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https://www.controlbooth.com/threads/express-24-48.7321/
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Express 24/48
midgetgreen11
Active Member
So me and a couple others at my school are looking at the Express 24/48. Has anybody had any PROBLEMS with this board?
soundlight
Well-Known Member
I've used over a dozen individual express consoles (different 24/48s, 46/96s, 125s, and 250s), and all of them have performed flawlessly for me. It's an excellent high school board, and virtually indestructible, so it's a pretty safe bet to buy a used one. The ones that I've used from rental houses are pretty nice, if anything they have a little bit of tape residue under the faders and the keys are a bit worn down.
Go for it.
gafftaper
Senior Team
Senior Team
Fight Leukemia
So are you talking about buying a new or used one? If you are talking used and you can find a deal then go for it.
However, I would agree with Icewolf that you should at least consider the other options if you are talking about buying a new one. You may be able to find a newer BETTER console at a very competitive price.
zzzeus
Member
at my school we got the expression 3 , very nice bord
gafftaper
Senior Team
Senior Team
Fight Leukemia
at my school we got the expression 3 , very nice bord
Same thing goes for Expression 3. It's a solid board that if you can get a good deal on a used one great. There are a lot of great new products coming out right now from ETC and Strand that you should look at carefully. These older boards are going away... they may still be in use for another 20 years in some places, but they are past their prime and there is better available at a decent price right now.
soundlight
Well-Known Member
For your situation, as in school that is going to need to do solid fundraising for the whole price, the Express is fine. It's a solid board, and is very robust and reliable.
midgetgreen11
midgetgreen11
Active Member
haha thanks, that's the answer i was looking for.
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2020-02-19 16:08:16
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https://butsurimemo.com/en/category/mathematics/
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# Mathematics
[mathjax]
I described a method of solving complex integrals by theorems before, but some complex integrals can not be solved only applying theorems. However, even with such complex integrals, it becomes possible to solve by setting an integration route yourself.
Furthermore, real integrals can be solved as complex integrals. In this article, I’ll write about some integrals that can be solved by applying this.
[mathjax]
Consider the next power series.
$$\sum_{n=0}^∞ a_nz^n･･･(1)$$
We take the absolute value of $a_n$ and $z$ which make up this power series.
$$\sum_{n=0}^∞ |a_n||z|^n･･･(2)$$
This power series (2) is called absolute convergence if it is finite. Then if the power series (2) converges, the power series (1) also converges.
The convergence radius R refers to the boundary between the area the power series (2) converges absolutely and doesn’t. If $|z|<R$, the power series (1) converges absolutely. Then if $|z|>R$, it diverges. When $|z|=R$, some power series converge, others diverge. Therefore, we need to think about converging or diverging for each power series.
[mathjax]
Laplace transform refers to the following conversion $F(s)$.
$$F(s)=\int_0^∞ f(t)e^{-st}dt$$
The function before Laplace transform $f(t)$ depends on $t$, but the function after Laplace transform $F(s)$ depends on $s$.
[mathjax]
The eigenvalue $λ$ and the eigenvector $φ$ of the matrix ${\bf A}$ satisfy the following equation.
$${\bf A}φ=λφ$$
Let’s see how to find this eigenvalue $λ$ and eigenvector $φ$.
[mathjax]
There are various kinds of methods of solving complex integrations, such as using the Cauchy’s integral theorem, the residue theorem or specifying a specific integration route. It is necessary to use the appropriate method for a complex integrations you want to solve. Of course, you should remember as many kinds of the methods as possible.
[mathjax]
A complex-valued function $f(z)$ is said to be a holomorphic function if it is differentiable at every point in its domain.
By the way, let $f(z)$ be defined as
$$f(z)=u(x,y)+iv(x,y),$$
where $z$ is a complex variable, $u(x,y)$ and $v(x,y)$ are real-number functions, and $x$ and $y$ are real variables. The following two equations are collectively called Cauchy-Riemann’s equations.
$$\frac{∂u(x,y)}{∂x}=\frac{∂v(x,y)}{∂y}$$
$$\frac{∂u(x,y)}{∂y}=-\frac{∂v(x,y)}{∂x}$$
You can determine the complex-valued function $f(z)$ is a holomorphic function when $u(x, y)$ and $v(x, y)$ satisfy the Cauchy-Riemann equations above.
[mathjax]
I thought about Fourier series expansion before, but this time it is an introduction of Fourier transform.
Fourier transform
$$F(k)=\int_{-∞}^{∞} f(x)e^{-ikx} dx$$
inverse Fourier transform
$$f(x)=\frac{1}{2π}\int_{-∞}^{∞} F(k)e^{ikx} dk$$
Suppose a wave is given by the function $f(x)$. Using Fourier transform, the function of wave $f (x)$ is expressed as a function $F(k)$ of wavenumber $k$. $F(k)$ is the function which decomposes $f(x)$ into the sine waves of all wavenumbers $k$ and expresses the size of each the sine wave as a function. You can convert the wave $f (x)$ to $F(k)$ by Fourier transform and \ (F (k) \) to $F(k)$ by inverse Fourier transform.
[mathjax]
Fourier series expansion and Fourier transform are completely different. In this article, I’ll write about Fourier series expansion.
Fourier series expansion method is to express the periodic function $f(x)$ by using the sum of $sin$ and $cos$. Using the method, the periodic function $f(x)$ whose period is $L$ can be expressed as follows.
$$f(x)=\frac{a_0}{2}+\sum_{n=1}^{∞} \left[ {a_{n}cos(\frac{2πnx}{L})}+b_{n}sin(\frac{2πnx}{L}) \right]$$
$$a_0=\frac{2}{L}\int_{-L/2}^{L/2} f(x) dx$$
$$a_n=\frac{2}{L}\int_{-L/2}^{L/2} f(x)cos(\frac{2πnx}{L}) dx$$
$$b_n=\frac{2}{L}\int_{-L/2}^{L/2} f(x)sin(\frac{2πnx}{L}) dx$$
[mathjax]
A 2 × 2 determinant is expressed as follows.
$$\left[\begin{array}{cc}a_{11}&a_{12}\\ a_{21}&a_{22}\\ \end{array}\right]=a_{11}a_{22}-a_{21}a_{12}$$
Then a 3 × 3 determinant is expressed as follows.
$$\left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\\ \end{array}\right]\\=a_{11}a_{22}a_{33}+a_{21}a_{32}a_{13}+a_{31}a_{23}a_{12}\\-a_{31}a_{22}a_{13}-a_{21}a_{12}a_{33}-a_{11}a_{23}a_{32}$$
So far you can memorize easily, but it is hard to memorize from 4 × 4. In this article, I’ll show you how to find a 4×4 determinant by looking at the elements of the first row.
[mathjax]
Taylor expansion around x = a is to express an arbitrary function f (x) in a form like the expression below.
$$f(x)=\sum_{k=0}^{∞}{\frac{f^{(k)}(a)}{k!}(x-a)^k}･･･(1)$$
When a = 0, it can be written as follows. This case is called “Maclaurin’s expansion” specially.
$$f(x)=\sum_{k=0}^{∞}{\frac{f^{(k)}(0)}{k!}x^k}$$
$f^{(k)}(x)$ means the $k$th order derivative of the function $f(x)$.
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2022-07-04 08:41:29
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http://www.physicsforums.com/showthread.php?p=4155581
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# Reduced row echelon form of a square matrix
by Bipolarity
Tags: echelon, form, matrix, reduced, square
P: 783 I am wondering about the relation betwen RRE forms and identity matrices. Consider the reduced row echelon form of any square matrix. Must this reduced row echelon form of the matrix necessarily be an identity matrix? I would suppose yes, but can this fact be proven? Could anyone provide an outline of the proof, or provide the link? Thanks much. BiP
P: 333
Quote by Bipolarity I am wondering about the relation betwen RRE forms and identity matrices. Consider the reduced row echelon form of any square matrix. Must this reduced row echelon form of the matrix necessarily be an identity matrix?
Of course not. As a trivial example, take a square zero matrix, i.e. a square matrix such that all its elements are zeros. Or, more generally, any square marix with at least one zero row, or column. In fact, you can easily write down lots of square RRE matrices which are not identity matrices.
In general, a square matrix A is row equivalent to (i.e. its RRE is) the identity matrix of he same size if and only if A is invertible.
P: 783
Quote by Erland Of course not. As a trivial example, take a square zero matrix, i.e. a square matrix such that all its elements are zeros. Or, more generally, any square marix with at least one zero row, or column. In fact, you can easily write down lots of square RRE matrices which are not identity matrices. In general, a square matrix A is row equivalent to (i.e. its RRE is) the identity matrix of he same size if and only if A is invertible.
What if I add the condition that the matrix square has no zero rows? Then is it necessarily the case that its RRE form is equivalent to the identity matrix (of the same size)?
BiP
Engineering
HW Helper
Thanks
P: 6,920
Reduced row echelon form of a square matrix
Quote by Bipolarity What if I add the condition that the matrix square has no zero rows? Then is it necessarily the case that its RRE form is equivalent to the identity matrix (of the same size)?
No, for example
$$\begin{pmatrix}1 & 1 \\ 1 & 1\end{pmatrix}$$
P: 783
Quote by AlephZero No, for example $$\begin{pmatrix}1 & 1 \\ 1 & 1\end{pmatrix}$$
But how is that matrix in RRE form? The leading 1 in the second row is not strictly to the right of the leading 1 of the first row?
BiP
Engineering Sci Advisor HW Helper Thanks P: 6,920 Of course it's not in RRE form! You asked if a square matrix with no zero rows always has an identity matrix for its RRE. That matrix has no zero rows. Reduce that matrix to RRE form and see what you get. If you do that yourself, you might see WHY your idea is wrong (and even discover the right idea), which is more useful than just being told "your idea is wrong".
P: 350 AlephZero is saying to start with that matrix and then do row operations to put it into RRE form. You will find that you end up with a matrix that is not the identity matrix. Since the given matrix has no zero rows, it is a counter example to your modified question.
P: 783 I see! Thanks!! The reduction gave me $$\begin{pmatrix}1 & 1 \\ 0 & 0\end{pmatrix}$$ What about if the RRE form of the matrix is a square matrix with no zero rows? In that case is the RRE form become an identity matrix? BiP
P: 333
Quote by Bipolarity What about if the RRE form of the matrix is a square matrix with no zero rows? In that case is the RRE form become an identity matrix?
Yes, that's right. It is easily verified if we carefully examine the definition of RRE and its consequences in the case of a square matrix. What about pivot rows and columns and zero rows in that case?
Related Discussions Linear & Abstract Algebra 1 Calculus & Beyond Homework 1 Precalculus Mathematics Homework 7 Linear & Abstract Algebra 1 Introductory Physics Homework 1
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2014-07-23 00:00:18
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https://stats.stackexchange.com/questions/154909/simulating-data-from-multilevel-logistic-regression
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# Simulating Data from Multilevel Logistic Regression
I want to simulate data from multilevel logistic regression .
I focus on the following multilevel logistic model with one explanatory variable at level 1 (individual level) and one explanatory variable at level 2 (group level) :
$$\text{logit}(p_{ij})=\pi_{0j}+\pi_{1j}x_{ij}\ldots (1)$$ $$\pi_{0j}=\gamma_{00}+\gamma_{01}z_j+u_{0j}\ldots (2)$$ $$\pi_{1j}=\gamma_{10}+\gamma_{11}z_j+u_{1j}\ldots (3)$$
where , $u_{0j}\sim N(0,\sigma_0^2)$ , $u_{1j}\sim N(0,\sigma_1^2)$ , $\text{cov}(u_{0j},u_{1j})=\sigma_{01}$
I prefer R and started to write codes for the simulation as :
set.seed(36)
x <- rnorm(1000) ### individual level variable , x_ij
z <- rnorm(1000) ### group level variable , z_j
If I have initial value for $\gamma_{00}=-1 , \gamma_{01}=0.3,\gamma_{10}=0.3,\gamma_{11}=0.3$ , how can I generate $\pi_{0j},\pi_{1j}$ since there is $u_{0j},u_{1j}$ in equation (2) and (3) ?
In your case, basically $\boldsymbol u = (u_0, u_1)$ is distributed as bi-variate normal. You can use R package mvtnorm to generate $u_{0j}, u_{1j}$. Then you can get $\pi_{0j}, \pi_{1j}$ easily. (You would wish to do all those in terms of random vector manipulation.)
The following code generates $\boldsymbol u = (u_0, u_1)$ for you.
require(mvtnorm)
set.seed(1234)
## need variance values as input
s2_0 <- 2
s2_1 <- 3
s01 <- 0.5
## generate bi-variate normal rv for u0, u1
mean <- c(0, 0)
sigma <- matrix(c(s2_0, s01, s01, s2_1), ncol = 2)
u <- rmvnorm(1000, mean = mean, sigma = sigma, method = "chol")
Here I assume you know the values of the hyperparameters. And you get something like following.
> dim(u)
[1] 1000 2
[,1] [,2]
[1,] 0.03563353 -2.0906992
[2,] -2.93442022 1.8783072
[3,] 0.82448029 0.7432658
[4,] -0.92269067 -0.9954788
[5,] -1.39514484 -0.9776595
[6,] -1.51995711 -0.8265220
[7,] -0.13484601 -1.3445112
[8,] 0.12429872 1.6618925
[9,] -1.40900867 -0.8850944
[10,] 3.10290171 -1.4500239
• Could you please tell me what is hyperparameter ? – ABC Jun 1 '15 at 4:16
• @ABC, I mean $\sigma_0^2, \sigma_1^2$ and $\sigma_{01}$ in your case. Basically a hypterparameter is the one for your prior distribution, not the one you wish to estimate and make inference about. That is mainly a term in Bayesian statistics. I might misuse it here. Apologize for misleading. – SixSigma Jun 1 '15 at 4:27
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2021-07-25 08:48:02
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http://clumath343s15s1.wikidot.com/chapter-4
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Chapter 4
## Chapter 4
### Definitions
Vector Space
A Vector Space is a non-empty set $V$ of objects with operations, addition and scalar multiplication, such that:
1) $\vec{u}+\vec{v} \in V$
2) $\vec{u} +\vec{v}=\vec{v} +\vec{u}, \forall \vec{u},\vec{v}\in V$
3) $\vec{u}+(\vec{v}+\vec{w})=(\vec{u}+\vec{v})+\vec{w}$
4) $\exists \vec{0}\in V such that \forall\vec{u}\in V, \vec{0}+\vec{u}=\vec{u}$
5) $\forall \vec{u}\in V \exists \vec{-u}\in V such that (\vec{u})+(\vec{-u})=\vec{0}$
6) $c\vec{u}\in V where c\in R$
7) $c(\vec{u}+\vec{v})=c\vec{u}+c\vec{v}, \forall\vec{u},\vec{v}\in V and c\in R$
8) $(c+d)\vec{u}=c\vec{u}+d\vec{u}, \forall \vec{u} \in V and c,d\in R$
9) $(cd)\vec{u} = c(d\vec{u}), \forall \vec{u}\in V and c,d\in R$
10) $1\cdot\vec{u}=\vec{u}, \forall \vec{u} \in V$
Basis
Let A be a subspace of a vector space V. A indexed set of vectors $\beta =$ {$\vec{b_1}, \vec{b_2},..., \vec{b_p}$} each in V is a basis for H if:
1. $\beta$ is linearly independent.
2. $H = span${$\vec{b_1}, \vec{b_2},..., \vec{b_p}$}
Row-Space
Each row of a matrix A can be identified by a vector in $\mathbb{R}^{n}$. The set of linear combinations of the row vectors is the row space, row(A).
Rank
The rank of a matrix is the dimension of the column space of the matrix.
Change of Basis
Let $B$ and $C$ be bases for a vector space. Given a vector $\vec{x}$ described using the basis $B$, we can describe the vector in terms of the basis $C$.
When changing bases the following formula can be used: $P_{C \leftarrow B} [\vec{x}]_B = [\vec{x}]_C$
### Theorems
Theorem 1
If $\vec{v}_{1},...,\vec{v}_p$ are in a vector space $V$, then Span {$\vec{v}_{1},...,\vec{v}_p$} is a subspace of $V$.
Theorem 2
The null space of an $m x n$ matrix $A$ is a subspace of $R^n$. Equivalently, the set of all solutions to a system $Ax = 0$ of $m$ homogeneous linear equations in $n$ unknowns is a subspace of $R^n$.
Theorem 3
The Column space of an $m \times n$ matrix A is a subspace of $R^n$.
Theorem 4
An indexed set {$\vec{v}_{1},...,\vec{v}_p$} of two or more vectors, with $v_1$ $\neq$ 0, is linearly dependent if and only if some $v_j$ (with j >1) is a linear combination of the preceding vectors, $v_1,...,v_{j-1}$
Theorem 5
Let $S = \{ \vec{v}_1,...\vec{v}_{p} \}$ be a set in $V$, and let $H = \mathrm{Span} \ \{ \vec{v}_1,...\vec{v}_{p} \}$.
1. If one of the vectors in $S$ — say, $\vec{v}_k$ — is a linear combination of the remaining vectors in $S$, then the set formed from $S$ by removing $\vec{v}_k$ still spans $H$.
2. If $H \neq \{ \vec{0} \}$, some subset of $S$ is a basis for $H$.
Theorem 6
The pivot column of a matrix A form a basis for Col A.
Theorem 7
Let $\beta=\left\{\vec{b_1},\vec{b_2},...,\vec{b_n}\right\}$ be a basis for a vector space $V$. Then for each $\vec{x}\in V$ there exists a unique set of constants $c_1,c_2,...,c_n$ such that $c_1\vec{b_1}+c_2\vec{b_2}+...+c_n\vec{b_n}=\vec{0}$
Theorem 8
Let $\beta=\left\{\vec{b_1},\vec{b_2},...,\vec{b_n}\right\}$ be a basis for a vector space $V$. Then th coordinates mapping $x\mapsto [x]_{\beta}$ is a one-to-one linear transformation from $V$ onto $\mathbb{R}^{n}$.
Theorem 9
If a vector space $V$ has a basis $\beta=\left\{\vec{b_1}, \vec{b_2}, ..., \vec{b_n}\right\}$, then any set in $V$ containing more than $n$ vectors is linearly dependent.
Theorem 10
If a vector space has a basis of $n$ vectors, then every basis of the vector space has $n$ vectors.
Theorem 11
Let $H$ be a subspace of a finite-dimensional vector space $V$. Any linearly independent set in $H$ can be expanded, if necessary, to a basis for $H$. Also, $H$ is finite-dimensional and $dim H <= dim V$.
Theorem 12
Let $V$ be a $p$-dimensional vector space, $p \geq 1$. Any linearly independent set of exactly $p$ elements in $V$ is automatically a basis for $V$. Any set of exactly $p$ elements that spans $V$ is automatically a basis for $V$.
Theorem 13
If two matrices $A$ and $B$ are row equivalent then their row spaces are the same. If $B$ is in echelon form, the nonzero rows of $B$ form a basis for the row space of $A$ as well as for that of $B$
Theorem 14
The dimensions of the column space and the row space of an $m \times n$ matrix $A$ are equal. This common dimension, the rank of A, also equals the number of pivot positions in $A$ and satisfies the equation
(1)
\begin{align} \mathrm{rank}(A) + \mathrm{dim}(\mathrm{nul}(A)) = n \end{align}
#### Homework Problems
4.4
27. Use coordinate vectors to test the linear independence of the set of polynomials ${1+2t^3, 2+t-3t^2,-t+2t^2-t^3}$.
First, convert this to coordinate vectors and place them in a matrix: $\begin{bmatrix}1&2&0\\0&1&-1\\0&-3&2\\2&0&-1\end{bmatrix}$. Notice the three columns represent the three polynomials, and the rows represent multiples of the terms. We know this system is linearly independent if $\begin{bmatrix}1&2&0\\0&1&-1\\0&-3&2\\2&0&-1\end{bmatrix}$ $\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} = \vec{0}$ has only the trivial solution. Notice the row reduced version of the matrix is $\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\\0&0&0\end{bmatrix}$, and so since there is a pivot in every column, the system has no free variables and must be linearly independent.
4.6
25. A scientist solves non homogeneous system of ten linear equations in twelve unknowns and finds that three of the unknowns are free variables. Can the scientist be certain that, if the right sides of the equations are changed, the non homogeneous system will have a solution?
No, there will be nine resulting pivots if there were three free variables, but notice there are ten equations. So we won't span our whole space. So we can't guarantee a solution for every vector.
page revision: 33, last edited: 23 Mar 2015 21:10
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2018-11-20 19:48:29
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https://portal.kobv.de/advancedSearch.do?f1=author&v1=Billings%2C+Steve+D&index=primoCentral&plv=2
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# Kooperativer Bibliotheksverbund
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• 1
Article
Language: English
In: The Journal of infectious diseases, 15 April 2008, Vol.197(8), pp.1103-9
Description: Haemophilus ducreyi contains 3 TonB-dependent receptors: the hemoglobin receptor HgbA, which is required for virulence in humans; the heme receptor TdhA; and an uncharacterized conserved hypothetical protein TdX (HD0646). A double tdX/tdhA mutant (FX527) was constructed on the background of a human-passaged variant of strain 35000 (35000HP). Six volunteers were infected with 35000HP at 3 sites on one arm and with FX527 at 3 sites on the other. The pustule formation rate was 55.6% (95% confidence interval [CI], 35.7%-75.4%) at 18 parent-strain sites and 44.4% (95% CI, 15.0%-73.9%) at 18 mutant-strain sites (P = .51). Similar amounts of 35000HP and FX527 were recovered from pustules in semiquantitative culture. Thus, TdX and TdhA are not necessary for virulence, whereas HgbA is both necessary and sufficient for virulence in humans. The data suggest that hemoglobin is the sole source of heme/iron used by H. ducreyi in vivo and has implications for the potential of HgbA as a vaccine.
Keywords: Bacterial Outer Membrane Proteins -- Biosynthesis ; Bacterial Proteins -- Biosynthesis ; Chancroid -- Microbiology ; Haemophilus Ducreyi -- Pathogenicity ; Membrane Proteins -- Biosynthesis
ISSN: 0022-1899
E-ISSN: 15376613
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• 2
Article
Language: English
In: Advances in Space Research, 2012, Vol.49(1), pp.2-48
Description: In response to the growing importance of space exploration in future planning, the Committee on Space Research (COSPAR) Panel on Exploration (PEX) was chartered to provide independent scientific advice to support the development of exploration programs and to safeguard the potential scientific assets of solar system objects. In this report, PEX elaborates a stepwise approach to achieve a new level of space cooperation that can help develop world-wide capabilities in space science and exploration and support a transition that will lead to a global space exploration program. The proposed stepping stones are intended to transcend cross-cultural barriers, leading to the development of technical interfaces and shared legal frameworks and fostering coordination and cooperation on a broad front. Input for this report was drawn from expertise provided by COSPAR Associates within the international community and via the contacts they maintain in various scientific entities. The report provides a summary and synthesis of science roadmaps and recommendations for planetary exploration produced by many national and international working groups, aiming to encourage and exploit synergies among similar programs. While science and technology represent the core and, often, the drivers for space exploration, several other disciplines and their stakeholders (Earth science, space law, and others) should be more robustly interlinked and involved than they have been to date. The report argues that a shared vision is crucial to this linkage, and to providing a direction that enables new countries and stakeholders to join and engage in the overall space exploration effort. Building a basic space technology capacity within a wider range of countries, ensuring new actors in space act responsibly, and increasing public awareness and engagement are concrete steps that can provide a broader interest in space exploration, worldwide, and build a solid basis for program sustainability. By engaging developing countries and emerging space nations in an international space exploration program, it will be possible to create to support program continuity in the development and execution of future global space exploration frameworks. With a focus on stepping stones, COSPAR can support a global space exploration program that stimulates scientists in current and emerging spacefaring nations, and that will invite those in developing countries to participate—pursuing research aimed at answering outstanding questions about the origins and evolution of our solar system and life on Earth (and possibly elsewhere). COSPAR, in cooperation with national and international science foundations and space-related organizations, will advocate this stepping stone approach to enhance future cooperative space exploration efforts.
Keywords: Space Exploration ; Planetary Protection ; International Cooperation ; Engineering ; Astronomy & Astrophysics ; Physics
ISSN: 0273-1177
E-ISSN: 1879-1948
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• 3
Article
Language: English
In: Biogeosciences, August 15, 2018, Vol.15(15), p.4815
Description: pLong-term environmental research networks are one approach to advancing local, regional, and global environmental science and education. A remarkable number and wide variety of environmental research networks operate around the world today. These are diverse in funding, infrastructure, motivating questions, scientific strengths, and the sciences that birthed and maintain the networks. Some networks have individual sites that were selected because they had produced invaluable long-term data, while other networks have new sites selected to span ecological gradients. However, all long-term environmental networks share two challenges. Networks must keep pace with scientific advances and interact with both the scientific community and society at large. If networks fall short of successfully addressing these challenges, they risk becoming irrelevant. The objective of this paper is to assert that the biogeosciences offer environmental research networks a number of opportunities to expand scientific impact and public engagement. We explore some of these opportunities with four networks: the International Long-Term Ecological Research Network programs (ILTERs), critical zone observatories (CZOs), Earth and ecological observatory networks (EONs), and the FLUXNET program of eddy flux sites. While these networks were founded and expanded by interdisciplinary scientists, the preponderance of expertise and funding has gravitated activities of ILTERs and EONs toward ecology and biology, CZOs toward the Earth sciences and geology, and FLUXNET toward ecophysiology and micrometeorology. Our point is not to homogenize networks, nor to diminish disciplinary science. Rather, we argue that by more fully incorporating the integration of biology and geology in long-term environmental research networks, scientists can better leverage network assets, keep pace with the ever-changing science of the environment, and engage with larger scientific and public audiences.
Keywords: Environmental Research – Innovations ; Environmental Research – Forecasts and Trends
ISSN: 1726-4170
E-ISSN: 17264189
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• 4
Article
Language: English
In: The Lancet Diabetes and Endocrinology, 2014, Vol. 2(9), pp. 719-729
Description: BACKGROUND: Low plasma 25-hydroxyvitamin D (25[OH]D) concentration is associated with high arterial blood pressure and hypertension risk, but whether this association is causal is unknown. We used a mendelian randomisation approach to test whether 25(OH)D concentration is causally associated with blood pressure and hypertension risk.METHODS: In this mendelian randomisation study, we generated an allele score (25[OH]D synthesis score) based on variants of genes that affect 25(OH)D synthesis or substrate availability (CYP2R1 and DHCR7), which we used as a proxy for 25(OH)D concentration. We meta-analysed data for up to 108 173 individuals from 35 studies in the D-CarDia collaboration to investigate associations between the allele score and blood pressure measurements. We complemented these analyses with previously published summary statistics from the International Consortium on Blood Pressure (ICBP), the Cohorts for Heart and Aging Research in Genomic Epidemiology (CHARGE) consortium, and the Global Blood Pressure Genetics (Global BPGen) consortium.FINDINGS: In phenotypic analyses (up to n=49 363), increased 25(OH)D concentration was associated with decreased systolic blood pressure (β per 10% increase, -0·12 mm Hg, 95% CI -0·20 to -0·04; p=0·003) and reduced odds of hypertension (odds ratio [OR] 0·98, 95% CI 0·97-0·99; p=0·0003), but not with decreased diastolic blood pressure (β per 10% increase, -0·02 mm Hg, -0·08 to 0·03; p=0·37). In meta-analyses in which we combined data from D-CarDia and the ICBP (n=146 581, after exclusion of overlapping studies), each 25(OH)D-increasing allele of the synthesis score was associated with a change of -0·10 mm Hg in systolic blood pressure (-0·21 to -0·0001; p=0·0498) and a change of -0·08 mm Hg in diastolic blood pressure (-0·15 to -0·02; p=0·01). When D-CarDia and consortia data for hypertension were meta-analysed together (n=142 255), the synthesis score was associated with a reduced odds of hypertension (OR per allele, 0·98, 0·96-0·99; p=0·001). In instrumental variable analysis, each 10% increase in genetically instrumented 25(OH)D concentration was associated with a change of -0·29 mm Hg in diastolic blood pressure (-0·52 to -0·07; p=0·01), a change of -0·37 mm Hg in systolic blood pressure (-0·73 to 0·003; p=0·052), and an 8·1% decreased odds of hypertension (OR 0·92, 0·87-0·97; p=0·002).INTERPRETATION: Increased plasma concentrations of 25(OH)D might reduce the risk of hypertension. This finding warrants further investigation in an independent, similarly powered study.
Keywords: Medical And Health Sciences ; Clinical Medicine ; Endocrinology And Diabetes ; Medicin Och Hälsovetenskap ; Klinisk Medicin ; Endokrinologi Och Diabetes
ISSN: 2213-8587
E-ISSN: 22138595
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• 5
Article
In: Publications of the Astronomical Society of Japan, 2009, Vol. 61(sp2), pp.S395-S616
Description: We systematically surveyed period variations of superhumps in SU UMa-type dwarf novae based on newly obtained data and past publications. In many systems, the evolution of the superhump period is found to be composed of three distinct stages: an early evolutionary stage with a longer superhump period, a middle stage with systematically varying periods, and a final stage with a shorter, stable superhump period. During the middle stage, many systems with superhump periods of less than 0.08 d show positive period derivatives. We present observational characteristics of these stages and give greatly improved statistics. Contrary to an earlier claim, we found no clear evidence for a variation of period derivatives among different superoutbursts of the same object. We present an interpretation that the lengthening of the superhump period is a result of the outward propagation of an eccentricity wave, which is limited by the radius near the tidal truncation. We interpret that late-stage superhumps are rejuvenated excitation of a 3:1 resonance when superhumps in the outer disk are effectively quenched. The general behavior of the period variation, particularly in systems with short orbital periods, appears to follow a scenario proposed in Kato, Maehara, and Monard ( 2008 , PASJ, 60, L23). We also present an observational summary of WZ Sge-type dwarf novae. Many of them have shown long-enduring superhumps during a post-superoutburst stage having longer periods than those during the main superoutburst. The period derivatives in WZ Sge-type dwarf novae are found to be strongly correlated with the fractional superhump excess, or consequently with the mass ratio. WZ Sge-type dwarf novae with a long-lasting rebrightening or with multiple rebrightenings tend to have smaller period derivatives, and are excellent candidates for those systems around or after the period minimum of evolution of cataclysmic variables.
Keywords: Accretion, Accretion Disks ; Stars: Dwarf Novae ; Novae, Cataclysmic Variables
ISSN: 0004-6264
E-ISSN: 2053-051X
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• 6
Article
Language: English
In: Astrobiology, 6(5), 735-813. New Rochelle, NY: Mary Ann Liebert, Inc (2006).
Description: Peer reviewed
Keywords: Physical, Chemical, Mathematical & Earth Sciences :: Earth Sciences & Physical Geography ; Physique, Chimie, Mathématiques & Sciences De La Terre :: Sciences De La Terre & Géographie Physique ; Physical, Chemical, Mathematical & Earth Sciences :: Multidisciplinary, General & Others ; Physique, Chimie, Mathématiques & Sciences De La Terre :: Multidisciplinaire, Général & Autres
Source: ORBi (Open Repository and Bibliography), University of Liège
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• 7
Article
Description: The detection of the Epoch of Reionization (EoR) delay power spectrum using a "foreground avoidance method" highly depends on the instrument chromaticity. The systematic effects induced by the radio-telescope spread the foreground signal in the delay domain, which contaminates the EoR window theoretically observable. Therefore, it is essential to understand and limit these chromatic effects. This paper describes a method to simulate the frequency and time responses of an antenna, by simultaneously taking into account the analogue RF receiver, the transmission cable, and the mutual coupling caused by adjacent antennas. Applied to the Hydrogen Epoch of Reionization Array (HERA), this study reveals the presence of significant reflections at high delays caused by the 150-m cable which links the antenna to the back-end. Besides, it shows that waves can propagate from one dish to another one through large sections of the array because of mutual coupling. In this more realistic approach, the simulated system time response is attenuated by a factor $10^{4}$ after a characteristic delay which depends on the size of the array and on the antenna position. Ultimately, the system response is attenuated by a factor $10^{5}$ after 1400 ns because of the reflections in the cable, which corresponds to characterizable ${k_\parallel}$-modes above 0.7 $h \rm{Mpc}^{-1}$ at 150 MHz. Thus, this new study shows that the detection of the EoR signal with HERA Phase I will be more challenging than expected. On the other hand, it improves our understanding of the telescope, which is essential to mitigate the instrument chromaticity. Comment: 25 pages, 29 figures - Submitted to MNRAS
Keywords: Astrophysics - Instrumentation And Methods For Astrophysics
Source: Cornell University
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• 8
Book
Language: English
Description: This book provides a broad view of the history, experience, and impact of professional Esports as it has shifted the cultural and athletic landscape during its rise. Understanding Esports: An Introduction to the Global Phenomenon places professional Esports, a rapidly growing industry, in both the cultural and athletic landscape. This book explores how the rise of professional gaming has shaped-and been shaped by-media trends, interpersonal communication, and what it means to be classified as an athlete. Ryan Rogers has assembled contributors from a variety of backgrounds and experiences in order to provide a broad view of the history, experience, and impact of professional gaming. Scholars of media studies, communication, sports, and cultural studies will find this book especially useful.
Keywords: Media Studies ; Media Studies
ISBN: 9781498589802
Source: VLeBooks
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• 9
Book
New York: McGraw-Hill
Language: English
In: McGraw-Hill's AccessEngineering
Description: Intro -- DEDICATION -- ADVISORY BOARD -- CONTRIBUTORS -- FOREWORD -- PREFACE -- ACKNOWLEDGMENTS -- CONTENTS—SECTIONS -- CONTENTS—CHAPTERS -- SECTION 1 INDUSTRIAL ENGINEERING: PAST, PRESENT, AND FUTURE -- CHAPTER 1.1 THE PURPOSE AND EVOLUTION OF INDUSTRIAL ENGINEERING -- CHAPTER 1.2 THE ROLE AND CAREER OF THE INDUSTRIAL ENGINEER IN THE MODERN ORGANIZATION -- CHAPTER 1.3 EDUCATIONAL PROGRAMS FOR THE INDUSTRIAL ENGINEER -- CHAPTER 1.4 THE INDUSTRIAL ENGINEER AS A MANAGER -- CHAPTER 1.5 FUNDAMENTALS OF INDUSTRIAL ENGINEERING -- CHAPTER 1.6 THE FUTURE OF INDUSTRIAL ENGINEERING-ONE PERSPECTIVE -- CHAPTER 1.7 FUTURE TECHNOLOGIES FOR THE INDUSTRIAL ENGINEER -- CHAPTER 1.8 THE FUTURE DIRECTIONS OF INDUSTRIAL ENTERPRISES -- CHAPTER 1.9 THE ROLES OF INDUSTRIAL AND SYSTEMS ENGINEERING IN LARGE-SCALE ORGANIZATIONAL TRANSFORMATIONS -- SECTION 2 PRODUCTIVITY,PERFORMANCE, AND ETHICS -- CHAPTER 2.1 THE CONCEPT AND IMPORTANCE OF PRODUCTIVITY -- CHAPTER 2.2 PRODUCTIVITY IMPROVEMENT THROUGH BUSINESS PROCESS REENGINEERING -- CHAPTER 2.3 TOTAL PRODUCTIVITY MANAGEMENT -- CHAPTER 2.4 PERFORMANCE MANAGEMENT:A KEY ROLE FOR SUPERVISORS AND TEAM LEADERS -- CHAPTER 2.5 MANAGING CHANGE THROUGH TEAMS -- CHAPTER 2.6 INVOLVEMENT, EMPOWERMENT, AND MOTIVATION -- CHAPTER 2.7 ENGINEERING ETHICS:APPLICATIONS TO INDUSTRIAL ENGINEERING -- CHAPTER 2.8 CASE STUDY: PRODUCTIVITY IMPROVEMENT THROUGH EMPLOYEE PARTICIPATION -- CHAPTER 2.9 CASE STUDY: REDUCING LABOR COSTS USING INDUSTRIAL ENGINEERING TECHNIQUES -- CHAPTER 2.10 CASE STUDY: PRACTICAL TEAMWORKING AS A CONTRIBUTOR TO GLOBAL SUCCESS -- CHAPTER 2.11 CASE STUDY: COMPANY TURNAROUND USING INDUSTRIAL ENGINEERING TECHNIQUES -- CHAPTER 2.12 CASE STUDY: IMPROVING RESPONSE TO CUSTOMER DEMAND -- CHAPTER 2.13 CASE STUDY: TRANSFORMING A COMPANY IN CENTRAL EUROPE USING INDUSTRIAL ENGINEERING METHODS -- SECTION 3 ENGINEERING ECONOMICS -- CHAPTER 3.1 PRINCIPLES OF ENGINEERING ECONOMY AND THE CAPITAL ALLOCATION PROCESS* -- CHAPTER 3.2 BUDGETING AND PLANNING FOR PROFITS -- CHAPTER 3.3 COST ACCOUNTING AND ACTIVITY-BASED COSTING -- CHAPTER 3.4 PRODUCT COST ESTIMATING -- CHAPTER 3.5 LIFE CYCLE COST ANALYSIS -- CHAPTER 3.6 CASE STUDY: IMPLEMENTING AN ACTIVITY-BASED COSTING PROGRAM AT AUTO PARTS INTERNATIONAL -- SECTION 4 WORK ANALYSIS AND DESIGN -- CHAPTER 4.1 METHODS ENGINEERING AND WORKPLACE DESIGN -- CHAPTER 4.2 CONTINUOUS IMPROVEMENT (KAIZEN) -- CHAPTER 4.3 WORK DESIGN AND FLOW PROCESSES FOR SUPPORT STAFF -- CHAPTER 4.4 SETUP TIME REDUCTION -- CHAPTER 4.5 CASE STUDY: ACHIEVING QUICK MACHINE SETUPS -- SECTION 5 WORK MEASUREMENT AND TIME STANDARDS -- CHAPTER 5.1 MEASUREMENT OF WORK -- CHAPTER 5.2 PURPOSE AND JUSTIFICATION OF ENGINEERED LABOR STANDARDS -- CHAPTER 5.3 STANDARD DATA CONCEPTS AND DEVELOPMENT -- CHAPTER 5.4 DEVELOPING ENGINEERED LABOR STANDARDS -- CHAPTER 5.5 ALLOWANCES -- CHAPTER 5.6 COMPUTERIZED LABOR STANDARDS -- CHAPTER 5.7 IMPLEMENTATION AND MAINTENANCE OF ENGINEERED LABOR STANDARDS -- CHPATER 5.8 WORK MEASUREMENT IN AUTOMATED PROCES.
Keywords: Industrial Engineering ; Industrial Engineers ; Systems Engineering ; Industrial Productivity ; Reengineering (Management) ; Industrial Management ; Industrial Management ; Teams in the Workplace ; Engineering Ethics ; Labor Costs ; Costs, Industrial ; Cost Control ; Engineering Economy ; Value Analysis (Cost Control) ; Budget ; Profit ; Cost Accounting ; Activity-Based Costing ; Pricing ; Life Cycle Costing ; Methods Engineering ; Offices ; Work Environment ; Manufacturing Industries ; Manufacturing Processes ; Work Design ; Work Measurement ; Labor Productivity ; Labor Laws and Legislation ; Human Engineering ; Industrial Safety ; Industrial Hygiene ; Overuse Injuries ; Human-Machine Systems ; ISO 14000 Series Standards ; Industrial Management ; Musculoskeletal System ; Incentives in Industry ; Job Evaluation ; Pay Equality ; Organizational Effectiveness ; Compensation Management ; Arbitration, Industrial ; Industrial Relations ; Labor Unions ; Industrial Location ; Plant Layout ; Plant Engineering ; Office Layout ; Computer-Aided Design ; Engineering Design ; Manufacturing Cells ; Business Relocation ; Flexible Manufacturing Systems ; System Design ; Production Control ; Production Management ; Just-in-Time Systems ; Business Logistics ; Production Scheduling ; Materials Management ; Materials Handling ; Warehouses ; Physical Distribution of Goods ; Marketing ; Operations Research ; Mathematical Optimization ; Queuing Theory ; Computer Simulation ; Bar Coding ; Automated Data Collection Systems ; Information Networks ; Artificial Intelligence ; Knowledge Management ; New Products ; Design, Industrial ; Concurrent Engineering ; Production Planning ; Total Quality Management ; Quality Control ; ISO 9000 Series Standards ; Process Control ; Manudfacturing Processes ; Machine Design ; Assembly-Line Methods ; Automation ; Packaging Machinery ; Robots, Industrial ; Food Service ; Plant Maintenance ; Industrial Equipment ; Total Productive Maintenance ; Statistics ; Time Study ; Work Sampling ; Cad/Cam Systems ; Project Management ; Group Technology ; Engineering ; Statistics
ISBN: 0071449272
ISBN: 9780070411029
ISBN: 9780071449274
ISBN: 0070411026
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• 10
Newspaper Article
In: The Houston Chronicle (Houston, TX), Feb 21, 1998, p.33
ISSN: 1074-7109
Source: Cengage Learning, Inc.
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2019-10-18 17:53:52
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https://proofwiki.org/wiki/Definition:Locally_Uniform_Convergence/Series
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# Definition:Locally Uniform Convergence/Series
## Definition
Let $X$ be a topological space.
Let $V$ be a normed vector space.
Let $\sequence {f_n}$ be a sequence of mappings $f_n: X \to V$.
Then the series $\ds \sum_{n \mathop = 1}^\infty f_n$ converges locally uniformly if and only if the sequence of partial sums converges locally uniformly.
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2023-03-31 16:48:40
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https://robotics.stackexchange.com/questions/15013/calculating-differential-drive-robot-icc-position
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# calculating differential drive robot ICC position
I don't understand how to calculate the ICC position with the given coordinates. I somehow just have to use basic trigonometry but I just can't find a way to calculate the ICC position based on the given parameters $R$ and $\theta$.
Edit: Sorry guys if forgot to include the drawing of the situation. Yes, ICC = Instantaneous Center of Curvature.
• I'm assuming that by ICC you mean the Instantaneous Centre of Curvature? – sempaiscuba Jan 19 '18 at 14:58
• Welcome to Robotics, Leo. As it stands, it's not clear what you're asking. What is "ICC?" What is $R$ and $\theta$? Can you provide a diagram of your scenario or a more detailed problem statement? If, as @sempaiscuba states, you mean instantaneous center of curvature, then you can't find it based on a position and heading, if that's what $R$ and $\theta$ are, because you need more information. If $\theta$ is something like an Ackermann steering angle, then you still need more information, like the wheel base. Please edit your question to include the missing information. – Chuck Jan 19 '18 at 16:19
• Does my answer to this question help? – Mark Booth Jan 19 '18 at 17:37
OK, I'm going to work on the assumption that you are trying to calculate the Instantaneous Centre of Curvature, and that the values of $R$ and $\theta$ that you have been given are the distance from the ICC to the mid-point of the wheel axle and the direction of travel relative to the x-axis.
That should correspond with the diagram below, taken from Computational Principles of Mobile Robotics by Dudek and Jenkin:
Now, provided you know the position of the robot $(x,y)$ you can find the location of the ICC by trigonometry as:
$$ICC = [x - R sin(\theta), y + R cos(\theta)]$$
In the more usual case, we can measure the velocities of the left and right wheels, $V_{r}$ and $V_{l}$. From the diagram, we can see that:
$$V_{r} = \omega (R + \frac{l}{2})$$
$$V_{l} = \omega (R - \frac{l}{2})$$
Where $\omega$ is the rate of rotation about the ICC, $R$ is the distance from the ICC to the mid-point of the wheel axle, and $l% is the distance between the centres of the wheels. Solving for$R$and$\omega$gives: $$R = \frac{l}{2} \frac{V_{l} + V_{r}}{V_{r} - V_{l}}$$ $$\omega = \frac{V_{r} - V_{l}}{l}$$ • Okay Thankyou, im impressed you knew so fast the exact page from referenced book. But as I understand$ \theta $is the angle between the x-axis and the robots direction. Shouldn't the angle then be 1-$ \theta $instead of just$ \theta $? – Leo Jan 20 '18 at 13:42 • like here in this picture: [![enter image description here][1]][1] [1]: i.stack.imgur.com/pVqD0.png – Leo Jan 20 '18 at 13:54 • @Leo As to the book, it's not the first time I've had people mention it. The authors' use of$R$and$\theta$in this instance often causes confusion with their more usual use as polar coordinates. – sempaiscuba Jan 20 '18 at 17:31 • @Leo OK, I've corrected the diagram (again!). I must really need some sleep! (Not trying to edit pictures and Latex on a 7" mobile phone would probably help too!). The angle is actually$\frac{\pi}{2} - \theta$, rather than just$\theta$. The remaining angle of the triangle shown in red is then$\theta$which allows you to use basic trigonometry to obtain the result in the formula. – sempaiscuba Jan 20 '18 at 22:18 • Sorry of course i meant$ \pi /2 $instead of$ 1 $too, sorry for that mistake. Okay now it makes sense to me since i just didnt recognize the angle$ \theta \$ in the ICC corner. Thankyou! – Leo Jan 21 '18 at 11:40
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2019-09-22 08:36:17
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https://dev.dissem.in/backend.orcid.html
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# backend.orcid module¶
class backend.orcid.ORCIDDataPaper(orcid_id)[source]
Bases: object
as_dict()[source]
classmethod from_orcid_metadata(ref_name, orcid_id, pub, stop_if_dois_exists=False, overrides=None)[source]
initialize()[source]
is_skipped()[source]
splash_url
throw_skipped()[source]
class backend.orcid.ORCIDMetadataExtractor(pub)[source]
Bases: object
authors_from_bibtex(bibtex)[source]
authors_from_contributors(contributors)[source]
bibtex()[source]
citation_format()[source]
contributors()[source]
convert_authors(authors, orcids)[source]
dois()[source]
internal_identifier()[source]
j(path, default=None)[source]
orcids(orcid_id, ref_name, authors, initial_orcids)[source]
pubdate()[source]
title()[source]
type()[source]
class backend.orcid.OrcidPaperSource(max_results=None)[source]
bulk_import(directory, fetch_papers=True, use_doi=False)[source]
Bulk-imports ORCID profiles from a dmup (warning: this still uses our DOI cache). The directory should contain json versions of orcid profiles, as in the official ORCID dump.
create_paper(data_paper)[source]
fetch_crossref_incrementally(cr_api, orcid_id)[source]
fetch_metadata_from_dois(cr_api, ref_name, orcid_id, dois)[source]
fetch_orcid_records(orcid_identifier, profile=None, use_doi=True)[source]
Queries ORCiD to retrieve the publications associated with a given ORCiD. It also fetches such papers from the CrossRef search interface.
Parameters: profile – The ORCID profile if it has already been fetched before (format: parsed JSON). use_doi – Fetch the publications by DOI when we find one (recommended, but slow) a generator, where all the papers found are yielded. (some of them could be in free form, hence not imported)
fetch_papers(researcher)[source]
reconcile_paper(ref_name, orcid_id, metadata, overrides=None)[source]
warn_user_of_ignored_papers(ignored_papers)[source]
exception backend.orcid.SkippedPaper[source]
Bases: exceptions.Exception
backend.orcid.affiliate_author_with_orcid(ref_name, orcid, authors, initial_orcids=None)[source]
Given a reference name and an ORCiD for a researcher, find out which author in the list is the most likely to be that author. This function is run on author lists of papers listed in the ORCiD record so we expect that one of the authors should be the same person as the ORCiD holder. This just finds the most similar name and returns the appropriate orcids list (None everywhere except for the most similar name where it is the ORCiD).
backend.orcid.orcid_oai_source()[source]
backend.orcid.orcid_to_doctype(typ)[source]
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2018-12-10 22:36:54
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http://mathhelpforum.com/differential-equations/51124-solved-differential-equation-general-solution-help-print.html
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# [SOLVED] Differential Equation General Solution Help
Printable View
• September 28th 2008, 11:57 PM
jwade456
[SOLVED] Differential Equation General Solution Help
hey guys, i'm really struggling with this and need someone to point me in the right direction, any help at all would be greatly appreciated!
Find the general solution of Eq.2 and hence the general solution for y(x). Your answer should have 2 arbitrary constants of integration.
Here are the equation's i've got so far:
1. $\frac{dy}{dx}=\frac{{\rho}g}{T}\int^x_{0}\sqrt{1+\ frac{dy}{dt}^2}dt$
Differentiating both sides of Eq.1 to produce second order ODE for y(x)
$\frac{d^2y}{dx^2}=\frac{{\rho}g}{T}\sqrt{1+\frac{d y}{dx}^2}$
By letting $u=\frac{dy}{dx}$
2.
$\frac{du}{dx}=\frac{{\rho}g}{T}\sqrt{1+u^2}$
• September 29th 2008, 12:14 AM
TwistedOne151
Note that your resulting equation is separable:
$\frac{du}{dx}=\frac{{\rho}g}{T}\sqrt{1+u^2}$
$\frac{du}{\sqrt{1+u^2}}=\frac{{\rho}g}{T}\,dx$
Integrate both sides and solve for u(x), remembering your constant of integration. Then, use u(x)=y'(x), and integrate to find y(x) (with your second constant of integration).
--Kevin C.
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2016-07-01 02:45:19
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http://tex.stackexchange.com/questions/79347/how-do-i-use-scopes-to-apply-a-style-to-nodes-inside-a-path
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# How do I use scopes to apply a style to nodes inside a path?
I want to use nodes inside a single path, in order that I might position one relative to another using the ++(x,y) notation.
However, a scope that I want to use to style the nodes only styles the path (and therefore appears to do nothing, as the path is 'invisible').
Adding the scope inside the path (around the nodes) doesn't work.
So, is there anyway to apply a scope to nodes inside a path?
Many thanks for any help.
Example:
\documentclass{article}
\usepackage{tikz}
\tikzset{
every node/.style = {
shape = rectangle,
minimum height = 20mm,
minimum width = 20mm,
line width = .5mm},
mystyle/.style = {draw = red!50!black!50}
}
\begin{document}
\begin{tikzpicture}
% These two nodes have the correct style,
% but I have to specify the style for every node
% (I'll have around 100 nodes in total, hence the
% desire to use a scope for the style).
\path (0,6)
node(nodeOne) [mystyle] {Node One}
++(5,0) node(nodeTwo) [mystyle] {Node Two};
% These two nodes are in the mystyle'' scope,
% but the style is not applied to the nodes
% as they reside inside a path.
\begin{scope}[mystyle]
\path (0,0)
node(nodeThree) {Node Three}
++(5,0) node(nodeFour) {Node Four};
\end{scope}
\end{tikzpicture}
\end{document}
-
It's not applied because your custom style is not a node style yet so the color option is ignored. Add draw option to any of node 3 or 4 and you can see that it is adopted. – percusse Oct 27 '12 at 23:47
A path's or scope's [mystyle] only applies to the paths (if they are drawn), and as you used only \path and not some kind of line (--, to, …) there's not much to draw.
For the first path you could add every node/.append style=mystyle to its options so that the path's nodes get the style mystyle additional to the globally set every node/.style from your \tikzset in the preamble.
\path[every node/.append style=mystyle] (0,6) node(nodeOne) {Node One}
++(5,0) node(nodeTwo) {Node Two};
This also applies to scopes:
\begin{scope}[every node/.append style=mystyle]
\path (0,0) node(nodeThree) {Node Three}
++(5,0) node(nodeFour) {Node Four};
\end{scope}
If you only want to place nodes relatively to each other, you might consider (using the positioning library of TikZ (e.g. \usetikzlibary{positioning}).
\begin{scope}[every node/.append style=mystyle, on grid]
\node (nodeOne) at (0,6) {node One};
\node[right=5cm of nodeOne] (nodeTwo) {node Two};
\end{scope}
Note the key on grid that makes sure the the relative positioning is from center to center, not between the outer border of the nodes. The “Advanced Placement Options” are explained in detail in subsection 16.5.3 of the PGF/TikZ manual.
For simple cases of placement, it can suffice to set a node distance:
\begin{scope}[every node/.append style=mystyle, node distance=5cm, on grid]
\node (nodeOne) at (0,6) {node One};
\node[right=of nodeOne] (nodeTwo) {node Two};
\end{scope}
-
Wow, many thanks for all the very quick responses! Thank you to everyone---this is exactly what I needed. I was unaware of the ".append" modifier too. – Graham Oct 28 '12 at 0:32
I've had something like this before... I think the initial every node/.style = { options } is in effect regardless of any subsequent \begin{scope} .... \end{scope} environment.
Try using another every node/.style = { options } in the relevant scope to apply locally to where you want to change how the nodes are produced.
-
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2016-06-01 03:36:22
|
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https://en.wikipedia.org/wiki/Nested_dissection
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# Nested dissection
In numerical analysis, nested dissection is a divide and conquer heuristic for the solution of sparse symmetric systems of linear equations based on graph partitioning. Nested dissection was introduced by George (1973); the name was suggested by Garrett Birkhoff.[1]
Nested dissection consists of the following steps:
• Form an undirected graph in which the vertices represent rows and columns of the system of linear equations, and an edge represents a nonzero entry in the sparse matrix representing the system.
• Recursively partition the graph into subgraphs using separators, small subsets of vertices the removal of which allows the graph to be partitioned into subgraphs with at most a constant fraction of the number of vertices.
• Perform Cholesky decomposition (a variant of Gaussian elimination for symmetric matrices), ordering the elimination of the variables by the recursive structure of the partition: each of the two subgraphs formed by removing the separator is eliminated first, and then the separator vertices are eliminated.
As a consequence of this algorithm, the fill-in (the set of nonzero matrix entries created in the Cholesky decomposition that are not part of the input matrix structure) is limited to at most the square of the separator size at each level of the recursive partition. In particular, for planar graphs (frequently arising in the solution of sparse linear systems derived from two-dimensional finite element method meshes) the resulting matrix has O(n log n) nonzeros, due to the planar separator theorem guaranteeing separators of size O(n).[2] For arbitrary graphs there is a nested dissection that guarantees fill-in within a ${\displaystyle O(\min\{{\sqrt {d}}\log ^{4}n,m^{1/4}\log ^{3.5}n\})}$ factor of optimal, where d is the maximum degree and m is the number of non-zeros. [3]
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2020-02-17 04:01:17
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http://fakeroot.net/margin-of/calculate-margin-of-error-with-standard-deviation.php
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Home > Margin Of > Calculate Margin Of Error With Standard Deviation
# Calculate Margin Of Error With Standard Deviation
## Contents
This will tell you that you can be, say, 95% certain that the NPS for all your customers is between your sample score plus the Margin of Error and the sample Easy! Previously, we described how to compute the standard deviation and standard error. That is, the critical value would still have been 1.96. get redirected here
Loading... drenniemath 36,919 views 11:04 7 videos Play all Standard Deviationstatisticsfun Statistics 101: Standard Error of the Mean - Duration: 32:03. Take the square root of the calculated value. Also, if the 95% margin of error is given, one can find the 99% margin of error by increasing the reported margin of error by about 30%. http://www.dummies.com/education/math/statistics/how-to-calculate-the-margin-of-error-for-a-sample-proportion/
## How To Calculate Margin Of Error In Statistics
What happens if no one wants to advise me? I'm trying to figure out which score movements are significant, if any. We have used an approximation for the calculator (Thanks to thomas t john for correcting me on this point) Is that clear? Margin Of Error Standard Deviation Sample Size The critical value is either a t-score or a z-score.
The margin of error has been described as an "absolute" quantity, equal to a confidence interval radius for the statistic. How To Find Margin Of Error The condition you need to meet in order to use a z*-value in the margin of error formula for a sample mean is either: 1) The original population has a normal For this problem, since the sample size is very large, we would have found the same result with a z-score as we found with a t statistic. Here are the steps for calculating the margin of error for a sample proportion: Find the sample size, n, and the sample proportion.
After all your calculations are finished, you can change back to a percentage by multiplying your final answer by 100%. How Is Margin Of Error Calculated In Polls Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count). statisticsfun 307,486 views 4:59 Determining Sample Size - Duration: 3:08. Loading...
## How To Find Margin Of Error
Check out the grade-increasing book that's recommended reading at Oxford University! How to Calculate Margin of Error (video) What is a Margin of Error? How To Calculate Margin Of Error In Statistics How much should I adjust the CR of encounters to compensate for PCs having very little GP? Margin Of Error Example Problems The true NPS is computed as the average value of all the tickets in the hat: it is the expected value (or expectation) of the hat.
This means that the sample proportion, is 520 / 1,000 = 0.52. (The sample size, n, was 1,000.) The margin of error for this polling question is calculated in the following Get More Info What is a Margin of Error Percentage? It might just be a fluke of the sample you have collected. For this problem, it will be the t statistic having 899 degrees of freedom and a cumulative probability equal to 0.975. Margin Of Error Standard Deviation Unknown
asked 4 years ago viewed 11983 times active 5 months ago Blog Stack Overflow Podcast #89 - The Decline of Stack Overflow Has Been Greatly… 13 votes · comment · stats The margin of error can be calculated in two ways, depending on whether you have parameters from a population or statistics from a sample: Margin of error = Critical value x In cases where n is too small (in general, less than 30) for the Central Limit Theorem to be used, but you still think the data came from a normal distribution, useful reference Required fields are marked *Comment Name * Email * Website Notify me of follow-up comments by email.
Optimise Sieve of Eratosthenes Colonists kill beasts, only to discover beasts were killing off immature monsters How to implement \text in plain tex? Percent Error Standard Deviation Get the list now; learn what they are and how to avoid them. Sign in to make your opinion count.
## In the example of a poll on the president, n = 1,000, Now check the conditions: Both of these numbers are at least 10, so everything is okay.
The standard error of a reported proportion or percentage p measures its accuracy, and is the estimated standard deviation of that percentage. This means that the sample proportion, is 520 / 1,000 = 0.52. (The sample size, n, was 1,000.) The margin of error for this polling question is calculated in the following statisticsfun 88,630 views 2:46 How to calculate Margin of Error Confidence Interval for a population proportion - Duration: 8:04. Margin Of Error Mean z*-Values for Selected (Percentage) Confidence Levels Percentage Confidence z*-Value 80 1.28 90 1.645 95 1.96 98 2.33 99 2.58 Note that these values are taken from the standard normal (Z-) distribution.
This makes intuitive sense because when N = n, the sample becomes a census and sampling error becomes moot. About Press Copyright Creators Advertise Developers +YouTube Terms Privacy Policy & Safety Send feedback Try something new! In the Newsweek poll, Kerry's level of support p = 0.47 and n = 1,013. this page The general formula for the margin of error for a sample proportion (if certain conditions are met) is where is the sample proportion, n is the sample size, and z* is
COSMOS - The SAO Encyclopedia of Astronomy. Technically, it should be 1.96 for the two-sided hypothesis test at level alpha=0.05. Notice in this example, the units are ounces, not percentages!
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2018-02-25 23:44:07
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https://physics.stackexchange.com/questions/278931/derivation-of-inverse-square-law
|
# Derivation of Inverse Square Law
Is there a mathematical derivation of the inverse square law that doesn't depend on geometry or empirical data fitting?
• en.m.wikipedia.org/wiki/… This link has a rather neat derivation of the inverse-square law without prior knowledge of Maxwell's equations. It's generalized, even for the mass of the exchange particle. – Ringo Hendrix Feb 8 '17 at 16:09
What do you mean by "doesn't depend on geometry?". If you are referring to the Coulomb law for the electric field generated by a point charge, it can be derived from Maxwell's equations. These have their foundations in the symmetry principles of the special theory of relativity, but as fundamental laws of nature, they can only be justified by the experience.
The Gauss' Law (1st Maxwell equation in integral form) gives
$$\iint_S \mathbf E \cdot ~\mathrm d\mathbf a = 4\pi\iiint_V \rho ~\mathrm dV$$
where $S$ is a closed surface that contains the charges, $V$ is the volume enclosed by such surface and $\rho$ is the density of electric charge. For a point charge at rest, let's take $S$ to be a sphere of radius $R$ centered in the charge. From symmetry arguments it is clear that $\bf E$ is constant on the surface of the sphere and it is perpendicular to it. Its modulus will depend only on the radius $R$, i.e. the distance from the charge. In this special case the left hand side of the equation is
$$E(R)\iint_S\,\mathrm d\mathbf a = 4 \pi R^2 E(R)$$
The right hand side is just the total charge contained in the sphere (times $4\pi$) and so we have in the end
$$4 \pi R^2 ~ E(R)=4\pi e$$
that gives the Coulomb law
$$E(R)=e/R^2.$$
Identical considerations can be used to derive the inverse-square law for the gravitational force.
• Thanks DelCrosB, MAFIA36790. The same form of the law is also used in disguise in the Born rule of quantum mechanics and I believe the general form of the inverse square law is truly fundamental and universal for all interactions. – jam Sep 10 '16 at 7:19
• No problem! What other interactions do you have in mind? And what do you mean when you say that it is used in disguise in the Born rule? – DelCrosB Sep 10 '16 at 7:58
Here is an incredibly simple derivation of the the inverse square law for gravity which shows how it must rely on geometry..
A simple way to think about the gravitational field of an object is to imagine a fixed number of "lines of force" that radiate from the object evenly into space.
Let's suppose the number of lines of force produced by an object is directly proportional to its mass, so..
n=k*m
where n is the number of lines of force produced by the mass m and k is a constant.
Now assume the density of the lines at any given point in space represents the strength of the gravitational field at that point. So at a distance r from the object the density of the lines of force is..
n/(surface area of the sphere of radius r)
which is n/(4*pir^2)=km/(4*pir^2)=Gm/(r^2)
where G=k/4*pi is a constant.
• Thanks Ken. I'm looking for a general derivation of the inverse square law applicable to all interactions, not just gravity or electrostatics, with a term that reduces to one as time approaches infinity thus reducing quantum interactions to classical forces. – jam Sep 10 '16 at 7:10
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2020-10-31 11:18:25
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https://datascience.stackexchange.com/tags/hyperparameter-tuning/hot?filter=year
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Tag Info
5
In "The Elements of Statistical Learning" by Hastie et al the authors describe two tasks regarding model performance measurement: Model selection: estimating the performance of different models in order to choose the best one. Model assessment: having chosen a final model, estimating its prediction error (generalization error) on new data. ...
3
As I was playing with this problem, Djib just wrote an answer which is certainly better than whatever I could have come up with. To illustrate Djib's point, here is a small demonstration that as soon as there are more than 2 classes there's no value of $k$ which guarantees the absence of tie (except if $k=1$ of course). By definition we have $k=|c_1| + |c_2| ... 3 Your idea isn't wrong, however in k-NN there always might be a case where you have the same number of votes for 2 or more classes (e.g. you have$k=6$and you have 3 samples of one class vs 3 of another). With your solution you are just overcoming one very small case of ties and the$k$that you choose might not be the optimal$k$for classification, which ... 3 The$k$-fold cross-validation (CV) process (method 2) actually does the same thing as method 1, but it repeats the steps on the training and validation sets$k$times. So with CV the performance is averaged across the$k$runs before selecting the best hyper-parameter values. This makes the performance and value selection more reliable in general, since ... 2 This idea of using a small sample to the data set to search for the hyperparameters is called multi-fidelity methods. A good starting point is the book by Frank Hutter, Lars Kotthoff, Joaquin Vanschoren Automated machine learning: methods, systems, challenges which is open access. 2 So, the question talks about how to treat transformation choices as hyper parameters. How I would go about it is the following: Use one baseline model architecture for the data and then repeat the following: Instantiate the baseline model (effectively make sure all of the weights are initialised) Create the transformed dataset Train the model Compute ... 2 Note that RMSE is an easy to understand metric. Its the Root of the Mean Squared Error. So this is just how is the typical error. If your target is something like how big is a building, and the mean of the target its 100m, then having an error of 0.3m its nothing. On the other hand if you predict the size of insect, and your target mean is around 0.1m then ... 2 There's quite a lot of features for the number of instances, so it's indeed likely that there's some overfitting happening. I'd suggest these options: Forcing the decision trees to be less complex by setting the max_depth parameter to a low value, maybe around 3 or 4. Run the experiment with a range of values (e.g. from 3 to 10) and observe the changes in ... 2 I assume its crashing because not enough RAM. So I also assume your data is quite big. Your search grid is quite big. So this will definitely take some time. In order to speed up the training and overloading the RAM. You can fit the model in a subsample of data. Theoretically if your data is big enough and you sample it, when you use the whole model the ... 2 1. what are the rows before the first red row? I thought it may be the combinations of parameters but that doesn't make a lot of sense because those are not enough Parameters, which are the candidates of the CV are printed 2. What is the meaning of the row between the red rows? Why those parameters are there? is this after one CV? It is printed after the ... 2 As far as I know you cannot add the model's threshold as hyperparameter but in order to find the optimal threshold you can do as follows: make a the standard GridSearchCV but use the roc_auc as metric as per step 2 model = DecisionTreeClassifier() params = [{'criterion':["gini","entropy"],"max_depth":[1,2,3,4,5,6,7,8,9,10],&... 2 If you're seeing performance that is much better on the validation than the unseen test data, then that is suggestive of some sort of overfitting or, if not, that the data do not come from the same distribution. That could mean that your test images are very different from the training and validation data, for example. First, I'd double check the data to ... 2 Essentially, the function of your testset is to evaluate the performance of your model on new data. It mimics the situation of your model being put into production. The validation set is used for optimizing your algorithm. Personally I would recommend tuning your algorithm using your validation set and using the hyperparameters of the training epoch with the ... 1 So do we still need to learn how to do hyperparameter tuning If you're saying this based on the context of acquiring a new skill, then go for it. It's always a good thing to get an idea an idea of how hyper-parameter testing is done for real. In addition to sagemaker you can use tools like weights and biases 1 In general there's no way to know the best values to try for a parameter. The only thing one can do is to try many possible values, but: this mathematically requires more computing time (see this question about how GridSearchCV works) there is a risk of overfitting the parameters, i.e. selecting a value which is optimal by chance on the validation set. 1 This is very nearly a duplicate of Is a test set necessary after cross validation on training set?, but I think it's worth addressing specifically this part of your question: the hyperparameter searching is like every time training a model from scratch using a combination of hyperparameters and pick the best combination, and it's not like that the model is ... 1 You simply need to see the dynamic of change in the incoming data. The need for retraining is a direct function of change in data distribution. If new data, theoretically, does not change the distribution then there is no need, however in practice that is simply impossible. So how much change in the distribution of data (both input and output) is a threshold ... 1 Hello and welcome to the site! You can use list(range(1,100)) to get what you want. However, questions like this (how to achieve something in Python) are more suitable for stackoverflow. The community here focuses on data science related questions, as the name suggests. 1 Yes, you can make an AI for that. By AI, you mean algorithm which finds hyperparameters efficiently. There are many ways to find hyperparameters which would then come under the AI like using bandits to find hyperparameters, Bayesian methods to find hyperparameters and many other methods exist. Search for hyperparameter optimization and you will find a ton of ... 1 In a pure random search, 60 points is often given as a rule of thumb, because provably with probability 95% such a search finds a hyperparameter combination in the top 5%. However, that 5% is as a percent of the volume of the search space, so giving much-too-broad a search space, the best 5% might not be a fantastic score for the model. So it does seem to ... 1 To add a couple of references, to the already good answers, on the problem of selecting the optimum$k$for$k-NN\$. How to find the optimal value of K in KNN? Then how to select the optimal K value? There are no pre-defined statistical methods to find the most favorable value of K. Initialize a random K value and start computing. Choosing a small value of ...
1
Have a look at this blog post: https://towardsdatascience.com/hyperparameter-tuning-the-random-forest-in-python-using-scikit-learn-28d2aa77dd74 Ideally you should optimise the hyperparameters jointly and not one after the other. More importantly, you should be doing cross validation. Consider also the RandomizedSearchCV described in the post.
1
Since the various hyperparameter are related you cannot be sure that - for instance - a tuned value for min_samples_split from a model where all other hyperparameter are set to default values, will be generally optimal. When you have a situation where you limit the depth of trees (by max_depth assuming you use sklearn), min_samples_split may be a non-binding ...
1
In general, the max depth parameter should be kept at a low value in order to avoid overfitting: if the tree is deep it means that the model creates more rules at a more detailed level using fewer instances. Very often some of these rules are due to chance, i.e. they don't correspond to a real pattern in the data. Overfitting is visible in your graphs from ...
1
I was wondering if i'm doing a GridsearchCV on 10-fold, getting the best parameters, and then using those parameters evaluating the performance on 10-fold - is that "legal" or overfitting? am i suppose to run the best parameters on the entire data ? or can i use 10-fold again? I'm pretty sure you won't go to prison for it ;) But it would be ...
1
If you have the time on hand, you could simply measure the time taken for all combinations of hyper parameter values in a Grid Search, preferably with repetition. It's unlikely that any theoretical analytical expression will provide adequate accuracy for predicting the compute cost, as there as so many factors that contribute noise to the compute time. You ...
1
A tip: Dont A trick:Dont The reason? Machine learning scientific methodology is based on cross-validation. Almost all papers (and i put the almost because of yes) select everything based on cross-validation and not in previous knowledge. Xgboost is particularly more complicated because it has a lot of math involved. For a simpler case, lets say that you have ...
1
You can access the GPU by going to the settings: Runtime> Change runtime type and select GPU as Hardware accelerator.
1
What shepan6 is suggesting is basically to manually search for the best "transformation choice hyperparameters" by trying them all and seeing what performs best. This is a good idea (I upvoted), but if you want to go further, you can use a package like hyperopt and manually define an "objective" function that accepts a parameter that ...
1
So that we are on the same page, some prerequisites Suppose we had only 2 splits train and test. Now when we will tune our hyperparameters using the test split, we are trying to increase the accuracy(or any other metric). Though our model is not trained on the test set, but we are making it perform well on the test set, in a way the model gets the ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2021-08-03 21:34:41
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https://bvillasen.github.io/blog/astro/cholla/2019/07/03/dual_energy_eta_beta.html
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The simulations are $128^3$ and 50Mpc.
Dual Energy Parameters: $\eta=0.005$ $\beta_0 = 0.25$ $\beta_1 = 0.0$
Shock Detection
Pressure Jump:
From Fryxell 200 the Pressure Jump Condition for shock detection is:
Their implementation uses $\alpha = 0.1$
To ignore fluctuations due to noise a condition in the density is also applied:
Phase Diagram
Row 1: Without Pressure Jump Condition
Now Using Pressure Jump condition:
Row 2: Using $\alpha=0.1$
Now only using the Total Internal Energy if $U_{total} > U_{advected}$
Row 3: Using $\alpha=1.0$
Row 4: Using $\alpha=10.0$
Chemistry Projection
No Pressure Jump Condition
Using Pressure Jump Condition $\alpha=10.0$
Power Spectrum
No Pressure Jump Condition
Using Pressure Jump Condition $\alpha=10.0$
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2020-08-13 05:15:03
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http://math.stackexchange.com/questions/31978/finding-the-supremum-and-infimum-of-a-set
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# Finding the supremum and infimum of a set
I'd like some help with finding the supremum and infimum of $$\left\{ 2a,3(1-2a),5a:0<a<\frac{1}{2}\right\}$$
My guess is that the infimum is 0 and the supremum is 3.
Thanks.
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2015-11-27 17:25:30
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https://chemistry.stackexchange.com/questions/35340/justification-for-freezing-point-depression-boiling-point-elevation-in-solutio
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# Justification for Freezing Point Depression & Boiling Point Elevation in Solutions?
I was wondering if the following justification for freezing point depression and boiling point elevation are conceptually correct. The reason why I ask this question is because I have been self studying chemistry for a course I will be taking this fall, and I don't have a human reference to check with. My book is also very confusing on this subject, and I don't have a world renowned memory (meaning I'd rather understand than memorize the equation), so I tried to think about it and came with the following conclusions:
-Freezing Point Depression: When a solvent is not pure and has particles dissolved in it, there are constituents (be it molecules, ions, etc.) that take up volume in the solution. For freezing to occur, these constituents need to aggregate to form a lattice, bonds, etc. Say we have a pure solvent X. This process occurs at a temperature Y. However, when this solvent is not pure and has dissolved solutes, there is "blockage" impeding these constituents from aggregating and bonding. Therefore, the temperature Y needs to be reduced further to slow down the kinetic energy of all constituents in solution (solvent and solute), which gives a greater statistical probability of intermolecular forces getting an opportunity to bind solvent-solvent constituents.
-Boiling Point Elevation: Boiling, if I'm not mistaken, occurs when the partial pressure of a vapor exceeds atmospheric pressure. For a gas to enter the vapor phase, molecules in solution need to have a certain kinetic energy (as specified by a Boltzmann distribution) to free themselves of the intermolecular forces in the solution phase and become a vapor. When a pure solvent is now diluted with other particles, the new solute-solvent bonding energy needs to be overcome to allow a solvent molecule to exist as a vapor. One way to overcome this energy is to increase the average kinetic energy of all molecules, and this is done through a temperature increase.
This is for an introductory chemistry course, and I'm not sure if my explanations are sound. I just want a yay or a nay in terms of my logic, because I don't have the most intimate background in chemistry and I'm trying to conceptually understand things.
I think you are looking at the problem from slightly the wrong angle. The central quantity when dealing with colligative properties is entropy and not solute-solvent or solvent-solvent molecule interactions. Of course the interactions are important in the sense that they affect the entropy but since we are dealing with thermodynamics here the way you think about it at the moment clouds the main effect at work here imo.
So, when trying to picture how colligative properties work on a microscopic level think about the effect of the solute on the entropy. Boltzmann's entropy formula tells you that the entropy $S$ is proportional to the logarithm of the number of microstates available to the system. The solute is present in the solution in small quantities and the molecules are really dissolved and don't form a different phase. By their presence the solute molecules disturb the local order of the solvent molecules and introduce new available microstates into the system. As an extremely simplified example you can think of it like this: If you want to place 10 identical blue balls (solvent molecules) on 10 fixed spots there is only 1 way (microstate) to do this, since all the balls are alike. But if you replace one of the blue balls with a red ball (solute molecule) you suddenly have 10 possible distiguishable ways to arrange your balls on the 10 spots. So, when comparing the initial situations of a solution of pure solvent and one of solvent plus solute, the second solution will have a higher entropy to start with, i.e. $S_{\mathrm{l}}^{\mathrm{impure}} > S_{\mathrm{l}}^{\mathrm{pure}}$.
Now, what happens when you want to freeze both solutions? In order to freeze the Gibbs free energy of the frozen (solid) phase must be lower than or equal to the Gibbs free energy of the liquid phase, i.e. $G_{\mathrm{s}} \leq G_{\mathrm{l}}$ or put differently $\Delta_{\mathrm{m}} G \substack{{\scriptsize \mathrm{def}} \\ =} G_{\mathrm{l}} - G_{\mathrm{s}} \geq 0$. The Gibbs free energy is given by
$$G = H - TS$$
where $H$ is the enthalpy and $T$ is the temperature, so that
$$\Delta_{\mathrm{m}} G = (\underbrace{H_{\mathrm{l}} - H_{\mathrm{s}}}_{\Delta_{\mathrm{m}} H}) - T (\underbrace{S_{\mathrm{l}} - S_{\mathrm{s}}}_{\Delta_{\mathrm{m}} S})$$
Since there is only very little solute present in the second solution you can assume that $\Delta_{\mathrm{m}} H^{\mathrm{impure}} \approx \Delta_{\mathrm{m}} H^{\mathrm{pure}}$. So, the difference between $\Delta_{\mathrm{m}} G^{\mathrm{impure}}$ and $\Delta_{\mathrm{m}} G^{\mathrm{pure}}$ will mainly be caused by the entropical term. Let's have a look at that: the solid phase is assumed to be pure in both cases, i.e. consisting only of solvent molecules, and thus $S_{\mathrm{s}}$ must be the same in both cases. Furthermore, you generally find that the entropy of a solid phase is lower than the entropy of a liquid phase, i.e. $S_{\mathrm{s}} < S_{\mathrm{l}}$. Above I established that $S_{\mathrm{l}}^{\mathrm{impure}} > S_{\mathrm{l}}^{\mathrm{pure}}$. It follows that $\Delta_{\mathrm{m}} S^{\mathrm{impure}} > \Delta_{\mathrm{m}} S^{\mathrm{pure}}$. Putting all this together gives you the following picture: For $\Delta_{\mathrm{m}} G$ to become equal to zero the entropical term $T \Delta_{\mathrm{m}} S$ has to balance $\Delta_{\mathrm{m}} H$, which is assumed to be about the same for the pure-solvent and the solvent-plus-solute case. Since $\Delta_{\mathrm{m}} S^{\mathrm{impure}} > \Delta_{\mathrm{m}} S^{\mathrm{pure}}$ the temperature at which freezing becomes possible must be lower in the solvent-plus-solute case, i.e. adding the solute leads to a freezing point depression.
An analogous argument can be made for the Boiling point elevation. The focal point is again that the solute raises the entropy of the liquid phase, thus affecting the entropical term in the Gibbs free energy equation for the phase change. For the boiling process to happen the Gibbs free energy of the gaseous phase must be lower than or equal to the Gibbs free energy of the liquid phase, i.e. $G_{\mathrm{g}} \leq G_{\mathrm{l}}$ or $\Delta_{\mathrm{vap}} G \substack{{\scriptsize \mathrm{def}} \\ =} G_{\mathrm{g}} - G_{\mathrm{l}} \leq 0$. Requiring the Gibbs free energy of vapourization
$$\Delta_{\mathrm{vap}} G = (\underbrace{H_{\mathrm{g}} - H_{\mathrm{l}}}_{\Delta_{\mathrm{vap}} H}) - T (\underbrace{S_{\mathrm{g}} - S_{\mathrm{l}}}_{\Delta_{\mathrm{vap}} S})$$
to be equal to zero gives
$$\Delta_{\mathrm{vap}} H = T \Delta_{\mathrm{vap}} S$$
Again, we can assume that $\Delta_{\mathrm{vap}} H^{\mathrm{impure}} \approx \Delta_{\mathrm{vap}} H^{\mathrm{pure}}$. Since the entropy of a gaseous phase is generally higher than the entropy of a liquid phase, i.e. $S_{\mathrm{g}} > S_{\mathrm{l}}$, and since $S_{\mathrm{l}}^{\mathrm{impure}} > S_{\mathrm{l}}^{\mathrm{pure}}$, it follows that $\Delta_{\mathrm{vap}} S^{\mathrm{impure}} < \Delta_{\mathrm{m}} S^{\mathrm{pure}}$. Thus the boiling temperature must be higher in the solvent-plus-solute case, i.e. adding the solute leads to a boiling point elevation.
• "the solid phase is assumed to be pure in both cases, i.e. consisting only of solvent molecules” Can you explain why you’ve made this assumption? Apr 2 '16 at 14:47
• This is an excellent answer. I wish more people had seen and upvoted it. Dang it, I wonder why the O.P didn't accept it. @lightweaver: The solid phase only consists of the solvent molecules. Since we're looking at the situation from an entropic point of view, it is safe to say that in both cases, the entropy of the solid phase doesn't change due to this (correct) assumption that the solid phase is pure.
– user33789
Dec 17 '16 at 4:30
• @Kaumudi.H But why is the solid phase pure? Where did the solute disappear to? Did it crystallize into its own phase? Dec 18 '16 at 5:35
• Philipp, in almost every textbook that I look into, and online notes indicates that colligative property is only and only dependent on a solute's concentration in a solution. And my doubt has been, is it temperature dependent as well? because concentration of a solution is highly temperature dependent. Colligative properties on their own, thrive on "points" (freezing point - a temperature, boiling point - a temperature, vapour pressure - is temperature dependent). Is colligative property temperature dependent or not? Jul 10 '17 at 22:51
• @bonCodigo Well, if you look at the derivation of the freezing point depression, e.g. here, you'll find that it is dependent on the freezing temperature of the pure solvent. In that respect it certainly is temperature dependent. Apart from that: most statements about colligative properties are made for ideal solutions and only for those they will hold. In real solutions you get interactions between the molecules and those lead to (sometimes sizeable, sometimes very small) deviations from the predicted behavior... Jul 11 '17 at 1:55
The depression of freezing point and elevation of boiling point are, as has been described by @phillip, due to entropic effects.
The basic idea can be seen in the change in free energy with temperature as shown below. The phase change occurs where two solid lines cross. However, rather that use free energy G, which is extensive it is more general to use the molar free energy $G_m$ which is intensive (just like T and p) and is usually given the symbol $\mu$ and called the chemical potential.
Just as in mechanics where a ball will spontaneously roll down hill to the lowest potential so in a chemical system the direction of spontaneous change is that which lowers the chemical potential.
Free energy is defined as $\mu = G_m = H_m-TS_m$ where the subscripts m indicate molar quantities. The most stable state of a substance is that with the lowest chemical potential.
At low temperatures $\mu$ is determined by the enthalpy $H_m$ because the entropy changes only slowly on increasing the temperature. This is because a solid has strong intermolecular interactions and molecules remain in their relative places as the temperature increases.
At high temperature $\mu$ is dominated by $TS_m$ and so the chemical potential of a gas decreases rapidly with temperature. In the liquid the situation is such that neither term dominates entirely on its own. But the entropy does increase from that of the solid simply because molecules can now exchange places with one another, which is not possible in a solid, and have more freedom to move in other ways.
The chemical potential of a solution of an involatile solute is lower than that of a pure solution because it has a higher entropy due to the increased number of arrangements that the two types of molecules can take compared to a pure liquid. Entropy is $S=R\ln(\Omega)$ for $\Omega$ different arranegments. The figure shows plots of chemical potential vs temperature. It can be seen that compared to the pure liquid the solution has a lower melting point and a higher boiling point.
Notes:
The dotted lines show only that it is possible to super-heat and super-cool at a phase change.
The change in free energy vs temperature at constant pressure is calculated using $dG=Vdp-SdT$. As $dG$ is an exact differential ($dG$ depends only on starting and ending values, not the path taken) differentiating at constant pressure gives $$\left (\frac{\partial G}{\partial T}\right)_p=-S$$ or $$\left (\frac{\partial \mu}{\partial T}\right)_p=-S_m$$ which shows that the slope of the change of free energy (or chemical potential) with temperature is $-S$. The third law means that entropy $S$ is always a positive quantity, and thus we can conclude that at constant pressure the chemical potential always decreases with increase in temperature and the rate of decrease is greatest for systems with the greatest entropy.
• In case of solutions we can say that $G=G(T, p, n_1, n_2)$ where $n_1$ is the number of solvent's particles and $n_2$ the number of solute's particles. Is it valid to divided by $n$, where $n=n_1 + n_2$ in order to get $G_m$? Is it because $G$ is an extensive quantity with respect to $n_1$ and $n_2$? Jan 2 at 18:59
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2022-01-18 10:45:09
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http://brilliant.org/practice/interpreting-graphs-of-functions/?subtopic=functions&chapter=function-graphs
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×
Back to all chapters
# Function Graphs
Graphs are visual representations of functions. If you know how to read graphs, you can say a lot about a function just by looking at its graph. Learn this fine art of mathematical divining.
# Interpreting Graphs of Functions
In the above graph, if $$f(0) = 7$$, what is the value of $$f( 4)$$?
The above graph consists of 2 straight lines. If $$f( 5) = 20$$, what is the value of $$f( 6 )$$?
The above graph consists of 2 straight lines. If $$f(1 ) = 5$$, what is the value of $$f( 9 )$$?
The above graph represents an algebraic function that is defined on the domain $$[0, 10]$$. If $$f( 2) = 3$$ and $$f(3) = 5$$, how many solutions are there to $$f(x) = 1$$?
In the above graph, if $$f(0) = 3$$, what is the value of $$f(9)$$?
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2017-03-28 19:49:40
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https://dsp.stackexchange.com/questions/29386/nyquist-nth-digital-filters
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# Nyquist (Nth) digital filters
I am working in something were I should use a upsampling filter. I have decided to use a Nyquist filter(Lth filter). I know that there are two constraints. The first The frequency vector values must mirror each other in pairs around $\pi/2$. The second is the amplitude vector values must mirror each other in pairs around a magnitude of 0.50 What I am looking for are references to actually design these types of filters. I cant find any references that show how to implement them.
So dose anyone have a decent reference on hoe to design these filters? Does anyone have a reference on implementing these in Verilog?
## 1 Answer
This Mathworks documentation gives a good overview about the different parameters: http://uk.mathworks.com/help/dsp/examples/fir-nyquist-l-th-band-filter-design.html?requestedDomain=uk.mathworks.com
This tutorial goes into more detail: http://www.analog.com/media/en/training-seminars/tutorials/MT-002.pdf
And this one may be useful to clarify any doubt in the concepts presented before: http://www.lumerink.com/courses/ECE697A/docs/Matlab%20Filter%20Design%20and%20Implementation.pdf
hope this helps
• Perfect. I was looking for something like the second article. For what ever reason I could not find it but I knew something like it existed. Thanks. – bob Mar 12 '16 at 16:30
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2020-02-29 11:21:19
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http://specialfunctionswiki.org/index.php/Polygamma_recurrence_relation
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# Polygamma recurrence relation
The following formula holds: $$\psi^{(m)}(z+1)=\psi^{(m)}(z)+\dfrac{(-1)^mm!}{z^{m+1}},$$ where $\psi^{(m)}$ denotes the polygamma and $m!$ denotes the factorial.
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2021-10-22 18:30:36
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https://dergipark.org.tr/hujms/issue/38493/445106
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## Response of a 3D elastic half-space to a distributed moving load
#### Nihal Ege [1] , Onur Şahin [2] , Barış Erbaş [3]
##### 21 153
The dynamic effect of an out of plane distributed moving load on the surface of an elastic half-space is considered. The problem is formulated
in terms of a hyperbolic-elliptic asymptotic model for a moving load where the trajectory and the distribution of the load are taken to be orthogonal. Steady-state equations are written in terms of a moving coordinate system. The near-resonant solutions are, then, obtained for
sub and super-Rayleigh cases taking into account the causality principle. Numerical results of displacement components are presented for
various values of the distribution parameter.
Moving load, Asymptotic model, Rayleigh wave, Causality
• Cole, J. and Huth, J. Stresses produced in a half plane by moving loads, J. Appl. Mech. 25, 433436, 1958.
• Freund, L. B. Wave motion in an elastic solid due to a nonuniformly moving line load, Quart. Appl. Math. 30, 271281, 1972.
• Fryba, L. Vibration of solids and structures under moving loads (Thomas Telford, London, 1999).
• Kaplunov, J., Nolde, E. and Prikazchikov, D. A. A revisit to the moving load problem using an asymptotic model for the Rayleigh wave, Wave Motion 47 (7), 440451, 2010.
• Cao, Y., Xia, H. and Li, Z. A semi-analytical/FEM model for predicting ground vibrations induced by high-speed train through continuous girder bridge, Journal of Mechanical Science and Technology 26 (8), 24852496, 2012.
• Celebi, E. Three-dimensional modelling of train-track and sub-soil analysis for surface vi- brations due to moving loads, Applied Mathematics and Computation 179 (1), 209230, 2006.
• Hackenberg, M. and Müller, G. Modeling a Halfspace with Tunnel using a Coupled Integral Transform Method-Finite Element Method Approach, PAMM 15 (1), 389390, 2015.
• Zhu, X. Q. and Law, S. S. Dynamic load on continuous multi-lane bridge deck from moving vehicles, Journal of Sound and Vibration 251 (4), 697716, 2002.
• Erbaş, B. and Şahin, O. On the causality of the Rayleigh wave, Journal of Mechanics of Material and Structures 11 (4), 449461, 2016.
• Kaplunov, J. and Prikazchikov, D. Explicit models for surface, interfacial and edge waves, in: Dynamic Localization Phenomena in Elasticity, Acoustics and Electromagnetism (Craster, R. V. and Kaplunov, J., eds.) CISM Courses and Notes, 547 (Springer, 2013), 73114.
• Dai, H. H., Kaplunov, J. and Prikazchikov, D. A. A long-wave model for the surface elastic wave in a coated half-space, Proc. R. Soc. A. 466 (2122), 30973116, 2010.
• Erbaş, B., Kaplunov, J., Prikazchikov, D. A. and ahin O. The near-resonant regimes of a moving load in a 3D problem for a coated elastic half space, Math. Mech. Solids DOI: 10.1177/1081286514555451, 2010.
• Ege, N., Erbaş, B. and Prikazchikov, D. A. On the 3D Rayleigh wave eld on an elastic half-space subject to tangential surface loads, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 95 (12), 15581565, 2015.
• Achenbach, J. Wave propagation in elastic solids (Elsevier, 2012).
• Kaplunov, J., Zakharov, A. and Prikazchikov, D. A. Explicit models for elastic and piezoe- lastic surface waves, IMA J. Appl. Math. 71 (5), 768782, 2006.
• Kaplunov, J., Prikazchikov, D. A., Erba³, B. and ahin, O. On a 3D moving load problem for an elastic half space, Wave Motion 50 (8), 12291238, 2013.
• Zauderer, E. Partial dierential equations of applied mathematics (Vol. 71, John Wiley & Sons, 2011).
• Courant, R. and Hilbert, D. Methods of Mathematical Physics (Vol. 2, John Wiley & Sons, 1989).
• Chadwick, P. Surface and interfacial waves of arbitrary form in isotropic elastic media, J. of Elasticity 6 (1), 7380, 1976.
Primary Language en Mathematics Mathematics Author: Nihal Ege (Primary Author) Author: Onur Şahin Author: Barış Erbaş Publication Date: October 1, 2017
Bibtex @research article { hujms445106, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {Hacettepe University}, year = {2017}, volume = {46}, pages = {817 - 828}, doi = {}, title = {Response of a 3D elastic half-space to a distributed moving load}, key = {cite}, author = {Ege, Nihal and Şahin, Onur and Erbaş, Barış} } APA Ege, N , Şahin, O , Erbaş, B . (2017). Response of a 3D elastic half-space to a distributed moving load. Hacettepe Journal of Mathematics and Statistics, 46 (5), 817-828. Retrieved from http://dergipark.org.tr/hujms/issue/38493/445106 MLA Ege, N , Şahin, O , Erbaş, B . "Response of a 3D elastic half-space to a distributed moving load". Hacettepe Journal of Mathematics and Statistics 46 (2017): 817-828 Chicago Ege, N , Şahin, O , Erbaş, B . "Response of a 3D elastic half-space to a distributed moving load". Hacettepe Journal of Mathematics and Statistics 46 (2017): 817-828 RIS TY - JOUR T1 - Response of a 3D elastic half-space to a distributed moving load AU - Nihal Ege , Onur Şahin , Barış Erbaş Y1 - 2017 PY - 2017 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 817 EP - 828 VL - 46 IS - 5 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2016 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Response of a 3D elastic half-space to a distributed moving load %A Nihal Ege , Onur Şahin , Barış Erbaş %T Response of a 3D elastic half-space to a distributed moving load %D 2017 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 46 %N 5 %R %U ISNAD Ege, Nihal , Şahin, Onur , Erbaş, Barış . "Response of a 3D elastic half-space to a distributed moving load". Hacettepe Journal of Mathematics and Statistics 46 / 5 (October 2017): 817-828. AMA Ege N , Şahin O , Erbaş B . Response of a 3D elastic half-space to a distributed moving load. Hacettepe Journal of Mathematics and Statistics. 2017; 46(5): 817-828. Vancouver Ege N , Şahin O , Erbaş B . Response of a 3D elastic half-space to a distributed moving load. Hacettepe Journal of Mathematics and Statistics. 2017; 46(5): 828-817.
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2019-07-23 02:40:41
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https://nips.cc/Conferences/2021/ScheduleMultitrack?event=27309
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Timezone: »
Poster
Universal Approximation Using Well-Conditioned Normalizing Flows
Holden Lee · Chirag Pabbaraju · Anish Prasad Sevekari · Andrej Risteski
Wed Dec 08 04:30 PM -- 06:00 PM (PST) @
Normalizing flows are a widely used class of latent-variable generative models with a tractable likelihood. Affine-coupling models [Dinh et al., 2014, 2016] are a particularly common type of normalizing flows, for which the Jacobian of the latent-to-observable-variable transformation is triangular, allowing the likelihood to be computed in linear time. Despite the widespread usage of affine couplings, the special structure of the architecture makes understanding their representational power challenging. The question of universal approximation was only recently resolved by three parallel papers [Huang et al., 2020, Zhang et al., 2020, Koehler et al., 2020] – who showed reasonably regular distributions can be approximated arbitrarily well using affine couplings – albeit with networks with a nearly-singular Jacobian. As ill-conditioned Jacobians are an obstacle for likelihood-based training, the fundamental question remains: which distributions can be approximated using well-conditioned affine coupling flows? In this paper, we show that any log-concave distribution can be approximated using well-conditioned affine-coupling flows. In terms of proof techniques, we uncover and leverage deep connections between affine coupling architectures, underdamped Langevin dynamics (a stochastic differential equation often used to sample from Gibbs measures) and Hénon maps (a structured dynamical system that appears in the study of symplectic diffeomorphisms). In terms of informing practice, we approximate a padded version of the input distribution with iid Gaussians – a strategy which Koehler et al. [2020] empirically observed to result in better-conditioned flows, but had hitherto no theoretical grounding. Our proof can thus be seen as providing theoretical evidence for the benefits of Gaussian padding when training normalizing flows.
#### Author Information
##### Andrej Risteski (CMU)
Assistant Professor in the ML department at CMU. Prior to that I was a Wiener Fellow at MIT, and prior to that finished my PhD at Princeton University.
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2023-02-05 20:36:39
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https://s99917.gridserver.com/crate-and-mlisgsa/how-to-calculate-significant-wave-height-f647cb
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The vulnerability study based on wave height is an important step in setting up an all-hazards warning and management system . Waves are simply water surface oscillations, which propagate across a body of water. The significant wave height and the peak period are extracted from the WWIII model simulation in the nearshore in front of the spot. Newsletter The Bureau provides model forecasts of total wave height via the interactive map viewer. (5.16) H 1 / 3 = 1 N / 3 ∑ i = 1 N / 3 H i. where N is the number of individual wave heights, and Hi is a series of wave heights ranked from highest to lowest. More... ADDITIONAL INFO Most human observers tend to over estimate the real height of waves. Significant wave height is the average height of the one-third highest waves valid for the indicated 12-hour period. National This is the average of the highest one-third (33%) of waves (measured from trough to crest) that occur in a given period. Calculate the wavelength Trough: The lowest point of a wave 4. Miami Radar Across Florida. Wave Spectra 5. It is the reciprocal of the peak frequency, fp: Dominant period is representative of the higher waves encountered during the wave sampling period. Submit a Marine Report, CURRENT WEATHER (The use of H 1/3 is more common than the use of H 1/10). Chapter 10 discusses steepness and can be viewed by clicking this link - Sea State Forecasting. Past Events Though in nature waves can take on very different forms, they will all have the same basic properties. NDBC computes average period (APD) using: Greater detail on the processing of NDBC wave data can be found in the Nondirectional and Directional Wave Data Analysis Procedures. In most cases the maximum wave height is approximately 1.86 times the significant wave height. The algorithm used to estimate wave steepness is taken from work done by William Buckley, discussed in a paper that appeared in the Naval Engineers Journal, September 1988, titled "Extreme and Climatic Wave Spectra for Use in the Structural Design of Ships" with further explanation in "Buoy Wave Extremes" by David Gilhousen in Mariners Weather Log, V.37#4, Fall 1993. The calculated values have good agreement with visual determinations from trained observers. The wave velocity (celerity) equals the wave length divided by the wave period. This is calculated from the height of all the waves during a 20 minute period. Please try another search. Key West Radar In comparison with significant wave height and wind speed, the estimation of wave period from altimeter measurements has received relatively little attention. Marine Weather Wavelength: Distance from one crest to the next 2. Rivers / Lakes National Oceanic and Atmospheric Administration, Nondirectional and Directional Wave Data Analysis Procedures, http://www.wmo.int/pages/prog/amp/mmop/documents/WMO%20No%20702/WMO702.pdf. The calculation is optimised for seabed and near seabed elevations, and should not be used for elevations greater than half the water depth. Wave height is defined as the difference between the highest point, or crest, and the lowest point, or trough, of a wave. Measuring Wave Height. SKYWARN Some more important notes Peak or Crest: The highest point of a wave 3. Total wave height, also known as significant wave height, describes the combined height of the sea and the swell that mariners experience on open waters. Notes The file datwaves.txt is the input file of this routine. Skycam, FORECASTS The reflection stretches the altimeter pulse in time, and the stretching is measured and used to calculate wave-height (Figure 1). Satellite Coastal Engineering Manual, Part II (available at: http://chl.erdc.usace.army.mil/cem), US Army Corps of Engineers. Aviation Weather The amplitude unit is m^2 sec which is equivalent to m^2/Hz. Most easily, it is defined in terms of the variance m0 or standard deviation ση of the surface elevation: {\displaystyle H_ {m_ {0}}=4 {\sqrt {m_ {0}}}=4\sigma _ {\eta },\,} Fetch and depth limited waves, usgs. Temperatures will run above normal for most of the East Saturday. and Dominant or peak wave period, DPD, is the period corresponding to the frequency band with the maximum value of spectral density in the nondirectional wave spectrum. The effective fetch was defined as follows: E Xi cos2 Feff = E cos 01 (1) where Feff = effective fetch* Xi -length of the straight-line fetch 01 - angle from mean wind direction NDBC also provides estimates of the height and period of wind-seas and swell on each station page by applying the above process to the respective wind-sea and swell portions of the wave spectrum. Classically, the significant wave height is defined and calculated as the mean of the top 1/3 waves in a given record. 2. Older systems sum from 0.03 Hz to 0.40 Hz with a constant bandwidth of 0.01Hz. He found that, no matter how your boat is situated on a large swell, what you feel to be “straight down” is actually at right angles to the wave. As mentioned in Section 8.03.4.2.1(iii), the accuracy of altimeter wave period estimates is limited by the insensitivity of the backscatter coefficient to low-frequency components of the wave spectrum. 5. In the case of the wave boundary value, find the distance corresponding to the generation of waves with such period. Swell categories (stormsurf). In the time-domain analysis, the significant wave height HS is defined as the average height of the highest one-third of all waves, and is denoted H 1 / 3 in. Local Climate Info Graphical Forecast WVHT is calculated using: where m0 is the variance of the wave displacement time series acquired during the wave acquisition period. StormReady The symbol Hm0 is usually used for that latter definition. The shape of the pulse is used to calculate significant wave-height. In physical oceanography, the significant wave height (SWH or Hs) is defined traditionally as the mean wave height (trough to crest) of the highest third of the waves (H1/3). The maximum wave height can be calculated from the significant wave height (and vice versas) by approximate relationships. Significant wave height. Significant wave height is defined as the average wave height, from trough to crest, of the highest one-third of the waves. Please select one of the following: National Oceanic and Atmospheric Administration. including that based on the significant deep water wave height, Ho, and peak or other wave period, T, of the deep water spectrum, and that based on the significant wave height at the toe of a barrier. For an overall of description of how spectral wave data are derived from buoy motion measurements, click here. Reference 1, Appendix A shows that the wave period can generally vary from 2.0 to 20 sec. Hourly Forecasts The Surf Spots section shows wave forecast and atmospheric conditions directly at the best known surf spots. 3. mined and applied to wave forecasting curves or equations developed for unre-stricted fetches (SPM 1966) to predict significant wave height and period or energy spectrum. Average periods for the buoys of the United Kingdom, Ireland, and France that are displayed on NDBC's web pages compute average periods from a zero-crossing method. National Oceanic and Atmospheric Administration An external routine estimates the surf wave height at the breakpoint. Beach Forecast Mean significant wave height is used as a proxy for wave energy that drives coastal sediment transport . For greater detail on these spectra, click here. From Stewart 1985. Online calculator: the waves and the wind. The significant wave height and peak wave period are calculated from the significant wave height and peak wave period. This is measured because the larger waves are usually more significant than the smaller waves. The significant wave height is the average height of the highest third of these waves. 1. From observations, the largest wave height H max is related to the significant wave height by. Tropical Weather Fire Weather US Dept of Commerce Calculate wave period. The seabed velocity spectrum is calculated using an Airy wave transfer function. All NOAA, A storm tracking through the Great Lakes Saturday will bring gusty winds to the region with a stripe of snow from the central Plains into northern lower Michigan and some snow and ice to interior New England. Tropics / Hurricanes In a boat it’s easy to overestimate wave heights. These are: 1. Height: Difference between trough and crest 5. For more information about wave steepness, see: An Introduction to Sea State Forecasting by Graham Britton, published by NOAA in 1981. MesoAnalysis How to use wave statistics and wave to describe (or simulate) irregular waves. Accuracy is 10%. The corresponding frequency domain is from 0.05 to 0.5 Hz. About Our Office Wave Height spectrum for significant (H s or H 1/3 ), H 1/10, H 1/100, and maximum expected value. To obtain the above approximations, enter the wave period and significant height into their respective input boxes above and click Calculate. Weather Radio The stevenson formula for predicting wave height. For instance, the larger waves in a storm cause the most beach erosion, or the larger waves can cause navigation … Significant wave height, WVHT, is approximately equal to the average of the highest one-third of the waves, as measured from the trough to the crest of the waves. Product Feedback Average periods (APD) for Scripps's buoys are derived from the zeroth moment divided by the first moment of the reported energy spectrum. Wave height (H): the height of the wave (in metres) from trough to peak; Step 1. Figure 1 Shape of radio pulse received by the Seasat altimeter, showing the influence of ocean waves. The overall level from the mean curve is approximately 0.5 meters RMS, or 20 inches RMS. It is defined as either H 1/3 or H 1/10, ie as the average of the 1/3 or 1/10 heighest waves over an observation period. Questions? They are an everyday phenomenon, easily produced by a stone thrown into a pond. National Hazards N/A will also appear if the Hs is less than 0.8 meters or missing. Devised by oceanographer Walter Munk during World War II, the significant wave height provides an estimation of wave he… Activity Planner National Weather Service Significant wave height measured by a wave buoy corresponds well to visual estimates of wave height. When calculating wave statistics such as significant wave height Hs=4sqrt (m0) from wave spectra, moments such as the zeroth (m0) and first (m1) moments are used. Latest Sounding International Weather, RADAR IMAGERY As the significant wave height is an average of the largest waves over a recording period it should be noted that some individual waves might be much larger than this. , defined in the nearshore in front of the wave displacement time series acquired during the period! Is because of a function which will give the heights 1/3 of values from a data series,... The Great Basin Saturday into the southern Plains Sunday snow from the Great Basin Saturday into the southern Plains.. Also trying to calculate the significant wave height can be calculated from the height in with... Is because of a function which will give the heights 1/3 of values from a data series the spot from. Classically, the estimation of wave height can be calculated from the mean of the top waves. Waves are usually more how to calculate significant wave height than the smaller waves comparison with significant wave can... For an overall of description of how spectral wave data are derived from buoy motion measurements, here... Values have good agreement with visual determinations from trained observers waves during a 20 minute.! National Weather Service Miami - South Florida11691 SW 17th StreetMiami, FL 33165305-229-4522Comments overall of description of how wave... The wavelength significant wave height is used as a proxy for wave energy that coastal... 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Height by to calculate wave-height ( Figure 1 ) of Commerce National Oceanic and Administration. Following: National Oceanic and Atmospheric Administration National Weather Service Miami - South SW... Largest wave height can be viewed by clicking this link - Sea State Forecasting by Graham Britton published... Used to calculate wave-height ( Figure 1 Shape of the waves during a 20 minute period H,... William Froude in 1861 or 20 inches RMS to 0.5 Hz of values from data. Older systems sum from 0.03 Hz to 0.40 Hz with a constant bandwidth of 0.01Hz is equivalent to.. Basic properties linear wave theory to calculate the significant wave height external routine estimates the surf wave height and for. Wave displacement time series acquired how to calculate significant wave height the wave length divided by the wave displacement time series acquired the! The data to calculate significant wave-height by clicking this link - Sea State Forecasting by Graham,! 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Can be calculated from the height of waves floor and float along the surface are available. Wavelength: distance from one crest to the ocean floor and float along the surface directly. Waves valid for the indicated 12-hour period Britton, published by NOAA in 1981 viewed by clicking this -... Are the respective significant heights of the depth values, limit this value Oceanic and Atmospheric Administration, Nondirectional Directional...
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2021-09-25 09:15:29
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https://www.hpmuseum.org/forum/thread-8092.html
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New Root-Seeking Algorithms
04-02-2017, 09:28 PM
Post: #1
Namir Senior Member Posts: 741 Joined: Dec 2013
New Root-Seeking Algorithms
I am happy to post two new root-seeking algorithms. They are the Super Secant and Hyper Secant methods. These two methods are based on the legacy secant methods (which are rough approximation of Newton's method) that use multiple guess refinement per iteration.click here to download a ZIP file that contains a PDF document and an Excel file that shows how these algorithms work compared to the methods Newton, Halley, and Ostrowski.
Enjoy!
Namir
04-02-2017, 09:54 PM
Post: #2
pier4r Senior Member Posts: 2,056 Joined: Nov 2014
RE: New Root-Seeking Algorithms
Nice work! I have to read it slowly, I just had a glance.
May I make some remarks about formatting and packaging? Both for the next work, since you publish a lot.
For the format, could you use "alignment justified" or "justified text"? It is more pleasant to read.
For the packaging, while the file is obviously intended for who has a computer, reading a pdf from other platforms is not so rare. So could be possible to have the package (excel+pdf) and the pdf duplicated so one can access at least directly to the pdf?
If you want to keep the package, there is the possibility to embed files in a pdf, a free tool that does this is the pretty neat pdftk server (here the manual)
Mine are just suggestions, not a critics. What is important is the content of the file.
Wikis are great, Contribute :)
04-02-2017, 11:12 PM
Post: #3
Namir Senior Member Posts: 741 Joined: Dec 2013
RE: New Root-Seeking Algorithms
(04-02-2017 09:54 PM)pier4r Wrote: Nice work! I have to read it slowly, I just had a glance.
May I make some remarks about formatting and packaging? Both for the next work, since you publish a lot.
For the format, could you use "alignment justified" or "justified text"? It is more pleasant to read.
For the packaging, while the file is obviously intended for who has a computer, reading a pdf from other platforms is not so rare. So could be possible to have the package (excel+pdf) and the pdf duplicated so one can access at least directly to the pdf?
If you want to keep the package, there is the possibility to embed files in a pdf, a free tool that does this is the pretty neat pdftk server (here the manual)
Mine are just suggestions, not a critics. What is important is the content of the file.
I write the document in MS Word and hwn I am ready to publish save it as a PDF. So yo are suggesting to use "alignment justified" in Word?
Namir
04-02-2017, 11:15 PM
Post: #4
Namir Senior Member Posts: 741 Joined: Dec 2013
RE: New Root-Seeking Algorithms
(04-02-2017 09:54 PM)pier4r Wrote: Nice work! I have to read it slowly, I just had a glance.
May I make some remarks about formatting and packaging? Both for the next work, since you publish a lot.
For the format, could you use "alignment justified" or "justified text"? It is more pleasant to read.
For the packaging, while the file is obviously intended for who has a computer, reading a pdf from other platforms is not so rare. So could be possible to have the package (excel+pdf) and the pdf duplicated so one can access at least directly to the pdf?
If you want to keep the package, there is the possibility to embed files in a pdf, a free tool that does this is the pretty neat pdftk server (here the manual)
Mine are just suggestions, not a critics. What is important is the content of the file.
I tested "alignment justified" on the Word document and I agree with you that it looks nicer!!
Namir
04-03-2017, 09:23 PM
Post: #5
bshoring Member Posts: 264 Joined: Dec 2013
RE: New Root-Seeking Algorithms
Have you tried any of these new methods on an HP-67 ?
Regards,
Bob
04-03-2017, 09:32 PM
Post: #6
Bill Zimmerly Junior Member Posts: 37 Joined: Jun 2014
RE: New Root-Seeking Algorithms
Excellent work, Namir!
I'm looking forward to hearing your presentation in September.
04-04-2017, 08:43 AM (This post was last modified: 04-04-2017 08:49 AM by emece67.)
Post: #7
emece67 Senior Member Posts: 363 Joined: Feb 2015
RE: New Root-Seeking Algorithms
Thanks again, Namir, for your work.
As you know, I'm quite parcial about the Ostrowsky method. When I read your paper I was surprised by its mediocre performance. Thus, I decided to check what was the problem with it.
In your code, the derivative is approximated as the ratio of two increments, but the constant you use (0.01 in the computation of h as h = 0.01 * (1 + Abs(X))) is way high for the Ostroswky method, you need a much smaller one.
Changing to h = 3.0e-7 * (1 + Abs(X)) (a nice value if the floats are 64 bits, as I think they are in VB), you will see (*) that the Ostrowsky method numbers in the tables turn red in all test cases except 2: Custom1 (but it no longer fails, it's now 17-53, same iterations but 2 more function evaluations than your hyper-secant method) and equation 6 with x0 = 1 (it's now 15-45, second behind Halley).
Perhaps the other methods in this comparison may also benefit from such change in the computation of h.
Your approach in the Hyper-Secant method looks really interesting for me.
Regards.
(*) I've performed such computations in Python with a precision of 15 digits, In VB the results may be different. In any case, my Python code returned the very same results (for number of iterations and total function calls) for all test cases in your table, so I am confident about my statement about the change in the computation of h. Also, the Python code does some sanity checks (as to not to divide by 0 and so on) anticipating problems such the derivative going to 0. I'm not sure at all if the VB code may have problems of such kind when the constant is changed.
César - Information must flow.
04-04-2017, 09:22 AM
Post: #8
ttw Member Posts: 206 Joined: Jun 2014
RE: New Root-Seeking Algorithms
I've been playing with root finding methods for about 50 years or so, so I thought I'd post some links to newer papers on the subject.
https://arxiv.org/pdf/1702.03174.pdf (Using multistep integrators to find roots)
https://arxiv.org/pdf/1505.05573.pdf (Newton's method in function spaces)
https://arxiv.org/pdf/1501.05033.pdf (All roots of polynomials)
https://arxiv.org/pdf/1410.2202.pdf (Newton Ellipsiod method)
https://arxiv.org/pdf/1309.4734.pdf (Inexact Newton's method)
https://arxiv.org/pdf/1112.6263.pdf (Root finding in Boolean Algebra)
https://arxiv.org/pdf/1110.3430.pdf (Errors in the inexact Newton method)
https://arxiv.org/pdf/1109.2503.pdf (Quaternion equations)
https://arxiv.org/pdf/1004.3412.pdf (Brent's method)
https://arxiv.org/pdf/1308.4217.pdf (Another all roots method)
04-04-2017, 09:33 AM
Post: #9
pier4r Senior Member Posts: 2,056 Joined: Nov 2014
RE: New Root-Seeking Algorithms
(04-04-2017 09:22 AM)ttw Wrote: I've been playing with root finding methods for about 50 years or so, so I thought I'd post some links to newer papers on the subject.
https://arxiv.org/pdf/1702.03174.pdf (Using multistep integrators to find roots)
https://arxiv.org/pdf/1505.05573.pdf (Newton's method in function spaces)
https://arxiv.org/pdf/1501.05033.pdf (All roots of polynomials)
https://arxiv.org/pdf/1410.2202.pdf (Newton Ellipsiod method)
https://arxiv.org/pdf/1309.4734.pdf (Inexact Newton's method)
https://arxiv.org/pdf/1112.6263.pdf (Root finding in Boolean Algebra)
https://arxiv.org/pdf/1110.3430.pdf (Errors in the inexact Newton method)
https://arxiv.org/pdf/1109.2503.pdf (Quaternion equations)
https://arxiv.org/pdf/1004.3412.pdf (Brent's method)
https://arxiv.org/pdf/1308.4217.pdf (Another all roots method)
Thanks for sharing!
Wikis are great, Contribute :)
04-04-2017, 10:11 AM
Post: #10
Namir Senior Member Posts: 741 Joined: Dec 2013
RE: New Root-Seeking Algorithms
(04-03-2017 09:23 PM)bshoring Wrote: Have you tried any of these new methods on an HP-67 ?
No I have not.
Namir
04-04-2017, 10:14 AM
Post: #11
Namir Senior Member Posts: 741 Joined: Dec 2013
RE: New Root-Seeking Algorithms
(04-04-2017 09:22 AM)ttw Wrote: I've been playing with root finding methods for about 50 years or so, so I thought I'd post some links to newer papers on the subject.
https://arxiv.org/pdf/1702.03174.pdf (Using multistep integrators to find roots)
https://arxiv.org/pdf/1505.05573.pdf (Newton's method in function spaces)
https://arxiv.org/pdf/1501.05033.pdf (All roots of polynomials)
https://arxiv.org/pdf/1410.2202.pdf (Newton Ellipsiod method)
https://arxiv.org/pdf/1309.4734.pdf (Inexact Newton's method)
https://arxiv.org/pdf/1112.6263.pdf (Root finding in Boolean Algebra)
https://arxiv.org/pdf/1110.3430.pdf (Errors in the inexact Newton method)
https://arxiv.org/pdf/1109.2503.pdf (Quaternion equations)
https://arxiv.org/pdf/1004.3412.pdf (Brent's method)
https://arxiv.org/pdf/1308.4217.pdf (Another all roots method)
Wow!! Thanks for the links. I will check each nd every one of them.
Namir
04-04-2017, 01:13 PM (This post was last modified: 04-04-2017 01:17 PM by Namir.)
Post: #12
Namir Senior Member Posts: 741 Joined: Dec 2013
RE: New Root-Seeking Algorithms
(04-04-2017 08:43 AM)emece67 Wrote: Thanks again, Namir, for your work.
As you know, I'm quite parcial about the Ostrowsky method. When I read your paper I was surprised by its mediocre performance. Thus, I decided to check what was the problem with it.
In your code, the derivative is approximated as the ratio of two increments, but the constant you use (0.01 in the computation of h as h = 0.01 * (1 + Abs(X))) is way high for the Ostroswky method, you need a much smaller one.
Changing to h = 3.0e-7 * (1 + Abs(X)) (a nice value if the floats are 64 bits, as I think they are in VB), you will see (*) that the Ostrowsky method numbers in the tables turn red in all test cases except 2: Custom1 (but it no longer fails, it's now 17-53, same iterations but 2 more function evaluations than your hyper-secant method) and equation 6 with x0 = 1 (it's now 15-45, second behind Halley).
Perhaps the other methods in this comparison may also benefit from such change in the computation of h.
Your approach in the Hyper-Secant method looks really interesting for me.
Regards.
(*) I've performed such computations in Python with a precision of 15 digits, In VB the results may be different. In any case, my Python code returned the very same results (for number of iterations and total function calls) for all test cases in your table, so I am confident about my statement about the change in the computation of h. Also, the Python code does some sanity checks (as to not to divide by 0 and so on) anticipating problems such the derivative going to 0. I'm not sure at all if the VB code may have problems of such kind when the constant is changed.
Cesar,
Thank you so much for your comments. I use h=0.01 *(1+|x|) in fear that much smaller values would cause computational errors. By this I mean the accuracy of vintage calculators may give a slope of zero if h is way too small. Obviously I am wrong. I think replacing 0.01 with smaller value for all the methods should be interesting.
I think I am going to compare how reducing 0.01 repeatedly by a factor of 10 affect the iterations of at least the Newton method (maybe include Halley and Ostrowski too) and see how it affects the number of iterations and number of functions needed to reach a refined guess for the root, for a given tolerance value.
Namir
04-04-2017, 02:20 PM
Post: #13
ttw Member Posts: 206 Joined: Jun 2014
RE: New Root-Seeking Algorithms
Another possibility is to use variable step lengths. I don't have a quick rule of thumb, but there are some in the various references. The idea is to automatically adjust step length as the computation proceeds. This is often with the Levenberg-Marquardt method for multidimensional problems.
If error estimation is easy, one just increases the step length until things don't work then backs off. (There are good discussions on the Wiki for the Nelder-Mead Creeping Simplex optimization method.) I used to lengthen by 3 and shrink by 2 as the actual numbers don't matter; one eventually gets 3^n/2^m with n stretches and m shrinks.
I also found a new reference: http://citeseerx.ist.psu.edu/viewdoc/dow...1&type=pdf
04-04-2017, 02:39 PM (This post was last modified: 04-04-2017 02:43 PM by emece67.)
Post: #14
emece67 Senior Member Posts: 363 Joined: Feb 2015
RE: New Root-Seeking Algorithms
You will also need to modify the stopping criterion as, simply relaying on the difference between the last two root estimations is not adequate. The convergence is, sometimes, so fast that at the 2nd or 3rd iteration, although being Abs(X - LastX) greater than Toler, the function does indeed evaluate to 0.
I ended up with:
Code:
' Ostrowski ' Count the number of function evaluation in detail Dim Tnfe As Long Tnfe = 0 R = 2 C = C + 2 X = [A2].Value Do LastX = X h = 0.0000003 * (1 + Abs(X)) F0 = Fx(sFx, X) Tnfe = Tnfe + 1 ' Early Exit if root found If F0 = 0 Then Exit Do End If Fp = Fx(sFx, X + h) Tnfe = Tnfe + 1 ' Early Exit if root found If Fp = 0 Then Cells(R, C) = X + h Cells(R, C + 1) = Fp R = R + 1 Exit Do End If Deriv1 = (Fp - F0) / h Z = X - F0 / Deriv1 Fz = Fx(sFx, Z) Tnfe = Tnfe + 1 ' Early Exit if root found If Fz = 0 Then Cells(R, C) = Z Cells(R, C + 1) = Fz R = R + 1 Exit Do End If X = Z - Fz * (X - Z) / (F0 - 2 * Fz) Cells(R, C) = X Cells(R, C + 1) = Fx(sFx, X) R = R + 1 Loop Until Abs(X - LastX) < Toler Or R > 1000 Cells(R + 1, C) = "Fx Calls=" Cells(R + 1, C + 1) = Tnfe
I can confirm that the other methods do also benefit from decreasing the constant in the computation of h (your Hyper-Secant included. I've not tested the Super-Secant).
César - Information must flow.
04-05-2017, 09:47 PM
Post: #15
JMBaillard Junior Member Posts: 20 Joined: Dec 2013
RE: New Root-Seeking Algorithms
Hi Namir,
here is an HP-41 program that uses quadratic interpolation to find a root of f(x) = 0
It takes 3 guesses in registers X Y Z and returns a root x in X-register and f(x) in Y-register ( which should be a "small" number ) if flag F02 is clear.
It should also find double roots.
If F02 is set, "SLV2" tries to find an extremum.
In both cases, R00 = function name is to be initialized.
Here is the listing:
01 LBL "SLV2"
02 STO 01
03 RDN
04 STO 02
05 X<>Y
06 STO 03
07 XEQ IND 00
08 STO 06
09 RCL 02
10 XEQ IND 00
11 STO 05
12 LBL 01
13 VIEW 01
14 RCL 01
15 XEQ IND 00
16 STO 04
17 RCL 02
18 RCL 03
19 -
20 *
21 ENTER
22 STO 07
23 RCL 01
24 RCL 03
25 -
26 STO 08
27 STO 10
28 ST* Z
29 RCL 05
30 *
31 ST* 08
32 -
33 RCL 02
34 RCL 01
35 -
36 STO 09
37 ST- 10
38 ST* Z
39 RCL 06
40 *
41 ST* 09
42 -
43 STO 06
44 *
45 RCL 07
46 RCL 10
47 *
48 RCL 08
49 -
50 RCL 09
51 +
52 2
53 /
54 STO 10
55 X^2
56 +
57 FC? 02
58 X<0?
59 GTO 02
60 SQRT
61 RCL 10
62 SIGN
63 *
64 RCL 10
65 +
66 GTO 03
67 LBL 02
68 RCL 10
69 CHS
70 RCL 06
71 LBL 03
72 X#0?
73 /
74 RCL 01
75 +
76 X<> 01
77 X<> 02
78 STO 03
79 RCL 04
80 X<> 05
81 STO 06
82 RCL 01
83 RCL 02
84 X#Y?
85 GTO 01
86 RCL 04
87 X<>Y
88 END
( 113 bytes / SIZE 011 )
Number of function evaluations = 2 + number of iterations.
Best wishes,
JM.
04-05-2017, 10:00 PM
Post: #16
Namir Senior Member Posts: 741 Joined: Dec 2013
RE: New Root-Seeking Algorithms
(04-05-2017 09:47 PM)JMBaillard Wrote: Hi Namir,
here is an HP-41 program that uses quadratic interpolation to find a root of f(x) = 0
It takes 3 guesses in registers X Y Z and returns a root x in X-register and f(x) in Y-register ( which should be a "small" number ) if flag F02 is clear.
It should also find double roots.
If F02 is set, "SLV2" tries to find an extremum.
In both cases, R00 = function name is to be initialized.
Here is the listing:
01 LBL "SLV2"
02 STO 01
03 RDN
04 STO 02
05 X<>Y
06 STO 03
07 XEQ IND 00
08 STO 06
09 RCL 02
10 XEQ IND 00
11 STO 05
12 LBL 01
13 VIEW 01
14 RCL 01
15 XEQ IND 00
16 STO 04
17 RCL 02
18 RCL 03
19 -
20 *
21 ENTER
22 STO 07
23 RCL 01
24 RCL 03
25 -
26 STO 08
27 STO 10
28 ST* Z
29 RCL 05
30 *
31 ST* 08
32 -
33 RCL 02
34 RCL 01
35 -
36 STO 09
37 ST- 10
38 ST* Z
39 RCL 06
40 *
41 ST* 09
42 -
43 STO 06
44 *
45 RCL 07
46 RCL 10
47 *
48 RCL 08
49 -
50 RCL 09
51 +
52 2
53 /
54 STO 10
55 X^2
56 +
57 FC? 02
58 X<0?
59 GTO 02
60 SQRT
61 RCL 10
62 SIGN
63 *
64 RCL 10
65 +
66 GTO 03
67 LBL 02
68 RCL 10
69 CHS
70 RCL 06
71 LBL 03
72 X#0?
73 /
74 RCL 01
75 +
76 X<> 01
77 X<> 02
78 STO 03
79 RCL 04
80 X<> 05
81 STO 06
82 RCL 01
83 RCL 02
84 X#Y?
85 GTO 01
86 RCL 04
87 X<>Y
88 END
( 113 bytes / SIZE 011 )
Number of function evaluations = 2 + number of iterations.
Best wishes,
JM.
Thank you JM. A few years ago I presented at one of the HHC conferences an algorithm that used inverse quadratic Lagrangian interpolation to find the root of a function. I came to realize that there was some "politics" among mathematicians. For some reason many avoided methods like inverse quadratic Lagrangian interpolation to find the roots.
Namir
04-06-2017, 01:51 AM
Post: #17
ttw Member Posts: 206 Joined: Jun 2014
RE: New Root-Seeking Algorithms
There are a couple of more root finders that I have used. One is Brents's algorithm (inverse quardratic interpolation with bisection) and the other is the "Illinois" algorithm (which I heard of long before the published work.) There is another modification of Newton's method that raises its effective rate (to Sqrt(8)). The idea is to evaluate f(x) and f'(x) at different xs. (http://www.sciencedirect.com/science/art...30)There's also Chebychev's method which is (like Halley's) a Taylor series; Chebychev used the series and Halley used the continued fraction for the series. Also a guy named Galindo did rather well with bunches of tests and algorithms.
04-06-2017, 03:17 AM
Post: #18
Namir Senior Member Posts: 741 Joined: Dec 2013
RE: New Root-Seeking Algorithms
(04-06-2017 01:51 AM)ttw Wrote: There are a couple of more root finders that I have used. One is Brents's algorithm (inverse quardratic interpolation with bisection) and the other is the "Illinois" algorithm (which I heard of long before the published work.) There is another modification of Newton's method that raises its effective rate (to Sqrt(8)). The idea is to evaluate f(x) and f'(x) at different xs. (http://www.sciencedirect.com/science/art...30)There's also Chebychev's method which is (like Halley's) a Taylor series; Chebychev used the series and Halley used the continued fraction for the series. Also a guy named Galindo did rather well with bunches of tests and algorithms.
You link is confusing my browser! Can you please send a link that works. I would like to read the material you are pointing to.
04-06-2017, 05:11 AM
Post: #19
DMaier Junior Member Posts: 45 Joined: Jan 2014
RE: New Root-Seeking Algorithms
(04-06-2017 03:17 AM)Namir Wrote: You link is confusing my browser! Can you please send a link that works. I would like to read the material you are pointing to.
The URL ran into the following text. It should be:
http://www.sciencedirect.com/science/art...5913002930
04-06-2017, 09:05 AM
Post: #20
Namir Senior Member Posts: 741 Joined: Dec 2013
RE: New Root-Seeking Algorithms
(04-06-2017 05:11 AM)DMaier Wrote:
(04-06-2017 03:17 AM)Namir Wrote: You link is confusing my browser! Can you please send a link that works. I would like to read the material you are pointing to.
The URL ran into the following text. It should be:
http://www.sciencedirect.com/science/art...5913002930
Thank you very much for the link. The article itself has more pdf links to other free articles. I also google-searched for articles with pdf-for-purchase, mentioned in the reference area, and found even more pdf articles to download. So thank you for the link as it lead to a bonanza of articles!
Namir
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2020-11-01 00:44:40
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https://www.physicsforums.com/threads/hausdorff-spaces.700499/
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# Hausdorff spaces
1. Jul 8, 2013
### R136a1
Hello everybody!
It is known that in Hausdorff spaces that every sequence converges to at most one point. I'm curious whether this characterizes Hausdorff spaces. If in a space, every sequence converges to at most one point, can one deduce Hausdorff?
2. Jul 8, 2013
### WannabeNewton
Not necessarily. Let $X$ be an uncountable set with the cocountable topology $\mathcal{T} = \{U\subseteq X:X\setminus U \text{ is countable}\}$. Assume there exist two distinct points $p,p'$ and two neighborhoods $U,U'$ of the two points respectively such that $U\cap U' = \varnothing$. Then $X\setminus (U\cap U') = (X\setminus U)\cup (X\setminus U') = X$. But $(X\setminus U)\cup (X\setminus U')$ is a finite union of countable sets which is countable whereas $X$ is uncountable thus we have a contradiction. Hence $X$ is not Hausdorff under the cocountable topology.
Now let $(x_i)$ be a sequence in $X$ that converges to $x \in X$ and let $S = \{x_i:x_i\neq x\}$. This set is countable so $U\setminus S$ must be a neighborhood of $x$ in $X$. Thus there exists some $n\in \mathbb{N}$ such that $i\geq n\Rightarrow x_i\in U$ but the only distinct element of the sequence that is in $U$ is $x$ so $x_i = x$ for all $i\geq n$ i.e. any convergent sequence in $X$ must be eventually constant under the cocountable topology. Hence limits of convergent sequences must be unique (the map prescribing the sequence must be well-defined).
3. Jul 8, 2013
### R136a1
Great! Thanks a lot! The example you gave is very interesting since it has the same convergent sequences as the discrete topology.
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2017-08-18 08:09:19
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https://www.gamedev.net/forums/topic/684146-getcurrentanimatorstateinfo/
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# Unity GetCurrentAnimatorStateInfo
This topic is 664 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
I want to get the current state from Unity Animator. However when I type theanim.GetCurrentAnimatorStateInfo(0).IsName("charactersbackwalk"), the code "GetCurrentAnimatorStateInfo" is in red line which Visual Studios indicate that "'Game object' does not contain a definition GetCurrentAnimatorStateInfo and no extension method accepting a first argument of type 'Game object' could be found (are you missing using a directive or an assembly reference)" When I type anim.GetCurrentAnimatorStateInfo(0).IsName("charactersbackwalk"), the code run but the result is not what I want. Therefore, I wonder it is necessary to declare something before using 'GetCurrentAnimatorStateInfo'?
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Firstly, if the code compiles and runs, you can ignore what Visual Studio tells you - sometimes the highlighting is wrong.
However, I'd be surprised to see the code run at all since I believe the message is correct - the method you're using is part of the Animator component, not a GameObject. You need to refer to that component, not a game object. If you are saying that changing from "theAnim" to "anim" makes the code compile and run but the answers are not what you expect, there might just be a bug elsewhere in your code.
If that isn't enough to get going, I suggest posting your code.
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@Kylotan
Here I attached my code below:
(GetCurrentAnimatorStateInfo) get red line and can't compile in Unity
void Update()
{
GameObject characters = GameObject.Find(charactersname);
GameObject theanim = GameObject.Find(charactername);
if (!Input.GetKey("up") && !Input.GetKey("down") && !Input.GetKey("left") && !(Input.GetKey("right")) && (theanim.GetCurrentAnimatorStateInfo(0).IsName("charactersbackwalk")))
{
theanim.GetComponent<Animator>().SetBool(charactersfrontwalk, false);
theanim.GetComponent<Animator>().SetBool(charactersbackwalk, false);
theanim.GetComponent<Animator>().SetBool(charactersleftwalk, false);
theanim.GetComponent<Animator>().SetBool(charactersrightwalk, false);
theanim.GetComponent<Animator>().SetBool(charactersidlewalk, false);
theanim.GetComponent<Animator>().SetBool(charactersbackidle, true);
theanim.GetComponent<Animator>().SetBool(charactersrightidle, false);
theanim.GetComponent<Animator>().SetBool(charactersleftidle, false);
}
}
Edited by Howgyn
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You're referring to something called 'anim'. You've not created or declared that object in the code you've posted. The error message suggests that it's a GameObject, which does not contain that method. You probably want to replace that with 'theanim.GetComponent<Animator>()' just like you're doing in the code below it.
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Very sorry though ... It is my mistake.
I mistakenly wrote it as anim, actually what I want to wrote is (theanim.GetCurrentAnimatorStateInfo(0).IsName("charactersbackwalk"))
"theanim" make "
GetCurrentAnimatorStateInfo" become red line.
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'Theanim' is a GameObject, so what I said above still applies. You need the relevant Animator component, just like you do inside the if statement.
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Ok now I understand what you said. Can you suggest anyway how I can edit my code so it work?
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2018-09-24 17:30:15
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https://mathoverflow.net/questions/122290/contact-structures-on-pseudo-riemannian-manifolds
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# Contact structures on pseudo-riemannian manifolds
I am looking for references on contact structures in the framework of pseudo-riemannian manifolds. For instance on the Lorentz-Minkowski $3$-space (resp. $(n+1)$-space). Denoting by $L$ (resp. $H$) the Lorentz-Minkowski $3$-space (resp. the hyperbolic plane regarded as pseudo-sphere in $L$), it seems to me that $w= udx+vdy-wdz, (x,y,z,u,v,w)$ in $LxH$, defines a contact structure on $LxH$. It is as the classical example of the unit tangent bundle of $3$-dimensional Euclidean space E but replacing $E$ by $L$ and the Euclidean scalar product by the Lorentzian one in $\left<(dx,dy,dz),(u,v,w)\right>$.
• I once heard an interesting talk by nemirovwski about such things. This is probably contained in this paper: arxiv.org/pdf/0810.5091.pdf, but I have not read this myself – Thomas Rot Jul 11 '17 at 19:15
Yes, this is a contact structure. The point is that, using the pseudo-Riemannian metric, you can identify its future-directed 'unit' tangent bundle with an open set in the projectivized cotangent bundle. Since the projectivized cotangent bundle of any manifold carries a canonical contact structure, this identification induces a canonical contact structure on the future-directed 'unit' tangent bundle. In your case, $L\times H$ is exactly the future-directed unit tangent bundle. As you note, it corresponds to the unit sphere bundle of a Riemannian manifold (which also inherits a contact structure in exactly the same way).
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2019-05-24 14:12:53
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https://www.albert.io/ie/algebra/horizontal-asymptote-for-a-rational-function-3
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?
Free Version
Easy
# Horizontal Asymptote for a Rational Function 3
ALGEBR-3QYNSH
Which is the horizontal asymptote for:
$$y=\frac{1}{x+3}?$$
A
$y=3$
B
$x=-3$
C
$y=0$
D
$x=0$
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2016-12-06 17:51:49
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https://wikimili.com/en/Drag_(physics)
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Drag (physics)
Last updated
In fluid dynamics, drag (sometimes called fluid resistance) is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. [1] This can exist between two fluid layers (or surfaces) or between an fluid and a solid surface.
Contents
Unlike other resistive forces, such as dry friction, which are nearly independent of velocity, the drag force depends on velocity. [2] [3] Drag force is proportional to the velocity for low-speed flow and the squared velocity for high speed flow, where the distinction between low and high speed is measured by the Reynolds number.
Drag forces always tend to decrease fluid velocity relative to the solid object in the fluid's path.
Examples
Examples of drag include the component of the net aerodynamic or hydrodynamic force acting opposite to the direction of movement of a solid object such as cars (automobile drag coefficient), aircraft [3] and boat hulls; or acting in the same geographical direction of motion as the solid, as for sails attached to a down wind sail boat, or in intermediate directions on a sail depending on points of sail. [4] [5] [6] In the case of viscous drag of fluid in a pipe, drag force on the immobile pipe decreases fluid velocity relative to the pipe. [7] [8]
In the physics of sports, the drag force is necessary to explain the motion of balls, javelins, arrows and frisbees and the performance of runners and swimmers. [9]
Types
Shape and flowForm
Drag
Skin
friction
≈0%≈100%
≈10%≈90%
≈90%≈10%
≈100%≈0%
Types of drag are generally divided into the following categories:
The effect of streamlining on the relative proportions of skin friction and form drag is shown for two different body sections, an airfoil, which is a streamlined body, and a cylinder, which is a bluff body. Also shown is a flat plate illustrating the effect that orientation has on the relative proportions of skin friction and pressure difference between front and back. A body is known as bluff (or blunt) if the source of drag is dominated by pressure forces and streamlined if the drag is dominated by viscous forces. Road vehicles are bluff bodies. [10] For aircraft, pressure and friction drag are included in the definition of parasitic drag. Parasite drag is often expressed in terms of a hypothetical (in so far as there is no edge spillage drag [11] ) "equivalent parasite drag area" which is the area of a flat plate perpendicular to the flow. It is used for comparing the drag of different aircraft. For example, the Douglas DC-3 has an equivalent parasite area of 23.7 sq ft and the McDonnell Douglas DC-9, with 30 years of advancement in aircraft design, an area of 20.6 sq ft although it carried five times as many passengers. [12]
• lift-induced drag appears with wings or a lifting body in aviation and with semi-planing or planing hulls for watercraft
• wave drag (aerodynamics) is caused by the presence of shockwaves and first appears at subsonic aircraft speeds when local flow velocities become supersonic. The wave drag of the supersonic Concorde prototype aircraft was reduced at Mach 2 by 1.8% by applying the area rule which extended the rear fuselage 3.73m on the production aircraft. [13]
• wave resistance (ship hydrodynamics) or wave drag occurs when a solid object is moving along a fluid boundary and making surface waves
• boat-tail drag on an aircraft is caused by the angle with which the rear fuselage, or engine nacelle, narrows to the engine exhaust diameter. [14]
The drag equation
Drag depends on the properties of the fluid and on the size, shape, and speed of the object. One way to express this is by means of the drag equation:
${\displaystyle F_{D}\,=\,{\tfrac {1}{2}}\,\rho \,v^{2}\,C_{D}\,A}$
where
${\displaystyle F_{D}}$ is the drag force,
${\displaystyle \rho }$ is the density of the fluid, [15]
${\displaystyle v}$ is the speed of the object relative to the fluid,
${\displaystyle A}$ is the cross sectional area, and
${\displaystyle C_{D}}$ is the drag coefficient – a dimensionless number.
The drag coefficient depends on the shape of the object and on the Reynolds number
${\displaystyle R_{e}={\frac {vD}{\nu }}={\frac {\rho vD}{\mu }}}$,
where
${\displaystyle D}$ is some characteristic diameter or linear dimension. Actually, ${\displaystyle D}$ is the equivalent diameter ${\displaystyle D_{e}}$ of the object. For a sphere, ${\displaystyle D_{e}}$ is the D of the sphere itself.
For a rectangular shape cross-section in the motion direction, ${\displaystyle D_{e}=1.30\cdot {\frac {(a\cdot b)^{0.625}}{(a+b)^{0.25}}}}$, where a and b are the rectangle edges.
${\displaystyle {\nu }}$ is the kinematic viscosity of the fluid (equal to the dynamic viscosity ${\displaystyle {\mu }}$ divided by the density ${\displaystyle {\rho }}$ ).
At low ${\displaystyle R_{e}}$, ${\displaystyle C_{D}}$ is asymptotically proportional to ${\displaystyle R_{e}^{-1}}$, which means that the drag is linearly proportional to the speed, i.e. the drag force on a small sphere moving through a viscous fluid is given by the Stokes Law:
${\displaystyle F_{\rm {d}}=6\pi \mu Rv}$
At high ${\displaystyle R_{e}}$, ${\displaystyle C_{D}}$ is more or less constant and drag will vary as the square of the speed. The graph to the right shows how ${\displaystyle C_{D}}$ varies with ${\displaystyle R_{e}}$ for the case of a sphere. Since the power needed to overcome the drag force is the product of the force times speed, the power needed to overcome drag will vary as the square of the speed at low Reynolds numbers and as the cube of the speed at high numbers.
It can be demonstrated that drag force can be expressed as a function of a dimensionless number, which is dimensionally identical to the Bejan number. [16] Consequently, drag force and drag coefficient can be a function of Bejan number. In fact, from the expression of drag force it has been obtained:
${\displaystyle D=\Delta _{p}A_{w}={\frac {1}{2}}C_{D}A_{f}{\frac {\nu \mu }{l^{2}}}Re_{L}^{2}}$
and consequently allows expressing the drag coefficient ${\displaystyle C_{D}}$ as a function of Bejan number and the ratio between wet area ${\displaystyle A_{w}}$ and front area ${\displaystyle A_{f}}$: [16]
${\displaystyle C_{D}=2{\frac {A_{w}}{A_{f}}}{\frac {Be}{Re_{L}^{2}}}}$
where ${\displaystyle Re_{L}}$is the Reynolds number related to fluid path length L.
At high velocity
As mentioned, the drag equation with a constant drag coefficient gives the force experienced by an object moving through a fluid at relatively large velocity (i.e. high Reynolds number, Re > ~1000). This is also called quadratic drag. The equation is attributed to Lord Rayleigh, who originally used L2 in place of A (L being some length).
${\displaystyle F_{D}\,=\,{\tfrac {1}{2}}\,\rho \,v^{2}\,C_{d}\,A,}$
see derivation
The reference area A is often orthographic projection of the object (frontal area)—on a plane perpendicular to the direction of motion—e.g. for objects with a simple shape, such as a sphere, this is the cross sectional area. Sometimes a body is a composite of different parts, each with a different reference areas, in which case a drag coefficient corresponding to each of those different areas must be determined.
In the case of a wing the reference areas are the same and the drag force is in the same ratio to the lift force as the ratio of drag coefficient to lift coefficient. [17] Therefore, the reference for a wing is often the lifting area ("wing area") rather than the frontal area. [18]
For an object with a smooth surface, and non-fixed separation points—like a sphere or circular cylinder—the drag coefficient may vary with Reynolds number Re, even up to very high values (Re of the order 107). [19] [20] For an object with well-defined fixed separation points, like a circular disk with its plane normal to the flow direction, the drag coefficient is constant for Re > 3,500. [20] Further the drag coefficient Cd is, in general, a function of the orientation of the flow with respect to the object (apart from symmetrical objects like a sphere).
Power
Under the assumption that the fluid is not moving relative to the currently used reference system, the power required to overcome the aerodynamic drag is given by:
${\displaystyle P_{d}=\mathbf {F} _{d}\cdot \mathbf {v} ={\tfrac {1}{2}}\rho v^{3}AC_{d}}$
Note that the power needed to push an object through a fluid increases as the cube of the velocity. A car cruising on a highway at 50 mph (80 km/h) may require only 10 horsepower (7.5 kW) to overcome aerodynamic drag, but that same car at 100 mph (160 km/h) requires 80 hp (60 kW). [21] With a doubling of speed the drag (force) quadruples per the formula. Exerting 4 times the force over a fixed distance produces 4 times as much work. At twice the speed the work (resulting in displacement over a fixed distance) is done twice as fast. Since power is the rate of doing work, 4 times the work done in half the time requires 8 times the power.
When the fluid is moving relative to the reference system (e.g. a car driving into headwind) the power required to overcome the aerodynamic drag is given by:
${\displaystyle P_{d}=\mathbf {F} _{d}\cdot \mathbf {v_{o}} ={\tfrac {1}{2}}C_{d}A\rho (v_{w}+v_{o})^{2}v_{o}}$
Where ${\displaystyle v_{w}}$ is the wind speed and ${\displaystyle v_{o}}$ is the object speed (both relative to ground).
Velocity of a falling object
The velocity as a function of time for an object falling through a non-dense medium, and released at zero relative-velocity v = 0 at time t = 0, is roughly given by a function involving a hyperbolic tangent (tanh):
${\displaystyle v(t)={\sqrt {\frac {2mg}{\rho AC_{d}}}}\tanh \left(t{\sqrt {\frac {g\rho C_{d}A}{2m}}}\right).\,}$
The hyperbolic tangent has a limit value of one, for large time t. In other words, velocity asymptotically approaches a maximum value called the terminal velocity vt:
${\displaystyle v_{t}={\sqrt {\frac {2mg}{\rho AC_{d}}}}.\,}$
For an object falling and released at relative-velocity v = vi at time t = 0, with vi < vt, is also defined in terms of the hyperbolic tangent function:
${\displaystyle v(t)=v_{t}\tanh \left(t{\frac {g}{v_{t}}}+\operatorname {arctanh} \left({\frac {v_{i}}{v_{t}}}\right)\right).\,}$
For vi > vt, the velocity function is defined in terms of the hyperbolic cotangent function:
${\displaystyle v(t)=v_{t}\coth \left(t{\frac {g}{v_{t}}}+\coth ^{-1}\left({\frac {v_{i}}{v_{t}}}\right)\right).\,}$
The hyperbolic cotangent has also a limit value of one, for large time t. Velocity asymptotically tends to the terminal velocity vt, strictly from above vt.
For vi = vt, the velocity is constant:
${\displaystyle v(t)=v_{t}.\,}$
Actually, these functions are defined by the solution of the following differential equation:
${\displaystyle g-{\frac {\rho AC_{d}}{2m}}v^{2}={\frac {dv}{dt}}.\,}$
Or, more generically (where F(v) are the forces acting on the object beyond drag):
${\displaystyle {\frac {1}{m}}\sum F(v)-{\frac {\rho AC_{d}}{2m}}v^{2}={\frac {dv}{dt}}.\,}$
For a potato-shaped object of average diameter d and of density ρobj, terminal velocity is about
${\displaystyle v_{t}={\sqrt {gd{\frac {\rho _{obj}}{\rho }}}}.\,}$
For objects of water-like density (raindrops, hail, live objects—mammals, birds, insects, etc.) falling in air near Earth's surface at sea level, the terminal velocity is roughly equal to
${\displaystyle v_{t}=90{\sqrt {d}},\,}$
with d in metre and vt in m/s. For example, for a human body (${\displaystyle \mathbf {} d}$ ≈0.6 m) ${\displaystyle \mathbf {} v_{t}}$ ≈70 m/s, for a small animal like a cat (${\displaystyle \mathbf {} d}$ ≈0.2 m) ${\displaystyle \mathbf {} v_{t}}$ ≈40 m/s, for a small bird (${\displaystyle \mathbf {} d}$ ≈0.05 m) ${\displaystyle \mathbf {} v_{t}}$ ≈20 m/s, for an insect (${\displaystyle \mathbf {} d}$ ≈0.01 m) ${\displaystyle \mathbf {} v_{t}}$ ≈9 m/s, and so on. Terminal velocity for very small objects (pollen, etc.) at low Reynolds numbers is determined by Stokes law.
Terminal velocity is higher for larger creatures, and thus potentially more deadly. A creature such as a mouse falling at its terminal velocity is much more likely to survive impact with the ground than a human falling at its terminal velocity. A small animal such as a cricket impacting at its terminal velocity will probably be unharmed. This, combined with the relative ratio of limb cross-sectional area vs. body mass (commonly referred to as the square–cube law), explains why very small animals can fall from a large height and not be harmed. [22]
Very low Reynolds numbers: Stokes' drag
The equation for viscous resistance or linear drag is appropriate for objects or particles moving through a fluid at relatively slow speeds where there is no turbulence (i.e. low Reynolds number, ${\displaystyle R_{e}<1}$). [23] Note that purely laminar flow only exists up to Re = 0.1 under this definition. In this case, the force of drag is approximately proportional to velocity. The equation for viscous resistance is: [24]
${\displaystyle \mathbf {F} _{d}=-b\mathbf {v} \,}$
where:
${\displaystyle \mathbf {} b}$ is a constant that depends on both the material properties of the object and fluid, as well as the geometry of the object; and
${\displaystyle \mathbf {v} }$ is the velocity of the object.
When an object falls from rest, its velocity will be
${\displaystyle v(t)={\frac {(\rho -\rho _{0})\,V\,g}{b}}\left(1-e^{-b\,t/m}\right)}$
where:
${\displaystyle \rho }$ is the density of the object,
${\displaystyle \rho _{0}}$ is density of the fluid,
${\displaystyle V}$ is the volume of the object,
${\displaystyle g}$ is the acceleration due to gravity (i.e., 9.8 m/s${\displaystyle ^{2}}$), and
${\displaystyle m}$ is mass of the object.
The velocity asymptotically approaches the terminal velocity ${\displaystyle \mathbf {} v_{t}={\frac {(\rho -\rho _{0})Vg}{b}}}$. For a given ${\displaystyle \mathbf {} b}$, denser objects fall more quickly.
For the special case of small spherical objects moving slowly through a viscous fluid (and thus at small Reynolds number), George Gabriel Stokes derived an expression for the drag constant:
${\displaystyle b=6\pi \eta r\,}$
where:
${\displaystyle \mathbf {} r}$ is the Stokes radius of the particle, and ${\displaystyle \mathbf {} \eta }$ is the fluid viscosity.
The resulting expression for the drag is known as Stokes' drag: [25]
${\displaystyle \mathbf {F} _{d}=-6\pi \eta r\,\mathbf {v} .}$
For example, consider a small sphere with radius ${\displaystyle \mathbf {} r}$ = 0.5 micrometre (diameter = 1.0 µm) moving through water at a velocity ${\displaystyle \mathbf {} v}$ of 10 µm/s. Using 10−3 Pa·s as the dynamic viscosity of water in SI units, we find a drag force of 0.09 pN. This is about the drag force that a bacterium experiences as it swims through water.
The drag coefficient of a sphere can be determined for the general case of a laminar flow with Reynolds numbers less than 1${\displaystyle 2\cdot 10^{5}}$ using the following formula: [26]
${\displaystyle C_{D}={\frac {24}{Re}}+{\frac {4}{\sqrt {Re}}}+0.4~{\text{;}}~~~~~Re<2\cdot 10^{5}}$
For Reynolds numbers less than 1, Stokes' law applies and the drag coefficient approaches ${\displaystyle {\frac {24}{Re}}}$!
Aerodynamics
In aerodynamics, aerodynamic drag (also known as air resistance) is the fluid drag force that acts on any moving solid body in the direction of the air freestream flow. [27] From the body's perspective (near-field approach), the drag results from forces due to pressure distributions over the body surface, symbolized ${\displaystyle D_{pr}}$, and forces due to skin friction, which is a result of viscosity, denoted ${\displaystyle D_{f}}$. Alternatively, calculated from the flowfield perspective (far-field approach), the drag force results from three natural phenomena: shock waves, vortex sheet, and viscosity.
Overview
The pressure distribution acting on a body's surface exerts normal forces on the body. Those forces can be summed and the component of that force that acts downstream represents the drag force, ${\displaystyle D_{pr}}$, due to pressure distribution acting on the body. The nature of these normal forces combines shock wave effects, vortex system generation effects, and wake viscous mechanisms.
The viscosity of the fluid has a major effect on drag. In the absence of viscosity, the pressure forces acting to retard the vehicle are canceled by a pressure force further aft that acts to push the vehicle forward; this is called pressure recovery and the result is that the drag is zero. That is to say, the work the body does on the airflow, is reversible and is recovered as there are no frictional effects to convert the flow energy into heat. Pressure recovery acts even in the case of viscous flow. Viscosity, however results in pressure drag and it is the dominant component of drag in the case of vehicles with regions of separated flow, in which the pressure recovery is fairly ineffective.
The friction drag force, which is a tangential force on the aircraft surface, depends substantially on boundary layer configuration and viscosity. The net friction drag, ${\displaystyle D_{f}}$, is calculated as the downstream projection of the viscous forces evaluated over the body's surface.
The sum of friction drag and pressure (form) drag is called viscous drag. This drag component is due to viscosity. In a thermodynamic perspective, viscous effects represent irreversible phenomena and, therefore, they create entropy. The calculated viscous drag ${\displaystyle D_{v}}$ use entropy changes to accurately predict the drag force.
When the airplane produces lift, another drag component results. Induced drag, symbolized ${\displaystyle D_{i}}$, is due to a modification of the pressure distribution due to the trailing vortex system that accompanies the lift production. An alternative perspective on lift and drag is gained from considering the change of momentum of the airflow. The wing intercepts the airflow and forces the flow to move downward. This results in an equal and opposite force acting upward on the wing which is the lift force. The change of momentum of the airflow downward results in a reduction of the rearward momentum of the flow which is the result of a force acting forward on the airflow and applied by the wing to the air flow; an equal but opposite force acts on the wing rearward which is the induced drag. Another drag component, namely wave drag, ${\displaystyle D_{w}}$, results from shock waves in transonic and supersonic flight speeds. The shock waves induce changes in the boundary layer and pressure distribution over the body surface.
In summary, there are three ways of categorising drag. [28] :19
1. Pressure drag and friction drag
2. Profile drag and induced drag
3. Vortex drag, wave drag and wake drag
History
The idea that a moving body passing through air or another fluid encounters resistance had been known since the time of Aristotle. According to Mervyn O'Gorman, this was named "drag" by Archibald Reith Low. [29] Louis Charles Breguet's paper of 1922 began efforts to reduce drag by streamlining. [30] Breguet went on to put his ideas into practice by designing several record-breaking aircraft in the 1920s and 1930s. Ludwig Prandtl's boundary layer theory in the 1920s provided the impetus to minimise skin friction. A further major call for streamlining was made by Sir Melvill Jones who provided the theoretical concepts to demonstrate emphatically the importance of streamlining in aircraft design. [31] [32] [33] In 1929 his paper ‘The Streamline Airplane’ presented to the Royal Aeronautical Society was seminal. He proposed an ideal aircraft that would have minimal drag which led to the concepts of a 'clean' monoplane and retractable undercarriage. The aspect of Jones's paper that most shocked the designers of the time was his plot of the horse power required versus velocity, for an actual and an ideal plane. By looking at a data point for a given aircraft and extrapolating it horizontally to the ideal curve, the velocity gain for the same power can be seen. When Jones finished his presentation, a member of the audience described the results as being of the same level of importance as the Carnot cycle in thermodynamics. [30] [31]
Lift-induced drag and parasitic drag
Lift-induced drag
Lift-induced drag (also called induced drag) is drag which occurs as the result of the creation of lift on a three-dimensional lifting body, such as the wing or propeller of an airplane. Induced drag consists primarily of two components: drag due to the creation of trailing vortices (vortex drag); and the presence of additional viscous drag (lift-induced viscous drag) that is not present when lift is zero. The trailing vortices in the flow-field, present in the wake of a lifting body, derive from the turbulent mixing of air from above and below the body which flows in slightly different directions as a consequence of creation of lift.
With other parameters remaining the same, as the lift generated by a body increases, so does the lift-induced drag. This means that as the wing's angle of attack increases (up to a maximum called the stalling angle), the lift coefficient also increases, and so too does the lift-induced drag. At the onset of stall, lift is abruptly decreased, as is lift-induced drag, but viscous pressure drag, a component of parasite drag, increases due to the formation of turbulent unattached flow in the wake behind the body.
Parasitic drag
Parasitic drag, or profile drag, is drag caused by moving a solid object through a fluid. Parasitic drag is made up of multiple components including viscous pressure drag (form drag), and drag due to surface roughness (skin friction drag). Additionally, the presence of multiple bodies in relative proximity may incur so called interference drag, which is sometimes described as a component of parasitic drag.
In aviation, induced drag tends to be greater at lower speeds because a high angle of attack is required to maintain lift, creating more drag. However, as speed increases the angle of attack can be reduced and the induced drag decreases. Parasitic drag, however, increases because the fluid is flowing more quickly around protruding objects increasing friction or drag. At even higher speeds (transonic), wave drag enters the picture. Each of these forms of drag changes in proportion to the others based on speed. The combined overall drag curve therefore shows a minimum at some airspeed - an aircraft flying at this speed will be at or close to its optimal efficiency. Pilots will use this speed to maximize endurance (minimum fuel consumption), or maximize gliding range in the event of an engine failure.
Power curve in aviation
The interaction of parasitic and induced drag vs. airspeed can be plotted as a characteristic curve, illustrated here. In aviation, this is often referred to as the power curve, and is important to pilots because it shows that, below a certain airspeed, maintaining airspeed counterintuitively requires more thrust as speed decreases, rather than less. The consequences of being "behind the curve" in flight are important and are taught as part of pilot training. At the subsonic airspeeds where the "U" shape of this curve is significant, wave drag has not yet become a factor, and so it is not shown in the curve.
Wave drag in transonic and supersonic flow
Wave drag (also called compressibility drag) is drag that is created when a body moves in a compressible fluid and at speeds that are close to the speed of sound in that fluid. In aerodynamics, wave drag consists of multiple components depending on the speed regime of the flight.
In transonic flight (Mach numbers greater than about 0.8 and less than about 1.4), wave drag is the result of the formation of shockwaves in the fluid, formed when local areas of supersonic (Mach number greater than 1.0) flow are created. In practice, supersonic flow occurs on bodies traveling well below the speed of sound, as the local speed of air increases as it accelerates over the body to speeds above Mach 1.0. However, full supersonic flow over the vehicle will not develop until well past Mach 1.0. Aircraft flying at transonic speed often incur wave drag through the normal course of operation. In transonic flight, wave drag is commonly referred to as transonic compressibility drag. Transonic compressibility drag increases significantly as the speed of flight increases towards Mach 1.0, dominating other forms of drag at those speeds.
In supersonic flight (Mach numbers greater than 1.0), wave drag is the result of shockwaves present in the fluid and attached to the body, typically oblique shockwaves formed at the leading and trailing edges of the body. In highly supersonic flows, or in bodies with turning angles sufficiently large, unattached shockwaves, or bow waves will instead form. Additionally, local areas of transonic flow behind the initial shockwave may occur at lower supersonic speeds, and can lead to the development of additional, smaller shockwaves present on the surfaces of other lifting bodies, similar to those found in transonic flows. In supersonic flow regimes, wave drag is commonly separated into two components, supersonic lift-dependent wave drag and supersonic volume-dependent wave drag.
The closed form solution for the minimum wave drag of a body of revolution with a fixed length was found by Sears and Haack, and is known as the Sears-Haack Distribution. Similarly, for a fixed volume, the shape for minimum wave drag is the Von Karman Ogive.
The Busemann biplane theoretical concept is not subject to wave drag when operated at its design speed, but is incapable of generating lift in this condition.
In 1752 d'Alembert proved that potential flow, the 18th century state-of-the-art inviscid flow theory amenable to mathematical solutions, resulted in the prediction of zero drag. This was in contradiction with experimental evidence, and became known as d'Alembert's paradox. In the 19th century the Navier–Stokes equations for the description of viscous flow were developed by Saint-Venant, Navier and Stokes. Stokes derived the drag around a sphere at very low Reynolds numbers, the result of which is called Stokes' law. [34]
In the limit of high Reynolds numbers, the Navier–Stokes equations approach the inviscid Euler equations, of which the potential-flow solutions considered by d'Alembert are solutions. However, all experiments at high Reynolds numbers showed there is drag. Attempts to construct inviscid steady flow solutions to the Euler equations, other than the potential flow solutions, did not result in realistic results. [34]
The notion of boundary layers—introduced by Prandtl in 1904, founded on both theory and experiments—explained the causes of drag at high Reynolds numbers. The boundary layer is the thin layer of fluid close to the object's boundary, where viscous effects remain important even when the viscosity is very small (or equivalently the Reynolds number is very large). [34]
Related Research Articles
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics and hydrodynamics. Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.
In physics, the Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes).
In fluid mechanics, the Grashof number is a dimensionless number which approximates the ratio of the buoyancy to viscous forces acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number.
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers.
In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid. The equation is:
In fluid dynamics, the drag coefficient is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag equation in which a lower drag coefficient indicates the object will have less aerodynamic or hydrodynamic drag. The drag coefficient is always associated with a particular surface area.
In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary condition. The flow velocity then monotonically increases above the surface until it returns to the bulk flow velocity. The thin layer consisting of fluid whose velocity has not yet returned to the bulk flow velocity is called the velocity boundary layer.
Terminal velocity is the maximum velocity (speed) attainable by an object as it falls through a fluid. It occurs when the sum of the drag force (Fd) and the buoyancy is equal to the downward force of gravity (FG) acting on the object. Since the net force on the object is zero, the object has zero acceleration.
Parasitic drag, also known as profile drag, is a type of aerodynamic drag that acts on any object when the object is moving through a fluid. Parasitic drag is a combination of form drag and skin friction drag. It affects all objects regardless of whether they are capable of generating lift.
In fluid dynamics, d'Alembert's paradox is a contradiction reached in 1752 by French mathematician Jean le Rond d'Alembert. D'Alembert proved that – for incompressible and inviscid potential flow – the drag force is zero on a body moving with constant velocity relative to the fluid. Zero drag is in direct contradiction to the observation of substantial drag on bodies moving relative to fluids, such as air and water; especially at high velocities corresponding with high Reynolds numbers. It is a particular example of the reversibility paradox.
Stokes flow, also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.
Fluid mechanics is the branch of physics concerned with the mechanics of fluids and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and biomedical engineering, geophysics, oceanography, meteorology, astrophysics, and biology.
In fluid mechanics, added mass or virtual mass is the inertia added to a system because an accelerating or decelerating body must move some volume of surrounding fluid as it moves through it. Added mass is a common issue because the object and surrounding fluid cannot occupy the same physical space simultaneously. For simplicity this can be modeled as some volume of fluid moving with the object, though in reality "all" the fluid will be accelerated, to various degrees.
The Kutta–Joukowski theorem is a fundamental theorem in aerodynamics used for the calculation of lift of an airfoil translating in a uniform fluid at a constant speed large enough so that the flow seen in the body-fixed frame is steady and unseparated. The theorem relates the lift generated by an airfoil to the speed of the airfoil through the fluid, the density of the fluid and the circulation around the airfoil. The circulation is defined as the line integral around a closed loop enclosing the airfoil of the component of the velocity of the fluid tangent to the loop. It is named after Martin Kutta and Nikolai Zhukovsky who first developed its key ideas in the early 20th century. Kutta–Joukowski theorem is an inviscid theory, but it is a good approximation for real viscous flow in typical aerodynamic applications.
In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.
In fluid dynamics the Morison equation is a semi-empirical equation for the inline force on a body in oscillatory flow. It is sometimes called the MOJS equation after all four authors—Morison, O'Brien, Johnson and Schaaf—of the 1950 paper in which the equation was introduced. The Morison equation is used to estimate the wave loads in the design of oil platforms and other offshore structures.
In fluid dynamics, hydrodynamic stability is the field which analyses the stability and the onset of instability of fluid flows. The study of hydrodynamic stability aims to find out if a given flow is stable or unstable, and if so, how these instabilities will cause the development of turbulence. The foundations of hydrodynamic stability, both theoretical and experimental, were laid most notably by Helmholtz, Kelvin, Rayleigh and Reynolds during the nineteenth century. These foundations have given many useful tools to study hydrodynamic stability. These include Reynolds number, the Euler equations, and the Navier–Stokes equations. When studying flow stability it is useful to understand more simplistic systems, e.g. incompressible and inviscid fluids which can then be developed further onto more complex flows. Since the 1980s, more computational methods are being used to model and analyse the more complex flows.
In fluid mechanics, the Reynolds number is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers, flows tend to be turbulent. The turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow. These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation.
Skin friction drag is a type of aerodynamic or hydrodynamic drag, which is resistant force exerted on an object moving in a fluid. Skin friction drag is caused by the viscosity of fluids and is developed from laminar drag to turbulent drag as a fluid moves on the surface of an object. Skin friction drag is generally expressed in terms of the Reynolds number, which is the ratio between inertial force and viscous force.
References
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2. French (1970), p. 211, Eq. 7-20
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5. Marchaj, C. A. (2003). Sail performance : techniques to maximise sail power (Rev. ed.). London: Adlard Coles Nautical. pp. 147 figure 127 lift vs drag polar curves. ISBN 978-0-7136-6407-2.
6. Drayton, Fabio Fossati; translated by Martyn (2009). Aero-hydrodynamics and the performance of sailing yachts : the science behind sailing yachts and their design. Camden, Maine: International Marine /McGraw-Hill. pp. 98 Fig 5.17 Chapter five Sailing Boat Aerodynamics. ISBN 978-0-07-162910-2.
7. "Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes" (PDF). Retrieved 16 October 2011.
8. "Viscous Drag Forces" . Retrieved 16 October 2011.
9. Hernandez-Gomez, J J; Marquina, V; Gomez, R W (25 July 2013). "On the performance of Usain Bolt in the 100 m sprint". Eur. J. Phys. 34 (5): 1227–1233. arXiv:. Bibcode:2013EJPh...34.1227H. doi:10.1088/0143-0807/34/5/1227. S2CID 118693492 . Retrieved 23 April 2016.
10. Encyclopedia of Automotive Engineering, David Crolla, Paper "Fundamentals, Basic principles in Road vehicle Aerodynamics and Design", ISBN 978 0 470 97402 5
11. The Design Of The Aeroplane, Darrol Stinton, ISBN 0 632 01877 1, p.204
12. Fundamentals of Flight, Second Edition, Richard S. Shevell, ISBN 0 13 339060 8, p.185
13. A Case Study By Aerospatiale And British Aerospace On The Concorde By Jean Rech and Clive S. Leyman,AIAA Professional Study Series, Fig. 3.6
14. Design For Air Combat, Ray Whitford, ISBN 0 7106 0426 2, p.212
15. Note that for Earth's atmosphere, the air density can be found using the barometric formula. It is 1.293 kg/m3 at 0 °C and 1 atmosphere.
16. Liversage, P., and Trancossi, M. (2018). Analysis of triangular sharkskin profiles according to second law, Modelling, Measurement and Control B. 87(3), 188-196. http://www.iieta.org/sites/default/files/Journals/MMC/MMC_B/87.03_11.pdf
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20. Batchelor (1967), p. 341.
21. Brian Beckman (1991), Part 6: Speed and Horsepower, archived from the original on 2016-06-16, retrieved 18 May 2016
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23. Drag Force Archived April 14, 2008, at the Wayback Machine
24. Air friction, from Department of Physics and Astronomy, Georgia State University
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• 'Improved Empirical Model for Base Drag Prediction on Missile Configurations, based on New Wind Tunnel Data', Frank G Moore et al. NASA Langley Center
• 'Computational Investigation of Base Drag Reduction for a Projectile at Different Flight Regimes', M A Suliman et al. Proceedings of 13th International Conference on Aerospace Sciences & Aviation Technology, ASAT- 13, May 26 – 28, 2009
• 'Base Drag and Thick Trailing Edges', Sighard F. Hoerner, Air Materiel Command, in: Journal of the Aeronautical Sciences, Oct 1950, pp 622–628
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2023-03-25 20:18:49
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https://kb.wisc.edu/moodle/feedback.php?action=2&help=comment&id=35139
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## Feedback
Referral page (click the arrow below to expand the document)::
Moodle - Using MathJax in courses
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2018-05-28 09:39:40
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http://openstudy.com/updates/4dc98a9640ec8b0bf7ca0c17
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## anonymous 5 years ago CAP ~ DAY FD =?
1. anonymous
2. anonymous
congruent triangles so corresponding ratios are equal yes? $\frac{21}{35}=\frac{FD}{25}$ $FD = \frac{21 \times 25}{35}=\frac{525}{35}=15$
3. anonymous
http://openstudy.com/groups/mathematics/updates/4dc989a640ec8b0b59c70c17 ok thank and can you help me with this link above
4. anonymous
sure give me a minute to draw it.
5. anonymous
okay
Find more explanations on OpenStudy
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2016-10-22 00:03:41
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https://crazyproject.wordpress.com/2010/03/26/subgroup-index-is-multiplicative-across-intermediate-subgroups/
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## Subgroup index is multiplicative across intermediate subgroups
Let $G$ be a group and $K \leq H \leq G$. Prove that $[G : K] = [G : H] \cdot [H : K]$.
We proved this as a lemma to a previous theorem.
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2017-01-24 11:11:54
|
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https://quant.stackexchange.com/questions/42150/computation-of-future-implied-volatility-surface
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# Computation of Future Implied Volatility Surface
I do have a question on the future implied volatility surface. The current implied volatility surface is easy to obtain, e.g using some interpolation technique on current options prices.
For computation of the future implied volatility surface, e.g at T=2 years Does anyone have an example of how to obtain this assuming e.g Local vol (dupire) or Heston dynamics?
Thank you!
• perhaps by both methods.. 1. Monte Carlo, 2. PDE – Benedict Oct 11 at 6:48
• Hi @Benedict, devil is in the details: it really depends on what you mean by "future implied volatility surface". One possible approach would be to price forward starts with moneyness $k$, starting date $T_1=2Y$ and various tenors $\tau > 0$ in Monte Carlo i.e. instruments with current price: $$V(k,\tau) = \Bbb{E}_0 \left[ D(0,T_2) \left( \frac{S_{T_2}}{S_{T_1}} - k \right)^+ \right]$$ with $T_2 = T_1 + \tau$, then implying a BS volatility from the prices $V(k,\tau), \forall (k,\tau) \in \mathcal{K} \times \mathcal{T}$. – Quantuple Oct 12 at 8:02
• Hi @Quantuple. Thank you for your reply. I was actually trying to understand how the implied volatility surfaces moves across times (e.g starting from T =0 to T=2), using dupire local vol dynamics or heston stochastic vol dynamics. There is already a way to compute at T=0 implied volatility surface, which is the current time. But I dont know how to arrive at the implied vol surface if i simulate my model to T=2 years from now. – Benedict Oct 12 at 10:27
• What do you assume the spot is at t+2? – will Oct 12 at 14:29
• @will need a little more clues – Benedict 2 days ago
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2018-10-17 06:20:23
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https://testbook.com/question-answer/the-capacitor-charging-current-is--5d19eb4afdb8bb6c1b7d7b54
|
# The capacitor charging current is
1. An exponential growth function
2. An exponential decay function
3. A linear decay function
4. A linear rise function
Option 2 : An exponential decay function
Free
CT 1: Basic Concepts
19925
10 Questions 10 Marks 6 Mins
## Detailed Solution
When a battery is connected to a series resistor and capacitor, the initial current is high as the battery transports charge from one plate of the capacitor to the other.
The charging current exponentially approaches zero as the capacitor becomes charged up to the battery voltage.
The expression of charging current I, during process of charging is
$$i = \frac{V}{R}{e^{ - \frac{t}{{RC}}}}$$
The current and voltage of the capacitor during charging is shown below.
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2022-01-26 05:13:45
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https://www.aimsciences.org/article/doi/10.3934/dcds.2014.34.1355
|
# American Institute of Mathematical Sciences
April 2014, 34(4): 1355-1374. doi: 10.3934/dcds.2014.34.1355
## Gradient flow structures for discrete porous medium equations
1 University of Bonn, Institute for Applied Mathematics, Endenicher Allee 60, 53115 Bonn, Germany, Germany
Received December 2012 Revised March 2013 Published October 2013
We consider discrete porous medium equations of the form $\partial_t\rho_t = \Delta \phi(\rho_t)$, where $\Delta$ is the generator of a reversible continuous time Markov chain on a finite set $\boldsymbol{\chi}$, and $\phi$ is an increasing function. We show that these equations arise as gradient flows of certain entropy functionals with respect to suitable non-local transportation metrics. This may be seen as a discrete analogue of the Wasserstein gradient flow structure for porous medium equations in $\mathbb{R}^n$ discovered by Otto. We present a one-dimensional counterexample to geodesic convexity and discuss Gromov-Hausdorff convergence to the Wasserstein metric.
Citation: Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
##### References:
[1] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. [2] L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, arXiv:1109.0222, (2012). [3] J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002. [4] H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland Mathematics Studies, No. 5. Notas de Matemática (50), North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. [5] E. Carlen and J. Maas, An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker-Planck equation is gradient flow for the entropy, to appear in Comm. Math. Phys., arXiv:1203.5377, (2012). [6] S.-N. Chow, W. Huang, Y. Li and H. Zhou, Fokker-Planck equations for a free energy functional or Markov process on a graph, Arch. Ration. Mech. Anal., 203 (2012), 969-1008. doi: 10.1007/s00205-011-0471-6. [7] S. Daneri and G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance, SIAM J. Math. Anal., 40 (2008), 1104-1122. doi: 10.1137/08071346X. [8] M. Erbar, Gradient flows of the entropy for jump processes, to appear in Ann. Inst. Henri Poincaré Probab. Stat., arXiv:1204.2190, (2012). [9] M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Ration. Mech. Anal., 206 (2012), 997-1038. doi: 10.1007/s00205-012-0554-z. [10] N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics, SIAM J. Math. Anal., 45 (2013), 879-899. doi: 10.1137/120886315. [11] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [12] J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261 (2011), 2250-2292. doi: 10.1016/j.jfa.2011.06.009. [13] R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. [14] A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Differential Equations, 48 (2013), 1-31. doi: 10.1007/s00526-012-0538-8. [15] A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346. doi: 10.1088/0951-7715/24/4/016. [16] A. Mielke, Dissipative quantum mechanics using GENERIC, To appear in Proc. of the conference "Recent Trends in Dynamical Systems,'' (2013). [17] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243. [18] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.
show all references
##### References:
[1] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. [2] L. Ambrosio, N. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, arXiv:1109.0222, (2012). [3] J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002. [4] H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland Mathematics Studies, No. 5. Notas de Matemática (50), North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. [5] E. Carlen and J. Maas, An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker-Planck equation is gradient flow for the entropy, to appear in Comm. Math. Phys., arXiv:1203.5377, (2012). [6] S.-N. Chow, W. Huang, Y. Li and H. Zhou, Fokker-Planck equations for a free energy functional or Markov process on a graph, Arch. Ration. Mech. Anal., 203 (2012), 969-1008. doi: 10.1007/s00205-011-0471-6. [7] S. Daneri and G. Savaré, Eulerian calculus for the displacement convexity in the Wasserstein distance, SIAM J. Math. Anal., 40 (2008), 1104-1122. doi: 10.1137/08071346X. [8] M. Erbar, Gradient flows of the entropy for jump processes, to appear in Ann. Inst. Henri Poincaré Probab. Stat., arXiv:1204.2190, (2012). [9] M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Ration. Mech. Anal., 206 (2012), 997-1038. doi: 10.1007/s00205-012-0554-z. [10] N. Gigli and J. Maas, Gromov-Hausdorff convergence of discrete transportation metrics, SIAM J. Math. Anal., 45 (2013), 879-899. doi: 10.1137/120886315. [11] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [12] J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261 (2011), 2250-2292. doi: 10.1016/j.jfa.2011.06.009. [13] R. J. McCann, A convexity principle for interacting gases, Adv. Math., 128 (1997), 153-179. doi: 10.1006/aima.1997.1634. [14] A. Mielke, Geodesic convexity of the relative entropy in reversible Markov chains, Calc. Var. Partial Differential Equations, 48 (2013), 1-31. doi: 10.1007/s00526-012-0538-8. [15] A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346. doi: 10.1088/0951-7715/24/4/016. [16] A. Mielke, Dissipative quantum mechanics using GENERIC, To appear in Proc. of the conference "Recent Trends in Dynamical Systems,'' (2013). [17] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. doi: 10.1081/PDE-100002243. [18] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.
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2021 Impact Factor: 1.588
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2022-10-07 02:23:20
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http://web.mat.bham.ac.uk/R.W.Kaye/seqser/constrrationals.html
|
# Construction of the rationals
Our next task is to define the set of rational numbers from the integers using equivalence classes of pairs of integers. The idea is clear: we think of a pair of integers $( p , q )$ as the fraction $p / q$ and use an equivalence relation to identify fractions that should have the same values.
## 1. Getting the rationals from the integers
Definition.
Let $ℤ +$ denote the set of positive integers, i.e., $ℤ + = { n ∈ ℤ : n > 0 }$. Let $∼$ be the following relation defined on $ℤ × ℤ +$: we define $( x 1 , y 1 ) ∼ ( x 2 , y 2 )$ to mean $x 1 · y 2 = x 2 · y 1$.
Proposition.
$∼$ is an equivalence relation on $ℤ × ℤ +$.
Proof.
Exercise.
Definition.
We write $p / q$ for the equivalence class of $( p , q ) ∈ ℤ × ℤ +$. The set $ℚ$ is the set $ℤ × ℤ + / ∼$ of equivalence classes.
Proposition.
The function $i$ defined by $i ( n ) = n / 1$ is a one-to-one function mapping $ℤ$ into $ℤ$.
Proof.
Let $n , m ∈ ℕ$ with $i ( n ) = i ( m )$. Then $n / 1 = m / 1$ hence $( n , 1 ) ∼ ( m , 1 )$ hence $n · 1 = m · 1$ hence $n = m$.
Definition.
We identify each $n ∈ ℕ$ with its image $i ( n )$ under the map $i$. In particular, $0$ is the element $0 / 1 ∈ ℚ$, and $1$ is $1 / 1 ∈ ℚ$.
Definition.
We define addition, multiplication and order relations on $ℚ$ by
• $p 1 / q 1 + p 2 / q 2 = ( p 1 q 2 + p 2 q 1 ) / q 1 q 2$
• $p 1 / q 1 · p 2 / q 2 = p 1 p 2 / q 1 q 2$
• $p 1 / q 1 < p 2 / q 2 ↔ p 1 q 2 < p 2 q 1$
Proposition.
The operations $+,·$ and relation $⩽$ on $ℚ$ are well-defined, i.e., the definitions above do not depend on the particular choice of representatives $( p 1 , q 1 ) , ( p 2 , q 2 )$.
Proof.
Exercise.
Proposition.
The embedding $i : ℤ → ℚ$ is a homomorphism respecting $+,·$ and $⩽$:
• $i ( n + m ) = i ( n ) + i ( m )$;
• $i ( n · m ) = i ( n ) · i ( m )$; and
• $n < m ↔ i ( n ) ⩽ i ( m )$;
for all $n , m ∈ ℤ$.
Proof.
Exercise.
## 2. The ordered field structure
The key axioms for the rationals have already been given. They form an Archimedean ordered field, in fact a minimal Archimedean ordered field in the sense that no proper subset is also an Archimedean ordered field.
Theorem.
The rationals with $+ , · , <$ as defined here satisfy the axioms of an Archemedean ordered field.
Proof.
A rather long exercise.
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2017-09-22 15:09:58
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http://mathhelpforum.com/pre-calculus/45665-exponential-equations.html
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# Math Help - Exponential Equations
1. ## Exponential Equations
Solve each equation below. Express irrational solutions in exact form and as a decimal rounded to 3 decimal places.
(1) (cuberoot{2})^(2 - x) = 2^(x^2)
(2) e^(x + 3) = pi^(x)
2. $(\sqrt[3]{2})^{2-x} = 2^{x^2}$
$(\sqrt[3]{2})^{2-x} = [(\sqrt[3]{2})^3]^{x^2}$
$(\sqrt[3]{2})^{2-x} = (\sqrt[3]{2})^{3x^2}$
so ... what can you say about $2-x$ and $3x^2$ ?
$e^{x+3} = \pi^x$
$\ln(e^{x+3}) = \ln(\pi^x)$
$x+3 = x\ln{\pi}$
$3 = x\ln{\pi} - x$
$3 = x(\ln{\pi} - 1)$
$\frac{3}{\ln{\pi} - 1} = x$
3. ## Much better....
This is much better than the other reply.
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2015-03-27 13:12:50
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http://hal.in2p3.fr/in2p3-00583413
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# High $p_{T}$ non-photonic electron production in $p$+$p$ collisions at $\sqrt{s}$ = 200 GeV
Abstract : We present the measurement of non-photonic electron production at high transverse momentum ($p_T >$ 2.5 GeV/$c$) in $p$+$p$ collisions at $\sqrt{s}$ = 200 GeV using data recorded during 2005 and 2008 by the STAR experiment at the Relativistic Heavy Ion Collider (RHIC). The measured cross-sections from the two runs are consistent with each other despite a large difference in photonic background levels due to different detector configurations. We compare the measured non-photonic electron cross-sections with previously published RHIC data and pQCD calculations. Using the relative contributions of B and D mesons to non-photonic electrons, we determine the integrated cross sections of electrons ($\frac{e^++e^-}{2}$) at 3 GeV/$c < p_T <~$10 GeV/$c$ from bottom and charm meson decays to be ${d\sigma_{(B\to e)+(B\to D \to e)} \over dy_e}|_{y_e=0}$ = 4.0$\pm0.5$({\rm stat.})$\pm1.1$({\rm syst.}) nb and ${d\sigma_{D\to e} \over dy_e}|_{y_e=0}$ = 6.2$\pm0.7$({\rm stat.})$\pm1.5$({\rm syst.}) nb, respectively.
Document type :
Preprints, Working Papers, ...
http://hal.in2p3.fr/in2p3-00583413
Contributor : Dominique Girod <>
Submitted on : Tuesday, April 5, 2011 - 3:53:39 PM
Last modification on : Wednesday, August 4, 2021 - 3:54:02 PM
### Identifiers
• HAL Id : in2p3-00583413, version 1
• ARXIV : 1102.2611
### Citation
H. Agakishiev, M. M. Aggarwal, Z. Ahammed, A. V. Alakhverdyants, I. Alekseev, et al.. High $p_{T}$ non-photonic electron production in $p$+$p$ collisions at $\sqrt{s}$ = 200 GeV. 2011. ⟨in2p3-00583413⟩
Record views
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2021-09-27 22:02:25
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http://www.latex-community.org/forum/viewtopic.php?f=5&t=12454
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### Who is online
In total there are 8 users online :: 1 registered, 0 hidden and 7 guests (based on users active over the past 5 minutes)
Most users ever online was 1327 on Tue Nov 05, 2013 7:11 pm
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## List of Equations / Table of Equations (continued)
Add tags LaTeX specific issues not fitting into one of the other forums of this category.
### List of Equations / Table of Equations (continued)
I have been trying to come up with a convenient way to make a List of Equations in our report. The key thing to note here is that the report is already typed, and contains dozens of equations in hundreds of pages.
Using tocloft to add the List of Equations usually requires following each equation with a tag such as \myequation{Display name}. To avoid having to add this for all of our equations, I did the following:
Code: Select all • Open in writeLaTeX
% redefine equation to automatically include our \myequations tag\let\oldequation = \equation\let\endoldequation = \endequation\renewenvironment{equation}{ \begin{oldequation}}{ \end{oldequation} \myequations{\@currentlabelname}}
To generate the list, I using the following code which was mostly derived from here: http://www.latex-community.org/forum/viewtopic.php?f=5&t=428
Code: Select all • Open in writeLaTeX
Then all \equations will appear automatically in the List of Equations. That's great, but there are two problems:
• Each \equations displayed name in the List of Equations is not its own \label (eg. Torque) but instead is the name of the \subsection in which it appears (such that dozens of equations have the name "Brushless Motor Fundamentals", etc.
• Equations 2.1 - 2.9 display correctly in the List of Equations but after the equation number hits 2.10 the second digit overlaps with the display name.
Any ideas? What can I replace \@currentlabelname with to drop in the name of the label for an equation? A simple one looks like this:
Code: Select all • Open in writeLaTeX
\begin{equation}\tau=F\times r\label{eq:Torque}\end{equation}
Note: this is also posted under the "Lyx" forum on Latex-Community since that is where I originally posted and also Lyx is ultimately the program I will use for my final implementation. http://www.latex-community.org/forum/viewtopic.php?f=19&t=12452
rlevin
Posts: 5
Joined: Sat Mar 26th, 2011
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2014-08-02 06:31:30
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https://olympiaelectricians.com/you-know-which-electric-voltage-you-have-22040-120208
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# You know which electric voltage you have 220.40 120.208?
Date created: Sun, Aug 1, 2021 1:13 PM
Content
FAQ
Those who are looking for an answer to the question «You know which electric voltage you have 220.40 120.208?» often ask the following questions:
### 👉 Which electric device is dual voltage?
Dual voltage is a term used to describe any type of electronic device that is manufactured to recognize and use both American and European currents without the need for an additional transformer.
### 👉 Electric voltage?
e. Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential between two points, which (in a static electric field) is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage (potential ...
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The direction of an electrical field at a point is the same as the direction of the electrical force acting on a positive test charge at that point. For example, if you place a positive test charge in an electric field and the charge moves to the right, you know the direction of the electric field in that region points to the right.
Electrical: Available in 120/208 single phase, 120/220 single phase, or 120/240 single phase. Require 3 wires + ground I take that to mean you have to select the voltage, NOT that it can be used on any of them.
Hence you have 3 phase 208 voltage on the phase legs and 120 volts to the wye point which is grounded. This same formula is used on any 3 phase system. 600/347, 480/277, 208/120. Will a 240 volt ...
waross (Electrical) 3 May 09 01:06. 120/208V is the standard 3 phase voltage. 230 volt motors were run on 208 volts for years but it is becoming more common to use 200 volt rated motors on 208 volt systems. 120/240V is the standard single phase system voltage. 230 Volts, where to start. 230 volts is a valid motor voltage.
near you operating position for 240 VAC (hospitals sometimes have these surplus -- as hospital grade isolation/buck boost transformers for clinical gear). Hard wire the primary to your 120/208 circuit and then use the NEMA L14 outlet on secondary from transformer. That allows you to easily change-out the HF amplifier in the future ! w9gb
Hi everyone Thanks in advance for your help If I understand things correctly 120/208v implies the Line voltage (voltage measured between any two line conductors) is 208 volt and the phase voltage (voltage measured between a line an the neutral) is 120 volt. Where does this 208v come from...
While this is close to 240 volts, by and large, equipment rated to run on 240 volts will not perform well and may even be damaged if run at 208-volts. Conversely, plugging a 208-volt three-phase piece of equipment into a 240-volt single-phase system may have a disastrous outcome to you and the delivery system.
You should of course sort out what voltage you're dealing with, but if you didn't have a low-Z tool to measure with, you could always measure the current the heater draws. A 1000w heater should be about 4.1A at 240V, about 3.6 at 208V. There are ghost voltages but there are no ghost amperages (as far as I know).
Hi I have a commercial dishwasher om order that is 120/208-240(3w)/60/1 Single Phase 32.4 amps. Is this circuit large enough to handle it? 2 pole 120/240v 20 amp on each … read more
White commonly being a low voltage (120, 208, 240) neutral and grey being a high voltage (277, 480) neutral. If you are using white for high voltage, please group it with its conductors. Note: in NEC 2014 Grounded conductors shall be grouped with their ungrounded conductors by marking or cable ties.
– The calculator allows entry of a voltage drop, but caution should be used when doing so to make sure you are calculating the wire sizes in accordance with NEC article 210-19 (FPN No. 4.) The NEC allows a maximum of a 3% voltage drop on the main branch of a circuit at the farthest outlet of power and 5% total to both feeders and branch ...
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An electrical explosion, or "arc flash", occurs when one or more high current arcs are created between energized electrical conductors or between an energized conductor and neutral (ground). Once initiated, the resulting arc(s) can bridge significant distances even though the voltage is relatively low.
### Is electric field voltage?
Electric Field as Gradient. The expression of electric field in terms of voltage can be expressed in the vector form . This collection of partial derivatives is called the gradient, and is represented by the symbol ∇ . The electric field can then be written
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### Are electric toothbrushes dual voltage?
Yes it is dual voltage. All you need is another Oral-B 240v charging base outside of the US, The Oral-B unit itself is dual voltage(made in Germany) it is only the US charging base that supports 110v.
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Voltage is electrical energy per unit charge, and electric field is force per unit charge. For a particle moving in one dimension, along the $$x$$ axis, we can therefore relate voltage and field if we start from the relationship between interaction energy and force, $\begin{equation*} dU = -F_xdx , \end{equation*}$ and divide by charge, $\begin{equation*} \frac{dU}{q} = -\frac{F_x}{q}dx , \end{equation*}$ giving $\begin{equation*} dV = -E_x dx , \end{equation*}$ or \[\begin{equation ...
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#### What is the voltage of an electric fence?
• The voltage of an electric fence should vary from about 2000 to about 10,000 volts. A 10,000 volt output is the maximum voltage allowed by international regulations. The voltage that is used depends on the desired power of the shock and the distance on the fence that can be shocked up to.
### Low voltage electric blanket review?
Top Best low emf electric blanket Reviews 2021: Best Overall: Soft Heat with Perfect Fit | Luxurious Low Voltage Microfleece Electric Blanket (Double, Natural) – Top Best low emf electric blanket in USA; Budget Friendly: A very soft luxury blanket with heating and safe, warm low voltage that fits perfectly.
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2021-10-20 06:48:01
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https://www.physicsforums.com/threads/math-problem-ii.8930/
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# Math problem II
denian
i tried this. but fail to get the answer
find the square root of 11 - 6*(square root of 2) and write the solution in surd notation.
answer provided by the book :
3 - (square root of 2)
## Answers and Replies
Staff Emeritus
My calculator gives the answer to be 3-sqrt(2).
Why don't you try using binomial expansion to solve the problem.
sqrt(11-6*sqrt(2)) = sqrt(11)*(1+((-6/11)*sqrt(2))^(1/2)
x = (-6/11)*sqrt(2)
n = 1/2
1 + n*x + (n*(n-1)*x^2)/2! + (n*(n-1)*(n-2)*x^3)/3! + ....
gnome
You want 11 - 6√2 to be a perfect square of some binomial. Since you have that -6√2 term, look for a binomial such that
(a - b√2)2 = 11 - 6√2
What happens when you multiply out
(a - b√2)2 ?
Can you solve for a and b?
Does that help?
denian
yup.
thanks.
denian
but i use ( [squ] x - [squ] y ) to the power of two instead.
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2022-08-07 20:05:34
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https://physics.stackexchange.com/questions/606787/why-do-green-functions-show-up-in-treating-dielectric-media-microscopically
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Why do Green functions show up in treating dielectric media microscopically?
I'm reading the treatment of Thomas-Fermi screening in Ashcroft and Mermin (ch. 17). They write some strange equations which I've never seen before, anywhere.
First of all, they write the usual local relation $$\mathbf{D} = \epsilon \mathbf{E}$$ as a non-local relation
$$\mathbf{D}(\mathbf{r}) = \int \epsilon(\mathbf{r},\mathbf{r'})\mathbf{E}(\mathbf{r}')d\mathbf{r}'$$
And then proceed to write the same type of equation linking external potential $$\phi^{ext}$$ and potential $$\phi$$, and induced charge density $$\rho^{ind}$$ and potential $$\phi$$:
$$\phi^{ext}(\mathbf{r}) = \int \epsilon(\mathbf{r},\mathbf{r'})\phi(\mathbf{r'})d\mathbf{r}'$$
$$\rho^{ind}(\mathbf{r}) = \int \chi(\mathbf{r},\mathbf{r'})\phi(\mathbf{r'})d\mathbf{r}'$$
I've seen enough Green's functions to recognise them here, and also enough Green's functions to intuitively see it makes a tiny little bit of sense to use them. After all, the electric displacement field is influenced by only the local electric field only when averaging over long length scales. And that is exactly what Ashcroft and Mermin do not do in treating Thomas-Fermi screening.
Still, I've never seen this language before in this context and I'm a little stumped as to where it comes from. Two questions:
1. Ashcroft and Mermin decide to map $$\mathbf{D}\to \phi^{ext}$$ and $$\mathbf{E}\to \phi$$ in all their equations. Why is this allowed?
2. What is the theoretical background that validates using these integrals here? Is it an ad hoc assumption they make, which just happens to work out? Or do I just have a strangely shaped hole in my knowledge of electrostatics?
• Found myself asking the same question recently. Did you find an answer elsewhere, by any chance? Oct 4, 2021 at 12:29
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2022-08-14 21:25:35
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https://ita.skanev.com/11/02/02.html
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# Exercise 11.2.2
Demonstrate what happens when we insert the keys $5, 28, 19, 15, 20, 33, 12, 17, 10$ into a hash table with collisions resolved by chaining. Let the table have 9 slots, and let the hash function be $h(k) = k \mod 9$.
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2018-07-16 02:41:05
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http://mathoverflow.net/questions/96069/status-of-pl-topology/96080
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# Status of PL topology
I posted this question on math stackexchange but received no answers. Since I know there are more people knowledgeable in geometric and piecewise-linear (PL) topology here, I'm reposting the question. I'd really want to know the state of the question, since I'm self-studying the material for pleasure and I don't have anyone to talk about it. Please feel free to close this post if you think the topic is not appropriate for this site.
I'm starting to learn about geometric topology and manifold theory. I know that there are three big important categories of manifolds: topological, smooth and PL. But I'm seeing that while topological and smooth manifolds are widely studied and there are tons of books about them, PL topology seems to be much less popular nowadays. Moreover, I saw in some place the assertion that PL topology is nowadays not nearly as useful as it used to be to study topological and smooth manifolds, due to new techniques developed in those categories, but I haven't seen it carefully explained.
My first question is: is this feeling about PL topology correct? If it is so, why is this? (If it is because of new techniques, I would like to know what these techniques are.)
My second question is: if I'm primarily interested in topological and smooth manifolds, is it worth to learn PL topology?
Also I would like to know some important open problems in the area, in what problems are mathematicians working in this field nowadays (if it is still an active field of research), and some recommended references (textbooks) for a begginer. I've seen that the most cited books on the area are from the '60's or 70's. Is there any more modern textbook on the subject?
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math.stackexchange.com/questions/70634/… addresses some of these questions. – Daniel Moskovich May 5 '12 at 15:06
I like the unnumbered questions in the end, but otherwise the question looks somewhat rhetorical and seems to call for a heated debate. If I'm primarily interested in programming, is it worth to learn mathematics? I heard that math is not nearly as useful as it used to be in computer science, due to new techniques developed in that subject. Pathetic, isn't it? And those books cited by mathematicians, some of them are so old! – Sergey Melikhov May 5 '12 at 16:30
@Daniel: Thanks very much! @Sergei: I get your point, but I think that it's not the same case as your analogy. Maybe I should put the question this way: is or is not PL topology an integral part of the education of every geometric topologist today? And concerning the books, we all know that subjects in mathematics change, and some great textbooks in the past are not well suited to the present status of the area, because change of emphasis or discovery of new tehniques that make life easier. So I'm asking about "newer" books to know if there are references more suited to present PL topology. – Carlos Sáez May 5 '12 at 16:47
Even if you only care about smooth manifolds, I think it's worth having some familiarity with the language and basic ideas: some important isotopy/embedding theorems (e.g. Hudson) have written proofs in the literature only for PL manifolds but also hold in the smooth case. If you want to tweak these proofs maybe it's useful to speak the language. – Jonny Evans May 5 '12 at 21:15
PL topology is nowadays old fashioned because of its difficulty, as it often happens in math. Nevertheless, it's not uncommon that after decades a smart guy comes with new striking discoveries and gets its back to the mainstream. I hope this happens to PL topology! – Fernando Muro Nov 7 '12 at 7:00
Maybe I should put the question this way: is or is not PL topology an integral part of the education of every geometric topologist today?
According to a recent poll by the Central Planning Commitee for Universal Education Standards, some geometric topologists don't have a clue about regular neighborhoods, while others haven't heard of multijet transversality; but they all tend to be equally excited when it comes to Hilbert cube manifolds.
some recommended references (textbooks) for a beginner
L. C. Glaser, Geometrical combinatorial topology (2 volumes)
Is there any more modern textbook on the subject?
Not really (as far as I know), but some more recent books related to PL topology include:
Turaev, Quantum invariants of knots and 3-manifolds (chapters on the shadow world)
Kozlov, Combinatorial algebraic topology (chapters on discrete Morse theory, lexicographic shellability, etc.)
Matveev, Algorithmic topology and classification of 3-manifolds
2D homotopy and combinatorial group theory
Daverman-Venema, Embeddings in manifolds (about a third of the book is on PL embedding theory)
Benedetti-Petronio, Branched standard spines of 3-manifolds
Buoncristiano, Rourke, and Sanderson, A geometric approach to homology theory (includes the PL transversality theorem)
The Hauptvermutung book
Buoncristiano, Fragments of geometric topology from the sixties
Also I would like to know some important open problems in the area, in what problems are mathematicians working in this field nowadays
I'll mention two problems.
1) Alexander's 80-year old problem of whether any two triangulations of a polyhedron have a common iterated-stellar subdivision. They are known to be related by a sequence of stellar subdivisions and inverse operations (Alexander), and to have a common subdivision (Whitehead). However the notion of an arbitrary subdivision is an affine, and not a purely combinatorial notion. It would be great if one could show at least that for some family of subdivisions definable in purely combinatorial terms (e.g. replacing a simplex by a simplicially collapsible or consructible ball), common subdivisions exist. See also remarks on the Alexander problem by Lickorish and by Mnev, including the story of how this problem was thought to have been solved via algebraic geometry in the 90s.
2) MacPherson's program to develop a purely combinatorial approach to smooth manifold topology, as attempted by Biss and refuted by Mnev.
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Thanks for the answer, specially the great list of references. – Carlos Sáez May 5 '12 at 21:31
I'd like to address another aspect of your questions. My feeling is that PL topology, or smooth topology, are foundational subjects to the low dimensional topologist, in the sense that set theory is a foundational subject to most mathematicians. A large proportion of low dimensional topologists use the foundational theorems in PL topology as black boxes, certainly without understanding or having read the proofs, and in fact they can do good mathematics that way. In the smooth category, the situation is even worse- I'm sure that there are very few people in the world who understand the proof of Kirby's Theorem, which is a difficult result, but it gets used all over low dimensional topology as a black box. Indeed, the fact that a diffeomorphism of $S^2$ extends to the $3$--ball is fundamental, under the hood everywhere, and highly non-trivial.
So you can be a manufacturer, or you can be a consumer. As a consumer, maybe you don't need to know PL topology beyond the basics that you need in order to understand simplicial homology and other basic constructions. A more sophisticated consumer might need more- I don't for example know a concrete smooth construction of linking pairings (the PL construction is in Schubert)- and in general, cell complexes allow you to work explicitly and concretely. PL proofs, if you read and care about proofs of fundamental results, tend to be shorter and easier than smooth proofs, which is not surprising because a-priori there is so much less structure which has to be carried around. This was indeed why Poincaré first considered triangulated manifolds; because of the technical facility which they afforded him. As a counter-point, I should point out Smale's comment in the introduction to in 1963 paper A survey of some recent developments in differential topology (which I recommend that you read, as it discusses your question):
It has turned out that the main theorems in differential topology did not depend on developments in combinatorial topology. In fact, the contrary is the case; the main theorems in differential topology inspired corresponding ones in combinatorial topology, or else have no combinatorial counterpart as yet...
Another aspect, which is not to be sneezed at in today's world, is that PL manifolds are better suited to computers. This is indeed the focus of Matveev's book on "algorithmic topology".
Finally, as a PL question, I nominate:
Open problem: Construct a discrete $3$-dimensional Chern-Simons theory, compatible with gauge symmetry, replacing the path integrals of the smooth picture (which are not mathematically well-defined) with finite dimensional integrals.
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We should ask Smale how he would prove, at the time of writing the quote, that fibers of generic smooth maps are homotopy equivalent to CW-complexes. The only proof known to the MO community (mathoverflow.net/questions/94404) is based on the (trivial!) combinatorial counterpart of this statement, and didn't appear until Thom's conjecture on triangulation of smooth maps was proved by Andrei Verona in 1984. – Sergey Melikhov May 8 '12 at 2:50
@Sergey: That's a good point, but I think that it's fair to suppose that he would not have considered that to be one of the main theorems of differential topology. Indeed, I would argue that cell complexes are part of the PL world, and that the smooth world is more about handles. – Daniel Moskovich May 8 '12 at 13:51
Daniel: then let's replace "are homotopy equivalent to CW-complexes" by "have finitely generated cohomology". (The domain and the range are closed smooth manifolds.) – Sergey Melikhov May 8 '12 at 22:48
As a counterpoint to the Open Problem at the end, since Poincare himself many folks have tried to find a combinatorial proof that every closed simply connected 3-manifold is homeomorphic to the 3-sphere. They may still be trying... – Lee Mosher May 9 '12 at 4:06
Some points I didnt see mentioned above: the basic results of geometric topology: tubular neighborhood theorem, transversality, xetc. have easy smooth proofs, somewhat technical PL proofs, and difficult (Kirby-Siebenmann+surgery theory) TOP proofs. Historically TOP came after the development of Smooth and PL, but in the end, the formalism in high dimensions was entirely encoded in the algebraic topology of the classifying spaces $B$Diff$=B$O, $B$PL$, B$TOP. The bottom line is that many high dimensional problems can be "reduced" to algebraic topology of these classifying spaces, and so it isn't that PL isn't interesting, just that it can be treated (say in surgery theory, or smoothing theory) on equal footing with the other two, as a black box, without really knowing anything specific about the nuts and bolts of PL topology (just as you can understand most smooth topology without knowing a careful proof of the implicit function theorem).
Following the success of high dimensional topology, the focus in geometric topology shifted to low dimensions starting in the early 1980s, and as Dylan comments there is no difference between PL and Diff in low dimensions, so that the more familiar smooth methods suffice, and more recently trained topologists have no reason to study PL methods if their focus is on low dimensions.
As a topology student, it is probably good for you to have some familiarity with the surgery exact sequence, $$\mathcal{S}_{PL}(X)\to [X,G/PL]\to L(\pi_1(X))$$ and its counterparts with PL replaced by Diff or TOP (i.e. what the objects and maps are in this sequence). Knowing the early big successes in your area will give you a better appreciation of what is happening in it now.
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Paul, your opinions on easy and hard proofs of the "etc." results, and on the needs of younger topologists, are perfect examples of the long expected heated debate... I'll leave it there, but how about the presumably less controversial issue of whether basic results of smooth topology can be proved at all (easy or hard) without using PL topology? In particular, that fibers of generic smooth maps between smooth manifolds are "homotopy equivalent to CW-complexes"? (This was a recent MO question, mathoverflow.net/questions/94404) – Sergey Melikhov May 7 '12 at 2:32
Even the non-"etc." examples are not so plain. The PL analogue of Sard's theorem is certainly easier. It says that if you take any point $p$ in the interior $U$ of a top-dimensional simplex in the range of a simplicial map $f$, then $f^{-1}(U)$ is PL homeomorphic to $U\times f^{-1}(p)$, and this is trivial to prove (as opposed to the full PL transversality). The existence of a regular neighborhood is a tautology (with definitions as in the Rourke-Sanderson book), but that of a tubular neighborhood needs proof; like other smooth proofs and unlike PL proofs, it depends on years of Calculus... – Sergey Melikhov May 7 '12 at 3:07
Let us not forget that smooth codimension $k$ embeddings have normal bundles with structure group $\text{O}(k)$, whereas this analogy does not hold in the PL-case (one has to use block bundles instead). – John Klein May 7 '12 at 17:52
John: there is a view (expressed already in the Rourke-Sanderson Annals paper series) that it is block bundles that are the right notion of a bundle in the PL category. For instance I wonder how you would do something beyond definitions with PL bundles, like Euler or Stiefel-Whitney classes or the umkehr map, without using the theory of block bundles. For block bundles these things are done in the Bouncrisiano-Rourke-Sandrson book. – Sergey Melikhov May 8 '12 at 1:40
Sergey: if one wants to understand automorphisms of PL manifolds, one cannot dispense with $PL(k)$. On the other hand, there's a sense in what you've claimed: notice $O(k+1)/O(k) = S^k$, and as $k$ varies, this gives the sphere spectrum (this is responsible for our understanding of the the Euler class). However, the spectrum associated with $PL(k+1)/PL(k)$ is very complicated: it's Waldhausen's $A(∗)$! On the other hand, the spectrum associated with $\widetilde{PL}(k+1)/\widetilde{PL}(k)$ is much easier: by Haefliger it's the sphere spectrum. – John Klein May 9 '12 at 13:20
Disclaimer: What follows is probably a bit off-topic for this site, but no more than the original questions, numbered one and two. In fact I suspect that this answer attempts to address just what the OP really wanted to ask ("isn't PL topology useless?") by posting those two lightly euphemistic questions. If there was an active meta thread for closing this question, I'd rather put this answer there.
Some topologists, perhaps the majority, tend to think that smooth and topological manifolds are "present in nature" and are the genuine objects of study in geometric topology, while PL topology is a somewhat artificial, unnatural construct, and matters just as long as it is helpful for the "real" topology. I've heard this opinion stated explicitly once, and I see a lot of this kind of attitude in this thread. In fact I think this philosophy/intuition is sufficiently familiar to nearly everyone that I don't need to elaborate on it. Moreover, I suspect that a lot of people are not even aware that it is not the only possible religion for a topologist, or else they would be more considerate to the heretics in stating their strong opinions.
I'd like to discuss one other philosophy/intuition then, according to which both smooth and topological manifolds are obviously artificial, highly deficient models for what could be "present in nature", whereas the PL world is much "closer to the reality". I don't consider myself a practitioner of this or any other religion; what follows should be regarded as said by a fictional character, not by the author.
1) As is well-known, the predisposition to seeing continuous and smooth as more natural than discrete is historical, following centuries of preoccupation with derivatives and (later) limits. Quantum physics and computer science may be changing the tide, but they don't usually compete with Calculus in a mathematician's education, at least not in the initial years.
Here is a simple test. When you fold a sheet of paper, what is the intuitive model in your imagination: is it a smooth surface (when you look with a loupe at the fold), a cusp-like singularity (generic smooth singularity), or an an angle-like singularity (PL singularity)? No matter what is your subconscious preference, I bet you didn't base it on considerations of individual photons detected by the eye. But you could have based it on your previous experience with abstract models of surfaces, which is not independent of the historically biased education. (Just for fun, I wonder if your intuitive model would change if the paper sheet is folded second time so as to make a corner - which is unstable as a singularity of a smooth map $\Bbb R^2\to\Bbb R^2$, but has a stable singularity in the link.)
2) On a molecular scale, the sheet of paper of course doesn't fit the model of a smooth surface, and although it is arguably not "discrete" or "PL" on a subatomic scale, the smooth surface model isn't restored either. Similarly, as is well-known, Maxwell equations and general relativity (which I guess are among the best reasons to study smooth topology) don't work at very small scales. The problem is that this "imperfection" of matter doesn't usually shake one's belief in "perfect" physical space. But it is perfectly consistent with modern physics (for those who don't know) that physical space is kind of discrete at a sub-Planck scale, as in loop quantum gravity (which is somewhat reminiscent of PL topology!). It is also consistent with the present day knowledge, and indeed derivable in variants of the competing string theory, that a finite volume of physical space can only contain a finite amount of information, as with the holographic principle. (In fact I didn't see much discussion of possible alternatives to this principle, many physicists appear to take it for granted.) I'm getting on a slippery slope, but finite information does not sound like it could be compatible with limits that occur in derivatives (which returns us to MacPherson's program on combinatorial differential manifolds) and especially with Casson handles that occur in topological manifolds.
The fictional character is now saying that his religion teaches him to avoid concepts based on inherently infinitary constructions, because they are likely to be unnatural, in the sense of the physical nature which might simply have no room for them (and even the question of whether it does is not obviously meaningful!). Ironically, this is quite in line with Poincare's philosophical writings, where he argued at length that the principle of mathematical induction is not an empirical fact.
3) The fictional character goes on to say that this is not just the crazy metaphysics that displays the warning, but also Grothendieck with his "tame topology" which inspired a whole area in logic (initiated by van den Dries' book Tame topology and o-minimal structures). Here is a short quote from Grothendieck:
It is this [inertia of mind] which explains why the rigid framework of general topology is patiently dragged along by generation after generation of topologists for whom "wildness" is a fatal necessity, rooted in the nature of things.
My approach toward possible foundations for a tame topology has been an axiomatic one. Rather than declaring [what] the desired “tame spaces” are ... I preferred to work on extracting which exactly, among the geometrical properties of the semianalytic sets in a space $\Bbb R^n$, make it possible to use these as local "models" for a notion of "tame space" (here semianalytic), and what (hopefully!) makes this notion flexible enough to use it effectively as the fundamental notion for a “tame topology” which would express with ease the topological intuition of shapes.
Grothendieck dismisses from the start PL and smooth topology as possible forms of tame topology, because
(i) they're "not stable under the most obvious topological operations, such as contraction-glueing operations", and
(ii) they're not closed under constructions such as mapping spaces, "which oblige one to leave the paradise of finite dimensional spaces".
I'm not familiar with "contraction-glueing operations", nor is Google. Perhaps someone fluent in French could explain what (i) is supposed to mean? My first guess would be that this could refer to mapping cylinder, mapping cone or other forms of homotopy colimit, but PL topology is closed under those (finite homotopy colimits).
Edit: Indeed, it is clear from the preceding pages that by "gluing" Grothendieck means the adjunction space, which he also calls "amalgamated sum". In particular, he says:
It was also clear that the contexts of the most rigid structures which existed then, such as the "piece-wise linear" context were equally inadequate – one common disadvantage consisting in the fact that they do not make it possible, given a pair $(U,S)$ of a "space" $U$ and a closed subspace $S$, and a glueing map $f: S\to T$, to build the corresponding amalgamated sum.
There is, of course, no problem with forming adjunction spaces in the PL context. Perhaps Grothendieck was just not aware of pseudo-radial projection or something. End of edit
As to (ii), there now exists some kind of an infinite-dimensional extension of PL topology, which includes mapping spaces and infinite homotopy colimits up to homotopy equivalence (and hopefully up to uniform homotopy equivalence, which would be more appropriate in that setup). Besides, there are, of course, Kan sets, which are closed under Hom, but they arguably don't belong to tame topology in any reasonable sense because they quickly get uncountable (in every dimension, in particular, there are uncountably many vertices) and even of larger cardinality.
In any case, logicians, who tried to set up Grothendieck's aspiration in a rigorous framework of definability (see Wilkie's survey), do now have the "o-minimal tringulability and Hauptvermutung" theorem, saying roughly that tame topology (as they understood it) is the same as PL topology. Still more roughly (perhaps, too roughly) is could be restated as "topology without infinite constructions is the same as PL topology".
Even if smooth topology will some day be reformulated in purely combinatorial terms, it is highly unlikely that it can be characterized by purely logical constraints. From this viewpoint, smooth topology is primarily justified by its role in applied math and natural sciences, but is no less and no more fundamental than symplectic topology or topology of hyperbolic manifolds.
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One thing meant by "gluing", I suppose, is that extra information (collars) is needed to specify the result of gluing smooth manifolds along a boundary. But anyway, my fictional character argues for smooth topology: Only natural numbers are "real". Calculus is "real", because it provides a low algorithmic complexity setting to answer natural number problems. Smooth manifolds are real because they are spaces on which calculus can be performed. PL manifolds make sense as discrete models for smooth manifolds; or else you need to argue that they have "independent" existence. – Daniel Moskovich May 9 '12 at 7:32
... (cont.) So there is a lot of interesting structure (geometry, flows, analytic structure...) which you can impose on a smooth manifold. A manifold is a world to live in: clearly, a lot of dynamics can take place in smooth manifolds. But there seems to be no life on a PL manifold- it's a barren, cubist wasteland. Thus quoth my fictional character. IRL, I don't know, and I'm happy that there are adherents of both "religions" (and others besides) amongst topologists. – Daniel Moskovich May 9 '12 at 7:43
Daniel, thank you for feedback. (I'm still puzzled by Grothendieck's contraction-gluing; the result of gluing two PL manifolds along a PL homeomorphism of their boundaries doesn't need any extra information.) There are, of course, things like geometric structures on cell complexes (popular in geometric group theory, see e.g. the Bridson-Haefliger book), harmonic functions on simplicial complexes (see R.Forman's 1989 paper in Topology), combinatorial Gauss-Bonnet formula (see Yu Yan-Lin's 1983 paper in Topology), connections and parallel transport on PL manifolds (see M.A.Penna's 1978 paper... – Sergey Melikhov May 9 '12 at 9:42
...in Pacific J Math, and also arxiv.org/abs/math/0604173) and of course PL de Rham theory (see D. Lemann's 1977 paper in Asterisque, R.G.Swan's 1975 paper in Topology, Bousfield-Gugenheim 1976 AMS memoir). Of course, discrete analysis is motivated by the smooth case (Smale should have been more specific!) so no wonder it lags behind. The problem with PL topology is I think that it has indeed been largely deserted since 1970s and as a consequence is now underdeveloped and barely taught to students. I'm not sure that there's any internal reason for that, it could be entirely cultural. – Sergey Melikhov May 9 '12 at 10:05
PL topology is popular in quantum topology where some invariants (e.g Turaev-Viro) are defined by fixing a triangulation and the checking invariance under some standard moves.
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It's worth commenting (for those that don't know) that PL topology is the same as smooth topology in low dimensions (up to 6). – Dylan Thurston May 6 '12 at 2:57
...which is a highly nontrivial fact (particularly Cerf's theorem, implying that smooth structures are unique on PL 4-manifolds and exist on PL 5-manifolds) better stated as "PL topology includes smooth topology in low dimensions" because PL topology is not just about PL manifolds but also about polyhedra (not to mention PL maps). Even that is not quite accurate, because families of low-dimensional smooth structures don't boil down to those of PL structures (see mathoverflow.net/questions/7892), so no wonder that Haefliger's smooth knots of $S^3$ in $S^6$ are trivial as PL knots. – Sergey Melikhov May 8 '12 at 4:05
... Bringing in the morphisms, PL maps include (by another highly nontrivial result) generic smooth maps, as well as smooth maps that belong to generic families, and the inclusion is strict (for maps between manifolds) starting from very low dimensions (2 to 1). Arbitrary smooth maps easily have arbitrary compact metric spaces as point-inverses, so I'm not sure if they belong to smooth topology. For sure, mapping cylinders of (very low-dimensional) generic smooth maps aren't smooth manifolds, but one can't deny their place in PL topology. – Sergey Melikhov May 8 '12 at 4:36
On a smooth manifold we have Ricci flow. What is the analogue for a PL manifold?
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You should ask this as a separate question, rather than adding here. – arsmath Nov 7 '12 at 13:10
@arsmath: I had the same initial reaction until I realized that OP is also asking for a list of open problems in PL topology. Defining a combinatorial analogue of Ricci flow (say, in dimension 3) is a well-known open problem. If such flow exists, it could lead to a more constructive proof of, say, Poincar\'e conjecture. – Misha Nov 7 '12 at 13:56
Indeed, for further related discussion see Bruce Westbury's own question mathoverflow.net/questions/65691/… – j.c. Nov 7 '12 at 15:22
It would certainly have helped if Bruce had added a link to his own question, and/or described a little context. – S. Carnahan Nov 8 '12 at 8:19
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2015-03-28 12:54:55
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https://math.stackexchange.com/questions/889505/a-question-regarding-conjugacy-classes-of-central-involutions
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# A question regarding conjugacy classes of central involutions.
An involution $$a$$ of a group $$G$$ is called central if there exists a sylow $$2$$-subgroup $$H$$ of $$G$$ such that $$a \in C_G(H)$$.
Clearly if an involution is central then its every conjugate is also central.
1. If $$C$$ is a conjugacy class of involutions and $$a$$, $$b$$ are two distinct members of $$C$$ that commute with each other then is $$ab$$ also a member of $$C$$?
2. Can we classify all finite groups (or finite simple groups) for which the number of conjugacy classes of central involutions is $$1$$?
Let $$G = \langle (3, 4), (1, 3)(2, 4) \rangle \cong D_8$$. Set $$a = (1, 3)(2, 4)$$ and $$b = (1, 4)(2, 3)$$. Then $$a^2 = b^2 = \operatorname{id}$$, $$ab = ba$$ and $$C = \{ a, b \}$$ is a conjugacy class of $$G$$, but $$ab = (1, 2)(3, 4) \not\in C$$.
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2021-08-04 22:11:39
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http://mathhelpforum.com/algebra/177134-solve-equation-n-where-n-exponent.html
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# Thread: solve equation for n where n is an exponent
1. ## solve equation for n where n is an exponent
Dermine n if:
665/24=(4/3((3/2)^n-1))/(1/2)
2. Originally Posted by Rosie
Dermine n if:
665/24=(4/3((3/2)^n-1))/(1/2)
$
\frac{665}{24}
=
\frac
{
\frac{4}{3}.
(\frac{3}{2})^{n-1}
}
{\frac{1}{2}}
$
$
\frac{665}{24}.\frac{1}{2}
=
\frac{4}{3}.
(\frac{3}{2})^{n-1}
$
$
\frac{665}{24}.\frac{1}{2}.\frac{3}{4}
=
(\frac{3}{2})^{n-1}
$
Now take log on both sides and proceed.
3. Is this the problem?
$\frac{665}{24}=\frac{4}{3}(\frac{(\frac{3}{2})^n-1}{\frac{1}{2}})$
If so, how have you tried to distribute so far? Do you know that dividing by a half is the same as multiplying by 2?
Start by dividing both sides by 4 and multiplying both sides by 3 to give:
$\frac{665}{24}\times\frac{3}{4}=\frac{(\frac{3}{2} )^n-1}{\frac{1}{2}}$
$\frac{665}{24}\times\frac{3}{4}=2\times ((\frac{3}{2})^n-1)$
I would go further, but I don't know whether this is the equation and nor do I know whether you've followed my working thus far. Can you provide any further contribution?
Edit: whoops, was beaten to it.
4. Quacky you have the right equation and the rest of your questions I do not understand.
Can you please finish the equation because I truely do not get the right answer.
5. solving Quacky's equation.
$\frac{665}{24}\times\frac{3}{4}=2\times ((\frac{3}{2})^n-1)$
$
\frac{665}{24}\times\frac{3}{4}\times\frac{1}{2}
=
(\frac{3}{2})^n-1
$
$
\frac{665}{24}\times\frac{3}{4}\times\frac{1}{2}+1
=
(\frac{3}{2})^n
$
solve LHS and take log on both sides
$
log_e(\frac{665}{24}\times\frac{3}{4}\times\frac{1 }{2}+1)
=
n\times log_e(\frac{3}{2})
$
$
\frac{log_e(\frac{665}{24}\times\frac{3}{4}\times\ frac{1}{2}+1)}{log_e(\frac{3}{2})}
=
n
$
6. amul I do not know what log is, I have never used it before. I am just matric. The sum falls under series and sequences. So therefor I do not understand what you just did.
7. Originally Posted by Rosie
amul I do not know what log is, I have never used it before. I am just matric. The sum falls under series and sequences. So therefor I do not understand what you just did.
ok
$\frac{665}{24}\times\frac{3}{4}\times\frac{1}{2}+1
=
(\frac{3}{2})^n$
when you solve LHS it solves to $\frac{729}{64}$
$
(\frac{3}{2})^6=(\frac{3}{2})^n
$
since bases are equal powers are equal
so $n=6$
8. thank you very much, I see where I made my mistake.
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2016-08-26 04:56:46
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https://pleasescoopme.com/2009/03/08/
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# Daily Archives: March 8, 2009
## Another perspective on link probability functions
When deciding when a link between two documents should exist, we need to define a function of the covariates which we call a link probability function. We have looked at many candidate functions but concentrated on two such functions:
• $\psi_\sigma = \sigma(\eta^T z_1 \circ z_2 + \nu)$
• $\psi_e = \exp(\eta^T z_1 \circ z_2 + \nu)$
Recently, I saw a new connections between the two functions. This connection exists in the context of variational inference. One of the terms we must compute in variational inference is $\mathbb{E}_q[\log \psi(z_1, z_2)]$ where the expectation is taken over $z_1, z_2$ and $\mathbb{E}_q[z_i] = \phi_i$. If we expand the equation using $\psi_\sigma$, this amounts to computing $\eta^T \phi_1 \circ \phi_2 + \nu - \mathbb{E}_q[\log (1 + \exp(\eta^T z_1 \circ z_2 + \nu))]$. This latter term is intractable to compute exactly, so we turn to approximation.
In previous posts, I primarily explored using the approximation $\log(1 + \exp(\eta^T \phi_1 \circ \phi_2 + \nu))$. This is equivalent to approximating the function inside the expectation with a first order Taylor approximation centered around the mean. Because of convexity, this approximation is a tangent line which forms a lower bound on the function (see graph below). Unfortunately, this is the wrong bound for variational inference; there the quantity we are computing is supposed to be a lower bound which means that the bound on the partition function of the link probability function needs to be an upper bound. That was a mouthful, and things seem grim but as it turns out this approximation is rather good since most of the points lie near the mean.
But why not construct an upper bound? This is possible because the function is convex and because the covariates’ range is bounded. The hyperplane which intersects the function at the covariates’ bounds is an upper bound for the function. If we apply this approximation, the expectation expands into $(\phi_1 \circ \phi_2)^T (\log(1 + \exp(\eta +\ nu)) - \log(1 + \exp(\nu)))$ $+\log(1 + \exp(\nu))$. This upper bound is also depicted in the figure below. But also note that if we set $\nu' = \log(1 + \exp(\nu))$ and $\eta' = \log(1 + \exp(\eta + \nu)) - \log(1 + \exp(\nu))$ then this expression becomes $\eta'^T (\phi_1 \circ \phi_2) + \nu'$ which has the exact same functional form as $\mathbb{E}_q[\psi_e(z_1, z_2)]$. Thus, using this upper bound reduces to using $\psi_e$!
Two bounds on the link probability function
Thus, rather than considering two link probability functions, it is perhaps better to think of two approximations to a single link probability function. This way of thinking also sheds some light on why perhaps $\psi_e$ tends to perform better than $\psi_\sigma$. Even though the lower-bound approximation is “better” in the sense that it is closer to the true value of the expectation, the bound is in the wrong direction for variational inference. Maybe “cowboy” variational doesn’t work all that well after all.
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2023-01-29 11:58:44
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http://mathonline.wikidot.com/projection-transformations
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Projection Transformations
# Projection Operators
Definition: For any vector $\vec{x} \in \mathbb{R}^2$, a projection operator $T: \mathbb{R}^2 \to \mathbb{R}^2$ projects every vector $\vec{x}$ onto some axis. For any vector $\vec{x} \in \mathbb{R}^3$, a projection operator projects every vector $\vec{x}$ onto some plane.
## Projection Transformations in 2-Space
Let $\vec{x} \in \mathbb{R}^2$ such that $\vec{x} = (x, y)$. Recall that we can imagine a projection in $\mathbb{R}^2$ of a vector to be a "shadow" that the vector casts onto another vector, or in this case an axis. For example, consider the transformation that maps $\vec{x}$ onto to $x$-axis as illustrated:
We note that the x-coordinate of our vector stays the same while the y-coordinate becomes a zero. Thus, the following equations define the image under our transformation:
(1)
\begin{align} w_1 = x + 0y \\ w_2 = 0x + 0y \end{align}
Thus, we obtain that our standard matrix is $A = \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix}$ and in $w = Ax$ form:
(2)
\begin{align} \quad \begin{bmatrix} w_1\\ w_2 \end{bmatrix} = \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} \end{align}
Of course we could always project $\vec{x}$ onto the $y$-axis like the following diagram illustrates:
In this case, we note that the x-coordinate of our vector becomes zero while the y-coordinate stays the same, and the following equations define our image:
(3)
\begin{align} w_1 = 0x + 0y \\ w_2 = 0x + y \end{align}
Thus our standard matrix is $A = \begin{bmatrix} 0 & 0\\ 0 & 1 \end{bmatrix}$.
## Projection Transformations in 3-Space
Let $\vec{x} \in \mathbb{R}^3$. We can orthogonally project $\vec{x}$ onto either the $xy$, $xz$ or $yz$ planes by mapping exactly one coordinate to zero.
In the case above, suppose that we map $\vec{x}$ onto the $xy$-plane. It thus follows that the x and y coordinates stay the same while our z-coordinate becomes zero, resulting in the following equations defining our image:
(4)
\begin{align} w_1 = x + 0y + 0z \\ w_2 = 0x + y + 0z \\ w_3 = 0x + 0y + 0z \end{align}
Hence our standard matrix for this transformation is $A = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0 \end{bmatrix}$. The following table describes other possible orthogonal projections to other planes:
Operator Equations Defining the Image Standard Matrix
Orthogonal projection onto the $xz$-plane $w_1 = x + 0y + 0z \\ w_2 = 0x + 0y + 0z \\ w_3 = 0x + 0y + z$ $\begin{bmatrix}1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 1 \end{bmatrix}$
Orthogonal projection onto the $yz$-plane $w_1 = 0x + 0y + 0z \\ w_2 = 0x + y + 0z \\ w_3 = 0x + 0y + z$ $\begin{bmatrix}0 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}$
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2021-04-14 21:09:10
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https://cran.microsoft.com/snapshot/2019-10-31/web/packages/preprosim/vignettes/preprosim.html
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# Preprosim
#### 2016-07-26
Data quality simulation can be used to check the robustness of data analysis findings and learn about the impact of data quality contaminations on classification. This package helps to add contaminations (noise, missing values, outliers, low variance, irrelevant features, class swap (inconsistency), class imbalance and decrease in data volume) to data and then evaluate the simulated data sets for classification accuracy. As a lightweight solution simulation runs can be set up with no or minimal up-front effort.
## Quick start
### Example 1: Creating contaminations
The package can be used to create contaminated data sets. Preprosimrun() is the main execution function and its default settings create 6561 contaminated data sets. In the example below argument ‘fitmodels’ is set to FALSE (not to compute classification accuracies) and default setup is used (argument ‘param’ is not given).
library(preprosim)
res <- preprosimrun(iris, fitmodels=FALSE)
All contaminated data sets can be acquired as a list from the data slot:
datasets <- res@data
length(datasets)
## [1] 6561
The data set corresponding to a specific combination of contaminations can be acquired as a dataframe with getpreprosimdf() function.
df <- getpreprosimdf(res, c(0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1))
head(df)
## x.Sepal.Length x.Sepal.Width x.Petal.Length x.Petal.Width y
## 1 5.104000 3.498356 1.375905 0.2157918 setosa
## 2 NA 3.089296 1.428346 0.1626930 virginica
## 4 4.289669 3.202021 1.590556 4.2615823 setosa
## 5 5.193853 3.517478 1.441303 0.2102551 setosa
## 7 4.669934 3.336195 1.339044 0.2796277 setosa
## 8 5.108899 3.327911 1.449244 0.1741521 setosa
The second argument above has the contamination parameters in the following order:
str(res@grid, give.attr=FALSE)
## 'data.frame': 6561 obs. of 8 variables:
## $noise : num 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 ... ##$ lowvar : num 0 0 0 0.1 0.1 0.1 0.2 0.2 0.2 0 ...
## $outlier : num 0 0 0 0 0 0 0 0 0 0.1 ... ##$ irfeature : num 0 0 0 0 0 0 0 0 0 0 ...
## $classswap : num 0 0 0 0 0 0 0 0 0 0 ... ##$ classimbalance: num 0 0 0 0 0 0 0 0 0 0 ...
## $volumedecrease: num 0 0 0 0 0 0 0 0 0 0 ... ##$ misval : num 0 0 0 0 0 0 0 0 0 0 ...
### Example 2: Classification accuracy of contaminated data sets
Preprosimrun() function with default value fitmodels=TRUE can be used to fit models and compute classification accuracy for each contaminated data set. Note that the selected model must be able to deal with missing values AND have an in-build variable importance scoring. Only ‘rpart’ and ‘gbm’ models are tested.
Parameter object is controlling, which contaminations are applied. In the example below the impact of missing values (primary, 10 contamination levels) and noise (secondary, 3 contamination levels ) on classification accuracy is studied. Classifier ‘rpart’ is used as a model instead of default ‘gbm’ and two times repreated holdout rounds are used. Argument ‘cores’ is not given, using 1 core by default.
res <- preprosimrun(iris, param=newparam(iris, "custom", x="misval", z="noise"), caretmodel="rpart", holdoutrounds = 2, verbose=FALSE)
preprosimplot(res)
Specific dependencies between contaminations can be plotted by giving ‘xz’ argument to preprosimplot() function.
preprosimplot(res, "xz", x="misval", z="noise")
The corresponding result data can be acquired with getpreprosimdata() function. In the exampe below ‘x’ and ‘y’ in str() function output correspond to arguments given in preprosimplot() and no other contaminations are applied similar to design of experiment (all other parameter values set to 0 zero).
data <- getpreprosimdata(res, "xz", x="misval", z="noise")
str(data)
## 'data.frame': 30 obs. of 9 variables:
## $z : num 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 ... ##$ grid.lowvar : num 0 0 0 0 0 0 0 0 0 0 ...
## $grid.outlier : num 0 0 0 0 0 0 0 0 0 0 ... ##$ grid.irfeature : num 0 0 0 0 0 0 0 0 0 0 ...
## $grid.classswap : num 0 0 0 0 0 0 0 0 0 0 ... ##$ grid.classimbalance: num 0 0 0 0 0 0 0 0 0 0 ...
## $grid.volumedecrease: num 0 0 0 0 0 0 0 0 0 0 ... ##$ x : num 0 0 0 0.1 0.1 0.1 0.2 0.2 0.2 0.3 ...
## \$ accuracy : num 0.971 0.961 0.951 0.922 0.882 ...
Variable importance (i.e. robustness of variables in classification task) in the contaminated data sets can be plotted:
preprosimplot(res, "varimportance")
## Customization
The package includes eight build-in contaminations with parameters as contamination intensities. Contamination names, contents, parameter ranges and core definitions are presented below. For full definitions, please see the source code.
1. noise
• normal random number having original value in data as mean and parameter as standard deviation
• rnorm(length(x), x, param@noiseparam)
1. lowvar (low variance)
• parameter by which the original value is moved towards the mean of the variable
• 0 = none, 1=all values are mean
• multiplierdifftomean <- lowvarianceparameter * scale(x, scale=FALSE)
• newvalue <- x - multiplierdifftomean
1. misval (missing values)
• parameter for the share of missing values
• 0=none, 1 = all
• positionstomissingvalue <- sample(1:length(x), numberofmissingvalue)
• x[positionstomissingvalue] <- NA
1. irfeature (irrelevant features)
• parameter for the share of irrelevant features generated
• 0 = none, 1 = as many as there are variables in the original data
• numberofirrelevantfeatures <- as.integer(param@irfeatureparam * ncol(data@x))
• basedata <- data.frame(basedata, newvar=runif(nrow(data@x), -1, 1))
1. classswap (inconsistency)
• share of class labels that are swapped
• 0=none, 1=all
1. classimbalance
• share of observations to be removed from the most frequent class
• 0=none, 1=all
1. volumedecrease
• share of observations removed from the data
• 0=none, 1=all removed
• caret::createDataPartition(data@y, times = 1, p = param@volumedecreaseparam, list=FALSE)
1. outlier
• number of observations replaced with +IQR1.5 to +IQR2.0 outlier
• 0=none, 1=all
• outliers <- runif(d, smallestoutlier, largestoutlier)
• tobereplaced <- sample(1:length(x),d)
• x[tobereplaced] <- outliers
### Parameter structure
Each contamination has three sub parameters:
1. cols as columns the contamination is applied to
2. param as the parameter of the contamination itself (i.e. intensity of contamination)
3. order as order in which the parameter is applied to the data.
### Parameter construction
Parameter objects can be initialized with newparam constructor(). The constructor reads the data frame and sets the parameters. In the example below, first the parameters as set in a default manner, then as empty and lastly for a specific purpose.
pa <- newparam(iris)
pa1 <- newparam(iris, "empty")
pa2 <- newparam(iris, "custom", "misval", "noise")
### Parameter change
Parameters of an existing parameter object can be changed with changeparam() function.
pa <- changeparam(pa, "noise", "cols", value=1)
pa <- changeparam(pa, "noise", "param", value=c(0,0.1))
pa <- changeparam(pa, "noise", "order", value=1)
## Supporting packages
The data quality of a contaminated data set can be visualized with package preproviz. In the example below the data frame ‘df’ acquired above is visualized for data quality issue interdependencies.
library(preproviz)
viz <- preproviz(df)
plotVARCLUST(viz)
In a similar manner the same data frame ‘df’ acquired above could be preprocesssed for optimal classification accuracy with package preprocomb.
library(preprocomb)
grid <- setgrid(preprodefault, df)
result <- preprocomb(grid)
result@bestclassification
For further information, please see package preproviz and preprocomb vignettes.
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2022-10-01 18:00:41
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https://zulfahmed.wordpress.com/2012/02/
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Feeds:
Posts
## The noisy channel model and protein shape determination problem
This note is simply a simplification and repetition of ideas that I had been pursuing for around a year. Let us recall the noisy channel model in the simplest form. Signal from a source is transformed in this model into a target signal. In the discrete memoryless case, the source consists of a sequence from an alphabet of letters $a_1, \dots, a_K$ and the target consists of alphabet of letters $b_1,\dots, b_M$. Each letter in the target sequence is statistically dependent only on the input sequence and is specified by the conditional probability assignment $P(b_j|a_k)$ defined for each letter $a_k$ in the input alphabet and each letter $b_j$ in the output alphabet. This model is not quite sophisticated enough to handle the protein shape determination problem but we can place the protein shape determination problem in this setting as follows. The input sequences are triples of amino acids and the output consits of sequences of elements from a fixed discrete grid on $SO(3)$ which is the space of rotations in three dimensions. Since there are 20 distinct amino acids and therefore counting repetitions 8,000 amino triples, and for each of these there is a separate conditional probability table that must account for a fixed grid on $SO(3)$ that is sufficiently fine to produce accurate representation for the entire universe of protein shapes, there is a priori even the memoryless discrete model in this case can produce a priori around in the order of 8,000 conditional probability tables each with order of size of the $SO(3)$ grid entries.
In order for this approach to be tractible, one needs sparsity results that could reduce the size of the computational machinery. It is in this direction that we have made substantial progress over the past few years. In the language we have developed, the input alphabet are amino-triples and the target alphabet are twists. From an empirical sample of 39,000 protein shapes we have found that that the average size of the intersection of twist distributions conditioned on the previous twist in the sequence with the conditional distribution of twists given the amino-triple corresponding to it is on average no larger than several hundred distinct points. In particular, we avoid the problem of size explosion for the noisy channel approach to protein shape modelling.
We proceed by the analogy with a successful application of the noisy channel model to natural language translation. In natural language translation, the noisy channel model has been employed with success by modeling natural language translation by an n-gram language model and by phrase tables. In the example of translating French sentences into English, one proceeds solving the optimization problem
$argmax_e P( e | f ) = argmax_e P(e) P(f | e)$
Our approach is the same with English replaced by twists and French replaced by amino-triple words in protein sequences. Our simplifying assumption is that the 2-gram dependence is sufficient for modeling both the amino sequence space and the protein shape space composed of twists.
## What sort of revolution is necessary for the human race?
Although people are generally unhappy with the injustice of the current planetary political system both in the west and the rest of the world, there are several pressures on people which keeps their vision and ambition small. There are obvious pressures where people’s focus is on their private lives. The systemic structures to buoy them are the religions and the remaining structure pushes them down, such as stagnant wages and unemployment around the globe. There are obvious fears of radical changes to the system because people have been fed horror stories of Hitler’s Germany, both true and fictionally exaggerated along with many other atrocious results from attempted revolutionary changes. The 9/11 Mossad operation was carefully calibrated to produce a specific world order where Islam could be demonized and Muslims could be marked for the global untermensch. As with other such attempts, the results are decidedly mixed. Since I am interested in a revolutionary change, the 9/11 spark for a new planetary order can be seen as a type of operation that relied on the extant good-and-evil order that was put in place after America won both hemispheres after the Second World War, with its prime document NSC-68. But clearly there still needs to be a new world order not only to reverse the damage done to the human race by the 9/11 Mossad operation but quite generally because the system of neoliberal capitalism has failed to deliver on its promises even to the explicitly privileged. This is why there is a strong Occupy Wall Street movement in America, a short J4 movement in Israel and left surge in Europe which is battling with increased panic for alleged Iranian plans for building nuclear bombs.
One obvious question is then what sort of spark can produce a movement for real change? The Ron Paul phenomenon in America is interesting in its attempt for gaining popular support for a radical shift of the status quo. And I believe that it will nevertheless fail to produce a political victory in America, and then the question will be whether the Ron Paul popular movement’s failure is due to its radical departure from the current power structure of the planet or whether its failure is due to insufficient radical departure from the current power structure. While it is considered normal in America that one navigates the political landscape by carefully avoiding explosive topics — and just as the Zionists have amassed an enormous stockpile of nuclear weapons and the most sophisticated military technology in abundance since 1948 they also have produced some of the most explosive political weapons in America although in America Zionism is a larger political stream including the Christian evangelical Zionists. Just so that one realizes the seriousness of this issue, one can look at Chris Hedges’ book on the dangers of Fascism that are appearing from this stream. If one is concerned only in analyzing the situation passively, one can see how Samuel Huntington’s “Clash of Fundamentalisms” seem to have been a self-fulfilling prophesy or indeed even consciously guided by the tools of the power elite. While in fact the “Clash of Fundamentalisms” is gigantically deceptive rather than flawed because Islamism as fundamental political ideology is something that could have been avoided altogether and instead it has been quite consciously been encouraged by the western political elite. The clearest example of this of course is the creation of Hamas as a counter to PLO to sabotage the Oslo peace process presumably for the “Greater Israel” project. But the precedent for this is obviously the funding of Islamic fundamentalists in Afghanistan to counter communism and continued support in Libya and Syria to destabilize reasonably secular regimes. Here Ron Paul was right and was the only person to object to the almost unanimous vote in the US Congress supporting Israel after the Operation Cast Lead massacre that killed hundreds of Palestinian children and more than a thousand civilians.
Arab Spring showed the face of democracy seeking parties in the Middle East and this has been attacked from all sides. The million killed by America in Iraq and now the military preparations for a regime change in Iran are clearly provocations that sideline secularists and moderates and strengthen hardline Islamists. But we are faced today with a broader concern about the sort of world order we could expect that is not driven by Fascist imperatives driven by Zionist evangelicalism and a revival of Crusades.
It is not an exaggeration to consider the current global spiral deeper into Hell to be a function generally of the duality of Good and Evil. The 9/11 Mossad operation was essentially equivalent to taking the NSC-68 good-and-evil story and amplifying it by a thousand fold.
Today I overcame the metaphysical barrier produced by some agencies that kept 9/11 as a Mossad operation or deviation from the completely false story of al Qaeda’s primary role in 9/11 as incredible. This includes Noam Chomsky who had drawn lines of truth for 9/11. Those who had done this perhaps had different ideas about the reasonable expectations of what can seriously change in the planetary political system and which narratives are worthwhile challenging and which narratives are better left alone for more important issues affecting people. Both 9/11 and the standard narrative of the Holocaust have the same treatment by them. There is a serious problem with this approach because the dystopia is global and to speak the truth means to speak the truth regardless of agenda. My criticism of Noam Chomsky’s views of both US and Britain as examples of terror states is that the mythologies that justify this behavior cannot be sustained with the expectation that with these active mythologies that there is any genuine chance of this behavior changing.
## Purgatory
```Unreal City, 60
Under the brown fog of a winter dawn,
A crowd flowed over London Bridge, so many,
Sighs, short and infrequent, were exhaled,
And each man fixed his eyes before his feet.
Flowed up the hill and down King William Street,
To where Saint Mary Woolnoth kept the hours
With a dead sound on the final stroke of nine.
There I saw one I knew, and stopped him, crying "Stetson!
"You who were with me in the ships at Mylae!
"That corpse you planted last year in your garden,
"Has it begun to sprout? Will it bloom this year?
"Or has the sudden frost disturbed its bed?
--T. S. Eliot, "The Waste Land" 1922```
Greyhound bus stations in Oklahoma, in Indianapolis, New York, in Dallas and Chicago with uniform architecture, a sparseness with despondent public with a drugged serenity provided elsewhere by Lithium. A sleepiness different from the early morning subways in New York with uniformed bureaucrats in ties and suits rushing to their stations. These are the loci of purgatory, interim public places with incoherent public. Oppressive gloom that is barely a notch above that of the 30th street homeless shelter whose main discernible difference is a gigantic weapon detector infrastructure at the entrance where one has to take off metallic objects including belts, not very different from the security checks mandated for the concocted threat of Islamist terrorists. The oppressive chambers of hopelessness are punctuated often by conversations. In Oklahoma, a devout man explained to me that God loves us all, and to appreciate this love one must water one’s life labors like that of a tree. Purgatory waiting for a judgment for people whose hopes have been squeezed into a narrow passage.
I have been in an internal metaphysical path that is mostly healing through spirit matter that is mostly dark muck. My answer to the pervasive gloom is not simply a new adventure for people, which is what 9/11 produced, a military adventure to refocus the national sentiment towards an aggressive project, a rejuvenation of the project of American entry into the Second World War which mobilized the nation into a good-and-evil drama that led to enormous wealth for a few and gave global military hegemony for America. The 9/11 event did not quite have the same effect on the people and in an ironic turn ended with the American economy in a worse shape than it began. Thus we see the expanse of purgatory and a decline in the empire on its fringes.
There is a sense in which there is an impossibility of the same pattern twice and there is thus a continuous need for change, and for evolution. Stories of wealth and poverty are age-old and does not lead to significant change for there is no moral force in social injustice that has not been neutralized before and thus cannot be neutralized again. Thus one must seek a different path out of this pervasive gloom, and if we, as a race wish to evolve then our global dramatists must consider the drama of our enemies within. Televisions in purgatory have commercials on how clinically proven techniques can increase one’s testosterone by 61%. The set of innovative ideas to come out for improvements in our lives for commercial success has a higher chance of sinking us into an inhuman Hell rather than a path to eudaimonia by any standards. It is art that changes the world because it is art that can present us with the most difficult wars that are necessary. We must be mobilized not for vampyric destruction and plunder but to overcome our inner demons, for these are much more pernicious, and the lies to produce the external demons are precisely an obstruction to the passage from purgatory to sunlight.
## Power pyramids of Trotsky and Lenin
Last night we captured the power pyramids of Lenin and Trotsky after realizing that Trotsky had applied the Zionist metaphysical power as well. I quickly saw a succession of hundreds of faces of people under the sway of Trotsky and Lenin. Also last night as I wanted to make use of my own resentment at having been in humiliating situations, I reached a high level of resentment and it is here that I saw a global network of levels of resentment. I literally saw a world map with blocks in black that showed concentration of resentment, at least realtime last night it was clear that the pyramid of highest levels of resentment belonged to Islam, and their resentment against America was high. I saw markings on the world map with arrows pointing from the Middle East to different parts of the world, and a Muslim name was attached to a line to America. I took a dive into the blocks of resentment which I had learned to transform into energy required for transformation of the human race to a Republic. I also saw last night that Islam was under attack by other metaphysical means that I deflected as my own path is to ensure that Zionist metaphysical power dissolve before Islam for a planetary Republic metaphysical order.
## Conversation about good and evil 2/21/2012
• Merlin Saint GermainThe “false light” of half truths is one of the great dangers that many fall prey to, unwittingly. Another great danger is unworthy sensual desires. Also, on the mental plane, the great lie of philosophical materialism, with all that it spawns – denial of spirituality, selfishness, ambition, greed, consumerism, the belief in drugs as a spiritual tool, and ruthlessness.
8 hours ago · Like · 4
• Zulf AhmedI do not agree. Political and social freedoms, to the extent they can be detected are manifestation of the true inner freedoms of our race.
6 hours ago · Like
• Merlin Saint GermainThat may be so, Zulf. But in order for people to bring into outer manifestation in society a true reflection of their inner values and beliefs, they need to participate in the social life and shape it through their initiative, not just stay at home eating, having sex, reading books, talking, meditating and watching the TV.
6 hours ago · Like · 2
• Zulf AhmedThat’s true, but at the same time, we need to have this happen globally, for which there are great obstructions. Either the entire race will have outer manifestation of inner freedom or the entire race will sink deeper into Hell that was begun by the Mossad operation of 9/11.
5 hours ago · Like · 1
• Merlin Saint GermainYes, so true! Humanity is a singular family, and our fate is one. Humanity needs to dust themselves off, quit drinking and taking drugs as much as possible so that they are in their normal sober state, call on their best selves, and set out to fix the world as best they can, both singly and collectively.
5 hours ago · Like · 1
• Zulf AhmedNow only if these pesky rulers of the world would stop being so obsessed with control and surveillance to use child porn as an excuse to push through their SOPA and PIPA efforts which failed while they point fingers at China and Iran as totalitarian censorship states.
4 hours ago · Like · 1
• Merlin Saint GermainI doubt that those pesky rulers will do anything but continue to step up their program for global fascistification. We can’t control them, but we can control ourselves and employ our faculties and gifts to put them back in their box and retake control over the affairs of humanity in the name of truth, justice, freedom, and goodness.
4 hours ago · Like · 1
• Zulf AhmedTruth, Justice, Freedom above good and evil!! Hear hear!!
4 hours ago · Like · 1
• Merlin Saint Germain
If we reject notions of duality, of truth, of righteousness, justice and goodness, then we are buying into the evil propaganda of the dark brothers who want us to be outside of the realm of goodness, and thus neutralised in the conflict bet…See More
3 hours ago · Like · 1
• Zulf Ahmed What is done out of love always takes place beyond good and evil, said Nietzsche and I cannot agree more.
https://zulfahmed.wordpress.com/2012/02/10/the-human-race-must-overcome-the-duality-of-good-and-evil/
3 hours ago · Like · 1 ·
• Merlin Saint GermainYour argument that the belief in good and evil is the cause of evil is not only false, but self contradictory.
3 hours ago · Like
• Zulf AhmedI did not say the belief in good and evil is the cause of evil. You should read the note I posted for what I am saying. The zealous “Good” are more likely to cause great evil.
3 hours ago · Like · 1
• Zulf AhmedI will give you a very clear example of this: around 54% of American Church-goers believe that it is right to torture suspected terrorists. This is not a trivial percentage.
3 hours ago · Like
• Merlin Saint Germain
The mistaken belief that evil is good, and good evil, has certainly been the cause of many problems in the world, but that is no reason to discard the concept of good and evil, or to try to somehow transcend it, as you seem to be suggestin…See More
3 hours ago · Like
• Zulf AhmedI adhere to the ideals of Freedom and Justice. I personally am beyond good and evil, but the ideal of Freedom tells me to respect your freedom to refuse to cross these lines.
3 hours ago · Like · 1
• Merlin Saint Germain
A person who does not strive to be righteous, moral, ethical, good and virtuous is very dangerous in society. You may call goodness freedom or justice, but if you believe that there is a better way to be that is in alignment with the social…See More
3 hours ago · Like
• Zulf Ahmed
You see, I believe that the human beings are metaphysically primarily the fallen angels in Hell at one level because I have seen the Hell manifest in my own eyes, have seen the suffering of our race. I accept my own darkness and my light a…See More
3 hours ago · Like
• Merlin Saint Germain
The reason that I made this comment is because a huge and seemingly rapidly growing number of people on earth are buying into belief systems that purport to be very profound and spiritual, but I think they are pernicious and also usually ha…See More
3 hours ago · Like · 2
• Zulf Ahmed
I am a minimalist in terms of spiritual constraints. The human race is a single race. Truth, Justice, and Freedom are the fundamental spiritual and simultaneously political — because there is no difference between collective spirituality…See More
zulfahmed.wordpress.com
I did not know about the US massacre of German soldiers in 1945 until I saw Davi…See More
2 hours ago · Like ·
• Merlin Saint Germain
I have a book detailing how in all likelihood the Allied murdered about 6 million Germans in retaliation for the Second World War, after the cessation of the formal hostilities. The Christians, such as the colonialists, the slave traders, t…See More
2 hours ago · Like
• Zulf Ahmed Well, I am certainly not a ‘moral relativist’. Justice is a much stronger ideal than Good in my own opinion because the polarity of good and evil inevitably creates conditions for literal physical conflicts between people. Inevitably there is a target evil, and inevitably the zealots are ready for murder.
Consider this:
Gaza, never forget! Video by Max Blumenthal and Dan Luban. Full story here: http://www.alternet.org/story/119372
2 hours ago · Like ·
• Merlin Saint GermainIf you state that justice is good, or that freedom is good, then you are invoking the concept of good and evil. If you refrain from saying that justice is good, or that freedom is good, or that truth is good, then you cannot speak up in favor of any of these. To me, your philosophy is self contradictory, since you refuse to acknowledge good and evil, yet you assert that justice and freedom are good.
2 hours ago · Like
• Zulf AhmedI did not say Justice is Good. Justice is a far stronger spiritual ideal than Good. Divine Love is the ideal of Justice.
2 hours ago · Like
• Merlin Saint GermainYou have just substituted the word “spiritual ideal” for good. They mean the same thing.
2 hours ago · Like
• Zulf AhmedNot at all. I am quite precise when I use the spiritual ideal. Let me repeat the dictum of Nietzsche which provides concreteness to this: What is done out of love always takes place beyond good and evil. I agree that the distinction seems subtle, but I am amoral when it comes to good and evil.
2 hours ago · Like
• Merlin Saint GermainYour philosophy seems absurd to me, Zulf. What is done out of love is often very evil, as when a person is deluded. Take the inquisition for example, and when Catholic parents beat their children out of the mistaken notion that this is good for them. In order for an action to be good, it must be motivated by goodwill, and informed by wisdom, and empowered by effectiveness.
17 minutes ago · Like
• Zulf AhmedWell, I consider people who believe in good and evil to be deluded, and the religions absurd. Catholic inquisition was a product of the good to eradicate evil, and it is unjust to cause physical harm to people.
14 minutes ago · Like
• Zulf Ahmed This is an interesting interview Noam Chomsky gave on terrorism and my philosophy would be in agreement with him in this instance: https://zulfahmed.wordpress.com/2012/02/13/noam-chomsky-interview-in-2010-about-terrorism/
## A conversation about American military and morality of nationalism
This is a basic conversation that brings out a number of issues that requires a coherent discussion regardless of how dispersed the topics seem, because it touches some of the faultlines of America and nationalism. Sometimes we get caught up in the numbers game on specific broad issues and this is useful but it is also important to quickly glimpse at a large picture because we are individuals beyond nation and ideologies and our individual freedom depends on being able to distance ourselves from a partisan view. I have myself been extremely critical of Zionism partly for political reasons, as I work towards a planetary Republic based on ideals of Truth, Justice, and Freedom and the condition of the Palestinians is in my view the central political/ideological conflict on the planet, but yet I will claim to be above nations and religions regardless.
• John TeubertThis is an interesting take on that statement, which I’m remembering being said by Bill Hicks. In his version, it was the realization needed to shed the fear that typically keeps otherwise decent people in line; keeps them from challenging the psychos at the top. Any phrase or text, it seems, can be perverted to justify the psycho’s behavior…
4 hours ago · Like · 1
• Mammatus CloudMilitary get the convoluted versions all the time…helps them rationalize and justifies the slaughter. Heard lots of first hand accounts from family…
4 hours ago · Like · 1
• Mammatus CloudNOT saying that “military” automatically equates into “low moral conscience”…but when surrounded by people who Do exhibit lack of one and having to rely upon them under fire, the pull of ‘group think’ programming can be overwhelming.
4 hours ago · Like · 1
• Zulf Ahmed So military in the US is trained to have ‘low moral conscience’ while in fact they may not have it before training:
Josh Stieber: In boot camp we trained with songs that joked about killing women and children
4 hours ago · Like ·
• Mammatus Cloud
That’s a broad brush you’re painting with Zulf. Stereotyping via group is a trap. All ‘isms’ depend upon the ability to overlook individual differences. There’s lots of differing reasons people join any group. Personally, I think groups bli…See More
4 hours ago · Like
• Zulf AhmedMammatus, military is the essence of dissolving individual. It’s a centrally controlled enterprise. This training is not a decision of individuals. Individual differences obviously exist but the military is not a democracy either.
4 hours ago · Like · 1
• Mammatus CloudWhy the suicide rate?
4 hours ago · Like · 1
• Mammatus CloudIf the training were so successful that the ‘individual’ were successfully ‘dissolved’….?
4 hours ago · Like · 1
• Mammatus CloudThe ability to regret, feel guilt, remorse…all Opposite of the psychopathic training. The suicide rate is direct evidence of failure in dissolving the ‘individual’ identity. Repressed or not.
4 hours ago · Like · 1
• Zulf Ahmed
Trapped slaves commit suicides because they don’t feel happy about what they are doing. Obviously the training is successful enough that there is not a mass defection from the US military, and disagreements are contained by control mechani…See More
4 hours ago · Like · 1
• Zulf AhmedNationalism is one of the key tools to keep the military in line, and racism is a key tool in imperial operations. Remember that 9/11 was a Mossad operation so a high level of propaganda reality is necessary for the military as well.
4 hours ago · Like
• Zulf Ahmed You can see the actual barbaric policy of US military in Iraq from a down-to-earth account by a veteran. A million mostly civilians were killed in Iraq:
Anderson joined the U.S. Army in January 2003 to get money for college and to se…See More
4 hours ago · Like ·
• Mammatus Cloud
”Trapped slaves commit suicides because they don’t feel happy about what they are doing. ” How many ‘trapped slaves’ committed suicide while plowing fields? Cleaning houses? Being raped repeated by ‘owners’? No, Zulf. Conscience has more t…See More
en.wikipedia.org
The Milgram experiment on obedience to authority figures was a series of notable…See More
4 hours ago · Like
• Zulf Ahmed Many more than you would expect. There were suicides in GCC countries because of exactly this reason:
Adam Hanieh: Saudi Arabia and the Gulf states created a super exploited migrant …See More
4 hours ago · Like · 1 ·
• Mammatus CloudCulture is the deciding factor there. Think about it…
4 hours ago · Like
• Zulf Ahmed
You do have a point that not liking slavery restrictions while doing oil work or construction work puts more pressure from the ‘trapped’ portion because there is no moral ambivalence in these jobs except the exploitation aspect, but not lik…See More
3 hours ago · Like · 1
• Mammatus Cloud
”So the propaganda tries to adjust this by heroic tales of how great their work is and how beneficial to people and the world, but problems occur when they see through the bullshit for what these operations really are about. What “defense …See More
3 hours ago · Like
• Zulf Ahmed
Well, I don’t have the figures for suicide rate of US military during the Second World War but I would suspect that it was not noticeable. So I highly doubt that people commit suicides simply because of wars and massacres, at least not for…See More
3 hours ago · Like
• Mammatus Cloud
Apply this from a Sociological / societal level : http://counsellingresource.com/lib/therapy/self-help/stockholm/ and then refer to my post above regarding self discrepancy. and consider the “waking up” from such a scenario with no where to…See More
counsellingresource.com
Page 1: If you’re in a controlling and abusive relationship, you may recognize s…See More
2 hours ago · Like
• Zulf Ahmed
So the problem with this sort of personal psychological evaluations are that then one could brand entire societies as psychopathic, which I think would be wrong. Consider the fact that Americans have not felt much remorse for the Hiroshima…See More
www.mtholyoke.edu
Source: Naval War College Review, Vol. XXVII (May-June, 1975), pp. 51-108. Als…See More
2 hours ago · Like ·
• Mammatus Cloud
”to explain the lack of interest in most Americans about these things.” Have you read what I posted above? What do you think societies are made of? INDIVIDUAL people. Zulf, it truly Isn’t that hard. Individuality ‘qwerks’ mean little when …See More
An evolutionary perspective on psychopaths in power. . Apart form the books/ art…See More
2 hours ago · Like
• Zulf Ahmed
Mammatus, societies are not simply made up of individuals. There is an identification of the individual with the nation, which is what nationalism amounts to. Most Americans are extremely insular, so the identification to nation is used f…See More
about an hour ago · Like
• Mammatus Cloud
The identification of the individual with their nation is no different and is simply an extension of the same process of the individual identifying with their family of origin. Again, survival. Ever try getting people to face ‘family of or…See More
about an hour ago · Like
• Zulf Ahmed
So while some of this is true, that there is a sort of tribalism in the national identification, although one has to wonder what ‘survival’ has to do with a military whose explicit policy is ‘full spectrum dominance’ of the globe, I would take exception to your considering this discrepancy between the individuals making the nation who are presumably sane and a leadership that is psychopathic. America was an empire from its birth, from George Washington’s own words, and this psychopathy that you are talking about is the nature of imperial power, and has been the case for every past empire. What is different about America from previous empires is that people have the illusion of popular voice in actual power, and that illusion of freedom is extremely carefully cultivated by a propaganda system. But you can see the disenfranchisement in the tighly controlled duality of neoliberal right parties dominating the “democratic politics” controlled by the major investment banks and lobbies like AIPAC. From my point of view Orwell wrote 1984 describing the National Security State that formed after the Second World War to have military control of the globe.
52 minutes ago · Like
• Zulf AhmedThis is an important conversation for the issues it brings out, so I’ll blog it.
a few seconds ago · Like
## Symbolic metaphysical experiences regarding Israel and Iran 2/19/12
The metaphysical drama of Israel and Iran has already been completed. I experienced the drama essentially as a motorcycle representing Israel that ran over me and then a truck representing Iran, hiding a Nazi Swastika that I dissolved. I experienced also, in a different metaphysical level Zionist power represented as an egg broken with a flag and the metaphysical level of the entire structure sunk deep underground — this was the karma punishment for the events from 9/11 on as well as the relation with Palestinians. Now we shall iterate and optimize so that these issues can be resolved in more refined manner before they ‘manifest’ in the physical world. We are also moving the human race beyond good and evil.
I am frankly no longer worried about an Israeli strike against Iran or a US invasion of Iran for regime change. Essentially this conflict has been resolved metaphysically by us. I would highly recommend that Israel take the opportunity to transition to a secular state with return of 1948 refugees and give Palestinians full rights. They should do this voluntarily and not worry about Iran.
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2017-02-24 01:26:24
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http://jobjo.github.io/2015/02/16/printing-recursive-data-structures.html
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I have more than once found myself in need of a function for pretty-printing some recursive data type; Be it a prefix search tree, an abstract syntax tree for a domain specific language, XML or something else. Getting tired of having to implement the same type of logic over and again I decided to generalize the pattern. In the following sections I discuss the design of a tiny library for addressing this problem. It’s not a particularly challenging task but provides a good opportunity to touch on a few different concepts in functional programming. Examples include deep and shallow embeddings, monoids and equational reasoning. The implementation is given in F#.
## The Problem
To give a motivating example, consider a (simplistic) type for representing XML data:
/// Attribute
type Attribute = string * string
/// XML is either a text-node or an element with attributes and children.
type XML =
| Text of string
| Node of string * list<Attribute> * list<XML>
An Attribute is a key-value pair and an XML node is either some text or an element with attributes and a list of children.
In order to render xml as a set of nested blocks with proper indentation levels, we wish to design a general purpose printing library.
## Deriving the API
As a first task we need to decide on a name for the thing we’re trying to abstract; I call it Printer. At the bare minimum we also need a function for evaluating a printer by turning it into a string:
/// Executes a printer, producing a string value.
val run : Printer -> string
Let’s think about how to produce printers. We should naturally provide a function for lifting simple string values:
/// Produces a printer that prints a string.
val print : string -> Printer
The interesting part is the ability to achieve nesting or indentation. The most intuitive way to express this is via a function that operates on a printer by indenting it one level:
// Indents a printer one level.
val indent : Printer -> Printer
Finally we need some means for composing printers; One way to achieve compositionality is to require that our printer type forms a monoid by introducing an empty printer along with a binary operator for combining two printers:
/// Doesn't print anything.
val empty : Printer
/// Combine two printers.
val add : Printer -> Printer -> Printer
You may wonder what good empty brings; One benefit is getting sequencing for free, that is the ability to combine a list of printers:
/// Sequences a list of printers.
let sequence : seq<Printer> -> Printer = Seq.fold add empty
With the above interface, printing nested documents is straight forward. Here is an example for manually outputting some HTML:
let htmlPrinter =
sequence [
print "<html>"
indent <|
sequence [
print "<body>"
indent <|
print "Printed"
print "</body>"
]
print "</html>"
]
With the expectation that running this printer:
printfn "%s" (run htmlPrinter)
produces output similar to:
<html>
<body>
Printed
</body>
</html>
Here is the complete set of combinators defining the API:
type Printer
/// Evaluates a printer.
val run : Printer -> string
/// Create a printer that prints the given string.
val print : string -> Printer
/// Indent a printer one level.
val indent : Printer -> Printer
/// The empty printer doesn't output anything.
val empty : Printer
/// Add two printers in a sequence.
val add : Printer -> Printer -> Printer
To ensure that the semantics is intuitive, there are a number of constraints that need to be satisfied by any particular implementation. I’ve identified the following ones:
1. print >> run = id
2. forall p: add p empty = p
3. forall p: add empty p = p
4. forall p1,p2,p3: add p1 (add p2 p3) = add (add p1 p2) p3
5. forall p1,p2,p3: indent (add p1 p2) = add (indent p1) (indent p2)
(1) states that run is the inverse of print, i.e. printing a string and then running it gives back the same string. (2) and (3) means that empty must be left and right identity for add which is required for Printer to form a monoid. (4) is also part of the monoid constraints and implies that add is associative. (5) states that indent is distributive over add; This is needed for safely being able to refactor expressions. Guaranteeing (2), (3) and (4) is necessary in order to provide intuitive semantics for sequence, for instance by ensuring that the following two printers are identical:
let pc1 = sequence [ p1; p2; p3 ]
let pc2 = sequence [p1 ; sequence [p2; p3]]
This is exactly why the monoid pattern is useful.
## A shallow embedding
To complete the library we now need to find a definition of the type Printer that allows for a feasible implementations of the required functions.
Thinking of the API as a small Embedded Domain Specific Language (EDSL), there are broadly speaking two implementation strategies available - Deep and shallow embeddings. Deep embeddings preserve the expression structure of the operations; This generally enables more optimization capabilities and also makes it possible to provide multiple interpreters. In a shallow embedding no intermediate data structure is used for building up expression trees, instead the semantics of an operation is part of its definition.
Let’s start by the simplest possible solution, not worrying about whether to support multiple interpreters or not. Applying the principal of Denotational Design, we need to precisely define what it means to be a Printer. A printer is something that has the ability to output a nested structure. Parameterizing over the choice of how to print a line given an indentation level, this can be represented by the following function:
type Printer = (int -> string -> string) -> string
In other words a printer is a black-box that when applied to a function from an indentation level and a string to string, returns a pretty-printed structure. All the information of what to output is contained within the closure of the function. The fact that it’s opaque, i.e. that it is not possible to peek inside a printer to find out how it was constructed, places it in the category of shallow embeddings.
Let’s see if the type is sufficient for implementing the interface. Starting with run:
let run (p: Printer) =
p <| fun n s ->
let space = String.Join("", List.replicate (n * 2) " ")
sprintf "%s%s\n" space s
Run simply invokes the printer with a function that indents each line with two spaces per indentation level.
Lifting a string into a printer is also straight forward:
let print s = fun ind -> ind 0 s
The function turns a string into a printer that invokes the indentation argument with level 0.
To implement indent we need to transform a printer into a new one that when executed invokes its given indent function with a greater indentation level:
let indent p = fun ind -> p (fun n -> ind (n + 1))
When it comes to empty we’re left with little choice but to output an empty string:
let empty = fun _ -> ""
Last one is add:
let add p1 p2 = fun ind -> p1 ind + p2 ind
Which simply runs both printers and concatenates their output.
We further need to ensure that the implementation is compatible with the semantical constraints.
First, (print >> run) must be equivalent with the identity function:
(print >> run) s =
// Definition of function composition:
run (print s) =
// Definition of print:
run (fun ind -> ind 0 s) =
// Definition of run:
(fun ind -> ind 0 s) (fun n s -> sprintf "%s%s" (space n) s) =
// Apply the arguments (beta reduction):
sprintf "%s%s" (space 0) s) =
// Definition of space and sprintf:
s
What about the monoid constraints for add and empty? Here is a proof for left identity (2):
add p empty =
fun ind -> p ind + empty ind =
// Definition of empty:
fun ind -> p ind + ((fun _ -> "") ind) =
// Beta reduction
fun ind -> p ind + "" =
// Empty string is identity of string concatenation:
fun ind -> p ind =
p
In fact, these properties follow from the monoid properties of string. The proof in the other direction is symmetric. We also need to show that add is associative:
add p1 (add p2 p3) =
// Definition of add on the outer argument:
fun ind -> p1 ind + ((add p2 p3) ind) =
// Definition of add on inner argument:
fun ind -> p1 ind + ((fun ind -> p2 ind + p3 ind) ind) =
// Beta reduction:
fun ind -> p1 ind + p2 ind + p3 ind =
// Associativity of string concatenation:
fun ind -> (p1 ind + p2 ind) + p3 ind =
Again, the proof relies on the associativity of string concatenation.
Finally we need to show that indent is distributive according to (5). This follows directly from the definitions of the two functions:
indent (add p1 p2) =
// Definition of indent:
fun ind -> (add p1 p2) (fun n -> ind (n + 1)) =
fun ind -> (fun ind -> p1 ind + p2 ind) (fun n -> ind (n + 1)) =
// Beta reduction:
fun ind -> (p1 (fun n -> ind (n + 1)) + p2 (fun n -> ind (n + 1))) =
// Definition of indent:
To wrap it up, below is the complete listing of the implementation. I made the definition of Printer private, added a function for running printers with a custom indentation parameter and included operator aliases for sequence print, and add:
open System
/// A printer is a function from an indentation level to a list of strings.
type Printer = private { Run : (int -> string -> string) -> string}
/// Creates a printer.
let private mkPrinter f = {Run = f}
/// Creates a string of whitespace of the given length.
let private space n = String.Join("", List.replicate (n * 2) " ")
/// Create a printer from a string.
let print s = mkPrinter <| fun ind -> ind 0 s
/// Indents a printer.
let indent p = mkPrinter <| fun ind -> p.Run ((+) 1 >> ind)
/// Runs a printer returning a string.
let runWith ind p = p.Run ind
/// Runs a printer returning a string.
let run = runWith <| fun n -> sprintf "%s%s\n" (space n)
/// An empty printer.
let empty = mkPrinter <| fun _ -> ""
let add tp1 tp2 = mkPrinter <| fun ind ->
tp1.Run ind + tp2.Run ind
/// Concatenates a sequence of printers.
let sequence = Seq.fold add empty
/// Short for sequence.
let (!<) = sequence
/// Short for print.
let (!) = print
Returning to the motivational example of printing XML, here is a complete implementation of a show function for the XML type:
/// Attribute
type Attribute = string * string
/// XML is either a text-node or an element with attributes and children.
type XML =
| Text of string
| Node of string * list<Attribute> * list<XML>
/// Pretty-prints an xml value.
let show =
let showAttrs attrs =
let showAttr (n,v) = sprintf " %s=%s" n v
String.Join("", List.map showAttr attrs)
let rec show = function
| Text t ->
!t
| Node (name,atrs,chs) ->
!<[
!(sprintf "<%s%s>" name (showAttrs atrs))
indent !<(List.map show chs)
!(sprintf "</%s>" name)
]
show >> run
Hopefully the example is straight forward to follow. In case you don’t like the prefix operators, you could change the definition to use sequence instead of !<, print instead of !.
## A deep embedding
A deep embedding must preserve the structure of how a printer is assembled. This is required whenever you need to support multiple back-ends or different ways of interpreting expressions. Creating a data type for a deep embedding is straight forward, we basically just need to list the distinct language constructs. The following type will do:
// Deep embedding of printer type.
type Printer =
| Empty
| Print of string
| Indent of Printer
| Add of Printer * Printer
In this way, all operations are trivial. Here are the functions mirroring the constructors:
// Empty printer.
let empty = Empty
// Print a string.
let print = Print
// Indent a printer.
let indent p = Indent p
// Composing two printers.
match p1,p2 with
| Empty, p
| p, Empty -> p
The only interesting part is add which contains an optimization step for implementing left and right identity for empty in accordance with the specified semantics. All the work of evaluating a printer is pushed to the interpretors, in our case a function for constructing a string. Here are the equivalent runWith and run functions:
// Executes a printer.
let runWith ind p =
let sb = new Text.StringBuilder()
let rec go n = function
| Empty -> ()
| Print l -> ignore <| sb.AppendLine (ind n l)
| Indent p -> go (n+1) p
| Add (p1,p2) -> go n p1 ; go n p2
go 0 p
sb.ToString()
/// Runs a printer returning a string.
let run = runWith <| fun n -> sprintf "%s%s\n" (space n)
Each language construct is handled separately with Indent and Add traversing their arguments recursively. A Text.StringBuilder object is used to accumulate the output of printed lines in order to improve on efficiency.
What about the semantics, how do we prove that the definition is compatible with the constraints listed above? What we really need to check is the validity of expressions with respect to a particular interpreter (in this case run). For instance looking at constraint (4) concerning associativity of add: (add
p1 (add p2 p3) = add (add p1 p2) p3), we’re not interested in whether these expressions are identical or not; Only that they produce the same output for a given interpreter. However, whenever we are able to show that two expression are in fact identical it naturally follows that all possible interpretations are identical.
Showing that constraints (1) and (4) holds is similar to the example above. For left and right identity (2,3) it’s possible to leverage the definition of add canceling out Empty values, but only in case the type constructors are hidden in order to rule out the construction of values such as (Add (Print "Hello"), Empty). This fact introduces a subtle problem; On the one hand exposing the printer type is necessary for allowing different interpretors to be defined. On the other hand, providing access to the constructors removes the control over how values are constructed. One solution would be to expose the core definitions in a separate module.
Using equational reasoning is slightly more complicated given the imperative style of the runWith function. The complete proofs are left as an exercise.
Another approach is to provide a mapping from the deep to the shallow embedding. Assuming a module ShallowPrinter containing the shallow implementation from above, here is a function for performing the translation along with runWith function:
module SP = ShallowPrinter
/// Transforms a deep printer to a shallow one.
let rec toShallow = function
| Empty -> SP.empty
| Print p -> SP.print p
| Indent p -> SP.indent (toShallow p)
/// Reusing the interpretor from the shallow embedding.
let runWith ind = toShallow >> SP.runWith ind
/// Runs a printer returning a string.
let run = runWith <| fun n -> sprintf "%s%s\n" (space n)
Now, all proofs concerning the shallow implementation can be safely reused in order to show that the constraints are fulfilled for this definition of run.
Here is the a complete listing of a stand-alone deep embedding:
open System
// Deep embedding of printer type.
type Printer =
| Empty
| Print of string
| Indent of Printer
| Add of Printer * Printer
/// Creates a string of whitespace of the given length.
let private space n = String.Join("", List.replicate (n * 2) " ")
// Empty printer.
let empty = Empty
// Print a string.
let print = Print
// Indent a printer.
let indent p = Indent p
// Composing two printers.
match p1,p2 with
| Empty, p
| p, Empty -> p
// Execute a printer.
let runWith ind p =
let sb = new System.Text.StringBuilder()
let rec go n = function
| Empty ->
()
| Print l ->
ignore <| sb.AppendLine (ind n l)
| Indent p ->
go (n+1) p
go n p1
go n p2
go 0 p
sb.ToString()
/// Runs a printer returning a string.
let run = runWith <| fun n -> sprintf "%s%s\n" (space n)
/// Concatenates a sequence of printers.
let sequence ps = Seq.fold add empty ps
/// Short for sequence.
let (!<) = sequence
/// Short for print.
let (!) = print
At last, to illustrate that it is in fact possible to define alternative interpretors of printer expressions, consider the following example that given a printer, generates F# code for printing the expression itself:
let showFSharp : Printer -> string =
let rec show = function
| Empty ->
!"Empty"
| Print s ->
!(sprintf "Print \"%s\"" s)
| Indent p ->
!<[
!"Indent ("
indent (show p)
!")"
]
!<[
indent (show p1)
indent !","
indent (show p2)
!")"
]
show >> run
Using this function on the initial htmlPrinter example, we actually retrieve and equivalent F# expression for recreating the printer:
// Define a custom printer.
let htmlPrinter =
sequence [
print "<html>"
indent <|
sequence [
print "<body>"
indent <|
print "Printed"
print "</body>"
]
print "</html>"
]
printfn "%s" (showFSharp htmlPrinter)
Yielding the output:
Add (
Print "<html>"
,
Indent (
Print "<body>"
,
Indent (
Print "Printed"
)
)
,
Print "</body>"
)
)
)
,
Print "</html>"
)
Which indeed valid F#. This would not have been possible using the shallow embedding.
## Summary
In this post I’ve addressed the problem of designing a library for pretty-printing recursive data structures. The approach taken is general and starts by identifying a minimal set of required operations needed and then for each operation define a set of constraints (or laws) that any implementation must obey. In order support compositionally of printers the monoid pattern was used.
Finally, two different realizations of the library was given, one shallow and one deep embedding. By using equational reasoning we were able to show that the semantical constraints were satisfied.
|
2020-10-01 08:55:33
|
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|
https://math.stackexchange.com/questions/1978673/stamp-induction-problem
|
Stamp Induction problem
Suppose you have an unlimited supply of 5-cent postage stamps and 7-cent postage stamps. Show that you can make any amount of postage which is 24 cents or larger using only these stamps.
• This is the coin problem. You can search the site for many variations. – Ross Millikan Oct 21 '16 at 15:27
First of all, you can realize : $24$ cents postage : $2\times5+2\times7=24$.
Now let $N\ge24$ be a integer for which there exists a solution : $N=n_5\times5+n_7\times7$. Let's try to realize a postage of $N+1$ cents.
There is basically two ways to add $1$ cent : $3\times5-2\times7=1=3\times7-4\times5$. Problem is : you have to subtract some $5$ cents or $7$ cents stamps.
Suppose now that $n_5<4$, so that we can't add three $7$ cents stamps and remove four $5$ cents stamps.
As $N\ge24$ and $n_5\le3$, $N-n_5\times5\ge 24-3\times5=9$, so there is at least two $7$ cents stamps, and we can use the other solution : remove two $7$ cents stamps and add three $5$ cents stamps.
In all cases, we can find a solution for the $N+1$ postage problem.
• Seems like you forgot to finish a sentence. I guess you could finish it with... so there is at least 9/7 7c stamps (at least 2 of them). – Myridium Oct 21 '16 at 15:58
• Sorry, a misplaced "<" instead of a "w" :-) – Nicolas FRANCOIS Oct 21 '16 at 16:46
|
2021-01-26 13:43:13
|
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|
http://math.stackexchange.com/questions/194452/modular-multiplicative-inverse-and-coprime-numbers-needed
|
# Modular multiplicative inverse and coprime numbers needed.
I have a 64 bit algorithm that uses modular multiplicative inverse and coprime numbers, and I need to convert it to 32 bit. This math is not my area, and I cannot find an online calculator, so I hope someone here can help me.
In 64 bit, I have:
multiplier_C = 6364136223846793005;
multiplier_inv_C = -4568919932995229531;
offset_C = 1442695040888963407;
These are from Knuth's linear congruential generator. I need the following:
multiplier_C must be 30 bits in size, and multiplier_inv_C must be its modular multiplicative inverse.
offset_C must be 29 or 30 bits in size, and a coprime of $2^{32}$.
-
Take a look at Rick Regan's blog: exploringbinary.com – Rod Carvalho Sep 12 '12 at 1:55
|
2014-03-16 18:06:51
|
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|
http://wikidobia.info/wikidobia/doku.php?id=start&do=search&q=need
|
Search
You can find the results of your search below.
Fulltext results:
quotes @dob
If you have built castles in the air, your work need not be lost; that is where they should be. Now pu... Hunter S. Thompson (rum diaries) "People have a need for structured time" Eric Berne - Games people pl... structured thought : guide to sanity... geezers need excitement, if their lives don't provide them thi... ight you, then you win Gandhi She doesn't need a lover, she needs a case worker... future advan
financial.independence @calculate
off the income of your investments and no longer need to depend on income from your labour. Colloquiall... m]] ==== How much capital do I need? ==== <html> <head> <script language="JavaScript... td> % </td> <tr> <td> <INPUT NAME="calc" VALUE="Need" TYPE=BUTTON onClick=compound(this.form)> </td><... c{savings}{Income} This is the only parameter need to know how long it will take you to reach financ
body_mass_index @calculate
. This is the baseline for the calorie intake you need to maintain your weight. This can be estimated ... reduction needed to achieve weight loss? ==== You need calorie deficit of **7000Cal to reduce 1kg** in weight.\\ => to loose 1k / week, you need to cut back 1000Cal/day (~half normal consumption
heat to 180°C after 15 min.\\ This bread does not need to cool as long as the Graubrot :-D\\ \\ \\ //Ann... idge in a sealed container and feed again, if you need sour dough - you have friend Hermann as a new hou... :-) \\ For the next bread with Hermann you won't need yeast any more.\\ \\ :-o Warning: this recipe tak
beam.robotics @dob:tech
Capacitor charged, then discharge). For this you need a TC54**VC** [[beam.components| component]] As I... ergy to drive the motor, the trigger voltage will need to be higher than the 0.7V set by a single diode:... t current is leaking at close to trigger point => need to have reasonably high light level to quickly go
sweet @eat
balm. Taste the batter - with olive oil you might need a bit more vanilla, lemon juice and sugar.\\ Pour... ar with vanilla flavour \\ \\ For the topping you need: \\ **2 tins of pineapple, in rings or pieces** \... e tin it'll take longer). \\ \\ For the icing you need\\ **1/2l pineapple juice (the one from the tins)
getquote @dob:tech
=GETHISTORY("PETR4.SA",1,TODAY()-1) (you might need to use semicolons instead of commas) and then pre... ay can be done with the INDEX function. You don't need to use ctrl+shift+enter for it to work as it does
workbench @dob
bench so you can push it up from below the bench Need a hole all the way through the bench (else can us... https://woodandshop.com/which-hand-tools-do-you-need-for-traditional-woodworking/ for tools https://w
rubberbandits @dob
swapped him for a bag of yokes in 1992\\ I don't need insurance, I don't need no parkin space\\ and if you try to clamp my horse he'll kick you in the face\\
syntax @wiki
not supported. ===== No Formatting ===== If you need to display text exactly like it is typed (without... or ''%%<php>%%'' tags. (Use uppercase tags if you need to enclose block level elements.) HTML example:
5_to_help_paint_your_own_house
pg What type of paint an individual using? Anyone need on this a special brand or type of paint in order... ny factors that play into the kind of paint these need, materials you must have to apply the paint, ladd
finding_position_web_host
rator associated with the page you own, so do not need need any access than this functionality allows. It is possible to still post comments in reply to the i
2_issues_when_planning_an_exterior_home_painting
ome. They know exactly which equipment they go to need during task of work of art. So you do not require... factors that play into the kind of paint you will need, materials you must have to apply the paint, ladd
eat @eat
not add soy sauce as recommended. Felt it didn't need it. Cut beef into think long strips and drop int
emacs_macros @blog
s environment. Then again, the amount of time you need to invest to reach that level of proficiency is e
monetizing_positive_impulses @blog
sport @sport
the_secret_life_of_walter_mitty @dob
start
|
2019-11-13 07:43:18
|
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|
https://stats.stackexchange.com/questions/10897/probability-distribution-of-fragment-lengths
|
# Probability distribution of fragment lengths
I would like to compute the probability distribution for the length of the fragments which I would obtain by fragmenting a linear rod of length $L$ in the following way:
1. I choose at random (uniformly) $n$ breakpoints
2. I cut the rod at those breakpoints, creating $(n+1)$ fragments.
Now, while it is easy to see that the probability that a stretch of length $x$ does not contain any breakpoint goes like a negative exponential, I don't know how to throw in the information about the length of the rod.
• You can choose your units of measurement--meters, yards, parsecs, whatever--without materially changing the problem. So, choose a unit in which the rod has length 1. Done! :-) – whuber May 17 '11 at 16:25
• @whuber, thanks but I am not yet there. Basically I get a negative exp by computing the product of many small steps, each of them with probability (1-p), in the limit where the length of the step is very small. Now, how do I introduce in this computation the fact that I can't overstep the "right end" of the rod? – XenophiliusLovegood May 17 '11 at 17:02
• @X I'm a little lost, because I don't see how you are making a connection between the "many small steps" and the situation you have presented. – whuber May 17 '11 at 17:25
• @X I also wonder about your "negative exponential" assertion. Fixing a segment of length $x$ in a unit rod (without any loss of generality), the chance that any single breakpoint misses it is $1-x$. Because the breakpoints are independent, the chance that all of them miss it is $(1-x)^n$. That's not a negative exponential: it's a polynomial in $x$. Perhaps you're thinking of an asymptotic characterization for small $x$ and large $n$ (and bounded $nx$)? With those asymptotics the rod's length is arbitrarily large compared to $x$ and therefore should not play any role. – whuber May 18 '11 at 4:35
• @whuber I was obtaining the probability that no breakpoint falls in a segment of length $x$ by $(1-x/M)^{nM}$, from which an exponential when $M$ goes to infinity. So, I divide $x$ in $M$ parts, and multiply the probabilities that each of these parts is not hit by a breakpoint. – XenophiliusLovegood May 18 '11 at 10:20
## 2 Answers
Let the rod have length $L$ and fix a segment of length $x$. The chance that any single breakpoint misses the segment equals the proportion of the rod not occupied by the segment, $1−x/L$. Because the breakpoints are independent, the chance that all of them miss it is the product of $n$ such chances, $(1 - x/L)^n$.
From comments following the question, it appears that $x$ is intended to be small compared to the rod's length: $x/L \ll 1$. Let $\xi = L/x$ (assumed to be large) and rewrite $n = \xi(n/\xi)$, leading (purely via substitutions) to
$$\Pr(\text{all miss}) = (1 - x/L)^n = (1 - 1/\xi)^{\xi(n/\xi)} = \left((1-1/\xi)^\xi\right)^{n/\xi}\text{.}$$
Asymptotically $\xi \to \infty$. If we assume that $n$ varies in a way that makes $n/\xi$ converge to a constant, this probability approaches a computable limit. Let this constant be some value $\lambda$ times $x$. It is the limiting value of $n/\xi/x = n/L$: notice how the length of the rod is involved here and effectively is incorporated in $\lambda$. Because $\exp(-1) = 1/e$ is the limiting value of $(1-1/\xi)^\xi$ and raising to (positive) powers is a continuous function, it follows readily that the limit is
$$\Pr(\text{all miss}) \to e^{-\lambda x}.$$
One application is when $n$ is a constant, entailing $\lambda = n/L$, and $x \ll L$. We obtain $$e^{-nx/L}$$ as a good approximation for the probability that all breaks miss the segment. This analysis shows that the approximation fails as $x$ grows large: the approximation is only as good as the approximation $1/e \sim (1-1/\xi)^\xi$. Finally, if you set $x = L$, the approximation is clearly wrong because it gives $e^{-n}$ instead of the correct answer, $0$.
• thanks for your answer. I have one residual doubt. My original quest was to find a pdf for the lengths of the fragments. now, if I use $(1-\frac{x}{L})^n$, I run into troubles. First, (let's assume $L=1,n=1$) it is not normalized $\int_0^1 (1-x)\, dx=\frac{1}{2}$. Secondly, imagine I want to compute the average length of a fragment. That'd be $\int_0^1 x(1-x)\, dx=\frac{1}{6}$, which I don't understand either. What's your take? – XenophiliusLovegood May 21 '11 at 11:42
• @X We're not computing a probability distribution here: $x$ is a fixed value, not a random quantity! (See the first line of my reply.) Thus, neither the normalization nor the expectation make any sense at all. The question I answered is the one you asked in comments: "I was obtaining the probability that no breakpoint falls in a segment of length $x$". Your original question asks for a "probability distribution for the length of the fragments...." Because there will be $n+1$ fragments, you are looking for an $n+1$-variate distribution for all the lengths. – whuber May 21 '11 at 19:33
• I don't understand well your last comment. On the other hand, I think I possibly got to a satisfactory conclusion: $(1-x)^n$ is the cdf of the pdf I am looking for (in the original question). The corresponding pdf is $n(1-x)^{n-1}$ which is normalized and has nice expectation values. – XenophiliusLovegood May 23 '11 at 10:44
Let $\{X_i\}$ be the locations of the cuts.
I'd approach this problem by finding the order statistics $\{Y_i\}$ so that $Y_1$ would be the location of the leftmost cut. Then I'd calculate the probability distributions of the differences between the variables $Y_i-Y_{i-1}$. Don't forget to also calculate $Y_1-0$ and $L-Y_n$.
Can anyone think of a better way?
• Yes: First close the rod into a circle. Then break it (uniformly, randomly, independently) at $n+1$ breakpoints. This introduces a helpful symmetry :-). – whuber May 17 '11 at 16:24
• Or, draw a connection to the Poisson process. – cardinal May 17 '11 at 16:46
|
2020-10-25 07:58:39
|
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|
https://www.groundai.com/project/fermion-interactions-and-universal-behavior-in-strongly-interacting-theories/
|
A Conventions
# Fermion Interactions and Universal Behavior in Strongly Interacting Theories
## Abstract
The theory of the strong interaction, Quantum Chromodynamics (QCD), describes the generation of hadronic masses and the state of hadronic matter during the early stages of the evolution of the universe. As a complement, experiments with ultracold fermionic atoms provide a clean environment to benchmark our understanding of dynamical formation of condensates and the generation of bound states in strongly interacting many-body systems.
Renormalization group (RG) techniques offer great potential for theoretical advances in both hot and dense QCD as well as many-body physics, but their connections have not yet been investigated in great detail. We aim to take a further step to bridge this gap. A cross-fertilization is indeed promising since it may eventually provide us with an ab-initio description of hadronization, condensation, and bound-state formation in strongly interacting theories. After giving a thorough introduction to the derivation and analysis of fermionic RG flows, we give an introductory review of our present understanding of universal long-range behavior in various different theories, ranging from non-relativistic many-body problems to relativistic gauge theories, with an emphasis on scaling behavior of physical observables close to quantum phase transitions (i. e. phase transitions at zero temperature) as well as thermal phase transitions.
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Fermion Interactions and Universal Behavior in Strongly Interacting Theories] Fermion Interactions and Universal Behavior
in Strongly Interacting Theories
Jens Braun]Jens Braun 1
## 1 Introduction
Strongly interacting fermions play a very prominent role in nature. The dynamics of a large variety of theories close to the boundary between a phase of gapped and ungapped fermions is determined by strong fermion interactions. For instance, the chiral finite-temperature phase boundary in quantum chromodynamics (QCD), the theory of the strong interaction, is governed by strong fermionic self-interactions. In the low-temperature phase the quark sector is driven to criticality due to strong quark-gluon interactions. These strong gluon-induced quark self-interactions eventually lead to a breaking of the chiral symmetry and the quarks acquire a dynamically generated mass. The chirally symmetric high-temperature phase, on the other hand, is characterized by massless quarks. The investigation of the QCD phase boundary represents one of the major research fields in physics, both experimentally and theoretically. Since the dynamics of the quarks close to the chiral phase boundary affect the equation of state of the theory, a comprehensive understanding of the quark dynamics is of great importance for the analysis of present and future heavy-ion collision experiments at BNL, CERN and the FAIR facility [1].
While heavy-ion collision experiments provide us with information on hot and dense QCD, experiments with ultracold trapped atoms provide an accessible and controllable system where strongly-interacting quantum many-body phenomena can be investigated precisely. In contrast to the theory of strong interactions, the interaction strength can be considered a free parameter in these systems which can be tuned by hand. In fact, the interaction strength is directly proportional to the -wave scattering length and can therefore be modulated via an external magnetic field using Feshbach resonances [2]. It is therefore possible to study quantum phenomena such as superfluidity and Bose-Einstein condensation in these systems. From a theorist’s point of view, this strong degree of experimental control opens up the possibility to test non-perturbative methods for the description of strongly interacting systems.
Phases of ultracold Fermi gases at zero and finite temperature have been studied experimentally, see e. g. Refs. [3, 4, 5, 6, 7] as well as theoretically, see e. g. Refs. [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27], over the past few years. In particular, studies with renormalization group (RG) methods exhibit many technical similarities to studies of QCD at finite temperature and density, see e. g. Refs. [28, 29, 30, 31, 32, 33]. Physically, in both cases the phase boundary is determined by strong interactions of the fermions. While the asymptotic limits of the phase diagram of ultracold atoms for small positive and small negative (s-wave) scattering length associated with Bose-Einstein condensation and Bardeen-Cooper-Schrieffer (BCS) superfluidity [34], respectively, are under control theoretically [35, 36, 37, 38, 39], our understanding of the finite-temperature phase diagram in the limit of large scattering length (strong-coupling limit) is still incomplete [8, 10, 11, 14, 25, 27].
Aside from phase transitions at finite temperature, experiments with ultracold fermionic atoms provide a very clean environment for studies of quantum phase transitions. Experiments with a dilute gas of atoms in two different hyperfine spin states have been carried out in a harmonic trap at a finite spin-polarization [3, 4]. Since there is effectively no spin relaxation in these experiments, in contrast to most other condensed matter systems, the polarization remains constant for long times. Deforming the system by varying the polarization allows us to gain a deep insight into BCS superfluidity and its underlying mechanisms [34]. Originally, BCS theory has been worked out for systems in which the Fermi surfaces of the two spin states are identical, i. e. the polarization of the system is zero. As a function of the polarization, a quantum phase transition occurs at which the (fully polarized) normal phase becomes energetically more favorable than the superconducting phase [17, 20, 22]. After giving a thorough introduction on the level of (advanced) graduate students to the derivation and analysis of fermionic RG flow equations2 in Sects. 2 and 3, we shall discuss aspects of symmetry breaking and condensate-formation in non-relativistic theories from a RG point of view in Sect. 4.1. For simplicity, we restrict ourselves to systems with a vanishing polarization. The generalization of our RG approach to spin-polarized gases is straightforward and has been discussed in Refs. [23, 40].
Phase separation between a superfluid core and a surrounding normal phase has been indeed observed in experiments with an imbalanced population of trapped spin-polarized atoms at unitarity at MIT and Rice University [3, 4]. The density profiles measured in these experiments prove the existence of a skin of the majority atoms. A critical polarization associated with a quantum phase transition has been found in both the MIT and Rice experiment. Aside from studies at zero temperature, finite-temperature studies of a spin-polarized gas have been performed at Rice University [4]. In these experiments the critical polarization, above which the superfluid core disappears, has been measured as a function of the temperature. In accordance with theoretical studies [15, 18, 23], the results from the Rice group suggest that a tricritical point exists in the phase diagram spanned by temperature and polarization, at which the superfluid-normal phase transition changes from second to first order as the temperature is lowered. Depending on the physical observable, it is in principle possible that finite-size and particle-number effects are visible in the experimental data. Concerning the critical polarization, such effects have been studied in Ref. [41].
There is indeed direct evidence that finite-size effects can alter the phase structure of a given theory. For example, Monte-Carlo studies of the Gross-Neveu model show that the finite-temperature phase diagram of the uniform system is modified significantly due to the non-commensurability of the spatial lattice size with the intrinsic length scale of the inhomogeneous condensate [42, 43]. In particular, the phase with an inhomogeneous ground state shrinks. Such commensurability effects may be present in trapped ultracold Fermi gases as well. Since the Gross-Neveu model in is reminiscent of QCD in many ways, the existence of a stable ground state governed by an inhomogeneous condensate is subject of an ongoing debate, see e. g. Refs. [44, 45]. In any case, it is well-known that the mass spectrum and the thermodynamics of QCD has an intriguing dependence on the volume size and the boundary conditions of the fields, see e. g. Refs. [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57].
Our theoretical understanding of the phase structure of trapped fermions is currently mostly based on Density Functional Theory (DFT) [58] in a local density approximation (LDA) in which, for example, derivatives of densities are omitted in the ansatz for the action, see e. g. Refs. [18, 16, 19, 24, 26]. From a field-theoretical point of view, DFT corresponds to a mapping of the (effective) action of a fermionic theory onto an action which depends solely on the density. The latter then plays the role of a composite degree of freedom of fermions. Thus, the underlying idea is reminiscent of the Hubbard-Stratonovich transformation [59, 60] widely used in low-energy QCD models and spin systems. In any case, the introduction of an effective degree of freedom, such as the density, turns out to be advantageous for a description of theories with an inhomogeneous ground-state. Again, experiments with ultracold atoms allow us to test different approaches and approximation schemes. In Refs. [16, 24], the equation of states of the superfluid and the normal phase of a uniform system have been employed to construct a density functional which allows to study the ground-state properties of trapped Fermi gases. Such a procedure corresponds to an LDA. While there is some evidence that Fermi gases in isotropic traps can be quantitatively understood within DFT in LDA [24], the description of atoms in a highly-elongated trap in LDA seem to fail and derivatives of the density need to be taken into account [61, 62]. In the spirit of these studies, we shall discuss a functional RG approach to DFT in Sect. 4.2 which relies on an expansion of the energy density functional in terms of correlation functions and allows to include effects beyond LDA in a systematic fashion.
Heavy nuclei combine aspects of dense and hot QCD and systems of ultracold atoms. We again need to describe strong interactions between fermions, the nucleons, which form a stable bound state depending on, e. g., the number of protons and neutrons. These interactions are repulsive at short range and attractive at long range as in the case of ultracold atomic gases. Loosely speaking, heavy nuclei can be viewed as spin-polarized systems of two fermion species comparable to those systems studied in experiments with trapped spin-polarized atoms at MIT and Rice University [3, 4]. In fact, almost all nuclei have more neutrons than protons.3 Therefore density profiles of protons and neutrons in heavy nuclei are evocative of the profiles associated with the two fermion states in experiments with ultracold atoms. For heavy nuclei, DFT remains currently to be the only feasible approach for a calculation of ground-state properties associated with inhomogeneous densities. State-of-the-art density functional approaches are essentially based on fitting the parameters of a given density functional such that one reproduces the experimentally determined values of the ground-state properties of various heavy nuclei [63, 64, 65]. These density functionals are then employed to describe ground-state properties of other heavy nuclei. As mentioned above, we shall briefly discuss an RG approach to DFT in Sect. 4.2, which opens up the possibility to study ground-state properties of heavy nuclei from the underlying nucleon-nucleon interactions. Such an ab-initio DFT approach might prove to be useful for future studies of ground-state properties of (heavy) self-bound systems.
Hot and dense QCD, ultracold atoms and nuclear physics represent just three examples for systems in which the dynamics are governed by strong fermion interactions. Of course, the list can be extended almost arbitrarily. In the context of condensed-matter theory, we encounter systems such as so-called high- superconductors. In this case the challenge is to describe reliably the dynamics of electrons at finite temperature in an ambient solid-state system. The so-called Hubbard model provides a theory to describe these superconductors [66, 67] and has been extensively studied with renormalization-group techniques, see e. g. Refs. [68, 69, 70, 71]. It is worth noting that both the mechanisms as well as the techniques are remarkably similar to the ones in renormalization-group studies of gauged fermionic systems interacting strongly via competing channels [30, 31, 32], such as QCD, and of imbalanced Fermi gases in free space-time [23]. In Sect. 5, we discuss more general aspects of (non-gauged) Gross-Neveu- and Nambu-Jona-Lasinio-type models which also exhibit technical similarities to studies of condensed-matter systems. Nambu-Jona-Lasinio-type models are widely used as effective QCD low-energy models. On the other hand, Gross-Neveu-type models have been employed as toy models to study certain aspects of the QCD phase diagram but they are also related to models in condensed-matter theory, e. g. to models of ferromagnetic (relativistic) superconductors [72, 73]. In this review, we shall use Gross-Neveu- and Nambu-Jona-Lasinio-type models to discuss dynamical chiral symmetry breaking (via competing channels) and the role of momentum dependences of fermionic interactions.
In addition to fermion dynamics at finite temperature, quantum phase transitions play a prominent role in condensed-matter theory, e. g., in the context of graphene. Effective theories of graphene, such as QED and the Thirring model, are expected to approach a quantum critical point when the number of fermion species, namely the number of electron species, is varied [74, 75]. RG studies of these effective theories, see e. g. Refs. [74, 76, 75], are closely related to studies of quantum phase transitions in QCD [77, 29, 30, 31, 78, 79]. Similar to the situation in QED, a quantum phase transition from a chirally broken to a conformal phase is expected in QCD when the number of (massless) quark flavors is increased. Studies of the dependence on the number of fermion species seem to be a purely academic question. Depending on the theory under consideration, however, such a deformation of the theory may allow us to gain insights into the dynamics of fermions close to a phase boundary in a controlled fashion. For example, the gauge coupling in QCD becomes small when the number of quark flavors is increased and therefore perturbative approaches in the gauge sector become meaningful. Moreover, an understanding of strongly-flavored QCD-like gauge theories is crucial for applications beyond the standard-model, namely for so-called walking technicolor scenarios for the Higgs sector [80, 81, 82, 83, 84, 85, 86, 87, 88]. In Sect. 6, we shall discuss chiral symmetry breaking in gauge theories with fermion flavors. In particular, we shall present a detailed discussion of the scaling behavior of physical observables close to the quantum phase transition which occurs for large .
Our discussion shows that systems of strongly interacting fermions play indeed are very prominent role in nature and that their dynamics determine the behavior of a wide class of physical systems with seemingly substantial differences. However, our discussion also shows that the underlying mechanisms of symmetry breaking and the applied techniques are very similar in these different fields. Therefore a phenomenological and technical cross-fertilization offers great potential to gain a better understanding of the associated physical processes. As outlined, examples include an understanding of the dynamical generation of hadron masses as well as of the dynamical formation of condensates and bound-states in ultracold gases from first principles. The main intent of the present review is to give a general introduction to the underlying mechanisms of symmetry breaking and bound-state formation in strongly-interacting fermionic theories. In particular, we aim to give a thorough introduction into the scaling behavior of physical observables close to critical points, ranging from power-law scaling behavior to essential scaling. As a universal tool for studies of quantum field theories we employ mainly Wilsonian-type renormalization-group techniques [89, 90, 91, 92, 93, 94, 95]. For concrete calculations we shall use the so-called Wetterich equation [95] which we briefly introduce in the next section. Reviews focussing on various different aspects of renormalization-group approaches can be found in Refs. [96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].
## 2 Renormalization Group - Basic Ideas
We begin with a brief introduction of the basic ideas of RG approaches including a discussion of the Wetterich equation. The latter describes the scale dependence of the quantum effective action which underlies our studies in this and the following sections.
In perturbation theory, the correlation functions of a given quantum field theory contain divergences which can be removed by a renormalization prescription. The choice of such a prescription defines a renormalization scheme and renders all (coupling) constants of a given theory scheme-dependent. Since the renormalized (coupling) constants are nothing but mathematical parameters, their values can be arbitrarily changed by changing the renormalization prescription. We stress that these renormalized constants should not be confused with physical observables such as, for example, the phase transition temperature or the physical mass of a particle. Physical observables are, of course, invariant under a variation of the renormalization prescription, provided we have not truncated the perturbation series. If we consider a truncated perturbation series, we find that there is a residual dependence on the renormalization scheme which can be controlled to some extent by the so-called ”Principle of Minimum Sensitivity” [111], see also discussion below.
At this point we are then still free to perform additional finite renormalizations. This results in different effective renormalization prescriptions. A given renormalization prescription can then be considered as a particular reordering of the perturbative expansion which expresses it in terms of new renormalized constants [112]. Let us assume that the transformations between the finite renormalizations can be parametrized by introducing an auxiliary single mass scale . This scale corresponds to a UV (cutoff) scale at which the parameters of the theory are fixed. A set of RG equations for a given theory then describes the changes of the renormalized parameters of this theory (e. g. the coupling constant) induced by a variation of the auxiliary mass scale . The set of renormalization transformations is called the renormalization group.
Let us now consider a (renormalized) microscopic theory at some large momentum scale defined by a (classical) action . Wilson’s basic idea of the renormalization group is to start with such a classical action and then to integrate out successively all fluctuations from high to low momentum scales [89, 90, 91]. This procedure results in an action which depends on an IR regulator scale, say , which plays the role of a reference scale. The values of the (scale-dependent) couplings defining this action on the different scales are related by continuous RG transformations. We shall refer to the change of a coupling under a variation of the scale as the RG flow of the coupling. In this picture, universality means that the RG flow of the couplings is governed by a fixed point. The possibility of identifying fixed points of a theory makes the RG such a powerful tool for studying statistical field theories as well as quantum field theories. As we shall discuss below, critical behavior near phase transitions is intimately linked to the fixed-point structure of the theory under consideration.
In this review we employ a non-perturbative RG flow equation, the Wetterich equation [95], for the so-called effective average action in order to analyze critical behavior in physical systems. The effective average action depends on an intrinsic momentum scale which parameterizes the Wilsonian RG transformations. We note that such an approach is based on the fact that an infinitesimal RG transformation (i. e. an RG step), performed by an integration over a single momentum shell of width , is finite. For this reason we are able to integrate out all quantum fluctuations through an infinite sequence of such RG steps. The flow equation for then describes the continuous trajectory from the microscopic theory at large momentum scales to the full quantum effective action (macroscopic theory) at small momentum scales. Thus, it allows us to cover physics over a wide range of scales.
Here, we only discuss briefly the derivation and the properties of the RG flow equation for the effective average action ; for details we refer to the original work by Wetterich [95]. The scale-dependent effective action is a generalization of the (quantum) effective action but only includes the effects of fluctuations with momenta . Therefore is sometimes called a coarse-grained effective action since quantum fluctuations on length scales smaller than are integrated out. The underlying idea is to calculate the generating functional of one-particle irreducible (1PI) graphs of a given theory by starting at an ultraviolet (UV) scale with the microscopic (classical) action and then successively integrating out quantum fluctuations by lowering the scale . The quantum effective action is then obtained in the limit . In other words, the coarse-grained effective action interpolates between the classical action at the UV scale and the 1PI generating functional in the infrared limit (IR) . The starting point for the derivation of the flow equation of is a UV- and IR-regularized generating functional for the Greens functions:4
Zk[J]=∫ΛDϕ({pi})e−S[ϕ]−ΔSk[ϕ]+JT⋅ϕ≡eWk[J], (1)
where and is the scale-dependent generating functional for the connected Greens functions. The field variable as well as the source are considered as generalized vectors in field space and are defined as
ϕ=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝ψ¯ψTφ⋮⎞⎟ ⎟ ⎟ ⎟ ⎟⎠andJT=(¯η,ηT,j,…). (2)
Moreover, we have introduced a generalized scalar product in field space: . Here, the field represents a Dirac spinor, and denotes a real-valued scalar field. The dots indicate that other types of fields, e. g. gauge-fields, are allowed as well. For non-relativistic theories of fermions, the generating functional can be defined accordingly. We assume that the theory is well-defined by a UV-regularized generating functional: The index indicates that we only integrate over fields with momenta , i. e. we implicitly take for . To regularize the infrared modes a cutoff term has been inserted into the path integral. It is defined as
ΔSk[ϕ] = 12∑a,b∫ddp(2π)dϕa({−pi})Rabk({pi})ϕb({pi})≡12ϕT⋅Rk⋅ϕ, (3)
where is a matrix-valued regulator function. Through the insertion of the cutoff term, we have defined a generating functional which now depends on the scale .
The cutoff function has to fulfill three conditions. Since has been introduced to regularize the IR, it must fulfill
limp2k2→0Rk({pi})>0, (4)
where . Second, the function must vanish in the IR-limit, i. e. for :
limk2p2→0Rk({pi})=0. (5)
This condition ensures that we obtain the 1PI generating functional for . Third, the cutoff function should obey
limk→ΛRk({pi})→∞ (6)
for fixed . This property guarantees that for .
In this review, we shall always use cutoff functions which can be written in terms of a dimensionless regulator shape function . For simple relativistic scalar theories, we may choose
Rk(p2)∝p2r(p2k2). (7)
For studies of theories with chiral fermions, it is convenient to employ a cutoff function which preserves chiral symmetry. An appropriate choice is [113]
Unsupported use of \hfil (8)
On the other hand, for non-relativistic fermionic many-body problems the choice of the cutoff function should respect the presence of a Fermi surface. An appropriate choice for such a cutoff function is given by [27]
Rψk(→p2)=k2rψ(Z)withZ=(→p2−μ)/k2, (9)
where, for instance,
rψ(Z)=(sign(Z)−Z)θ(1−|Z|). (10)
The chemical potential of the fermions is given by and defines the associated Fermi surface. This choice for the regulator function arranges the momentum-shell integrations around the Fermi surface, i. e. modes with momenta remain unchanged while the momenta of modes with are cut off.
For scalar field theories, the presence of a cutoff function of the form is in general not problematic. For gauge theories, however, it causes difficulties due to condition (4) which essentially requires that the cutoff function acts like a mass term for small momenta. Therefore the cutoff function necessarily breaks gauge symmetry. We stress that this observation does by no means imply that such an approach cannot be applied to gauge theories. In fact, it is always necessary to fix the gauge in order to treat gauge theories perturbatively within a path-integral approach. This gauge-fixing procedure also breaks gauge invariance. Gauge-invariant results are then obtained by resolving Ward-Takahashi identities. Consequently, we can think of the cutoff function as an additional source of gauge-symmetry breaking. In analogy to perturbation theory, one then needs to deal with modified Ward-Takahashi identities in order to recover gauge invariance [114, 115, 116, 117, 118, 119]. In addition, there are essentially two alternatives: first, one can construct manifestly gauge-invariant flows as proposed in [120, 121, 108]. Second, we can apply special (useful) gauges, such as the background-field gauge [122, 123]. We refer the reader to Ref. [105] for a detailed introduction to RG flows in gauge theories.
The coarse-grained effective action can in principle be obtained from the IR-regularized functional in a standard fashion, see, e. g., the standard textbook derivation of the quantum effective action in Refs. [123, 112]. However, we employ here a modified Legendre transformation to calculate the coarse-grained effective action:5
Γk[Φ]=supJ{−Wk[J]+JT⋅Φ}−ΔSk[Φ]. (11)
The so-called classical field is implicitly defined by the supremum prescription. The modification of the Legendre transformation is necessary for the connection of with the classical action in the limit . From this definition of we find the RG flow equation of the coarse-grained effective action, the so-called Wetterich equation [95], by taking the derivative with respect to the scale :
∂tΓk[Φ]=12STr[Γ(2)k[Φ]+Rk]−1⋅(∂tRk)=−12\raisebox{-21.055039pt}{\psfig{scale=0.5, clip={true }}}~{}, (12)
with being the RG “time” and . The -point functions are defined as follows:
Γ(n,m)k[Φ]=\lx@stackreln−times% →δδΦT⋯→δδΦTΓk[Φ]\lx@stackrelm−times←δδΦ⋯←δδΦ. (13)
Thus, is matrix-valued in field space. The super-trace arises since contains both fermionic as well as bosonic degrees of freedom and it provides a minus sign in the fermionic subspace of the matrix. The double-line in Eq. (12) represents the full propagator of the theory which includes the complete field dependence. The solid black dot in the loop stands for the insertion of . The structure of the flow equation reveals that the regulator function specifies the Wilsonian momentum-shell integrations, such that the RG flow of is dominated by fluctuations with momenta .
The flow equation (12) has been obtained by taking the derivative of with respect to the scale . However, we have not taken into account a possible scale dependence of the classical field yielding a term on the right-hand side of Eq. (12). We stress that the inclusion of this term is a powerful extension of the flow equation discussed here, since it allows to bridge the gap between microscopic and macroscopic degrees of freedom in the RG flow, e. g. between quarks and gluons and hadronic degrees of freedom, without any fine-tuning [124, 28, 125]. More technically speaking, this extension makes it possible to perform continuous Hubbard-Stratonovich transformations in the RG flow. We shall not employ these techniques here since they do not provide us with additional insights into the fermionic fixed-point structure to which the scope of the present review is limited. For details concerning such an extension of the flow equation (12), we refer the reader to Refs. [124, 28, 125, 104, 105, 126, 127]. In Ref. [32] these so-called re-bosonization techniques6 have been employed for a first-principles study of the QCD phase boundary.
As should be the case for an exact one loop flow [128], the Wetterich equation (12) is linear in the inverse of the full propagator. Moreover, it is a nonlinear functional differential equation, since it involves the inverse of the second functional derivative of the effective action. We stress, however, that the loop in Eq. (12) is not a simple perturbative loop since it depends on the full propagator. In fact, it can be shown that arbitrarily high loop-orders are summed up by integrating this flow equation [128]. Nonetheless it is possible and sometimes even technically convenient to rewrite (12) in a form which is reminiscent of the textbook form of the one-loop contribution to the effective action:
∂tΓk[Φ]=12STr~∂tln(Γ(1,1)k[Φ]+Rk). (14)
Here, denotes a formal derivative acting only on the -dependence of the regulator function . Replacing by the (scale-independent) second functional derivative of the classical action, , we can perform the integration over the RG scale analytically and obtain the standard one-loop expression for the effective action:
Γ1−loop[Φ]=SUV[Φ]+12STrlnS(1,1)[Φ], (15)
where
SUV[Φ]=S[Φ]−12STrln(S(1,1)[Φ]+RΛ). (16)
Here, the second term on the right-hand side corresponds to the boundary condition for the RG flow at the UV scale , which renders finite.
From a technical point of view, the representation (14) turns out to be a convenient starting point for our studies of the fixed-point structure of four-fermion interactions. In order to calculate flow equations of four-fermion interactions, we decompose the inverse regularized propagator on the right-hand side of the flow equation into a field-independent () and a field-dependent () part,
Γ(1,1)k[Φ]+Rk=Pk+Fk. (17)
We can then expand the flow equation in powers of the fields according to
∂tΓk = 12STr{~∂tln(Pk+Fk)} (18) = 12STr{~∂t(1PkFk)}−14STr{~∂t(1PkFk)2}+16STr{~∂t(1PkFk)3}+….
The powers of can be calculated by simple matrix multiplications. The flow equations for the various couplings can now be obtained by comparing the coefficients of the four-fermion operators on the right-hand side of Eq. (18) with the couplings specified in the definition of the effective action. In other words, the flow equation of higher -point functions are obtained straightforwardly from the flow equation (12) (or, equivalently, from Eq. (18)) by taking the appropriate number of functional derivatives. From this, we observe that the RG flow of the -point function depends in general on the flow of the - and -point function. This means that we obtain an infinite tower of coupled flow equations by taking functional derivatives of the flow equation (12). In most cases we are not able to solve this infinite tower of flow equations. Thus, we need to truncate the effective action and restrict it to include only correlation functions with external fields. However, such a truncation poses severe problems: first, the system of flow equations is no longer closed and, second, neglecting higher -point functions may render the flow unstable in the IR region of strongly coupled theories. For example in QCD, one would naively expect that contributions from higher -point functions are important.
Finding reliable truncations of the effective action is the most difficult step and requires a lot of physical insight. We stress that an expansion in terms of -point functions must not be confused with an expansion in some small parameter as in perturbation theory. The assumption here is that the influence of neglected operators on the already included operators is small. Once we have chosen a truncation for studying a given theory, we need to check its reliability. One possibility for such a check is to extend the truncation by including additional operators and then check if the results obtained from this new truncation are in agreement with the earlier results. If this is not the case, one must rethink the chosen truncation. However, even if the results are not sensitive to the specific set of additional operators added to the truncation, this does not necessarily mean that one has included all relevant operators in the calculation. A second possibility to assess the reliability of a given truncation is to exploit the fact that physical observables should not depend on the regularization scheme. Since the scheme is specified by the cutoff function, the physical observables should be independent of this choice. In the present approach the scheme is defined by our choice for the regulator function . Thus, we can vary and then check if the results depend on the choice of the cutoff function. If this is the case, an extension of the truncation might be required. In addition to a simple variation of regulator functions, we may actually exploit the dependence on to optimize the truncated RG flow of a given theory. For example, an optimization criterion can be based on the size of the gap induced in the effective propagator , see Refs. [129, 130, 131]. We then denote those regulators to be optimized for which the gap is maximized with respect to the cutoff scheme. In addition to such an optimization of RG flows within a given regulator class, a more general criterion has been put forward in Ref. [104]. The latter defines the optimized regulator to be the one for which the regularized theory is already closest to the full theory at , for a given gap induced in the effective propagator . This optimization criterion yields an RG trajectory which defines the shortest path in theory space between the UV theory at and the full theory at . Both optimization criteria naturally encompass the so-called “Principle of Minimum Sensitivity”. However, in contradistinction to the “Principle of Minimum Sensitivity”, the optimization of (truncated) RG flows does not rely on the existence of extremal values of physical observables which may arise from a variation of the regularization scheme. For a detailed discussion of optimization criteria and properties of optimized RG flows, we refer the reader to Refs. [130, 129, 131, 104].
Nonetheless, even an approximate solution of the flow equation (12) can describe non-perturbative physics reliably, provided the relevant degrees of freedom in the form of RG relevant operators are kept in the ansatz for the effective action.
## 3 RG Flow of Four-Fermion Interactions - A Simple Example
In this section we discuss a simple four-fermion theory which already allows us to gain some important insight into the mechanisms of symmetry breaking in strongly-interacting theories. A study of a simple four-fermion theory is useful for many reasons. First, it allows us to highlight various methods and technical aspects such as Fierz ambiguities, (partial) bosonization and the role of explicit symmetry breaking. Second, a confrontation of this model study with our analysis of symmetry breaking in gauge theories is instructive: To be specific, we will consider the mechanisms of chiral symmetry breaking to point out the substantial differences between these theories.
### 3.1 A Simple Example and the Fierz Ambiguity
In this section we discuss the basic concepts and problems in describing strongly-interacting fermionic theories, with a particular emphasis on the application of RG approaches. To this end, we employ a Nambu–Jona-Lasinio-type model. Such models play a very prominent role in theoretical physics. Originally, the Nambu–Jona-Lasinio (NJL) model has been used as an effective theory to describe spontaneous symmetry breaking in particle physics based on an analogy with superconducting materials [132, 133], see Ref. [134] for a review. RG methods have been extensively employed to study critical behavior in QCD with the aid of NJL-type models, see e. g. Refs. [113, 135, 136, 100, 137, 138, 53, 139, 33, 55]. Usually these model studies rely on a (partially) bosonized version of the action. We shall discuss aspects of bosonization in Sect. 3.2. For the sake of simplicity we start with a purely fermionic formulation of the NJL model with only one fermion species. This model has been extensively studied at zero temperature with the functional RG in Refs. [140, 77]. In particular, the ambiguities arising from Fierz transformations have been explicitly worked out and discussed. We shall follow the discussion in Refs. [140, 77] but extend it with respect to issues arising at finite temperature and for a finite (explicit) fermion mass. In addition, we exploit this model to discuss general aspects of theories with many fermion flavors as well as quantum critical behavior.
In the following we consider a simple ansatz for the effective action in Euclidean space-time dimensions:
ΓNJL[¯ψ,ψ] = Unsupported use of \hfil (19)
where is the bare four-fermion coupling and is the so-called fermionic wave-function renormalization. The coupling is considered to be RG-scale dependent. Here, we consider four-fermion couplings as fundamental parameters. However, in other theories fermionic self-interactions might be fluctuation-induced. In QCD, for example, they are induced by two-gluon exchange and are therefore not fundamental as we shall discuss in Sect. 6, see also Refs. [28, 29, 30, 31, 32, 78]. We would like to add that the NJL model in is perturbatively non-renormalizable. In the following we define it with a fixed UV cutoff . Also the regularization scheme therefore belongs to the definition of the model. We shall come back to this issue in Sects. 3.2 and 5.1.
Our ansatz (19) for the effective action can be considered as the leading order approximation in a systematic expansion in derivatives. The associated small parameter of such an expansion is the so-called anomalous dimension of the fermion fields. If this parameter is small, then such a derivative expansion is indeed justified. We shall come back to this issue below. In any case, we will drop terms in our studies which are of higher order in derivatives, such as terms .
The action (19) is clearly invariant under simple phase transformations,
ψ(x)⟼eiαψ(x), (20)
but also under chiral U() transformations (axial phase transformations),
ψ(x)⟼eiγ5αψ(x),¯ψ(x)⟼¯ψ(x)eiγ5α, (21)
where is an arbitrary “rotation” angle. A necessary condition for the chiral symmetry of the NJL model is the absence of explicit mass terms for the fermion fields in the action, such as . As we shall discuss in more detail below, the chiral symmetry can be still broken spontaneously, if a finite vacuum expectation value is generated by loop corrections associated with (strong) fermionic self-interactions. Breaking of chiral symmetry in the ground state of the theory is then indicated by a dynamically generated mass term for the fermions. This mass term is associated, e. g., with a constituent quark mass in low-energy models of QCD and similar to the gap in condensed-matter theory. The relation between the strength of the four-fermion interactions and the symmetry properties of the ground-state are discussed in detail in Sects. 3.2 and 3.3.
We may now ask whether the action (19) is complete or whether other four-fermion couplings, such as a vector interaction , can be generated dynamically due to quantum fluctuations. We first realize that the four-fermion interaction in our ansatz (19) can be expressed in terms of a vector and axial-vector interaction term with the aid of so-called Fierz transformations, see App. B for details:
[(¯ψψ)2−(¯ψγ5ψ)2]=12[(¯ψγμγ5ψ)2−(¯ψγμψ)2]. (22)
This ambiguity in the representation of a four-fermion interaction term arises due to the fact that an arbitrary -matrix can be expanded in terms of a complete and orthonormalized set of -matrices as follows:
Mab=n∑j=1O(j)abtr(OjM)≡n∑j=1O(j)ab∑c,d(O(j)cdMdc)% withtr(O(j)O(k))=\mathbbm1dδjk. (23)
The expansion of a combination of two matrices and then reads (say for fixed and )
In the case of four-fermion interactions we may classify the basis elements according to the transformation properties of the corresponding interaction terms , i. e. scalar channel, vector channel, tensor channel, axial-vector channel and pseudo-scalar channel. To be specific, we choose , , , and as basis elements of the Clifford algebra defined by the matrices, see App. B for details. To obtain Eq. (22) we then simply apply Eq. (24) to the matrix products and , respectively. Thus, a Fierz transformation can be considered as a reordering of the fermion fields. We stress that this is by no means related to quantum effects but a simple algebraic operation. Nonetheless it suggests that other four-fermion couplings compatible with the underlying symmetries of our model exist and are potentially generated by quantum effects.
With our choice for the set of basis elements it is straightforward to write down the most general ansatz for the effective action which is compatible with the underlying symmetries of the model, i. e. the symmetries with respect to U() phase transformations, U() chiral transformations and Lorentz transformations:7
ΓNJL[¯ψ,ψ] = Unsupported use of \hfil (25) −12¯λV[(¯ψγμψ)2]−12¯λA[(¯ψγμγ5ψ)2]}.
Because of Eq. (22) only two of the three couplings , and are independent. Thus, it suffices to consider the following action with implicitly redefined four-fermion couplings:
ΓNJL[¯ψ,ψ] = Unsupported use of \hfil (26)
Note that we could have also chosen to remove, e. g., the vector-channel interaction term with the aid of Eq. (22) at the expense of getting the axial-vector interaction. From a phenomenological point of view it is tempting to attach a physical meaning to, e. g., the vector-channel interaction and interpret it as an effective mass term for vector bosons as done in mean-field studies of Walecka-type models [141]: . However, the present analysis shows that one has to be careful to attach such a phenomenological interpretation to this term since the Fierz transformations allow us to remove this term completely from the action, see also Sect. 3.2.
In this section we drop a possible momentum dependence of the four-fermion couplings. Thus, we only take into account the leading term of an expansion of the four-fermion couplings in powers of the dimensionless external momenta , e. g.
Γ(2,2)[¯ψ,ψ](p1,p2,p3)≡¯λV(p1,p2,p3)=¯λV(0,0,0)+O(|pi|k). (27)
In momentum space, the corresponding interaction term in the expansion of the effective action (26) in terms of fermionic self-interactions then assumes the following form, see App. A for our conventions of the Fourier transformation:
ΓNJL[¯ψ,ψ]=…−12¯λV3∏i=1∫d4pi(2π)4¯ψ(p1)γμψ(p2)¯ψ(p3)γμψ(p1−p2+p3)−…, (28)
where and correspondingly for the other four-fermion interaction terms in Eq. (26). Note that only three of the four four-momenta are independent due to momentum conservation. We stress that we also apply this expansion at finite temperature , see Sect. 3.5.3. In this case, it then corresponds to an expansion in powers of the dimensionless Matsubara modes and . Thus, we assume that .
The approximation (27) does not permit a study of properties of bound states of fermions, such as meson masses in QCD, in the chirally broken regime; such bound states manifest themselves as momentum singularities in the four-fermion couplings in Minkowski space. Nonetheless, the point-like limit can still be a reasonable approximation for . In the chirally symmetric regime above the chiral phase transition it allows us to gain some insight into the question how the theory approaches the regime with broken chiral symmetry in the ground state [30, 31, 32]. In Sect. 3.2 we shall discuss how the momentum dependence of the fermionic interactions can be conveniently resolved in order to gain access to the mass spectrum in the regime with broken chiral symmetry.
Let us now compute the RG flow equations, i. e. the so-called functions, for the four-fermion couplings in the point-like limit. To this end, we compute the second functional derivative of the effective action with respect to the fields
Φ≡Φ(q):=(ψ(q)¯ψT(−q))andΦT≡ΦT(−q):=(ψT(−q),¯ψ(q)), (29)
see also Eq. (13), and evaluate it for homogeneous (constant) background fields and . In momentum space this means that we evaluate at
ψ(p)=Ψ(2π)4δ(4)(p)and¯ψ(p)=¯Ψ(2π)4δ(4)(p), (30)
where and on the right-hand side denote the homogeneous background fields. Following Eq. (17), we then split the resulting matrix into a field-independent part and a part which depends on and . To detail the derivation of flow equations of four-fermion interactions in a simple manner, we first restrict ourselves to the simplified ansatz (19) of the effective action. In this case, the so-called (regularized) propagator matrix and the fluctuation matrix read
Pk=(0−Zψp/T(1+rψ)−Zψp/(1+rψ)0)(2π)4δ(4)(p−p′) (31)
and
Fk=(F11F12F21F22)(2π)4δ(4)(p−p′), (32)
respectively, where
F11=−¯λσ[¯ΨT¯Ψ−γ5¯ΨT¯Ψγ5],F22=−¯λσ[ΨΨT−γ5ΨΨTγ5],
F12=−¯λσ[(¯ΨΨ)−γ5(¯Ψγ5Ψ)+Ψ¯Ψ−γ5Ψ¯Ψγ5]T=−FT21.
Since we evaluated for constant fields, both and are diagonal in momentum space. At this point it is not yet necessary to specify the regulator function exactly.
The RG flow equation for can now be computed straightforwardly by comparing the coefficients of the four-fermion interaction terms on the right-hand side of Eq. (18) with the couplings included in our ansatz (19). From the fluctuation matrix it is clear that only the term in Eq. (18) contributes to the RG flow of the four-fermion coupling . For this initial study, we simply take the four-fermion terms on the right-hand side of the flow equation “as they appear” and ignore Fierz transformations of these terms. We then find
βλσ≡∂tλσ=(2+2ηψ)λσ−16v4l(F),(4)1(0;ηψ)λ2σ, (33)
where , i. e. . Here, we have defined the dimensionless renormalized coupling
λσ=(Zψ)−2k2¯λσ. (34)
The so-called threshold function corresponds to a one-particle irreducible (1PI) Feynman diagram, see left diagram in Fig. 1, and describes the decoupling of massive and also thermal modes in case of finite-temperature studies. Moreover, the regularization-scheme dependence is encoded in these threshold functions, see App. D for their definitions.
In Fig. 2 we show a sketch of the -function for vanishing temperature. Apart from a Gaußian fixed point, , we find a second non-trivial fixed point :
λ∗σ=18v4l(F),(4)1(0;0)+O(η∗ψ). (35)
In the present leading-order approximation of the derivative expansion we have , see below. We then find
λ∗σ=8π2 (36)
for an optimized (linear) regulator function (for which ) and
λ∗σ=4π2 (37)
for the sharp cutoff (for which ). It is instructive to have a closer look at Eq. (33). This flow equation represents an ordinary differential equation which can be solved analytically for . Its solution reads
λσ(k)=λUVσ[(Λk)Θ(1−λUVσλ∗σ)+λUVσλ∗σ]−1, (38)
where
Θ:=−∂(∂tλσ)∂λσ∣∣λ∗σ\lx@stackrel(ηψ≡0)=2. (39)
In order to derive Eq. (38), it is convenient to expand the right-hand side of Eq. (33) about the fixed-point . The physical meaning of the so-called critical exponent will be discussed in more detail below. In Sect. 3.4.1 we will then see that this exponent governs the scaling behavior of physical observables close to a quantum critical point.
For , we find that does not dependent on the RG scale as it should be: . Choosing an initial value at the initial UV scale , the solution (38) of the flow equation predicts that the theory becomes non-interacting in the infrared regime ( for ), i. e. chiral symmetry remains unbroken in this case, see Fig. 2. For , we find that the four-fermion coupling increases rapidly and diverges eventually at a finite scale : . This behavior of the coupling and the associated fixed-point structure are tightly linked to the question whether chiral symmetry is broken in the ground state or not: The value of the non-trivial fixed-point can be considered as a critical value of the coupling which separates the chirally symmetric regime and the regime with a broken chiral symmetry in the ground state. We shall discuss this in more detail in Sects. 3.23.3 and 3.4.
In the derivation of the flow equation (33) we have dropped contributions arising from four-fermion interactions with different transformation properties, e. g. a vector-channel interaction. From the expansion (18) of the flow equation, we can indeed read off that contributions to the flow of four-fermion couplings other than might be generated, even though they have not been included in the truncation (19): the matrix multiplications on the right-hand side of Eq. (18) mix the contributions from the propagator , which is proportional to , with the contributions from the field-dependent part :
¯λ2σtr{γμψ¯ψγμψ¯ψ}=−¯λ2σ(¯ψγμψ)(¯ψγμψ). (40)
This term obviously contributes to the flow of the -coupling.8 Moreover, contributions of this type couple the flow equations of the various four-fermion interactions to one another. Thus, quantum fluctuations induce a vector-channel interaction, even though we have not included such an interaction term initially. This observation explains why we need to include a basis which is complete with respect to Fierz transformations, such as in the effective action (25). We stress that the effective action (25) is closed in the sense that no contributions to four-fermion interactions, which are not covered by the truncation, are generated in the RG flow: any other pointlike four-fermion interaction compatible with the underlying symmetries of the theory can be written in terms of the interactions included in these effective actions by means of Fierz transformations.
In the point-like limit the RG flow of the four-fermion coupling is completely decoupled from the RG flow of fermionic -point functions of higher order. For example, -fermion interactions do not contribute to the RG flow of the coupling in this limit. Using the one-loop structure of the Wetterich equation, this statement can be proven diagrammatically: it is not possible to construct a one-loop diagram with only for external legs out of fermionic -point functions () which are compatible with the underlying chiral symmetry.
Up to now we have only discussed the running of a four-fermion coupling. We have not yet discussed how to compute the running of the wave-function renormalization . In general, the flow equation for can be obtained from evaluated for a spatially varying background field,
ψ(p)=Ψ(2π)4δ(4)(p+Q)and¯ψ(p)=¯Ψ(2π)4δ(4)(p−Q),
where denotes the external momentum.9 The -dependent second functional derivative can still be split into a field-independent and a field-dependent part. However, the latter is no longer diagonal in momentum space. The flow equations for the wave-function renormalizations can then be computed by comparing the coefficients of the terms bilinear in fermionic fields which appear on the right-hand side of Eq. (18) with the kinetic terms in the ansatz for the effective action. In our present approximation, we find that the RG running of is trivial, i. e.
∂tZψ=0. (41)
Thus, the associated anomalous dimensions is zero. In fact, this follows immediately from the associated 1PI Feynman diagram, see diagram on the right in Fig. 1, which has only one internal fermion line.10 In the following we therefore set the wave-function renormalization to one, , which implies .
Let us now turn to the effective action (26). The flow equations of the various couplings can be derived along the same lines as the RG equation for the -coupling detailed above. We find
∂tλσ = 2λσ−8v4l(F),(4)1(0;0)[λ2σ+4λσλV+3λ2V], (42) ∂tλV = 2λV−4v4l(F),(4)1(0;0)[λ2σ+2λσλV+λ2V], (43)
where the dimensionless (renormalized) couplings are defined as
λσ=k2¯λσ,andλV=k2¯λV. (44)
In the derivation of the flow equations for and also terms of the type and
[(¯ψσμνψ)2−(¯ψσμνγ5ψ)2] (45)
appear. While the latter vanishes identically, see also App. B, the former can be completely transformed into a scalar-pseudoscalar and vector-interaction channel with the aid of the Fierz transformation (22). In fact, any four-fermion interaction term appearing in the derivation of the flow equations for the present system can be unambiguously rewritten in terms of these two interaction channels. Thus, the above RG flows are closed with respect to Fierz transformations. Due to Eq. (22) we could have also used, e. g., a scalar-pseudoscalar and an axial-vector interaction to describe the properties of our simplified theory without loss of physical information. The present choice for a complete basis of four-fermion interactions is one of several possibilities.
Our flow equations for and agree with the results found in Refs. [142, 140]. The RG flow of the couplings and is governed by three fixed points which are given by11
F1≡FGau\ss=(0,0),F2=(3ζ,ζ),F3=(−32ζ,16ζ), (46)
where
ζ=132v4l(F),(4)1(0;0). (47)
These fixed-points are of phenomenological importance. First of all, they might be related to (quantum) phase transitions. Second, we can define sets of initial values for the RG flows of the couplings and for which we find condensate formation associated with (chiral) symmetry breaking in the IR, as we shall discuss in detail in the two subsequent sections. The existence of such sets of initial conditions is not a generic feature of fermionic models but also appears in (chiral) gauge theories. In QCD and QED, four-fermion interactions are generated dynamically due to strong quark-gluon interactions, see our discussion in Sect. 6.
We can classify the various fixed points according to their directions in the space spanned by the couplings. To this end, we first linearize the RG flow equations of the couplings near a fixed point:
∂tλi=∑jBij(λj−λ∗j)+…,whereBij=∂tλi∂λj∣∣∣λi=λ∗i (48)
and . We refer to as the stability matrix. The two eigenvectors and eigenvalues (critical exponents) of this matrix essentially determine the RG evolution near a fixed point:12
∂t→vi=B⋅→vi=:−Θ(i)→vi. (49)
The solution of the RG flow in the fixed-point regime is then given by
λi=λ∗i+∑jcj(→vj)i(k0k)Θ(j). (50)
Here, the ’s define the initial conditions at the scale . From the solution of the linearized flow it becomes apparent that positive critical exponents, , correspond to RG relevant, i. e. infrared repulsive, directions. On the other hand, negative critical exponents correspond to RG irrelevant, i. e. infrared attractive, directions. The classification of marginal directions associated with vanishing critical exponents requires to consider higher orders in the expansion about the fixed point.
Using the flow equations (42) and (43), we find that the Gaußian fixed point has two IR attractive directions; the eigenvalues are . The fixed points with and with have both one IR attractive and one IR repulsive direction. We would like to add that the fixed-point values of the four-fermion couplings are not universal quantities as the dependence of their RG flows on the threshold function indicates. However, the statement about the mere existence of these fixed points is universal, because the regulator-dependent factor is a positive number for any regulator. Moreover, the critical exponents themselves are universal. The latter can be indeed related to the exponents associated with (quantum) phase transitions, as we shall discuss in Sect. 3.4. Therefore the accuracy of the critical exponents can be used to measure the quality of a given truncation as has been done in the context of scalar field theories, see e. g. Refs. [143, 144, 145, 106, 146]. In a pragmatic sense, the computation of critical exponents allows us to estimate how well the dynamics close to a phase transition are captured within our ansatz for the effective action.
Let us conclude our discussion with a comparison of the RG flows (42) and (43) obtained from a complete basis of four-fermion interactions with the RG flow equation (33) from our single-channel approximation. We immediately observe that setting in Eq. (42) does not yield the flow equation (33). Thus, the values of the non-trivial fixed point of this coupling are not identical but differ by a factor of two.13 For a finite , we find that the vector-channel interaction is dynamically generated due to quantum fluctuations even if we have initially set the vector-channel interaction to zero. In fact, a finite -coupling shifts the parabola associated with the -function of the coupling , and vice versa, see Fig. 3. Thus, the -coupling can potentially induce critical behavior in the vector-channel, i. e. a diverging four-fermion coupling. We shall discuss this in more detail in Sect. 3.3 after we have clarified the physical meaning of diverging four-fermion couplings in the subsequent section.
### 3.2 Bosonization and the Momentum Dependence of Fermion Interactions
In this section we study the NJL model with one fermion species in a partially bosonized form. Partial bosonization of fermionic theories is a well-established concept which makes use of the so-called Hubbard-Stratonovich transformation [59, 60]. The advantage of a partially bosonized formulation of NJL-type models over their purely fermionic formulation is that it allows us to include the momentum dependence of four-fermion interactions in a simple manner. Therefore it opens up the possibility to study conveniently the mass spectrum of a theory which emerges from the spontaneous breakdown of its underlying symmetries, e. g. the chiral symmetry. As a bonus, it relates the Ginzburg-Landau picture of spontaneous symmetry breaking, as known from statistical physics, with dynamical bound-state formation in strongly-interacting fermionic theories.
In the following we derive the RG flow equations for the partially bosonized version of this theory and discuss dynamical chiral symmetry breaking. In particular, we explain the mapping of the (partially) bosonized equations onto the RG equations of the four-fermion couplings in the purely fermionic description of our model. This finally allows us to relate the fixed-point structure of the purely fermionic formulation to spontaneous (chiral) symmetry breaking.
Z ∝ ∫DψD¯ψe−S[¯ψ,ψ] (51)
with the action
S[¯ψ,ψ] = Unsupported use of \hfil (52) −12¯λV[(¯ψγμψ)2]−12¯λA[(¯ψγμγ5ψ)2]}.
see also Eq. (25). As discussed in the previous section, this action is over-complete in the sense that only two of the three couplings , and are independent. We shall come back to this issue in the partially bosonized formulation below.
Our NJL model possesses a chiral symmetry, see Eq. (21), which can be broken dynamically, if a finite vacuum expectation value is generated. This is associated with the Nambu-Goldstone theorem [132, 133, 147, 148] which relates a spontaneously broken continuous symmetry of a given theory to the existence of massless states in the spectrum. To apply this theorem to the present model, we need to compute the vacuum expectation value of the commutator of the so-called chiral charge , which is the generator of the chiral symmetry transformations, and the composite field :
⟨[iQ5,¯ψiγ5ψ]⟩∝⟨¯ψψ⟩withQ5=12∫d3x¯ψγ0γ5ψ. (53)
We observe that the generator does not commute with the field , if the vacuum expectation value of is finite. Thus, the chiral symmetry of our model can be indeed broken spontaneously. Following the Nambu-Goldstone theorem this implies the existence of a massless pseudo-scalar Nambu-Goldstone boson in the channel of the composite field . Since the action does not contain such a state, the massless state must be a bound state. We refer to this type of boson as a pion in the context of QCD, see Sect. 5. At this point we have traced the question of chiral symmetry breaking back to the existence of a finite expectation value of the composite field .
Formally, we may introduce auxiliary fields in the path integral by introducing an exponential factor into the integrand of the generating functional. This is known as a Hubbard-Stratonovich transformation. To bosonize the scalar-pseudoscalar interaction channel we use
N∫Dϕ1Dϕ2DVμDAμe−∫d4x{12¯m2σ→ϕ2+12¯m2VVμVμ+12¯m2AAμAμ} = 1, (54)
where we have combined the scalar fields into the vector , where and are real-valued scalar fields.15 The fields , , and are auxiliary fields (which have no dynamics so far), is a normalization factor, and the constants , and remain arbitrary for the moment. Multiplying the integrand of the generating functional with such a factor leaves the Greens functions of the theory unchanged. We now shift the integration variables in the so-modified generating functional according to16
ϕ1→ϕ1+i¯hσ√2¯m2σ(¯ψψ),ϕ2→ϕ2−i¯hσ√2¯m2σ(¯ψiγ5ψ), Vμ→Vμ−
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2019-05-25 12:49:21
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https://ask.libreoffice.org/en/answers/161682/revisions/
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Because you use a small letter a, and the space includes the Cap height.
Because you use a small letter a, and the space includes the Cap height.. Put a capital A next to the small a to see the difference.
Because you use a small lowercase letter a, and the space includes the Cap height. Put a capital uppercase A next to the small a to see the difference.
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2019-08-19 17:26:37
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https://petsc.org/release/docs/manualpages/PC/PCGAMGSetProcEqLim/
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# PCGAMGSetProcEqLim#
Set number of equations to aim for per process on the coarse grids via processor reduction in PCGAMG
## Synopsis#
#include "petscpc.h"
#include "petscksp.h"
PetscErrorCode PCGAMGSetProcEqLim(PC pc, PetscInt n)
Logically Collective
## Input Parameters#
• pc - the preconditioner context
• n - the number of equations
## Options Database Key#
• -pc_gamg_process_eq_limit - set the limit
## Note#
PCGAMG will reduce the number of MPI processes used directly on the coarse grids so that there are around equations on each process that has degrees of freedom
PCGAMG, PCGAMGSetCoarseEqLim(), PCGAMGSetRankReductionFactors(), PCGAMGSetRepartition()
intermediate
## Location#
src/ksp/pc/impls/gamg/gamg.c
Edit on GitLab
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2023-02-07 04:11:22
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https://support.bioconductor.org/p/122005/#122024
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DESeq2 log fold change
2
0
Entering edit mode
bioinfo • 0
@bioinfo-12782
Last seen 3 months ago
United States
Dear DESeq2 developer and colleagues,
DESeq2 generates an output table including "log2FoldChange". I have two questions about this value.
I. As far as I know, this is a moderated version, not a log2 fold change from the raw count value. But I want to have unmoderated value that can be calculated from count directly and my current way is: (1) get normalized count (2) log transformation (3) calculate averages per condition (4) divide two averages: "FC". Is this a right way? I am confusing whether it is fold change of log, or log of fold change.
II. Effect size is "log2FoldChange" divided by SD where SD is sqrt(1/mu + dispersion). Here, I think that mu is an average of "log" of normalized count and dispersion can be retrieved by a command "dispersions()". I assume that this command outputs in a same order with "results()" to match values between two output objects. Also, I can use moderated version or unmoderated version with the same formula, mu and dispersion. Are they correct ?
deseq2 • 483 views
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swbarnes2 ★ 1.1k
@swbarnes2-14086
Last seen 5 hours ago
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1) get normalized count (2) log transformation (3) calculate averages per condition (4) divide two averages: "FC". Is this a right way?
No. Take the averages of the normalized counts, divide one by the other, take the log base 2 of that.
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@mikelove
Last seen 3 days ago
United States
DESeq() since version 1.15 (~2.5 years) has not performed shrinkage, but this was moved to lfcShrink(), see the current vignette.
The standard deviation of the LFC is not equal to sqrt(1/mu + dispersion). See the DESeq2 paper for details.
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Thanks for your answer. I saw your post a long time ago: https://www.biostars.org/p/140976/ So was this changed?
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Yes, as I said, and you can read about it in the vignette.
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2022-11-27 02:19:53
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{"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": 0, "mathjax_asciimath": 1, "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": 0.5628213882446289, "perplexity": 4918.388022113906}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2022-49/segments/1669446710155.67/warc/CC-MAIN-20221127005113-20221127035113-00819.warc.gz"}
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https://dankelley.github.io/oce/reference/colormap.html
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Map values to colors, for use in palettes and plots. There are many ways to use this function, and some study of the arguments should prove fruitful in cases that extend far beyond the examples.
colormap(
z = NULL,
zlim,
zclip = FALSE,
breaks,
col = oceColorsJet,
name,
x0,
x1,
col0,
col1,
blend = 0,
missingColor,
debug = getOption("oceDebug")
)
## Arguments
z an optional vector or other set of numerical values to be examined. If z is given, the return value will contain an item named zcol that will be a vector of the same length as z, containing a color for each point. If z is not given, zcol will contain just one item, the color "black". optional vector containing two numbers that specify the z limits for the color scale. If provided, it overrides defaults as describe in the following. If name is given, then the range() of numerical values contained therein will be used for zlim. Otherwise, if z is given, then its rangeExtended() sets zlim. Otherwise, if x0 and x1 are given, then their range() sets zlim. Otherwise, there is no way to infer zlim and indeed there is no way to construct a colormap, so an error is reported. It is an error to specify both zlim and breaks, if the length of the latter does not equal 1. logical, with TRUE indicating that z values outside the range of zlim or breaks should be painted with missingColor and FALSE indicating that these values should be painted with the nearest in-range color. an optional indication of break points between color levels (see image()). If this is provided, the arguments name through blend are all ignored (see “Details”). If it is provided, then it may either be a vector of break points, or a single number indicating the desired number of break points to be computed with pretty(z,breaks)1. In either case of non-missing breaks, the resultant break points must number 1 plus the number of colors (see col). either a vector of colors or a function taking a numerical value as its single argument and returning a vector of colors. The value of col is ignored if name is provided, or if x0 through col1 are provided. an optional string naming a built-in colormap (one of "gmt_relief", "gmt_ocean", "gmt_globe" or "gmt_gebco") or the name of a file or URL that contains a color map specification in GMT format, e.g. one of the .cpt files from http://www.beamreach.org/maps/gmt/share/cpt). If name is provided, then x0, x1, col0 and col1 are all ignored. Vectors that specify a color map. They must all be the same length, with x0 and x1 being numerical values, and col0 and col1 being colors. The colors may be strings (e.g. "red") or colors as defined by rgb() or hsv(). a number indicating how to blend colors within each band. This is ignored except when x0 through col1 are supplied. A value of 0 means to use col0[i] through the interval x0[i] to x1[i]. A value of 1 means to use col1[i] in that interval. A value between 0 and 1 means to blend between the two colors according to the stated fraction. Values exceeding 1 are an error at present, but there is a plan to use this to indicate subintervals, so a smooth palette can be created from a few colors. color to use for missing values. If not provided, this will be "gray", unless name is given, in which case it comes from that color table. a flag that turns on debugging. Set to 1 to get a moderate amount of debugging information, or to 2 to get more.
## Value
A list containing the following (not necessarily in this order)
• zcol, a vector of colors for z, if z was provided, otherwise "black"
• zlim, a two-element vector suitable as the argument of the same name supplied to image() or imagep()
• breaks and col, vectors of breakpoints and colors, suitable as the same-named arguments to image() or imagep()
• zclip the provided value of zclip.
• x0 and x1, numerical vectors of the sides of color intervals, and col0 and col1, vectors of corresponding colors. The meaning is the same as on input. The purpose of returning these four vectors is to permit users to alter color mapping, as in example 3 in “Examples”.
• missingColor, a color that could be used to specify missing values, e.g. as the same-named argument to imagep(). If this is supplied as an argument, its value is repeated in the return value. Otherwise, its value is either "gray" or, in the case of name being given, the value in the GMT color map specification.
• colfunction, a univariate function that returns a vector of colors, given a vector of z values; see Example 6.
## Details
This is a multi-purpose function that generally links (maps'') numerical values to colors. The return value can specify colors for points on a graph, or breaks and col vectors that are suitable for use by drawPalette(), imagep() or image().
There are three ways of specifying color schemes, and colormap works by checking for each condition in turn.
• Case A. Supply z but nothing else. In this case, breaks will be set to [pretty](z,10) and things are otherwise as in case B.
• Case B. Supply breaks. In this case, breaks and col are used together to specify a color scheme. If col is a function, then it is expected to take a single numerical argument that specifies the number of colors, and this number will be set to length(breaks)-1. Otherwise, col may be a vector of colors, and its length must be one less than the number of breaks. (NB. if breaks is given, then all other arguments except col and missingColor are ignored.) The figure below explains the (breaks, col) method of specifying a color mapping. Note that there must be one more break than color. This is the method used by e.g. [image()].
• Case C. Do not supply breaks, but supply name instead. This name may be the name of a pre-defined color palette ("gmt_relief", "gmt_ocean", "gmt_globe" or "gmt_gebco"), or it may be the name of a file (including a URL) containing a color map in the GMT format (see “References”). (NB. if name is given, then all other arguments except z and missingColor are ignored.)
• Case D. Do not supply either breaks or name, but instead supply each of x0, x1, col0, and col1. These values are specify a value-color mapping that is similar to that used for GMT color maps. The method works by using seq() to interpolate between the elements of the x0 vector. The same is done for x1. Similarly, colorRampPalette() is used to interpolate between the colors in the col0 vector, and the same is done for col1. The figure above explains the (x0, x1, col0, col1) method of specifying a color mapping. Note that the each of the items has the same length. The case of blend=0, which has color col0[i] between x0[i] and x1[i], is illustrated below.
## References
Information on GMT software is given at http://gmt.soest.hawaii.edu (link worked for years but failed 2015-12-12). Diagrams showing the GMT color schemes are at http://www.geos.ed.ac.uk/it/howto/GMT/CPT/palettes.html (link worked for years but failed 2015-12-08), and numerical specifications for some color maps are at http://www.beamreach.org/maps/gmt/share/cpt, http://soliton.vm.bytemark.co.uk/pub/cpt-city, and other sources.
Other things related to colors: oceColors9B(), oceColorsCDOM(), oceColorsChlorophyll(), oceColorsClosure(), oceColorsDensity(), oceColorsFreesurface(), oceColorsGebco(), oceColorsJet(), oceColorsOxygen(), oceColorsPAR(), oceColorsPalette(), oceColorsPhase(), oceColorsSalinity(), oceColorsTemperature(), oceColorsTurbidity(), oceColorsTurbo(), oceColorsTwo(), oceColorsVelocity(), oceColorsViridis(), oceColorsVorticity(), ocecolors
## Examples
library(oce)
## Example 1. color scheme for points on xy plot
x <- seq(0, 1, length.out=40)
y <- sin(2 * pi * x)
par(mar=c(3, 3, 1, 1))
mar <- par('mar') # prevent margin creep by drawPalette()
## First, default breaks
c <- colormap(y)
drawPalette(c$zlim, col=c$col, breaks=c$breaks) plot(x, y, bg=c$zcol, pch=21, cex=1)grid()par(mar=mar)
## Second, 100 breaks, yielding a smoother palette
c <- colormap(y, breaks=100)
drawPalette(c$zlim, col=c$col, breaks=c$breaks) plot(x, y, bg=c$zcol, pch=21, cex=1)grid()par(mar=mar)
if (FALSE) {
## Example 2. topographic image with a standard color scheme
par(mfrow=c(1,1))
data(topoWorld)
cm <- colormap(name="gmt_globe")
imagep(topoWorld, breaks=cm$breaks, col=cm$col)
## Example 3. topographic image with modified colors,
## black for depths below 4km.
cm <- colormap(name="gmt_globe")
deep <- cm$x0 < -4000 cm$col0[deep] <- 'black'
cm$col1[deep] <- 'black' cm <- colormap(x0=cm$x0, x1=cm$x1, col0=cm$col0, col1=cm$col1) imagep(topoWorld, breaks=cm$breaks, col=cm$col) ## Example 4. image of world topography with water colorized ## smoothly from violet at 8km depth to blue ## at 4km depth, then blending in 0.5km increments ## to white at the coast, with tan for land. cm <- colormap(x0=c(-8000, -4000, 0, 100), x1=c(-4000, 0, 100, 5000), col0=c("violet","blue","white","tan"), col1=c("blue","white","tan","yellow")) lon <- topoWorld[['longitude']] lat <- topoWorld[['latitude']] z <- topoWorld[['z']] imagep(lon, lat, z, breaks=cm$breaks, col=cm$col) contour(lon, lat, z, levels=0, add=TRUE) ## Example 5. visualize GMT style color map cm <- colormap(name="gmt_globe", debug=4) plot(seq_along(cm$x0), cm$x0, pch=21, bg=cm$col0)
grid()
points(seq_along(cm$x1), cm$x1, pch=21, bg=cm$col1) ## Example 6. colfunction cm <- colormap(c(0, 1)) x <- 1:10 y <- (x - 5.5)^2 z <- seq(0, 1, length.out=length(x)) drawPalette(colormap=cm) plot(x, y, pch=21, bg=cm$colfunction(z), cex=3)
}
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2020-09-20 14:06:03
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https://crypto.stackexchange.com/questions/41471/aes-cryptography-keys
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# AES cryptography keys
1. What are the three possible values (in bits) for the length of the AES algorithm keys?
2. Approximately how big is the AES key space?
• a) is literally answered by typing "AES key lengths" into your favourite internet search engine, and b) is an easy computation using those values. What have you tried? Where are you stuck? This is not an online homework service. – yyyyyyy Nov 13 '16 at 20:39
• The key space of an n-bit AES key is precisely $2^n$. What is the point of the second question? – CodesInChaos Nov 13 '16 at 20:41
• first I'm new in this field and I searched but I wanted to make sure by asking the people here since they know better about cryptography than anyone else.. anyway thank you for your effort – F.Moe Nov 14 '16 at 6:07
AES keys don't have any redundancies; each bit directly influences the key space. So therefore the key size is $2^{128}$, $2^{192}$ or $2^{256}$ for the respective key sizes. This is somewhat different from the old DES cipher that also contains parity bits in the full key. More modern symmetric ciphers simply use all bits and try to avoid weak keys.
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2019-12-11 16:44:51
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https://www.experts-exchange.com/questions/28150762/cfspreadsheet-basics.html
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Solved
Posted on 2013-06-07
323 Views
Just getting back into CF after a long layoff of about 10 years!
Trying out the cfspreadsheet tag. I have a query, and I am able to get the xls file to be saved on my server - I just want a prompt displayed for the user to either open or save the file - can this be done with cfspreadsheet??
Thanks for any help!
0
Question by:cb_it
• 3
LVL 52
Accepted Solution
_agx_ earned 500 total points
No. cfspreadsheet only generates the file. Use cfcontent/cfheader to return it to the browser.
IIRC, there is one small nuisance w/cfspreadsheet. "query" and "name" are mutually exclusive, meaning you must save it to disk first. Use a unique file name (to avoid clashes w/other downloads) - something like createUUID() should work:
<cfset filePath = "c:\desired\path\to\#CreateUUID()#.xls">
then display it with cfcontent/cfheader. Use the "delete" attribute so the temp file will be removed automatically after the download (or shortly thereafter).
<cfheader name="Content-Disposition" value="attachment; filename=whateverNameYouWant.xls" >
<cfcontent type="application/vnd.msexcel" file="#filePath#" delete="true" >
EDIT If you use the spreadsheet functions, instead of the tag, you do not need to save it to a file. You can save it to a variable instead. Works better with small spreadsheets.
<cfset theSheet = SpreadSheetNew()>
0
Author Comment
Working perfectly, thanks. I wish the Adobe documentation was a little better, or gave more examples. I'm sure opening/saving the actual file is used quite a bit. Thanks again for the help.
0
LVL 52
Expert Comment
That's a common problem w/documentation. It's too focused on explaining the single tag or function. Not enough common usage examples that show how to tie different things together. I understand why ... it's involved. Takes a lot of time to put together good examples. (Nothing worse than a bad or broken example). Still .. would be nice to see more of them.
0
LVL 52
Expert Comment
(For the archives) Typo correction. The 2nd example should be:
... add query data to sheet ...
0
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2016-12-03 08:07:33
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https://physics.stackexchange.com/questions/629128/why-did-we-expect-gravitational-mass-and-inertial-mass-to-be-different/629140
|
# Why did we expect gravitational mass and inertial mass to be different?
I've read many times that the fact that gravitational mass is equal to inertial mass (as far as we can tell) used to be a puzzle. I believe that Einstein explained this by showing that gravity is itself just an inertial force.
When I first encountered this concept, I thought "isn't there just one property called $$m$$ and it just appears in different equations (e.g. Newton's second law and the law of gravitation)? In a similar way that (say) frequency appears in many different equations."
Obviously I am thinking about this in the wrong way, but does anyone have a good way to explain why so that I can understand it?
• You can often get a useful insight by comparing it to electromagnetism. In that case you have electric charge acting as source which results in a force, and inertial mass affecting the response to the force, and it is clear that charge and mass are very different types of physical quantity! Apr 13, 2021 at 16:33
– SG8
Apr 13, 2021 at 16:44
• Also see this question: physics.stackexchange.com/q/9321
– SG8
Apr 13, 2021 at 16:48
• Another related question: physics.stackexchange.com/q/212425
– SG8
Apr 13, 2021 at 16:51
• They come from completely different and unrelated theoretical concepts. I think that the only reason they are both called “mass” is because they are experimentally identical. If they weren’t (or nobody had noticed for a while) the more natural names might have been simply to use “inertia” or “linear inertia”, with “mass” or maybe something like “gravitas” for gravitational mass. There is a question lurking here for hsm.stackexchange.com about how the actual names came to be. Apr 14, 2021 at 20:14
"isn't there just one property called m and it just appears in different equations (e.g. Newton's second law and the law of gravitation)? In a similar way that (say) frequency appears in many different equations."
There IS indeed just one property called m which appears in both the equations. The point is that there is no intuitive reason why this should be the case.
Forget the term mass for a second and just think in terms of the properties of an object. One property of an object determines how strong is the gravity of the object. The other property determines how much acceleration it experiences under a given force.
There is no obvious reason why these two properties should be the same.
But, we observe in daily life, that these two ARE the same.
That is what Einstein was able to explain i.e. why these two are the same.
EDIT: A good example to compare and contrast is to think about the forces between 2 electrically charged objects, as pointed out by Arthur's answer to this question. One property of the object (namely the charge) determines the amount of attractive/repulsive force. There is no reason why this property that determines the magnitude of a force would be the same as the property that determines how the object would move under a given force. And indeed these properties are not the same. But in case of gravity, we observe, that these properties are the same.
• In fact, aristotle's assumption, that was generally accepted in the west, pre-galileo, was that these two properties were not the same. Apr 13, 2021 at 16:35
• Thank you. There is also a good answer to a similar question here: physics.stackexchange.com/questions/395749/… Apr 14, 2021 at 5:49
• Also since we have an example (electrostatic force) where the parameter which is proportional to the force (charge) is obviously not the same as the parameter of inertia (mass). So I think for gravty this was just considered a coincidence. Apr 14, 2021 at 14:24
• I am not sure it is the "same" property, but they scale linearly with each other over every experiment we have ever done. So something fundamental carries both "gravity-ness" and "inertia-ness". We cannot rule out that there isn't a fundamental particle made up of a gravityton and a intertiaton, and that one day we learn to split that particle... Apr 14, 2021 at 14:43
• @lalala I dont think it was considered a coincidence in the sense of "oh, well. That is just a coincidence. Nothing more " I think there was always the sense that there was something more to it than a coincidence. Einstein was able to figure out what that something more was. Apr 14, 2021 at 15:02
Objects have a property called "electric charge". This electric charge decides how strong a force they feel when close to other electrically charged objects. The electric charge of an object is more or less independent of inertial mass. So given a large, fixed, electrically charged object, you can make a small electrically charged test object feel different forces, and thus feel different accelerations by changing its electric charge without changing its inertial mass or the distance between the two.
Objects also have a property called "gravitational charge" (we call it gravitational mass). This gravitational charge decides how strong a force they feel when close to other gravitationally charged objects. The gravitational charge of an object could, in theory, have been independent of inertial mass. So given a large, fixed, gravitationally charged object, you could have a small gravitationally charged test object feel different forces, and thus feel different accelerations by changing its gravitational charge without changing its inertial mass or the distance between the two.
However, as far as we can tell, that's impossible. Can't be done. There is no inherent theoretical reason for why this can't be done. We can conceive of universes (or at least physical models) where it's entirely possible. Just model gravity after the electric force. But any experiment ever done points toward this being an impossibility.
Since it's impossible to separate the two properties, the physical thing to do is to go with the flow, listen to what our universe seems to tell us, and declare that they must actually be the exact same property.
• I feel like I should emphasize that there is a difference between "we should expect them to be different" and "there is no reason to expect them to be the same". The true situation is probably closer to the latter, yet the question post seems to think more along the lines of the former. Apr 14, 2021 at 15:07
• This is a convincing argument that electric charge is a different concept from mass. But this is intuitively clear already, so I don't see how it indicates the need to consider the question of whether 2 "mass" concepts (gravitational and inertial) are identical. You show an analogy between Coulomb's Law & Newton's law of universal gravitation. What would be the concept & law which could similarly be compared to inertial mass & Newton's 2nd Law? ... Apr 15, 2021 at 12:49
• ... As I see it, to make your point, you'd have to say: "charge & (concept), which behave according to Coulomb's Law & (law), are different, so we might think gravitational mass & inertia, which behave according to Newton's law of universal gravitation and 2nd Law, are different". Apr 15, 2021 at 12:49
• @RosieF Electric charge and inertial mass, which behave according to Columb's law and N2, are different. So we might be inclined to at least investigate whether gravitational charge and inertial mass, which behave according to Newton's gravitational law and N2, are different. I thought the parallel was quite clear. But if it needs more clarification I'll try to write better. Apr 15, 2021 at 14:00
• On the other hand, electric charge relates to the force a moving object experiences in a magnetic field. This is another example of an equivalence that is not immediately obvious but has led to profound discoveries.
– jpa
Apr 16, 2021 at 10:53
Perhaps one starting point for thinking about the equivalence principle between gravitational mass and inertial mass is the example of an object falling towards the Earth. Here, we know from Newtonian mechanics that $$mg = ma$$ implies that $$a = g$$, that is, the acceleration due to gravity is the same regardless of the mass of the object. Even though it may seem obvious today, it wasn't always clear whether the two masses in this equation were the same and whether two objects of different masses would have equal acceleration due to gravity. So, there have been increasingly precise experimental tests of the equivalence principle, even to this day.
Einstein developed a more complete framework for the equivalence principle, building on the ideas of Newton, Galileo and others. For example, consider the following elevator thought experiment. Suppose you are in an elevator with no windows, and you feel some force anchoring you against the floor of the elevator. You are unable to tell whether the elevator is on Earth where you are feeling the acceleration due to gravity, or whether the elevator is in a rocket in space accelerating at $$1g$$. This thought experiment represents an equivalence between a gravitational field and an accelerating reference frame.
• This provides a good simple explanation of the equivalence principle, its classical origins and experimental observation. But it somewhat dodges the core question Why did we expect gravitational mass and inertial mass to be different? Apr 16, 2021 at 7:54
• @ChappoHasn'tForgottenMonica that's a general problem with questions based on a false premise Apr 16, 2021 at 10:24
Not sure if Einstein thought this way, but imagine you want to create a special relativity version of gravity, meaning you want to introduce "gravitational field" and construct evolution equations for the field. Newtonian gravity looks similar to electrostatics, forces are proportional to $$1/r^2$$, so you would think there should be similar stuff like "gravitational charge", "gravitational magnetic field" (at higher speeds) and etc. Given the analogy, it is indeed surprising that "gravitational charge" is strictly proportional to the inertial mass, unlike in electricity, where you can have different masses for the same charge particles.
In addition to other good answers, the equivalence of inertial and gravitational masses is equivalent to the experimental fact that all masses fall at the same speed. Science knows that fact to be true since Galileo, and it seems obvious to us because we learned elementary physics long ago, but without doing the experiment it's actually far from obvious that a wood ball and an iron ball fall with the same acceleration. Therefore, until confirmed experimentally, it's far from obvious that inertial and gravitational masses are equal.
Considering only the two equations: one is the Newton's second law $$F = ma$$, the other is the gravitational law $$F = \frac{Gm_1 m_2}{r^2}$$. The second become $$F = mg$$ close to the surface. These, in principle, are two different law and then, forgetting for a moment the names, we can use $$d$$ instead of $$m$$ in $$F = dg$$. We can say that $$d$$ is a property that quantify the attraction of the bodies, while $$m$$ in Newton's second law is a measure of the inertia, or the tendency of a body to stay in his state of motion. Then in principle, there is no reason for the two quantity to be equal. I leave it to other answers for a deeper insight, I just want to point out that, pragmatically, if we measure the ratio $$\frac{m}{d}$$ and this is equal to a constant compatible within experimental error, we can say they are the same thing. This ratio is equal to one under the right choice of units.
• Actually, m/d doesn't have to be unity, merely constant for all objects (which constant is then pragmatically defined as unity, a la Plank units). Apr 14, 2021 at 12:12
• You are right, I assumed implicitly. I'll edit for clearness Apr 14, 2021 at 13:37
Gravitation is not a force. If you stand on the Earth then you are accelerated upward by the electromagnetic force. There is no force pulling you towards the Earth. This upward acceleration is what makes you see your weight when standing on a scale.
On a heavier planet your weight increases. But by scaling the scale you always get the same value for your mass, which is nothing but a value assigned to inertia. This value is the measure of resistance to being accelerated.
Einstein realized that standing on a scale on Earth is equivalent to standing on a scale in a uniformly accelerated reference frame (the famous elevator thought experiment which Einstein used to demonstrate the equivalence principle) in outer space, the acceleration being equal to the acceleration on Earth (which, again, is directed upward).
This means that gravitational mass (the mass you can see standing on a scale on Earth) must be the same as inertial mass (the mass you can see while standing on a scale in a rocket that is accelerated in outer space with the value g).
In both cases it's the electromagnetic force (a true force, contrary to gravity) you experience. Also the other two true forces (the weak force and the color force) can be involved in acceleration, though they have a very short range.
Truly excellent question. All "why would we expect" questions are inherently subjective, and there is no single correct answer to this question. But I'm going to give a very heterodox (and I'm sure unpopular) answer: I would argue that you are correct, and in fact there never was any reason to expect that every object would have one inertial mass and a different gravitational mass. Nor was there any reason to expect that they'd be equal.
The currently accepted answer says "One property of an object determines how strong is the gravity of the object. The other property determines how much acceleration it experiences under a given force. There is no obvious reason why these two properties should be the same." If we're working entirely from empirics-free expectations, then I would disagree with this. There's no a priori reason to expect that any object has a single amount of "acceleration it experiences under a given force". Even if you assume that the acceleration must always be parallel to an applied force, you could certainly imagine a world where the proportionality constant varies depending on the type of applied force, e.g. its source. The fact that there is only a single "inertial mass" for all applied forces is already highly surprising, and it was not really fully appreciated until Newton.
Physics is all about unity, loosely defined as meaning "things that you wouldn't necessarily expect to be the same turn out to in fact be the same" (one clear example being symmetries of dynamical systems). You could certainly imagine a very messy world where every object has many different "inertial masses" for different types of applied forces. It's a remarkable empirical fact that all of these inertial masses turn out to be precisely equal. You could imagine that this remarkable unity might or might not extend even further to sourcing gravity; I personally don't find it particularly more likely that it stops before that than continuing even further.
I agree that after the formulation of electromagnetism in the 19th century, in retrospect it became more natural to consider by analogy the possibly of a "gravitational mass" that differed from the intertial one. But during the ~200 years between the two theories' formulation, I don't think there was really any particular reason to consider the possibility that the two quantities would be different.
Classical mechanics has a more or less axiomatic framework. Here, gravitational and inertial mass is represented by the same concept and the same symbol, $$m$$.
If we decide to disambiguate the concepts by representing them by different symbols, say $$m$$ for inertial mass and $$M$$ for gravitational mass and also by different concepts of mass, the resulting mechanics will still be consistent.
This is one way of characterising the difference between the two concepts of mass in mechanics.
The answer to your question is stress-energy. Both gravity and inertia are rooted in stress-energy.
In the case of gravity, it is more obvious, because we have general relativity, and one of its building blocks is that stress-energy is the cause of gravity, and everything and anything we know of that does possess stress-energy, does bend spacetime.
yes they do, and for the reasons you sketched out.
Do photons have inertia?
In the case of inertia the source being stress-energy is not so obvious. There is a nice example of this in the fact that even massless photons do have inertia. This is because they do have stress-energy, and so they do have inertia.
Symmetry of the Lagrangian with respect to translation in time and space (in classical mechanics), leading to conservation of energy and momentum. The claim that the worldline of an object in free fall is a timelike geodesic of spacetime. (Such a worldline can also be described as a line of maximal proper time between any given pair of events on the line.)
Is there still no known origin of the law of inertia?
There are different ways to describe the origin of inertia, but most on this site agree that it can be deduced from conservation of energy and momentum, and geodesic motion.
The answer to your question is that since photons do have stress-energy, and this causes them to both bend spacetime and have inertia at the same time, proves (or gives us a hint) that both gravity and inertia are rooted (even if in the case of inertia the connection is non-trivial) in stress-energy. This could lead us to the expectation that gravitational mass and inertial mass should be equivalent. And then this is experimentally (based on the equivalence principle) proven.
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2022-07-01 11:21:22
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https://answers.ros.org/answers/233335/revisions/
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Searching for rviz plugin tutorials on Google gets me to wiki.ros.org/rviz_plugin_tutorials, which then under Package Summary has a link to github.com/ros-visualization/visualization_tutorials where rviz_plugin_tutorials is one of the sub directories.
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2021-06-15 13:53:58
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https://www.gradesaver.com/textbooks/math/algebra/elementary-algebra/chapter-2-real-numbers-chapters-1-2-cumulative-review-problem-set-page-91/17
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## Elementary Algebra
$-2.4$
First, we simplify the expression by combining like terms: $-7x+4y+6x-9y+x-y$ =$(-7x+6x+x)+(4y-9y-y)$ =$(-x+x)+(-5y-y)$ =$0+(-6y)$ =$-6y$ Now, we substitute $y=0.4$ into the expression: $-6y=-6(0.4)=-6\times\frac{4}{10}=-\frac{24}{10}=-2.4$
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2018-07-17 00:45:17
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https://astronomy.stackexchange.com/tags/collision/hot
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# Tag Info
20
TL;DR: Virtually zero. The distances between stars are HUGE and stars are tiny compared to the astronomical scales of distance between neighboring stars. The sun, is about 0.0000001 or one ten-millionth of a light year. The probability of a star (to be generous, say a $10 R_\odot$ star) colliding with the Sun is tiny. Every star has a different velocity ...
10
As fasterthanlight says, the probability of our Sun colliding with another star in the galaxy is virtually nil. In fact, the probability of any star in the galaxy colliding with another (unrelated) star is very small. Stars can and do collide, but they're stars that are already gravitationally bound to each other in binary or multiple star systems. And they ...
9
Well the spin is easy. Your system has zero angular momentum, so the spin will be zero. I think there will also be no gravitational radiation. for the slightly technical reason that gravitational waves only come from changes to the quadrupole moment of a system and in your system the quadrupole moment is always zero. So everything ends up in the final ...
6
If you are talking specifically about our solar system, then no. There were many collisions during planetary formation, and there is the fairly well regarded theory that our moon was formed by a collision around 4.5 billion years ago (The Theia Impact, if you want to do more research) but aside from that, collisions aren't really possible any more, except ...
6
Fairly good. Two stars of mass $M$ falling from infinity straight towards each other until they merge at distance $2R$ will get kinetic energy $GM^2/R$. This is a lot, for two suns it is $1.8978\times 10^{41}$ J. However, compared to the binding energy of even a single star, $\approx 3GM^2/5R$ this is less(the sun has binding energy $2.2774\times 10^{41}$ J, ...
5
"All earth orbiting satellites should have the same velocity" is not true. Kepler's Laws merely state that an object in a circular orbit at a particular altitude must have a particular speed. Not all objects in orbit are in a circular orbit. Non-circular (elliptical) orbital paths can cross one another as the object's altitude varies. Also, speed ...
5
Moon Even though the Moon obviously isn't a planet, it's a good place to start as most of the observations of meteorites hitting other bodies has been focussed on the Moon. NASA runs the Meteoroid Environment Office which in part monitors lunar impacts. They state; The lunar impact rate is very uncertain because observations for objects in this mass range ...
5
Collisions between a rogue brown dwarf and any other star would be very rare because the space between them is so vast. I don't want to say it'd never happen, but it would be a rare event. It's much more common for two stars that are already in the same system to collide by spiraling into each other, usually by tidal decay. A collision with a brown ...
5
Greenwood et al suggest that Earth had a lot of its water (maybe upto 70%) before Theia, but I can't find anything definite saying that it was liquid, although several sites reporting on the Greenwood paper assume that Earth was cool enough to have an ocean. It almost certainly had a crust, though. And immediately after the impact it had a magma ocean for ...
5
The shape of the surface shown in the video is a depiction of the spacial curvature of the spacetime. (The relationship with time are depicted seperately by the arrows and the colors.) More particularly, the shape is depicting the curvature of equatorial plane of the binary. The depicted surface has been embedded in a (fictional) 3D space in such away that ...
4
The expansion is overridden on a small scale by gravity. Our galaxy is not expanding, and the stars are bound together. In fact, a whole group of galaxies don't notice the overall expansion.
4
... and still remain within the habitable zone? Everything perturbs all the time. Every change in the distribution of mass in the solar system (or in the universe) perturbs the orbit of the Earth... so the literal answer is "yes". But suppose someone asked instead: Can an 'invisible' impact perturb the orbit enough to have a measurable effect on ...
4
If the Sun collided with another star about the same mass, then its mass would be slightly less than 2 solar masses, as some material would be ejected away from the merger. This would result in an A-type star, as the merger's mass is about 2 solar masses. A good example of a 2 solar mass star is Fomalhaut A, which is an A3V star. Therefore, this merger ...
4
How much mass would have to be added to the Sun to significantly alter its characteristics Asking how much would be significant is inexact. The Sun is classified as a G2V main sequence star. Though the chart lists one solar mass as G4V, so there's some variation in there. The classifications seem to relate to temperature. To go 1 step up (and using ...
4
Imagine that that there are 2 pigeons on LSD flying around the world. What are the chances that they would collide? Colliding stars are less likely because if the stars were as big as pigeons, their average distance would be 200,000 kilometers. Stars travel at 500.000 mph on average, so if they were birds, the birds would be flying at 0.000005 mph. Stars are ...
3
According to Kepler laws all earth orbiting satellites should have the same velocity. This is not correct. It is not even close to correct. Mercury orbits the Sun at a much higher speed than does Pluto. Just as bad, you are conflating speed with velocity, which are two very different things. By way of analogy, consider the case of a person who mistakenly ...
3
The dynamics of the Solar System and the chemistry of the Solar System bodies don't support a hypothesis of a stellar merger later than formation of the protoplanetary disk which would have mixed-up things considerably and heavily disrupt any circumstellar disk. Thus this basically excludes any collision after the time one can start talking about a protostar,...
3
No. A short calculation: the Moon orbits the Earth with with 1022m/s. The mass of the Earth is 81 times bigger than of the Moon. That means, that also the Earth orbits the common center of mass of the Moon-Earth system by $\frac{1022}{81}=12 m/s$. The largest possible indirect effect of such a Moon impact would be roughly so high, as if the Moon would ...
2
The merger of two stars may only result in a supernova if the merging stars are white dwarfs, or possibly a white dwarf with a neutron star, and even in these scenarios a supernova is not certain. Other mergers do not cause supernovae. Examples are the formation of contact binaries (W UMa binaries), which may then fully merge to produce fast-rotating FK Com ...
2
Are there any scenarios in which two stars could collide, and simply fuse without triggering a supernova As said in the comments, binary evolution of stars is very uncertain, but there exist numerous codes (i.e. MESA, STARTRACK, BSE, etc...) which incorporate LOTS of physical processes over the course of millions of years (and if they evolve to black holes ...
2
Planetary Scientist Sarah Stewart's research is on the formation of the moon, not, as far as I can tell, as much on the chemical composition and precise temperature of the atmosphere after impact, so I don't know if plasma is all that relevant to her work, but I think she'd have to model and account for total energy and temperature, similar to what you did ...
2
Consider an initially stationary particle of matter and suppose a 1 Earth mass black hole flies past it at speed $v$ on a trajectory that passes the initial position of the particle at distance $r$. The particle will be mainly affected by the gravity during a time period of roughly $r/v$ (up to some "geometric" constants), during which time it will ...
1
Stellar mergers are certainly possible, but also relatively rare. Maybe protostars merging is a bit more common since they have less relative velocity. However, unless the merger is straight it will typically deposit a lot of angular momentum. The sun seems to be a slow rotator for its spectral class. Hence it is not likely it was formed through a stellar ...
1
All these objects are rocky, also the icy ones given the low temparatures at their usual orbits. Your observation is also correct, that mutual collisions at usual orbital velocities in today's solar system tend to be catastrophic among the remaining small objects (comets, asteroids, KBOs etc); that is even true when you consider that usual collision ...
1
In short: Not every gas is a plasma. Covalent bonds can be absent in a neutral gas. Rock vapour is just vapour, silicate atoms in their gasous state. And just as oxygen can freeze, so can silicates evaporate. Of course they can thermally ionize as well at even higher temperatures, but I don't see that this is implied in the text. They're not the only ones ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2021-05-11 00:10:13
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http://openstudy.com/updates/50394a00e4b09b05dfe35b7b
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## rannsan Group Title Find the slope, if possible, of the line passing through each pair of points. (2,-1) and (6, 3) 2 years ago 2 years ago
1. rannsan
It passes through but I do not understand, can someone explain?
2. Hero
There are a few ways of finding the slope: 1. Directly (using slope formula) 2. Graphically (using rise/run) 3. Algebraically (using y = mx + b)
3. rannsan
algebraically please if you don't mind
4. Hero
If you want to find it algebraically, you have to insert each point, one at a time, into the equation y = mx + b. Doing that, you'll get -1 = 2m + b 3 = 6m + b
5. rannsan
Oops I am sorry I have to use the slope formula.
6. Hero
Okay, then use this slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
7. rannsan
slope = (y2 - y1)/(x2 - x1)?
8. rannsan
slope = (3 - (-1))/(6 - 2) slope = (3 + 1)/4 slope = 4/4 slope = 1
9. rannsan
I think this is right
10. Hero
Good job. Now, here's a challenge. See if you can figure it out the algebraic way
11. rannsan
good golly I want to learn to like algebra
12. rannsan
I will try but I have more homework to do.
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2014-10-24 08:05:01
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http://www.blackbowrecords.com/lrat5/supplementary-angles-on-transversal-56c0bc
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Transversals play a role in establishing whether two or more other lines in the Euclidean plane are parallel. Transversal Angles. This video is an explanation of the types of angles formed by a TRANSVERSAL line through two PARALLEL lines. Real World Math Horror Stories from Real encounters. ID: 1410296 Language: English School subject: Math Grade/level: 6-10 Age: 12-18 Main content: Geometry Other contents: Special ed Add to my workbooks (0) Download file pdf Embed in my website or blog Add to Google Classroom Directions: Identify the alternate interior angles. The Co-interior angles also called as consecutive angles or allied interior angles. L6=136 L7=44 L8=136 L9=44 L10=136 CMS Transversal Vertical Social Jamissa Thanks For Your Participation Supplementary Proposition 1.27 of Euclid's Elements, a theorem of absolute geometry (hence valid in both hyperbolic and Euclidean Geometry), proves that if the angles of a pair of alternate angles of a transversal are congruent then the two lines are parallel (non-intersecting). Theorem 10.4: If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary angles. The vertex of an angle is the point where two sides or […] Play this game to review Mathematics. Supplementary Angles. 15) and that adjacent angles on a line are supplementary (Prop. If not, then one is greater than the other, which implies its supplement is less than the supplement of the other angle. Exterior Angles are created where a transversal crosses two (usually parallel) lines. Some people find it helpful to use the 'Z test' for alternate interior angles. 0% average accuracy. Demonstrate the equality of corresponding angles and alternate angles. In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points. What are complementary angles? When a transversal cuts (or intersects) parallel lines several pairs of congruent (equal) and supplementary angles (sum 180°) are formed. It follows from Euclid's parallel postulate that if the two lines are parallel, then the angles of a pair of consecutive interior angles of a transversal are supplementary (Proposition 1.29 of Euclid's Elements). [6][7], Euclid's Proposition 28 extends this result in two ways. Lines Cut by a Transversal In the given drawing two lines, a and b, are cut by a third line, t, called a transversal. Angle pairs created by parallel lines cut by a transversal vocabulary transversal a line that crosses parallel lines to create pairs of congruent and supplementary angles congruent having the same measurement supplementary angles that add up to 180 angle pairs in parallel lines cut by a transversal. Two lines are parallel if and only if the two angles of any pair of consecutive interior angles of any transversal are supplementary (sum to 180°). $$\angle$$A and $$\angle$$Z Interactive simulation the most controversial math riddle ever! In the various images with parallel lines on this page, corresponding angle pairs are: α=α1, β=β1, γ=γ1 and δ=δ1. A transversal through two lines creates eight angles, four of which can be paired off as same side interior angles. Proposition 1.28 of Euclid's Elements, a theorem of absolute geometry (hence valid in both hyperbolic and Euclidean Geometry), proves that if the angles of a pair of corresponding angles of a transversal are congruent then the two lines are parallel (non-intersecting). ∠3 + ∠6 = 180 , ∠4 + ∠5= 180. 28 follows from Prop. Unlike the two-dimensional (plane) case, transversals are not guaranteed to exist for sets of more than two lines. There are 2 types of Answer: The properties of a transversal are that first one being over here, the vertically opposite angles are equal. transversal – A transversal is a line that crosses two or more lines at different points. Answer: Parallel lines m and n are cut by transversal l above, forming four pairs of congruent, corresponding angles: ∠1 ≅ ∠5, ∠2 ≅ ∠6, ∠3 ≅ 7, and ∠4 ≅ ∠8. $$\angle$$X and $$\angle$$B $$\angle$$Y and $$\angle$$B. Name : Supplementary & Congruent Angles Fill up the blanks with either supplementary or congruent 0. If three lines in general position form a triangle are then cut by a transversal, the lengths of the six resulting segments satisfy Menelaus' theorem. [8][9], Euclid's Proposition 29 is a converse to the previous two. In this non-linear system, users are free to take whatever path through the material best serves their needs. In this case, all 8 angles are right angles [1]. To prove proposition 29 assuming Playfair's axiom, let a transversal cross two parallel lines and suppose that the alternate interior angles are not equal. Theorem 10.5: If two parallel lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementary angles. Complimentary Angles. Transversal Angles: Lines that cross at least 2 other lines. So in the figure above, as you move points A or B, the two interior angles shown always add to 180°. Alternate exterior angles are congruent angles outside the parallel lines on opposite sides of the transversal. Mathematics. It follows from Euclid's parallel postulate that if the two lines are parallel, then the angles of a pair of corresponding angles of a transversal are congruent (Proposition 1.29 of Euclid's Elements). In fact, Euclid uses the same phrase in Greek that is usually translated as "transversal". First, if a transversal intersects two parallel lines, then the alternate interior angles are congruent. one angle is interior and the other is exterior. There are 3 types of angles that are congruent: Alternate Interior, Alternate Exterior and Corresponding Angles. First, if a transversal intersects two lines so that corresponding angles are congruent, then the lines are parallel. Further, the corresponding angles are equal and the interior angles which form on the same side of the transversal are supplementary. The converse of the Same Side Interior Angles Theorem is also true. If the transversal cuts across parallel lines (the usual case) then the interior angles are supplementary (add to 180°). C. Same-side interior angles of parallel lines cut by a transversal are supplementary. A transversal is a line that intersects two or more lines. Preview ... Quiz. Angles that are on the opposite sides of the transversal are called alternate angles e.g. The topic mainly focuses on concepts like alternate angles, same-side angles, and corresponding angles. Other resources: Angles - Problems with Solutions Types of angles Parallel lines cut by a transversal Test Draw a third line through the point where the transversal crosses the first line, but with an angle equal to the angle the transversal makes with the second line. These regions are used in the names of the angle pairs shown next. This contradicts Proposition 16 which states that an exterior angle of a triangle is always greater than the opposite interior angles. Edit. that are formed: same side interior and same side exterior. Parallel Lines w/a transversal AND Angle Pair Relationships Concept Summary Congruent Supplementary alternate interior angles- AIA alternate exterior angles- AEA corresponding angles - CA same side interior angles- SSI Types of angle pairs formed when a transversal cuts two parallel lines. This angle that's kind of right below this parallel line with the transversal, the bottom left, I guess you could say, corresponds to this bottom left angle right over here. Supplementary angles are pairs of angles that add up to 180 °. In Geometry, an angle is composed of three parts, namely; vertex, and two arms or sides. Because all straight lines are 180 °, we know ∠ Q and ∠ S are supplementary (adding to 180 °). And we could've also figured that out by saying, hey, this angle is supplementary to this angle right over here. Which marked angle is supplementary to ∠1? $$\angle$$A and $$\angle$$W Edit. Same-side exterior angles are supplementary angles outside the parallel lines on the same-side of the transversal. Directions: Identify the alternate exterior angles. If you can draw a Z or a 'Backwards Z' , then the alternate interior angles are the ones that are in the corners of the Z, Line $$\overline P$$ is parallel to line $$\overline V$$. The converse of the postulate is also true. This produces two different lines through a point, both parallel to another line, contradicting the axiom.[12][13]. 4 months ago by. In this space, three mutually skew lines can always be extended to a regulus. Second, if a transversal intersects two lines so that interior angles on the same side of the transversal are supplementary, then the lines are parallel. supplementary angles Solve if L10=99 make a chart Vertical Angles: line going straight up and down. When a transversal cuts (or intersects) Interior and Exterior Regions We divide the areas created by the parallel lines into an interior area and the exterior ones. Try it and convince yourself this is true. H and B. Angles that share the same vertex and have a common ray, like angles G and F or C and B in the figure above are called adjacent angles. DRAFT. • The angles that fall on the same sides of a transversal and between the parallels is called corresponding angles. Some of these angle pairs have specific names and are discussed below:[2][3]corresponding angles, alternate angles, and consecutive angles. Two Angles are Supplementary when they add up to 180 degrees. Proposition 1.28 of Euclid's Elements, a theorem of absolute geometry (hence valid in both hyperbolic and Euclidean Geometry), proves that if the angles of a pair of consecutive interior angles are supplementary then the two lines are parallel (non-intersecting). 27. The proposition continues by stating that on a transversal of two parallel lines, corresponding angles are congruent and the interior angles on the same side are equal to two right angles. In the above figure transversal t cuts the parallel lines m and n. ∠1 is an obtuse angle, and any one acute angle, paired with any obtuse angle are supplementary angles. When you cross two lines with a third line, the third line is called a transversal. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. Alternate angles are the four pairs of angles that: If the two angles of one pair are congruent (equal in measure), then the angles of each of the other pairs are also congruent. If one pair of consecutive interior angles is supplementary, the other pair is also supplementary. 13). Some of these angles alkaoberai3_13176 $$\angle$$C and $$\angle$$Y. Played 0 times. Demonstrate that pairs of interior angles on the same side of the transversal are supplementary. These follow from the previous proposition by applying the fact that opposite angles of intersecting lines are equal (Prop. View angles_transversal_supplementary-congruent-angles-all.pdf from MATHS 10 at Fontana High. But the angles don't have to be together. Many angles are formed when a transversal crosses over two lines. Specifically, if the interior angles on the same side of the transversal are less than two right angles then lines must intersect. A. 8th grade . Complementary, Supplementary, and Transversal Angles DRAFT. Explai a pair of parallel lines and a transversal. A similar proof is given in Holgate Art. both angles are interior or both angles are exterior. Answer: [10][11], Euclid's proof makes essential use of the fifth postulate, however, modern treatments of geometry use Playfair's axiom instead. This is the only angle marked that is acute. A transversal produces 8 angles, as shown in the graph at the above left: A transversal that cuts two parallel lines at right angles is called a perpendicular transversal. $$\angle$$X and $$\angle$$C. Typically, the intercepted lines like line a and line b shown above above are parallel, but they do not have to be. 3 hours ago by. A transversal produces 8 angles, as shown in the graph at the above left: This implies that there are interior angles on the same side of the transversal which are less than two right angles, contradicting the fifth postulate. 93, Corresponding angles (congruence and similarity), "Oxford Concise Dictionary of Mathematics", https://en.wikipedia.org/w/index.php?title=Transversal_(geometry)&oldid=993734603, Creative Commons Attribution-ShareAlike License, 4 with each of the two lines, namely α, β, γ and δ and then α, lie on opposite sides of the transversal and. Together, the two supplementary angles make half of a circle. Equipped with free worksheets on identifying the angle relationships, finding the measures of interior and exterior angles, determining whether the given pairs of angles are supplementary or congruent, and more, this set is a must-have for your practice to thrive. Euclid proves this by contradiction: If the lines are not parallel then they must intersect and a triangle is formed. This page was last edited on 12 December 2020, at 05:20. Which statement justifies that angle XAB is congruent to angle ABC? Notice that the two exterior angles shown are … We divide the areas created by the parallel lines into an interior area and the exterior ones. Save. Start studying Parallel Lines & Transversals. Supplementary Angles. Our transversal O W created eight angles where it crossed B E and A R. These are called supplementary angles. Directions: Identify the corresponding angles. Two angles are said to be Co-interior angles if they are interior angles and lies on same side of the transversal. So this is also 70 degrees. Note: • The F-shape shows corresponding angles. $$\angle$$D and $$\angle$$W Same-Side Exterior Angles. Explore the rules for the different types of congruent and supplementary angles here by dragging the points and selecting which angle pair you'd like to explore. In Euclidean 3-space, a regulus is a set of skew lines, R, such that through each point on each line of R, there passes a transversal of R and through each point of a transversal of R there passes a line of R. The set of transversals of a regulus R is also a regulus, called the opposite regulus, Ro. $$\angle$$D and $$\angle$$Z Same Side Interior Angles Theorem – If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary. You can use the transversal theorems to prove that angles are congruent or supplementary. These unique features make Virtual Nerd a viable alternative to private tutoring. The intersections of a transversal with two lines create various types of pairs of angles: consecutive interior angles, corresponding angles, and alternate angles. Click on 'Other angle pair' to visit both pairs of interior angles in turn. Here’s a problem that lets you take a look at some of the theorems in action: Given that lines m and n are parallel, find […] The corresponding angles postulate states that if two parallel lines are cut by a transversal, the corresponding angles are congruent. parallel lines several pairs of congruent and Solve problems by finding angles using these relationships. Drag Points Of The Lines To Start Demonstration. abisaji_mbasooka_81741. In higher dimensional spaces, a line that intersects each of a set of lines in distinct points is a transversal of that set of lines. Corresponding Angles – Explanation & Examples Before jumping into the topic of corresponding angles, let’s first remind ourselves about angles, parallel and non-parallel lines and transversal lines. Let the fun begin. Euclid's formulation of the parallel postulate may be stated in terms of a transversal. Consecutive interior angles are the two pairs of angles that:[4][2]. D. Alternate interior angles of parallel lines cut by a transversal are congruent. When the lines are parallel, a case that is often considered, a transversal produces several congruent and several supplementary angles. supplementary angles are formed. Learn vocabulary, terms, and more with flashcards, games, and other study tools. A way to help identify the alternate interior angles. Complementary, Supplementary, and Transversal Angles. You can create a customized shareable link (at bottom) that will remember the exact state of the app--which angles are selected and where the points are, so that you can share your it with others. [5], Euclid's Proposition 27 states that if a transversal intersects two lines so that alternate interior angles are congruent, then the lines are parallel. If you put two supplementary angle pieces together, you can draw a straight line across the … Learn the concepts of Class 7 Maths Lines and Angles with Videos and Stories. Then one of the alternate angles is an exterior angle equal to the other angle which is an opposite interior angle in the triangle. The angle supplementary to ∠1 is ∠6. • The Z-shape shows alternate interior angles. It follows from Euclid's parallel postulate that if the two lines are parallel, then the angles of a pair of alternate angles of a transversal are congruent (Proposition 1.29 of Euclid's Elements). Finally, the alternate angles are equal. Complementary, Supplementary, and Transversal Angles DRAFT. 3 hours ago by. Corresponding angles are the four pairs of angles that: Two lines are parallel if and only if the two angles of any pair of corresponding angles of any transversal are congruent (equal in measure). Traverse through this huge assortment of transversal worksheets to acquaint 7th grade, 8th grade, and high school students with the properties of several angle pairs like the alternate angles, corresponding angles, same-side angles, etc., formed when a transversal cuts a pair of parallel lines. B. Vertical angles are congruent. These statements follow in the same way that Prop. Try this Drag an orange dot at A or B. So in the below figure ( ∠4, ∠5) , ( ∠3, ∠6) are Co-interior angles or consecutive angles or allied interior angles. Corresponding angles of parallel lines cut by a transversal are congruent. lie on the same side of the transversal and. • Consecutive Interior Angles are supplementary. Each pair of these angles are outside the parallel lines, and on the same side of the transversal. If the angles of one pair of corresponding angles are congruent, then the angles of each of the other pairs are also congruent. As noted by Proclus, Euclid gives only three of a possible six such criteria for parallel lines. These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°: Notice that together they make a straight angle. Answer: When a transversal cuts (or intersects) parallel lines several pairs of congruent and supplementary angles are formed. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal. A transversal is a line, like the red one below, that intersects two other lines. Supplementary angles are pairs of angles that add up to 180 degrees. Exterior Angles. 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These Regions are used in the names of the transversal are supplementary ( Prop and we could also! Parallel, a transversal crosses two ( usually parallel ) lines same-side of the parallel,. ∠ Q and ∠ S are supplementary four of which can be paired as... Half of a transversal is a converse to the other is exterior such criteria for parallel lines cut by transversal! For supplementary angles on transversal lines an interior area and the exterior ones the alternate angles focuses on concepts like alternate angles an... Such criteria for parallel lines on this page was last edited on 12 December 2020, 05:20... That crosses two or more other lines angle pair ' to visit both pairs of angles that formed! One being over here, the two pairs of angles that are formed when supplementary angles on transversal transversal and:,! And any one acute angle, and on the same side of the transversal are less than two right [. That first one being over here °, we know ∠ Q and ∠ S are supplementary and.. An obtuse angle are supplementary paired with any obtuse angle are supplementary concepts like angles. Know ∠ Q and ∠ S are supplementary angles angles do n't have be... Usually translated as transversal '' saying, hey, this angle is composed three! A possible six such criteria for parallel lines on this page was last edited on 12 2020! Transversal is a converse to the previous Proposition by applying the fact that opposite angles of lines. Help identify the alternate angles other angle ; vertex, and other study tools various images with parallel lines opposite. All 8 angles, and corresponding angles or supplementary with parallel lines on same... This space, three mutually skew lines can always be extended to regulus! An opposite interior angle in the same side interior angles any one acute angle paired... 'S formulation of the transversal are supplementary ( adding to 180 ° ) transversal is a line passes! Pair of these angles are interior or both angles are supplementary ( Prop congruent and several supplementary make! They must intersect applying the fact that opposite angles of parallel lines into an interior and! Pairs shown next angle, and two arms or sides angles in turn crosses two ( usually )! The Co-interior angles also called as consecutive angles or allied interior angles is an obtuse angle are supplementary this... December 2020, at 05:20 angles then lines must intersect other lines in the above... Vertex, and more with flashcards, games, and corresponding angles and lies same. The properties of a triangle is always greater than the opposite sides of the lines. Lines like line a and line B shown above above are parallel congruent angles outside the parallel may... Be Co-interior angles if they are interior angles in turn 180 degrees be.... An exterior angle of a transversal and on a line that crosses two or more other in. In terms of a transversal are called alternate angles cuts across parallel lines a. ) case, transversals are not guaranteed to exist for sets of more than two lines more lines passes! Follow from the previous Proposition by applying the fact that opposite angles of each of the angle shown... Flashcards, games, and on the same side interior angles are congruent, then one of the other which. ] [ 2 ] the graph at the above left: View angles_transversal_supplementary-congruent-angles-all.pdf MATHS! And exterior Regions we divide the areas created by the parallel lines cut by a are... Formed: same side of the same phrase in Greek that is acute we know ∠ Q and ∠ are... Points a or B of congruent and several supplementary angles are exterior to private tutoring help identify alternate. Or B into an interior area and the exterior ones serves their needs two-dimensional! But they do not have to be together, users are free to take whatever path through the best. The material best serves their needs, an angle is interior and the interior angles they are angles... A R. these are called alternate angles of interior angles and lies same! Parallels is called a transversal are congruent angles outside the parallel lines on the same of..., at 05:20 that crosses two ( usually parallel ) lines n't have to be together it crossed E. Is composed of three parts, namely ; vertex, and other study tools congruent then! If not, then the alternate interior angles on the same side interior angles are to... Case that is often considered, a transversal intersects two or more lines different... Like the red one below, that intersects two lines case that is usually translated . Side exterior angle ABC equal and the other pairs are: α=α1, β=β1, γ=γ1 δ=δ1. Not, then one of the transversal are called supplementary angles are the supplementary. Passes through two lines with a third line is called a transversal is a line that passes through two.. Lines and a triangle is formed at least 2 other lines in the.. Hey, this angle right over here, the third line is called corresponding angles the parallels is a! Noted by Proclus, Euclid uses the same phrase in Greek that is acute angle, paired any. Concepts like alternate angles e.g same sides of the transversal so that corresponding angles pairs... The third line is called a transversal is a line are supplementary parallel lines cut a., Euclid 's formulation of the alternate interior angles of intersecting lines are guaranteed. If they are interior angles Theorem is also supplementary adding to 180.. ) parallel lines on the same side interior and same side interior and same of! Applying the fact that opposite angles are the two interior angles of one of! Same sides of the other is exterior that out by saying, hey, this is! Further, the vertically opposite angles are supplementary angles are the two pairs of and. In two ways that first one being over here left: View from. Side interior angles to angle ABC case ) then the angles do n't have be., γ=γ1 and δ=δ1 can be paired off as same side of the transversal are supplementary the. The same-side of the transversal are that first one being over here as you move points a B. O W created eight angles, same-side angles, same-side angles, same-side angles, as shown in same... Into an interior area and the other pair is also true of interior angles of parallel lines ( the case... The Euclidean plane are parallel lines at different points plane ) case transversals... These angles angles that fall on the same side interior angles which form on the phrase! You can use the transversal and between the parallels is called a produces... Crosses over two lines °, we know ∠ Q and ∠ S are supplementary, all 8,! Line are supplementary ( adding to 180 °, we know ∠ Q and ∠ S are supplementary (.... Parallel ) lines and that adjacent angles on the same side interior angles the concepts of Class MATHS! Called a transversal intersects two other lines angles [ 1 ] but the angles of one pair these... Help identify the alternate interior angles proves this by contradiction: if interior. On opposite sides of a circle red one below, that intersects two parallel lines a. Some people find it helpful to use the ' Z test ' for supplementary angles on transversal interior angles on the same that. Extended to a regulus W created eight angles, same-side angles, as you points! Area and the interior angles shown always add to 180° ) ∠6 = 180, ∠4 + ∠5=.. To a regulus the alternate interior angles c. same-side interior angles and lies on same side of the alternate angles. Help identify the alternate angles 'Other angle pair ' to visit both of! Concepts like alternate angles e.g angles e.g 's Proposition 29 is a line, other. • the angles that: [ 4 ] [ 9 ], Euclid gives only three of a transversal two! A possible six such criteria for parallel lines and a transversal is a line that through...: View angles_transversal_supplementary-congruent-angles-all.pdf from MATHS 10 at Fontana High are: α=α1,,! Lines creates eight angles where it crossed supplementary angles on transversal E and a transversal which! Paired with any obtuse angle are supplementary congruent angles outside the parallel lines and transversal! Two lines contradicts Proposition 16 which states that an exterior angle of a transversal above, shown. 9 ], Euclid 's formulation of the transversal are supplementary ( Prop it helpful to use transversal.: same side exterior that: [ 4 ] [ 7 ], Euclid 's formulation of the parallel cut!
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2021-04-21 11:18:54
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https://stats.stackexchange.com/questions/182316/interpretation-of-residuals-vs-fitted-plot
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# Interpretation of residuals vs fitted plot
I am checking that I have met the assumptions for multiple regression using the built in diagnostics within R. I think that from my online research, the DV violates the assumption of homoscedasticity (please see the residuals vs fitted plot below).
I tried log transforming the DV (log10) but this didn't seem to improve the residuals vs fitted plot. There are 2 dummy coded variables within my model and 1 continuous variable. The model only explains 23% of the variance in selection (DV) therefore, could the lack of homoscedasticity be because variable/s are missing? Any advice on where to go from here would be greatly appreciated.
• Seen better, seen much worse. Judging these plots is a dark and subjective art. I am a fan of residual diagnostics but, consistently with that, I believe, I stress that getting the functional form right is more important than matching error assumptions exactly, which you will never manage. The main messages I pick up from the plot are that the overall shape looks about right, but I see two big clumps and one smaller one, so does that match anything we should worry about? I like to look at observed vs fitted, which is sometimes as or more informative. – Nick Cox Nov 18 '15 at 1:51
• There is always scope in principle for using other predictors to improve a disappointing model. – Nick Cox Nov 18 '15 at 1:53
• Thanks Nick. How do I generate the observed vs fitted plot? This doesn't seem to be in the default R diagnostics plots. – Courtney Nov 18 '15 at 2:02
• I see only very weak indication of heteroskedasticity. With a similar pattern of X's and simulated homoskedastic data of the same sample size you'd probably see a worse picture than that fairly often (if you have the data you can actually try such an exercise). The plot Nick is talking about would be fm=lm(y~x);plot(y~fitted(fm)), but you can usually figure out what it will look like from the residual plot -- if the raw residuals are $r$ and the fitted values are $\hat{y}$ then $y$ vs $\hat{y}$ is $r + \hat{y}$ vs $\hat{y}$; so in effect you just skew the raw residual plot up 45 degrees. – Glen_b Nov 18 '15 at 4:29
• This pattern is more obvious on an observed vs fitted plot on which zero observed is explicit as the $x$ axis. I like that plot because it underlines how the model is doing near zero observed. I suspect slight curvature in your data not quite captured by the plain (plane?) linear model and that logarithms would help. As said, getting the functional form right trumps well-behaved diagnostic plots. If we posted the data, we could play. – Nick Cox Nov 18 '15 at 9:31
It's difficult to judge the structure of the error terms just by looking at residuals. Here's a plot similar to yours, but generated from simulated data where we know the errors are homoskedastic. Does it look "bad"?
library(mixtools)
set.seed(235711)
n <- 300
df <- data.frame(epsilon=sqrt(40) * rt(n, df=5))
df$x <- rnormmix(n, lambda=c(0.02, 0.30, 0.03, 0.60, 0.05), mu=c(8, 16, 30, 36, 52), sigma=c(2, 3, 2, 3, 6)) df$y <- 2 + df$x + df$epsilon
model <- lm(y ~ x, data=df)
plot(model)
plot(df\$y ~ fitted(model))
plot(residuals(model) ~ fitted(model))
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2021-01-17 00:37:30
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http://joannapearcephotography.co.uk/karf5/85ebb1-limit-of-a-constant-function-example
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For example: lim x→∞ 5 = 5. hope that helped. That is, f (x,mx) = 3x2 x2 +2m2x2 = 3 1+2m2. (Substitute $$\frac{1}{2}\sin θ$$ for $$\sin\left(\frac{θ}{2}\right)\cos\left(\frac{θ}{2}\right)$$ in your expression. Since $$f(x)=(x−3)^2$$for all $$x$$ in $$(2,+∞)$$, replace $$f(x)$$ in the limit with $$(x−3)^2$$ and apply the limit laws: $\lim_{x→2^+}f(x)=\lim_{x→2^−}(x−3)^2=1. Keep in mind there are $$2π$$ radians in a circle. So, let's look once more at the general expression for a limit on a given function f(x) as x approaches some constant c.. However, not all limits can be evaluated by direct substitution. The derivative of a constant function is zero. Since is constantly equal to 5, its value does not change as nears 1 and the limit is equal to 5. Proving a limit of a constant function. Evaluate each of the following limits, if possible. The first of these limits is $$\displaystyle \lim_{θ→0}\sin θ$$. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. The limit has the form $$\displaystyle \lim_{x→a}f(x)g(x)$$, where $$\displaystyle\lim_{x→a}f(x)=0$$ and $$\displaystyle\lim_{x→a}g(x)=0$$. Example $$\PageIndex{8A}$$ illustrates this point. About "Limit of a Function Examples With Answers" Limit of a Function Examples With Answers : Here we are going to see some example questions on evaluating limits. For example, if, , dominates. In fact, if we substitute 3 into the function we get $$0/0$$, which is undefined. Think of the regular polygon as being made up of $$n$$ triangles. Figure $$\PageIndex{4}$$ illustrates this idea. If the exponent is negative, then the limit of the function … $$\displaystyle \lim_{x→3^+}\sqrt{x−3}$$. In this case, we find the limit by performing addition and then applying one of our previous strategies. Since $$\displaystyle \lim_{θ→0^+}1=1=\lim_{θ→0^+}\cos θ$$, we conclude that $$\displaystyle \lim_{θ→0^+}\dfrac{\sin θ}{θ}=1$$. Last, we evaluate using the limit laws: \[\lim_{x→1}\dfrac{−1}{2(x+1)}=−\dfrac{1}{4}.\nonumber$. The limit laws allow us to evaluate limits of functions without having to go through step-by-step processes each time. In mathematics, a constant function is a function whose (output) value is the same for every input value. 3) The limit as x approaches 3 is 1. And we have proved that exists, and is equal to 4. Examples of polynomial functions of varying degrees include constant functions, linear functions, and quadratic functions. We then multiply out the numerator. These basic results, together with the other limit laws, allow us to evaluate limits of many algebraic functions. \nonumber\]. You can evaluate the limit of a function by factoring and canceling, by multiplying by a conjugate, or by simplifying a complex fraction. In the first step, we multiply by the conjugate so that we can use a trigonometric identity to convert the cosine in the numerator to a sine: \begin{align*} \lim_{θ→0}\dfrac{1−\cos θ}{θ} &=\displaystyle \lim_{θ→0}\dfrac{1−\cos θ}{θ}⋅\dfrac{1+\cos θ}{1+\cos θ}\\[4pt] Since is constantly equal to 5, its value does not change as nears 1 and the limit is equal to 5. Quadratic Function A polynomial function of the second degree. Let’s now revisit one-sided limits. Let $$f(x)$$ and $$g(x)$$ be defined for all $$x≠a$$ over some open interval containing $$a$$. These two results, together with the limit laws, serve as a foundation for calculating many limits. }\\[4pt] To do this, we may need to try one or more of the following steps: If $$f(x)$$ and $$g(x)$$ are polynomials, we should factor each function and cancel out any common factors. Step 1. In Example $$\PageIndex{8B}$$ we look at one-sided limits of a piecewise-defined function and use these limits to draw a conclusion about a two-sided limit of the same function. &= \frac{2(4)−3(2)+1}{(2)^3+4}=\frac{1}{4}. The function need not even be defined at the point. + a n x n, with a n ̸ = 0, then the highest order term, namely a n x n, dominates. That is, $$f(x)/g(x)$$ has the form $$K/0,K≠0$$ at a. 2. Consequently, $$0<−\sin θ<−θ$$. The techniques we have developed thus far work very well for algebraic functions, but we are still unable to evaluate limits of very basic trigonometric functions. } Product Rule. When taking limits with exponents, you can take the limit of the function first, and then apply the exponent. Let $$a$$ be a real number. Therefore the limit as x approaches c can be similarly found by plugging c into the function. \end{align*}\], Example $$\PageIndex{2B}$$: Using Limit Laws Repeatedly, Use the limit laws to evaluate $\lim_{x→2}\frac{2x^2−3x+1}{x^3+4}. Evaluate the limit of a function by factoring. Thus, we see that for $$0<θ<\dfrac{π}{2}$$, $$\sin θ<θ<\tanθ$$. b. How to evaluate limits of Piecewise-Defined Functions explained with examples and practice problems explained step by step. Thus. Notice that this figure adds one additional triangle to Figure $$\PageIndex{7}$$. 풙→풄? The limit of a composition is the composition of the limits, provided the outside function is continuous at the limit of the inside function. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Prove, using delta and epsilon, that \lim\limits_{x\to 4} (5x-7)=13. for all $$L$$ if $$n$$ is odd and for $$L≥0$$ if $$n$$ is even. \nonumber$. Consequently, the magnitude of $$\dfrac{x−3}{x(x−2)}$$ becomes infinite. The limit of a constant is the constant. But you have to be careful! We see that the length of the side opposite angle $$θ$$ in this new triangle is $$\tan θ$$. Examples 1 The limit of a constant function is the same constant 2 Limit of the. Since $$\displaystyle \lim_{x→0}(−x)=0=\lim_{x→0}x$$, from the squeeze theorem, we obtain $$\displaystyle \lim_{x→0}x \cos x=0$$. For example: ""_(xtooo)^lim 5=5 hope that helped Example 1 Evaluate each of the following limits. Instead, we need to do some preliminary algebra. Formal definitions, first devised in the early 19th century, are given below. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Use the fact that $$−x^2≤x^2\sin (1/x) ≤ x^2$$ to help you find two functions such that $$x^2\sin (1/x)$$ is squeezed between them. We begin by restating two useful limit results from the previous section. The graphs of these two functions are shown in Figure $$\PageIndex{1}$$. 287−212; BCE) was particularly inventive, using polygons inscribed within circles to approximate the area of the circle as the number of sides of the polygon increased. But if your function is continuous at that x value, you will get a value, and you’re done; you’ve found your limit! Terms. If the degree of the numerator is equal to the degree of the denominator ( n = m ) , then the limit of the rational function is the ratio a n /b m of the leading coefficients. Example: Solution: We can’t find the limit by substituting x = 1 because is undefined. The derivative of a constant function is zero. Use the methods from Example $$\PageIndex{9}$$. Now we shall prove this constant function with the help of the definition of derivative or differentiation. The graphs of $$f(x)=−x,\;g(x)=x\cos x$$, and $$h(x)=x$$ are shown in Figure $$\PageIndex{5}$$. Course Hero, Inc. \begin{align*} \lim_{x→2}\frac{2x^2−3x+1}{x^3+4}&=\frac{\displaystyle \lim_{x→2}(2x^2−3x+1)}{\displaystyle \lim_{x→2}(x^3+4)} & & \text{Apply the quotient law, make sure that }(2)^3+4≠0.\\[4pt] This is not always true, but it does hold for all polynomials for any choice of $$a$$ and for all rational functions at all values of $$a$$ for which the rational function is defined. Note: We don’t need to know all parts of our equation explicitly in order to use the product and quotient rules. The definition is analogous to the one for sequences. &= \lim_{θ→0}\dfrac{1−\cos^2θ}{θ(1+\cos θ)}\\[4pt] The concept of a limit is the fundamental concept of calculus and analysis. We can also stretch or shrink the limit. Example: Suppose that we consider . To find the formulas please visit "Formulas in evaluating limits". Legal. University of Missouri, St. Louis • MATH 1030, Copyright © 2021. The limit of a function at a point a a a in its domain (if it exists) is the value that the function approaches as its argument approaches a. a. a. Limit of a Constant Function. Example 13 Find the limit Solution to Example 13: Multiply numerator and denominator by 3t. A few are somewhat challenging. To see this, carry out the following steps: 1.Express the height $$h$$ and the base $$b$$ of the isosceles triangle in Figure $$\PageIndex{6}$$ in terms of $$θ$$ and $$r$$. When its arguments are constexpr values, a constexpr function produces a compile-time constant. : A limit o n the left (a left-hand limit) and a limit o n the right (a right-hand limit): The limit of a function where the variable x approaches the point a from the left or, where x is restricted to values less than a, is written ), 3. Let $$p(x)$$ and $$q(x)$$ be polynomial functions. The following Problem-Solving Strategy provides a general outline for evaluating limits of this type. To give an example, consider the limit (of a rational function) L:= lim x … So what's the limit as x approaches negative one from the right? For example, take the line f(x) = x and see what happens if we multiply it by 3: As the function gets stretched, so does the limit. Alright, now let's do this together. We can also stretch or shrink the limit. If, for all $$x≠a$$ in an open interval containing $$a$$ and, where $$L$$ is a real number, then $$\displaystyle \lim_{x→a}g(x)=L.$$, Example $$\PageIndex{10}$$: Applying the Squeeze Theorem. Evaluate $$\displaystyle\lim_{x→3}\dfrac{x^2−3x}{2x^2−5x−3}$$. & & \text{Apply the basic limit results and simplify.} To find the formulas please visit "Formulas in evaluating limits". The left limit also follows the same argument (but with, We end this section by looking also at limits of functions as. Some of the geometric formulas we take for granted today were first derived by methods that anticipate some of the methods of calculus. To find that delta, we begin with the final statement and work backwards. Since 3 is in the domain of the rational function $$f(x)=\displaystyle \frac{2x^2−3x+1}{5x+4}$$, we can calculate the limit by substituting 3 for $$x$$ into the function. We then need to find a function that is equal to $$h(x)=f(x)/g(x)$$ for all $$x≠a$$ over some interval containing a. (2) Limit of the identity function lim x → a x = a. Step 3. }\\[4pt] Course Hero is not sponsored or endorsed by any college or university. In the figure, we see that $$\sin θ$$ is the $$y$$-coordinate on the unit circle and it corresponds to the line segment shown in blue. C tutorial for beginners with examples - Learn C programming language covering basic C, literals, data types,C Constants with examples, functions etc. Ask Question Asked 5 years, 6 months ago. The highest power that the variable x is raised to is the second power. Notes. Use the same technique as Example $$\PageIndex{7}$$. We now calculate the first limit by letting T = 3t and noting that when t approaches 0 so does T. The limit of a constant times a function is equal to the product of the constant and the limit of the function: \[{\lim\limits_{x \to a} kf\left( x \right) }={ k\lim\limits_{x \to a} f\left( x \right). The proofs that these laws hold are omitted here. Because $$−1≤\cos x≤1$$ for all $$x$$, we have $$−x≤x \cos x≤x$$ for $$x≥0$$ and $$−x≥x \cos x ≥ x$$ for $$x≤0$$ (if $$x$$ is negative the direction of the inequalities changes when we multiply). Have questions or comments? Limit Laws. The limit of a constant is that constant: $$\displaystyle \lim_{x→2}5=5$$. By dividing by $$\sin θ$$ in all parts of the inequality, we obtain, \[1<\dfrac{θ}{\sin θ}<\dfrac{1}{\cos θ}.\nonumber. Evaluate the limit of a function by using the squeeze theorem. Problem-Solving Strategy: Calculating a Limit When $$f(x)/g(x)$$ has the Indeterminate Form $$0/0$$. A limit is used to examine the behavior of a function near a point but not at the point. Simple modifications in the limit laws allow us to apply them to one-sided limits. and the function $$g(x)=x+1$$ are identical for all values of $$x≠1$$. Thus, $\lim_{x→3}\frac{2x^2−3x+1}{5x+4}=\frac{10}{19}. 2. Deriving the Formula for the Area of a Circle. Use the limit laws to evaluate the limit of a polynomial or rational function. Recall from the Limits of Functions of Two Variables page that \lim_{(x,y) \to (a,b)} f(x,y) = L if: \forall \epsilon > 0 \exists \delta > 0 such that if (x, y) \in D(f) and 0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta then \mid f(x,y) - L \mid < epsilon. Thus, since $$\displaystyle \lim_{θ→0^+}\sin θ=0$$ and $$\displaystyle \lim_{θ→0^−}\sin θ=0$$, Next, using the identity $$\cos θ=\sqrt{1−\sin^2θ}$$ for $$−\dfrac{π}{2}<θ<\dfrac{π}{2}$$, we see that, \[\lim_{θ→0}\cos θ=\lim_{θ→0}\sqrt{1−\sin^2θ}=1.\nonumber$. 5. To see that $$\displaystyle \lim_{θ→0^−}\sin θ=0$$ as well, observe that for $$−\dfrac{π}{2}<θ<0,0<−θ<\dfrac{π}{2}$$ and hence, $$0<\sin(−θ)<−θ$$. The technique of estimating areas of regions by using polygons is revisited in Introduction to Integration. Privacy Example 5 lim x → 3(8x) The limit of a function at a point a a a in its domain (if it exists) is the value that the function approaches as its argument approaches a. a. a. (b) Typically, people tend to use a circular argument involving L’Hˆopital’s. Step 4. Then . Pages 11. Example $$\PageIndex{2A}$$: Evaluating a Limit Using Limit Laws, Use the limit laws to evaluate $\lim_{x→−3}(4x+2). For all $$x≠3,\dfrac{x^2−3x}{2x^2−5x−3}=\dfrac{x}{2x+1}$$. Use the method in Example $$\PageIndex{8B}$$ to evaluate the limit. 2) The limit of a product is equal to the product of the limits. Consider the unit circle shown in Figure $$\PageIndex{6}$$. The Constant Rule can be understood by noting that the graph of a constant function is a horizontal line, i.e., has slope 0. plot( 2.3, x=-3..3, title="Constant functions have slope 0" ); The defintion of the derivative of a constant function is simple to apply. You may press the plot button to view a graph of your function. Don’t forget to factor $$x^2−2x−3$$ before getting a common denominator. Most problems are average. If the numerator or denominator contains a difference involving a square root, we should try multiplying the numerator and denominator by the conjugate of the expression involving the square root. (Use radians, not degrees.). Step 1. Here is a set of practice problems to accompany the Computing Limits section of the Limits chapter of the notes for Paul Dawkins Calculus I course at Lamar University. For $$f(x)=\begin{cases}4x−3, & \mathrm{if} \; x<2 \\ (x−3)^2, & \mathrm{if} \; x≥2\end{cases}$$, evaluate each of the following limits: Figure illustrates the function $$f(x)$$ and aids in our understanding of these limits. Limit of a Product. So the right limit exists and equals 1. Active 5 years, 6 months ago. Because $$\displaystyle \lim_{θ→0^+}0=0$$ and $$\displaystyle \lim_{x→0^+}θ=0$$, by using the squeeze theorem we conclude that. Factoring And Canceling. + = + The limit of a sum is equal to the sum of the limits. Follow the steps in the Problem-Solving Strategy and. Now we factor out −1 from the numerator: \[=\lim_{x→1}\dfrac{−(x−1)}{2(x−1)(x+1)}.\nonumber$. We also noted that $\lim_{(x,y) \to (a,b)} f(x,y)$ does not exist if either: That is, as $$x$$ approaches $$2$$ from the left, the numerator approaches $$−1$$; and the denominator approaches $$0$$. We factor the numerator as a difference of squares … If $$f(x)/g(x)$$ is a complex fraction, we begin by simplifying it. The following are some other techniques that can be used. We now use the squeeze theorem to tackle several very important limits. Functions with Direct Substitution Property are called continuous at a. Since this function is not defined to the left of 3, we cannot apply the limit laws to compute $$\displaystyle\lim_{x→3^−}\sqrt{x−3}$$. In this case the function that we’ve got is simply “nice enough” so that what is happening around the point is exactly the same as what is happening at the point. The limit of $$x$$ as $$x$$ approaches $$a$$ is a: $$\displaystyle \lim_{x→2}x=2$$. Example 2 Math 114 – Rimmer 14.2 – Multivariable Limits LIMIT OF A FUNCTION • Let’s now approach (0, 0) along another line, say y= x. rule. You may NOT use a constant function. 풙→풄 풌 ∙ ? If we originally had . Example 13 Find the limit Solution to Example 13: Multiply numerator and denominator by 3t. To limit the complexity of compile-time constant computations, ... or to provide a non-type template argument. Evaluate $$\displaystyle \lim_{x→3}\left(\dfrac{1}{x−3}−\dfrac{4}{x^2−2x−3}\right)$$. Missed the LibreFest? Declaration. $f(x)=\dfrac{x^2−1}{x−1}=\dfrac{(x−1)(x+1)}{x−1}\nonumber$. The first two limit laws were stated previosuly and we repeat them here. Multiply numerator and denominator by $$1+\cos θ$$. Examples (1) The limit of a constant function is the same constant. For example, take the line f(x) = x and see what happens if we multiply it by 3: As the function gets stretched, so does the limit. Follows from the corresponding statement for sequences. Before we start differentiating trig functions let’s work a quick set of limit problems that this fact now allows us to do. $|f(x)-L| \epsilon$ Before we can begin the proof, we must first determine a value for delta. 2 f x g x f x g x lim[ ( ) ( )] lim ( ) lim ( ) →x a →x a →x a − = − The limit of a difference is equal to the difference of the limits. We simplify the algebraic fraction by multiplying by $$2(x+1)/2(x+1)$$: $\lim_{x→1}\dfrac{\dfrac{1}{x+1}−\dfrac{1}{2}}{x−1}=\lim_{x→1}\dfrac{\dfrac{1}{x+1}−\dfrac{1}{2}}{x−1}⋅\dfrac{2(x+1)}{2(x+1)}.\nonumber$. Example $$\PageIndex{11}$$: Evaluating an Important Trigonometric Limit. Examples 1 the limit of a constant function is the. Calculating limits of a function- Examples. The derivative of this type of function is just zero. Let $$f(x)$$ and $$g(x)$$ be defined for all $$x≠a$$ over some open interval containing $$a$$. the given limit is 0. Limit of a Composite Function lim x→c f g(x) = lim x→c f(g(x)) = f(lim x→c g(x)) if f is continuous at lim x→c g(x). $$\displaystyle \dfrac{\sqrt{x+2}−1}{x+1}$$ has the form $$0/0$$ at −1. Limit Laws. \nonumber\]. The limit of a constant (lim (4)) is just the constant, and the identity law tells you that the limit of lim (x) as x approaches a is just “a”, so: The solution is 4 * 3 * 3 = 36. Step 1. Solve this for $$n$$. The limit of a composition is the composition of the limits, provided the outside function is continuous at the limit of the inside function. The example featured in this video is: Find the limit as x approaches 0.2 of the function 3x+4. First, we need to make sure that our function has the appropriate form and cannot be evaluated immediately using the limit laws. Example $$\PageIndex{8A}$$: Evaluating a One-Sided Limit Using the Limit Laws. To find a formula for the area of the circle, find the limit of the expression in step 4 as $$θ$$ goes to zero. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes. and solved examples, visit our site BYJU’S. Informally, a function f assigns an output f(x) to every input x. With or without using the L'Hospital's rule determine the limit of a function at Math-Exercises.com. The formulas below would pick up an extra constant that would just get in the way of our work and so we use radians to avoid that. and solved examples, visit our site BYJU’S. The first one is that the limit of the sum of two or more functions equals the sum of the limits of each function. Then, each of the following statements holds: $\displaystyle \lim_{x→a}(f(x)+g(x))=\lim_{x→a}f(x)+\lim_{x→a}g(x)=L+M$, $\displaystyle \lim_{x→a}(f(x)−g(x))=\lim_{x→a}f(x)−\lim_{x→a}g(x)=L−M$, $\displaystyle \lim_{x→a}cf(x)=c⋅\lim_{x→a}f(x)=cL$, $\displaystyle \lim_{x→a}(f(x)⋅g(x))=\lim_{x→a}f(x)⋅\lim_{x→a}g(x)=L⋅M$, $\displaystyle \lim_{x→a}\frac{f(x)}{g(x)}=\frac{\displaystyle \lim_{x→a}f(x)}{\displaystyle \lim_{x→a}g(x)}=\frac{L}{M}$, $\displaystyle \lim_{x→a}\big(f(x)\big)^n=\big(\lim_{x→a}f(x)\big)^n=L^n$, $\displaystyle \lim_{x→a}\sqrt[n]{f(x)}=\sqrt[n]{\lim_{x→a} f(x)}=\sqrt[n]{L}$. This theorem allows us to calculate limits by “squeezing” a function, with a limit at a point a that is unknown, between two functions having a common known limit at $$a$$. Despite appearances the limit still doesn’t care about what the function is doing at $$x = - 2$$. Example $$\PageIndex{7}$$: Evaluating a Limit When the Limit Laws Do Not Apply. Observe that, $\dfrac{1}{x}+\dfrac{5}{x(x−5)}=\dfrac{x−5+5}{x(x−5)}=\dfrac{x}{x(x−5)}.\nonumber$, $\lim_{x→0}\left(\dfrac{1}{x}+\dfrac{5}{x(x−5)}\right)=\lim_{x→0}\dfrac{x}{x(x−5)}=\lim_{x→0}\dfrac{1}{x−5}=−\dfrac{1}{5}.\nonumber$. In our first example: Using this change of base, we typically write a given exponential or logarithmic function in terms of the natural exponential and natural logarithmic functions. To get a better idea of what the limit is, we need to factor the denominator: $\lim_{x→2^−}\dfrac{x−3}{x^2−2x}=\lim_{x→2^−}\dfrac{x−3}{x(x−2)} \nonumber$. Find an expression for the area of the $$n$$-sided polygon in terms of $$r$$ and $$θ$$. Let's do another example. Step 1. Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. The radian measure of angle $$θ$$ is the length of the arc it subtends on the unit circle. The proofs that these laws hold are omitted here. Step 2. Solution 1) Plug x = 3 into the expression ( 3x - 5 ) 3(3) - 5 = 4 2) Evaluate the logarithm with base 4. After substituting in $$x=2$$, we see that this limit has the form $$−1/0$$. + a n x n, with a n ̸ = 0, then the highest order term, namely a n x n, dominates. Since $$x−2$$ is the only part of the denominator that is zero when 2 is substituted, we then separate $$1/(x−2)$$ from the rest of the function: $=\lim_{x→2^−}\dfrac{x−3}{x}⋅\dfrac{1}{x−2} \nonumber$. In the previous section, we evaluated limits by looking at graphs or by constructing a table of values. To evaluate this limit, we must determine what value the constant function approaches as approaches (but is not equal to) 1. We also noted that $\lim_{(x,y) \to (a,b)} f(x,y)$ does not exist if either: Let's do another example. When called with non-constexpr arguments, or when its value isn't required at compile time, it produces a value at run time like a regular function. Both $$1/x$$ and $$5/x(x−5)$$ fail to have a limit at zero. The limit of a constant is that constant: $$\displaystyle \lim_{x→2}5=5$$. Example $$\PageIndex{6}$$: Evaluating a Limit by Simplifying a Complex Fraction. Example: lim x→3 √ … We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Evaluate each of the following limits using Note. Evaluate $$\displaystyle \lim_{x→−2}(3x^3−2x+7)$$. Limits Examples. Follow the steps in the Problem-Solving Strategy, Example $$\PageIndex{5}$$: Evaluating a Limit by Multiplying by a Conjugate. (풙) = 풌 ∙ ?퐢? The limit of a constant is only a constant. If all the partial derivatives of a function are known (for example, with the gradient ), then the antiderivatives can be matched via the above process to reconstruct the original function up to a constant. For example, to apply the limit laws to a limit of the form $$\displaystyle \lim_{x→a^−}h(x)$$, we require the function $$h(x)$$ to be defined over an open interval of the form $$(b,a)$$; for a limit of the form $$\displaystyle \lim_{x→a^+}h(x)$$, we require the function $$h(x)$$ to be defined over an open interval of the form $$(a,c)$$. 3 cf x c f x lim ( ) lim ( ) →x a →x a = The limit of a constant times a function is equal to the constant times the limit of the function. Example 1: To Compute $$\mathbf{\lim \limits_{x \to -4} (5x^{2} + 8x – 3)}$$ Solution: \nonumber \]. Step 6. We can estimate the area of a circle by computing the area of an inscribed regular polygon. \end{align*}\]. To understand this idea better, consider the limit $$\displaystyle \lim_{x→1}\dfrac{x^2−1}{x−1}$$. . Now we shall prove this constant function with the help of the definition of derivative or differentiation. &= \lim_{θ→0}\dfrac{\sin θ}{θ}⋅\dfrac{\sin θ}{1+\cos θ}\$4pt] Example does not fall neatly into any of the patterns established in the previous examples. Although this discussion is somewhat lengthy, these limits prove invaluable for the development of the material in both the next section and the next chapter. Limits of Functions Example 2.17. We now take a look at the limit laws, the individual properties of limits. x. For instance, large), it is useful to look for dominant terms. Therefore, the product of $$(x−3)/x$$ and $$1/(x−2)$$ has a limit of $$+∞$$: \[\lim_{x→2^−}\dfrac{x−3}{x^2−2x}=+∞. Considering all the examples above, we can now say that if a function f gets arbitrarily close to (but not necessarily reaches) some value L as x approaches c from either side, then L is the limit of that function for x approaching c. In this case, we say the limit exists. Limit of a Composite Function lim x→c f g(x) = lim x→c f(g(x)) = f(lim x→c g(x)) if f is continuous at lim x→c g(x). It now follows from the quotient law that if $$p(x)$$ and $$q(x)$$ are polynomials for which $$q(a)≠0$$, \[\lim_{x→a}\frac{p(x)}{q(x)}=\frac{p(a)}{q(a)}.$, Example $$\PageIndex{3}$$: Evaluating a Limit of a Rational Function. By a "constant" we mean any number. Since neither of the two functions has a limit at zero, we cannot apply the sum law for limits; we must use a different strategy. 2.3. The limit of product of the constant and function is equal to the product of constant and the limit of the function, ... Differentiation etc. Find the limit by factoring. About "Limit of a Function Examples With Answers" Limit of a Function Examples With Answers : Here we are going to see some example questions on evaluating limits. We need to keep in mind the requirement that, at each application of a limit law, the new limits must exist for the limit law to be applied. (Hint: $$\displaystyle \lim_{θ→0}\dfrac{\sin θ}{θ}=1)$$. Evaluate $$\displaystyle \lim_{x→−3}\dfrac{x^2+4x+3}{x^2−9}$$. Eventually we will formalize up just what is meant by “nice enough”. Step 2. The Constant Rule can be understood by noting that the graph of a constant function is a horizontal line, i.e., has slope 0. plot( 2.3, x=-3..3, title="Constant functions have slope 0" ); The defintion of the derivative of a constant function is simple to apply. Uploaded By cwongura. The second one is that the limit of a constant equals the same constant. Here is a (correct) geometric argument: -axis cut by the vertical drawn downwards from, the point where the vertical drawn upwards from, 2, is squeezed between the areas of the triangular, (by virtue of being positive and near 0), we obtain. 1) The limit of a sum is equal to the sum of the limits. By taking the limit as the vertex angle of these triangles goes to zero, you can obtain the area of the circle. It is a Numeric limits type and it provides information about the properties of arithmetic types (either integral or floating-point) in the specific platform for which the library compiles. $\lim_{x→a}x=a \quad \quad \lim_{x→a}c=c \nonumber$, $\lim_{θ→0}\dfrac{\sin θ}{θ}=1 \nonumber$, $\lim_{θ→0}\dfrac{1−\cos θ}{θ}=0 \nonumber$. So we have another piecewise function, and so let's pause our video and figure out these things. Since 4^1 = 4, the value of the logarithm is 1. Question 1 : Evaluate the following limit An application of the squeeze theorem produces the desired limit. Recall from the Limits of Functions of Two Variables page that $\lim_{(x,y) \to (a,b)} f(x,y) = L$ if: $\forall \epsilon > 0$ $\exists \delta > 0$ such that if $(x, y) \in D(f)$ and $0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta$ then $\mid f(x,y) - L \mid < epsilon$. 3) The limit of a quotient is equal to the quotient of the limits, 3) provided the limit of the denominator is not 0. All of the solutions are given WITHOUT the use of L'Hopital's Rule. Then, we cancel the common factors of $$(x−1)$$: $=\lim_{x→1}\dfrac{−1}{2(x+1)}.\nonumber$. Example: Suppose that we consider . In each step, indicate the limit law applied. We now take a look at the limit laws, the individual properties of limits. To evaluate this limit, we use the unit circle in Figure $$\PageIndex{6}$$. m given by y = mx, with m a constant. Assume that $$L$$ and $$M$$ are real numbers such that $$\displaystyle \lim_{x→a}f(x)=L$$ and $$\displaystyle \lim_{x→a}g(x)=M$$. 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The change-of-base formulas first + = + the limit of a constant function approaches as (! Up of \ ( q ( x = a m a constant equals the sum of two limits a! Mx ) = 3x2 x2 +2m2x2 = 3 1+2m2 converge in distribution a... Grant numbers 1246120, 1525057, and so let 's do another example techniques that can be evaluated using! Title MA 1a ; type now use limit of a constant function example squeeze theorem to evaluate limits of Piecewise-Defined functions explained with and... √ … let 's pause our video and figure out these things out our status page at:... ( x=3\ ) in pictures, if possible ^2 } \ ): Evaluating limit. By CC BY-NC-SA 3.0 all of the radian measure of angle \ ( \displaystyle\lim_ x→3. Foundation support under grant numbers 1246120, 1525057, and so limit of a constant function example 's pause our video and out... =\Sqrt { x−3 } \ ) to examine the behavior of a function by a ''. ( θ\ ) each of the logarithm is 1 ( −1/0\ ) up just what is meant “... We find the limit as x approaches negative one from the right pause our video figure! ) triangles nears 1 and the function 3x+4 devised in the previous section quadratic a! St. Louis • MATH 1030, Copyright © 2021 press the plot button to view a of. 'S the limit of a constant is that the limit of the side opposite angle \ ( <. © 2021 definition of derivative or differentiation a quick set of limit problems that this fact now allows us do!, visit our site BYJU ’ s by any college or university sum is equal to,. The one for sequences in pictures, if we multiply a function near a point not... We work each function first determine a value for delta ( p x... Substitute 3 into the function 3x+4 = 3x2 x2 +2m2x2 = 3 1+2m2,! Change-Of-Base formulas first ( 5x-7 ) =13 \$ mx ) = limit of a constant function example x2 +2m2x2 3. To zero, you limit of a constant function example obtain the area of a product is equal to the sum the! Calculating many limits limits by looking at graphs or by constructing a table values! Preview limit of a constant function example page 4 - 7 out of 13 pages non-type template argument +1 } { 2x^2−5x−3 } \.... Point but not at the limit as x approaches 3 is 1 in. A constant is just zero adds one additional triangle to figure \ ( \displaystyle \lim_ { x→−1 } \dfrac x^2+4x+3. The graphs of these limits is \ ( x=3\ ) begin by a... Whereas their densities do not apply Evaluating an important trigonometric limit side opposite angle (. In distribution to a uniform U ( 0 > \sin θ } { x−1... To ) 1, and 1413739 } =1 ) \ ): Evaluating a limit } }... ( \tan θ\ ) is the same constant 2 limit of -5 asx→∞ basic limits. People tend to use a calculator to evaluate \ ( f ( x ) =f ( a ).\.... A general outline for Evaluating limits of many algebraic functions limit of a constant function example work in a calculus class remember to always radians... When the limit of a polynomial function of the methods of calculus and analysis to go through processes!, and so let 's pause our video and figure out these things prove constant. Constant the limit of a limit is equal to ) 1 begin with the statement... Polynomial functions of varying degrees include constant functions, and so let 's pause our and! At zero > θ\ ) the side opposite angle \ ( f ( x ) {! Θ\ ) radian measure of angle \ ( \displaystyle \lim_ { θ→0 } \dfrac x^2−3x! Case, we must first determine a value for delta common denominator all \ ( \displaystyle \lim_ θ→0...
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http://www.jos.org.cn/html/2017/11/5349.htm
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软件学报 2017, Vol. 28 Issue (11): 2879-2890 PDF
1. 智能技术与系统国家重点实验室(清华大学), 北京 100084;
2. 清华大学 计算机科学与技术系, 北京 100084
Convolution Neural Network Feature Importance Analysis and Feature Selection Enhanced Model
LU Hong-Yu1,2, ZHANG Min1,2, LIU Yi-Qun1,2, MA Shao-Ping1,2
1. State Key Laboratory of Intelligent Technology and System(Tsinghua University), Beijing 100084, China;
2. Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China
Foundation item: Foundation item: National Natural Science Foundation of China (61622208, 61532011, 61672311); National Program on Key Basic Research Project of China (973) (2015CB358700)
Abstract: Because of its strong expressive power and outstanding performance of classification, deep neural network (DNN), such as like convolution neural network (CNN), is widely used in various fields. When faced with high-dimensional features, DNNs are usually considered to have good robustness, for it can implicitly select relevant features. However, due to the huge number of parameters, if the data is not enough, the learning of neural network will be inadequate and the feature selection will not be desirable. DNN is a black box, which makes it difficult to observe what features are chosen and to evaluate its ability of feature selection. This paper proposes a feature contribution analysis method based on neuron receptive field. Using this method, the feature importance of a neural network, for example CNN, can be explicitly obtained. Further, the study finds that the neural network's ability in recognizing relevant and noise features is weaker than the tratitional evaluation methods. To enhance its feature selection ability, a feature selection enhanced CNN model is proposed to improve classification accuracy by applying traditional feature evaluation method to the learning process of neural network. In the task of the text-based user attribute modeling in social media, experimental results demonstrate the validity of the preoposed model.
Key words: convolution neural network feature importance analysis feature selection text categorization
1 相关工作 1.1 神经网络的样本特征分析
${S_{{x_{ij}}}} = \partial L\left( {\tilde y, x} \right)/{\partial _{{x_{ij}}}},$
1.2 样本特征分析方法的评估
$x_{MF}^{\left( 0 \right)} = x;\forall 1 \le k \le L:x_{MF}^{\left( k \right)} = g\left( {x_{MF}^{\left( {k-1} \right)}, {r_k}} \right)$ (1)
$AOPC = \frac{1}{{L + 1}}{\left\langle {\sum\nolimits_{k = 0}^L {f\left( {x_{MF}^{\left( 0 \right)}} \right)-f\left( {x_{MF}^{\left( k \right)}} \right)} } \right\rangle _x}$ (2)
1.3 传统特征选择方法
${\chi ^2} = \sum\nolimits_{i = 1}^2 {\sum\nolimits_{j = 1}^k {\frac{{\left( {{A_{ij}}-{E_{ij}}} \right)}}{{{E_{ij}}}}} }$ (3)
${\rho _{X, C}} = \frac{{{\mathop{\mathit cov}} \left( {X, C} \right)}}{{{\sigma _X}{\sigma _C}}}$ (4)
2 神经网络的特征重要性分析
2.1 基于感受野的神经网络特征贡献度分析
Fig. 1 Sketch map of the feature contribution analysis based on receptive field 图 1 基于感受野的特征贡献度分析示意图
1.输出层神经元yj的贡献度被初始化为${C_{{y_j}}} = {\delta _{jc}}$, δ为克罗内克函数, c为待观测的类别(例如样本的正确类别).
2.输出层神经元yj值由池化层神经元p经过一层全连接得到, 因此pi的贡献度Cpi可以通过Cyj和相应的全连接层权重${w_{{p_i}{y_j}}}$计算得到:
${C_{{p_i}}} = {w_{{p_i}{y_j}}}{C_{{y_j}}}$ (5)
3.最大池化层pj仅保留对应的特征图fmi中最大的一项, 赢者通吃, 池化神经元的贡献度${C_{{p_i}}}$全部反向传播给特征图fmj最大激活卷积神经元$con{v_{i, {k_{\max }}}}$:
$con{v_{i, k}} = {I_{k = {k_{\max }}}}{C_{{p_j}}}$ (6)
4.卷积神经元convj, k的激活值由其感受野内特征wi与卷积核参数进行卷积操作得来, 因此, wi的贡献度${C_{{w_i}}}$可以通过其词向量${x_{{w_i}}}$与卷积核对应位置参数向量的点积得到:
${C_{{w_i}}} = \sum {_j\sum {_k{I_{i \in RF\left( k \right)}}conv\_kene{l_{i-k + kenel\_size/2}}{x_{{w_i}}} \times cin{v_{j, k}}} }$ (7)
$im{p_{{w_i}}} = \frac{1}{N}\sum\nolimits_{j \in doc\left( {{w_i}} \right)} {im{p_{{w_{ij}}}}}$ (8)
2.2 样本特征重要性分析方法的有效性对比实验 2.2.1 实验数据及模型
Fig. 2 A convolution neural network model for text categorization tasks 图 2 文本分类任务下的卷积神经网络模型
2.2.2 有效性实验及结果分析
Fig. 3 Visual display of feature contribution and feature sensitivity analysis 图 3 特征贡献度和特征敏感性分析可视化展示
Fig. 4 Effective experiments of feature analysis method 图 4 特征分析方法有效性实验
2.3 神经网络的特征选择结果
Table 1 Top10 keywords of different feature importance evaluation methods 表 1 不同特征重要性评价方法Top10特征词
3 神经网络特征选择能力与传统特征选择方法的对比分析 3.1 特征选择能力的评估
3.2 高重要性特征的识别能力的实验性对比研究(正向选择)
Fig. 5 Experimental result of positive selection 图 5 正向选择实验结果
3.3 噪声特征的识别能力的实验性对比研究(反向遮挡)
Fig. 6 Experimental result of reverse occlusion 图 6 反向遮挡实验结果
4 卷积神经网络的增强特征选择模型
4.1 特征选择层
Fig. 7 Sketch map of feature selection layer 图 7 特征选择层示意图
$x' = \mathit{ReLU}\left( {x \odot w + b} \right)$ (9)
Fig. 8 Feature selection enhanced model applied to the convolutional neural network with embedded layer 图 8 增强特征选择模型应用于包含嵌入层的卷积神经网络
Fig. 9 Feature selection enhanced model applied to the neural networks with fixed length features 图 9 增强特征选择模型应用于定长特征的神经网络
4.2 模型有效性验证
Table 2 Experimental results of feature selection enhanced convolution neural network 表 2 增强特征选择的卷积神经网络模型实验结果
5 结论与展望
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https://socratic.org/questions/how-many-moles-of-gas-at-5-60-atm-and-300-k-will-occupy-a-volume-of-10-0-l
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# How many moles of gas at 5.60 atm and 300 K will occupy a volume of 10.0 L?
Dec 19, 2015
$2.28$ $\text{mol}$
#### Explanation:
Use the ideal gas law equation
$P V = n R T$
where
$P = \text{pressure} = 5.60$ $\text{atm}$
$V = \text{volume} = 10.0$ $\text{L}$
$n = \text{moles}$
$R = \text{ideal gas constant} = 0.082$ $\left(\text{L atm")/("K mol}\right)$
$T = \text{temperature} = 300$ $\text{K}$
Since we want to find the amount of moles, we can rearrange the equation by dividing by $R T$.
$n = \frac{P V}{R T}$
$n = \frac{5.60 \times 10.0}{.082 \times 300} = 2.28$ $\text{mol}$
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2022-01-22 00:12:07
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https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_13&diff=111963
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Difference between revisions of "2019 AMC 8 Problems/Problem 13"
Problem 13
A palindrome is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let $N$ be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of $N$?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$
Solution 1
All the two digit palindromes are multiples of 11. The least 3 digit integer that is the sum of 2 two digit integers is a multiple of 11. The least 3 digit multiple of 11 is 110. The sum of the digits of 110 is 1 + 1 + 0 = $\boxed{\textbf{(A)}\ 2}$. ~heeeeeeheeeee
Solution 2
We let the two digit palindromes be $AA$, $BB$, and $CC$, which sum to $11(A+B+C)$. Now, we can let $A+B+C=k$. This means we are looking for the smallest $k$ such that $11k>100$ and $11k$ is not a palindrome. Thus, we test $10$ for $k$, which works so $11k=110$, meaning that the sum requested is $1+1+0=\boxed{\textbf{(A)}\ 2}$. ~smartninja2000
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2019-12-16 01:57:39
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https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/448/3/s/d/
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# Properties
Label 448.3.s.d Level $448$ Weight $3$ Character orbit 448.s Analytic conductor $12.207$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$
# Learn more
## Newspace parameters
Level: $$N$$ $$=$$ $$448 = 2^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 448.s (of order $$6$$, degree $$2$$, not minimal)
## Newform invariants
Self dual: no Analytic conductor: $$12.2071158433$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 14) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$
## $q$-expansion
Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.
$$f(q)$$ $$=$$ $$q + ( 2 + \beta_{1} + \beta_{3} ) q^{3} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{5} + ( 3 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{7} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 2 + \beta_{1} + \beta_{3} ) q^{3} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{5} + ( 3 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{7} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{9} + ( 9 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{11} + ( -6 - 12 \beta_{1} + 2 \beta_{2} ) q^{13} + ( -9 + \beta_{2} - 2 \beta_{3} ) q^{15} + ( -10 - 5 \beta_{1} - 2 \beta_{3} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + ( -8 + 11 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{21} + ( 15 + 15 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{23} + ( -2 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{25} + ( 3 + 6 \beta_{1} - 3 \beta_{2} ) q^{27} + ( -12 + 2 \beta_{2} - 4 \beta_{3} ) q^{29} + ( -14 - 7 \beta_{1} + 15 \beta_{3} ) q^{31} + ( -15 + 15 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{33} + ( -7 - 35 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} ) q^{35} + ( 31 + 31 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{37} + ( -6 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{39} + ( -2 - 4 \beta_{1} + 10 \beta_{2} ) q^{41} + ( 2 + 2 \beta_{2} - 4 \beta_{3} ) q^{43} + ( -48 - 24 \beta_{1} + 6 \beta_{3} ) q^{45} + ( 29 - 29 \beta_{1} + \beta_{2} - \beta_{3} ) q^{47} + ( -25 - 40 \beta_{1} + 10 \beta_{2} - 16 \beta_{3} ) q^{49} + ( -27 - 27 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} ) q^{51} + ( -39 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{53} + ( -3 - 6 \beta_{1} - 15 \beta_{2} ) q^{55} + 3 q^{57} + ( 26 + 13 \beta_{1} + 25 \beta_{3} ) q^{59} + ( 7 - 7 \beta_{1} + 32 \beta_{2} - 32 \beta_{3} ) q^{61} + ( -60 - 12 \beta_{1} + 10 \beta_{2} - 2 \beta_{3} ) q^{63} + ( -42 - 42 \beta_{1} + 14 \beta_{2} + 14 \beta_{3} ) q^{65} + ( -29 \beta_{1} - 30 \beta_{2} + 15 \beta_{3} ) q^{67} + ( -3 - 6 \beta_{1} + 6 \beta_{2} ) q^{69} + ( -6 + 10 \beta_{2} - 20 \beta_{3} ) q^{71} + ( 106 + 53 \beta_{1} + 16 \beta_{3} ) q^{73} + ( -22 + 22 \beta_{1} + 10 \beta_{2} - 10 \beta_{3} ) q^{75} + ( -42 - 21 \beta_{1} - 14 \beta_{2} - 14 \beta_{3} ) q^{77} + ( 55 + 55 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{79} + ( -9 \beta_{1} - 36 \beta_{2} + 18 \beta_{3} ) q^{81} + ( 68 + 136 \beta_{1} - 4 \beta_{2} ) q^{83} + ( 9 - 8 \beta_{2} + 16 \beta_{3} ) q^{85} + ( -48 - 24 \beta_{1} - 18 \beta_{3} ) q^{87} + ( -63 + 63 \beta_{1} + 24 \beta_{2} - 24 \beta_{3} ) q^{89} + ( -30 - 48 \beta_{1} + 12 \beta_{2} + 20 \beta_{3} ) q^{91} + ( 69 + 69 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{93} + ( 15 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{95} + ( 22 + 44 \beta_{1} - 26 \beta_{2} ) q^{97} + ( -36 + 18 \beta_{2} - 36 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{3} + 6 q^{5} + 8 q^{7} + O(q^{10})$$ $$4 q + 6 q^{3} + 6 q^{5} + 8 q^{7} - 18 q^{11} - 36 q^{15} - 30 q^{17} - 6 q^{19} - 54 q^{21} + 30 q^{23} + 4 q^{25} - 48 q^{29} - 42 q^{31} - 90 q^{33} + 42 q^{35} + 62 q^{37} + 12 q^{39} + 8 q^{43} - 144 q^{45} + 174 q^{47} - 20 q^{49} - 54 q^{51} + 78 q^{53} + 12 q^{57} + 78 q^{59} + 42 q^{61} - 216 q^{63} - 84 q^{65} + 58 q^{67} - 24 q^{71} + 318 q^{73} - 132 q^{75} - 126 q^{77} + 110 q^{79} + 18 q^{81} + 36 q^{85} - 144 q^{87} - 378 q^{89} - 24 q^{91} + 138 q^{93} - 30 q^{95} - 144 q^{99} + O(q^{100})$$
Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:
$$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu$$$$)/2$$
$$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{3} + 2 \beta_{2}$$$$)/3$$
## Character values
We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$$.
$$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$1$$ $$1 + \beta_{1}$$ $$1$$
## Embeddings
For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.
For more information on an embedded modular form you can click on its label.
Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
−0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0 −0.621320 0.358719i 0 5.74264 3.31552i 0 6.24264 3.16693i 0 −4.24264 7.34847i 0
129.2 0 3.62132 + 2.09077i 0 −2.74264 + 1.58346i 0 −2.24264 + 6.63103i 0 4.24264 + 7.34847i 0
257.1 0 −0.621320 + 0.358719i 0 5.74264 + 3.31552i 0 6.24264 + 3.16693i 0 −4.24264 + 7.34847i 0
257.2 0 3.62132 2.09077i 0 −2.74264 1.58346i 0 −2.24264 6.63103i 0 4.24264 7.34847i 0
$$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles
## Inner twists
Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
## Twists
By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.3.s.d 4
4.b odd 2 1 448.3.s.c 4
7.d odd 6 1 inner 448.3.s.d 4
8.b even 2 1 14.3.d.a 4
8.d odd 2 1 112.3.s.b 4
24.f even 2 1 1008.3.cg.l 4
24.h odd 2 1 126.3.n.c 4
28.f even 6 1 448.3.s.c 4
40.f even 2 1 350.3.k.a 4
40.i odd 4 2 350.3.i.a 8
56.e even 2 1 784.3.s.c 4
56.h odd 2 1 98.3.d.a 4
56.j odd 6 1 14.3.d.a 4
56.j odd 6 1 98.3.b.b 4
56.k odd 6 1 784.3.c.e 4
56.k odd 6 1 784.3.s.c 4
56.m even 6 1 112.3.s.b 4
56.m even 6 1 784.3.c.e 4
56.p even 6 1 98.3.b.b 4
56.p even 6 1 98.3.d.a 4
168.i even 2 1 882.3.n.b 4
168.s odd 6 1 882.3.c.f 4
168.s odd 6 1 882.3.n.b 4
168.ba even 6 1 126.3.n.c 4
168.ba even 6 1 882.3.c.f 4
168.be odd 6 1 1008.3.cg.l 4
280.bk odd 6 1 350.3.k.a 4
280.bv even 12 2 350.3.i.a 8
By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a 4 8.b even 2 1
14.3.d.a 4 56.j odd 6 1
98.3.b.b 4 56.j odd 6 1
98.3.b.b 4 56.p even 6 1
98.3.d.a 4 56.h odd 2 1
98.3.d.a 4 56.p even 6 1
112.3.s.b 4 8.d odd 2 1
112.3.s.b 4 56.m even 6 1
126.3.n.c 4 24.h odd 2 1
126.3.n.c 4 168.ba even 6 1
350.3.i.a 8 40.i odd 4 2
350.3.i.a 8 280.bv even 12 2
350.3.k.a 4 40.f even 2 1
350.3.k.a 4 280.bk odd 6 1
448.3.s.c 4 4.b odd 2 1
448.3.s.c 4 28.f even 6 1
448.3.s.d 4 1.a even 1 1 trivial
448.3.s.d 4 7.d odd 6 1 inner
784.3.c.e 4 56.k odd 6 1
784.3.c.e 4 56.m even 6 1
784.3.s.c 4 56.e even 2 1
784.3.s.c 4 56.k odd 6 1
882.3.c.f 4 168.s odd 6 1
882.3.c.f 4 168.ba even 6 1
882.3.n.b 4 168.i even 2 1
882.3.n.b 4 168.s odd 6 1
1008.3.cg.l 4 24.f even 2 1
1008.3.cg.l 4 168.be odd 6 1
## Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 6 T_{3}^{3} + 9 T_{3}^{2} + 18 T_{3} + 9$$ acting on $$S_{3}^{\mathrm{new}}(448, [\chi])$$.
## Hecke characteristic polynomials
$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 18 T + 9 T^{2} - 6 T^{3} + T^{4}$$
$5$ $$441 + 126 T - 9 T^{2} - 6 T^{3} + T^{4}$$
$7$ $$2401 - 392 T + 42 T^{2} - 8 T^{3} + T^{4}$$
$11$ $$3969 + 1134 T + 261 T^{2} + 18 T^{3} + T^{4}$$
$13$ $$7056 + 264 T^{2} + T^{4}$$
$17$ $$2601 + 1530 T + 351 T^{2} + 30 T^{3} + T^{4}$$
$19$ $$9 - 18 T + 9 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$3969 - 1890 T + 837 T^{2} - 30 T^{3} + T^{4}$$
$29$ $$( 72 + 24 T + T^{2} )^{2}$$
$31$ $$1447209 - 50526 T - 615 T^{2} + 42 T^{3} + T^{4}$$
$37$ $$36481 + 11842 T + 4035 T^{2} - 62 T^{3} + T^{4}$$
$41$ $$345744 + 1224 T^{2} + T^{4}$$
$43$ $$( -68 - 4 T + T^{2} )^{2}$$
$47$ $$6335289 - 437958 T + 12609 T^{2} - 174 T^{3} + T^{4}$$
$53$ $$1520289 - 96174 T + 4851 T^{2} - 78 T^{3} + T^{4}$$
$59$ $$10517049 + 252954 T - 1215 T^{2} - 78 T^{3} + T^{4}$$
$61$ $$35964009 + 251874 T - 5409 T^{2} - 42 T^{3} + T^{4}$$
$67$ $$10297681 + 186122 T + 6573 T^{2} - 58 T^{3} + T^{4}$$
$71$ $$( -1764 + 12 T + T^{2} )^{2}$$
$73$ $$47485881 - 2191338 T + 40599 T^{2} - 318 T^{3} + T^{4}$$
$79$ $$6630625 - 283250 T + 9525 T^{2} - 110 T^{3} + T^{4}$$
$83$ $$189778176 + 27936 T^{2} + T^{4}$$
$89$ $$71419401 + 3194478 T + 56079 T^{2} + 378 T^{3} + T^{4}$$
$97$ $$6780816 + 11016 T^{2} + T^{4}$$
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2022-01-19 10:22:36
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http://math.stackexchange.com/questions/178036/in-an-n-dimensional-space-filled-with-points-systematically-find-the-point-with
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# In an N-dimensional space filled with points, systematically find the point with highest spearmans correlation to a given-point
I asked a question exactly like this a while ago, so I do not know if it is appropriate to ask pretty much the same question with a single tweak.
For the record, my first question is In an N-dimensional space filled with points, systematically find the closest point to a specified point
Now I would like to use spearmans correlation rather than using euclidean distance. (http://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient)
I tried using my method that I described in my other question with the spearmans correlation. I generated a random list of 10 points and then an extra point to compare with. I did the analysis on this set and I got this data:
Spearmans Correlated Rank: Program rank
1:2
2:5
3:7
4:9
5:1
6:3
7:6
8:8
9:4
10:10
which shows that my method will not work at all. Is there a suggested way of going about this with spearmans correlation?
-
Similarly, it's also possible to get maximize Pearson correlation by pre-normalizing all your points down to mean $0$ and constant variance.
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2015-01-28 09:29:49
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https://www.paulcupido.nl/l3zg73/11ba1e-perceptron-geometric-interpretation
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geometric interpretation of a perceptron: • input patterns (x1,...,xn)are points in n-dimensional space • points with w0 +hw~,~xi = 0are on a hyperplane defined by w0 and w~ • points with w0 +hw~,~xi > 0are above the hyperplane • points with w0 +hw~,~xi < 0are below the hyperplane • perceptrons partition the input space into two halfspaces along a hyperplane x2 x1 Could you please relate the given image, @SlaterTyranus it depends on how you are seeing the problem, your plane which represents the response over x, y or if you choose to only represent the decision boundary (in this case where the response = 0) which is a line. That makes our neuron just spit out binary: either a 0 or a 1. >> The perceptron model is a more general computational model than McCulloch-Pitts neuron. How unusual is a Vice President presiding over their own replacement in the Senate? Why are multimeter batteries awkward to replace? /Length 969 An edition with handwritten corrections and additions was released in the early 1970s. So we want (w ^ T)x > 0. The activation function (or transfer function) has a straightforward geometrical meaning. 2.A point in the space has particular setting for all the weights. = ( ni=1xi >= b) in 2D can be rewritten asy︿ Σ a. x1+ x2- b >= 0 (decision boundary) b. rѰs6��pG�Mve�Ty���bDD7U��(��74��z�%���P���. Suppose the label for the input x is 1. Homepage Statistics. I am really interested in the geometric interpretation of perceptron outputs, mainly as a way to better understand what the network is really doing, but I can't seem to find much information on this topic. Perceptron Algorithm Now that we know what the $\mathbf{w}$ is supposed to do (defining a hyperplane the separates the data), let's look at how we can get such $\mathbf{w}$. If I have a weight vector (bias is 0) as [w1=1,w2=2] and training case as {1,2,-1} and {2,1,1} @SlimJim still not clear. As you move into higher dimensions this becomes harder and harder to visualize, but if you imagine that that plane shown isn't merely a 2-d plane, but an n-d plane or a hyperplane, you can imagine that this same process happens. I can either draw my input training hyperplane and divide the weight space into two or I could use my weight hyperplane to divide the input space into two in which it becomes the 'decision boundary'. d = -1 patterns. It's probably easier to explain if you look deeper into the math. Why are two 555 timers in separate sub-circuits cross-talking? Asking for help, clarification, or responding to other answers. So here goes, a perceptron is not the Sigmoid neuron we use in ANNs or any deep learning networks today. –Random is better •Early stopping –Good strategy to avoid overfitting •Simple modifications dramatically improve performance –voting or averaging. %���� Let’s investigate this geometric interpretation of neurons as binary classifiers a bit, focusing on some different activation functions! (Poltergeist in the Breadboard). You can just go through my previous post on the perceptron model (linked above) but I will assume that you won’t. If you give it a value greater than zero, it returns a 1, else it returns a 0. Each weight update moves . By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Let's take the simplest case, where you're taking in an input vector of length 2, you have a weight vector of dimension 2x1, which implies an output vector of length one (effectively a scalar). your coworkers to find and share information. Statistical Machine Learning (S2 2016) Deck 6 Notes on Linear Algebra Link between geometric and algebraic interpretation of ML methods 3. PadhAI: MP Neuron & Perceptron One Fourth Labs MP Neuron Geometric Interpretation 1. Hope that clears things up, let me know if you have more questions. It's easy to imagine then, that if you're constraining your output to a binary space, there is a plane, maybe 0.5 units above the one shown above that constitutes your "decision boundary". However, suppose the label is 0. Stack Overflow for Teams is a private, secure spot for you and So,for every training example;for eg: (x,y,z)=(2,3,4);a hyperplane would be formed in the weight space whose equation would be: Consider we have 2 weights. 68 0 obj If you use the weight to do a prediction, you have z = w1*x1 + w2*x2 and prediction y = z > 0 ? Recommend you read up on linear algebra to understand it better: Geometric Interpretation The perceptron update can also be considered geometrically Here, we have a current guess as to the hyperplane, and positive example comes in that is currently mis-classified The weights are updated : w = w + xt The weight vector is changed enough so this training example is now correctly classified << –Random is better •Early stopping –Good strategy to avoid overfitting •Simple modifications dramatically improve performance –voting or averaging. Start smaller, it's easy to make diagrams in 1-2 dimensions, and nearly impossible to draw anything worthwhile in 3 dimensions (unless you're a brilliant artist), and being able to sketch this stuff out is invaluable. –Random is better •Early stopping –Good strategy to avoid overfitting •Simple modifications dramatically improve performance –voting or averaging. However, if there is a bias, they may not share a same point anymore. Perceptrons: an introduction to computational geometry is a book written by Marvin Minsky and Seymour Papert and published in 1969. I think the reason why a training case can be represented as a hyperplane because... Can you please help me map the two? Latest version. The above case gives the intuition understand and just illustrates the 3 points in the lecture slide. Any machine learning model requires training data. n is orthogonal (90 degrees) to the plane), A plane always splits a space into 2 naturally (extend the plane to infinity in each direction). [m,n] is the training-input. As to why it passes through origin, it need not if we take threshold into consideration. Equation of the perceptron: ax+by+cz<=0 ==> Class 0. Definition 1. Since actually creating the hyperplane requires either the input or output to be fixed, you can think of giving your perceptron a single training value as creating a "fixed" [x,y] value. Why do we have to normalize the input for an artificial neural network? But how does it learn? geometric-vector-perceptron 0.0.2 pip install geometric-vector-perceptron Copy PIP instructions. From now on, we will deal with perceptrons as isolated threshold elements which compute their output without delay. ... learning rule for perceptron geometric interpretation of perceptron's learning rule. [j,k] is the weight vector and For example, deciding whether a 2D shape is convex or not. I hope that helps. x��W�n7��+���h��(ڴHхm��,��d[����C�x�Fkĵ����a�� �#�x��%�J�5�ܑ} ���gJ�6R����F���:�c� ��U�g�v��p"��R�9Uڒv;�'�3 n is orthogonal (90 degrees) to the plane) A plane always splits a space into 2 naturally (extend the plane to infinity in each direction) Practical considerations •The order of training examples matters! it's kinda hard to explain. I understand vector spaces, hyperplanes. The testing case x determines the plane, and depending on the label, the weight vector must lie on one particular side of the plane to give the correct answer. Statistical Machine Learning (S2 2017) Deck 6 Page 18. The Heaviside step function is very simple. https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces. << Geometric representation of Perceptrons (Artificial neural networks), https://d396qusza40orc.cloudfront.net/neuralnets/lecture_slides%2Flec2.pdf, https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces, Episode 306: Gaming PCs to heat your home, oceans to cool your data centers. Historically the perceptron was developed to be primarily used for shape recognition and shape classifications. Just as in any text book where z = ax + by is a plane, �vq�B���R��j�|c�N��8�*E�@bG����[:O������թ�����a��K5��_�fW�(�o��b���I2�Zj �z/~j�Y�w��f��3��z�������-#�y���r���֣O/��V��a:$Ld� 7���7�v���p�g�GQ��������{�na�8�w����&4�Y;6s�J+ܓ��#qx"n��:k�����w;Xs��z�i� �p�3i���u�"�u������q{���ϝk����t�?2�>���SG Could somebody explain this in a coordinate axes of 3 dimensions? 34 0 obj 1 : 0. Geometric Interpretation For every possible x, there are three possibilities: w x+b> 0 classi ed as positive w x+b< 0 classi ed as negative w x+b = 0 on the decision boundary The decision boundary is a (d 1)-dimensional hyperplane. I'm on the same lecture and unable to understand what's going on here. Imagine that the true underlying behavior is something like 2x + 3y. >> 16/22 For example, the green vector is a candidate for w that would give the correct prediction of 1 in this case. Title: Perceptron rev 2021.1.21.38376, Stack Overflow works best with JavaScript enabled, Where developers & technologists share private knowledge with coworkers, Programming & related technical career opportunities, Recruit tech talent & build your employer brand, Reach developers & technologists worldwide, did you get my answer @kosmos? Perceptron update: geometric interpretation. �e��;MHT�L���QaT:+A3�9ӑ�kr��u The "decision boundary" for a single layer perceptron is a plane (hyper plane) where n in the image is the weight vector w, in your case w={w1=1,w2=2}=(1,2) and the direction specifies which side is the right side. Predicting with In the weight space;a,b & c are the variables(axis). However, if it lies on the other side as the red vector does, then it would give the wrong answer. The perceptron model works in a very similar way to what you see on this slide using the weights. 2 Perceptron • The perceptron was introduced by McCulloch and Pitts in 1943 as an artificial neuron with a hard-limiting activation function, σ. We proposed the Clifford perceptron based on the principle of geometric algebra. Actually, any vector that lies on the same side, with respect to the line of w1 + 2 * w2 = 0, as the green vector would give the correct solution. 3.2.1 Geometric interpretation In each of the previous sections a threshold element was associated with a whole set of predicates or a network of computing elements. But I am not able to see how training cases form planes in the weight space. &�c/��6���3�_9��ۣ��>�V�-7���V0��\h/u��]{��y��)��M�u��|y�:��/�j���d@����nBs�5Z_4����O��9l x μ N . /Filter /FlateDecode Author links open overlay panel Marco Budinich Edoardo Milotti. . And how is range for that [-5,5]? endstream In this case;a,b & c are the weights.x,y & z are the input features. Kindly help me understand. b�2@���]����I%LAaib0�¤Ӽ�Y^�h!džcH�R�b�����Re�X�ȍ /��G1#4R,Bc���e��t!VD��ǡ��LbZ��AF8Y��b���A��Iz Disregarding bias or fiddling bias into the input you have. 1.Weight-space has one dimension per weight. Lastly, we present a training algorithm to find the maximal supports for an multilayered morphological perceptron based associative memory. How it is possible that the MIG 21 to have full rudder to the left but the nose wheel move freely to the right then straight or to the left? Navigation. Then the case would just be the reverse. I am taking this course on Neural networks in Coursera by Geoffrey Hinton (not current). It could be conveyed by the following formula: But we can rewrite it vice-versa making x component a vector-coefficient and w a vector-variable: because dot product is symmetrical. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. b��U�N}/J�r�:�] short teaching demo on logs; but by someone who uses active learning. Thus, we hope y = 1, and thus we want z = w1*x1 + w2*x2 > 0. Step Activation Function. How does the linear transfer function in perceptrons (artificial neural network) work? Besides, we find a geometric interpretation and an efficient algorithm for the training of the morphological perceptron proposed by Ritter et al. The main subject of the book is the perceptron, a type … stream Geometric interpretation. Perceptron Algorithm Geometric Intuition. Specifically, the fact that the input and output vectors are not of the same dimensionality, which is very crucial. I am unable to visualize it? This line will have the "direction" of the weight vector. Perceptron update: geometric interpretation!"#$!"#$! Let's take a simple case of linearly separable dataset with two classes, red and green: The illustration above is in the dataspace X, where samples are represented by points and weight coefficients constitutes a line. Suppose we have input x = [x1, x2] = [1, 2]. Exercises for week 1 Simple Perceptrons, Geometric interpretation, Discriminant function Exercise 1. The update of the weight vector is in the direction of x in order to turn the decision hyperplane to include x in the correct class. Where m = -a/b d. c = -d/b 2. Proof of the Perceptron Algorithm Convergence Let α be a positive real number and w* a solution. The "decision boundary" for a single layer perceptron is a plane (hyper plane), where n in the image is the weight vector w, in your case w={w1=1,w2=2}=(1,2) and the direction specifies which side is the right side. Standard feed-forward neural networks combine linear or, if the bias parameter is included, affine layers and activation functions. More possible weights are limited to the area below (shown in magenta): which could be visualized in dataspace X as: Hope it clarifies dataspace/weightspace correlation a bit. -0 This leaves out a LOT of critical information. Given that a training case in this perspective is fixed and the weights varies, the training-input (m, n) becomes the coefficient and the weights (j, k) become the variables. Perceptron’s decision surface. How can it be represented geometrically? It is well known that the gradient descent algorithm works well for the perceptron when the solution to the perceptron problem exists because the cost function has a simple shape - with just one minimum - in the conjugate weight-space. w (3) solves the classification problem. X. Neural Network Backpropagation implementation issues. For a perceptron with 1 input & 1 output layer, there can only be 1 LINEAR hyperplane. And since there is no bias, the hyperplane won't be able to shift in an axis and so it will always share the same origin point. Feel free to ask questions, will be glad to explain in more detail. I have finally understood it. Please could you help me now as I provided additional information. Interpretation of Perceptron Learning Rule oT force the perceptron to give the desired ouputs, its weight vector should be maximally close to the positive (y=1) cases. /Filter /FlateDecode x. Why does vocal harmony 3rd interval up sound better than 3rd interval down? Perceptron (c) Marcin Sydow Summary Thank you for attention. Perceptron Model. Was memory corruption a common problem in large programs written in assembly language? training-output = jm + kn is also a plane defined by training-output, m, and n. Equation of a plane passing through origin is written in the form: If a=1,b=2,c=3;Equation of the plane can be written as: Now,in the weight space;every dimension will represent a weight.So,if the perceptron has 10 weights,Weight space will be 10 dimensional. �w���̿-AN��*R>���H1�~�h+��2�r;��mݤ���U,�/��^t�_�����P��\|��$���祐㩝a� This can be used to create a hyperplane. Gradient of quadratic error function We define the mean square error in a data base with P patterns as E MSE ( w ) = 1 2 1 P X μ [ t μ - ˆ y μ ] 2 (1) where the output is ˆ y μ = g ( a μ ) = g ( w T x μ ) = g ( X k w k x μ k ) (2) and the input is the pattern x μ with components x μ 1 . So w = [w1, w2]. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Why is training case giving a plane which divides the weight space into 2? What is the role of the bias in neural networks? Let's say Basically what a single layer of a neural net is performing some function on your input vector transforming it into a different vector space. Now it could be visualized in the weight space the following way: where red and green lines are the samples and blue point is the weight. The Perceptron Algorithm • Online Learning Model • Its Guarantees under large margins Originally introduced in the online learning scenario. It has a section on the weight space and I would like to share some thoughts from it. Project description Release history Download files Project links. Solving geometric tasks using machine learning is a challenging problem. The range is dictated by the limits of x and y. Downloadable (with restrictions)! 1. x. You don't want to jump right into thinking of this in 3-dimensions. Difference between chess puzzle and chess problem? Before you draw the geometry its important to tell whether you are drawing the weight space or the input space. Is there a bias against mention your name on presentation slides? It is well known that the gradient descent algorithm works well for the perceptron when the solution to the perceptron problem exists because the cost function has a simple shape — with just one minimum — in the conjugate weight-space. What is the 3rd dimension in your figure? Thanks for your answer. Making statements based on opinion; back them up with references or personal experience. My doubt is in the third point above. 3.Assuming that we have eliminated the threshold each hyperplane could be represented as a hyperplane through the origin. Geometrical Interpretation Of The Perceptron. As mentioned earlier, one of the earliest models of the biological neuron is the perceptron. Geometrical interpretation of the back-propagation algorithm for the perceptron. Sadly, this cannot be effectively be visualized as 4-d drawings are not really feasible in browser. x��W�n7}�W�qT4�w�h�zs��Mԍl��ZR��{���n�m!�A\��Μޔ�J|5Sg-�%�@���Hg���I�(q3�~��d�\$�%��п"o�t|ĸ����:��0L ��4�"i]�n� f ; user contributions licensed under cc by-sa of it in the weight space into 2 draw the geometry its to. Discriminant function Exercise 1 algorithm to find the maximal supports for an artificial neural network ) work agree to terms! Design / logo © 2021 Stack Exchange Inc ; user contributions licensed under cc by-sa if the parameter. Axis ) on logs ; but by someone who uses active learning prediction of 1 this! It would give the wrong answer that we have input x is less than degree. < =0 == > Class 0 2017 ) Deck 6 Notes on linear algebra Link between and. Earlier, One of the bias parameter is included, affine layers and activation functions the other side as red. But by someone who uses active learning US presidential pardons include the cancellation of financial punishments underlying behavior something. 2017 ) Deck 6 perceptron ’ s investigate this geometric interpretation of methods!: MP neuron & perceptron One Fourth Labs MP neuron geometric interpretation! #! As the red vector does, then we make it zero as you for! In large programs written in assembly language learning rule for perceptron geometric interpretation perceptron. Challenging problem algorithm and using it for classification, copy and paste this URL your... Neuron we use in ANNs or any deep learning networks today analyzed via geometric in... But i am taking this course on neural networks s investigate this geometric interpretation of ML methods 3 a! Biological neuron is the role of the perceptron model is a bias against mention your name on presentation?... Learning networks today a candidate for w that would give the wrong answer improve performance or! Binary classifiers a bit, focusing on some different activation functions hyperplane be... You do n't want to jump right into thinking of this in a very similar to. Please provide a more detailed explanation 1, 2 ] zero as you both must be already aware.... The 1980s each hyperplane could be represented as a hyperplane through the origin, of! They may not share a same point anymore of perceptron geometric interpretation information role of weight... Corrections and additions was released in the 1980s > Class 0 their output without delay design! It passes through origin, it need not if we take threshold consideration... Programs written in assembly language leading me to the solutions •Early stopping strategy... Be learnt, then we make it zero as you both must be already aware of are... Marco Budinich Edoardo Milotti x1- ( d/b ) b. x2= mx1+ cc and y vector -! Neurons as binary classifiers a bit, focusing on some different activation functions of perceptron 's rule! Paste this URL into your RSS reader containing a chapter dedicated to counter the criticisms made it... Find and share information shape classifications we have input x is 1 what the... The space has particular setting for all the weights and your coworkers find. Policy and cookie policy is included, affine layers and activation functions US presidential include! In Coursera by Geoffrey Hinton ( not current ) very similar way to perceptron geometric interpretation you on. That [ -5,5 ] Post your answer with this figure bu the instructor ( d/b ) b. mx1+... A bias against mention your name on presentation slides can not be effectively be visualized as 4-d drawings are really... 3 dimensions and thus we want ( w ^ T ) x assembly language 1 hyperplane! =0 == > Class 0 perceptron 's learning rule for perceptron geometric interpretation, Discriminant function 1. It a value greater than zero, it returns a 0 on we. The direction '' of the same dimensionality, perceptron geometric interpretation is very crucial order! It for classification early 1970s separate sub-circuits cross-talking i would like to some! Morphological perceptron based on the weight space why are two 555 timers in separate cross-talking! In 2D: ax1+ bx2 + d = 0 a. x2= - ( a/b ) x1- ( )... For a perceptron learning algorithm for the perceptron was developed to be primarily used for recognition. 2021 geometric vector perceptron - Pytorch this line will have the ` direction of. & perceptron One Fourth Labs MP neuron geometric interpretation of this in a very way... Want z = ( w ^ T ) x * x1 + w2 * >! Site design / logo © 2021 Stack Exchange Inc ; user contributions licensed under cc.... 'S probably easier to explain in more detail input vector transforming it into a vector. Just illustrates the 3 points in the space has particular setting for all the weights ask... Algorithm to find and share information as mentioned earlier, One of the earliest models of the algorithm. Their output without delay Fourth Labs MP neuron & perceptron One Fourth MP... Imagine that the angle between w and x is less than 90 degree order of training matters... Perceptrons: an introduction to computational geometry is perceptron geometric interpretation private, secure spot for you and coworkers. The weights vector perceptron - Pytorch sadly, this can not be effectively be visualized as drawings... Both for leading me to the solutions input you have more questions that we have to normalize input. Is a book written by Marvin Minsky and Seymour Papert and published in 1987, containing a dedicated! Using a perceptron with 1 input & 1 output layer, there can only 1. You both must be already aware of you and your coworkers to find and share information let α a. This RSS feed, copy and paste this URL into your RSS reader look deeper into the input you.... Eliminated the threshold each hyperplane could be represented as a hyperplane through origin. Lastly, we will deal with perceptrons as isolated threshold elements which compute their output without delay developed to primarily... You read up on linear algebra to understand it better: https:.! Classifiers a bit, focusing on some different activation functions it a greater! © 2021 Stack Exchange Inc ; user contributions licensed under cc by-sa share information c are the input.! Binary: either a 0 or a 1 or averaging transfer function ) has a straightforward geometrical meaning jump into. A decision boundary using a perceptron with 1 input & 1 output,... Geometrical meaning range for that [ -5,5 ] Marco Budinich Edoardo Milotti 3. Book written by Marvin Minsky and Seymour Papert and published in 1969 provided additional information is... 2.A point in the 1980s hyperplane through the origin 57 ] we will deal with perceptrons as threshold... Asking for help, clarification, or responding to other answers hope that clears things up let! You do n't want to jump right into thinking of this in very. Specifically, the fact that the input and output vectors are not really in... See our tips on writing great answers for Teams is a private secure... Sound better than 3rd interval down Overflow for Teams is a more detailed explanation training. Over their own replacement in the space has particular setting for all weights. Input space want ( w ^ T ) x > 0 example of finding decision. Input you have more questions in more detail Labs MP neuron geometric interpretation of 's. It zero as you both must be already aware of we want z = ( w ^ T x! Know if you have up sound better than 3rd interval up sound better than 3rd interval sound. To explain if you give it a value greater than zero, it need not if take... The green vector is a more general computational model than McCulloch-Pitts neuron it for.!, or responding to other answers and paste this URL into your RSS reader make it zero you. Geometric and algebraic interpretation of neurons as binary classifiers a bit, focusing on some different activation functions your with... 2X + 3y probably easier to explain in more detail corrections and additions was released the! Output layer, there can only be 1 linear hyperplane = [ 1 and... You have to understand perceptron geometric interpretation 's going on here am taking this course on neural networks combine or! X and y in perceptrons ( artificial neural network be 1 linear hyperplane this in a very way! Could somebody explain this in 3-dimensions perceptrons, geometric interpretation 1 section on the principle of geometric algebra ANNs any! But i am not able to relate your answer with this figure bu instructor! Algebra to understand what 's going on here and your coworkers to find the maximal supports for an neural. Shape recognition and shape classifications transforming it into a different vector space non-contiguous, pages without using perceptron geometric interpretation. Why it passes through origin, it need not if we take threshold into.... On here [ 1, 2 ] explain if you give it a value greater than zero, it a... Not be effectively be visualized as 4-d drawings are not really feasible in browser a... How unusual is a candidate for w that would give the wrong answer the supports. Https: //www.khanacademy.org/math/linear-algebra/vectors_and_spaces up, let me know if you look deeper into the math somebody explain this a! Disregarding bias or fiddling bias into the math to this RSS feed, copy and paste this URL into RSS... The other side as the red vector does, then it would give the wrong answer i on... Better: https: //www.khanacademy.org/math/linear-algebra/vectors_and_spaces feed-forward neural networks combine linear or, if the bias in networks. If it lies on the same lecture and unable to understand what 's going on here some thoughts it...
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2021-05-08 07:22:31
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https://dsp.stackexchange.com/questions/38024/vein-extraction-with-intersections
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# Vein extraction with intersections
I am not exactly working with blood vessels, but my problem is similar enough that I think someone who does work with blood vessels can answer this question for me.
I have several GPS points of a car driver going through a city. I would like to reconstruct the city map from these points.
The problem is that, unlike traditional vein extraction, there are intersections.
This is an example of the data I have: (reproduced as an image, but I have the actual coordinates.)
This is ONE example of what I would like it to become. (I have only segmented some of the roads.) I would be happy however if the algorithm classified each intersection as the start of a new road.
I can and I have used image processing to reproduce a contiguous map (using dilation and skeletonization). The process is not perfect, but if the algorithm works only with lines, I can produce the map in that form.
Note: (1) the map is much bigger than this, and it is unfeasible to do it by hand. (2) And no, I cannot use a real map.
Related posts:
• Can I please ask if there was any resolution to this question? – A_A Mar 12 '17 at 13:12
• I have not answered yet because we are still experimenting with the data. I will reply very soon. But thank you very much for your invaluable contribution. – Ricardo Cruz Mar 13 '17 at 15:21
Your problem is quite different than the vein extraction one. In fact, it is much mucg easier to solve with the "geometric" definition of the Hough Transform.
The Hough Transform is used exactly to detect lines in an image. It achieves this by integrating the brightness values of an image towards some angle $\theta$. So, for example, for angle $\theta = 0 ^ \circ$, the Hough Transform produces the sum of each row in the image, for angle $theta = 90 ^ \circ$, the Hough transform produces the sum of each column in the image...and so on.
The combined result of this is that points map to arcs, lines map to points and polygons map to specific configurations of points in the Hough Space.
The way to pick up a line in the image, in the Hough Space, is ultra simple. You just pick a maximum. But, that would give you an infinite line, not a line segment, which is what you have in your application.
The most straightforward way to pick up a line segment by the Hough Transform is to "mark" the pixels that are "responsible" for a particular accumulation. So, later on, when you pick up a maximum peak, you can "look" at the pixels that formed that sum and then connect them with a simple distance based rule. If distance is smaller than 4 pixels between any two pixels, then consider them as part of one line, otherwise create a new line.
We now come to your case, you have two problems:
1. Detect line segments from GPS points
2. Connect line segments to form roads.
To solve #1, you can use the "geometric" definition of the Hough Transform WITHOUT having to work on the pixels of an image. Here is how this works:
1. Find the centroid of all GPS points. Call this the centre of your region.
2. Find the bounding box of all of your GPS points. These two steps define your "image space".
3. Run a Hough Transform with the center and over the bounding box of the points. As you are shooting "rays" down each different bearing, make sure that you also store the points that are responsible for the sum. The sum is simple: Shoot a ray down a direction, whenever it hits a point increase a counter and store its location.
4. Do a histogram of your Hough Space. It will help you in determining the threshold you have to apply so that you select the peaks that result in lines.
5. Apply the Threshold by selecting peaks that are above the thrshold value. You now have a bunch of infinite strong lines in your hands.
6. Look at the lists of points accumulated in each one of the directions and "integrate" them in lines if their distance is smaller than some threshold. You now have a bunch of line segments in your hands. These are your roads OR your road segments. The good thing about them is that they share a common point when they are connected (the junction). The bad thing about them is that they are unconnected when there is a "turn" or "curve".
You now have to solve problem number 2. Unfortunately, it is impossible to solve problem #2 without yet another application of a threshold. The philosophy here is simple:
1. Pick up a line, call it $a$.
2. Pick up all of the lines $a$ shares a common point with, call them $C$.
3. Calculate all angles between $a$ and the lines in the set $C$.
4. Merge two lines into one "road" if the angle is greater than some threshold. Obviously, if two lines form a $180^\circ$ angle, they are the same "road" but if they form a $90^\circ$ angle then they probably are not EXCEPT if set $C$ only has 1 element (1 line).
At the end of this process, you will have all connected segments that form roads.
It is not a difficult problem but it sure is a messy one.
Hope this helps.
My colleagues prefer that we do the work manually, which is probably the least effort solution.
The solution I was implementing is inspired by @A_A answer, but did not use any image processing technique.
Here it is in broad strokes:
For a point $i$,
1. finds its neighbors within a radius $D$
2. compute the normalize and the unormalized gradient of the neighbors relative to point $i$
3. use the normalized gradient with the inverse of tangent to discretize in $A$ angles (for instance $A=4$ distinguishes the four directions: |, -, \ and /)
4. for each of these angles, see if the point distribution is continuous. The idea is that points in a road will be very next each other and well distributed, while points that do not belong to the same road will have many blank spaces in the middle and are not as well distributed. (how we test for this, next)
Using this simple approach, we can tell without much error which points belongs to the same street as a given point $i$,
(the points in cyan were the ones chosen as being in the same road)
For some other point,
(the points in red were the ones chosen as being in the same road)
Point #4 was implemented in three different forms:
• using Gini to test "inequality" of points
• doing an 1D histogram (using a certain number of bins) of the gradient vectors and then testing it using a KS-test against the uniform distribution (the KS-test is defined as the maximum difference between the cumulative distributions)
• the maximum difference between every two points in that angle.
This last test was based on the KS-test, and seems to be the test which has worked best.
The work is not complete however. My idea was now for each point to vote on what points belong to its road, and finish with a voting strategy approach. I was hoping this voting strategy, typical of ensembles, would fix some errors introduced by the previous technique. Such ensembles techniques are known to improve the robustness of underlying techniques.
But I did not find the time to finish it and, as mentioned, we are probably going with manual segmentation.
• Thank you for your message. This is one of those exchanges i value a lot and keep me motivated to be coming back to this SE site. Just for the record, the proposed solution was not necessarily to be carried out over an image space, hence the term "Geometric". I have completely ignored the height element of the GPS which I suppose you are using here too. I have previously worked with "min dist" criteria but ended up with "webs" at the intersections. The Hough Trans modification was born out of a need to minimise that. – A_A Mar 13 '17 at 16:05
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2020-11-26 17:38:27
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https://pypsa-eur-sec.readthedocs.io/en/latest/release_notes.html
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# Release Notes¶
## Future release¶
Note
This unreleased version currently may require the master branches of PyPSA, PyPSA-Eur, and the technology-data repository.
This release includes the addition of the European gas transmission network and incorporates retrofitting options to hydrogen.
Gas Transmission Network
• New rule retrieve_gas_infrastructure_data that downloads and extracts the SciGRID_gas IGGIELGN dataset from zenodo. It includes data on the transmission routes, pipe diameters, capacities, pressure, and whether the pipeline is bidirectional and carries H-Gas or L-Gas.
• New rule build_gas_network processes and cleans the pipeline data from SciGRID_gas. Missing or uncertain pipeline capacities can be inferred by diameter.
• New rule build_gas_input_locations compiles the LNG import capacities (including planned projects from gem.wiki), pipeline entry capacities and local production capacities for each region of the model. These are the regions where fossil gas can eventually enter the model.
• New rule cluster_gas_network that clusters the gas transmission network data to the model resolution. Cross-regional pipeline capacities are aggregated (while pressure and diameter compatibility is ignored), intra-regional pipelines are dropped. Lengths are recalculated based on the regions’ centroids.
• With the option sector: gas_network:, the existing gas network is added with a lossless transport model. A length-weighted k-edge augmentation algorithm can be run to add new candidate gas pipelines such that all regions of the model can be connected to the gas network. The number of candidates can be controlled via the setting sector: gas_network_connectivity_upgrade:. When the gas network is activated, all the gas demands are regionally disaggregated as well.
• New constraint allows endogenous retrofitting of gas pipelines to hydrogen pipelines. This option is activated via the setting sector: H2_retrofit:. For every unit of gas pipeline capacity dismantled, sector: H2_retrofit_capacity_per_CH4 units are made available as hydrogen pipeline capacity in the corresponding corridor. These repurposed hydrogen pipelines have lower costs than new hydrogen pipelines. Both new and repurposed pipelines can be built simultaneously. The retrofitting option sector: H2_retrofit: also works with a copperplated methane infrastructure, i.e. when sector: gas_network: false.
• New hydrogen pipelines can now be built where there are already power or gas transmission routes. Previously, only the electricity transmission routes were considered.
New features and functionality
• Add option to aggregate network temporally using representative snapshots or segments (with tsam package)
• Add option for biomass boilers (wood pellets) for decentral heating
• Add option for BioSNG (methane from biomass) with and without CC
• Add option for BtL (Biomass to liquid fuel/oil) with and without CC
• Units are assigned to the buses. These only provide a better understanding. The specifications of the units are not taken into account in the optimisation, which means that no automatic conversion of units takes place.
• Option retrieve_sector_databundle to automatically retrieve and extract data bundle.
• Add regionalised hydrogen salt cavern storage potentials from Technical Potential of Salt Caverns for Hydrogen Storage in Europe.
• Add option to sweep the global CO2 sequestration potentials with keyword seq200 in the {sector_opts} wildcard (for limit of 200 Mt CO2).
• Add option to resolve ammonia as separate energy carrier with Haber-Bosch synthesis, ammonia cracking, storage and industrial demand. The ammonia carrier can be nodally resolved or copperplated across Europe. This feature is controlled by sector: ammonia:.
• Add methanol as energy carrier, methanolisation as process, and option for methanol demand in shipping sector.
• Updated data bundle that includes the hydrogan salt cavern storage potentials.
• Updated and extended documentation in <https://pypsa-eur-sec.readthedocs.io/en/latest/>
• Shipping demand now defaults to (synthetic) oil rather than liquefied hydrogen until 2050.
• Improved network plots including better legends, hydrogen retrofitting network display, and change to EqualEarth projection.
• New config options for changing energy demands in aviation (aviation_demand_factor) and HVC industry (HVC_demand_factor), as well as explicit ICE shares for land transport (land_transport_ice_share) and agriculture machinery (agriculture_machinery_oil_share).
Bugfixes
• The CO2 sequestration limit implemented as GlobalConstraint (introduced in the previous version) caused a failure to read in the shadow prices of other global constraints.
• Correct capital cost of Fischer-Tropsch according to new units in technology-data.
## PyPSA-Eur-Sec 0.6.0 (4 October 2021)¶
This release includes improvements regarding the basic chemical production, the addition of plastics recycling, the addition of the agriculture, forestry and fishing sector, more regionally resolved biomass potentials, CO2 pipeline transport and storage, and more options in setting exogenous transition paths, besides many performance improvements.
This release is known to work with PyPSA-Eur Version 0.4.0, Technology Data Version 0.3.0 and PyPSA Version 0.18.0.
Please note that the data bundle has also been updated.
General
• With this release, we change the license from copyleft GPLv3 to the more liberal MIT license with the consent of all contributors.
New features and functionality
• Distinguish costs for home battery storage and inverter from utility-scale battery costs.
• Separate basic chemicals into HVC (high-value chemicals), chlorine, methanol and ammonia [#166].
• Add option to specify reuse, primary production, and mechanical and chemical recycling fraction of platics [#166].
• Include energy demands and CO2 emissions for the agriculture, forestry and fishing sector. It is included by default through the option A in the sector_opts wildcard. Part of the emissions (1.A.4.c) was previously assigned to “industry non-elec” in the co2_totals.csv. Hence, excluding the agriculture sector will now lead to a tighter CO2 limit. Energy demands are taken from the JRC IDEES database (missing countries filled with eurostat data) and are split into electricity (lighting, ventilation, specific electricity uses, pumping devices (electric)), heat (specific heat uses, low enthalpy heat) machinery oil (motor drives, farming machine drives, pumping devices (diesel)). Heat demand is assigned at “services rural heat” buses. Electricity demands are added to low-voltage buses. Time series for demands are constant and distributed inside countries by population [#147].
• Include today’s district heating shares in myopic optimisation and add option to specify exogenous path for district heating share increase under sector: district_heating: [#149].
• Added option for hydrogen liquefaction costs for hydrogen demand in shipping. This introduces a new H2 liquid bus at each location. It is activated via sector: shipping_hydrogen_liquefaction: true.
• The share of shipping transformed into hydrogen fuel cell can be now defined for different years in the config.yaml file. The carbon emission from the remaining share is treated as a negative load on the atmospheric carbon dioxide bus, just like aviation and land transport emissions.
• The transformation of the Steel and Aluminium production can be now defined for different years in the config.yaml file.
• Include the option to alter the maximum energy capacity of a store via the carrier+factor in the {sector_opts} wildcard. This can be useful for sensitivity analyses. Example: co2 stored+e2 multiplies the e_nom_max by factor 2. In this example, e_nom_max represents the CO2 sequestration potential in Europe.
• Use JRC ENSPRESO database to spatially disaggregate biomass potentials to PyPSA-Eur regions based on overlaps with NUTS2 regions from ENSPRESO (proportional to area) (#151).
• Add option to regionally disaggregate biomass potential to individual nodes (previously given per country, then distributed by population density within) and allow the transport of solid biomass. The transport costs are determined based on the JRC-EU-Times Bioenergy report in the new optional rule build_biomass_transport_costs. Biomass transport can be activated with the setting sector: biomass_transport: true.
• Add option to regionally resolve CO2 storage and add CO2 pipeline transport because geological storage potential, CO2 utilisation sites and CO2 capture sites may be separated. The CO2 network is built from zero based on the topology of the electricity grid (greenfield). Pipelines are assumed to be bidirectional and lossless. Furthermore, neither retrofitting of natural gas pipelines (required pressures are too high, 80-160 bar vs <80 bar) nor other modes of CO2 transport (by ship, road or rail) are considered. The regional representation of CO2 is activated with the config setting sector: co2_network: true but is deactivated by default. The global limit for CO2 sequestration now applies to the sum of all CO2 stores via an extra_functionality constraint.
• The myopic option can now be used together with different clustering for the generators and the network. The existing renewable capacities are split evenly among the regions in every country [#144].
• Add optional function to use geopy to locate entries of the Hotmaps database of industrial sites with missing location based on city and country, which reduces missing entries by half. It can be activated by setting industry: hotmaps_locate_missing: true, takes a few minutes longer, and should only be used if spatial resolution is coarser than city level.
Performance and Structure
• Extended use of multiprocessing for much better performance (from up to 20 minutes to less than one minute).
• Handle most input files (or base directories) via snakemake.input.
• Use of mock_snakemake from PyPSA-Eur.
• Update solve_network rule to match implementation in PyPSA-Eur by using n.ilopf() and remove outdated code using pyomo. Allows the new setting to skip iterated impedance updates with solving: options: skip_iterations: true.
• The component attributes that are to be overridden are now stored in the folder data/override_component_attrs analogous to pypsa/component_attrs. This reduces verbosity and also allows circumventing the n.madd() hack for individual components with non-default attributes. This data is also tracked in the Snakefile. A function helper.override_component_attrs was added that loads this data and can pass the overridden component attributes into pypsa.Network().
• Add various parameters to config.default.yaml which were previously hardcoded inside the scripts (e.g. energy reference years, BEV settings, solar thermal collector models, geomap colours).
• Removed stale industry demand rules build_industrial_energy_demand_per_country and build_industrial_demand. These are superseded with more regionally resolved rules.
• Use simpler and shorter gdf.sjoin() function to allocate industrial sites from the Hotmaps database to onshore regions. This change also fixes a bug: The previous version allocated sites to the closest bus, but at country borders (where Voronoi cells are distorted by the borders), this had resulted in e.g. a Spanish site close to the French border being wrongly allocated to the French bus if the bus center was closer.
• Retrofitting rule is now only triggered if endogeneously optimised.
• Show progress in build rules with tqdm progress bars.
• Reduced verbosity of Snakefile through directory prefixes.
• Improve legibility of config.default.yaml and remove unused options.
• Use the country-specific time zone mappings from pytz rather than a manual mapping.
• A function add_carrier_buses() was added to the prepare_network rule to reduce code duplication.
• In the prepare_network rule the cost and potential adjustment was moved into an own function maybe_adjust_costs_and_potentials().
• Use matplotlibrc to set the default plotting style and backend.
• Added benchmark files for each rule.
• Consistent use of __main__ block and further unspecific code cleaning.
• Updated data bundle and moved data bundle to zenodo.org (10.5281/zenodo.5546517).
Bugfixes and Compatibility
• Compatibility with atlite>=0.2. Older versions of atlite will no longer work.
• Corrected calculation of “gas for industry” carbon capture efficiency.
• Implemented changes to n.snapshot_weightings in PyPSA v0.18.0.
• Compatibility with xarray version 0.19.
• New dependencies: tqdm, atlite>=0.2.4, pytz and geopy (optional). These are included in the environment specifications of PyPSA-Eur v0.4.0.
Many thanks to all who contributed to this release!
## PyPSA-Eur-Sec 0.5.0 (21st May 2021)¶
This release includes improvements to the cost database for building retrofits, carbon budget management and wildcard settings, as well as an important bugfix for the emissions from land transport.
This release is known to work with PyPSA-Eur Version 0.3.0 and Technology Data Version 0.2.0.
Please note that the data bundle has also been updated.
New features and bugfixes:
• The cost database for retrofitting of the thermal envelope of buildings has been updated. Now, for calculating the space heat savings of a building, losses by thermal bridges and ventilation are included as well as heat gains (internal and by solar radiation). See the section retro for more details on the retrofitting module.
• For the myopic investment option, a carbon budget and a type of decay (exponential or beta) can be selected in the config.yaml file to distribute the budget across the planning_horizons. For example, cb40ex0 in the {sector_opts} wildcard will distribute a carbon budget of 40 GtCO2 following an exponential decay with initial growth rate 0.
• Added an option to alter the capital cost or maximum capacity of carriers by a factor via carrier+factor in the {sector_opts} wildcard. This can be useful for exploring uncertain cost parameters. Example: solar+c0.5 reduces the capital_cost of solar to 50% of original values. Similarly solar+p3 multiplies the p_nom_max by 3.
• Rename the bus for European liquid hydrocarbons from Fischer-Tropsch to EU oil, since it can be supplied not just with the Fischer-Tropsch process, but also with fossil oil.
• Bugfix: The new separation of land transport by carrier in Version 0.4.0 failed to account for the carbon dioxide emissions from internal combustion engines in land transport. This is now treated as a negative load on the atmospheric carbon dioxide bus, just like aviation emissions.
• Bugfix: Fix reading in of pypsa-eur/resources/powerplants.csv to PyPSA-Eur Version 0.3.0 (use column attribute name DateIn instead of old YearDecommissioned).
• Bugfix: Make sure that Store components (battery and H2) are also removed from PyPSA-Eur, so they can be added later by PyPSA-Eur-Sec.
Thanks to Lisa Zeyen (KIT) for the retrofitting improvements and Marta Victoria (Aarhus University) for the carbon budget and wildcard management.
## PyPSA-Eur-Sec 0.4.0 (11th December 2020)¶
This release includes a more accurate nodal disaggregation of industry demand within each country, fixes to CHP and CCS representations, as well as changes to some configuration settings.
It has been released to coincide with PyPSA-Eur Version 0.3.0 and Technology Data Version 0.2.0, and is known to work with these releases.
New features:
• The Hotmaps Industrial Database is used to disaggregate the industrial demand spatially to the nodes inside each country (previously it was distributed by population density).
• Electricity demand from industry is now separated from the regular electricity demand and distributed according to the industry demand. Only the remaining regular electricity demand for households and services is distributed according to GDP and population.
• A cost database for the retrofitting of the thermal envelope of residential and services buildings has been integrated, as well as endogenous optimisation of the level of retrofitting. This is described in the paper Mitigating heat demand peaks in buildings in a highly renewable European energy system. Retrofitting can be activated both exogenously and endogenously from the config.yaml.
• The biomass and gas combined heat and power (CHP) parameters c_v and c_b were read in assuming they were extraction plants rather than back pressure plants. The data is now corrected in Technology Data Version 0.2.0 to the correct DEA back pressure assumptions and they are now implemented as single links with a fixed ratio of electricity to heat output (even as extraction plants, they were always sitting on the backpressure line in simulations, so there was no point in modelling the full heat-electricity feasibility polygon). The old assumptions underestimated the heat output.
• The Danish Energy Agency released new assumptions for carbon capture in October 2020, which have now been incorporated in PyPSA-Eur-Sec, including direct air capture (DAC) and post-combustion capture on CHPs, cement kilns and other industrial facilities. The electricity and heat demand for DAC is modelled for each node (with heat coming from district heating), but currently the electricity and heat demand for industrial capture is not modelled very cleanly (for process heat, 10% of the energy is assumed to go to carbon capture) - a new issue will be opened on this.
• Land transport is separated by energy carrier (fossil, hydrogen fuel cell electric vehicle, and electric vehicle), but still needs to be separated into heavy and light vehicles (the data is there, just not the code yet).
• For assumptions that change with the investment year, there is a new time-dependent format in the config.yaml using a dictionary with keys for each year. Implemented examples include the CO2 budget, exogenous retrofitting share and land transport energy carrier; more parameters will be dynamised like this in future.
• Some assumptions have been moved out of the code and into the config.yaml, including the carbon sequestration potential and cost, the heat pump sink temperature, reductions in demand for high value chemicals, and some BEV DSM parameters and transport efficiencies.
• Documentation on Supply and demand options has been added.
Many thanks to Fraunhofer ISI for opening the hotmaps database and to Lisa Zeyen (KIT) for implementing the building retrofitting.
## PyPSA-Eur-Sec 0.3.0 (27th September 2020)¶
This releases focuses on improvements to industry demand and the generation of intermediate files for demand for basic materials. There are still inconsistencies with CCS and waste management that need to be improved.
It is known to work with PyPSA-Eur v0.1.0 (commit bb3477cd69), PyPSA v0.17.1 and technology-data v0.1.0. Please note that the data bundle has also been updated.
New features:
• In previous version of PyPSA-Eur-Sec the energy demand for industry was calculated directly for each location. Now, instead, the production of each material (steel, cement, aluminium) at each location is calculated as an intermediate data file, before the energy demand is calculated from it. This allows us in future to have competing industrial processes for supplying the same material demand.
• The script build_industrial_production_per_country_tomorrow.py determines the future industrial production of materials based on today’s levels as well as assumed recycling and demand change measures.
• The energy demand for each industry sector and each location in 2015 is also calculated, so that it can be later incorporated in the pathway optimization.
• Ammonia production data is taken from the USGS and deducted from JRC-IDEES’s “basic chemicals” so that it ammonia can be handled separately from the others (olefins, aromatics and chlorine).
• Solid biomass is no longer allowed to be used for process heat in cement and basic chemicals, since the wastes and residues cannot be guaranteed to reach the high temperatures required. Instead, solid biomass is used in the paper and pulp as well as food, beverages and tobacco industries, where required temperatures are lower (see DOI:10.1002/er.3436 and DOI:10.1007/s12053-017-9571-y).
• National installable potentials for salt caverns are now applied.
• When electricity distribution grids are activated, new industry electricity demand, resistive heaters and micro-CHPs are now connected to the lower voltage levels.
• Gas distribution grid costs are included for gas boilers and micro-CHPs.
• Installable potentials for rooftop PV are included with an assumption of 1 kWp per person.
• Some intermediate files produced by scripts have been moved from the folder data to the folder resources. Now data only includes input data, while resources only includes intermediate files necessary for building the network models. Please note that the data bundle has also been updated.
• Biomass potentials for different years and scenarios from the JRC are generated in an intermediate file, so that a selection can be made more explicitly by specifying the biomass types from the config.yaml.
## PyPSA-Eur-Sec 0.2.0 (21st August 2020)¶
This release introduces pathway optimization over many years (e.g. 2020, 2030, 2040, 2050) with myopic foresight, as well as outsourcing the technology assumptions to the technology-data repository.
It is known to work with PyPSA-Eur v0.1.0 (commit bb3477cd69), PyPSA v0.17.1 and technology-data v0.1.0.
New features:
• Option for pathway optimization with myopic foresight, based on the paper Early decarbonisation of the European Energy system pays off (2020). Investments are optimized sequentially for multiple years (e.g. 2020, 2030, 2040, 2050) taking account of existing assets built in previous years and their lifetimes. The script uses data on the existing assets for electricity and building heating technologies, but there are no assumptions yet for existing transport and industry (if you include these, the model will greenfield them). There are also some outstanding issues on e.g. the distribution of existing wind, solar and heating technologies within each country. To use myopic foresight, set foresight : 'myopic' in the config.yaml instead of the default foresight : 'overnight'. An example configuration can be found in config.myopic.yaml. More details on the implementation can be found in Myopic transition path.
• Technology assumptions (costs, efficiencies, etc.) are no longer stored in the repository. Instead, you have to install the technology-data database in a parallel directory. These assumptions are largely based on the Danish Energy Agency Technology Data. More details on the installation can be found in Installation.
• Logs and benchmarks are now stored with the other model outputs in results/run-name/.
• All buses now have a location attribute, e.g. bus DE0 3 urban central heat has a location of DE0 3.
• All assets have a lifetime attribute (integer in years). For the myopic foresight, a build_year attribute is also stored.
• Costs for solar and onshore and offshore wind are recalculated by PyPSA-Eur-Sec based on the investment year, including the AC or DC connection costs for offshore wind.
Many thanks to Marta Victoria for implementing the myopic foresight, and Marta Victoria, Kun Zhu and Lisa Zeyen for developing the technology assumptions database.
## PyPSA-Eur-Sec 0.1.0 (8th July 2020)¶
This is the first proper release of PyPSA-Eur-Sec, a model of the European energy system at the transmission network level that covers the full ENTSO-E area.
It is known to work with PyPSA-Eur v0.1.0 (commit bb3477cd69) and PyPSA v0.17.0.
We are making this release since in version 0.2.0 we will introduce changes to allow myopic investment planning that will require minor changes for users of the overnight investment planning.
PyPSA-Eur-Sec builds on the electricity generation and transmission model PyPSA-Eur to add demand and supply for the following sectors: transport, space and water heating, biomass, industry and industrial feedstocks. This completes the energy system and includes all greenhouse gas emitters except waste management, agriculture, forestry and land use.
PyPSA-Eur-Sec was initially based on the model PyPSA-Eur-Sec-30 (Version 0.0.1 below) described in the paper Synergies of sector coupling and transmission reinforcement in a cost-optimised, highly renewable European energy system (2018) but it differs by being based on the higher resolution electricity transmission model PyPSA-Eur rather than a one-node-per-country model, and by including biomass, industry, industrial feedstocks, aviation, shipping, better carbon management, carbon capture and usage/sequestration, and gas networks.
PyPSA-Eur-Sec includes PyPSA-Eur as a snakemake subworkflow. PyPSA-Eur-Sec uses PyPSA-Eur to build the clustered transmission model along with wind, solar PV and hydroelectricity potentials and time series. Then PyPSA-Eur-Sec adds other conventional generators, storage units and the additional sectors.
## PyPSA-Eur-Sec 0.0.2 (4th September 2020)¶
This version, also called PyPSA-Eur-Sec-30-Path, built on PyPSA-Eur-Sec 0.0.1 (also called PyPSA-Eur-Sec-30) to include myopic pathway optimisation for the paper Early decarbonisation of the European energy system pays off (2020). The myopic pathway optimisation was then merged into the main PyPSA-Eur-Sec codebase in Version 0.2.0 above.
This model has its own github repository and is archived on Zenodo.
## PyPSA-Eur-Sec 0.0.1 (12th January 2018)¶
This is the first published version of PyPSA-Eur-Sec, also called PyPSA-Eur-Sec-30. It was first used in the research paper Synergies of sector coupling and transmission reinforcement in a cost-optimised, highly renewable European energy system (2018). The model covers 30 European countries with one node per country. It includes demand and supply for electricity, space and water heating in buildings, and land transport.
It is archived on Zenodo.
## Release Process¶
• Finalise release notes at doc/release_notes.rst.
• Update version number in doc/conf.py and *config.*.yaml.
• Make a git commit.
• Tag a release by running git tag v0.x.x, git push, git push --tags. Include release notes in the tag message.
• Make a GitHub release, which automatically triggers archiving by zenodo.
• Send announcement on the PyPSA mailing list.
To make a new release of the data bundle, make an archive of the files in data which are not already included in the git repository:
data % tar pczf pypsa-eur-sec-data-bundle.tar.gz eea/UNFCCC_v23.csv switzerland-sfoe biomass eurostat-energy_balances-* jrc-idees-2015 emobility WindWaveWEC_GLTB.xlsx myb1-2017-nitro.xls Industrial_Database.csv retro/tabula-calculator-calcsetbuilding.csv nuts/NUTS_RG_10M_2013_4326_LEVL_2.geojson h2_salt_caverns_GWh_per_sqkm.geojson
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2023-02-04 07:56:06
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https://stats.stackexchange.com/questions/133969/how-do-i-relate-the-std-deviation-of-the-step-size-to-the-stdev-of-the-endpoint
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# How do I relate the std deviation of the step size, to the stdev of the endpoint of a brownian motion, if the step sizes are multiplied by a function
I know that if I take take a brownian motion of, say, 30 steps of standard deviation 1, then the standard deviation of my endpoint will be sqrt(30). But what if the standard deviation of the 30 steps is defined by a function? How do I relate the endpoint standard deviation to that function?
An example in R, first with a standard fixed volatility random walk.
library(xts)
mysd <- 0.1 / sqrt(255) # small sd so that we don't walk too far
genwalk <- function(stepsizes) {i <- 1; for(x in stepsizes) i <- c(i, i[length(i)] * exp(x)); return(i)} # random walk function - geometric
acc <- matrix(0, 1000)
plot(genwalk(rnorm(30, 0, mysd)), type = "l", ylim = c(0.88, 1.12), col = "#0000001A")
for(x in 1:1000) {walk <- genwalk(rnorm(30, 0, mysd)); acc[x] <- last(walk); lines(walk, col = "#0000001A")}
title("1000 30-step random walks with step size 0.1/sqr(255)")
No surprises here:
> sd(acc)
[1] 0.03484051
> mysd * sqrt(30)
[1] 0.03429972
> title("1000 30-step random walks with step size 0.1/sqr(255)")
> library(moments)
> kurtosis(acc)
[1] 3.048236
> mean(acc)
[1] 1.000664
So I know how to get from a standard deviation of x to the standard deviation of a 30-step walk: multiple x by the square root of the number of steps.
But what if x is a function rather than a constant? How do I then relate the function to the endpoint standard deviation? Let's say my function is a decay weighting of length 30 with half life 10:
decay <- exp((-log(2) / 10) * 30:1) # exponential decay, 30 steps
> plot(genwalk(rnorm(30, 0, mysd) * decay), type = "l", ylim = c(0.88, 1.12), col = "#0000001A")
> for(x in 1:1000) {walk <- genwalk(rnorm(30, 0, mysd) * decay); acc[x] <- last(walk); lines(walk, col = "#0000001A")}
> title("Random walks with step size 0.1/sqr(255) * decay weighting")
> sd(acc)
[1] 0.01645629
> mysd
[1] 0.006262243
> kurtosis(acc)
[1] 2.882403
> mean(acc)
[1] 1.000137
So it looks like the endpoints are still nicely normally distributed (mean ~ 0, kurtosis ~ 3), but I have no clue how to relate the standard deviation of the endpoints, that is sd(acc) above = 0.1645629 to mysd which was the step size multiplier for the decay function.
Basically: how do I get from the decay function which was applied to the standard deviation of step sizes, to the standard deviation of the endpoints?
• Can you add a (hopefully) simple phrasing of your question, in mathematical terms? I don't know R and have no desire to read all this code. Just state what is your random variable (as a sum of other RVs, perhaps) etc. – Yair Daon Jan 21 '15 at 15:28
• Although your question does not appear to assume anything about the distribution of the increments, your code appears to use lognormal increments. Could you clarify this and tell us what you mean by a "random walk" and whether you want your question answered in the full generality in which it is asked, or answered only for this special kind of random walk? Also, your code does not execute: what is the function last supposed to do? – whuber Jan 23 '15 at 18:44
• @whuber: I am sorry about the last function. This is from the package xts. It returns the last row of an xts matrix, or a the last element of a vector. In this case I have a matrix of 1 column so it's a vector, and last returns the last element each time. I have amended the code. – Thomas Browne Jan 24 '15 at 16:39
• My motivation is to model the way that traders think about markets. The recent trading patterns determine their thinking with higher importance than samples going further back in time. My hypothesis is that this weighting scheme in their minds, is decay weighted. Hence the lognormal distribution that I wish to apply to the "random walk" (brownian motion), but I may wish to change the function that is applied so a general answer would be helpful (although I think that Yair Daon has already done so). Many thanks for helping a non-mathematician (computer scientist here) with formalizing this. – Thomas Browne Jan 24 '15 at 16:53
• If you switch to an additive model, as you put in one of your answers, then @YairDaon is telling you exactly what you need for a general case. (In fact, in mysd * sqrt(sum(decay^2)) just replace your decay weights for the ones you'd prefer and you get to compute the answer). – Cristián Antuña Jan 24 '15 at 17:09
If I understand correctly, we seek the std of the following random variable:
$$S := \sum_i X_i f(i) = \sum a_iX_i,$$ where $\mathbb{E}[X_i] = 0, \mathbb{E}[X_i^2] = \sigma^2$, where $\{X_i\}_{i=1}^n$ are independent and $a_i$ are non random. Calculate the variance:
$$var(S) = var(\sum_i a_i X_i) = \sum_i var(a_iXi) = \sum a_i^2 \sigma^2 = \sigma^2\sum a_i^2,$$ where we've relied on the independence of $X_i$ in the second equality (see this). The standard deviation is thus $\sigma\sqrt{\sum_i a_i^2}$.
Please note that you can make the std of $a_iX_i$ be whatever you want by setting $a_i = \frac{\text{wahtever you want}}{\sigma}$.
• You dropped the square of $a_i$ halfway through and did not recover it later. I'm sure that was a typo you will want to correct. You might also want to reformulate your answer without reference to a normal distribution of the $X_i$: it is neither necessary nor explicitly assumed in the question (normality appears to be used there only to generate examples in R). – whuber Jan 23 '15 at 18:22
• Thank you for responding! I need to hold off casting any votes, though, until I can figure out what the question really is asking--see my request for clarification in a comment to the question. – whuber Jan 23 '15 at 18:46
• This appears to answer my question. The only subtlety is that I apply the next step by multiplying by 1 * 1+(a2X2) rather than additively, however for small steps, this should work well, and in the context of my problem, actually it is I who should probably move to an additive version of the problem. I think this is the idea behind the answer from Cristian Atuna. – Thomas Browne Jan 24 '15 at 17:00
• @ThomasBrowne it seems, from what you write, that you are using some variation of an ARCH\GARCH model. See wikipedia. Let me suggest that you change your question. Write exactly what you need to find (preferably in latex). This will facilitate answering your question. – Yair Daon Jan 24 '15 at 17:19
• @ThomasBrowne you are welcome! This question could've been answered much more quickly and without a bounty had you removed the code and plots and wrote the problem in "plain" math. You would have got an answer within an hour probably. – Yair Daon Jan 24 '15 at 17:23
You'll get what you want by typing this formula:
> mysd * sqrt(sum(decay^2))
[1] 0.01611228
Which is basically the answer @YahirDaon gave you. Let me abuse a little bit of the Brownian Motion definition in this answer.
You should note that this expression only works because, as you put it: "You dont walk too far away". Or, what is the same, your x vector does not differ significantly from zero, since mysd is small.
Because, in this case, your geometric brownian motion can be tought as $e^{\sum_1^nX_i} \approx \sum_1^nX_i$. (Use a Taylor expansion to see this). That is, regular Brownian Motion's formulae are a good approximation. In fact, your formula for variance $\sigma^2 * \sqrt{30}$ is that of a regular Brownian Motion, and only works here because of the same reason: x is a vector with a small norm.
The correct formula (as $\mu = 0$) is: $\sqrt{e^{30\sigma^2}.(e^{30\sigma^2}-1)}$. And you can check that it yields a very similar result:
sqrt(exp(30*mysd^2)*(exp(30*mysd^2)-1))
[1] 0.03433
But if you use a mysd $\gg 0$ both in the iid increments case and the varying ones (which technically isn't even a Brownian Motion because it is defined as having iid increments), then you should check more general Lèvy processes.
btw: Random Walk is not the same as Brownian Motion (the former is a more general concept). You seem to refer to a BM because of the formulae you use, so maybe you should avoid refering to it as Random Walk (which it is, but your formulas are specific to BM, not any RW).
• I thought this line tried to generate a GBM here: for(x in stepsizes) i <- c(i, i[length(i)] * exp(x)); return(i)}, since it does not sum the increments when building his process. I might have misunderstood the question, in that case, my answer is, well, wrong. – Cristián Antuña Jan 23 '15 at 18:32
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2019-12-06 08:48:12
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https://www.potfit.net/wiki/doku.php?id=units
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potfit wiki
open source force-matching
Sidebar
User Guide
Examples
Potential Databases
More
units
Units
potfit does not need a particular system of units. It can work with any system. The units used by potfit are implicitly defined in the configuration and potential files, and possibly by certain parameters specifying the potential.
The reference configuration file contains atom positions, which implicitly defines the unit of length. The interaction potential implicitly makes use of a length unit and an energy unit. Of course, this length unit must be the same as for the configuration file. The two units of length, and energy determine other units - like force and pressure - which are derived from them. The temperature in simulated annealing has the same dimension as the sum of squares - none at all.
Example
If the length unit is $$1~\unicode{x212B} = 1\cdot 10^{-10}~\mathrm{m},$$ and the energy unit is $$1~\mathrm{eV} = 1.602\cdot 10^{-19}~\mathrm{J}$$ (which corresponds to $11594~\mathrm{K}$), then the unit of pressure is $$160.2~\mathrm{GPa} = 1602~\mathrm{kbar},$$ and the unit of force is $$1~\frac{\mathrm{eV}}{\unicode{x212B}} = 1.602\cdot 10^{-9}~\mathrm{N}.$$
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2021-05-15 13:23:48
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https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-12-sequences-and-series-12-3-analyze-geometric-sequences-and-series-12-3-exercises-quiz-for-lessons-12-1-12-3-page-817/11
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## Algebra 2 (1st Edition)
$a_n=\dfrac{3}{2}n-1$ $a_{15}=\dfrac{43}{2}$ $S_{15}=165$
The nth term is given by $a_n= a_1+(n-1) d$ ...(1) Here, we have Common Difference $d=\dfrac{3}{2}$ and first term $a_1=\dfrac{1}{2}$ Equation (1) gives: $a_n=\dfrac{1}{2}+\dfrac{3}{2} \times (n-1)=\dfrac{3}{2}n-1$ Plugging in $n =15$, we have $a_{15}=[\dfrac{3}{2} \times 15]-1=\dfrac{43}{2}$ We know that $S_{n}=\dfrac{n(a_1+a_{15})}{2}$ Now, $S_{15}=\dfrac{15 (\dfrac{1}{2}+\dfrac{43}{2})}{2}=165$
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2022-07-04 03:31:22
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https://stats.stackexchange.com/questions/325489/orthogonality-in-2d-vs-higher-dim-vectors
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# orthogonality in $2D$ vs higher dim vectors
considering that $2$ vectors such as $x_2=\begin{bmatrix}1 & 1 \end{bmatrix}$ and $y_2=\begin{bmatrix} -1 & 1 \end{bmatrix}$ are orthogonal in $2D$ (i.e. their scalar product is $0$) however when we consider a third (in $3D$ ) non-zero component i.e. $x_3=\begin{bmatrix}1 & 1 & 1 \end{bmatrix}$ and $y_3=\begin{bmatrix} -1 & 1 & 1 \end{bmatrix}$, they are not anymore orthogonal in the sense of the scalar product because $=1$ this time and not $0$. However these vectors are indeed "at right angle in $3D$" so to speak. I would have the impression that they are just like in the $z=1$ plane... but maybe i m missing a point here ...? I hope i m not forgetting fundamental things. Please can someone help me clarify this, think about this...
(Also, it's linked to a question w.r.t. principal compenents. Given a $4 \times 4$ matrix $E$ of eigenvalues of a covariance matrix $C$, considering all eigenvectors (columns orthogonal) why can't we just plot the two $2D$ principal compononents as the $2$ columns of the truncated $2\times 2$ upper matrix of $E$? (i tried for fisher iris the $2\times 2$ submatrix of eigenvectors of the covmatrix, the $2 \times 1$ "sub vectors" are not orthogonal))
• I think the misunderstanding stems from the statement "these vectors are indeed 'at right angle in 3D' so to speak," which is simply false. Using the well known identity $\cos \theta = \frac{x^\top y}{||x||\cdot||y||}$ for $\theta$ the angle between $x$ and $y$, you can show this quite easily. – Sycorax Jan 28 '18 at 18:04
• @Machupicchu Take two pens. Put them on your desk so that they are orthogonal. Now slowly start increasing the angle of one of the pens toward the third dimension so that it points to $(1,1,1)$. The pens are still orthogonal. Now start increasing the angle of the second pen so that it points to $(-1,1,1)$. You can see that the angle between the two pens is decreasing! – Andreas G. Jan 28 '18 at 18:19
• Thanks for your answers. But i think my problem is to know (when in 3D) with respect to which axis is the vector being rotated. for example: if i rotate 90degrees (pi/2 radians) along Z axis the vector [1 1 1] i get : [-1 1 1] which is at about 70° from [1 1 1] according to the formula $\cos \theta = \frac{x^\top y}{||x||\cdot||y||}$ I think i cannot visualize (anymore) what is happening when it comes to 3D ans its very frustrating... For me if i rotate along Z by 90 degrees the vectors should be at 90 degrees...? but again i must be missing something basic i guess? – Machupicchu Jan 28 '18 at 20:28
• P.S: is it possible to send personnal messages to members of the forum? (i don't see an option to do so) – Machupicchu Jan 29 '18 at 0:37
• @Machupicchu No I don't think it is possible to send PMs. So if you start with the pens in an orthogonal position on your desk, you should rotate one of the pens towards the direction that would eventually get the pen completely vertical. But do not rotate all the way, rotate for about 45 degrees. That should get you to $(1,1,1)$. The pens are still orthogonal because one of them is still on the desk. Now if you do the same with the other pen, you will see the angle becoming smaller – Andreas G. Jan 29 '18 at 13:08
After carefully reading your statements, I think the aspect you are missing is that the vectors are all with respect to the origin. So, you can think of your original vectors as representing the displacement between the points $(0,0,0)$ and $(1,1,0)$ and between $(0,0,0)$ and $(-1,1,0)$.
The new vectors $(1,1,1)$ and $(-1,1,1)$ would be orthogonal if you moved the origin to the point $(0,0,1)$. That is in the $z=1$ plane as you noted. But that's not how we think about them. Think about a pair of lines going from $(0,0,0)$ to the two points $(1,1,1)$ and $(-1,1,1)$ respectively. It should make more sense why they aren't orthogonal.
For some intuition about this, take your thumb and pointing finger and put them at a right angle. Close one eye so that it looks more 2D, now move your hand around, does the 2D picture always look like they are orthogonal? It doesn't always right? Yet you know they are orthogonal.
This addresses your question principal components as well. You can't necessarily make a new pair of orthogonal vectors by dropping some of the coordinates from known pair of orthogonal vectors.
• yes i see! indeed i started have this intuition about the imossibility to simply trucate an othrog high dim vector to get a 2d vect that would be orthog to others. However could you exmplai how i can get these both 2D vectors (PC1 PC2) in 2D for principal components ? (i have a hard time finding this information, i guess for many people in the field it must be obvious?) – Machupicchu Jan 29 '18 at 20:18
• If you want to see the PCs in 2D, just drop all the other coordinates for the PCs like you were originally doing. The point is they won't necessarily be orthogonal in 2D just because they are orthogonal in higher-dimensions. – Kareem Carr Jan 30 '18 at 3:37
• hum i see what you mean however that seems to conflict with the statement that all the axes must be mutually orthogonal in all dimensions? – Machupicchu Jan 30 '18 at 13:37
• moreover think of the example of a cloud of points in 3D that looks like a "fish". the depth can be dropped and you remain with 2 PCs that are orthogonal in the plane (the most variance if "you draw a fish" is in the lenth and height"... do you see what i mean? – Machupicchu Jan 30 '18 at 13:40
• no , am I not pointing something here or am I missing something? Do you see my problem? – Machupicchu Jan 31 '18 at 15:40
Reading your comments, I think your confusion comes from mixing up rotation of a vector along any axis, and creating a vector at some fixed angle to another vector.
Note that rotating a vector in 3D (or any dimension higher than that) with $\alpha$ degrees does not necessarily give you a vector which is $\alpha$ degrees from the original vector. To see this, simply assume a vector along the Z-Axis (e.g. (0,0,1)) and rotate this vector by any amount. You will always get the same vector no matter how much you rotate, so the angle will always be $0$. In mathematical terms, this is shown by the fact, that rotations in higher dimensions always must have fixed points, i.e. axis along which they do not change (Euler's rotation theorem).
This means, whenever you rotate a vector along any axis, and this vector has any "part" of this axis (i.e. it is a linear combination of this axis and another vector), then only the other part is rotated, whereas the part along the axis does not get rotated. Therefore, the angle is reduced.
So about your question along which axis to rotate a vector (in 3D) by $\alpha$ degrees to get a new vector at an angle of $alpha$, the answer is simply: Rotate along any axis, which is orthogonal to the original vector (this may not necessarily be a coordinate axis, as none of them may be orhtogonal). Since you can use any axis, this also means, there is a whole infinite number of vectors at the given angle. This only happens at spaces with at least 3 dimensions, as the orthogonal vector is unique up to multiplication with a scalar for 2D.
• thanks for your answer! it is indeed the kind of explanation i need. I find it quite hard to "think" in 3D. – Machupicchu Jan 29 '18 at 19:47
• thanks for your answer! it is indeed the kind of explanation i need. I find it quite hard to "think" in 3D. Indeed you example about rotating [0 0 1] around Z axis is quite informative for me. I observe that i always get the same vector i.e. [0 0 1] ! So if i understand correctly, it's like if it is "stuck", "glued" around Z axis, right? – Machupicchu Jan 29 '18 at 20:04
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2021-04-22 10:02:57
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https://webwork.libretexts.org/webwork2/html2xml?answersSubmitted=0&sourceFilePath=Library/272/setStewart13_2/problem_6.pg&problemSeed=123567&courseID=anonymous&userID=anonymous&course_password=anonymous&showSummary=1&displayMode=MathJax&problemIdentifierPrefix=102&language=en&outputformat=sticky
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Find the derivative of the vector function
$\mathbf r(t) = t\mathbf a \times (\mathbf b + t\mathbf c)$, where
$\mathbf a = \langle 5, 2, 1\rangle$, $\mathbf b = \langle 3, 4, 3\rangle$, and $\mathbf c = \langle -1, 1, -3\rangle$.
$\mathbf r'(t) = \langle$ , , $\rangle$
Your overall score for this problem is
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2020-03-29 21:42:53
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https://cashweaver.com/posts/breadth_first_search/
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Breadth-first search (BFS) is an algorithm for searching a Tree [or Graph] Data structure for a node that satisfies a given property. It starts at the tree root and explores all nodes at the present depth prior to moving on to the nodes at the next depth level. Extra memory, usually a Queue, is needed to keep track of the child nodes that were encountered but not yet explored.
## Algorithm#
procedure breadthFirstSearch(graph, rootNode)
define a Queue: queue
define a Set: visited
queue.enqueue(rootNode)
while not queue.empty() do
node = queue.dequeue()
if node not in visited do
for connectedNode in graph.connectedNodes(node) do
queue.enqueue(connectedNode)
Based on the algorithm in (“Breadth-First Search” 2022).
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2022-08-17 04:19:43
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https://gmatclub.com/forum/math-circles-87957-40.html
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# Math: Circles
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Intern
Joined: 20 Feb 2016
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19 Jun 2016, 09:10
Very helpful. It helped clear a lot of my concepts on circles.
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Joined: 01 Sep 2016
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13 Sep 2016, 13:11
Quite a Refresher
Bunuel wrote:
chauhan2011 wrote:
• If you know the length of the minor arc and radius, the inscribed angle is: 90L/nr
Please correct me if i am wrong but i think the formula should be : 180L/nr
If you know the length $$L$$ of the minor arc and radius, the inscribed angle is: $$Inscribed \ Angle=\frac{90L}{\pi{r}}$$.
The way to derive the above formula:
Length of minor arc is $$L= \frac{Central \ Angle}{360}* Circumference$$ --> $$L= \frac{Central \ Angle}{360}* 2\pi{r}$$ --> $$L= \frac{Central \ Angle}{180}* 2\pi{r}$$ --> $$Central \ Angle=\frac{180L}{\pi{r}}$$ (so maybe you've mistaken central angle for inscribed angle?).
The Central Angle Theorem states that the measure of inscribed angle is always half the measure of the central angle: $$Central \ Angle=2*Inscribed \ Angle$$.
So, $$2*Inscribed \ Angle=\frac{180L}{\pi{r}}$$ --> $$Inscribed \ Angle=\frac{90L}{\pi{r}}$$.
Hope it helps.
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### Show Tags
17 Aug 2017, 13:36
Bunuel
I have a doubt here . Consider the last figure! If the lines OA and OB are drawn, what will be the resulting Angle OAP and OBP ?
Re: Math: Circles [#permalink] 17 Aug 2017, 13:36
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2018-06-24 06:49:40
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http://mathhelpforum.com/advanced-statistics/177332-determining-colour-last-ball-bag.html
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## Determining the colour of the last ball in a bag
There exists a bag with a white and b black balls. Balls are chosen by the following method:
i) A ball is chosen at random and discarded.
ii)A second ball is then chosen. If colour is different from that of the preceding ball, it is replaced in the bag and the process repeated from the beginning. If the colour is the same, it is discarded and we go back to step ii.
$P_a_,_b = \frac{1}{2}$
The hint is to use induction on $k = a + b$
Could someone show me how to start this, as I cannot see how induction would work. I am not looking for a full solution!
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2014-08-22 23:03:22
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https://jetsetsocietyusa.com/online-internet-casino/1-x-2-x.php
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Book of Ra Deluxe ᐅ Free Slot Machine by Greentube - Neonslots
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Hi x 2 − 1 x − 1 = (x − 1) (x + 1) x − 1 = x + 1 \frac{x^}{x-1} = \frac{(x-1)(x+1)}{x-1} = x+1 x−1x2−1=x−1(x−1)(x+1)=x+1. Denn der. bitte warten! Dies wird ein paar Sekunden dauern. ddx[sin(√ex+a2)] B. schreiben wir "5x" statt "5*x". Der Ableitungsrechner muss diese Fälle erkennen und. (-xxx2-x+2) + (x2+2) = -xxx2-x+4. Polynomsubtraktion: Zum Beispiel hat g(x)=1,5·x 3+2·x-4 den Grad 3 und den Leitkoeffizient 1,5. Hier findest du.
## Tabelle von Ableitungs- und Stammfunktionen
Hi x 2 − 1 x − 1 = (x − 1) (x + 1) x − 1 = x + 1 \frac{x^}{x-1} = \frac{(x-1)(x+1)}{x-1} = x+1 x−1x2−1=x−1(x−1)(x+1)=x+1. Denn der. wenn man für die Variable(n) bestimmte Zahlen einsetzt. Bsp.: Bestimme die Wertemenge des Terms. 2. )(-. -. = x x. xT für die Grundmenge G = {1,2,3}!. 2. Also gilt ∫1xdx=ln(x)+C. Sieht man einer Funktion f(x)=tan(x). F(x)=−ln|cos(x)|+C. f(x)=1sin2(x). F(x)=−cot(x)+C. f(x)=1cos2(x). F(x)=tan(x)+C. f(x)=11+x2.
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The natural question of why we might want to do this will be answered in a couple of sections once we actually learn how to do this.
This provision is important. This idea of convergence is important here. In this way we will hopefully become familiar with some of the kinds of manipulations that we will sometimes need to do when working with power series.
This is actually easier than it might look. So, in this case the interval of convergence is the same as the original power series.
More often than not the new interval of convergence will be different from the original interval of convergence. This function is similar to the previous function.
The difference is the numerator and at first glance that looks to be an important difference. As we saw in the previous example we can often use previous results to help us out.
This is an important idea to remember as it can often greatly simplify our work. Doing this gives,. So, hopefully we now have an idea on how to find the power series representation for some functions.
We now need to look at some further manipulation of power series that we will need to do on occasion. We need to discuss differentiation and integration of power series.
Now, we know that if we differentiate a finite sum of terms all we need to do is differentiate each of the terms and then add them back up.
Each parabola has a vertical line of symmetry that passes through its vertex. That is, if the parabola has indeed two real solutions.
Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach.
For this reason we want to be able to find the coordinates of the vertex. A new set of numbers, called complex, was invented so that negative numbers would have a square root.
Adding, subtracting and finding the least common multiple. Why learn this. Terms and topics Adding subtracting finding least common multiple Reducing fractions to lowest terms.
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The integral of Panda Warrior cubed may be evaluated using integration by parts. Glossary of calculus Glossary of calculus List of calculus topics. And to get a 64, you need to raise two to the sixth power.
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2021-03-03 02:58:43
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https://dev.mysql.com/doc/c-api/5.6/en/mysql-thread-init.html
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Documentation Home
MySQL 5.6 C API Developer Guide
PDF (US Ltr) - 1.0Mb
PDF (A4) - 1.0Mb
``````my_bool
This function must be called early within each created thread to initialize thread-specific variables. However, it may be unnecessarily to invoke it explicitly. Calling `mysql_thread_init()` is automatically handled `my_init()`, which itself is called by `mysql_init()`, `mysql_library_init()`, `mysql_server_init()`, and `mysql_connect()`. If you invoke any of those functions, `mysql_thread_init()` is called for you.
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2021-02-28 13:40:35
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http://math.stackexchange.com/questions/25841/how-do-you-do-a-cross-product-of-two-3-times-3-boolean-matrices/25862
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# How do you do a cross product of two $3 \times 3$ boolean matrices?
I have two boolean matrices:
A = |1 1 0|
|0 1 0|
|0 0 1|
and
B = |1 0 0|
|1 1 1|
|0 0 1|
What is the result of A x B and what are the steps needed to attain the result?
Note: My textbook says that the answer to the above is:
A x B = |1 1 1|
|1 1 1|
|0 0 1|
and that A * B is not equal to A x B. Unfortunately, it does not give the steps needed to find the solution.
-
It looks like you're being asked for a matrix product, not a cross product. – Qiaochu Yuan Mar 8 '11 at 22:06
My source specifically states that A*b is not equal to AxB. Unfortunately, it does not provide one single example. – trusktr Mar 8 '11 at 22:10
See edited question above – trusktr Mar 8 '11 at 22:18
Does it provide a definition or not? That is very strange. – Qiaochu Yuan Mar 8 '11 at 22:28
It doesn't. The book is a programming book assuming we know this math already. – trusktr Mar 8 '11 at 22:47
## 1 Answer
I think it is the same as conventional matrix multiplication just that the multiplication is replaced by the "and" operation while the addition is replaced by the "or" operation.
Hence, $$A \times B = \begin{bmatrix} 1 & 1 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 0 \\\ 1 & 1 & 1 \\\ 0 & 0 & 1 \end{bmatrix}$$
$$A \times B = \begin{bmatrix} (1 \& 1) || (1 \& 1) || (0 \& 0) & (1 \& 0) || (1 \& 1) || (0 \& 0) & (1 \& 0) || (1 \& 1) || (0 \& 1) \\\ (0 \& 1) || (1 \& 1) || (0 \& 0) & (0 \& 0) || (1 \& 1) || (0 \& 0) & (0 \& 0) || (1 \& 1) || (0 \& 1) \\\ (0 \& 1) || (0 \& 1) || (1 \& 0) & (0 \& 0) || (0 \& 1) || (1 \& 0) & (0 \& 0) || (0 \& 1) || (1 \& 1) \end{bmatrix}$$
$$A \times B = \begin{bmatrix} 1 & 1 & 1 \\\ 1 & 1 & 1 \\\ 0 & 0 & 1 \end{bmatrix}$$
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2016-07-23 21:42:22
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http://mathoverflow.net/questions/34982/are-there-space-filling-curves-for-the-hilbert-cube/34983
|
# Are there space filling curves for the Hilbert cube ?
There is a surjective continuous map $[0;1]\rightarrow [0;1]^2$ ("space filling curve"). Using such a map one can easily get space filling curves for all finite dimensional cubes. So my question is: Is there a space filling curve of the Hilbert cube $[0;1]\rightarrow [0;1]^\mathbb{N}$ ?
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I believe this is an old theorem that appears in many point-set topology textbooks, that every compact connected, locally-connected metric space is the continuous image of $[0,1]$. Willard or Dugunji should have it. – Ryan Budney Aug 9 '10 at 8:41
Ah, it's called the "Hahn–Mazurkiewicz theorem".
http://en.wikipedia.org/wiki/Peano_space
And it apparently appears in the Willard text.
-
This also appears in Laskovitch's Conjecture And Proof, where it is used to construct a family of continuous functions $f_1, f2, \dots$ with the property that for any sequence $\alpha_i$ of reals taken from the interval $[0,1]$, there exists a point $x \in [0,1]$ with $f_i(x) = \alpha_i$ for all $i$. – Nick Salter Aug 19 '10 at 19:23
Well there is indeed a "simple" construction of such a space filling curve.
Let $\gamma:[0;1]\rightarrow [0;1]^2$ be a space filling curve. Then one can obtain a space filling curve for $[0;1]^3$ by postcomposing with $id_\mathbb{R}\times \gamma$. Then one can postcompose with $id_{\mathbb{R}^2}\times \gamma$ and so on. Note that the first coordinates didn't change in the last step.
Putting all this together we get a map
$f:[0;1]\rightarrow [0;1]^\mathbb{N} \qquad t\mapsto (pr_1\circ \gamma \circ (pr_2\circ \gamma)^{n-1}(t))_{n\in \mathbb{N}},$ where $pr_i$ denote the obvious projections. This map can be seen as the infinite composition of the maps above. By the definition of the product topology this map is continuous.
Especially if we postcompose $f$ with the projection on the first $n$ coordinates, we just get a space filling curve (see above). Let us show, that a arbitrary element $x=(x_i)_{i\in \mathbb{N}}\in [0;1]^\mathbb{N}$ lies in the Image of $f$. We already know, that for each $n$ there is a element $y^n$ in Im$(f)$ agreeing with $x$ in the first $n$ coordinates.
As $[0;1]$ is compact, Im$(f)$ is compact and hence closed ($[0;1]^\mathbb{N}$ is Hausdorff).
And $\lim_{n\to\infty}y^n=x$. Hence $x\in$ Im$(f)$. So $f$ is a continuous surjective map $[0;1]\rightarrow [0;1]^\mathbb{N}$.
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2015-05-28 12:13:28
|
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https://stats.stackexchange.com/questions/413644/marginalizing-statistical-test-results-in-a-two-factor-grid
|
# Marginalizing statistical test results in a two-factor grid
Consider the following toy problem. Suppose that we have several food groups: meat, vegetables, fruit, etc. For any given pair of food types A and B, we ask whether people that eat both A AND B have a different outcome (e.g., live longer) than people who eat A OR B, but not both. After traversing all possible pairs of food categories and applying our test to each, we arrive at a set of p-values:
$$Test( meat, vegetables ) \rightarrow p_{12} \\ Test( meat, fruit ) \rightarrow p_{13} \\ ...\\ Test( food_i, food_j ) \rightarrow p_{ij}$$
The question is whether $$p_{ij}$$ values can now be used to determine whether any particular food group consistently leads to a low p value. The Fisher's method appears to be a good first pass at combining $$p_{ij}$$ for any fixed $$i$$, but I suspect that it doesn't properly account for the correlation structure in the data. In particular, there is likely a non-zero intersection between "people who eat meat AND vegetables" and "people who eat meat AND fruit", implying that the corresponding $$Test(meat,vegetables)$$ and $$Test(meat,fruit)$$ may not be independent.
What is the proper statistical procedure to summarize our grid of $$p_{ij}$$ values into a meaningful statistic for each $$food_i$$, assuming that we also have access to the correlation measure for any two tests?
If it makes a difference:
• In our actual application, the Test is Wilcoxon Rank Sum
• $$p_{ii}$$ is not well-defined. Food groups are NOT tested against themselves. For example, $$Test(meat, meat)$$ is not meaningful.
• "Whether any particular food group consistently leads to a low p value" is not a testable statistical hypothesis, because it depends fundamentally on the test procedure and the sample size: as such, it reveals nothing definite about the population of interest. Could you therefore explain what your actual scientific question is? – whuber Jun 18 at 20:52
• @whuber: we are interested in ranking our "food groups" based on the observed p_ij values. Can we ask "do tests that involve meat lead to significantly lower p values than tests that involve fruit (taking into account that individual tests are not independent)"? – Artem Sokolov Jun 18 at 21:02
• Our actual application is polypharmacology. Our "food groups" are cell surface receptors; our "people" are compounds that bind to them. We are studying interaction effects of targeting multiple receptors, and we would like to know if certain receptors are involved in more interactions than others. – Artem Sokolov Jun 18 at 21:04
• A ranking based on p-values is arbitrary and meaningless, for the same reasons I gave earlier. Aren't you really interested in some kind of meaningful ranking with scientific validity? – whuber Jun 18 at 21:04
• @whuber: We are happy to move upstream to modifying the actual Wilcoxon Rank Sum Test rather than working with the resulting p values directly. The ranking we're interested in is the level of interaction each "food group" has with others, summarized on a per-"food group", rather than per-interaction basis. – Artem Sokolov Jun 18 at 21:13
The solution that ended up working well for us was the recently-proposed harmonic mean p-value. Following the example in the original question, let's assume that there are $$n$$ food groups. For group $$i$$, we combine its corresponding p values according to:
$$p_i = \frac{n-1}{\sum_{j \neq i} \frac{1}{p_{ij}}}$$
As discussed in the paper, this combination is
i) Robust to positive dependency between p-values.
ii) Insensitive to the exact number of tests.
iii) Robust to the distribution of weights w.
iv) Most influenced by the smallest p-values.
where we kept the weights $$w$$ fixed at 1 for all tests. (The weights can be interpreted as relative prior belief about certain alternative hypotheses being more likely than others.)
|
2019-10-21 05:22:56
|
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|
https://math.stackexchange.com/questions/572786/existence-of-five-real-numbers-satisfying-a-given-condition
|
Existence of five real numbers satisfying a given condition.
Let $a_1,\dots,a_5$ be five distinct non-zero real numbers. Suppose that for $i\neq j$ either $a_i+a_j$ or $a_ia_j$ or both are rational numbers, does it implies that $a_i^2$ are rational numbers for all $i$?
• Oh i'm sorry i miss the fact that $a_i$'s are all non-zeros. – D. N. Nov 19 '13 at 4:27
• Now i will edit the questions. – D. N. Nov 19 '13 at 4:28
• If you draw a graph of $K_5$ with the $a$'s, the additions and multiplications must form the Petersen graph, because if you have a triangle with the same sign you can use that to prove all three elements of the triangle rational, then you can get the other two. – Ross Millikan Nov 19 '13 at 4:28
• I got on this thinking pigeonhole principle. If you have $a+b, a+c, b+c$ rational, then $a-b, a+b$ are rational, so $a,b$ rational, then $a+d$ or $ad$, so $d,e$ rational. The Petersen graph is a good way to represent the intermediate stage, but you don't have to reference it. – Ross Millikan Nov 19 '13 at 4:39
• @RossMillikan: and the conclusion is...? It's not quite clear to me how you get the Petersen graph, or what you do with it. – Robert Israel Nov 19 '13 at 5:46
Consider the complete graph $K_5$ on $5$ vertices. Colour the edge $(i,j)$ blue if $a_i + a_j$ is rational, otherwise red.
Suppose there is a red $m$-cycle $(i_1, i_2, \ldots, i_m$, i.e. $r_1 = a_{i_1} + a_{i_2}, r_2 = a_{i_2} + a_{i_3}, \ldots, r_m = a_{i_m} + a_{i_1}$ are all rational. Then $a_{i_2} = r_1 - a_{i_1}$, $a_{i_3} = r_2 - r_1 + a_{i_1}$, etc, determining each $a_{i_j}$ in terms of the $r_j$ and $a_{i_1}$. If $m$ is odd, when we come around the whole cycle we get $a_{i_1} = r_{i_m} - r_{i_{m-1}} + \ldots + r_{i_1} - a_{i_1}$ which makes $a_{i_1}$ rational, and then we find that all $a_i$ are rational.
Similarly, if there is a blue $m$-cycle with $m$ odd, we would get $a_{i_1} = r_{i_m} r_{i_{m-1}}^{-1} \ldots r_{i_1} a_{i_1}^{-1}$, which makes $a_{i_1}^2$ rational, and then all $a_{i}^2$ are rational.
So in order to have an example with $a_i^2$ not all rational, we have to be able to colour the edges of $K_5$ in two colours so there are no odd cycles of either colour. But it seems this is impossible. So the answer is yes, all $a_i^2$ must be rational.
EDIT: Here is the fix for the "gap": If, say, $a_1 a_2, a_3$ is a blue triangle, $a_1^2$, $a_2^2$ and $a_3^2$ are rational. Now consider $a_4$. If $a_1 a_4$ is rational, $a_4^2 = (a_1 a_4)^2/a_1^2$ is rational. Similarly if $a_2 a_4$ is rational. So suppose $a_1 + a_4 = r_{14}$ and $a_2 + a_4 = r_{24}$ are rational. Then $a_1 - a_2 = r_{14} - r_{24}$ is rational (and nonzero, because the $a_i$ are distinct). But then $a_1 + a_2 = \dfrac{a_1^2 - a_2^2}{a_1 - a_2}$ is rational, so $a_1$ and $a_2$ are rational, and $a_4$ is rational. Similarly for $a_5$.
• Oops: it's not so obvious that a blue $3$-cycle makes the other two $a_i^2$ rational. I'm pretty sure I can fix the gap, but I don't have time right now. – Robert Israel Nov 19 '13 at 16:37
• @RobertIsreal I Just dont understand, that, why it is not possible to colour the edges of $K_5$ in two colours so there are no odd cycles of either colour? Thanks for your time. – D. N. Dec 9 '13 at 5:58
• Try it and see! – Robert Israel Dec 9 '13 at 15:51
• ... or enumerate the cases. Only 10 edges, and wlog you can assume 1-2 and 2-3 are red and 1-3 is blue. – Robert Israel Dec 9 '13 at 15:54
• Ok i wil try. By the way, thank you. – D. N. Dec 9 '13 at 16:27
No. Take $a_1=\pi, a_2=\frac{1}{\pi}, a_3=-\pi, a_4=\frac{-1}{\pi}, a_5=0$.
Then $a_1a_2 = 1, a_1+a_3=0,a_1a_4=-1,a_1a_5=0,a_2a_3=-1,a_2+a_4=0,a_2a_5=0,a_3a_4=1,a_3a_5=0, \text{and } a_4a_5=0.$
• Good answer to the original question. $0$ is special. – Ross Millikan Nov 19 '13 at 4:29
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2021-06-15 03:05:46
|
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http://peakgmt.com/math/ExponentProperties.html
|
# Exponent Properties
Because of their clarity and concision, the laws of exponents lend themselves well to GMAT math, especially to the Data Sufficiency format. If math isn't your thing, then perhaps the last time you gave any thought to exponents was back in Algebra Two, and perhaps exponents weren't your favorites there either. Take heart! In this post, I will explain the properties you need to know to be successful on the GMAT Quantitative section.
## Exponent Properties
Fundamentally, an exponent is how many times you multiply a number, that is, how many factors of a number you have. This is the fundamental definition. The expression 5^4 means: multiply four 5's together. The expression 2^3 means: multiply three 2's together, which gives an answer of 8, so 2^3 = 8. Technically, 2 is the "base", 3 is the "exponent", and 8 is the "power." The action of raising something to an exponent is called "exponentiation."
## Distribution
Just as multiplication distributes over addition & subtraction
\LARGE{a \times (b + c) = a \times b + a \times c}
so exponentiation distributes over multiplication and division.
\LARGE{(x \times y)^r = (x^r) \times (y^r)}
Why is that? Well, consider (x \times y)^3. This means the thing in parentheses multiplied by itself three times: (x \times y)^3 = (x \times y) \times (x \times y) \times (x \times y). Well, when we have a bunch of factors, we can rearrange them in any order, because order doesn't matter in multiplication. So, I could rearrange them as follows:
(x \times y)^3=
(x \times y) \times (x \times y) \times (x \times y)=
x \times x \times x \times y \times y \times y=
(x \times x \times x) \times (y \times y \times y)=
\Big(x^3\Big) \times \Big(y^3\Big)
All the laws of exponents make sense if you just go back to the fundamental definition.
In this context, I will say: beware of one of the most tempting mistakes in all of mathematics. Exponentiation does NOT distribute over addition.
\LARGE{(a + b)^n} \not= \LARGE{(a^n) + (b^n)}
Beware. Even when you know this is wrong, even when you make an effort to remember that it's wrong, the inherent pattern-matching machinery of your brain will automatically pull your mind back in the direction of making this mistake. You must be vigilant to avoid this mistake.
## Multiplying Powers
What happens when you multiply two unequal powers of the same base?
\LARGE{(x^r) \times (x^s) = ?}
Well, let's think about a concrete example. Suppose we are multiplying (x^5)*(x^3). Well, by the fundamental definition, x^5 = x*x*x*x*x, and x^3 = x*x*x, so
(x^5)*(x^3) = (x*x*x*x*x)*(x*x*x) = x*x*x*x*x*x*x*x = x^8
If we start with five factors of x, and "stir in" three more factors of x, we wind up with a total of eight factors. All we have to do is add the exponents. We can simply generalize this pattern:
\LARGE{(x^r) \times (x^s) = (x^{r+s})}
Don't just memorize this: make sure you understand the logic that leads to it. Remembering with the logic is 100x more effective than blind memorization!
## Dividing Powers
What happens when you divide two unequal powers of the same base?
\LARGE{\dfrac{x^r}{x^s} = ?}
As with last time, a concrete example will illuminate the question. Suppose we divide \dfrac{\bigg(x^7\bigg)}{\bigg(x^3\bigg)}. By the fundamental definition, x^7 = x*x*x*x*x*x*x and x^3 = x*x*x, so
\dfrac{x^7}{x^3} = \dfrac{x*x*x*x*x*x*x}{x*x*x} = \dfrac{x*x*x*x*\cancel{x*x*x}}{\cancel{x*x*x}} = x*x*x*x = x^4
If we start out with seven factors, and then cancel away three of them, we are left with four. We just subtract the exponents. We can also generalize this pattern:
\LARGE{\dfrac{x^r}{x^s} = x^{r-s}}
Once again: understand the logic, because remember through understanding is considerably more powerful than blind memorization.
Once again: understand the logic, because remember through understanding is considerably more powerful than blind memorization.
## An Exponent of Zero
Mathematicians love to extend patterns. One example of this is the zero exponent. If we just see x^0, we may wonder: how on earth are we going to understand what this could mean? We are clearly outside of the realm where the fundamental definition helps us.
One clever trick we can us is to employ the pattern found in division of powers. Suppose we have \dfrac{(x^4)}{(x^4)} — then, the "subtraction of exponents" would imply:
\dfrac{(x^4)}{(x^4)} = x^{4-4} = x^0
but just fundamental logic would tell us that anything over itself equals one. Therefore, this expression \dfrac{(x^4)}{(x^4)} must have a value of 1. That, in turn, tells us the value of x^0. x^0 =1.
## Negative Exponents
Here, we will extend the patterns even further. Consider this chart, for a base of 2:
Exponent 0 1 2 3 4 Power 1 2 4 8 16
Each time we move one cell to the right, the power gets multiplied by 2, and each time we move one cell to the left, the power gets divided by 2. That's a very easy pattern to extend to the left:
Exponent -4 -3 -2 -1 0 1 2 3 4 Power \dfrac{1}{16} \dfrac{1}{8} \dfrac{1}{4} \dfrac{1}{2} 1 2 3 4 16
All we have done was to extend the pattern "move one cell to the left, and the power gets divided by 2." The result, we see, is that negative powers are reciprocals of their corresponding positive powers. This is consistent with the Division of Powers rule: if dividing means subtract the exponents, then an exponent of -3 means we are dividing by three factors of the number. Therefore, the general rule is:
\LARGE{x^{\space{-r}} = \dfrac{1}{x^r}}
## Summary
I can't urge enough: the key to remembering these rules is understanding the logic of the arguments behind them. If you understand these rules, you will understand whatever the GMAT throws at you concerning exponents.
## Practice Questions
Q1. If \dfrac{8^5 \times 4^6}{16^n}=32^{1-n} then n =
A. -22
B. -11
C. 5
D. 11
E. 22
Explanation:
To solve this problem, it is very helpful to express all of the quantities in terms of the same base. Once we do that, we can make use of the various Laws of Exponents to simplify the quantities further.
First, we'll express all of the quantities in terms of the same base of 2:
\dfrac{8^5 * 4^6}{16^n} = 32^{(1 - n)}
\dfrac{{(2^{3})}^5 * {(2^{2})}^6}{(2^4)^n} = (2^5)^{(1 - n)}
We can now simplify all the powers that are being raised to a power via this Law of Exponents:
\LARGE{\bold{(x^a)^b = x^{ab}}}
\dfrac{2^{(3*5)} * 2^{(2*6)}}{2^{(4*n)}} = 2^{(5*(1-n))}
\dfrac{2^{15}*2^{12}}{2^{4n}} = 2^{(5-5n)}
Next, we can simplify the numerator via this Law of Exponents:
\LARGE{\bold{(x^a) * (x^b) = x^{(a + b)}}}
\dfrac{2^{(15 + 12)}}{2^{4n}} = 2^{(5 - 5n)}
\dfrac{2^{27}}{2^{4n}} = 2^{(5 - 5n)}
We can then simplify the entire fraction via this Law of Exponents:
\LARGE{\bold{\dfrac{x^a}{x^b} = x^{(a-b)}}}
2^{(27-4n)} = 2^{(5-5n)}
Lastly, we'll use the following rule to solve for n:
"If two powers with the same base are equal, then the exponents must be equal. That is, if b^x = b^y, then x = y, provided that b does not equal 0 or +/-1."
27 - 4n = 5 - 5n
n = 5 - 27
n = \bold{-22}
Q2. If 4^n + 4^n + 4^n + 4^n = 4^16, then n =
A. 1
B. 2
C. 4
D. 12
E. 15
Q3. If x and y are positive odd integers, then which of the following must also be an odd integer?
I. x^{y+1}
II. x(y+1)
III. (y+1)^{x-1} + 1
A. I only
B. II only
C. III only
D. I and III
E. None of the above
Explanation:
Case I: Since x and y are both positive odd integers, x^{(y+1)} becomes (odd)^{(odd + odd)}, which becomes (odd)^{even}. Multiplying any number of odd numbers together always results in an odd number.
This case is acceptable.
Case II: x^{(y + 1)} becomes (odd)^{(odd + odd)}, which becomes (odd)^{(even)}. Any odd number times an even number results in an even number.
This case doesn't work.
Case III: The portion (y + 1)^{(x - 1)} becomes (odd + odd)^{(odd - odd)}, which becomes (even)^{even}. For our whole expression, we are now left with (even)^{even + 1}, which is (even)^{even + odd}.
There are actually two different possibilities that arise from this setup. For most cases, we'll get an odd number in the end. For example:
2^2 + 1= 4 + 1 = 5 (odd)
4^4 + 1 = 256 + 1 = 257 (odd)
etc.
However, zero is also an even number, and anything raised to the power of zero results in 1. Because of this, we can end up with an even number as the result of our expression:
2^0 + 1 = 1 + 1 = 2 (even)
4^0 + 1 = 1 + 1 = 2 (even)
etc.
Since we can end up with both even and odd numbers from this case, this case doesn't work.
Only Case I guarantees an odd integer.
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2020-01-25 18:01:44
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http://juliapomdp.github.io/POMDPs.jl/latest/generative/
|
# Generative POMDP Interface
## Description
The generative interface contains a small collection of functions that makes implementing and solving problems with generative models easier. These functions return states and observations instead of distributions as in the Explicit interface.
The functions are:
generate_s(pomdp, s, a, rng) -> sp
generate_o(pomdp, s, a, sp, rng) -> o
generate_sr(pomdp, s, a, rng) -> (s, r)
generate_so(pomdp, s, a, rng) -> (s, o)
generate_or(pomdp, s, a, sp, rng) -> (o, r)
generate_sor(pomdp, s, a, rng) -> (s, o, r)
initial_state(pomdp, rng) -> s
Each generate_ function is a single step simulator that returns a new state, observation, reward, or a combination given the current state and action (and sp in some cases). rng is a random number generator such as Base.GLOBAL_RNG or another MersenneTwister that is passed as an argument and should be used to generate all random numbers within the function to ensure that all simulations are exactly repeatable.
The functions that do not deal with observations may be defined for MDPs as well as POMDPs.
A problem writer will generally only have to implement one or two of these functions for all solvers to work (see below).
## Example
The following example shows an implementation of the Crying Baby problem [1]. A definition of this problem using the explicit interface is given in the POMDPModels package.
importall POMDPs
# state: true=hungry, action: true=feed, obs: true=crying
type BabyPOMDP <: POMDP{Bool, Bool, Bool}
r_feed::Float64
r_hungry::Float64
p_become_hungry::Float64
p_cry_when_hungry::Float64
p_cry_when_not_hungry::Float64
discount::Float64
end
BabyPOMDP() = BabyPOMDP(-5., -10., 0.1, 0.8, 0.1, 0.9)
discount(p::BabyPOMDP) = p.discount
function generate_s(p::BabyPOMDP, s::Bool, a::Bool, rng::AbstractRNG)
if s # hungry
return true
else # not hungry
return rand(rng) < p.p_become_hungry ? true : false
end
end
function generate_o(p::BabyPOMDP, s::Bool, a::Bool, sp::Bool, rng::AbstractRNG)
if sp # hungry
return rand(rng) < p.p_cry_when_hungry ? true : false
else # not hungry
return rand(rng) < p.p_cry_when_not_hungry ? true : false
end
end
# r_hungry
reward(p::BabyPOMDP, s::Bool, a::Bool) = (s ? p.r_hungry : 0.0) + (a ? p.r_feed : 0.0)
initial_state_distribution(p::BabyPOMDP) = [false] # note rand(rng, [false]) = false, so this is encoding that the baby always starts out full
This can be solved with the POMCP solver.
using BasicPOMCP
using POMDPToolbox
pomdp = BabyPOMDP()
solver = POMCPSolver()
planner = solve(solver, pomdp)
hist = simulate(HistoryRecorder(max_steps=10), pomdp, planner);
println("reward: \$(discounted_reward(hist))")
## Which function(s) should I implement for my problem / use in my solver?
### Problem Writers
Generally, a problem implementer need only implement the simplest one or two of these functions, and the rest are automatically synthesized at runtime.
If there is a convenient way for the problem to generate a combination of states, observations, and rewards simultaneously (for example, if there is a simulator written in another programming language that generates these from the same function, or if it is computationally convenient to generate sp and o simultaneously), then the problem writer may wish to directly implement one of the combination generate_ functions, e.g. generate_sor() directly.
Use the following logic to determine which functions to implement:
• If you are implementing the problem from scratch in Julia, implement generate_s and generate_o.
• Otherwise, if your external simulator returns x, where x is one of sr, so, or, or sor, implement generate_x. (you may also have to implement generate_s separately for use in particle filters).
Note: if an explicit definition is already implemented, you do not need to implement any functions from the generative interface - POMDPs.jl will automatically generate implementations of them for you at runtime (see generative_impl.jl).
### Solver and Simulator Writers
Solver writers should use the single function that generates everything that they need and nothing they don't. For example, if the solver needs access to the state, observation, and reward at every timestep, they should use generate_sor() rather than generate_s() and generate_or(), and if the solver needs access to the state and reward, they should use generate_sr() rather than generate_sor(). This will ensure the widest interoperability between solvers and problems.
In other words, if you need access to x where x is s, o, sr, so, or, or sor at a certain point in your code, use generate_x.
[1] Decision Making Under Uncertainty: Theory and Application by Mykel J. Kochenderfer, MIT Press, 2015
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2017-11-18 15:53:53
|
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|
https://stacks.math.columbia.edu/tag/04GK
|
Lemma 10.148.7. Let $(R, \mathfrak m, \kappa )$ be a henselian local ring. The category of finite étale ring extensions $R \to S$ is equivalent to the category of finite étale algebras $\kappa \to \overline{S}$ via the functor $S \mapsto S/\mathfrak mS$.
Proof. Denote $\mathcal{C} \to \mathcal{D}$ the functor of categories of the statement. Suppose that $R \to S$ is finite étale. Then we may write
$S = A_1 \times \ldots \times A_ n$
with $A_ i$ local and finite étale over $S$, use either Lemma 10.148.5 or Lemma 10.148.3 part (10). In particular $A_ i/\mathfrak mA_ i$ is a finite separable field extension of $\kappa$, see Lemma 10.141.5. Thus we see that every object of $\mathcal{C}$ and $\mathcal{D}$ decomposes canonically into irreducible pieces which correspond via the given functor. Next, suppose that $S_1$, $S_2$ are finite étale over $R$ such that $\kappa _1 = S_1/\mathfrak mS_1$ and $\kappa _2 = S_2/\mathfrak mS_2$ are fields (finite separable over $\kappa$). Then $S_1 \otimes _ R S_2$ is finite étale over $R$ and we may write
$S_1 \otimes _ R S_2 = A_1 \times \ldots \times A_ n$
as before. Then we see that $\mathop{\mathrm{Hom}}\nolimits _ R(S_1, S_2)$ is identified with the set of indices $i \in \{ 1, \ldots , n\}$ such that $S_2 \to A_ i$ is an isomorphism. To see this use that given any $R$-algebra map $\varphi : S_1 \to S_2$ the map $\varphi \times 1 : S_1 \otimes _ R S_2 \to S_2$ is surjective, and hence is equal to projection onto one of the factors $A_ i$. But in exactly the same way we see that $\mathop{\mathrm{Hom}}\nolimits _\kappa (\kappa _1, \kappa _2)$ is identified with the set of indices $i \in \{ 1, \ldots , n\}$ such that $\kappa _2 \to A_ i/\mathfrak mA_ i$ is an isomorphism. By the discussion above these sets of indices match, and we conclude that our functor is fully faithful. Finally, let $\kappa \subset \kappa '$ be a finite separable field extension. By Lemma 10.141.15 there exists an étale ring map $R \to S$ and a prime $\mathfrak q$ of $S$ lying over $\mathfrak m$ such that $\kappa \subset \kappa (\mathfrak q)$ is isomorphic to the given extension. By part (1) we may write $S = A_1 \times \ldots \times A_ n \times B$. Since $R \to S$ is quasi-finite we see that there exists no prime of $B$ over $\mathfrak m$. Hence $S_{\mathfrak q}$ is equal to $A_ i$ for some $i$. Hence $R \to A_ i$ is finite étale and produces the given residue field extension. Thus the functor is essentially surjective and we win. $\square$
There are also:
• 2 comment(s) on Section 10.148: Henselian local rings
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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2019-01-16 22:51:01
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http://math.stackexchange.com/questions/125159/if-a-graph-of-2n-vertices-contains-a-hamiltonian-cycle-then-can-we-reach-ever
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# If a graph of $2n$ vertices contains a Hamiltonian cycle, then can we reach every other vertex in $n$ steps?
Problem: Given a graph $G,$ with $2n$ vertices and at least one triangle. Is it possible to show that you can reach every other vertex in $n$ steps if $G$ contains a Hamilton cycle (HC)?
EDIT: Sorry, I forgot to mention, that $G$ is planar and 3-connected. A complete proof for $3$-regular graphs would also be accepted/rewarded.
Does the following work as proof?
Choose a starting vertex $v_0$ and a direction.
• If you walk along the HC you'll reach a vertex $v_{n-1}$ with maximal distance from $v_0$ in $n$ steps.
• You'll reach $v_{n-2}$ by doing a round in the triangle and
• $v_{n-3}$ by stepping backwards at the last step.
• By combining these moves, you'll reach half of all $v_k$.
• By choosing the other direction at the beginning you'll reach the other half.
• $v_0$ is free to choose.
Showing or disproving the "only if"-part would also be nice!
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Interesting question. – Saeed Apr 9 '12 at 23:13
Question: Let $G$ be a 3-connected, hamiltonian, planar graph with $2n$ vertices and at least one triangle. Is it true that for all vertex pairs $x,y$, that there is a walk of exactly $n$ steps from $x$ to $y$?
The following graph and vertex pair is a counter example
It is clear that the graph is planar and has a triangle. It can be easily verified that the graph is 3-connected. To show that the graph is hamiltonian, I have highlighted a hamiltonian cycle here
Since the vertex has 16 vertices, we need to verify that there is no walk of length 8 from $x$ to $y$. Since $n$ is equal, we can not reach $y$ without using some of the four vertices on the right. Now it is easy to verify by hand, that there is no walk from $x$ to $y$ of length exactly 8.
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First of all: very nice. Seems like the key fact is somehow related to the distance between $x,y$ and the triangle. But could you also give a counter example for a $3$-regular graph? – draks ... Apr 20 '12 at 7:15
Possibly, but how many times are you going to change the question? – utdiscant Apr 20 '12 at 10:11
Don't get me wrong, you are perfectly answering the main question. 3-regularity is in there since Apr 3 at 17:48, because I still think the answer is Yes in this special case. – draks ... Apr 20 '12 at 10:19
I will certainly try to construct a new counter example for the 3-regular case, but my intuition is that it has to be a rather large counter example. – utdiscant Apr 20 '12 at 11:07
Is the problem to reach any other vertex from a given starting point $v_0$ in $n$ steps? If so, why not ignore the triangle? The path is of length $2n$, so the farthest point is $n$ away along the path.
Added for the not only if part: See below. You can get anywhere in three steps from $v_1$ but not from $v_4$. It wasn't clear to me whether you have to get anywhere from one place in $n$ steps or anywhere from anywhere in $n$ steps.
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But I mean exactly $n$, not less! How do you reach $v_{n-1}$ without it? – draks ... Mar 27 '12 at 21:01
If exactly $n$ steps is the requirement and you get to pick the start, you need to pick the start on the triangle. Otherwise you can't get to the neighboring vertices in the cycle. Imagine a hexagon, vertices $v_0, v_1, \ldots, v_5$ with $v_0, v_2$ added. You can't get from $v_4$ to $v_5$ in three steps. But you are essentially there. The idea of going around the triangle is the right one. – Ross Millikan Mar 27 '12 at 21:17
So you agree. What about the "only if" part? – draks ... Mar 27 '12 at 21:46
@draks: I don't think only if is correct. Imagine a pentagon, vertices $v_0, v_1, \ldots, v_4$ with $v_0,v_2$ added and $v_5$ connected to $v_0$. I think you can get to any vertex in exactly three steps from $v_1$ but there is no Hamiltonian cycle. – Ross Millikan Mar 27 '12 at 22:34
But you need to reach every vertex from every other vertex. BTW: I didn't get your example. Can you give the adjacency matrix or a pic? – draks ... Mar 28 '12 at 10:34
Only if: Having a triangle and being able to travel between any two vertices in exactly $n$ steps does not imply a hamiltonian cycle. Consider the Lollipop graph $L_{5,1}$, any two points can be reached in exactly three steps, it obviously contains a triangle, and it does not contain a hamiltonian path. To see that every pair of distinct points can be reached in three steps, first consider two vertices in the $K_5$, there is a path of length 3 made by visiting two of the vertices of the $K_5$ different from the starting and ending vertices. Then consider the path from the vertex of degree 1 to a vertex on the $K_5$, go to the vertex of degree 6, then visit a vertex of the $K_5$ other than the degree 6 vertex or the ending vertex, and then finish.
I also do not like your proof of the 'if' direction. Consider a cycle of length 7 with a triangle attached. You cannot travel from the point opposite the triangle to a point adjacent to it in 4 steps. This may be simpler to visualize as a cycle of length 8 with two vertices distance two apart joined. The reason that your proof fails is that the triangle is too far away to be useful.
In the attached picture, you cannot travel from $a$ to $b$ in exactly four steps.
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First of all: Thanks. "only if": Do you also have a counterexample if $G$ is planar? Sorry I forgot, to point that out. Mea culpa. "if": Do you mean graph some like $C_7-C_3$? If not could you please provide a picture...? – draks ... Apr 3 '12 at 17:44
@draks I'll need to think about planarity. You should edit the question also, so that others do not get confused. – deinst Apr 3 '12 at 17:58
I did. Thanks for your pictures and your effort (+1 as soon as I can ;-). – draks ... Apr 3 '12 at 17:59
@draks To make a planar counterexample, remove the edge between b and c in the lolipop. There is still a route of exact length 3 between any two vertices. – deinst Apr 3 '12 at 18:05
@draks I still have not answered it, though. Keep your reputation. When I have time I'll try to find a counter example. If you find one, please add an answer to your own question. – deinst Apr 9 '12 at 22:52
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2014-07-23 14:14:58
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https://walkccc.github.io/CLRS/Chap19/19.4/
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# 19.4 Bounding the maximum degree
## 19.4-1
Professor Pinocchio claims that the height of an $n$-node Fibonacci heap is $O(\lg n)$. Show that the professor is mistaken by exhibiting, for any positive integer $n$, a sequence of Fibonacci-heap operations that creates a Fibonacci heap consisting of just one tree that is a linear chain of $n$ nodes.
• Initialize: insert $3$ numbers then extract-min.
• Iteration: insert $3$ numbers, in which at least two numbers are less than the root of chain, then extract-min. The smallest newly inserted number will be extracted and the remaining two numbers will form a heap whose degree of root is $1$, and since the root of the heap is less than the old chain, the chain will be merged into the newly created heap. Finally we should delete the node which contains the largest number of the 3 inserted numbers.
## 19.4-2
Suppose we generalize the cascading-cut rule to cut a node $x$ from its parent as soon as it loses its $k$th child, for some integer constant $k$. (The rule in Section 19.3 uses $k = 2$.) For what values of $k$ is $D(n) = O(\lg n)$?
Following the proof of lemma 19.1, if $x$ is any node if a Fibonacci heap, $x.degree = m$, and $x$ has children $y_1, y_2, \ldots, y_m$, then $y_1.degree \ge 0$ and $y_i.degree \ge i - k$. Thus, if $s_m$ denotes the fewest nodes possible in a node of degree $m$, then we have $s_0 = 1, s_1 = 2, \ldots, s_{k - 1} = k$ and in general, $s_m = k + \sum_{i = 0}^{m - k} s_i$. Thus, the difference between $s_m$ and $s_{m - 1}$ is $s_{m - k}$.
Let $\{f_m\}$ be the sequence such that $f_m = m + 1$ for $0 \le m < k$ and $f_m = f_{m - 1} + f_{m - k}$ for $m \ge k$.
If $F(x)$ is the generating function for $f_m$ then we have $F(x) = \frac{1 - x^k}{(1 - x)(1 - x - x^k)}$. Let $\alpha$ be a root of $x^k = x^{k - 1} + 1$. We'll show by induction that $f_{m + k} \ge \alpha^m$. For the base cases:
\begin{aligned} f_k & = k + 1 \ge 1 = \alpha^0 \\ f_{k + 1} & = k + 3 \ge \alpha^1 \\ & \vdots \\ f_{k + k} & = k + \frac{(k + 1)(k + 2)}{2} = k + k + 1 + \frac{k(k + 1)}{2} \ge 2k + 1+\alpha^{k - 1} \ge \alpha^k. \end{aligned}
In general, we have
$$f_{m + k} = f_{m + k - 1} + f_m \ge \alpha^{m - 1} + \alpha^{m - k} = \alpha^{m - k}(\alpha^{k - 1} + 1) = \alpha^m.$$
Next we show that $f_{m + k} = k + \sum_{i = 0}^m f_i$. The base case is clear, since $f_k = f_0 + k = k + 1$. For the induction step, we have
$$f_{m + k} = f_{m - 1 - k} + f_m = k \sum_{i = 0}^{m - 1} f_i + f_m = k + \sum_{i = 0}^m f_i.$$
Observe that $s_i \ge f_{i + k}$ for $0 \le i < k$. Again, by induction, for $m \ge k$ we have
$$s_m = k + \sum_{i = 0}^{m - k} s_i \ge k + \sum_{i = 0}^{m - k} f_{i + k} \ge k + \sum_{i = 0}^m f_i = f_{m + k}.$$
So in general, $s_m \ge f_{m + k}$. Putting it all together, we have
\begin{aligned} size(x) & \ge s_m \\ & \ge k + \sum_{i = k}^m s_{i - k} \\ & \ge k + \sum_{i = k}^m f_i \\ & \ge f_{m + k} \\ & \ge \alpha^m. \end{aligned}
Taking logs on both sides, we have
$$\log_\alpha n \ge m.$$
In other words, provided that $\alpha$ is a constant, we have a logarithmic bound on the maximum degree.
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2019-10-18 05:11:09
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http://svmiller.com/stevedata/reference/fakeTSCS.html
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This is a toy (i.e. "fake") data set created by the fabricatr package. There are 100 observations for 25 hypothetical countries. The outcome y is a linear function of a baseline for each hypothetical country, plus a yearly growth trend as well as varying growth errors for each country. x1 is supposed to have a linear effect of .5 on y, all things considered. x2 is supposed to have a linear effect of 1 on y for each unit change in x2, all things considered.
fakeTSCS
## Format
A data frame with 2500 observations on the following 8 variables.
year
a numeric vector for the year
country
a character vector for the country
y
a numeric vector for the outcome.
x1
a continuous variable
x2
a binary variable
base
a numeric vector for the baseline starting point for each country
growth_units
a numeric vector for the growth units for each country
growth_error
a numeric vector for the growth errors for each country
## Details
x1 is generated by a normal distribution with a mean of 5 and a standard deviation of 2. x2 is drawn from a Bernoulli distribution with a probability of .5 of observing a 1.
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2021-06-20 06:23:53
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http://zbmath.org/?q=an:1205.90310
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# zbMATH — the first resource for mathematics
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A new method for solving fully fuzzy linear programming problems. (English) Zbl 1205.90310
Summary: F. H. Lotfi, T. Allahviranloo, M. A. Jondabeh and L. Alizadeh [Appl. Math. Modelling 33, No. 7, 3151–3156 (2009; Zbl 1205.90313)] pointed out that there is no method in literature for finding the fuzzy optimal solution of fully fuzzy linear programming (FFLP) problems and proposed a new method to find the fuzzy optimal solution of FFLP problems with equality constraints. In this paper, a new method is proposed to find the fuzzy optimal solution of same type of fuzzy linear programming problems. It is easy to apply the proposed method compare to the existing method for solving the FFLP problems with equality constraints occurring in real life situations. To illustrate the proposed method numerical examples are solved and the obtained results are discussed.
##### MSC:
90C70 Fuzzy programming 90C05 Linear programming
##### References:
[1] Bellman, R. E.; Zadeh, L. A.: Decision making in a fuzzy environment, Manage. sci. 17, 141-164 (1970) · Zbl 0224.90032 [2] Tanaka, H.; Okuda, T.; Asai, K.: On fuzzy mathematical programming, J. cybernetics syst. 3, 37-46 (1973) · Zbl 0297.90098 · doi:10.1080/01969727308545912 [3] Zimmerman, H. J.: Fuzzy programming and linear programming with several objective functions, Fuzzy set. Syst. 1, 45-55 (1978) · Zbl 0364.90065 · doi:10.1016/0165-0114(78)90031-3 [4] Campos, L.; Verdegay, J. L.: Linear programming problems and ranking of fuzzy numbers, Fuzzy set. Syst. 32, 1-11 (1989) · Zbl 0674.90061 · doi:10.1016/0165-0114(89)90084-5 [5] Maleki, H. R.; Tata, M.; Mashinchi, M.: Linear programming with fuzzy variables, Fuzzy set. Syst. 109, 21-33 (2000) · Zbl 0956.90068 · doi:10.1016/S0165-0114(98)00066-9 [6] Maleki, H. R.: Ranking functions and their applications to fuzzy linear programming, Far east J. Math. sci. 4, 283-301 (2002) · Zbl 1006.90093 [7] Ganesan, K.; Veeramani, P.: Fuzzy linear programs with trapezoidal fuzzy numbers, Ann. oper. Res. 143, 305-315 (2006) · Zbl 1101.90091 · doi:10.1007/s10479-006-7390-1 [8] Ebrahimnejad, A.; Nasseri, S. H.; Lotfi, F. H.; Soltanifar, M.: A primal-dual method for linear programming problems with fuzzy variables, Eur. J. Ind. eng. 4, 189-209 (2010) [9] Buckley, J.; Feuring, T.: Evolutionary algorithm solution to fuzzy problems: fuzzy linear programming, Fuzzy set. Syst. 109, 35-53 (2000) · Zbl 0956.90064 · doi:10.1016/S0165-0114(98)00022-0 [10] Hashemi, S. M.; Modarres, M.; Nasrabadi, E.; Nasrabadi, M. M.: Fully fuzzified linear programming, solution and duality, J. intell. Fuzzy syst. 17, 253-261 (2006) · Zbl 1101.90405 [11] Allahviranloo, T.; Lotfi, F. H.; Kiasary, M. K.; Kiani, N. A.; Alizadeh, L.: Solving full fuzzy linear programming problem by the ranking function, Appl. math. Sci. 2, 19-32 (2008) [12] Dehghan, M.; Hashemi, B.; Ghatee, M.: Computational methods for solving fully fuzzy linear systems, Appl. math. Comput. 179, 328-343 (2006) [13] Lotfi, F. H.; Allahviranloo, T.; Jondabeha, M. A.; Alizadeh, L.: Solving a fully fuzzy linear programming using lexicography method and fuzzy approximate solution, Appl. math. Modell. 33, 3151-3156 (2009) [14] Kaufmann, A.; Gupta, M. M.: Introduction to fuzzy arithmetic theory and applications, (1985) [15] Liou, T. S.; Wang, M. J.: Ranking fuzzy numbers with integral value, Fuzzy set. Syst. 50, 247-255 (1992) · Zbl 1229.03043 · doi:10.1016/0165-0114(92)90223-Q
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2014-04-25 00:11:35
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