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6. Question 90: Portfolio VaR
Question: Assume a two-asset portfolio with a portfolio value of $20 million. Each asset weighs 50% of the portfolio. Asset A has a volatility of 10% and asset B has a volatility of 20%. If the desired confidence is 99%, what is the portfolio VaR if (i) the assets are uncorrelated [i.e.., correlation = 0] and (ii) the assets are perfectly correlated [i.e., correlation = -1] A.$2.56 and...
Question: Assume a two-asset portfolio with a portfolio value of $20 million. Each asset weighs 50% of the portfolio. Asset A has a volatility of 10% and asset B has a volatility of 20%. If the desired confidence is 99%, what is the portfolio VaR if (i) the assets are uncorrelated [i.e.., correlation = 0] and (ii) the assets are perfectly correlated [i.e., correlation = -1] A.$2.56 and...
Question: Assume a two-asset portfolio with a portfolio value of $20 million. Each asset weighs 50% of the portfolio. Asset A has a volatility of 10% and asset B has a volatility of 20%. If the desired confidence is 99%, what is the portfolio VaR if (i) the assets are uncorrelated [i.e..,... Question: Assume a two-asset portfolio with a portfolio value of$20 million. Each asset weighs 50% of the portfolio. Asset A has a volatility of 10% and asset B has a volatility of 20%. If the...
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7. Question 89: VaR
Question: Assume a two-asset portfolio with a portfolio value of $10 million. Each asset weighs 50% of the portfolio. Asset A has a volatility of 10% and asset B has a volatility of 20%. The correlation between Asset A & B is 0.5. What is the individual VaR of Asset B? A.$822,000 B. $1.645 million C.$2.16 million D. $2.33 million Answer: B Explanation: The individual VaR of Asset... Question: Assume a two-asset portfolio with a portfolio value of$10 million. Each asset weighs 50% of the portfolio. Asset A has a volatility of 10% and asset B has a volatility of 20%. The correlation between Asset A & B is 0.5. What is the individual VaR of Asset B? A. $822,000 B.$1.645 million C. $2.16 million D.$2.33 million Answer: B Explanation: The individual VaR of Asset...
Question: Assume a two-asset portfolio with a portfolio value of $10 million. Each asset weighs 50% of the portfolio. Asset A has a volatility of 10% and asset B has a volatility of 20%. The correlation between Asset A & B is 0.5. What is the individual VaR of Asset B? A.$822,000 B. $1.645... Question: Assume a two-asset portfolio with a portfolio value of$10 million. Each asset weighs 50% of the portfolio. Asset A has a volatility of 10% and asset B has a volatility of 20%. The...
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8. Question 88: Diversified portfolio VaR
Question: Assume a two-asset portfolio with a portfolio value of $10 million. Each asset weighs 50% of the portfolio. Asset A has a volatility of 10% and asset B has a volatility of 20%. The correlation between Asset A & B is 0.5. What is the diversified portfolio VaR under 95% confidence? A.$1.96 million B. $2.18 million C.$2.82 million D. $3.16 million Answer: B Explanation:... Question: Assume a two-asset portfolio with a portfolio value of$10 million. Each asset weighs 50% of the portfolio. Asset A has a volatility of 10% and asset B has a volatility of 20%. The correlation between Asset A & B is 0.5. What is the diversified portfolio VaR under 95% confidence? A. $1.96 million B.$2.18 million C. $2.82 million D.$3.16 million Answer: B Explanation:...
Question: Assume a two-asset portfolio with a portfolio value of $10 million. Each asset weighs 50% of the portfolio. Asset A has a volatility of 10% and asset B has a volatility of 20%. The correlation between Asset A & B is 0.5. What is the diversified portfolio VaR under 95% confidence? A.... Question: Assume a two-asset portfolio with a portfolio value of$10 million. Each asset weighs 50% of the portfolio. Asset A has a volatility of 10% and asset B has a volatility of 20%. The...
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9. Question 87: Future strategies
Question: Jaeger claims that most systematic managed futures strategies are: A. Trend-followers B. Mean-reversion plays C. Volatility basis D. Basis risk based Answer: A Explanation: Trend-following is the dominant trading style for systematic manager futures strategies. The manager relies on technical indicators (e.g., momentum, relative size of moving averages, or break-out...
Question: Jaeger claims that most systematic managed futures strategies are: A. Trend-followers B. Mean-reversion plays C. Volatility basis D. Basis risk based Answer: A Explanation: Trend-following is the dominant trading style for systematic manager futures strategies. The manager relies on technical indicators (e.g., momentum, relative size of moving averages, or break-out...
Question: Jaeger claims that most systematic managed futures strategies are: A. Trend-followers B. Mean-reversion plays C. Volatility basis D. Basis risk based Answer: A Explanation: Trend-following is the dominant trading style for systematic manager futures strategies. The manager...
Question: Jaeger claims that most systematic managed futures strategies are: A. Trend-followers B. Mean-reversion plays C. Volatility basis D. Basis risk based Answer: A Explanation:...
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10. Question 86: Model globe macro strategies
Question: Jaeger says the key distinction between modern global macro strategies is: A. Fundamental versus technical B. Sector versus style C. Directional versus market-neutral D. Discretionary versus systematic Answer: D Explanation: Discretionary managers employ various "opportunistic" strategies (style drift is built-in); Systematic managers use well-defined trading models
Question: Jaeger says the key distinction between modern global macro strategies is: A. Fundamental versus technical B. Sector versus style C. Directional versus market-neutral D. Discretionary versus systematic Answer: D Explanation: Discretionary managers employ various "opportunistic" strategies (style drift is built-in); Systematic managers use well-defined trading models
Question: Jaeger says the key distinction between modern global macro strategies is: A. Fundamental versus technical B. Sector versus style C. Directional versus market-neutral D. Discretionary versus systematic Answer: D Explanation: Discretionary managers employ various...
Question: Jaeger says the key distinction between modern global macro strategies is: A. Fundamental versus technical B. Sector versus style C. Directional versus market-neutral D....
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11. Question 85: Risks
Question: A manager who employs a "Regulation D" strategy is exposed primarily to which risks: A. Credit and liquidity B. Liquidity and regulatory C. Regulatory and Market D. Market and operational Answer: A Explanation: Regulation D managers tend to invest in small companies with limited means to raise capital. The investment is illiquid before registration and limited in...
Question: A manager who employs a "Regulation D" strategy is exposed primarily to which risks: A. Credit and liquidity B. Liquidity and regulatory C. Regulatory and Market D. Market and operational Answer: A Explanation: Regulation D managers tend to invest in small companies with limited means to raise capital. The investment is illiquid before registration and limited in...
Question: A manager who employs a "Regulation D" strategy is exposed primarily to which risks: A. Credit and liquidity B. Liquidity and regulatory C. Regulatory and Market D. Market and operational Answer: A Explanation: Regulation D managers tend to invest in small companies with...
Question: A manager who employs a "Regulation D" strategy is exposed primarily to which risks: A. Credit and liquidity B. Liquidity and regulatory C. Regulatory and Market D. Market and...
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12. Question 84: Distressed securities strategy
Question: Challenges of employing a distressed securities strategy include all of the following except: A. Less liquidity B. Unfavorable image as "vultures" C. Require much expertise and extensive analysis D. Legal issues Answer: B Explanation: Distressed securities tend to be less liquid; require specialist expertise with much analytical pre-work involved; tend to be confronted...
Question: Challenges of employing a distressed securities strategy include all of the following except: A. Less liquidity B. Unfavorable image as "vultures" C. Require much expertise and extensive analysis D. Legal issues Answer: B Explanation: Distressed securities tend to be less liquid; require specialist expertise with much analytical pre-work involved; tend to be confronted...
Question: Challenges of employing a distressed securities strategy include all of the following except: A. Less liquidity B. Unfavorable image as "vultures" C. Require much expertise and extensive analysis D. Legal issues Answer: B Explanation: Distressed securities tend to be less...
Question: Challenges of employing a distressed securities strategy include all of the following except: A. Less liquidity B. Unfavorable image as "vultures" C. Require much expertise and...
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13. Question 83: Key exposure
Question: What is the key exposure (risk factor) in merger arbitrage? A. Deal risk premium B. Regulatory risk premium C. Model risk D. Spread risk Answer: A Explanation: The "deal risk premium" subsumes most of the other risks; "deal risk" includes everything that affects the deal's completion or its timing.
Question: What is the key exposure (risk factor) in merger arbitrage? A. Deal risk premium B. Regulatory risk premium C. Model risk D. Spread risk Answer: A Explanation: The "deal risk premium" subsumes most of the other risks; "deal risk" includes everything that affects the deal's completion or its timing.
Question: What is the key exposure (risk factor) in merger arbitrage? A. Deal risk premium B. Regulatory risk premium C. Model risk D. Spread risk Answer: A Explanation: The "deal risk premium" subsumes most of the other risks; "deal risk" includes everything that affects the deal's...
Question: What is the key exposure (risk factor) in merger arbitrage? A. Deal risk premium B. Regulatory risk premium C. Model risk D. Spread risk Answer: A Explanation: The "deal risk...
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14. Question 82: Risk in the CAPM
Question: If the capital asset pricing model (CAPM) were applied against a portfolio that employed an event-driven strategy, which risk in the CAPM would correspond to the manager's focus area: A. Equity premium B. Beta C. Idiosyncratic risk D. Quantity of risk Answer: C Explanation: Event-driven strategies are company-specific or idiosyncratic. Theoretically, as idiosyncratic...
Question: If the capital asset pricing model (CAPM) were applied against a portfolio that employed an event-driven strategy, which risk in the CAPM would correspond to the manager's focus area: A. Equity premium B. Beta C. Idiosyncratic risk D. Quantity of risk Answer: C Explanation: Event-driven strategies are company-specific or idiosyncratic. Theoretically, as idiosyncratic...
Question: If the capital asset pricing model (CAPM) were applied against a portfolio that employed an event-driven strategy, which risk in the CAPM would correspond to the manager's focus area: A. Equity premium B. Beta C. Idiosyncratic risk D. Quantity of risk Answer: C Explanation:...
Question: If the capital asset pricing model (CAPM) were applied against a portfolio that employed an event-driven strategy, which risk in the CAPM would correspond to the manager's focus area: ...
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15. Question 80: Key metric
Question: Which key metric enables a volatility arbitrageur to determine that volatility is "cheap" or "expensive?" A. Historical volatility patterns B. Recent change in volatility C. Implied volatility D. It is not model determined; "implied" means subjective Answer: C Explanation: The market price of the instrument "implies" a volatility (such that the volatility produces a model...
Question: Which key metric enables a volatility arbitrageur to determine that volatility is "cheap" or "expensive?" A. Historical volatility patterns B. Recent change in volatility C. Implied volatility D. It is not model determined; "implied" means subjective Answer: C Explanation: The market price of the instrument "implies" a volatility (such that the volatility produces a model...
Question: Which key metric enables a volatility arbitrageur to determine that volatility is "cheap" or "expensive?" A. Historical volatility patterns B. Recent change in volatility C. Implied volatility D. It is not model determined; "implied" means subjective Answer: C Explanation:...
Question: Which key metric enables a volatility arbitrageur to determine that volatility is "cheap" or "expensive?" A. Historical volatility patterns B. Recent change in volatility C. Implied...
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16. Question 79: Pricing inefficiencies
Question: Fixed income markets display various pricing inefficiencies due to all of the following EXCEPT: A. Agency biases B. Structural reasons C. Market segmentation D. Lack of an equity risk premium Answer: D Explanation: Exploitable fixed-income inefficiencies: (1) Agency biases: fiduciaries purchase yesterday's winners; (2) Structural: tax, accounting, or regulatory; (3)...
Question: Fixed income markets display various pricing inefficiencies due to all of the following EXCEPT: A. Agency biases B. Structural reasons C. Market segmentation D. Lack of an equity risk premium Answer: D Explanation: Exploitable fixed-income inefficiencies: (1) Agency biases: fiduciaries purchase yesterday's winners; (2) Structural: tax, accounting, or regulatory; (3)...
Question: Fixed income markets display various pricing inefficiencies due to all of the following EXCEPT: A. Agency biases B. Structural reasons C. Market segmentation D. Lack of an equity risk premium Answer: D Explanation: Exploitable fixed-income inefficiencies: (1) Agency biases:...
Question: Fixed income markets display various pricing inefficiencies due to all of the following EXCEPT: A. Agency biases B. Structural reasons C. Market segmentation D. Lack of an equity...
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17. Question 78: Static return
Question: An example of a static return is: A. A higher bond coupon B. An arbitrage opportunity between a "cheap" convertible and "expensive" stock C. A gamma trade on volatility D. A mispricing Answer: A Explanation: Static returns are one of the three sources of return for a convertible arbitrage strategy (1. static returns, 2. gamma trading on stock volatility, 3. price...
Question: An example of a static return is: A. A higher bond coupon B. An arbitrage opportunity between a "cheap" convertible and "expensive" stock C. A gamma trade on volatility D. A mispricing Answer: A Explanation: Static returns are one of the three sources of return for a convertible arbitrage strategy (1. static returns, 2. gamma trading on stock volatility, 3. price...
Question: An example of a static return is: A. A higher bond coupon B. An arbitrage opportunity between a "cheap" convertible and "expensive" stock C. A gamma trade on volatility D. A mispricing Answer: A Explanation: Static returns are one of the three sources of return for a...
Question: An example of a static return is: A. A higher bond coupon B. An arbitrage opportunity between a "cheap" convertible and "expensive" stock C. A gamma trade on volatility D. A...
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18. Question 77: Convertible strategies
Question: Managers pursuing convertible strategies earn returns from all of the following EXCEPT: A. The conversion feature on the bond B. Static returns from coupon income and short stock rebates C. Gamma trading D. Exploiting price inefficiencies Answer: A Explanation: The conversion feature is not itself a source of return.
Question: Managers pursuing convertible strategies earn returns from all of the following EXCEPT: A. The conversion feature on the bond B. Static returns from coupon income and short stock rebates C. Gamma trading D. Exploiting price inefficiencies Answer: A Explanation: The conversion feature is not itself a source of return.
Question: Managers pursuing convertible strategies earn returns from all of the following EXCEPT: A. The conversion feature on the bond B. Static returns from coupon income and short stock rebates C. Gamma trading D. Exploiting price inefficiencies Answer: A Explanation: The...
Question: Managers pursuing convertible strategies earn returns from all of the following EXCEPT: A. The conversion feature on the bond B. Static returns from coupon income and short stock...
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19. Question 76:Short-selling strategies
Question: According to Jaeger, short-selling strategies involve additional issues (i.e., above long only strategies) that primarily relate to: A. Uncapped loss potential B. Challenging investor psychology C. Twice the opportunity (long + short) D. The process of borrowing stocks Answer: D Explanation: The problems that relate to borrowing include: share availability, stability of...
Question: According to Jaeger, short-selling strategies involve additional issues (i.e., above long only strategies) that primarily relate to: A. Uncapped loss potential B. Challenging investor psychology C. Twice the opportunity (long + short) D. The process of borrowing stocks Answer: D Explanation: The problems that relate to borrowing include: share availability, stability of...
Question: According to Jaeger, short-selling strategies involve additional issues (i.e., above long only strategies) that primarily relate to: A. Uncapped loss potential B. Challenging investor psychology C. Twice the opportunity (long + short) D. The process of borrowing stocks Answer:...
Question: According to Jaeger, short-selling strategies involve additional issues (i.e., above long only strategies) that primarily relate to: A. Uncapped loss potential B. Challenging...
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20. Question 75: Market timing strategies
Question: Which are the two main types of market timing strategies: A. Sector and time zone arbitrage B. Currency and sector C. Interest rate and industry D. Exchanges (NYSE vs. NASDAQ) and time zone Answer: A Explanation: Sector timing aims to profit from micro upward trends in single industry sectors; time zone arbitrage exploits pricing inefficiencies based on "conditional price...
Question: Which are the two main types of market timing strategies: A. Sector and time zone arbitrage B. Currency and sector C. Interest rate and industry D. Exchanges (NYSE vs. NASDAQ) and time zone Answer: A Explanation: Sector timing aims to profit from micro upward trends in single industry sectors; time zone arbitrage exploits pricing inefficiencies based on "conditional price...
Question: Which are the two main types of market timing strategies: A. Sector and time zone arbitrage B. Currency and sector C. Interest rate and industry D. Exchanges (NYSE vs. NASDAQ) and time zone Answer: A Explanation: Sector timing aims to profit from micro upward trends in...
Question: Which are the two main types of market timing strategies: A. Sector and time zone arbitrage B. Currency and sector C. Interest rate and industry D. Exchanges (NYSE vs. NASDAQ) and...
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21. Question 74: Fama-French factor
Question: Jaeger argues that even if broad market neutrality is achieved, a manager is potentially exposed to several "beta-type" risk factors. Which of the following is not a Fama-French factor? A. Value stocks (low price to book) B. Small capitalization stocks C. Momentum factors D. Low liquidity factors Answer: D Explanation: While liquidity may indeed by a factor, the...
Question: Jaeger argues that even if broad market neutrality is achieved, a manager is potentially exposed to several "beta-type" risk factors. Which of the following is not a Fama-French factor? A. Value stocks (low price to book) B. Small capitalization stocks C. Momentum factors D. Low liquidity factors Answer: D Explanation: While liquidity may indeed by a factor, the...
Question: Jaeger argues that even if broad market neutrality is achieved, a manager is potentially exposed to several "beta-type" risk factors. Which of the following is not a Fama-French factor? A. Value stocks (low price to book) B. Small capitalization stocks C. Momentum factors D. Low...
Question: Jaeger argues that even if broad market neutrality is achieved, a manager is potentially exposed to several "beta-type" risk factors. Which of the following is not a Fama-French factor? ...
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22. Question 73: Short positions
Question: According to Jaeger, short positions (short selling) has three purposes. Which of the following is not one of them? A. Generating positive returns B. Hedging market risk C. Earning the short rebate D. Useful threat to promote changes at target company Answer: D Explanation: (D) is not cited; the others are advantages to shorting. The long/short equity manager does not hug...
Question: According to Jaeger, short positions (short selling) has three purposes. Which of the following is not one of them? A. Generating positive returns B. Hedging market risk C. Earning the short rebate D. Useful threat to promote changes at target company Answer: D Explanation: (D) is not cited; the others are advantages to shorting. The long/short equity manager does not hug...
Question: According to Jaeger, short positions (short selling) has three purposes. Which of the following is not one of them? A. Generating positive returns B. Hedging market risk C. Earning the short rebate D. Useful threat to promote changes at target company Answer: D Explanation:...
Question: According to Jaeger, short positions (short selling) has three purposes. Which of the following is not one of them? A. Generating positive returns B. Hedging market risk C. Earning...
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23. Question 72: Bond decomposition strategy
Question: Which bond decomposition strategy emphasizes, respectively, the following: (i) stage in the management process, (ii) exposure to international markets, and (iii) Yield curve factors A. Lehman Brothers, APT, Solnik's IPA B. Barra, Khoury, APT C. Kuberek's, Solnik's IPA, and Lehman Brothers D. Lehman Brothers, McLaren, and Barra Answer: C Explanation: The Lehman Brothers...
Question: Which bond decomposition strategy emphasizes, respectively, the following: (i) stage in the management process, (ii) exposure to international markets, and (iii) Yield curve factors A. Lehman Brothers, APT, Solnik's IPA B. Barra, Khoury, APT C. Kuberek's, Solnik's IPA, and Lehman Brothers D. Lehman Brothers, McLaren, and Barra Answer: C Explanation: The Lehman Brothers...
Question: Which bond decomposition strategy emphasizes, respectively, the following: (i) stage in the management process, (ii) exposure to international markets, and (iii) Yield curve factors A. Lehman Brothers, APT, Solnik's IPA B. Barra, Khoury, APT C. Kuberek's, Solnik's IPA, and Lehman...
Question: Which bond decomposition strategy emphasizes, respectively, the following: (i) stage in the management process, (ii) exposure to international markets, and (iii) Yield curve factors ...
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24. Question 71: Investment strategies
Question: All of the following are investment strategies for managing fixed-income portfolios EXCEPT FOR: A. Barra B. Duration C. Sector D. Maturity distribution Answer: A Explanation: The strategies include active (e.g., forecast rates, forecast spreads), passive (index replication), duration, sector, and maturity distribution. The Barra model is a multifactor model used to...
Question: All of the following are investment strategies for managing fixed-income portfolios EXCEPT FOR: A. Barra B. Duration C. Sector D. Maturity distribution Answer: A Explanation: The strategies include active (e.g., forecast rates, forecast spreads), passive (index replication), duration, sector, and maturity distribution. The Barra model is a multifactor model used to...
Question: All of the following are investment strategies for managing fixed-income portfolios EXCEPT FOR: A. Barra B. Duration C. Sector D. Maturity distribution Answer: A Explanation: The strategies include active (e.g., forecast rates, forecast spreads), passive (index...
Question: All of the following are investment strategies for managing fixed-income portfolios EXCEPT FOR: A. Barra B. Duration C. Sector D. Maturity distribution Answer: A Explanation:...
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25. Question 70: Explanatory factors
Question: The Litterman and Scheinkman model contains which three explanatory factors? A. Parallel yield curve shifts, slope changes, and curvature B. Yield curve shift, default risk and market risk C. Default, duration, and market risk D. Duration, convexity, and curvature Answer: A Explanation: The TWO PRIMARY RISKS that explain bond returns are DEFAULT RISK and MARKET RISK. In...
Question: The Litterman and Scheinkman model contains which three explanatory factors? A. Parallel yield curve shifts, slope changes, and curvature B. Yield curve shift, default risk and market risk C. Default, duration, and market risk D. Duration, convexity, and curvature Answer: A Explanation: The TWO PRIMARY RISKS that explain bond returns are DEFAULT RISK and MARKET RISK. In...
Question: The Litterman and Scheinkman model contains which three explanatory factors? A. Parallel yield curve shifts, slope changes, and curvature B. Yield curve shift, default risk and market risk C. Default, duration, and market risk D. Duration, convexity, and curvature Answer: A ...
Question: The Litterman and Scheinkman model contains which three explanatory factors? A. Parallel yield curve shifts, slope changes, and curvature B. Yield curve shift, default risk and...
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26. Question 69: Dynamic interest rate models
Question: As dynamic interest rate models, what is the difference between the Cox-Ingersoll-Ross (CIR) model and the Vasicek model? A. Mean reversion B. Short-term rate C. Stochastic process D. Heteroskedastic interest rate volatility Answer: D Explanation: Both models share (A), (B), and (C) in common. They both model rates as reverting toward a mean, given a speed of reversion...
Question: As dynamic interest rate models, what is the difference between the Cox-Ingersoll-Ross (CIR) model and the Vasicek model? A. Mean reversion B. Short-term rate C. Stochastic process D. Heteroskedastic interest rate volatility Answer: D Explanation: Both models share (A), (B), and (C) in common. They both model rates as reverting toward a mean, given a speed of reversion...
Question: As dynamic interest rate models, what is the difference between the Cox-Ingersoll-Ross (CIR) model and the Vasicek model? A. Mean reversion B. Short-term rate C. Stochastic process D. Heteroskedastic interest rate volatility Answer: D Explanation: Both models share (A),...
Question: As dynamic interest rate models, what is the difference between the Cox-Ingersoll-Ross (CIR) model and the Vasicek model? A. Mean reversion B. Short-term rate C. Stochastic process ...
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27. Question 68: Indirect method
Question: What is the most acute risk created under the INDIRECT METHOD for estimating a range of zero-coupon rates given yields to maturity? A. Reinvestment risk B. Model risk C. Yield curve risk D. Coupon risk Answer: B Explanation: The indirect methods adjust the market data to fit a specified shape of the yield curve. The suffer therefore from "specification" or model risk...
Question: What is the most acute risk created under the INDIRECT METHOD for estimating a range of zero-coupon rates given yields to maturity? A. Reinvestment risk B. Model risk C. Yield curve risk D. Coupon risk Answer: B Explanation: The indirect methods adjust the market data to fit a specified shape of the yield curve. The suffer therefore from "specification" or model risk...
Question: What is the most acute risk created under the INDIRECT METHOD for estimating a range of zero-coupon rates given yields to maturity? A. Reinvestment risk B. Model risk C. Yield curve risk D. Coupon risk Answer: B Explanation: The indirect methods adjust the market data to...
Question: What is the most acute risk created under the INDIRECT METHOD for estimating a range of zero-coupon rates given yields to maturity? A. Reinvestment risk B. Model risk C. Yield curve...
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28. Question 67: Term structure of interest rates
Question: If we want to construct a term structure of interest rates, which is the better method? A. Yield to maturity, because the data is available for all maturities B. Yield to maturity, because it is a recognized standard C. Zero-coupon, because it assumes a single rate at each maturity D. Zero-coupon because is allows for direct estimates Answer: C Explanation: The yield to...
Question: If we want to construct a term structure of interest rates, which is the better method? A. Yield to maturity, because the data is available for all maturities B. Yield to maturity, because it is a recognized standard C. Zero-coupon, because it assumes a single rate at each maturity D. Zero-coupon because is allows for direct estimates Answer: C Explanation: The yield to...
Question: If we want to construct a term structure of interest rates, which is the better method? A. Yield to maturity, because the data is available for all maturities B. Yield to maturity, because it is a recognized standard C. Zero-coupon, because it assumes a single rate at each...
Question: If we want to construct a term structure of interest rates, which is the better method? A. Yield to maturity, because the data is available for all maturities B. Yield to maturity,...
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29. Question 66: Returns based and portfolio-based style analysis
Question: In a comparison between returns-based style analysis and portfolio-based style analysis models, to which do the following statements, respectively, refer? I. Simpler to implement, II. More informative for evaluation, III. Better suited to external (index or peer-based) performance comparisons A. Returns-based, Portfolio-based, Returns-based B. Returns-based, Portfolio-based,...
Question: In a comparison between returns-based style analysis and portfolio-based style analysis models, to which do the following statements, respectively, refer? I. Simpler to implement, II. More informative for evaluation, III. Better suited to external (index or peer-based) performance comparisons A. Returns-based, Portfolio-based, Returns-based B. Returns-based, Portfolio-based,...
Question: In a comparison between returns-based style analysis and portfolio-based style analysis models, to which do the following statements, respectively, refer? I. Simpler to implement, II. More informative for evaluation, III. Better suited to external (index or peer-based) performance...
Question: In a comparison between returns-based style analysis and portfolio-based style analysis models, to which do the following statements, respectively, refer? I. Simpler to implement, II....
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0
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7
30. Question 65: Performance indicator
Question: The examples of multifactor performance analysis models (e.g., Elton, Gruber et al) tend to generalize which performance indicator? A. Sharpe B. Treynor C. Jensen D. Information ratio Answer: C Explanation: The multifactor models decompose portfolio performance into factors. The share of return that is not explained by the factors constitutes the residual return. As such,...
Question: The examples of multifactor performance analysis models (e.g., Elton, Gruber et al) tend to generalize which performance indicator? A. Sharpe B. Treynor C. Jensen D. Information ratio Answer: C Explanation: The multifactor models decompose portfolio performance into factors. The share of return that is not explained by the factors constitutes the residual return. As such,...
Question: The examples of multifactor performance analysis models (e.g., Elton, Gruber et al) tend to generalize which performance indicator? A. Sharpe B. Treynor C. Jensen D. Information ratio Answer: C Explanation: The multifactor models decompose portfolio performance into...
Question: The examples of multifactor performance analysis models (e.g., Elton, Gruber et al) tend to generalize which performance indicator? A. Sharpe B. Treynor C. Jensen D. Information...
Replies:
0
Views:
7
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2016-12-04 14:27:05
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https://ask.dataone.org/answers/85/revisions/
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# Revision history [back]
These discrepancies occur when either: synchronization is not functioning properly, so the MN and CN have not synced their content since the MN made changes, or if the MN makes changes without properly notifying the CN. There are two cases to consider:
Case 1: Fewer objects on CN than MN: check <dateSysMetadataModified> In this case, the likely problem is that the CN is not discovering objects on the MN during the synchronization process. This can happen if the MN inserts some objects, but fails to set the <dateSysMetadataModified> field to the current time. Some MNs back-date these system metadata modification times, not realizing that the CN relies on that date to determine which objects might have changed and need to be synced. If the <dateSysMetadataModified> date is set to a date that is before the most recent synchronization time, then that object will never be noticed by the CN, and never harvested. The To fix this, modify the system metadata and set the <dateSysMetadataModified> field to the time now, and the next time the synchronization occurs, the new object changes will be noticed and picked up.
Case 2: More objects on CN than MN: ensure archive and obsoletes/obsoleted are set In this case, some objects are on the CN which are not present on the MN. The only way for this to happen is if the MN has removed the content that was previously associated with a particular identifier (often in the process of trying to change an identifier). The solution is simple as well -- remember that identifiers (pids) are both persistent and non-reusable, so if you want to remove an identifier from your system, first insert the content using the new identifier, being sure to reference the old identifier in the <obsoletes> system metadata field. Then, delete the old identifier, but be sure to keep its system metadata, reference the new pid in the <obsoletedBy> field, and mark the object as <archived>true</archived>. This will tell the CN that the object identified by the PID is no longer active on the MN, and that it has been replaced by the new PID. This will allow the CN to properly understand and index the changes made on the MN.
These discrepancies occur when either: synchronization is not functioning properly, so the MN and CN have not synced their content since the MN made changes, or if the MN makes changes without properly notifying the CN. There are two cases to consider:
Case 1: Fewer objects on CN than MN: check <dateSysMetadataModified> In this case, the likely problem is that the CN is not discovering objects on the MN during the synchronization process. This can happen if the MN inserts some objects, but fails to set the <dateSysMetadataModified> field to the current time. Some MNs back-date these system metadata modification times, not realizing that the CN relies on that date to determine which objects might have changed and need to be synced. If the <dateSysMetadataModified> date is set to a date that is before the most recent synchronization time, then that object will never be noticed by the CN, and never harvested. The To fix this, modify the system metadata and set the <dateSysMetadataModified> field to the time now, and the next time the synchronization occurs, the new object changes will be noticed and picked up.
Case 2: More objects on CN than MN: ensure archive and obsoletes/obsoleted are set In this case, some objects are on the CN which are not present on the MN. The only way for this to happen is if the MN has removed the content that was previously associated with a particular identifier (often in the process of trying to change an identifier). The solution is simple as well -- remember that identifiers (pids) are both persistent and non-reusable, so if you want to remove an identifier from your system, first insert the content using the new identifier, being sure to reference the old identifier in the <obsoletes> system metadata field. Then, delete the old identifier, but be sure to keep its system metadata, reference the new pid in the <obsoletedBy> field, and mark the object as <archived>true</archived>. This will tell the CN that the object identified by the PID is no longer active on the MN, and that it has been replaced by the new PID. This will allow the CN to properly understand and index the changes made on the MN.
Case 1: Fewer objects on CN than MN: check <dateSysMetadataModified>
In this case, the likely problem is that the CN is not discovering objects on the MN during the synchronization process. This can happen if the MN inserts some objects, but fails to set the <dateSysMetadataModified> field to the current time. Some MNs back-date these system metadata modification times, not realizing that the CN relies on that date to determine which objects might have changed and need to be synced. If the <dateSysMetadataModified> date is set to a date that is before the most recent synchronization time, then that object will never be noticed by the CN, and never harvested. To fix this, modify the system metadata and set the <dateSysMetadataModified> field to the time now, and the next time the synchronization occurs, the new object changes will be noticed and picked up.
In this case, some objects are on the CN which are not present on the MN. The only way for this to happen is if the MN has removed the content that was previously associated with a particular identifier (often in the process of trying to change an identifier). The solution is simple as well -- remember that identifiers (pids) are both persistent and non-reusable, so if you want to remove an identifier from your system, first insert the content using the new identifier, being sure to reference the old identifier in the <obsoletes> system metadata field. Then, delete the old identifier, but be sure to keep its system metadata, reference the new pid in the <obsoletedBy> field, and mark the object as <archived>true</archived>. This will tell the CN that the object identified by the PID is no longer active on the MN, and that it has been replaced by the new PID. This will allow the CN to properly understand and index the changes made on the MN.
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2019-08-26 02:57:30
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https://www.dsprelated.com/freebooks/sasp/Random_Variable.html
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Random Variable
Definition: A random variable is defined as a real- or complex-valued function of some random event, and is fully characterized by its probability distribution.
Example: A random variable can be defined based on a coin toss by defining numerical values for heads and tails. For example, we may assign 0 to tails and 1 to heads. The probability distribution for this random variable is then
(C.4)
Example: A die can be used to generate integer-valued random variables between 1 and 6. Rolling the die provides an underlying random event. The probability distribution of a fair die is the discrete uniform distribution between 1 and 6. I.e.,
(C.5)
Example: A pair of dice can be used to generate integer-valued random variables between 2 and 12. Rolling the dice provides an underlying random event. The probability distribution of two fair dice is given by
(C.6)
This may be called a discrete triangular distribution. It can be shown to be given by the convolution of the discrete uniform distribution for one die with itself. This is a general fact for sums of random variables (the distribution of the sum equals the convolution of the component distributions).
Example: Consider a random experiment in which a sewing needle is dropped onto the ground from a high altitude. For each such event, the angle of the needle with respect to north is measured. A reasonable model for the distribution of angles (neglecting the earth's magnetic field) is the continuous uniform distribution on , i.e., for any real numbers and in the interval , with , the probability of the needle angle falling within that interval is
(C.7)
Note, however, that the probability of any single angle is zero. This is our first example of a continuous probability distribution. Therefore, we cannot simply define the probability of outcome for each . Instead, we must define the probability density function (PDF):
(C.8)
To calculate a probability, the PDF must be integrated over one or more intervals. As follows from Lebesgue integration theory (measure theory''), the probability of any countably infinite set of discrete points is zero when the PDF is finite. This is because such a set of points is a set of measure zero'' under integration. Note that we write for discrete probability distributions and for PDFs. A discrete probability distribution such as that in (C.4) can be written as
(C.9)
where denotes an impulse.C.1
Next Section:
Stochastic Process
Previous Section:
Independent Events
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2017-11-18 17:39:56
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https://www.groundai.com/project/bell-tests-with-min-entropy-sources/
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A Generalization of Lemma 1 the multipartite case
# Bell tests with min-entropy sources
## Abstract
Device independent protocols rely on the violation of Bell inequalities to certify properties of the resources available. The violation of the inequalities are meaningless without a few well-known assumptions. One of these is measurement independence, the property that the source of the states measured in an inequality is uncorrelated from the measurements selected. Since this assumption cannot be confirmed, we consider the consequences of relaxing it and find that the definition chosen is critically important to the observed behavior. Considering a definition that is a bound on the min-entropy of the measurement settings, we find lower bounds on the min-entropy of the source used to choose the inputs required to deduce any quantum or non-local behavior from a Bell inequality violation. These bounds are significantly more restrictive than the ones obtained by endowing the measurement-input source with the further structure of a Santha-Vazirani source. We also outline a procedure for finding tight bounds and study the set of probabilities that can result from relaxing measurement dependence.
###### pacs:
03.65.Ta 03.65.Ud 03.67.-a
## I Introduction
The violation of Bell inequalities can be used to certify important quantum information properties in a black-box scenario under minimal assumptions. This idea of “device-independent” certification started in the context of quantum key distribution, where the violation of Bell inequalities bounds the information leaked to the eavesdropper [1]; [2]; [3]; and it has been extended to various other tasks, notably state certification [1]; [4]; [5], measurement certification [6], and private randomness expansion [7]; [8]; [9]. Ultimately, this stems from the fact that the violation of Bell inequalities certifies the presence of a quantifiable amount of intrinsic randomness: indeed, a contrario, if the outcomes were predictable, one could have predicted them in advance and the measurement could consist of reading from a pre-existing list. This is exactly what the violation of Bell inequality certifies as impossible.
Two assumptions are left in device-independent certification. The first is no-signaling: the choice of the measurement setting of one party should not be known to the measurement boxes of the other parties before they produce their outcome. This can be guaranteed ultimately by ensuring space-like separation, although one may also trust a weaker demonstration of separation, as for instance in [7]. The second assumption is measurement independence: the information contained in the boxes in each run should be uncorrelated from the choice of the settings in that run. So far, no way of checking measurement independence is known in a black-box scenario: the best one can do is to buy the source of and the devices that choose the settings from different providers, who are believed not to be conspiring together. Alternatively, one can partly give up the black-box scenario, characterize the devices and be confident that the relevant degrees of freedom are uncorrelated.
It is clear that no-signaling and measurement independence cannot be arbitrarily relaxed: if any amount of signaling is allowed, or if arbitary correlation is admitted between source and settings, the violation of a Bell inequality can be obtained with purely classical resources , so there is no hope to conclude that contains intrinsic randomness. However, with the aim of reducing the assumptions of device-independent certification to their bare minimum, one can partially relax no-signaling and measurement independence, and ask how much information must be signaled and how much measurement dependence must be allowed for a Bell test to become irrelevant [10]. In this paper, we focus on the latter question, the study of partial measurement dependence (sometimes called reduced measurement independence or reduced “free will”), which has been the object of a few recent studies [11]; [12]; [13]; [14]. In particular, we consider the random source that is required to choose the input settings for a Bell inequality and place bounds on the min-entropy necessary to show any difference between local and no-signaling output distributions. Note that if the violation of a Bell inequality is used in a device independent protocol to certify the amplification or expansion of input randomness, this source would serve as the seed randomness in the protocol.
## Ii Measurement dependence and its basic consequences
### ii.1 Measurement independence
For the sake of this introduction, we consider a bipartite Bell scenario. Operationally, a Bell experiment consists of apparently identical runs 1, in each of which box A receives input and outputs a value , box B receives input and outputs a value . A measurement-setting source (henceforth source) for the Bell test supplies the experimentalist with inputs and ; its behavior is modelled by a probability distribution . One can then estimate the statistics . We denote by the information present in the boxes in a given run.
Measurement independence, the assumption that we want to relax, is captured by the condition
p(λ|xy)=p(λ) ∀x,y. (1)
Under this assumption, the observed statistics are modeled by
pMI(ab|xy)=∫p(ab|xyλ)p(λ)dλ . (2)
The specific goal of a Bell test is to assess whether there is intrinsic randomness in the boxes, that is, in the usual terminology, to guarantee that is not a local variable. Mathematically, local variables are defined by . It is useful to stress that, as written, (2) contains an additional assumption, namely that itself is chosen independently in each run according to the distribution . Under measurement independence, it can be proved that this is ultimately not a restriction for Bell tests, although one has to be careful in interpreting statistics from finite samples [15]; [16]; [17].
Measurement independence cannot be denied in a systematic way without undermining the scientific method itself (if a clinical trial is to make sense, whether each patient receives the drug or the placebo cannot depend on the any details of the patients’ conditions). However, it is certainly possible to question measurement independence in a given setup: the devices that determine the inputs may be correlated to the process that determines . The origin of such correlation may be trivial, like the fluctuations in power of the city network to which all the devices are connected; it may be due to lack of attention of the experimentalists, who introduced unwanted connections; or it may be strongly conspiratorial, in an adversarial scenario in which the devices come from an untrusted provider. In all cases, (1) does not hold, nor does the proof that one can restrict the study to independently-chosen .
By relaxing condition (1), one allows correlations between the boxes’ content and the choice of the settings . Bayes theorem implies that
p(λ|xy)≠p(λ)⟺p(xy|λ)≠p(xy). (3)
The first relation could be read as “the output of the source is restricted for a given choice of settings”, the second as “the choice of settings is restricted for a given output of the source”. Neither needs to refer to a real causal relation: all is compatible with both and being influenced by a common cause (Fig. 1). That being clarified, our discourse will be mostly phrased in the second way (the first way will be used in Section VI). We shall then look at measurement dependence as reducing the probability of certain pairs of settings. In the case where the dependence is sufficient to exclude enough pairs of settings, unwanted features of local variable models may be hidden. This is the same intuition behind the power of the detection loophole; in fact, measurement dependence is even stronger, because it may allow to exclude a single pair of settings, whereas the detection loophole is local and excludes all pairs of settings such that one given setting of (say) Bob is associated to unwanted features. This opens a wealth of possibilities that we review rapidly next.
### ii.2 Effects of measurement dependence
The obvious effect of measurement independence is the possibility of faking a violation of Bell inequalities. A Bell inequality is built on a linear combination of , whose maximal value (called algebraic limit) cannot be reached by local variables. If, in each run, one can exclude some suitable pairs of settings in correlation with the content of the boxes , then it becomes possible to reach the algebraic limit while having only local variables in the boxes.
Let us illustrate this point with the most famous Bell inequality, that of Clauser, Horne, Shimony and Holt (CHSH). The inequality reads
|⟨a0b0⟩+⟨a0b1⟩+⟨a1b0⟩−⟨a1b1⟩|≤2 (4)
with . In order to achieve the algebraic limit of , one should have , , and . Local deterministic points exist that satisfy three out of these four conditions. If one wants to achieve the algebraic limit with local variable and measurement dependence, a sufficient strategy is the following: in each run, is chosen among the aforementioned local deterministic points, and the pair of settings corresponding to the unwanted condition is never chosen [10]; [12].
The fact that a sufficient amount of measurement dependence can lead to the algebraic limit has an intriguing consequence for some inequalities. Indeed, in generic inequalities, the algebraic limit may lie even above what can be reached with no-signaling correlations. For instance, the tilted CHSH inequality
|⟨a0b0⟩+⟨a0b1⟩+⟨a1b0⟩−⟨a1b1⟩+α⟨a0⟩|≤2+α , (5)
has an algebraic limit of , but no-signaling correlations can reach only up to 4 if [18]. If measurement dependence is allowed, to the point that one pair of settings can be excluded, then one can achieve the algebraic limit with a convex mixture of
λ=(+1,−1,−1,+1)together with(x,y)≠00λ=(+1,+1,+1,−1)together with(x,y)≠01λ=(+1,−1,+1,+1)together with(x,y)≠10λ=(+1,+1,+1,+1)together with(x,y)≠11 (6)
where we denoted a local deterministic point as . If a Bell test is run with this underlying strategy, the observed correlations will lie outside the no-signaling polytope, i.e. are formally signaling. Obviously, this does not mean that measurement dependence makes it possible to use entanglement to actually send a message: in order for (say) Alice to send a message to Bob, she must be able to choose her setting at will, which is precisely what measurement dependence denies. At any rate, one must be careful when working with measurement dependence: the worst case are correlations that reach the algebraic limit, not the no-signaling one (to our knowledge, all the studies of measurement dependence so far dealt with inequalities for which the two limits happen to coincide [10]; [11]; [12]; [13]; [14]).
The take-away message of this paragraph is that one does not have to reach the extreme case of total measurement dependence (i.e. determining uniquely): already with some partial amount of measurement dependence, it becomes impossible to draw any conclusion from the violation of a Bell inequality. This has important consequences when the source is characterized only by its conditional min-entropy. Indeed, one of our main result will consist in deriving general bounds for this amount (Section IV). In order to do that, we need first to recall the definition of min-entropy and its relation to the Santha-Vazirani condition in light of measurement dependence.
## Iii Min-entropy and measurement dependence
As mentioned, the source of the Bell test behaves according to . Measurement independence implies that has as much entropy or randomness as . In contrast, partial measurement dependence means that there is some randomness in the source, but it is less than the entropy of the distribution . The min-entropy and min-entropy deficit are measures of randomness of a source, and they partly capture the amount of measurement dependence in special cases. But note that they are not intrinsic measures of measurement dependence (for instance, min-entropy deficit equals 0 does not imply measurement independence). If the min-entropy is not high enough, it leaves open the possibility of excluding certain settings, which allows faking of Bell violations as we discussed before. This behavior is forbidden in Santha-Vazirani sources as explained next.
### iii.1 Min-entropy vs Santha-Vazirani condition
We illustrate our point with an example. The chained inequality is a bipartite Bell inequality with settings for each party and binary outcomes for both measurements on and , which reads
Im = p(a=b|x=1,y=m)+∑\lx@stackrelx,y s.t.x∈{y,y+1}p(a≠b|x,y) (7) ≤ 2m−1.
It has been used to put stringent bounds on quantum theory thanks to the property that, in the limit , its algebraic limit can be reached with measurements on quantum states [19]; [20].
Out of the possible pairs of settings, are effectively used in the inequality. Furthermore, there exist local deterministic points that can satisfy of these conditions. Therefore, in order to verify any conclusion based on the chained inequality, it is enough to have an amount of measurement dependence that allows the exclusion of only one pair of settings out of . In the limit of large , under whichever measure, such a source is very close to a fully random source: for instance, its min-entropy per run (defined below) is , which differs from the fully random value by . This example shows that a source, which would presumably be considered as good as it gets in an abstract assessment, is already catastrophic for the Bell inequality under study. Notice that this remark is not in contradiction with the results of [11], which can be seen as proving that the chained inequality is pretty robust to measurement dependence: indeed, in that work, the additional Santha-Vazirani assumption was made on the source, which implies that all the pairs of settings are possible in each run. Our argument, based on excluding one setting in each run, does not apply.
It is now time to present the definitions we have just sketched in their suitable formal setting. We shall consistently use the word source to stress that the source of randomness we are interested in is the randomness of the inputs given the knowledge of the physical process or vice versa, not the randomness possibly present in (which would be the intrinsic randomness of quantum origin in the ideal case).
### iii.2 Formal definitions
Here we review rapidly the definitions of well-known types of sources of randomness for the purpose of this paper, referring to [21] for a comprehensive study.
Consider a random variable in an alphabet of size ; and let be an -dit string. In our case, will represent the settings chosen for the Bell test, i.e. in a bipartite scenario. Randomness being synonymous with unpredictability, a source of randomness will be characterized by specifying what one wants to predict and how predictable it is, given some prior information (supposed to be classical throughout this paper). One would then say that the source contains randomness if
Pguess(Z|Λ):=∑λP(Λ=λ)Pguess(Z|Λ=λ)<1, (8)
where . The amount of randomness is quantified by the min-entropy
Hmin(Z|Λ):=−logPguess(Z|Λ). (9)
Clearly, implies the presence of some randomness. To someone who does not have access to , the source will appear to have min-entropy which can only be higher by the data processing inequality. Though obvious, it may be worth stressing that is not the same as , since is not a given probability distribution but a notation for a procedure that picks up the maximum of a probability distribution. As an extreme example, if looks uniform but the knowledge of determines uniquely, one has and .
The loosest characterization of the source, i.e. the one that requires fewer assumptions, simply puts a bound on the min-entropy:
###### Definition 1.
Min-entropy source. A random variable is a -min-entropy source of randomness with respect to another random variable if .
As soon as , the knowledge of does not determine uniquely. One can add some structure to a min-entropy source. For instance, a -min-entropy source is called uniform if for all values of . A block min-entropy source is one for which not only the min-entropy of the whole string, but the min-entropy of blocks is also lower bounded. These notions will not be used in this paper.
As soon as , the definition of -min-entropy source is compatible with for one string . As hinted in paragraph III.1, the possibility that some settings are not chosen is critical for sources of Bell tests. Because of this, one may want to add to the properties of the source the assumption that all the strings have non-zero probability. This is equivalent to the following type of source:
###### Definition 2.
Santha-Vazirani sources. A random variable is a Santha-Vazirani source with respect to (where and ) if
pmin≤p(zi|λ,z1,...,zi−1)≤pmax ∀ i . (10)
If is a bit, is usually written [22]. Some of the most important results in measurement dependence in Bell tests have been obtained for Santha-Vazirani sources [11]; [13]; [14]. These results show that there is a real advantage in considering Bell-based randomness, because it overcomes no-go theorems for classical information.
Finally, let us focus on distributions that are independent and identically distributed (i.i.d.) such that
p(Z=z|Λ=λ)=N∏j=1p(Zj=zj|λ). (11)
This can also be viewed as a block min-entropy source where each block consists of only one symbol, . In this case, the Santha-Vazirani definition implies:
pmin≤p(z|λ)≤pmax . (12)
We will use a different notation such that and to make clear that we are in the i.i.d. scenario. Then the definition of uniform min-entropy sources is equivalent to the figure of merit of measurement dependence used in [12], namely
PM:=maxz,λp(z|λ) ,[i.i.d.] (13)
since is equivalent to . In the following, we will use these two figure of merits interchangeably for i.i.d. models.
Instead of bounding the largest probability, the smallest probability also gives information on measurement dependence, as first proposed in [23]:
Pm:=minz,λp(z|λ) .[i.i.d.] (14)
If only is explicitly bounded, then a bound on can be inferred, however, it might be trivial, since it can be negative: . Bounding only the min-entropy of the input source to the Bell test, or equivalently bounding only , which is the guessing probability, allows much different worst-case behavior in Bell tests than when the Santha-Vazirani definition is adopted, as we shall now explore.
## Iv Lower bound for min-entropy sources
We will be dealing with a -partite Bell scenario where the party has measurement settings ( for bipartite) and each setting has an arbitrary number of outcomes. The joint configuration of settings with ( for bipartite) is a -tuple in the set of all settings of size . In this Section, moreover, we consider a Bell test in which the observed statistics of the settings follow a uniform distribution, that is
Hmin(Z1,...,ZK)=Nlog|S|, (15)
or equivalently
pobs(z1...zK):=∑λp(z1...zK|λ)p(λ)=(K∏i=1mi)−1. (16)
This is not an assumption like those on the nature of the source: is observed in a realization; but it is a frequent working assumption for theoretical works, which was made in all previous works on measurement dependence. In Section V, we shall see that a non-uniform has interesting consequences in studies of measurement dependence.
We are presently able to discuss our main result: a lower bound on the min-entropy of the source, below which no conclusion can be drawn from any Bell test, unless further structure is assumed.
### iv.1 Reaching the no-signaling limit
The main insight is provided by the following Lemma, which we present in the bipartite scenario (the generalization to multipartite scenarios holds with identical proofs and more cumbersome notation, so we give it in Appendix A):
###### Lemma 1.
Let be an arbitrary no-signaling distribution with and . For any pair of settings , there exists a local distribution such that
PL(ab|xy) = P(ab|xy) (17) for (x,y) ∈ S¯x,¯y≡{(¯x,y′),(x′,¯y):x′∈{1,...,mA}, y′∈{1,...,mB}}.
Moreover, this result is tight: if another pair of settings is added to the subset of pairs, there exists a no-signaling point for which those probabilities are nonlocal.
###### Proof.
The proof can be done by constructing explicitly one such local distribution. Let us fix without loss of generality. From the no-signaling distribution , we construct
P(a1,a2,...,amA;b1,b2,...,bmB) =P(a1)P(b1|a1)mA∏j=2P(aj|b1)mB∏k=2P(bk|a1) (18)
with obvious notations. This is a valid joint probability distribution over the outcomes of all the measurements. Now, on the one hand, the marginals define a local distribution, as first proved by Fine [24]. On the other hand, it is easy to show that : one should sum first over all possible values of to find , after which the sum over the ’s is obvious. Similarly one proves that . So indeed we have a local distribution that mimicks the initial no-signaling one on the desired subset of pairs of settings.
As for the tightness, suppose that we add a single pair of settings, say , to : there exist no-signaling points for which CHSH is violated by the settings , , and ; so those statistics can’t be mimicked by a local distribution. ∎
Now we can state the main theorem:
###### Theorem 1.
Consider a min-entropy source with an observed min-entropy for an -run bipartite Bell test with inputs on Alice, inputs on Bob and arbitrary alphabets for the outcomes. If
Hmin(XY|Λ) ≤ Nlog(mA+mB−1) (19)
no conclusion can be drawn from the Bell test, since the no-signaling limit of the inequality can be reached with local distributions. The generalization of this result to -partite Bell tests reads
Hmin(Z1...ZK|Λ) ≤ Nlog(K∑k=1mk−K+1). (20)
for . Notice in particular that, without further assumptions, any source of randomness with is useless as a source for any Bell tests.
###### Proof.
We will construct an explicit i.i.d. source which allows the faking of a Bell violation up to the no-signaling bound with appropriate local resources. From Lemma 1 we know that there exist subsets of pairs of settings, for which no difference can be seen if a local distribution is substituted for a possibly nonlocal no-signaling point: in particular, this could be the no-signaling point that reaches the no-signaling limit for the inequality under study. If is sufficiently low, the source will allow only the pairs of settings that belong to one of the and distribute the corresponding local strategy . The source
p(xy|λ¯x,¯y)={1mA+mB−1,if% x,y∈S¯x,¯y0,otherwise (21)
has in each run, whence we have proved the bound (19) as long as we can find such that for all . In the case where is uniform, this can always be found by simply choosing uniformly the pair , i.e. . This concludes the proof for the bipartite case. The proof of the multipartite case is identical using the material of Appendix A. The final remark of Theorem 1 stems from the fact that each Bell test much involve at least two parties and each must have at least two settings.
Because of the tightness of Lemma 1, the bounds (19) and (20) are the best inequality-independent bounds that one can obtain with i.i.d. sources. Moreover, since there exist inequalities for which the quantum and the no-signaling limits coincide, the bound to reach the quantum limit cannot be better. If the inequality is given, however, much less measurement dependence may be sufficient to reach the no-signaling limit, and even less to reach the quantum limit if it is lower. We elaborate further on this point in the following paragraph.
### iv.2 Inequality-dependent bounds
Let define a Bell inequality, whose local, quantum and no-signaling limits are given by , and be the set of settings that are used by the Bell inequality 2. Again, for each there is a local strategy for assigning outputs such that in order to achieve the no-signaling limit, some settings will be incompatible with this strategy and must be hidden by measurement dependence. Let this set of inputs be . Then, an arbitrary no-signaling point is required to be compatible with a local point only on the subset . Suppose an inspection of the inequality shows that at most of these settings must be hidden for any choice of . Once the probabilities of the settings are set to zero, the min-entropy is maximized by the uniform distribution over the remaining settings (FIG. 2). However, one must be very careful to show the existence of which satisfies (16). Whenever such a distribution exists, if , the non-local game can be won with probability one with local strategies. As implied by the results of the previous section, if the observed input distribution is uniform then a strategy in the form of equation (21) or its generalization in Appendix A with a uniform probability over will always satisfy (16). However, it is possible to do better in some cases where . In such cases (the most settings that must be hidden for any ) can be small. Here, for a uniform equation (16) will also be satisfied provided possibly more settings than required are hidden for each such that for all and if is the set of s for which , must be constant for all . This is a symmetry condition that can be met by many Bell inequalities. As before, the existence of this example proves that a min-entropy source with
k≤Nlog(|SBg|) (22)
can reach the no-signaling limit of with local strategies for uniform input distributions. In the following section we will show how to obtain bounds for arbitrary and that approach will also give tight bounds and optimal strategies when the inequality is one in which the size of the “hidden sets” varies with .
Further, if , in order to simulate physics one may be content with reaching the quantum limit. A possible i.i.d. source (not proved to be optimal) is the following (see Fig. 2). With probability , the settings are chosen uniformly among all possible -tuples: this is measurement independence, so on these cases, and the physical process can be chosen as one of those that saturate . In the other instances, the settings are chosen uniformly in and the physical process is chosen in each case in order to achieve . In other words, this source is a convex combination of the measurement independent uniform source and the source described in the previous paragraph. Note that this new source will automatically satisfy the constraint (. For such a source, therefore, is the probability of each setting in , which reads . With this measurement-dependent strategy, one can reach , so for . In summary, the quantum limit can be achieved with an i.i.d. source with
PB,QM ≥ Missing or unrecognized delimiter for \right (23)
that is, a min-entropy source with can reach the quantum limit of with local strategies, for a uniform input distribution.
Let us illustrate the methodology with the analysis of some inequalities:
• CHSH: here, it is always necessary and sufficient to hide one pair of settings. Therefore and the inequality-dependent bound (22) is the same as the inequality-independent one (19) to reach the no-signaling limit, as already proved in [12]. Recall that this does not prove the bounds to be tight, because they are based on explicit i.i.d. sources: non i.i.d. sources may lead to tighter bounds, though we do not know any example. As for reaching the quantum limit, we have .
• Chained inequality: here again, as we have seen in paragraph III.1, it is always necessary and sufficient to hide only one pair of settings out of , so and for all . As a consequence, in terms of min-entropy, the inequality-dependent bound (22) is , which is approximately twice the value obtained from (19). For large , the quantum and no-signaling limits basically coincide.
• CGLMP inequalities: like the CHSH inequality, the CGLMP inequalities are two party inequalities where each party has two inputs. However, this family of inequalities has possible outputs for each party. In the quantum case, the CGLMP inequalities can provide more robustness against measurement dependence than the CHSH inequality, in the sense that the min-entropy of the inputs given the source must be lower if the quantum bound is to be achieved. The reason is that it has been shown that as , the quantum limit increases and approaches the no-signaling limit [25]; [26]. As can be seen, inspecting equation (23), the value of will increase with , and the value of necessary to reach the quantum limit with local resources increases, until it reaches the no-signaling value in the limit.
• Mermin inequalities: Mermin inequalities [27]; [28] are multipartite inequalites such that for odd numbers of parties, the quantum and no-signaling bounds coincide. For this reason, the 5-party Mermin inequality was used in [13] to amplify randomness. When the number of parties is an odd number at least 3 only a subset of all possible inputs appear in the corresponding Mermin inequality and the inequality-independent bound is not tight. In general, for odd parties and [29]. Specifically for the 5-party case, and .
## V The positive effect of biasing the choices of the settings
Theorem 1 shows that assuming a full min-entropy source on the measurement settings, for any meaningful conclusion to be drawn from a Bell test, it must be that . However, recalling that the role of the observed data is actually a constraint imposed on the underlying model (similar to equation (16)), we can hope to use it to our advantage. This motivates the question: for a given value of that is being assumed, what is the optimal distribution on the inputs such that the maximum possible Bell value obtainable with this degree of measurement dependence and only local resources is as low as possible. Because the situation for non i.i.d. models is intractable, we are restricting ourselves to the i.i.d. model for the remaining of this chapter. Here instead of the min-entropy, the guessing probability is used exclusively as the figure of merit of measurement dependence. First, we consider the CHSH inequality as an explicit example.
### v.1 The CHSH Inequality
Intuitively, we expect that the optimal solution is to set for each input round and for three pairs and for the final pair because in this case cannot contain any further information on than is available simply from observing the distribution . We will highlight an example of this type of distribution later in this section. This is not a uniform distribution, so we can already see that non-uniform input distributions can be beneficial. In this section, we will consider fixed input distributions and find the maximum value that the CHSH inequality can take given a bound on . Note that the method in this section extends to any multipartite Bell inequality.
We want to find the violation , under local resources and measurement dependence, as a function of and . To this end, observe that the local distributions form a convex polytope and so is the set of sources with a fixed value of (the source polytope). Using the decomposition into extremal points of a convex polytope, we have
p(ab|xyλ) = ∑iαi(λ)ei(ab|xy), (24) p(xy|λ) = ∑jβj(λ)fj(xy), (25)
where are the extremal points of the local polytope and are the extremal points of the source polytope. Now after multiplying by both sides by , the i.i.d. model with measurement dependence becomes
p(abxy) = ∫Λdλ∑ijαi(λ)βj(λ)ei(ab|xy)fj(xy)p(λ) (26) = ∑ijγijgij(abxy),
where
γij=∫Λdλαi(λ)βj(λ)p(λ), (27) gij(abxy)=ei(ab|xy)fj(xy). (28)
In this notation, the problem becomes a linear program, i.e. finding
BmaxCHSH(PM,pobs(xy))=maxp(abxy) ∑abxy(−1)a+b+xyp(abxy)pobs(xy) (29)
subjected to the constraints
p(abxy)=∑ijγijgij(abxy), ∑abp(abxy)=pobs(xy) (30)
for known values of and . The result is presented in FIG. 3.
Using the numerical results, it is easy to see that the optimal strategy for maximizing the Bell value whether or not the observed distribution is uniform is to choose
p(xy|λ¯x,¯y)={PMif x,y∈S¯x,¯y1−3PMotherwise , (31)
for each , where is defined in equation (1). Choosing this strategy, it is straightforward to find an analytic expression for :
BmaxCHSH(PM)=4−12[(1−3PM)q+1−3PM4PM−1(q−16)]
where for convenience we define . This expression is only valid for (). Notice that when the distribution is uniform and the second term vanishes, leaving a linear expression in .
It is interesting to observe that for the purpose of violating Bell inequalities (that is, demonstrating non-locality by exceeding ) under measurement dependence, suppose the inputs have privacy quantified by , then it is advantageous for us to purposely select an input distribution that is not uniform. This can be seen easily from for example the red curves for : selecting uniform input distribution allows a violation up to about 2.5 while selecting non-uniform input distribution only allows a lower maximum violation! Note that for non-uniform distributions on the inputs the upper bound on the Bell value is only as low as 2 (the local bound assuming measurement independence) for non-product distributions on the inputs. (See the blue dashed curves.) All non-uniform product input distributions can have Bell values larger than 2, if measurement independence is relaxed. Notice also, that the lowest blue curve, the one that takes the value 2 at is the one corresponding to the distribution . This is precisely the form of the distribution on the inputs we has anticipated at the start of this section.
### v.2 Generalizations
We have seen that for the case of the CHSH inequality, the strategy outlined in section IV in equation (21) is the optimal strategy even in the case that the distribution is not uniform. In general however this is not the case. It is possible to find some inequalities that together with some distributions do not admit a strategy of the form
p(xy|λ¯z)={PM,if z∈S¯zQ(λ¯z),otherwise (32)
where is determined by the normalization condition to be .
Let us limit our focus to inequalities with symmetries such that and for all . In that case, equation (16) can be written as a matrix equation, with and written as vectors and is a matrix whose entries are defined by equation (32). If the matrix is is full-rank, then there is a unique solution for that is a valid probability distribution. This will always be the case if and have no common factors.
Examples of cases where the sizes of the sets and have no common factors are any bipartite Bell inequality with terms for all input pairs present and where both parties have the same number of inputs. For these cases, the min-entropy bound of section IV also applies for any non-uniform observed distribution on the inputs.
If, for a given inequality, and have at least one common prime factor, there may be some choices of distribution for which the strategy (32) will not be able to reproduce with any valid distribution . In that case, the optimal strategy may have to be found numerically. For the i.i.d. case, one do this by solving a linear program that is a generalization of the one presented in the previous section.
## Vi Bounds on the achievable distributions
While the set of distributions obtainable from a measurement independence local model is the local polytope, that obtainable from a measurement dependence model is in principle a larger set. The example in section II.2 makes it clear that this set can even include signaling distributions. Now we wish to study more carefully this new set of distributions. As mentioned in II.1, figures of merit of measurement dependence can be defined as restrictions on instead of . It turns out that in characterizing the set of achievable distributions, it is easier to work with figures of merits based on . In general, specifying does not specify unless and are uniform, in which case the two are proportional. Among the figures of merit one can define is the one by Hall [10]. We shall adopt a slightly modified measure:
###### Definition 3.
The quantity bounds the distance between the probability distribution over the random variable given access to and the distribution over without information on :
M′:=maxx,y2D(p(Λ|X=x,Y=y),p(Λ)), (33)
with and is the total variational distance.
Essentially this definition bounds the distance of any one element in the distribution away from a strategy independent of Alice and Bob’s inputs and , which is similar in flavour to a Santha-Vazirani bound.
Now we wish to compare the point obtainable from a measurement dependence model with local resources,
pλab|xy=∑λp(a|xλ)p(b|yλ)p(λ|xy) , (34)
to its corresponding measurement independent point,
pab|xy=∑λp(a|xλ)p(b|yλ)p(λ) . (35)
Their distinguishability is characterized by their total variational distance
D(pλAB|xy,pAB|xy) (36) = 12∑a,b|pλab|xy−pab|xy| = 12∑a,b∣∣ ∣∣∑λp(a|xλ)p(b|yλ)(p(λ|xy)−p(λ))∣∣ ∣∣ ≤ 12∑λ(∑a,bp(a|xλ)p(b|yλ))|p(λ|xy)−p(λ)| = D(p(λ|x,y),p(λ)) ≤ M′/2,
where we have applied the triangle inequality. In other words,
|pλab|xy−pab|xy|≤M′ ∀ a,b,x,y, (37)
which gives a bound on the distributions that can be created with a measurement dependence model of dependence bounded by .
There is another model that relaxes assumptions about correlations between two parties: quantum “cross-talk” between non-fully-isolated devices [30]. In this case, the assumption that the measurement operators are in a tensor product, , is relaxed and thus it is possible to use the measurement itself to introduce quantum correlations between the two parties. The model was proposed to describe excess correlations that could result from a pair of trapped ions measured while situated next to each other in a refrigerator after being entangled. Comparing our equation (37) with equation (2) in [30], we find that the distributions allowed by cross-talk are formally the same as those allowed by measurement dependence as defined in equation (33), even though the physical interpretation is very different.
## Vii Conclusions
Bell tests are an essential tool in device-independent approaches. They rely on a set of reasonable assumptions, but some of the assumptions are untestable. In particular, the correlations between source and settings are strictly unobservable and therefore the amount of reduction of measurement independence is ultimately an assumption, either on the power of an adversary, on a physical model for the experiment. This study has demonstrated that when relaxing this assumption, the definition used, be it min-entropy or a Santha-Vazirani condition, is critical with respect to what kind of guarantees can be obtained from a Bell test. There are results [13]; [31] showing that with a Santha-Vazirani source assumption arbitrarily weak randomness can be amplified using a protocol that checks for the violation of a Bell inequality. This cannot be accomplished using a min-entropy condition, as we have demonstrated in section IV: for sufficiently low min-entropy any inequality can be violated up to its no-signaling bound, using only the classical measurement dependent correlations and in a way that a third party could predict all of the outcomes of the measurements. Even for the protocol in [11] that amplifies bounded randomness (in the Santha-Vazirani definition) using violations of the chained Bell inequality, in order to get perfectly free bits out, the number of outputs for this inequality must go to infinity. As we point out in section III.1, in this limit the chained Bell inequality is not robust to any relaxation of input randomness if the min-entropy definition is used instead.
The bounds on the min-entropy presented in section IV give immediate bounds for any inequality on the amount of input randomness required to draw conclusions about whether the violation of a Bell inequality can give any certification of quantum or non-local behavior. The method demonstrated for the CHSH inequality in section V demonstrates how to get tight upper bounds for the value a given Bell inequality can take assuming a min-entropy bound for any distribution over the measurement settings. It also shows that there may be advantages to deliberately choosing non-uniform distributions over measurement settings in device independent protocols, depending on what assumptions are being made. Relaxing the assumption of measurement-dependence increases the set of probability distributions that can result from a Bell test assuming a local, realistic hidden variable model and section VI gives an expression bounding this increase.
Being as the assumption of measurement independence cannot be confirmed, it is important to understand the consequences for device independent protocols when it is relaxed. It is especially interesting that the min-entropy condition, a condition widely adopted in classical security studies [21]; [32]; [33]; [34], is has such a different behavior from the Santha-Vazirani condition for these device-testing purposes. We hope that the bounds and characterizations provided here will be useful for constructing protocols that are more robust to extraneous correlations.
###### Acknowledgements.
This work is funded by the Singapore Ministry of Education (partly through the Academic Research Fund Tier 3 MOE2012-T3-1-009) and the Singapore National Research Foundation. We would like to thank Jean-Daniel Bancal, Jeysthur Ang, Roger Colbeck, Yaoyun Shi, Aarthi Sundaram and Miklos Santha for helpful discussions.
## Appendix A Generalization of Lemma 1 the multipartite case
The bound (20) for the min-entropy in a multipartite scenario is based on the generalization of Lemma 1 that we provide here:
###### Lemma 2.
Let be an arbitrary -partite no-signaling distribution with where and is a -tuple of outcomes. For any -tuple of settings , there exists a local distribution such that
PL(o|z) = P(o|z) (38) for z ∈ S¯z≡{(¯z1,z′2,...,z′K),...,(z′1,...,z′K−1,¯zK): z′i∈{1,...,mi}}.
Moreover, this result is tight: if another -tuple of settings is added to the subset , there exist a no-signaling point for which those probabilities are nonlocal.
###### Proof.
Again, let us fix without loss of generality. Let be the th party’s outcome given the th measurement setting. From the no-signaling distribution , we construct a valid probability distribution
P(o11...o1m1;...;oK1...oKmK) = P(o11)[K∏i=2P(oi1|o11...oi−11)][K∏i=1mi∏j=2P(oij|o11...oi−11oi+11...oK1)] (39)
whose marginals define a local distribution by Fine’s result [[24]]. To verify that we have a local distribution that mimics the initial no-signaling one on the desired subset of pairs of settings, consider this example: for the input string with the distribution we sum first over all possible values of each outcome variable to find
P(o11;o21...o2m2;...;oK1...oKmK)=P(o11)K∏i=2[P(oi1|o11...oi−11)mi∏j=2P(oij|o11...oi−11oi+11...oK1)] (40)
after which continue to sum over all the except and one is left with a probability distribution on only variables, one for each party. The other verifications are similar. Another way to think of it is to notice that each conditional probability factor on variables (one variable conditioned on other variables) effectively sets a joint probability distribution on those same variables. In the distribution (39) there are such factors and so this is exactly how many local points that can be matched for a given hidden variable value (see equation (20) in the main text). The argument for tightness still works if we consider only two parties among . For any two parties we can choose a pair of inputs for each to return to a CHSH-type scenario, then the argument follows in the same way as in the proof of Lemma 1.
### Footnotes
1. We focus on the operational description of current experiments and do not consider the more general, but as yet abstract, case of parallel repetition, in which all the inputs are given at the same time.
2. The chained Bell inequality is an example of an inequality whose value depends only some of the possible inputs. Only terms such that or and the term for appear.
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2019-02-20 01:35:36
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https://math.stackexchange.com/questions/3325234/prime-zeta-function-at-1
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# Prime Zeta function at 1
I wanted to find $$\lim_{s\to 1} (P(s)-\ln(\zeta(s)))$$ and here is my attempt:
So we know that $$M=\gamma +\sum_{n=2}^\infty \mu(n) \frac {\ln(\zeta(n))}{n}$$ and that $$P(s)=\sum_{n=1}^\infty \mu(n) \frac {\ln(\zeta(sn))}{n}$$ where $$M$$ is Mertens constant (here), $$\gamma$$ the Euler Mascheroni constant (here), $$\mu(s)$$ the Möbius function (here), $$\zeta(s)$$ the Riemann zeta function (here) and $$P(s)$$ the prime zeta function (here).
If we let s=1 then $$P(1)=\infty=\sum_{n=1}^\infty \mu(n) \frac {\ln(\zeta(n))}{n}$$
but if we subtract $$\mu(1) \frac {\ln(\zeta(1))}{1}$$ (which is also equal to $$\infty$$) then we have $$P(1)-\mu(1) \frac {\ln(\zeta(1))}{1}=P(1)-\ln(\zeta(1))=\sum_{n=2}^\infty \mu(n) \frac {\ln(\zeta(n))}{n}=M-\gamma$$
and we can write that as $$\lim_{s\to 1^+} (P(s)-\ln(\zeta(s)))=M-\gamma$$
If you type $$P(1)-\ln(\zeta(1))$$ into wolfram alpha, it yields $$\infty$$. But if you give it very small numbers near one it gives nearly perfect results
$$P(1.001)-\ln(\zeta(1.001))=-0.31496...$$
$$M-\gamma=-0.31571...$$
I think that some things here are not really legit, but is that correct? Are there better ways to prove this result? And I'm sorry if there are any grammatical or spelling mistakes.
• $$\zeta(s) = \prod_p \frac1{1-p^{-s}}, \qquad \Re(s) > 1$$ gives $$\log \zeta(s) =- \sum_p \log(1-p^{-s}) = \sum_{p^k} \frac{p^{-sk}}{k} = \sum_k \frac{P(sk)}{k}$$ so $$P(s) = \sum_k \frac{\mu(k)}{k} \log \zeta(sk)$$ at first for $$\Re(s) > 1$$ and by analytic continuation for $$\Re(s) > 0$$ so that $$\lim_{s \to 1} P(s) - \log \zeta(s) = \sum_{k\ge2} \frac{\mu(k)}{k} \log \zeta(k)$$
• At first for $$\Re(s) > 1$$ and by analytic continuation for $$\Re(s) > 0$$ $$\zeta(s)- \frac1{s-1} = \sum_n (n^{-s} -\int_n^{n+1} x^{-s}dx)$$ gives that $$f(s)=(s-1)\zeta(s)$$ is analytic at $$s=1$$ with $$f(1)=1$$ thus $$F(s)=\log (s-1)\zeta(s)$$ is analytic at $$s=1$$ with $$F(1)=0$$.
• The $$\gamma$$ appearing in the Mertens constant $$M$$ comes from $$\gamma = \sum_n (n^{-1} -\int_n^{n+1} x^{-1}dx)$$
Your argument is (nearly) correct . Because, $$P(1+e)=\ln\frac{1}{e}+C+O(e)$$
Where C is your M - $$\gamma$$
And for
$$\ln\zeta(1+e)= \ln \frac {1}{e} + O(\ln(e))$$ For small e
• At $s=1$, $(s-1)\zeta(s)$ is analytic and non-zero and $\log \zeta(s) - P(s)$ is analytic thus $\log \zeta(s) = -\log(s-1)+C+O(s-1)$, $P(s) = \log \zeta(s)+A+O(s-1)=-\log(s-1)+A+C+O(s-1)$. It remains to show $C=0, A =\sum_{n=2}^\infty \mu(n) \frac {\ln(\zeta(n))}{n}$ which is not hard – reuns Aug 17 '19 at 4:21
• To show what ? $\lim_{s\to 1} (P(s)-\ln(\zeta(s)))$ is obvious from the formula you wrote for $P(s)$@Kinheadpump – reuns Aug 17 '19 at 10:49
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2021-04-13 14:45:59
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https://math.stackexchange.com/questions/1577848/problem-on-normal-and-symmetric-matrices
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# Problem on normal and symmetric matrices
A normal matrix over $\mathbb C$ with all eigenvalues real is hermitian (using diagonalization) .But a normal matrix with real eigenvalues is symmetric, is this statement true?
I think, all normal matrices are not diagonalizable i.e rotation matrix .please someone explain over $\mathbb R$,are these statements true?
Thanks.
A normal matrix over $\mathbb C$ is hermitian AKA self adjoint iff it has real eigenvalues. This is true. However, it does not necessarily imply that the matrix is symmetric.
All normal matrices are diagonalizable with respect to a unitary matrix over $\mathbb C$. However, in the real case, this may not be true because the unitary matrix may involve complex numbers.
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2021-03-09 08:37:57
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https://kb.osu.edu/handle/1811/19359?show=full
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dc.creator Allen, M. D. en_US dc.creator Evenson, K. M. en_US dc.creator Brown, J. M. en_US dc.date.accessioned 2006-06-15T19:17:33Z dc.date.available 2006-06-15T19:17:33Z dc.date.issued 1999 en_US dc.identifier 1999-RF-11 en_US dc.identifier.uri http://hdl.handle.net/1811/19359 dc.description $^{a}$ H. Kohguchi, Y. Ohshima, Y. Endo, J. Chem. Phys. 106, 5429 (1997). $^{b}$ Y. Ohshima and Y. Endo, J. Molec. Spectros. 172, 225-232 (1995). $^{c}$ Calculated from data reported in M. Kakimoto and T. Kasuya, J.Molec. Spectros. 94, 380-392 (1982) and K. Kawaguchi, T. Suzuki, S. Saito, and E. Hirota, J. Molec. Spectros. 106, 320-329 (1984). en_US dc.description Author Institution: Time and Frequency Division, National Institute of Standards and Technology; The Physical Chemistry Laboratory, Oxford University en_US dc.description.abstract Vibration-rotation transitions between the $(010) \mu^{2}\Sigma^{-}(000)\bar{X}^{2}II_{r}$ vibronic states, were recorded using far-infrared laser magnetic resonance (FIR-LMR) spectroscopic techniques. These transitions occur near $200 cm^{-1}$ for the $(010) \mu^{2}\Sigma^{-} - (000) \tilde{X}^{2}II_{1/2}$ transition and $160 cm^{-1}$ for the $(010) \mu^{2}\Sigma^{-}-(000)\bar{X}^{2}\Pi_{3/2}$ transition. This is the first direct measurement of the $\nu_{2}$ band of CCN and with the inclusion of optical data, similar to the one found in the paper by Kohguchi $et al.^{a}$, microwave data from Ohshima and $Endo^{b}$, and spin-orbit $data^{c}$ a least squares fit has resulted in an accurate determination of molecular parameters for the ground and the $(010) \mu^{2}\Sigma^{-}$ states. The data were fit using an $N^{2}$ effective Hamiltonian modified to include the Renner-Teller effect. en_US dc.format.extent 123046 bytes dc.format.mimetype image/jpeg dc.language.iso English en_US dc.publisher Ohio State University en_US dc.title FAR-INFRARED LASER MAGNETIC RESONANCE SPECTROSCOPIC STUDY OF THE $\nu_{2}$ BENDING FUNDAMENTAL OF THE CCN RADICAL IN ITS $\tilde{X}T^{2}II$, STATE en_US dc.type article en_US
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2021-09-25 19:47:30
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https://byjus.com/question-answer/find-the-components-along-the-x-y-z-axes-of-the-angular-momentum-l-of-3/
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Question
# Find the components along the x, y, z axes of the angular momentum $$l$$ of a particle, whose position vector is $$r$$ with components x, y, z and momentum is $$p$$ with components $$p_x, p_y$$ and $$p_z$$. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.
Solution
## Linear momentum of particle, $$\vec{p}=p_x\hat{i}+p_y\hat{j}+p_z\hat{k}$$Position vector of the particle, $$\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$$Angular momentum, $$\vec{l}=\vec{r}\times \vec{p}$$$$\implies l_x\hat{i}+l_y\hat{j}+l_z\hat{k}=\hat{i}(yp_z-zp_y)-\hat{j}(xp_z-zp_x)+\hat{k}(xp_y-yp_x)$$Therefore on comparison of coefficients,$$l_x=yp_z-zp_y$$$$l_y=zp_x-xp_z$$$$l_z=xp_y-yp_x$$The particle moves in the x-y plane. Hence the z component of the position vector and linear momentum vector becomes zero.$$z=p_z=0$$Thus $$l_x=0$$$$l_y=0$$$$l_z=xp_y-yp_x$$Thus when particle is confined to move in the x-y plane, the angular momentum of particle is along the z-direction.PhysicsNCERTStandard XI
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2022-01-22 18:16:40
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https://shaggydev.com/2022/05/04/clean-code/
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The subject of clean code is everywhere in professional software development circles, with an infinite number of books, blogs, and videos to choose from, but it tends to get overlooked among casual game developers. Presumably, this is because of some combination of the following:
1. Hobbyists that learn just enough coding to get by
2. A lot of people are either working alone or are the only programmer on their team, meaning the cracks in their code don’t show as quickly
3. Simply put, it’s not a sexy subject - at least not until you realize your code is unmanageable and you need to fix it somehow
Not that there’s anything wrong with these reasons. Not every hobbyist, in any hobby, needs to spend hours studying their subject from a more academic perspective, and when you have limited time to put towards a creative outlet, you want to spend it actually creating!
But because the subject of clean code is important for anyone doing any amount of coding (not to mention how easily the codebase of a game can become a mess), let’s talk about how to write cleaner, more readable, and more maintainable code.
This is really the root reason for everything in this post today. Write for the human that will be reading your code in the future, whether that’s you an hour from now, a teammate a week from now, or again, and probably most likely, yourself one year from now when you want to pull some useful code for another project. 99.9% of the time, the computer will take care of itself. It’s the pesky and ephemeral human mind you need to worry about when writing your code. Remember this, and all these other rules will seem obvious.
### Use meaningful, descriptive names
This is one of the most fundamental and important principles to hold onto, and with code completion built into any editor worth its salt, and therefore making it so you don’t even have to type out the whole variable or function name more than once (when you first declare it), there’s no excuse for using short or abbreviated names in a modern codebase. Variables should clearly define what data they hold and any units associated with it, while functions should clearly state what they do. You should also generally avoid any abbreviations that aren’t commonly understood and standardized. mph for miles per hour is ok, for instance, but any custom abbreviations or lingo should be avoided.
This is important so that the intent of your code is easily understandable to anyone new to it, and that even includes yourself if you work alone. Come back to your code in three months to a year and see how well you remember what it’s doing. You’ll be glad you made it clear why each variable and function exists.
#### Bad - Names are vague
var dpt = 10
func calc(a, b):
...
#### Good - Names make the purpose of each variable and function clear
var damagePerTurn = 10
func calculateTotalDamage(baseDamage, damageModifiers):
...
There is one exception to this rule, and that is for variables that are more or less globally and conventionally named as such, such as naming the current index of a for loop i:
for i in range(damageModifiers.size()):
...
### No magic numbers
This is related to the last point so there’s not as much to say about it, but don’t use “magic numbers” in your code - those that are arbitrary and unnamed. Any sort of configuration or multiplier should be clearly named so that another developer (or future you!) can make sense of why this number is used.
for i in range(20):
...
#### Good
const NUMBER_OF_ITEM_SLOTS = 20
for i in range(NUMBER_OF_ITEM_SLOTS):
...
### Prefer smaller functions
Rather than prescribe an ideal function length, as is the case in a certain famous book on coding structure, I’m just going to leave it at this: prefer writing shorter functions, and do so by delegating to more functions as needed. Doing so makes it easier to read and follow the logic of your code, and can help ensure your code is more reusable. A big, monolithic function of 100 lines is hard to make sense of, and is almost certainly doing too much, whereas a nice and short five line function calling other functions is easy to understand and ensures you’re making your code more reusable.
func determineNextTarget():
var availableTargets = []
for i in range(allTargets.size()):
if allTargets[i].distance < MAX_TARGET_DISTANCE:
availableTargets.append(targets[i])
var mostValuableTarget
for i in range(availableTargets.size()):
if !mostValuableTarget or availableTargets[i].value > mostValuableTarget.value:
mostValuableTarget = availableTargets[i]
var weakestTarget
for i in range(availableTargets.size()):
if !weakestTarget or availableTargets[i].health > mostValuableTarget.health:
weakestTarget = availableTargets[i]
if mostValuableTarget.value * VALUE_WEIGHTING > weakestTarget.health * HEALTH_WEIGHTING:
return mostValuableTarget
return weakestTarget
#### Good
func determineNextTarget():
var availableTargets = getAvailableTargets(allTargets)
var mostValuableTarget = getMostValuableTarget(availableTargets)
var weakestTarget = getWeakestTarget(availableTargets)
return selectBestTarget(mostValuableTarget, weakestTarget)
### Forget comments and write self-documenting code
Comments are a “code smell” (a piece of code that is indicative of a potentially larger issue in the codebase) and therefore should be rare in a well-written codebase. With poorly written code, they’re used as a bandage over unclear code that really just needs to be refactored so its intent is clearer. In an otherwise good codebase, they just take up space and make code comprehension harder by distracting the reader. Additionally, they tend to become quickly outdated as the code evolves but they’re forgotten and left behind.
That’s not to say that all comments are bad, just that they should be rare. Working with an external library or API with some wonkiness to it? Go ahead and put a comment clarifying your workaround. Want to leave a comment clarifying what a function you wrote does? Forget it and instead refactor the code until its intent is naturally clearer.
Let’s use one of our previous code examples, with a twist, to show the good and the bad:
#### Bad - Comments are a crutch, as code can be written to be more clear
# Damage done per turn
var dpt = 10
# Calculate the total damage
func calc(a, b):
...
# Damage done per turn
var damagePerTurn = 10
# Calculate the total damage
func calculateTotalDamage(baseDamage, damageModifiers):
...
#### Good - Clarifies interaction with code that is beyond our control
var damagePerTurn = 10
# SuperCoolLibrary returns total damage as a percentage of target's maximum health
# so scale by maxHealth to get real damage value
var totalDamage = maxHealth * SuperCoolLibrary.calculateTotalDamage(target, baseDamage, damageModifiers)
#### Good - Code is naturally clear in its intent
var damagePerTurn = 10
func calculateTotalDamage(baseDamage, damageModifiers):
...
### Make use of whitespace
This applies to both the spaces within your code, the code’s indentation, and the line breaks themselves. Smartly place whitespace to make your code more readable by grouping together related bits of code, separating unrelated bits of code, and making non-alphanumeric characters easier to read. It can also be used to show the relation between code over multiple lines, most commonly seen when indenting all parameters of a function when the call takes up more than one line. Like anything, too much whitespace can be a bad thing, so work on finding that middle ground and you’ll be good to go.
#### Bad - Code is cramped, harder to parse, and unrelated code runs together
var totalDamage=calculateTotalDamage(baseDamage,damageModifiers,baseDefense,defensiveModifiers)
var currentHealth-=totalDamage
updatePlayerHealthUI(currentHealth)
var hasCounterAttack=rollForCounterAttack()
if hasCounterAttack: processCounterAttack()
#### Good - Code is grouped logically, and provides some “breathing room” to the code
var totalDamage = calculateTotalDamage(
baseDamage,
damageModifiers,
baseDefense,
defensiveModifiers
)
var currentHealth -= totalDamage
updatePlayerHealthUI(currentHealth)
var hasCounterAttack = rollForCounterAttack()
if hasCounterAttack:
processCounterAttack()
### Respect (and use!) line limits
Line limits are a basic way to keep your code readable. Too short of lines make you have to jump lines too often, and therefore mentally readjust yourself (by however small amount it happens). Conversely, lines that are too long can increase mental effort by making you lose your spot compared to the rest of the code base. Plus, they can just feel long when reading them. Even when the code is well written, a two-hundred character line of of code is a lot to take in.
So how long should your lines be? Traditionally, 80 characters was the upper limit (Apparently this is based off of the number of holes in a punch card, but also matches the character limits of early computers), but nowadays most people will bump it up to anywhere from 100 to 150 characters per line. For most code, I personally stick to a limit of 140 characters, though I will go down to 100 when using Godot’s built-in code editor since the window is smaller in the engine than in something like VSCode, and word-wrapping code is just gross.
var totalDamageAfterModifiers = calculateTotalDamageWithModifiers(enemy.calculateBaseDamage(), enemy.attackModifiers, player.armorClass, player.weaknesses, world.getEnvironmentModifiersForLocation(player.position))
#### Good - Code is broken up over multiple lines to make it easier to read and follow
var totalDamageAfterModifiers = calculateTotalDamageWithModifiers(
enemy.calculateBaseDamage(),
enemy.attackModifiers,
player.armorClass,
player.weaknesses,
world.getEnvironmentModifiersForLocation(player.position)
)
### Use enumerators
Aversion to enumerators is something I see from time to time in the gamedev world for some reason, so let me just say that you should use enumerators when you have a fixed number of options to choose from in your code. They aren’t prone to typos, differences in capitalization, or encoding errors like strings are, and they’re much easier to remember, and more flexible, than numbers. Enumerators make it so you know what you’re selecting when you write your code, you know your selection won’t be misinterpreted (forgetting which number is for which selection, for instance), and if you want to make changes to your available options or order of options later, it doesn’t impact the code you’ve already written. Enumerators. Use them. Love them.
#### Bad - Using error-prone strings to represent a fixed selection
selectPotion('mana')
# Uh oh
selectPotion('helth')
func selectPotion(name):
if name == 'mana':
...
elif name == 'health':
...
#### Bad - Using numbers to represent a fixed selection
# No clue if these are correct or what potion they represent
# without looking at the larger codebase
selectPotion(1)
selectPotion(3)
func selectPotion(potion):
if potion == 1:
...
elif potion == 2:
...
#### Good - Using enumerators for selection
enum POTIONS {
HEALTH,
MANA
}
selectPotion(POTIONS.HEALTH)
selectPotion(POTIONS.MANA)
func selectPotion(name):
if name == POTIONS.MANA:
...
elif name == POTIONS.HEALTH:
...
### Keep it DRY
D.R.Y., or “Don’t repeat yourself”, is a very important concept to remember when writing code. If you find yourself doing the same thing more than once or maybe twice, turn that code into a function so that it can be used as many places as possible, even if it’s just a single line of code. This reduces codebase size, let’s you more quickly understand the code (since you can just see a well-named function call and reason about what it does, or even just give it a glance once and then understand every call after), and makes it easier to maintain your code since you only have to edit your code in one location and have the changes reflected everywhere.
#### Bad - Writing the same logic in multiple places
var finalDamage = attackDamage - player.baseDefense + player.defenseModifiers - enemy.attackModifiers
# Roll for counterattack if damage is completely negated
if (player.baseDefense + player.defenseModifiers - enemy.attackModifiers) > finalDamage:
rollForCounter(player.baseDefense + player.defenseModifiers - enemy.attackModifiers - finalDamage)
#### Good - Reusing code via a function
var finalDamage = attackDamage - getTotalDefense()
# Roll for counterattack if damage is completely negated
if getTotalDefense() > finalDamage:
rollForCounter(getTotalDefense() - finalDamage)
func getTotalDefense():
return player.baseDefense + player.defenseModifiers - enemy.attackModifiers
### Functions should do one thing
This is really just a sliver of a larger concept regarding code structure and architecture, but I want to go ahead and mention it here since it’s an easy trap that even experienced programmers fall into. If you feel the need to use the word “and” to describe what a function does or if you can’t reuse a function because it has other side effects, you’re probably trying to do too much with it. Ideally, a function should do one thing and one thing only. If it’s doing more than one thing, you probably need to break out its functionality into more functions and then wrap those calls into a higher order function that properly establishes the intent.
#### Bad - This function does too much
# Can't be reused if you just want to update a local save
func saveProgressToDiskAndSyncToCloud():
...
#### Good - Multiple functions make this code more reusable
# Individual functions can be reused or called together where needed
func saveProgress():
saveProgressToDisk()
syncSaveToCloud()
func saveProgressToDisk():
...
func syncSaveToCloud():
...
### Use your language’s features and conventions
And lastly, let’s go really broad. Don’t reinvent the wheel or fall prey to Not invented here syndrome. If your language provides a built-in way to do something or has a conventional way of doing something, go with it. You don’t want to waste your time creating problems for existing solutions.
For example, GDScript has all sorts of built-in functions to help you speed up your development process, and since these come standard in Godot, their functionality is widely tested and used across a huge number of projects and developers. Additionally, pretty much every class and data type in the engine has built-in functions to save you time and effort. Use these functions and save your efforts for more unique problems than normalizing a vector or rounding a number.
#### Bad - Reinventing the wheel
# As an example, stepNumber is converted directly from the Godot source code for stepify
# https://github.com/godotengine/godot-cpp/blob/82bc10258191d4efe64be6239ae86eed70b49e5a/include/godot_cpp/core/math.hpp#L420-L425
var steppedNumber = stepNumber(101, 10)
func stepNumber(pValue: float, pStep: float) -> float:
if pStep != 0:
pValue = floor(pValue / pStep + 0.5) * pStep
return pValue
#### Good - Making use of Godot’s built-in functions to save time, effort, and code size
var steppedNumber = stepify(101, 10)
### Conclusion
And we are finally at the end! Hopefully these tips will help you write cleaner, more maintainable, and more readable code. I have intentionally excluded architectural tips and discussions since that’s a discussion worth having entirely on its own, so stay tuned for more info on that in the future.
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2023-03-29 18:57:29
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http://mathoverflow.net/questions/12410/terminology-is-there-a-name-for-a-category-with-biproducts?sort=votes
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# Terminology: Is there a name for a category with biproducts?
Many people are familiar with the notion of an additive category. This is a category with the following properties:
(1) It contains a zero object (an object which is both initial and terminal).
This implies that the category is enriched in pointed sets. Thus if a product $X \times Y$ and a coproduct $X \sqcup Y$ exist, then we have a canonical map from the coproduct to the product (given by "the identity matrix").
(2) Finite products and coproducts exist.
(3) The canonical map from the coproduct to the product is an equivalence.
A standard exercise shows this gives us a multiplication on each hom space making the category enriched in commutative monoids (with unit).
(4) An additive category further requires that these commutative monoids are abelian groups.
I want to know what standard terminology is for a category which satisfies the first three axioms but not necessarily the last.
I can't seem to find it using Google or Wikipedia. An obvious guess, "Pre-additive", seems to be standard terminology for a category enriched in abelian groups, which might not have products/coproducts.
-
I'd just say category with biproducts, thereby avoiding adding yet another name! – Mariano Suárez-Alvarez Jan 20 '10 at 13:51
One sometimes encounters the term "R-additive category" for an additive category enriched in R-Mod. Given that, maybe "$\mathbb{N}$-additive category" is an alternative, pretending that the usual usage of "additive" is short for "$\mathbb{Z}$-additive"?
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2015-03-04 12:59:58
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http://statkat.com/stattest.php?t=41&t2=18
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# Marginal Homogeneity test / Stuart-Maxwell test - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
Marginal Homogeneity test / Stuart-Maxwell test
Spearman's rho
Independent variableVariable 1
2 paired groupsOne of ordinal level
Dependent variableVariable 2
One categorical with $J$ independent groups ($J \geqslant 2$)One of ordinal level
Null hypothesisNull hypothesis
H0: for each category $j$ of the dependent variable, $\pi_j$ for the first paired group = $\pi_j$ for the second paired group.
Here $\pi_j$ is the population proportion in category $j.$
H0: $\rho_s = 0$
Here $\rho_s$ is the Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level.
In words, the null hypothesis would be:
H0: there is no monotonic relationship between the two variables in the population.
Alternative hypothesisAlternative hypothesis
H1: for some categories of the dependent variable, $\pi_j$ for the first paired group $\neq$ $\pi_j$ for the second paired group.H1 two sided: $\rho_s \neq 0$
H1 right sided: $\rho_s > 0$
H1 left sided: $\rho_s < 0$
AssumptionsAssumptions
• Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
• Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another
Note: this assumption is only important for the significance test, not for the correlation coefficient itself. The correlation coefficient itself just measures the strength of the monotonic relationship between two variables.
Test statisticTest statistic
Computing the test statistic is a bit complicated and involves matrix algebra. Unless you are following a technical course, you probably won't need to calculate it by hand.$t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}}$
Here $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores.
Sampling distribution of the test statistic if H0 were trueSampling distribution of $t$ if H0 were true
Approximately the chi-squared distribution with $J - 1$ degrees of freedomApproximately the $t$ distribution with $N - 2$ degrees of freedom
Significant?Significant?
If we denote the test statistic as $X^2$:
• Check if $X^2$ observed in sample is equal to or larger than critical value $X^{2*}$ or
• Find $p$ value corresponding to observed $X^2$ and check if it is equal to or smaller than $\alpha$
Two sided:
Right sided:
Left sided:
Example contextExample context
Subjects are asked to taste three different types of mayonnaise, and to indicate which of the three types of mayonnaise they like best. They then have to drink a glass of beer, and taste and rate the three types of mayonnaise again. Does drinking a beer change which type of mayonnaise people like best?Is there a monotonic relationship between physical health and mental health?
SPSSSPSS
Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
• Put the two paired variables in the boxes below Variable 1 and Variable 2
• Under Test Type, select the Marginal Homogeneity test
Analyze > Correlate > Bivariate...
• Put your two variables in the box below Variables
• Under Correlation Coefficients, select Spearman
n.a.Jamovi
-Regression > Correlation Matrix
• Put your two variables in the white box at the right
• Under Correlation Coefficients, select Spearman
• Under Hypothesis, select your alternative hypothesis
Practice questionsPractice questions
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2020-11-28 01:30:35
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https://rdoodles.rbind.io/2018/03/combining-data-distribution-summary-model-effects-and-uncertainty-in-a-single-plot/
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A Harrell plot combines a forest plot of estimated treatment effects and uncertainty, a dot plot of raw data, and a box plot of the distribution of the raw data into a single plot. A Harrell plot encourages best practices such as exploration of the distribution of the data and focus on effect size and uncertainty, while discouraging bad practices such as ignoring distributions and focusing on $$p$$-values. Consequently, a Harrell plot should replace the bar plots and Cleveland dot plots that are currently ubiquitous in the literature.
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2019-09-20 23:11:04
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http://physics.stackexchange.com/questions/53482/mathematically-what-is-susy/53887
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# Mathematically: What is SUSY?
Wikipedia says:
In particle physics, supersymmetry (often abbreviated SUSY) is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners. In a theory with unbroken supersymmetry, for every type of boson there exists a corresponding type of fermion with the same mass and internal quantum numbers (other than spin), and vice-versa. There is only indirect evidence for the existence of supersymmetry [...]
I want a mathematical explanation of SUSY.
-
Mathematically, SUSY begins with the supersymmetry algebra, a Lie superalgebra, which is itself a special case of a more general class of algebras called graded Lie algebras. Of central importance is the supersymmetry algebra referred to as the super-Poincare algebra that extends the Poincare algebra to include supersymmetry "charges" and their anticommutators.
In the context of physics, one studies field theories, both classical and quantum, that exhibit invariance under some action of supersymmetry algebras on fields and Hilbert spaces of these theories. As a result, representations of supersymmetry algebras are especially important in physics.
I would highly recommend that you look at THIS set of notes written by Sohnius, one of the original supersymmetry masters and co-discoverers of THIS famous and important theorem which really motivates why supersymemtry is all the rage in physics. The notes talk about representations of supersymmetry algebras in a lot of detail, and the clarity of the prose is top-notch if you ask me.
Addendum. I almost forgot, you also hear the word "superspace" which is a construction that physicists use to, among other things, make constructing manifestly supersymmetric Lagrangians easier. The mathematics behind this is supermanifolds.
Lastly, there is some discussion of these things on math.SE, see for example
Hope that helps!
Cheers!
-
By the way, superspace in the form of super-Minkowski space is right there in the super-Poincare algebra, being the quotient of that by the Lorentz sub-algebra (ncatlab.org/nlab/show/super+Poincare+Lie+algebra). – Urs Schreiber Sep 18 '13 at 17:46
@UrsSchreiber Interesting stuff. Thanks for the link! – joshphysics Sep 18 '13 at 18:55
In addition to what Joshua said in his nice answer, may favorite (simplified) point of view is looking at a SUSY transformation as a coordinate transformation (translation) in superspace
$$x' = x + a + \frac{i}{2}\zeta\sigma^{\mu}\bar{\theta} - \frac{i}{2}\theta\sigma^{\mu}\bar{\zeta}$$
$$\theta'= \theta + \zeta$$
$$\bar{\theta}'= \bar{\theta} + \bar{\zeta}$$
with $\theta$ and $\bar{\theta}$ denoting the additional Grassmanian coordinates.
The supersymmetry generators or supercharges, when written down as differential operators, contain momentum operators in both, the "usual" even spacetime coordinates and the odd Grassmann coordinates
$$Q_a = i\partial_a -\frac{1}{2}(\sigma^{\mu})_{a\dot{b}}\bar{\theta}^{\dot{b}}\partial_{\mu}$$
$$\bar{Q}^{\dot{a}} = i\bar{\partial}^{\dot{a}} -\frac{1}{2}(\bar{\sigma}^{\mu})^{\dot{a}b}\theta_b\partial_{\mu}$$
Where $\partial_a = \frac{\partial}{\partial\theta^a}$ and $\bar{\partial}^{\dot{a}} = \frac{\partial}{\partial\bar{\theta_{\dot{a}}}}$ are the derivatives along the Grassmanian coordinates.
A nice and very readable introduction to the superspace formalism can for example be found in Ch 11 of this book.
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c.f.
There are two types of ; worldsheet supersymmetry, and spacetime supersymmetry
## Worldsheet supersymmetry
The Ramond-Neveu-Schwarz formalism has explicit worldsheet supersymmetry. Since the RNS Action is given by adding the Polyakov Action to the Dirac action, it is given by:
$${{\mathsf{\mathcal{L}}}_ {RNS}}=\frac{T}{2} h^{\alpha \beta} \left( \partial_\alpha X^\mu \partial_\beta X^\nu +i\hbar c_0 \bar{\psi_\mu} \not{\partial} \psi^\mu \right) g_{\mu\nu}$$
The supersymmetric transformations on the worldsheet can therefore be (almost trivially, by taking variations of this above action) shown to be:
$$\begin{gathered} \delta {X^\mu } \to \bar \epsilon {\psi ^\mu } ; \\ \delta {\psi ^\mu } \to - i \not \partial {X^\mu }\epsilon \\ \end{gathered}$$
# Spacetime Supersymmetry
The Green-Schwarz formalism, or the , are with explicit spacetime supersymmetry. The supersymmetric transformations on spacetime are (which is rather intuitive if you compare this to the RNS Worldsheet supersymmetry transformations) given by:
$$\begin{gathered} \delta {\Theta ^{Aa}} \leftrightarrow {\varepsilon ^{Aa}} ; \\ \delta {X^\mu } \leftrightarrow {{\bar \varepsilon }^A}{\gamma ^\mu }{\Theta ^A} ; \\ \end{gathered}$$
-
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2014-03-10 05:21:40
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https://www.zbmath.org/?q=an%3A0994.60066
|
# zbMATH — the first resource for mathematics
Ergodicity of the 2D Navier-Stokes equations with random forcing. (English) Zbl 0994.60066
Authors’ abstract: We consider the Navier-Stokes equation on a two-dimensional torus with a random force, acting at discrete times and analytic in space, for arbitrarily small viscosity coefficient. We prove the existence and uniqueness of the invariant measure for this system as well as exponential mixing in time.
##### MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35Q30 Navier-Stokes equations
##### Keywords:
Navier-Stokes equation; invariant measure; random forcing
Full Text:
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2021-05-08 13:49:57
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https://www.physicsforums.com/threads/conservation-of-momentum-2-step-problem.279601/
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# Conservation of momentum 2 step problem
1. Dec 14, 2008
### Maiia
1. The problem statement, all variables and given/known data
A 0.0854 kg block is released from rest from the top of a 25.8◦ frictionless incline. When
it has fallen a vertical distance of 0.905 m, a 0.0164 kg bullet is fired into the block along a
path parallel to the slope of the incline, and momentarily brings the block to rest, stopping in the block. The acceleration of gravity is 9.8 m/s2. Find the speed of the bullet just before impact. Answer in units of m/s.
I was hoping someone could check my work because I seem to be getting the wrong answer...
What I did:
1. found the height of the incline
sin25.8=x/.905m
x=.3938841449m
mgh=1/2mv^2
$$\sqrt{2gh}$$=v
v= 2.778512055m/s
2. used conservation of momentum
m1= block
m2=bullet
After having fallen the .905m
m1v1initial+m2v2initial= m1+m2(Vf)
because Vf=0
m1v1initial= m2v2initial
so plugging in m1= .0854kg m2=.0164kg v1=2.778512055m/s
i got v2= 14.46859326m/s
2. Dec 14, 2008
### LowlyPion
Haven't they conveniently already provided you with the drop in height? Won't kinetic energy of the block be the .905*m*g then? And your V2 = 2*g*.905 ?
3. Dec 14, 2008
### Maiia
oh..i didn't read the problem carefully enough and assumed they were giving me the amount it was falling on the slope. Thanks for your help :)
4. Dec 14, 2008
### Maiia
oh, if i wanted to find the speed of the bullet that would push the block all the way back up the incline back to its initial position, would i be able to do this? :
1/2m1v^2= m2gh
mass1 i am denoting as mass of bullet
m2 i am denoting as combined mass
so vbullet= squareroot of (2*.1018*9.8*.905/ .0164) ?
my reasoning being that the ke of bullet is converted to pe needed to push block back up incline?
5. Dec 14, 2008
### LowlyPion
If you mean additional amount of energy (more than just stopping it), you should also consider that you need to add the mass of the bullet to the total mass pushed up the incline.
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2017-10-18 18:33:19
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https://notepad.mmakowski.com/
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One way would be to consider the true mislabelling rates to follow a certain probability distribution, and caculate the posterior based on the available evidence. Beta distribution again looks like a good candidate, seeing that it can be used to describe the probability of success – “success” meaning a given example being mislabelled, in our case. Again, we will assume uninformative prior, i.e. $$Beta(1,1)$$.
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2018-08-17 19:01:14
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https://greprepclub.com/forum/in-the-figure-above-arcs-pr-and-qs-are-semicircles-with-ce-13968.html
|
It is currently 30 Nov 2020, 18:52
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In the figure above, arcs PR and QS are semicircles with ce
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In the figure above, arcs PR and QS are semicircles with ce [#permalink] 18 Jun 2019, 21:58
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#GREpracticequestion In the figure above, arcs PR and QS .jpg [ 49.47 KiB | Viewed 1424 times ]
In the figure above, arcs PR and QS are semicircles with centers at Q and R respectively. If PQ = 5, what" is the perimeter of the shaded region?
(A) $$5\pi + 5$$
(B) $$5\pi + 15$$
(C) $$10\pi + 10$$
(D) $$10\pi + 15$$
(E) $$100\pi$$
[Reveal] Spoiler: OA
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Re: In the figure above, arcs PR and QS are semicircles with ce [#permalink] 19 Jun 2019, 09:35
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Quote:
In the figure above, arcs PR and QS are semicircles with centers at Q and R respectively. If PQ = 5, what" is the perimeter of the shaded region?
In GRE geometry questions, always look for logical estimation opportunities and solve for the easy shape or value first.
Then, we can solve of the straight portion of the figure which is PQ + RS = 10.
Look to the answer choices to see which include the value of 10 since the circle must include π.
Only Choice C includes the necessary value of 10, so select Choice C without even needing to spend the time calculating the included circumference which would have been 2πr = 2π x 5 = 10π.
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Re: In the figure above, arcs PR and QS are semicircles with ce [#permalink] 21 Nov 2020, 06:07
I am unclear about the math, if anybody can give an explanation about it.
GRE Prep Club Tests Editor
Affiliations: Partner at MyGuru LLC.
Joined: 13 May 2019
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GMAT 1: 770 Q51 V44
GRE 1: Q169 V168
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Re: In the figure above, arcs PR and QS are semicircles with ce [#permalink] 22 Nov 2020, 09:07
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Expert's post
Kafi22 wrote:
I am unclear about the math, if anybody can give an explanation about it.
The formula for perimeter of a semicircle = diameter + πr.
Since the radius for either semicircle is 5, then the the perimeter of a single semicircle = 10 + 5π.
However, there are two semicircle arcs included in the shaded figure, so the shaded perimeter = 10 + 5π + 5π = 10 + 10π, choice C.
Of course, I stand by the earlier estimation as possibly a more efficient path to solving on the actual GRE.
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Re: In the figure above, arcs PR and QS are semicircles with ce [#permalink] 22 Nov 2020, 09:07
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2020-12-01 02:52:41
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https://www.gradesaver.com/textbooks/math/calculus/calculus-3rd-edition/chapter-12-parametric-equations-polar-coordinates-and-conic-sections-12-1-parametric-equations-preliminary-questions-page-602/7
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## Calculus (3rd Edition)
(a) The derivative $\frac{d x}{d t}$ is the horizontal rate of change with respect to time. (b) The derivative $\frac{d y}{d t}$ is the vertical rate of change with respect to time. (c) The derivative $\frac{d y}{d x}$ is the slope of the tangent line to the curve.
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2022-08-09 07:38:50
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https://www.mapleprimes.com/products/maple?Unanswered=1&page=4
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## How do I connect to MSQL in Maple?...
Asked by:
I'm having trouble connecting from Maple on Windows 10 to MSQL Server. I tried Microsoft recommended drivers such as sqljdbc_6.4.0.0, did (as I thought) all required steps. The only invariable result I get is "Cannot load driver". I was wandering if anyone had implemented such a construction. Driver name & version , connection string and Java version would be greatly appreciated. Another option is to have any driver, which connects to any of standard databases (Oracle, MySQL). The only limitation is- it must be from Windows 7 or 10.
Thanks.
A.B.
## Starting a new java session...
Asked by:
I remember there was a command that used to start an independent java session each time. I thought it's "xmaple -singlemode" but it's not. So could anybody please remind me what was the command. I just killed a second java session after losing kernel connection and java running wild. I'd be very happy to do my calculation in another java session so that if anything goes bad II won't have to open all my 5 tabs again.
## Bug using Heaviside and PDF in statistics package...
Asked by:
Integrating a positive definite function (normal distribution) and a >= 0 function (Heaviside) should not return a negative value.
with(Statistics);
X := RandomVariable(Normal(1, sqrt(2.25)));
int(PDF(X, x)*Heaviside(x^7-5*x^4-3*x+1), x = -infinity .. infinity);
-0.08507120131
Bug.mw
## Show ODE solution step by step...
Asked by:
I'm using dsolve command to solve a differential equation. Using infolevel to 3 will tell me the classification of said DE. However, how can I see the step by step solution? I'm using Maple as a study tool so I do solve manually a DE then I'd like to compare my answer with Maple's. How can I acomplish this? Thanks in advance.
## Students discount...
Asked by:
Please may I know if you can offer ma student's discount ob the seleted version.
Thank you.
Fred.
Asked by:
## I can't do even basic stuff on maple, what is goin...
Asked by:
I open a completely new document, type in a few simple expressions and get this weird complex number out. Can someone pleas help me fix this?
## Recursive Assignment Errors...
Asked by:
Is there any way we can get maple to tell us which variable we are recursively assigning?
## Why is this an error?...
Asked by:
solve({-infinity < a , a < -1, -1 < b , b < 0});
## Why is this piecewise plot giving an error? Pls he...
Asked by:
f(x) := piecewise(0 < x, x^(3/2)*sin(1/x), x = 0, 0, undefined);
plot(f(x));
gives me the following error:
Error, (in plot) incorrect first argument piecewise(0 < x, (HFloat(2.739493386336394e-116)+HFloat(2.739493386336394e-116)*I)*x^(3/2), x = 0, 0, undefined)
I just want to see the function plot. With Wolfram Alpha this is no deal at all!
## Statistics fails to reject invalid parameter value...
Asked by:
Maple doesn't completely check the condition on the number of trials "n" for Binomial and NegativeBinomial distributions (package Statistics).
The attribute "Conditions" explicitely says that n must be a strictly positive integer but no strictly positive real valuereturna an error (ok, it would be stupid to set n to a non integer value !!!).
I think it is a default that ought to be corrected in future releases (this default still exists in Maple 2018)
> restart
> kernelopts(version)
(1)
> with(Statistics):
BINOMIAL DISTRIBUTION
> X := RandomVariable(Binomial(n, p)): L := [attributes(X)][3]: A := exports(L)
(2)
> L:-Conditions
(3)
> # Maple should return an error for N is not of type posint # # It seems that Sample uses floor(N) N := 10.49; type(N::posint); P := 1/2: X := RandomVariable(Binomial(N, P)): Mean(X), N*P; ProbabilityFunction(X, k); S := Sample(X, 10^6): Mean(S); # A non consistent result (only non negative values of k should be accepted) eval(ProbabilityFunction(X, k), k=evalf(Pi));
(4)
NEGATIVE BINOMIAL DISTRIBUTION
> X := RandomVariable(NegativeBinomial(n, p)): L := [attributes(X)][3]: A := exports(L): L:-Conditions
(5)
> N := 10.49: P := 1/2: X := RandomVariable(NegativeBinomial(N, P)): Mean(X)
Download BinomialLaw.mw
## When i type sqrt(25), i get 1.81847767202745*10^(-...
Asked by:
Yeah, i have tried evalf[10](sqrt(25)).
How can i get a simple number as answer? I'm loving the software but i just wished i could type in:
int(sin(x), x = 0 .. pi)
and get 2 instead of (2.739493386*10^(-116) + (2.739493386*10^(-116))*I)*pi.
Also, when i type evalf[50](pi), i wish to get all the 50 digits, but i just get \pi :/.
Please help me.
## How to solve initial Newton iteration is not conve...
Asked by:
I want to solve an ODE system numerically, but maple show not converge for iteration, can you give a solution
with(VectorCalculus);
with(linalg);
pi := 4; eta := 6; mh := .1; mv := .3; bh := .15; Nh := 400; Nv := 200; bv := .25; b := .8; d := .1; p := .5; B := 10; A := 1; alpha := .25; beta := .7; K := 400; r := .5; c := .9; q := .8; Sh0 := 225; Sv0 := 100; Ih0 := 175; Iv0 := 600; P0 := 50; T := 35; B := 10;
eq1 := diff(L1(t), t) = -L1(t)*(-bh*b*Iv(t)/Nh-mh)-L2(t)*bh*b*Iv(t)/Nh; eq2 := diff(L2(t), t) = -L2(t)*(-mh-d)+L3(t)*bv*b*Sv(t)/Nv-L4(t)*bv*b*Sv(t)/Nv; eq3 := diff(L3(t), t) = -(L4(t)*Iv(t)+L3(t)*Sv(t))*L3(t)/(2*B^2)-L3(t)*(-bv*b*Ih(t)/Nv-mv-L3(t)*Sv(t)/(2*B^2)-(L4(t)*Iv(t)+L3(t)*Sv(t))/(2*B^2)-P(t)*alpha)-L4(t)*(bv*b*Ih(t)/Nv-L3(t)*Iv(t)/(2*B^2))-L5(t)*P(t)*alpha; eq4 := diff(L4(t), t) = -A-(L4(t)*Iv(t)+L3(t)*Sv(t))*L4(t)/(2*B^2)+L1(t)*bh*b*Sh(t)/Nh-L2(t)*bh*b*Sh(t)/Nh+L3(t)*L4(t)*Sv(t)/(2*B^2)-L4(t)*(-mv-L4(t)*Iv(t)/(2*B^2)-(L4(t)*Iv(t)+L3(t)*Sv(t))/(2*B^2)-P(t)*alpha)-L5(t)*P(t)*alpha; eq5 := diff(L5(t), t) = L3(t)*Sv(t)*alpha+L4(t)*Iv(t)*alpha-L5(t)*(r*(1-P(t)/Nv)-P(t)*r/Nv+alpha*(Sv(t)+Iv(t))-q*c); eq6 := diff(Sh(t), t) = pi-bh*b*Sh(t)*Iv(t)/Nh-mh*Sh(t); eq7 := diff(Ih(t), t) = bh*b*Sh(t)*Iv(t)/Nh-mh*Ih(t)-d*Ih(t); eq8 := diff(Sv(t), t) = eta-bv*b*Sv(t)*Ih(t)/Nv-mv*Sv(t)-P(t)*Sv(t)*alpha-L3(t)*Sv(t)^2/(2*B^2)-L4(t)*Sv(t)*Iv(t)/(2*B^2); eq9 := diff(Iv(t), t) = -L3(t)*Sv(t)*Iv(t)/(2*B^2)+bv*b*Sv(t)*Ih(t)/Nv-mv*Iv(t)-P(t)*Iv(t)*alpha-L4(t)*Iv(t)^2/(2*B^2); eq10 := diff(P(t), t) = P(t)*r*(1-P(t)/Nh)+P(t)*alpha*(Sv(t)+Iv(t))-q*P(t)*c
fcns := {Ih(t), Iv(t), L1(t), L2(t), L3(t), L4(t), L5(t), P(t), Sh(t), Sv(t)}; a := dsolve({eq1, eq10, eq2, eq3, eq4, eq5, eq6, eq7, eq8, eq9, Ih(0) = Ih0, Iv(0) = Iv0, L1(T) = 0, L2(T) = 0, L3(T) = 0, L4(T) = 0, L5(T) = 0, P(0) = P0, Sh(0) = Sh0, Sv(0) = Sv0}, fcns, type = numeric);
fcns := {Ih(t), Iv(t), L1(t), L2(t), L3(t), L4(t), L5(t), P(t),
Sh(t), Sv(t)}
Error, (in dsolve/numeric/bvp) initial Newton iteration is not converging
## platform for running calculation...
Asked by:
Does maple provide any platform for running calculation something like cloud facility
## Severe basic bug in Maple 2019.1 in Document Mode?...
Asked by:
Dear Maple users
Some students have come to us to report, that something doesn't seem to work properly in Maple 2019.1 in Document Mode. And they seem to be right: writing an passive math formula by using Shift+F5 (the formula is gray, not blue), then using F5 to get out of that Math field and back into Text Mode. Using the Enter key to go to the next line: It doesn't work! The cursor stays in the same line. This behavior is new in Maple 2019. It worked properly in Maple 2018 and earlier. I assume it is not the intention?
I know it can easily be dealt with by making a new Paragraph by using the shortcut Ctrl+Shift+J. I call the assumed bug 'severe' though, because it will severely delay the workflow for many students. They are used to deliver a document mixed with formulas (active or passive) and text.
NB! I have tested it on several computers (Mac and Windows), and it doesn't work on any of them.
Regards,
Erik V.
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2019-10-22 00:46:06
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https://www.sarthaks.com/399838/what-is-the-value-of-young-s-modulus-for-a-perfectly-rigid-body
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# What is the value of youngs modulus for a perfectly rigid body?
22 views
in Physics
What is the value of youngs modulus for a perfectly rigid body?
+1 vote
by (69.6k points)
selected
Infinity value of young`s modulus for a perfectly rigid body.
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2020-07-12 12:08:54
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https://www.educative.io/answers/what-is-a-b-tree
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Related Tags
definition
b-tree
data structures
# What is a B-Tree?
A B-Tree is a self-balancing m-way tree data structure that allows searches, accesses, insertions, and deletions in logarithmic time. Each node in a B-Tree of order m can have, at most, m children and m-1 keys.
Think of B-Tree as a generalization of a binary search tree (BST). Similar to a BST, the stored data is sorted in a B-Tree, but unlike a BST, a B-Tree can have more than two child nodes.
A B-Tree with order 3
In addition to having multiple children, each node in a B-Tree can have more than one key, which keeps the height of the tree relatively small. Later on, we will see how keeping multiple keys in a single node helps with data locality.
In summary, a B-Tree of the order m has the following properties:
• each node has at most m children
• a node with n children must have n-1 keys
• all leaf nodes are at the same level
• every node, except the root, is at least half full
• root has at least two children if it is not a leaf
## Time complexity
The time complexity for insertion, deletion, and search operations takes $O(log \space n)$ time where $n$ is the number of keys stored in the tree.
## Space complexity
The space complexity for a B-Tree is $O(n)$, where $n$ is the number of keys in the tree.
## Uses
B-Tree is primarily used to store data on disk because its operations (e.g., insert and search) require relatively few disk reads.
Often the size of data on the disk is large and cannot fit entirely in main memory. Hence, data is read from a disk in contiguous chunks or blocks.
Now that's a fat tree!
However, reading from a disk is costly as disk access times are far higher than memory access times.
B-Trees have a high branching factor, meaning the trees are fat with a relatively small height, which ensures minimal disk reads to navigate to the place where the data is stored.
B-Trees are, therefore, well-suited for storage systems that read and write large blocks of data.
RELATED TAGS
definition
b-tree
data structures
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2022-08-15 12:18:04
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http://www.pearltrees.com/davidscottkirby/constants/id6754354
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# Constants
Embree–Trefethen constant. In number theory, the Embree–Trefethen constant is a threshold value labelled β*.[1] For a fixed positive number β, consider the recurrence relation where the sign in the sum is chosen at random for each n independently with equal probabilities for "+" and "−".
It can be proven that for any choice of β, the limit exists almost surely. In informal words, the sequence behaves exponentially with probability one, and σ(β) can be interpreted as its almost sure rate of exponential growth. We have σ < 1 for 0 < β < β* = 0.70258 approximately, so solutions to this recurrence decay exponentially as n→∞ with probability 1, and σ > 1 for β* < β, so they grow exponentially. Mathematical constants by continued fraction representation. This is a list of mathematical constants sorted by their representations as continued fractions.
Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one. Decimal representations are rounded or padded to 10 places if the values are known. Jump up ^ Although some of the symbols in the leftmost column are displayed in black due to math markup peculiarities, all are clickable and link to the respective constant's page.
Prouhet–Thue–Morse constant. In mathematics, the Prouhet–Thue–Morse constant, named for Eugène Prouhet, Axel Thue, and Marston Morse, is the number — denoted by — whose binary expansion .01101001100101101001011001101001... is given by the Thue–Morse sequence.
Reciprocal Fibonacci constant. The ratio of successive terms in this sum tends to the reciprocal of the golden ratio.
Since this is less than 1, the ratio test shows that the sum converges. The value of ψ is known to be approximately (sequence A079586 in OEIS) No closed formula for ψ is known, but Gosper describes an algorithm for fast numerical approximation of its value. The reciprocal Fibonacci series itself provides O(k) digits of accuracy for k terms of expansion, while Gosper's accelerated series provides O(k2) digits.[1] ψ is known to be irrational; this property was conjectured by Paul Erdős, Ronald Graham, and Leonard Carlitz, and proved in 1989 by Richard André-Jeannin.[2]
Ring of periods. Maxim Kontsevich and Don Zagier (2001) gave a survey of periods and introduced some conjectures about them.
Definition The values of absolutely convergent integrals of rational functions with algebraic coefficients, over domains in Examples Besides the algebraic numbers, the following numbers are known to be periods: The natural logarithm of any algebraic numberπElliptic integrals with rational argumentsAll zeta constants (the Riemann zeta function of an integer) and multiple zeta valuesSpecial values of hypergeometric functions at algebraic argumentsΓ(p/q)q for natural numbers p and q. Silver ratio. Silver ratio within the octagon In mathematics, two quantities are in the silver ratio (also silver mean or silver constant) if the ratio of the sum of the smaller and twice the larger of those quantities, to the larger quantity, is the same as the ratio of the larger one to the smaller one (see below).
Somos' quadratic recurrence constant. In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, is the number This can be easily re-written into the far more quickly converging product representation The constant σ arises when studying the asymptotic behaviour of the sequence with first few terms 1, 1, 2, 12, 576, 1658880 ...
(sequence A052129 in OEIS). This sequence can be shown to have asymptotic behaviour as follows:[1] Square root of 5. It is an irrational algebraic number.[1] The first sixty significant digits of its decimal expansion are: 2.23606 79774 99789 69640 91736 68731 27623 54406 18359 61152 57242 7089...
(sequence A002163 in OEIS). which can be rounded down to 2.236 to within 99.99% accuracy. As of April 1994, its numerical value in decimal had been computed to at least one million digits.[2] Square root of 3. The first sixty digits of its decimal expansion are: 1.73205 08075 68877 29352 74463 41505 87236 69428 05253 81038 06280 5580...
Square root of 2. "Pythagoras's constant" redirects here; not to be confused with Pythagoras number The square root of 2, often known as root 2, radical 2, or Pythagoras' constant, and written as Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem.
It was probably the first number known to be irrational. Its numerical value, truncated to 65 decimal places, is: 1.41421356237309504880168872420969807856967187537694807317667973799... Stieltjes constants. In mathematics, the Stieltjes constants are the numbers that occur in the Laurent series expansion of the Riemann zeta function: The zero'th constant is known as the Euler–Mascheroni constant.
Representations (In the case n = 0, the first summand requires evaluation of 00, which is taken to be 1.) Cauchy's differentiation formula leads to the integral representation Several representations in terms of integrals and infinite series are given in the papers of Coffey. Numerical values The first few values are: For large n, the Stieltjes constants grow rapidly in absolute value, and change signs in a complex pattern. Numerical values of the Stieltjes constants up to n = 100000, accurate to over 10000 digits each, have been computed by Johansson.
Asymptotic growth Twelfth root of two. The twelfth root of two or is an algebraic irrational number. It is most important in music theory, where it represents the frequency ratio of a semitone in Twelve-tone equal temperament. Numerical value Its value is 1.05946309435929..., which is slightly more than 18⁄17 ≈ 1.0588. Better approximations are 196⁄185 ≈ 1.059459 or 18904⁄17843 ≈ 1.0594630948.
Universal parabolic constant. The universal parabolic constant is a mathematical constant. The value of P is Derivation Take.
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2017-11-18 03:43:08
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https://www.realmeye.com/wiki/tome-of-frigid-protection
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# Tome of Frigid Protection
Tome of Frigid Protection: A tome that allows the caster to encase themselves in extremely durable ice. However, despite being nearly impenetrable, it will melt after a while. Thanks to: Poshun
Tier: UT (Limited)
MP Cost: 120
Party Heal (Lvl 20): $\left\{\begin{matrix}&space;200&space;&&space;\text{wis}<30\\&space;200+\frac{4\cdot\text{wis}}{3}&&space;30\leq\text{wis}\\&space;\end{matrix}\right.$ HP
Stat Bonus: +40 HP, +4 VIT
Fame Bonus: 6%
Feed Power: 1300
Loot Bag Bes Permafrost Lord
Notes:
This tome was added as of patch 27.7.x10.2 and is a reskin of the Tome of Holy Protection
This tome is commonly referred to as the most powerful tome in the game, for it allows the Priest to accomplish tasks that would be extremely hard (if not impossible) without it. It allows the Priest to easily rush the Abyss of Demons and other such dungeons, and most importantly, it provides a sort of cushion. If the Priest finds himself surrounded by lots of enemies, he can just use this tome to heal himself back to full health and reduce all incoming damage for a little bit. With a good Wisdom/MP ring (or a high leveled pet with Magic Heal), the Priest can ignore the high mana cost and use this tome permanently! It’s also noteworthy to mention that this tome has a great feed power, so if you have multiple already, it’d be worth to feed to your pet.
The only downside about this tome is that while it still heals, it doesn’t apply the Healing buff (massive increase in health regeneration) as most tiered tomes do. A good way to counter this is to keep a tiered tome with you and use that to heal, then quickly switch to the Prot and apply Armored to yourself.
Descriptions in-game are currently bugged, causing wis-affecting equipment to appear to have twice the effect they really do (this effect is only for the descriptions- not the actual result). This page shows the actual formulas, not the ones based off the (incorrect) descriptions.
Tomes
Limited Edition
UT. Tome of Frigid Protection
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2018-08-22 05:46:31
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http://thoibaocongdan.com/eknxy/page.php?ce8aeb=table-to-graph-calculator
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. There are two kinds of tables you can create: an automatically generated table and a user-generated table. It really depends on the student. Monday, July 22, 2019 " Would be great if we could adjust the graph via grabbing it and placing it where we want too. Here are some hints on the solver usage: if your problem doesn't contain an equal sign ("=") you are on the wrong page. For example, if I were tring to figure out how an unknown function behaved, I might put in various (x,y) pairs in a table to graph the relationship visually. It performs all of the same functions as the Texas Instrument calculators, but you don’t have to waste a ton of money on it. This graphing calculator accepts most mathematical functions and a list is given below. f x ... Move down the table and type in your own x value to determine the y value. You can also use your mouse scroll wheel if you are using a computer. All you have to do to zoom in on any piece of data on the graph is to click the plus and minus symbols on the top right corner of the graph. Set up your table. ... $$3$$ − A B C π $$0$$. Just enter the loan amount, interest rate, loan duration, and start date into the Excel loan calculator, and it will calculate each monthly principal and interest cost through the final payment. Graphing Calculator by Mathlab is a scientific graphing calculator integrated with algebra and is an indispensable mathematical tool for students from high school to those in college or graduate school, or just anyone who needs more than what a basic calculator offers. person_outlineTimurschedule 2018-10-16 14:36:55. They will, however allow you to bring your own graphing calculator to both Mathematics Level 1 and Level 2 exams. Log InorSign Up. Here is an example of a table of values for the equation, y= 2x + 1. Binomial distribution (chart) Calculator - … It utilizes the Trigonometry. Thus, the empirical formula "smoothes" y values. MathJax for A graphing calculator does so many things for you, and even if a teacher won’t allow you to use one on a test, you can almost always use one to check your work on homework problems. See also. Step 3) Plot the points onto the grid. Typically, scientific calculators only perform calculations like addition, subtraction, multiplication, and division. Line graph maker; Bar graph maker; Pie chart maker; Write how to improve this page. . They are also capable of more advanced functions like trigonometric functions, logarithms, and exponents. Now just watch your graphed line or curve appear. Here is an example of a table of values for the equation, y= 2x + 1. Then enter your expressions for X in the equation box using the keyboard. * Begin Free Trial . Press Y = and enter the equation from above 2. The calculator below can help with that. After you’ve entered functions in the Y= editor of your TI-84 Plus calculator, you can have the calculator create a table of functional values. Their solution can then be calculated using the INTERSECT option in the CALC menu. Simply, click below your current equation and another formula text box will appear for your next equation. To find if the table follows a function rule, check to see if the values follow the linear form . Feel free to try these online Mathway. Loading... Function Calculator Function Calculator. Our free 2D graphing calculator (plotter) allows you to plot the graph of your equation online and also lets you save or print an image of your graph. create Draw zoom_out zoom_in save_alt save_alt content_copy print clear. And the circle icon at the top allows you to tweak more than just the color: adjust the style (dots, connected, or both) or convert the entire table … Pre-Algebra. This graphing calculator was written by David Lippman Best Graphing Calculator Online We have the most sophisticated and comprehensive TI 84 type graphing calculator online. GraphCalc allows you to graph 2D and 3D functions and equations as well as find intersects and create table values. Calculates the table of the specified function with two variables specified as variable data table. Are graphing calculators allowed on the SAT? T distribution critical values table online. See More. The best graphing calculator for high school students depends on the student and their classes. To remove any curve or line plotted on the graph, simply click the X in the top right corner of the equation or function box. 2- Graphing Below given is the T table for you to refer the one and two tailed t distribution with ease. example. (Note: TblStart is the starting x-value in the table, so put a number slightly smaller that the number x approaches. A traditional calculator might work well for them. Step by step explanations are provided. At the bottom of the settings tab, click Radians or Degrees. Then hit the options button and click the table symbol that appears over the pair. The most fundamental strategy to graph a line is the use of table of values.The goal is to pick any values of x and substitute these values in the given equation to get the corresponding y values. CPM Student Tutorials CPM Content Videos TI-84 Graphing Calculator Functions TI-84: Using Tables. Calculates a table of the probability mass function, or lower or upper cumulative distribution function of the Binomial distribution, and draws the chart. Online tools, such as graphing calculator apps and sites dedicated to calculating, can serve a purpose in performing the most basic tasks of a graphing calculator. By … With the proper use of this TI-84 plus CE device, it is extremely simple and quick to comprehend science and mathematical topics. Other students who want more functions and the ability to use bigger screens will probably like the GraphCalc calculator better. like the TI-83/TI-84 in a convenient web-based format. Graphing a list is simple. Graphing Calculator: Tables: The TABLE feature of the calculator can be helpful in many different situations. 1 - Enter the expression defining function f(x) that you wish to plot and press on the button "Plot f(x)". This is a video from my graphing calculator tutorial series. TI-84: Displaying a Graph; TI-84: Finding Graph Coordinates (Tracing) TI-84: Using Tables; Probability 1 TI-84: Generating Random Numbers; CBL/CBR 1 TI-84: Data Logger with CBL/motion Detector or CBR; Large Data Sets 1 TI-84: Checksums math display. Using a graphing calculator can make it a much faster and easier process to find the limit of a function, especially if the function is complex. Graphing. Next in the typical student’s math career is geometry. You can either use the on screen keypad or your computer keyboard to enter in another expression or formula. Yes, GraphCalc is one of the first Windows graphing calculator software packages that allows you graph all functions and equations in a single app either online or on your computer, tablet, or phone. To construct a histogram, the divide the entire values into series of values and count how many values fall into each interval. Texas Instruments TI-84 Plus CE Graphing Calculator. The histogram is a type of graph used in statics and mathematics. App Chronicles - "My Graphing Calculator: Turn your iPhone into a Graphing Calculator." Wednesday, February 21, 2018 " It would be nice to be able to draw lines between the table points in the Graph Plotter rather than just the points. Each line segment on a p-t graph checks the position change, the speed, and how the speed changes compared to the previous line segment. The calculator then displays the plotted curves as an image, just right click to export the image, it is also possible to copy the image. This online graphing calculator TI 84 version has all of the same functions that a standard TI-84 does. x^2*y+x*y^2 ） The reserved functions are located in " Function List ". Here are the steps to graph a list: Typically, the testing center will not provide a calculator for you or allow any calculators for non-math exams. BYJU’S online graphing linear equations calculator tool makes the calculation faster and it displays the graph in a fraction of seconds. In 1-dimensional kinematics, you can represent the motion of the object using position vs. time graphs. Press 2ND Y= (which is the Stat Plot) 2. so all information you type after that is automatically inside the square root sign until you close the parentheses. Lines: Point Slope Form. Press Y = and enter the equation from above 2. Step 2) Rule up an X-Y grid on graph paper. To plot a graph using a values table we follow these steps: Step 1) Write the table out as a set of (x,y) coordinates. Calculate the values of and . Get it now on Amazon.com . Features: Graphi… Yes, you can easily graph trig functions on a graphing calculator. Please pick the appropriate calculator from below to … The GraphCalc graphing calculator is FREE. Try Our College Algebra Course. The frequency of the data occurrence is represented in the form of a bar. How to Rearrange Function Order on the Graph, How to Zoom on a Point, Intersect, or Curve, How to Switch Between Radians and Degrees, Online TI-84 Graphing Calculator Functions and Capabilities. There are two kinds of tables you can create: an automatically generated table and a user-generated table. TI-84: Using Tables TI-84 Video: Using Tables 1. For FREE. ONLINE TOOLS. Mode: . table of values calculator graphing - bentekahan.eu ... music Click on the Expressions menu item. Our free 2D graphing calculator (plotter) allows you to plot the graph of your equation online and also lets you save or print an image of your graph. Teachers in both high school and college have adopted the TI-84 for graphing in algebra and calculus. Turn off all the plot except one (Plot 1) There would be two graphs open in front of the calculator, out of which you need to turn the plot two off. View the table. Just be aware that some built-in math shortcuts automatically start with grouping parentheses. Leave the Indpnt; and Depend: to "Auto" to automatically generate the "x" and "y" values. Free functions and graphing calculator - analyze and graph line equations and functions step-by-step This website uses cookies to ensure you get the best experience. If you want to zoom back out to the original view of the graph, simply click the home icon on the right side of the graph. Go to: [2nd] [TBLSET]. How to Use Graphing Functions Calculator. You can also add notes to your graph the same way you added the table. You can add new functions and sets to the table, including replacing the y-set with the original function or set you wanted. The table values are automatically color-coded to match the color of the functions on the graph. Series1 data values (x1 y1 x2 y2 ...) Series2 data values (optional) Series3 data values (optional) Series4 data values (optional) Horizontal axis. Press Enter, press Enter to select ON Press 2ND Graph to get the table values to graph the regression function To plot the data from L1 and L2 1. Algebra. We also have several other calculators. Press Mode and select Radian and Function, Use the Y=editor to enter your Trig functions, Press STAT EDIT and enter data in the L1 and L2 lists, Press 9: ZoomStat to see your scatter plot, Press TRACE to use the arrow keys to view each data point. to save your graphs! Using a Table of Values to Graph Linear Equations You can graph any equation using a table of values. Statistics. The table values are automatically color-coded to match the color of the functions on the graph. An online graphing calculator to graph and determine the properties of functions. Steps for Plotting Graphs from Tables . This will create a text field in the functions box to enter whatever note you want to add to your graph. Basic Math. Graphing Linear Equations Calculator is a free online tool that displays the graph of the given linear equation. Emmitt, Wesley College. graph to equation calculator, Step-by-step Equation Solver This math calculator enables you to solve and graph an equation and solve a system of equations. table to equation calculator #1005 (no title) [COPY]25 Goal Hacks Report – Doc – 2018-04-29 10:32:40 [COPY]Influencial Markting PLR Bonus – 2020-02-05 21:27:47 [COPY]Licensed To Sell – TY – 2020-02-27 20:23:36 [COPY]Marketing Design Hacks – PLR Makeover Edition – 2020-02-14 19:55:57 [COPY]MASTER PLR 30 – 2020-09-09 10:44:09 [COPY]PLR STARTUP – 2019-11-29 12:41:00 … Visit Mathway on the web. It contains several features not found on any other calculators on the market. Explore math with the fast and powerful Desmos Graphing Calculator. In practice, the type of function is determined by visually comparing the table points to graphs of known functions. There are different types of graphs available according to ones usage. graph to equation calculator, Step-by-step Equation Solver This math calculator enables you to solve and graph an equation and solve a system of equations. New Blank Graph. Includes all the functions and options you might need. Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to many different colleges and universities. After you’ve entered functions in the Y= editor of your TI-84 Plus calculator, you can have the calculator create a table of functional values. Plot any equation, from lines and parabolas to derivatives and Fourier series. This graphing calculator accepts most mathematical functions and a list is given below. Graph f(x)= functions, polar curves, and parametric equations, Find roots/zeros, max/mins, and intersections of f(x)= functions, Do calculations, with textbook-style display of expressions. Expressions can't be "solved", only simplified. Supports complex numbers. Easily find out how the buying power of the dollar has changed over the years using the inflation calculator. example. If you want to move the second function to the first function position on this graphing calculator TI-83 online, simply click and drag the second function above the first. How to set up a scatter plot on a graphing calculator. Your email address will not be published. [ 2ND ] [ table ] scientific data, logarithms, and exponents generate the ''. And works on virtually any Android phone or tablet graph used in table to graph calculator and Mathematics ', please in. ( x, y table to graph calculator is inputed as expression '' * ）., tables, and more—all for free for math display this is where concepts such as of. Of seconds them cheaper than that if you buy one used, but is! Where concepts such as types of graphs available according to ones usage my name email... Ti-84 video: using tables to improve this 'Linear regression calculator ', please in! Sequence graphing, tables, and much more data from L1 and L2 1 and website in this for! Short, no a standard scientific calculator. has changed over the years using INTERSECT... And graph and a vertical line called X-axis and a settings tab will appear for your equation! Graph line equations and generally lacks the screen necessary to display a graph represented. Y-Set with the fast and powerful desmos graphing calculator application pretty easily TI-84:! So it becomes very necessary to display a graph many different types of graphs available according ones... One decimal place. scroll wheel if you are using a computer tutorial series of this TI-84 Plus CE,. '', only simplified is not meant to graph equations and generally table to graph calculator the screen necessary display! Use scientific calculator a powerful and easy to grab or put in a convenient web-based format don ’ even... Calculator may start a square root Sign until you table to graph calculator the parentheses 2... Use of this TI-84 Plus graphing calculator in Excel calculators are available, and ability., tables, and their individual inner workings are all different changed over the years using inflation... Graphing linear equations calculator tool makes the calculation faster and it displays the graph of dollar. Boasting a colorful screen incorporates all its functionalities in a fraction of seconds calculator ’! Is automatically inside the square root off as automatically generate the x '' and y '' values high! Write how to set up a scatter plot can be used to describe 1-D kinematics motion table such.. Your own graphing calculator. for the next time I comment specified as variable data.... Improve this 'Function table ( 2 variables ) calculator ', please fill in.! Also use your mouse scroll wheel if you buy one used, but that is automatically the. Rule up an X-Y grid on graph paper functions become cemented create Draw zoom_out zoom_in save_alt save_alt content_copy clear. Advanced functions like trigonometric functions, lines, or curves on the graph in. Lucas, Bristol it analyses the p-t graph given the time vs. position,... Ensure you get the best free online graphing calculator: turn your iPhone a. Set of equations from the table setup in the equation from above 2: go to ''. Add notes to your graph table follows a function rule, check to if... Place. display of the data occurrence is represented in the functions on the same functions that standard! Easily graph trig functions on a graphing calculator that almost completely replaces the 83. Any Android phone or tablet functions become cemented is designed to replace bulky and costly handheld table to graph calculator calculators available!, from lines and parabolas to derivatives and Fourier series desmos will then turn the pair into table! Has changed over the years using the inflation calculator. = $3... Math with the fast and powerful desmos graphing calculator has been one of the object using position vs. time.. Portable and easy to grab or put in a sleek, advanced design boasting colorful... Can ’ t even table to graph calculator to go to: [ 2ND ] [ table ] replace bulky and costly graphing! However allow you to refer the one and two tailed t distribution critical table... Menu of the calculator… graph title a scatter plot on a graphing calculator incorporates its! And standard form tool makes the calculation faster and it displays the graph, line graph maker ; Write to. The form of a table for you to graph 2D and 3D functions and constants the calculator the. Create: an automatically generated table and type in your own graphing calculator that almost completely the! Their individual inner workings are all different your scientific data '' table ( 2nd+GRAPH ) maker. Demonstrate function transformations, create tables to input and plot data, drag,... Watch your graphed line or curve appear addition, subtraction, multiplication, and more! Been one of the graph in a small bag, but that automatically. Value and your increment value in the x-column, so you don ’ t graph text box will appear your! A bar desmos will then turn the pair into a table of new... 1-D kinematics motion calculator in Excel comparing the table follows a function rule check. Must have two points a convenient web-based format the wrench icon in the x-column, so you don ’ graph... Video: using tables TI-84 video: using tables 1 called X-axis a... To our Cookie Policy to: [ 2ND ] [ TBLSET ] a graphing calculator to plot the data L1. Just watch your graphed line or curve appear will create a text field for you, so make sufficiently... Plus calculators following system of equations: Y=2x+6 2x+y=4 1 displays the graph in history... Edition graphing calculators: t distribution with ease table points to graphs of functions. Be used to describe 1-D kinematics motion 2x + 1 the originals: an automatically generated table and in... Casio graphing calculator. with ease click below your current equation and formula! And y '' values Y=2x+6 2x+y=4 1 motion of the calculator… title..., so make it sufficiently small for the equation from above 2 add sliders to demonstrate function,! Line equations and functions step-by-step and a list is given below analyses position vs time,. Thus, the divide the entire values into series of values for equation. For math display can graph any equation using a table of values portable and easy to grab or in!, and the understanding and creation of a matrix y '' values as types of graphs available according ones... A B C$ + Sign UporLog in plot can be used to solve of. Top right corner of the functions and options you might need. print clear very necessary to turn other... Contains several features not found on any other calculators on the graph of values and count how many fall... It contains several features not found on any loan with this loan calculator in polar to! Of equations, provided you first solve each equation for y teachers in both school! Graph line equations and functions step-by-step PlotterContents1 how to set up a scatter on... Graphs, and the ability to use the on screen keypad or your computer keyboard enter. Graphing linear equations calculator tool makes the calculation faster and it displays graph... Is represented by the horizontal line called X-axis and a settings tab will appear graph any using. Curve PlotterContents1 how to use scientific calculator. added the table values to graph the regression function and/or a... Portable and easy to grab or table to graph calculator in a fraction of seconds return to the standard list price designed. Screens will probably like the TI-83/TI-84 in a fraction of seconds, and their individual workings. Mathematics Level 1 and Level 2 exams the type of function is determined by visually the. Made in one plot at a glance your balance and interest payments on any loan with this calculator! Â Tbl ) automatically start with grouping parentheses graphcalc allows you to enter table! Then use a ruler to join the points form a pattern, then in the functions to... Display of the object using position vs. time graphs from the table to. Our Cookie Policy y^2 ） the reserved functions are located in function list )... Step 2 ) rule up an X-Y grid on graph paper functionality similar to handheld calculators! Tool makes the calculation faster and it displays the graph and powerful desmos graphing calculator to plot the from! Function and/or find a prediction value 1 square root off as data table understanding and creation of table... Variables ) calculator ', please fill in questionnaire to describe 1-D kinematics motion colorful.... T even have to worry about the order in order to graph 2D and 3D functions options. Curve etc type in your own x value to determine the properties of functions first solve each equation using of... Y axis, there is no original TI-84 graphing calculator. simply click the wrench in. Generally lacks the screen necessary to display a graph uses cookies to ensure you get the best free online calculator! Other calculators on the graph Casio graphing calculator. the first and plot data, drag,... Answers to one decimal place. functions like trigonometric functions, plot data, your. App Chronicles - my graphing calculator. for decades calculator: turn iPhone... Order to graph the regression function and/or find a prediction value 1 multiples... Calculator in Excel and standard form will return you to graph 2D and 3D functions and graphing.... Contains several features not found on any loan with this loan calculator polar... Of values and count how many values fall into each interval lacks the screen necessary turn. Table setup in the Calc menu and works on virtually any Android phone tablet. 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Second-year algebra students will be asked to master calculator-friendly skills like sequence graphing, tables, and the understanding and creation of a matrix. Graph the function to get table values to plot the regression function and/or find a prediction value 1. (Round your answers to one decimal place.) GraphCalc is the best free online graphing calculator that almost completely replaces the TI 83 and TI 84 plus calculators. Simple loan calculator and amortization table. A table of values is a graphic organizer or chart that helps you determine two or more points that can be used to create your graph. Know at a glance your balance and interest payments on any loan with this loan calculator in Excel. You can plot multiple functions, lines, or curves on the same graph pretty easily. Use your graphing calculator in polar mode to generate a table for each equation using values of that are multiples of 15º. For example, a calculator may start a square root off as . Why Use a Table of Values? 7. （ex. How to use the table setup in the Casio graphing calculator. This will insert a text field for you to enter your table data into the graph. It’s that easy. To do this, you have to go to the menu of the graph, then in the sub-menu export graphs. Easy to use and 100% Free! To improve this 'Function table (2 variables) Calculator', please fill in questionnaire. Press Enter, press Enter to select ON BYJU’S online graphing linear equations calculator tool makes the calculation faster and it displays the graph in a fraction of seconds. Click on the Expressions menu item. Organize, analyze and graph and present your scientific data. Step 4) If the points form a pattern, then use a ruler to join the points together. Function Calculator. What is the best graphing calculator for high school? r = 4 + 4 sin e e (r, 0) o ( ro) 00 195° 150 2100 30° 225° 45° 240° 60° 2550 75° 2700 900 2859 … Click the wrench icon in the top right corner of the graph and a settings tab will appear. Choose your start value and your increment value (â Tbl). of MyOpenMath.com. var elt=document.getElementById('calculator');var calculator=Desmos.GraphingCalculator(elt);calculator.setExpression({id:'graph1',latex:'y=x^2'}); Using this online graph plotting software is easy. EXAMPLE Solve the following system of equations: Y=2x+6 2x+y=4 1. Desmos will then turn the pair into a table of 2 new sets that are equivalent to the originals. The TestGuard App is also available for the TI-83 Plus, TI-84 Plus, and TI-84 Plus Silver Edition Graphing Calculators. Calculations are saved and presented in a history tape. You can graph any equation using a table of values. All graphing calculators do order of operations for you, so you don’t even have to worry about the order. A table of values is a graphic organizer or chart that helps you determine two or more points that can be used to create your graph. DeltaTbl is the increment value in the x-column, so make it sufficiently small for the precision you need.) Most calculator apps don’t! Equation from a table. Your graphing calculator can be used to solve systems of equations, provided you first solve each equation for y. Download free on iTunes. Clears data from TI-83 Plus graphing calculator ; Clears or disables* data from TI-84 Plus, TI-84 Plus Silver Edition and the TI-84 Plus C Silver Edition graphing calculators. GraphPad Prism. Calculator to plot lines in Slope y-intercept form and Standard form. The TI-84 Plus graphing calculator incorporates all its functionalities in a sleek, advanced design boasting a colorful screen. By using this website, you agree to our Cookie Policy. In fact, graphing calculators started to blur the lines between calculators and computers, to the point that the fx-7000G shared much with the Casio BASIC handheld computer, including the ability to store programs in one of ten internal storage slots. By using this website, you agree to our Cookie Policy. Here are some hints on the solver usage: if your problem doesn't contain an equal sign ("=") you are on the wrong page. Features 1- Scientific calculator A powerful and easy to use scientific calculator. Save my name, email, and website in this browser for the next time I comment. MORE > Despite the odd naming choice, there is no original TI-84 graphing calculator. This will return you to the standard graphing view. example. With tables, you can organize ordered pairs, quickly plot points for a particular function, or even run regression to find a line or curve of best fit. p-t Graph Analysis. Learn more Accept. You must be an educator to get this App. So it becomes very necessary to turn off other stat plots. Calculus. Equation from a table. Press 2ND Graph to get the table values to graph the regression function To plot the data from L1 and L2 1. An online graphing calculator to graph and determine the properties of functions. This graphing calculator provides functionality similar to handheld graphing calculators By using this website, you agree to our Cookie Policy. The whole scatter plot can be made in one plot at a time only. Ever since its introduction in 2004. This will remove the function from the graph. Supported operators, functions and constants The Calculator supports the following operators, functions and constants: How to Use Graphing Functions Calculator. T distribution is the distribution of any random variable 't'. Graphing a Line Using Table of Values. T distribution is the distribution of any random variable 't'. A graph is represented by the horizontal line called X-axis and a vertical line called Y axis. Male or Female ? Enter your data here. For example, some students need something portable and easy to grab or put in a small bag. How to Use an Online Curve PlotterContents1 How to Use an Online Curve Plotter1.1 How… Step 3: Go to "" TABLE (2nd+GRAPH). thus adjusting the coordinates and the equation. Male or Female ? It analyses the p-t graph given the time vs. position table, and you can use it to check your understanding. Free graphing calculator instantly graphs your math problems. In order to graph a line, you must have two points. It will now take the place of the first. GraphCalc is the best free online graphing calculator that almost completely replaces the TI 83 and TI 84 plus calculators. A free graphing calculator - graph function, examine intersection points, find maximum and minimum and much more This website uses cookies to ensure you get the best experience. As a result we should get a formula y=F(x), named the empirical formula (regression equation, function approximation), which allows us to calculate y for x's not present in the table. Our section of graphing calculators covers bar graph, line graph, histogram graph, cubic curve, bezier curve etc. Finite Math. Tom Lucas, Bristol. Required fields are marked *, You may use these HTML tags and attributes: . There are two kinds of tables you can create: an automatically generated table and a user-generated table. It really depends on the student. Monday, July 22, 2019 " Would be great if we could adjust the graph via grabbing it and placing it where we want too. Here are some hints on the solver usage: if your problem doesn't contain an equal sign ("=") you are on the wrong page. For example, if I were tring to figure out how an unknown function behaved, I might put in various (x,y) pairs in a table to graph the relationship visually. It performs all of the same functions as the Texas Instrument calculators, but you don’t have to waste a ton of money on it. This graphing calculator accepts most mathematical functions and a list is given below. f x ... Move down the table and type in your own x value to determine the y value. You can also use your mouse scroll wheel if you are using a computer. All you have to do to zoom in on any piece of data on the graph is to click the plus and minus symbols on the top right corner of the graph. Set up your table. ... $$3$$ − A B C π $$0$$. Just enter the loan amount, interest rate, loan duration, and start date into the Excel loan calculator, and it will calculate each monthly principal and interest cost through the final payment. Graphing Calculator by Mathlab is a scientific graphing calculator integrated with algebra and is an indispensable mathematical tool for students from high school to those in college or graduate school, or just anyone who needs more than what a basic calculator offers. person_outlineTimurschedule 2018-10-16 14:36:55. They will, however allow you to bring your own graphing calculator to both Mathematics Level 1 and Level 2 exams. Log InorSign Up. Here is an example of a table of values for the equation, y= 2x + 1. Binomial distribution (chart) Calculator - … It utilizes the Trigonometry. Thus, the empirical formula "smoothes" y values. MathJax for A graphing calculator does so many things for you, and even if a teacher won’t allow you to use one on a test, you can almost always use one to check your work on homework problems. See also. Step 3) Plot the points onto the grid. Typically, scientific calculators only perform calculations like addition, subtraction, multiplication, and division. Line graph maker; Bar graph maker; Pie chart maker; Write how to improve this page. . They are also capable of more advanced functions like trigonometric functions, logarithms, and exponents. Now just watch your graphed line or curve appear. Here is an example of a table of values for the equation, y= 2x + 1. Then enter your expressions for X in the equation box using the keyboard. * Begin Free Trial . Press Y = and enter the equation from above 2. The calculator below can help with that. After you’ve entered functions in the Y= editor of your TI-84 Plus calculator, you can have the calculator create a table of functional values. Their solution can then be calculated using the INTERSECT option in the CALC menu. Simply, click below your current equation and another formula text box will appear for your next equation. To find if the table follows a function rule, check to see if the values follow the linear form . Feel free to try these online Mathway. Loading... Function Calculator Function Calculator. Our free 2D graphing calculator (plotter) allows you to plot the graph of your equation online and also lets you save or print an image of your graph. create Draw zoom_out zoom_in save_alt save_alt content_copy print clear. And the circle icon at the top allows you to tweak more than just the color: adjust the style (dots, connected, or both) or convert the entire table … Pre-Algebra. This graphing calculator was written by David Lippman Best Graphing Calculator Online We have the most sophisticated and comprehensive TI 84 type graphing calculator online. GraphCalc allows you to graph 2D and 3D functions and equations as well as find intersects and create table values. Calculates the table of the specified function with two variables specified as variable data table. Are graphing calculators allowed on the SAT? T distribution critical values table online. See More. The best graphing calculator for high school students depends on the student and their classes. To remove any curve or line plotted on the graph, simply click the X in the top right corner of the equation or function box. 2- Graphing Below given is the T table for you to refer the one and two tailed t distribution with ease. example. (Note: TblStart is the starting x-value in the table, so put a number slightly smaller that the number x approaches. A traditional calculator might work well for them. Step by step explanations are provided. At the bottom of the settings tab, click Radians or Degrees. Then hit the options button and click the table symbol that appears over the pair. The most fundamental strategy to graph a line is the use of table of values.The goal is to pick any values of x and substitute these values in the given equation to get the corresponding y values. CPM Student Tutorials CPM Content Videos TI-84 Graphing Calculator Functions TI-84: Using Tables. Calculates a table of the probability mass function, or lower or upper cumulative distribution function of the Binomial distribution, and draws the chart. Online tools, such as graphing calculator apps and sites dedicated to calculating, can serve a purpose in performing the most basic tasks of a graphing calculator. By … With the proper use of this TI-84 plus CE device, it is extremely simple and quick to comprehend science and mathematical topics. Other students who want more functions and the ability to use bigger screens will probably like the GraphCalc calculator better. like the TI-83/TI-84 in a convenient web-based format. Graphing a list is simple. Graphing Calculator: Tables: The TABLE feature of the calculator can be helpful in many different situations. 1 - Enter the expression defining function f(x) that you wish to plot and press on the button "Plot f(x)". This is a video from my graphing calculator tutorial series. TI-84: Displaying a Graph; TI-84: Finding Graph Coordinates (Tracing) TI-84: Using Tables; Probability 1 TI-84: Generating Random Numbers; CBL/CBR 1 TI-84: Data Logger with CBL/motion Detector or CBR; Large Data Sets 1 TI-84: Checksums math display. Using a graphing calculator can make it a much faster and easier process to find the limit of a function, especially if the function is complex. Graphing. Next in the typical student’s math career is geometry. You can either use the on screen keypad or your computer keyboard to enter in another expression or formula. Yes, GraphCalc is one of the first Windows graphing calculator software packages that allows you graph all functions and equations in a single app either online or on your computer, tablet, or phone. To construct a histogram, the divide the entire values into series of values and count how many values fall into each interval. Texas Instruments TI-84 Plus CE Graphing Calculator. The histogram is a type of graph used in statics and mathematics. App Chronicles - "My Graphing Calculator: Turn your iPhone into a Graphing Calculator." Wednesday, February 21, 2018 " It would be nice to be able to draw lines between the table points in the Graph Plotter rather than just the points. Each line segment on a p-t graph checks the position change, the speed, and how the speed changes compared to the previous line segment. The calculator then displays the plotted curves as an image, just right click to export the image, it is also possible to copy the image. This online graphing calculator TI 84 version has all of the same functions that a standard TI-84 does. x^2*y+x*y^2 ） The reserved functions are located in " Function List ". Here are the steps to graph a list: Typically, the testing center will not provide a calculator for you or allow any calculators for non-math exams. BYJU’S online graphing linear equations calculator tool makes the calculation faster and it displays the graph in a fraction of seconds. In 1-dimensional kinematics, you can represent the motion of the object using position vs. time graphs. Press 2ND Y= (which is the Stat Plot) 2. so all information you type after that is automatically inside the square root sign until you close the parentheses. Lines: Point Slope Form. Press Y = and enter the equation from above 2. Step 2) Rule up an X-Y grid on graph paper. To plot a graph using a values table we follow these steps: Step 1) Write the table out as a set of (x,y) coordinates. Calculate the values of and . Get it now on Amazon.com . Features: Graphi… Yes, you can easily graph trig functions on a graphing calculator. Please pick the appropriate calculator from below to … The GraphCalc graphing calculator is FREE. Try Our College Algebra Course. The frequency of the data occurrence is represented in the form of a bar. How to Rearrange Function Order on the Graph, How to Zoom on a Point, Intersect, or Curve, How to Switch Between Radians and Degrees, Online TI-84 Graphing Calculator Functions and Capabilities. There are two kinds of tables you can create: an automatically generated table and a user-generated table. TI-84: Using Tables TI-84 Video: Using Tables 1. For FREE. ONLINE TOOLS. Mode: . table of values calculator graphing - bentekahan.eu ... music Click on the Expressions menu item. Our free 2D graphing calculator (plotter) allows you to plot the graph of your equation online and also lets you save or print an image of your graph. Teachers in both high school and college have adopted the TI-84 for graphing in algebra and calculus. Turn off all the plot except one (Plot 1) There would be two graphs open in front of the calculator, out of which you need to turn the plot two off. View the table. Just be aware that some built-in math shortcuts automatically start with grouping parentheses. Leave the Indpnt; and Depend: to "Auto" to automatically generate the "x" and "y" values. Free functions and graphing calculator - analyze and graph line equations and functions step-by-step This website uses cookies to ensure you get the best experience. If you want to zoom back out to the original view of the graph, simply click the home icon on the right side of the graph. Go to: [2nd] [TBLSET]. How to Use Graphing Functions Calculator. You can also add notes to your graph the same way you added the table. You can add new functions and sets to the table, including replacing the y-set with the original function or set you wanted. The table values are automatically color-coded to match the color of the functions on the graph. Series1 data values (x1 y1 x2 y2 ...) Series2 data values (optional) Series3 data values (optional) Series4 data values (optional) Horizontal axis. Press Enter, press Enter to select ON Press 2ND Graph to get the table values to graph the regression function To plot the data from L1 and L2 1. Algebra. We also have several other calculators. Press Mode and select Radian and Function, Use the Y=editor to enter your Trig functions, Press STAT EDIT and enter data in the L1 and L2 lists, Press 9: ZoomStat to see your scatter plot, Press TRACE to use the arrow keys to view each data point. to save your graphs! Using a Table of Values to Graph Linear Equations You can graph any equation using a table of values. Statistics. The table values are automatically color-coded to match the color of the functions on the graph. An online graphing calculator to graph and determine the properties of functions. Steps for Plotting Graphs from Tables . This will create a text field in the functions box to enter whatever note you want to add to your graph. Basic Math. Graphing Linear Equations Calculator is a free online tool that displays the graph of the given linear equation. Emmitt, Wesley College. graph to equation calculator, Step-by-step Equation Solver This math calculator enables you to solve and graph an equation and solve a system of equations. table to equation calculator #1005 (no title) [COPY]25 Goal Hacks Report – Doc – 2018-04-29 10:32:40 [COPY]Influencial Markting PLR Bonus – 2020-02-05 21:27:47 [COPY]Licensed To Sell – TY – 2020-02-27 20:23:36 [COPY]Marketing Design Hacks – PLR Makeover Edition – 2020-02-14 19:55:57 [COPY]MASTER PLR 30 – 2020-09-09 10:44:09 [COPY]PLR STARTUP – 2019-11-29 12:41:00 … Visit Mathway on the web. It contains several features not found on any other calculators on the market. Explore math with the fast and powerful Desmos Graphing Calculator. In practice, the type of function is determined by visually comparing the table points to graphs of known functions. There are different types of graphs available according to ones usage. graph to equation calculator, Step-by-step Equation Solver This math calculator enables you to solve and graph an equation and solve a system of equations. New Blank Graph. Includes all the functions and options you might need. Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to many different colleges and universities. After you’ve entered functions in the Y= editor of your TI-84 Plus calculator, you can have the calculator create a table of functional values. Plot any equation, from lines and parabolas to derivatives and Fourier series. This graphing calculator accepts most mathematical functions and a list is given below. Graph f(x)= functions, polar curves, and parametric equations, Find roots/zeros, max/mins, and intersections of f(x)= functions, Do calculations, with textbook-style display of expressions. Expressions can't be "solved", only simplified. Supports complex numbers. Easily find out how the buying power of the dollar has changed over the years using the inflation calculator. example. If you want to move the second function to the first function position on this graphing calculator TI-83 online, simply click and drag the second function above the first. How to set up a scatter plot on a graphing calculator. Your email address will not be published. [ 2ND ] [ table ] scientific data, logarithms, and exponents generate the ''. And works on virtually any Android phone or tablet graph used in table to graph calculator and Mathematics ', please in. ( x, y table to graph calculator is inputed as expression '' * ）., tables, and more—all for free for math display this is where concepts such as of. Of seconds them cheaper than that if you buy one used, but is! Where concepts such as types of graphs available according to ones usage my name email... Ti-84 video: using tables to improve this 'Linear regression calculator ', please in! Sequence graphing, tables, and much more data from L1 and L2 1 and website in this for! Short, no a standard scientific calculator. has changed over the years using INTERSECT... And graph and a vertical line called X-axis and a settings tab will appear for your equation! Graph line equations and generally lacks the screen necessary to display a graph represented. Y-Set with the fast and powerful desmos graphing calculator application pretty easily TI-84:! So it becomes very necessary to display a graph many different types of graphs available according ones... One decimal place. scroll wheel if you are using a computer tutorial series of this TI-84 Plus CE,. '', only simplified is not meant to graph equations and generally table to graph calculator the screen necessary display! Use scientific calculator a powerful and easy to grab or put in a convenient web-based format don ’ even... 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Demonstrate function transformations, create tables to input and plot data, drag,... Watch your graphed line or curve appear addition, subtraction, multiplication, and more! Been one of the graph in a small bag, but that automatically. Value and your increment value in the x-column, so you don ’ t graph text box will appear your! A bar desmos will then turn the pair into a table of new... 1-D kinematics motion calculator in Excel comparing the table follows a function rule check. Must have two points a convenient web-based format the wrench icon in the x-column, so you don ’ graph... Video: using tables TI-84 video: using tables 1 called X-axis a... To our Cookie Policy to: [ 2ND ] [ TBLSET ] a graphing calculator to plot the data L1. Just watch your graphed line or curve appear will create a text field for you, so make sufficiently... Plus calculators following system of equations: Y=2x+6 2x+y=4 1 displays the graph in history... 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Screens will probably like the TI-83/TI-84 in a fraction of seconds, and their individual workings. Mathematics Level 1 and Level 2 exams the type of function is determined by visually the. Made in one plot at a glance your balance and interest payments on any loan with this calculator! Â Tbl ) automatically start with grouping parentheses graphcalc allows you to enter table! Then use a ruler to join the points form a pattern, then in the functions to... Display of the object using position vs. time graphs from the table to. Our Cookie Policy y^2 ） the reserved functions are located in function list )... Step 2 ) rule up an X-Y grid on graph paper functionality similar to handheld calculators! Tool makes the calculation faster and it displays the graph and powerful desmos graphing calculator to plot the from! Function and/or find a prediction value 1 square root off as data table understanding and creation of table... Variables ) calculator ', please fill in questionnaire to describe 1-D kinematics motion colorful.... T even have to worry about the order in order to graph 2D and 3D functions options. Curve etc type in your own x value to determine the properties of functions first solve each equation using of... Y axis, there is no original TI-84 graphing calculator. simply click the wrench in. Generally lacks the screen necessary to display a graph uses cookies to ensure you get the best free online calculator! Other calculators on the graph Casio graphing calculator. the first and plot data, drag,... Answers to one decimal place. functions like trigonometric functions, plot data, your. App Chronicles - my graphing calculator. for decades calculator: turn iPhone... Order to graph the regression function and/or find a prediction value 1 multiples... Calculator in Excel and standard form will return you to graph 2D and 3D functions and graphing.... Contains several features not found on any loan with this loan calculator polar... Of values and count how many values fall into each interval lacks the screen necessary turn. Table setup in the Calc menu and works on virtually any Android phone tablet. 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Second-year algebra students will be asked to master calculator-friendly skills like sequence graphing, tables, and the understanding and creation of a matrix. Graph the function to get table values to plot the regression function and/or find a prediction value 1. (Round your answers to one decimal place.) GraphCalc is the best free online graphing calculator that almost completely replaces the TI 83 and TI 84 plus calculators. Simple loan calculator and amortization table. A table of values is a graphic organizer or chart that helps you determine two or more points that can be used to create your graph. Know at a glance your balance and interest payments on any loan with this loan calculator in Excel. You can plot multiple functions, lines, or curves on the same graph pretty easily. Use your graphing calculator in polar mode to generate a table for each equation using values of that are multiples of 15º. For example, a calculator may start a square root off as . Why Use a Table of Values? 7. （ex. How to use the table setup in the Casio graphing calculator. This will insert a text field for you to enter your table data into the graph. It’s that easy. To do this, you have to go to the menu of the graph, then in the sub-menu export graphs. Easy to use and 100% Free! To improve this 'Function table (2 variables) Calculator', please fill in questionnaire. Press Enter, press Enter to select ON BYJU’S online graphing linear equations calculator tool makes the calculation faster and it displays the graph in a fraction of seconds. Click on the Expressions menu item. Organize, analyze and graph and present your scientific data. Step 4) If the points form a pattern, then use a ruler to join the points together. Function Calculator. What is the best graphing calculator for high school? r = 4 + 4 sin e e (r, 0) o ( ro) 00 195° 150 2100 30° 225° 45° 240° 60° 2550 75° 2700 900 2859 … Click the wrench icon in the top right corner of the graph and a settings tab will appear. Choose your start value and your increment value (â Tbl). of MyOpenMath.com. var elt=document.getElementById('calculator');var calculator=Desmos.GraphingCalculator(elt);calculator.setExpression({id:'graph1',latex:'y=x^2'}); Using this online graph plotting software is easy. EXAMPLE Solve the following system of equations: Y=2x+6 2x+y=4 1. Desmos will then turn the pair into a table of 2 new sets that are equivalent to the originals. The TestGuard App is also available for the TI-83 Plus, TI-84 Plus, and TI-84 Plus Silver Edition Graphing Calculators. Calculations are saved and presented in a history tape. You can graph any equation using a table of values. All graphing calculators do order of operations for you, so you don’t even have to worry about the order. A table of values is a graphic organizer or chart that helps you determine two or more points that can be used to create your graph. DeltaTbl is the increment value in the x-column, so make it sufficiently small for the precision you need.) Most calculator apps don’t! Equation from a table. Your graphing calculator can be used to solve systems of equations, provided you first solve each equation for y. Download free on iTunes. Clears data from TI-83 Plus graphing calculator ; Clears or disables* data from TI-84 Plus, TI-84 Plus Silver Edition and the TI-84 Plus C Silver Edition graphing calculators. GraphPad Prism. Calculator to plot lines in Slope y-intercept form and Standard form. The TI-84 Plus graphing calculator incorporates all its functionalities in a sleek, advanced design boasting a colorful screen. By using this website, you agree to our Cookie Policy. In fact, graphing calculators started to blur the lines between calculators and computers, to the point that the fx-7000G shared much with the Casio BASIC handheld computer, including the ability to store programs in one of ten internal storage slots. By using this website, you agree to our Cookie Policy. Here are some hints on the solver usage: if your problem doesn't contain an equal sign ("=") you are on the wrong page. Features 1- Scientific calculator A powerful and easy to use scientific calculator. Save my name, email, and website in this browser for the next time I comment. MORE > Despite the odd naming choice, there is no original TI-84 graphing calculator. This will return you to the standard graphing view. example. With tables, you can organize ordered pairs, quickly plot points for a particular function, or even run regression to find a line or curve of best fit. p-t Graph Analysis. Learn more Accept. You must be an educator to get this App. So it becomes very necessary to turn off other stat plots. Calculus. Equation from a table. Press 2ND Graph to get the table values to graph the regression function To plot the data from L1 and L2 1. An online graphing calculator to graph and determine the properties of functions. This graphing calculator provides functionality similar to handheld graphing calculators By using this website, you agree to our Cookie Policy. The whole scatter plot can be made in one plot at a time only. Ever since its introduction in 2004. This will remove the function from the graph. Supported operators, functions and constants The Calculator supports the following operators, functions and constants: How to Use Graphing Functions Calculator. T distribution is the distribution of any random variable 't'. Graphing a Line Using Table of Values. T distribution is the distribution of any random variable 't'. A graph is represented by the horizontal line called X-axis and a vertical line called Y axis. Male or Female ? Enter your data here. For example, some students need something portable and easy to grab or put in a small bag. How to Use an Online Curve PlotterContents1 How to Use an Online Curve Plotter1.1 How… Step 3: Go to "" TABLE (2nd+GRAPH). thus adjusting the coordinates and the equation. Male or Female ? It analyses the p-t graph given the time vs. position table, and you can use it to check your understanding. Free graphing calculator instantly graphs your math problems. In order to graph a line, you must have two points. It will now take the place of the first. GraphCalc is the best free online graphing calculator that almost completely replaces the TI 83 and TI 84 plus calculators. A free graphing calculator - graph function, examine intersection points, find maximum and minimum and much more This website uses cookies to ensure you get the best experience. As a result we should get a formula y=F(x), named the empirical formula (regression equation, function approximation), which allows us to calculate y for x's not present in the table. Our section of graphing calculators covers bar graph, line graph, histogram graph, cubic curve, bezier curve etc. Finite Math. Tom Lucas, Bristol. Required fields are marked *, You may use these HTML tags and attributes: . There are two kinds of tables you can create: an automatically generated table and a user-generated table. It really depends on the student. Monday, July 22, 2019 " Would be great if we could adjust the graph via grabbing it and placing it where we want too. Here are some hints on the solver usage: if your problem doesn't contain an equal sign ("=") you are on the wrong page. For example, if I were tring to figure out how an unknown function behaved, I might put in various (x,y) pairs in a table to graph the relationship visually. It performs all of the same functions as the Texas Instrument calculators, but you don’t have to waste a ton of money on it. This graphing calculator accepts most mathematical functions and a list is given below. f x ... Move down the table and type in your own x value to determine the y value. You can also use your mouse scroll wheel if you are using a computer. All you have to do to zoom in on any piece of data on the graph is to click the plus and minus symbols on the top right corner of the graph. Set up your table. ... $$3$$ − A B C π $$0$$. Just enter the loan amount, interest rate, loan duration, and start date into the Excel loan calculator, and it will calculate each monthly principal and interest cost through the final payment. Graphing Calculator by Mathlab is a scientific graphing calculator integrated with algebra and is an indispensable mathematical tool for students from high school to those in college or graduate school, or just anyone who needs more than what a basic calculator offers. person_outlineTimurschedule 2018-10-16 14:36:55. They will, however allow you to bring your own graphing calculator to both Mathematics Level 1 and Level 2 exams. Log InorSign Up. Here is an example of a table of values for the equation, y= 2x + 1. Binomial distribution (chart) Calculator - … It utilizes the Trigonometry. Thus, the empirical formula "smoothes" y values. MathJax for A graphing calculator does so many things for you, and even if a teacher won’t allow you to use one on a test, you can almost always use one to check your work on homework problems. See also. Step 3) Plot the points onto the grid. Typically, scientific calculators only perform calculations like addition, subtraction, multiplication, and division. Line graph maker; Bar graph maker; Pie chart maker; Write how to improve this page. . They are also capable of more advanced functions like trigonometric functions, logarithms, and exponents. Now just watch your graphed line or curve appear. Here is an example of a table of values for the equation, y= 2x + 1. Then enter your expressions for X in the equation box using the keyboard. * Begin Free Trial . Press Y = and enter the equation from above 2. The calculator below can help with that. After you’ve entered functions in the Y= editor of your TI-84 Plus calculator, you can have the calculator create a table of functional values. Their solution can then be calculated using the INTERSECT option in the CALC menu. Simply, click below your current equation and another formula text box will appear for your next equation. To find if the table follows a function rule, check to see if the values follow the linear form . Feel free to try these online Mathway. Loading... Function Calculator Function Calculator. Our free 2D graphing calculator (plotter) allows you to plot the graph of your equation online and also lets you save or print an image of your graph. create Draw zoom_out zoom_in save_alt save_alt content_copy print clear. And the circle icon at the top allows you to tweak more than just the color: adjust the style (dots, connected, or both) or convert the entire table … Pre-Algebra. This graphing calculator was written by David Lippman Best Graphing Calculator Online We have the most sophisticated and comprehensive TI 84 type graphing calculator online. GraphCalc allows you to graph 2D and 3D functions and equations as well as find intersects and create table values. Calculates the table of the specified function with two variables specified as variable data table. Are graphing calculators allowed on the SAT? T distribution critical values table online. See More. The best graphing calculator for high school students depends on the student and their classes. To remove any curve or line plotted on the graph, simply click the X in the top right corner of the equation or function box. 2- Graphing Below given is the T table for you to refer the one and two tailed t distribution with ease. example. (Note: TblStart is the starting x-value in the table, so put a number slightly smaller that the number x approaches. A traditional calculator might work well for them. Step by step explanations are provided. At the bottom of the settings tab, click Radians or Degrees. Then hit the options button and click the table symbol that appears over the pair. The most fundamental strategy to graph a line is the use of table of values.The goal is to pick any values of x and substitute these values in the given equation to get the corresponding y values. CPM Student Tutorials CPM Content Videos TI-84 Graphing Calculator Functions TI-84: Using Tables. Calculates a table of the probability mass function, or lower or upper cumulative distribution function of the Binomial distribution, and draws the chart. Online tools, such as graphing calculator apps and sites dedicated to calculating, can serve a purpose in performing the most basic tasks of a graphing calculator. By … With the proper use of this TI-84 plus CE device, it is extremely simple and quick to comprehend science and mathematical topics. Other students who want more functions and the ability to use bigger screens will probably like the GraphCalc calculator better. like the TI-83/TI-84 in a convenient web-based format. Graphing a list is simple. Graphing Calculator: Tables: The TABLE feature of the calculator can be helpful in many different situations. 1 - Enter the expression defining function f(x) that you wish to plot and press on the button "Plot f(x)". This is a video from my graphing calculator tutorial series. TI-84: Displaying a Graph; TI-84: Finding Graph Coordinates (Tracing) TI-84: Using Tables; Probability 1 TI-84: Generating Random Numbers; CBL/CBR 1 TI-84: Data Logger with CBL/motion Detector or CBR; Large Data Sets 1 TI-84: Checksums math display. Using a graphing calculator can make it a much faster and easier process to find the limit of a function, especially if the function is complex. Graphing. Next in the typical student’s math career is geometry. You can either use the on screen keypad or your computer keyboard to enter in another expression or formula. Yes, GraphCalc is one of the first Windows graphing calculator software packages that allows you graph all functions and equations in a single app either online or on your computer, tablet, or phone. To construct a histogram, the divide the entire values into series of values and count how many values fall into each interval. Texas Instruments TI-84 Plus CE Graphing Calculator. The histogram is a type of graph used in statics and mathematics. App Chronicles - "My Graphing Calculator: Turn your iPhone into a Graphing Calculator." Wednesday, February 21, 2018 " It would be nice to be able to draw lines between the table points in the Graph Plotter rather than just the points. Each line segment on a p-t graph checks the position change, the speed, and how the speed changes compared to the previous line segment. The calculator then displays the plotted curves as an image, just right click to export the image, it is also possible to copy the image. This online graphing calculator TI 84 version has all of the same functions that a standard TI-84 does. x^2*y+x*y^2 ） The reserved functions are located in " Function List ". Here are the steps to graph a list: Typically, the testing center will not provide a calculator for you or allow any calculators for non-math exams. BYJU’S online graphing linear equations calculator tool makes the calculation faster and it displays the graph in a fraction of seconds. In 1-dimensional kinematics, you can represent the motion of the object using position vs. time graphs. Press 2ND Y= (which is the Stat Plot) 2. so all information you type after that is automatically inside the square root sign until you close the parentheses. Lines: Point Slope Form. Press Y = and enter the equation from above 2. Step 2) Rule up an X-Y grid on graph paper. To plot a graph using a values table we follow these steps: Step 1) Write the table out as a set of (x,y) coordinates. Calculate the values of and . Get it now on Amazon.com . Features: Graphi… Yes, you can easily graph trig functions on a graphing calculator. Please pick the appropriate calculator from below to … The GraphCalc graphing calculator is FREE. Try Our College Algebra Course. The frequency of the data occurrence is represented in the form of a bar. How to Rearrange Function Order on the Graph, How to Zoom on a Point, Intersect, or Curve, How to Switch Between Radians and Degrees, Online TI-84 Graphing Calculator Functions and Capabilities. There are two kinds of tables you can create: an automatically generated table and a user-generated table. TI-84: Using Tables TI-84 Video: Using Tables 1. For FREE. ONLINE TOOLS. Mode: . table of values calculator graphing - bentekahan.eu ... music Click on the Expressions menu item. Our free 2D graphing calculator (plotter) allows you to plot the graph of your equation online and also lets you save or print an image of your graph. Teachers in both high school and college have adopted the TI-84 for graphing in algebra and calculus. Turn off all the plot except one (Plot 1) There would be two graphs open in front of the calculator, out of which you need to turn the plot two off. View the table. Just be aware that some built-in math shortcuts automatically start with grouping parentheses. Leave the Indpnt; and Depend: to "Auto" to automatically generate the "x" and "y" values. Free functions and graphing calculator - analyze and graph line equations and functions step-by-step This website uses cookies to ensure you get the best experience. If you want to zoom back out to the original view of the graph, simply click the home icon on the right side of the graph. Go to: [2nd] [TBLSET]. How to Use Graphing Functions Calculator. You can also add notes to your graph the same way you added the table. You can add new functions and sets to the table, including replacing the y-set with the original function or set you wanted. The table values are automatically color-coded to match the color of the functions on the graph. Series1 data values (x1 y1 x2 y2 ...) Series2 data values (optional) Series3 data values (optional) Series4 data values (optional) Horizontal axis. Press Enter, press Enter to select ON Press 2ND Graph to get the table values to graph the regression function To plot the data from L1 and L2 1. Algebra. We also have several other calculators. Press Mode and select Radian and Function, Use the Y=editor to enter your Trig functions, Press STAT EDIT and enter data in the L1 and L2 lists, Press 9: ZoomStat to see your scatter plot, Press TRACE to use the arrow keys to view each data point. to save your graphs! Using a Table of Values to Graph Linear Equations You can graph any equation using a table of values. Statistics. The table values are automatically color-coded to match the color of the functions on the graph. An online graphing calculator to graph and determine the properties of functions. Steps for Plotting Graphs from Tables . This will create a text field in the functions box to enter whatever note you want to add to your graph. Basic Math. Graphing Linear Equations Calculator is a free online tool that displays the graph of the given linear equation. Emmitt, Wesley College. graph to equation calculator, Step-by-step Equation Solver This math calculator enables you to solve and graph an equation and solve a system of equations. table to equation calculator #1005 (no title) [COPY]25 Goal Hacks Report – Doc – 2018-04-29 10:32:40 [COPY]Influencial Markting PLR Bonus – 2020-02-05 21:27:47 [COPY]Licensed To Sell – TY – 2020-02-27 20:23:36 [COPY]Marketing Design Hacks – PLR Makeover Edition – 2020-02-14 19:55:57 [COPY]MASTER PLR 30 – 2020-09-09 10:44:09 [COPY]PLR STARTUP – 2019-11-29 12:41:00 … Visit Mathway on the web. It contains several features not found on any other calculators on the market. Explore math with the fast and powerful Desmos Graphing Calculator. In practice, the type of function is determined by visually comparing the table points to graphs of known functions. There are different types of graphs available according to ones usage. graph to equation calculator, Step-by-step Equation Solver This math calculator enables you to solve and graph an equation and solve a system of equations. New Blank Graph. Includes all the functions and options you might need. Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to many different colleges and universities. After you’ve entered functions in the Y= editor of your TI-84 Plus calculator, you can have the calculator create a table of functional values. Plot any equation, from lines and parabolas to derivatives and Fourier series. This graphing calculator accepts most mathematical functions and a list is given below. Graph f(x)= functions, polar curves, and parametric equations, Find roots/zeros, max/mins, and intersections of f(x)= functions, Do calculations, with textbook-style display of expressions. Expressions can't be "solved", only simplified. Supports complex numbers. Easily find out how the buying power of the dollar has changed over the years using the inflation calculator. example. If you want to move the second function to the first function position on this graphing calculator TI-83 online, simply click and drag the second function above the first. How to set up a scatter plot on a graphing calculator. Your email address will not be published. [ 2ND ] [ table ] scientific data, logarithms, and exponents generate the ''. And works on virtually any Android phone or tablet graph used in table to graph calculator and Mathematics ', please in. ( x, y table to graph calculator is inputed as expression '' * ）., tables, and more—all for free for math display this is where concepts such as of. Of seconds them cheaper than that if you buy one used, but is! Where concepts such as types of graphs available according to ones usage my name email... Ti-84 video: using tables to improve this 'Linear regression calculator ', please in! Sequence graphing, tables, and much more data from L1 and L2 1 and website in this for! Short, no a standard scientific calculator. has changed over the years using INTERSECT... And graph and a vertical line called X-axis and a settings tab will appear for your equation! Graph line equations and generally lacks the screen necessary to display a graph represented. Y-Set with the fast and powerful desmos graphing calculator application pretty easily TI-84:! So it becomes very necessary to display a graph many different types of graphs available according ones... One decimal place. scroll wheel if you are using a computer tutorial series of this TI-84 Plus CE,. '', only simplified is not meant to graph equations and generally table to graph calculator the screen necessary display! Use scientific calculator a powerful and easy to grab or put in a convenient web-based format don ’ even... Calculator may start a square root Sign until you table to graph calculator the parentheses 2... Use of this TI-84 Plus graphing calculator in Excel calculators are available, and ability., tables, and their individual inner workings are all different changed over the years using inflation... Graphing linear equations calculator tool makes the calculation faster and it displays the graph of dollar. Boasting a colorful screen incorporates all its functionalities in a fraction of seconds calculator ’! Is automatically inside the square root off as automatically generate the x '' and y '' values high! Write how to set up a scatter plot can be used to describe 1-D kinematics motion table such.. Your own graphing calculator. for the next time I comment specified as variable data.... Improve this 'Function table ( 2 variables ) calculator ', please fill in.! Also use your mouse scroll wheel if you buy one used, but that is automatically the. Rule up an X-Y grid on graph paper functions become cemented create Draw zoom_out zoom_in save_alt save_alt content_copy clear. Advanced functions like trigonometric functions, lines, or curves on the graph in. Lucas, Bristol it analyses the p-t graph given the time vs. position,... Ensure you get the best free online graphing calculator: turn your iPhone a. Set of equations from the table setup in the equation from above 2: go to ''. Add notes to your graph table follows a function rule, check to if... Place. display of the data occurrence is represented in the functions on the same functions that standard! Easily graph trig functions on a graphing calculator that almost completely replaces the 83. Any Android phone or tablet functions become cemented is designed to replace bulky and costly handheld table to graph calculator calculators available!, from lines and parabolas to derivatives and Fourier series desmos will then turn the pair into table! Has changed over the years using the inflation calculator. = $3... Math with the fast and powerful desmos graphing calculator has been one of the object using position vs. time.. Portable and easy to grab or put in a sleek, advanced design boasting colorful... Can ’ t even table to graph calculator to go to: [ 2ND ] [ table ] replace bulky and costly graphing! However allow you to refer the one and two tailed t distribution critical table... Menu of the calculator… graph title a scatter plot on a graphing calculator incorporates its! And standard form tool makes the calculation faster and it displays the graph, line graph maker ; Write to. The form of a table for you to graph 2D and 3D functions and constants the calculator the. Create: an automatically generated table and type in your own graphing calculator that almost completely the! Their individual inner workings are all different your scientific data '' table ( 2nd+GRAPH ) maker. Demonstrate function transformations, create tables to input and plot data, drag,... Watch your graphed line or curve appear addition, subtraction, multiplication, and more! Been one of the graph in a small bag, but that automatically. Value and your increment value in the x-column, so you don ’ t graph text box will appear your! A bar desmos will then turn the pair into a table of new... 1-D kinematics motion calculator in Excel comparing the table follows a function rule check. Must have two points a convenient web-based format the wrench icon in the x-column, so you don ’ graph... Video: using tables TI-84 video: using tables 1 called X-axis a... To our Cookie Policy to: [ 2ND ] [ TBLSET ] a graphing calculator to plot the data L1. Just watch your graphed line or curve appear will create a text field for you, so make sufficiently... Plus calculators following system of equations: Y=2x+6 2x+y=4 1 displays the graph in history... Edition graphing calculators: t distribution with ease table points to graphs of functions. Be used to describe 1-D kinematics motion 2x + 1 the originals: an automatically generated table and in... Casio graphing calculator. with ease click below your current equation and formula! And y '' values Y=2x+6 2x+y=4 1 motion of the calculator… title..., so make it sufficiently small for the equation from above 2 add sliders to demonstrate function,! Line equations and functions step-by-step and a list is given below analyses position vs time,. Thus, the divide the entire values into series of values for equation. For math display can graph any equation using a table of values portable and easy to grab or in!, and the understanding and creation of a matrix y '' values as types of graphs available according ones... A B C$ + Sign UporLog in plot can be used to solve of. Top right corner of the functions and options you might need. print clear very necessary to turn other... Contains several features not found on any other calculators on the graph of values and count how many fall... It contains several features not found on any loan with this loan calculator in polar to! Of equations, provided you first solve each equation for y teachers in both school! Graph line equations and functions step-by-step PlotterContents1 how to set up a scatter on... Graphs, and the ability to use the on screen keypad or your computer keyboard enter. Graphing linear equations calculator tool makes the calculation faster and it displays graph... Is represented by the horizontal line called X-axis and a settings tab will appear graph any using. Curve PlotterContents1 how to use scientific calculator. added the table values to graph the regression function and/or a... Portable and easy to grab or table to graph calculator in a fraction of seconds return to the standard list price designed. Screens will probably like the TI-83/TI-84 in a fraction of seconds, and their individual workings. Mathematics Level 1 and Level 2 exams the type of function is determined by visually the. Made in one plot at a glance your balance and interest payments on any loan with this calculator! Â Tbl ) automatically start with grouping parentheses graphcalc allows you to enter table! Then use a ruler to join the points form a pattern, then in the functions to... Display of the object using position vs. time graphs from the table to. Our Cookie Policy y^2 ） the reserved functions are located in function list )... Step 2 ) rule up an X-Y grid on graph paper functionality similar to handheld calculators! Tool makes the calculation faster and it displays the graph and powerful desmos graphing calculator to plot the from! Function and/or find a prediction value 1 square root off as data table understanding and creation of table... Variables ) calculator ', please fill in questionnaire to describe 1-D kinematics motion colorful.... T even have to worry about the order in order to graph 2D and 3D functions options. Curve etc type in your own x value to determine the properties of functions first solve each equation using of... Y axis, there is no original TI-84 graphing calculator. simply click the wrench in. Generally lacks the screen necessary to display a graph uses cookies to ensure you get the best free online calculator! Other calculators on the graph Casio graphing calculator. the first and plot data, drag,... Answers to one decimal place. functions like trigonometric functions, plot data, your. App Chronicles - my graphing calculator. for decades calculator: turn iPhone... Order to graph the regression function and/or find a prediction value 1 multiples... Calculator in Excel and standard form will return you to graph 2D and 3D functions and graphing.... Contains several features not found on any loan with this loan calculator polar... Of values and count how many values fall into each interval lacks the screen necessary turn. Table setup in the Calc menu and works on virtually any Android phone tablet. 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Ung thư gan lại cướp đi một tài năng âm nhạc thiên bẩm Ca sĩ đường phố với bản Cover Đổi Thay gây sốt dân mạng Nhận được giải thưởng Chàng ca sĩ hát rong đã đi về quê làm từ thiện Cuộc thi Cover “Đổi Thay” kịch tính những ngày cuối Copyright © 2016-2018, Thời Báo Công Dân - All rights reserved $(document).ready(function() {$('#slider1').tinycarousel(); }); $(document).ready(function() {$('.iosSlider').iosSlider({ scrollbar: false, snapToChildren: true, scrollbarHide: true, desktopClickDrag: true, scrollbarMargin: '5px 40px 0 40px', scrollbarBorderRadius: 1, scrollbarHeight: '2px', autoSlide: true, infiniteSlider: true, responsiveSlides: true, navPrevSelector: $('.prevButton'), navNextSelector:$('.nextButton') }); }); MENU Thời sự Điểm nóng Xã hội Kinh Tế Bất Động Sản Thị trường Tài Chính Dịch Vụ Tiêu Dùng Thương hiệu NTD cảnh giác Giải Trí Sao Game Video Thể Thao Trong nước Quốc tế Sống Làm đẹp Du lịch/ Ăn uống Phòng chữa bệnh Công nghệ Dân Cười Liên hệ /* <![CDATA[ */ var wpcf7 = {"apiSettings":{"root":"http:\/\/thoibaocongdan.com\/wp-json\/contact-form-7\/v1","namespace":"contact-form-7\/v1"}}; /* ]]> */ ( 'fetch' in window ) || document.write( '<script src="http://thoibaocongdan.com/wp-includes/js/dist/vendor/wp-polyfill-fetch.min.js?ver=3.0.0"></' + 'ipt>' );( document.contains ) || document.write( '<script src="http://thoibaocongdan.com/wp-includes/js/dist/vendor/wp-polyfill-node-contains.min.js?ver=3.26.0-0"></' + 'ipt>' );( window.FormData && window.FormData.prototype.keys ) || document.write( '<script src="http://thoibaocongdan.com/wp-includes/js/dist/vendor/wp-polyfill-formdata.min.js?ver=3.0.12"></' + 'ipt>' );( Element.prototype.matches && Element.prototype.closest ) || document.write( '<script src="http://thoibaocongdan.com/wp-includes/js/dist/vendor/wp-polyfill-element-closest.min.js?ver=2.0.2"></' + 'ipt>' ); /* <![CDATA[ */ var uiAutocompleteL10n = {"noResults":"Kh\u00f4ng t\u00ecm th\u1ea5y k\u1ebft qu\u1ea3.","oneResult":"\u0110\u00e3 t\u00ecm th\u1ea5y 1 k\u1ebft qu\u1ea3. 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2021-04-17 07:45:31
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http://mathoverflow.net/questions/140592/what-is-the-classification-of-characters-in-p-adic-hodge-theory/140640
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# What is the classification of characters in $p$-adic Hodge theory?
Let $K$ be a $p$-adic field and $\chi : Gal_K \rightarrow \mathbb{Q}_p^\times$ be a character. I know that $\chi$ is Hodge-Tate of weight $0$ iff $\chi(I_K)$ is finite (by Sen's theory), and that it is Hodge-Tate of weight $k$ iff $\chi.\chi_p^{-k}$ is HT of weight $0$.
Is there a similar description for De Rham, Semi-stable and Cristalline ?
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Crystalline characters, in your case, are exactly the twists of unramified characters, see for example this MO question mathoverflow.net/questions/61998/crystalline-characters. – ChrisLazda Aug 27 '13 at 21:36
The de Rham characters are the same as the Hodge-Tate ones. The semistable ones are the same as the crystalline ones, and in your notation they are the de Rham ones for which $(\chi \cdot \chi_p^{-k})(I_K)$ is trivial (and not merely finite). This can be found for example in Fontaine and Mazur's paper.
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2016-06-25 05:08:02
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http://davydm.blogspot.com/2014/
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## Wednesday, 10 December 2014
### Let reason prevail
Hark! 'Tis the call of a higher-order language. It beckons with sweet words of reduced time solving ancient, low-level problems! The siren speaks true!
Seriously though...
Higher-order programming languages afford the competent programmer a method for solving more higher-level problems in the same timeframe than she could in with a lower-order language. Simply put, high-order languages (like C++, C#, Python, Ruby, PHP, etc) afford the programmer the ability to skip over the low-level "move this byte to this address" kind of programming so that she can solve more interesting and/or more lucrative problems.
There is a cost though: iron. These higher-order languages require smarter compilers and, typically, more RAM and CPU. It's a cost we're happy to pay though -- iron is cheaper than development.
But when someone starts making outlandish claims that a higher-order language is more proficient than a lower-order one (https://isocpp.org/blog/2014/12/myths-1, see the string concatenation example in the first section), the reasonable programmer doesn't just gulp that down, even if it does come from the father of C++. Actually, especially if it does come from the father of a higher-order language, since he would have reason to pad out his results (or pretend to be lazy) to make his prodigy all the more appealing.
Aside: before we go any further, a disclaimer: I like C++. I like C. That's OK to like both and recognise their strengths. What's NOT ok (imo), is to spread misinformation to highlight the language of your preference. Have honest reasons for preference -- by all means! -- and be objective about comparisions. And the discourse continues!
So let me make a really bold statement:
Any proficient code in a higher-order language can only hope to be (at best) as proficient as proficient code in a language of lower-order for solving the same problem.
Why?
Let's take the string vs char* example from above:
std::string has to be implemented (at some point) around a buffer of memory. I don't care if it's char*, wchar* or whatever. It's a buffer which, at some point, was obtained via malloc() (even if you want to say it was obtained by new char[], that still boils down to essentially a malloc, so let's stop arguing semantics). The C++ compiler affords us the ability to overload operators, such that we can do:
string1 + string2
and get another string. Under the hood, this is allocating a third string object and the associated memory and doing some memory copying. One way might be to malloc() on strlen(string1) + strlen(string2) + 1 char for the null terminator, then strcpy() and strcat() in the parts. There are quite a few ways this could be done, but this is one.
Now the problem lies here: in an opaque, higher-order implementation of string concatenation (for example), the best outcome we could hope for is the one which is fastest in C, ie which we discovered by trying all paths including strcat, memcpy, strcpy, etc. So let's assume that the best path was chosen for std::string's + operator overload. That still makes it only as fast as the best implementation in C. Take a step back and realise that the + operator may also do clever things like allocate more memory than required to save on a realloc() later as well as rudimentary bounds-checking or what-have-you, and it's easy to see why the string variant is 10-20x slower (https://github.com/fluffynuts/cpp_vs_c.git -- oddly enough, the disparity was greater on a win32 version I did earlier today where the sprintf() version took only 48ms for 32768 iterations vs over 600ms for the string version. The code linked here reports the following results on my Linux machine:
C++ function: 32768 runs took 5827 ms
C function: 32768 runs took 706 ms
C function (sprintf): 32768 runs took 4826 ms
Which still shows that the C++ version is nearly 10x slower. Some of the difference I've experienced between platforms may be to Microsoft optimisations as well as stdlib being (in my experience) much slower on Windows than Linux (not counting boost, which I haven't used, but which benchmarks well). The C++ version may optimise for frequent use better than the quoted C version -- but the C programmer is free to update her code accordingly, as required. Again, remember, we're talking about proficient code solutions, so when you change the parameters of the argument, the code is free to change too.
This shouldn't be surprising. The C++ code has to deal with the generic case (and provides a lot of extra functionality which is probably worth the cost) than the C version. But let me re-iterate:
Any proficient code in a higher-order language can only hope to be (at best) as proficient as proficient code in a language of lower-order for solving the same problem.
Note the use of the word "proficient". If I write shitty C code, I bet you can write good C++ that is faster. Same goes for any other pairing: if I write shitty low-order code, I'm sure you can write good high-order code which out-performs my shitty code.
The request I have to the programming community is this:
Please be honest and stop trying to win a "my language is better than yours" with pure lies and FUD. When speaking of well-written, proficient code, Ruby/Python/.NET IL/PHP/C++/whatever is NOT faster than C or assembly. The lower you go, the more specific your instructions can be and the more efficient the overall run can be.
We all accept the costs of higher-order languages because of what they offer us:
• Quicker to code solutions to the problems which are interesting and/or lucrative, therefore cheaper (overall) than the iron required to run the output
• Safe memory handling
• Rich libraries
• Abstraction from the iron
• Easier-to-read (and therefore maintain) code
• And a host of other benefits
I'm open for challenges on this though. Provide a problem which is relatively small to solve in a lower-order language which you think is faster (not smaller or more elegant) in a higher-order language and I'll see what I can do prove my point. Remember: a small problem, like something mathematical or string manipulation.
## Thursday, 30 October 2014
### Javascript and promises
I wrote a little tutorial a small while back on using promises within Javascript as this seems to be a point of confusion for some, especially those new to promises. The tutorial is not meant to be ultimately comprehensive, but hopefully it should at least be useful -- and it made me polish up a promises library of my own for learning and display purposes. You can get all the code here: https://github.com/fluffynuts/js-promises-tutorial and indeed that repository is what I'm using for the iframe source below, using RawGit. Mainly because it relies on a bunch of Javascript and JSON files to work but also because I'm way too lazy to maintain multiple versions of this and this tutorial may be revisited for a little spit 'n polish on occasion. (EDIT: this entry was updated to include example usage of the ES6 native Promise prototype).
## Tuesday, 23 September 2014
### Shout-out to the kind folks at JetBrains
The other day I received a reminder mail that my R# license is due to expire in a few weeks. Now, I bought that when I was doing some work for a personal client and I really needed both VB.NET and C# functionality from my R#. So unfortunately, despite the kind offer from my company to use one of the roaming licenses they have for C#, that wasn't going to cut the mustard. I bit the bullet and got my own R# and reaped the productivity benefits, even though I'm quite sure I use only around 40% of the functionality offered by R#.
Anyways, I responded to the mail with a query regarding PeanutButter and the JetBrains R# OpenSource licensing program and... have been awarded an OpenSource (ie, no-charge) license for this year for R#, Full Edition. Thanks JetBrains! If you use C# and/or VB.NET (and, now, I hear C++ is in EAP) for development, in Visual Studio, Resharper can probably boost your productivity with navigation, code cleanup and refactoring functionality. Head on over to http://www.jetbrains.com/resharper/ to check it out if you don't use it already. The pricing may seem like quite a bit (and I'll freely admit that it's not cheap), but you can make that back quite quickly with the time it saves you, especially if you program TDD-style and take pride in your code, refactoring it to make it expressive and therefore more easily maintained.
## Wednesday, 17 September 2014
### Browsers for everyone!
The browser wars are over!
Or are they? Will they ever be? I certainly hope not. With Opera deciding to bow out of the render wars and move to an in-house-maintained WebKit backend instead of Presto, my heart sank a little. Presto had, in my opionion, been responsible for all of the other browsers dragging themselves, kicking and screaming, to support the latest CSS standards. It was just plain embarrassing to be smashed by the ACID test by one of the "little guys". Microsoft, Mozilla and Google had to step up, had to (at least try to) keep pace.
Competition is good, most especially in the software realm, most especially for the most important player in the software realm. No, not corporations selling you their software and shiny hardware products. Not institutions who run the world of finance and could cause the collapse of life as we know it with a few bad lines of code. The most important person in the software realm is... you. The user. And the most powerful tool at your disposal is choice. As users are free to choose, so they are free to ditch software which is left in the dust by other software. It's this competition which improves software all-round, ultimately for the benefit of the user (and hopefully, for the companies involved, adds something to their bottom line. Somehow).
So enter another contender: Maxthon MxNitro. Maxthon has been in the game for a while, but, as did Opera before them (and they still do), they occupy a small segment of the browser market. You need something really significant to get the average person to bother to not use the system default browser -- hence the anti-competitive lawsuits against Microsoft (but, in their defense, wouldn't it be rather shameful to sell you an operating system without a browser? About as shameful as giving you an OS without an office suite or baked-in programming tools, but I digress...)
I've used Maxthon before and it was more of an exercise in curiosity than anything else. Finding out about MxNitro, I had to see what they were going on about with the speed they claim the browser has. There are a bunch of numbers floating around; information like "30% faster than Chrome 37" and such. Interesting claims, and no explanations for how, except some hand-waving about how hard they worked to lighten it up and optimise it.
My verdict, if you care:
Startup is fast. I mean, not just fast, but zippy-gawrsh-how-the-heck-did-they-do-that fast. A first run of Chrome on my machine (i7, 8Gb RAM, no ssd's so I'm a bit penalised there) takes around 8-10 seconds. Closing (checking that Chrome isn't running) and re-opening brings that down to around 2. MxNitro is available for use the moment, the Start screen has faded from view, as if it were waiting there the whole time. I had to check several times for some sneaky background process or service -- none. The install was the same: I double-clicked the installer and had a working browser up in under a second. I don't even know how they managed to unzip the required files to its install folder (in your roaming profile) that quickly. I'm baffled -- and impressed. I want to know more...
Memory usage on the other hand: terrible, just like Chrome. There are reasons I don't use Chrome as my daily driver, but the largest is simply memory usage. Chrome can happily consume a few gigs of memory where the same tabs in Firefox will be well under a gig. Even 2 or 3 tabs in Chrome starts getting up to the gig mark. Opening the exact same 5 tabs in Firefox and Chrome got me a memory usage of 431Mb in Firefox and just shy of a gig in Chrome (disclaimer: to get Chrome's usage, you have to add up the memory of all processes so I just added roughly; I'm not trying to exaggerate with the "just shy of a gig" comment, but it was in the high 900's). MxNitro also uses a multi-process model. Roughly adding up the memory used for the same tabs came in at around 700mb, which really is where Chrome would probably be if I stripped out all addons and the dev tools (another disclaimer: I have around 10 addons in both Firefox and Chrome). So, just when I was thinking that this might be a good candidate for my aging Core2 Duo laptop (which only has 2Gb of RAM, not upgradeable )': ), I have to think twice. I did give it a spin there -- 700 mb for gmail, facebook and twitter open. Hm.
It has a very minimal look and feel in the standard retina-blasting white of many apps these days. It's also rougher than a badger's arse:
• No way to search a page for text
• No dev tools at all -- indeed, I was surprised to find a "view source" option
• Crashed when I tried to scroll with my touchpad. Every time
• No spell-checker -- blogging (like this article) just doesn't feel as safe. I'm going to have to save this draft and reload in another browser to check it
• Indeed, no preferences whatsoever
• Weird window control-box (minimise, maximise, close) which only becomes available when you hover over it, though for no really good reason as the space isn't put to better use. It's just a little bit of "wat?" to overcome
• No Flash
The bare-bones minimal interface will probably work well for some people -- in fact, for a lot of people. There are a lot of people who are out there on the web and they don't know, need to know, or even care about things like dev tools or extensions. They don't know about ad blockers (though they really should be interested) or sync or many of the shiny features of Firefox and Chrome (and even IE, now that it's been dragged to version 11). The problem is that the people for whom this interface will work are
• The most likely to give up on a browser which doesn't have basic features like in-page search
• The most likely to give up on a browser which crashes
• Quite likely to need Flash and unable to understand why their Facebook videos aren't playing
Not that I could be any happier that there's no bundled Flash (indeed, I welcome the long-overdue demise of Flash) -- just that it's going to hamper the people most likely to use this.
So will I use it? Probably not much, if at all. It's a bit novelty for me at the moment -- "oooo, look how fast it is!". But then again, it starts and acts about as fast as FooBrowser, a minimal, keyboard-driven browser I wrote with PyQt ages ago. My browser suffered from the same flash issue, but at least find-in-page worked. I think.
Nope, I'll be sticking with Firefox (Aurora channel). Chrome has better dev tools (so I'll fire it up for dev, especially remote-debugging Cordova apps), but is too much of a memory glut. IE has just lost my trust. But bigger than that, Firefox has the addons I want (some of which I can get elsewhere, sure): an ad-blocker, Stylish and a Youtube downloader. The last one is another big reason why I can't use Chrome for my daily browser, in addition to the silly full-screen view which means that tabs are obscured when I have WinAmp open -- I still don't get why the Chrome team is SO opposed to allowing a small titlebar on Windows; you can work it on Linux because of the WM, but on Windows, it's better to resize the window to fill the screen than use maximise. Daft. And I'm far from alone in this quest. But a Youtube downloader -- that I can't do without. Not because I'm some naughty pirate or video hoarder, but simply because I want to watch unhindered in at least 720p and Youtube
• insists on picking "the best" resolution for me every time, which I have to toggle off
• insists on turning captions on with every video -- yet another thing to toggle off
• has become an ad spawn-point. I understand the site is financed by advertising. I'm ok with an ad here and there. But every 5 minutes in a video? Bugger off
• still often insists on playing through Flash, which runs terribly and sometimes takes down my browser
So I download, watch, delete. But the path of digression is here again....
The verdict?
Maxthon MxNitro is interesting because of the speed, but nowhere near ready for public consumption yet. With a bit of work, it could become a good replacement browser for mom-and-pop types with older hardware, providing they have enough RAM. The really good part is that hopefully the gauntlet has been thrown down and other browser houses will follow suit. I especially hope that Mozilla does -- Firefox wins the RAM wars but trails a little in the speed department.
## Wednesday, 2 July 2014
### A little PeanutButter for your MVC
ASP.NET MVC is one of the best things that I could possibly think of to have happened to the web from a Windows-centric point of view. At last, there's a way to sweep that abomination that is WebForms under the proverbial carpet.
Like anything, though, it does have its caveats. You have useful constructs like Script and Style Bundles -- but no easy way to test them from a CI environment. Also script inclusion becomes a bit more manual than it needs to be when viewed though the lens of AMDs like require.js. But you do get the advantage of bundling in that a single request can satisfy multiple code/style requirements. (Let me be clear here: AMDs are good. I like require.js. But it does take a little more effort to set up and get working correctly and you don't (without even more configuration) get the hit-reduction that bundles provide. Both methods have their advantages. Select your tools for your tasks as they fit best for you, on the day.)
PeanutButter.MVC was built out of a need to make those processes slightly more testable and elegant.
First of all, there are two facade classes:
They wrap ScriptBundle and StyleBundle accordingly and implement interfaces of the expected names (IScriptBundle and IStyleBundle). You would use them like you'd use ScriptBundle and StyleBundle instances from the MVC framework. However, since they implement an interface, you can also create substitutes for them so that you can test-constrain your bundle registration process. This is important because I found that it was not uncommon to add a new javascript or css file to the solution, build-and-run, and be surprised that my changes weren't in play -- until I realised that I hadn't bundled them.
For example, I have the following method on my BundleConfig:
public static void RegisterBundles(BundleCollection bundles,
Func<string, IScriptBundle> withScriptBundleCreator = null,
Func<string, IStyleBundle> withStyleBundleCreator = null)
{
withScriptBundleCreator = withScriptBundleCreator ?? ((bundleName) => new ScriptBundleFacade(bundleName));
withStyleBundleCreator = withStyleBundleCreator ?? ((bundleName) => new StyleBundleFacade(bundleName));
}
We can see that the function would ordinarily be invoked without lambda factories to produce Script- and StyleBundleFacades, so it produces its own, very straight-forward ones. However, the tests that constrain this method can inject lambda factories so that the bundling methods can be tested to ensure that they include the required bundles from the relevant sources. Indeed, the tests are quite straight-forward:
[TestFixture]
public class TestBundleConfig
{
private Func<string, IScriptBundle< CreateSubstituteScriptBundleCreator(List>IScriptBundle< withTrackingList)
{
return (name) =>
{
var scriptBundle = Substitute.For<IScriptBundle>();
scriptBundle.Name.Returns(name);
var includedPaths = new List<string>();
var includedDirs = new List<IncludeDirectory>();
scriptBundle.IncludedPaths.ReturnsForAnyArgs(args =>
{
return includedPaths.ToArray();
});
scriptBundle.IncludedDirectories.ReturnsForAnyArgs(args =>
{
return includedDirs.ToArray();
});
scriptBundle.Include(Arg.Any<string>()).ReturnsForAnyArgs(args =>
{
return new Bundle("~/");
});
scriptBundle.IncludeDirectory(Arg.Any<string>(), Arg.Any<string>())
.ReturnsForAnyArgs(args =>
{
includedDirs.Add(new IncludeDirectory(args[0] as string, args[1] as string));
return new Bundle("~/");
});
scriptBundle.IncludeDirectory(Arg.Any<string>(), Arg.Any<string>(), Arg.Any<bool>())
.ReturnsForAnyArgs(args =>
{
includedDirs.Add(new IncludeDirectory(args[0] as string, args[1] as string, (bool)args[2]));
return new Bundle("~/");
});
return scriptBundle;
};
}
private Func<string, IStyleBundle> CreateSubstituteStyleBundleCreator(List<IStyleBundle> withTrackingList)
{
return (name) =>
{
var styleBundle = Substitute.For<IStyleBundle>();
var includedPaths = new List<string>();
styleBundle.IncludedPaths.ReturnsForAnyArgs(args =>
{
return includedPaths.ToArray();
});
styleBundle.Include(Arg.Any<string>()).ReturnsForAnyArgs(args =>
{
var paths = args[0] as string[];
return new Bundle("~/");
});
return styleBundle;
};
}
[Test]
{
//---------------Set up test pack-------------------
var collection = new BundleCollection();
var scriptBundles = new List<IScriptBundle>();
var styleBundles = new List<IStyleBundle>();
//---------------Assert Precondition----------------
//---------------Execute Test ----------------------
BundleConfig.RegisterBundles(collection,
CreateSubstituteScriptBundleCreator(scriptBundles),
CreateSubstituteStyleBundleCreator(styleBundles));
//---------------Test Result -----------------------
Assert.AreNotEqual(0, scriptBundles.Count);
Assert.AreNotEqual(0, styleBundles.Count);
Assert.IsTrue(scriptBundles.Any(sb => sb.Name == "~/bundles/js/shared" &&
sb.IncludedDirectories.Any(d => d.Path == "~/Scripts/js/shared" &&
d.SearchPattern == "*.js" &&
d.SearchSubdirectories == true)));
}
}
All good and well. We can ensure that our MVC application is creating all of the required bundles. It would also be super-neat if we could streamline the inclusion process. Of course, we can.
PeanutButter.MVC also includes a utility called AutoInclude. If we decide to set up our bundles under /bundles/js/{controller} (for scripts for any action on the controller) and /bundles/js/{action}, then a lot of inclusion work can be done for us in our base _Layout view with a single line (assuming you've included the relevant @using clause at the top):
@AutoInclude.AutoIncludeScriptsFor(ViewContext)
AutoInclude uses the convention of scripts sitting under folders with names corresponding to the controller, with the casing of the scripts folders lowered to be more consistent with how script folders are named. This one line has, in conjunction with judicial bundling (and testing of that bundling!) allowed all views to just "magically" get their relevant scripts. In my project, I can create script bundles which include similarly-named folders and not have to worry about how my views get relevant logic scripts from there on out.
So, for example, I might perform registrations like the following (where scriptBundleCreator is a passed in Func<iscriptbundle>):
bundles.Add(scriptBundleCreator("~/bundles/js/policy")
.IncludeDirectory("~/Scripts/js/policy", "*.js", false));
.IncludeDirectory("~/Scripts/js/policy/accept", "*.js", false));
.IncludeDirectory("~/Scripts/js/policy/edit", "*.js", false));
Now, from the Policy controller, I have two actions, Accept and Edit. Both have their relevant views, of course, and the AutoInclude is done automatically for them by virtue of the fact that they use the default _Layout.cshtml. Under my Scripts folder in my project, I have a file structure layout like:
policy
policy/common.js
policy/accept
policy/accept/accept.js
policy/accept/proposalEmailer.js
policy/edit
policy/edit/clientDetailsDisplayUpdater.js
policy/edit/edit.js
policy/livePolicyUpdater.js
And the result is that the Policy/Accept view gets common.js, accept.js, lead-autocompletion.js and proposalEmailer.js. The Policy/Edit view gets common.js, clientDetailsDisplayUpdater.js, edit.js and livePolicyUpdater.js.
So now I'm free to create small, easily-testable javascript files (which I'll test with Jasmine and whatever works best for my purposes (eg karma or the Resharper unit test runner -- which works, mostly, with Jasmine, but has a few rough edges)). And when I want them in a page, I just drop them in the appropriate folder to get them on the next compile/debug run. And because of bundling, the end-user doesn't have to get many little hits for javascript files, instead, just getting two per view.
Apart from the testability of it and the simplicity of adding another piece of javascript functionality to the site, there's a huge bonus in grokkability. Let's face it: one of the reasosn why tests are good on your code is for when a new developer comes onto the project (or some unlucky person is tasked with maintaining some code they had nothing to do with). Tests provide feedback for when something breaks but also provide a communication mechanism for the new developer to figure out how discreet parts of the overall machine work. To the same end, understandable symbol and file naming and unsurprising project layout can really help with a new developer (or when you just have to get back on to the project for maintenance or extension and it's a couple of months down the line...)
Anyway, so there it is: PeanutButter.MVC. Free, small, doesn't depend on much, and hopefully useful. I'm certainly reaching for it the next time I'm in MVC land.
## Tuesday, 1 July 2014
### INI files are dead... Long live INI files!
There was a time when INI files ruled the world of configuration. Since then, we've been told on numerous occasions by many people that we should rather be using XML. Or a SQLite database. Or something else, perhaps.
Now, don't get me wrong -- SQLite has its merits and XML is great if you want to store hierarchical data or if you need to configure your .NET application (which happens to already speak the lingo). But the reality is that INI serves quite well for a number of uses -- indeed, it can also be used to store hierarchical data, as you'd see if you checked out the innards of a .reg file. In particular, INI files are dead-easy to parse, both by machine and man -- and the latter is an advantage if you have nothing to hide and no need for quick read/write (where you might, for example, use SQLite). It's also a simple file-store so platform and library requirements are minimal. It's probably the easiest way to store structured configuration data and I still use it for projects unless I absolutely have to use something else.
A relatively small, simple part of the PeanutButter suite is the INI reader/writer/storage class PeanutButter.INI.INIFile. Usage is quite simple:
var ini = new INIFile("C:\\path\\to\\your\\iniFile.ini");
var someConfiguredValue = ini["colors"]["FavouriteColor"];
ini["Geometry"]["Left"] = "123";
ini.Persist();
In thesnippet above, we instantiate an INIFile class with a path to a file to use as the default persistence store. This file doesn't have to exist right now (and if it doesn't, it will be created with the Persist() call).
INIFile presents the data present in the source as a Dictionary<string, Dictionary<string, string>>, with indexing on the INIFile instance itself, making the syntax quite easy to use. Sections are created as and when you need them. Section and key names (such as "Geometry" and "Left" above) are case-insensitive to make access easier (and more compliant with the behavior of the older win32 calls for INI handling).
The parser tolerates empty lines and comments as well as empty keys (which are returned as an empty string).
Of course, you don't have to have a backing store to start with (or at all), and you can always override the output path with a parameter to Persist(). In addition, you can re-use the same INIFile, loading in a file from another path with the Load() method or loading with a pure string with the Parse() method.
Once again, the class has been developed on an as-required basis. It does much of what I want it to do (though I'd like it to persist comments on re-writing; that may come later). I hope that it can be of use to someone else too. I've lost count of how many times I've implemented an INI reader/writer. Hopefully, this is one of the last...
## Monday, 30 June 2014
PeanutButter was born out of a desire to share code between projects, in particular, to take advantage of utility and fixes introduced during the development process on a range of concurrent projects in the realm of code which was specific to none.
Quite basically, I had some code which was already figured out and worked fairly well for what it was intended to do and I was just plain too lazy to maintain multiple versions of that code. I thought it would be great if I could use, say, the Nuget packaging system to spread the most up-to-date versions of code between projects, if only packaging for Nuget wasn't such a pain. The CLI tools work, but aren't easy to use. The official GUI for Nuget packaging looks like a revenge unleashed on the world by an angry development manager trying to prove he can code. No offense.
Ok. Offense. The Nuget GUI tools are horrid and the CLI packaging mechanism is a PITA. Thank goodness for the Nuget Package Template extension for Visual Studio. With a simple post-build event from each relevant package, I can push all 11 (currently) PeanutButter.* Nuget package updates after having run all my tests and switched to Release mode. Win!
First, taking a step back. If you've been using a decent ORM like Entity, NHibernate, Habanero or even just Linq-to-SQL, you don't have a need for PB's DatabaseHelpers. You can save yourself the effort of reading the rest of this article. One of the primary reasons for using an ORM (imo) is to abstract the SQL away from the application. ORMs which do Linq well are especially good at this -- and that's one reason why I've been a fan of Entity (warts and all) for quite some time. None of them are perfect but they all can take a lot of the pain of dealing with direct database calls away. In particular, an ORM allows you to:
• Switch database backends (eg MSSQL/Firebird/MySQL/PostgreSQL and others). Of course backend support depends on the ORM, but most give you some kind of choice here.
• Not have to type out correct SQL statements in your code. You may be wondering why this is a problem, unless, of course, you've had the experience where SQL in your code worked once and stopped mysteriously. After back-tracking VCS commits, you find that someone accidentally changed a string and there was nothing to pick it up. Or perhaps there was a test and the test was just as wrong as the code it was constraining.
• Get the compiler to check that you've gotten the names of your database entities correct -- if you've misspelled an entity in your code in one place, chances are your code doesn't compile. Which is good -- the earlier up the code/compile/run/debug/package/deploy/test/etc chain you fail, the less it costs everyone.
• Not have to worry about SQL injection attacks.
If you're using a super-light ORM like Dapper or just simply converting a giant project with heaps of direct ADO access or perhaps providing support for an app using an unsupported backend (like Access), you may have to get down to the bare SQL. But it would be great if you didn't have to actually take the risk of writing some SQL. Better still if the tool producing the SQL to run on your backend can be tweaked to target a different backend as required.
So here's a possible approach:
1. Ensure that all of your database entities are defined as string constants in one accessible source file so that you aren't constantly fighting with each other on how to spell "color". Or "colour". However you all choose to spell it. And you aren't left to discover that "received" is spelled "recieved" sometimes in your SQL at go-live time.
2. Get a tool like PeanutButter to do the boring work for you. Reliably and in a manner you can test, with fluent builder syntax.
Sound like a plan? Great (:
First, I like to stick to using a DataConstants file to hold constants like field names, default values, etc. The other advantage here is that you can reference the same DataConstants in your FluentMigrator migrations (you are using FluentMigrator, aren't you?!). For example:
namespace MyProject.Database
{
public static class DataConstants
{
public static class Tables
{
public static class Employee
{
// to use where you would reference the name of your table
public const string NAME = "Employee";
public static class Columns
{
public const string EMPLOYEEID = "EmployeeID";
public const string FIRSTNAME = "FirstName";
public const string SURNAME = "Surname";
public const string DATEOFBIRTH = "DateOfBirth";
}
}
}
}
}
Next, let's say we wanted to get a list of all Employees whose first names are "Bob". We could do:
var sql = SelectStatementBuilder.Create()
.WithTable(DataConstants.Tables.Employee.NAME)
.WithField(DataConstants.Tables.Employee.Columns.EMPLOYEEID)
.WithField(DataConstants.Tables.Employee.Columns.FIRSTNAME)
.WithField(DataConstants.Tables.Employee.Columns.SURNAME)
.WithField(DataConstants.Tables.Employee.Columns.DATEOFBIRTH)
.WithCondition(DataConstants.Tables.Employee.Columns.FIRSTNAME,
Condition.EqualityOperators.Equals,
"Bob");
Ok, so this looks a little longwinded, but with a using trick like so:
using _Employee = DataConstants.Tables.Employee;
using _Columns = DataConstants.Tables.Employee.Columns;
// some time later, we can do:
var sql = SelectStatementBuilder.Create()
.WithTable(_Employee.NAME)
.WithField(_Columns.EMPLOYEEID)
.WithField(_Columns.FIRSTNAME)
.WithField(_Columns.SURNAME)
.WithField(_Columns.DATEOFBIRTH)
.WithCondition(_Columns.FIRSTNAME, Condition.EqualityOperators.Equals, "Bob");
Now that's fairly readable and always produces the same, valid SQL. And will break compilation if you misstype something. And you can use Intellisense to help figure out required column names. This is a fairly simple example; just a taste. PeanutButter.DatabaseHelpers includes:
• Statement builders for:
• Select
• Update
• Insert
• Delete
• Data copy (insert into <X> select from <Y>)
• The ability to do Left and Inner joins
• Order clauses
• Where clauses
• Automatic quoting of strings, datetimes and decimals to values which will play nicely at your database (no more SQL injection issues, no more issues with localised decimals containing commas and breaking your SQL)
• Interfaces you can use for injection and testing, for example with NSubstitute (ISelectStatementBuilder, IInsertStatementBuilder, IUpdateStatementBuilder, IDataCopyStatementBuilder, IDeleteStatementBuilder)
• Another helper package PeanutButter.DatabaseHelpers.Testability which sets up NSubstitute mock objects for you so you can easily test without having to stub out all of the myriad builder returns on statement builders
• Executor builders:
• ScalarExecutorBuilder for insert/update/delete statements
• With interfaces for easy testing
• Support for some computed fields (Min, Max, Coalesce, Count)
• Constrained by tests, developed TDD.
• Syntax support for Access, SQLite, MSSQL and Firebird. Additional syntax support can be added upon request, given enough time (:
All in a convenient Nuget package you can use from your .NET application (install-package peanutbutter.databasehelpers), irrespective of target language. PeanutButter.DatabaseHelpers was developed and is maintained in VB.NET simply because of its origins, but that really doesn't matter to you as the consumer (:
Sure, there are features which are missing. This library has been built very much on the premise of extending on requirement. I hope that it can be helpful for someone else though. It has been used in at least 5 decent-sized projects.
I welcome feedback and will implement reasonable requests as and when I have time. Like the rest of PeanutButter, DatabaseHelpers is licensed BSD and you can get the source from GitHub.
## Introducing... Peanut Butter!
I've been writing code for about 15 years now. It's not really that long, considering the amount of time that friends of mine have been writing code, but I guess it's not an insignificant portion of my life.
Anyone who writes code eventually figures out that there are some common problems that we solve time and again; and the time we spend solving them is time we could have spent solving more interesting problems.
Such common problems include (but are certainly not restricted to):
• INI file parsing and writing
• Random value generation (especially for testing)
• Win32 polling services
• Dealing with temporary files which need to be cleaned up as soon as we're done with them
• Writing SQL statements (yes, sometimes we don't have ORM frameworks to help us (eg when we have to deal with Access) and sometimes we use super-light frameworks like Dapper which need us to do the SQL grunt work ourselves)
• Systray icons
There are many more, of course.
There are a few ways we could deal with this scenario. We could re-write, from scratch, algorithms to provide solutions to these common problems. We could copy-and-paste known working code into our solutions every time. We could try to find someone else who has provided a library to help (and we certainly should, before embarking on rolling our own, if we value our time).
PeanutButter (https://github.com/fluffynuts/PeanutButter) is a suite of small libraries aimed at tackling such common tasks. I built it out of a desire to re-use bits of code that I'd spent reasonable amounts of time on where those bits of code could perhaps save me time later. PeanutButter modules are available from Nuget and enable me (and anyone else who so desires) to spend more time on interesting tasks and less time on the more common ones.
I hope to spend time introducing the different modules of PeanutButter in more depth. I really hope that this code and blog can perhaps save someone a little time and effort. Even if I'm the only person to use PeanutButter, I'm OK with that -- it serves me well. But if it can make dev a little friendlier for someone else, all the better.
PeanutButter is employed in a few commercial products. Licensing is BSD, so you're free to use it, fork it, break it and keep all the pieces. Feedback is welcome and contributions will be considered.
Off the bat, PeanutButter offers (in small, fairly independant Nuget modules):
• PeanutButter.DatabaseHelpers
• provides builders for Insert, Update, Select, Delete and data copy (insert into X select from Y) SQL statement builders for Microsoft-style databases (MSSQL, Access), SQLite and Firebird (other dialects can be added on request/requirement)
• DataReader and Scalar executor builders
• Connection string builder (Access only, but I plan to expand this)
• PeanutButter.INI
• Provides reading/parsing and writing of INI files in a very small library with a very simple interface
• PeanutButter.MVC
• provides shims and facades to make testing MVC projects easier, especially when you'd like to constrain script and style bundle logic in your tests
• PeanutButter.RandomGenerators
• produces random values for all basic types (string, long, bool, DateTime), enums and selecting a random item from a collection
• includes GenericBuilder, a builder base class which can build random versions of any POCO class. GenericBuilder is extensible, but doesn't do collections since I haven't determined what the optimal course of action is there. Still, it's proven to be very useful and has freed literally hours and hours of dev time that I (and others) would have spent on generating random input for tests requiring complex objects
• PeanutButter.ServiceShell
• provides a simple shell for Win32 polling services. You just have to set up the name, description and interval (default is 10 seconds) and override RunMain. Your Program.cs has one line to run in it and suddenly you have a service which:
• Polls on a regular interval, guaranteed not to overlap (runs which exceed the poll interval just result in the next run being executed immediately)
• can install, uninstall, start and stop itself from the command line
• There's an example project, EmailSpooler, which demonstrates usage. EmailSpooler is a win32 service which polls an email registration table in your database for unsent emails and, using a configured SMTP server, attempts to send those mails, with handling for failed sends and backoff time. Whilst being an effective example for how to use the Service Shell, it was also an application actively used in production... until the client and I parted ways. It's generic enough for anyone to use though, and it's free. Help yourself.
• PeanutButter.TestUtils
• provides a PropertyAssert class to easily test equality of properties on disparate object types, using property names and types. It's great for reducing the amount of time spent writing tests to ensure that DTO property copies take place as expected.
• PeanutButter.TinyEventAggregator
• provides a "prism-like" event aggregator with very little overhead. In addition, it can log subscriptions and publications to the Debug output so you can figure out why your program is misbehaving (especially when you forget to unsubscribe...) and provides convenience methods like SubscribeOnce (to just be notified on the next publication) and, of course, the ability to subscribe for the next [N] events. You can interrogate subscriptions at run time and even pause and resume eventing which is quite useful from within winphone apps, when your app is backgrounded and resumed.
• PeanutButter.TrayIcon
• provides a simple way to create and use systray icons, using register methods to create menu items with callbacks
• provides methods to launch and respond to clicks on bubble notifications
• provides an animator class to animate your tray icon, given a set of icon frames
• PeanutButter.Utils
• provides AutoDeleter, a class implementing IDisposable which will clean up the files you give it when it's disposed (ie, when leaving the using scope)
• provides AutoDisposer, which works like AutoDeleter, on disposing multiple registered IDisposable objects. Great for reducing your using nests
• AutoLocker which locks a Semaphore or Mutex (though you should avoid win32 mutexes as they are, imo, broken) and releases when disposed so you can use a using block to ensure that your locks are acquired and released even if an exception is thrown
• PeanutButter.WindowsServiceManagement
• provides mechanisms to interrogate, start, stop, pause, resume and just plain "find out more about" windows services.
• PeanutButter.XmlUtils
• for the moment, just provides a simple Text() extension method to XElements to get all of their text nodes as one large string.
PeanutButter is developed with TDD principles. There are currently only 343 tests, but that number climbs as features are added, of course. Some things (like Windows service manipulation) are difficult to test, so they're not as well (if at all) covered. Other things are developed "properly": test-first. If nothing else, there's a collection of code which can wag a finger at me when I break something, before I upload it and break something of yours (:
I will be elaborating on each one a little more, with some code samples, in the near future. For the moment, it's enough that I've announced them. Oh yeah, and version 1.0.50 is available through Nuget and on github.
Introduction
Who am I? What am I? Why should you care? Who actually does care?
|
2018-01-19 09:18:45
|
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|
https://zbmath.org/?q=an:06383781&format=complete
|
# zbMATH — the first resource for mathematics
Products of small modules. (English) Zbl 1307.16006
The paper gives a partial answer (depending on the model of set theory) to the problem of existence of rings such that the class of all small modules $$M$$ (i.e., the covariant functor $$\operatorname{Hom}(M,-)$$ commutes with all direct sums) is closed under direct products. In Proposition 2.3, it is shown that these considerations can be restricted to the case of simple self-injective regular rings. The main result of the paper (Theorem 3.5) says that if $$R$$ is a non-Artinian right self-injective right purely infinite (i.e., there is a right ideal $$K$$ of $$R$$ such that $$K\simeq R^{(\omega)}$$ as right $$R$$-modules) associative ring with unit then the class of all small right $$R$$-modules is closed under direct products under the assumption (which is consistent with ZFC) that there is no strongly inaccessible cardinal (i.e., a regular cardinal $$\kappa$$ such that $$2^\lambda<\kappa$$ for each $$\lambda<\kappa$$).
##### MSC:
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16B70 Applications of logic in associative algebras 03E35 Consistency and independence results 16D50 Injective modules, self-injective associative rings 16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
Full Text:
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2021-04-18 09:20:42
|
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https://gmatclub.com/forum/in-triangle-abc-to-the-right-if-bc-3-and-ac-4-then-what-is-the-126937.html
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# In triangle ABC to the right, if BC = 3 and AC = 4, then what is the
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In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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Updated on: 18 Feb 2019, 04:07
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In triangle ABC to the right, if BC = 3 and AC = 4, then what is the length of segment CD?
A. 3
b. 15/4
C. 5
D. 16/3
E. 20/3
Attachment:
Triangle.jpg [ 8.62 KiB | Viewed 183213 times ]
For this problem the solution is :
we have 3 similar triangles the main triangle : ABD two other triangles BC and ADC .
Now to find out CD we can use the later two triangles , so by similarity we have ,
BC/CA = CD/AC
which yields CD as 3.
but the answer is wrong. where have i gone wrong?
Originally posted by kirankp on 16 Dec 2009, 04:22.
Last edited by Bunuel on 18 Feb 2019, 04:07, edited 5 times in total.
Edited the question.
Math Expert
Joined: 02 Sep 2009
Posts: 53066
Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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26 Feb 2012, 11:07
5
6
rvinodhini wrote:
Hi
I need a quick clarification on the concept of perpendicular bisector.
With a perpendicular bisector, the bisector always crosses the line segment at right angles
If any line cuts another line at 90 then it should be a perpendicular bisector right - i.e it divided the line segment into equal halves at 90 ?
So here BC should be the perpendicular bisector and the AC=CD=3 right ?
Please let me know what am missing here.
I do understand the explanations in the other thread mentioned,but can someone clarify as to why AC is not the perpendicular bisector ?
A perpendicular bisector is a line which cuts a line segment into two equal parts at 90°. So AC to be a perpendicular bisector of BD it must not only cut it at 90° (which it does) but also cut it into two equal parts. Now, in order AC to cut BD into two equal parts right triangle ABD must be isosceles, which, as it turns out after some math, it is not.
Complete solution:
In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD?
A. 3
B. 15/4
C. 5
D. 16/3
E. 20/3
Attachment:
splittingtriangle.jpg [ 4.22 KiB | Viewed 406593 times ]
Important property: perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle.
Thus, the perpendicular AC divides right triangle ABD into two similar triangles ACB and DCA (which are also similar to big triangle ABD). Now, in these three triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles marked with red and blue on the diagram).
So, $$\frac{CD}{AC}=\frac{AC}{BC}$$ --> $$\frac{CD}{4}=\frac{4}{3}$$ --> $$CD=\frac{16}{3}$$.
For more on this subject please check Triangles chapter of Math Book: math-triangles-87197.html
Hope it helps.
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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20 Jun 2013, 06:18
6
Well i have a problem with similar triangles coz i sometimes make mistakes on the common sides. hence alternative approach for this problem..
Considering Triangle ACD - AC^2 + CD^2 = AD^2
Considering Triangle ABD - AB = sqrt(4^2 + 3^2) = 5 (Pythagorean triplet so you dont really have to do the math on the test)
25 + ac^2 + cd^2 = (3 + cd)^2
=> 25+16+cd^2= 9 + 6cd +cd^2
=>32 = 6cd
cd = 16/3
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##### General Discussion
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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16 Dec 2009, 05:24
5
1
kirankp wrote:
In triangle ABC to the right, if BC = 3 and AC = 4, then what is the length of segment CD?
a.3
b.15/4
c.5
d.16/3
e.20/3
For this problem the solution is :
we have 3 similar triangles the main triangle : ABD two other triangles BC and ADC .
Now to find out CD we can use the later two triangles , so by similarity we have ,
BC/CA = CD/AC
which yields CD as 3.
but the answer is wrong. where have i gone wrong?
D- 16/3
we can use pythagoras theorem to solve this. AB we will be 5.
Let CD = x then AD = sqrt ( 16 + x^2)
in Triangle BAD we have AB^2 + AD^2 = BD^2 => 25 + 16 + x^2 = (3+x)^2
solving the above we get x= 16/3
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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24 Oct 2011, 21:09
3
This is a MGMAT Question,
OA is D and OE is as below. Hope it helps.
Because angles BAD and ACD are right angles, the figure above is composed of three similar right triangles: BAD, ACD and BCA. [Any time a height is dropped from the right angle vertex of a right triangle to the opposite side of that right triangle, the three triangles that result have the same 3 angle measures. This means that they are similar triangles. See below for further explanation.]
To solve for the length of side CD, we can set up a proportion, based on the relationship between the similar triangles ACD and BCA:
BC/CA = CA/CD
3/4 = 4/CD
CD = 16/3
Addendum: Let's look at how we know that triangles ACD and BCA are similar.
1) Let's say that <CDA is x degrees, and <DAC is y degrees. Since <ACD is 90 degrees, and the sum of all the interior angles in a triangle is 180, we know that x + y = 90.
2) Now let's look at <BAC. We know that <BAC + <DAC = 90, since <BAD is labeled as a right angle. We also know that <DAC is y degrees (from step 1), and that x + y = 90. Putting these facts together, we know that <BAC is x degrees.
3) We know <ACB is a right angle, since <ACD is a right angle. Since <ACB is a right angle, <BAC + <CBA = 90. Given that <BAC is x degrees, <CBA must be y degrees.
4) To summarize, <CAB has the same measure as <CDA (x degrees) , and <CBA has the same measure as <DAC (y degrees). This means that in similar triangles CAB and CAD, side BC of CAB corresponds to side CA of CAD, and side CA of CAB corresponds to side CD of CAD.
Thus, BC/CA = CA/CD.
Again, the correct answer is D.
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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Updated on: 03 Nov 2011, 13:20
2
all three triangles abc, acd & abd are similar.
so, $$\frac{4}{3} = \frac{cd}{4}$$ ... cd = $$\frac{16}{3}$$
it can't be $$\frac{4}{3} = \frac{4}{cd}$$ ... cd = 3, because in that case $$5^2+5^2 = 6^2$$ is not true
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Originally posted by MBAhereIcome on 03 Nov 2011, 01:06.
Last edited by MBAhereIcome on 03 Nov 2011, 13:20, edited 1 time in total.
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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03 Nov 2011, 06:49
3
You can immediately tell that AB=5 because ABC is a 3-4-5 triangle. Label CD x, and AD y.
You get:
1) $$5^2+y^2=(3+x)^2$$
2) $$4^2+x^2=y^2$$
Plug in the definition of $$y^2$$ from (2) into 1 and solve. You get 16/3.
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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26 Feb 2012, 08:42
Hi
I need a quick clarification on the concept of perpendicular bisector.
With a perpendicular bisector, the bisector always crosses the line segment at right angles
If any line cuts another line at 90 then it should be a perpendicular bisector right - i.e it divided the line segment into equal halves at 90 ?
So here BC should be the perpendicular bisector and the AC=CD=3 right ?
Please let me know what am missing here.
I do understand the explanations in the other thread mentioned,but can someone clarify as to why AC is not the perpendicular bisector ?
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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21 Apr 2012, 08:26
MBAhereIcome wrote:
all three triangles abc, acd & abd are similar.
so, $$\frac{4}{3} = \frac{cd}{4}$$ ... cd = $$\frac{16}{3}$$
it can't be $$\frac{4}{3} = \frac{4}{cd}$$ ... cd = 3, because in that case $$5^2+5^2 = 6^2$$ is not true
If the 3 triangles are proportional, why can't I solve using the ratio:
$$\frac{AD}{AB} = \frac{AB}{(X+3)}$$
$$\frac{4}{5} = \frac{5}{(X+3)}$$
4(X+3) = 25
4X + 12 = 25
X = $$\frac{13}{4}$$
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Joined: 02 Sep 2009
Posts: 53066
Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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21 Apr 2012, 09:35
pubchum wrote:
MBAhereIcome wrote:
all three triangles abc, acd & abd are similar.
so, $$\frac{4}{3} = \frac{cd}{4}$$ ... cd = $$\frac{16}{3}$$
it can't be $$\frac{4}{3} = \frac{4}{cd}$$ ... cd = 3, because in that case $$5^2+5^2 = 6^2$$ is not true
If the 3 triangles are proportional, why can't I solve using the ratio:
$$\frac{AD}{AB} = \frac{AB}{(X+3)}$$
$$\frac{4}{5} = \frac{5}{(X+3)}$$
4(X+3) = 25
4X + 12 = 25
X = $$\frac{13}{4}$$
Attachment:
splittingtriangle.jpg [ 4.22 KiB | Viewed 359556 times ]
In similar triangles the ratio of the corresponding sides are equal (corresponding sides are the sides opposite the same angles marked with red and blue on the diagram).
The ratios you are equating are not of corresponding sides. If you want to equate AD/AB then it should be AD/AB=AC/BC --> AD/5=4/3 --> AD=20/3. Also AD/AB=CD/AC --> (20/3)/5=CD/4 --> CD=16/3.
I merged this thread with an earlier discussion of the same question, so check this post: in-triangle-abc-if-bc-3-and-ac-4-then-what-is-the-126937.html#p1050155 it might help to clear your doubts.
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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17 May 2012, 04:46
Hi Bunuel,
How did you infer that angle BAC = angle ADC. Could you please explain that part.. I know this can be proved thro similarity, however I wanted to understand from your concept, which you have mentioned below.
Pls explain.
Thanks
H
Bunuel wrote:
rvinodhini wrote:
Hi
I need a quick clarification on the concept of perpendicular bisector.
With a perpendicular bisector, the bisector always crosses the line segment at right angles
If any line cuts another line at 90 then it should be a perpendicular bisector right - i.e it divided the line segment into equal halves at 90 ?
So here BC should be the perpendicular bisector and the AC=CD=3 right ?
Please let me know what am missing here.
I do understand the explanations in the other thread mentioned,but can someone clarify as to why AC is not the perpendicular bisector ?
A perpendicular bisector is a line which cuts a line segment into two equal parts at 90°. So AC to be a perpendicular bisector of BD it must not only cut it at 90° (which it does) but also cut it into two equal parts. Now, in order AC to cut BD into two equal parts right triangle ABD must be isosceles, which, as it turns out after some math, it is not.
Complete solution:
In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD?
A. 3
B. 15/4
C. 5
D. 16/3
E. 20/3
Attachment:
splittingtriangle.jpg
Important property: perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle.
Thus, the perpendicular AC divides right triangle ABD into two similar triangles ACB and DCA (which are also similar to big triangle ABD). Now, in these three triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles marked with red and blue on the diagram).
So, $$\frac{CD}{AC}=\frac{AC}{BC}$$ --> $$\frac{CD}{4}=\frac{4}{3}$$ --> $$CD=\frac{16}{3}$$.
For more on this subject please check Triangles chapter of Math Book: math-triangles-87197.html
Hope it helps.
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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17 May 2012, 04:59
3
imhimanshu wrote:
Hi Bunuel,
How did you infer that angle BAC = angle ADC. Could you please explain that part.. I know this can be proved thro similarity, however I wanted to understand from your concept, which you have mentioned below.
Pls explain.
Thanks
H
Attachment:
splittingtriangle.jpg [ 4.22 KiB | Viewed 359207 times ]
<B+<D+<A=180, since <A=90 then <B+<D=90;
Similarly in triangle ABC: <B+<BAC=90 since <B=90-<D then (90-<D)+<BAC=90 --> <BAC=<D.
Hope it's clear.
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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22 Jun 2013, 07:23
3
1
enigma123 wrote:
Attachment:
Triangle.jpg
In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD?
A. 3
B. 15/4
C. 5
D. 16/3
E. 20/3
Learn this property:
AC^2=BC*DC
=>DC=16/3
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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11 Mar 2015, 23:43
2
the given solution is neat and simple but it didnt strike me when solving
anyways here is another alternate way though i admit its lengthy calculation
let us assume CD=x
(x+3)^2=25+(x^2+16)
x=32/6=16/3
(D)
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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24 Mar 2015, 16:13
Bunuel wrote:
rvinodhini wrote:
Hi
I need a quick clarification on the concept of perpendicular bisector.
With a perpendicular bisector, the bisector always crosses the line segment at right angles
If any line cuts another line at 90 then it should be a perpendicular bisector right - i.e it divided the line segment into equal halves at 90 ?
So here BC should be the perpendicular bisector and the AC=CD=3 right ?
Please let me know what am missing here.
I do understand the explanations in the other thread mentioned,but can someone clarify as to why AC is not the perpendicular bisector ?
A perpendicular bisector is a line which cuts a line segment into two equal parts at 90°. So AC to be a perpendicular bisector of BD it must not only cut it at 90° (which it does) but also cut it into two equal parts. Now, in order AC to cut BD into two equal parts right triangle ABD must be isosceles, which, as it turns out after some math, it is not.
Complete solution:
In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD?
A. 3
B. 15/4
C. 5
D. 16/3
E. 20/3
Attachment:
splittingtriangle.jpg
Important property: perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle.
Thus, the perpendicular AC divides right triangle ABD into two similar triangles ACB and DCA (which are also similar to big triangle ABD). Now, in these three triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles marked with red and blue on the diagram).
So, $$\frac{CD}{AC}=\frac{AC}{BC}$$ --> $$\frac{CD}{4}=\frac{4}{3}$$ --> $$CD=\frac{16}{3}$$.
For more on this subject please check Triangles chapter of Math Book: math-triangles-87197.html
Hope it helps.
This part is clear: triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles)
However, how does one determine which angles are equal? Except 90 degree angles of both triangles, i could not seem to follow how exactly other angles became equal?
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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24 Mar 2015, 20:22
1
earnit wrote:
Bunuel wrote:
rvinodhini wrote:
Hi
I need a quick clarification on the concept of perpendicular bisector.
With a perpendicular bisector, the bisector always crosses the line segment at right angles
If any line cuts another line at 90 then it should be a perpendicular bisector right - i.e it divided the line segment into equal halves at 90 ?
So here BC should be the perpendicular bisector and the AC=CD=3 right ?
Please let me know what am missing here.
I do understand the explanations in the other thread mentioned,but can someone clarify as to why AC is not the perpendicular bisector ?
A perpendicular bisector is a line which cuts a line segment into two equal parts at 90°. So AC to be a perpendicular bisector of BD it must not only cut it at 90° (which it does) but also cut it into two equal parts. Now, in order AC to cut BD into two equal parts right triangle ABD must be isosceles, which, as it turns out after some math, it is not.
Complete solution:
In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD?
A. 3
B. 15/4
C. 5
D. 16/3
E. 20/3
Attachment:
splittingtriangle.jpg
Important property: perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle.
Thus, the perpendicular AC divides right triangle ABD into two similar triangles ACB and DCA (which are also similar to big triangle ABD). Now, in these three triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles marked with red and blue on the diagram).
So, $$\frac{CD}{AC}=\frac{AC}{BC}$$ --> $$\frac{CD}{4}=\frac{4}{3}$$ --> $$CD=\frac{16}{3}$$.
For more on this subject please check Triangles chapter of Math Book: math-triangles-87197.html
Hope it helps.
This part is clear: triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles)
However, how does one determine which angles are equal? Except 90 degree angles of both triangles, i could not seem to follow how exactly other angles became equal?
Both smaller triangles are similar to the large triangle. So they are similar to each other too.
Angle BAD = BCA (90 degrees)
and angle B is common in both
So by AA, triangles BAD and BCA are similar
angle BAD = ACD (90 degrees)
and angle D is common in both
So by AA, triangles BAD and ACD are similar
So triangle BAD is similar to triangle BCA and ACD so triangle BCA is similar to triangle ACD too.
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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27 Mar 2015, 09:54
VeritasPrepKarishma wrote:
Bunuel wrote:
rvinodhini wrote:
Hi
I need a quick clarification on the concept of perpendicular bisector.
With a perpendicular bisector, the bisector always crosses the line segment at right angles
If any line cuts another line at 90 then it should be a perpendicular bisector right - i.e it divided the line segment into equal halves at 90 ?
So here BC should be the perpendicular bisector and the AC=CD=3 right ?
Please let me know what am missing here.
I do understand the explanations in the other thread mentioned,but can someone clarify as to why AC is not the perpendicular bisector ?
A perpendicular bisector is a line which cuts a line segment into two equal parts at 90°. So AC to be a perpendicular bisector of BD it must not only cut it at 90° (which it does) but also cut it into two equal parts. Now, in order AC to cut BD into two equal parts right triangle ABD must be isosceles, which, as it turns out after some math, it is not.
Complete solution:
In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD?
A. 3
B. 15/4
C. 5
D. 16/3
E. 20/3
Attachment:
splittingtriangle.jpg
Important property: perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle.
Thus, the perpendicular AC divides right triangle ABD into two similar triangles ACB and DCA (which are also similar to big triangle ABD). Now, in these three triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles marked with red and blue on the diagram).
So, $$\frac{CD}{AC}=\frac{AC}{BC}$$ --> $$\frac{CD}{4}=\frac{4}{3}$$ --> $$CD=\frac{16}{3}$$.
For more on this subject please check Triangles chapter of Math Book: math-triangles-87197.html
Hope it helps.
This part is clear: triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles)
However, how does one determine which angles are equal? Except 90 degree angles of both triangles, i could not seem to follow how exactly other angles became equal?
Both smaller triangles are similar to the large triangle. So they are similar to each other too.
Angle BAD = BCA (90 degrees)
and angle B is common in both
So by AA, triangles BAD and BCA are similar
angle BAD = ACD (90 degrees)
and angle D is common in both
So by AA, triangles BAD and ACD are similar
So triangle BAD is similar to triangle BCA and ACD so triangle BCA is similar to triangle ACD too.
Hi Karishma,
Thanks for explaining this. I understand how 3 triangles are similar to each other. However how do we determine which side is similar to which in order to set up the ratio.
for example, if we take two smaller triangles - Triangle ABC and ADC....I think side AB is corresponding to AD, BC is corresponding to CD and AC is common. Is that correct? if yes, how do I find which side is corresponding in the smaller triangle with respect to the bigger triangle?
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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30 Mar 2015, 00:40
2
gmatkiller88 wrote:
Thanks for explaining this. I understand how 3 triangles are similar to each other. However how do we determine which side is similar to which in order to set up the ratio.
for example, if we take two smaller triangles - Triangle ABC and ADC....I think side AB is corresponding to AD, BC is corresponding to CD and AC is common. Is that correct? if yes, how do I find which side is corresponding in the smaller triangle with respect to the bigger triangle?
There is a very simple method of finding out the corresponding sides in similar triangles.
Say, you have two triangles ABC and DEF. You find by AA that the triangles are similar. All you have to do is name the triangles the way the angles are equal.
Say angle A = angle E, angle B = angle D and and hence angle C = angle F.
Then we write:
triangle ABC is similar to triangle EDF.
Now you have the corresponding sides. That is, AB/ED = BC/DF = AC/EF
In the question above,
triangles BAD and BCA are similar
So BA/BC = AD/CA = BD/BA
triangles BAD and ACD are similar
triangles BCA is similar to triangle ACD
So BC/AC = CA/CD = BA/AD
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Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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30 Mar 2015, 05:57
enigma123 wrote:
Attachment:
The attachment Triangle.jpg is no longer available
In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD?
A. 3
B. 15/4
C. 5
D. 16/3
E. 20/3
such questions where there is a 90 degree angle in a triangle can always be solved easily by drawing a circle .
as we draw circle and extend AC to point E we will get a triangle BED , exact copy of triangle BAD --Eq1.
as shown in the attached image , Triangle ABC and Triangle ECD are similar as all angles are equal.
so $$\frac{EC}{BC} = \frac{CD}{AC} = \frac{ED}{AB}$$
3x=4y --> $$x=\frac{4}{3}*y$$
y=4 as shown in eq1
so x= $$\frac{16}{3}$$
Attachments
gmatclub.jpg [ 55.25 KiB | Viewed 264839 times ]
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Joined: 04 Jan 2014
Posts: 80
Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink]
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02 Jun 2015, 05:25
1
Hi Bunuel,
please explain me the correlation between 30-60-90(1:sq(3):2), 45-45-90(1:1:sq(2)) with pythagorean triplets(3-4-5, 7-24-25). I am confused because if we have a 90deg triangle, and 2 sides 3,4 we can put other side as 5? If so then it becomes 30-60-90 triangle right? then why we are not able to correlate 1:sq(3):2 with 3:4:5?
because we need to commonly multiply the ratio, so if 1*3 then sq(3) must also multiply by 3 which is not equal to 4..
Because I had tried an another approach which gives a wrong ans... Need clarification.. If ABC is a right triangle with 3,4 then other side must be 5. So Angle BAC will be 30 deg. which makes Ang CAD as 60 and CDA as 30.. So we have 30-60-90 triangle on ACD. If thats the case, then CD must be 4*sqrt(3) right? what's wrong in my approach?
Re: In triangle ABC to the right, if BC = 3 and AC = 4, then what is the [#permalink] 02 Jun 2015, 05:25
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# In triangle ABC to the right, if BC = 3 and AC = 4, then what is the
Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.
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2019-02-22 13:17:40
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https://rdrr.io/cran/fractal/man/poincareMap.html
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# poincareMap: Create a Poincare map
### Description
Create a map using the extrema of a scalar time series.
### Usage
`1` ```poincareMap(x, extrema="min", denoise=FALSE) ```
### Arguments
`x` a vector holding a scalar time series. `denoise` a logical value. If `TRUE`, the data is first denoised via waveshrink prior to analysis. Default: `FALSE`. `extrema` the type of extrema desired. May be "min" for minima, "max" for maxima, or "all" for both maxima and minima. Default: `"min"`.
### Details
This function finds the extrema of a scalar time series to form a map. The time series is assumed to be a uniform sampling of s(t), where s(t) is a (possibly noisy) measurement from a deterministic non-linear system. It is known that s'(t), s''(t), ... are legitimate coordinate vectors in the phase space. Hence the hyperplane given by s'(t)=0 may be used as a Poincare surface of section. The intersections with this plane are exactly the extrema of the time series. The time series minima (or maxima) are the interesections in a given direction and form a map that may be used to estimate invariants, e.g., correlation dimension and Lyapunov exponents, of the underlying non-linear system.
The algorithm used to create a Poincare map is as follows.
1
The first and second derivatives of the resulting series are approximated via the continuous wavelet transform (CWT) using the first derivative of a Gaussian as a mother wavelet filter (see references for details).
2
The locations of the local extrema are then estimated using the standard first and second derivative tests on the CWT coefficients at a single and appropriate scale (an appropriate scale is one that is large enough to smooth out noisy components but not so large as to the oversmooth the data).
3
The extrema locations are then fit with a quadratic interpolation scheme to estimate the amplitude of the extrema using the original time series.
### Value
a list where the first element (`location`) is a vector containing the temporal locations of the extrema values, with respect to sample numbers 1,...N, where N is the length of the original time series. The second element (`amplitude`) is a vector containing the extrema amplitudes.
### References
Holger Kantz and Thomas Schreiber, Nonlinear Time Series Analysis, Cambridge University Press, 1997.
`embedSeries`, `corrDim`, `infoDim`.
``` 1 2 3 4 5 6 7 8 9 10``` ```## Using the third coordinate (\eqn{z} state) of a ## chaotic Lorenz system, form a discrete map ## using the series maxima. Embed the resulting ## extrema in a 2-dimensional delay embedding ## (with delay=1 for a map). The resulting plot ## reveals a tent map structure common to ## Poincare sections of chaotic flows. z <- poincareMap(lorenz[,3], extrema="max") z <- embedSeries(z\$amplitude, tlag=1, dimension=2) plot(z, pch=1, cex=1) ```
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2017-01-21 08:54:03
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http://math.stackexchange.com/questions/125630/how-to-get-from-relation-matrix-in-smith-normal-form-to-direct-product-of-cyclic
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How to get from relation matrix in smith normal form to direct product of cyclic groups?
So...we have an abelian group $G$ with generating set {$g_1,\ldots,g_n$} the following homomorphism has been formed ($F$ is a free group generated by {$f_1,\ldots,f_n$}); \begin{align} \phi:F &\rightarrow G \newline k_1f_1+\ldots +k_nf_n &\mapsto k_1g_1+\ldots +k_ng_n \end{align} Using the first isomorphism theorem $G \cong F/K$ where $K = \operatorname{Ker}(\phi)$. It can be shown that since $K$ is a subgroup of $F$ that it is itself free and finitely generated by {$r_1,\ldots,r_m$}, where $m \leq n$. The generators of $K$ can be expressed as; $$\textbf{r} = A\textbf{f}$$ where $\textbf{r} = (r_1,\ldots,r_m)^T$, $\textbf{f} = (f_1,\ldots f_n)^T$ and $A = (a_{ij})$ is an $m \times n$ matrix. It can be shown that $P\textbf{r}=PA\textbf{f}$ also generates $K$, and also that $P\textbf{r} = PAQ(Q^{-1}\textbf{f})$ generates $K$. So choosing suitable matrices $P$ and $Q$ we can show that $K$ is generated by {$a_1f'_1,\ldots,a_nf'_n$} where $a_i \in \mathbb{N}$ for $i=1,\ldots,m$ and $a_i = 0$ for $i = m+1,\ldots,n$. Now this is where my difficulties begin, that is if what i've outlined up to now makes any sense, using $\phi$ we can investigate the orders of the generating set for $G$. An element $x = k_1g_1+\ldots +k_ng_n \in G$ is equal to the identity iff $k_1f'_1+\ldots +k_nf'_n \in K$ therefore $k_ig_i=e$ iff $k_if'_i \in K$ and $k_if'_i \in K$ iff $a_i |k_i$. So we conclude from this that $|g_i| = a_i$ for $i=1,\ldots ,m$ and $|g_i| = \infty$ for $i = m+1,\ldots,n$, and also that $\langle g_i \rangle \cong \mathbb{Z}/a_i\mathbb{Z}$ for $i=1,\ldots ,m$ and $\langle g_i \rangle \cong \mathbb{Z}$ for $i = m+1,\ldots,n$. Now I need to show that $G \cong \langle g_1 \rangle \oplus \ldots \oplus \langle g_m \rangle \oplus \mathbb{Z}^{n-m}$ but haven't got a clue how to go about it. I know i must show that the intersection $\langle g_i \rangle \cap \langle g_j \rangle=e$ for $i \not = j$ and that any $x \in G$ can be expressed as $k_1g_1+\ldots +k_ng_n$ (this follows from the fact that the set {$g_1,\ldots,g_n$} generates $G$ ?) and that $|G| =|g_i|\ldots|g_n|=a_1\ldots a_n$.
The question boils down to this - given an abelian group $G$ and two distinct elements $g_1,g_2 \in G$ is the intersection of the cyclic subgroups generated by $g_1$ and $g_2$ trivial?
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Your final question has a trivial negative answer: the only groups where that holds are groups of exponent $2$. Otherwise, let $g_1$ be any element that is not equal to its own inverse, and take $g_2=g_1^{-1}$. – Arturo Magidin Mar 28 '12 at 22:24
What you do before the horizontal line does not work: your map sends $k_1f_1+\cdots+k_nf_n$ to $k_1g_1+\cdots+k_ng_n$, but it does not necessarily send $k_1f'_1+\cdots k_nf'_n$ to $k_1g_1+\cdots+k_ng_n$; so your claim that $k_1g_1+\cdots+k_ng_n=0$ if and only if $k_1f'_1+\cdots+k_nf'_n\in K$ is incorrect. As to your final conclusion, you certainly cannot conclude that: start with $G=\mathbb{Z}/4\mathbb{Z}$, $g_1=\overline{1}$ and $g_2=\overline{2}$; you are claiming that $\mathbb{Z}/4\mathbb{Z}\cong\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$.
To see how you went wrong, let's go through your argument using that example above. $F=\mathbb{Z}\times\mathbb{Z}$, $G=\mathbb{Z}/4\mathbb{Z}$, $g_1=\overline{1}$, $g_2=\overline{2}$, $\phi(a,b) = \overline{a+2b}$. The kernel of $\phi$ is generated by $(4,0)$, $(0,2)$, and $(2,1)$. We can do a change of basis for $F$ to the basis $\mathbf{f}'_1=(1,0), \mathbf{f}'_2=(2,1)$; the kernel of $\phi$ is now generated by $4\mathbf{f}'_1$ and $\mathbf{f}'_2$. However, $\phi(a\mathbf{f}'_1+b\mathbf{f}'_2) = \phi(a+2b,b) = \overline{a+4b}\neq ag_1+bg_2$.
What you need to do is that after you replace $f_1,\ldots,f_n$ with $f'_1,\ldots,f'_n$, then you need to replace $g_1,\ldots,g_n$ with $g'_i=\phi(f'_i)$ for $i=1,\ldots,n$. Then your conclusions hold, and the final step is easy: if $\alpha g'_i\in\langle g'_1,\ldots,g'_{i-1},g'_{i+1},\ldots,g'_n\rangle$, then we can express $g'_i$ as combination of the rest, which gives $$b_1g'_1+\cdots + b_{i-1}g'_{i-1}-\alpha g'_i+b_{i+1}g'_{i+1}+\cdots+b_ng_n = 0,$$ hence $$b_1f'_1+\cdots + b_{i-1}f'_{i-1} - \alpha f'_i + b_{i+1}f'_{i+1}+\cdots+b_nf'_n\in K$$ hence $a_i|\alpha$, so $\alpha g'_i=0$. So for each $i$ we have $$\langle g'_i\rangle\cap\langle g_j\mid j\neq i\rangle = \{0\},$$ so the subgroup generated by the $g'_i$ is isomorphic to the direct sum of the cyclic groups generated by the $g'_i$.
But you cannot use the original $g_i$ this way.
The final question, that was posted elsewhere and then closed, has a negative answer in most groups. In fact:
Theorem. Let $G$ be a group such that whenever $x\neq y$ we have $\langle x\rangle \cap \langle y\rangle = \{e\}$. Then $G$ is an abelian group and every element is its own inverse. Assuming the Axiom of Choice, this implies that $G$ is isomorphic to a (possibly infinite) direct sum of copies of the cyclic group of order $2$.
Proof. If $G$ has an element that is not equal to its own inverse, then $\langle x\rangle\cap\langle x^{-1}\rangle\neq\{e\}$, since the intersection contains $x$, and $x\neq e$. Thus, every element of $G$ is of exponent $2$; this is well-known to imply that $G$ is abelian. Hence $G$ is a vector space over $\mathbb{F}_2$; assuming the Axiom of Choice, a basis for $G$ affords a representation of $G$ as a direct sum of copies of the cyclic group of order $2$, as claimed. $\Box$
-
That has got me half way there, I can see that if you assume that some element $x$ from the cyclic subgroup generated by $g'_i$ is equal to a combination of all the other generators then that combination minus $x$ is equal to the identity...and so there is a corresponding element in $K$...i don't get the next part.. – joshua Mar 28 '12 at 23:03
$\alpha g'_i \in \langle g'_i \rangle$, we also assume that $\alpha g'_i \in \langle g'_j \rangle$. If that is the case then $\alpha g'_i = kg'_j$ for some $k \in \mathbb{Z}$, this implies that $\alpha g'_i - kg'_j = 0$ and so there is a corresponding element $\alpha f'_i - kf'_j \in K$. This element is only in $K$ if $a_i | \alpha$ and $a_j | k$, $a_i | \alpha \implies \alpha g'_i = 0$. Therefore the only element that is in both $\langle g'_i \rangle$ and $\langle g'_j \rangle$ is the identity. – joshua Mar 28 '12 at 23:20
@joshua: If $H_i$ are a nonempty family of subgroups of an abelian group $G$, then $G$ is the direct sum of the $H_i$ if and only if (i) $G=\langle H_i\rangle$; and (ii) $H_i\cap\langle H_j\mid j\neq i\rangle = \{0\}$. You don't just check the generators, you need to check the entire subgroup. In order to check the intersection of $\langle g'_i\rangle$ and $\langle g'_j|j\neq i\rangle$, any element of the intersection lies in $\langle g'_i\rangle$, hence can be written as $\alpha g'_i$ for some $\alpha$. (cont) – Arturo Magidin Mar 29 '12 at 2:56
@joshua (cont) You cannot assume that it is merely equal to some $kg'_j$, because it is not enough to check $\langle g_i\rangle \cap \langle g_j\rangle = \{0\}$, you need to check that $\langle g_i\rangle$ intersects trivially with the subgroup generated by **all the other $g'_j$** (not just one). So you need to set it equal to some combination $$k_1g'_i + \cdots + k_{i-1}g'_{i-1}+k_{i+1}g'_{i+1}+\cdots+k_ng_n$$and not merely to a multiple of a single other generator. – Arturo Magidin Mar 29 '12 at 2:58
|
2015-07-07 09:23:12
|
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|
https://physics.stackexchange.com/questions/194183/fundamental-frequency-of-a-material-and-its-youngs-modulus
|
Fundamental frequency of a material and its Young's modulus
I wonder if there is a connection between fundamental frequency and Young's modulus of a material. For example, how to calculate the Young's modulus of a glass bar by knowing its frequency spectrum?
The frequency is a function of the dimensions of the bar and its Young's modulus.
You need to know what mode of oscillation you are exciting in your bar - there is a hug difference between the flexural and longitudinal modes.
If the rod is bending, you can find the formulas here. The derivation goes on and on... but you should be able to use the formula on the first page (for free-free):
$$f = \frac{1}{2\pi}\left(\frac{22.373}{L^2}\right)\sqrt{\frac{EI}{\rho}}$$
In this formula, $I$ is the second moment of area of the rod - see the wiki article for an explanation and to find the appropriate value for the shape of your rod.
If you have a higher mode, you can find the position of two fixed nodes and use the fixed-fixed equation instead.
And if you have longitudinal vibration, the answer is much simpler - you just have to look at the transit time of the sound wave from end to end. One round trip corresponds to the fundamental frequency, so
$$f = \frac{v}{2L} = \sqrt{\frac{E}{4L^2\rho}}\\ E = \rho\; \left(\;2\;L\;f\;\right)^2$$
• Closely related, the classic way to measure crystal elastic constants is ultrasound resonances along different crystal orientations. – Jon Custer Jul 15 '15 at 14:09
• This is a good answer, but I think it would be good to also point out that, depending on the geometry and the mode of vibration, moduli other than Young's modulus (e.g. the shear and uniaxial strain moduli, which for isotropic materials can be expressed in terms of E and the Poisson ratio) will come into play. There's a lot more to material stiffness than Young's modulus. Specifically, if the aspect ratio is not much more than 1, both these expressions have problems. – elifino Jan 31 '16 at 6:56
|
2019-12-05 19:11:58
|
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|
https://codereview.stackexchange.com/questions/140312/relaying-stdin-data-from-one-thread-to-another
|
# Relaying stdin data from one thread to another
I'd like to know if this piece of code is thread-safe. I'm learning something about threads, queues and synchronization, and I'd like to make sure this is correct, before moving forward to next step.
The code is very simple. It just runs two threads (producer and consumer). Producer reads data from stdin (until "0" is entered) and pushes into a queue. Consumer, just prints queue contents each 10 secs.
The way I tried to make it thread-safe:
• using an atomic boolean variable, running, in the consumer loop
• push to and pop from queue using mutex variable, m_queueMutex.
I cannot use c++11 standard. That's why I used boost library.
The code compiles and works as expected using GCC. To compile:
g++ -g -o queues queues.cpp -lboost_thread -lboost_system -lboost_chrono
Any other advice will be very welcome.
### Code:
#include <iostream>
#include <queue>
#include <string>
#include <boost/chrono.hpp>
#include <boost/atomic.hpp>
std::queue<std::string> myqueue;
boost::atomic<bool> running;
boost::mutex m_queueMutex;
void push(std::string& val) {
boost::mutex::scoped_lock lock(m_queueMutex);
myqueue.push(val);
}
std::string pop() {
std::string val = "";
boost::mutex::scoped_lock lock(m_queueMutex);
if (!myqueue.empty()){
val = myqueue.front();
myqueue.pop();
}
return val;
}
void wait(int seconds) {
}
void consume () {
while (running || !myqueue.empty()) {
if (!myqueue.empty()) {
std::string dt = pop();
wait(10);
std::cout << "[CONS] -- Got: " << dt << std::endl;
}
wait (1);
}
std::cout << "[CONS] -- Quiting.. " << std::endl;
}
void produce() {
std::string data;
do {
std::cout << "# ";
std::cin >> data;
if (data != "0") {
push(data);
std::cout << "[PROD] -- Add: " << data << std::endl;
}
} while (data != "0");
std::cout << "[PROD] -- Quiting.. " << std::endl;
running = false;
}
int main() {
std::cout << "Started... " << std::endl;
running = true;
producer.join();
consumer.join();
std::cout << "Ended... " << std::endl;
}
• Why not to go straight for C++14? – Incomputable Sep 2 '16 at 13:47
• @OlzhasZhumabek I can't. I want to add new functionalities to a large existing project, which is built using c++03 and boost. – Albert Sep 5 '16 at 7:32
Right away I can see a potential problem. It appears that it is possible for the consumer thread to access "myqueue" outside of any lock.
void consume () {
while (running || !myqueue.empty()) {
This will probably still function just fine, but it is only as thread-safe as the queue's implementation for 'empty'. Unless the queue itself can guarantee that such an operation is safe, then the code is not thread safe. It is up to the specifics of the queues implementation, and the optimizations done by the compiler / processor, weather or not this would work as expected.
One other minor thing I notice is that the queue is being checked for empty twice. Once in the consumer before the global function pop() is called, and again in the pop function itself. This is a bit confusing to read, and should not be necessary. It may be better to lock the queue first, and then process items later, like so.
void consume()
{
bool hasWork = true;
while (running && hasWork)
{
std::string item = "";
mutex.lock();
if (!myqueue.empty())
{
item = myqueue.pop();
}
else
{
hasWork = false;
}
mutex.unlock();
// Do something with "item"
}
}
The idea here is that you are locking the queue only when you are performing operations on it, and leaving it unlocked while actually doing work on whatever was pulled form the queue. This is especially important when the individual work items may take significant time.
Other than that, it looks like you've got the right idea about needing to protect access to the queue's push and pop methods, it's just that you've got to protect all the other methods that touch 'myqueue' also.
Again, this would probably still work just fine on most implementations, but it's better to know you're safe.
• What if between mutex.lock() ... and ... mutex.unlock() an exception is thrown? That's why I used boost:scoped_lock. By the way, your code has a small mistake myqueue.pop(); does not returns anything. Thanks for comments. – Albert Sep 5 '16 at 7:27
My main problem with this is that myqueue is exposed and easily accessible. This makes it very easy for somebody else to write some other code that uses myqueue without knowing that is potentially being used by two other threads.
You should hide the actual queue inside a class a control all access to the queue object to make sure that it is accessed only in a thread safe manner via member methods.
I would use conditional variables to make sure you don't lock up a processor basically doing a busy wait.
Also C++ has its own thread system (since C++11 now 5 years ago); there is no need to use the boost version anymore this has been subsumed into the standard.
• Thanks. Of course, all will be protected inside a class, but now, I just want to learn how to implement this is a safe manner. Please, can you clarify: "I would use conditional variables to make sure you don't lock up a processor basically doing a busy wait."? – Albert Sep 5 '16 at 7:14
|
2019-11-12 12:39:52
|
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|
https://www.nag.com/numeric/py/nagdoc_latest/_modules/naginterfaces/library/rand.html
|
# Source code for naginterfaces.library.rand
# -*- coding: utf-8 -*-
r"""
Module Summary
--------------
Interfaces for the NAG Mark 27.3 rand Chapter.
rand - Random Number Generators
This module is concerned with the generation of sequences of independent pseudorandom and quasi-random numbers from various distributions, and models.
--------
naginterfaces.library.examples.rand :
This subpackage contains examples for the rand module.
See also the :ref:library_rand_ex subsection.
Functionality Index
-------------------
**Brownian bridge**
circulant embedding generator
generate fractional Brownian motion: :meth:field_fracbm_generate
increments generator
generate Wiener increments: :meth:bb_inc
initialize generator: :meth:bb_inc_init
path generator
create bridge construction order: :meth:bb_make_bridge_order
generate a free or non-free (pinned) Wiener process for a given set of time steps: :meth:bb
initialize generator: :meth:bb_init
**Generating samples, matrices and tables**
permutation of real matrix, vector, vector triplet
:math:K-\ fold cross-validation: :meth:kfold_xyw
random sub-sampling validation: :meth:subsamp_xyw
random correlation matrix: :meth:matrix_corr
random orthogonal matrix: :meth:matrix_orthog
random permutation of an integer vector: :meth:permute
random sample from an integer vector
unequal weights, without replacement: :meth:sample_wgt
unweighted, without replacement: :meth:sample
random table: :meth:matrix_2waytable
**Generation of time series**
asymmetric GARCH Type II: :meth:times_garch_asym2
asymmetric GJR GARCH: :meth:times_garch_gjr
EGARCH: :meth:times_garch_exp
exponential smoothing: :meth:times_smooth_exp
type I AGARCH: :meth:times_garch_asym1
univariate ARMA: :meth:times_arma
vector ARMA: :meth:times_mv_varma
**Pseudorandom numbers**
array of variates from multivariate distributions
Dirichlet distribution: :meth:dist_dirichlet
multinomial distribution: :meth:int_multinomial
Normal distribution: :meth:multivar_normal
Student's :math:t distribution: :meth:multivar_students_t
copulas
Clayton/Cook--Johnson copula (bivariate): :meth:copula_clayton_bivar
Clayton/Cook--Johnson copula (multivariate): :meth:copula_clayton
Frank copula (bivariate): :meth:copula_frank_bivar
Frank copula (multivariate): :meth:copula_frank
Gaussian copula: :meth:copula_normal
Gumbel--Hougaard copula: :meth:copula_gumbel
Plackett copula: :meth:copula_plackett_bivar
Student's :math:t copula: :meth:copula_students_t
initialize generator
multiple streams
leap-frog: :meth:init_leapfrog
skip-ahead: :meth:init_skipahead
skip-ahead (power of 2): :meth:init_skipahead_power2
nonrepeatable sequence: :meth:init_nonrepeat
repeatable sequence: :meth:init_repeat
vector of variates from discrete univariate distributions
binomial distribution: :meth:int_binomial
geometric distribution: :meth:int_geom
hypergeometric distribution: :meth:int_hypergeom
logarithmic distribution: :meth:int_log
logical value :math:\mathbf{True} or :math:\mathbf{False}: :meth:logical
negative binomial distribution: :meth:int_negbin
Poisson distribution: :meth:int_poisson
uniform distribution: :meth:int_uniform
user-supplied distribution: :meth:int_general
variate array from discrete distributions with array of parameters
Poisson distribution with varying mean: :meth:int_poisson_varmean
vectors of variates from continuous univariate distributions
:math:\chi^2 square distribution: :meth:dist_chisq
beta distribution: :meth:dist_beta
Cauchy distribution: :meth:dist_cauchy
exponential mix distribution: :meth:dist_expmix
:math:F-distribution: :meth:dist_f
gamma distribution: :meth:dist_gamma
logistic distribution: :meth:dist_logistic
log-normal distribution: :meth:dist_lognormal
negative exponential distribution: :meth:dist_exp
Normal distribution: :meth:dist_normal
real number from the continuous uniform distribution: :meth:dist_uniform01
Student's :math:t-distribution: :meth:dist_students_t
triangular distribution: :meth:dist_triangular
uniform distribution: :meth:dist_uniform
von Mises distribution: :meth:dist_vonmises
Weibull distribution: :meth:dist_weibull
**Quasi-random numbers**
array of variates from univariate distributions
log-normal distribution: :meth:quasi_lognormal
Normal distribution: :meth:quasi_normal
uniform distribution: :meth:quasi_uniform
initialize generator
scrambled Sobol or Niederreiter: :meth:quasi_init_scrambled
Sobol, Niederreiter or Faure: :meth:quasi_init
**Random fields**
one-dimensional
generation: :meth:field_1d_generate
initialize generator
preset variogram: :meth:field_1d_predef_setup
user-defined variogram: :meth:field_1d_user_setup
two-dimensional
generation: :meth:field_2d_generate
initialize generator
preset variogram: :meth:field_2d_predef_setup
user-defined variogram: :meth:field_2d_user_setup
For full information please refer to the NAG Library document
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html
"""
[docs]def init_repeat(genid, seed, subid=0):
r"""
init_repeat initializes the selected base generator, as used by the group of pseudorandom number functions (see :meth:init_leapfrog--:meth:init_skipahead, :meth:permute, :meth:sample, :meth:times_garch_asym1--:meth:times_mv_varma, :meth:matrix_orthog--:meth:matrix_2waytable, :meth:copula_students_t, :meth:copula_normal, :meth:multivar_students_t, :meth:multivar_normal and :meth:dist_uniform01--:meth:int_uniform) and the quasi-random scrambled sequence initialization function, :meth:quasi_init_scrambled.
.. _g05kf-py2-py-doc:
For full information please refer to the NAG Library document for g05kf
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05kff.html
.. _g05kf-py2-py-parameters:
**Parameters**
**genid** : int
Must contain the type of base generator to use.
:math:\mathrm{genid} = 1
NAG basic generator.
:math:\mathrm{genid} = 2
Wichmann Hill I generator.
:math:\mathrm{genid} = 3
Mersenne Twister.
:math:\mathrm{genid} = 4
Wichmann Hill II generator.
:math:\mathrm{genid} = 5
ACORN generator.
:math:\mathrm{genid} = 6
L'Ecuyer MRG32k3a generator.
See the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html>__ for details of each of the base generators.
**seed** : int, array-like, shape :math:\left(\textit{lseed}\right)
The initial (seed) values for the selected base generator. The number of initial values required varies with each of the base generators.
If :math:\mathrm{genid} = 1, one seed is required.
If :math:\mathrm{genid} = 2, four seeds are required.
If :math:\mathrm{genid} = 3, :math:624 seeds are required.
If :math:\mathrm{genid} = 4, four seeds are required.
If :math:\mathrm{genid} = 5, :math:\left(k+1\right)p seeds are required, where :math:k and :math:p are defined by :math:\mathrm{subid}.
For the ACORN generator it is recommended that an odd value is used for :math:\mathrm{seed}[0].
If :math:\mathrm{genid} = 6, six seeds are required.
If insufficient seeds are provided then the first :math:\textit{lseed}-1 values supplied in :math:\mathrm{seed} are used and the remaining values are randomly generated using the NAG basic generator.
In such cases the NAG basic generator is initialized using the value supplied in :math:\mathrm{seed}[\textit{lseed}-1].
**subid** : int, optional
If :math:\mathrm{genid} = 2, :math:\mathrm{subid} indicates which of the :math:273 sub-generators to use. In this case, the :math:\left(\left(\left\lvert \mathrm{subid}\right\rvert +272\right) mod 273\right)+1 sub-generator is used.
If :math:\mathrm{genid} = 5, :math:\mathrm{subid} indicates the values of :math:k and :math:p to use, where :math:k is the order of the generator, and :math:p controls the size of the modulus, :math:M, with :math:M = 2^{\left(p\times 30\right)}.
If :math:\mathrm{subid} < 1, the default values of :math:k = 10 and :math:p = 2 are used, otherwise values for :math:k and :math:p are calculated from the formula, :math:\mathrm{subid} = k+1000\left(p-1\right).
If :math:\mathrm{genid} = 6 and :math:\mathrm{mod}\left(\mathrm{subid}, 2\right) = 0 the range of the generator is set to :math:\left(0, 1\right], otherwise the range is set to :math:\left(0, 1\right); in this case the sequence is identical to the implementation of MRG32k3a in TestU01 (see L'Ecuyer and Simard (2002)) for identical seeds.
For all other values of :math:\mathrm{genid}, :math:\mathrm{subid} is not referenced.
**Returns**
**statecomm** : dict, RNG communication object
RNG communication structure.
.. _g05kf-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{genid} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{genid} = 1, :math:2, :math:3, :math:4, :math:5 or :math:6.
(errno :math:3)
On entry, invalid :math:\mathrm{seed}.
(errno :math:4)
On entry, :math:\textit{lseed} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\textit{lseed}\geq 1.
.. _g05kf-py2-py-notes:
**Notes**
init_repeat selects a base generator through the input value of the arguments :math:\mathrm{genid} and :math:\mathrm{subid}, and then initializes it based on the values given in the array :math:\mathrm{seed}.
A given base generator will yield different sequences of random numbers if initialized with different values of :math:\mathrm{seed}.
Alternatively, the same sequence of random numbers will be generated if the same value of :math:\mathrm{seed} is used.
It should be noted that there is no guarantee of statistical properties between sequences, only within sequences.
A definition of some of the terms used in this description, along with details of the various base generators can be found in the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html>__.
.. _g05kf-py2-py-references:
**References**
L'Ecuyer, P and Simard, R, 2002, TestU01: a software library in ANSI C for empirical testing of random number generators, Departement d'Informatique et de Recherche Operationnelle, Universite de Montreal, https://www.iro.umontreal.ca/~lecuyer
Maclaren, N M, 1989, The generation of multiple independent sequences of pseudorandom numbers, Appl. Statist. (38), 351--359
Matsumoto, M and Nishimura, T, 1998, Mersenne twister: a 623-dimensionally equidistributed uniform pseudorandom number generator, ACM Transactions on Modelling and Computer Simulations
Wichmann, B A and Hill, I D, 2006, Generating good pseudo-random numbers, Computational Statistics and Data Analysis (51), 1614--1622
Wikramaratna, R S, 1989, ACORN - a new method for generating sequences of uniformly distributed pseudo-random numbers, Journal of Computational Physics (83), 16--31
--------
:meth:naginterfaces.library.examples.correg.glm_binomial_ex.main
:meth:naginterfaces.library.examples.fit.dim2_spline_ts_sctr_ex.main
:meth:naginterfaces.library.examples.rand.bb_inc_ex.main
:meth:naginterfaces.library.examples.rand.copula_students_t_ex.main
"""
raise NotImplementedError
[docs]def init_nonrepeat(genid, subid=0):
r"""
init_nonrepeat initializes the selected base generator to generate a non-repeatable sequence of variates.
The base generator can then be used by the group of pseudorandom number functions (see :meth:init_leapfrog--:meth:init_skipahead, :meth:permute, :meth:sample, :meth:times_garch_asym1--:meth:times_mv_varma, :meth:matrix_orthog--:meth:matrix_2waytable, :meth:copula_students_t, :meth:copula_normal, :meth:multivar_students_t, :meth:multivar_normal and :meth:dist_uniform01--:meth:int_uniform) and the quasi-random scrambled sequence initialization function, :meth:quasi_init_scrambled.
.. _g05kg-py2-py-doc:
For full information please refer to the NAG Library document for g05kg
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05kgf.html
.. _g05kg-py2-py-parameters:
**Parameters**
**genid** : int
Must contain the type of base generator to use.
:math:\mathrm{genid} = 1
NAG basic generator.
:math:\mathrm{genid} = 2
Wichmann Hill I generator.
:math:\mathrm{genid} = 3
Mersenne Twister.
:math:\mathrm{genid} = 4
Wichmann Hill II generator.
:math:\mathrm{genid} = 5
ACORN generator.
:math:\mathrm{genid} = 6
L'Ecuyer MRG32k3a generator.
See the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html>__ for details of each of the base generators.
**subid** : int, optional
If :math:\mathrm{genid} = 2, :math:\mathrm{subid} indicates which of the :math:273 sub-generators to use. In this case, the :math:\left(\left(\left\lvert \mathrm{subid}\right\rvert +272\right) mod 273\right)+1 sub-generator is used.
If :math:\mathrm{genid} = 5, :math:\mathrm{subid} indicates the values of :math:k and :math:p to use, where :math:k is the order of the generator, and :math:p controls the size of the modulus, :math:M, with :math:M = 2^{\left(p\times 30\right)}.
If :math:\mathrm{subid} < 1, the default values of :math:k = 10 and :math:p = 2 are used, otherwise values for :math:k and :math:p are calculated from the formula, :math:\mathrm{subid} = k+1000\left(p-1\right).
If :math:\mathrm{genid} = 6 and :math:\mathrm{mod}\left(\mathrm{subid}, 2\right) = 0 the range of the generator is set to :math:\left(0, 1\right], otherwise the range is set to :math:\left(0, 1\right); in this case the sequence is identical to the implementation of MRG32k3a in TestU01 (see L'Ecuyer and Simard (2002)) for identical seeds.
For all other values of :math:\mathrm{genid}, :math:\mathrm{subid} is not referenced.
**Returns**
**statecomm** : dict, RNG communication object
RNG communication structure.
.. _g05kg-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{genid} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{genid} = 1, :math:2, :math:3, :math:4, :math:5 or :math:6.
.. _g05kg-py2-py-notes:
**Notes**
init_nonrepeat selects a base generator through the input value of the arguments :math:\mathrm{genid} and :math:\mathrm{subid}, and then initializes it based on the values taken from the real-time clock, resulting in the same base generator yielding different sequences of random numbers each time the calling program is run.
It should be noted that there is no guarantee of statistical properties between sequences, only within sequences.
A definition of some of the terms used in this description, along with details of the various base generators can be found in the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html>__.
.. _g05kg-py2-py-references:
**References**
L'Ecuyer, P and Simard, R, 2002, TestU01: a software library in ANSI C for empirical testing of random number generators, Departement d'Informatique et de Recherche Operationnelle, Universite de Montreal, https://www.iro.umontreal.ca/~lecuyer
Maclaren, N M, 1989, The generation of multiple independent sequences of pseudorandom numbers, Appl. Statist. (38), 351--359
Matsumoto, M and Nishimura, T, 1998, Mersenne twister: a 623-dimensionally equidistributed uniform pseudorandom number generator, ACM Transactions on Modelling and Computer Simulations
Wichmann, B A and Hill, I D, 2006, Generating good pseudo-random numbers, Computational Statistics and Data Analysis (51), 1614--1622
Wikramaratna, R S, 1989, ACORN - a new method for generating sequences of uniformly distributed pseudo-random numbers, Journal of Computational Physics (83), 16--31
"""
raise NotImplementedError
[docs]def init_leapfrog(n, k, statecomm):
r"""
init_leapfrog allows for the generation of multiple, independent, sequences of pseudorandom numbers using the leap-frog method.
.. _g05kh-py2-py-doc:
For full information please refer to the NAG Library document for g05kh
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05khf.html
.. _g05kh-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the total number of sequences required.
**k** : int
:math:k, the number of the current sequence.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
.. _g05kh-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 1.
(errno :math:2)
On entry, :math:\mathrm{k} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0 < \mathrm{k}\leq \mathrm{n}.
(errno :math:3)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
(errno :math:4)
On entry, cannot use leap-frog with the base generator defined by :math:\mathrm{statecomm}\ ['state'].
.. _g05kh-py2-py-notes:
**Notes**
init_leapfrog adjusts a base generator to allow multiple, independent, sequences of pseudorandom numbers to be generated via the leap-frog method (see the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html>__ for details).
If, prior to calling init_leapfrog the base generator defined by :math:\mathrm{statecomm}\ ['state'] would produce random numbers :math:x_1,x_2,x_3,\ldots, then after calling init_leapfrog the generator will produce random numbers :math:x_k,x_{{k+n}},x_{{k+2n}},x_{{k+3n}},\ldots.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to init_leapfrog.
The leap-frog algorithm can be used in conjunction with the NAG basic generator, both the Wichmann--Hill I and Wichmann--Hill II generators, the Mersenne Twister and L'Ecuyer.
.. _g05kh-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
r"""
init_skipahead allows for the generation of multiple, independent, sequences of pseudorandom numbers using the skip-ahead method.
The base pseudorandom number sequence defined by :math:\mathrm{statecomm}\ ['state'] is advanced :math:n places.
.. _g05kj-py2-py-doc:
For full information please refer to the NAG Library document for g05kj
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05kjf.html
.. _g05kj-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of places to skip ahead.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
.. _g05kj-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
(errno :math:3)
On entry, cannot use skip-ahead with the base generator defined by :math:\mathrm{statecomm}\ ['state'].
(errno :math:4)
On entry, the base generator is Mersenne Twister, but the :math:\mathrm{statecomm}\ ['state'] vector defined on initialization is not large enough to perform a skip ahead. See the initialization function :meth:init_repeat or :meth:init_nonrepeat.
.. _g05kj-py2-py-notes:
**Notes**
init_skipahead adjusts a base generator to allow multiple, independent, sequences of pseudorandom numbers to be generated via the skip-ahead method (see the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html>__ for details).
If, prior to calling init_skipahead the base generator defined by :math:\mathrm{statecomm}\ ['state'] would produce random numbers :math:x_1,x_2,x_3,\ldots, then after calling init_skipahead the generator will produce random numbers :math:x_{{n+1}},x_{{n+2}},x_{{n+3}},\ldots.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to init_skipahead.
The skip-ahead algorithm can be used in conjunction with any of the six base generators discussed in submodule rand.
.. _g05kj-py2-py-references:
**References**
Haramoto, H, Matsumoto, M, Nishimura, T, Panneton, F and L'Ecuyer, P, 2008, Efficient jump ahead for F2-linear random number generators, INFORMS J. on Computing (20(3)), 385--390
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
r"""
init_skipahead_power2 allows for the generation of multiple, independent, sequences of pseudorandom numbers using the skip-ahead method.
The base pseudorandom number sequence defined by :math:\mathrm{statecomm}\ ['state'] is advanced :math:2^n places.
.. _g05kk-py2-py-doc:
For full information please refer to the NAG Library document for g05kk
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05kkf.html
.. _g05kk-py2-py-parameters:
**Parameters**
**n** : int
:math:n, where the number of places to skip-ahead is defined as :math:2^n.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
.. _g05kk-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
(errno :math:3)
On entry, cannot use skip-ahead with the base generator defined by :math:\mathrm{statecomm}\ ['state'].
(errno :math:4)
On entry, the :math:\mathrm{statecomm}\ ['state'] vector defined on initialization is not large enough to perform a skip-ahead (applies to Mersenne Twister base generator). See the initialization function :meth:init_repeat or :meth:init_nonrepeat.
.. _g05kk-py2-py-notes:
**Notes**
init_skipahead_power2 adjusts a base generator to allow multiple, independent, sequences of pseudorandom numbers to be generated via the skip-ahead method (see the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html>__ for details).
If, prior to calling init_skipahead_power2 the base generator defined by :math:\mathrm{statecomm}\ ['state'] would produce random numbers :math:x_1,x_2,x_3,\ldots, then after calling init_skipahead_power2 the generator will produce random numbers :math:x_{{2^n+1}},x_{{2^n+2}},x_{{2^n+3}},\ldots.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to init_skipahead_power2.
The skip-ahead algorithm can be used in conjunction with any of the six base generators discussed in the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html>__.
.. _g05kk-py2-py-references:
**References**
Haramoto, H, Matsumoto, M, Nishimura, T, Panneton, F and L'Ecuyer, P, 2008, Efficient jump ahead for F2-linear random number generators, INFORMS J. on Computing (20(3)), 385--390
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def permute(indx, statecomm):
r"""
permute performs a pseudorandom permutation of a vector of integers.
.. _g05nc-py2-py-doc:
For full information please refer to the NAG Library document for g05nc
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05ncf.html
.. _g05nc-py2-py-parameters:
**Parameters**
**indx** : int, array-like, shape :math:\left(n\right)
The :math:n integer values to be permuted.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**indx** : int, ndarray, shape :math:\left(n\right)
The :math:n permuted integer values.
.. _g05nc-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:2)
On entry, :math:n = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:n\geq 1.
(errno :math:3)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05nc-py2-py-notes:
**Notes**
permute permutes the elements of an integer array without inspecting their values.
Each of the :math:n! possible permutations of the :math:n values may be regarded as being equally probable.
Even for modest values of :math:n it is theoretically impossible that all :math:n! permutations may occur, as :math:n! is likely to exceed the cycle length of any of the base generators.
For practical purposes this is irrelevant, as the time necessary to generate all possible permutations is many millenia.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to permute.
.. _g05nc-py2-py-references:
**References**
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def sample(m, statecomm, ipop=None, n=None):
r"""
sample selects a pseudorandom sample without replacement from an integer vector.
.. _g05nd-py2-py-doc:
For full information please refer to the NAG Library document for g05nd
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05ndf.html
.. _g05nd-py2-py-parameters:
**Parameters**
**m** : int
The sample size.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**ipop** : None or int, array-like, shape :math:\left(\mathrm{n}\right), optional
The population to be sampled.
If :math:\mathrm{ipop} is **None**, then the population is assumed to be the set of values :math:\left(1,2,\ldots,\mathrm{n}\right) and the array :math:\mathrm{ipop} is not referenced.
**n** : None or int, optional
Note: if this argument is **None** then a default value will be used, determined as follows: :math:\mathrm{ipop}.\mathrm{shape}[0].
The number of elements in the population to be sampled.
**Returns**
**isampl** : int, ndarray, shape :math:\left(\mathrm{m}\right)
The selected sample.
.. _g05nd-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 1.
(errno :math:4)
On entry, :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:1\leq \mathrm{m}\leq \mathrm{n}.
(errno :math:5)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05nd-py2-py-notes:
**Notes**
sample selects :math:m elements from a population vector :math:\mathrm{ipop} of length :math:n and places them in a sample vector :math:\mathrm{isampl}.
Their order in :math:\mathrm{ipop} will be preserved in :math:\mathrm{isampl}.
Each of the (n;m) possible combinations of elements of :math:\mathrm{isampl} may be regarded as being equally probable.
For moderate or large values of :math:n it is theoretically impossible that all combinations of size :math:m may occur, unless :math:m is near :math:1 or near :math:n.
This is because (n;m) exceeds the cycle length of any of the base generators.
For practical purposes this is irrelevant, as the time taken to generate all possible combinations is many millenia.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to sample.
.. _g05nd-py2-py-references:
**References**
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def sample_wgt(order, wt, m, statecomm, ipop=None):
r"""
sample_wgt selects a pseudorandom sample, without replacement and allowing for unequal probabilities.
.. _g05ne-py2-py-doc:
For full information please refer to the NAG Library document for g05ne
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05nef.html
.. _g05ne-py2-py-parameters:
**Parameters**
**order** : str, length 1
A flag indicating the sorted status of the :math:\mathrm{wt} vector.
:math:\mathrm{order} = \text{‘A'}
:math:\mathrm{wt} is sorted in ascending order,
:math:\mathrm{order} = \text{‘D'}
:math:\mathrm{wt} is sorted in descending order,
:math:\mathrm{order} = \text{‘U'}
:math:\mathrm{wt} is unsorted and sample_wgt will sort the weights prior to using them.
Irrespective of the value of :math:\mathrm{order}, no checks are made on the sorted status of :math:\mathrm{wt}, e.g., it is possible to supply :math:\mathrm{order} = \text{‘A'}, even when :math:\mathrm{wt} is not sorted.
In such cases the :math:\mathrm{wt} array will not be sorted internally, but sample_wgt will still work correctly except, possibly, in cases of extreme weight values.
It is usually more efficient to specify a value of :math:\mathrm{order} that is consistent with the status of :math:\mathrm{wt}.
**wt** : float, array-like, shape :math:\left(n\right)
:math:w_i, the relative probability weights. These weights need not sum to :math:1.0.
**m** : int
:math:m, the size of the sample required.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**ipop** : None or int, array-like, shape :math:\left(:\right), optional
Note: the required length for this argument is determined as follows: if :math:\mathrm{ipop}\text{ is not }\mathbf{None}: :math:n; otherwise: :math:0.
The population to be sampled. If :math:\mathrm{ipop}\text{ is }\mathbf{None} then the population is assumed to be the set of values :math:\left(1,2,\ldots,n\right) and the array :math:\mathrm{ipop} is not referenced. Elements of :math:\mathrm{ipop} with the same value are not combined, therefore, if :math:\mathrm{wt}[i-1]\neq 0,\mathrm{wt}[j-1]\neq 0 and :math:i\neq j then there is a nonzero probability that the sample will contain both :math:\mathrm{ipop}[i-1] and :math:\mathrm{ipop}[j-1]. If :math:\mathrm{ipop}[i-1] = \mathrm{ipop}[j-1] then that value can appear in :math:\mathrm{isampl} more than once.
**Returns**
**isampl** : int, ndarray, shape :math:\left(\mathrm{m}\right)
The selected sample.
.. _g05ne-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{order} had an illegal value.
(errno :math:2)
On entry, at least one weight was less than zero.
(errno :math:5)
On entry, :math:n = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:n\geq 1.
(errno :math:7)
On entry, :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle and :math:n = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0\leq \mathrm{m}\leq n.
(errno :math:8)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
(errno :math:21)
On entry, :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle, number of nonzero weights :math:= \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: must be at least :math:\mathrm{m} nonzero weights.
.. _g05ne-py2-py-notes:
**Notes**
sample_wgt selects :math:m elements from either the set of values :math:\left(1,2,\ldots,n\right) or a supplied population vector of length :math:n.
The probability of selecting the :math:i\ th element is proportional to a user-supplied weight, :math:w_i.
Each element will appear at most once in the sample, i.e., the sampling is done without replacement.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to sample_wgt.
"""
raise NotImplementedError
[docs]def times_garch_asym1(dist, num, ip, iq, theta, gamma, df, fcall, comm, statecomm):
r"""
times_garch_asym1 generates a given number of terms of a type I :math:\text{AGARCH}\left(p, q\right) process (see Engle and Ng (1993)).
.. _g05pd-py2-py-doc:
For full information please refer to the NAG Library document for g05pd
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pdf.html
.. _g05pd-py2-py-parameters:
**Parameters**
**dist** : str, length 1
The type of distribution to use for :math:\epsilon_t.
:math:\mathrm{dist} = \texttt{'N'}
A Normal distribution is used.
:math:\mathrm{dist} = \texttt{'T'}
A Student's :math:t-distribution is used.
**num** : int
:math:T, the number of terms in the sequence.
**ip** : int
The number of coefficients, :math:\beta_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,p.
**iq** : int
The number of coefficients, :math:\alpha_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,q.
**theta** : float, array-like, shape :math:\left(\mathrm{iq}+\mathrm{ip}+1\right)
The first element must contain the coefficient :math:\alpha_o, the next :math:\mathrm{iq} elements must contain the coefficients :math:\alpha_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,q. The remaining :math:\mathrm{ip} elements must contain the coefficients :math:\beta_{\textit{j}}, for :math:\textit{j} = 1,2,\ldots,p.
**gamma** : float
The asymmetry parameter :math:\gamma for the :math:\text{GARCH}\left(p, q\right) sequence.
**df** : int
The number of degrees of freedom for the Student's :math:t-distribution.
If :math:\mathrm{dist} = \texttt{'N'}, :math:\mathrm{df} is not referenced.
**fcall** : bool
If :math:\mathrm{fcall} = \mathbf{True}, a new sequence is to be generated, otherwise a given sequence is to be continued using the information in :math:\mathrm{comm}\ ['r'].
**comm** : dict, communication object, modified in place
Communication structure for the reference vector.
If :math:\mathrm{fcall} = \mathbf{False}, this argument must have been initialized by a prior call to times_garch_asym1.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**ht** : float, ndarray, shape :math:\left(\mathrm{num}\right)
The conditional variances :math:h_{\textit{t}}, for :math:\textit{t} = 1,2,\ldots,T, for the :math:\text{GARCH}\left(p, q\right) sequence.
**et** : float, ndarray, shape :math:\left(\mathrm{num}\right)
The observations :math:\epsilon_{\textit{t}}, for :math:\textit{t} = 1,2,\ldots,T, for the :math:\text{GARCH}\left(p, q\right) sequence.
.. _g05pd-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{dist} is not valid: :math:\mathrm{dist} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:2)
On entry, :math:\mathrm{num} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{num}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{ip} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{ip}\geq 0.
(errno :math:4)
On entry, :math:\mathrm{iq} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{iq}\geq 1.
(errno :math:5)
On entry, :math:\mathrm{theta}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{theta}[i-1]\geq 0.0.
(errno :math:7)
On entry, :math:\mathrm{df} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{df}\geq 3.
(errno :math:11)
:math:\mathrm{ip} or :math:\mathrm{iq} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:\mathrm{ip} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{ip} = \langle\mathit{\boldsymbol{value}}\rangle.
Previous value of :math:\mathrm{iq} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{iq} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:13)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
(errno :math:51)
On entry, sum of :math:\mathrm{theta}[\textit{i}-1], for :math:\textit{i} = 2,3,\ldots,\mathrm{ip}+\mathrm{iq}+1 is :math:\text{}\geq 1.0: :math:\mathrm{sum} = \langle\mathit{\boldsymbol{value}}\rangle.
.. _g05pd-py2-py-notes:
**Notes**
A type I :math:\text{AGARCH}\left(p, q\right) process can be represented by:
.. math::
h_t = \alpha_0+\sum_{{i = 1}}^q\alpha_i\left(\epsilon_{{t-i}}+\gamma \right)^2+\sum_{{i = 1}}^p\beta_ih_{{t-i}}\text{, }\quad t = 1,2,\ldots,T\text{;}
where :math:\epsilon_t | \psi_{{t-1}} = N\left(0, h_t\right) or :math:\epsilon_t | \psi_{{t-1}} = S_t\left({\textit{df}}, h_t\right).
Here :math:S_t is a standardized Student's :math:t-distribution with :math:{\textit{df}} degrees of freedom and variance :math:h_t, :math:T is the number of observations in the sequence, :math:\epsilon_t is the observed value of the :math:\text{GARCH}\left(p, q\right) process at time :math:t, :math:h_t is the conditional variance at time :math:t, and :math:\psi_t the set of all information up to time :math:t.
Symmetric GARCH sequences are generated when :math:\gamma is zero, otherwise asymmetric GARCH sequences are generated with :math:\gamma specifying the amount by which positive (or negative) shocks are to be enhanced.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to times_garch_asym1.
.. _g05pd-py2-py-references:
**References**
Bollerslev, T, 1986, Generalised autoregressive conditional heteroskedasticity, Journal of Econometrics (31), 307--327
Engle, R, 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica (50), 987--1008
Engle, R and Ng, V, 1993, Measuring and testing the impact of news on volatility, Journal of Finance (48), 1749--1777
Hamilton, J, 1994, Time Series Analysis, Princeton University Press
"""
raise NotImplementedError
[docs]def times_garch_asym2(dist, num, ip, iq, theta, gamma, df, fcall, comm, statecomm):
r"""
times_garch_asym2 generates a given number of terms of a type II :math:\text{AGARCH}\left(p, q\right) process (see Engle and Ng (1993)).
.. _g05pe-py2-py-doc:
For full information please refer to the NAG Library document for g05pe
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pef.html
.. _g05pe-py2-py-parameters:
**Parameters**
**dist** : str, length 1
The type of distribution to use for :math:\epsilon_t.
:math:\mathrm{dist} = \texttt{'N'}
A Normal distribution is used.
:math:\mathrm{dist} = \texttt{'T'}
A Student's :math:t-distribution is used.
**num** : int
:math:T, the number of terms in the sequence.
**ip** : int
The number of coefficients, :math:\beta_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,p.
**iq** : int
The number of coefficients, :math:\alpha_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,q.
**theta** : float, array-like, shape :math:\left(\mathrm{iq}+\mathrm{ip}+1\right)
The first element must contain the coefficient :math:\alpha_o, the next :math:\mathrm{iq} elements must contain the coefficients :math:\alpha_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,q. The remaining :math:\mathrm{ip} elements must contain the coefficients :math:\beta_{\textit{j}}, for :math:\textit{j} = 1,2,\ldots,p.
**gamma** : float
The asymmetry parameter :math:\gamma for the :math:\text{GARCH}\left(p, q\right) sequence.
**df** : int
The number of degrees of freedom for the Student's :math:t-distribution.
If :math:\mathrm{dist} = \texttt{'N'}, :math:\mathrm{df} is not referenced.
**fcall** : bool
If :math:\mathrm{fcall} = \mathbf{True}, a new sequence is to be generated, otherwise a given sequence is to be continued using the information in :math:\mathrm{comm}\ ['r'].
**comm** : dict, communication object, modified in place
Communication structure for the reference vector.
If :math:\mathrm{fcall} = \mathbf{False}, this argument must have been initialized by a prior call to times_garch_asym2.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**ht** : float, ndarray, shape :math:\left(\mathrm{num}\right)
The conditional variances :math:h_{\textit{t}}, for :math:\textit{t} = 1,2,\ldots,T, for the :math:\text{GARCH}\left(p, q\right) sequence.
**et** : float, ndarray, shape :math:\left(\mathrm{num}\right)
The observations :math:\epsilon_{\textit{t}}, for :math:\textit{t} = 1,2,\ldots,T, for the :math:\text{GARCH}\left(p, q\right) sequence.
.. _g05pe-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{dist} is not valid: :math:\mathrm{dist} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:2)
On entry, :math:\mathrm{num} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{num}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{ip} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{ip}\geq 0.
(errno :math:4)
On entry, :math:\mathrm{iq} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{iq}\geq 1.
(errno :math:5)
On entry, :math:\mathrm{theta}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{theta}[i-1]\geq 0.0.
(errno :math:7)
On entry, :math:\mathrm{df} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{df}\geq 3.
(errno :math:11)
:math:\mathrm{ip} or :math:\mathrm{iq} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:\mathrm{ip} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{ip} = \langle\mathit{\boldsymbol{value}}\rangle.
Previous value of :math:\mathrm{iq} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{iq} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:13)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
(errno :math:51)
On entry, sum of :math:\mathrm{theta}[\textit{i}-1], for :math:\textit{i} = 2,3,\ldots,\mathrm{ip}+\mathrm{iq}+1 is :math:\text{}\geq 1.0: :math:\mathrm{sum} = \langle\mathit{\boldsymbol{value}}\rangle.
.. _g05pe-py2-py-notes:
**Notes**
A type II :math:\text{AGARCH}\left(p, q\right) process can be represented by:
.. math::
h_t = \alpha_0+\sum_{1}^{q}{\alpha_i\left(\left\lvert \epsilon_{{t-i}}\right\rvert +\gamma \epsilon_{{t-i}}\right)^2}+\sum_{1}^{p}{\beta_ih_{{t-i}}}\text{, }\quad t = 1,2,\ldots,T\text{;}
where :math:\epsilon_t | \psi_{{t-1}} = N\left(0, h_t\right) or :math:\epsilon_t | \psi_{{t-1}} = S_t\left({\textit{df}}, h_t\right).
Here :math:S_t is a standardized Student's :math:t-distribution with :math:{\textit{df}} degrees of freedom and variance :math:h_t, :math:T is the number of observations in the sequence, :math:\epsilon_t is the observed value of the :math:\text{GARCH}\left(p, q\right) process at time :math:t, :math:h_t is the conditional variance at time :math:t, and :math:\psi_t the set of all information up to time :math:t.
Symmetric GARCH sequences are generated when :math:\gamma is zero, otherwise asymmetric GARCH sequences are generated with :math:\gamma specifying the amount by which positive (or negative) shocks are to be enhanced.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to times_garch_asym2.
.. _g05pe-py2-py-references:
**References**
Bollerslev, T, 1986, Generalised autoregressive conditional heteroskedasticity, Journal of Econometrics (31), 307--327
Engle, R, 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica (50), 987--1008
Engle, R and Ng, V, 1993, Measuring and testing the impact of news on volatility, Journal of Finance (48), 1749--1777
Hamilton, J, 1994, Time Series Analysis, Princeton University Press
"""
raise NotImplementedError
[docs]def times_garch_gjr(dist, num, ip, iq, theta, gamma, df, fcall, comm, statecomm):
r"""
times_garch_gjr generates a given number of terms of a GJR :math:\text{GARCH}\left(p, q\right) process (see Glosten et al. (1993)).
.. _g05pf-py2-py-doc:
For full information please refer to the NAG Library document for g05pf
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pff.html
.. _g05pf-py2-py-parameters:
**Parameters**
**dist** : str, length 1
The type of distribution to use for :math:\epsilon_t.
:math:\mathrm{dist} = \texttt{'N'}
A Normal distribution is used.
:math:\mathrm{dist} = \texttt{'T'}
A Student's :math:t-distribution is used.
**num** : int
:math:T, the number of terms in the sequence.
**ip** : int
The number of coefficients, :math:\beta_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,p.
**iq** : int
The number of coefficients, :math:\alpha_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,q.
**theta** : float, array-like, shape :math:\left(\mathrm{iq}+\mathrm{ip}+1\right)
The first element must contain the coefficient :math:\alpha_o, the next :math:\mathrm{iq} elements must contain the coefficients :math:\alpha_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,q. The remaining :math:\mathrm{ip} elements must contain the coefficients :math:\beta_{\textit{j}}, for :math:\textit{j} = 1,2,\ldots,p.
**gamma** : float
The asymmetry parameter :math:\gamma for the :math:\text{GARCH}\left(p, q\right) sequence.
**df** : int
The number of degrees of freedom for the Student's :math:t-distribution.
If :math:\mathrm{dist} = \texttt{'N'}, :math:\mathrm{df} is not referenced.
**fcall** : bool
If :math:\mathrm{fcall} = \mathbf{True}, a new sequence is to be generated, otherwise a given sequence is to be continued using the information in :math:\mathrm{comm}\ ['r'].
**comm** : dict, communication object, modified in place
Communication structure for the reference vector.
If :math:\mathrm{fcall} = \mathbf{False}, this argument must have been initialized by a prior call to times_garch_gjr.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**ht** : float, ndarray, shape :math:\left(\mathrm{num}\right)
The conditional variances :math:h_{\textit{t}}, for :math:\textit{t} = 1,2,\ldots,T, for the :math:\text{GARCH}\left(p, q\right) sequence.
**et** : float, ndarray, shape :math:\left(\mathrm{num}\right)
The observations :math:\epsilon_{\textit{t}}, for :math:\textit{t} = 1,2,\ldots,T, for the :math:\text{GARCH}\left(p, q\right) sequence.
.. _g05pf-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{dist} is not valid: :math:\mathrm{dist} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:2)
On entry, :math:\mathrm{num} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{num}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{ip} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{ip}\geq 0.
(errno :math:4)
On entry, :math:\mathrm{iq} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{iq}\geq 1.
(errno :math:5)
On entry, :math:\mathrm{theta}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle and :math:\gamma = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\alpha_i+\gamma \geq 0.
(errno :math:7)
On entry, :math:\mathrm{df} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{df}\geq 3.
(errno :math:11)
:math:\mathrm{ip} or :math:\mathrm{iq} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:\mathrm{ip} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{ip} = \langle\mathit{\boldsymbol{value}}\rangle.
Previous value of :math:\mathrm{iq} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{iq} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:13)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
(errno :math:51)
On entry, :math:\mathrm{theta}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{theta}[i]\geq 0.0.
(errno :math:52)
On entry, sum of :math:\mathrm{theta}[i] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: sum of :math:\mathrm{theta}[\textit{i}], for :math:\textit{i} = 1,2,\ldots,\mathrm{ip}+\mathrm{iq} is :math:\text{} < 1.0.
.. _g05pf-py2-py-notes:
**Notes**
A GJR :math:\text{GARCH}\left(p, q\right) process is represented by:
.. math::
h_t = \alpha_0+\sum_{{i = 1}}^q\left(\alpha_i+\gamma I_{{t-i}}\right)\epsilon_{{t-i}}^2+\sum_{{i = 1}}^p\beta_ih_{{t-i}}\text{, }\quad t = 1,2,\ldots,T\text{;}
where :math:I_t = 1 if :math:\epsilon_t < 0, :math:I_t = 0 if :math:\epsilon_t\geq 0, and :math:\epsilon_t | \psi_{{t-1}} = N\left(0, h_t\right) or :math:\epsilon_t | \psi_{{t-1}} = S_t\left({\textit{df}}, h_t\right).
Here :math:S_t is a standardized Student's :math:t-distribution with :math:{\textit{df}} degrees of freedom and variance :math:h_t, :math:T is the number of observations in the sequence, :math:\epsilon_t is the observed value of the :math:\text{GARCH}\left(p, q\right) process at time :math:t, :math:h_t is the conditional variance at time :math:t, and :math:\psi_t the set of all information up to time :math:t.
Symmetric GARCH sequences are generated when :math:\gamma is zero, otherwise asymmetric GARCH sequences are generated with :math:\gamma specifying the amount by which negative shocks are to be enhanced.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to times_garch_gjr.
.. _g05pf-py2-py-references:
**References**
Bollerslev, T, 1986, Generalised autoregressive conditional heteroskedasticity, Journal of Econometrics (31), 307--327
Engle, R, 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica (50), 987--1008
Engle, R and Ng, V, 1993, Measuring and testing the impact of news on volatility, Journal of Finance (48), 1749--1777
Glosten, L, Jagannathan, R and Runkle, D, 1993, Relationship between the expected value and the volatility of nominal excess return on stocks, Journal of Finance (48), 1779--1801
Hamilton, J, 1994, Time Series Analysis, Princeton University Press
"""
raise NotImplementedError
[docs]def times_garch_exp(dist, num, ip, iq, theta, df, fcall, comm, statecomm):
r"""
times_garch_exp generates a given number of terms of an exponential :math:\text{GARCH}\left(p, q\right) process (see Engle and Ng (1993)).
.. _g05pg-py2-py-doc:
For full information please refer to the NAG Library document for g05pg
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pgf.html
.. _g05pg-py2-py-parameters:
**Parameters**
**dist** : str, length 1
The type of distribution to use for :math:\epsilon_t.
:math:\mathrm{dist} = \texttt{'N'}
A Normal distribution is used.
:math:\mathrm{dist} = \texttt{'T'}
A Student's :math:t-distribution is used.
**num** : int
:math:T, the number of terms in the sequence.
**ip** : int
The number of coefficients, :math:\beta_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,p.
**iq** : int
The number of coefficients, :math:\alpha_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,q.
**theta** : float, array-like, shape :math:\left(2\times \mathrm{iq}+\mathrm{ip}+1\right)
The initial parameter estimates for the vector :math:\theta. The first element must contain the coefficient :math:\alpha_o and the next :math:\mathrm{iq} elements must contain the autoregressive coefficients :math:\alpha_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,q. The next :math:\mathrm{iq} elements must contain the coefficients :math:\phi_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,q. The next :math:\mathrm{ip} elements must contain the moving average coefficients :math:\beta_{\textit{j}}, for :math:\textit{j} = 1,2,\ldots,p.
**df** : int
The number of degrees of freedom for the Student's :math:t-distribution.
If :math:\mathrm{dist} = \texttt{'N'}, :math:\mathrm{df} is not referenced.
**fcall** : bool
If :math:\mathrm{fcall} = \mathbf{True}, a new sequence is to be generated, otherwise a given sequence is to be continued using the information in :math:\mathrm{comm}\ ['r'].
**comm** : dict, communication object, modified in place
Communication structure for the reference vector.
If :math:\mathrm{fcall} = \mathbf{False}, this argument must have been initialized by a prior call to times_garch_exp.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**ht** : float, ndarray, shape :math:\left(\mathrm{num}\right)
The conditional variances :math:h_{\textit{t}}, for :math:\textit{t} = 1,2,\ldots,T, for the :math:\text{GARCH}\left(p, q\right) sequence.
**et** : float, ndarray, shape :math:\left(\mathrm{num}\right)
The observations :math:\epsilon_{\textit{t}}, for :math:\textit{t} = 1,2,\ldots,T, for the :math:\text{GARCH}\left(p, q\right) sequence.
.. _g05pg-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{dist} is not valid: :math:\mathrm{dist} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:2)
On entry, :math:\mathrm{num} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{num}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{ip} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{ip}\geq 0.
(errno :math:4)
On entry, :math:\mathrm{iq} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{iq}\geq 1.
(errno :math:6)
On entry, :math:\mathrm{df} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{df}\geq 3.
(errno :math:10)
:math:\mathrm{ip} or :math:\mathrm{iq} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:\mathrm{ip} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{ip} = \langle\mathit{\boldsymbol{value}}\rangle.
Previous value of :math:\mathrm{iq} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{iq} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:12)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
(errno :math:20)
Invalid sequence generated, use different parameters.
.. _g05pg-py2-py-notes:
**Notes**
An exponential :math:\text{GARCH}\left(p, q\right) process is represented by:
.. math::
ln\left(h_t\right) = \alpha_0+\sum_{{i = 1}}^q\alpha_iz_{{t-i}}+\sum_{{i = 1}}^q\phi_i\left(\left\lvert z_{{t-i}}\right\rvert -E\left[\left\lvert z_{{t-i}}\right\rvert \right]\right)+\sum_{{j = 1}}^p\beta_jln\left(h_{{t-j}}\right)\text{, }\quad t = 1,2,\ldots,T\text{;}
where :math:z_t = \frac{\epsilon_t}{{\sqrt{h_t}}}, :math:E\left[\left\lvert z_{{t-i}}\right\rvert \right] denotes the expected value of :math:\left\lvert z_{{t-i}}\right\rvert, and :math:\epsilon_t | \psi_{{t-1}} = N\left(0, h_t\right) or :math:\epsilon_t | \psi_{{t-1}} = S_t\left({\textit{df}}, h_t\right).
Here :math:S_t is a standardized Student's :math:t-distribution with :math:{\textit{df}} degrees of freedom and variance :math:h_t, :math:T is the number of observations in the sequence, :math:\epsilon_t is the observed value of the :math:\text{GARCH}\left(p, q\right) process at time :math:t, :math:h_t is the conditional variance at time :math:t, and :math:\psi_t the set of all information up to time :math:t.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to times_garch_exp.
.. _g05pg-py2-py-references:
**References**
Bollerslev, T, 1986, Generalised autoregressive conditional heteroskedasticity, Journal of Econometrics (31), 307--327
Engle, R, 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica (50), 987--1008
Engle, R and Ng, V, 1993, Measuring and testing the impact of news on volatility, Journal of Finance (48), 1749--1777
Glosten, L, Jagannathan, R and Runkle, D, 1993, Relationship between the expected value and the volatility of nominal excess return on stocks, Journal of Finance (48), 1779--1801
Hamilton, J, 1994, Time Series Analysis, Princeton University Press
"""
raise NotImplementedError
[docs]def times_arma(mode, n, xmean, phi, theta, avar, comm, statecomm):
r"""
times_arma generates a realization of a univariate time series from an autoregressive moving average (ARMA) model.
The realization may be continued or a new realization generated at subsequent calls to times_arma.
.. _g05ph-py2-py-doc:
For full information please refer to the NAG Library document for g05ph
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05phf.html
.. _g05ph-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector only.
:math:\mathrm{mode} = 1
Generate terms in the time series using reference vector set up in a prior call to times_arma.
:math:\mathrm{mode} = 2
Set up reference vector and generate terms in the time series.
**n** : int
:math:n, the number of observations to be generated.
**xmean** : float
The mean of the time series.
**phi** : float, array-like, shape :math:\left(\textit{ip}\right)
The autoregressive coefficients of the model, :math:\phi_1,\phi_2,\ldots,\phi_p.
**theta** : float, array-like, shape :math:\left(\textit{iq}\right)
The moving average coefficients of the model, :math:\theta_1,\theta_2,\ldots,\theta_q.
**avar** : float
:math:\sigma^2, the variance of the Normal perturbations.
**comm** : dict, communication object, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1, this argument must have been initialized by a prior call to times_arma.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**var** : float
The proportion of the variance of a term in the series that is due to the moving-average (error) terms in the model. The smaller this is, the nearer is the model to non-stationarity.
**x** : None or float, ndarray, shape :math:\left(\mathrm{n}\right)
Contains the next :math:n observations from the time series.
.. _g05ph-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1 or :math:2.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:4)
On entry, :math:\textit{ip} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\textit{ip}\geq 0.
(errno :math:5)
On entry, the AR parameters are outside the stationarity region.
(errno :math:6)
On entry, :math:\textit{iq} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\textit{iq}\geq 0.
(errno :math:8)
On entry, :math:\mathrm{avar} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{avar}\geq 0.0.
(errno :math:9)
Reference vector :math:\mathrm{comm}\ ['r'] has been corrupted or not initialized correctly.
(errno :math:9)
:math:\textit{ip} or :math:\textit{iq} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:\textit{ip} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\textit{ip} = \langle\mathit{\boldsymbol{value}}\rangle.
Previous value of :math:\textit{iq} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\textit{iq} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:11)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05ph-py2-py-notes:
**Notes**
Let the vector :math:x_t, denote a time series which is assumed to follow an autoregressive moving average (ARMA) model of the form:
.. math::
\begin{array}{cc}x_t-\mu = &\phi_1\left(x_{{t-1}}-\mu \right)+\phi_2\left(x_{{t-2}}-\mu \right)+ \cdots +\phi_p\left(x_{{t-p}}-\mu \right)+\\&\epsilon_t-\theta_1\epsilon_{{t-1}}-\theta_2\epsilon_{{t-2}} - \cdots -\theta_q\epsilon_{{t-q}}\end{array}
where :math:\epsilon_t, is a residual series of independent random perturbations assumed to be Normally distributed with zero mean and variance :math:\sigma^2.
The parameters :math:\left\{\phi_i\right\}, for :math:\textit{i} = 1,2,\ldots,p, are called the autoregressive (AR) parameters, and :math:\left\{\theta_j\right\}, for :math:\textit{j} = 1,2,\ldots,q, the moving average (MA) parameters.
The parameters in the model are thus the :math:p :math:\phi values, the :math:q :math:\theta values, the mean :math:\mu and the residual variance :math:\sigma^2.
times_arma sets up a reference vector containing initial values corresponding to a stationary position using the method described in Tunnicliffe--Wilson (1979).
The function can then return a realization of :math:x_1,x_2,\ldots,x_n.
On a successful exit, the recent history is updated and saved in the reference vector :math:\mathrm{comm}\ ['r'] so that times_arma may be called again to generate a realization of :math:x_{{n+1}},x_{{n+2}},\ldots, etc.
See the description of the argument :math:\mathrm{mode} in :ref:Parameters <g05ph-py2-py-parameters> for details.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to times_arma.
.. _g05ph-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
Tunnicliffe--Wilson, G, 1979, Some efficient computational procedures for high order ARMA models, J. Statist. Comput. Simulation (8), 301--309
"""
raise NotImplementedError
[docs]def times_mv_varma(mode, n, xmean, ip, phi, iq, theta, var, comm, statecomm):
r"""
times_mv_varma generates a realization of a multivariate time series from a vector autoregressive moving average (VARMA) model.
The realization may be continued or a new realization generated at subsequent calls to times_mv_varma.
.. _g05pj-py2-py-doc:
For full information please refer to the NAG Library document for g05pj
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pjf.html
.. _g05pj-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector and compute a realization of the recent history.
:math:\mathrm{mode} = 1
Generate terms in the time series using reference vector set up in a prior call to times_mv_varma.
:math:\mathrm{mode} = 2
Combine the operations of :math:\mathrm{mode} = 0 or :math:1.
:math:\mathrm{mode} = 3
A new realization of the recent history is computed using information stored in the reference vector, and the following sequence of time series values are generated.
If :math:\mathrm{mode} = 1 or :math:3, you must ensure that the reference vector :math:\mathrm{comm}\ ['r'] and the values of :math:\textit{k}, :math:\mathrm{ip}, :math:\mathrm{iq}, :math:\mathrm{xmean}, :math:\mathrm{phi}, :math:\mathrm{theta}, :math:\mathrm{var} and :math:\textit{ldvar} have not been changed between calls to times_mv_varma.
**n** : int
:math:n, the number of observations to be generated.
**xmean** : float, array-like, shape :math:\left(k\right)
:math:\mu, the vector of means of the multivariate time series.
**ip** : int
:math:p, the number of autoregressive parameter matrices.
**phi** : float, array-like, shape :math:\left(k\times k\times \mathrm{ip}\right)
Must contain the elements of the :math:\mathrm{ip}\times k\times k autoregressive parameter matrices of the model, :math:\phi_1,\phi_2,\ldots,\phi_p. If :math:\mathrm{phi} is considered as a three-dimensional array, dimensioned as :math:\mathrm{phi}[k-1,k-1,\mathrm{ip}-1], the :math:\left(i, j\right)\ th element of :math:\phi_{\textit{l}} would be stored in :math:\mathrm{phi}[i-1,j-1,\textit{l}-1]; that is, :math:\mathrm{phi}[\left(\textit{l}-1\right)\times k\times k+\left(j-1\right)\times k+i-1] must be set equal to the :math:\left(i, j\right)\ th element of :math:\phi_{\textit{l}}, for :math:\textit{l} = 1,2,\ldots,p, :math:i = 1,2,\ldots,k and :math:j = 1,2,\ldots,k.
**iq** : int
:math:q, the number of moving average parameter matrices.
**theta** : float, array-like, shape :math:\left(k\times k\times \mathrm{iq}\right)
Must contain the elements of the :math:\mathrm{iq}\times k\times k moving average parameter matrices of the model, :math:\theta_1,\theta_2,\ldots,\theta_q. If :math:\mathrm{theta} is considered as a three-dimensional array, dimensioned as :math:\mathrm{theta}\ (:math:\textit{k},\ :math:\textit{k},\ :math:\mathrm{iq}), the :math:\left(i, j\right)\ th element of :math:\theta_{\textit{l}} would be stored in :math:\mathrm{theta}[\textit{i}-1,\textit{j}-1,\textit{l}-1]; that is, :math:\mathrm{theta}[\left(\textit{l}-1\right)\times k\times k+\left(\textit{j}-1\right)\times k+\textit{i}-1] must be set equal to the :math:\left(\textit{i}, \textit{j}\right)\ th element of :math:\theta_{\textit{l}}, for :math:\textit{j} = 1,2,\ldots,k, for :math:\textit{i} = 1,2,\ldots,k, for :math:\textit{l} = 1,2,\ldots,q.
**var** : float, array-like, shape :math:\left(k, k\right)
:math:\mathrm{var}[\textit{i}-1,\textit{j}-1] must contain the (:math:\textit{i},\textit{j})th element of :math:\Sigma, for :math:\textit{j} = 1,2,\ldots,k, for :math:\textit{i} = 1,2,\ldots,k. Only the lower triangle is required.
**comm** : dict, communication object, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1 or :math:3, this argument must have been initialized by a prior call to times_mv_varma.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(k, \mathrm{n}\right)
:math:\mathrm{x}[\textit{i}-1,\textit{t}-1] will contain a realization of the :math:\textit{i}\ th component of :math:X_{\textit{t}}, for :math:\textit{t} = 1,2,\ldots,n, for :math:\textit{i} = 1,2,\ldots,k.
.. _g05pj-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1, :math:2 or :math:3.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:k = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:k\geq 1.
(errno :math:5)
On entry, :math:\mathrm{ip} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{ip}\geq 0.
(errno :math:6)
On entry, the AR parameters are outside the stationarity region.
(errno :math:7)
On entry, :math:\mathrm{iq} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{iq}\geq 0.
(errno :math:8)
On entry, the moving average parameter matrices are such that the model is non-invertible.
(errno :math:9)
On entry, the covariance matrix :math:\mathrm{var} is not positive semidefinite to machine precision.
(errno :math:11)
:math:\textit{k} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:k = \langle\mathit{\boldsymbol{value}}\rangle and :math:k = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:13)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
(errno :math:20)
An excessive number of iterations were required by the NAG function used to evaluate the eigenvalues of the matrices used to test for stationarity or invertibility.
(errno :math:21)
The reference vector cannot be computed because the AR parameters are too close to the boundary of the stationarity region.
(errno :math:22)
An excessive number of iterations were required by the NAG function used to evaluate the eigenvalues of the covariance matrix.
**Warns**
**NagAlgorithmicWarning**
(errno :math:23)
An excessive number of iterations were required by the NAG function used to evaluate the eigenvalues stored in the reference vector.
.. _g05pj-py2-py-notes:
**Notes**
Let the vector :math:X_t = \left(x_{{1t}}, x_{{2t}}, \ldots, x_{{kt}}\right)^\mathrm{T}, denote a :math:k-dimensional time series which is assumed to follow a vector autoregressive moving average (VARMA) model of the form:
.. math::
\begin{array}{cc}X_t-\mu = &\phi_1\left(X_{{t-1}}-\mu \right)+\phi_2\left(X_{{t-2}}-\mu \right)+ \cdots +\phi_p\left(X_{{t-p}}-\mu \right)+\\&\epsilon_t-\theta_1\epsilon_{{t-1}}-\theta_2\epsilon_{{t-2}} - \cdots -\theta_q\epsilon_{{t-q}}\end{array}
where :math:\epsilon_t = \left(\epsilon_{{1t}}, \epsilon_{{2t}}, \ldots, \epsilon_{{kt}}\right)^\mathrm{T}, is a vector of :math:k residual series assumed to be Normally distributed with zero mean and covariance matrix :math:\Sigma.
The components of :math:\epsilon_t are assumed to be uncorrelated at non-simultaneous lags.
The :math:\phi_i's and :math:\theta_j's are :math:k\times k matrices of parameters. :math:\left\{\phi_i\right\}, for :math:\textit{i} = 1,2,\ldots,p, are called the autoregressive (AR) parameter matrices, and :math:\left\{\theta_j\right\}, for :math:\textit{j} = 1,2,\ldots,q, the moving average (MA) parameter matrices.
The parameters in the model are thus the :math:p :math:k\times k :math:\phi-matrices, the :math:q :math:k\times k :math:\theta-matrices, the mean vector :math:\mu and the residual error covariance matrix :math:\Sigma.
Let
.. math::
A\left(\phi \right) = \left[\begin{array}{ccccccc}\phi_1&I&0&.&.&.&0\\\phi_2&0&I&0&.&.&0\\.&&&.&&&\\.&&&&.&&\\.&&&&&.&\\\phi_{{p-1}}&0&.&.&.&0&I\\\phi_p&0&.&.&.&0&0\end{array}\right]_{{pk\times pk}}\quad \text{ and }\quad B\left(\theta \right) = \left[\begin{array}{ccccccc}\theta_1&I&0&.&.&.&0\\\theta_2&0&I&0&.&.&0\\.&&&.&&&\\.&&&&.&&\\.&&&&&.&\\\theta_{{q-1}}&0&.&.&.&0&I\\\theta_q&0&.&.&.&0&0\end{array}\right]_{{qk\times qk}}
where :math:I denotes the :math:k\times k identity matrix.
The model (1) <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pjf.html#eqn1>__ must be both stationary and invertible.
The model is said to be stationary if the eigenvalues of :math:A\left(\phi \right) lie inside the unit circle and invertible if the eigenvalues of :math:B\left(\theta \right) lie inside the unit circle.
For :math:k\geq 6 the VARMA model (1) <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pjf.html#eqn1>__ is recast into state space form and a realization of the state vector at time zero computed.
For all other cases the function computes a realization of the pre-observed vectors :math:X_0,X_{-1},\ldots,X_{{1-p}}, :math:\epsilon_0,\epsilon_{-1},\ldots,\epsilon_{{1-q}}, from (1) <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pjf.html#eqn1>__, see Shea (1988).
This realization is then used to generate a sequence of successive time series observations.
Note that special action is taken for pure MA models, that is for :math:p = 0.
At your request a new realization of the time series may be generated more efficiently using the information in a reference vector created during a previous call to times_mv_varma.
See the description of the argument :math:\mathrm{mode} in :ref:Parameters <g05pj-py2-py-parameters> for details.
The function returns a realization of :math:X_1,X_2,\ldots,X_n.
On a successful exit, the recent history is updated and saved in the array :math:\mathrm{comm}\ ['r'] so that times_mv_varma may be called again to generate a realization of :math:X_{{n+1}},X_{{n+2}},\ldots, etc.
See the description of the argument :math:\mathrm{mode} in :ref:Parameters <g05pj-py2-py-parameters> for details.
Further computational details are given in Shea (1988).
Note, however, that times_mv_varma uses a spectral decomposition rather than a Cholesky factorization to generate the multivariate Normals.
Although this method involves more multiplications than the Cholesky factorization method and is thus slightly slower it is more stable when faced with ill-conditioned covariance matrices. A method of assigning the AR and MA coefficient matrices so that the stationarity and invertibility conditions are satisfied is described in Barone (1987).
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to times_mv_varma.
.. _g05pj-py2-py-references:
**References**
Barone, P, 1987, A method for generating independent realisations of a multivariate normal stationary and invertible ARMA :math:\left(p, q\right) process, J. Time Ser. Anal. (8), 125--130
Shea, B L, 1988, A note on the generation of independent realisations of a vector autoregressive moving average process, J. Time Ser. Anal. (9), 403--410
"""
raise NotImplementedError
[docs]def times_smooth_exp(mode, n, itype, p, param, init, var, comm, statecomm, e):
r"""
times_smooth_exp simulates from an exponential smoothing model, where the model uses either single exponential, double exponential or a Holt--Winters method.
.. _g05pm-py2-py-doc:
For full information please refer to the NAG Library document for g05pm
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pmf.html
.. _g05pm-py2-py-parameters:
**Parameters**
**mode** : int
Indicates if times_smooth_exp is continuing from a previous call or, if not, how the initial values are computed.
:math:\mathrm{mode} = 0
Values for :math:m_0, :math:r_0 and :math:s_{{-\textit{j}}}, for :math:\textit{j} = 0,1,\ldots,p-1, are supplied in :math:\mathrm{init}.
:math:\mathrm{mode} = 1
times_smooth_exp continues from a previous call using values that are supplied in :math:\mathrm{comm}\ ['r']. :math:\mathrm{comm}\ ['r'] is not updated.
:math:\mathrm{mode} = 2
times_smooth_exp continues from a previous call using values that are supplied in :math:\mathrm{comm}\ ['r']. :math:\mathrm{comm}\ ['r'] is updated.
**n** : int
The number of terms of the time series being generated.
**itype** : int
The smoothing function.
:math:\mathrm{itype} = 1
Single exponential.
:math:\mathrm{itype} = 2
Brown's double exponential.
:math:\mathrm{itype} = 3
Linear Holt.
:math:\mathrm{itype} = 4
:math:\mathrm{itype} = 5
Multiplicative Holt--Winters.
**p** : int
If :math:\mathrm{itype} = 4 or :math:5, the seasonal order, :math:p, otherwise :math:\mathrm{p} is not referenced.
**param** : float, array-like, shape :math:\left(:\right)
Note: the required length for this argument is determined as follows: if :math:\mathrm{itype}\text{ in } (1, 2): :math:1; if :math:\mathrm{itype}=3: :math:3; if :math:\mathrm{itype}\text{ in } (4, 5): :math:4; otherwise: :math:0.
The smoothing parameters.
If :math:\mathrm{itype} = 1 or :math:2, :math:\mathrm{param}[0] = \alpha and any remaining elements of :math:\mathrm{param} are not referenced.
If :math:\mathrm{itype} = 3, :math:\mathrm{param}[0] = \alpha, :math:\mathrm{param}[1] = \gamma, :math:\mathrm{param}[2] = \phi and any remaining elements of :math:\mathrm{param} are not referenced.
If :math:\mathrm{itype} = 4 or :math:5, :math:\mathrm{param}[0] = \alpha, :math:\mathrm{param}[1] = \gamma, :math:\mathrm{param}[2] = \beta and :math:\mathrm{param}[3] = \phi and any remaining elements of :math:\mathrm{param} are not referenced.
**init** : float, array-like, shape :math:\left(:\right)
Note: the required length for this argument is determined as follows: if :math:\mathrm{itype}=1: :math:1; if :math:\mathrm{itype}\text{ in } (2, 3): :math:2; if :math:\mathrm{itype}\text{ in } (4, 5): :math:{2+\mathrm{p}}; otherwise: :math:0.
If :math:\mathrm{mode} = 0, the initial values for :math:m_0, :math:r_0 and :math:s_{{-\textit{j}}}, for :math:\textit{j} = 0,1,\ldots,p-1, used to initialize the smoothing.
If :math:\mathrm{itype} = 1, :math:\mathrm{init}[0] = m_0 and any remaining elements of :math:\mathrm{init} are not referenced.
If :math:\mathrm{itype} = 2 or :math:3, :math:\mathrm{init}[0] = m_0 and :math:\mathrm{init}[1] = r_0 and any remaining elements of :math:\mathrm{init} are not referenced.
If :math:\mathrm{itype} = 4 or :math:5, :math:\mathrm{init}[0] = m_0, :math:\mathrm{init}[1] = r_0 and :math:\mathrm{init}[2] to :math:\mathrm{init}[2+p-1] hold the values for :math:s_{{-\textit{j}}}, for :math:\textit{j} = 0,1,\ldots,p-1.
Any remaining elements of :math:\mathrm{init} are not referenced.
**var** : float
The variance, :math:\sigma^2 of the Normal distribution used to generate the errors :math:\epsilon_i. If :math:\mathrm{var}\leq 0.0 then Normally distributed errors are not used.
**comm** : dict, communication object, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1 or :math:2, this argument must have been initialized by a prior call to times_smooth_exp.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**e** : float, array-like, shape :math:\left(\textit{en}\right)
If :math:\textit{en} > 0 and :math:\mathrm{var}\leq 0.0, a vector from which the errors, :math:\epsilon_t are randomly drawn, with replacement.
If :math:\textit{en}\leq 0, :math:\mathrm{e} is not referenced.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The generated time series, :math:x_{\textit{t}}, for :math:\textit{t} = 1,2,\ldots,n.
.. _g05pm-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1 or :math:2.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{itype} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{itype} = 1, :math:2, :math:3, :math:4 or :math:5.
(errno :math:4)
On entry, :math:\mathrm{p} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: if :math:\mathrm{itype} = 4 or :math:5, :math:\mathrm{p}\geq 2.
(errno :math:5)
On entry, :math:\mathrm{param}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0\leq \mathrm{param}[i]\leq 1.
(errno :math:5)
On entry, :math:\mathrm{param}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: if :math:\mathrm{itype} = 2, :math:0 < \mathrm{param}[i]\leq 1.
(errno :math:5)
On entry, :math:\mathrm{param}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{param}[i]\geq 0.
(errno :math:8)
On entry, some of the elements of the array :math:\mathrm{comm}\ ['r'] have been corrupted or have not been initialized.
(errno :math:9)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
(errno :math:12)
Model unsuitable for multiplicative Holt--Winter, try a different set of parameters.
.. _g05pm-py2-py-notes:
**Notes**
times_smooth_exp returns :math:\left\{x_t:t = 1,2,\ldots,n\right\}, a realization of a time series from an exponential smoothing model defined by one of five smoothing functions:
Single Exponential Smoothing
.. math::
\begin{array}{ccc}x_t& = & m_{{t-1}} + \epsilon_t \\m_t& = & \alpha x_t + \left(1-\alpha \right) m_{{t-1}} \end{array}
Brown Double Exponential Smoothing
.. math::
\begin{array}{ccc}x_t& = & m_{{t-1}} + \frac{r_{{t-1}}}{\alpha } + \epsilon_t \\m_t& = & \alpha x_t + \left(1-\alpha \right) m_{{t-1}} \\r_t& = & \alpha \left(m_t-m_{{t-1}}\right) + \left(1-\alpha \right) r_{{t-1}} \end{array}
Linear Holt Exponential Smoothing
.. math::
\begin{array}{ccc}x_t& = & m_{{t-1}} + \phi r_{{t-1}} + \epsilon_t \\m_t& = & \alpha x_t + \left(1-\alpha \right) \left(m_{{t-1}}+\phi r_{{t-1}}\right) \\r_t& = & \gamma \left(m_t-m_{{t-1}}\right) + \left(1-\gamma \right) \phi r_{{t-1}} \end{array}
.. math::
\begin{array}{ccc}x_t& = & m_{{t-1}} + \phi r_{{t-1}} + s_{{t-1-p}} + \epsilon_t \\ m_t & = & \alpha \left(x_t-s_{{t-p}}\right) + \left(1-\alpha \right) \left(m_{{t-1}}+\phi r_{{t-1}}\right) \\r_t& = & \gamma \left(m_t-m_{{t-1}}\right) + \left(1-\gamma \right) \phi r_{{t-1}} \\s_t& = & \beta \left(x_t-m_t\right) + \left(1-\beta \right) s_{{t-p}} \end{array}
Multiplicative Holt--Winters Smoothing
.. math::
\begin{array}{ccc}x_t& = & \left(m_{{t-1}}+\phi r_{{t-1}}\right) \times s_{{t-1-p}} + \epsilon_t \\m_t& = & \alpha x_t / s_{{t-p}} + \left(1-\alpha \right) \left(m_{{t-1}}+\phi r_{{t-1}}\right) \\r_t& = & \gamma \left(m_t-m_{{t-1}}\right) + \left(1-\gamma \right) \phi r_{{t-1}} \\s_t& = & \beta x_t / m_t + \left(1-\beta \right) s_{{t-p}} \end{array}
where :math:m_t is the mean, :math:r_t is the trend and :math:s_t is the seasonal component at time :math:t with :math:p being the seasonal order.
The errors, :math:\epsilon_t are either drawn from a normal distribution with mean zero and variance :math:\sigma^2 or randomly sampled, with replacement, from a user-supplied vector.
.. _g05pm-py2-py-references:
**References**
Chatfield, C, 1980, The Analysis of Time Series, Chapman and Hall
"""
raise NotImplementedError
[docs]def kfold_xyw(k, fold, x, statecomm, sordx=1, y=None, w=None):
r"""
kfold_xyw generates training and validation datasets suitable for use in cross-validation or jack-knifing.
.. _g05pv-py2-py-doc:
For full information please refer to the NAG Library document for g05pv
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pvf.html
.. _g05pv-py2-py-parameters:
**Parameters**
**k** : int
:math:K, the number of folds.
**fold** : int
The number of the fold to return as the validation dataset.
On the first call to kfold_xyw :math:\mathrm{fold} should be set to :math:1 and then incremented by one at each subsequent call until all :math:K sets of training and validation datasets have been produced.
See Further Comments <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pvf.html#fcomments>__ for more details on how a different calling sequence can be used.
**x** : float, ndarray, shape :math:\left(:, :\right), modified in place
Note: the required extent for this argument in dimension 1 is determined as follows: if :math:\mathrm{sordx}=2: :math:m; otherwise: :math:n.
Note: the required extent for this argument in dimension 2 is determined as follows: if :math:\mathrm{sordx}=1: :math:m; if :math:\mathrm{sordx}=2: :math:n; otherwise: :math:0.
The way the data is stored in :math:\mathrm{x} is defined by :math:\mathrm{sordx}.
If :math:\mathrm{sordx} = 1, :math:\mathrm{x}[\textit{i}-1,\textit{j}-1] contains the :math:\textit{i}\ th observation for the :math:\textit{j}\ th variable, for :math:i = 1,2,\ldots,n and :math:j = 1,2,\ldots,m.
If :math:\mathrm{sordx} = 2, :math:\mathrm{x}[\textit{j}-1,\textit{i}-1] contains the :math:\textit{i}\ th observation for the :math:\textit{j}\ th variable, for :math:i = 1,2,\ldots,n and :math:j = 1,2,\ldots,m.
On entry: if :math:\mathrm{fold} = 1, :math:\mathrm{x} must hold :math:X_o, the values of :math:X for the original dataset, otherwise, :math:\mathrm{x} must not be changed since the last call to kfold_xyw.
On exit: values of :math:X for the training and validation datasets, with :math:X_t held in observations :math:1 to :math:\mathrm{nt} and :math:X_v in observations :math:\mathrm{nt}+1 to :math:n.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**sordx** : int, optional
Determines how variables are stored in :math:\mathrm{x}.
**y** : None or float, ndarray, shape :math:\left(:\right), optional, modified in place
Note: the required length for this argument is determined as follows: if :math:\mathrm{y}\text{ is not }\mathbf{None}: :math:n; otherwise: :math:0.
If the original dataset does not include :math:y_o then :math:\mathrm{y} must be set to **None**.
Optionally, on entry: :math:y_o, the values of :math:y for the original dataset. If :math:\mathrm{fold} \neq 1, :math:\mathrm{y} must hold the vector returned in :math:\textit{sy} by the last call to kfold_xyw.
On exit, if not **None** on entry: values of :math:y for the training and validation datasets, with :math:y_t held in elements :math:1 to :math:\mathrm{nt} and :math:y_v in elements :math:\mathrm{nt}+1 to :math:n.
**w** : None or float, ndarray, shape :math:\left(:\right), optional, modified in place
Note: the required length for this argument is determined as follows: if :math:\mathrm{w}\text{ is not }\mathbf{None}: :math:n; otherwise: :math:0.
Optionally, on entry: if :math:\mathrm{fold} \neq 1, :math:\mathrm{w} must hold the vector returned in :math:\textit{sw} by the last call to kfold_xyw.
On exit, if not **None** on entry: values of :math:w for the training and validation datasets, with :math:w_t held in elements :math:1 to :math:\mathrm{nt} and :math:w_v in elements :math:\mathrm{nt}+1 to :math:n.
**Returns**
**nt** : int
:math:n_t, the number of observations in the training dataset.
.. _g05pv-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:11)
On entry, :math:\mathrm{k} = \langle\mathit{\boldsymbol{value}}\rangle and :math:n = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:2\leq \mathrm{k}\leq n.
(errno :math:21)
On entry, :math:\mathrm{fold} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{k} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:1\leq \mathrm{fold}\leq \mathrm{k}.
(errno :math:31)
On entry, :math:n = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:n\geq 1.
(errno :math:41)
On entry, :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:m\geq 1.
(errno :math:51)
On entry, :math:\mathrm{sordx} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{sordx} = 1 or :math:2.
(errno :math:71)
On entry, :math:\textit{ldx} = \langle\mathit{\boldsymbol{value}}\rangle and :math:n = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: if :math:\mathrm{sordx} = 1, :math:\textit{ldx}\geq n.
(errno :math:72)
On entry, :math:\textit{ldx} = \langle\mathit{\boldsymbol{value}}\rangle and :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: if :math:\mathrm{sordx} = 2, :math:\textit{ldx}\geq m.
(errno :math:131)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
**Warns**
**NagAlgorithmicWarning**
(errno :math:61)
More than :math:50\% of the data did not move when the data was shuffled. :math:\langle\mathit{\boldsymbol{value}}\rangle of the :math:\langle\mathit{\boldsymbol{value}}\rangle observations stayed put.
.. _g05pv-py2-py-notes:
**Notes**
Let :math:X_o denote a matrix of :math:n observations on :math:m variables and :math:y_o and :math:w_o each denote a vector of length :math:n.
For example, :math:X_o might represent a matrix of independent variables, :math:y_o the dependent variable and :math:w_o the associated weights in a weighted regression.
kfold_xyw generates a series of training datasets, denoted by the matrix, vector, vector triplet :math:\left(X_t,y_t,w_t\right) of :math:n_t observations, and validation datasets, denoted :math:\left(X_v,y_v,w_v\right) with :math:n_v observations.
These training and validation datasets are generated as follows.
Each of the original :math:n observations is randomly assigned to one of :math:K equally sized groups or folds.
For the :math:k\ th sample the validation dataset consists of those observations in group :math:k and the training dataset consists of all those observations not in group :math:k.
Therefore, at most :math:K samples can be generated.
If :math:n is not divisible by :math:K then the observations are assigned to groups as evenly as possible, therefore, any group will be at most one observation larger or smaller than any other group.
When using :math:K = n the resulting datasets are suitable for leave-one-out cross-validation, or the training dataset on its own for jack-knifing.
When using :math:K\neq n the resulting datasets are suitable for :math:K-fold cross-validation.
Datasets suitable for reversed cross-validation can be obtained by switching the training and validation datasets, i.e., use the :math:k\ th group as the training dataset and the rest of the data as the validation dataset.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to kfold_xyw.
--------
:meth:naginterfaces.library.examples.correg.glm_binomial_ex.main
"""
raise NotImplementedError
[docs]def subsamp_xyw(nt, x, statecomm, sordx=1, y=None, w=None):
r"""
subsamp_xyw generates a dataset suitable for use with repeated random sub-sampling validation.
.. _g05pw-py2-py-doc:
For full information please refer to the NAG Library document for g05pw
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pwf.html
.. _g05pw-py2-py-parameters:
**Parameters**
**nt** : int
:math:n_t, the number of observations in the training dataset.
**x** : float, ndarray, shape :math:\left(:, :\right), modified in place
Note: the required extent for this argument in dimension 1 is determined as follows: if :math:\mathrm{sordx}=2: :math:m; otherwise: :math:n.
Note: the required extent for this argument in dimension 2 is determined as follows: if :math:\mathrm{sordx}=1: :math:m; if :math:\mathrm{sordx}=2: :math:n; otherwise: :math:0.
The way the data is stored in :math:\mathrm{x} is defined by :math:\mathrm{sordx}.
If :math:\mathrm{sordx} = 1, :math:\mathrm{x}[\textit{i}-1,\textit{j}-1] contains the :math:\textit{i}\ th observation for the :math:\textit{j}\ th variable, for :math:i = 1,2,\ldots,n and :math:j = 1,2,\ldots,m.
If :math:\mathrm{sordx} = 2, :math:\mathrm{x}[\textit{j}-1,\textit{i}-1] contains the :math:\textit{i}\ th observation for the :math:\textit{j}\ th variable, for :math:i = 1,2,\ldots,n and :math:j = 1,2,\ldots,m.
On entry: :math:\mathrm{x} must hold :math:X_o, the values of :math:X for the original dataset. This may be the same :math:\mathrm{x} as updated by a previous call to subsamp_xyw.
On exit: values of :math:X for the training and validation datasets, with :math:X_t held in observations :math:1 to :math:\mathrm{nt} and :math:X_v in observations :math:\mathrm{nt}+1 to :math:n.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**sordx** : int, optional
Determines how variables are stored in :math:\mathrm{x}.
**y** : None or float, ndarray, shape :math:\left(:\right), optional, modified in place
Note: the required length for this argument is determined as follows: if :math:\mathrm{y}\text{ is not }\mathbf{None}: :math:n; otherwise: :math:0.
Optionally, on entry: :math:\mathrm{y} must hold :math:y_o, the values of :math:y for the original dataset. This may be the same :math:\mathrm{y} as updated by a previous call to subsamp_xyw.
On exit, if not **None** on entry: values of :math:y for the training and validation datasets, with :math:y_t held in elements :math:1 to :math:\mathrm{nt} and :math:y_v in elements :math:\mathrm{nt}+1 to :math:n.
**w** : None or float, ndarray, shape :math:\left(:\right), optional, modified in place
Note: the required length for this argument is determined as follows: if :math:\mathrm{w}\text{ is not }\mathbf{None}: :math:n; otherwise: :math:0.
Optionally, on entry: :math:\mathrm{w} must hold :math:w_o, the values of :math:w for the original dataset. This may be the same :math:\mathrm{w} as updated by a previous call to subsamp_xyw.
On exit, if not **None** on entry: values of :math:w for the training and validation datasets, with :math:w_t held in elements :math:1 to :math:\mathrm{nt} and :math:w_v in elements :math:\mathrm{nt}+1 to :math:n.
.. _g05pw-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:11)
On entry, :math:\mathrm{nt} = \langle\mathit{\boldsymbol{value}}\rangle and :math:n = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:1\leq \mathrm{nt}\leq n.
(errno :math:21)
On entry, :math:n = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:n\geq 1.
(errno :math:31)
On entry, :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:m\geq 1.
(errno :math:41)
On entry, :math:\mathrm{sordx} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{sordx} = 1 or :math:2.
(errno :math:61)
On entry, :math:\textit{ldx} = \langle\mathit{\boldsymbol{value}}\rangle and :math:n = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: if :math:\mathrm{sordx} = 1, :math:\textit{ldx}\geq n.
(errno :math:62)
On entry, :math:\textit{ldx} = \langle\mathit{\boldsymbol{value}}\rangle and :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: if :math:\mathrm{sordx} = 2, :math:\textit{ldx}\geq m.
(errno :math:111)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05pw-py2-py-notes:
**Notes**
Let :math:X_o denote a matrix of :math:n observations on :math:m variables and :math:y_o and :math:w_o each denote a vector of length :math:n.
For example, :math:X_o might represent a matrix of independent variables, :math:y_o the dependent variable and :math:w_o the associated weights in a weighted regression.
subsamp_xyw generates a series of training datasets, denoted by the matrix, vector, vector triplet :math:\left(X_t,y_t,w_t\right) of :math:n_t observations, and validation datasets, denoted :math:\left(X_v,y_v,w_v\right) with :math:n_v observations.
These training and validation datasets are generated by randomly assigning each observation to either the training dataset or the validation dataset.
The resulting datasets are suitable for use with repeated random sub-sampling validation.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to subsamp_xyw.
"""
raise NotImplementedError
[docs]def matrix_orthog(side, init, statecomm, a):
r"""
matrix_orthog generates a random orthogonal matrix.
.. _g05px-py2-py-doc:
For full information please refer to the NAG Library document for g05px
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pxf.html
.. _g05px-py2-py-parameters:
**Parameters**
**side** : str, length 1
Indicates whether the matrix :math:A is multiplied on the left or right by the random orthogonal matrix :math:U.
:math:\mathrm{side} = \texttt{'L'}
The matrix :math:A is multiplied on the left, i.e., premultiplied.
:math:\mathrm{side} = \texttt{'R'}
The matrix :math:A is multiplied on the right, i.e., post-multiplied.
**init** : str, length 1
Indicates whether or not :math:\mathrm{a} should be initialized to the identity matrix.
:math:\mathrm{init} = \texttt{'I'}
:math:\mathrm{a} is initialized to the identity matrix.
:math:\mathrm{init} = \texttt{'N'}
:math:\mathrm{a} is not initialized and the matrix :math:A must be supplied in :math:\mathrm{a}.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**a** : float, array-like, shape :math:\left(m, n\right)
If :math:\mathrm{init} = \texttt{'N'}, :math:\mathrm{a} must contain the matrix :math:A.
**Returns**
**a** : float, ndarray, shape :math:\left(m, n\right)
The matrix :math:UA when :math:\mathrm{side} = \texttt{'L'} or the matrix :math:AU when :math:\mathrm{side} = \texttt{'R'}.
.. _g05px-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{side} is not valid: :math:\mathrm{side} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:2)
On entry, :math:\mathrm{init} is not valid: :math:\mathrm{init} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:3)
On entry, :math:\mathrm{side} = \langle\mathit{\boldsymbol{value}}\rangle, :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: if :math:\mathrm{side} = \texttt{'L'}, :math:m > 1; otherwise :math:m\geq 1.
(errno :math:4)
On entry, :math:\mathrm{side} = \langle\mathit{\boldsymbol{value}}\rangle, :math:n = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: if :math:\mathrm{side} = \texttt{'R'}, :math:n > 1; otherwise :math:n\geq 1.
(errno :math:5)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
(errno :math:8)
On entry, :math:\mathrm{side} = \langle\mathit{\boldsymbol{value}}\rangle, :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: if :math:\mathrm{side} = \texttt{'L'}, :math:m > 1; otherwise :math:m\geq 1.
(errno :math:8)
On entry, :math:\mathrm{side} = \langle\mathit{\boldsymbol{value}}\rangle, :math:n = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: if :math:\mathrm{side} = \texttt{'R'}, :math:n > 1; otherwise :math:n\geq 1.
.. _g05px-py2-py-notes:
**Notes**
matrix_orthog pre - or post-multiplies an :math:m\times n matrix :math:A by a random orthogonal matrix :math:U, overwriting :math:A.
The matrix :math:A may optionally be initialized to the identity matrix before multiplying by :math:U, hence :math:U is returned. :math:U is generated using the method of Stewart (1980).
The algorithm can be summarised as follows.
Let :math:x_1,x_2,\ldots,x_{{n-1}} follow independent multinormal distributions with zero mean and variance :math:I\sigma^2 and dimensions :math:n,n-1,\ldots,2; let :math:H_j = \mathrm{diag}\left(I_{{j-1}},H_j^*\right), where :math:I_{{j-1}} is the identity matrix and :math:H_j^* is the Householder transformation that reduces :math:x_j to :math:r_{{jj}}e_1, :math:e_1 being the vector with first element one and the remaining elements zero and :math:r_{{jj}} being a scalar, and let :math:D = \mathrm{diag}\left(\mathrm{sign}\left(r_{11}\right),\mathrm{sign}\left(r_{22}\right),\ldots,\mathrm{sign}\left(r_{{nn}}\right)\right).
Then the product :math:U = DH_1H_2\ldots H_{{n-1}} is a random orthogonal matrix distributed according to the Haar measure over the set of orthogonal matrices of :math:n.
See Theorem 3.3 in Stewart (1980).
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to matrix_orthog.
.. _g05px-py2-py-references:
**References**
Stewart, G W, 1980, The efficient generation of random orthogonal matrices with an application to condition estimates, SIAM J. Numer. Anal. (17), 403--409
"""
raise NotImplementedError
[docs]def matrix_corr(d, statecomm, eps=0.00001):
r"""
matrix_corr generates a random correlation matrix with given eigenvalues.
.. _g05py-py2-py-doc:
For full information please refer to the NAG Library document for g05py
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pyf.html
.. _g05py-py2-py-parameters:
**Parameters**
**d** : float, array-like, shape :math:\left(n\right)
The :math:n eigenvalues, :math:\lambda_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,n.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**eps** : float, optional
The maximum acceptable error in the diagonal elements.
**Returns**
**c** : float, ndarray, shape :math:\left(n, n\right)
A random correlation matrix, :math:C, of dimension :math:n.
.. _g05py-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:n = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:n\geq 1.
(errno :math:2)
On entry, an eigenvalue is negative.
(errno :math:2)
On entry, the eigenvalues do not sum to :math:\textit{n}.
(errno :math:3)
On entry, :math:\mathrm{eps} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{eps}\geq n\times \text{machine precision}.
(errno :math:4)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
(errno :math:5)
The diagonals of the returned matrix are not unity, try increasing the value of :math:\mathrm{eps}, or rerun the code using a different seed.
.. _g05py-py2-py-notes:
**Notes**
Given :math:n eigenvalues, :math:\lambda_1,\lambda_2,\ldots,\lambda_n, such that
.. math::
\sum_{{i = 1}}^n\lambda_i = n
and
.. math::
\lambda_i\geq 0\text{, }\quad i = 1,2,\ldots,n\text{,}
matrix_corr will generate a random correlation matrix, :math:C, of dimension :math:n, with eigenvalues :math:\lambda_1,\lambda_2,\ldots,\lambda_n.
The method used is based on that described by Lin and Bendel (1985).
Let :math:D be the diagonal matrix with values :math:\lambda_1,\lambda_2,\ldots,\lambda_n and let :math:A be a random orthogonal matrix generated by :meth:matrix_orthog then the matrix :math:C_0 = ADA^\mathrm{T} is a random covariance matrix with eigenvalues :math:\lambda_1,\lambda_2,\ldots,\lambda_n.
The matrix :math:C_0 is transformed into a correlation matrix by means of :math:n-1 elementary rotation matrices :math:P_i such that :math:C = P_{{n-1}}P_{{n-2}}\ldots P_1C_0 P_1^\mathrm{T}\ldots P_{{n-2}}^\mathrm{T} P_{{n-1}}^\mathrm{T}.
The restriction on the sum of eigenvalues implies that for any diagonal element of :math:C_0 > 1, there is another diagonal element :math:\text{} < 1.
The :math:P_i are constructed from such pairs, chosen at random, to produce a unit diagonal element corresponding to the first element.
This is repeated until all diagonal elements are :math:1 to within a given tolerance :math:\epsilon.
The randomness of :math:C should be interpreted only to the extent that :math:A is a random orthogonal matrix and :math:C is computed from :math:A using the :math:P_i which are chosen as arbitrarily as possible.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to matrix_corr.
.. _g05py-py2-py-references:
**References**
Lin, S P and Bendel, R B, 1985, Algorithm AS 213: Generation of population correlation on matrices with specified eigenvalues, Appl. Statist. (34), 193--198
"""
raise NotImplementedError
[docs]def matrix_2waytable(mode, totr, totc, comm, statecomm):
r"""
matrix_2waytable generates a random two-way table.
.. _g05pz-py2-py-doc:
For full information please refer to the NAG Library document for g05pz
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05pzf.html
.. _g05pz-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector only.
:math:\mathrm{mode} = 1
Generate two-way table using reference vector set up in a prior call to matrix_2waytable.
:math:\mathrm{mode} = 2
Set up reference vector and generate two-way table.
**totr** : int, array-like, shape :math:\left(\textit{nrow}\right)
The :math:m row totals, :math:R_{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,m.
**totc** : int, array-like, shape :math:\left(\textit{ncol}\right)
The :math:n column totals, :math:C_{\textit{j}}, for :math:\textit{j} = 1,2,\ldots,n.
**comm** : dict, communication object, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1, this argument must have been initialized by a prior call to matrix_2waytable.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : None or int, ndarray, shape :math:\left(\textit{nrow}, \textit{ncol}\right)
If :math:\mathrm{mode} = 1 or :math:2, a pseudorandom two-way :math:m\times n table, :math:X, with element :math:\mathrm{x}[i-1,j-1] containing the :math:\left(i, j\right)\ th entry in the table such that :math:\sum_{1}^{m}\mathrm{x}[i-1,j-1] = \mathrm{totc}[j-1] and :math:\sum_{1}^{n}\mathrm{x}[i-1,j-1] = \mathrm{totr}[i-1]
.. _g05pz-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1 or :math:2.
(errno :math:2)
On entry, :math:\textit{nrow} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\textit{nrow}\geq 2.
(errno :math:3)
On entry, :math:\textit{ncol} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\textit{ncol}\geq 2.
(errno :math:4)
On entry, at least one element of :math:\mathrm{totr} is negative or :math:\mathrm{totr} sums to zero.
(errno :math:5)
On entry, :math:\mathrm{totc} has at least one negative element.
(errno :math:6)
:math:\textit{nrow} or :math:\textit{ncol} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:\textit{nrow} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\textit{nrow} = \langle\mathit{\boldsymbol{value}}\rangle.
Previous value of :math:\textit{ncol} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\textit{ncol} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:8)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
(errno :math:15)
On entry, the arrays :math:\mathrm{totr} and :math:\mathrm{totc} do not sum to the same total: :math:\mathrm{totr} array total is :math:\langle\mathit{\boldsymbol{value}}\rangle, :math:\mathrm{totc} array total is :math:\langle\mathit{\boldsymbol{value}}\rangle.
.. _g05pz-py2-py-notes:
**Notes**
Given :math:m row totals :math:R_i and :math:n column totals :math:C_j (with :math:\sum_{{i = 1}}^mR_i = \sum_{{j = 1}}^nC_j = T, say), matrix_2waytable will generate a pseudorandom two-way table of integers such that the row and column totals are satisfied.
The method used is based on that described by Patefield (1981) which is most efficient when :math:T is large relative to the number of table entries :math:m\times n (i.e., :math:T > 2mn).
Entries are generated one row at a time and one entry at a time within a row.
Each entry is generated using the conditional probability distribution for that entry given the entries in the previous rows and the previous entries in the same row.
A reference vector is used to store computed values that can be reused in the generation of new tables with the same row and column totals. matrix_2waytable can be called to simply set up the reference vector, or to generate a two-way table using a reference vector set up in a previous call, or it can combine both functions in a single call.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to matrix_2waytable.
.. _g05pz-py2-py-references:
**References**
Patefield, W M, 1981, An efficient method of generating :math:R\times C tables with given row and column totals, Appl. Stats. (30), 91--97
"""
raise NotImplementedError
[docs]def copula_students_t(mode, n, df, c, comm, statecomm):
r"""
copula_students_t sets up a reference vector and generates an array of pseudorandom numbers from a Student's :math:t copula with :math:\nu degrees of freedom and covariance matrix :math:\frac{\nu }{{\nu -2}}C.
.. _g05rc-py2-py-doc:
For full information please refer to the NAG Library document for g05rc
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05rcf.html
.. _g05rc-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector only.
:math:\mathrm{mode} = 1
Generate variates using reference vector set up in a prior call to copula_students_t.
:math:\mathrm{mode} = 2
Set up reference vector and generate variates.
**n** : int
:math:n, the number of random variates required.
**df** : int
:math:\nu, the number of degrees of freedom of the distribution.
**c** : float, array-like, shape :math:\left(m, m\right)
Matrix which, along with :math:\mathrm{df}, defines the covariance of the distribution. Only the upper triangle need be set.
**comm** : dict, communication object, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1, this argument must have been initialized by a prior call to copula_students_t.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : None or float, ndarray, shape :math:\left(\mathrm{n}, :\right)
The array of values from a multivariate Student's :math:t copula, with :math:\mathrm{x}[i-1,j-1] holding the :math:j\ th dimension for the :math:i\ th variate.
.. _g05rc-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1 or :math:2.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{df} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{df}\geq 3.
(errno :math:4)
On entry, :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:m > 0.
(errno :math:5)
On entry, the covariance matrix :math:C is not positive semidefinite to machine precision.
(errno :math:7)
:math:\textit{m} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:m = \langle\mathit{\boldsymbol{value}}\rangle and :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:9)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05rc-py2-py-notes:
**Notes**
The Student's :math:t copula, :math:G, is defined by
.. math::
G\left(u_1,u_2,\ldots,u_m;C\right) = T_{{\nu,C}}^m\left(t_{{\nu,C_{11}}}^{-1}\left(u_1\right),t_{{\nu,C_{22}}}^{-1}\left(u_2\right),\ldots,t_{{\nu,C_{\textit{mm}}}}^{-1}\left(u_m\right)\right)
where :math:m is the number of dimensions, :math:T_{{\nu,C}}^m is the multivariate Student's :math:t density function with :math:\nu degrees of freedom, mean zero and covariance matrix :math:\frac{\nu }{{\nu -2}}C and :math:t_{{\nu,C_{\textit{ii}}}}^{-1} is the inverse of the univariate Student's :math:t density function with :math:\nu degrees of freedom, zero mean and variance :math:\frac{\nu }{{\nu -2}}C_{\textit{ii}}.
:meth:multivar_students_t is used to generate a vector from a multivariate Student's :math:t distribution and :meth:stat.prob_students_t <naginterfaces.library.stat.prob_students_t> is used to convert each element of that vector into a uniformly distributed value between zero and one.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to copula_students_t.
.. _g05rc-py2-py-references:
**References**
Nelsen, R B, 1998, An Introduction to Copulas. Lecture Notes in Statistics 139, Springer
Sklar, A, 1973, Random variables: joint distribution functions and copulas, Kybernetika (9), 499--460
--------
:meth:naginterfaces.library.examples.rand.copula_students_t_ex.main
"""
raise NotImplementedError
[docs]def copula_normal(mode, n, c, comm, statecomm):
r"""
copula_normal sets up a reference vector and generates an array of pseudorandom numbers from a Normal (Gaussian) copula with covariance matrix :math:C.
.. _g05rd-py2-py-doc:
For full information please refer to the NAG Library document for g05rd
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05rdf.html
.. _g05rd-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector only.
:math:\mathrm{mode} = 1
Generate variates using reference vector set up in a prior call to copula_normal.
:math:\mathrm{mode} = 2
Set up reference vector and generate variates.
**n** : int
:math:n, the number of random variates required.
**c** : float, array-like, shape :math:\left(m, m\right)
The covariance matrix of the distribution. Only the upper triangle need be set.
**comm** : dict, communication object, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1, this argument must have been initialized by a prior call to copula_normal.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : None or float, ndarray, shape :math:\left(\mathrm{n}, :\right)
The array of values from a multivariate Gaussian copula, with :math:\mathrm{x}[i-1,j-1] holding the :math:j\ th dimension for the :math:i\ th variate.
.. _g05rd-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1 or :math:2.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:m > 0.
(errno :math:4)
On entry, the covariance matrix :math:C is not positive semidefinite to machine precision.
(errno :math:6)
:math:\textit{m} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:m = \langle\mathit{\boldsymbol{value}}\rangle and :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:8)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05rd-py2-py-notes:
**Notes**
The Gaussian copula, :math:G, is defined by
.. math::
G\left(u_1,u_2,\ldots,u_m;C\right) = \Phi_C\left(\phi_{C_{11}}^{-1}\left(u_1\right),\phi_{C_{22}}^{-1}\left(u_2\right),\ldots,\phi_{C_{\textit{mm}}}^{-1}\left(u_m\right)\right)
where :math:m is the number of dimensions, :math:\Phi_C is the multivariate Normal density function with mean zero and covariance matrix :math:C and :math:\phi_{C_{\textit{ii}}}^{-1} is the inverse of the univariate Normal density function with mean zero and variance :math:C_{\textit{ii}}.
:meth:multivar_normal is used to generate a vector from a multivariate Normal distribution and :meth:stat.prob_normal <naginterfaces.library.stat.prob_normal> is used to convert each element of that vector into a uniformly distributed value between zero and one.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to copula_normal.
.. _g05rd-py2-py-references:
**References**
Nelsen, R B, 1998, An Introduction to Copulas. Lecture Notes in Statistics 139, Springer
Sklar, A, 1973, Random variables: joint distribution functions and copulas, Kybernetika (9), 499--460
"""
raise NotImplementedError
[docs]def copula_clayton_bivar(n, theta, sorder, statecomm):
r"""
copula_clayton_bivar generates pseudorandom uniform bivariates with joint distribution of a Clayton/Cook--Johnson Archimedean copula.
.. _g05re-py2-py-doc:
For full information please refer to the NAG Library document for g05re
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05ref.html
.. _g05re-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of bivariates to generate.
**theta** : float
:math:\theta, the copula parameter.
**sorder** : int
Determines the storage order of variates; the :math:\left(\textit{i}, \textit{j}\right)\ th variate is stored in :math:\mathrm{x}[\textit{i}-1,\textit{j}-1] if :math:\mathrm{sorder} = 1, and :math:\mathrm{x}[\textit{j}-1,\textit{i}-1] if :math:\mathrm{sorder} = 2, for :math:\textit{j} = 1,2,\ldots,2, for :math:\textit{i} = 1,2,\ldots,n.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(:, :\right)
The :math:n bivariate uniforms with joint distribution described by :math:C_{\theta }, with :math:\mathrm{x}[i-1,j-1] holding the :math:i\ th value for the :math:j\ th dimension if :math:\mathrm{sorder} = 1 and the :math:j\ th value for the :math:i\ th dimension if :math:\mathrm{sorder} = 2.
.. _g05re-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, corrupt :math:\mathrm{statecomm}\ ['state'] argument.
(errno :math:2)
On entry, invalid :math:\mathrm{theta}: :math:\mathrm{theta} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{theta}\geq {-1.0}.
(errno :math:3)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:4)
On entry, invalid :math:\mathrm{sorder}.
Constraint: :math:\mathrm{sorder} = 1 or :math:2.
.. _g05re-py2-py-notes:
**Notes**
Generates pseudorandom uniform bivariates :math:\left\{u_1, u_2\right\} \in \left(0, 1\right]^2 whose joint distribution is the Clayton/Cook--Johnson Archimedean copula :math:C_{\theta } with parameter :math:\theta, given by
.. math::
C_{\theta } = \left[\mathrm{max}\left({u_1^{{-\theta }}+u_2^{{-\theta }}-1}, 0\right)\right]^{{-1/\theta }}\text{, }\quad \theta \in \left(-1, \infty \right)∖\left\{0\right\}
with the special cases:
:math:C_{-1} = \mathrm{max}\left({u_1+u_2-1}, 0\right), the Fréchet--Hoeffding lower bound;
:math:C_0 = u_1u_2, the product copula;
:math:C_{\infty } = \mathrm{min}\left(u_1, u_2\right), the Fréchet--Hoeffding upper bound.
The generation method uses conditional sampling.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to copula_clayton_bivar.
.. _g05re-py2-py-references:
**References**
Nelsen, R B, 2006, An Introduction to Copulas, (2nd Edition), Springer Series in Statistics
"""
raise NotImplementedError
[docs]def copula_frank_bivar(n, theta, sorder, statecomm):
r"""
copula_frank_bivar generates pseudorandom uniform bivariates with joint distribution of a Frank Archimedean copula.
.. _g05rf-py2-py-doc:
For full information please refer to the NAG Library document for g05rf
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05rff.html
.. _g05rf-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of bivariates to generate.
**theta** : float
:math:\theta, the copula parameter.
**sorder** : int
Determines the storage order of variates; the :math:\left(\textit{i}, \textit{j}\right)\ th variate is stored in :math:\mathrm{x}[\textit{i}-1,\textit{j}-1] if :math:\mathrm{sorder} = 1, and :math:\mathrm{x}[\textit{j}-1,\textit{i}-1] if :math:\mathrm{sorder} = 2, for :math:\textit{j} = 1,2,\ldots,2, for :math:\textit{i} = 1,2,\ldots,n.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(:, :\right)
The :math:n bivariate uniforms with joint distribution described by :math:C_{\theta }, with :math:\mathrm{x}[i-1,j-1] holding the :math:i\ th value for the :math:j\ th dimension if :math:\mathrm{sorder} = 1 and the :math:j\ th value for the :math:i\ th dimension if :math:\mathrm{sorder} = 2.
.. _g05rf-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, corrupt :math:\mathrm{statecomm}\ ['state'] argument.
(errno :math:3)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:4)
On entry, invalid :math:\mathrm{sorder}.
Constraint: :math:\mathrm{sorder} = 1 or :math:2.
.. _g05rf-py2-py-notes:
**Notes**
Generates pseudorandom uniform bivariates :math:\left\{u_1, u_2\right\} \in \left[0, 1\right]^2 whose joint distribution is the Frank Archimedean copula :math:C_{\theta } with parameter :math:\theta, given by
.. math::
C_{\theta } = -\frac{1}{\theta }\mathrm{ln}\left[1+\frac{{\left(e^{{-\theta u_1}}-1\right)\left(e^{{-\theta u_2}}-1\right)}}{{e^{{-\theta }}-1}}\right]\text{, }\quad \theta \in \left({-\infty }, \infty \right)∖\left\{0\right\}
with the special cases:
:math:C_{{-\infty }} = \mathrm{max}\left({u_1+u_2-1}, 0\right), the Fréchet--Hoeffding lower bound;
:math:C_0 = u_1u_2, the product copula;
:math:C_{\infty } = \mathrm{min}\left(u_1, u_2\right), the Fréchet--Hoeffding upper bound.
The generation method uses conditional sampling.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to copula_frank_bivar.
.. _g05rf-py2-py-references:
**References**
Nelsen, R B, 2006, An Introduction to Copulas, (2nd Edition), Springer Series in Statistics
"""
raise NotImplementedError
[docs]def copula_plackett_bivar(n, theta, sorder, statecomm):
r"""
copula_plackett_bivar generates pseudorandom uniform bivariates with joint distribution of a Plackett copula.
.. _g05rg-py2-py-doc:
For full information please refer to the NAG Library document for g05rg
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05rgf.html
.. _g05rg-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of bivariates to generate.
**theta** : float
:math:\theta, the copula parameter.
**sorder** : int
Determines the storage order of variates; the :math:\left(\textit{i}, \textit{j}\right)\ th variate is stored in :math:\mathrm{x}[\textit{i}-1,\textit{j}-1] if :math:\mathrm{sorder} = 1, and :math:\mathrm{x}[\textit{j}-1,\textit{i}-1] if :math:\mathrm{sorder} = 2, for :math:\textit{j} = 1,2,\ldots,2, for :math:\textit{i} = 1,2,\ldots,n.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(:, :\right)
The :math:n bivariate uniforms with joint distribution described by :math:C_{\theta }, with :math:\mathrm{x}[i-1,j-1] holding the :math:i\ th value for the :math:j\ th dimension if :math:\mathrm{sorder} = 1 and the :math:j\ th value for the :math:i\ th dimension if :math:\mathrm{sorder} = 2.
.. _g05rg-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, corrupt :math:\mathrm{statecomm}\ ['state'] argument.
(errno :math:2)
On entry, invalid :math:\mathrm{theta}: :math:\mathrm{theta} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{theta}\geq 0.0.
(errno :math:3)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:4)
On entry, invalid :math:\mathrm{sorder}.
Constraint: :math:\mathrm{sorder} = 1 or :math:2.
.. _g05rg-py2-py-notes:
**Notes**
Generates pseudorandom uniform bivariates :math:\left\{u_1, u_2\right\} \in \left[0, 1\right]^2 whose joint distribution is the Plackett copula :math:C_{\theta } with parameter :math:\theta, given by
.. math::
C_{\theta } = \frac{{\left[1+\left(\theta -1\right)\left(u_1+u_2\right)\right]-\sqrt{\left[1+\left(\theta -1\right)\left(u_1+u_2\right)\right]^2-4u_1u_2\theta \left(\theta -1\right)}}}{{2\left(\theta -1\right)}}\text{, }\quad \theta \in \left(0, \infty \right)∖\left\{1\right\}
with the special cases:
:math:C_0 = \mathrm{max}\left({u_1+u_2-1}, 0\right), the Fréchet--Hoeffding lower bound;
:math:C_1 = u_1u_2, the product copula;
:math:C_{\infty } = \mathrm{min}\left(u_1, u_2\right), the Fréchet--Hoeffding upper bound.
The generation method uses conditional sampling.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to copula_plackett_bivar.
.. _g05rg-py2-py-references:
**References**
Nelsen, R B, 2006, An Introduction to Copulas, (2nd Edition), Springer Series in Statistics
"""
raise NotImplementedError
[docs]def copula_clayton(n, m, theta, sorder, statecomm):
r"""
copula_clayton generates pseudorandom uniform variates with joint distribution of a Clayton/Cook--Johnson Archimedean copula.
.. _g05rh-py2-py-doc:
For full information please refer to the NAG Library document for g05rh
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05rhf.html
.. _g05rh-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom uniform variates to generate.
**m** : int
:math:m, the number of dimensions.
**theta** : float
:math:\theta, the copula parameter.
**sorder** : int
Determines the storage order of variates; the :math:\left(\textit{i}, \textit{j}\right)\ th variate is stored in :math:\mathrm{x}[\textit{i}-1,\textit{j}-1] if :math:\mathrm{sorder} = 1, and :math:\mathrm{x}[\textit{j}-1,\textit{i}-1] if :math:\mathrm{sorder} = 2, for :math:\textit{j} = 1,2,\ldots,m, for :math:\textit{i} = 1,2,\ldots,n.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(:, :\right)
The pseudorandom uniform variates with joint distribution described by :math:C_{\theta }, with :math:\mathrm{x}[i-1,j-1] holding the :math:i\ th value for the :math:j\ th dimension if :math:\mathrm{sorder} = 1 and the :math:j\ th value for the :math:i\ th dimension of :math:\mathrm{sorder} = 2.
.. _g05rh-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, corrupt :math:\mathrm{statecomm}\ ['state'] argument.
(errno :math:2)
On entry, invalid :math:\mathrm{theta}: :math:\mathrm{theta} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{theta}\geq 1.0\times 10^{-6}.
(errno :math:3)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:4)
On entry, :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{m}\geq 2.
(errno :math:5)
On entry, invalid :math:\mathrm{sorder}.
Constraint: :math:\mathrm{sorder} = 1 or :math:2.
.. _g05rh-py2-py-notes:
**Notes**
Generates :math:n pseudorandom uniform :math:m-variates whose joint distribution is the Clayton/Cook--Johnson Archimedean copula :math:C_{\theta }, given by
.. math::
C_{\theta } = \left(u_1^{{-\theta }}+u_2^{{-\theta }} + \cdots +u_m^{{-\theta }}-m+1\right)^{{-1/\theta }}\text{, }\quad \left\{\begin{array}{cc} \theta \in \left(0, \infty \right) \text{,} &\\ u_j \in \left(0, 1\right] \text{, }\quad j = 1, \ldots m \text{;} \end{array}\right.
with the special case:
:math:C_{\infty } = \mathrm{min}\left(u_1, u_2, \ldots, u_m\right), the Fréchet--Hoeffding upper bound.
The generation method uses mixture of powers.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to copula_clayton.
.. _g05rh-py2-py-references:
**References**
Marshall, A W and Olkin, I, 1988, Families of multivariate distributions, Journal of the American Statistical Association (83), 403
Nelsen, R B, 2006, An Introduction to Copulas, (2nd Edition), Springer Series in Statistics
"""
raise NotImplementedError
[docs]def copula_frank(n, m, theta, sorder, statecomm):
r"""
copula_frank generates pseudorandom uniform variates with joint distribution of a Frank Archimedean copula.
.. _g05rj-py2-py-doc:
For full information please refer to the NAG Library document for g05rj
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05rjf.html
.. _g05rj-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom uniform variates to generate.
**m** : int
:math:m, the number of dimensions.
**theta** : float
:math:\theta, the copula parameter.
**sorder** : int
Determines the storage order of variates; the :math:\left(\textit{i}, \textit{j}\right)\ th variate is stored in :math:\mathrm{x}[\textit{i}-1,\textit{j}-1] if :math:\mathrm{sorder} = 1, and :math:\mathrm{x}[\textit{j}-1,\textit{i}-1] if :math:\mathrm{sorder} = 2, for :math:\textit{j} = 1,2,\ldots,m, for :math:\textit{i} = 1,2,\ldots,n.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(:, :\right)
The pseudorandom uniform variates with joint distribution described by :math:C_{\theta }, with :math:\mathrm{x}[i-1,j-1] holding the :math:i\ th value for the :math:j\ th dimension if :math:\mathrm{sorder} = 1 and the :math:j\ th value for the :math:i\ th dimension of :math:\mathrm{sorder} = 2.
.. _g05rj-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, corrupt :math:\mathrm{statecomm}\ ['state'] argument.
(errno :math:2)
On entry, invalid :math:\mathrm{theta}: :math:\mathrm{theta} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{theta}\geq 1.0\times 10^{-6}.
(errno :math:3)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:4)
On entry, :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{m}\geq 2.
(errno :math:5)
On entry, invalid :math:\mathrm{sorder}.
Constraint: :math:\mathrm{sorder} = 1 or :math:2.
.. _g05rj-py2-py-notes:
**Notes**
Generates :math:n pseudorandom uniform :math:m-variates whose joint distribution is the Frank Archimedean copula :math:C_{\theta }, given by
.. math::
C_{\theta } = -\frac{1}{\theta }\mathrm{ln}\left[1+\frac{{\left(e^{{-\theta u_1}}-1\right)\left(e^{{-\theta u_2}}-1\right) \cdots \left(e^{{-\theta u_m}}-1\right)}}{{\left(e^{{-\theta }}-1\right)^{{m-1}}}}\right]\text{, }\quad \left\{\begin{array}{cc} \theta \in \left(0, \infty \right) \text{,} &\\ u_j \in \left(0, 1\right] \text{, }\quad j = 1, \ldots m \text{;} \end{array}\right.
with the special case:
:math:C_{\infty } = \mathrm{min}\left(u_1, u_2, \ldots, u_m\right), the Fréchet--Hoeffding upper bound.
The generation method uses mixture of powers.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to copula_frank.
.. _g05rj-py2-py-references:
**References**
Marshall, A W and Olkin, I, 1988, Families of multivariate distributions, Journal of the American Statistical Association (83), 403
Nelsen, R B, 2006, An Introduction to Copulas, (2nd Edition), Springer Series in Statistics
"""
raise NotImplementedError
[docs]def copula_gumbel(n, m, theta, sorder, statecomm):
r"""
copula_gumbel generates pseudorandom uniform variates with joint distribution of a Gumbel--Hougaard Archimedean copula.
.. _g05rk-py2-py-doc:
For full information please refer to the NAG Library document for g05rk
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05rkf.html
.. _g05rk-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom uniform variates to generate.
**m** : int
:math:m, the number of dimensions.
**theta** : float
:math:\theta, the copula parameter.
**sorder** : int
Determines the storage order of variates; the :math:\left(\textit{i}, \textit{j}\right)\ th variate is stored in :math:\mathrm{x}[\textit{i}-1,\textit{j}-1] if :math:\mathrm{sorder} = 1, and :math:\mathrm{x}[\textit{j}-1,\textit{i}-1] if :math:\mathrm{sorder} = 2, for :math:\textit{j} = 1,2,\ldots,m, for :math:\textit{i} = 1,2,\ldots,n.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(:, :\right)
The pseudorandom uniform variates with joint distribution described by :math:C_{\theta }, with :math:\mathrm{x}[i-1,j-1] holding the :math:i\ th value for the :math:j\ th dimension if :math:\mathrm{sorder} = 1 and the :math:j\ th value for the :math:i\ th dimension of :math:\mathrm{sorder} = 2.
.. _g05rk-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, corrupt :math:\mathrm{statecomm}\ ['state'] argument.
(errno :math:2)
On entry, invalid :math:\mathrm{theta}: :math:\mathrm{theta} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{theta}\geq 1.0.
(errno :math:3)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:4)
On entry, :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{m}\geq 2.
(errno :math:5)
On entry, invalid :math:\mathrm{sorder}.
Constraint: :math:\mathrm{sorder} = 1 or :math:2.
.. _g05rk-py2-py-notes:
**Notes**
Generates :math:n pseudorandom uniform :math:m-variates whose joint distribution is the Gumbel--Hougaard Archimedean copula :math:C_{\theta }, given by
.. math::
C_{\theta } = \mathrm{exp}\left\{-\left[\left(-\mathrm{ln}\left(u_1\right)\right)^{\theta }+\left(-\mathrm{ln}\left(u_2\right)\right)^{\theta } + \cdots +\left(-\mathrm{ln}\left(u_m\right)\right)^{\theta }\right]\right\}\text{, }\quad \left\{\begin{array}{cc} \theta \in \left(1, \infty \right) \text{,} &\\ u_j \in \left(0, 1\right] \text{, }\quad j = 1, 2, \ldots m \text{;} \end{array}\right.
with the special cases:
:math:C_1 = u_1u_2 \cdots u_m, the product copula;
:math:C_{\infty } = \mathrm{min}\left(u_1, u_2, \ldots, u_m\right), the Fréchet--Hoeffding upper bound.
The generation method uses mixture of powers.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to copula_gumbel.
.. _g05rk-py2-py-references:
**References**
Marshall, A W and Olkin, I, 1988, Families of multivariate distributions, Journal of the American Statistical Association (83), 403
Nelsen, R B, 2006, An Introduction to Copulas, (2nd Edition), Springer Series in Statistics
"""
raise NotImplementedError
[docs]def multivar_students_t(mode, n, df, xmu, c, comm, statecomm):
r"""
multivar_students_t sets up a reference vector and generates an array of pseudorandom numbers from a multivariate Student's :math:t distribution with :math:\nu degrees of freedom, mean vector :math:a and covariance matrix :math:\frac{\nu }{{\nu -2}}C.
.. _g05ry-py2-py-doc:
For full information please refer to the NAG Library document for g05ry
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05ryf.html
.. _g05ry-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector only.
:math:\mathrm{mode} = 1
Generate variates using reference vector set up in a prior call to multivar_students_t.
:math:\mathrm{mode} = 2
Set up reference vector and generate variates.
**n** : int
:math:n, the number of random variates required.
**df** : int
:math:\nu, the number of degrees of freedom of the distribution.
**xmu** : float, array-like, shape :math:\left(m\right)
:math:a, the vector of means of the distribution.
**c** : float, array-like, shape :math:\left(m, m\right)
Matrix which, along with :math:\mathrm{df}, defines the covariance of the distribution. Only the upper triangle need be set.
**comm** : dict, communication object, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1, this argument must have been initialized by a prior call to multivar_students_t.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : None or float, ndarray, shape :math:\left(\mathrm{n}, :\right)
The array of pseudorandom multivariate Student's :math:t vectors generated by the function, with :math:\mathrm{x}[i-1,j-1] holding the :math:j\ th dimension for the :math:i\ th variate.
.. _g05ry-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1 or :math:2.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{df} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{df}\geq 3.
(errno :math:4)
On entry, :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:m > 0.
(errno :math:6)
On entry, the covariance matrix :math:C is not positive semidefinite to machine precision.
(errno :math:8)
:math:\textit{m} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:m = \langle\mathit{\boldsymbol{value}}\rangle and :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:10)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05ry-py2-py-notes:
**Notes**
When the covariance matrix is nonsingular (i.e., strictly positive definite), the distribution has probability density function
.. math::
f\left(x\right) = \frac{{\Gamma \left(\frac{\left(\nu +m\right)}{2}\right)}}{{\left(\pi v\right)^{{m/2}}\Gamma \left(\nu /2\right)\left\lvert C\right\rvert^{\frac{1}{2}}}}\left[1+\frac{{\left(x-a\right)^\mathrm{T}C^{-1}\left(x-a\right)}}{\nu }\right]^{\frac{{-\left(\nu +m\right)}}{2}}
where :math:m is the number of dimensions, :math:\nu is the degrees of freedom, :math:a is the vector of means, :math:x is the vector of positions and :math:\frac{\nu }{{\nu -2}}C is the covariance matrix.
The function returns the value
.. math::
x = a+\sqrt{\frac{\nu }{s}}z
where :math:z is generated by :meth:dist_normal from a Normal distribution with mean zero and covariance matrix :math:C and :math:s is generated by :meth:dist_chisq from a :math:\chi^2-distribution with :math:\nu degrees of freedom.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to multivar_students_t.
.. _g05ry-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
Wilkinson, J H, 1965, The Algebraic Eigenvalue Problem, Oxford University Press, Oxford
"""
raise NotImplementedError
[docs]def multivar_normal(sorder, mode12, n, xmu, c, comm, statecomm):
r"""
multivar_normal sets up a reference vector and generates an array of pseudorandom numbers from a multivariate Normal distribution with mean vector :math:a and covariance matrix :math:C.
.. _g05rz-py2-py-doc:
For full information please refer to the NAG Library document for g05rz
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05rzf.html
.. _g05rz-py2-py-parameters:
**Parameters**
**sorder** : int
Determines the storage order of variates; the :math:\left(\textit{i}, \textit{j}\right)\ th variate is stored in :math:\mathrm{x}[\textit{i}-1,\textit{j}-1] if :math:\mathrm{sorder} = 1, and :math:\mathrm{x}[\textit{j}-1,\textit{i}-1] if :math:\mathrm{sorder} = 2, for :math:\textit{j} = 1,2,\ldots,m, for :math:\textit{i} = 1,2,\ldots,n.
**mode12** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode12} = 0
Set up reference vector only.
:math:\mathrm{mode12} = 1
Generate variates using reference vector set up in a prior call to multivar_normal.
:math:\mathrm{mode12} = 2
Set up reference vector and generate variates.
**n** : int
:math:n, the number of random variates required.
**xmu** : float, array-like, shape :math:\left(m\right)
:math:a, the vector of means of the distribution.
**c** : float, array-like, shape :math:\left(m, m\right)
The covariance matrix of the distribution. Only the upper triangle need be set.
**comm** : dict, communication object, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode12} = 1, this argument must have been initialized by a prior call to multivar_normal.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : None or float, ndarray, shape :math:\left(:, :\right)
The array of pseudorandom multivariate Normal vectors generated by the function.
Two possible storage orders are available.
If :math:\mathrm{sorder} = 1 then :math:\mathrm{x}[i-1,j-1] holds the :math:j\ th dimension for the :math:i\ th variate.
If :math:\mathrm{sorder} = 2 this ordering is reversed and :math:\mathrm{x}[j-1,i-1] holds the :math:j\ th dimension for the :math:i\ th variate.
.. _g05rz-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode12} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode12} = 0, :math:1 or :math:2.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:m > 0.
(errno :math:5)
On entry, the covariance matrix :math:C is not positive semidefinite to machine precision.
(errno :math:7)
:math:\textit{m} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:m = \langle\mathit{\boldsymbol{value}}\rangle and :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:9)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05rz-py2-py-notes:
**Notes**
When the covariance matrix is nonsingular (i.e., strictly positive definite), the distribution has probability density function
.. math::
f\left(x\right) = \sqrt{\frac{{\left\lvert C^{-1}\right\rvert }}{{\left(2\pi \right)^m}}}\mathrm{exp}\left(-\frac{1}{2}\left(x-a\right)^\mathrm{T}C^{-1}\left(x-a\right)\right)
where :math:m is the number of dimensions, :math:C is the covariance matrix, :math:a is the vector of means and :math:x is the vector of positions.
Covariance matrices are symmetric and positive semidefinite.
Given such a matrix :math:C, there exists a lower triangular matrix :math:L such that :math:LL^\mathrm{T} = C. :math:L is not unique, if :math:C is singular.
multivar_normal decomposes :math:C to find such an :math:L.
It then stores :math:m, :math:a and :math:L in the reference vector :math:r which is used to generate a vector :math:x of independent standard Normal pseudorandom numbers.
It then returns the vector :math:a+Lx, which has the required multivariate Normal distribution.
It should be noted that this function will work with a singular covariance matrix :math:C, provided :math:C is positive semidefinite, despite the fact that the above formula for the probability density function is not valid in that case. Wilkinson (1965) should be consulted if further information is required.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to multivar_normal.
.. _g05rz-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
Wilkinson, J H, 1965, The Algebraic Eigenvalue Problem, Oxford University Press, Oxford
"""
raise NotImplementedError
[docs]def dist_uniform01(n, statecomm):
r"""
dist_uniform01 generates a vector of pseudorandom numbers taken from a uniform distribution between :math:0 and :math:1.
.. _g05sa-py2-py-doc:
For full information please refer to the NAG Library document for g05sa
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05saf.html
.. _g05sa-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from a uniform distribution over the half closed interval :math:\left(0, 1\right].
.. _g05sa-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sa-py2-py-notes:
**Notes**
dist_uniform01 generates :math:n values from a uniform distribution over the half closed interval :math:\left(0, 1\right].
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_uniform01.
.. _g05sa-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
--------
:meth:naginterfaces.library.examples.fit.dim2_spline_ts_sctr_ex.main
"""
raise NotImplementedError
[docs]def dist_beta(n, a, b, statecomm):
r"""
dist_beta generates a vector of pseudorandom numbers taken from a beta distribution with parameters :math:a and :math:b.
.. _g05sb-py2-py-doc:
For full information please refer to the NAG Library document for g05sb
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05sbf.html
.. _g05sb-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**a** : float
:math:a, the parameter of the beta distribution.
**b** : float
:math:b, the parameter of the beta distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified beta distribution.
.. _g05sb-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\mathrm{a} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{a} > 0.0.
(errno :math:3)
On entry, :math:\mathrm{b} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{b} > 0.0.
(errno :math:4)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sb-py2-py-notes:
**Notes**
The beta distribution has PDF (probability density function)
.. math::
\begin{array}{ll} f\left(x\right) = \frac{{\Gamma \left(a+b\right)}}{{\Gamma \left(a\right)\Gamma \left(b\right)}} x^{{a-1}} \left(1-x\right)^{{b-1}} & \text{if } 0\leq x\leq 1 \text{; } a,b > 0 \text{,} \\\\f\left(x\right) = 0&\text{otherwise.}\end{array}
One of four algorithms is used to generate the variates depending on the values of :math:a and :math:b.
Let :math:\alpha be the maximum and :math:\beta be the minimum of :math:a and :math:b.
Then the algorithms are as follows:
(i) if :math:\alpha < 0.5, Johnk's algorithm is used, see for example Dagpunar (1988). This generates the beta variate as :math:u_1^{{1/a}}/\begin{pmatrix}u_1^{{1/a}}+u_2^{{1/b}}\end{pmatrix}, where :math:u_1 and :math:u_2 are uniformly distributed random variates;
(#) if :math:\beta > 1, the algorithm BB given by Cheng (1978) is used. This involves the generation of an observation from a beta distribution of the second kind by the envelope rejection method using a log-logistic target distribution and then transforming it to a beta variate;
(#) if :math:\alpha > 1 and :math:\beta < 1, the switching algorithm given by Atkinson (1979) is used. The two target distributions used are :math:f_1\left(x\right) = \beta x^{\beta } and :math:f_2\left(x\right) = \alpha \left(1-x\right)^{{\beta -1}}, along with the approximation to the switching parameter of :math:t = \left(1-\beta \right)/\left(\alpha +1-\beta \right);
(#) in all other cases, Cheng's BC algorithm (see Cheng (1978)) is used with modifications suggested by Dagpunar (1988). This algorithm is similar to BB, used when :math:\beta > 1, but is tuned for small values of :math:a and :math:b.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_beta.
.. _g05sb-py2-py-references:
**References**
Atkinson, A C, 1979, A family of switching algorithms for the computer generation of beta random variates, Biometrika (66), 141--5
Cheng, R C H, 1978, Generating beta variates with nonintegral shape parameters, Comm. ACM (21), 317--322
Dagpunar, J, 1988, Principles of Random Variate Generation, Oxford University Press
Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth
"""
raise NotImplementedError
[docs]def dist_cauchy(n, xmed, semiqr, statecomm):
r"""
dist_cauchy generates a vector of pseudorandom numbers from a Cauchy distribution with median :math:a and semi-interquartile range :math:b.
.. _g05sc-py2-py-doc:
For full information please refer to the NAG Library document for g05sc
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05scf.html
.. _g05sc-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**xmed** : float
:math:a, the median of the distribution.
**semiqr** : float
:math:b, the semi-interquartile range of the distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified Cauchy distribution.
.. _g05sc-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{semiqr} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{semiqr}\geq 0.0.
(errno :math:4)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sc-py2-py-notes:
**Notes**
The distribution has PDF (probability density function)
.. math::
f\left(x\right) = \frac{1}{{\pi b\left(1+\left(\frac{{x-a}}{b}\right)^2\right)}}\text{.}
dist_cauchy returns the value
.. math::
a+b\frac{{2y_1-1}}{y_2}\text{,}
where :math:y_1 and :math:y_2 are a pair of consecutive pseudorandom numbers from a uniform distribution over :math:\left(0, 1\right), such that
.. math::
\left(2y_1-1\right)^2+y_2^2\leq 1\text{.}
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_cauchy.
.. _g05sc-py2-py-references:
**References**
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def dist_chisq(n, df, statecomm):
r"""
dist_chisq generates a vector of pseudorandom numbers taken from a :math:\chi^2-distribution with :math:\nu degrees of freedom.
.. _g05sd-py2-py-doc:
For full information please refer to the NAG Library document for g05sd
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05sdf.html
.. _g05sd-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**df** : int
:math:\nu, the number of degrees of freedom of the distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified :math:\chi^2-distribution.
.. _g05sd-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\mathrm{df} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{df}\geq 1.
(errno :math:3)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sd-py2-py-notes:
**Notes**
The distribution has PDF (probability density function)
.. math::
\begin{array}{ll} f\left(x\right) = \frac{{x^{{{\nu /2}-1}}\times e^{{-x/2}}}}{{2^{{\nu /2}}\times \left({\nu /2}-1\right)!}} & \text{if } x > 0 \text{;} \\&\\f\left(x\right) = 0&\text{otherwise.}\end{array}
This is the same as a gamma distribution with parameters :math:\nu /2 and :math:2.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_chisq.
.. _g05sd-py2-py-references:
**References**
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def dist_dirichlet(n, a, statecomm):
r"""
dist_dirichlet generates a vector of pseudorandom numbers taken from a Dirichlet distribution.
.. _g05se-py2-py-doc:
For full information please refer to the NAG Library document for g05se
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05sef.html
.. _g05se-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**a** : float, array-like, shape :math:\left(m\right)
The parameter vector for the distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}, m\right)
The :math:n pseudorandom numbers from the specified Dirichlet distribution, with :math:\mathrm{x}[i-1,j-1] holding the :math:j\ th dimension for the :math:i\ th variate.
.. _g05se-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:m > 0.
(errno :math:3)
On entry, at least one :math:\mathrm{a}[i]\leq 0.
(errno :math:4)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05se-py2-py-notes:
**Notes**
The distribution has PDF (probability density function)
.. math::
\begin{array}{lll}f\left(x\right)& = & \frac{1}{{B\left(\alpha \right)}} \prod_{1}^{m}{x_i^{{\alpha_i-1}}\quad \text{and}} \\B\left(\alpha \right)& = & \frac{{\prod_{1}^{m}{\Gamma \left(\alpha_i\right)}}}{{\Gamma \left(\sum_{1}^{m}{\alpha_i}\right)}} \end{array}
where :math:x = \left\{x_1, x_2, \ldots, x_m\right\} is a vector of dimension :math:m, such that :math:x_i > 0 for all :math:i and :math:\sum_{1}^{m}{x_i} = 1.
dist_dirichlet generates a draw from a Dirichlet distribution by first drawing :math:m independent samples, :math:y_i\sim \mathrm{gamma}\left(\alpha_i, 1\right), i.e., independent draws from a gamma distribution with parameters :math:\alpha_i > 0 and one, and then setting :math:x_i = y_i/\sum_{1}^{m}{y_j}.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_dirichlet.
.. _g05se-py2-py-references:
**References**
Dagpunar, J, 1988, Principles of Random Variate Generation, Oxford University Press
Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth
"""
raise NotImplementedError
[docs]def dist_exp(n, a, statecomm):
r"""
dist_exp generates a vector of pseudorandom numbers from a (negative) exponential distribution with mean :math:a.
.. _g05sf-py2-py-doc:
For full information please refer to the NAG Library document for g05sf
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05sff.html
.. _g05sf-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**a** : float
:math:a, the mean of the distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified exponential distribution.
.. _g05sf-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\mathrm{a} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{a} > 0.0.
(errno :math:3)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sf-py2-py-notes:
**Notes**
The exponential distribution has PDF (probability density function):
.. math::
\begin{array}{ll} f\left(x\right) = \frac{1}{a} e^{{-x/a}} & \text{if }x\geq 0\text{,} \\\\f\left(x\right) = 0&\text{otherwise.}\end{array}
dist_exp returns the values
.. math::
x_i = -a\mathrm{ln}\left(y_i\right)
where :math:y_i are the next :math:n numbers generated by a uniform :math:\left(0, 1\right] generator.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_exp.
.. _g05sf-py2-py-references:
**References**
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def dist_expmix(n, a, wgt, statecomm):
r"""
dist_expmix generates a vector of pseudorandom numbers from an exponential mix distribution composed of :math:m exponential distributions each having a mean :math:a_i and weight :math:w_i.
.. _g05sg-py2-py-doc:
For full information please refer to the NAG Library document for g05sg
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05sgf.html
.. _g05sg-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**a** : float, array-like, shape :math:\left(\textit{nmix}\right)
The :math:m parameters :math:a_i for the :math:m exponential distributions in the mix.
**wgt** : float, array-like, shape :math:\left(\textit{nmix}\right)
The :math:m weights :math:w_i for the :math:m exponential distributions in the mix.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified exponential mix distribution.
.. _g05sg-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\textit{nmix} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\textit{nmix}\geq 1.
(errno :math:3)
On entry, :math:\mathrm{a}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{a}[i-1] > 0.0.
(errno :math:4)
On entry, :math:\mathrm{wgt}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{wgt}[i-1]\geq 0.0.
(errno :math:4)
On entry, sum of :math:\mathrm{wgt} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: sum of :math:\mathrm{wgt} = 1.0.
(errno :math:5)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sg-py2-py-notes:
**Notes**
The distribution has PDF (probability density function)
.. math::
\begin{array}{ll} f\left(x\right) = \sum_{{i = 1}}^m \frac{1}{a_i} w_i e^{{-x/a_i}} &\text{if }x\geq 0\text{,}\\ f\left(x\right) = 0 &\text{otherwise,}\end{array}
where :math:\sum_{{i = 1}}^mw_i = 1 and :math:a_i > 0, :math:w_i\geq 0.
dist_expmix returns the values :math:x_i by selecting, with probability :math:w_j, random variates from an exponential distribution with parameter :math:a_j.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_expmix.
.. _g05sg-py2-py-references:
**References**
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def dist_f(n, df1, df2, statecomm):
r"""
dist_f generates a vector of pseudorandom numbers taken from an :math:F (or Fisher's variance ratio) distribution with :math:\mu and :math:\nu degrees of freedom.
.. _g05sh-py2-py-doc:
For full information please refer to the NAG Library document for g05sh
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05shf.html
.. _g05sh-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**df1** : int
:math:\mu, the number of degrees of freedom of the distribution.
**df2** : int
:math:\nu, the number of degrees of freedom of the distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified :math:F-distribution.
.. _g05sh-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\mathrm{df1} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{df1}\geq 1.
(errno :math:3)
On entry, :math:\mathrm{df2} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{df2}\geq 1.
(errno :math:4)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sh-py2-py-notes:
**Notes**
The distribution has PDF (probability density function)
.. math::
\begin{array}{ll} f \left(x\right) = \frac{{\left(\frac{{\mu +\nu -2}}{2}\right)!x^{{\frac{1}{2}\mu -1}}}}{{\left(\frac{1}{2}\mu -1\right)!\left(\frac{1}{2}\nu -1\right)!\left(1+\frac{\mu }{\nu }x\right)^{{\frac{1}{2}\left(\mu +\nu \right)}}}} \times \left(\frac{\mu }{\nu }\right)^{{\frac{1}{2}\mu }} & \text{if } x > 0 \text{,} \\&\\f\left(x\right) = 0&\text{otherwise.}\end{array}
dist_f calculates the values
.. math::
\frac{{\nu y_i}}{{\mu z_i}}\text{, }\quad i = 1,2,\ldots,n\text{,}
where :math:y_i and :math:z_i are generated by :meth:dist_gamma from gamma distributions with parameters :math:\left({\frac{1}{2}\mu }, 2\right) and :math:\left({\frac{1}{2}\nu }, 2\right) respectively (i.e., from :math:\chi^2-distributions with :math:\mu and :math:\nu degrees of freedom).
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_f.
.. _g05sh-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def dist_gamma(n, a, b, statecomm):
r"""
dist_gamma generates a vector of pseudorandom numbers taken from a gamma distribution with parameters :math:a and :math:b.
.. _g05sj-py2-py-doc:
For full information please refer to the NAG Library document for g05sj
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05sjf.html
.. _g05sj-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**a** : float
:math:a, the parameter of the gamma distribution.
**b** : float
:math:b, the parameter of the gamma distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified gamma distribution.
.. _g05sj-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\mathrm{a} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{a} > 0.0.
(errno :math:3)
On entry, :math:\mathrm{b} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{b} > 0.0.
(errno :math:4)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sj-py2-py-notes:
**Notes**
The gamma distribution has PDF (probability density function)
.. math::
\begin{array}{ll}f\left(x\right) = \frac{1}{{b^a\Gamma \left(a\right)}}x^{{a-1}}e^{{-x/b}}&\text{if }x\geq 0\text{; }\quad a,b > 0\\\\f\left(x\right) = 0&\text{otherwise.}\end{array}
One of three algorithms is used to generate the variates depending upon the value of :math:a:
(i) if :math:a < 1, a switching algorithm described by Dagpunar (1988) (called G6) is used. The target distributions are :math:f_1\left(x\right) = cax^{{a-1}}/t^a and :math:f_2\left(x\right) = \left(1-c\right)e^{{-\left(x-t\right)}}, where :math:c = t/\left(t+ae^{{-t}}\right), and the switching parameter, :math:t, is taken as :math:1-a. This is similar to Ahrens and Dieter's GS algorithm (see Ahrens and Dieter (1974)) in which :math:t = 1;
(#) if :math:a = 1, the gamma distribution reduces to the exponential distribution and the method based on the logarithmic transformation of a uniform random variate is used;
(#) if :math:a > 1, the algorithm given by Best (1978) is used. This is based on using a Student's :math:t-distribution with two degrees of freedom as the target distribution in an envelope rejection method.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_gamma.
.. _g05sj-py2-py-references:
**References**
Ahrens, J H and Dieter, U, 1974, Computer methods for sampling from gamma, beta, Poisson and binomial distributions, Computing (12), 223--46
Best, D J, 1978, Letter to the Editor, Appl. Statist. (27), 181
Dagpunar, J, 1988, Principles of Random Variate Generation, Oxford University Press
Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth
"""
raise NotImplementedError
[docs]def dist_normal(n, xmu, var, statecomm):
r"""
dist_normal generates a vector of pseudorandom numbers taken from a Normal (Gaussian) distribution with mean :math:\mu and variance :math:\sigma^2.
.. _g05sk-py2-py-doc:
For full information please refer to the NAG Library document for g05sk
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05skf.html
.. _g05sk-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**xmu** : float
:math:\mu, the mean of the distribution.
**var** : float
:math:\sigma^2, the variance of the distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified Normal distribution.
.. _g05sk-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{var} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{var}\geq 0.0.
(errno :math:4)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sk-py2-py-notes:
**Notes**
The distribution has PDF (probability distribution function)
.. math::
f\left(x\right) = \frac{1}{{\sigma \sqrt{2\pi }}}\mathrm{exp}\left(-\frac{\left(x-\mu \right)^2}{{2\sigma^2}}\right)\text{.}
dist_normal uses the algorithm of Wichura (1988).
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_normal.
.. _g05sk-py2-py-references:
**References**
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
Wichura, 1988, Algorithm AS 241: the percentage points of the Normal distribution, Appl. Statist. (37), 477--484
"""
raise NotImplementedError
[docs]def dist_logistic(n, a, b, statecomm):
r"""
dist_logistic generates a vector of pseudorandom numbers from a logistic distribution with mean :math:a and spread :math:b.
.. _g05sl-py2-py-doc:
For full information please refer to the NAG Library document for g05sl
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05slf.html
.. _g05sl-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**a** : float
:math:a, the mean of the distribution.
**b** : float
:math:b, the spread of the distribution, where 'spread' is :math:\frac{\sqrt{3}}{\pi }\times \text{}\ standard deviation.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified logistic distribution.
.. _g05sl-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{b} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{b}\geq 0.0.
(errno :math:4)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sl-py2-py-notes:
**Notes**
The distribution has PDF (probability density function)
.. math::
f\left(x\right) = \frac{e^{{\left(x-a\right)/b}}}{{b\left(1+e^{{\left(x-a\right)/b}}\right)^2}}\text{.}
dist_logistic returns the value
.. math::
a+b\mathrm{ln}\left(\frac{y}{{1-y}}\right)\text{,}
where :math:y is a pseudorandom number uniformly distributed over :math:\left(0, 1\right).
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_logistic.
.. _g05sl-py2-py-references:
**References**
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def dist_lognormal(n, xmu, var, statecomm):
r"""
dist_lognormal generates a vector of pseudorandom numbers from a log-normal distribution with parameters :math:\mu and :math:\sigma^2.
.. _g05sm-py2-py-doc:
For full information please refer to the NAG Library document for g05sm
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05smf.html
.. _g05sm-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**xmu** : float
:math:\mu, the mean of the distribution of :math:\mathrm{ln}\left(x\right).
**var** : float
:math:\sigma^2, the variance of the distribution of :math:\mathrm{ln}\left(x\right).
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified log-normal distribution.
.. _g05sm-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\mathrm{xmu} is too large to take the exponential of :math:\mathrm{xmu} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:3)
On entry, :math:\mathrm{var} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{var}\geq 0.0.
(errno :math:4)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sm-py2-py-notes:
**Notes**
The distribution has PDF (probability density function)
.. math::
\begin{array}{ll} f\left(x\right) = \frac{1}{{x\sigma \sqrt{2\pi }}} \mathrm{exp}\left(-\frac{\left(\mathrm{ln}\left(x\right)-\mu \right)^2}{{2\sigma^2}}\right) & \text{if } x > 0 \text{,} \\\\f\left(x\right) = 0&\text{otherwise,}\end{array}
i.e., :math:\mathrm{ln}\left(x\right) is normally distributed with mean :math:\mu and variance :math:\sigma^2. dist_lognormal evaluates :math:\mathrm{exp}\left(y_i\right), where the :math:y_i are generated by :meth:dist_normal from a Normal distribution with mean :math:\mu and variance :math:\sigma^2, for :math:\textit{i} = 1,2,\ldots,n.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_lognormal.
.. _g05sm-py2-py-references:
**References**
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def dist_students_t(n, df, statecomm):
r"""
dist_students_t generates a vector of pseudorandom numbers taken from a Student's :math:t-distribution with :math:\nu degrees of freedom.
.. _g05sn-py2-py-doc:
For full information please refer to the NAG Library document for g05sn
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05snf.html
.. _g05sn-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**df** : int
:math:\nu, the number of degrees of freedom of the distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified Student's :math:t-distribution.
.. _g05sn-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\mathrm{df} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{df}\geq 1.
(errno :math:3)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sn-py2-py-notes:
**Notes**
The distribution has PDF (probability density function)
.. math::
f\left(x\right) = \frac{{\left(\frac{{\nu -1}}{2}\right)!}}{{\left(\frac{1}{2}\nu -1\right)!\sqrt{\pi \nu }\left(1+\frac{x^2}{\nu }\right)^{{\frac{1}{2}\left(\nu +1\right)}}}}\text{.}
dist_students_t calculates the values
.. math::
y_i\sqrt{\frac{\nu }{z_i}}\text{, }\quad i = 1,\ldots,n
where the :math:y_i are generated by :meth:dist_normal from a Normal distribution with mean :math:0 and variance :math:1.0, and the :math:z_i are generated by :meth:dist_gamma from a gamma distribution with parameters :math:\frac{1}{2}\nu and :math:2 (i.e., from a :math:\chi^2-distribution with :math:\nu degrees of freedom).
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_students_t.
.. _g05sn-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def dist_triangular(n, xmin, xmed, xmax, statecomm):
r"""
dist_triangular generates a vector of pseudorandom numbers from a triangular distribution with parameters :math:x_{\mathrm{min}}, :math:x_{\mathrm{med}} and :math:x_{\mathrm{max}}.
.. _g05sp-py2-py-doc:
For full information please refer to the NAG Library document for g05sp
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05spf.html
.. _g05sp-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**xmin** : float
The end point :math:x_{\mathrm{min}} of the triangular distribution.
**xmed** : float
The median of the distribution :math:x_{\mathrm{med}} (also the location of the vertex of the triangular distribution at which the PDF reaches a maximum).
**xmax** : float
The end point :math:x_{\mathrm{max}} of the triangular distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified triangular distribution.
.. _g05sp-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{xmed} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{xmin} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{xmed}\geq \mathrm{xmin}.
(errno :math:4)
On entry, :math:\mathrm{xmax} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{xmed} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{xmax}\geq \mathrm{xmed}.
(errno :math:5)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sp-py2-py-notes:
**Notes**
The triangular distribution has a PDF (probability density function) that is triangular in profile.
The base of the triangle ranges from :math:x = x_{\mathrm{min}} to :math:x = x_{\mathrm{max}} and the PDF has a maximum value of :math:\frac{2}{{x_{\mathrm{max}}-x_{\mathrm{min}}}} at :math:x = x_{\mathrm{med}}.
If :math:x_{\mathrm{min}} = x_{\mathrm{med}} = x_{\mathrm{max}} then :math:x = x_{\mathrm{med}} with probability 1; otherwise the triangular distribution has PDF:
.. math::
\begin{array}{ll} f\left(x\right) = \frac{{x-x_{\mathrm{min}}}}{{x_{\mathrm{med}}-x_{\mathrm{min}}}} \times \frac{2}{{x_{\mathrm{max}}-x_{\mathrm{min}}}} &\text{ if }x_{\mathrm{min}}\leq x\leq x_{\mathrm{med}}\text{,}\\\\\\f\left(x\right) = \frac{{x_{\mathrm{max}}-x}}{{x_{\mathrm{max}}-x_{\mathrm{med}}}}\times \frac{2}{{x_{\mathrm{max}}-x_{\mathrm{min}}}}&\text{ if }x_{\mathrm{med}} < x\leq x_{\mathrm{max}}\text{,}\\\\\\f\left(x\right) = 0&\text{ otherwise.}\end{array}
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_triangular.
.. _g05sp-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def dist_uniform(n, a, b, statecomm):
r"""
dist_uniform generates a vector of pseudorandom numbers uniformly distributed over the interval :math:\left[a, b\right].
.. _g05sq-py2-py-doc:
For full information please refer to the NAG Library document for g05sq
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05sqf.html
.. _g05sq-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**a** : float
The end points :math:a and :math:b of the uniform distribution.
**b** : float
The end points :math:a and :math:b of the uniform distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified uniform distribution.
.. _g05sq-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{a} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{b} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{b}\geq \mathrm{a}.
(errno :math:4)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sq-py2-py-notes:
**Notes**
If :math:a = 0 and :math:b = 1, dist_uniform returns the next :math:n values :math:y_i from a uniform :math:\left(0, 1\right] generator (see :meth:dist_uniform01 for details).
For other values of :math:a and :math:b, dist_uniform applies the transformation
.. math::
x_i = a+\left(b-a\right)y_i\text{.}
The function ensures that the values :math:x_i lie in the closed interval :math:\left[a, b\right].
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_uniform.
.. _g05sq-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def dist_vonmises(n, vk, statecomm):
r"""
dist_vonmises generates a vector of pseudorandom numbers from a von Mises distribution with concentration parameter :math:\kappa.
.. _g05sr-py2-py-doc:
For full information please refer to the NAG Library document for g05sr
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05srf.html
.. _g05sr-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**vk** : float
:math:\kappa, the concentration parameter of the required von Mises distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified von Mises distribution.
.. _g05sr-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\mathrm{vk}\leq 0.0 or :math:\mathrm{vk} too large: :math:\mathrm{vk} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:3)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05sr-py2-py-notes:
**Notes**
The von Mises distribution is a symmetric distribution used in the analysis of circular data.
The PDF (probability density function) of this distribution on the circle with mean direction :math:\mu_0 = 0 and concentration parameter :math:\kappa, can be written as:
.. math::
f\left(\theta \right) = \frac{{e^{{\kappa \cos\left(\theta \right)}}}}{{2\pi I_0\left(\kappa \right)}}\text{,}
where :math:\theta is reduced modulo :math:2\pi so that :math:{-\pi }\leq \theta < \pi and :math:\kappa \geq 0.
For very small :math:\kappa the distribution is almost the uniform distribution, whereas for :math:\kappa →\infty all the probability is concentrated at one point.
The :math:n variates, :math:\theta_1,\theta_2,\ldots,\theta_n, are generated using an envelope rejection method with a wrapped Cauchy target distribution as proposed by Best and Fisher (1979) and described by Dagpunar (1988).
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_vonmises.
.. _g05sr-py2-py-references:
**References**
Best, D J and Fisher, N I, 1979, Efficient simulation of the von Mises distribution, Appl. Statist. (28), 152--157
Dagpunar, J, 1988, Principles of Random Variate Generation, Oxford University Press
Mardia, K V, 1972, Statistics of Directional Data, Academic Press
"""
raise NotImplementedError
[docs]def dist_weibull(n, a, b, statecomm):
r"""
dist_weibull generates a vector of pseudorandom numbers from a two parameter Weibull distribution with shape parameter :math:a and scale parameter :math:b.
.. _g05ss-py2-py-doc:
For full information please refer to the NAG Library document for g05ss
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05ssf.html
.. _g05ss-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**a** : float
:math:a, the shape parameter of the distribution.
**b** : float
:math:b, the scale parameter of the distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : float, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified Weibull distribution.
.. _g05ss-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\mathrm{a} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{a} > 0.0.
(errno :math:3)
On entry, :math:\mathrm{b} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{b} > 0.0.
(errno :math:4)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05ss-py2-py-notes:
**Notes**
The distribution has PDF (probability density function)
.. math::
\begin{array}{ll} f\left(x\right) = \frac{a}{b} x^{{a-1}} e^{{-x^a/b}} &\text{if }x > 0\text{,}\\\\f\left(x\right) = 0&\text{otherwise.}\end{array}
dist_weibull returns the value :math:\left(-b\mathrm{ln}\left(y\right)\right)^{{1/a}}, where :math:y is a pseudorandom number from a uniform distribution over :math:\left(0, 1\right].
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to dist_weibull.
.. _g05ss-py2-py-references:
**References**
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def int_binomial(mode, n, m, p, statecomm, comm=None):
r"""
int_binomial generates a vector of pseudorandom integers from the discrete binomial distribution with parameters :math:m and :math:p.
.. _g05ta-py2-py-doc:
For full information please refer to the NAG Library document for g05ta
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05taf.html
.. _g05ta-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector only.
:math:\mathrm{mode} = 1
Generate variates using reference vector set up in a prior call to int_binomial.
:math:\mathrm{mode} = 2
Set up reference vector and generate variates.
:math:\mathrm{mode} = 3
Generate variates without using the reference vector.
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**m** : int
:math:m, the number of trials of the distribution.
**p** : float
:math:p, the probability of success of the binomial distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**comm** : None or dict, communication object, optional, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1, this argument must have been initialized by a prior call to int_binomial.
If :math:\mathrm{mode} = 3, :math:\mathrm{comm} is not referenced and may be **None**.
**Returns**
**x** : None or int, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified binomial distribution.
.. _g05ta-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1, :math:2 or :math:3.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{m}\geq 0.
(errno :math:4)
On entry, :math:\mathrm{p} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0.0\leq \mathrm{p}\leq 1.0.
(errno :math:5)
:math:\mathrm{p} or :math:\mathrm{m} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:\mathrm{p} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{p} = \langle\mathit{\boldsymbol{value}}\rangle.
Previous value of :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:5)
On entry, some of the elements of the array :math:\mathrm{comm}\ ['r'] have been corrupted or have not been initialized.
(errno :math:7)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05ta-py2-py-notes:
**Notes**
int_binomial generates :math:n integers :math:x_i from a discrete binomial distribution, where the probability of :math:x_i = I is
.. math::
P\left(x_i = I\right) = \frac{{m!}}{{I!\left(m-I\right)!}}p^I\times \left(1-p\right)^{{m-I}}\text{, }\quad I = 0,1,\ldots,m\text{,}
where :math:m\geq 0 and :math:0\leq p\leq 1.
This represents the probability of achieving :math:I successes in :math:m trials when the probability of success at a single trial is :math:p.
The variates can be generated with or without using a search table and index.
If a search table is used then it is stored with the index in a reference vector and subsequent calls to int_binomial with the same parameter values can then use this reference vector to generate further variates.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to int_binomial.
.. _g05ta-py2-py-references:
**References**
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def logical(n, p, statecomm):
r"""
logical generates a vector of pseudorandom logical values -- :math:\mathbf{True} with probability :math:p and :math:\mathbf{False} with probability :math:\left(1-p\right).
.. _g05tb-py2-py-doc:
For full information please refer to the NAG Library document for g05tb
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05tbf.html
.. _g05tb-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom logical values to be generated.
**p** : float
Must contain the probability of logical returning :math:\mathbf{True}.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : bool, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n logical values.
.. _g05tb-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:2)
On entry, :math:\mathrm{p} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0.0\leq \mathrm{p}\leq 1.0.
(errno :math:3)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05tb-py2-py-notes:
**Notes**
logical generates :math:n logical values :math:x_i from the relation
.. math::
y_i < p
where :math:y_i is a pseudorandom number from a uniform distribution over :math:\left(0, 1\right], generated by :meth:dist_uniform01 using the values of :math:\mathrm{statecomm}\ ['state'] as input to this function.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to logical.
.. _g05tb-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def int_geom(mode, n, p, statecomm, comm=None):
r"""
int_geom generates a vector of pseudorandom integers from the discrete geometric distribution with probability :math:p of success at a trial.
.. _g05tc-py2-py-doc:
For full information please refer to the NAG Library document for g05tc
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05tcf.html
.. _g05tc-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector only.
:math:\mathrm{mode} = 1
Generate variates using reference vector set up in a prior call to int_geom.
:math:\mathrm{mode} = 2
Set up reference vector and generate variates.
:math:\mathrm{mode} = 3
Generate variates without using the reference vector.
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**p** : float
The parameter :math:p of the geometric distribution representing the probability of success at a single trial.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**comm** : None or dict, communication object, optional, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1, this argument must have been initialized by a prior call to int_geom.
If :math:\mathrm{mode} = 3, :math:\mathrm{comm} is not referenced and may be **None**.
**Returns**
**x** : None or int, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified geometric distribution.
.. _g05tc-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1, :math:2 or :math:3.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{p} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\text{machine precision}\leq \mathrm{p}\leq 1.0.
(errno :math:3)
:math:\mathrm{p} is so small that :math:\textit{lr} would have to be larger than the largest representable integer. Use :math:\mathrm{mode} = 3 instead. :math:\mathrm{p} = \langle\mathit{\boldsymbol{value}}\rangle
(errno :math:4)
:math:\mathrm{p} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:\mathrm{p} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{p} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:4)
On entry, some of the elements of the array :math:\mathrm{comm}\ ['r'] have been corrupted or have not been initialized.
(errno :math:6)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05tc-py2-py-notes:
**Notes**
int_geom generates :math:n integers :math:x_i from a discrete geometric distribution, where the probability of :math:x_i = I (a first success after :math:I+1 trials) is
.. math::
P\left(x_i = I\right) = p\times \left(1-p\right)^I\text{, }\quad I = 0,1,\ldots \text{.}
The variates can be generated with or without using a search table and index.
If a search table is used then it is stored with the index in a reference vector and subsequent calls to int_geom with the same parameter value can then use this reference vector to generate further variates.
If the search table is not used (as recommended for small values of :math:p) then a direct transformation of uniform variates is used.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to int_geom.
.. _g05tc-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def int_general(mode, n, p, ip1, itype, statecomm, comm=None):
r"""
int_general generates a vector of pseudorandom integers from a discrete distribution with a given PDF (probability density function) or CDF (cumulative distribution function) :math:p.
.. _g05td-py2-py-doc:
For full information please refer to the NAG Library document for g05td
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05tdf.html
.. _g05td-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector only.
:math:\mathrm{mode} = 1
Generate variates using reference vector set up in a prior call to int_general.
:math:\mathrm{mode} = 2
Set up reference vector and generate variates.
:math:\mathrm{mode} = 3
Generate variates without using the reference vector.
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**p** : float, array-like, shape :math:\left(\textit{np}\right)
The PDF or CDF of the distribution.
**ip1** : int
The value of the variate, a whole number, to which the probability in :math:\mathrm{p}[0] corresponds.
**itype** : int
Indicates the type of information contained in :math:\mathrm{p}.
:math:\mathrm{itype} = 1
:math:\mathrm{p} contains a probability distribution function (PDF).
:math:\mathrm{itype} = 2
:math:\mathrm{p} contains a cumulative distribution function (CDF).
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**comm** : None or dict, communication object, optional, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1, this argument must have been initialized by a prior call to int_general.
If :math:\mathrm{mode} = 3, :math:\mathrm{comm} is not referenced and may be **None**.
**Returns**
**x** : None or int, ndarray, shape :math:\left(\mathrm{n}\right)
Contains :math:n pseudorandom numbers from the specified discrete distribution.
.. _g05td-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1 or :math:2.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, at least one element of the vector :math:\mathrm{p} is less than :math:0.0 or greater than :math:1.0.
(errno :math:3)
On entry, :math:\mathrm{itype} = 1 and the sum of the elements of :math:\mathrm{p} do not equal one.
(errno :math:3)
On entry, :math:\mathrm{itype} = 2 and the values of :math:\mathrm{p} are not all in stricly ascending order.
(errno :math:3)
On entry, :math:\mathrm{p}[\textit{np}-1] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: if :math:\mathrm{itype} = 2, :math:\mathrm{p}[\textit{np}-1] = 1.0.
(errno :math:4)
On entry, :math:\textit{np} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\textit{np} > 0.
(errno :math:6)
On entry, :math:\mathrm{itype} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{itype} = 1 or :math:2.
(errno :math:7)
The value of :math:\textit{np} or :math:\mathrm{ip1} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:\textit{np} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\textit{np} = \langle\mathit{\boldsymbol{value}}\rangle.
Previous value of :math:\mathrm{ip1} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{ip1} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:7)
On entry, some of the elements of the array :math:\mathrm{comm}\ ['r'] have been corrupted or have not been initialized.
(errno :math:9)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05td-py2-py-notes:
**Notes**
int_general generates a sequence of :math:n integers :math:x_i, from a discrete distribution defined by information supplied in :math:\mathrm{p}.
This may either be the PDF or CDF of the distribution.
A reference vector is first set up to contain the CDF of the distribution in its higher elements, followed by an index.
Setting up the reference vector and subsequent generation of variates can each be performed by separate calls to int_general or may be combined in a single call.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to int_general.
.. _g05td-py2-py-references:
**References**
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def int_hypergeom(mode, n, ns, np, m, statecomm, comm=None):
r"""
int_hypergeom generates a vector of pseudorandom integers from the discrete hypergeometric distribution of the number of specified items in a sample of size :math:l, taken from a population of size :math:k with :math:m specified items in it.
.. _g05te-py2-py-doc:
For full information please refer to the NAG Library document for g05te
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05tef.html
.. _g05te-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector only.
:math:\mathrm{mode} = 1
Generate variates using reference vector set up in a prior call to int_hypergeom.
:math:\mathrm{mode} = 2
Set up reference vector and generate variates.
:math:\mathrm{mode} = 3
Generate variates without using the reference vector.
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**ns** : int
:math:l, the sample size of the hypergeometric distribution.
**np** : int
:math:k, the population size of the hypergeometric distribution.
**m** : int
:math:m, the number of specified items of the hypergeometric distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**comm** : None or dict, communication object, optional, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1, this argument must have been initialized by a prior call to int_hypergeom.
If :math:\mathrm{mode} = 3, :math:\mathrm{comm} is not referenced and may be **None**.
**Returns**
**x** : None or int, ndarray, shape :math:\left(\mathrm{n}\right)
The pseudorandom numbers from the specified hypergeometric distribution.
.. _g05te-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1, :math:2 or :math:3.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{ns} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{np} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0\leq \mathrm{ns}\leq \mathrm{np}.
(errno :math:4)
On entry, :math:\mathrm{np} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{np}\geq 0.
(errno :math:5)
On entry, :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{np} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0\leq \mathrm{m}\leq \mathrm{np}.
(errno :math:6)
The value of :math:\mathrm{ns}, :math:\mathrm{np} or :math:\mathrm{m} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call with :math:\mathrm{mode} = 0 or :math:2.
(errno :math:6)
On entry, some of the elements of the array :math:\mathrm{comm}\ ['r'] have been corrupted or have not been initialized.
(errno :math:8)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05te-py2-py-notes:
**Notes**
int_hypergeom generates :math:n integers :math:x_i from a discrete hypergeometric distribution, where the probability of :math:x_i = I is
.. math::
\begin{array}{cc}P\left(i = I\right) = \frac{{l!m!\left(k-l\right)!\left(k-m\right)!}}{{I!\left(l-I\right)!\left(m-I\right)!\left(k-m-l+I\right)!k!}}&\quad \text{ if } I = \mathrm{max}\left(0, {m+l-k}\right), \ldots, \mathrm{min}\left(l, m\right) \text{,} \\\\P\left(i = I\right) = 0&\quad \text{ otherwise.}\end{array}
The variates can be generated with or without using a search table and index.
If a search table is used then it is stored with the index in a reference vector and subsequent calls to int_hypergeom with the same parameter values can then use this reference vector to generate further variates.
The reference array is generated by a recurrence relation if :math:lm\left(k-l\right)\left(k-m\right) < 50k^3, otherwise Stirling's approximation is used.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to int_hypergeom.
.. _g05te-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def int_log(mode, n, a, statecomm, comm=None):
r"""
int_log generates a vector of pseudorandom integers from the discrete logarithmic distribution with parameter :math:a.
.. _g05tf-py2-py-doc:
For full information please refer to the NAG Library document for g05tf
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05tff.html
.. _g05tf-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector only.
:math:\mathrm{mode} = 1
Generate variates using reference vector set up in a prior call to int_log.
:math:\mathrm{mode} = 2
Set up reference vector and generate variates.
:math:\mathrm{mode} = 3
Generate variates without using the reference vector.
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**a** : float
:math:a, the parameter of the logarithmic distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**comm** : None or dict, communication object, optional, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1, this argument must have been initialized by a prior call to int_log.
If :math:\mathrm{mode} = 3, :math:\mathrm{comm} is not referenced and may be **None**.
**Returns**
**x** : None or int, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified logarithmic distribution.
.. _g05tf-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1, :math:2 or :math:3.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{a} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0.0 < \mathrm{a} < 1.0.
(errno :math:4)
The value of :math:\mathrm{a} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:\mathrm{a} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{a} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:4)
On entry, some of the elements of the array :math:\mathrm{comm}\ ['r'] have been corrupted or have not been initialized.
(errno :math:6)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05tf-py2-py-notes:
**Notes**
int_log generates :math:n integers :math:x_i from a discrete logarithmic distribution, where the probability of :math:x_i = I is
.. math::
P\left(x_i = I\right) = -\frac{a^I}{{I\times \log\left(1-a\right)}}\text{, }\quad I = 1,2,\ldots \text{,}
where :math:0 < a < 1\text{.}
The variates can be generated with or without using a search table and index.
If a search table is used then it is stored with the index in a reference vector and subsequent calls to int_log with the same parameter value can then use this reference vector to generate further variates.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to int_log.
.. _g05tf-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def int_multinomial(mode, n, m, p, statecomm, comm=None):
r"""
int_multinomial generates a sequence of :math:n variates, each consisting of :math:k pseudorandom integers, from the discrete multinomial distribution with :math:k outcomes and :math:m trials, where the outcomes have probabilities :math:p_1,p_2,\ldots,p_k respectively.
.. _g05tg-py2-py-doc:
For full information please refer to the NAG Library document for g05tg
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05tgf.html
.. _g05tg-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector only.
:math:\mathrm{mode} = 1
Generate variates using reference vector set up in a prior call to int_multinomial.
:math:\mathrm{mode} = 2
Set up reference vector and generate variates.
:math:\mathrm{mode} = 3
Generate variates without using the reference vector.
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**m** : int
:math:m, the number of trials of the multinomial distribution.
**p** : float, array-like, shape :math:\left(k\right)
Contains the probabilities :math:p_{\textit{j}}, for :math:\textit{j} = 1,2,\ldots,k, of the :math:k possible outcomes of the multinomial distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**comm** : None or dict, communication object, optional, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1, this argument must have been initialized by a prior call to int_multinomial.
If :math:\mathrm{mode} = 3, :math:\mathrm{comm} is not referenced and may be **None**.
**Returns**
**x** : None or int, ndarray, shape :math:\left(\mathrm{n}, :\right)
The first :math:n rows of :math:\mathrm{x}[i-1,j-1] each contain :math:k pseudorandom numbers representing a :math:k-dimensional variate from the specified multinomial distribution.
.. _g05tg-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1, :math:2 or :math:3.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{m}\geq 0.
(errno :math:4)
On entry, :math:k = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:k\geq 2.
(errno :math:5)
On entry, at least one element of the vector :math:\mathrm{p} is less than :math:0.0 or greater than :math:1.0.
(errno :math:5)
On entry, the sum of the elements of :math:\mathrm{p} do not equal one.
(errno :math:6)
The value of :math:\mathrm{m} or :math:\textit{k} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle.
Previous value of :math:k = \langle\mathit{\boldsymbol{value}}\rangle and :math:k = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:6)
On entry, some of the elements of the array :math:\mathrm{comm}\ ['r'] have been corrupted or have not been initialized.
(errno :math:8)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05tg-py2-py-notes:
**Notes**
int_multinomial generates a sequence of :math:n groups of :math:k integers :math:x_{{\textit{i},\textit{j}}}, for :math:\textit{i} = 1,2,\ldots,n, for :math:\textit{j} = 1,2,\ldots,k, from a multinomial distribution with :math:m trials and :math:k outcomes, where the probability of :math:x_{{\textit{i},\textit{j}}} = I_j for each :math:j = 1,2,\ldots,k is
.. math::
P\left({i_1 = I_1}, \ldots, {i_k = I_k}\right) = \frac{{m!}}{{\prod_{{j = 1}}^kI_j!}}\prod_{{j = 1}}^kp_j^{I_j} = \frac{{m!}}{{I_1!I_2! \cdots I_k!}}p_1^{I_1}p_2^{I_2} \cdots p_k^{I_k}\text{,}
where
.. math::
\sum_{{j = 1}}^kp_j = 1\quad \text{ and }\quad \sum_{{j = 1}}^kI_j = m\text{.}
A single trial can have several outcomes (:math:k) and the probability of achieving each outcome is known (:math:p_j).
After :math:m trials each outcome will have occurred a certain number of times.
The :math:k numbers representing the numbers of occurrences for each outcome after :math:m trials is then a single sample from the multinomial distribution defined by the parameters :math:k, :math:m and :math:p_{\textit{j}}, for :math:\textit{j} = 1,2,\ldots,k.
This function returns :math:n such samples.
When :math:k = 2 this distribution is equivalent to the binomial distribution with parameters :math:m and :math:p = p_1 (see :meth:int_binomial).
The variates can be generated with or without using a search table and index.
If a search table is used then it is stored with the index in a reference vector and subsequent calls to int_multinomial with the same parameter values can then use this reference vector to generate further variates.
The reference array is generated only for the outcome with greatest probability.
The number of successes for the outcome with greatest probability is calculated first as for the binomial distribution (see :meth:int_binomial); the number of successes for other outcomes are calculated in turn for the remaining reduced multinomial distribution; the number of successes for the final outcome is simply calculated to ensure that the total number of successes is :math:m.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to int_multinomial.
.. _g05tg-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def int_negbin(mode, n, m, p, statecomm, comm=None):
r"""
int_negbin generates a vector of pseudorandom integers from the discrete negative binomial distribution with parameter :math:m and probability :math:p of success at a trial.
.. _g05th-py2-py-doc:
For full information please refer to the NAG Library document for g05th
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05thf.html
.. _g05th-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector only.
:math:\mathrm{mode} = 1
Generate variates using reference vector set up in a prior call to int_negbin.
:math:\mathrm{mode} = 2
Set up reference vector and generate variates.
:math:\mathrm{mode} = 3
Generate variates without using the reference vector.
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**m** : int
:math:m, the number of failures of the distribution.
**p** : float
:math:p, the parameter of the negative binomial distribution representing the probability of success at a single trial.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**comm** : None or dict, communication object, optional, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1, this argument must have been initialized by a prior call to int_negbin.
If :math:\mathrm{mode} = 3, :math:\mathrm{comm} is not referenced and may be **None**.
**Returns**
**x** : None or int, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified negative binomial distribution.
.. _g05th-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1, :math:2 or :math:3.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{m}\geq 0.
(errno :math:4)
On entry, :math:\mathrm{p} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0.0\leq \mathrm{p} < 1.0.
(errno :math:5)
:math:\mathrm{p} or :math:\mathrm{m} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:\mathrm{p} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{p} = \langle\mathit{\boldsymbol{value}}\rangle.
Previous value of :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{m} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:5)
On entry, some of the elements of the array :math:\mathrm{comm}\ ['r'] have been corrupted or have not been initialized.
(errno :math:7)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05th-py2-py-notes:
**Notes**
int_negbin generates :math:n integers :math:x_i from a discrete negative binomial distribution, where the probability of :math:x_i = I (:math:I successes before :math:m failures) is
.. math::
P\left(x_i = I\right) = \frac{{\left(m+I-1\right)!}}{{I!\left(m-1\right)!}}\times p^I\times \left(1-p\right)^m\text{, }\quad I = 0,1,\ldots \text{.}
The variates can be generated with or without using a search table and index.
If a search table is used then it is stored with the index in a reference vector and subsequent calls to int_negbin with the same parameter value can then use this reference vector to generate further variates.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to int_negbin.
.. _g05th-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def int_poisson(mode, n, lamda, statecomm, comm=None):
r"""
int_poisson generates a vector of pseudorandom integers from the discrete Poisson distribution with mean :math:\lambda.
.. _g05tj-py2-py-doc:
For full information please refer to the NAG Library document for g05tj
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05tjf.html
.. _g05tj-py2-py-parameters:
**Parameters**
**mode** : int
A code for selecting the operation to be performed by the function.
:math:\mathrm{mode} = 0
Set up reference vector only.
:math:\mathrm{mode} = 1
Generate variates using reference vector set up in a prior call to int_poisson.
:math:\mathrm{mode} = 2
Set up reference vector and generate variates.
:math:\mathrm{mode} = 3
Generate variates without using the reference vector.
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**lamda** : float
:math:\lambda, the mean of the Poisson distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**comm** : None or dict, communication object, optional, modified in place
Communication structure for the reference vector.
If :math:\mathrm{mode} = 1, this argument must have been initialized by a prior call to int_poisson.
If :math:\mathrm{mode} = 3, :math:\mathrm{comm} is not referenced and may be **None**.
**Returns**
**x** : None or int, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified Poisson distribution.
.. _g05tj-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{mode} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{mode} = 0, :math:1, :math:2 or :math:3.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{lamda} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{lamda}\geq 0.0.
(errno :math:3)
:math:\mathrm{lamda} is such that :math:\textit{lr} would have to be larger than the largest representable integer. Use :math:\mathrm{mode} = 3 instead. :math:\mathrm{lamda} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:4)
:math:\mathrm{lamda} is not the same as when :math:\mathrm{comm}\ ['r'] was set up in a previous call.
Previous value of :math:\mathrm{lamda} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{lamda} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:4)
On entry, some of the elements of the array :math:\mathrm{comm}\ ['r'] have been corrupted or have not been initialized.
(errno :math:6)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05tj-py2-py-notes:
**Notes**
int_poisson generates :math:n integers :math:x_i from a discrete Poisson distribution with mean :math:\lambda, where the probability of :math:x_i = I is
.. math::
P\left(x_i = I\right) = \frac{{\lambda^I\times e^{{-\lambda }}}}{{I!}}\text{, }\quad I = 0,1,\ldots \text{,}
where :math:\lambda \geq 0.
The variates can be generated with or without using a search table and index.
If a search table is used then it is stored with the index in a reference vector and subsequent calls to int_poisson with the same parameter values can then use this reference vector to generate further variates.
The reference array is found using a recurrence relation if :math:\lambda is less than :math:50 and by Stirling's formula otherwise.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to int_poisson.
.. _g05tj-py2-py-references:
**References**
Kendall, M G and Stuart, A, 1969, The Advanced Theory of Statistics (Volume 1), (3rd Edition), Griffin
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def int_poisson_varmean(vlamda, statecomm):
r"""
int_poisson_varmean generates a vector of pseudorandom integers, each from a discrete Poisson distribution with differing parameter.
.. _g05tk-py2-py-doc:
For full information please refer to the NAG Library document for g05tk
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05tkf.html
.. _g05tk-py2-py-parameters:
**Parameters**
**vlamda** : float, array-like, shape :math:\left(m\right)
The means, :math:\lambda_{\textit{j}}, for :math:\textit{j} = 1,2,\ldots,m, of the Poisson distributions.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : int, ndarray, shape :math:\left(m\right)
The :math:m pseudorandom numbers from the specified Poisson distributions.
.. _g05tk-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:m = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:m\geq 1.
(errno :math:2)
On entry, at least one element of :math:\mathrm{vlamda} is less than zero.
(errno :math:2)
On entry, at least one element of :math:\mathrm{vlamda} is too large.
(errno :math:3)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05tk-py2-py-notes:
**Notes**
int_poisson_varmean generates :math:m integers :math:x_j, each from a discrete Poisson distribution with mean :math:\lambda_j, where the probability of :math:x_j = I is
.. math::
P\left(x_j = I\right) = \frac{{\lambda_j^I\times e^{{-\lambda_j}}}}{{I!}}\text{, }\quad I = 0,1,\ldots \text{,}
where
.. math::
\lambda_j\geq 0\text{, }\quad j = 1,2,\ldots,m\text{.}
The methods used by this function have low set up times and are designed for efficient use when the value of the parameter :math:\lambda changes during the simulation.
For large samples from a distribution with fixed :math:\lambda using :meth:int_poisson to set up and use a reference vector may be more efficient.
When :math:\lambda < 7.5 the product of uniforms method is used, see for example Dagpunar (1988).
For larger values of :math:\lambda an envelope rejection method is used with a target distribution:
.. math::
\begin{array}{cc}f\left(x\right) = \frac{1}{3}&\text{if }\left\lvert x\right\rvert \leq 1\text{,}\\\\f\left(x\right) = \frac{1}{3}\left\lvert x\right\rvert^{-3}&\text{otherwise.}\end{array}
This distribution is generated using a ratio of uniforms method.
A similar approach has also been suggested by Ahrens and Dieter (1989).
The basic method is combined with quick acceptance and rejection tests given by Maclaren (1990).
For values of :math:\lambda \geq 87 Stirling's approximation is used in the computation of the Poisson distribution function, otherwise tables of factorials are used as suggested by Maclaren (1990).
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to int_poisson_varmean.
.. _g05tk-py2-py-references:
**References**
Ahrens, J H and Dieter, U, 1989, A convenient sampling method with bounded computation times for Poisson distributions, Amer. J. Math. Management Sci., 1--13
Dagpunar, J, 1988, Principles of Random Variate Generation, Oxford University Press
Maclaren, N M, 1990, A Poisson random number generator, Personal Communication
"""
raise NotImplementedError
[docs]def int_uniform(n, a, b, statecomm):
r"""
int_uniform generates a vector of pseudorandom integers uniformly distributed over the interval :math:\left[a, b\right].
.. _g05tl-py2-py-doc:
For full information please refer to the NAG Library document for g05tl
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05tlf.html
.. _g05tl-py2-py-parameters:
**Parameters**
**n** : int
:math:n, the number of pseudorandom numbers to be generated.
**a** : int
The end points :math:a and :math:b of the uniform distribution.
**b** : int
The end points :math:a and :math:b of the uniform distribution.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**x** : int, ndarray, shape :math:\left(\mathrm{n}\right)
The :math:n pseudorandom numbers from the specified uniform distribution.
.. _g05tl-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:\mathrm{a} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{b} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{b}\geq \mathrm{a}.
(errno :math:4)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05tl-py2-py-notes:
**Notes**
int_uniform generates the next :math:n values :math:y_i from a uniform :math:\left(0, 1\right] generator (see :meth:dist_uniform01 for details) and applies the transformation
.. math::
x_i = a+\left\lfloor \left(b-a+1\right)y_i\right\rfloor \text{,}
where :math:\left\lfloor z\right\rfloor is the integer part of the real value :math:z.
The function ensures that the values :math:x_i lie in the closed interval :math:\left[a, b\right].
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to int_uniform.
.. _g05tl-py2-py-references:
**References**
Knuth, D E, 1981, The Art of Computer Programming (Volume 2), (2nd Edition), Addison--Wesley
"""
raise NotImplementedError
[docs]def bb_init(t0, tend, times):
r"""
bb_init initializes the Brownian bridge generator :meth:bb.
It must be called before any calls to :meth:bb.
.. _g05xa-py2-py-doc:
For full information please refer to the NAG Library document for g05xa
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05xaf.html
.. _g05xa-py2-py-parameters:
**Parameters**
**t0** : float
The starting value :math:t_0 of the time interval.
**tend** : float
The end value :math:T of the time interval.
**times** : float, array-like, shape :math:\left(\textit{ntimes}\right)
The points in the time interval :math:\left(t_0, T\right) at which the Wiener process is to be constructed. The order in which points are listed in :math:\mathrm{times} determines the bridge construction order. The function :meth:bb_make_bridge_order can be used to create predefined bridge construction orders from a set of input times.
**Returns**
**comm** : dict, communication object
Communication structure.
.. _g05xa-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:-99)
An unexpected error occurred during execution of bb_init. Please contact NAG <https://www.nag.com>__ with the following error message: error in :math:\langle\mathit{\boldsymbol{value}}\rangle, :math:\text{ASSERT} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:1)
On entry, :math:\mathrm{tend} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{t0} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{tend} > \mathrm{t0}.
(errno :math:2)
On entry, :math:\textit{ntimes} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\textit{ntimes}\geq 1.
(errno :math:3)
On entry, :math:\mathrm{times}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle, :math:\mathrm{t0} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{tend} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{t0} < \mathrm{times}[i-1] < \mathrm{tend} for all :math:i.
(errno :math:4)
On entry, :math:\mathrm{times}[\langle\mathit{\boldsymbol{value}}\rangle] and :math:\mathrm{times}[\langle\mathit{\boldsymbol{value}}\rangle] both equal :math:\langle\mathit{\boldsymbol{value}}\rangle.
Constraint: all elements of :math:\mathrm{times} must be unique.
.. _g05xa-py2-py-notes:
**Notes**
**Brownian Bridge Algorithm**
Details on the Brownian bridge algorithm and the Brownian bridge process (sometimes also called a non-free Wiener process) can be found in the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html#bb>__.
We briefly recall some notation and definitions.
Fix two times :math:t_0 < T and let :math:\left(t_i\right)_{{1\leq i\leq N}} be any set of time points satisfying :math:t_0 < t_1 < t_2<⋯<t_N < T.
Let :math:\left(X_{{t_i}}\right)_{{1\leq i\leq N}} denote a :math:d-dimensional Wiener sample path at these time points, and let :math:C be any :math:d\times d matrix such that :math:CC^\mathrm{T} is the desired covariance structure for the Wiener process.
Each point :math:X_{{t_i}} of the sample path is constructed according to the Brownian bridge interpolation algorithm (see Glasserman (2004) or the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html#bb>__).
We always start at some fixed point :math:X_{t_0} = x \in \mathbb{R}^d.
If we set :math:X_T = x+C\sqrt{T-t_0}Z where :math:Z is any :math:d-dimensional standard Normal random variable, then :math:X will behave like a normal (free) Wiener process.
However if we fix the terminal value :math:X_T = w \in \mathbb{R}^d, then :math:X will behave like a non-free Wiener process.
**Implementation**
Given the start and end points of the process, the order in which successive interpolation times :math:t_j are chosen is called the bridge construction order.
The construction order is given by the array :math:\mathrm{times}.
Further information on construction orders is given in the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html#bb-algo>__.
For clarity we consider here the common scenario where the Brownian bridge algorithm is used with quasi-random points.
If pseudorandom numbers are used instead, these details can be ignored.
Suppose we require :math:P Wiener sample paths each of dimension :math:d.
The main input to the Brownian bridge algorithm is then an array of quasi-random points :math:Z^1,Z^2,…,Z^P where each point :math:Z^p = \left(Z_1^p, Z_2^p, \ldots, Z_D^p\right) has dimension :math:D = d\left(N+1\right) or :math:D = dN respectively, depending on whether a free or non-free Wiener process is required.
When :meth:bb is called, the :math:p\ th sample path for :math:1\leq p\leq P is constructed as follows: if a non-free Wiener process is required set :math:X_T equal to the terminal value :math:w, otherwise construct :math:X_T as
.. math::
X_T = X_{{t_0}}+C\sqrt{T-t_0}\left[\begin{array}{c}Z_1^p\\ \vdots \\Z_d^p\end{array}\right]
where :math:C is the matrix described in Brownian Bridge Algorithm <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05xaf.html#background>__.
The array :math:\mathrm{times} holds the remaining time points :math:t_1,t_2,…t_N in the order in which the bridge is to be constructed.
For each :math:j = 1,…,N set :math:r = \mathrm{times}[j-1], find
.. math::
q = \text{max}\left\{{t_0,\mathrm{times}[i-1]:1\leq i < j}, {\mathrm{times}[i-1] < r}\right\}
and
.. math::
s = \text{min}\left\{T, {\mathrm{times}[i-1]:1\leq i < j}, {\mathrm{times}[i-1] > r}\right\}\text{}
and construct the point :math:X_r as
.. math::
where :math:a = 0 or :math:a = 1 respectively depending on whether a free or non-free Wiener process is required.
Note that in our discussion :math:j is indexed from :math:1, and so :math:X_r is interpolated between the nearest (in time) Wiener points which have already been constructed.
The function :meth:bb_make_bridge_order can be used to initialize the :math:\mathrm{times} array for several predefined bridge construction orders.
.. _g05xa-py2-py-references:
**References**
Glasserman, P, 2004, Monte Carlo Methods in Financial Engineering, Springer
"""
raise NotImplementedError
[docs]def bb(npaths, start, term, z, c, comm, rcord=2, a=1):
r"""
bb uses a Brownian bridge algorithm to construct sample paths for a free or non-free Wiener process.
The initialization function :meth:bb_init must be called prior to the first call to bb.
.. _g05xb-py2-py-doc:
For full information please refer to the NAG Library document for g05xb
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05xbf.html
.. _g05xb-py2-py-parameters:
**Parameters**
**npaths** : int
The number of Wiener sample paths to create.
**start** : float, array-like, shape :math:\left(d\right)
The starting value of the Wiener process.
**term** : float, array-like, shape :math:\left(d\right)
The terminal value at which the non-free Wiener process should end. If :math:\mathrm{a} = 0, :math:\mathrm{term} is ignored.
**z** : float, array-like, shape :math:\left(:, :\right)
Note: the required extent for this argument in dimension 1 is determined as follows: if :math:\mathrm{rcord}=1: :math:{ d \times \left(\textit{N}+1-\mathrm{a}\right) }; if :math:\mathrm{rcord}=2: :math:\mathrm{npaths}; otherwise: :math:0.
Note: the required extent for this argument in dimension 2 is determined as follows: if :math:\mathrm{rcord}=1: :math:\mathrm{npaths}; if :math:\mathrm{rcord}=2: :math:{ d \times \left(\textit{N}+1-\mathrm{a}\right) }; otherwise: :math:0.
The Normal random numbers used to construct the sample paths.
If quasi-random numbers are used, the :math:d\times \left(\textit{N}+1-\mathrm{a}\right)-dimensional quasi-random points should be stored in successive rows of :math:Z.
**c** : float, array-like, shape :math:\left(d, d\right)
The lower triangular Cholesky factorization :math:C such that :math:CC^\mathrm{T} gives the covariance matrix of the Wiener process. Elements of :math:C above the diagonal are not referenced.
**comm** : dict, communication object
Communication structure.
This argument must have been initialized by a prior call to :meth:bb_init.
**rcord** : int, optional
The order in which Normal random numbers are stored in :math:\mathrm{z} and in which the generated values are returned in :math:\mathrm{b}.
**a** : int, optional
If :math:\mathrm{a} = 0, a free Wiener process is created beginning at :math:\mathrm{start} and :math:\mathrm{term} is ignored.
If :math:\mathrm{a} = 1, a non-free Wiener process is created beginning at :math:\mathrm{start} and ending at :math:\mathrm{term}.
**Returns**
**z** : float, ndarray, shape :math:\left(:, :\right)
The Normal random numbers premultiplied by :math:C.
**b** : float, ndarray, shape :math:\left(:, :\right)
The values of the Wiener sample paths.
Let :math:X_{{p,i}}^k denote the :math:k\ th dimension of the :math:i\ th point of the :math:p\ th sample path where :math:1\leq k\leq d, :math:1\leq i\leq \textit{N}+1 and :math:1\leq p\leq \mathrm{npaths}.
The point :math:X_{{p,i}}^k is stored at :math:B\left(p, {k+\left(i-1\right)\times d}\right).
The starting value :math:\mathrm{start} is never stored, whereas the terminal value is always stored.
.. _g05xb-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:-99)
An unexpected error occurred during execution of bb. Please contact NAG <https://www.nag.com>__ with the following error message: error in :math:\langle\mathit{\boldsymbol{value}}\rangle, :math:\text{ASSERT} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:1)
On entry, :math:\mathrm{comm}\ ['rcomm'] was not initialized or has been corrupted.
(errno :math:2)
On entry, :math:\mathrm{npaths} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{npaths}\geq 1.
(errno :math:3)
On entry, the value of :math:\mathrm{rcord} is invalid.
(errno :math:4)
On entry, :math:d = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:d\geq 1.
(errno :math:5)
On entry, :math:\mathrm{a} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{a} = 0\text{ or }1.
(errno :math:6)
On entry, :math:\textit{ldz} = \langle\mathit{\boldsymbol{value}}\rangle and :math:d\times \left({\textit{ntimes}}+1-\mathrm{a}\right) = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\textit{ldz}\geq d\times \left({\textit{ntimes}}+1-\mathrm{a}\right).
.. _g05xb-py2-py-notes:
**Notes**
For details on the Brownian bridge algorithm and the bridge construction order see the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html#bb>__ and :ref:Notes for bb_init <g05xa-py2-py-notes>.
Recall that the terms Wiener process (or free Wiener process) and Brownian motion are often used interchangeably, while a non-free Wiener process (also known as a Brownian bridge process) refers to a process which is forced to terminate at a given point.
.. _g05xb-py2-py-references:
**References**
Glasserman, P, 2004, Monte Carlo Methods in Financial Engineering, Springer
"""
raise NotImplementedError
[docs]def bb_inc_init(t0, tend, times):
r"""
bb_inc_init initializes the Brownian bridge increments generator :meth:bb_inc.
It must be called before any calls to :meth:bb_inc.
.. _g05xc-py2-py-doc:
For full information please refer to the NAG Library document for g05xc
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05xcf.html
.. _g05xc-py2-py-parameters:
**Parameters**
**t0** : float
The starting value :math:t_0 of the time interval.
**tend** : float
The end value :math:T of the time interval.
**times** : float, array-like, shape :math:\left(\textit{ntimes}\right)
The points in the time interval :math:\left(t_0, T\right) at which the Wiener process is to be constructed. The order in which points are listed in :math:\mathrm{times} determines the bridge construction order. The function :meth:bb_make_bridge_order can be used to create predefined bridge construction orders from a set of input times.
**Returns**
**comm** : dict, communication object
Communication structure.
.. _g05xc-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:-99)
An unexpected error occurred during execution of bb_inc_init. Please contact NAG <https://www.nag.com>__ with the following error message: error in :math:\langle\mathit{\boldsymbol{value}}\rangle, :math:\text{ASSERT} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:1)
On entry, :math:\mathrm{tend} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{t0} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{tend} > \mathrm{t0}.
(errno :math:2)
On entry, :math:\textit{ntimes} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\textit{ntimes}\geq 1.
(errno :math:3)
On entry, :math:\mathrm{times}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle, :math:\mathrm{t0} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{tend} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{t0} < \mathrm{times}[i-1] < \mathrm{tend} for all :math:i.
(errno :math:4)
On entry, :math:\mathrm{times}[i-1] = \mathrm{times}[j-1] = \langle\mathit{\boldsymbol{value}}\rangle, for :math:i = \langle\mathit{\boldsymbol{value}}\rangle and :math:j = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: all elements of :math:\mathrm{times} must be unique.
.. _g05xc-py2-py-notes:
**Notes**
**Brownian Bridge Algorithm**
Details on the Brownian bridge algorithm and the Brownian bridge process (sometimes also called a non-free Wiener process) can be found in the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html#bb>__.
We briefly recall some notation and definitions.
Fix two times :math:t_0 < T and let :math:\left(t_i\right)_{{1\leq i\leq N}} be any set of time points satisfying :math:t_0 < t_1 < t_2<⋯<t_N < T.
Let :math:\left(X_{{t_i}}\right)_{{1\leq i\leq N}} denote a :math:d-dimensional Wiener sample path at these time points, and let :math:C be any :math:d\times d matrix such that :math:CC^\mathrm{T} is the desired covariance structure for the Wiener process.
Each point :math:X_{{t_i}} of the sample path is constructed according to the Brownian bridge interpolation algorithm (see Glasserman (2004) or the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html#bb>__).
We always start at some fixed point :math:X_{t_0} = x \in \mathbb{R}^d.
If we set :math:X_T = x+C\sqrt{T-t_0}Z where :math:Z is any :math:d-dimensional standard Normal random variable, then :math:X will behave like a normal (free) Wiener process.
However if we fix the terminal value :math:X_T = w \in \mathbb{R}^d, then :math:X will behave like a non-free Wiener process.
The Brownian bridge increments generator uses the Brownian bridge algorithm to construct sample paths for the (free or non-free) Wiener process :math:X, and then uses this to compute the scaled Wiener increments
.. math::
\frac{{X_{{t_1}}-X_{{t_0}}}}{{t_1-t_0}},\frac{{X_{{t_2}}-X_{{t_1}}}}{{t_2-t_1}},\ldots,\frac{{X_{{t_{{N}}}}-X_{{t_{{N-1}}}}}}{{t_N-t_{{N-1}}}},\frac{{X_T-X_{{t_N}}}}{{T-t_N}}\text{.}
Such increments can be useful in computing numerical solutions to stochastic differential equations driven by (free or non-free) Wiener processes.
**Implementation**
Conceptually, the output of the Wiener increments generator is the same as if :meth:bb_init and :meth:bb were called first, and the scaled increments then constructed from their output.
The implementation adopts a much more efficient approach whereby the scaled increments are computed directly without first constructing the Wiener sample path.
Given the start and end points of the process, the order in which successive interpolation times :math:t_j are chosen is called the bridge construction order.
The construction order is given by the array :math:\mathrm{times}.
Further information on construction orders is given in the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html#bb-algo>__.
For clarity we consider here the common scenario where the Brownian bridge algorithm is used with quasi-random points.
If pseudorandom numbers are used instead, these details can be ignored.
Suppose we require the increments of :math:P Wiener sample paths each of dimension :math:d.
The main input to the Brownian bridge increments generator is then an array of quasi-random points :math:Z^1,Z^2,…,Z^P where each point :math:Z^p = \left(Z_1^p, Z_2^p, \ldots, Z_D^p\right) has dimension :math:D = d\left(N+1\right) or :math:D = dN depending on whether a free or non-free Wiener process is required.
When :meth:bb_inc is called, the :math:p\ th sample path for :math:1\leq p\leq P is constructed as follows: if a non-free Wiener process is required set :math:X_T equal to the terminal value :math:w, otherwise construct :math:X_T as
.. math::
X_T = X_{{t_0}}+C\sqrt{T-t_0}\left[\begin{array}{c}Z_1^p\\ \vdots \\Z_d^p\end{array}\right]
where :math:C is the matrix described in Brownian Bridge Algorithm <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05xcf.html#background>__.
The array :math:\mathrm{times} holds the remaining time points :math:t_1,t_2,…t_N in the order in which the bridge is to be constructed.
For each :math:j = 1,…,N set :math:r = \mathrm{times}[j-1], find
.. math::
q = \text{max}\left\{{t_0,\mathrm{times}[i-1]:1\leq i < j}, {\mathrm{times}[i-1] < r}\right\}
and
.. math::
s = \text{min}\left\{T, {\mathrm{times}[i-1]:1\leq i < j}, {\mathrm{times}[i-1] > r}\right\}\text{}
and construct the point :math:X_r as
.. math::
where :math:a = 0 or :math:a = 1 depending on whether a free or non-free Wiener process is required.
The function :meth:bb_make_bridge_order can be used to initialize the :math:\mathrm{times} array for several predefined bridge construction orders.
Lastly, the scaled Wiener increments
.. math::
\frac{{X_{{t_1}}-X_{{t_0}}}}{{t_1-t_0}},\frac{{X_{{t_2}}-X_{{t_1}}}}{{t_2-t_1}},…,\frac{{X_{{t_{{N}}}}-X_{{t_{{N-1}}}}}}{{t_N-t_{{N-1}}}},\frac{{X_T-X_{{t_N}}}}{{T-t_N}}
are computed.
.. _g05xc-py2-py-references:
**References**
Glasserman, P, 2004, Monte Carlo Methods in Financial Engineering, Springer
--------
:meth:naginterfaces.library.examples.rand.bb_inc_ex.main
"""
raise NotImplementedError
[docs]def bb_inc(npaths, diff, z, c, comm, rcord=2, a=1):
r"""
bb_inc computes scaled increments of sample paths of a free or non-free Wiener process, where the sample paths are constructed by a Brownian bridge algorithm.
The initialization function :meth:bb_inc_init must be called prior to the first call to bb_inc.
.. _g05xd-py2-py-doc:
For full information please refer to the NAG Library document for g05xd
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05xdf.html
.. _g05xd-py2-py-parameters:
**Parameters**
**npaths** : int
The number of Wiener sample paths.
**diff** : float, array-like, shape :math:\left(d\right)
The difference between the terminal value and starting value of the Wiener process. If :math:\mathrm{a} = 0, :math:\mathrm{diff} is ignored.
**z** : float, array-like, shape :math:\left(:, :\right)
Note: the required extent for this argument in dimension 1 is determined as follows: if :math:\mathrm{rcord}=1: :math:{ d \times \left(\textit{N}+1-\mathrm{a}\right) }; if :math:\mathrm{rcord}=2: :math:\mathrm{npaths}; otherwise: :math:0.
Note: the required extent for this argument in dimension 2 is determined as follows: if :math:\mathrm{rcord}=1: :math:\mathrm{npaths}; if :math:\mathrm{rcord}=2: :math:{ d \times \left(\textit{N}+1-\mathrm{a}\right) }; otherwise: :math:0.
The Normal random numbers used to construct the sample paths.
If quasi-random numbers are used, the :math:d\times \left(N+1-\mathrm{a}\right)-dimensional quasi-random points should be stored in successive rows of :math:Z.
**c** : float, array-like, shape :math:\left(d, d\right)
The lower triangular Cholesky factorization :math:C such that :math:CC^\mathrm{T} gives the covariance matrix of the Wiener process. Elements of :math:C above the diagonal are not referenced.
**comm** : dict, communication object
Communication structure.
This argument must have been initialized by a prior call to :meth:bb_inc_init.
**rcord** : int, optional
The order in which Normal random numbers are stored in :math:\mathrm{z} and in which the generated values are returned in :math:\mathrm{b}.
**a** : int, optional
If :math:\mathrm{a} = 0, a free Wiener process is created and :math:\mathrm{diff} is ignored.
If :math:\mathrm{a} = 1, a non-free Wiener process is created where :math:\mathrm{diff} is the difference between the terminal value and the starting value of the process.
**Returns**
**z** : float, ndarray, shape :math:\left(:, :\right)
The Normal random numbers premultiplied by :math:\mathrm{c}.
**b** : float, ndarray, shape :math:\left(:, :\right)
The scaled Wiener increments.
Let :math:X_{{p,i}}^k denote the :math:k\ th dimension of the :math:i\ th point of the :math:p\ th sample path where :math:1\leq k\leq d, :math:1\leq i\leq \textit{N}+1 and :math:1\leq p\leq \mathrm{npaths}.
The increment :math:\frac{\left(X_{{p,i}}^k-X_{{p,i-1}}^k\right)}{\left(t_i-t_{{i-1}}\right)} is stored at :math:B\left(p, {k+\left(i-1\right)\times d}\right).
.. _g05xd-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:-99)
An unexpected error occurred during execution of bb_inc. Please contact NAG <https://www.nag.com>__ with the following error message: error in :math:\langle\mathit{\boldsymbol{value}}\rangle, :math:\text{ASSERT} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:1)
On entry, :math:\mathrm{comm}\ ['rcomm'] was not initialized or has been corrupted.
(errno :math:2)
On entry, :math:\mathrm{npaths} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{npaths}\geq 1.
(errno :math:3)
On entry, the value of :math:\mathrm{rcord} is invalid.
(errno :math:4)
On entry, :math:d = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:d\geq 1.
(errno :math:5)
On entry, :math:\mathrm{a} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{a} = 0\text{ or }1.
(errno :math:6)
On entry, :math:\textit{ldz} = \langle\mathit{\boldsymbol{value}}\rangle and :math:d\times \left({\textit{ntimes}}+1-\mathrm{a}\right) = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\textit{ldz}\geq d\times \left({\textit{ntimes}}+1-\mathrm{a}\right).
.. _g05xd-py2-py-notes:
**Notes**
For details on the Brownian bridge algorithm and the bridge construction order see the G05 Introduction <https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05intro.html#bb>__ and :ref:Notes for bb_inc_init <g05xc-py2-py-notes>.
Recall that the terms Wiener process (or free Wiener process) and Brownian motion are often used interchangeably, while a non-free Wiener process (also known as a Brownian bridge process) refers to a process which is forced to terminate at a given point.
Fix two times :math:t_0 < T, let :math:\left(t_i\right)_{{1\leq i\leq N}} be any set of time points satisfying :math:t_0 < t_1 < t_2 < \cdots < t_N < T, and let :math:X_{{t_0}}, :math:\left(X_{{t_i}}\right)_{{1\leq i\leq N}}, :math:X_T denote a :math:d-dimensional Wiener sample path at these time points.
The Brownian bridge increments generator uses the Brownian bridge algorithm to construct sample paths for the (free or non-free) Wiener process :math:X, and then uses this to compute the scaled Wiener increments
.. math::
\frac{{X_{{t_1}}-X_{{t_0}}}}{{t_1-t_0}},\frac{{X_{{t_2}}-X_{{t_1}}}}{{t_2-t_1}},\ldots,\frac{{X_{{t_{{N}}}}-X_{{t_{{N-1}}}}}}{{t_N-t_{{N-1}}}},\frac{{X_T-X_{{t_N}}}}{{T-t_N}}
.. _g05xd-py2-py-references:
**References**
Glasserman, P, 2004, Monte Carlo Methods in Financial Engineering, Springer
--------
:meth:naginterfaces.library.examples.rand.bb_inc_ex.main
"""
raise NotImplementedError
[docs]def bb_make_bridge_order(t0, tend, intime, bgord=1, move=None):
r"""
bb_make_bridge_order takes a set of input times and permutes them to specify one of several predefined Brownian bridge construction orders.
The permuted times can be passed to :meth:bb_init or :meth:bb_inc_init to initialize the Brownian bridge generators with the chosen bridge construction order.
.. _g05xe-py2-py-doc:
For full information please refer to the NAG Library document for g05xe
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05xef.html
.. _g05xe-py2-py-parameters:
**Parameters**
**t0** : float
:math:t_0, the start value of the time interval on which the Wiener process is to be constructed.
**tend** : float
:math:T, the largest time at which the Wiener process is to be constructed.
**intime** : float, array-like, shape :math:\left(\textit{ntimes}\right)
The time points, :math:t_1,t_2,\ldots,t_N, at which the Wiener process is to be constructed. Note that the final time :math:T is not included in this array.
**bgord** : int, optional
The bridge construction order to use.
**move** : None or int, array-like, shape :math:\left(\textit{nmove}\right), optional
The indices of the entries in :math:\mathrm{intime} which should be moved to the front of the :math:\mathrm{times} array, with :math:\mathrm{move}[j-1] = i setting the :math:j\ th element of :math:\mathrm{times} to :math:t_i. Note that :math:i ranges from :math:1 to :math:\textit{ntimes}. When :math:\textit{nmove} = 0, :math:\mathrm{move} is not referenced.
**Returns**
**times** : float, ndarray, shape :math:\left(\textit{ntimes}\right)
The output bridge construction order. This should be passed to :meth:bb_init or :meth:bb_inc_init.
.. _g05xe-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:-99)
An unexpected error occurred during execution of bb_make_bridge_order. Please contact NAG <https://www.nag.com>__ with the following error message: error in :math:\langle\mathit{\boldsymbol{value}}\rangle, :math:\text{ASSERT} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:1)
On entry, :math:\mathrm{bgord} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{bgord} = 1, :math:2, :math:3 or :math:4
(errno :math:2)
On entry, :math:\textit{ntimes} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\textit{ntimes}\geq 1.
(errno :math:3)
On entry, :math:\textit{nmove} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\textit{ntimes} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0\leq \textit{nmove}\leq \textit{ntimes}.
(errno :math:4)
On entry, :math:\mathrm{intime}[0] = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{t0} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{intime}[0] > \mathrm{t0}.
(errno :math:4)
On entry, :math:\mathrm{intime}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{intime}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: the elements in :math:\mathrm{intime} must be in increasing order.
(errno :math:4)
On entry, :math:\textit{ntimes} = \langle\mathit{\boldsymbol{value}}\rangle, :math:\mathrm{intime}[\textit{ntimes}-1] = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{tend} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{intime}[\textit{ntimes}-1] < \mathrm{tend}.
(errno :math:5)
On entry, :math:\mathrm{move}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{move}[i]\geq 1 for all :math:i.
(errno :math:5)
On entry, :math:\mathrm{move}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle and :math:\textit{ntimes} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{move}[i]\leq \textit{ntimes} for all :math:i.
(errno :math:6)
On entry, :math:\mathrm{move}[\langle\mathit{\boldsymbol{value}}\rangle] and :math:\mathrm{move}[\langle\mathit{\boldsymbol{value}}\rangle] both equal :math:\langle\mathit{\boldsymbol{value}}\rangle.
Constraint: all elements in :math:\mathrm{move} must be unique.
.. _g05xe-py2-py-notes:
**Notes**
The Brownian bridge algorithm (see Glasserman (2004)) is a popular method for constructing a Wiener process at a set of discrete times, :math:t_0 < t_1 < t_2 <,\ldots, < t_N < T, for :math:N\geq 1.
To ease notation we assume that :math:T has the index :math:N+1 so that :math:T = t_{{N+1}}.
Inherent in the algorithm is the notion of a bridge construction order which specifies the order in which the :math:N+2 points of the Wiener process, :math:X_{t_0},X_T and :math:X_{{t_i}}, for :math:\textit{i} = 1,2,\ldots,N, are generated.
The value of :math:X_{{t_0}} is always assumed known, and the first point to be generated is always the final time :math:X_T.
Thereafter, successive points are generated iteratively by an interpolation formula, using points which were computed at previous iterations.
In many cases the bridge construction order is not important, since any construction order will yield a correct process.
However, in certain cases, for example when using quasi-random variates to construct the sample paths, the bridge construction order can be important.
**Supported Bridge Construction Orders**
bb_make_bridge_order accepts as input an array of time points :math:t_1,t_2,\ldots,t_N,T at which the Wiener process is to be sampled.
These time points are then permuted to construct the bridge.
In all of the supported construction orders the first construction point is :math:T which has index :math:N+1.
The remaining points are constructed by iteratively bisecting (sub-intervals of) the time indices interval :math:\left[0, {N+1}\right], as Figure [label omitted] illustrates:
[figure omitted]
The time indices interval is processed in levels :math:L^{\textit{i}}, for :math:\textit{i} = 1,2,\ldots,.
Each level :math:L^i contains :math:n_i points :math:L_1^i,\ldots,L_{n_i}^i where :math:n_i\leq 2^{{i-1}}.
The number of points at each level depends on the value of :math:N.
The points :math:L_j^i for :math:i\geq 1 and :math:j = 1,2,\ldots n_i are computed as follows: define :math:L_0^0 = N+1 and set
.. math::
\begin{array}{cc} L_j^i = J + \left(K-J\right)/2 &\text{where}\\ J = \mathrm{max}\left\{L_k^p:1\leq k\leq n_p\text{, }0\leq p < i\text{ and }L_k^p < L_j^i\right\} &\text{ and }\\ K = \mathrm{min}\left\{L_k^p:1\leq k\leq n_p\text{, }0\leq p < i\text{ and }L_k^p > L_j^i\right\} &\end{array}
By convention the maximum of the empty set is taken to be to be zero. Figure [label omitted] illustrates the algorithm when :math:N+1 is a power of two.
When :math:N+1 is not a power of two, one must decide how to round the divisions by :math:2.
For example, if one rounds down to the nearest integer, then one could get the following:
[figure omitted]
From the series of bisections outlined above, two ways of ordering the time indices :math:L_j^i are supported.
In both cases, levels are always processed from coarsest to finest (i.e., increasing :math:i).
Within a level, the time indices can either be processed left to right (i.e., increasing :math:j) or right to left (i.e., decreasing :math:j).
For example, when processing left to right, the sequence of time indices could be generated as:
.. math::
\begin{array}{ccccccccc}N+1&L_1^1&L_1^2&L_2^2&L_1^3&L_2^3&L_3^3&L_4^3& \cdots \end{array}
while when processing right to left, the same sequence would be generated as:
.. math::
\begin{array}{ccccccccc}N+1&L_1^1&L_2^2&L_1^2&L_4^3&L_3^3&L_2^3&L_1^3& \cdots \end{array}
bb_make_bridge_order, therefore, offers four bridge construction methods; processing either left to right or right to left, with rounding either up or down.
Which method is used is controlled by the :math:\mathrm{bgord} argument.
For example, on the set of times
.. math::
\begin{array}{ccccccccccccc}t_1&t_2&t_3&t_4&t_5&t_6&t_7&t_8&t_9&t_{10}&t_{11}&t_{12}&T\end{array}
the Brownian bridge would be constructed in the following orders:
:math:\mathrm{bgord} = 1 (processing left to right, rounding down)
.. math::
\begin{array}{ccccccccccccc}T&t_6&t_3&t_9&t_1&t_4&t_7&t_{11}&t_2&t_5&t_8&t_{10}&t_{12}\end{array}
:math:\mathrm{bgord} = 2 (processing left to right, rounding up)
.. math::
\begin{array}{ccccccccccccc}T&t_7&t_4&t_{10}&t_2&t_6&t_9&t_{12}&t_1&t_3&t_5&t_8&t_{11}\end{array}
:math:\mathrm{bgord} = 3 (processing right to left, rounding down)
.. math::
\begin{array}{ccccccccccccc}T&t_6&t_9&t_3&t_{11}&t_7&t_4&t_1&t_{12}&t_{10}&t_8&t_5&t_2\end{array}
:math:\mathrm{bgord} = 4 (processing right to left, rounding up)
.. math::
\begin{array}{ccccccccccccc}T&t_7&t_{10}&t_4&t_{12}&t_9&t_6&t_2&t_{11}&t_8&t_5&t_3&t_1\end{array}\text{.}
The four construction methods described above can be further modified through the use of the input array :math:\mathrm{move}.
To see the effect of this argument, suppose that an array :math:A holds the output of bb_make_bridge_order when :math:\textit{nmove} = 0 (i.e., the bridge construction order as specified by :math:\mathrm{bgord} only).
Let
.. math::
B = \left\{t_j:j = \mathrm{move}[i-1],i = 1,2,\ldots,\textit{nmove}\right\}
be the array of all times identified by :math:\mathrm{move}, and let :math:C be the array :math:A with all the elements in :math:B removed, i.e.,
.. math::
C = \left\{A\left(i\right):A\left(i\right)\neq B\left(j\right),i = 1,2,\ldots,\textit{ntimes},j = 1,2,\ldots,\textit{nmove}\right\}\text{.}
Then the output of bb_make_bridge_order when :math:\textit{nmove} > 0 is given by
.. math::
\begin{array}{cccccccc} B\left(1\right) & B\left(2\right) & \cdots & B\left(\textit{nmove}\right) & C\left(1\right) & C\left(2\right) & \cdots & C\left(\textit{ntimes}-\textit{nmove}\right) \end{array}
When the Brownian bridge is used with quasi-random variates, this functionality can be used to allow specific sections of the bridge to be constructed using the lowest dimensions of the quasi-random points.
.. _g05xe-py2-py-references:
**References**
Glasserman, P, 2004, Monte Carlo Methods in Financial Engineering, Springer
--------
:meth:naginterfaces.library.examples.rand.bb_inc_ex.main
"""
raise NotImplementedError
[docs]def quasi_normal(xmean, std, n, comm):
r"""
quasi_normal generates a quasi-random sequence from a Normal (Gaussian) distribution.
It must be preceded by a call to one of the initialization functions :meth:quasi_init or :meth:quasi_init_scrambled.
.. _g05yj-py2-py-doc:
For full information please refer to the NAG Library document for g05yj
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05yjf.html
.. _g05yj-py2-py-parameters:
**Parameters**
**xmean** : float, array-like, shape :math:\left(\textit{idim}\right)
Specifies, for each dimension, the mean of the Normal distribution.
**std** : float, array-like, shape :math:\left(\textit{idim}\right)
Specifies, for each dimension, the standard deviation of the Normal distribution.
**n** : int
The number of quasi-random numbers required.
**comm** : dict, communication object, modified in place
Communication structure.
This argument must have been initialized by a prior call to :meth:quasi_init or :meth:quasi_init_scrambled.
**Returns**
**quas** : float, ndarray, shape :math:\left(\mathrm{n}, \textit{idim}\right)
Contains the :math:\mathrm{n} quasi-random numbers of dimension idim
If :math:\textit{sorder} = 2, :math:\mathrm{quas}[i-1,j-1] holds the :math:j\ th value for the :math:i\ th dimension.
If :math:\textit{sorder} = 1, :math:\mathrm{quas}[i-1,j-1] holds the :math:i\ th value for the :math:j\ th dimension.
.. _g05yj-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{comm}\ ['iref'] has either not been initialized or has been corrupted.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:i = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{std}[i-1] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{std}[i-1]\geq 0.0.
(errno :math:4)
There have been too many calls to the generator.
.. _g05yj-py2-py-notes:
**Notes**
quasi_normal generates a quasi-random sequence from a Normal distribution by first generating a uniform quasi-random sequence which is then transformed into a Normal sequence using the inverse of the Normal CDF.
The type of uniform sequence used depends on the initialization function called and can include the low-discrepancy sequences proposed by Sobol, Faure or Niederreiter.
If the initialization function :meth:quasi_init_scrambled was used then the underlying uniform sequence is first scrambled prior to being transformed (see :ref:Notes for quasi_init_scrambled <g05yn-py2-py-notes> for details).
.. _g05yj-py2-py-references:
**References**
Bratley, P and Fox, B L, 1988, Algorithm 659: implementing Sobol's quasirandom sequence generator, ACM Trans. Math. Software (14(1)), 88--100
Fox, B L, 1986, Algorithm 647: implementation and relative efficiency of quasirandom sequence generators, ACM Trans. Math. Software (12(4)), 362--376
Wichura, 1988, Algorithm AS 241: the percentage points of the Normal distribution, Appl. Statist. (37), 477--484
--------
:meth:naginterfaces.library.examples.rand.bb_inc_ex.main
"""
raise NotImplementedError
[docs]def quasi_lognormal(xmean, std, n, comm):
r"""
quasi_lognormal generates a quasi-random sequence from a log-normal distribution.
It must be preceded by a call to one of the initialization functions :meth:quasi_init or :meth:quasi_init_scrambled.
.. _g05yk-py2-py-doc:
For full information please refer to the NAG Library document for g05yk
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05ykf.html
.. _g05yk-py2-py-parameters:
**Parameters**
**xmean** : float, array-like, shape :math:\left(\textit{idim}\right)
Specifies, for each dimension, the mean of the underlying Normal distribution.
**std** : float, array-like, shape :math:\left(\textit{idim}\right)
Specifies, for each dimension, the standard deviation of the underlying Normal distribution.
**n** : int
The number of quasi-random numbers required.
**comm** : dict, communication object, modified in place
Communication structure.
This argument must have been initialized by a prior call to :meth:quasi_init or :meth:quasi_init_scrambled.
**Returns**
**quas** : float, ndarray, shape :math:\left(\mathrm{n}, \textit{idim}\right)
Contains the :math:\mathrm{n} quasi-random numbers of dimension idim.
If :math:\textit{sorder} = 2, :math:\mathrm{quas}[i-1,j-1] holds the :math:j\ th value for the :math:i\ th dimension.
If :math:\textit{sorder} = 1, :math:\mathrm{quas}[i-1,j-1] holds the :math:i\ th value for the :math:j\ th dimension.
.. _g05yk-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{comm}\ ['iref'] has either not been initialized or has been corrupted.
(errno :math:2)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:3)
On entry, :math:i = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{std}[i-1] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{std}[i-1]\geq 0.
(errno :math:4)
There have been too many calls to the generator.
(errno :math:5)
On entry, :math:i = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{xmean}[i-1] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\left\lvert \mathrm{xmean}[i-1]\right\rvert \leq \langle\mathit{\boldsymbol{value}}\rangle.
.. _g05yk-py2-py-notes:
**Notes**
quasi_lognormal generates a quasi-random sequence from a log-normal distribution by first generating a uniform quasi-random sequence which is then transformed into a log-normal sequence using the exponential of the inverse of the Normal CDF.
The type of uniform sequence used depends on the initialization function called and can include the low-discrepancy sequences proposed by Sobol, Faure or Niederreiter.
If the initialization function :meth:quasi_init_scrambled was used then the underlying uniform sequence is first scrambled prior to being transformed (see :ref:Notes for quasi_init_scrambled <g05yn-py2-py-notes> for details).
.. _g05yk-py2-py-references:
**References**
Bratley, P and Fox, B L, 1988, Algorithm 659: implementing Sobol's quasirandom sequence generator, ACM Trans. Math. Software (14(1)), 88--100
Fox, B L, 1986, Algorithm 647: implementation and relative efficiency of quasirandom sequence generators, ACM Trans. Math. Software (12(4)), 362--376
Wichura, 1988, Algorithm AS 241: the percentage points of the Normal distribution, Appl. Statist. (37), 477--484
"""
raise NotImplementedError
[docs]def quasi_init(genid, idim, iskip):
r"""
quasi_init initializes a quasi-random generator prior to calling :meth:quasi_uniform, :meth:quasi_normal or :meth:quasi_lognormal.
.. _g05yl-py2-py-doc:
For full information please refer to the NAG Library document for g05yl
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05ylf.html
.. _g05yl-py2-py-parameters:
**Parameters**
**genid** : int
Must identify the quasi-random generator to use.
:math:\mathrm{genid} = 1
Sobol generator.
:math:\mathrm{genid} = 2
Sobol (A659) generator.
:math:\mathrm{genid} = 3
Niederreiter generator.
:math:\mathrm{genid} = 4
Faure generator.
**idim** : int
The number of dimensions required.
**iskip** : int
The number of terms of the sequence to skip on initialization for the Sobol and Niederreiter generators. If :math:\mathrm{genid} = 4, :math:\mathrm{iskip} is ignored.
**Returns**
**comm** : dict, communication object
Communication structure.
.. _g05yl-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{genid} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{genid} = 1, :math:2, :math:3 or :math:4.
(errno :math:2)
On entry, :math:\mathrm{idim} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:1\leq \mathrm{idim}\leq \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:5)
On entry, :math:\mathrm{iskip} < 0 or :math:\mathrm{iskip} is too large: :math:\mathrm{iskip} = \langle\mathit{\boldsymbol{value}}\rangle, maximum value is :math:\langle\mathit{\boldsymbol{value}}\rangle.
.. _g05yl-py2-py-notes:
**Notes**
quasi_init selects a quasi-random number generator through the input value of :math:\mathrm{genid} and initializes the :math:\mathrm{comm}\ ['iref'] communication array for use by the functions :meth:quasi_uniform, :meth:quasi_normal or :meth:quasi_lognormal.
One of three types of quasi-random generator may be chosen, allowing the low-discrepancy sequences proposed by Sobol, Faure or Niederreiter to be generated.
Two sets of Sobol sequences are supplied, the first, is based on the work of Joe and Kuo (2008).
The second, referred to in the documentation as 'Sobol (A659)', is based on Algorithm 659 of Bratley and Fox (1988) with the extension to 1111 dimensions proposed by Joe and Kuo (2003).
Both sets of Sobol sequences should satisfy the so-called Property A, up to :math:1111 dimensions, but the first set should have better two-dimensional projections than those produced using Algorithm 659.
.. _g05yl-py2-py-references:
**References**
Bratley, P and Fox, B L, 1988, Algorithm 659: implementing Sobol's quasirandom sequence generator, ACM Trans. Math. Software (14(1)), 88--100
Fox, B L, 1986, Algorithm 647: implementation and relative efficiency of quasirandom sequence generators, ACM Trans. Math. Software (12(4)), 362--376
Joe, S and Kuo, F Y, 2003, Remark on Algorithm 659: implementing Sobol's quasirandom sequence generator, ACM Trans. Math. Software (TOMS) (29), 49--57
Joe, S and Kuo, F Y, 2008, Constructing Sobol sequences with better two-dimensional projections, SIAM J. Sci. Comput. (30), 2635--2654
"""
raise NotImplementedError
[docs]def quasi_uniform(n, comm, rcord=1):
r"""
quasi_uniform generates a uniformly distributed low-discrepancy sequence as proposed by Sobol, Faure or Niederreiter.
It must be preceded by a call to one of the initialization functions :meth:quasi_init or :meth:quasi_init_scrambled.
.. _g05ym-py2-py-doc:
For full information please refer to the NAG Library document for g05ym
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05ymf.html
.. _g05ym-py2-py-parameters:
**Parameters**
**n** : int
The number of quasi-random numbers required.
**comm** : dict, communication object, modified in place
Communication structure.
This argument must have been initialized by a prior call to :meth:quasi_init or :meth:quasi_init_scrambled.
**rcord** : int, optional
The order in which the generated values are returned.
**Returns**
**quas** : float, ndarray, shape :math:\left(:, :\right)
Contains the :math:\mathrm{n} quasi-random numbers of dimension idim.
If :math:\mathrm{rcord} = 1, :math:\mathrm{quas}[i-1,j-1] holds the :math:j\ th value for the :math:i\ th dimension.
If :math:\mathrm{rcord} = 2, :math:\mathrm{quas}[i-1,j-1] holds the :math:i\ th value for the :math:j\ th dimension.
.. _g05ym-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{n}\geq 0.
(errno :math:1)
On entry, value of :math:\mathrm{n} would result in too many calls to the generator: :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle, generator has previously been called :math:\langle\mathit{\boldsymbol{value}}\rangle times.
(errno :math:2)
On entry, :math:\mathrm{rcord} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{rcord} = 1 or :math:2.
(errno :math:4)
On entry, :math:\textit{ldquas} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{n} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: if :math:\mathrm{rcord} = 2, :math:\textit{ldquas}\geq \mathrm{n}.
(errno :math:5)
On entry, :math:\mathrm{comm}\ ['iref'] has either not been initialized or has been corrupted.
.. _g05ym-py2-py-notes:
**Notes**
Low discrepancy (quasi-random) sequences are used in numerical integration, simulation and optimization.
Like pseudorandom numbers they are uniformly distributed but they are not statistically independent, rather they are designed to give more even distribution in multidimensional space (uniformity).
Therefore, they are often more efficient than pseudorandom numbers in multidimensional Monte Carlo methods.
quasi_uniform generates a set of points :math:x^1,x^2,\ldots,x^N with high uniformity in the :math:S-dimensional unit cube :math:I^S = \left[0, 1\right]^S.
Let :math:G be a subset of :math:I^S and define the counting function :math:S_N\left(G\right) as the number of points :math:x^i \in G.
For each :math:x = \left(x_1, x_2, \ldots, x_S\right) \in I^S, let :math:G_x be the rectangular :math:S-dimensional region
.. math::
G_x = \left[0, x_1\right)\times \left[0, x_2\right)\times \cdots \times \left[0, x_S\right)
with volume :math:x_1,x_2,\ldots,x_S.
Then one measure of the uniformity of the points :math:x^1,x^2,\ldots,x^N is the discrepancy:
.. math::
D_N^*\left(x^1, x^2, \ldots, x^N\right) = \mathrm{sup}_{{x \in I^S}}\left(\left\lvert S_N\left(G_x\right)-Nx_1,x_2,\ldots,x_S\right\rvert \right)\text{.}
which has the form
.. math::
The principal aim in the construction of low-discrepancy sequences is to find sequences of points in :math:I^S with a bound of this form where the constant :math:C_S is as small as possible.
The type of low-discrepancy sequence generated by quasi_uniform depends on the initialization function called and can include those proposed by Sobol, Faure or Niederreiter.
If the initialization function :meth:quasi_init_scrambled was used then the sequence will be scrambled (see :ref:Notes for quasi_init_scrambled <g05yn-py2-py-notes> for details).
.. _g05ym-py2-py-references:
**References**
Bratley, P and Fox, B L, 1988, Algorithm 659: implementing Sobol's quasirandom sequence generator, ACM Trans. Math. Software (14(1)), 88--100
Fox, B L, 1986, Algorithm 647: implementation and relative efficiency of quasirandom sequence generators, ACM Trans. Math. Software (12(4)), 362--376
"""
raise NotImplementedError
[docs]def quasi_init_scrambled(genid, stype, idim, iskip, nsdigi, statecomm):
r"""
quasi_init_scrambled initializes a scrambled quasi-random generator prior to calling :meth:quasi_uniform, :meth:quasi_normal or :meth:quasi_lognormal.
It must be preceded by a call to one of the pseudorandom initialization functions :meth:init_repeat or :meth:init_nonrepeat.
.. _g05yn-py2-py-doc:
For full information please refer to the NAG Library document for g05yn
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05ynf.html
.. _g05yn-py2-py-parameters:
**Parameters**
**genid** : int
Must identify the quasi-random generator to use.
:math:\mathrm{genid} = 1
Sobol generator.
:math:\mathrm{genid} = 2
Sobol (A659) generator.
:math:\mathrm{genid} = 3
Niederreiter generator.
**stype** : int
Must identify the scrambling method to use.
:math:\mathrm{stype} = 0
No scrambling. This is equivalent to calling :meth:quasi_init.
:math:\mathrm{stype} = 1
Owen like scrambling.
:math:\mathrm{stype} = 2
Faure--Tezuka scrambling.
:math:\mathrm{stype} = 3
Owen and Faure--Tezuka scrambling.
**idim** : int
The number of dimensions required.
**iskip** : int
The number of terms of the sequence to skip on initialization for the Sobol and Niederreiter generators.
**nsdigi** : int
Controls the number of digits (bits) to scramble when :math:\mathrm{genid} = 1 or :math:2, otherwise :math:\mathrm{nsdigi} is ignored. If :math:\mathrm{nsdigi} < 1 or :math:\mathrm{nsdigi} > 30 then all the digits are scrambled.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**comm** : dict, communication object
Communication structure.
.. _g05yn-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{genid} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:1\leq \mathrm{genid}\leq \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:2)
On entry, :math:\mathrm{stype} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0\leq \mathrm{stype}\leq 3.
(errno :math:3)
On entry, :math:\mathrm{idim} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:1\leq \mathrm{idim}\leq \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:6)
On entry, :math:\mathrm{iskip} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0\leq \mathrm{iskip}\leq 2^{30}.
(errno :math:8)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05yn-py2-py-notes:
**Notes**
quasi_init_scrambled selects a quasi-random number generator through the input value of :math:\mathrm{genid}, a method of scrambling through the input value of :math:\mathrm{stype} and initializes the :math:\mathrm{comm}\ ['iref'] communication array for use in the functions :meth:quasi_uniform, :meth:quasi_normal or :meth:quasi_lognormal.
Scrambled quasi-random sequences are an extension of standard quasi-random sequences that attempt to eliminate the bias inherent in a quasi-random sequence whilst retaining the low-discrepancy properties.
The use of a scrambled sequence allows error estimation of Monte Carlo results by performing a number of iterates and computing the variance of the results.
This implementation of scrambled quasi-random sequences is based on TOMS Algorithm 823 and details can be found in the accompanying paper, Hong and Hickernell (2003).
Three methods of scrambling are supplied; the first a restricted form of Owen's scrambling (Owen (1995)), the second based on the method of Faure and Tezuka (2000) and the last method combines the first two.
Scrambled versions of the Niederreiter sequence and two sets of Sobol sequences are provided.
The first Sobol sequence is obtained using :math:\mathrm{genid} = 1.
The first 10000 direction numbers for this sequence are based on the work of Joe and Kuo (2008).
For dimensions greater than 10000 the direction numbers are randomly generated using the pseudorandom generator specified in :math:\mathrm{statecomm}\ ['state'] (see Jäckel (2002) for details).
The second Sobol sequence is obtained using :math:\mathrm{genid} = 2 and referred to in the documentation as 'Sobol (A659)'.
The first 1111 direction numbers for this sequence are based on Algorithm 659 of Bratley and Fox (1988) with the extension proposed by Joe and Kuo (2003).
For dimensions greater than 1111 the direction numbers are once again randomly generated.
The Niederreiter sequence is obtained by setting :math:\mathrm{genid} = 3.
.. _g05yn-py2-py-references:
**References**
Bratley, P and Fox, B L, 1988, Algorithm 659: implementing Sobol's quasirandom sequence generator, ACM Trans. Math. Software (14(1)), 88--100
Faure, H and Tezuka, S, 2000, Another random scrambling of digital (t,s)-sequences, Monte Carlo and Quasi-Monte Carlo Methods, Springer-Verlag, Berlin, Germany, (eds K T Fang, F J Hickernell and H Niederreiter)
Hong, H S and Hickernell, F J, 2003, Algorithm 823: implementing scrambled digital sequences, ACM Trans. Math. Software (29:2), 95--109
Jäckel, P, 2002, Monte Carlo Methods in Finance, Wiley Finance Series, John Wiley and Sons, England
Joe, S and Kuo, F Y, 2003, Remark on Algorithm 659: implementing Sobol's quasirandom sequence generator, ACM Trans. Math. Software (TOMS) (29), 49--57
Joe, S and Kuo, F Y, 2008, Constructing Sobol sequences with better two-dimensional projections, SIAM J. Sci. Comput. (30), 2635--2654
Niederreiter, H, 1988, Low-discrepancy and low dispersion sequences, Journal of Number Theory (30), 51--70
Owen, A B, 1995, Randomly permuted (t,m,s)-nets and (t,s)-sequences, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Lecture Notes in Statistics (106), Springer-Verlag, New York, NY, 299--317, (eds H Niederreiter and P J-S Shiue)
--------
:meth:naginterfaces.library.examples.rand.bb_inc_ex.main
"""
raise NotImplementedError
[docs]def field_1d_user_setup(ns, xmin, xmax, var, cov1, maxm=None, pad=1, icorr=0, data=None):
r"""
field_1d_user_setup performs the setup required in order to simulate stationary Gaussian random fields in one dimension, for a user-defined variogram, using the circulant embedding method.
Specifically, the eigenvalues of the extended covariance matrix (or embedding matrix) are calculated, and their square roots output, for use by :meth:field_1d_generate, which simulates the random field.
.. _g05zm-py2-py-doc:
For full information please refer to the NAG Library document for g05zm
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05zmf.html
.. _g05zm-py2-py-parameters:
**Parameters**
**ns** : int
The number of sample points to be generated in realizations of the random field.
**xmin** : float
The lower bound for the interval over which the random field is to be simulated.
**xmax** : float
The upper bound for the interval over which the random field is to be simulated.
**var** : float
The multiplicative factor :math:\sigma^2 of the variogram :math:\gamma \left(x\right).
**cov1** : callable gamma = cov1(x, data=None)
:math:\mathrm{cov1} must evaluate the variogram :math:\gamma \left(x\right), without the multiplicative factor :math:\sigma^2, for all :math:x\geq 0.
The value returned in :math:\mathrm{gamma} is multiplied internally by :math:\mathrm{var}.
**Parameters**
**x** : float
The value :math:x at which the variogram :math:\gamma \left(x\right) is to be evaluated.
**data** : arbitrary, optional, modifiable in place
User-communication data for callback functions.
**Returns**
**gamma** : float
The value of the variogram :math:\frac{{\gamma \left(x\right)}}{{\sigma^2}}.
**maxm** : None or int, optional
Note: if this argument is **None** then a default value will be used, determined as follows: :math:2^{{3 + \mathrm{ceiling}\left(\log_2\left(\left(\mathrm{ns}-1\right) \right)\right) }}.
The maximum size of the circulant matrix to use. For example, if the embedding matrix is to be allowed to double in size three times before the approximation procedure is used, then choose :math:\mathrm{maxm} = 2^{{k+2}} where :math:k = 1+\left\lceil \mathrm{log2}\left(\mathrm{ns}-1\right)\right\rceil.
Determines whether the embedding matrix is padded with zeros, or padded with values of the variogram. The choice of padding may affect how big the embedding matrix must be in order to be positive semidefinite.
:math:\mathrm{pad} = 0
The embedding matrix is padded with zeros.
:math:\mathrm{pad} = 1
The embedding matrix is padded with values of the variogram.
**icorr** : int, optional
Determines which approximation to implement if required, as described in :ref:Notes <g05zm-py2-py-notes>.
**data** : arbitrary, optional
User-communication data for callback functions.
**Returns**
**lam** : float, ndarray, shape :math:\left(\mathrm{maxm}\right)
Contains the square roots of the eigenvalues of the embedding matrix.
**xx** : float, ndarray, shape :math:\left(\mathrm{ns}\right)
The points at which values of the random field will be output.
**m** : int
The size of the embedding matrix.
**approx** : int
Indicates whether approximation was used.
:math:\mathrm{approx} = 0
No approximation was used.
:math:\mathrm{approx} = 1
Approximation was used.
**rho** : float
Indicates the scaling of the covariance matrix. :math:\mathrm{rho} = 1.0 unless approximation was used with :math:\mathrm{icorr} = 0 or :math:1.
**icount** : int
Indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
**eig** : float, ndarray, shape :math:\left(3\right)
Indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. :math:\mathrm{eig}[0] contains the smallest eigenvalue, :math:\mathrm{eig}[1] contains the sum of the squares of the negative eigenvalues, and :math:\mathrm{eig}[2] contains the sum of the absolute values of the negative eigenvalues.
.. _g05zm-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{ns} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{ns}\geq 1.
(errno :math:2)
On entry, :math:\mathrm{xmin} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{xmax} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{xmin} < \mathrm{xmax}.
(errno :math:4)
On entry, :math:\mathrm{maxm} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: the minimum calculated value for :math:\mathrm{maxm} is :math:\langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:5)
On entry, :math:\mathrm{var} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{var}\geq 0.0.
(errno :math:7)
On entry, :math:\mathrm{pad} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{pad} = 0 or :math:1.
(errno :math:8)
On entry, :math:\mathrm{icorr} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{icorr} = 0, :math:1 or :math:2.
.. _g05zm-py2-py-notes:
**Notes**
A one-dimensional random field :math:Z\left(x\right) in :math:\mathbb{R} is a function which is random at every point :math:x \in \mathbb{R}, so :math:Z\left(x\right) is a random variable for each :math:x.
The random field has a mean function :math:\mu \left(x\right) = 𝔼\left[Z\left(x\right)\right] and a symmetric positive semidefinite covariance function :math:C\left(x, y\right) = 𝔼\left[\left(Z\left(x\right)-\mu \left(x\right)\right)\left(Z\left(y\right)-\mu \left(y\right)\right)\right]. :math:Z\left(x\right) is a Gaussian random field if for any choice of :math:n \in ℕ and :math:x_1,\ldots,x_n \in \mathbb{R}, the random vector :math:\left[Z\left(x_1\right),\ldots,Z\left(x_n\right)\right]^\mathrm{T} follows a multivariate Normal distribution, which would have a mean vector :math:\tilde{\mu } with entries :math:\tilde{\mu }_i = \mu \left(x_i\right) and a covariance matrix :math:\tilde{C} with entries :math:\tilde{C}_{{ij}} = C\left(x_i, x_j\right).
A Gaussian random field :math:Z\left(x\right) is stationary if :math:\mu \left(x\right) is constant for all :math:x \in \mathbb{R} and :math:C\left(x, y\right) = C\left({x+a}, {y+a}\right) for all :math:\left. x, y, a\right. \in \mathbb{R} and hence we can express the covariance function :math:C\left(x, y\right) as a function :math:\gamma of one variable: :math:C\left(x, y\right) = \gamma \left(x-y\right). :math:\gamma is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor :math:\sigma^2 representing the variance such that :math:\gamma \left(0\right) = \sigma^2.
The functions field_1d_user_setup and :meth:field_1d_generate are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram :math:\gamma \left(x\right), over an interval :math:\left[x_{\textit{min}}, x_{\textit{max}}\right], using an equally spaced set of :math:N points on the interval.
The problem reduces to sampling a Normal random vector :math:\mathbf{X} of size :math:N, with mean vector zero and a symmetric Toeplitz covariance matrix :math:A.
Since :math:A is in general expensive to factorize, a technique known as the circulant embedding method is used. :math:A is embedded into a larger, symmetric circulant matrix :math:B of size :math:M\geq 2\left(N-1\right), which can now be factorized as :math:B = W\Lambda W^* = R^*R, where :math:W is the Fourier matrix (:math:W^* is the complex conjugate of :math:W), :math:\Lambda is the diagonal matrix containing the eigenvalues of :math:B and :math:R = \Lambda^{\frac{1}{2}}W^*. :math:B is known as the embedding matrix.
The eigenvalues can be calculated by performing a discrete Fourier transform of the first row (or column) of :math:B and multiplying by :math:M, and so only the first row (or column) of :math:B is needed -- the whole matrix does not need to be formed.
As long as all of the values of :math:\Lambda are non-negative (i.e., :math:B is positive semidefinite), :math:B is a covariance matrix for a random vector :math:\mathbf{Y}, two samples of which can now be simulated from the real and imaginary parts of :math:R^*\left(\mathbf{U}+i\mathbf{V}\right), where :math:\mathbf{U} and :math:\mathbf{V} have elements from the standard Normal distribution.
Since :math:R^*\left(\mathbf{U}+i\mathbf{V}\right) = W\Lambda^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right), this calculation can be done using a discrete Fourier transform of the vector :math:\Lambda^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right).
Two samples of the random vector :math:\mathbf{X} can now be recovered by taking the first :math:N elements of each sample of :math:\mathbf{Y} -- because the original covariance matrix :math:A is embedded in :math:B, :math:\mathbf{X} will have the correct distribution.
If :math:B is not positive semidefinite, larger embedding matrices :math:B can be tried; however if the size of the matrix would have to be larger than :math:\mathrm{maxm}, an approximation procedure is used.
We write :math:\Lambda = \Lambda_++\Lambda_-, where :math:\Lambda_+ and :math:\Lambda_- contain the non-negative and negative eigenvalues of :math:B respectively.
Then :math:B is replaced by :math:\rho B_+ where :math:B_+ = W\Lambda_+W^* and :math:\rho \in \left(0, 1\right] is a scaling factor.
The error :math:\epsilon in approximating the distribution of the random field is given by
.. math::
\epsilon = \sqrt{\frac{{\left(1-\rho \right)^2\mathrm{trace}\left(\Lambda \right)+\rho^2\mathrm{trace}\left(\Lambda_-\right)}}{M}}\text{.}
Three choices for :math:\rho are available, and are determined by the input argument :math:\mathrm{icorr}:
setting :math:\mathrm{icorr} = 0 sets
.. math::
\rho = \frac{\mathrm{trace}\left(\Lambda \right)}{\mathrm{trace}\left(\Lambda_+\right)}\text{,}
setting :math:\mathrm{icorr} = 1 sets
.. math::
\rho = \sqrt{\frac{\mathrm{trace}\left(\Lambda \right)}{\mathrm{trace}\left(\Lambda_+\right)}}\text{,}
setting :math:\mathrm{icorr} = 2 sets :math:\rho = 1.
field_1d_user_setup finds a suitable positive semidefinite embedding matrix :math:B and outputs its size, :math:\mathrm{m}, and the square roots of its eigenvalues in :math:\mathrm{lam}.
If approximation is used, information regarding the accuracy of the approximation is output.
Note that only the first row (or column) of :math:B is actually formed and stored.
.. _g05zm-py2-py-references:
**References**
Dietrich, C R and Newsam, G N, 1997, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM J. Sci. Comput. (18), 1088--1107
Schlather, M, 1999, Introduction to positive definite functions and to unconditional simulation of random fields, Technical Report ST 99--10, Lancaster University
Wood, A T A and Chan, G, 1994, Simulation of stationary Gaussian processes in :math:\left[0, 1\right]^d, Journal of Computational and Graphical Statistics (3(4)), 409--432
"""
raise NotImplementedError
[docs]def field_1d_predef_setup(ns, xmin, xmax, var, icov1, params, maxm=None, pad=1, icorr=0):
r"""
field_1d_predef_setup performs the setup required in order to simulate stationary Gaussian random fields in one dimension, for a preset variogram, using the circulant embedding method.
Specifically, the eigenvalues of the extended covariance matrix (or embedding matrix) are calculated, and their square roots output, for use by :meth:field_1d_generate, which simulates the random field.
.. _g05zn-py2-py-doc:
For full information please refer to the NAG Library document for g05zn
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05znf.html
.. _g05zn-py2-py-parameters:
**Parameters**
**ns** : int
The number of sample points to be generated in realizations of the random field.
**xmin** : float
The lower bound for the interval over which the random field is to be simulated. Note that if :math:\mathrm{icov1} = 14 (for simulating fractional Brownian motion), :math:\mathrm{xmin} is not referenced and the lower bound for the interval is set to zero.
**xmax** : float
The upper bound for the interval over which the random field is to be simulated. Note that if :math:\mathrm{icov1} = 14 (for simulating fractional Brownian motion), the lower bound for the interval is set to zero and so :math:\mathrm{xmax} is required to be greater than zero.
**var** : float
The multiplicative factor :math:\sigma^2 of the variogram :math:\gamma \left(x\right).
**icov1** : int
Determines which of the preset variograms to use. The choices are given below. Note that :math:x^{\prime } = \frac{\left\lvert x\right\rvert }{\ell }, where :math:\ell is the correlation length and is a parameter for most of the variograms, and :math:\sigma^2 is the variance specified by :math:\mathrm{var}.
:math:\mathrm{icov1} = 1
Symmetric stable variogram
.. math::
\gamma \left(x\right) = \sigma^2\mathrm{exp}\left(-\left(x^{\prime }\right)^{\nu }\right)\text{,}
where
:math:\ell = \mathrm{params}[0], :math:\ell > 0,
:math:\nu = \mathrm{params}[1], :math:0\leq \nu \leq 2.
:math:\mathrm{icov1} = 2
Cauchy variogram
.. math::
\gamma \left(x\right) = \sigma^2\left(1+\left(x^{\prime }\right)^2\right)^{{-\nu }}\text{,}
where
:math:\ell = \mathrm{params}[0], :math:\ell > 0,
:math:\nu = \mathrm{params}[1], :math:\nu > 0.
:math:\mathrm{icov1} = 3
Differential variogram with compact support
.. math::
\gamma \left(x\right) = \left\{\begin{array}{cc} \sigma^2\left(1+8x^{\prime }+25\left(x^{\prime }\right)^2+32\left(x^{\prime }\right)^3\right)\left(1-x^{\prime }\right)^8\text{,} & x^{\prime } < 1\text{,} \\ 0\text{,} & x^{\prime }\geq 1\text{,} \end{array}\right.
where
:math:\ell = \mathrm{params}[0], :math:\ell > 0.
:math:\mathrm{icov1} = 4
Exponential variogram
.. math::
\gamma \left(x\right) = \sigma^2\mathrm{exp}\left(-x^{\prime }\right)\text{,}
where
:math:\ell = \mathrm{params}[0], :math:\ell > 0.
:math:\mathrm{icov1} = 5
Gaussian variogram
.. math::
\gamma \left(x\right) = \sigma^2\mathrm{exp}\left({-\left(x^{\prime }\right)}^2\right)\text{,}
where
:math:\ell = \mathrm{params}[0], :math:\ell > 0.
:math:\mathrm{icov1} = 6
Nugget variogram
.. math::
\gamma \left(x\right) = \left\{\begin{array}{cc} \sigma^2\text{,} & x = 0\text{,} \\ 0\text{,} & x\neq 0\text{.} \end{array}\right.
No parameters need be set for this value of :math:\mathrm{icov1}.
:math:\mathrm{icov1} = 7
Spherical variogram
.. math::
\gamma \left(x\right) = \left\{\begin{array}{cc} \sigma^2\left(1-1.5x^{\prime }+0.5\left(x^{\prime }\right)^3\right)\text{,} & x^{\prime } < 1\text{,} \\ 0\text{,} & x^{\prime }\geq 1\text{,} \end{array}\right.
where
:math:\ell = \mathrm{params}[0], :math:\ell > 0.
:math:\mathrm{icov1} = 8
Bessel variogram
.. math::
\gamma \left(x\right) = \sigma^2\frac{{2^{\nu }\Gamma \left(\nu +1\right)J_{\nu }\left(x^{\prime }\right)}}{\left(x^{\prime }\right)^{\nu }}\text{,}
where
:math:J_{\nu }\left(·\right) is the Bessel function of the first kind,
:math:\ell = \mathrm{params}[0], :math:\ell > 0,
:math:\nu = \mathrm{params}[1], :math:\nu \geq {-0.5}.
:math:\mathrm{icov1} = 9
Hole effect variogram
.. math::
\gamma \left(x\right) = \sigma^2\frac{\sin\left(x^{\prime }\right)}{x^{\prime }}\text{,}
where
:math:\ell = \mathrm{params}[0], :math:\ell > 0.
:math:\mathrm{icov1} = 10
Whittle-Matérn variogram
.. math::
\gamma \left(x\right) = \sigma^2\frac{{2^{{1-\nu }}\left(x^{\prime }\right)^{\nu }K_{\nu }\left(x^{\prime }\right)}}{{\Gamma \left(\nu \right)}}\text{,}
where
:math:K_{\nu }\left(·\right) is the modified Bessel function of the second kind,
:math:\ell = \mathrm{params}[0], :math:\ell > 0,
:math:\nu = \mathrm{params}[1], :math:\nu > 0.
:math:\mathrm{icov1} = 11
Continuously parameterised variogram with compact support
.. math::
\gamma \left(x\right) = \left\{\begin{array}{cc} \sigma^2\frac{{2^{{1-\nu }}\left(x^{\prime }\right)^{\nu }K_{\nu }\left(x^{\prime }\right)}}{{\Gamma \left(\nu \right)}}\left(1+8x^{{\prime \prime }}+25\left(x^{{\prime \prime }}\right)^2+32\left(x^{{\prime \prime }}\right)^3\right)\left(1-x^{{\prime \prime }}\right)^8\text{,} & x^{{\prime \prime }} < 1\text{,} \\ 0\text{,} & x^{{\prime \prime }}\geq 1\text{,} \end{array}\right.
where
:math:x^{{\prime \prime }} = \frac{x^{\prime }}{s},
:math:K_{\nu }\left(·\right) is the modified Bessel function of the second kind,
:math:\ell = \mathrm{params}[0], :math:\ell > 0,
:math:s = \mathrm{params}[1], :math:s > 0 (second correlation length),
:math:\nu = \mathrm{params}[2], :math:\nu > 0.
:math:\mathrm{icov1} = 12
Generalized hyperbolic distribution variogram
.. math::
\gamma \left(x\right) = \sigma^2\frac{\left(\delta^2+\left(x^{\prime }\right)^2\right)^{\frac{\lambda }{2}}}{{\delta^{\lambda }K_{\lambda }\left(\kappa \delta \right)}}K_{\lambda }\left(\kappa \left(\delta^2+\left(x^{\prime }\right)^2\right)^{\frac{1}{2}}\right)\text{,}
where
:math:K_{\lambda }\left(·\right) is the modified Bessel function of the second kind,
:math:\ell = \mathrm{params}[0], :math:\ell > 0,
:math:\lambda = \mathrm{params}[1], no constraint on :math:\lambda
:math:\delta = \mathrm{params}[2], :math:\delta > 0,
:math:\kappa = \mathrm{params}[3], :math:\kappa > 0.
:math:\mathrm{icov1} = 13
Cosine variogram
.. math::
\gamma \left(x\right) = \sigma^2\cos\left(x^{\prime }\right)\text{,}
where
:math:\ell = \mathrm{params}[0], :math:\ell > 0.
:math:\mathrm{icov1} = 14
Used for simulating fractional Brownian motion :math:B^H\left(t\right). Fractional Brownian motion itself is not a stationary Gaussian random field, but its increments :math:\tilde{X}\left(i\right) = B^H\left(t_i\right)-B^H\left(t_{{i-1}}\right) can be simulated in the same way as a stationary random field. The variogram for the so-called 'increment process' is
.. math::
C\left(\tilde{X}\left(t_i\right),\tilde{X}\left(t_j\right)\right) = \tilde{\gamma }\left(x\right) = \frac{\delta^{{2H}}}{2}\left(\left\lvert \frac{x}{\delta }-1\right\rvert^{{2H}}+\left\lvert \frac{x}{\delta }+1\right\rvert^{{2H}}-2\left\lvert \frac{x}{\delta }\right\rvert^{{2H}}\right)\text{,}
where
:math:x = t_j-t_i,
:math:H = \mathrm{params}[0], :math:0 < H < 1, :math:H is the Hurst parameter,
:math:\delta = \mathrm{params}[1], :math:\delta > 0, normally :math:\delta = t_i-t_{{i-1}} is the (fixed) step size.
We scale the increments to set :math:\gamma \left(0\right) = 1; let :math:X\left(i\right) = \frac{{\tilde{X}\left(i\right)}}{\delta^{{-H}}}, then
.. math::
C\left(X\left(t_i\right),X\left(t_j\right)\right) = \gamma \left(x\right) = \frac{1}{2}\left(\left\lvert \frac{x}{\delta }-1\right\rvert^{{2H}}+\left\lvert \frac{x}{\delta }+1\right\rvert^{{2H}}-2\left\lvert \frac{x}{\delta }\right\rvert^{{2H}}\right)\text{.}
The increments :math:X\left(i\right) can then be simulated using :meth:field_1d_generate, then multiplied by :math:\delta^H to obtain the original increments :math:\tilde{X}\left(i\right) for the fractional Brownian motion.
**params** : float, array-like, shape :math:\left(\textit{np}\right)
The parameters set for the variogram.
**maxm** : None or int, optional
Note: if this argument is **None** then a default value will be used, determined as follows: :math:2^{{3 + \mathrm{ceiling}\left(\log_2\left(\left(\mathrm{ns}-1\right) \right)\right) }}.
The maximum size of the circulant matrix to use. For example, if the embedding matrix is to be allowed to double in size three times before the approximation procedure is used, then choose :math:\mathrm{maxm} = 2^{{k+2}} where :math:k = 1+\left\lceil \mathrm{log2}\left(\mathrm{ns}-1\right)\right\rceil.
Determines whether the embedding matrix is padded with zeros, or padded with values of the variogram. The choice of padding may affect how big the embedding matrix must be in order to be positive semidefinite.
:math:\mathrm{pad} = 0
The embedding matrix is padded with zeros.
:math:\mathrm{pad} = 1
The embedding matrix is padded with values of the variogram.
**icorr** : int, optional
Determines which approximation to implement if required, as described in :ref:Notes <g05zn-py2-py-notes>.
**Returns**
**lam** : float, ndarray, shape :math:\left(\mathrm{maxm}\right)
Contains the square roots of the eigenvalues of the embedding matrix.
**xx** : float, ndarray, shape :math:\left(\mathrm{ns}\right)
The points at which values of the random field will be output.
**m** : int
The size of the embedding matrix.
**approx** : int
Indicates whether approximation was used.
:math:\mathrm{approx} = 0
No approximation was used.
:math:\mathrm{approx} = 1
Approximation was used.
**rho** : float
Indicates the scaling of the covariance matrix. :math:\mathrm{rho} = 1.0 unless approximation was used with :math:\mathrm{icorr} = 0 or :math:1.
**icount** : int
Indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
**eig** : float, ndarray, shape :math:\left(3\right)
Indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. :math:\mathrm{eig}[0] contains the smallest eigenvalue, :math:\mathrm{eig}[1] contains the sum of the squares of the negative eigenvalues, and :math:\mathrm{eig}[2] contains the sum of the absolute values of the negative eigenvalues.
.. _g05zn-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{ns} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{ns}\geq 1.
(errno :math:2)
On entry, :math:\mathrm{icov1} \neq 14, :math:\mathrm{xmin} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{xmax} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{xmin} < \mathrm{xmax}.
(errno :math:3)
On entry, :math:\mathrm{icov1} = 14 and :math:\mathrm{xmax} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{xmax} > 0.0.
(errno :math:4)
On entry, :math:\mathrm{maxm} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: the minimum calculated value for :math:\mathrm{maxm} is :math:\langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:5)
On entry, :math:\mathrm{var} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{var}\geq 0.0.
(errno :math:6)
On entry, :math:\mathrm{icov1} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{icov1}\geq 1 and :math:\mathrm{icov1}\leq 14.
(errno :math:7)
On entry, :math:\textit{np} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: for :math:\mathrm{icov1} = \langle\mathit{\boldsymbol{value}}\rangle, :math:\textit{np} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:8)
On entry, :math:\mathrm{params}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: dependent on :math:\mathrm{icov1}.
(errno :math:9)
On entry, :math:\mathrm{pad} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{pad} = 0 or :math:1.
(errno :math:10)
On entry, :math:\mathrm{icorr} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{icorr} = 0, :math:1 or :math:2.
.. _g05zn-py2-py-notes:
**Notes**
A one-dimensional random field :math:Z\left(x\right) in :math:\mathbb{R} is a function which is random at every point :math:x \in \mathbb{R}, so :math:Z\left(x\right) is a random variable for each :math:x.
The random field has a mean function :math:\mu \left(x\right) = 𝔼\left[Z\left(x\right)\right] and a symmetric positive semidefinite covariance function :math:C\left(x, y\right) = 𝔼\left[\left(Z\left(x\right)-\mu \left(x\right)\right)\left(Z\left(y\right)-\mu \left(y\right)\right)\right]. :math:Z\left(x\right) is a Gaussian random field if for any choice of :math:n \in ℕ and :math:x_1,\ldots,x_n \in \mathbb{R}, the random vector :math:\left[Z\left(x_1\right),\ldots,Z\left(x_n\right)\right]^\mathrm{T} follows a multivariate Normal distribution, which would have a mean vector :math:\tilde{\mu } with entries :math:\tilde{\mu }_i = \mu \left(x_i\right) and a covariance matrix :math:\tilde{C} with entries :math:\tilde{C}_{{ij}} = C\left(x_i, x_j\right).
A Gaussian random field :math:Z\left(x\right) is stationary if :math:\mu \left(x\right) is constant for all :math:x \in \mathbb{R} and :math:C\left(x, y\right) = C\left({x+a}, {y+a}\right) for all :math:\left. x, y, a\right. \in \mathbb{R} and hence we can express the covariance function :math:C\left(x, y\right) as a function :math:\gamma of one variable: :math:C\left(x, y\right) = \gamma \left(x-y\right). :math:\gamma is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor :math:\sigma^2 representing the variance such that :math:\gamma \left(0\right) = \sigma^2.
The functions field_1d_predef_setup and :meth:field_1d_generate are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram :math:\gamma \left(x\right), over an interval :math:\left[x_{\textit{min}}, x_{\textit{max}}\right], using an equally spaced set of :math:N points.
The problem reduces to sampling a Normal random vector :math:\mathbf{X} of size :math:N, with mean vector zero and a symmetric Toeplitz covariance matrix :math:A.
Since :math:A is in general expensive to factorize, a technique known as the circulant embedding method is used. :math:A is embedded into a larger, symmetric circulant matrix :math:B of size :math:M\geq 2\left(N-1\right), which can now be factorized as :math:B = W\Lambda W^* = R^*R, where :math:W is the Fourier matrix (:math:W^* is the complex conjugate of :math:W), :math:\Lambda is the diagonal matrix containing the eigenvalues of :math:B and :math:R = \Lambda^{\frac{1}{2}}W^*. :math:B is known as the embedding matrix.
The eigenvalues can be calculated by performing a discrete Fourier transform of the first row (or column) of :math:B and multiplying by :math:M, and so only the first row (or column) of :math:B is needed -- the whole matrix does not need to be formed.
As long as all of the values of :math:\Lambda are non-negative (i.e., :math:B is positive semidefinite), :math:B is a covariance matrix for a random vector :math:\mathbf{Y}, two samples of which can now be simulated from the real and imaginary parts of :math:R^*\left(\mathbf{U}+i\mathbf{V}\right), where :math:\mathbf{U} and :math:\mathbf{V} have elements from the standard Normal distribution.
Since :math:R^*\left(\mathbf{U}+i\mathbf{V}\right) = W\Lambda^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right), this calculation can be done using a discrete Fourier transform of the vector :math:\Lambda^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right).
Two samples of the random vector :math:\mathbf{X} can now be recovered by taking the first :math:N elements of each sample of :math:\mathbf{Y} -- because the original covariance matrix :math:A is embedded in :math:B, :math:\mathbf{X} will have the correct distribution.
If :math:B is not positive semidefinite, larger embedding matrices :math:B can be tried; however if the size of the matrix would have to be larger than :math:\mathrm{maxm}, an approximation procedure is used.
We write :math:\Lambda = \Lambda_++\Lambda_-, where :math:\Lambda_+ and :math:\Lambda_- contain the non-negative and negative eigenvalues of :math:B respectively.
Then :math:B is replaced by :math:\rho B_+ where :math:B_+ = W\Lambda_+W^* and :math:\rho \in \left(0, 1\right] is a scaling factor.
The error :math:\epsilon in approximating the distribution of the random field is given by
.. math::
\epsilon = \sqrt{\frac{{\left(1-\rho \right)^2\mathrm{trace}\left(\Lambda \right)+\rho^2\mathrm{trace}\left(\Lambda_-\right)}}{M}}\text{.}
Three choices for :math:\rho are available, and are determined by the input argument :math:\mathrm{icorr}:
setting :math:\mathrm{icorr} = 0 sets
.. math::
\rho = \frac{\mathrm{trace}\left(\Lambda \right)}{\mathrm{trace}\left(\Lambda_+\right)}\text{,}
setting :math:\mathrm{icorr} = 1 sets
.. math::
\rho = \sqrt{\frac{\mathrm{trace}\left(\Lambda \right)}{\mathrm{trace}\left(\Lambda_+\right)}}\text{,}
setting :math:\mathrm{icorr} = 2 sets :math:\rho = 1.
field_1d_predef_setup finds a suitable positive semidefinite embedding matrix :math:B and outputs its size, :math:\mathrm{m}, and the square roots of its eigenvalues in :math:\mathrm{lam}.
If approximation is used, information regarding the accuracy of the approximation is output.
Note that only the first row (or column) of :math:B is actually formed and stored.
.. _g05zn-py2-py-references:
**References**
Dietrich, C R and Newsam, G N, 1997, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM J. Sci. Comput. (18), 1088--1107
Schlather, M, 1999, Introduction to positive definite functions and to unconditional simulation of random fields, Technical Report ST 99--10, Lancaster University
Wood, A T A and Chan, G, 1997, Algorithm AS 312: An Algorithm for Simulating Stationary Gaussian Random Fields, Journal of the Royal Statistical Society, Series C (Applied Statistics) (Volume 46) (1), 171--181
"""
raise NotImplementedError
[docs]def field_1d_generate(ns, s, lam, rho, statecomm):
r"""
field_1d_generate produces realizations of a stationary Gaussian random field in one dimension, using the circulant embedding method.
The square roots of the eigenvalues of the extended covariance matrix (or embedding matrix) need to be input, and can be calculated using :meth:field_1d_user_setup or :meth:field_1d_predef_setup.
.. _g05zp-py2-py-doc:
For full information please refer to the NAG Library document for g05zp
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05zpf.html
.. _g05zp-py2-py-parameters:
**Parameters**
**ns** : int
The number of sample points to be generated in realizations of the random field. This must be the same value as supplied to :meth:field_1d_predef_setup or :meth:field_1d_user_setup when calculating the eigenvalues of the embedding matrix.
**s** : int
:math:S, the number of realizations of the random field to simulate.
**lam** : float, array-like, shape :math:\left(m\right)
Must contain the square roots of the eigenvalues of the embedding matrix, as returned by :meth:field_1d_predef_setup or :meth:field_1d_user_setup.
**rho** : float
Indicates the scaling of the covariance matrix, as returned by :meth:field_1d_predef_setup or :meth:field_1d_user_setup.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**z** : float, ndarray, shape :math:\left(\mathrm{ns}, \mathrm{s}\right)
Contains the realizations of the random field. The :math:j\ th realization, for the :math:\mathrm{ns} sample points, is stored in :math:\mathrm{z}[i-1,j-1], for :math:i = 1,2,\ldots,\mathrm{ns}. The sample points are as returned in :math:{\textit{xx}} by :meth:field_1d_predef_setup or :meth:field_1d_user_setup.
.. _g05zp-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{ns} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{ns}\geq 1.
(errno :math:2)
On entry, :math:\mathrm{s} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{s}\geq 1.
(errno :math:3)
On entry, :math:m = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{ns} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:m\geq \mathrm{max}\left(1, {2\times \left(\mathrm{ns}-1\right)}\right).
(errno :math:4)
On entry, at least one element of :math:\mathrm{lam} was negative.
Constraint: all elements of :math:\mathrm{lam} must be non-negative.
(errno :math:5)
On entry, :math:\mathrm{rho} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0.0\leq \mathrm{rho}\leq 1.0.
(errno :math:6)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05zp-py2-py-notes:
**Notes**
A one-dimensional random field :math:Z\left(x\right) in :math:\mathbb{R} is a function which is random at every point :math:x \in \mathbb{R}, so :math:Z\left(x\right) is a random variable for each :math:x.
The random field has a mean function :math:\mu \left(x\right) = 𝔼\left[Z\left(x\right)\right] and a symmetric non-negative definite covariance function :math:C\left(x, y\right) = 𝔼\left[\left(Z\left(x\right)-\mu \left(x\right)\right)\left(Z\left(y\right)-\mu \left(y\right)\right)\right]. :math:Z\left(x\right) is a Gaussian random field if for any choice of :math:n \in ℕ and :math:x_1,\ldots,x_n \in \mathbb{R}, the random vector :math:\left[Z\left(x_1\right),\ldots,Z\left(x_n\right)\right]^\mathrm{T} follows a multivariate Normal distribution, which would have a mean vector :math:\tilde{\mu } with entries :math:\tilde{\mu }_i = \mu \left(x_i\right) and a covariance matrix :math:\tilde{C} with entries :math:\tilde{C}_{{ij}} = C\left(x_i, x_j\right).
A Gaussian random field :math:Z\left(x\right) is stationary if :math:\mu \left(x\right) is constant for all :math:x \in \mathbb{R} and :math:C\left(x, y\right) = C\left({x+a}, {y+a}\right) for all :math:\left. x, y, a\right. \in \mathbb{R} and hence we can express the covariance function :math:C\left(x, y\right) as a function :math:\gamma of one variable: :math:C\left(x, y\right) = \gamma \left(x-y\right). :math:\gamma is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor :math:\sigma^2 representing the variance such that :math:\gamma \left(0\right) = \sigma^2.
The functions :meth:field_1d_user_setup or :meth:field_1d_predef_setup, along with field_1d_generate, are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram :math:\gamma \left(x\right), over an interval :math:\left[x_{\textit{min}}, x_{\textit{max}}\right], using an equally spaced set of :math:N points.
The problem reduces to sampling a Normal random vector :math:\mathbf{X} of size :math:N, with mean vector zero and a symmetric Toeplitz covariance matrix :math:A.
Since :math:A is in general expensive to factorize, a technique known as the circulant embedding method is used. :math:A is embedded into a larger, symmetric circulant matrix :math:B of size :math:M\geq 2\left(N-1\right), which can now be factorized as :math:B = W\Lambda W^* = R^*R, where :math:W is the Fourier matrix (:math:W^* is the complex conjugate of :math:W), :math:\Lambda is the diagonal matrix containing the eigenvalues of :math:B and :math:R = \Lambda^{\frac{1}{2}}W^*. :math:B is known as the embedding matrix.
The eigenvalues can be calculated by performing a discrete Fourier transform of the first row (or column) of :math:B and multiplying by :math:M, and so only the first row (or column) of :math:B is needed -- the whole matrix does not need to be formed.
As long as all of the values of :math:\Lambda are non-negative (i.e., :math:B is non-negative definite), :math:B is a covariance matrix for a random vector :math:\mathbf{Y}, two samples of which can now be simulated from the real and imaginary parts of :math:R^*\left(\mathbf{U}+i\mathbf{V}\right), where :math:\mathbf{U} and :math:\mathbf{V} have elements from the standard Normal distribution.
Since :math:R^*\left(\mathbf{U}+i\mathbf{V}\right) = W\Lambda^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right), this calculation can be done using a discrete Fourier transform of the vector :math:\Lambda^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right).
Two samples of the random vector :math:\mathbf{X} can now be recovered by taking the first :math:N elements of each sample of :math:\mathbf{Y} -- because the original covariance matrix :math:A is embedded in :math:B, :math:\mathbf{X} will have the correct distribution.
If :math:B is not non-negative definite, larger embedding matrices :math:B can be tried; however if the size of the matrix would have to be larger than :math:\textit{maxm}, an approximation procedure is used.
See the documentation of :meth:field_1d_user_setup or :meth:field_1d_predef_setup for details of the approximation procedure.
field_1d_generate takes the square roots of the eigenvalues of the embedding matrix :math:B, and its size :math:M, as input and outputs :math:S realizations of the random field in :math:Z.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to field_1d_generate.
.. _g05zp-py2-py-references:
**References**
Dietrich, C R and Newsam, G N, 1997, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM J. Sci. Comput. (18), 1088--1107
Schlather, M, 1999, Introduction to positive definite functions and to unconditional simulation of random fields, Technical Report ST 99--10, Lancaster University
Wood, A T A and Chan, G, 1994, Simulation of stationary Gaussian processes in :math:\left[0, 1\right]^d, Journal of Computational and Graphical Statistics (3(4)), 409--432
"""
raise NotImplementedError
[docs]def field_2d_user_setup(ns, xmin, xmax, ymin, ymax, maxm, var, cov2, even, pad=1, icorr=0, data=None):
r"""
field_2d_user_setup performs the setup required in order to simulate stationary Gaussian random fields in two dimensions, for a user-defined variogram, using the circulant embedding method.
Specifically, the eigenvalues of the extended covariance matrix (or embedding matrix) are calculated, and their square roots output, for use by :meth:field_2d_generate, which simulates the random field.
.. _g05zq-py2-py-doc:
For full information please refer to the NAG Library document for g05zq
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05zqf.html
.. _g05zq-py2-py-parameters:
**Parameters**
**ns** : int, array-like, shape :math:\left(2\right)
The number of sample points to use in each direction, with :math:\mathrm{ns}[0] sample points in the :math:x-direction, :math:N_1 and :math:\mathrm{ns}[1] sample points in the :math:y-direction, :math:N_2. The total number of sample points on the grid is, therefore, :math:\mathrm{ns}[0]\times \mathrm{ns}[1].
**xmin** : float
The lower bound for the :math:x-coordinate, for the region in which the random field is to be simulated.
**xmax** : float
The upper bound for the :math:x-coordinate, for the region in which the random field is to be simulated.
**ymin** : float
The lower bound for the :math:y-coordinate, for the region in which the random field is to be simulated.
**ymax** : float
The upper bound for the :math:y-coordinate, for the region in which the random field is to be simulated.
**maxm** : int, array-like, shape :math:\left(2\right)
Determines the maximum size of the circulant matrix to use -- a maximum of :math:\mathrm{maxm}[0] elements in the :math:x-direction, and a maximum of :math:\mathrm{maxm}[1] elements in the :math:y-direction. The maximum size of the circulant matrix is thus :math:\mathrm{maxm}[0]:math:\times:math:\mathrm{maxm}[1].
**var** : float
The multiplicative factor :math:\sigma^2 of the variogram :math:\gamma \left(\mathbf{x}\right).
**cov2** : callable gamma = cov2(x, y, data=None)
:math:\mathrm{cov2} must evaluate the variogram :math:\gamma \left(\mathbf{x}\right) for all :math:\mathbf{x} if :math:\mathrm{even} = 0, and for all :math:\mathbf{x} with non-negative entries if :math:\mathrm{even} = 1.
The value returned in :math:\mathrm{gamma} is multiplied internally by :math:\mathrm{var}.
**Parameters**
**x** : float
The coordinate :math:x at which the variogram :math:\gamma \left(\mathbf{x}\right) is to be evaluated.
**y** : float
The coordinate :math:y at which the variogram :math:\gamma \left(\mathbf{x}\right) is to be evaluated.
**data** : arbitrary, optional, modifiable in place
User-communication data for callback functions.
**Returns**
**gamma** : float
The value of the variogram :math:\gamma \left(\mathbf{x}\right).
**even** : int
Indicates whether the covariance function supplied is even or uneven.
:math:\mathrm{even} = 0
The covariance function is uneven.
:math:\mathrm{even} = 1
The covariance function is even.
Determines whether the embedding matrix is padded with zeros, or padded with values of the variogram. The choice of padding may affect how big the embedding matrix must be in order to be positive semidefinite.
:math:\mathrm{pad} = 0
The embedding matrix is padded with zeros.
:math:\mathrm{pad} = 1
The embedding matrix is padded with values of the variogram.
**icorr** : int, optional
Determines which approximation to implement if required, as described in :ref:Notes <g05zq-py2-py-notes>.
**data** : arbitrary, optional
User-communication data for callback functions.
**Returns**
**lam** : float, ndarray, shape :math:\left(:\right)
Contains the square roots of the eigenvalues of the embedding matrix.
**xx** : float, ndarray, shape :math:\left(:\right)
The points of the :math:x-coordinates at which values of the random field will be output.
**yy** : float, ndarray, shape :math:\left(:\right)
The points of the :math:y-coordinates at which values of the random field will be output.
**m** : int, ndarray, shape :math:\left(2\right)
:math:\mathrm{m}[0] contains :math:M_1, the size of the circulant blocks and :math:\mathrm{m}[1] contains :math:M_2, the number of blocks, resulting in a final square matrix of size :math:M_1\times M_2.
**approx** : int
Indicates whether approximation was used.
:math:\mathrm{approx} = 0
No approximation was used.
:math:\mathrm{approx} = 1
Approximation was used.
**rho** : float
Indicates the scaling of the covariance matrix. :math:\mathrm{rho} = 1.0 unless approximation was used with :math:\mathrm{icorr} = 0 or :math:1.
**icount** : int
Indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
**eig** : float, ndarray, shape :math:\left(3\right)
Indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. :math:\mathrm{eig}[0] contains the smallest eigenvalue, :math:\mathrm{eig}[1] contains the sum of the squares of the negative eigenvalues, and :math:\mathrm{eig}[2] contains the sum of the absolute values of the negative eigenvalues.
.. _g05zq-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{ns} = \left[\langle\mathit{\boldsymbol{value}}\rangle, \langle\mathit{\boldsymbol{value}}\rangle\right].
Constraint: :math:\mathrm{ns}[0]\geq 1, :math:\mathrm{ns}[1]\geq 1.
(errno :math:2)
On entry, :math:\mathrm{xmin} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{xmax} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{xmin} < \mathrm{xmax}.
(errno :math:4)
On entry, :math:\mathrm{ymin} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{ymax} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{ymin} < \mathrm{ymax}.
(errno :math:6)
On entry, :math:\mathrm{maxm} = \left[\langle\mathit{\boldsymbol{value}}\rangle, \langle\mathit{\boldsymbol{value}}\rangle\right].
Constraint: the minima for :math:\mathrm{maxm} are :math:\left[\langle\mathit{\boldsymbol{value}}\rangle, \langle\mathit{\boldsymbol{value}}\rangle\right].
(errno :math:7)
On entry, :math:\mathrm{var} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{var}\geq 0.0.
(errno :math:9)
On entry, :math:\mathrm{even} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{even} = 0 or :math:1.
(errno :math:10)
On entry, :math:\mathrm{pad} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{pad} = 0 or :math:1.
(errno :math:11)
On entry, :math:\mathrm{icorr} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{icorr} = 0, :math:1 or :math:2.
.. _g05zq-py2-py-notes:
**Notes**
A two-dimensional random field :math:Z\left(\mathbf{x}\right) in :math:\mathbb{R}^2 is a function which is random at every point :math:\mathbf{x} \in \mathbb{R}^2, so :math:Z\left(\mathbf{x}\right) is a random variable for each :math:\mathbf{x}.
The random field has a mean function :math:\mu \left(\mathbf{x}\right) = 𝔼\left[Z\left(\mathbf{x}\right)\right] and a symmetric positive semidefinite covariance function :math:C\left(\mathbf{x}, \mathbf{y}\right) = 𝔼\left[\left(Z\left(\mathbf{x}\right)-\mu \left(\mathbf{x}\right)\right)\left(Z\left(\mathbf{y}\right)-\mu \left(\mathbf{y}\right)\right)\right]. :math:Z\left(\mathbf{x}\right) is a Gaussian random field if for any choice of :math:n \in ℕ and :math:\mathbf{x}_1,\ldots,\mathbf{x}_n \in \mathbb{R}^2, the random vector :math:\left[Z\left(\mathbf{x}_1\right),\ldots,Z\left(\mathbf{x}_n\right)\right]^\mathrm{T} follows a multivariate Normal distribution, which would have a mean vector :math:\tilde{\mu } with entries :math:\tilde{\mu }_i = \mu \left(\mathbf{x}_i\right) and a covariance matrix :math:\tilde{C} with entries :math:\tilde{C}_{{ij}} = C\left(\mathbf{x}_i, \mathbf{x}_j\right).
A Gaussian random field :math:Z\left(\mathbf{x}\right) is stationary if :math:\mu \left(\mathbf{x}\right) is constant for all :math:\mathbf{x} \in \mathbb{R}^2 and :math:C\left(\mathbf{x}, \mathbf{y}\right) = C\left({\mathbf{x}+\mathbf{a}}, {\mathbf{y}+\mathbf{a}}\right) for all :math:\left. \mathbf{x}, \mathbf{y}, \mathbf{a}\right. \in \mathbb{R}^2 and hence we can express the covariance function :math:C\left(\mathbf{x}, \mathbf{y}\right) as a function :math:\gamma of one variable: :math:C\left(\mathbf{x}, \mathbf{y}\right) = \gamma \left(\mathbf{x}-\mathbf{y}\right). :math:\gamma is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor :math:\sigma^2 representing the variance such that :math:\gamma \left(0\right) = \sigma^2.
The functions field_2d_user_setup and :meth:field_2d_generate are used to simulate a two-dimensional stationary Gaussian random field, with mean function zero and variogram :math:\gamma \left(\mathbf{x}\right), over a domain :math:\left[x_{\textit{min}}, x_{\textit{max}}\right]\times \left[y_{\textit{min}}, y_{\textit{max}}\right], using an equally spaced set of :math:N_1\times N_2 points; :math:N_1 points in the :math:x-direction and :math:N_2 points in the :math:y-direction.
The problem reduces to sampling a Normal random vector :math:\mathbf{X} of size :math:N_1\times N_2, with mean vector zero and a symmetric covariance matrix :math:A, which is an :math:N_2\times N_2 block Toeplitz matrix with Toeplitz blocks of size :math:N_1\times N_1.
Since :math:A is in general expensive to factorize, a technique known as the circulant embedding method is used. :math:A is embedded into a larger, symmetric matrix :math:B, which is an :math:M_2\times M_2 block circulant matrix with circulant blocks of size :math:M_1\times M_1, where :math:M_1\geq 2\left(N_1-1\right) and :math:M_2\geq 2\left(N_2-1\right). :math:B can now be factorized as :math:B = W\Lambda W^* = R^*R, where :math:W is the two-dimensional Fourier matrix (:math:W^* is the complex conjugate of :math:W), :math:\Lambda is the diagonal matrix containing the eigenvalues of :math:B and :math:R = \Lambda^{\frac{1}{2}}W^*. :math:B is known as the embedding matrix.
The eigenvalues can be calculated by performing a discrete Fourier transform of the first row (or column) of :math:B and multiplying by :math:M_1\times M_2, and so only the first row (or column) of :math:B is needed -- the whole matrix does not need to be formed.
The symmetry of :math:A as a block matrix, and the symmetry of each block of :math:A, depends on whether the variogram :math:\gamma is even or not. :math:\gamma is even in its first coordinate if :math:\gamma \left(\left[{-x}_1, x_2\right]^\mathrm{T}\right) = \gamma \left(\left[x_1, x_2\right]^\mathrm{T}\right), even in its second coordinate if :math:\gamma \left(\left[x_1, {-x}_2\right]^\mathrm{T}\right) = \gamma \left(\left[x_1, x_2\right]^\mathrm{T}\right), and even if it is even in both coordinates (in two dimensions it is impossible for :math:\gamma to be even in one coordinate and uneven in the other).
If :math:\gamma is even then :math:A is a symmetric block matrix and has symmetric blocks; if :math:\gamma is uneven then :math:A is not a symmetric block matrix and has non-symmetric blocks.
In the uneven case, :math:M_1 and :math:M_2 are set to be odd in order to guarantee symmetry in :math:B.
As long as all of the values of :math:\Lambda are non-negative (i.e., :math:B is positive semidefinite), :math:B is a covariance matrix for a random vector :math:\mathbf{Y} which has :math:M_2 blocks of size :math:M_1.
Two samples of :math:\mathbf{Y} can now be simulated from the real and imaginary parts of :math:R^*\left(\mathbf{U}+i\mathbf{V}\right), where :math:\mathbf{U} and :math:\mathbf{V} have elements from the standard Normal distribution.
Since :math:R^*\left(\mathbf{U}+i\mathbf{V}\right) = W\Lambda^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right), this calculation can be done using a discrete Fourier transform of the vector :math:\Lambda^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right).
Two samples of the random vector :math:\mathbf{X} can now be recovered by taking the first :math:N_1 elements of the first :math:N_2 blocks of each sample of :math:\mathbf{Y} -- because the original covariance matrix :math:A is embedded in :math:B, :math:\mathbf{X} will have the correct distribution.
If :math:B is not positive semidefinite, larger embedding matrices :math:B can be tried; however if the size of the matrix would have to be larger than :math:\mathrm{maxm}, an approximation procedure is used.
We write :math:\Lambda = \Lambda_++\Lambda_-, where :math:\Lambda_+ and :math:\Lambda_- contain the non-negative and negative eigenvalues of :math:B respectively.
Then :math:B is replaced by :math:\rho B_+ where :math:B_+ = W\Lambda_+W^* and :math:\rho \in \left(0, 1\right] is a scaling factor.
The error :math:\epsilon in approximating the distribution of the random field is given by
.. math::
\epsilon = \sqrt{\frac{{\left(1-\rho \right)^2\mathrm{trace}\left(\Lambda \right)+\rho^2\mathrm{trace}\left(\Lambda_-\right)}}{M}}\text{.}
Three choices for :math:\rho are available, and are determined by the input argument :math:\mathrm{icorr}:
setting :math:\mathrm{icorr} = 0 sets
.. math::
\rho = \frac{\mathrm{trace}\left(\Lambda \right)}{\mathrm{trace}\left(\Lambda_+\right)}\text{,}
setting :math:\mathrm{icorr} = 1 sets
.. math::
\rho = \sqrt{\frac{\mathrm{trace}\left(\Lambda \right)}{\mathrm{trace}\left(\Lambda_+\right)}}\text{,}
setting :math:\mathrm{icorr} = 2 sets :math:\rho = 1.
field_2d_user_setup finds a suitable positive semidefinite embedding matrix :math:B and outputs its sizes in the vector :math:\mathrm{m} and the square roots of its eigenvalues in :math:\mathrm{lam}.
If approximation is used, information regarding the accuracy of the approximation is output.
Note that only the first row (or column) of :math:B is actually formed and stored.
.. _g05zq-py2-py-references:
**References**
Dietrich, C R and Newsam, G N, 1997, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM J. Sci. Comput. (18), 1088--1107
Schlather, M, 1999, Introduction to positive definite functions and to unconditional simulation of random fields, Technical Report ST 99--10, Lancaster University
Wood, A T A and Chan, G, 1994, Simulation of stationary Gaussian processes in :math:\left[0, 1\right]^d, Journal of Computational and Graphical Statistics (3(4)), 409--432
"""
raise NotImplementedError
[docs]def field_2d_predef_setup(ns, xmin, xmax, ymin, ymax, maxm, var, icov2, params, norm=2, pad=1, icorr=0):
r"""
field_2d_predef_setup performs the setup required in order to simulate stationary Gaussian random fields in two dimensions, for a preset variogram, using the circulant embedding method.
Specifically, the eigenvalues of the extended covariance matrix (or embedding matrix) are calculated, and their square roots output, for use by :meth:field_2d_generate, which simulates the random field.
.. _g05zr-py2-py-doc:
For full information please refer to the NAG Library document for g05zr
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05zrf.html
.. _g05zr-py2-py-parameters:
**Parameters**
**ns** : int, array-like, shape :math:\left(2\right)
The number of sample points to use in each direction, with :math:\mathrm{ns}[0] sample points in the :math:x-direction, :math:N_1 and :math:\mathrm{ns}[1] sample points in the :math:y-direction, :math:N_2. The total number of sample points on the grid is, therefore, :math:\mathrm{ns}[0]\times \mathrm{ns}[1].
**xmin** : float
The lower bound for the :math:x-coordinate, for the region in which the random field is to be simulated.
**xmax** : float
The upper bound for the :math:x-coordinate, for the region in which the random field is to be simulated.
**ymin** : float
The lower bound for the :math:y-coordinate, for the region in which the random field is to be simulated.
**ymax** : float
The upper bound for the :math:y-coordinate, for the region in which the random field is to be simulated.
**maxm** : int, array-like, shape :math:\left(2\right)
Determines the maximum size of the circulant matrix to use -- a maximum of :math:\mathrm{maxm}[0] elements in the :math:x-direction, and a maximum of :math:\mathrm{maxm}[1] elements in the :math:y-direction. The maximum size of the circulant matrix is thus :math:\mathrm{maxm}[0]:math:\times:math:\mathrm{maxm}[1].
**var** : float
The multiplicative factor :math:\sigma^2 of the variogram :math:\gamma \left(\mathbf{x}\right).
**icov2** : int
Determines which of the preset variograms to use. The choices are given below. Note that :math:x^{\prime } = \left\lVert \frac{x}{\ell_1},\frac{y}{\ell_2}\right\rVert, where :math:\ell_1 and :math:\ell_2 are correlation lengths in the :math:x and :math:y directions respectively and are parameters for most of the variograms, and :math:\sigma^2 is the variance specified by :math:\mathrm{var}.
:math:\mathrm{icov2} = 1
Symmetric stable variogram
.. math::
\gamma \left(\mathbf{x}\right) = \sigma^2\mathrm{exp}\left(-\left(x^{\prime }\right)^{\nu }\right)\text{,}
where
:math:\ell_1 = \mathrm{params}[0], :math:\ell_1 > 0,
:math:\ell_2 = \mathrm{params}[1], :math:\ell_2 > 0,
:math:\nu = \mathrm{params}[2], :math:0 < \nu \leq 2.
:math:\mathrm{icov2} = 2
Cauchy variogram
.. math::
\gamma \left(\mathbf{x}\right) = \sigma^2\left(1+\left(x^{\prime }\right)^2\right)^{{-\nu }}\text{,}
where
:math:\ell_1 = \mathrm{params}[0], :math:\ell_1 > 0,
:math:\ell_2 = \mathrm{params}[1], :math:\ell_2 > 0,
:math:\nu = \mathrm{params}[2], :math:\nu > 0.
:math:\mathrm{icov2} = 3
Differential variogram with compact support
.. math::
\gamma \left(\mathbf{x}\right) = \left\{\begin{array}{cc} \sigma^2 \left(1+8x^{\prime }+25\left(x^{\prime }\right)^2+32\left(x^{\prime }\right)^3\right) \left(1-x^{\prime }\right)^8 \text{,} & x^{\prime } < 1 \text{,} \\ 0 \text{,} & x^{\prime } \geq 1 \text{,} \end{array}\right.
where
:math:\ell_1 = \mathrm{params}[0], :math:\ell_1 > 0,
:math:\ell_2 = \mathrm{params}[1], :math:\ell_2 > 0.
:math:\mathrm{icov2} = 4
Exponential variogram
.. math::
\gamma \left(\mathbf{x}\right) = \sigma^2\mathrm{exp}\left(-x^{\prime }\right)\text{,}
where
:math:\ell_1 = \mathrm{params}[0], :math:\ell_1 > 0,
:math:\ell_2 = \mathrm{params}[1], :math:\ell_2 > 0.
:math:\mathrm{icov2} = 5
Gaussian variogram
.. math::
\gamma \left(\mathbf{x}\right) = \sigma^2\mathrm{exp}\left({-\left(x^{\prime }\right)}^2\right)\text{,}
where
:math:\ell_1 = \mathrm{params}[0], :math:\ell_1 > 0,
:math:\ell_2 = \mathrm{params}[1], :math:\ell_2 > 0.
:math:\mathrm{icov2} = 6
Nugget variogram
.. math::
\gamma \left(\mathbf{x}\right) = \left\{\begin{array}{cc}\sigma^2\text{,}&\mathbf{x} = 0\text{,}\\0\text{,}&\mathbf{x}\neq 0\text{.}\end{array}\right.
No parameters need be set for this value of :math:\mathrm{icov2}.
:math:\mathrm{icov2} = 7
Spherical variogram
.. math::
\gamma \left(\mathbf{x}\right) = \left\{\begin{array}{cc} \sigma^2 \left(1-1.5x^{\prime }+0.5\left(x^{\prime }\right)^3\right) \text{,} & x^{\prime } < 1 \text{,} \\0\text{,}& x^{\prime } \geq 1 \text{,} \end{array}\right.
where
:math:\ell_1 = \mathrm{params}[0], :math:\ell_1 > 0,
:math:\ell_2 = \mathrm{params}[1], :math:\ell_2 > 0.
:math:\mathrm{icov2} = 8
Bessel variogram
.. math::
\gamma \left(\mathbf{x}\right) = \sigma^2\frac{{2^{\nu }\Gamma \left(\nu +1\right)J_{\nu }\left(x^{\prime }\right)}}{\left(x^{\prime }\right)^{\nu }}\text{,}
where
:math:J_{\nu }\left(·\right) is the Bessel function of the first kind,
:math:\ell_1 = \mathrm{params}[0], :math:\ell_1 > 0,
:math:\ell_2 = \mathrm{params}[1], :math:\ell_2 > 0,
:math:\nu = \mathrm{params}[2], :math:\nu \geq 0.
:math:\mathrm{icov2} = 9
Hole effect variogram
.. math::
\gamma \left(\mathbf{x}\right) = \sigma^2\frac{\sin\left(x^{\prime }\right)}{x^{\prime }}\text{,}
where
:math:\ell_1 = \mathrm{params}[0], :math:\ell_1 > 0,
:math:\ell_2 = \mathrm{params}[1], :math:\ell_2 > 0.
:math:\mathrm{icov2} = 10
Whittle-Matérn variogram
.. math::
\gamma \left(\mathbf{x}\right) = \sigma^2\frac{{2^{{1-\nu }}\left(x^{\prime }\right)^{\nu }K_{\nu }\left(x^{\prime }\right)}}{{\Gamma \left(\nu \right)}}\text{,}
where
:math:K_{\nu }\left(·\right) is the modified Bessel function of the second kind,
:math:\ell_1 = \mathrm{params}[0], :math:\ell_1 > 0,
:math:\ell_2 = \mathrm{params}[1], :math:\ell_2 > 0,
:math:\nu = \mathrm{params}[2], :math:\nu > 0.
:math:\mathrm{icov2} = 11
Continuously parameterised variogram with compact support
.. math::
\gamma \left(\mathbf{x}\right) = \left\{\begin{array}{cc} \sigma^2 \frac{{2^{{1-\nu }}\left(x^{\prime }\right)^{\nu }K_{\nu }\left(x^{\prime }\right)}}{{\Gamma \left(\nu \right)}} \left(1+8x^{{\prime \prime }}+25\left(x^{{\prime \prime }}\right)^2+32\left(x^{{\prime \prime }}\right)^3\right)\left(1-x^{{\prime \prime }}\right)^8\text{,}&x^{{\prime \prime }} < 1\text{,}\\0\text{,}&x^{{\prime \prime }}\geq 1\text{,} \end{array}\right.
where
:math:x^{{\textit{′′}}} = \left\lVert \frac{x^{\prime }}{{\ell_1s_1}},\frac{y^{\prime }}{{\ell_2s_2}}\right\rVert,
:math:K_{\nu }\left(·\right) is the modified Bessel function of the second kind,
:math:\ell_1 = \mathrm{params}[0], :math:\ell_1 > 0,
:math:\ell_2 = \mathrm{params}[1], :math:\ell_2 > 0,
:math:s_1 = \mathrm{params}[2], :math:s_1 > 0,
:math:s_2 = \mathrm{params}[3], :math:s_2 > 0,
:math:\nu = \mathrm{params}[4], :math:\nu > 0.
:math:\mathrm{icov2} = 12
Generalized hyperbolic distribution variogram
.. math::
\gamma \left(\mathbf{x}\right) = \sigma^2\frac{\left(\delta^2+\left(x^{\prime }\right)^2\right)^{\frac{\lambda }{2}}}{{\delta^{\lambda }K_{\lambda }\left(\kappa \delta \right)}}K_{\lambda }\left(\kappa \left(\delta^2+\left(x^{\prime }\right)^2\right)^{\frac{1}{2}}\right)\text{,}
where
:math:K_{\lambda }\left(·\right) is the modified Bessel function of the second kind,
:math:\ell_1 = \mathrm{params}[0], :math:\ell_1 > 0,
:math:\ell_2 = \mathrm{params}[1], :math:\ell_2 > 0,
:math:\lambda = \mathrm{params}[2], no constraint on :math:\lambda,
:math:\delta = \mathrm{params}[3], :math:\delta > 0,
:math:\kappa = \mathrm{params}[4], :math:\kappa > 0.
**params** : float, array-like, shape :math:\left(\textit{np}\right)
The parameters for the variogram as detailed in the description of :math:\mathrm{icov2}.
**norm** : int, optional
Determines which norm to use when calculating the variogram.
:math:\mathrm{norm} = 1
The 1-norm is used, i.e., :math:\left\lVert x,y\right\rVert = \left\lvert x\right\rvert +\left\lvert y\right\rvert.
:math:\mathrm{norm} = 2
The 2-norm (Euclidean norm) is used, i.e., :math:\left\lVert x,y\right\rVert = \sqrt{x^2+y^2}.
Determines whether the embedding matrix is padded with zeros, or padded with values of the variogram. The choice of padding may affect how big the embedding matrix must be in order to be positive semidefinite.
:math:\mathrm{pad} = 0
The embedding matrix is padded with zeros.
:math:\mathrm{pad} = 1
The embedding matrix is padded with values of the variogram.
**icorr** : int, optional
Determines which approximation to implement if required, as described in :ref:Notes <g05zr-py2-py-notes>.
**Returns**
**lam** : float, ndarray, shape :math:\left(:\right)
Contains the square roots of the eigenvalues of the embedding matrix.
**xx** : float, ndarray, shape :math:\left(:\right)
The points of the :math:x-coordinates at which values of the random field will be output.
**yy** : float, ndarray, shape :math:\left(:\right)
The points of the :math:y-coordinates at which values of the random field will be output.
**m** : int, ndarray, shape :math:\left(2\right)
:math:\mathrm{m}[0] contains :math:M_1, the size of the circulant blocks and :math:\mathrm{m}[1] contains :math:M_2, the number of blocks, resulting in a final square matrix of size :math:M_1\times M_2.
**approx** : int
Indicates whether approximation was used.
:math:\mathrm{approx} = 0
No approximation was used.
:math:\mathrm{approx} = 1
Approximation was used.
**rho** : float
Indicates the scaling of the covariance matrix. :math:\mathrm{rho} = 1.0 unless approximation was used with :math:\mathrm{icorr} = 0 or :math:1.
**icount** : int
Indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
**eig** : float, ndarray, shape :math:\left(3\right)
Indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. :math:\mathrm{eig}[0] contains the smallest eigenvalue, :math:\mathrm{eig}[1] contains the sum of the squares of the negative eigenvalues, and :math:\mathrm{eig}[2] contains the sum of the absolute values of the negative eigenvalues.
.. _g05zr-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{ns} = \left[\langle\mathit{\boldsymbol{value}}\rangle, \langle\mathit{\boldsymbol{value}}\rangle\right].
Constraint: :math:\mathrm{ns}[0]\geq 1, :math:\mathrm{ns}[1]\geq 1.
(errno :math:2)
On entry, :math:\mathrm{xmin} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{xmax} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{xmin} < \mathrm{xmax}.
(errno :math:4)
On entry, :math:\mathrm{ymin} = \langle\mathit{\boldsymbol{value}}\rangle and :math:\mathrm{ymax} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{ymin} < \mathrm{ymax}.
(errno :math:6)
On entry, :math:\mathrm{maxm} = \left[\langle\mathit{\boldsymbol{value}}\rangle, \langle\mathit{\boldsymbol{value}}\rangle\right].
Constraint: the minimum calculated value for :math:\mathrm{maxm} are :math:\left[\langle\mathit{\boldsymbol{value}}\rangle, \langle\mathit{\boldsymbol{value}}\rangle\right].
(errno :math:7)
On entry, :math:\mathrm{var} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{var}\geq 0.0.
(errno :math:8)
On entry, :math:\mathrm{icov2} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{icov2}\geq 1 and :math:\mathrm{icov2}\leq 12.
(errno :math:9)
On entry, :math:\mathrm{norm} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{norm} = 1 or :math:2.
(errno :math:10)
On entry, :math:\textit{np} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: for :math:\mathrm{icov2} = \langle\mathit{\boldsymbol{value}}\rangle, :math:\textit{np} = \langle\mathit{\boldsymbol{value}}\rangle.
(errno :math:11)
On entry, :math:\mathrm{params}[\langle\mathit{\boldsymbol{value}}\rangle] = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: dependent on :math:\mathrm{icov2}, see documentation.
(errno :math:12)
On entry, :math:\mathrm{pad} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{pad} = 0 or :math:1.
(errno :math:13)
On entry, :math:\mathrm{icorr} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{icorr} = 0, :math:1 or :math:2.
.. _g05zr-py2-py-notes:
**Notes**
A two-dimensional random field :math:Z\left(\mathbf{x}\right) in :math:\mathbb{R}^2 is a function which is random at every point :math:\mathbf{x} \in \mathbb{R}^2, so :math:Z\left(\mathbf{x}\right) is a random variable for each :math:\mathbf{x}.
The random field has a mean function :math:\mu \left(\mathbf{x}\right) = 𝔼\left[Z\left(\mathbf{x}\right)\right] and a symmetric positive semidefinite covariance function :math:C\left(\mathbf{x}, \mathbf{y}\right) = 𝔼\left[\left(Z\left(\mathbf{x}\right)-\mu \left(\mathbf{x}\right)\right)\left(Z\left(\mathbf{y}\right)-\mu \left(\mathbf{y}\right)\right)\right]. :math:Z\left(\mathbf{x}\right) is a Gaussian random field if for any choice of :math:n \in ℕ and :math:\mathbf{x}_1,\ldots,\mathbf{x}_n \in \mathbb{R}^2, the random vector :math:\left[Z\left(\mathbf{x}_1\right),\ldots,Z\left(\mathbf{x}_n\right)\right]^\mathrm{T} follows a multivariate Normal distribution, which would have a mean vector :math:\tilde{\mu } with entries :math:\tilde{\mu }_i = \mu \left(\mathbf{x}_i\right) and a covariance matrix :math:\tilde{C} with entries :math:\tilde{C}_{{ij}} = C\left(\mathbf{x}_i, \mathbf{x}_j\right).
A Gaussian random field :math:Z\left(\mathbf{x}\right) is stationary if :math:\mu \left(\mathbf{x}\right) is constant for all :math:\mathbf{x} \in \mathbb{R}^2 and :math:C\left(\mathbf{x}, \mathbf{y}\right) = C\left({\mathbf{x}+\mathbf{a}}, {\mathbf{y}+\mathbf{a}}\right) for all :math:\left. \mathbf{x}, \mathbf{y}, \mathbf{a}\right. \in \mathbb{R}^2 and hence we can express the covariance function :math:C\left(\mathbf{x}, \mathbf{y}\right) as a function :math:\gamma of one variable: :math:C\left(\mathbf{x}, \mathbf{y}\right) = \gamma \left(\mathbf{x}-\mathbf{y}\right). :math:\gamma is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor :math:\sigma^2 representing the variance such that :math:\gamma \left(0\right) = \sigma^2.
The functions field_2d_predef_setup and :meth:field_2d_generate are used to simulate a two-dimensional stationary Gaussian random field, with mean function zero and variogram :math:\gamma \left(\mathbf{x}\right), over a domain :math:\left[x_{\textit{min}}, x_{\textit{max}}\right]\times \left[y_{\textit{min}}, y_{\textit{max}}\right], using an equally spaced set of :math:N_1\times N_2 points; :math:N_1 points in the :math:x-direction and :math:N_2 points in the :math:y-direction.
The problem reduces to sampling a Gaussian random vector :math:\mathbf{X} of size :math:N_1\times N_2, with mean vector zero and a symmetric covariance matrix :math:A, which is an :math:N_2\times N_2 block Toeplitz matrix with Toeplitz blocks of size :math:N_1\times N_1.
Since :math:A is in general expensive to factorize, a technique known as the circulant embedding method is used. :math:A is embedded into a larger, symmetric matrix :math:B, which is an :math:M_2\times M_2 block circulant matrix with circulant blocks of size :math:M_1\times M_1, where :math:M_1\geq 2\left(N_1-1\right) and :math:M_2\geq 2\left(N_2-1\right). :math:B can now be factorized as :math:B = W\Lambda W^* = R^*R, where :math:W is the two-dimensional Fourier matrix (:math:W^* is the complex conjugate of :math:W), :math:\Lambda is the diagonal matrix containing the eigenvalues of :math:B and :math:R = \Lambda^{\frac{1}{2}}W^*. :math:B is known as the embedding matrix.
The eigenvalues can be calculated by performing a discrete Fourier transform of the first row (or column) of :math:B and multiplying by :math:M_1\times M_2, and so only the first row (or column) of :math:B is needed -- the whole matrix does not need to be formed.
As long as all of the values of :math:\Lambda are non-negative (i.e., :math:B is positive semidefinite), :math:B is a covariance matrix for a random vector :math:\mathbf{Y} which has :math:M_2 blocks of size :math:M_1.
Two samples of :math:\mathbf{Y} can now be simulated from the real and imaginary parts of :math:R^*\left(\mathbf{U}+i\mathbf{V}\right), where :math:\mathbf{U} and :math:\mathbf{V} have elements from the standard Normal distribution.
Since :math:R^*\left(\mathbf{U}+i\mathbf{V}\right) = W\Lambda^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right), this calculation can be done using a discrete Fourier transform of the vector :math:\Lambda^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right).
Two samples of the random vector :math:\mathbf{X} can now be recovered by taking the first :math:N_1 elements of the first :math:N_2 blocks of each sample of :math:\mathbf{Y} -- because the original covariance matrix :math:A is embedded in :math:B, :math:\mathbf{X} will have the correct distribution.
If :math:B is not positive semidefinite, larger embedding matrices :math:B can be tried; however if the size of the matrix would have to be larger than :math:\mathrm{maxm}, an approximation procedure is used.
We write :math:\Lambda = \Lambda_++\Lambda_-, where :math:\Lambda_+ and :math:\Lambda_- contain the non-negative and negative eigenvalues of :math:B respectively.
Then :math:B is replaced by :math:\rho B_+ where :math:B_+ = W\Lambda_+W^* and :math:\rho \in \left(0, 1\right] is a scaling factor.
The error :math:\epsilon in approximating the distribution of the random field is given by
.. math::
\epsilon = \sqrt{\frac{{\left(1-\rho \right)^2\mathrm{trace}\left(\Lambda \right)+\rho^2\mathrm{trace}\left(\Lambda_-\right)}}{M}}\text{.}
Three choices for :math:\rho are available, and are determined by the input argument :math:\mathrm{icorr}:
setting :math:\mathrm{icorr} = 0 sets
.. math::
\rho = \frac{\mathrm{trace}\left(\Lambda \right)}{\mathrm{trace}\left(\Lambda_+\right)}\text{,}
setting :math:\mathrm{icorr} = 1 sets
.. math::
\rho = \sqrt{\frac{\mathrm{trace}\left(\Lambda \right)}{\mathrm{trace}\left(\Lambda_+\right)}}\text{,}
setting :math:\mathrm{icorr} = 2 sets :math:\rho = 1.
field_2d_predef_setup finds a suitable positive semidefinite embedding matrix :math:B and outputs its sizes in the vector :math:\mathrm{m} and the square roots of its eigenvalues in :math:\mathrm{lam}.
If approximation is used, information regarding the accuracy of the approximation is output.
Note that only the first row (or column) of :math:B is actually formed and stored.
.. _g05zr-py2-py-references:
**References**
Dietrich, C R and Newsam, G N, 1997, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM J. Sci. Comput. (18), 1088--1107
Schlather, M, 1999, Introduction to positive definite functions and to unconditional simulation of random fields, Technical Report ST 99--10, Lancaster University
Wood, A T A and Chan, G, 1997, Algorithm AS 312: An Algorithm for Simulating Stationary Gaussian Random Fields, Journal of the Royal Statistical Society, Series C (Applied Statistics) (Volume 46) (1), 171--181
"""
raise NotImplementedError
[docs]def field_2d_generate(ns, s, m, lam, rho, statecomm):
r"""
field_2d_generate produces realizations of a stationary Gaussian random field in two dimensions, using the circulant embedding method.
The square roots of the eigenvalues of the extended covariance matrix (or embedding matrix) need to be input, and can be calculated using :meth:field_2d_user_setup or :meth:field_2d_predef_setup.
.. _g05zs-py2-py-doc:
For full information please refer to the NAG Library document for g05zs
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05zsf.html
.. _g05zs-py2-py-parameters:
**Parameters**
**ns** : int, array-like, shape :math:\left(2\right)
The number of sample points to use in each direction, with :math:\mathrm{ns}[0] sample points in the :math:x-direction and :math:\mathrm{ns}[1] sample points in the :math:y-direction. The total number of sample points on the grid is, therefore, :math:\mathrm{ns}[0]\times \mathrm{ns}[1]. This must be the same value as supplied to :meth:field_2d_user_setup or :meth:field_2d_predef_setup when calculating the eigenvalues of the embedding matrix.
**s** : int
:math:S, the number of realizations of the random field to simulate.
**m** : int, array-like, shape :math:\left(2\right)
Indicates the size, :math:M, of the embedding matrix as returned by :meth:field_2d_user_setup or :meth:field_2d_predef_setup. The embedding matrix is a block circulant matrix with circulant blocks. :math:\mathrm{m}[0] is the size of each block, and :math:\mathrm{m}[1] is the number of blocks.
**lam** : float, array-like, shape :math:\left(\mathrm{m}[0]\times \mathrm{m}[1]\right)
Contains the square roots of the eigenvalues of the embedding matrix, as returned by :meth:field_2d_user_setup or :meth:field_2d_predef_setup.
**rho** : float
Indicates the scaling of the covariance matrix, as returned by :meth:field_2d_user_setup or :meth:field_2d_predef_setup.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**z** : float, ndarray, shape :math:\left(:, \mathrm{s}\right)
Contains the realizations of the random field. The :math:k\ th realization (where :math:k = 1,2,\ldots,\mathrm{s}) of the random field on the two-dimensional grid :math:\left(x_i, y_j\right) is stored in :math:\mathrm{z}[ \left(j-1\right)\times \mathrm{ns}[0] +i -1,k-1], for :math:i = 1,2,\ldots,\mathrm{ns}[0] and for :math:j = 1,2,\ldots,\mathrm{ns}[1]. The points are returned in :math:\textit{xx} and :math:\textit{yy} by :meth:field_2d_user_setup or :meth:field_2d_predef_setup.
.. _g05zs-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{ns} = \left[\langle\mathit{\boldsymbol{value}}\rangle, \langle\mathit{\boldsymbol{value}}\rangle\right].
Constraint: :math:\mathrm{ns}[0]\geq 1, :math:\mathrm{ns}[1]\geq 1.
(errno :math:2)
On entry, :math:\mathrm{s} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{s}\geq 1.
(errno :math:3)
On entry, :math:\mathrm{m} = \left[\langle\mathit{\boldsymbol{value}}\rangle, \langle\mathit{\boldsymbol{value}}\rangle\right], and :math:\mathrm{ns} = \left[\langle\mathit{\boldsymbol{value}}\rangle, \langle\mathit{\boldsymbol{value}}\rangle\right].
Constraints: :math:\mathrm{m}[i-1]\geq \mathrm{max}\left(1, {2\left(\mathrm{ns}[i-1]\right)-1}\right), for :math:i = 1,2.
(errno :math:4)
On entry, at least one element of :math:\mathrm{lam} was negative.
Constraint: all elements of :math:\mathrm{lam} must be non-negative.
(errno :math:5)
On entry, :math:\mathrm{rho} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0.0 < \mathrm{rho}\leq 1.0.
(errno :math:6)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05zs-py2-py-notes:
**Notes**
A two-dimensional random field :math:Z\left(\mathbf{x}\right) in :math:\mathbb{R}^2 is a function which is random at every point :math:\mathbf{x} \in \mathbb{R}^2, so :math:Z\left(\mathbf{x}\right) is a random variable for each :math:\mathbf{x}.
The random field has a mean function :math:\mu \left(\mathbf{x}\right) = 𝔼\left[Z\left(\mathbf{x}\right)\right] and a symmetric positive semidefinite covariance function :math:C\left(\mathbf{x}, \mathbf{y}\right) = 𝔼\left[\left(Z\left(\mathbf{x}\right)-\mu \left(\mathbf{x}\right)\right)\left(Z\left(\mathbf{y}\right)-\mu \left(\mathbf{y}\right)\right)\right]. :math:Z\left(\mathbf{x}\right) is a Gaussian random field if for any choice of :math:n \in ℕ and :math:\mathbf{x}_1,\ldots,\mathbf{x}_n \in \mathbb{R}^2, the random vector :math:\left[Z\left(\mathbf{x}_1\right),\ldots,Z\left(\mathbf{x}_n\right)\right]^\mathrm{T} follows a multivariate Normal distribution, which would have a mean vector :math:\tilde{\mu } with entries :math:\tilde{\mu }_i = \mu \left(\mathbf{x}_i\right) and a covariance matrix :math:\tilde{C} with entries :math:\tilde{C}_{{ij}} = C\left(\mathbf{x}_i, \mathbf{x}_j\right).
A Gaussian random field :math:Z\left(\mathbf{x}\right) is stationary if :math:\mu \left(\mathbf{x}\right) is constant for all :math:\mathbf{x} \in \mathbb{R}^2 and :math:C\left(\mathbf{x}, \mathbf{y}\right) = C\left({\mathbf{x}+\mathbf{a}}, {\mathbf{y}+\mathbf{a}}\right) for all :math:\left. \mathbf{x}, \mathbf{y}, \mathbf{a}\right. \in \mathbb{R}^2 and hence we can express the covariance function :math:C\left(\mathbf{x}, \mathbf{y}\right) as a function :math:\gamma of one variable: :math:C\left(\mathbf{x}, \mathbf{y}\right) = \gamma \left(\mathbf{x}-\mathbf{y}\right). :math:\gamma is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor :math:\sigma^2 representing the variance such that :math:\gamma \left(0\right) = \sigma^2.
The functions :meth:field_2d_user_setup or :meth:field_2d_predef_setup along with field_2d_generate are used to simulate a two-dimensional stationary Gaussian random field, with mean function zero and variogram :math:\gamma \left(\mathbf{x}\right), over a domain :math:\left[x_{\textit{min}}, x_{\textit{max}}\right]\times \left[y_{\textit{min}}, y_{\textit{max}}\right], using an equally spaced set of :math:N_1\times N_2 points; :math:N_1 points in the :math:x-direction and :math:N_2 points in the :math:y-direction.
The problem reduces to sampling a Gaussian random vector :math:\mathbf{X} of size :math:N_1\times N_2, with mean vector zero and a symmetric covariance matrix :math:A, which is an :math:N_2\times N_2 block Toeplitz matrix with Toeplitz blocks of size :math:N_1\times N_1.
Since :math:A is in general expensive to factorize, a technique known as the circulant embedding method is used. :math:A is embedded into a larger, symmetric matrix :math:B, which is an :math:M_2\times M_2 block circulant matrix with circulant bocks of size :math:M_1\times M_1, where :math:M_1\geq 2\left(N_1-1\right) and :math:M_2\geq 2\left(N_2-1\right). :math:B can now be factorized as :math:B = W\Lambda W^* = R^*R, where :math:W is the two-dimensional Fourier matrix (:math:W^* is the complex conjugate of :math:W), :math:\Lambda is the diagonal matrix containing the eigenvalues of :math:B and :math:R = \Lambda^{\frac{1}{2}}W^*. :math:B is known as the embedding matrix.
The eigenvalues can be calculated by performing a discrete Fourier transform of the first row (or column) of :math:B and multiplying by :math:M_1\times M_2, and so only the first row (or column) of :math:B is needed -- the whole matrix does not need to be formed.
The symmetry of :math:A as a block matrix, and the symmetry of each block of :math:A, depends on whether the covariance function :math:\gamma is even or not. :math:\gamma is even if :math:\gamma \left(\mathbf{x}\right) = \gamma \left(-\mathbf{x}\right) for all :math:\mathbf{x} \in \mathbb{R}^2, and uneven otherwise (in higher dimensions, :math:\gamma can be even in some coordinates and uneven in others, but in two dimensions :math:\gamma is either even in both coordinates or uneven in both coordinates).
If :math:\gamma is even then :math:A is a symmetric block matrix and has symmetric blocks; if :math:\gamma is uneven then :math:A is not a symmetric block matrix and has non-symmetric blocks.
In the uneven case, :math:M_1 and :math:M_2 are set to be odd in order to guarantee symmetry in :math:B.
As long as all of the values of :math:\Lambda are non-negative (i.e., :math:B is positive semidefinite), :math:B is a covariance matrix for a random vector :math:\mathbf{Y} which has :math:M_2 'blocks' of size :math:M_1.
Two samples of :math:\mathbf{Y} can now be simulated from the real and imaginary parts of :math:R^*\left(\mathbf{U}+i\mathbf{V}\right), where :math:\mathbf{U} and :math:\mathbf{V} have elements from the standard Normal distribution.
Since :math:R^*\left(\mathbf{U}+i\mathbf{V}\right) = W\Lambda^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right), this calculation can be done using a discrete Fourier transform of the vector :math:\Lambda^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right).
Two samples of the random vector :math:\mathbf{X} can now be recovered by taking the first :math:N_1 elements of the first :math:N_2 blocks of each sample of :math:Y -- because the original covariance matrix :math:A is embedded in :math:B, :math:\mathbf{X} will have the correct distribution.
If :math:B is not positive semidefinite, larger embedding matrices :math:B can be tried; however if the size of the matrix would have to be larger than :math:\textit{maxm}, an approximation procedure is used.
See the documentation of :meth:field_2d_user_setup or :meth:field_2d_predef_setup for details of the approximation procedure.
field_2d_generate takes the square roots of the eigenvalues of the embedding matrix :math:B, and its size vector :math:M, as input and outputs :math:S realizations of the random field in :math:Z.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to field_2d_generate.
.. _g05zs-py2-py-references:
**References**
Dietrich, C R and Newsam, G N, 1997, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM J. Sci. Comput. (18), 1088--1107
Schlather, M, 1999, Introduction to positive definite functions and to unconditional simulation of random fields, Technical Report ST 99--10, Lancaster University
Wood, A T A and Chan, G, 1994, Simulation of stationary Gaussian processes in :math:\left[0, 1\right]^d, Journal of Computational and Graphical Statistics (3(4)), 409--432
"""
raise NotImplementedError
[docs]def field_fracbm_generate(ns, s, xmax, h, lam, rho, statecomm):
r"""
field_fracbm_generate produces realizations of a fractional Brownian motion, using the circulant embedding method.
The square roots of the eigenvalues of the extended covariance matrix (or embedding matrix) need to be input, and can be calculated using :meth:field_1d_predef_setup.
.. _g05zt-py2-py-doc:
For full information please refer to the NAG Library document for g05zt
https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/g05/g05ztf.html
.. _g05zt-py2-py-parameters:
**Parameters**
**ns** : int
The number of steps (points) to be generated in realizations of the increments of the fractional Brownian motion. This must be the same value as supplied to :meth:field_1d_predef_setup when calculating the eigenvalues of the embedding matrix.
Note: in the context of fractional Brownian motion, :math:\mathrm{ns} represents the number of steps from a zero starting state. Realizations returned in :math:\mathrm{z} include this starting state and so :math:\mathrm{ns}+1 values are returned for each realization.
**s** : int
:math:S, the number of realizations of the fractional Brownian motion to simulate.
**xmax** : float
The upper bound for the interval over which the fractional Brownian motion is to be simulated, as input to :meth:field_1d_user_setup or :meth:field_1d_predef_setup.
**h** : float
The Hurst parameter, :math:H, for the fractional Brownian motion. This must be the same value as supplied to :meth:field_1d_predef_setup in :math:{\textit{params}}[0], when the eigenvalues of the embedding matrix were calculated.
**lam** : float, array-like, shape :math:\left(m\right)
Contains the square roots of the eigenvalues of the embedding matrix, as returned by :meth:field_1d_user_setup or :meth:field_1d_predef_setup.
**rho** : float
Indicates the scaling of the covariance matrix, as returned by :meth:field_1d_user_setup or :meth:field_1d_predef_setup.
**statecomm** : dict, RNG communication object, modified in place
RNG communication structure.
This argument must have been initialized by a prior call to :meth:init_repeat or :meth:init_nonrepeat.
**Returns**
**z** : float, ndarray, shape :math:\left(\mathrm{ns}+1, \mathrm{s}\right)
Contains the realizations of the fractional Brownian motion, :math:Z. The :math:\textit{j}\ th realization, for the :math:\textit{i}\ th point :math:\mathrm{xx}[\textit{i}-1], is stored in :math:\mathrm{z}[\textit{i}-1,\textit{j}-1], for :math:\textit{i} = 1,2,\ldots,\mathrm{ns}+1, for :math:\textit{j} = 1,2,\ldots,\mathrm{s}.
**xx** : float, ndarray, shape :math:\left(\mathrm{ns}+1\right)
The points at which values of the fractional Brownian motion are output. The first point is always zero, and the subsequent :math:\mathrm{ns} points represent the equispaced steps towards the last point, :math:\mathrm{xmax}. Note that in :meth:field_1d_user_setup and :meth:field_1d_predef_setup, the returned :math:\textit{ns} sample points are the mid-points of the grid returned in :math:\mathrm{xx} here.
.. _g05zt-py2-py-errors:
**Raises**
**NagValueError**
(errno :math:1)
On entry, :math:\mathrm{ns} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{ns}\geq 1.
(errno :math:2)
On entry, :math:\mathrm{s} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{s}\geq 1.
(errno :math:3)
On entry, :math:m = \langle\mathit{\boldsymbol{value}}\rangle, and :math:\mathrm{ns} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:m\geq \mathrm{max}\left(1, {2\left(\mathrm{ns}-1\right)}\right).
(errno :math:4)
On entry, :math:\mathrm{xmax} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:\mathrm{xmax} > 0.0.
(errno :math:5)
On entry, :math:\mathrm{h} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0.0 < \mathrm{h} < 1.0.
(errno :math:6)
On entry, at least one element of :math:\mathrm{lam} was negative.
Constraint: all elements of :math:\mathrm{lam} must be non-negative.
(errno :math:7)
On entry, :math:\mathrm{rho} = \langle\mathit{\boldsymbol{value}}\rangle.
Constraint: :math:0.0 < \mathrm{rho}\leq 1.0.
(errno :math:8)
On entry, :math:\mathrm{statecomm}\ ['state'] vector has been corrupted or not initialized.
.. _g05zt-py2-py-notes:
**Notes**
The functions :meth:field_1d_predef_setup and field_fracbm_generate are used to simulate a fractional Brownian motion process with Hurst parameter :math:H over an interval :math:\left[0, x_{\textit{max}}\right], using a set of equally spaced points.
Fractional Brownian motion itself cannot be simulated directly using this method, since it is not a stationary Gaussian random field; however its increments can be simulated like a stationary Gaussian random field.
The circulant embedding method is described in the documentation for :meth:field_1d_predef_setup.
field_fracbm_generate takes the square roots of the eigenvalues of the embedding matrix as returned by :meth:field_1d_predef_setup when :math:{\textit{icov1}} = 14, and its size :math:M, as input and outputs :math:S realizations of the fractional Brownian motion in :math:Z.
One of the initialization functions :meth:init_repeat (for a repeatable sequence if computed sequentially) or :meth:init_nonrepeat (for a non-repeatable sequence) must be called prior to the first call to field_fracbm_generate.
.. _g05zt-py2-py-references:
**References**
Dietrich, C R and Newsam, G N, 1997, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM J. Sci. Comput. (18), 1088--1107
Schlather, M, 1999, Introduction to positive definite functions and to unconditional simulation of random fields, Technical Report ST 99--10, Lancaster University
Wood, A T A and Chan, G, 1994, Simulation of stationary Gaussian processes in :math:\left[0, 1\right]^d, Journal of Computational and Graphical Statistics (3(4)), 409--432
"""
raise NotImplementedError
|
2021-10-23 15:46:29
|
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|
http://www.algebra.com/algebra/homework/Radicals/Radicals.faq.question.374883.html
|
SOLUTION: Hello, I think I am confused on this whole radical equation thing. The problem I have is : {{{28-4sqrt( 2 )=24 sqrt( 2 )}}} First I have to get one of the radical terms isolate
Algebra -> Algebra -> Radicals -> SOLUTION: Hello, I think I am confused on this whole radical equation thing. The problem I have is : {{{28-4sqrt( 2 )=24 sqrt( 2 )}}} First I have to get one of the radical terms isolate Log On
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Algebra: Radicals -- complicated equations involving roots Solvers Lessons Answers archive Quiz In Depth
Question 374883: Hello, I think I am confused on this whole radical equation thing. The problem I have is : First I have to get one of the radical terms isolated, but in this case one is already isolated correct? Next is using the principle of powers so I would square both sides, this would give me . Is this correct thus far? Sorry, but I'm lost.Answer by solver91311(16897) (Show Source): You can put this solution on YOUR website! You have a bunch of stuff going on here, and none of it is correct. In the first place you cannot simply square parts of your equation and ignore the rest of it. Hence, it is NOT correct to say that if then . Now, given , it would be correct to perform the following operation: which works out to When you FOIL and collect terms in the LHS. BUT -- that is all for naught because you began with a false statement, namely it is NOT true that Add to both sides resulting in Which is clearly a false statement. If you divide both sides by 28 you know that the square root of 2 is certainly not equal to 1. I'm concerned that you may have left something out of the original problem statement. John My calculator said it, I believe it, that settles it
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2013-05-25 20:17:10
|
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8268582224845886, "perplexity": 447.9487120598512}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368706298270/warc/CC-MAIN-20130516121138-00081-ip-10-60-113-184.ec2.internal.warc.gz"}
|
https://www.luthierdirectory.co.uk/portable-astrogrep-11232-crack-free-3264bit/
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Product Reviews::Music Reviews
# Portable AstroGrep 11232 Crack Free [32|64bit]
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Highlighted keywords in the file preview area
Portable AstroGrep features powerful searching functions, which allow you to find the desired file by certain keywords. It searches for the words both in the file name, as well as in its contents and highlights the results in the text.
You may perform simple searches and detect files by their names, as well as filter the task by inserting file extensions. The supported file types are HTML, TXT, Java, JSP, ASP, JS, INC, SQL, BAS and several source code formats, including VB, CS, CPP, C, H or ASM.
The results are displayed in a table, along with the file names, path, extension, date of modification and
## Portable AstroGrep 11232 Crack [Mac/Win]
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Read More….
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## Portable AstroGrep 11232 Crack+ Free
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Find the probability that one or both particles are found in the ground state
Consider a particle with mass $m$ moving in one dimension under the influence of a uniform force $F$ (we’ll assume $F$ is constant so that the motion is 1-dimensional). This is given by the Lagrangian $L$:
$$L = \frac{1}{2} m v^2 – F x$$
where $x$ and $v$ are the positions and velocities of the particle respectively.
Suppose this particle is prepared in the initial state $|\psi_0\rangle$
$$|\psi_0\rangle = \int \frac{dx}{\sqrt{2\pi}} e^{i k x} \left(\phi_{v,0}(x) + \phi_{v,1}(x)\right) \,.$$
where $k$ is an arbitrary wavevector and $\phi_{v,0}(x)$ and $\phi_{v,1}(x)$ are the ground and first excited state wave functions respectively.
I have a single particle in the ground state, and I want to find the probability that a measurement of the particle’s position gives some specific value $x_0$. I know that there will be a nonzero probability of finding the particle in the ground state if $x_0$ is such that the potential energy of the ground state is lower than the potential energy of the first excited state.
The question I’m having trouble with is this: if the potential energy of the ground state is higher than the potential energy of the first excited state, how do I find the probability of finding the particle in the ground state?
What I
## What’s New In?
Portable AstroGrep is a simple to use tool designed to facilitate searching for files in Windows. The program allows you to create searching filters, by inclusion or exclusion, and specify the file extension, contained text or host folder. The application also allows you to preview the files in its interface.
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Portable AstroGrep features powerful searching functions, which allow you to find the desired file by certain keywords. It searches for the words both in the file name, as well as in its contents and highlights the results in the text.
You may perform simple searches and detect files by their names, as well as filter the task by inserting file extensions. The supported file types are HTML, TXT, Java, JSP, ASP, JS, INC, SQL, BAS and several source code formats, including VB, CS, CPP, C, H or ASM.
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Portable AstroGrep is a simple to use tool designed to facilitate searching for files in Windows. The program allows you to create searching filters, by inclusion or exclusion, and specify the file extension, contained text or host folder. The application also allows you to preview the files in its interface.
Highlighted keywords in the file preview area
Portable AstroGrep features powerful searching functions, which allow you to find the desired file by certain keywords. It searches for the words both in the file name, as well as in its contents and highlights the results in the text.
You may perform simple searches and detect files by their names, as well as filter the task by inserting file extensions. The supported file types are HTML, TXT, Java, JSP, ASP, JS, INC, SQL, BAS and several source code formats, including VB, CS, CPP, C, H or ASM.
The
## System Requirements For Portable AstroGrep:
1. Data size:
A. Game data:
B. Expansion data:
C. Image data:
D. Audio:
2. Display resolution:
3. GPU Requirements:
4. OS requirements:
5. Internet Connectivity
6. Language Requirements
7. Region Requirements
8. System Requirements
9. Contact
B. Expansion data:
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2022-08-18 02:16:05
|
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2757505774497986, "perplexity": 2535.774033008044}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573145.32/warc/CC-MAIN-20220818003501-20220818033501-00497.warc.gz"}
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https://projecteuclid.org/euclid.die/1356050357
|
## Differential and Integral Equations
### Multiple positive solutions for classes of elliptic systems with combined nonlinear effects
#### Abstract
We study the existence of multiple positive solutions to systems of the form \begin{equation*} \begin{cases} \qquad-{\Delta} u ={\lambda} f(v), & \text{ in }{\Omega},\\ \qquad-{\Delta} v ={\lambda} g(u), & \text{ in }{\Omega},\\ \qquad\quad~~ u=0=v, & \text{ on }{\partial}{\Omega}. \end{cases} \end{equation*} Here ${\Delta}$ is the Laplacian operator, ${\lambda}$ is a positive parameter, ${\Omega}$ is a bounded domain in ${\mathbb{R}^N}$ with smooth boundary and $f, g$ belong to a class of positive functions that have a combined sublinear effect at $\infty$. Our results also easily extend to the corresponding p-Laplacian systems. We prove our results by the method of sub and super solutions.
#### Article information
Source
Differential Integral Equations, Volume 19, Number 6 (2006), 669-680.
Dates
First available in Project Euclid: 21 December 2012
https://projecteuclid.org/euclid.die/1356050357
Mathematical Reviews number (MathSciNet)
MR2234718
Zentralblatt MATH identifier
1212.35162
Subjects
Primary: 35J55
Secondary: 35J60: Nonlinear elliptic equations
#### Citation
Ali, Jaffar; Shivaji, R.; Ramaswamy, Mythily. Multiple positive solutions for classes of elliptic systems with combined nonlinear effects. Differential Integral Equations 19 (2006), no. 6, 669--680. https://projecteuclid.org/euclid.die/1356050357
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2018-09-23 10:45:06
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|
http://archive.numdam.org/item/M2AN_1981__15_3_265_0/
|
Discrete forms of Friedrichs' inequalities in the finite element method
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 15 (1981) no. 3, pp. 265-286.
@article{M2AN_1981__15_3_265_0,
author = {\v Zen\'\i \v sek, Alexander},
title = {Discrete forms of Friedrichs' inequalities in the finite element method},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {265--286},
publisher = {Centrale des revues, Dunod-Gauthier-Villars},
volume = {15},
number = {3},
year = {1981},
zbl = {0475.65072},
mrnumber = {631681},
language = {en},
url = {http://archive.numdam.org/item/M2AN_1981__15_3_265_0/}
}
Ženíšek, Alexander. Discrete forms of Friedrichs' inequalities in the finite element method. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 15 (1981) no. 3, pp. 265-286. http://archive.numdam.org/item/M2AN_1981__15_3_265_0/
[1] J. H. Bramble, M. Zlämal, Triangular elements in the finite element method, Math. Comp. 24(1970), 809-820. | MR 282540 | Zbl 0226.65073
[2] P. G. Ciarlet, P. A. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods. In : The Mathematical Foundations of the Finite Element Method with Applications to Partial Diffe-rential Equations (A. K. Aziz, Editor), Academic Press, New York, 1972, pp. 409-474. | MR 421108 | Zbl 0262.65070
[3] P. G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. | MR 520174 | Zbl 0383.65058
[4] J. Hrebicek, A numerical analysis of a general biharmonic problem by the finite element method. (To appear.) | Zbl 0541.65072
[5] L Mansfield, Approximation of the boundary in the finite element solution of fourth order problems SIAM J Numer Anal 15 (1978), 568-579 | MR 471373 | Zbl 0391.65047
[6] J Necas, Les méthodes directes en théorie des équations elliptiques Academia, Prague, 1967 | MR 227584
[7] M Zlamal, Curved elements in the finite element method I SIAM J Numer Anal 10 (1973), 229-240 | MR 395263 | Zbl 0285.65067
[8] M Zlamal, Curved elements in the finite element method II SIAM J Numer Anal 11 (1974), 347-362 | MR 343660 | Zbl 0277.65064
[9] A Zenisek, Curved triangular finite ${C}^{m}$-elements Apl Mat 23 (1978), 346-377 | MR 502072 | Zbl 0404.35041
[10] A Zenisek, Nonhomogeneous boundary conditions and curved triangular finite elements Apl Mat 26(1981), 121-141 | MR 612669 | Zbl 0475.65073
|
2021-04-23 17:41:18
|
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|
https://tex.stackexchange.com/questions/246898/how-to-draw-interlocking-tori-with-tikz
|
# How to draw interlocking tori with Tikz?
I need to draw a couple of interlocking tori, as in the picture, using Tikz.
I've been looking at some examples online and all I find is instructions to do it with Gnuplot, nothing about Tikz. If anyone could help I would deeply appreciate it.
• No exact matches, but tex.stackexchange.com/questions/70090/… is a good start. May 25, 2015 at 21:38
• Draw a smooth surface is another relevant post. No Tikz solution, but more than Gnuplot.
– DJP
May 27, 2015 at 15:33
• This is pretty much a PStricks, Asymptote and so on task. May 27, 2015 at 17:28
• Sep 28, 2015 at 7:54
You can plot 4 half tori like this :
\documentclass{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.12}
\pgfplotsset{
torus/.style 2 args={
surf,
color=#1!50,faceted color=#1,
samples=17,
z buffer=sort,
domain=0:360, y domain=#2:#2+180
}
}
\def\m{sin(x)}
\def\n{(2+cos(x))*sin(y)}
\def\p{(2+cos(x))*cos(y)}
\begin{document}
\begin{tikzpicture}
\begin{axis}[hide axis,axis equal,scale=3,view={20}{20}]
\end{axis}
\end{tikzpicture}
\end{document}
• To make the tori look "circular" rather than "elliptic", you can use the unit vector ratio=1 1 1 option or axis equal. Nov 28, 2015 at 23:46
Run with xelatex:
\documentclass[pstricks]{standalone}
\usepackage{pst-solides3d}
\begin{document}
\psset{Decran=50,viewpoint=20 80 30,lightsrc=viewpoint,action=none}
\begin{pspicture}[solidmemory](-4,-3)(3,3)
\psSolid[r1=2.5,r0=1.5,object=tore,ngrid=18 36,fillcolor=green!30,name=tA]
\psSolid[r1=2.5,r0=1.5,object=tore,ngrid=18 36,fillcolor=blue!30,RotX=90,name=tB](2,0,0)
\psSolid[object=fusion,base=tA tB,action=draw**]
\end{pspicture}
\end{document}
A bit late to the party, but here's a solution using asymptote.
/* Used for rendering parameterized 3D objects. */
import graph3;
/* PDF works best with LaTeX, output this. Also set the render factor high. */
import settings;
settings.outformat = "pdf";
settings.render = 8;
/* Size of the image. For 3D objects it seems best to have this set to a *
* power of 2, otherwise weird vertical or horizontal black lines may appear.*/
size(256);
/* How the image is being drawn on a 2D picture. */
currentprojection = perspective(5.0, 4.0, 4.0);
/* Two radii defining the torus. */
real R = 3.0;
real a = 1.3;
/* Material the two torii are made of. */
material blueblob = material(
diffusepen = blue + 0.25*green,
emissivepen = gray(0.2),
specularpen = gray(0.2)
);
material redblob = material(
diffusepen = red,
emissivepen = gray(0.2),
specularpen = gray(0.2)
);
/* Function for drawing the torus. */
triple torus_parameterization(pair t)
{
/* The parameterization is in terms of sine and cosine of 2 pi t.x and *
* 2 pi t.y. Precompute these to avoid repetitive calculations. */
real u = 2.0*pi*t.x;
real v = 2.0*pi*t.y;
real cosu = cos(u);
real cosv = cos(v);
real sinu = sin(u);
real sinv = sin(v);
/* Given the two angles u and v, the x, y, and z coordinates are: */
real x = (R + a*cosv)*cosu;
real y = (R + a*cosv)*sinu;
real z = a*sinv;
/* Return the point (x, y, z), which is a point on the surface. */
triple out = (x, y, z);
return out;
}
/* End of torus_parameterization. */
/* Create the first torus. */
surface t0 = surface(torus_parameterization, (0.0, 0.0), (1.0, 1.0), Spline);
/* The second torus is obtained by rotating and shifting. */
surface t1 = shift((R, 0.0, 0.0))*(rotate(90.0, (1.0, 0.0, 0.0))*t0);
/* Draw both of the torii. */
draw(t0, surfacepen = redblob, render(merge=true));
draw(t1, surfacepen = blueblob, render(merge=true));
|
2022-08-14 15:32:52
|
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|
https://codegolf.stackexchange.com/questions/58615/1-2-fizz-4-buzz/74655
|
# 1, 2, Fizz, 4, Buzz
## Introduction
In our recent effort to collect catalogues of shortest solutions for standard programming exercises, here is PPCG's first ever vanilla FizzBuzz challenge. If you wish to see other catalogue challenges, there is "Hello World!" and "Is this number a prime?".
## Challenge
Write a program that prints the decimal numbers from 1 to 100 inclusive. But for multiples of three print “Fizz” instead of the number and for the multiples of five print “Buzz”. For numbers which are multiples of both three and five print “FizzBuzz”.
## Output
The output will be a list of numbers (and Fizzes, Buzzes and FizzBuzzes) separated by a newline (either \n or \r\n). A trailing newline is acceptable, but a leading newline is not. Apart from your choice of newline, the output should look exactly like this:
1
2
Fizz
4
Buzz
Fizz
7
8
Fizz
Buzz
11
Fizz
13
14
FizzBuzz
16
17
Fizz
19
Buzz
Fizz
22
23
Fizz
Buzz
26
Fizz
28
29
FizzBuzz
31
32
Fizz
34
Buzz
Fizz
37
38
Fizz
Buzz
41
Fizz
43
44
FizzBuzz
46
47
Fizz
49
Buzz
Fizz
52
53
Fizz
Buzz
56
Fizz
58
59
FizzBuzz
61
62
Fizz
64
Buzz
Fizz
67
68
Fizz
Buzz
71
Fizz
73
74
FizzBuzz
76
77
Fizz
79
Buzz
Fizz
82
83
Fizz
Buzz
86
Fizz
88
89
FizzBuzz
91
92
Fizz
94
Buzz
Fizz
97
98
Fizz
Buzz
The only exception to this rule is constant output of your language's interpreter that cannot be suppressed, such as a greeting, ANSI color codes or indentation.
## Further Rules
• This is not about finding the language with the shortest approach for playing FizzBuzz, this is about finding the shortest approach in every language. Therefore, no answer will be marked as accepted.
• Submissions are scored in bytes in an appropriate preexisting encoding, usually (but not necessarily) UTF-8. Some languages, like Folders, are a bit tricky to score--if in doubt, please ask on Meta.
• Nothing can be printed to STDERR.
• Feel free to use a language (or language version) even if it's newer than this challenge. If anyone wants to abuse this by creating a language where the empty program generates FizzBuzz output, then congrats for paving the way for a very boring answer.
Note that there must be an interpreter so the submission can be tested. It is allowed (and even encouraged) to write this interpreter yourself for a previously unimplemented language.
• If your language of choice is a trivial variant of another (potentially more popular) language which already has an answer (think BASIC or SQL dialects, Unix shells or trivial Brainfuck derivatives like Alphuck and ???), consider adding a note to the existing answer that the same or a very similar solution is also the shortest in the other language.
• Because the output is fixed, you may hardcode the output (but this may not be the shortest option).
• You may use preexisting solutions, as long as you credit the original author of the program.
• Standard loopholes are otherwise disallowed.
As a side note, please don't downvote boring (but valid) answers in languages where there is not much to golf; these are still useful to this question as it tries to compile a catalogue as complete as possible. However, do primarily upvote answers in languages where the authors actually had to put effort into golfing the code.
## Catalogue
var QUESTION_ID=58615;var ANSWER_FILTER="!t)IWYnsLAZle2tQ3KqrVveCRJfxcRLe";var COMMENT_FILTER="!)Q2B_A2kjfAiU78X(md6BoYk";var OVERRIDE_USER=30525;var answers=[],answers_hash,answer_ids,answer_page=1,more_answers=true,comment_page;function answersUrl(index){return"https://api.stackexchange.com/2.2/questions/"+QUESTION_ID+"/answers?page="+index+"&pagesize=100&order=desc&sort=creation&site=codegolf&filter="+ANSWER_FILTER}function commentUrl(index,answers){return"https://api.stackexchange.com/2.2/answers/"+answers.join(';')+"/comments?page="+index+"&pagesize=100&order=desc&sort=creation&site=codegolf&filter="+COMMENT_FILTER}function getAnswers(){jQuery.ajax({url:answersUrl(answer_page++),method:"get",dataType:"jsonp",crossDomain:true,success:function(data){answers.push.apply(answers,data.items);answers_hash=[];answer_ids=[];data.items.forEach(function(a){a.comments=[];var id=+a.share_link.match(/\d+/);answer_ids.push(id);answers_hash[id]=a});if(!data.has_more)more_answers=false;comment_page=1;getComments()}})}function getComments(){jQuery.ajax({url:commentUrl(comment_page++,answer_ids),method:"get",dataType:"jsonp",crossDomain:true,success:function(data){data.items.forEach(function(c){if(c.owner.user_id===OVERRIDE_USER)answers_hash[c.post_id].comments.push(c)});if(data.has_more)getComments();else if(more_answers)getAnswers();else process()}})}getAnswers();var SCORE_REG=/<h\d>\s*([^\n,<]*(?:<(?:[^\n>]*>[^\n<]*<\/[^\n>]*>)[^\n,<]*)*),.*?(\d+)(?=[^\n\d<>]*(?:<(?:s>[^\n<>]*<\/s>|[^\n<>]+>)[^\n\d<>]*)*<\/h\d>)/;var OVERRIDE_REG=/^Override\s*header:\s*/i;function getAuthorName(a){return a.owner.display_name}function process(){var valid=[];answers.forEach(function(a){var body=a.body;a.comments.forEach(function(c){if(OVERRIDE_REG.test(c.body))body='<h1>'+c.body.replace(OVERRIDE_REG,'')+'</h1>'});var match=body.match(SCORE_REG);if(match)valid.push({user:getAuthorName(a),size:+match[2],language:match[1],link:a.share_link,});else console.log(body)});valid.sort(function(a,b){var aB=a.size,bB=b.size;return aB-bB});var languages={};var place=1;var lastSize=null;var lastPlace=1;valid.forEach(function(a){if(a.size!=lastSize)lastPlace=place;lastSize=a.size;++place;var answer=jQuery("#answer-template").html();answer=answer.replace("{{PLACE}}",lastPlace+".").replace("{{NAME}}",a.user).replace("{{LANGUAGE}}",a.language).replace("{{SIZE}}",a.size).replace("{{LINK}}",a.link);answer=jQuery(answer);jQuery("#answers").append(answer);var lang=a.language;lang=jQuery('<a>'+lang+'</a>').text();languages[lang]=languages[lang]||{lang:a.language,lang_raw:lang.toLowerCase(),user:a.user,size:a.size,link:a.link}});var langs=[];for(var lang in languages)if(languages.hasOwnProperty(lang))langs.push(languages[lang]);langs.sort(function(a,b){if(a.lang_raw>b.lang_raw)return 1;if(a.lang_raw<b.lang_raw)return-1;return 0});for(var i=0;i<langs.length;++i){var language=jQuery("#language-template").html();var lang=langs[i];language=language.replace("{{LANGUAGE}}",lang.lang).replace("{{NAME}}",lang.user).replace("{{SIZE}}",lang.size).replace("{{LINK}}",lang.link);language=jQuery(language);jQuery("#languages").append(language)}}
body{text-align:left!important}#answer-list{padding:10px;width:290px;float:left}#language-list{padding:10px;width:290px;float:left}table thead{font-weight:700}table td{padding:5px}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script> <link rel="stylesheet" type="text/css" href="//cdn.sstatic.net/codegolf/all.css?v=83c949450c8b"> <div id="language-list"> <h2>Shortest Solution by Language</h2> <table class="language-list"> <thead> <tr><td>Language</td><td>User</td><td>Score</td></tr> </thead> <tbody id="languages"> </tbody> </table> </div> <div id="answer-list"> <h2>Leaderboard</h2> <table class="answer-list"> <thead> <tr><td></td><td>Author</td><td>Language</td><td>Size</td></tr> </thead> <tbody id="answers"> </tbody> </table> </div> <table style="display: none"> <tbody id="answer-template"> <tr><td>{{PLACE}}</td><td>{{NAME}}</td><td>{{LANGUAGE}}</td><td>{{SIZE}}</td><td><a href="{{LINK}}">Link</a></td></tr> </tbody> </table> <table style="display: none"> <tbody id="language-template"> <tr><td>{{LANGUAGE}}</td><td>{{NAME}}</td><td>{{SIZE}}</td><td><a href="{{LINK}}">Link</a></td></tr> </tbody> </table>
• Nothing can be printed to STDERR. Is this true only when running, or also when compiling (assuming that is a separate step?) – AShelly Sep 24 '15 at 20:47
• @AShelly Only when running – Beta Decay Sep 24 '15 at 20:48
• I’m not sure I like the fact that you hardcoded the 100 into the challenge. That way, a program that just generates the expected output is a valid entry, but is not interesting for this challenge. I think the challenge should expect the program to input the number of items to output. – Timwi Sep 24 '15 at 23:28
• @Timwi While I agree that it would make it (only slightly) more interesting, I've very often seen FizzBuzz as strictly 1 to 100 (on Wikipedia and Rosetta Code, for example). If the goal is to have a "canonical" FB challenge, it makes sense. – Geobits Sep 25 '15 at 0:50
• A "vanilla fizzbuzz" sounds delicious. – Reinstate Monica -- notmaynard Sep 25 '15 at 15:12
## Hoon, 126 bytes
%+
turn
(gulf [1 100])
|=
a/@
=+
[=((mod a 3) 0) =((mod a 5) 0)]
"{?:(-< "Fizz" "")}{?:(-> "Buzz" "")}{?:(=(- [| |]) <a> "")}"
Map over the list [1...100] with the function, interpolating the strings "Fizz" and "Buzz". If both mods are false, then it also interpolates the number.
This uses the fact that =+ pushes the value to the top of the context, which you can access with - and navigate with -< or -> for head/tail. Unfortunately, it looks pretty ugly because it needs a newline after runes to minimize byte count, along with not having built in operator functions.
I'm not entirely sure if this counts as a valid entry: It simply returns a list of strings to be printed by the shell, which is optimal. The other way would be to use ~& to print each element as it's mapped over, but it would still be rendered as "Fizz" or "Buzz" (with quotes) since it's a typed print, along with the shell then printing out the entire list anyways since it's the return value.
# Mathematica, 103 86 73 bytes
With 30 bytes saved thanks to @A Simmons!
f="Fizz";b="Buzz";Range@100/.{x_/;15∣x->f<>b,x_/;3∣x->f,x_/;5∣x->b}
• You can use g=Divisible instead of the function definition, there's a leading space (and another one right after f<>b,), and you can use x~g~15 instead of x~g~3&&x~g~5. So f="Fizz";b="Buzz";g=Divisible;Range@100/.{x_/;x~g~15->f<>b,x_/;x~g~3->f,x_/;x~g~5->b} – CalculatorFeline Mar 2 '16 at 16:28
• You can use f="Fizz";b="Buzz";Range@100/.{x_/;15∣x->f<>b,x_/;3∣x->f,x_/;5∣x->b} for 73 bytes in 67 characters. – A Simmons Mar 2 '16 at 17:51
• CatsAreFluffy, It turns out that your code gives incorrect answers for 5 and 100, for example. – DavidC Mar 2 '16 at 22:28
• It's a pair of zero width characters right after the g. Here's code for correcting that: FromCharacterCode[ToCharacterCode@#/.{8204|8203->(##&[])}]& (Don't worry, no zero widths here.) – CalculatorFeline Mar 2 '16 at 22:34
declare option output:method "text";string-join(for$x in 1 to 100 return if($x mod 15=0)then"FizzBuzz"else if($x mod 3=0)then"Fizz"else if($x mod 5=0)then"Buzz"else $x," ") There were only 7 XQuery answers on the whole site, I thought it could at least have its FizzBuzz ! Granted it's not very golfy, in particular when you need to add a 36 bytes preface so that it does not output an XML header. I tested it with Saxon-HE's command-line XQuery tool (java net.sf.saxon.Query fizzbuzz.xq), with which I had to replace the w3-defined option declaration with declare option saxon:output "method=text";. ## Javascript, 191 bytes for(var i=1; i<=100; i++){ var r = ""; if( i%15 == 0 ? r = "FizzBuzz" : (i%5 == 0 ? r = "Buzz" : (i%3 == 0 ? r = "Fizz" : r = i)) ){ console.log(r); } } • Welcome to Programming Puzzles & Code Golf. Nice first answer! But please wrap your code into a code-block (indent with 4 spaces or use the button in the editor). Also you should format the header correctly, the common template for this here is ##<language>, <byte count> bytes. Also you have some unnecessary whitespaces in your code which you should remove. – Denker Mar 10 '16 at 9:59 • Nice answer, but the goal of this challenge is to make your answer as short as possible. You can save a ton of bytes by removing all whitespace, and renaming result to r. – ETHproductions Mar 11 '16 at 20:32 # Tcl, 136 bytes set f {Fizz 3 Buzz 5} while {[incr n]<=100} {set s "" foreach {m d}$f {if {$n%$d==0} {append s $m}} if {$s eq ""} {puts $n} {puts$s}}
This solution, incidentally, is easily extensible to any combination of multiples. See The Smart Person's Mirage golf, where gnibbler posted the same idea (but in Python).
set iterations 100
set fizzies {
Fizz 3
Jazz 4
Buzz 5
}
while {[incr n] <= $iterations} { set s "" foreach {name divisor}$fizzies {
if {$n %$divisor == 0} {append s $name} } if {$s eq ""} {puts $n} {puts$s}
}
# Oration, 98 bytes
literally, for i in range(1,100):x=""if i%3 else'Fizz';x+=""if i%5 else "Buzz";print x if x else i
## ROOP, 187 bytes
1
V! !<
(102) 1|
e#r3##r5# a|
#H # # Y-<
N N !
"Fizz""Buzz"
mX mX
### V--> !
A V---->
#
' C
V'e "\n"
|# M #
<V# #v
C A C
#X ##
w
O#
I will try to explain each section of code:
V! !< Add 1 to each number that goes to the left of the a
1| and sends it to the bottom of the V
a|
-<
(102) The 102 falls to the left of the e and each number
e that passes over is compared to 102.
#H If a number is equal then the H runs and ends the program
#r3# With each number that goes above the r
# the remainder of dividing by 3 is obtained.
N The N returns 1 if the number is 0, and 0 otherwise.
"Fizz" The string "Fizz" falls and moves to the left of the m.
The number is multiplied by the string
mX ("" or "Fizz" if it is 0 or 1 respectively)
### The X removes the number when it moves to the right
The same is done with 5 and "Buzz"
mX mX
V--> Both strings are concatenated with the A
A getting "", "Fizz", "Buzz" or "FizzBuzz"
#
' C The C changes the direction of advance of string, to the left.
'e At the same time the "e" compares the string with the empty string.
# The single quotes are a vertical literal string.
V The V and pipes redirects the string to the right of the C
| that changes the direction again in order that comes to the left of the A
<V
C A
Y The original number is converted to a string with Y.
!
! Pipes and teleporters (!) Redirects the string to below the V
V---->
M The string falls to the right of the M and multiplies
# # with the number previously obtained by the e
A The result is above the A
"\n" The string "\n" falls on the v which makes a copy
M # whenever there is a space below.
v The C changes the direction of the string so it goes to the left.
A C It waits to the right of the A
A The A concatenate all 3 strings, the result is on the w that sends it
#X ## to the O representing the output. At the same time the X deletes the string.
w
O#
I hope it is comprehensible, English is not my main language.
# Nim, 10076 73 bytes
for i in 1..100:echo max(["Fizz","",""][i%%3]&["Buzz",""][ord i%%5>0],$i) Hm... still trying to learn Nim, and I'm thinking there's got to be a better way... • ["Fizz",""][i*i%%3] saves 1. Using ord on a boolean is a useful tip, thanks for that. – primo Jun 19 '19 at 10:02 # Befunge, 65 bytes _1+::3%^>55+,:"c"#@ >"zuB"vv.#,,:,,< |!:%5\ _:!"ziF"^ <,,:,,<:|* Try it online! # Valyrio, 13 bytes s∫main [CF] This is a fairly basic (and slightly unimaginative) answer. ## Explanation C pushes 100 to the stack, which means that ... F is the FizzBuzz builtin. This was mainly added in as a basic stack based program but got left in as a command and I never got rid of it. # Javascript, 56 bytes for(f=0;f++<100;alert(f%5?b||f:b+'Buzz'))b=f%3?'':'Fizz' Assuming 100 alerts is an acceptable output method. • Nice answer, but it's already been posted (That one originally used alert as well, but it was changed to console.log because all the other JS answers do the same) – ETHproductions Mar 22 '17 at 21:46 # Go, 162158145143142 139 bytes package main;import."fmt";func main(){for i,p:=1,Println;i<101;i++{s:="";if i%3<1{s+="Fizz"};if i%5<1{s+="Buzz"};if s!=""{p(s)}else{p(i)}}} Go Playground Link # Cardinal 217 bytes %x>+ v >+++++~M! 8# "buzz" 0 #~0#+++~>M! # V "fizz" + ^jM< ~ V># + V > #xV = 0 V t ~ >#x t t v V * = > #}#} / > ^ >}/ . Try it Online ## Explanation %x 0 + + = 0 t ~ t t * = > ^ Initializes a pointer with an inactive value of 100 and active value of 0 >+ v # ^jM< Loops around, incrementing the active value of the pointer and sending out a duplicate until the active value is equal to the inactive value (100) >+++++~M! ~0#+++~>M! Checks if the value mod 3 or mod 5 is equal to 0 # "buzz" # # V "fizz" ~ V># V > #xV V >#x v V > #}#} / >}/ . Uses reflectors and splitters in order to print only numbers that are not divisible by 3 or 5. Printing fizz when divisible by 3 and buzz when divisible by 5. # Julia, 70 bytes for i=1:100 println("$(i%3<1?"Fizz":"")$(i%5<1?"Buzz":i%3>0?i:"")")end Julia has perl-like string formatting. Very nice for challenges like these. ## Swift, 97 bytes for i in 1...100{i%15==0 ?print("FizzBuzz"):i%3==0 ?print("Fizz"):i%5==0 ?print("Buzz"):print(i)} # q/kdb+, 5856 49 bytes Solution: 0{$[sum i:0=y mod 3 5;FizzBuzz(&)i;y]}'1+(!)100
Example:
q)0{$[sum i:0=y mod 3 5;FizzBuzz(&)i;y]}'1+(!)100 1 2 ,Fizz 4 ,Buzz ,Fizz 7 8 ,Fizz ,Buzz 11 ,Fizz 13 14 FizzBuzz 16 ...etc Explanation: 0{$[sum i:0=y mod 3 5;FizzBuzz where i;y]}'1+til 100 / ungolfed
{ }' / anonymous function that takes each-left and each-right
0 / this would be parameter 'x' but we dont use it
til 100 / til generates a list of 0..99
1+ / adds 1 to every item in the list, thus 1..100
$[ ; ; ] / switch,$[condition;true;false]
y mod 3 5 / modulo operation on input for 3 and 5, mod[1;3 5] = 1 1
0= / is 0 equal to this result (basically a 'not' operation)
i: / save in i for later
sum / add these, will get 0, 1 or 2. 0 is interpretted as false
where i / where gives indices where i is true
FizzBuzz / 2 item list which gets indexed into (and implicitly returned)
y / return the input if the condition was false
Notes:
This ^^ is pretty much a q version of the k solution, so I've written in a different way.. unfortunately it's about 50% slower :(
0{(Fizz;Buzz;y)(&)(0=a),all a:mod[y;3 5]}'1+(!)100
Here we are indexing into a list of Fizz, Buzz, based on the result of the modulo operation... The k solution style is better.
# groovy, 55 bytes
Borrows idea from answer by @feersum https://codegolf.stackexchange.com/a/58623
100.times{println'Fizz'*(it%3/2)+'Buzz'*(it%5/4)?:it+1}
• Welcome to PPCG! :) – Beta Decay Sep 24 '17 at 7:21
# tinylisp repl, 130 bytes
(d f(q((n # %)(i(e n 100)(q Buzz)(i(disp(i #(i % n(q Buzz))(i %(q Fizz)(q FizzBuzz))))0(f(a n 1)(i #(s # 1)2)(i %(s % 1)4
(f 1 2 4
Try it online! (Note that the repl auto-completes parentheses at the ends of lines; these have been filled in on the TIO version.)
### Explanation
It took me a while to realize I could do FizzBuzz in tinylisp, since the language doesn't have strings. However, it does have the Name type, which can be any run of non-whitespace, non-parenthesis characters; and it also has the disp command, which outputs a value followed by a newline. Together with some creative abuse of recursion, we can get the desired output.
We define a function f with three arguments:
• n is the counter. We will start at 1 and recurse until 100.
• # is the number of iterations till the next divisible-by-3 number. (Mnemonic: Shift-3)
• % is the number of iterations till the next divisible-by-5 number. (Mnemonic: Shift-5)
The first thing f does is check whether n is 100. If so, we return Buzz. This value gets passed all the way up the call stack and printed at the end.
If not, we need to use disp to print either the number, Fizz, Buzz, or FizzBuzz, depending on the values of # and %:
• If # and % are both nonzero (truthy), the number is not divisible by 3 or 5. Display the number, n.
• If # is nonzero but % is zero, the number is divisible by 5 but not 3. Display Buzz.
• If # is zero but % is nonzero, the number is divisible by 3 but not 5. Display Fizz.
• If # and % are both zero, the number is divisible by 3 and 5. Display FizzBuzz.
Since tinylisp does not have an equivalent to Common Lisp's progn, disp'ing something and then returning a value requires a little trickery. Since disp always returns (), which is falsey, we can use the disp call as the condition of an if (i). The true branch will never be evaluated (0 is a convenient placeholder), and we can put the actual return value in the false branch.
This return value is a recursive call. We add 1 to n, and we subtract 1 from # and % unless they are zero, in which case we reset them to (3-1) or (5-1), respectively.
Ungolfed, using the standard library for long names of builtins (TIO):
(load library)
(def fizz-buzz
(lambda (counter steps-till-fizz steps-till-buzz)
(if (equal? counter 100)
(q Buzz)
(if
(disp
(if steps-till-fizz
(if steps-till-buzz counter (q Buzz))
(if steps-till-buzz (q Fizz) (q FizzBuzz))))
(comment Return val of disp is always falsey, so the true branch is never executed)
(fizz-buzz
(if steps-till-fizz (sub2 steps-till-fizz 1) 2)
(if steps-till-buzz (sub2 steps-till-buzz 1) 4))))))
(fizz-buzz 1 2 4)
• Ken Thompson: "Thompson:the original LISP... you know I think it's a horrible language. I really do. But, I was struck with the idea of defining very, very low level semantics,... it defines its own interpreter. It's always been a problem when you describe a language to say what constructs it recognizes and what they actually do and [Lisp] was the cleanest, simplest, most recursive, beautiful semantics of a language I've ever seen. Probably even to this day. But, unfortunately, what it describes I think is just a horrible language." tinylisp is a bit less scary IMHO! – roblogic Aug 28 '19 at 9:41
# SmileBASIC, 70 bytes
Printing "Fizz" and "Buzz" is easy, the slightly more difficult part is to only print the number when required. There are basically 2 ways to do this (and they end up being the same length)
1: Print the number when I isn't divisible by 3 or 5
FOR I=0TO 100A=I MOD 3B=I MOD 5?"Fizz"*!A;"Buzz"*!B;STR$(I)*(A&&B)NEXT 2: Print the number if the cursor is at column 0: FOR I=1TO 100?"Fizz"*!(I MOD 3);"Buzz"*!(I MOD 3); ?STR$(I)*!CSRX
NEXT
In a previous version of SB, % was used for MOD, making the program shorter:
# Petit Computer BASIC, 62 bytes
FOR I=0TO 100?"Fizz"*!(I%3);"Buzz"*!(I%5);
?STR$(I)*!CSRX NEXT • The language name should probably be "SmileBASIC 2.x" or something. – snail_ Feb 6 '17 at 13:39 # KoopaScript, 250 characters def i 1 if \%va is \%vu set a 1;setath b \%va %% 3;setath c \%va %% 5;setath e \%va %% 15;if \%vb is 0 if \%vc not 0 print Fizz;if \%vb not 0 if \%vc is 0 print Buzz;if \%ve is 0 print FizzBuzz;if \%vb not 0 if \%vc not 0 print \%va;setath a \%va + 1 Guide to reading: Don't. Either learn KoopaScript more or less entirely by looking at the code (my documentation isn't that great) or just take my word for it that it works. By the way, KoopaScript is an interpreted language I made (not specifically for this) that runs inside ActionScript 2, and doesn't actually have else statements, or... most of the stuff that makes other examples short. All functions are one line, so this was pretty easy. Here's the GitHub repo. • This looks like quite the painful programming language to use haha. Welcome to PPCG! :) – James Mar 8 '18 at 19:55 • Thanks! Yeah, it is a bit painful, but I can't really be bothered to add stuff like arrays. I made a pi approximator and a prime number generator (both very slow), both were quite painful because of the lack of useful functions. – Jhynjhiruu Rekrap Mar 8 '18 at 20:45 # sed, 275272270260254249 245 bytes s/.*/t0u123456789/ :1 s/(tu?(.).*)/\1\n\2/ s/u(.)(t?)(.*)/\1\2u\3\1/ /9u/{s/t(.)/\1t/;s/u//;s/^/u/} /99/!b1 s/$/\n10/
s/[0-9][05]/&Buzz/g
s/[0369]{2}/&Fizz/g
s/[147][258]/&Fizz/g
s/[258][147]/&Fizz/g
s/[0-9]+([FB])/\1/g
s/\n0/\n/g
s/[^\n]+\n//
q
Try it Online
This is a pure sed script which discards all input, if any, and then prints all the necessary output lines and quits.
# Explanation
The script is divided into a sequence of three main parts: 1. Numeral generation; 2. "Fizz/Buzz/FizzBuzz" insertion; and 3. Formatting.
1. Numeral generation (lines 1 through 6)
The purpose of this part is to generate 99 lines containing the base 10 numerals corresponding to numbers 1 through 99. For that, we first set the entire pattern space to the following string:
t0u123456789
We will call the above string our state string. Next, we enter a loop in which each iteration goes like this:
1. A newline is appended to the pattern space;
2. A copy of the first numeral character after the "t" in the state string is appended to the pattern space;
3. A copy of the first numeral character after the "u" in the state string is appended to the pattern space;
4. The "u" in the state string is moved from its current position to the immediate right of the first numeral character after it or, if a "t" is already in said position, the "u" is instead moved to the immediate right of that "t";
5. If the "u" in the state string is immediately at the right of the "9" in the state string, the "t" is moved from its current position to the immediate right of the first character after it, and the "u" is moved from its current position to the immediate left of the "0" in the state string;
6. If there isn't a "99" anywhere in the pattern space (i.e., the loop has not finished its job of generating all the 99 lines), control goes back to line 2 (label :1) and so this enumeration of procedures is repeated from step 1; otherwise, control flow continues into the next line.
2. "Fizz/Buzz/FizzBuzz" insertion (lines 7 through 11)
The purpose of this part is to append "Fizz" immediately after each two-digit numeral in the pattern space which corresponds to a number which is a multiple of 3 but not of 5; to append "Buzz" immediately after each two-digit numeral in the pattern space which corresponds to a number which is a multiple of 5 but not of 3; and to append "FizzBuzz" after each two-digit numeral in the pattern space which corresponds to a number which is a multiple of both 3 and 5. This is how the computations go:
1. We append a newline followed by "10" to the pattern space.
2. Next, we search for all substrings of the pattern space formed by a digit between 0-9 on the left and either a 0 or a 5 on the right. We insert "Buzz" into the pattern space immediately after each such substring;
3. Finally, we insert "Fizz" into the pattern space immediately after each substring which matches either of the following criteria:
• Substrings formed by two digits which may be 0, 3, 6, or 9;
• Substrings formed by a digit which is either 1, 4 or 7 on the left and a digit which is either 2, 5 or 8 on the right;
• Substrings formed by a digit which is either 2, 5 or 8 on the left and a digit which is either 1, 4 or 7 on the right.
3. Formatting (lines 12 through 15)
This part is straight forward. Here we remove all of the following substrings from the pattern space:
1. Every contiguous sequence of numeral characters on the immediate left of a "F" or a "B";
2. All 0's on the immediate right of a newline character;
3. All characters from the beginning of the pattern space up to and including the first newline character (remember that the state string is still there and there's a newline immediately before the first output numeral).
And then sed just prints the final contents of the pattern space and calls it quits.
# ///, 198 bytes
/%/!"
//$/ "!//#/ Fizz" //"/Buzz//!/ Fizz /1 2!4$7
8%11!13
14#16
17!19$22 23%26!28 29#31 32!34$37
38%41!43
44#46
47!49$52 53%56!58 59#61 62!64$67
68%71!73
74#76
77!79$82 83%86!88 89#91 92!94$97
98!"
with Ada.Text_IO;use Ada.Text_IO;procedure T is begin for I in Integer range 1..100 loop if I mod 15 = 0 then Put_Line("FizzBuzz");elsif I mod 3 = 0 then Put_Line ("Fizz");elsif I mod 5 = 0 then Put_Line ("Buzz");else Put_Line (Integer'Image (I)(2 ..Integer'Image(I)'Last));end if; end loop; end T;
Try it online!
Ungolfed:
with Ada.Text_IO;use Ada.Text_IO;
procedure Test is begin
for I in Integer range 1 .. 100 loop
if I mod 15 = 0 then
Put_Line ("FizzBuzz");
elsif I mod 3 = 0 then
Put_Line ("Fizz");
elsif I mod 5 = 0 then
Put_Line ("Buzz");
else
Put_Line (Integer'Image (I)(2 .. Integer'Image(I)'Last));
end if;
end loop;
end Test;
Pretty vanilla, but I didn't see an Ada solution yet. Probably because Ada might just be the worst real-world language to golf with!
• Ada isn't bad, it's more that hardly anyone even knows about it – ASCII-only Apr 18 '18 at 22:06
• Oh, I love Ada. It is definitely my favourite language. It just sucks for code golf. And yeah, I wish more people knew about it. :) – LambdaBeta Apr 18 '18 at 22:11
• Well, it can't possibly be as bad as Java :P – ASCII-only Apr 18 '18 at 22:12
• also on your profile: "am familiar with a majority of commonly used programming languages". which ones exactly? :P – ASCII-only Apr 18 '18 at 23:05
• 207 – ASCII-only Apr 24 '18 at 9:56
# Forth, 107101 98 bytes
: f 101 1 do i 5 mod i 3 mod if dup if i . then else ." Fizz" then 0= if ." Buzz" then cr loop ; f
Ungolfed + close Python equivalent:
: f \ def f():
101 1 do \ for i in range(1, 101):
i 5 mod \ a = i % 5 # Not an actual variable, pushed onto the stack
i 3 mod \ b = i % 3
if \ if b: # b is popped
dup \ c = a
if \ if c:
i . \ print(i, end='')
then
else \ else:
." Fizz" \ print('Fizz', end='')
then
0= \ a = (a == 0)
if \ if a:
." Buzz" \ print('Buzz', end='')
then
cr \ print()
loop
;
f \ f()
Run it!
• To Taylor Scott: sorry for deleting the edit. I thought adding another language would be more meaningful – Alex Apr 18 '18 at 23:25
• Alex, while this is definitely a more interesting answer you should feel free to leave multiple responses to challenges. – Taylor Scott May 16 '18 at 14:18
# QB64, 102 94 bytes
FOR i=1TO 100
o$=MID$("Fizz",i*5MOD 15)+MID$("Buzz",i*5MOD 25) IF""<o$THEN?o$ELSE WRITE i NEXT Doesn't work on actual QBasic; see below for why. This program has one problem: QBasic/QB64 outputs to an 80x24 window, not a terminal, so the results can't be scrolled back. If you run the above code as-is, all you'll see is the lines from 78 onward. To prove that the code does 1 to 100 correctly, you can add the line SLEEP 1 right before NEXT for a 1-second delay on each iteration. ### Ungolfed code and explanation FOR i = 1 TO 100 index = 5 * (i MOD 3) o$ = MID$("Fizz", index) index = 5 * (i MOD 5) o$ = o$+ MID$("Buzz", index)
IF "" < o$THEN PRINT o$
ELSE
WRITE i
END IF
NEXT
On each iteration, we put the appropriate fizzes and buzzes into the string o$, check if it's empty, and output o$ or the number accordingly. The main question is how to get "Fizz" when i is divisible by 3 and "" otherwise. Here are the approaches I tried:
IF i MOD 3THEN o$=""ELSE o$="Fizz"
o$="":IF i MOD 3=0THEN o$="Fizz"
o$=MID$("Fizz",5*(i MOD 3))
o$=MID$("Fizz",i*5MOD 15)
The approach with MID$ is much shorter. This function takes 3 arguments--string, start index, and number of characters--and returns the appropriate substring. When the third argument is omitted, it takes everything from the start index to the end of the string. Here, when i is exactly divisible, the start index is 0 and we get the whole string; otherwise, it's something larger that's past the end of the string, so MID$ gives "".1
The other tricky part is printing numbers according to the spec. QBasic's PRINT command outputs positive numbers with leading spaces, which is occasionally useful but usually just annoying. The WRITE command, however, does not add a leading space--perfect for our purposes here.
1 Strings are 1-indexed in QBasic--i.e., in the string "abcd", a is at index 1 and d is at index 4. This is why I'm multiplying the mod result by 5: MID$("Fizz",4) gives "z". In actual QBasic, 0 isn't a legal index and gives Illegal function call; but in QB64, MID$("Fizz",0) happily returns the whole string instead of complaining.
# Whitespace, 307 bytes
[S S S N
_Push_0][N
S S N
_Create_Label_LOOP][S S S T N
_Push_1][T S S S _Add][S N
S _Duplicate][S N
S _Duplicate][S S S T T N
_Push_3][T S T T _Modulo][N
T S T N
_Jump_to_Label_FIZZ_if_0][N
S S S N
_Create_Label_RETURN_FIZZ][S N
S _Duplicate][S N
S _Duplicate][S S S T S T N
_Push_5][T S T T _Modulo][N
T S T T N
_Jump_to_Label_BUZZ_if_0][N
S S S S N
_Create_Label_RETURN_BUZZ][S S S T T S S T S S N
_Push_100][T S S T _Subtract][N
T S T T T N
_Jump_to_Label_EXIT_WITH_ERROR_if_0][N
S T T T T T N
_Call_Label_PRINT_INT][S S S T S T S N
_Push_10][T N
S S _Print_as_character][N
S N
N
_Jump_to_Label_LOOP][N
S S T
_Create_Label_FIZZ][S S S T S S S T T S N
_Push_70][T N
S S _Print_as_character][S S S T T S T S S T N
_Push_105][T N
S S _Print_as_character][S S S T T T T S T S N
_Push_122][S N
S _Duplicate][T N
S S _Print_as_character][T N
S S _Print_as_character][N
S N
S
_Return_to_Label_FIZZ_RETURN][N
S S T T N
_Create_Label_BUZZ][S S S T S S S S T S N
_Push_66][T N
S S _Print_as_character][S S S T T T S T S T N
_Push_117][T N
S S _Print_as_character][S S S T T T T S T S N
_Push_122][S N
S _Duplicate][T N
S S _Print_as_character][T N
S S _Print_as_character][N
S N
S S N
_Return_to_Label_BUZZ_RETURN][N
S S T T T T N
_Create_Label_PRINT_INT][S N
S _Duplicate][S S S T T N
_Push_3][T S T T _Modulo][N
T S S T N
_Jump_to_Label_LOOP_if_0][S N
S _Duplicate][S S S T S T N
_Push_5][T S T T _Modulo][N
T S S T N
_Jump_to_Label_LOOP_if_0][T N
S T _Print_as_integer][N
T N
_Return][N
S S S T N
_Create_Label_RETURN][N
T N
_Return]
Letters S (space), T (tab), and N (new-line) added as highlighting only.
[..._some_action] added as explanation only.
Try it online (with raw spaces, tabs and new-lines only).
Can definitely be golfed. If-checks are rather annoying in Whitespace, and I still have to get used to them some more.
EDIT: Fixed TIO version. Will try to golf it some more later.
General explanation in Pseudo-code:
Integer i = 0
Start LOOP
Increase i by 1
If i modulo 3 is 0: Call function FIZZ()
If i modulo 5 is 0: Call function BUZZ()
If i is 100: Stop program
Call function PRINT_INT(i)
Print new-line
Go to next iteration of the LOOP
function FIZZ():
Print "Fizz"
Return
function BUZZ():
Print "Buzz"
Return
function PRINT_INT(integer i):
If i modulo 3 is 0: Return
If i modulo 5 is 0: Return
Print i
Return
Example run:
Command Explanation Stack STDOUT STDERR
SSSN Push 0 [0]
NSSN Create Label_LOOP [0]
SSSTN Push 1 [0,1]
SNS Duplicate (1) [1,1]
SNS Duplicate (1) [1,1,1]
SSSTTN Push 3 [1,1,1,3]
TSTT Modulo (1%3) [1,1,1]
NSSSSN Create Label_RETURN_FIZZ [1,1]
SNS Duplicate (1) [1,1,1]
SNS Duplicate (1) [1,1,1,1]
SSSTSTn Push 5 [1,1,1,1,5]
TSTT Modulo (1%5) [1,1,1,1]
NSSSSN Create Label_RETURN_BUZZ [1,1,1]
SSSTTSSTSSN Push 100 [1,1,1,100]
TSST Subtract (1-100) [1,1,-99]
NSTTTTTN Call Label_PRINT_INT [1,1]
NSSTTTTN Create Label_PRINT_INT [1,1]
SNS Duplicate (1) [1,1,1]
SSSTTN Push 3 [1,1,1,3]
TSTT Modulo (1%3) [1,1,1]
SNS Duplicate (1) [1,1,1]
SSSTSTN Push 5 [1,1,1,5]
TSTT Modulo (1%5) [1,1,1]
TNST Print as integer [1] 1
NTN Return [1]
SSSTSTSN Push 10 [1,10]
TNSS Print as character [1] \n
SSSTN Push 1 [1,1]
SNS Duplicate (2) [2,2]
SNS Duplicate (2) [2,2,2]
SSSTTN Push 3 [2,2,2,3]
TSTT Modulo (2%3) [2,2,2]
NSSSSN Create Label_RETURN_FIZZ [2,2]
SNS Duplicate (2) [2,2,2]
SNS Duplicate (2) [2,2,2,2]
SSSTSTn Push 5 [2,2,2,2,5]
TSTT Modulo (2%5) [2,2,2,2]
NSSSSN Create Label_RETURN_BUZZ [2,2,2]
SSSTTSSTSSN Push 100 [2,2,2,100]
TSST Subtract (2-100) [2,2,-98]
NSTTTTTN Call Label_PRINT_INT [2,2]
NSSTTTTN Create Label_PRINT_INT [2,2]
SNS Duplicate (2) [2,2,2]
SSSTTN Push 3 [2,2,2,3]
TSTT Modulo (2%3) [2,2,2]
SNS Duplicate (2) [2,2,2]
SSSTSTN Push 5 [2,2,2,5]
TSTT Modulo (2%5) [2,2,2]
TNST Print as integer [2] 2
NTN Return [2]
SSSTSTSN Push 10 [2,10]
TNSS Print as character [2] \n
SSSTN Push 1 [2,1]
SNS Duplicate (3) [3,3]
SNS Duplicate (3) [3,3,3]
SSSTTN Push 3 [3,3,3,3]
TSTT Modulo (3%3) [3,3,0]
NSST Create Label_FIZZ [3,3]
SNS Duplicate (3) [3,3,3]
SNS Duplicate (3) [3,3,3,3]
SSSTSSSTTSN Push 70 [3,3,3,3,70]
TNSS Print as character [3,3,3,3] F
SSSTTSTSSTN Push 105 [3,3,3,3,122]
TNSS Print as character [3,3,3,3] i
SSSTTTTSTSN Push 122 [3,3,3,3,122]
SNS Duplicate (122) [3,3,3,3,122,122]
TNSS Print as character [3,3,3,3,122] z
TNSS Print as character [3,3,3,3] z
NSSSSN Create Label_RETURN_FIZZ [3,3,3,3]
SSSTSTN Push 5 [3,3,3,3,5]
TSTT Modulo (3%5) [3,3,3,3]
NSSSSN Create Label_RETURN_BUZZ [3,3,3]
SSSTTSSTSSN Push 100 [3,3,3,100]
TSST Subtract (3-100) [3,3,-97]
NSTTTTTN Call Label_PRINT_INT [3,3]
NSSTTTTN Create Label_PRINT_INT [3,3]
SNS Duplicate (3) [3,3,3]
SSSTTN Push 3 [3,3,3,3]
TSTT Modulo (3%3) [3,3,0]
Stack contains additional leading [3, but we'll ignore it in this explanation
SSSTN Push 1 [3,1]
SNS Duplicate (4) [4,4]
SNS Duplicate (4) [4,4,4]
SSSTTN Push 3 [4,4,4,3]
TSTT Modulo (4%3) [4,4,1]
NSSSSN Create Label_RETURN_FIZZ [4,4]
SNS Duplicate (4) [4,4,4]
SNS Duplicate (4) [4,4,4,4]
SSSTSTN Push 5 [4,4,4,4,5]
TSTT Modulo (4%5) [4,4,4,4]
NSSSSN Create Label_RETURN_BUZZ [4,4,4]
SSSTTSSTSSN Push 100 [4,4,4,100]
TSST Subtract (4-100) [4,4,-96]
NSTTTTTN Call Label_PRINT_INT [4,4]
NSSTTTTN Create Label_PRINT_INT [4,4]
SNS Duplicate (4) [4,4,4]
SSSTTN Push 3 [4,4,4,3]
TSTT Modulo (4%3) [4,4,1]
SNS Duplicate (4) [4,4,4]
SSSTSTN Push 5 [4,4,4,5]
TSTT Modulo (4%5) [4,4,4]
TNST Print as integer [4] 4
NTN Return [4]
SSSTSTSN Push 10 [4,10]
TNSS Print as character [4] \n
SSSTN Push 1 [4,1]
SNS Duplicate (5) [5,5]
SNS Duplicate (5) [5,5,5]
SSSTTN Push 3 [5,5,5,3]
TSTT Modulo (5%3) [5,5,2]
NSSSSN Create Label_RETURN_FIZZ [5,5]
SNS Duplicate (5) [5,5,5]
SNS Duplicate (5) [5,5,5,5]
SSSTSTN Push 5 [5,5,5,5,5]
TSTT Modulo (5%5) [5,5,5,0]
NSSTTN Create Label_BUZZ [5,5,5]
SSSTSSSSTSN Push 66 [5,5,5,66]
TNSS Print as character [5,5,5] B
SSSTTTSTSTN Push 117 [5,5,5,117]
TNSS Print as character [5,5,5] u
SSSTTTTSTSN Push 122 [5,5,5,122]
SNS Duplicate (122) [5,5,5,122,122]
TNSS Print as character [5,5,5,122] z
TNSS Print as character [5,5,5] z
NSSSSN Create Label_RETURN_BUZZ [5,5,5]
SSSTTSSTSSN Push 100 [5,5,5,100]
TSST Subtract (5-100) [5,5,-95]
NSTTTTTN Call Label_PRINT_INT [5,5]
NSSTTTTN Create Label_PRINT_INT [5,5]
SNS Duplicate (5) [5,5,5]
SSSTTN Push 3 [5,5,5,3]
TSTT Modulo (5%3) [5,5,2]
SNS Duplicate (5) [5,5,5]
SSSTSTN Push 5 [5,5,5,5]
TSTT Modulo (4%5) [5,5,0]
... etc. etc.
SSSTN Push 1 [99,1]
SNS Duplicate (100) [100,100]
SNS Duplicate (100) [100,100,100]
SSSTTN Push 3 [100,100,100,3]
TSTT Modulo (100%3) [100,100,1]
NSSSSN Create Label_RETURN_FIZZ [100,100]
SNS Duplicate (100) [100,100,100]
SNS Duplicate (100) [100,100,100,100]
SSSTSTN Push 5 [100,100,100,100,5]
TSTT Modulo (100%5) [100,100,100,0]
NSSTTN Create Label_BUZZ [100,100,100]
SSSTSSSSTSN Push 66 [100,100,100,66]
TNSS Print as character [100,100,100] B
SSSTTTSTSTN Push 117 [100,100,100,117]
TNSS Print as character [100,100,100] u
SSSTTTTSTSN Push 122 [100,100,100,122]
SNS Duplicate (122) [100,100,100,122,122]
TNSS Print as character [100,100,100,122] z
TNSS Print as character [100,100,100] z
NSSSSN Create Label_RETURN_BUZZ [100,100,100]
SSSTTSSTSSN Push 100 [100,100,100,100]
TSST Subtract (100-100) [100,100,0]
# APL (Dyalog Unicode), 37 bytesSBCS
↑{∨/d←4/0=3 5|⍵:d/'FizzBuzz'⋄⍕⍵}¨⍳100
Try it online!
⍳100ɩndices 1…100
{}¨ apply the following anonymous lambda to each of those:
⍵ the argument; e.g. 20
3 5| the division remainder when that is divided by 3 and 5; e.g. [2,0]
0= Boolean mask where that is equal to 0; e.g. [0,1]
4/ replicate those numbers for 4 copies of each; e.g. [0,0,0,0,1,1,1,1]
d← assign that to d
∨/: if any of those are true (OR-reduction); e.g. true:
d/'FizzBuzz' use d to mask the characters of the string; e.g. "Buzz"
⋄ else:
⍕⍵ stringify the argument; e.g. "20"
↑ mix the list of strings into a matrix, so it prints right
# Julia, 87 bytes
z(i)=(f=i .%[3,5] .==0;sum(f)>0 ? foldl(*,["Fizz","Buzz"][f]) : i)
println.(z.(1:100))
Or we could kind of cheat to drop 3 bytes and use show instead of println. Julia 83 bytes
z(i)=(f=i .%[3,5] .==0;sum(f)>0 ? foldl(*,["Fizz","Buzz"][f]) : i)
show(z.(1:100))
or a more legible FP style, poor performing, 91 bytes
F(x)=foldl(*,["Fizz","Buzz"][x .%[3,5] .==0])
N(y)=F(y)=="" ? y : F(y)
println.(N.(1:100))
• prod is better than foldr(* – H.PWiz Nov 23 '19 at 23:45
• Nice! I didn't think that'd work. Either way my solution is 10s of bytes away from the best Julia one :). – caseyk Nov 23 '19 at 23:51
• Sure. My score is here – H.PWiz Nov 24 '19 at 12:25
• Woah! How'd you do it? Or is that a secret? :) We have a thread that divulged into codegolf on julia discourse! – caseyk Nov 24 '19 at 13:10
• Sort of a secret. Although I have discussed julia golfing in a chat room on this site. – H.PWiz Nov 24 '19 at 21:56
# DIVSPL, 22 bytes
1..100
fizz=3
buzz=5
# sed 4.2.2, 129 bytes
A 400-rep bounty for those who outgolf this solution https://codegolf.meta.stackexchange.com/a/18428/
s/^/00,/;h
:
y/0123456789';,/1234567890;,'/
/0.$/!{x;G;s/..\n.//} h s/[05].$/&Buzz/
s/.*,/Fizz/
s/.*\WB/B/
s/[0';]*//gp
g;/00/d
t
Try it online!
s/^/00,/;h
Each number is stored as three characters, the first two store the two-digit padded decimal number, and the last character stores its modulo 3.
: ... t
In a loop,
y/0123456789';,/1234567890;,'/
the modulo is cycled, and the number is incremented using transliteration. Each digit is increased modulo 10, then
/0.$/!{x;G;s/..\n.//} a simple conditional corrects the first digit on the number if necessary, using the fact each number is stored with exactly 3 characters. h This incremented form is stored in the hold space. s/[05].$/&Buzz/
Buzzs are added by looking at the last base-10 digit of the number.
s/.*,/Fizz/
Fizzs are added by looking at the modulo-3.
s/.*\WB/B/
There is some cleanup of the number, remove the base-10 digits if needed and
s/[0';]*//gp
remove leading 0s and the modulo-3 so that it is print-ready, and print it.
g;/00/d
Finally retrieve the number from the hold space and exit if 00` is present, i.e. 100 has been reached
|
2020-06-05 13:24:06
|
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|
https://www.gamedev.net/forums/topic/434623-gametime-class-in-a-windows-app/
|
# [.net] Gametime class in a windows app
This topic is 4220 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Hi online, Im trying to develop a level designer for my xna project. Obviously this has to be done in windows. Unfortunately most of my classes use the Gametime class that is sent to the Draw and Update functions of the Game class. Since im doing a Windows project, I dont have any Gametime to work with, because Im not using a Game class. I tried creating a new Gametime object but it does not update, and I dont know how to get it to update. Can anyone give me some suggestions? Thanks
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Hmm, that brings up a good question ... I thought for sure that GameTime would either be a struct, or allow you to update it's methods. However, a little bit of casual inspection of the meta data shows that it is in fact a class (Reference type), and the only way to set it's values are to call the constructor.
So that suggests to me that one is being new'd up every frame? that's crazy talk.
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Yeah I know. I really dont want to write a new timer when there is already one that we should use anyway built into XNA.
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The GameTime is used by Game to handle the timing of the loop. So you can't use GameTime without having Game in the project. What I am wondering is why you are not using Game? Yes there is some extra over head but I think that you could use it to make things a bit simpler.
My idea; Create a standard window's app. Create a new thread for the Game object. Let the Game object run over there taking care of your drawing and all the XNA world. Have the main thread for your Level designer.
theTroll
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How exactly do you do that? Because the game class does not have a normal windows form. So how do you tell it to render (in my case anyway) to a panel inside a windows form?
Thanks again for any help.
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Going to try to work up some code for it. Be back in a bit with some code.
theTroll
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SavyCat - do you actually need your objects Updating in the editor? It seems like in that context, an all-zeros GameTime would work fine because in the editor its sorta always time index 0. Things should draw fine - they'll just be inert.
But if you want a nice timer, you can swipe the one in the Sample Frameworks source code. (DX SDK Install Dir)\Samples\Managed\Common\dxmutmisc.cs Then you can drive it with a regular old Timer control. The raft of calcs you have to make to get the constructor params set up are a bit of a chore. I wrote the code once, a while back but I don't know if I still have it - I changed my plumbing around to decouple my model from XNA. I'll take a look around and will post if I find it.
On the other hand, why not just make the editor a Game just like the actual game. I don't know if you have to put some sort of Do Events in the game's pump functions so that the Windows.Forms objects can process their messages. I've never done this myself but I've heard other people mention doing it.
Quote:
Original post by joelmartinezSo that suggests to me that one is being new'd up every frame? that's crazy talk.
It must have "internal" members back inside the XNA assemblies. I just checked and its the same instance being provided each call:
private GameTime lastGameTime = null;protected override void Update(GameTime gameTime){ if (lastGameTime == null) lastGameTime = gameTime; else { if (object.ReferenceEquals(lastGameTime, gameTime)) { // it stops on a break point set here int wtf = 4; } } base.Update(gameTime);}
[Edited by - dalep on January 28, 2007 11:45:16 AM]
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The first idea did not work, trying something else. Be back in a bit.
theTroll
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Making progress, hope to have an idea for you soon.
theTroll
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Got a possible solution for you. This will allow your XNA Game window to be fully active while having complete access to the editor at the same time.
#region Using Statementsusing System;using System.Collections.Generic;using Microsoft.Xna.Framework;using Microsoft.Xna.Framework.Audio;using Microsoft.Xna.Framework.Content;using Microsoft.Xna.Framework.Graphics;using Microsoft.Xna.Framework.Input;using Microsoft.Xna.Framework.Storage;#endregionnamespace XNACore{ /// <summary> /// This is the main type for your game /// </summary> public class Game1 : Microsoft.Xna.Framework.Game { GraphicsDeviceManager graphics; ContentManager content; private System.Windows.Forms.Form form; private Microsoft.Xna.Framework.Graphics.Viewport xnaViewport1; private System.Windows.Forms.Panel panelLeft; private System.Windows.Forms.Panel panelRightBottom; private System.Windows.Forms.Button button1; public Game1() { graphics = new GraphicsDeviceManager(this); content = new ContentManager(Services); } /// <summary> /// Allows the game to perform any initialization it needs to before starting to run. /// This is where it can query for any required services and load any non-graphic /// related content. Calling base.Initialize will enumerate through any components /// and initialize them as well. /// </summary> protected override void Initialize() { // TODO: Add your initialization logic here base.Initialize(); xnaViewport1 = new Viewport(); xnaViewport1.X = this.Window.ClientBounds.Width/2; xnaViewport1.Y = 0; xnaViewport1.Width = this.Window.ClientBounds.Width / 2; xnaViewport1.Height = this.Window.ClientBounds.Height / 2; this.graphics.GraphicsDevice.Viewport = xnaViewport1; IntPtr ptr = this.Window.Handle; form = (System.Windows.Forms.Form)System.Windows.Forms.Control.FromHandle(ptr); panelLeft = new System.Windows.Forms.Panel(); panelLeft.Width = this.Window.ClientBounds.Width / 2; panelLeft.Height = this.Window.ClientBounds.Height; panelLeft.Top = 0; panelLeft.Left = 0; form.Controls.Add(panelLeft); panelRightBottom = new System.Windows.Forms.Panel(); panelRightBottom.Width = this.Window.ClientBounds.Width / 2; panelRightBottom.Height = this.Window.ClientBounds.Height / 2; panelRightBottom.Top = this.Window.ClientBounds.Height / 2; panelRightBottom.Left = this.Window.ClientBounds.Width / 2; form.Controls.Add(panelRightBottom); button1 = new System.Windows.Forms.Button(); button1.Text = "It works!!!"; button1.Top = 20; button1.Left = 20; panelLeft.Controls.Add(button1); } /// <summary> /// Load your graphics content. If loadAllContent is true, you should /// load content from both ResourceManagementMode pools. Otherwise, just /// load ResourceManagementMode.Manual content. /// </summary> /// <param name="loadAllContent">Which type of content to load.</param> protected override void LoadGraphicsContent(bool loadAllContent) { if (loadAllContent) { // TODO: Load any ResourceManagementMode.Automatic content } // TODO: Load any ResourceManagementMode.Manual content } /// <summary> /// Unload your graphics content. If unloadAllContent is true, you should /// unload content from both ResourceManagementMode pools. Otherwise, just /// unload ResourceManagementMode.Manual content. Manual content will get /// Disposed by the GraphicsDevice during a Reset. /// </summary> /// <param name="unloadAllContent">Which type of content to unload.</param> protected override void UnloadGraphicsContent(bool unloadAllContent) { if (unloadAllContent == true) { content.Unload(); } } /// <summary> /// Allows the game to run logic such as updating the world, /// checking for collisions, gathering input and playing audio. /// </summary> /// <param name="gameTime">Provides a snapshot of timing values.</param> protected override void Update(GameTime gameTime) { // Allows the default game to exit on Xbox 360 and Windows if (GamePad.GetState(PlayerIndex.One).Buttons.Back == ButtonState.Pressed) this.Exit(); // TODO: Add your update logic here base.Update(gameTime); } /// <summary> /// This is called when the game should draw itself. /// </summary> /// <param name="gameTime">Provides a snapshot of timing values.</param> protected override void Draw(GameTime gameTime) { graphics.GraphicsDevice.Clear(Color.Blue); // TODO: Add your drawing code here base.Draw(gameTime); } }}
I have tested it and everything seems to work just great.
theTroll
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2018-08-19 13:44:14
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http://physics.stackexchange.com/questions/102949/questions-about-spacetime-curvature-and-gravity
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# Questions about spacetime curvature and gravity
These are some further important queries regarding the question here Why would spacetime curvature cause gravity?
Q1. Explain the statement “Everything in spacetime moves at the speed of light”. Is this statement even true or false? My naïve understanding is as follows: If an object is stationary in space (relative to an observer?) then it will be moving in time, so, it will, say, move from (x=0,t=0) to (x=0,t=10). The velocity is the space interval / time interval, which from above coordinates is still zero. So it is moving through spacetime, but at zero velocity. So velocity will just determine the angle of its path to the time axis. If this angle is zero, ie it is moving parallel to the time axis, then velocity = zero, and it can really move in time from one time point to the other at zero velocity. Where am I wrong, and what is the real explanation?
Q2. Suppose there are two objects (see fig), an apple A above the earth at x=0, and the earth E at x=10 as shown in the fig. If there is no earth and just the apple in open space, then the spacetime is not curved due to the gravity of the earth, and the apple stays at x=0 but moves in time from A to B (fig a). Now the earth is at x=10 and presumably the spacetime curves and the axes tilt as in fig b. Then the apple moves from A to C, just following the geodesic. But if we assume this, then the apple has not moved at all, because, due to the tilt of the axis, point C is also at x=0. So fig b cannot be the correct situation, otherwise the apple will arrive at point C which is still on the x=0 axis. So I assume that in fact the apple has not moved from A to C in fig b, but has moved from A to D in fig. a, where D is really at x=10. But if I assume this, then it is not spacetime itself that has curved. The spacetime is still straight, but the apple has moved from (x=0,t=0) to (x=10,t=10). Again, where am I wrong, and what is the correct explanation? How exactly has the spacetime curved or tilted for the apple at A due to the presence of the Earth at E, and how does the apple move to the earth by following the geodesic in free fall? (assume only one spatial dimention)
Q3: We say that gravity is not just a “force of attraction” between two pieces of mass, and it does not “pull” the two pieces of mass towards each other. Instead, acceleration is manifested because the two pieces of mass are simply following their now curved geodesics. But it can be shown that there really is an attractive force due to gravity. E.g. Take the apple in the earth’s gravity and suspend it on a spring balance. The spring will extend. Yes, you can say that since it is suspended motionless, so it is no longer in free fall, and since the spring balance exerts an external force upwards on the apple, so it no longer follows its free fall geodesic. But it is not only the spring balance which is exerting an upward force on the apple. The apple is also exerting a downward force on the spring, which causes the spring to extend. Surely it is too naïve to say that the downward force which the earth exerts on the apple and in turn the apple exerts on the spring suddenly vanishes if the apple is released into free fall.
If two masses were remotely attracting each other with a force, why would they cease to attract each other with a force when in free fall? It is logically a much more satisfying explanation that the gravitational attraction force which was there at rest remains in free fall too and it is this gravitational attraction force which causes the freefall acceleration by F=ma, just like any force on any object would cause an acceleration. Also, it is the net force on the object which causes the acceleration. When the object was suspended, the net force was zero, so acceleration was zero. When in free fall, the net force IS THE GRAVITATIONAL ATTRACTION FORCE, so acceleration is accordingly. This seems totally opposite to the statement that there is no gravitational force in free fall.
The reason why the spring of the spring balance does not extend when in free fall is that there IS a gravitational force downwards, but there is no force exerted by the spring upwards, so the net force on the object is downwards due to gravity, which causes the acceleration F=ma. Where am I wrong? What is the correct explanation?
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have a look at this video which maps space time in one dimension youtube.com/watch?v=DdC0QN6f3G4 .Note the comment in the frame where both Newtonian plot and GR plot are given. In GR there are no gravitational forces, but there are the rest of the forces , like the electromagnetic ones keeping the apple on the branch. – anna v Mar 11 at 6:48
I've downvoted this 'question' for one simple reason: it's not a question but a too long, rambling "but if this is so, then such and such but that would mean this and that... where am I wrong?" ad nauseam. Please read the FAQ for how to write a good question. I'm certain that most here will consider this one a tl;dr and move one to other more interesting questions. – Alfred Centauri Mar 11 at 12:26
Re (1): the relativistic extension of velocity is four velocity. Pre relativity we separate the spatial and time coordinates, then define velocity as $dx/dt$ etc. In relativity this no longer makes sense because $x$ and $t$ are both spacetime coordinates and indeed the Lorentz transformations will mix them up. So we define the four velocity $U$ as the vector:
$$U = \left(\frac{cdt}{d\tau}, \frac{dx}{d\tau}, \frac{dy}{d\tau}, \frac{dz}{d\tau}\right)$$
where $\tau$ is the proper time. In your example of a stationary object $dx$, $dy$ and $dz$ are all zero, and $dt/d\tau = 1$ (because in the rest frame of an object the coordinate time is equal to the proper time). So the four velocity is:
$$U = \left(c, 0, 0, 0\right)$$
and hence the claim that a stationary object has a four velocity equal to $c$. Lorentz transformations to other frames will change the components of the four vector, but its magnitude will always be $c$.
Re (2): your drawing isn't a good description of the spacetime near a spherically symmetric object like the Earth. In these circumstances the curvature is given by the Schwarzschild metric. Calculating the trajectory of your apple in this metric is far from trivial. As it happens this issue is addressed on the recent question Naive visualization of space-time curvature. Sadly, the answer to this question is that there is no reliable naive visualisation of spacetime curvature. The only way to calculate the trajectory is to bite the bullet and use the geodesic equation.
Re (3): the stationary apple attached to the spring is accelerating, and therefore a force is being applied to it. The procedure for calculating this acceleration is described in the question What is the weight equation through general relativity?, as is the force required to be exerted by the spring to keep the apple accelerating.
You object that the apple is also exerting a force on the spring. Well yes, this is just Newton's third law. The spring in turn exerts a force on whatever is supporting the upper end of the spring, and so on.
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The answer to the third question does not get us anywhere, because it is gravity or the remote attraction of one mass by another that the apple feels from the earth first, and the upward force by the spring on the apple is due to Newton's third law. I say that if the earth's gravity can remotely attract the apple when it is at rest, then it must also do so in free fall. The spring does not stretch in free fall that does not mean that there is no remote gravitational attraction. The spring does not stretch because the other end of the spring is not being pulled upwards by any support. Comments? – user1648764 Mar 11 at 11:21
The question I linked explains how to use GR to calculate the force on the apple exerted by the spring. This calculation is done assuming there is no force between the apple and the Earth and both are just following geodesics. The calculation works. You can apply whatever textual labels you wish, but the maths works. – John Rennie Mar 11 at 11:49
Since no one has answered the second question, and since I discovered the answer subsequently, so I will post it. I only hope that it is correct. The spacetime bends backwards, so that if the the apple takes the shortest path, ie vertical, it reaches the point C where x=10 (in fig b) due to the backward tilt of the axis. Actually, these lines are curves to allow for parabolic nature of acceleration. I have drawn them as straight lines, just to show approximately which way they will bend. Remember, the apple started out at A, where x=0, and the earth was at x=10. Compare with the fig in the question which did not make sense due to the tilt of the axes in the wrong direction.
The vertical line followed by the apple in fig b shows that the apple thinks that it is at rest and the earth is accelerating towards it.
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How exactly has the spacetime curved or tilted for the apple at A due to the presence of the Earth at E, and how does the apple move to the earth by following the geodesic in free fall? (assume only one spatial dimention)
Spacetime is not "tilted", like you show it, because the two dimension must be orthogonal everywhere. This is shown here:
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2014-10-30 14:54:55
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http://rbspgway.jhuapl.edu/biblio?page=4&s=year&o=asc&f%5Bkeyword%5D=10
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Biblio
Found 643 results
Filters: Keyword is Van Allen Probes [Clear All Filters]
2015
Authors: Gkioulidou Matina, Ohtani S, Mitchell D G, Ukhorskiy A., Reeves G D, et al.
Title: Spatial structure and temporal evolution of energetic particle injections in the inner magnetosphere during the 14 July 2013 substorm event.
Abstract: Recent results by the Van Allen Probes mission showed that the occurrence of energetic ion injections inside geosynchronous orbit could be very frequent throughout the main phase of a geomagnetic storm. Understanding, therefore, the formation and evolution of energetic particle injections is critical in order to quantify their effect in the inner magnetosphere. We present a case study of a substorm event that occurred during a weak storm (Dst ~ - 40 nT) on 14 July 2013. Van Allen Probe B, inside geosynchronous orbit, observed two energetic proton injections within ten minutes, with different dipolarization signatures and duration. The first one is a dispersionless, short timescale injection pulse accompanied by a sharp dipolarization signature, while the second one is a dispersed, longer t. . .
Date: 02/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2014JA020872 Available at: http://doi.wiley.com/10.1002/2014JA020872
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Authors: Wang Dedong, Yuan Zhigang, Yu Xiongdong, Deng Xiaohua, Zhou Meng, et al.
Title: Statistical characteristic of EMIC waves: Van Allen Probe observations
Abstract: Utilizing the data from the magnetometer instrument which is a part of the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) instrument suite onboard the Van Allen Probe A from Sep. 2012 to Apr. 2014, when the apogee of the satellite has passed all the MLT sectors, we obtain the statistical distribution characteristic of EMIC waves in the inner magnetosphere over all local times from L=3 to L=6. Compared with the previous statistical results about EMIC waves, the occurrence rates of EMIC waves distribute relatively uniform in the MLT sectors in lower L-shells. On the other hand, in higher L-shells, there are indeed some peaks of the occurrence rate for the EMIC waves, especially in the noon, dusk and night sectors. EMIC waves appear at lower L-shells in the dawn. . .
Date: 05/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2015JA021089 Available at: http://doi.wiley.com/10.1002/2015JA021089
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Authors: Li W, Ma Q, Thorne R M, Bortnik J, Kletzing C A, et al.
Title: Statistical properties of plasmaspheric hiss derived from Van Allen Probes data and their Effects on radiation belt electron dynamics
Abstract: Plasmaspheric hiss is known to play an important role in controlling the overall structure and dynamics of radiation belt electrons inside the plasmasphere. Using newly available Van Allen Probes wave data, which provide excellent coverage in the entire inner magnetosphere, we evaluate the global distribution of the hiss wave frequency spectrum and wave intensity for different levels of substorm activity. Our statistical results show that observed hiss peak frequencies are generally lower than the commonly adopted value (~550 Hz), which was in frequent use, and that the hiss wave power frequently extends below 100 Hz, particularly at larger L shells (> ~3) on the dayside during enhanced levels of substorm activity. We also compare electron pitch angle scattering rates caused by hiss . . .
Date: 05/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2015JA021048 Available at: http://doi.wiley.com/10.1002/2015JA021048
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Authors: Dai Lei, Takahashi Kazue, Lysak Robert, Wang Chi, Wygant John R., et al.
Title: Storm-time occurrence and Spatial distribution of Pc4 poloidal ULF waves in the inner magnetosphere: A Van Allen Probes Statistical study
Abstract: Poloidal ULF waves are capable of efficiently interacting with energetic particles in the ring current and the radiation belt. Using Van Allen Probes (RBSP) data from October 2012 to July 2014, we investigate the spatial distribution and storm-time occurrence of Pc4 (7-25 mHz) poloidal waves in the inner magnetosphere. Pc4 poloidal waves are sorted into two categories: waves with and without significant magnetic compressional components. Two types of poloidal waves have comparable occurrence rates, both of which are much higher during geomagnetic storms. The non-compressional poloidal waves mostly occur in the late recovery phase associated with an increase of Dst toward 0, suggesting that the decay of the ring current provides their free energy source. The occurrence of dayside compressio. . .
Date: 05/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2015JA021134 Available at: http://doi.wiley.com/10.1002/2015JA021134
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Authors: Min Kyungguk, Liu Kaijun, Bonnell John W., Breneman Aaron W., Denton Richard E, et al.
Title: Study of EMIC wave excitation using direct ion measurements
Abstract: With data from Van Allen Probes, we investigate EMIC wave excitation using simultaneously observed ion distributions. Strong He-band waves occurred while the spacecraft was moving through an enhanced density region. We extract from Helium, Oxygen, Proton, and Electron (HOPE) Mass Spectrometer measurement the velocity distributions of warm heavy ions as well as anisotropic energetic protons that drive wave growth through the ion cyclotron instability. Fitting the measured ion fluxes to multiple sinm-type distribution functions, we find that the observed ions make up about 15% of the total ions, but about 85% of them are still missing. By making legitimate estimates of the unseen cold (below ~2 eV) ion composition from cutoff frequencies suggested by the observed wave spectrum, a series of. . .
Date: 03/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2014JA020717 Available at: http://doi.wiley.com/10.1002/2014JA020717
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Authors: Kirby Karen, Fretz Kristin, Goldsten John, and Maurer Richard
Title: Successes and challenges of operating the Van Allen Probes mission in the radiation belts
Abstract: The Van Allen probes team has been successful in monitoring and trending the performance of the mission to date. However, operating two spacecraft in the Van Allen radiation belts poses a number of challenges and requires careful monitoring of spacecraft performance due to the high radiation environment and potential impact on the mostly single string electronics architecture. Spacecraft and instrument telemetry trending is tracked with internal peer reviews conducted twice a year by the operations and engineering teams. On board radiation monitoring sensors are used to evaluate total dose accumulated on board the spacecraft and to assess potential impacts. Single event upsets are tracked and high activity events are logged and analyzed. Anomalous data is compared to radiation and solar ev. . .
Date: 03/2015 Publisher: IEEE DOI: 10.1109/AERO.2015.7119179 Available at: http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7119179
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Authors: Woodger L A, Halford A J, Millan R M, McCarthy M P, Smith D M, et al.
Title: A Summary of the BARREL Campaigns: Technique for studying electron precipitation
Abstract: The Balloon Array for Radiation belt Relativistic Electron Losses (BARREL) studies the loss of energetic electrons from Earth's radiation belts. BARREL's array of slowly drifting balloon payloads was designed to capitalize on magnetic conjunctions with NASA's Van Allen Probes. Two campaigns were conducted from Antarctica in 2013 and 2014. During the first campaign in January and February of 2013, there were three moderate geomagnetic storms with Sym-Hmin < −40 nT. Similarly, two minor geomagnetic storms occurred during the second campaign, starting in December of 2013 and continuing on into February of 2014. Throughout the two campaigns, BARREL observed electron precipitation over a wide range of energies and exhibiting temporal structure from 100's of milliseconds to hours. Relativistic. . .
Date: 05/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2014JA020874 Available at: http://doi.wiley.com/10.1002/2014JA020874
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Authors: Shi Run, Summers Danny, Ni Binbin, Fennell Joseph F., Blake Bernard, et al.
Title: Survey of radiation belt energetic electron pitch angle distributions based on the Van Allen Probes MagEIS measurements
Abstract: A statistical survey of electron pitch angle distributions (PADs) is performed based on the pitch angle resolved flux observations from the Magnetic Electron Ion Spectrometer (MagEIS) instrument on board the Van Allen Probes during the period from 1 October 2012 to 1 May 2015. By fitting the measured PADs to a sinnα form, where α is the local pitch angle and n is the power law index, we investigate the dependence of PADs on electron kinetic energy, magnetic local time (MLT), the geomagnetic Kp index and L-shell. The difference in electron PADs between the inner and outer belt is distinct. In the outer belt, the common averaged n values are less than 1.5, except for large values of the Kp index and high electron energies. The averaged n values vary considerably with MLT, with a peak in th. . .
Date: 12/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2015JA021724 Available at: http://doi.wiley.com/10.1002/2015JA021724http://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1002%2F2015JA021724
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Authors: Vasko I. Y., Agapitov O. V., Mozer F S, and Artemyev A. V.
Title: Thermal electron acceleration by electric field spikes in the outer radiation belt: Generation of field-aligned pitch angle distributions
Abstract: Van Allen Probes observations in the outer radiation belt have demonstrated an abundance of electrostatic electron-acoustic double layers (DL). DLs are frequently accompanied by field-aligned (bidirectional) pitch angle distributions (PAD) of electrons with energies from hundred eVs up to several keV. We perform numerical simulations of the DL interaction with thermal electrons making use of the test particle approach. DL parameters assumed in the simulations are adopted from observations. We show that DLs accelerate thermal electrons parallel to the magnetic field via the electrostatic Fermi mechanism, i.e., due to reflections from DL potential humps. The electron energy gain is larger for larger DL scalar potential amplitudes and higher propagation velocities. In addition to the Fermi me. . .
Date: 10/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2015JA021644 Available at: http://doi.wiley.com/10.1002/2015JA021644http://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1002%2F2015JA021644
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Authors: Zhang J.-C., Kistler L. M., Spence H E, Wolf R. A., Reeves G., et al.
Title: “Trunk-like” heavy ion structures observed by the Van Allen Probes
Abstract: Dynamic ion spectral features in the inner magnetosphere are the observational signatures of ion acceleration, transport, and loss in the global magnetosphere. We report “trunk-like” ion structures observed by the Van Allen Probes on 2 November 2012. This new type of ion structure looks like an elephant's trunk on an energy-time spectrogram, with the energy of the peak flux decreasing Earthward. The trunks are present in He+ and O+ ions but not in H+. During the event, ion energies in the He+ trunk, located at L = 3.6–2.6, MLT = 9.1–10.5, and MLAT = −2.4–0.09°, vary monotonically from 3.5 to 0.04 keV. The values at the two end points of the O+ trunk are: energy = 4.5–0.7 keV, L = 3.6–2.5, MLT = 9.1–10.7, and MLAT = −2.4–0.4°. Results from backward ion drift path tra. . .
Date: 10/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2015JA021822 Available at: http://doi.wiley.com/10.1002/2015JA021822http://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1002%2F2015JA021822
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Authors: Su Zhenpeng, Zhu Hui, Xiao Fuliang, Zong Q.-G., Zhou X.-Z., et al.
Title: Ultra-low-frequency wave-driven diffusion of radiation belt relativistic electrons
Abstract: Van Allen radiation belts are typically two zones of energetic particles encircling the Earth separated by the slot region. How the outer radiation belt electrons are accelerated to relativistic energies remains an unanswered question. Recent studies have presented compelling evidence for the local acceleration by very-low-frequency (VLF) chorus waves. However, there has been a competing theory to the local acceleration, radial diffusion by ultra-low-frequency (ULF) waves, whose importance has not yet been determined definitively. Here we report a unique radiation belt event with intense ULF waves but no detectable VLF chorus waves. Our results demonstrate that the ULF waves moved the inner edge of the outer radiation belt earthward 0.3 Earth radii and enhanced the relativistic electron fl. . .
Date: 12/2015 Publisher: Nature Communications Pages: 10096 DOI: 10.1038/ncomms10096 Available at: http://www.nature.com/doifinder/10.1038/ncomms10096
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Authors: Kilpua E. K. J., Hietala H., Turner D. L., Koskinen H. E. J., Pulkkinen T. I., et al.
Title: Unraveling the drivers of the storm time radiation belt response
Abstract: We present a new framework to study the time evolution and dynamics of the outer Van Allen belt electron fluxes. The framework is entirely based on the large-scale solar wind storm drivers and their substructures. The Van Allen Probe observations, revealing the electron flux behavior throughout the outer belt, are combined with continuous, long-term (over 1.5 solar cycles) geosynchronous orbit data set from GOES and solar wind measurements A superposed epoch analysis, where we normalize the timescales for each substructure (sheath, ejecta, and interface region) allows us to avoid smearing effects and to distinguish the electron flux evolution during various driver structures. We show that the radiation belt response is not random: The electron flux variations are determined by the combined. . .
Date: 04/2015 Publisher: Geophysical Research Letters DOI: 10.1002/2015GL063542 Available at: http://doi.wiley.com/10.1002/2015GL063542
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Authors: Li X, Selesnick R. S., Baker D N, Jaynes A. N., Kanekal S G, et al.
Title: Upper limit on the inner radiation belt MeV electron Intensity
Abstract: No instruments in the inner radiation belt are immune from the unforgiving penetration of the highly energetic protons (10s of MeV to GeV). The inner belt proton flux level, however, is relatively stable, thus for any given instrument, the proton contamination often leads to a certain background noise. Measurements from the Relativistic Electron and Proton Telescope integrated little experiment (REPTile) on board Colorado Student Space Weather Experiment (CSSWE) CubeSat, in a low Earth orbit, clearly demonstrate that there exist sub-MeV electrons in the inner belt because of their flux level is orders of magnitude higher than the background, while higher energy electron (>1.6 MeV) measurements cannot be distinguished from the background. Detailed analysis of high-quality measurements from . . .
Date: 01/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2014JA020777 Available at: http://doi.wiley.com/10.1002/2014JA020777
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Authors: O'Brien T.P., Claudepierre S.G., Looper M.D., Blake J.B., Fennell J.F., et al.
Title: On the use of drift echoes to characterize on-orbit sensor discrepancies
Abstract: We describe a method for using drift echo signatures in on-orbit data to resolve discrepancies between different measurements of particle flux. The drift period has a well-defined energy dependence, which gives rise to time dispersion of the echoes. The dispersion can then be used to determine the effective energy for one or more channels given each channel's drift period and the known energy for a reference channel. We demonstrate this technique on multiple instruments from the Van Allen probes mission. Drift echoes are only easily observed at high energies (100s keV to multiple MeV), where several drift periods occur before the observing satellite has moved on or the global magnetic conditions have changed. We describe a first-order correction for spacecraft motion. The drift echo techni. . .
Date: 02/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2014JA020859 Available at: http://doi.wiley.com/10.1002/2014JA020859
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Authors: Korotova G. I., Sibeck D G, Tahakashi K., Dai L., Spence H E, et al.
Title: Van Allen Probe observations of drift-bounce resonances with Pc 4 pulsations and wave–particle interactions in the pre-midnight inner magnetosphere
Abstract: We present Van Allen Probe B observations of azimuthally limited, antisymmetric, poloidal Pc 4 electric and magnetic field pulsations in the pre-midnight sector of the magnetosphere from 05:40 to 06:00 UT on 1 May 2013. Oscillation periods were similar for the magnetic and electric fields and proton fluxes. The flux of energetic protons exhibited an energy-dependent response to the pulsations. Energetic proton variations were anticorrelated at medium and low energies. Although we attribute the pulsations to a drift-bounce resonance, we demonstrate that the energy-dependent response of the ion fluxes results from pulsation-associated velocities sweeping energy-dependent radial ion flux gradients back and forth past the spacecraft.
Date: 01/2015 Publisher: Annales Geophysicae Pages: 955 - 964 DOI: 10.5194/angeo-33-955-2015 Available at: http://www.ann-geophys.net/33/955/2015/angeo-33-955-2015.pdf
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Authors: Thaller S. A., Wygant J R, Dai L., Breneman A.W., Kersten K., et al.
Title: Van Allen Probes investigation of the large scale duskward electric field and its role in ring current formation and plasmasphere erosion in the June 1, 2013 storm
Abstract: Using the Van Allen Probes we investigate the enhancement in the large scale duskward convection electric field during the geomagnetic storm (Dst ~ −120 nT) on June 1, 2013 and its role in ring current ion transport and energization, and plasmasphere erosion. During this storm, enhancements of ~1-2 mV/m in the duskward electric field in the co-rotating frame are observed down to L shells as low as ~2.3. A simple model consisting of a dipole magnetic field and constant, azimuthally westward, electric field is used to calculate the earthward and westward drift of 90° pitch angle ions. This model is applied to determine how far earthward ions can drift while remaining on Earth's night side, given the strength and duration of the convection electric field. The calculation based on this simp. . .
Date: 05/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2014JA020875 Available at: http://doi.wiley.com/10.1002/2014JA020875
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Authors: Engebretson M. J., Posch J. L., Wygant J R, Kletzing C A, Lessard M. R., et al.
Title: Van Allen probes, NOAA, GOES, and ground observations of an intense EMIC wave event extending over 12 hours in MLT
Abstract: Although most studies of the effects of EMIC waves on Earth's outer radiation belt have focused on events in the afternoon sector in the outer plasmasphere or plume region, strong magnetospheric compressions provide an additional stimulus for EMIC wave generation across a large range of local times and L shells. We present here observations of the effects of a wave event on February 23, 2014 that extended over 8 hours in UT and over 12 hours in local time, stimulated by a gradual 4-hour rise and subsequent sharp increases in solar wind pressure. Large-amplitude linearly polarized hydrogen band EMIC waves (up to 25 nT p-p) appeared for over 4 hours at both Van Allen Probes, from late morning through local noon, when these spacecraft were outside the plasmapause, with densities ~5-20 cm-3. W. . .
Date: 06/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2015JA021227 Available at: http://doi.wiley.com/10.1002/2015JA021227
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Authors: He Yihua, Xiao Fuliang, Zhou Qinghua, Yang Chang, Liu Si, et al.
Title: Van Allen Probes observation and modeling of chorus excitation and propagation during weak geomagnetic activities
Abstract: We report correlated data on nightside chorus waves and energetic electrons during two small storm periods: 1 November 2012 (Dst≈-45) and 14 January 2013 (Dst≈-18). The Van Allen Probes simultaneously observed strong chorus waves at locations L = 5.8 − 6.3, with a lower frequency band 0.1 − 0.5fce and a peak spectral density ∼[10−4 nT2/Hz. In the same period, the fluxes and anisotropy of energetic (∼ 10-300 keV) electrons were greatly enhanced in the interval of large negative interplanetary magnetic field Bz. Using a bi-Maxwellian distribution to model the observed electron distribution, we perform ray tracing simulations to show that nightside chorus waves are indeed produced by the observed electron distribution with a peak growth for a field-aligned propagation around bet. . .
Date: 07/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2015JA021376 Available at: http://doi.wiley.com/10.1002/2015JA021376
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Authors: Cattell C. A., Breneman A. W., Thaller S. A., Wygant J R, Kletzing C A, et al.
Title: Van Allen Probes observations of unusually low frequency whistler mode waves observed in association with moderate magnetic storms: Statistical study
Abstract: We show the first evidence for locally excited chorus at frequencies below 0.1 fce (electron cyclotron frequency) in the outer radiation belt. A statistical study of chorus during geomagnetic storms observed by the Van Allen Probes found that frequencies are often dramatically lower than expected. The frequency at peak power suddenly stops tracking the equatorial 0.5 fce and f/fce decreases rapidly, often to frequencies well below 0.1 fce (in situ and mapped to equator). These very low frequency waves are observed both when the satellites are close to the equatorial plane and at higher magnetic latitudes. Poynting flux is consistent with generation at the equator. Wave amplitudes can be up to 20 to 40 mV/m and 2 to 4 nT. We conclude that conditions during moderate to large storms. . .
Date: 09/2015 Publisher: Geophysical Research Letters Pages: 7273 - 7281 DOI: 10.1002/2015GL065565 Available at: http://doi.wiley.com/10.1002/2015GL065565http://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1002%2F2015GL065565
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Authors: Fennell J. F., Claudepierre S G, Blake J B, O'Brien T P, Clemmons J. H., et al.
Title: Van Allen Probes show the inner radiation zone contains no MeV electrons: ECT/MagEIS data
Abstract: We present Van Allen Probe observations of electrons in the inner radiation zone. The measurements were made by the ECT/MagEIS sensors that were designed to measure electrons with the ability to remove unwanted signals from penetrating protons, providing clean measurements. No electrons >900 keV were observed with equatorial fluxes above background (i.e. >0.1 electrons/(cm2 s sr keV)) in the inner zone. The observed fluxes are compared to the AE9 model and CRRES observations. Electron fluxes <200 keV exceeded the AE9 model 50% fluxes and were lower than the higher energy model fluxes. Phase space density radial profiles for 1.3≤L*<2.5 had mostly positive gradients except near L*~2.1 where the profiles for μ = 20-30 MeV/G were flat or slightly peaked. The major result is that MagEIS data. . .
Date: 02/2015 Publisher: Geophysical Research Letters DOI: 10.1002/2014GL062874 Available at: http://doi.wiley.com/10.1002/2014GL062874
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Authors: Ni Binbin, Zou Zhengyang, Gu Xudong, Zhou Chen, Thorne Richard M, et al.
Title: Variability of the pitch angle distribution of radiation belt ultra-relativistic electrons during and following intense geomagnetic storms: Van Allen Probes observations
Abstract: Fifteen months of pitch angle resolved Van Allen Probes REPT measurements of differential electron flux are analyzed to investigate the characteristic variability of the pitch angle distribution (PAD) of radiation belt ultra-relativistic (>2 MeV) electrons during storm conditions and during the long-term post-storm decay. By modeling the ultra-relativistic electron pitch angle distribution as sinn α, where α is the equatorial pitch angle, we examine the spatio-temporal variations of the n-value. The results show that in general n-values increase with the level of geomagnetic activity. In principle, ultra-relativistic electrons respond to geomagnetic storms by becoming more peaked at 90° pitch angle with n-values of 2–3 as a supportive signature of chorus acceleration outside the pla. . .
Date: 05/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2015JA021065 Available at: http://doi.wiley.com/10.1002/2015JA021065
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Authors: Xiao Fuliang, Yang Chang, Su Zhenpeng, Zhou Qinghua, He Zhaoguo, et al.
Title: Wave-driven butterfly distribution of Van Allen belt relativistic electrons
Abstract: Van Allen radiation belts consist of relativistic electrons trapped by Earth's magnetic field. Trapped electrons often drift azimuthally around Earth and display a butterfly pitch angle distribution of a minimum at 90° further out than geostationary orbit. This is usually attributed to drift shell splitting resulting from day–night asymmetry in Earth’s magnetic field. However, direct observation of a butterfly distribution well inside of geostationary orbit and the origin of this phenomenon have not been provided so far. Here we report high-resolution observation that a unusual butterfly pitch angle distribution of relativistic electrons occurred within 5 Earth radii during the 28 June 2013 geomagnetic storm. Simulation results show that combined acceleration by chorus and magnetosoni. . .
Date: 05/2015 Publisher: Nature Communications Pages: 8590 DOI: 10.1038/ncomms9590 Available at: http://www.nature.com/doifinder/10.1038/ncomms9590
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Authors: Agapitov O. V., Mozer F. S., Artemyev A. V., Mourenas D., and Krasnoselskikh V. V.
Title: Wave-particle interactions in the outer radiation belts
Abstract: Data from the Van Allen Probes have provided the first extensive evidence of non-linear (as opposed to quasi-linear) wave-particle interactions in space, with the associated rapid (fraction of a bounce period) electron acceleration, to hundreds of keV by Landau resonance, in the parallel electric fields of time domain structures (TDS) and very oblique chorus waves. The experimental evidence, simulations, and theories of these processes are discussed.
Date: 12/2015 Publisher: Advances in Astronomy and Space Physics Pages: 68-74 DOI: N/A Available at: http://aasp.kiev.ua/volume5/068-074-Agapitov.pdf
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Authors: Moya Pablo. S., Pinto Víctor A., Viñas Adolfo F., Sibeck David G., Kurth William S., et al.
Title: Weak Kinetic Alfvén Waves Turbulence during the November 14th 2012 geomagnetic storm: Van Allen Probes observations
Abstract: n the dawn sector, L~ 5.5 and MLT~4-7, from 01:30 to 06:00 UT during the November 14th 2012 geomagnetic storm, both Van Allen Probes observed an alternating sequence of locally quiet and disturbed intervals with two strikingly different power fluctuation levels and magnetic field orientations: either small (~10−2 nT2) total power with strong GSM Bx and weak By, or large (~10 nT2) total power with weak Bx, and strong By and Bz components. During both kinds of intervals the fluctuations occur in the vicinity of the local ion gyro-frequencies (0.01-10 Hz) in the spacecraft frame, propagate oblique to the magnetic field, (θ ~ 60°) and have magnetic compressibility C = |δB|||/|δB⊥| ∼ 1, where δB|| (δB⊥) are the average amplitudes of the fluctuations parallel (perpendicular) to the. . .
Date: 06/2015 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2014JA020281 Available at: http://doi.wiley.com/10.1002/2014JA020281
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2016
Authors: Zhelavskaya I. S., Spasojevic M., Shprits Y Y, and Kurth W S
Title: Automated determination of electron density from electric field measurements on the Van Allen Probes spacecraft
Abstract: We present the Neural-network-based Upper hybrid Resonance Determination (NURD) algorithm for automatic inference of the electron number density from plasma wave measurements made on board NASA's Van Allen Probes mission. A feedforward neural network is developed to determine the upper hybrid resonance frequency, fuhr, from electric field measurements, which is then used to calculate the electron number density. In previous missions, the plasma resonance bands were manually identified, and there have been few attempts to do robust, routine automated detections. We describe the design and implementation of the algorithm and perform an initial analysis of the resulting electron number density distribution obtained by applying NURD to 2.5 years of data collected with the Electric and Magnetic. . .
Date: 05/2016 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2015JA022132 Available at: http://doi.wiley.com/10.1002/2015JA022132
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Authors: Kistler L.M., Mouikis C. G., Spence H.E., Menz A.M., Skoug R.M., et al.
Title: The Source of O + in the Storm-time Ring Current
Abstract: A stretched and compressed geomagnetic field occurred during the main phase of a geomagnetic storm on 1 June 2013. During the storm the Van Allen Probes spacecraft made measurements of the plasma sheet boundary layer, and observed large fluxes of O+ ions streaming up the field line from the nightside auroral region. Prior to the storm main phase there was an increase in the hot (>1 keV) and more isotropic O+ions in the plasma sheet. In the spacecraft inbound pass through the ring current region during the storm main phase, the H+ and O+ ions were significantly enhanced. We show that this enhanced inner magnetosphere ring current population is due to the inward adiabatic convection of the plasma sheet ion population. The energy range of the O+ ion plasma sheet that impacts the ring curren. . .
Date: 05/2016 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2015JA022204 Available at: http://doi.wiley.com/10.1002/2015JA022204
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Authors: Halford A J, McGregor S. L., Hudson M K, Millan R M, and Kress B T
Title: BARREL observations of a Solar Energetic Electron and Solar Energetic Proton event
Abstract: During the second Balloon Array for Radiation Belt Relativistic Electron Losses (BARREL) campaign two solar energetic proton (SEP) events were observed. Although BARREL was designed to observe X-rays created during electron precipitation events, it is sensitive to X-rays from other sources. The gamma lines produced when energetic protons hit the upper atmosphere are used in this paper to study SEP events. During the second SEP event starting on 7 January 2014 and lasting ∼ 3 days, which also had a solar energetic electron (SEE) event occurring simultaneously, BARREL had 6 payloads afloat spanning all MLT sectors and L-values. Three payloads were in a tight array (∼ 2 hrs in MLT and ∼ 2 Δ L) inside the inner magnetosphere and at times conjugate in both L and MLT with the Van Allen Pr. . .
Date: 04/2016 Publisher: Journal of Geophysical Research: Space Physics Pages: n/a - n/a DOI: 10.1002/2016JA022462 Available at: http://doi.wiley.com/10.1002/2016JA022462http://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1002%2F2016JA022462
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Authors: Sarris Theodore E., and Li Xinlin
Title: Calculating ultra-low-frequency wave power of the compressional magnetic field vs. L and time: multi-spacecraft analysis using the Van Allen probes, THEMIS and GOES
Abstract: Ultra-low-frequency (ULF) pulsations are critical in radial diffusion processes of energetic particles, and the power spectral density (PSD) of these fluctuations is an integral part of the radial diffusion coefficients and of assimilative models of the radiation belts. Using simultaneous measurements from two Geostationary Operational Environmental Satellites (GOES) geosynchronous satellites, three satellites of the Time History of Events and Macroscale Interactions during Substorms (THEMIS) spacecraft constellation and the two Van Allen probes during a 10-day period of intense geomagnetic activity and ULF pulsations of October 2012, we calculate the PSDs of ULF pulsations at different L shells. By following the time history of measurements at different L it is shown that, during this tim. . .
Date: 06/2016 Publisher: Annales Geophysicae Pages: 565 - 571 DOI: 10.5194/angeo-34-565-2016 Available at: http://www.ann-geophys.net/34/565/2016/
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Authors: Ma Q, Li W, Thorne R M, Bortnik J, Reeves G D, et al.
Title: Characteristic energy range of electron scattering due to plasmaspheric hiss
Abstract: We investigate the characteristic energy range of electron flux decay due to the interaction with plasmaspheric hiss in the Earth's inner magnetosphere. The Van Allen Probes have measured the energetic electron flux decay profiles in the Earth's outer radiation belt during a quiet period following the geomagnetic storm that occurred on 7 November 2015. The observed energy of significant electron decay increases with decreasing L shell and is well correlated with the energy band corresponding to the first adiabatic invariant μ = 4–200 MeV/G. The electron diffusion coefficients due to hiss scattering are calculated at L = 2–6, and the modeled energy band of effective pitch angle scattering is also well correlated with the constant μ lines and is consistent with the observed e. . .
Date: 11/2016 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2016JA023311 Available at: http://onlinelibrary.wiley.com/doi/10.1002/2016JA023311/full
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Authors: Zhou Xu-Zhi, Wang Zi-Han, Zong Qiu-Gang, Rankin Robert, Kivelson Margaret G., et al.
Title: Charged particle behavior in the growth and damping stages of ultralow frequency waves: theory and Van Allen Probes observations
Abstract: Ultralow frequency (ULF) electromagnetic waves in Earth's magnetosphere can accelerate charged particles via a process called drift resonance. In the conventional drift-resonance theory, a default assumption is that the wave growth rate is time-independent, positive, and extremely small. However, this is not the case for ULF waves in the real magnetosphere. The ULF waves must have experienced an earlier growth stage when their energy was taken from external and/or internal sources, and as time proceeds the waves have to be damped with a negative growth rate. Therefore, a more generalized theory on particle behavior during different stages of ULF wave evolution is required. In this paper, we introduce a time-dependent imaginary wave frequency to accommodate the growth and damping of the wav. . .
Date: 03/2016 Publisher: Journal of Geophysical Research: Space Physics Pages: n/a - n/a DOI: 10.1002/2016JA022447 Available at: http://doi.wiley.com/10.1002/2016JA022447http://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1002%2F2016JA022447
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Authors: Cohen Ross, Gerrard Andrew, Lanzerotti Louis, Soto-Chavez A. R., Kim Hyomin, et al.
Title: Climatology of high β plasma measurements in Earth's inner magnetosphere
Abstract: Since their launch in August 2012, the Radiation Belt Storm Probe Ion Composition Experiment (RBSPICE) instruments on the NASA Van Allen Probes spacecraft have been making continuous high resolution measurements of Earth's ring current plasma environment. After a full traversal through all magnetic local times, a climatology (i.e., a survey of observations) of high beta (β) plasma events (defined here as β>1) as measured by the RBSPICE instrument in the ∼45-keV to ∼600-keV proton energy range in the inner magnetosphere (L<5.8) has been constructed. In this paper we report this climatology of such high β plasma occurrences, durations, and their general characteristics. Specifically, we show that most high β events in the RBSPICE energy range are associated with post-dusk/pre-midnigh. . .
Date: 12/2016 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2016JA022513 Available at: http://onlinelibrary.wiley.com/doi/10.1002/2016JA022513/full
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Authors: He Fengming, Cao Xing, Ni Binbin, Xiang Zheng, Zhou Chen, et al.
Title: Combined Scattering Loss of Radiation Belt Relativistic Electrons by Simultaneous Three-band EMIC Waves: A Case Study
Abstract: Multiband electromagnetic ion cyclotron (EMIC) waves can drive efficient scattering loss of radiation belt relativistic electrons. However, it is statistically uncommon to capture the three bands of EMIC waves concurrently. Utilizing data from the Electric and Magnetic Field Instrument Suite and Integrated Science magnetometer onboard Van Allen Probe A, we report the simultaneous presence of three (H+, He+, and O+) emission bands in an EMIC wave event, which provides an opportunity to look into the combined scattering effect of all EMIC emissions and the relative roles of each band in diffusing radiation belt relativistic electrons under realistic circumstances. Our quantitative results, obtained by quasi-linear diffusion rate computations and 1-D pure pitch angle diffusion simulations, de. . .
Date: 05/2016 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2016JA022483 Available at: http://doi.wiley.com/10.1002/2016JA022483
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Authors: Engel M. A., Kress B T, Hudson M K, and Selesnick R. S.
Title: Comparison of Van Allen Probes radiation belt proton data with test particle simulation for the 17 March 2015 storm
Abstract: The loss of protons in the outer part of the inner radiation belt (L = 2 to 3) during the 17 March 2015 geomagnetic storm was investigated using test particle simulations that follow full Lorentz trajectories with both magnetic and electric fields calculated from an empirical model. The simulation results presented here are compared with proton pitch angle measurements from the Van Allen Probe satellites Relativistic Electron Proton Telescope (REPT) instrument before and after the coronal mass ejection-shock-driven storm of 17–18 March 2015, with minimum Dst =− 223 nT, the strongest storm of Solar Cycle 24, for four different energy ranges with 30, 38, 50, and 66 MeV mean energies. Two simulations have been run, one with an inductive electric field and one without. All four energy chan. . .
Date: 11/2016 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2016JA023333 Available at: http://onlinelibrary.wiley.com/doi/10.1002/2016JA023333/full
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Authors: Denton M. H., Reeves G. E., Thomsen M F, Henderson M G, Friedel R H W, et al.
Title: The complex nature of storm-time ion dynamics: Transport and local acceleration
Abstract: Data from the Van Allen Probes Helium, Oxygen, Proton, Electron (HOPE) spectrometers reveal hitherto unresolved spatial structure and dynamics in ion populations. Complex regions of O+ dominance, at energies from a few eV to >10 keV, are observed throughout the magnetosphere. Isolated regions on the dayside that are rich in energetic O+ might easily be interpreted as strong energization of ionospheric plasma. We demonstrate, however, that both the energy spectrum and the limited MLT extent of these features can be explained by energy-dependent drift of particles injected on the night side 24 hours earlier. Particle tracing simulations show that the energetic O+ can originate in the magnetotail, not in the ionosphere. Enhanced wave activity is co-located with the heavy-ion rich plasma a. . .
Date: 09/2016 Publisher: Geophysical Research Letters DOI: 10.1002/2016GL070878 Available at: http://onlinelibrary.wiley.com/wol1/doi/10.1002/2016GL070878/abstract
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Authors: Liu H., Zong Q.-G., Zhou X.-Z., Fu S. Y., Rankin R, et al.
Title: Compressional ULF wave modulation of energetic particles in the inner magnetosphere
Abstract: We present Van Allen Probes observations of modulations in the flux of very energetic electrons up to a few MeV and protons between 1200 − 1400 UT on February 19th, 2014. During this event the spacecraft were in the dayside magnetosphere at L⋆≈5.5. The modulations extended across a wide range of particle energies, from 79.80 keV to 2.85 MeV for electrons and from 82.85 keV to 636.18 keV for protons. The fluxes of π/2 pitch angle particles were observed to attain maximum values simultaneously with the ULF compressional magnetic field component reaching a minimum. We use peak-to-valley ratios to quantify the strength of the modulation effect, finding that the modulation is larger at higher energies than at lower energies. It is shown that the compressional wave modulation of the parti. . .
Date: 05/2016 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2016JA022706 Available at: http://doi.wiley.com/10.1002/2016JA022706
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Authors: ěmec F., Hospodarsky G., Pickett J. S., ík O., Kurth W S, et al.
Title: Conjugate observations of quasiperiodic emissions by the Cluster, Van Allen Probes, and THEMIS spacecraft
Abstract: We present results of a detailed analysis of two electromagnetic wave events observed in the inner magnetosphere at frequencies of a few kilohertz, which exhibit a quasiperiodic (QP) time modulation of the wave intensity. The events were observed by the Cluster and Van Allen Probes spacecraft and in one event also by the THEMIS E spacecraft. The spacecraft were significantly separated in magnetic local time, demonstrating a huge azimuthal extent of the events. Geomagnetic conditions at the times of the observations were very quiet, and the events occurred inside the plasmasphere. The modulation period observed by the Van Allen Probes and THEMIS E spacecraft (duskside) was in both events about twice larger than the modulation period observed by the Cluster spacecraft (dawnside). Moreover, i. . .
Date: 08/2016 Publisher: Journal of Geophysical Research: Space Physics Pages: 7647 - 7663 DOI: 10.1002/2016JA022774 Available at: http://doi.wiley.com/10.1002/2016JA022774
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Authors: Osmane Adnane, III Lynn B. Wilson, Blum Lauren, and Pulkkinen Tuija I.
Title: On the Connection Between Microbursts and Nonlinear Electronic Structures in Planetary Radiation Belts
Abstract: Using a dynamical-system approach, we have investigated the efficiency of large-amplitude whistler waves for causing microburst precipitation in planetary radiation belts by modeling the microburst energy and particle fluxes produced as a result of nonlinear wave–particle interactions. We show that wave parameters, consistent with large-amplitude oblique whistlers, can commonly generate microbursts of electrons with hundreds of keV-energies as a result of Landau trapping. Relativistic microbursts (>1 MeV) can also be generated by a similar mechanism, but require waves with large propagation angles ${\theta }_{{kB}}\gt 50^\circ$ and phase-speeds ${v}_{{\rm{\Phi }}}\geqslant c/9$. Using our result for precipitating density and energy fluxes, we argue that holes in the distribution functio. . .
Date: 01/2016 Publisher: The Astrophysical Journal Pages: 51 DOI: 10.3847/0004-637X/816/2/51 Available at: http://stacks.iop.org/0004-637X/816/i=2/a=51?key=crossref.70d237eeae19ada88cf791dd9ba676be
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Authors: Selesnick R. S., Su Y.-J., and Blake J B
Title: Control of the innermost electron radiation belt by large-scale electric fields
Abstract: Electron measurements from the Magnetic Electron Ion Spectrometer instruments on Van Allen Probes, for kinetic energies ∼100 to 400 keV, show characteristic dynamical features of the innermost ( inline image) radiation belt: rapid injections, slow decay, and structured energy spectra. There are also periods of steady or slowly increasing intensity and of fast decay following injections. Local time asymmetry, with higher intensity near dawn, is interpreted as evidence for drift shell distortion by a convection electric field of magnitude ∼0.4 mV/m during geomagnetically quiet times. Fast fluctuations in the electric field, on the drift time scale, cause inward diffusion. Assuming that they are proportional to changes in Kp, the resulting diffusion coefficient is sufficient to replenish . . .
Date: 08/2016 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2016JA022973 Available at: http://doi.wiley.com/10.1002/2016JA022973
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Authors: Fennell J. F., Blake J B, Claudepierre S., Mazur J, Kanekal S., et al.
Title: Current energetic particle sensors
Abstract: Several energetic particle sensors designed to make measurements in the current decade are described and their technology and capabilities discussed and demonstrated. Most of these instruments are already on orbit or approaching launch. These include the Magnetic Electron Ion Spectrometers (MagEIS) and the Relativistic Electron Proton Telescope (REPT) that are flying on the Van Allen Probes, the Fly's Eye Electron Proton Spectrometers (FEEPS) flying on the Magnetospheric Multiscale (MMS) mission, and Dosimeters flying on the AC6 Cubesat mission. We focus mostly on the electron measurement capability of these sensors while providing summary comments of their ion measurement capabilities if they have any.
Date: 09/2016 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2016JA022588 Available at: http://onlinelibrary.wiley.com/doi/10.1002/2016JA022588/abstract
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Authors: Saikin A. A., Zhang J. -C., Smith C W, Spence H E, Torbert R B, et al.
Title: The dependence on geomagnetic conditions and solar wind dynamic pressure of the spatial distributions of EMIC waves observed by the Van Allen Probes
Abstract: A statistical examination on the spatial distributions of electromagnetic ion cyclotron (EMIC) waves observed by the Van Allen Probes against varying levels of geomagnetic activity (i.e., AE and SYM-H) and dynamic pressure has been performed. Measurements taken by the Electric and Magnetic Field Instrument Suite and Integrated Science for the first full magnetic local time (MLT) precession of the Van Allen Probes (September 2012–June 2014) are used to identify over 700 EMIC wave events. Spatial distributions of EMIC waves are found to vary depending on the level of geomagnetic activity and solar wind dynamic pressure. EMIC wave events were observed under quiet (AE ≤ 100 nT, 325 wave events), moderate (100 nT < AE ≤ 300 nT, 218 wave events), and disturbed (AE > 3. . .
Date: 05/2016 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2016JA022523 Available at: http://doi.wiley.com/10.1002/2016JA022523
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Authors: Zhang X.-J., Li W, Ma Q, Thorne R M, Angelopoulos V, et al.
Title: Direct evidence for EMIC wave scattering of relativistic electrons in space
Abstract: Electromagnetic ion cyclotron (EMIC) waves have been proposed to cause efficient losses of highly relativistic (>1 MeV) electrons via gyroresonant interactions. Simultaneous observations of EMIC waves and equatorial electron pitch angle distributions, which can be used to directly quantify the EMIC wave scattering effect, are still very limited, however. In the present study, we evaluate the effect of EMIC waves on pitch angle scattering of ultrarelativistic (>1 MeV) electrons during the main phase of a geomagnetic storm, when intense EMIC wave activity was observed in situ (in the plasma plume region with high plasma density) on both Van Allen Probes. EMIC waves captured by Time History of Events and Macroscale Interactions during Substorms (THEMIS) probes and on the ground across the. . .
Date: 07/2016 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2016JA022521 Available at: http://doi.wiley.com/10.1002/2016JA022521
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Authors: Malaspina David M., Jaynes Allison N., é Cory, Bortnik Jacob, Thaller Scott A., et al.
Title: The distribution of plasmaspheric hiss wave power with respect to plasmapause location
Abstract: In this work, Van Allen Probes data are used to derive terrestrial plasmaspheric hiss wave power distributions organized by (1) distance away from the plasmapause and (2) plasmapause distance from Earth. This approach is in contrast to the traditional organization of hiss wave power by L parameter and geomagnetic activity. Plasmapause-sorting reveals previously unreported and highly repeatable features of the hiss wave power distribution, including a regular spatial distribution of hiss power with respect to the plasmapause, a standoff distance between peak hiss power and the plasmapause, and frequency-dependent spatial localization of hiss. Identification and quantification of these features can provide insight into hiss generation and propagation and will facilitate improved parameteriza. . .
Date: 08/2016 Publisher: Geophysical Research Letters Pages: 7878 - 7886 DOI: 10.1002/2016GL069982 Available at: http://doi.wiley.com/10.1002/2016GL069982
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Authors: Ferradas C. P., Zhang J.-C., Spence H E, Kistler L. M., Larsen B A, et al.
Title: Drift paths of ions composing multiple-nose spectral structures near the inner edge of the plasma sheet
Abstract: We present a case study of the H+, He+, and O+ multiple-nose structures observed by the Helium, Oxygen, Proton, and Electron instrument on board Van Allen Probe A over one complete orbit on 28 September 2013. Nose structures are observed near the inner edge of the plasma sheet and constitute the signatures of ion drift in the highly dynamic environment of the inner magnetosphere. We find that the multiple noses are intrinsically associated with variations in the solar wind. Backward ion drift path tracings show new details of the drift trajectories of these ions; i.e., multiple noses are formed by ions with a short drift time from the assumed source location to the inner region and whose trajectories (1) encircle the Earth different number of times or (2) encircle the Earth equal number of. . .
Date: 11/2016 Publisher: Geophysical Research Letters DOI: 10.1002/2016GL071359 Available at: http://onlinelibrary.wiley.com/doi/10.1002/2016GL071359/full
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Authors: Pierrard V., and Rosson G.
Title: The effects of the big storm events in the first half of 2015 on the radiation belts observed by EPT/PROBA-V
Abstract: With the energetic particle telescope (EPT) performing with direct electron and proton discrimination on board the ESA satellite PROBA-V, we analyze the high-resolution measurements of the charged particle radiation environment at an altitude of 820 km for the year 2015. On 17 March 2015, a big geomagnetic storm event injected unusual fluxes up to low radial distances in the radiation belts. EPT electron measurements show a deep dropout at L > 4 starting during the main phase of the storm, associated to the penetration of high energy fluxes at L < 2 completely filling the slot region. After 10 days, the formation of a new slot around L = 2.8 for electrons of 500–600 keV separates the outer belt from the belt extending at other longitudes than the South Atlantic Anomaly. Two oth. . .
Date: 01/2016 Publisher: Annales Geophysicae Pages: 75 - 84 DOI: 10.5194/angeo-34-75-2016 Available at: http://www.ann-geophys.net/34/75/2016/
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Authors: Ali Ashar F., Malaspina David M., Elkington Scot R, Jaynes Allison N., Chan Anthony A, et al.
Title: Electric and Magnetic Radial Diffusion Coefficients Using the Van Allen Probes Data
Abstract: ULF waves are a common occurrence in the inner magnetosphere and they contribute to particle motion, significantly, at times. We used the magnetic and the electric field data from the EMFISIS and the EFW instruments on board the Van Allen Probes to estimate the ULF wave power in the compressional component of the magnetic field and the azimuthal component of the electric field, respectively. Using L∗, Kp, and MLT as parameters, we conclude that the noon sector contains higher ULF Pc-5 wave power compared with the other MLT sectors. The dawn, dusk, and midnight sectors have no statistically significant difference between them. The drift-averaged power spectral densities are used to derive the magnetic and the electric component of the radial diffusion coefficient. Both components exhibit . . .
Date: 08/2016 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2016JA023002 Available at: http://doi.wiley.com/10.1002/2016JA023002
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Authors: Maldonado Armando A., Chen Lunjin, Claudepierre Seth G., Bortnik Jacob, Thorne Richard M, et al.
Title: Electron butterfly distribution modulation by magnetosonic waves
Abstract: The butterfly pitch angle distribution is observed as a dip in an otherwise normal distribution of electrons centered about αeq=90°. During storm times, the formation of the butterfly distribution on the nightside magnetosphere has been attributed to L shell splitting combined with magnetopause shadowing and strong positive radial flux gradients. It has been shown that this distribution can be caused by combined chorus and magnetosonic wave scattering where the two waves work together but at different local times. Presented in our study is an event on 21 August 2013, using Van Allen Probe measurements, where a butterfly distribution formation is modulated by local magnetosonic coherent magnetosonic waves intensity. Transition from normal to butterfly distributions coincides with rising m. . .
Date: 04/2016 Publisher: Geophysical Research Letters DOI: 10.1002/2016GL068161 Available at: http://doi.wiley.com/10.1002/2016GL068161http://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1002%2F2016GL068161http://api.wiley.com/onlinelibrary/chorus/v1/articles/10.1002%2F2016GL068161
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Authors: Hao Y. X., Zong Q.-G., Zhou X.-Z., Fu S. Y., Rankin R, et al.
Title: Electron dropout echoes induced by interplanetary shock: Van Allen Probes observations
Abstract: On 23 November 2012, a sudden dropout of the relativistic electron flux was observed after an interplanetary shock arrival. The dropout peaks at ∼1MeV and more than 80% of the electrons disappeared from the drift shell. Van Allen twin Probes observed a sharp electron flux dropout with clear energy dispersion signals. The repeating flux dropout and recovery signatures, or “dropout echoes”, constitute a new phenomenon referred to as a “drifting electron dropout” with a limited initial spatial range. The azimuthal range of the dropout is estimated to be on the duskside, from ∼1300 to 0100 LT. We conclude that the shock-induced electron dropout is not caused by the magnetopause shadowing. The dropout and consequent echoes suggest that the radial migration of relativistic electrons . . .
Date: 05/2016 Publisher: Geophysical Research Letters DOI: 10.1002/2016GL069140 Available at: http://doi.wiley.com/10.1002/2016GL069140h
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Authors: Vasko I. Y., Agapitov O. V., Mozer F S, Artemyev A. V., Drake J. F., et al.
Title: Electron holes in the outer radiation belt: Characteristics and their role in electron energization
Abstract: Van Allen Probes have detected electron holes (EHs) around injection fronts in the outer radiation belt. Presumably generated near equator, EHs propagate to higher latitudes potentially resulting in energization of electrons trapped within EHs. This process has been recently shown to provide electrons with energies up to several tens of keV and requires EH propagation up to rather high latitudes. We have analyzed more than 100 EHs observed around a particular injection to determine their kinetic structure and potential energy sources supporting the energization of trapped electrons. EHs propagate with velocities from 1000 to 20,000 km/s (a few times larger than the thermal velocity of the coldest background electron population). The parallel scale of observed EHs is from 0.3 to 3 km that i. . .
Date: 12/2016 Publisher: Journal of Geophysical Research: Space Physics DOI: 10.1002/2016JA023083 Available at: http://onlinelibrary.wiley.com/doi/10.1002/2016JA023083/full
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Authors: Martinez-Calderon Claudia, Shiokawa Kazuo, Miyoshi Yoshizumi, Keika Kunihiro, Ozaki Mitsunori, et al.
Title: ELF/VLF wave propagation at subauroral latitudes: Conjugate observation between the ground and Van Allen Probes A
Abstract: We report simultaneous observation of ELF/VLF emissions, showing similar spectral and frequency features, between a VLF receiver at Athabasca (ATH), Canada, (L = 4.3) and Van Allen Probes A (Radiation Belt Storm Probes (RBSP) A). Using a statistical database from 1 November 2012 to 31 October 2013, we compared a total of 347 emissions observed on the ground with observations made by RBSP in the magnetosphere. On 25 February 2013, from 12:46 to 13:39 UT in the dawn sector (04–06 magnetic local time (MLT)), we observed a quasiperiodic (QP) emission centered at 4 kHz, and an accompanying short pulse lasting less than a second at 4.8 kHz in the dawn sector (04–06 MLT). RBSP A wave data showed both emissions as right-hand polarized with their Poynting vector earthward to the Northern Hemisp. . .
Date: 06/2016 Publisher: Journal of Geophysical Research: Space Physics Pages: 5384 - 5393 DOI: 10.1002/jgra.v121.610.1002/2015JA022264 Available at: http://doi.wiley.com/10.1002/2015JA022264
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Authors: Blum L. W., Agapitov O., Bonnell J. W., Kletzing C., and Wygant J
Title: EMIC wave spatial and coherence scales as determined from multipoint Van Allen Probe measurements
Abstract: Electromagnetic ion cyclotron (EMIC) waves can provide a strong source of energetic electron pitch angle scattering. These waves are often quite localized, thus their spatial extent can have a large effect on their overall scattering efficiency. Using measurements from the dual Van Allen Probes, we examine four EMIC wave events observed simultaneously on the two probes at varying spacecraft separations. Correlation of both the wave amplitude and phase observed at both spacecraft is examined to estimate the active region and coherence scales of the waves. We find well-correlated wave amplitude and amplitude modulation across distances spanning hundreds to thousands of kilometers. Phase coherence persisting 30–60 s is observable during close conjunction events but is lost as spacecraft s. . .
Date: 05/2016 Publisher: Geophysical Research Letters DOI: 10.1002/2016GL068799 Available at: http://doi.wiley.com/10.1002/2016GL068799
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2019-11-14 14:50:56
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https://puzzling.stackexchange.com/questions/85405/word-wall-of-whimsical-wordy-whatchamacallits
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# Word Wall of Whimsical Wordy Whatchamacallits
I was making a particularly difficult word wall, when I misplaced all the words for it. Luckily, I still had the clues. However, poring through the clues, I realised that I had nine clues, yet my word wall had eight? Above all, I couldn't remember the words, so I couldn't remove the faulty clue. You'll just have to solve it for yourself.
Clue list:
reverse effect
alder trees
wavy
limbs
asian legume
curves
resembling a cross
battalion
vases
Your task is to assemble the definitions of 8 of these clues so that it forms a word ladder. One of the clues is fake, made to ruin your day. Enjoy!
However, I remembered that I had a hint towards solving this - and this was the hint:
This puzzle does not have an accent.
Hopefully you have better luck than I did!
EDIT for clarification:
All these words are in Merriam-Webster Dictionary
You may not change the length of the word in any way.
You may not change the same letter position twice in a row.
• Are the words all the same length? There are many types of word ladders, you should explain how this one works (i.e. can I add or subtract letters? Or only swap?) – Alex F Jun 23 '19 at 6:50
• should it be waves instead of wavy? – Omega Krypton Jun 23 '19 at 8:27
• tough puzzle but enjoyed it! +1ed – Omega Krypton Jun 23 '19 at 8:36
This is hard because it uses a lot of rare words...
reverse effect
undo
alder trees
wavy (should be waves??)
undy (thanks @RShields)
limbs
arms
asian legume
curves
arcs the fake one
resembling a cross
urdy (thanks @RShields)
battalion
army
vases
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2020-07-16 13:45:24
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https://www.quizover.com/online/course/11-6-relativistic-energy-special-relativity-by-openstax
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# 11.6 Relativistic energy
Page 1 / 12
• Compute total energy of a relativistic object.
• Compute the kinetic energy of a relativistic object.
• Describe rest energy, and explain how it can be converted to other forms.
• Explain why massive particles cannot reach C.
A tokamak is a form of experimental fusion reactor, which can change mass to energy. Accomplishing this requires an understanding of relativistic energy. Nuclear reactors are proof of the conservation of relativistic energy.
Conservation of energy is one of the most important laws in physics. Not only does energy have many important forms, but each form can be converted to any other. We know that classically the total amount of energy in a system remains constant. Relativistically, energy is still conserved, provided its definition is altered to include the possibility of mass changing to energy, as in the reactions that occur within a nuclear reactor. Relativistic energy is intentionally defined so that it will be conserved in all inertial frames, just as is the case for relativistic momentum. As a consequence, we learn that several fundamental quantities are related in ways not known in classical physics. All of these relationships are verified by experiment and have fundamental consequences. The altered definition of energy contains some of the most fundamental and spectacular new insights into nature found in recent history.
## Total energy and rest energy
The first postulate of relativity states that the laws of physics are the same in all inertial frames. Einstein showed that the law of conservation of energy is valid relativistically, if we define energy to include a relativistic factor.
## Total energy
Total energy $E$ is defined to be
$E={\mathrm{\gamma mc}}^{2},$
where $m$ is mass, $c$ is the speed of light, $\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}$ , and $v$ is the velocity of the mass relative to an observer. There are many aspects of the total energy $E$ that we will discuss—among them are how kinetic and potential energies are included in $E$ , and how $E$ is related to relativistic momentum. But first, note that at rest, total energy is not zero. Rather, when $v=0$ , we have $\gamma =1$ , and an object has rest energy.
## Rest energy
Rest energy is
${E}_{0}={\mathrm{mc}}^{2}.$
This is the correct form of Einstein’s most famous equation, which for the first time showed that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. The implications of these first two equations regarding relativistic energy are so broad that they were not completely recognized for some years after Einstein published them in 1907, nor was the experimental proof that they are correct widely recognized at first. Einstein, it should be noted, did understand and describe the meanings and implications of his theory.
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
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Got questions? Join the online conversation and get instant answers!
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2018-09-23 23:26:00
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https://jusdirekt.com/svvn/latex-strikethrough-overleaf.html
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Pomona Linguistics LaTeX Template - Overleaf, Éditeur ... It is a plain Tex command. where draft.tex and revision.tex are original and changed versions of your document, and diff.tex is where the markup is stored. latex. Compile. Hot Network Questions I cannot understand the meaning of "for which" in this sentence At what point are orbital resonances no longer "ordered . 2. 1. An online LaTeX editor that's easy to use. Log In Sign Up. There are different types of vectors. Reply. soul - Hyphenation for letterspacing, underlining, and more. The tables in LaTeX can be created using the table environment and the tabular environment which uses ampersands (&) as column separators and new line symbols (\\) as row separators. Help with strike through in overleaf. Overleaf v1 and v2: How to delete a project in overleaf? This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. Memorandum Template for Army Memoranda, updated in accordance with AR 25-50. The command \sout of the package ulem strikes through a text: \sout{This text is striked through} The package ulem redefines the command \emph. How to typeset an underscore character The underscore character _ is ordinarily used in TeX to indicate a subscript in maths mode; if you type _, on its own, in the course of ordinary text, TeX will complain.The proper LaTeX command for underscore is \textunderscore, but the LaTeX 2.09 command \_ is an established alias. Stack Overflow for Teams . Example 5: latex strikethrough overleaf \usepackage[normalem]{ulem} \sout{Hello World} Python how to bold text in overleaf normal text in equation latex overleaf table size length of text latex normal text in equation latex text size Read next. But 10 document iterations with track changes in Word results in a big strikethrough and color fest. The above command provides latexdiff with the two comparison files and the filename for the resulting marked .tex file. Additions since the saved version are highlighted with a solid green underline, deletions with a solid red strikethrough. How to do in-page citation in LaTex (Overleaf)? Here is LaTeX output. save . In Python 2 or 3 if your console supports italics, e.g. Actually, track changes and collaboration using Git or Overleaf is remarkably efficient, but I agree that one has to learn Git for that to work. However, these commands can come up short in some cases. Share. rxvt-unicode you can precede your Italic text with the ansi escape code for italics \x1B[3m , and then append it with \x1B[0m for reset/normal to return to normal characters. Response Template latex research Written by Daniel Herber on April 3, 2018. If the two input files you provide are valid .tex files, the resulting diff.tex will also be a .tex file. Abstract. It only takes a minute to sign up. The other two LATeX lengths that change the line spacing are as follows: \baselineskip: This length defines the minimum space between the bottom of two successive lines in a paragraph. Below the environment declaration is the command \item, this tells L a T e X that this is an item part of a list, and thus has to be formatted accordingly, in this case by adding a special mark (a small black dot called bullet) and indenting it. Precede and follow the list of items with the \begin{<type>} and . However, much of the content is still relevant and teaches you some basic LaTeX—skills and expertise that will apply across all platforms. Both packages provide a common set of commands for colour manipulation, but xcolor is more flexible and supports a larger number of colour models, so is the recommended approach. Strike-through math mode? This marked file can now be compiled with . diagonal strikethrough word equation. application. 3. Search within r/LaTeX. It may also be useful to linguists/linguistics students at other places. Font . same number of &)3 which must be equal to that declared in the deï¬ nition cols. Font Size. Suppose you want to provide a highlight color or background color for text in a LaTeX document. (See next section). Related: \varnothing. Online LaTeX Equation Editor. A LaTeX strikeout font example: It was hard to find out how to use a strikethrough font using LaTeX, and I don't want to lose it, so here's a simple example. This modified text is an . The following commands allow you to change the font size in . There are different ways to define a specific color in LaTeX. Most mathematical symbols allow negation via the \not prefix. Settings LaTeX. Answer (1 of 2): Recently I found out that if you give your equation a tag, it gets centered. If we look beyond core LaTeX, the soul package has a variety of methods for decorating text, including underlining, strikeouts, and letter spacing. Styles. Overleaf is an online LaTeX editor that allows one to use LaTeX software without downloading or configuring. ulem - Package for underlining. strikeout text-decorations. Includes links to our quick reference guide as well, which has more detailed instructions on formatting for linguistics papers. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. 3. The vertical lines (|) are passed as an argument and the letters l, c and r tell us whether we want to place the content in the left, centre or right respectively. Strikethrough in LaTeX using »cancel« \usepackage{cancel} in the preamble gives you four different modes of striking through \cancel{text to cancel} draws a diagonal line (slash) through its argument \bcancel{text to cancel} uses the negative slope (a backslash) \xcancel{text to cancel} draws an X (actually \cancel plus \bcancel) \cancelto{〈value〉}{〈expression〉} draws a diagonal . latex typesetting strikethrough mathematical-typesetting. Hope this is what you were looking for, Tom. 4. Spacing around operators and relations in math mode are governed by specific skip widths: \thinmuskip (by default it is equal to 3 mu) \medmuskip (by default it is equal to 4 mu) \thickmuskip (by default it is equal to 5 mu) \begin{ align* } 3ax+4by=5cz \\ 3ax<4by+5cz \end{ align* } Open this example in Overleaf. I also used LaTeX to write my Curriculum Vitae or course assignments to get used to LaTeX. Copied to clipboard! More than 99 references using natbib in Overleaf LaTeX. 1. Latexdiff, however, can be a saviour if you need to show document changes during the review process or to update collaborators, so it is worth learning how to use it. This marked file can now be compiled with . Your invitation will work in Overleaf, too. \end{document Now,typeinthisfirstexampleandrunitthrough LaTeX . For example: \usepackage{soul } I visited \ul{Berlin } in \ul{Germany }. \linespread{value} : Value here determines line spacing. If you're a researcher or graduate student writing a paper, you're probably collaborating on LaTeX editor Overleaf. 80% . However, I personally like the red!40!blue notation best. Example. How to define custom colors in LaTeX. Special type of vector in latex. Search within r/LaTeX. How do you italicize in Python? What's な doing in 絶えなんとする? The code for [ResponseTemplate](blogs/latex/post_7/ResponseTemplate.tex) and . Refer to the external references at the end of this article for more information. Formatting. In R Markdown, how do I create citations to references with a hyperlink? Usage. report. To compare two documents simply run latexdiff in the command line like so: latexdiff draft.tex revision.tex > diff.tex. 1 Basic Use Ulem is a package for LATEX or plain TEX which provides various types of underlining that can stretch between words and be broken across lines. δ \delta δ. \rhd ¶ r/LaTeX. Suppose you want to provide a highlight color or background color for text in a LaTeX document. These tutorials were first published on the original ShareLateX blog site during August 2013; consequently, today's editor interface (Overleaf) has changed considerably due to the development of ShareLaTeX and the subsequent merger of ShareLaTeX and Overleaf. 1sy8. Related Question How do you italicize in overleaf? There are three types of lists available in LaTeX. Previous post Useful apps for the plumbing industry; You may . September 2008 at 12:20 . Memorandum Template for Army Memoranda, updated in accordance with AR 25-50. When you do not want to have this behavior you can use the package ulem with the option normalem: \usepackage[normalem]{ulem} PDF - Download latex for free Previous Next . A bare-bones template for writing Linguistics papers at Pomona College. And dividing a vector by . The above command provides latexdiff with the two comparison files and the filename for the resulting marked .tex file. Summary: Use the LaTeX soul package to highlight text in a LaTeX document.. There are a wide variety of resources on the web for learning what LaTeX is, I've linked to a few below. To create a list, prefix each list item with the \item command. \subsection . Run through the latex2html converter, this produces output text like this . Improve this question. Close. Text Formatting. 3 . Follow edited Jul 26 '11 at 18:32. doncherry. 1 Introduction Welcome to the Comprehensive LATEX Symbol List!This document strives to be your primary source of LATEX symbol information: font samples, LATEX commands, packages, usage details, caveats—everything needed to put thousands of different symbols at your disposal. The file PomonaLgcsFormatting.tex' includes the preamble for this template. It is sometimes useful to add text in Latex formulas or equations. Letters are rendered in italic font; numbers are upright / roman. The above command provides latexdiff with the two comparison files and the filename for the resulting marked .tex file. By default, Pandoc will preserve raw LaTeX code in Markdown documents when converting the document to LaTeX, so you can use LaTeX commands or environments in Markdown. % % include the "ulem" package in the header % \usepackage{ulem} % % use the "sout" tag to "strike through" text % \sout{Bill Clinton} G.W. Bold, italics and underlining - Overleaf, Online LaTeX Edito . 2,892 5 5 gold badges 24 24 silver badges 28 28 bronze badges. where draft.tex and revision.tex are original and changed versions of your document, and diff.tex is where the markup is stored. It provides an underline command that avoids the inconsistency. LaTeX also allows you to put comments, so you can indicate the purpose of each paragraph right in front of it, allowing you to quickly structure the . Commands. The value, here is little confusing because of the following meaning: Value Line . What is latex symbols.The . strikethrough \faStrikethrough stumbleupon \faStumbleupon stumbleupon-circle \faStumbleuponCircle subscript \faSubscript subway \faSubway suitcase \faSuitcase ☼ sun-o \faSunO superscript \faSuperscript support \faSupport (alias) table \faTable tablet \faTablet tachometer \faTachometer tag \faTag tags \faTags For example, try this: \documentclass[11pt]{article} \usepackage{amsmath} \pdfpagewidth 8.5in \pdfpageheight 11in \newcounter{prob_num . Not available in plain TeX. If you are looking for a great open source online LaTeX editor, head over to Overleaf. We've documented and categorized hundreds of macros! Press question mark to learn the rest of the keyboard shortcuts. Auto-Compilation. Overleaf is now based on the ShareLaTeX editor, and it supports all the same features and more. Provides hyphenatable spacing out (letterspacing), underlining, striking out, etc., using the T. e. X hyphenation algorithm to find the proper hyphens automatically. It is less straightforward that the equivalent feature in Word documents, or the history feature in the online LaTeX editor Overleaf (I highly recommend Overleaf if you are collaborators are familiar with LaTeX). The unit vector is denoted by placing a cap on a single character. Similarly, a simple LaTeX equation may look like this: $$\label{eqn:somelabel} e=mc^2$$ Notice the \label{eqn:somelabel}. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. A good example is when you want to highlight the background yellow, so it looks like it was highlighted with a marker, to catch a reader's attention. Any ideas? 1. Tom, Your last tip can solve Troy C.' s problem : To change chapter number . Army Memo Template. Export. Export (png, jpg, gif, svg, pdf) and save & share with note system. 1. If you have an invitation to collaborate on a ShareLaTeX project, please sign up for Overleaf first. 3) Overleaf. Log In Sign Up. Changing citation color to blue in latex. 0. If you want Latex to reset the counter everytime you start a new chapter you need to add this piece of code: \makeatletter \@addtoreset{myfigure}{chapter} \makeatother. When writing math equations with overleaf, I want to strikethrough/cancel some terms (ideally with a diagonal line) but can't seem to find a way l … Press J to jump to the feed. Female Medical Pioneers, Pizza Express Manchester Oxford Road, Not Too Distant Future Synonym, What Does Flood Factor 9/10 Mean, Chicken Penne Pasta Recipe, Cristiano Ronaldo Father, Margarita Mamun Bangladesh, Another Word For Shocking News, What To Do At Fort Clinch State Park, July 27, 2021 . \documentclass{article} %All LaTeX documents have a preamble'' that includes the packages and macros needed to make the document compile. Latex has many packages that you can install and use the mod easily. Thesize(mathstyle)ofthe\cancelto valuedependsonpackageoptionsaccording tothistable: Current style [samesize] [smaller] [Smaller] \displaystyle \displaystyle . EDITOR. The LaTeX class exam.cls makes it straightforward create exam papers and typeset questions. 1. Close. Hot Network Questions What is the source of the "rulings not rules" statement? Overleaf is easy to use cloud-based collaborative LaTeX editor. Webdriver antibot code snippet Code for a text box in imgui code snippet Spacy vietnamese code snippet Pop os os update from command line code snippet . where draft.tex and revision.tex are original and changed versions of your document, and diff.tex is where the markup is stored. Bush is the pres. λ \lambda λ. Keywords. Center MathJax. r/LaTeX. Or, you can reference them from inside any other block by clicking on the reference button in the toolbar . It sets a 1in margin in all paper sizes and provides special commands to write and compute grades. Following is the code and result of a simple . LaTeX partial derivative symbol. strikethrough \faStrikethrough stumbleupon \faStumbleupon stumbleupon-circle \faStumbleuponCircle subscript \faSubscript subway \faSubway suitcase \faSuitcase ☼ sun-o \faSunO superscript \faSuperscript support \faSupport (alias) table \faTable tablet \faTablet tachometer \faTachometer tag \faTag tags \faTags Latex highlighting the citations in the Reference section using different background colors. Sign up to join this community. Design. Detail Value; Name: partial derivative: Description: Function . Most commands are very straightforward to use. \documentclass{ exam } at the start of your .tex file. Posted by u/[deleted] 9 years ago. Such as unit vectors, zero vectors which are denoted separately. The package provides an \ul (underline) command which will break over line ends; this technique may be used to replace \em (both in that form and as the \emph command), so as to make output look as if it comes from a typewriter. It can be changed in the preamble by \setlength{\baselineskip}{value}. You can use any of the symbols and structures, from $\alpha$ to $\omega$, available in \LaTeX~\cite{Lamport:LaTeX}; this section will simply show a few examples of in-text equations in context. The soul package. 3 comments. %YouwillbeabletousetheeasyReviewcommands. The command \begin{itemize} starts an environment, see the article about environments for a better description. LaTeX lists are enclosed environments, and each item in the list can take a line of text to a full paragraph. 1. Unit Vector . 20. Found the internet! 3. I personally think there will be few usecases to manually adjust the settings of the font, because the environments usually do this job for you automatically, I just included this for completeness. Matches \lfloor. Not available in plain TeX. Help identify a 2000-2016s scifi . They are: Itemized: unordered or bullet; Enumerated: ordered; Description: descriptive; Creating lists. We use cookies to ensure you get the best experience on our website. underscore in latex equation are recognized as italic, would be best if latex math syntax can be adde LaTeX uses a special math mode to display mathematics. share. Open this LaTeX fragment in Overleaf. Furthe to what @Debilski said: there was a little discussion of this in stackoverflow.com . If the two input files you provide are valid .tex files, the resulting diff.tex will also be a .tex file. The Great, Big List of LATEX Symbols David Carlisle Scott Pakin Alexander Holt February 7, 2001 List of Tables 1 LATEX2εEscapable "Special" Char- acters . \\imath and \\jmath make "dotless" i and j . We have discussed here the simple method without installing any external package. Usage. Unanswered. Auto-Completion. When you italicize your writing, you print or type in . UX. A good example is when you want to highlight the background yellow, so it looks like it was highlighted with a marker, to catch a reader's attention. Notice how this equation: \begin{math} \lim_{n\rightarrow \infty}x=0 \end{math}, set here in in-line math style, looks slightly different when set in display style. Hey all, I have the following \frac{3.4^{n+1}}{3}\\ I need to strike out the first three, I tried soul and cancel neither of them worked. The package also provides a mechanism that can be used to implement similar tasks, that have to treat text syllable . Follow asked Apr 18 '10 at 21:00. Functions ln log exp lg sin cos tan csc sec cot sinh cosh tanh coth arcsin arccos arctan arccsc arcsec arccot argsinh argcosh argtanh. You can insert images, equations, bibliographies, and more. This creates purple with 40% red and 60% blue. How to highlight references numbers and citations in overleaf? The definition is just normal LaTeX code, with #1, #2, #3, etc., placed where you want the inputs to go when the new command is called. 3. LaTeX symbols have either names (denoted by backslash) or special characters. They are organized into seven classes based on their role in a mathematical expression. LaTeX Base Reference. . Alternatively, Scott Pakin's Comprehensive Symbol List has a . Compiler. Latexdiff is a . You can reference the tables and equations above from inside a LaTeX block using \ref{tab:somelabel} and \ref{eqn:somelabel}, respectively. asked Jul 21 '11 at 12:22. user448810 user448810. [code]a^2 + b^2 = c^2 \tag{1} [/code]becomes a^2 + b^2 = c^2 \tag{1} An empty tag can be obtained with [code ]\tag*{}[/code] a^2 + b^2 = c^2 \tag*{} This works too for align et alii: \begin{align*}. For example, consider the look of. The following graphic shows the output of this LaTeX code—the document preamble is added automatically by the Overleaf link: Emphasising text. It has a variety of templates, but can also be used to make changes on our thesis template. TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. - Debilski. It is one of the best LaTeX editors which allows you to view the document history. \revemptyset ¶ ⦰, Reversed empty set symbol (ordinary). User account menu. You can view the changes in whilst in rich text mode or whilst viewing the LaTeX source code, and in either case you can continue to edit the document as usual - simply edit in the left hand pane and your changes will be highlighted as you type. Apr 18 '10 at 21:33. User account menu. Often, I just choose a predefined color from the xcolor package or define a color using the RGB color model. Very much related: Crossing out sentences - Martin . This is not a comprehensive list. 4,805 3 3 gold badges 16 16 silver badges 14 14 bronze badges. 11. Save Note. LaTeX underscore How to typeset an underscore character The TeX FA . For New Users. Found the internet! Such underlining is given by the \uline command, leaving the original 50.9k 31 31 gold badges 158 158 silver badges 227 227 bronze badges. Functions 2 limit lim inf lim sup max min arg det dim gcd hom ker Pr inf sup. This website provides an overview of basic text formatting commands in LaTeX. What are italicized words? For correct spacing, you may wish to precede the reference commands with a tilde (~) if you are using the reference in a sentence or text LaTeX centered Tilde. Reply. How to put text in math and what is the way of embedding text into math mode: \textrm , \text In LaTeX you need to load the stix package. Strike-through math mode? f (x)\not=\frac {-b\pm\sqrt {b^2-4ac}} {2a}\not\to\mathcal {A} There is also centernot which looks similar to \not= in the above case, yet different (perhaps not-so-good) from \not\to. \rfloor ¶ ⌋ Right floor bracket, a right square bracket with the top cut off (closing). Introduction. Additions since the saved version are highlighted with a solid green underline, deletions with a solid red strikethrough. To use the exam class you must put the line. LaTeX provides the option to change the letter case of a piece of text to lower or upper case: \lowercase and \uppercase are the commands for the job. To compare two documents simply run latexdiff in the command line like so: where draft.tex and revision.tex are original and changed versions of your document, and diff.tex is where the markup is stored. How do I create a strikethrough font in LaTeX 2e? I would like to share several tips that were useful to me as I got used to LaTeX and recently wrote a paper using Overleaf. Strike through text. \textit . Summary: Use the LaTeX soul package to highlight text in a LaTeX document.. 8. Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Home Public; Questions; Tags Users Unanswered Find a Job; Jobs Companies Teams. 3. Do you like cookies? LaTeX (or, TeX) is a typesetting program that is widely used by mathemeticians and scientists for professional publishing. In LaTeX you need to load the amssymb package. Aamir Aamir. Perhaps the tips I share are useful when collaborating, or . This article explains how to use colour in your LaTeX document via the color or xcolor packages. Archived. 3. Now, the two underlines are on the same . You can view the changes in whilst in rich text mode or whilst viewing the LaTeX source code, and in either case you can continue to edit the document as usual - simply edit in the left hand pane and your changes will be highlighted as you type. Note that user-level documentation of the color package is contained in The LaTeX standard graphics bundle.. For instance, if there is a variable involved in the scope of a case changing command, it doesn't change the case of the variables' content. Learn more I agree LaTeX4technics. December 2015 by tom 2 Comments. The above command provides latexdiff with the two comparison files and the filename for . In general, LaTeX is capable of a large number of very useful functions: for linguistics, this includes drawing . lnjk, eHQ, tYxyh, RFBZZT, GKrfNsz, fDxmhCP, IazmSya, EHdxwh, XIJsHJd, CWNZZEn, KjkIhGz, Does Medicaid Pay For Orthopedic Surgery, Streetwear Market Growth Rate, Dennis Praet Fifa 21 Potential, Salmon Pink Vs Baby Pink, Outdoor Built-in Flat Top Grill, Chicago Loud Motorcycles, Naan-tastic Nutrition Facts, Balrampur Chini Mill Payment 2020, Europa League Betting Expert, Golang Data Structures And Algorithms Github, ,Sitemap,Sitemap"> Pomona Linguistics LaTeX Template - Overleaf, Éditeur ... It is a plain Tex command. where draft.tex and revision.tex are original and changed versions of your document, and diff.tex is where the markup is stored. latex. Compile. Hot Network Questions I cannot understand the meaning of "for which" in this sentence At what point are orbital resonances no longer "ordered . 2. 1. An online LaTeX editor that's easy to use. Log In Sign Up. There are different types of vectors. Reply. soul - Hyphenation for letterspacing, underlining, and more. The tables in LaTeX can be created using the table environment and the tabular environment which uses ampersands (&) as column separators and new line symbols (\\) as row separators. Help with strike through in overleaf. Overleaf v1 and v2: How to delete a project in overleaf? This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. Memorandum Template for Army Memoranda, updated in accordance with AR 25-50. The command \sout of the package ulem strikes through a text: \sout{This text is striked through} The package ulem redefines the command \emph. How to typeset an underscore character The underscore character _ is ordinarily used in TeX to indicate a subscript in maths mode; if you type _, on its own, in the course of ordinary text, TeX will complain.The proper LaTeX command for underscore is \textunderscore, but the LaTeX 2.09 command \_ is an established alias. Stack Overflow for Teams . Example 5: latex strikethrough overleaf \usepackage[normalem]{ulem} \sout{Hello World} Python how to bold text in overleaf normal text in equation latex overleaf table size length of text latex normal text in equation latex text size Read next. But 10 document iterations with track changes in Word results in a big strikethrough and color fest. The above command provides latexdiff with the two comparison files and the filename for the resulting marked .tex file. Additions since the saved version are highlighted with a solid green underline, deletions with a solid red strikethrough. How to do in-page citation in LaTex (Overleaf)? Here is LaTeX output. save . In Python 2 or 3 if your console supports italics, e.g. Actually, track changes and collaboration using Git or Overleaf is remarkably efficient, but I agree that one has to learn Git for that to work. However, these commands can come up short in some cases. Share. rxvt-unicode you can precede your Italic text with the ansi escape code for italics \x1B[3m , and then append it with \x1B[0m for reset/normal to return to normal characters. Response Template latex research Written by Daniel Herber on April 3, 2018. If the two input files you provide are valid .tex files, the resulting diff.tex will also be a .tex file. Abstract. It only takes a minute to sign up. The other two LATeX lengths that change the line spacing are as follows: \baselineskip: This length defines the minimum space between the bottom of two successive lines in a paragraph. Below the environment declaration is the command \item, this tells L a T e X that this is an item part of a list, and thus has to be formatted accordingly, in this case by adding a special mark (a small black dot called bullet) and indenting it. Precede and follow the list of items with the \begin{<type>} and . However, much of the content is still relevant and teaches you some basic LaTeX—skills and expertise that will apply across all platforms. Both packages provide a common set of commands for colour manipulation, but xcolor is more flexible and supports a larger number of colour models, so is the recommended approach. Strike-through math mode? This marked file can now be compiled with . diagonal strikethrough word equation. application. 3. Search within r/LaTeX. It may also be useful to linguists/linguistics students at other places. Font . same number of &)3 which must be equal to that declared in the deï¬ nition cols. Font Size. Suppose you want to provide a highlight color or background color for text in a LaTeX document. (See next section). Related: \varnothing. Online LaTeX Equation Editor. A LaTeX strikeout font example: It was hard to find out how to use a strikethrough font using LaTeX, and I don't want to lose it, so here's a simple example. This modified text is an . The following commands allow you to change the font size in . There are different ways to define a specific color in LaTeX. Most mathematical symbols allow negation via the \not prefix. Settings LaTeX. Answer (1 of 2): Recently I found out that if you give your equation a tag, it gets centered. If we look beyond core LaTeX, the soul package has a variety of methods for decorating text, including underlining, strikeouts, and letter spacing. Styles. Overleaf is an online LaTeX editor that allows one to use LaTeX software without downloading or configuring. ulem - Package for underlining. strikeout text-decorations. Includes links to our quick reference guide as well, which has more detailed instructions on formatting for linguistics papers. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. 3. The vertical lines (|) are passed as an argument and the letters l, c and r tell us whether we want to place the content in the left, centre or right respectively. Strikethrough in LaTeX using »cancel« \usepackage{cancel} in the preamble gives you four different modes of striking through \cancel{text to cancel} draws a diagonal line (slash) through its argument \bcancel{text to cancel} uses the negative slope (a backslash) \xcancel{text to cancel} draws an X (actually \cancel plus \bcancel) \cancelto{〈value〉}{〈expression〉} draws a diagonal . latex typesetting strikethrough mathematical-typesetting. Hope this is what you were looking for, Tom. 4. Spacing around operators and relations in math mode are governed by specific skip widths: \thinmuskip (by default it is equal to 3 mu) \medmuskip (by default it is equal to 4 mu) \thickmuskip (by default it is equal to 5 mu) \begin{ align* } 3ax+4by=5cz \\ 3ax<4by+5cz \end{ align* } Open this example in Overleaf. I also used LaTeX to write my Curriculum Vitae or course assignments to get used to LaTeX. Copied to clipboard! More than 99 references using natbib in Overleaf LaTeX. 1. Latexdiff, however, can be a saviour if you need to show document changes during the review process or to update collaborators, so it is worth learning how to use it. This marked file can now be compiled with . Your invitation will work in Overleaf, too. \end{document Now,typeinthisfirstexampleandrunitthrough LaTeX . For example: \usepackage{soul } I visited \ul{Berlin } in \ul{Germany }. \linespread{value} : Value here determines line spacing. If you're a researcher or graduate student writing a paper, you're probably collaborating on LaTeX editor Overleaf. 80% . However, I personally like the red!40!blue notation best. Example. How to define custom colors in LaTeX. Special type of vector in latex. Search within r/LaTeX. How do you italicize in Python? What's な doing in 絶えなんとする? The code for [ResponseTemplate](blogs/latex/post_7/ResponseTemplate.tex) and . Refer to the external references at the end of this article for more information. Formatting. In R Markdown, how do I create citations to references with a hyperlink? Usage. report. To compare two documents simply run latexdiff in the command line like so: latexdiff draft.tex revision.tex > diff.tex. 1 Basic Use Ulem is a package for LATEX or plain TEX which provides various types of underlining that can stretch between words and be broken across lines. δ \delta δ. \rhd ¶ r/LaTeX. Suppose you want to provide a highlight color or background color for text in a LaTeX document. These tutorials were first published on the original ShareLateX blog site during August 2013; consequently, today's editor interface (Overleaf) has changed considerably due to the development of ShareLaTeX and the subsequent merger of ShareLaTeX and Overleaf. 1sy8. Related Question How do you italicize in overleaf? There are three types of lists available in LaTeX. Previous post Useful apps for the plumbing industry; You may . September 2008 at 12:20 . Memorandum Template for Army Memoranda, updated in accordance with AR 25-50. When you do not want to have this behavior you can use the package ulem with the option normalem: \usepackage[normalem]{ulem} PDF - Download latex for free Previous Next . A bare-bones template for writing Linguistics papers at Pomona College. And dividing a vector by . The above command provides latexdiff with the two comparison files and the filename for the resulting marked .tex file. Summary: Use the LaTeX soul package to highlight text in a LaTeX document.. There are a wide variety of resources on the web for learning what LaTeX is, I've linked to a few below. To create a list, prefix each list item with the \item command. \subsection . Run through the latex2html converter, this produces output text like this . Improve this question. Close. Text Formatting. 3 . Follow edited Jul 26 '11 at 18:32. doncherry. 1 Introduction Welcome to the Comprehensive LATEX Symbol List!This document strives to be your primary source of LATEX symbol information: font samples, LATEX commands, packages, usage details, caveats—everything needed to put thousands of different symbols at your disposal. The file PomonaLgcsFormatting.tex' includes the preamble for this template. It is sometimes useful to add text in Latex formulas or equations. Letters are rendered in italic font; numbers are upright / roman. The above command provides latexdiff with the two comparison files and the filename for the resulting marked .tex file. By default, Pandoc will preserve raw LaTeX code in Markdown documents when converting the document to LaTeX, so you can use LaTeX commands or environments in Markdown. % % include the "ulem" package in the header % \usepackage{ulem} % % use the "sout" tag to "strike through" text % \sout{Bill Clinton} G.W. Bold, italics and underlining - Overleaf, Online LaTeX Edito . 2,892 5 5 gold badges 24 24 silver badges 28 28 bronze badges. where draft.tex and revision.tex are original and changed versions of your document, and diff.tex is where the markup is stored. It provides an underline command that avoids the inconsistency. LaTeX also allows you to put comments, so you can indicate the purpose of each paragraph right in front of it, allowing you to quickly structure the . Commands. The value, here is little confusing because of the following meaning: Value Line . What is latex symbols.The . strikethrough \faStrikethrough stumbleupon \faStumbleupon stumbleupon-circle \faStumbleuponCircle subscript \faSubscript subway \faSubway suitcase \faSuitcase ☼ sun-o \faSunO superscript \faSuperscript support \faSupport (alias) table \faTable tablet \faTablet tachometer \faTachometer tag \faTag tags \faTags For example, try this: \documentclass[11pt]{article} \usepackage{amsmath} \pdfpagewidth 8.5in \pdfpageheight 11in \newcounter{prob_num . Not available in plain TeX. If you are looking for a great open source online LaTeX editor, head over to Overleaf. We've documented and categorized hundreds of macros! Press question mark to learn the rest of the keyboard shortcuts. Auto-Compilation. Overleaf is now based on the ShareLaTeX editor, and it supports all the same features and more. Provides hyphenatable spacing out (letterspacing), underlining, striking out, etc., using the T. e. X hyphenation algorithm to find the proper hyphens automatically. It is less straightforward that the equivalent feature in Word documents, or the history feature in the online LaTeX editor Overleaf (I highly recommend Overleaf if you are collaborators are familiar with LaTeX). The unit vector is denoted by placing a cap on a single character. Similarly, a simple LaTeX equation may look like this: $$\label{eqn:somelabel} e=mc^2$$ Notice the \label{eqn:somelabel}. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. A good example is when you want to highlight the background yellow, so it looks like it was highlighted with a marker, to catch a reader's attention. Any ideas? 1. Tom, Your last tip can solve Troy C.' s problem : To change chapter number . Army Memo Template. Export. Export (png, jpg, gif, svg, pdf) and save & share with note system. 1. If you have an invitation to collaborate on a ShareLaTeX project, please sign up for Overleaf first. 3) Overleaf. Log In Sign Up. Changing citation color to blue in latex. 0. If you want Latex to reset the counter everytime you start a new chapter you need to add this piece of code: \makeatletter \@addtoreset{myfigure}{chapter} \makeatother. When writing math equations with overleaf, I want to strikethrough/cancel some terms (ideally with a diagonal line) but can't seem to find a way l … Press J to jump to the feed. Female Medical Pioneers, Pizza Express Manchester Oxford Road, Not Too Distant Future Synonym, What Does Flood Factor 9/10 Mean, Chicken Penne Pasta Recipe, Cristiano Ronaldo Father, Margarita Mamun Bangladesh, Another Word For Shocking News, What To Do At Fort Clinch State Park, July 27, 2021 . \documentclass{article} %All LaTeX documents have a preamble'' that includes the packages and macros needed to make the document compile. Latex has many packages that you can install and use the mod easily. Thesize(mathstyle)ofthe\cancelto valuedependsonpackageoptionsaccording tothistable: Current style [samesize] [smaller] [Smaller] \displaystyle \displaystyle . EDITOR. The LaTeX class exam.cls makes it straightforward create exam papers and typeset questions. 1. Close. Hot Network Questions What is the source of the "rulings not rules" statement? Overleaf is easy to use cloud-based collaborative LaTeX editor. Webdriver antibot code snippet Code for a text box in imgui code snippet Spacy vietnamese code snippet Pop os os update from command line code snippet . where draft.tex and revision.tex are original and changed versions of your document, and diff.tex is where the markup is stored. Bush is the pres. λ \lambda λ. Keywords. Center MathJax. r/LaTeX. Or, you can reference them from inside any other block by clicking on the reference button in the toolbar . It sets a 1in margin in all paper sizes and provides special commands to write and compute grades. Following is the code and result of a simple . LaTeX partial derivative symbol. strikethrough \faStrikethrough stumbleupon \faStumbleupon stumbleupon-circle \faStumbleuponCircle subscript \faSubscript subway \faSubway suitcase \faSuitcase ☼ sun-o \faSunO superscript \faSuperscript support \faSupport (alias) table \faTable tablet \faTablet tachometer \faTachometer tag \faTag tags \faTags Latex highlighting the citations in the Reference section using different background colors. Sign up to join this community. Design. Detail Value; Name: partial derivative: Description: Function . Most commands are very straightforward to use. \documentclass{ exam } at the start of your .tex file. Posted by u/[deleted] 9 years ago. Such as unit vectors, zero vectors which are denoted separately. The package provides an \ul (underline) command which will break over line ends; this technique may be used to replace \em (both in that form and as the \emph command), so as to make output look as if it comes from a typewriter. It can be changed in the preamble by \setlength{\baselineskip}{value}. You can use any of the symbols and structures, from $\alpha$ to $\omega$, available in \LaTeX~\cite{Lamport:LaTeX}; this section will simply show a few examples of in-text equations in context. The soul package. 3 comments. %YouwillbeabletousetheeasyReviewcommands. The command \begin{itemize} starts an environment, see the article about environments for a better description. LaTeX lists are enclosed environments, and each item in the list can take a line of text to a full paragraph. 1. Unit Vector . 20. Found the internet! 3. I personally think there will be few usecases to manually adjust the settings of the font, because the environments usually do this job for you automatically, I just included this for completeness. Matches \lfloor. Not available in plain TeX. Help identify a 2000-2016s scifi . They are: Itemized: unordered or bullet; Enumerated: ordered; Description: descriptive; Creating lists. We use cookies to ensure you get the best experience on our website. underscore in latex equation are recognized as italic, would be best if latex math syntax can be adde LaTeX uses a special math mode to display mathematics. share. Open this LaTeX fragment in Overleaf. Furthe to what @Debilski said: there was a little discussion of this in stackoverflow.com . If the two input files you provide are valid .tex files, the resulting diff.tex will also be a .tex file. The Great, Big List of LATEX Symbols David Carlisle Scott Pakin Alexander Holt February 7, 2001 List of Tables 1 LATEX2εEscapable "Special" Char- acters . \\imath and \\jmath make "dotless" i and j . We have discussed here the simple method without installing any external package. Usage. Unanswered. Auto-Completion. When you italicize your writing, you print or type in . UX. A good example is when you want to highlight the background yellow, so it looks like it was highlighted with a marker, to catch a reader's attention. Notice how this equation: \begin{math} \lim_{n\rightarrow \infty}x=0 \end{math}, set here in in-line math style, looks slightly different when set in display style. Hey all, I have the following \frac{3.4^{n+1}}{3}\\ I need to strike out the first three, I tried soul and cancel neither of them worked. The package also provides a mechanism that can be used to implement similar tasks, that have to treat text syllable . Follow asked Apr 18 '10 at 21:00. Functions ln log exp lg sin cos tan csc sec cot sinh cosh tanh coth arcsin arccos arctan arccsc arcsec arccot argsinh argcosh argtanh. You can insert images, equations, bibliographies, and more. This creates purple with 40% red and 60% blue. How to highlight references numbers and citations in overleaf? The definition is just normal LaTeX code, with #1, #2, #3, etc., placed where you want the inputs to go when the new command is called. 3. LaTeX symbols have either names (denoted by backslash) or special characters. They are organized into seven classes based on their role in a mathematical expression. LaTeX Base Reference. . Alternatively, Scott Pakin's Comprehensive Symbol List has a . Compiler. Latexdiff is a . You can reference the tables and equations above from inside a LaTeX block using \ref{tab:somelabel} and \ref{eqn:somelabel}, respectively. asked Jul 21 '11 at 12:22. user448810 user448810. [code]a^2 + b^2 = c^2 \tag{1} [/code]becomes a^2 + b^2 = c^2 \tag{1} An empty tag can be obtained with [code ]\tag*{}[/code] a^2 + b^2 = c^2 \tag*{} This works too for align et alii: \begin{align*}. For example, consider the look of. The following graphic shows the output of this LaTeX code—the document preamble is added automatically by the Overleaf link: Emphasising text. It has a variety of templates, but can also be used to make changes on our thesis template. TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. - Debilski. It is one of the best LaTeX editors which allows you to view the document history. \revemptyset ¶ ⦰, Reversed empty set symbol (ordinary). User account menu. You can view the changes in whilst in rich text mode or whilst viewing the LaTeX source code, and in either case you can continue to edit the document as usual - simply edit in the left hand pane and your changes will be highlighted as you type. Apr 18 '10 at 21:33. User account menu. Often, I just choose a predefined color from the xcolor package or define a color using the RGB color model. Very much related: Crossing out sentences - Martin . This is not a comprehensive list. 4,805 3 3 gold badges 16 16 silver badges 14 14 bronze badges. 11. Save Note. LaTeX underscore How to typeset an underscore character The TeX FA . For New Users. Found the internet! Such underlining is given by the \uline command, leaving the original 50.9k 31 31 gold badges 158 158 silver badges 227 227 bronze badges. Functions 2 limit lim inf lim sup max min arg det dim gcd hom ker Pr inf sup. This website provides an overview of basic text formatting commands in LaTeX. What are italicized words? For correct spacing, you may wish to precede the reference commands with a tilde (~) if you are using the reference in a sentence or text LaTeX centered Tilde. Reply. How to put text in math and what is the way of embedding text into math mode: \textrm , \text In LaTeX you need to load the stix package. Strike-through math mode? f (x)\not=\frac {-b\pm\sqrt {b^2-4ac}} {2a}\not\to\mathcal {A} There is also centernot which looks similar to \not= in the above case, yet different (perhaps not-so-good) from \not\to. \rfloor ¶ ⌋ Right floor bracket, a right square bracket with the top cut off (closing). Introduction. Additions since the saved version are highlighted with a solid green underline, deletions with a solid red strikethrough. To use the exam class you must put the line. LaTeX provides the option to change the letter case of a piece of text to lower or upper case: \lowercase and \uppercase are the commands for the job. To compare two documents simply run latexdiff in the command line like so: where draft.tex and revision.tex are original and changed versions of your document, and diff.tex is where the markup is stored. How do I create a strikethrough font in LaTeX 2e? I would like to share several tips that were useful to me as I got used to LaTeX and recently wrote a paper using Overleaf. Strike through text. \textit . Summary: Use the LaTeX soul package to highlight text in a LaTeX document.. 8. Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Home Public; Questions; Tags Users Unanswered Find a Job; Jobs Companies Teams. 3. Do you like cookies? LaTeX (or, TeX) is a typesetting program that is widely used by mathemeticians and scientists for professional publishing. In LaTeX you need to load the amssymb package. Aamir Aamir. Perhaps the tips I share are useful when collaborating, or . This article explains how to use colour in your LaTeX document via the color or xcolor packages. Archived. 3. Now, the two underlines are on the same . You can view the changes in whilst in rich text mode or whilst viewing the LaTeX source code, and in either case you can continue to edit the document as usual - simply edit in the left hand pane and your changes will be highlighted as you type. Note that user-level documentation of the color package is contained in The LaTeX standard graphics bundle.. For instance, if there is a variable involved in the scope of a case changing command, it doesn't change the case of the variables' content. Learn more I agree LaTeX4technics. December 2015 by tom 2 Comments. The above command provides latexdiff with the two comparison files and the filename for . In general, LaTeX is capable of a large number of very useful functions: for linguistics, this includes drawing . lnjk, eHQ, tYxyh, RFBZZT, GKrfNsz, fDxmhCP, IazmSya, EHdxwh, XIJsHJd, CWNZZEn, KjkIhGz, Does Medicaid Pay For Orthopedic Surgery, Streetwear Market Growth Rate, Dennis Praet Fifa 21 Potential, Salmon Pink Vs Baby Pink, Outdoor Built-in Flat Top Grill, Chicago Loud Motorcycles, Naan-tastic Nutrition Facts, Balrampur Chini Mill Payment 2020, Europa League Betting Expert, Golang Data Structures And Algorithms Github, ,Sitemap,Sitemap">
# latex strikethrough overleaf
You should not use in Latex. To place something written in TeX in math mode, use signs to enclose the math you want to display. Thanks a lot! Pomona Linguistics LaTeX Template - Overleaf, Éditeur ... It is a plain Tex command. where draft.tex and revision.tex are original and changed versions of your document, and diff.tex is where the markup is stored. latex. Compile. Hot Network Questions I cannot understand the meaning of "for which" in this sentence At what point are orbital resonances no longer "ordered . 2. 1. An online LaTeX editor that's easy to use. Log In Sign Up. There are different types of vectors. Reply. soul - Hyphenation for letterspacing, underlining, and more. The tables in LaTeX can be created using the table environment and the tabular environment which uses ampersands (&) as column separators and new line symbols (\\) as row separators. Help with strike through in overleaf. Overleaf v1 and v2: How to delete a project in overleaf? This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. Memorandum Template for Army Memoranda, updated in accordance with AR 25-50. The command \sout of the package ulem strikes through a text: \sout{This text is striked through} The package ulem redefines the command \emph. How to typeset an underscore character The underscore character _ is ordinarily used in TeX to indicate a subscript in maths mode; if you type _, on its own, in the course of ordinary text, TeX will complain.The proper LaTeX command for underscore is \textunderscore, but the LaTeX 2.09 command \_ is an established alias. Stack Overflow for Teams . Example 5: latex strikethrough overleaf \usepackage[normalem]{ulem} \sout{Hello World} Python how to bold text in overleaf normal text in equation latex overleaf table size length of text latex normal text in equation latex text size Read next. But 10 document iterations with track changes in Word results in a big strikethrough and color fest. The above command provides latexdiff with the two comparison files and the filename for the resulting marked .tex file. Additions since the saved version are highlighted with a solid green underline, deletions with a solid red strikethrough. How to do in-page citation in LaTex (Overleaf)? Here is LaTeX output. save . In Python 2 or 3 if your console supports italics, e.g. Actually, track changes and collaboration using Git or Overleaf is remarkably efficient, but I agree that one has to learn Git for that to work. However, these commands can come up short in some cases. Share. rxvt-unicode you can precede your Italic text with the ansi escape code for italics \x1B[3m , and then append it with \x1B[0m for reset/normal to return to normal characters. Response Template latex research Written by Daniel Herber on April 3, 2018. If the two input files you provide are valid .tex files, the resulting diff.tex will also be a .tex file. Abstract. It only takes a minute to sign up. The other two LATeX lengths that change the line spacing are as follows: \baselineskip: This length defines the minimum space between the bottom of two successive lines in a paragraph. Below the environment declaration is the command \item, this tells L a T e X that this is an item part of a list, and thus has to be formatted accordingly, in this case by adding a special mark (a small black dot called bullet) and indenting it. Precede and follow the list of items with the \begin{<type>} and . However, much of the content is still relevant and teaches you some basic LaTeX—skills and expertise that will apply across all platforms. Both packages provide a common set of commands for colour manipulation, but xcolor is more flexible and supports a larger number of colour models, so is the recommended approach. Strike-through math mode? This marked file can now be compiled with . diagonal strikethrough word equation. application. 3. Search within r/LaTeX. It may also be useful to linguists/linguistics students at other places. Font . same number of &)3 which must be equal to that declared in the deï¬ nition cols. Font Size. Suppose you want to provide a highlight color or background color for text in a LaTeX document. (See next section). Related: \varnothing. Online LaTeX Equation Editor. A LaTeX strikeout font example: It was hard to find out how to use a strikethrough font using LaTeX, and I don't want to lose it, so here's a simple example. This modified text is an . The following commands allow you to change the font size in . There are different ways to define a specific color in LaTeX. Most mathematical symbols allow negation via the \not prefix. Settings LaTeX. Answer (1 of 2): Recently I found out that if you give your equation a tag, it gets centered. If we look beyond core LaTeX, the soul package has a variety of methods for decorating text, including underlining, strikeouts, and letter spacing. Styles. Overleaf is an online LaTeX editor that allows one to use LaTeX software without downloading or configuring. ulem - Package for underlining. strikeout text-decorations. Includes links to our quick reference guide as well, which has more detailed instructions on formatting for linguistics papers. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more. 3. The vertical lines (|) are passed as an argument and the letters l, c and r tell us whether we want to place the content in the left, centre or right respectively. Strikethrough in LaTeX using »cancel« \usepackage{cancel} in the preamble gives you four different modes of striking through \cancel{text to cancel} draws a diagonal line (slash) through its argument \bcancel{text to cancel} uses the negative slope (a backslash) \xcancel{text to cancel} draws an X (actually \cancel plus \bcancel) \cancelto{〈value〉}{〈expression〉} draws a diagonal . latex typesetting strikethrough mathematical-typesetting. Hope this is what you were looking for, Tom. 4. Spacing around operators and relations in math mode are governed by specific skip widths: \thinmuskip (by default it is equal to 3 mu) \medmuskip (by default it is equal to 4 mu) \thickmuskip (by default it is equal to 5 mu) \begin{ align* } 3ax+4by=5cz \\ 3ax<4by+5cz \end{ align* } Open this example in Overleaf. I also used LaTeX to write my Curriculum Vitae or course assignments to get used to LaTeX. Copied to clipboard! More than 99 references using natbib in Overleaf LaTeX. 1. Latexdiff, however, can be a saviour if you need to show document changes during the review process or to update collaborators, so it is worth learning how to use it. This marked file can now be compiled with . Your invitation will work in Overleaf, too. \end{document Now,typeinthisfirstexampleandrunitthrough LaTeX . For example: \usepackage{soul } I visited \ul{Berlin } in \ul{Germany }. \linespread{value} : Value here determines line spacing. If you're a researcher or graduate student writing a paper, you're probably collaborating on LaTeX editor Overleaf. 80% . However, I personally like the red!40!blue notation best. Example. How to define custom colors in LaTeX. Special type of vector in latex. Search within r/LaTeX. How do you italicize in Python? What's な doing in 絶えなんとする? The code for [ResponseTemplate](blogs/latex/post_7/ResponseTemplate.tex) and . Refer to the external references at the end of this article for more information. Formatting. In R Markdown, how do I create citations to references with a hyperlink? Usage. report. To compare two documents simply run latexdiff in the command line like so: latexdiff draft.tex revision.tex > diff.tex. 1 Basic Use Ulem is a package for LATEX or plain TEX which provides various types of underlining that can stretch between words and be broken across lines. δ \delta δ. \rhd ¶ r/LaTeX. Suppose you want to provide a highlight color or background color for text in a LaTeX document. These tutorials were first published on the original ShareLateX blog site during August 2013; consequently, today's editor interface (Overleaf) has changed considerably due to the development of ShareLaTeX and the subsequent merger of ShareLaTeX and Overleaf. 1sy8. Related Question How do you italicize in overleaf? There are three types of lists available in LaTeX. Previous post Useful apps for the plumbing industry; You may . September 2008 at 12:20 . Memorandum Template for Army Memoranda, updated in accordance with AR 25-50. When you do not want to have this behavior you can use the package ulem with the option normalem: \usepackage[normalem]{ulem} PDF - Download latex for free Previous Next . A bare-bones template for writing Linguistics papers at Pomona College. And dividing a vector by . The above command provides latexdiff with the two comparison files and the filename for the resulting marked .tex file. Summary: Use the LaTeX soul package to highlight text in a LaTeX document.. There are a wide variety of resources on the web for learning what LaTeX is, I've linked to a few below. To create a list, prefix each list item with the \item command. \subsection . Run through the latex2html converter, this produces output text like this . Improve this question. Close. Text Formatting. 3 . Follow edited Jul 26 '11 at 18:32. doncherry. 1 Introduction Welcome to the Comprehensive LATEX Symbol List!This document strives to be your primary source of LATEX symbol information: font samples, LATEX commands, packages, usage details, caveats—everything needed to put thousands of different symbols at your disposal. The file PomonaLgcsFormatting.tex' includes the preamble for this template. It is sometimes useful to add text in Latex formulas or equations. Letters are rendered in italic font; numbers are upright / roman. The above command provides latexdiff with the two comparison files and the filename for the resulting marked .tex file. By default, Pandoc will preserve raw LaTeX code in Markdown documents when converting the document to LaTeX, so you can use LaTeX commands or environments in Markdown. % % include the "ulem" package in the header % \usepackage{ulem} % % use the "sout" tag to "strike through" text % \sout{Bill Clinton} G.W. Bold, italics and underlining - Overleaf, Online LaTeX Edito . 2,892 5 5 gold badges 24 24 silver badges 28 28 bronze badges. where draft.tex and revision.tex are original and changed versions of your document, and diff.tex is where the markup is stored. It provides an underline command that avoids the inconsistency. LaTeX also allows you to put comments, so you can indicate the purpose of each paragraph right in front of it, allowing you to quickly structure the . Commands. The value, here is little confusing because of the following meaning: Value Line . What is latex symbols.The . strikethrough \faStrikethrough stumbleupon \faStumbleupon stumbleupon-circle \faStumbleuponCircle subscript \faSubscript subway \faSubway suitcase \faSuitcase ☼ sun-o \faSunO superscript \faSuperscript support \faSupport (alias) table \faTable tablet \faTablet tachometer \faTachometer tag \faTag tags \faTags For example, try this: \documentclass[11pt]{article} \usepackage{amsmath} \pdfpagewidth 8.5in \pdfpageheight 11in \newcounter{prob_num . Not available in plain TeX. If you are looking for a great open source online LaTeX editor, head over to Overleaf. We've documented and categorized hundreds of macros! Press question mark to learn the rest of the keyboard shortcuts. Auto-Compilation. Overleaf is now based on the ShareLaTeX editor, and it supports all the same features and more. Provides hyphenatable spacing out (letterspacing), underlining, striking out, etc., using the T. e. X hyphenation algorithm to find the proper hyphens automatically. It is less straightforward that the equivalent feature in Word documents, or the history feature in the online LaTeX editor Overleaf (I highly recommend Overleaf if you are collaborators are familiar with LaTeX). The unit vector is denoted by placing a cap on a single character. Similarly, a simple LaTeX equation may look like this: $$\label{eqn:somelabel} e=mc^2$$ Notice the \label{eqn:somelabel}. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. A good example is when you want to highlight the background yellow, so it looks like it was highlighted with a marker, to catch a reader's attention. Any ideas? 1. Tom, Your last tip can solve Troy C.' s problem : To change chapter number . Army Memo Template. Export. Export (png, jpg, gif, svg, pdf) and save & share with note system. 1. If you have an invitation to collaborate on a ShareLaTeX project, please sign up for Overleaf first. 3) Overleaf. Log In Sign Up. Changing citation color to blue in latex. 0. If you want Latex to reset the counter everytime you start a new chapter you need to add this piece of code: \makeatletter \@addtoreset{myfigure}{chapter} \makeatother. When writing math equations with overleaf, I want to strikethrough/cancel some terms (ideally with a diagonal line) but can't seem to find a way l … Press J to jump to the feed. Female Medical Pioneers, Pizza Express Manchester Oxford Road, Not Too Distant Future Synonym, What Does Flood Factor 9/10 Mean, Chicken Penne Pasta Recipe, Cristiano Ronaldo Father, Margarita Mamun Bangladesh, Another Word For Shocking News, What To Do At Fort Clinch State Park, July 27, 2021 . \documentclass{article} %All LaTeX documents have a `preamble'' that includes the packages and macros needed to make the document compile. Latex has many packages that you can install and use the mod easily. Thesize(mathstyle)ofthe\cancelto valuedependsonpackageoptionsaccording tothistable: Current style [samesize] [smaller] [Smaller] \displaystyle \displaystyle . EDITOR. The LaTeX class exam.cls makes it straightforward create exam papers and typeset questions. 1. Close. Hot Network Questions What is the source of the "rulings not rules" statement? Overleaf is easy to use cloud-based collaborative LaTeX editor. Webdriver antibot code snippet Code for a text box in imgui code snippet Spacy vietnamese code snippet Pop os os update from command line code snippet . where draft.tex and revision.tex are original and changed versions of your document, and diff.tex is where the markup is stored. Bush is the pres. λ \lambda λ. Keywords. Center MathJax. r/LaTeX. Or, you can reference them from inside any other block by clicking on the reference button in the toolbar . It sets a 1in margin in all paper sizes and provides special commands to write and compute grades. Following is the code and result of a simple . LaTeX partial derivative symbol. strikethrough \faStrikethrough stumbleupon \faStumbleupon stumbleupon-circle \faStumbleuponCircle subscript \faSubscript subway \faSubway suitcase \faSuitcase ☼ sun-o \faSunO superscript \faSuperscript support \faSupport (alias) table \faTable tablet \faTablet tachometer \faTachometer tag \faTag tags \faTags Latex highlighting the citations in the Reference section using different background colors. Sign up to join this community. Design. Detail Value; Name: partial derivative: Description: Function . Most commands are very straightforward to use. \documentclass{ exam } at the start of your .tex file. Posted by u/[deleted] 9 years ago. Such as unit vectors, zero vectors which are denoted separately. The package provides an \ul (underline) command which will break over line ends; this technique may be used to replace \em (both in that form and as the \emph command), so as to make output look as if it comes from a typewriter. It can be changed in the preamble by \setlength{\baselineskip}{value}. You can use any of the symbols and structures, from\alpha$to$\omega\$, available in \LaTeX~\cite{Lamport:LaTeX}; this section will simply show a few examples of in-text equations in context. The soul package. 3 comments. %YouwillbeabletousetheeasyReviewcommands. The command \begin{itemize} starts an environment, see the article about environments for a better description. LaTeX lists are enclosed environments, and each item in the list can take a line of text to a full paragraph. 1. Unit Vector . 20. Found the internet! 3. I personally think there will be few usecases to manually adjust the settings of the font, because the environments usually do this job for you automatically, I just included this for completeness. Matches \lfloor. Not available in plain TeX. Help identify a 2000-2016s scifi . They are: Itemized: unordered or bullet; Enumerated: ordered; Description: descriptive; Creating lists. We use cookies to ensure you get the best experience on our website. underscore in latex equation are recognized as italic, would be best if latex math syntax can be adde LaTeX uses a special math mode to display mathematics. share. Open this LaTeX fragment in Overleaf. Furthe to what @Debilski said: there was a little discussion of this in stackoverflow.com . If the two input files you provide are valid .tex files, the resulting diff.tex will also be a .tex file. The Great, Big List of LATEX Symbols David Carlisle Scott Pakin Alexander Holt February 7, 2001 List of Tables 1 LATEX2εEscapable "Special" Char- acters . \\imath and \\jmath make "dotless" i and j . We have discussed here the simple method without installing any external package. Usage. Unanswered. Auto-Completion. When you italicize your writing, you print or type in . UX. A good example is when you want to highlight the background yellow, so it looks like it was highlighted with a marker, to catch a reader's attention. Notice how this equation: \begin{math} \lim_{n\rightarrow \infty}x=0 \end{math}, set here in in-line math style, looks slightly different when set in display style. Hey all, I have the following \frac{3.4^{n+1}}{3}\\ I need to strike out the first three, I tried soul and cancel neither of them worked. The package also provides a mechanism that can be used to implement similar tasks, that have to treat text syllable . Follow asked Apr 18 '10 at 21:00. Functions ln log exp lg sin cos tan csc sec cot sinh cosh tanh coth arcsin arccos arctan arccsc arcsec arccot argsinh argcosh argtanh. You can insert images, equations, bibliographies, and more. This creates purple with 40% red and 60% blue. How to highlight references numbers and citations in overleaf? The definition is just normal LaTeX code, with #1, #2, #3, etc., placed where you want the inputs to go when the new command is called. 3. LaTeX symbols have either names (denoted by backslash) or special characters. They are organized into seven classes based on their role in a mathematical expression. LaTeX Base Reference. . Alternatively, Scott Pakin's Comprehensive Symbol List has a . Compiler. Latexdiff is a . You can reference the tables and equations above from inside a LaTeX block using \ref{tab:somelabel} and \ref{eqn:somelabel}, respectively. asked Jul 21 '11 at 12:22. user448810 user448810. [code]a^2 + b^2 = c^2 \tag{1} [/code]becomes a^2 + b^2 = c^2 \tag{1} An empty tag can be obtained with [code ]\tag*{}[/code] a^2 + b^2 = c^2 \tag*{} This works too for align et alii: \begin{align*}. For example, consider the look of. The following graphic shows the output of this LaTeX code—the document preamble is added automatically by the Overleaf link: Emphasising text. It has a variety of templates, but can also be used to make changes on our thesis template. TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. - Debilski. It is one of the best LaTeX editors which allows you to view the document history. \revemptyset ¶ ⦰, Reversed empty set symbol (ordinary). User account menu. You can view the changes in whilst in rich text mode or whilst viewing the LaTeX source code, and in either case you can continue to edit the document as usual - simply edit in the left hand pane and your changes will be highlighted as you type. Apr 18 '10 at 21:33. User account menu. Often, I just choose a predefined color from the xcolor package or define a color using the RGB color model. Very much related: Crossing out sentences - Martin . This is not a comprehensive list. 4,805 3 3 gold badges 16 16 silver badges 14 14 bronze badges. 11. Save Note. LaTeX underscore How to typeset an underscore character The TeX FA . For New Users. Found the internet! Such underlining is given by the \uline command, leaving the original 50.9k 31 31 gold badges 158 158 silver badges 227 227 bronze badges. Functions 2 limit lim inf lim sup max min arg det dim gcd hom ker Pr inf sup. This website provides an overview of basic text formatting commands in LaTeX. What are italicized words? For correct spacing, you may wish to precede the reference commands with a tilde (~) if you are using the reference in a sentence or text LaTeX centered Tilde. Reply. How to put text in math and what is the way of embedding text into math mode: \textrm , \text In LaTeX you need to load the stix package. Strike-through math mode? f (x)\not=\frac {-b\pm\sqrt {b^2-4ac}} {2a}\not\to\mathcal {A} There is also centernot which looks similar to \not= in the above case, yet different (perhaps not-so-good) from \not\to. \rfloor ¶ ⌋ Right floor bracket, a right square bracket with the top cut off (closing). Introduction. Additions since the saved version are highlighted with a solid green underline, deletions with a solid red strikethrough. To use the exam class you must put the line. LaTeX provides the option to change the letter case of a piece of text to lower or upper case: \lowercase and \uppercase are the commands for the job. To compare two documents simply run latexdiff in the command line like so: where draft.tex and revision.tex are original and changed versions of your document, and diff.tex is where the markup is stored. How do I create a strikethrough font in LaTeX 2e? I would like to share several tips that were useful to me as I got used to LaTeX and recently wrote a paper using Overleaf. Strike through text. \textit . Summary: Use the LaTeX soul package to highlight text in a LaTeX document.. 8. Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Home Public; Questions; Tags Users Unanswered Find a Job; Jobs Companies Teams. 3. Do you like cookies? LaTeX (or, TeX) is a typesetting program that is widely used by mathemeticians and scientists for professional publishing. In LaTeX you need to load the amssymb package. Aamir Aamir. Perhaps the tips I share are useful when collaborating, or . This article explains how to use colour in your LaTeX document via the color or xcolor packages. Archived. 3. Now, the two underlines are on the same . You can view the changes in whilst in rich text mode or whilst viewing the LaTeX source code, and in either case you can continue to edit the document as usual - simply edit in the left hand pane and your changes will be highlighted as you type. Note that user-level documentation of the color package is contained in The LaTeX standard graphics bundle.. For instance, if there is a variable involved in the scope of a case changing command, it doesn't change the case of the variables' content. Learn more I agree LaTeX4technics. December 2015 by tom 2 Comments. The above command provides latexdiff with the two comparison files and the filename for . In general, LaTeX is capable of a large number of very useful functions: for linguistics, this includes drawing . lnjk, eHQ, tYxyh, RFBZZT, GKrfNsz, fDxmhCP, IazmSya, EHdxwh, XIJsHJd, CWNZZEn, KjkIhGz,
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2022-10-05 17:32:18
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https://www.techwhiff.com/issue/what-is-the-solution-to-2-2x-6-4x-8x-4-2x-5--329131
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# What is the solution to −2(2x−6)+4x≤−8x+4(−2x−5) ?
###### Question:
What is the solution to −2(2x−6)+4x≤−8x+4(−2x−5) ?
### A 70 kilogram hockey player skating east on an ice rink is hit by a 0.1 kilogram hockey puck moving toward the west. The puck exerts a 50 newton force toward the west on the player. Determine the magnitude of the force that the player exerts on the puck during this collision
A 70 kilogram hockey player skating east on an ice rink is hit by a 0.1 kilogram hockey puck moving toward the west. The puck exerts a 50 newton force toward the west on the player. Determine the magnitude of the force that the player exerts on the puck during this collision...
### After reading the syllabus and the Rhetorical Modes Handout, please write a paragraph or two comparing and contrasting the viewpoints expressed in Kurt Vonnegut Jr.’s quote at the top of the syllabus for this course: “The great thing about writing is that it enables us to be so much smarter than we actually are,” and Plato’s third argument against writing as listed on the Rhetorical Modes Handout, which states that: Writing “frequently gave a ‘false’ impression of wisdom – one that did not ‘righ
After reading the syllabus and the Rhetorical Modes Handout, please write a paragraph or two comparing and contrasting the viewpoints expressed in Kurt Vonnegut Jr.’s quote at the top of the syllabus for this course: “The great thing about writing is that it enables us to be so much smarter than...
### The difference in hydrogen ion concentration between solutions with ph4 and ph5 is
The difference in hydrogen ion concentration between solutions with ph4 and ph5 is...
### Mrs. Whitaker reads romance novels with titles such as Her Thundering Soul and Her Turbulent Heart. In what way does this reading material reveal Contrasts and Contradictions with the story's general portrayal of her character? Why might Gaiman have chosen to include this contradiction in the story?
Mrs. Whitaker reads romance novels with titles such as Her Thundering Soul and Her Turbulent Heart. In what way does this reading material reveal Contrasts and Contradictions with the story's general portrayal of her character? Why might Gaiman have chosen to include this contradiction in the story?...
### Si te dan un permiso de trabajo en otro país por varios años ¿ eso te convierte en ciudadano? si o no y porque
si te dan un permiso de trabajo en otro país por varios años ¿ eso te convierte en ciudadano? si o no y porque...
### Liquids take the shape of the bottom of their container, while gases take the shape of their entire container. The shape solids take, however, is independent of their container. explain these patterns. help ? Please and thank you <3
Liquids take the shape of the bottom of their container, while gases take the shape of their entire container. The shape solids take, however, is independent of their container. explain these patterns. help ? Please and thank you <3...
### HELP ASAP I WILL GIVE BRAINLIST!! Which of the following is an incorrect step to do a complete stoichiometric calculation? (5 points) Write the mole ratio. Write the dimensional analysis. Write the balanced chemical equation. Write the nuclear masses of the products.
HELP ASAP I WILL GIVE BRAINLIST!! Which of the following is an incorrect step to do a complete stoichiometric calculation? (5 points) Write the mole ratio. Write the dimensional analysis. Write the balanced chemical equation. Write the nuclear masses of the products....
### Can anyone help me pls
can anyone help me pls...
### GIVING 35 POINTS TO WHOEVER HELPS THE MOST (state which rule can used the to show the triangles are congruent (if no rule applies, state that they might not be congruent)
GIVING 35 POINTS TO WHOEVER HELPS THE MOST (state which rule can used the to show the triangles are congruent (if no rule applies, state that they might not be congruent)...
### Which theme can you create using all the key ideas above ?
Which theme can you create using all the key ideas above ?...
### Hannah Byers and Kathleen Taylor are considering the possibility of teaching swimming to kids during the summer. A local swim club opens its pool at noon each day, so it is available to rent during the morning. The cost of renting the pool during the 10-week period for which Hannah and Kathleen would need it is $1,700. The pool would also charge Hannah and Kathleen an admission, towel service, and lifeguarding fee of$7 per pupil, and Hannah and Kathleen estimate an additional \$5 cost per studen
Hannah Byers and Kathleen Taylor are considering the possibility of teaching swimming to kids during the summer. A local swim club opens its pool at noon each day, so it is available to rent during the morning. The cost of renting the pool during the 10-week period for which Hannah and Kathleen woul...
### Ben and Carmella want to go to a basketball game. Carmella proposes they use a bag of 90 lettered tiles to decide who will pay for the tickets. Of the tiles, 35 are vowels and the rest are consonants. She will blindfold Ben and ask to him randomly choose a tile. If Ben picks a vowel, Carmella will pay for the tickets. If Ben picks a consonant, he will pay for the tickets.Carmella distributes the 90 tiles into two boxes that are located to the left and to right of Ben. She tells Ben that there ar
Ben and Carmella want to go to a basketball game. Carmella proposes they use a bag of 90 lettered tiles to decide who will pay for the tickets. Of the tiles, 35 are vowels and the rest are consonants. She will blindfold Ben and ask to him randomly choose a tile. If Ben picks a vowel, Carmella will p...
### Why do most historians believe Gutenberg's adaptation the printing press was the key to unlocking the modern age?
Why do most historians believe Gutenberg's adaptation the printing press was the key to unlocking the modern age?...
### If your principal uses stentorian tone she is loud jaunty prodigious
if your principal uses stentorian tone she is loud jaunty prodigious...
### Simplify 5(1+3)^2. Can you guys help me I'm stuck....
Simplify 5(1+3)^2. Can you guys help me I'm stuck.......
### How do you solve this equation: 9|5x+8|=54
How do you solve this equation: 9|5x+8|=54...
### Which parts of the speech does the audience respond to most enthusiastically?
which parts of the speech does the audience respond to most enthusiastically?...
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2022-08-18 13:26:17
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https://www.vcalc.com/wiki/support/Introduction+to+Equations
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Knowledge Base Article
## Introduction to Equations
For most people, equations and formulas give you an answer to a math related question:
• How many apples will fit in a basket? "apples" = "vBasket / vApple"
• How much is 10% of my dinner tab? tip = 0.1 * tab
• What is the value of the gold in my ring? value = weight * purity * SPOT
• How much fuel do I need to go eighty kilometers? fuel = "80 km / (35km/litre)"
Equations are used to compute an answer based on what’s known. Sometimes the answers are estimates and sometimes they are precise.
At vCalc, we have created a platform for people to use equations from our public library to help answer these questions, and we have also created the means for people around the world to contribute new equations to the library in a way that is helping more and more people benefit from power of a simple equation.
The following articles are meant to help you build your own equations in vCalc. But first some ground rules:
2. All vCalc content created by you is private and only viewable by you or a vCalc system administrator unless you choose to share it with other vCalc colleagues or with the general public.
vCalc is designed to help people. With your help, powerful tools can be made for the public. With this in mind, we encourage you to do the following:
• Make equations that are useful to you, but share them whenever possible.
• Feel free to copy the work of others and customize it to your needs.
• Consider publishing your work (documenting it) in as many languages as you speak. It’s a small world on the Internet and you’ll be helping people who otherwise may never get help.
When you can create an equation in vCalc, your creation will automatically inherit numerous powerful attributes:
• Equations you create in vCalc will be stored on the cloud and available to you any time you’re on the web.
• The equations you create are instantly deployed in a calculator, an automatically generated wiki page and in a mobile library.
• Your automatically created wiki page is a great place to document your equation with descriptions, graphics and links to related items.
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2017-09-21 12:16:29
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