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# Cumulant
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Title: Cumulant Author: World Heritage Encyclopedia Language: English Subject: Collection: Theory of Probability Distributions Publisher: World Heritage Encyclopedia Publication Date:
### Cumulant
In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have identical cumulants as well, and similarly the cumulants determine the moments. In some cases theoretical treatments of problems in terms of cumulants are simpler than those using moments.
Just as for moments, where joint moments are used for collections of random variables, it is possible to define joint cumulants.
## Contents
• Definition 1
• Alternative definition of the cumulant generating function 1.1
• Uses in statistics 2
• Cumulants of some discrete probability distributions 3
• Cumulants of some continuous probability distributions 4
• Some properties of the cumulant generating function 5
• Some properties of cumulants 6
• Invariance and equivariance 6.1
• Homogeneity 6.2
• A negative result 6.4
• Cumulants and moments 6.5
• Cumulants and set-partitions 6.6
• Cumulants and combinatorics 6.7
• Joint cumulants 7
• Conditional cumulants and the law of total cumulance 7.1
• Relation to statistical physics 8
• History 9
• Cumulants in generalized settings 10
• Formal cumulants 10.1
• Bell numbers 10.2
• Cumulants of a polynomial sequence of binomial type 10.3
• Free cumulants 10.4
• References 12
## Definition
The cumulants κn of a random variable X are defined via the cumulant-generating function K(t), which is the natural logarithm of the moment-generating function:
K(t)=\log\mathbb{E}\!\left[e^{tX}\right].
The cumulants κn are obtained from a power series expansion of the cumulant generating function:
K(t)=\sum_{n=1}^\infty \kappa_{n} \frac{t^{n}}{n!}.
This expansion is a Maclaurin series, so that nth cumulant can be obtained by differentiating the above expansion n times and evaluating the result at zero:[1]
\kappa_{n} = K^{(n)}(0).
If the moment-generating function does not exist, the cumulants can be defined in terms of the relationship between cumulants and moments discussed later.
### Alternative definition of the cumulant generating function
Some writers[2][3] prefer to define the cumulant-generating function as the natural logarithm of the characteristic function, which is sometimes also called the second characteristic function,[4][5]
H(t)=\log\mathbb{E}\!\left[e^{i t X}\right]=\sum_{n=1}^\infty \kappa_n \frac{(it)^n}{n!}=\mu it - \sigma^2 \frac{ t^2}{2} + \cdots
An advantage of H(t)—in some sense the function K(t) evaluated for purely imaginary arguments—is that E(eitX) is well defined for all real values of t even when E(etX) is not well defined for all real values of t, such as can occur when there is "too much" probability that X has a large magnitude. Although the function H(t) will be well defined, it nonetheless may mimic K(t) by not having a Maclaurin series beyond (or, rarely, even to) linear order in the argument t. Thus, many cumulants may still not be well defined. Nevertheless, even when H(t) does not have a long Maclaurin series, it can be used directly in analyzing and, particularly, adding random variables. Both the Cauchy distribution (also called the Lorentzian) and stable distribution (related to the Lévy distribution) are examples of distributions for which the power-series expansions of the generating functions have only finitely many well-defined terms.
## Uses in statistics
Working with cumulants can have an advantage over using moments because for statistically independent random variables X and Y,
\begin{align} K_{X+Y}(t) & =\log\mathbb{E}\!\left[e^{t(X+Y)}\right] \\ &= \log\left(\mathbb{E}\left[e^{tX}\right]\mathbb{E}\left[e^{tY}\right]\right) \\ & = \log\mathbb{E}\left[e^{tX}\right] + \log\mathbb{E}\left[e^{tY}\right] \\ &= K_X(t) + K_Y(t), \end{align}
so that each cumulant of a sum of independent random variables is the sum of the corresponding cumulants of the addends. A related result is that a cumulant-generating function for a product of independent random variables (defined as a random sum of independent realizations) is the composition of the respective cumulant-generating functions:[6]
\begin{align} K_{XY}(t) &= \log \mathbb{E} \left[ e^{t \sum_{i=1}^{X} Y_i} \right] \\ &= K_X(K_Y(t)). \end{align}
A distribution with given cumulants κn can be approximated through an Edgeworth series.
## Cumulants of some discrete probability distributions
• The constant random variables X = μ. The cumulant generating function is K(t) =μt. The first cumulant is κ1 = K '(0) = μ and the other cumulants are zero, κ2 = κ3 = κ4 = ... = 0.
• The Bernoulli distributions, (number of successes in one trial with probability p of success). The cumulant generating function is K(t) = log(1−p+pet). The first cumulants are κ1 = K '(0) = p and κ2 = K′′(0) = p·(1 − p). The cumulants satisfy a recursion formula
\kappa_{n+1}=p (1-p) \frac{d\kappa_n}{dp}.
• The geometric distributions, (number of failures before one success with probability p of success on each trial). The cumulant generating function is K(t) = log(p / (1 + (p−1)et)). The first cumulants are κ1 = K′(0) = p−1 − 1, and κ2 = K′′(0) = κ1p−1. Substituting p = (μ + 1)−1 gives K(t) = −log(1 + μ(1−et)) and κ1 = μ.
• The Poisson distributions. The cumulant generating function is K(t) = μ(et−1). All cumulants are equal to the parameter: κ1 = κ2 = κ3 = ... = μ.
• The binomial distributions, (number of successes in n independent trials with probability p of success on each trial). The special case n = 1 is a Bernoulli distribution. Every cumulant is just n times the corresponding cumulant of the corresponding Bernoulli distribution. The cumulant generating function is K(t) = n log(1−p+pet). The first cumulants are κ1 = K′(0) = np and κ2 = K′′(0) = κ1(1 − p). Substituting p = μ·n−1 gives K '(t) = ((μ−1n−1)·et + n−1)−1 and κ1 = μ. The limiting case n−1 = 0 is a Poisson distribution.
• The negative binomial distributions, (number of failures before n successes with probability p of success on each trial). The special case n = 1 is a geometric distribution. Every cumulant is just n times the corresponding cumulant of the corresponding geometric distribution. The derivative of the cumulant generating function is K '(t) = n·((1−p)−1·et−1)−1. The first cumulants are κ1 = K '(0) = n·(p−1−1), and κ2 = K ' '(0) = κ1·p−1. Substituting p = (μ·n−1+1)−1 gives K′(t) = ((μ−1 + n−1)etn−1)−1 and κ1 = μ. Comparing these formulas to those of the binomial distributions explains the name 'negative binomial distribution'. The limiting case n−1 = 0 is a Poisson distribution.
Introducing the variance-to-mean ratio
\varepsilon=\mu^{-1}\sigma^2=\kappa_1^{-1}\kappa_2,
the above probability distributions get a unified formula for the derivative of the cumulant generating function:
K'(t)=\mu\cdot(1+\varepsilon\cdot (e^{-t}-1))^{-1}.
The second derivative is
K''(t)=g'(t)\cdot(1+e^t\cdot (\varepsilon^{-1}-1))^{-1}
confirming that the first cumulant is κ1 = K′(0) = μ and the second cumulant is κ2 = K′′(0) = με. The constant random variables X = μ have ε = 0. The binomial distributions have ε = 1 − p so that 0 < ε < 1. The Poisson distributions have ε = 1. The negative binomial distributions have ε = p−1 so that ε > 1. Note the analogy to the classification of conic sections by eccentricity: circles ε = 0, ellipses 0 < ε < 1, parabolas ε = 1, hyperbolas ε > 1.
## Cumulants of some continuous probability distributions
• For the normal distribution with expected value μ and variance σ2, the cumulant generating function is K(t) = μt + σ2t2/2. The first and second derivatives of the cumulant generating function are K '(t) = μ + σ2·t and K"(t) = σ2. The cumulants are κ1 = μ, κ2 = σ2, and κ3 = κ4 = ... = 0. The special case σ2 = 0 is a constant random variable X = μ.
## Some properties of the cumulant generating function
The cumulant generating function K(t), if it exists, is infinitely differentiable and convex, and passes through the origin. Its first derivative ranges monotonically in the open interval from the infimum to the supremum of the support of the probability distribution, and its second derivative is strictly positive everywhere it is defined, except for the degenerate distribution of a single point mass. The cumulant-generating function exists if and only if the tails of the distribution are majorized by an exponential decay, that is, (see Big O notation,)
\exists c>0, F(x)=O(e^{cx}), x\to-\infty; and
\exists d>0, 1-F(x)=O(e^{-dx}),x\to+\infty;
where F is the cumulative distribution function. The cumulant-generating function will have vertical asymptote(s) at the infimum of such c, if such an infimum exists, and at the supremum of such d, if such a supremum exists, otherwise it will be defined for all real numbers.
If the support of a random variable X has finite upper or lower bounds, then its cumulant-generating function y=K(t), if it exists, approaches asymptote(s) whose slope is equal to the supremum and/or infimum of the support,
y=(t+1)\inf \mathrm{supp}X-\mu(X), and
y=(t-1)\sup\mathrm{supp}X+\mu(X),
respectively, lying above both these lines everywhere. (The integrals
\int_{-\infty}^0 \left[t\inf \mathrm{supp}X-K'(t)\right]dt, \qquad \int_{\infty}^0 \left[t\inf \mathrm{supp}X-K'(t)\right]dt
yield the y-intercepts of these asymptotes, since K(0)=0.)
For a shift of the distribution by c, K_{X+c}(t)=K_X(t)+ct. For a degenerate point mass at c, the cgf is the straight line K_c(t)=ct, and more generally, K_{X+Y}=K_X+K_Y if and only if X and Y are independent and their cgfs exist; (subindependence and the existence of second moments sufficing to imply independence.[7])
The natural exponential family of a distribution may be realized by shifting or translating K(t), and adjusting it vertically so that it always passes through the origin: if f is the pdf with cgf K(t)=\log M(t), and f|\theta is its natural exponential family, then f(x|\theta)=\frac1{M(\theta)}e^{\theta x} f(x), and K(t|\theta)=K(t+\theta)-K(\theta).
If K(t) is finite for a range t1 < Re(t) < t2 then if t1 < 0 < t2 then K(t) is analytic and infinitely differentiable for t1 < Re(t) < t2. Moreover for t real and t1 < t < t2 K(t) is strictly convex, and K'(t) is strictly increasing.
## Some properties of cumulants
### Invariance and equivariance
The first cumulant is shift-equivariant; all of the others are shift-invariant. This means that, if we denote by κn(X) the nth cumulant of the probability distribution of the random variable X, then for any constant c:
• \kappa_1(X + c) = \kappa_1(X) + c ~ \text{ and}
• \kappa_n(X + c) = \kappa_n(X) ~ \text{ for } ~ n \ge 2.
In other words, shifting a random variable (adding c) shifts the first cumulant (the mean) and doesn't affect any of the others.
### Homogeneity
The nth cumulant is homogeneous of degree n, i.e. if c is any constant, then
\kappa_n(cX)=c^n\kappa_n(X). \,
If X and Y are independent random variables then κn(X + Y) = κn(X) + κn(Y).
### A negative result
Given the results for the cumulants of the normal distribution, it might be hoped to find families of distributions for which κm = κm+1 = ... = 0 for some m > 3, with the lower-order cumulants (orders 3 to m − 1) being non-zero. There are no such distributions.[8] The underlying result here is that the cumulant generating function cannot be a finite-order polynomial of degree greater than 2.
### Cumulants and moments
The moment generating function is given by:
M(t) = 1+\sum_{n=1}^\infty \frac{\mu'_n t^n}{n!}=\exp\left(\sum_{n=1}^\infty \frac{\kappa_n t^n}{n!}\right) = \exp(K(t)).
So the cumulant generating function is the logarithm of the moment generating function
K(t) = \log M(t).
The first cumulant is the expected value; the second and third cumulants are respectively the second and third central moments (the second central moment is the variance); but the higher cumulants are neither moments nor central moments, but rather more complicated polynomial functions of the moments.
The moments can be recovered in terms of cumulants by evaluating the n-th derivative of \exp(K(t)) at t = 0,
\mu'_n = M^{(n)}(0) = \frac{\mathrm{d}^n \exp (K(t))}{\mathrm{d}t^n}\Big|_{t=0}.
Likewise, the cumulants can be recovered in terms of moments by evaluating the n-th derivative of \log M(t) at t=0,
\kappa_n = K^{(n)}(0) = \frac{\mathrm{d}^n \log M(t)}{\mathrm{d}t^n}\Big|_{t=0}.
The explicit expression for the n-th moment in terms of the first n cumulants, and vice versa, can be obtained by using Faà di Bruno's formula for higher derivatives of composite functions. In general, we have
\mu'_n = \sum_{k=1}^n B_{n,k}(\kappa_1,\ldots,\kappa_{n-k+1})
\kappa_n = \sum_{k=1}^n (-1)^{k-1} (k-1)! B_{n,k}(\mu'_1,\ldots,\mu'_{n-k+1}),
where B_{n,k} are incomplete (or partial) Bell polynomials.
In the like manner, if the mean is given by \mu, the central moment generating function is given by
C(t) = \mathbb{E}[e^{t(x-\mu)}] = e^{-\mu t} M(t) = \exp(K(t) - \mu t),
and the n-th central moment is obtained in terms of cumulants as
\mu_n = C^{(n)}(0) = \frac{\mathrm{d}^n}{\mathrm{d}t^n} \exp (K(t) - \mu t)\Big|_{t=0} = \sum_{k=1}^n B_{n,k}(0,\kappa_2,\ldots,\kappa_{n-k+1}).
Also, for n>1, the n-th cumulant in terms of the central moments is,
\kappa_n = K^{(n)}(0) = \frac{\mathrm{d}^n}{\mathrm{d}t^n} (\log C(t) + \mu t) \Big|_{t=0} = \sum_{k=1}^n (-1)^{k-1} (k-1)! B_{n,k}(0,\mu_2,\ldots,\mu_{n-k+1}).
The nth moment μ′n is an nth-degree polynomial in the first n cumulants. The first few expressions are:
\mu'_1=\kappa_1\,
\mu'_2=\kappa_2+\kappa_1^2\,
\mu'_3=\kappa_3+3\kappa_2\kappa_1+\kappa_1^3\,
\mu'_4=\kappa_4+4\kappa_3\kappa_1+3\kappa_2^2+6\kappa_2\kappa_1^2+\kappa_1^4\,
\mu'_5=\kappa_5+5\kappa_4\kappa_1+10\kappa_3\kappa_2 +10\kappa_3\kappa_1^2+15\kappa_2^2\kappa_1 +10\kappa_2\kappa_1^3+\kappa_1^5\,
\mu'_6=\kappa_6+6\kappa_5\kappa_1+15\kappa_4\kappa_2+15\kappa_4\kappa_1^2 +10\kappa_3^2+60\kappa_3\kappa_2\kappa_1+20\kappa_3\kappa_1^3+15\kappa_2^3 +45\kappa_2^2\kappa_1^2+15\kappa_2\kappa_1^4+\kappa_1^6.\,
The "prime" distinguishes the moments μ′n from the central moments μn. To express the central moments as functions of the cumulants, just drop from these polynomials all terms in which κ1 appears as a factor:
\mu_1=0\,
\mu_2=\kappa_2\,
\mu_3=\kappa_3\,
\mu_4=\kappa_4+3\kappa_2^2\,
\mu_5=\kappa_5+10\kappa_3\kappa_2\,
\mu_6=\kappa_6+15\kappa_4\kappa_2+10\kappa_3^2+15\kappa_2^3.\,
Similarly, the nth cumulant κn is an nth-degree polynomial in the first n non-central moments. The first few expressions are:
\kappa_1=\mu'_1\,
\kappa_2=\mu'_2-{\mu'_1}^2\,
\kappa_3=\mu'_3-3\mu'_2\mu'_1+2{\mu'_1}^3\,
\kappa_4=\mu'_4-4\mu'_3\mu'_1-3{\mu'_2}^2+12\mu'_2{\mu'_1}^2-6{\mu'_1}^4\,
\kappa_5=\mu'_5-5\mu'_4\mu'_1-10\mu'_3\mu'_2+20\mu'_3{\mu'_1}^2+30{\mu'_2}^2\mu'_1-60\mu'_2{\mu'_1}^3+24{\mu'_1}^5\,
\kappa_6=\mu'_6-6\mu'_5\mu'_1-15\mu'_4\mu'_2+30\mu'_4{\mu'_1}^2-10{\mu'_3}^2+120\mu'_3\mu'_2\mu'_1-120\mu'_3{\mu'_1}^3+30{\mu'_2}^3-270{\mu'_2}^2{\mu'_1}^2+360\mu'_2{\mu'_1}^4-120{\mu'_1}^6\,.
To express the cumulants κn for n > 1 as functions of the central moments, drop from these polynomials all terms in which μ'1 appears as a factor:
\kappa_2=\mu_2\,
\kappa_3=\mu_3\,
\kappa_4=\mu_4-3{\mu_2}^2\,
\kappa_5=\mu_5-10\mu_3\mu_2\,
\kappa_6=\mu_6-15\mu_4\mu_2-10{\mu_3}^2+30{\mu_2}^3\,.
To express the cumulants κn for n > 2 as functions of the standardized central moments, also set μ'2=1 in the polynomials:
\kappa_3=\mu_3\,
\kappa_4=\mu_4-3\,
\kappa_5=\mu_5-10\mu_3\,
\kappa_6=\mu_6-15\mu_4-10{\mu_3}^2+30\,.
The cumulants are also related to the moments by the following recursion formula:
\kappa_n=\mu'_n-\sum_{m=1}^{n-1}{n-1 \choose m-1}\kappa_m \mu_{n-m}'.
### Cumulants and set-partitions
These polynomials have a remarkable combinatorial interpretation: the coefficients count certain partitions of sets. A general form of these polynomials is
\mu'_n=\sum_\Pi \prod_{B\in\pi}\kappa_{\left|B\right|}
where
• π runs through the list of all partitions of a set of size n;
• "B ∈ π" means B is one of the "blocks" into which the set is partitioned; and
• |B| is the size of the set B.
Thus each monomial is a constant times a product of cumulants in which the sum of the indices is n (e.g., in the term κ3 κ22 κ1, the sum of the indices is 3 + 2 + 2 + 1 = 8; this appears in the polynomial that expresses the 8th moment as a function of the first eight cumulants). A partition of the integer n corresponds to each term. The coefficient in each term is the number of partitions of a set of n members that collapse to that partition of the integer n when the members of the set become indistinguishable.
### Cumulants and combinatorics
Further connection between cumulants and combinatorics can be found in the work of Gian-Carlo Rota and Jianhong (Jackie) Shen, where links to invariant theory, symmetric functions, and binomial sequences are studied via umbral calculus.[9]
## Joint cumulants
The joint cumulant of several random variables X1, ..., Xn is defined by a similar cumulant generating function
K(t_1,t_2,\dots,t_n)=\log E(\mathrm e^{\sum_{j=1}^n t_j X_j}).
A consequence is that
\kappa(X_1,\dots,X_n) =\sum_\pi (|\pi|-1)!(-1)^{|\pi|-1}\prod_{B\in\pi}E\left(\prod_{i\in B}X_i\right)
where π runs through the list of all partitions of { 1, ..., n }, B runs through the list of all blocks of the partition π, and |π| is the number of parts in the partition. For example,
\kappa(X,Y,Z)=E(XYZ)-E(XY)E(Z)-E(XZ)E(Y)-E(YZ)E(X)+2E(X)E(Y)E(Z).\,
If any of these random variables are identical, e.g. if X = Y, then the same formulae apply, e.g.
\kappa(X,X,Z)=E(X^2Z)-2E(XZ)E(X)-E(X^2)E(Z)+2E(X)^2E(Z),\,
although for such repeated variables there are more concise formulae. For zero-mean random vectors,
\kappa(X,Y,Z)=E(XYZ).\,
\kappa(X,Y,Z,W) = E(XYZW)-E(XY)E(ZW)-E(XZ)E(YW)-E(XW)E(YZ).\,
The joint cumulant of just one random variable is its expected value, and that of two random variables is their covariance. If some of the random variables are independent of all of the others, then any cumulant involving two (or more) independent random variables is zero. If all n random variables are the same, then the joint cumulant is the nth ordinary cumulant.
The combinatorial meaning of the expression of moments in terms of cumulants is easier to understand than that of cumulants in terms of moments:
E(X_1\cdots X_n)=\sum_\pi\prod_{B\in\pi}\kappa(X_i : i \in B).
For example:
E(XYZ)=\kappa(X,Y,Z)+\kappa(X,Y)\kappa(Z)+\kappa(X,Z)\kappa(Y) +\kappa(Y,Z)\kappa(X)+\kappa(X)\kappa(Y)\kappa(Z).\,
Another important property of joint cumulants is multilinearity:
\kappa(X+Y,Z_1,Z_2,\dots)=\kappa(X,Z_1,Z_2,\dots)+\kappa(Y,Z_1,Z_2,\dots).\,
Just as the second cumulant is the variance, the joint cumulant of just two random variables is the covariance. The familiar identity
\operatorname{var}(X+Y)=\operatorname{var}(X)+2\operatorname{cov}(X,Y)+\operatorname{var}(Y)\,
generalizes to cumulants:
\kappa_n(X+Y)=\sum_{j=0}^n {n \choose j} \kappa(\,\underbrace{X,\dots,X}_j,\underbrace{Y,\dots,Y}_{n-j}\,).\,
### Conditional cumulants and the law of total cumulance
The law of total expectation and the law of total variance generalize naturally to conditional cumulants. The case n = 3, expressed in the language of (central) moments rather than that of cumulants, says
\mu_3(X)=E(\mu_3(X\mid Y))+\mu_3(E(X\mid Y)) +3\,\operatorname{cov}(E(X\mid Y),\operatorname{var}(X\mid Y)).
In general,[10]
\kappa(X_1,\dots,X_n)=\sum_\pi \kappa(\kappa(X_{\pi_1}\mid Y),\dots,\kappa(X_{\pi_b}\mid Y))
where
• the sum is over all partitions π of the set { 1, ..., n } of indices, and
• π1, ..., πb are all of the "blocks" of the partition π; the expression κ(Xπm) indicates that the joint cumulant of the random variables whose indices are in that block of the partition.
## Relation to statistical physics
In statistical physics many extensive quantities – that is quantities that are proportional to the volume or size of a given system – are related to cumulants of random variables. The deep connection is that in a large system an extensive quantity like the energy or number of particles can be thought of as the sum of (say) the energy associated with a number of nearly independent regions. The fact that the cumulants of these nearly independent random variables will (nearly) add make it reasonable that extensive quantities should be expected to be related to cumulants.
A system in equilibrium with a thermal bath at temperature T can occupy states of energy E. The energy E can be considered a random variable, having the probability density. The partition function of the system is
Z(\beta) = \langle\exp(-\beta E)\rangle,\,
where β = 1/(kT) and k is Boltzmann's constant and the notation \langle A \rangle has been used rather than \mathbb{E}\!\left[A\right] for the expectation value to avoid confusion with the energy, E. The Helmholtz free energy is then
F(\beta) = -\beta^{-1}\log Z \,
and is clearly very closely related to the cumulant generating function for the energy. The free energy gives access to all of the thermodynamics properties of the system via its first second and higher order derivatives, such as its internal energy, entropy, and specific heat. Because of the relationship between the free energy and the cumulant generating function, all these quantities are related to cumulants e.g. the energy and specific heat are given by
E = \langle E \rangle_c
C= dE/dT = k \beta^2\langle E^2 \rangle_c = k \beta^2(\langle E^2\rangle - \langle E\rangle ^2)
and \langle E^2\rangle_c symbolizes the second cumulant of the energy. Other free energy is often also a function of other variables such as the magnetic field or chemical potential \mu, e.g.
\Omega=-\beta^{-1}\log(\langle \exp(-\beta E -\beta\mu N) \rangle),\,
where N is the number of particles and \Omega is the grand potential. Again the close relationship between the definition of the free energy and the cumulant generating function implies that various derivatives of this free energy can be written in terms of joint cumulants of E and N.
## History
The history of cumulants is discussed by Anders Hald.[11][12]
Cumulants were first introduced by Thorvald N. Thiele, in 1889, who called them semi-invariants.[13] They were first called cumulants in a 1932 paper[14] by Ronald Fisher and John Wishart. Fisher was publicly reminded of Thiele's work by Neyman, who also notes previous published citations of Thiele brought to Fisher's attention.[15] Stephen Stigler has said that the name cumulant was suggested to Fisher in a letter from Harold Hotelling. In a paper published in 1929,[16] Fisher had called them cumulative moment functions. The partition function in statistical physics was introduced by Josiah Willard Gibbs in 1901. The free energy is often called Gibbs free energy. In statistical mechanics, cumulants are also known as Ursell functions relating to a publication in 1927.
## Cumulants in generalized settings
### Formal cumulants
More generally, the cumulants of a sequence { mn : n = 1, 2, 3, ... }, not necessarily the moments of any probability distribution, are, by definition,
1+\sum_{n=1}^\infty m_n t^n/n!=\exp\left(\sum_{n=1}^\infty\kappa_n t^n/n!\right) ,
where the values of κn for n = 1, 2, 3, ... are found formally, i.e., by algebra alone, in disregard of questions of whether any series converges. All of the difficulties of the "problem of cumulants" are absent when one works formally. The simplest example is that the second cumulant of a probability distribution must always be nonnegative, and is zero only if all of the higher cumulants are zero. Formal cumulants are subject to no such constraints.
### Bell numbers
In combinatorics, the nth Bell number is the number of partitions of a set of size n. All of the cumulants of the sequence of Bell numbers are equal to 1. The Bell numbers are the moments of the Poisson distribution with expected value 1.
### Cumulants of a polynomial sequence of binomial type
For any sequence { κn : n = 1, 2, 3, ... } of scalars in a field of characteristic zero, being considered formal cumulants, there is a corresponding sequence { μ ′ : n = 1, 2, 3, ... } of formal moments, given by the polynomials above. For those polynomials, construct a polynomial sequence in the following way. Out of the polynomial
\mu'_6 = \kappa_6+6\kappa_5\kappa_1+15\kappa_4\kappa_2+15\kappa_4\kappa_1^2 +10\kappa_3^2+60\kappa_3\kappa_2\kappa_1 + 20\kappa_3\kappa_1^3+15\kappa_2^3 +45\kappa_2^2\kappa_1^2+15\kappa_2\kappa_1^4+\kappa_1^6
make a new polynomial in these plus one additional variable x:
p_6(x) = \kappa_6 \,x + (6\kappa_5\kappa_1 + 15\kappa_4\kappa_2 + 10\kappa_3^2)\,x^2m+(15\kappa_4\kappa_1^2+60\kappa_3\kappa_2\kappa_1+15\kappa_2^3)\,x^3 +(45\kappa_2^2\kappa_1^2)\,x^4+(15\kappa_2\kappa_1^4)\,x^5 +(\kappa_1^6)\,x^6,
and then generalize the pattern. The pattern is that the numbers of blocks in the aforementioned partitions are the exponents on x. Each coefficient is a polynomial in the cumulants; these are the Bell polynomials, named after Eric Temple Bell.
This sequence of polynomials is of binomial type. In fact, no other sequences of binomial type exist; every polynomial sequence of binomial type is completely determined by its sequence of formal cumulants.
### Free cumulants
In the identity
E(X_1\cdots X_n)=\sum_\pi\prod_{B\in\pi}\kappa(X_i : i\in B)
one sums over all partitions of the set { 1, ..., n }. If instead, one sums only over the noncrossing partitions, then one gets "free cumulants" rather than conventional cumulants treated above. These play a central role in free probability theory.[17] In that theory, rather than considering independence of random variables, defined in terms of Cartesian products of algebras of random variables, one considers instead "freeness" of random variables, defined in terms of free products of algebras rather than Cartesian products of algebras.
The ordinary cumulants of degree higher than 2 of the normal distribution are zero. The free cumulants of degree higher than 2 of the Wigner semicircle distribution are zero.[17] This is one respect in which the role of the Wigner distribution in free probability theory is analogous to that of the normal distribution in conventional probability theory.
## References
1. ^ Weisstein, Eric W. "Cumulant". From MathWorld – A Wolfram Web Resource. http://mathworld.wolfram.com/Cumulant.html
2. ^ Kendall, M. G., Stuart, A. (1969) The Advanced Theory of Statistics, Volume 1 (3rd Edition). Griffin, London. (Section 3.12)
3. ^ Lukacs, E. (1970) Characteristic Functions (2nd Edition). Griffin, London. (Page 27)
4. ^ Lukacs, E. (1970) Characteristic Functions (2nd Edition). Griffin, London. (Section 2.4)
5. ^ Aapo Hyvarinen, Juha Karhunen, and Erkki Oja (2001) Independent Component Analysis, John Wiley & Sons. (Section 2.7.2)
6. ^ Reluga, T. (2009). "Branching Processes and Noncommuting Random Variables in Population Biology". Canadian Applied Mathematics Quarterly 17 (2): 387–408.
7. ^ Hamedani, G. G.; Volkmer, Hans; Behboodian, J. (2012-03-01). "A note on sub-independent random variables and a class of bivariate mixtures". Studia Scientiarum Mathematicarum Hungarica 49 (1): 19–25.
8. ^ Lukacs, E. (1970) Characteristic Functions (2nd Edition), Griffin, London. (Theorem 7.3.5)
9. ^ Rota, G.-C.; Shen, J. (2000). "On the Combinatorics of Cumulants". Journal of Combinatorial Theory. Series A 91 (1–2): 283–304.
10. ^ Brillinger, D.R. (1969). "The Calculation of Cumulants via Conditioning". Annals of the Institute of Statistical Mathematics 21: 215–218.
11. ^ )
12. ^
13. ^ H. Cramér (1946) Mathematical Methods of Statistics, Princeton University Press, Section 15.10, p. 186.
14. ^ Fisher, R.A. , John Wishart, J.. (1932) The derivation of the pattern formulae of two-way partitions from those of simpler patterns, Proceedings of the London Mathematical Society, Series 2, v. 33, pp. 195–208 doi:10.1112/plms/s2-33.1.195
15. ^ Neyman, J. (1956): ‘Note on an Article by Sir Ronald Fisher,’ Journal of the Royal Statistical Society, Series B (Methodological), 18, pp. 288–94.
16. ^ Fisher, R. A. (1929). "Moments and Product Moments of Sampling Distributions". Proceedings of the London Mathematical Society 30: 199-238.
17. ^ a b Novak, Jonathan; Śniady, Piotr (2011). "What Is a Free Cumulant?". | 2021-04-11 08:02:35 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9003044962882996, "perplexity": 1069.1327337202345}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038061562.11/warc/CC-MAIN-20210411055903-20210411085903-00053.warc.gz"} |
http://mathhelpforum.com/pre-calculus/167906-limits-indeterminant-form.html | # Math Help - Limits of indeterminant form.
1. ## Limits of indeterminant form.
2. Originally Posted by niazk90
For Q3 note that $\displaystyle \frac{\sqrt{x^2 - x}}{x-9} = \frac{\sqrt{1 - \frac{1}{x}}}{1 - \frac{9}{x}}$.
For Q6 note that by multiplying by $\displaystyle \frac{\sqrt{2x+3} + \sqrt{x - 1}}{\sqrt{2x+3} + \sqrt{x - 1}}$ the expression becomes $\displaystyle \frac{x + 4}{\sqrt{2x+3} + \sqrt{x - 1}}$. | 2015-10-07 07:59:23 | {"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": 0, "img_math": 0, "codecogs_latex": 3, "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": 0.9863636493682861, "perplexity": 3170.3552638260626}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736682773.27/warc/CC-MAIN-20151001215802-00197-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://taoofmac.com/space/blog/2021/11/28/2230 | # My Quest for Home Automation, Part 5
We don’t celebrate Thanksgiving here, but it tends to be the time around which I take out our Tasmota-based smart outlets and set up the heaters again, and I guess it’s also a good time to document all the latest changes.
## Previous Instalments
• - Sonoff/Tasmota S20 outlets
• - Node-RED, Broadlink IR (since discontinued) and security
• - debugging and Xiaomi/Aqara ZigBee sensors
• - a detour through Azure IoT (since discontinued, but fun)
• - old Docker setup, Node-RED wizardry and accessory spoofing
• - the recent switch to a Pi 4 and away from Docker
## TL;DR:
My core setup is still based on (i.e., homebridge, a few plugins and mosquitto for glue to everything else) and ZigBee for security reasons (no public Internet services to work) and I have progressively cut down on the number of custom Node-RED flows to a point where I only need to spoof a couple of devices to have them show up on the Home app properly.
The only other protocol in use is WeMo, and that because a few Tasmota devices have WeMo emulation on so I can control them from an Echo Listen (HomePods are not officially available in Portugal and lack support for external speakers, so I can’t use them).
This is only acceptable because the WeMo TCP protocol also works without any outside access (other than Alexa voice recognition, the Echo talks directly to the devices), but I hope to do away with that next year since I haven’t fully given up on rolling my own smart speaker – I just haven’t had the time to fiddle with it, and all the OpenSource projects I was considering using have either stalled or gone cloud-centric/proprietary, which is a bit sad.
## Configuring a Tasmota Push-Button Timer
Besides setting up the heaters (which were also reconfigured to use HomeAssistant auto-discovery using SetOption19 on, which is now supported by my homebridge setup), another thing I had to set up (which was not documented elsewhere) was a three-minute push-button timer for one of the outlets, which can be coded into Tasmota itself like this (just paste the whole thing into the Tasmota console in a single line):
Rule1 On Button1#state Do
Backlog Power1 %value%;
RuleTimer1 180
EndOn
On Rules#Timer=1 do
Power1 off
EndOn
Don’t forget to also issue the Rule1 On command to actually enable the rule.
## Going ZigBee 3.0
A month or so ago, in an attempt to improve connectivity with faraway sensors, I replaced the CC2531 dongles with CC2652R ones, which are ZigBee 3.0 compatible:
I also added a dedicated router to my office to decrease latency when controlling lights (more on that later), and re-paired a couple of recalcitrant devices to their nearest router (something made a lot easier due to the new, built-in zigbee2mqtt web UI, which also replaced my own Node-RED equivalent).
There have been a few quirks (sometimes door sensor notifications - not all, just a couple - come a full minute after the fact for no reason), but I’ve seen sensors drop off the network much less and things have improved in general – or at least as much as possible given the challenge of getting ZigBee to work through a bunch of walls amidst the cacophony of noise in the 2.4GHz band around these parts.
## Better Maintenance
I have also done a few software updates to homebridge, zigbee2mqtt and node-red in the meantime, and they have worked out fine: overall much less hassle than the previous Docker setups, because I can now use their own self-update mechanisms inside piku environments without having to build custom images (or pull them down, which tends to be slow either way).
As always, everything is backed up to Azure using restic, plus I’m using Node-Red‘s built-in git support to keep track of (now very infrequent) changes there, so things are pretty stable and all daily interactions are done through the Home app.
## HomeKit Bugs
There are a few kinks, however – for some reason, the Home app allows me to define automations based on temperature sensors but completely fails to trigger them when temperature crosses those thresholds, which is… disappointing, and completely outside my control (unless I hack some Node-RED flows to do it, which I would rather not just yet).
But everything else has been rock solid, and I’m still very happy with the results – the base setup has now been working for nearly four years, and just keeps getting simpler to run. | 2022-11-29 13:59:32 | {"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": 0.3064873516559601, "perplexity": 3814.297178253624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710698.62/warc/CC-MAIN-20221129132340-20221129162340-00763.warc.gz"} |
http://wmbriggs.com/page/2/ | ## William M. Briggs
### Statistician to the Stars!
#### Page 2 of 728
A Logical Probabilist (note the large forehead) explains that the interocitor has three states.
Back to our Edge series. Sean Carroll says Bayes’s Theorem should be better known. He outlines the theorem in the familiar updating-prior-belief formula. But, as this modified classic article shows, this is not the most important facet of Bayesian theory.
Below we learn all probabilities fit into the schema $\Pr(\mbox{Y}|\mbox{X})$, where X is the totality of evidence we have for proposition Y. It does not matter how this final number is computed (if indeed it can be): it can sometimes be computed directly, and sometimes by busting X apart into “prior” and “new” information, or sometimes by busting X apart into ways that are convenient for the mechanics of the calculation. That’s all Bayes’s theorem is: a way to ease calculation in some but not all instances. An example is given below. The real innovation—the real magic—comes in understanding all probability is conditional, i.e. that it fits into the schema. As shown in this talk-of-the-town book.
This post is modified version of one that was been restored after The Hacking. All original comments were lost.
Bayesian theory probably isn’t what you think. Most have the idea that it’s all about “prior beliefs” and “updating” probabilities, or perhaps a way of encapsulating “feelings” quantitatively. The real innovation is something much more profound. And really, when it comes down to it, Bayes’s theorem isn’t even necessary for Bayesian theory. Here’s why.
Any probability is denoted by the schematic equation $\Pr(\mbox{Y}|\mbox{X})$ (all probability is conditional), which is the probability the proposition Y is true given the premise X. X may be compound, complex or simple. Bayes’s theorem looks like this:
$\Pr(\mbox{Y}|\mbox{W}\mbox{X}) = \frac{\Pr(\mbox{W}|\mbox{YX})\Pr(\mbox{Y}|\mbox{X})}{\Pr(\mbox{W}|\mbox{X})}$.
We start knowing or accepting the premise X, then later assume or learn W, and are able to calculate, or “update”, the probability of Y given this new information WX (read as “W and X are true or assumed true”). Bayes’s theorem is a way to compute $\Pr(\mbox{Y}|\mbox{W}\mbox{X})$. But it isn’t strictly needed. We could compute $\Pr(\mbox{Y}|\mbox{W}\mbox{X})$ directly from knowledge of W and X themselves. Sometimes the use of Bayes’s theorem can hinder.
Given X = “This machine must take one of states S1, S2, or S3”, we want the probability Y = “The machine is in state S1.” The deduced answer is 1/3. We then learn W = “The machine is malfunctioning and cannot take state S3”. The probability of Y given W and X is deduced as 1/2, as is trivial to see.
Now let’s find the result by applying Bayes’s theorem, the results of which must match. We know that $\Pr(\mbox{W}|\mbox{YX})/\Pr(\mbox{W}|\mbox{X}) = 3/2$, because $\Pr(\mbox{Y}|\mbox{X}) = 1/3$. But it’s difficult at first to tell how this comes about. What exactly is \Pr(\mbox{W}|\mbox{X}), the probability the machine malfunctions such that it cannot take state S3 given only the knowledge that it must take one of S1, S2, or S3? If we argue that if the machine is going to malfunction, given the premises we have (X), it is equally likely to be any of the three states, thus the probability is 1/3. Then $\Pr(\mbox{W}|\mbox{YX})$ must equal 1/2, but why? Given we know the machine is in state S1, and that it can take any of the three, the probability state S3 is the malfunction is 1/2, because we know the malfunctioning state cannot be S1, but can be S2 or S3. Using Bayes works, as it must, but in this case it added considerably to the burden of the calculation. In Uncertainty, I have other examples.
Most scientific, which is to say empirical, propositions start with the premise that they are contingent. This knowledge is usually left tacit; it rarely (or never) appears in equations. But it could: we could compute $\Pr(\mbox{Y}|\mbox{Y is contingent})$, which even is quantifiable (the open interval (0,1)). We then “update” this to $\Pr(\mbox{Y}|\mbox{X \& Y is contingent})$, which is 1/3 as above. Bayes’s theorem is again not needed.
Of course, there are many instances in which Bayes facilitates. Without this tool we would be more than hard pressed to calculate some probabilities. But the point is the theorem can but doesn’t have to be invoked as a computational aide. The theorem is not the philosophy.
The real innovation in Bayesian philosophy, whether it is recognized or not, came with the idea that any uncertain proposition can and must be assigned a probability, not in how the probabilities are calculated. (This dictum is not always assiduously followed.) This is contrasted with frequentist theory which assigns probabilities to some unknown propositions while forbidding this assignment in others, and where the choice is ad hoc. Given premises, a Bayesian can and does put a probability on the truth of an hypothesis (which is a proposition), a frequentist cannot—at least not formally. Mistakes and misinterpretations made by users of frequentist theory are legion.
The problem with both philosophies is misdirection, the unreasonable fascination with questions nobody asks, which is to say, the peculiar preoccupation with parameters. About that, another time.
This may be proved in three ways. The first…
See the first post in this series for an explanation and guide of our tour of Summa Contra Gentiles. All posts are under the category SAMT.
Previous post.
Two more Chapters this week, and fairly easy ones (for those who have studied since the beginning). We come to an exciting Chapter next week: how does the immaterial intellect contact the body?
Chapter 66 Against those who maintain that intellect and sense are the same. (alternate translation) We’re still using the alternate translation this week.
1 Thinking that there was no difference between intellect and sense, some of the early philosophers were close to the persons referred to above. But that notion of theirs is impossible.
2 For sense is found in all animals, whereas animals other than man have no intellect. This is evident from the fact that the latter perform diverse and opposite actions, not as though they possessed intellect, but as moved by nature, carrying out certain determinate operations of uniform character within the same species; every swallow builds its nest in the same way. Therefore, intellect is not the same as sense.
Notes It is a good joke to say you never see swallows down the Home Depot on a weekend. See also this Ed Feser article “Da Ya Think I’m Sphexy?” about the determinate behavior of animals.
3 Moreover, sense is cognizant only of singulars; for every sense power knows through individual species, since it receives the species of things in bodily organs. But the intellect is cognizant of universals, as experience proves. Therefore, intellect differs from sense.
Notes The “common sense” takes the input from the disparate senses and paints a picture, a unified whole, which the intellect considers. The intellect knows universals, which cannot be sensed. As the next paragraph emphasizes.
4 Then, too, sense-cognition is limited to corporeal things. This is clear from the fact that sensible qualities, which are the proper objects of the senses, exist only in such things; and without them the senses know nothing. On the other hand, the intellect knows incorporeal things, such as wisdom, truth, and the relations of things. Therefore, intellect and sense are not the same.
5 Likewise, a sense knows neither itself nor its operation; for instance, sight neither sees itself nor sees that it sees. This self-reflexive power belongs to a higher faculty, as is proved in the De anima [III, 2]. But the intellect knows itself, and knows that it knows. Therefore, intellect and sense are not the same.
6 Sense, furthermore, is corrupted by excess in the sensible object. But intellect is not corrupted by the exceedingly high rank of an intelligible object; for, indeed, he who understands greater things is more able afterwards to understand lesser things. The sensitive power therefore differs from the intellective.
Notes Bright lights overwhelm, blinding insights do not (and now you understand the metaphor).
Chapter 67 Against those who hold that the possible intellect is the imagination. (alternate translation) We’re still using the alternate translation this week.
1 The opinion of those who asserted that the possible intellect is not distinct from the imagination was akin to the notion just discussed. And that opinion is evidently false.
2 For imagination is present in non-human animals as well as in man. This is indicated by the fact that in the absence of sensible things, such animals shun or seek them; which would not be the case unless they retained an imaginative apprehension of them. But non-human animals are devoid of intellect, since no work of intellect is evident in them. Therefore imagination and intellect are not the same.
Notes A mental picture produced by sensation of a possibility is not apprehension of a universal. As the next paragraph emphasizes. If you’re stuck for something incorporeal to use as an example, pick a number, any number.
3 Moreover, imagination has to do with bodily and singular things only; as is said in the De anima [3], imagination is a movement caused by actual sensation. The intellect, however, grasps objects universal and incorporeal. Therefore, the possible intellect is not the imagination.
4 Again, it is impossible for the same thing to be mover and moved. But the phantasms move the possible intellect as sensibles move the senses, as Aristotle says in De anima III [7]. Therefore, the possible intellect cannot be the same as the imagination.
Notes Phantasm, the mental image provided by distillation of the senses using the bodily apparatus. (Wow.)
5 And again. It is proved in De anima III [4] that the intellect is not the act of any part of the body. Now the imagination has a determinate bodily organ. Therefore, the imagination is not the same as the possible intellect.
6 So it is that we read in the Book of Job (35:11): “Who teaches us more than the beasts of the earth, and instructs us more than the fowls of the air.” And by this we are given to understand that man is possessed of a power of knowledge superior to sense and imagination, which are shared by the other animals.
What do you think about a guy who goes around writing things like this? Recalling, as you read, Jesus and the Lord God and the Holy Ghost are one:
Now we may glimpse the vast sweep of the condemnation of sodomy as leveled in Leviticus. “You shall not lie down with a man as with a woman,” says the Lord, for “it is to’evah,” typically translated as “an abomination.” But the word and its near relations in Hebrew suggest three things: going badly astray, that is, wandering to your confusion and ruin; repugnant filth, as of excrement and loathsome disease; and idol-worship, with its combination of the bizarre and the disgusting—think of Moloch and the charred little ones. To lie with a man as with a woman is to engage in unreality, un-creation; it is like fouling yourself with excrement, or like eating filth not meant for food; it is like falling in adoration of the idols that are tohu w’vohu, waste and void, like the emptiness of the world before God said, “Let there be light.”
The reminder about the Trinity was to show that Jesus, who is one with the Lord God, said some fairly, well, merciless things to say about certain acts, things the modern world would rather not hear about. Rather, the world doesn’t mind hearing things like this, or like anything, so much as has an intense dislike knowing there are actual people who believe these words.
Anthony Esolen, the author of the passage, believes the words. Diverse words, too, given their variance with the spirit of the day. And we all love diversity, don’t we?
Well, there’s diversity, as in Webster’s “multiplicity of difference; multiformity; variance”, fine things all (up to a point). And then there’s Diversity, as in rigid strict mandatory unbending ruthless quota-bearing uniformity. Capital-D Diversity is the not so much the idol of the age, but a blunt instrument of political power.
Esolen penned an essay on the Big D, which was given by its editors the appropriate title “My College Succumbed to the Totalitarian Diversity Cult“. His college is the ostensibly Catholic Providence College. In it he asked:
Is not diversity as it is now preached a solvent for any culture? That is, supposing that the people of a tribe in the interior of Brazil are compelled to accept cultural diversity for its own sake, rather than merely adopting and adapting this or that beneficent feature of another culture (something that people have always done), will that not mean that their own culture must eventually vanish, or be reduced to the superficialities of food and dress?
Is not diversity, as currently promoted, at odds with the foundational diversity built into the nature of the human race, the diversity of male and female, to be resolved most dynamically and creatively in the union of man and woman in marriage?
These were the beginnings of the questions and intelligent comments. And the start of his troubles.
You know what happened. A brace of brats filled their diapers and went bawling to the Providence President who, befitting his sober and stately and priestly office, promptly capitulated. Perhaps he couldn’t deal with the stink. Or perhaps he felt it his duty to toss, albeit softly, Esolen to the baying mob, this being the default and reflexive action of college presidents everywhere since the Sixties.
That’s the briefest summary. Read about the full hilarity here and here. Don’t miss the petulance of the Faculty, which is circulating the petition and document “Breaking the Silence“. It begins:
As PC Faculty, we pledge to break the silence around systemic racism and discrimination on Providence College’s campus. While we vigorously support free expression, recent publications on the part of PC faculty have involved racist, xenophobic, misogynist, homophobic, and religiously chauvinist statements.
The poor bloated pink-faced Faculty! I can’t speak for the reader, but my awareness has just been raised. No Siberian gulag came close to the horrors present at Providence. It’s worse than you might have thought. In Siberia, Stalin & Co. at least let dissidents forage for scraps of pine bark to eat. At Providence, that pestilential hotbed of racism and X-phobia, people have to pay some \$60,000 per annum for the privilege to be abused—and that figure includes room and board. I wonder if they have meatless Fridays. Skip it.
“All Faculty, Briggs? Come now. There must be some Traditionalist Reactionaries remaining who haven’t self-emasculated and who defend poor Esolen.”
We know of one. A female creature (as Mike Royko would have said) by the name of Holly Taylor Coolman, who teaches theology. She gave an interview at Crux where she wondered aloud about the diminishing Catholic identify at Providence. About the row she said, “Our campus has seen increasing frustrations in the last few years, and I came to feel that a big blow-up was almost inevitable.” Asked if the disruption was between secularists and Catholics, she answered:
Not exactly. Another group immediately involved here are some of the people who tend to fall on the margins in our community-and also those supporting them. They have serious concerns about systemic forms of exclusion. (And here, too, are a number of concerns that I myself share.)
They can see, for example, that Providence College’s 100-year history includes almost nothing of the African-American experience, or of Hispanic culture and tradition. In the last few years, the college has made a concerted effort to recruit more students, faculty, and staff from underrepresented groups, but frankly, it hasn’t always succeeded in offering needed support once they arrive.
Coolwoman is right. A diligent search reveals no history of Hottentots, Maori, or Samoans at Providence. Damn few Finns and Latvians, neither. About horse lovers and other underrepresented groups, a count cannot be easily discovered, but it’s good money these folks were shunned.
But then, you didn’t see a lot of Irish learning to click in the Kalahari, an observed and undeniable fact which can only be, so Diversity theory assures, the result of systematic exclusion and racism. Probably sexism, homophobia, and every other manner of intellectual vice, too.
After all, should not a black man amble up to Providence to learn all about being black in the current climate instead of reading Shakespeare, Newton, Newman, Thucydides, Dante (translated by Esolen), Euclid, et cetera? Is not college about finding others who share your identify and reveling in that identity, however limited in place or time it is, making others acutely aware of that identity, and making that identity the sole basis and purpose of your life, and not about learning the best that was thought and said?
Esolen doesn’t think so. That fine gentleman is aware that you don’t need to go to college to know what you already know or believe what you already believe and can’t be talked out of. If all you care about is “social justice”, skip college, go right into “activism”, or stare at your identity in a reflection at the lake like that Greek fellow, and save yourself, or your parents, a bundle.
Problem is, Esolen is surrounded by social justice warriors who, though they lack in intellect, fortitude, and cleverness, are great in number and abound in indefatigable self-righteousness. The strain of defending Truth and Commonsense might be getting to him. This we gather from his recent essay in The Catholic Thing. Pardon the extensive quote.
Because of recent events at the school where I teach, Providence College, I have come to see that the winning side of the so-called culture wars has no interest in rational or equable conversation about the neuralgic issues of our time. I use the word interest advisedly. They have nothing to gain by it.
We can ask, till we are exhausted from asking, what they mean by “marriage,” if the thing is not rooted in the fundamental biology of the human race, and exactly what justifies any boundaries at all wherewith they suppose they can limit the definition. If man and man, why not man and woman and woman?…
It won’t matter. The aim was never rational coherence, or even a concern for the common good. The aim was power: to get what they wanted, to keep it, and to crush those who would question their right to it.
So they have the power now, power gained not by argument, whereof there has been very little, but by a combination of political force, mass media sentimentalism, public lassitude, and an anti-culture of licentiousness and the neglect of children.
Why bother to argue?…
Why indeed? Arguing only exposes you as an enemy of The People, and targets you as problem that must be dealt with—as Esolen learned with his Diversity essay. Argument and question are taken as political attacks, which, in a sense, they are. The Faculty who have surrendered to the World “are by nature no better and no worse than anyone else. It’s just that they have, whether they acknowledge it or not, exchanged the God of heaven for a god of prestige and power. Politics is the god.”
As long as you possess the “right” politics, you are like the pagan who has secured divine favor by the “right” sacrificial rituals.
You may then do as you please. You may, for example, go out of your way to ruin reputations and careers and turn families upside down; all justified, all for the good of the “cause.”
It is obvious our friend needs cheering up. What practical things can we do to help?
Pray, for one. It is still and always the best weapon. If you’re up for more earthly activities, you can buy his books. His Dante is excellent, and his The Politically Incorrect Guide to Western Civilization is great fun. There are many more.
This is posting here a day late, because I had already promised the Vox Day post yesterday. Yet there is still some fun to be had at the Stream: Is the Trump/Russia ‘Dossier’ the Fake News of the Decade?
Fake news or conspiracy theory? Or the most epic troll since Dan Rather was conned into accepting forged documents about George Bush? Or a hilarious amalgam of all three?
All elements of this story are as yet unknown, but what is unfolding has the makings of historical high comedy. Here’s a rundown.
Buzzfeed, a website whose specialty is celebrity tittle-tattle, asinine quizzes such as “Which ‘Pixie Hollow Fairy Are You?“, and get-skinny-quick-by-petting-cats articles, published a document, which they gave the graduated title of dossier, which purported to show how Russia, under the devious and genius scheming of Vladimir Putin, had been grooming and bribing Donald Trump for many years, and blackmailing him by threatening to reveal perverted sexual practices, so that Trump would be induced to enter the US Presidential election, win it by secret dirt supplied by Russian intelligence agents, and so place the once United States of America under the control of a foreign government.
Yes, really.
Go there to read the rest.
The last lines of the piece: “The story isn’t over. The news on why Steele wrote the document, if he wrote it, and why, including who paid him for it, is bound to generate even more fun.”
Assuming (it’s not a stretch) that Steele wrote the document, how did he arrive at the contents? Did he make it up whole cloth, filling in bits with suppositions he knew could never be checked because many of the events took place long ago in Russia? Was he given dirt by Republican NeverTrumpers and assured by them it was true? Was Steele himself duped by actual Russians who couldn’t believe their luck?
Did Steele, or whomever, do it to scam McCain and other NeverTrumpers? Would a real live MI-6 agent really think fictional scribblings would fool real live CIA and FBI agents? Maybe he thought the document would never be made public, because the Republicans who asked for it during the election would never release it, since it would paint the GOP in a bad light. But now that it has become public, Steele has scarpered, to use a Britishism. Is he now in Moscow sharing a flat with Edward Snowden?
If Steele was duped, are we going to hear of tales of a wigged Debbie Wasserman Schultz faking a Russian accent whispering in Steele’s ear? “Listen very carefully…I shall say this only once! | 2017-01-21 08:30:08 | {"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": 0, "img_math": 10, "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": 0.49073952436447144, "perplexity": 2992.5472343715655}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281001.53/warc/CC-MAIN-20170116095121-00550-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://leanprover-community.github.io/mathlib_docs/group_theory/abelianization.html | # mathlibdocumentation
group_theory.abelianization
# The abelianization of a group #
This file defines the commutator and the abelianization of a group. It furthermore prepares for the result that the abelianization is left adjoint to the forgetful functor from abelian groups to groups, which can be found in algebra/category/Group/adjunctions.
## Main definitions #
• commutator: defines the commutator of a group G as a subgroup of G.
• abelianization: defines the abelianization of a group G as the quotient of a group by its commutator subgroup.
• abelianization.map: lifts a group homomorphism to a homomorphism between the abelianizations
• mul_equiv.abelianization_congr: Equivalent groups have equivalent abelianizations
@[protected, instance]
def commutator.normal (G : Type u) [group G] :
def commutator (G : Type u) [group G] :
The commutator subgroup of a group G is the normal subgroup generated by the commutators [p,q]=p*q*p⁻¹*q⁻¹.
Equations
def abelianization (G : Type u) [group G] :
Type u
The abelianization of G is the quotient of G by its commutator subgroup.
Equations
@[protected, instance]
def abelianization.comm_group (G : Type u) [group G] :
Equations
@[protected, instance]
def abelianization.inhabited (G : Type u) [group G] :
Equations
@[protected, instance]
def abelianization.fintype (G : Type u) [group G] [fintype G] [decidable_pred (λ (_x : G), _x commutator G)] :
Equations
def abelianization.of {G : Type u} [group G] :
of is the canonical projection from G to its abelianization.
Equations
@[simp]
theorem abelianization.mk_eq_of {G : Type u} [group G] (a : G) :
quot.mk setoid.r a =
theorem abelianization.commutator_subset_ker {G : Type u} [group G] {A : Type v} [comm_group A] (f : G →* A) :
f.ker
def abelianization.lift {G : Type u} [group G] {A : Type v} [comm_group A] :
(G →* A) →* A)
If f : G → A is a group homomorphism to an abelian group, then lift f is the unique map from the abelianization of a G to A that factors through f.
Equations
@[simp]
theorem abelianization.lift.of {G : Type u} [group G] {A : Type v} [comm_group A] (f : G →* A) (x : G) :
= f x
theorem abelianization.lift.unique {G : Type u} [group G] {A : Type v} [comm_group A] (f : G →* A) (φ : →* A) (hφ : ∀ (x : G), φ = f x) {x : abelianization G} :
φ x =
@[simp]
theorem abelianization.lift_of {G : Type u} [group G] :
@[ext]
theorem abelianization.hom_ext {G : Type u} [group G] {A : Type v} [monoid A] (φ ψ : →* A) (h : = ) :
φ = ψ
def abelianization.map {G : Type u} [group G] {H : Type v} [group H] (f : G →* H) :
The map operation of the abelianization functor
Equations
@[simp]
theorem abelianization.map_of {G : Type u} [group G] {H : Type v} [group H] (f : G →* H) (x : G) :
@[simp]
theorem abelianization.map_id {G : Type u} [group G] :
@[simp]
theorem abelianization.map_comp {G : Type u} [group G] {H : Type v} [group H] (f : G →* H) {I : Type w} [group I] (g : H →* I) :
@[simp]
theorem abelianization.map_map_apply {G : Type u} [group G] {H : Type v} [group H] (f : G →* H) {I : Type w} [group I] {g : H →* I} {x : abelianization G} :
def mul_equiv.abelianization_congr {G : Type u} [group G] {H : Type v} [group H] (e : G ≃* H) :
Equivalent groups have equivalent abelianizations
Equations
@[simp]
theorem abelianization_congr_of {G : Type u} [group G] {H : Type v} [group H] (e : G ≃* H) (x : G) :
@[simp]
theorem abelianization_congr_refl {G : Type u} [group G] :
@[simp]
theorem abelianization_congr_symm {G : Type u} [group G] {H : Type v} [group H] (e : G ≃* H) :
@[simp]
theorem abelianization_congr_trans {G : Type u} [group G] {H : Type v} [group H] (e : G ≃* H) {I : Type v} [group I] (e₂ : H ≃* I) :
@[simp]
theorem abelianization.equiv_of_comm_symm_apply {H : Type u_1} [comm_group H] (ᾰ : abelianization H) :
def abelianization.equiv_of_comm {H : Type u_1} [comm_group H] :
An Abelian group is equivalent to its own abelianization.
Equations
@[simp]
theorem abelianization.equiv_of_comm_apply {H : Type u_1} [comm_group H] (ᾰ : H) : | 2022-01-25 19:36:58 | {"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": 0.5780400037765503, "perplexity": 795.8760603614559}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304872.21/warc/CC-MAIN-20220125190255-20220125220255-00103.warc.gz"} |
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# Main problems you encounter as a structural engineer48
## Main problems you encounter as a structural engineer
(OP)
Top 3 problems you encounter and have to overcome working in structural engineering.... GO!
### RE: Main problems you encounter as a structural engineer
Small contractors doing something they’ve never seen before.. “we’re not trying to hold up the empire state building...” yawn...
### RE: Main problems you encounter as a structural engineer
1) everything is too big and to heavy and to expensive.....
2) structural engineers just cause problems in the team
3) always too late
### RE: Main problems you encounter as a structural engineer
2
1) No one wants to think for themselves. Not that many years ago, a contractor would think through an issue and have a solution ready before they submit an RFI, today, they just say there's a problem, fix it. It's much easier to have an idea of what they want to do. The good contractors still do this, but they're becoming fewer and further between.
2) Timelines for completing drawings has shrunk to the point that almost zero coordination is done prior to issuance by the Prime Consultant. This leads to many more inconsistencies in the drawings, and therefore more RFIs, and you can then refer to my number 1 for my frustrations there.
3) Proper give and take on contract changes. Also in the not to distant past, if there was a minor change to something that would cost less than say $1000, no one requested a change to the contract amount. And in return, when pricing for larger changes was submitted we wouldn't be overly particular and let them get their money back there. More and more often I am getting proposed change notice pricing for less than$500. When a project goes that route, we review all of their costs with a fine toothed comb and bring them down at every chance possible.
### RE: Main problems you encounter as a structural engineer
Jayrod12, per your #1, don't you just love it when a contractor issues an RFI, you answer it and then they come back with, "Well can't we just do it this way instead?". That kills me. Tell me you have a preference and I'll do my best to make that work!
### RE: Main problems you encounter as a structural engineer
Green SE, the biggest issue you'll face is people questioning what you did and why you did it. Many times these people have no clue why we do what we do.
### RE: Main problems you encounter as a structural engineer
#### Quote (Rabbit12)
Tell me you have a preference and I'll do my best to make that work!
I have resorted to having this conversation with the GC on all my projects at the kickoff meeting. Even then, we rarely get a proposed solution, or when we do it's absolutely ludicrous. I've given an outright no a couple of times to a proposal on to get the response "Yeah we knew you weren't going to go for that". Then why FFS would you propose it.
### RE: Main problems you encounter as a structural engineer
From the perspective of a connection engineer working for a heavy structural steel fabricator in the United States:
1) Coordinating with teams of detailers on the other side of the world, and then reviewing their god-awful shop drawing submittals.
2) The damn sales department bidding jobs with unreasonable time constraints.
3) COMMUNICATION with Engineers of Record. The industry is filled with engineers who have trouble with basic communication skills. This increasingly applies to the large firms we work with who can offer H1b Visas to engineers with foreign credentials.
### RE: Main problems you encounter as a structural engineer
2
1. Hey can you stamp these plans....smh
2. I want to knock down all these walls and put windows everywhere.... but I dont want any steel or new concrete.
3. I need plans by Friday....Wait how much?
### RE: Main problems you encounter as a structural engineer
1. Gravity
2. Extreme feast and famine workload cycles
3. Scope creep
### RE: Main problems you encounter as a structural engineer
1. Lack of communication
2. Engineers living in "model world" (having blind faith in the model and not paying attention to how structures go together - the details)
3. Problems working with clients (lack of information, changes after design, unrealistic schedules, expectation that redesign (due cost estimates being over budget) should be done for free.
### RE: Main problems you encounter as a structural engineer
This thread is depressing. I thought I was the only one going through this and there must be hope somewhere, sometime...guess not...
### RE: Main problems you encounter as a structural engineer
Sorry, phamENG. Welcome to the world of being legally a profession, but priced as a commodity.
When friends tell me their kids want to be engineers and ask me about my job, I always have a hard time not overemphasizing the negatives.
As for the OP:
1) Schedule
2) Schedule
3) Schedule
All of the other items everyone else mentioned apply, but the brutal schedules on most of my projects are what drive the frustration. It's usually a cycle of no info, no info, no info, no info, "oh, here's that info you need (that was promised 2 weeks ago), can I get check sets today?"
### RE: Main problems you encounter as a structural engineer
Jayrod12, per your #1, some contractors today use RFI's as a weapon. They scour the drawings at the beginning of the project ask every imaginable question to set the tone that, 1. the drawings are incomplete or unclear, 2. we need answers immediately or else you are holding up the job, 3. any answer you give will result in a change order. I call these "assault RFI's". In fairness to contractors, sometimes the drawings they see are incomplete and ambiguous. At the firm where I work we strive to issue complete, correct and buildable designs and contract documents. It's a challenge.
### RE: Main problems you encounter as a structural engineer
D) All of the above
You can have it fast, right or cheap... pick two
Analog spoken here...
### RE: Main problems you encounter as a structural engineer
Kidding aside, I had often wondered if these were symptoms of the market segment we were operating in. The firm where I used to work full time (I recently went to work directly for a former client to get away from most of these issues!) serviced mostly small to medium clients. Everything from sagging floor joists to mid-rise buildings, light industrial, commercial, etc. But none of it was what I would consider "high end" work (you know - runner up for maybe being considered for the being listed on the last page of an obscure trade publication). Most of the bigger jobs were with clients who shopped the engineers for the lowest possible fee and then negotiated the lowest bidder even lower, and the owners always seemed to pay more for "value engineering" after the fact than they were willing to pay for the original design.
The result always seemed to be a shoe-string budget and an unattainable timeline. If you pushed back on the timeline, somebody else was waiting to steal your shoe string.
As you move up the chain in size and complexity of the projects, does the quality of the client improve or get worse? Or is it all just relative?
### RE: Main problems you encounter as a structural engineer
phamENG. Short answer: No. It doesn't improve with the size and complexity of the projects. If anything, you run into the corporate scenario where some people you deal with have been promoted to their level of incompetence. I think it's called the Peter Principle. And god forbid you work on a nuclear construction site in the US. Never again.
### RE: Main problems you encounter as a structural engineer
2
1) Aggressive schedule driven projects
2) "Screen blindness"
3) Construction standards
The Eureka moment is that you (a structural engineer) never actually work for structural engineers. You work for people that attempt to know structural engineering by proxy: architects, managers, drafters, sellers, financiers, and builders.
### RE: Main problems you encounter as a structural engineer
4
Architects
Architects, and
Architects
Mike McCann, PE, SE (WA, HI)
### RE: Main problems you encounter as a structural engineer
Budget, schedule, decent drafting help, etc ,etc
### RE: Main problems you encounter as a structural engineer
What is "screen blindness?" I feel like I'm missing some euphemism.
### RE: Main problems you encounter as a structural engineer
@azcats:
Putting something on-screen and justifying it is correct.
- This could apply to a CAD drawing and not checking for readability or correctness.
- This could also apply to analysis where the result on-screen is trusted without confirmation using off-screen methods.
- This could also apply to the opposite: using off-screen methods as a lone verification without regards to on-screen results that could improve the design, or suggest that something may be missing.
### RE: Main problems you encounter as a structural engineer
"Green SE, the biggest issue you'll face is people questioning what you did and why you did it. Many times these people have no clue why we do what we do."
The more I get questioned, the more detailed my answers get. That'll teach 'em.
### RE: Main problems you encounter as a structural engineer
Project managers... “ok, thanks for the design... now we need to value engineer it!”
THERE’S NO FAT IN IT!!! ITS Fu£kING CORRECT!!!! I CAN’T REDUCE IT!!
### RE: Main problems you encounter as a structural engineer
I knew some EEs who took to specifically adding obvious flaws in their submittals to one program lead because if he did not see a flaw, he would start looking for anything else he could change just to feel like he contributed. Anyone turning in well thought out work got screwed over, sometimes by arbitrarily increasing performance requirements ("I know it meets what the customer wants, but if it's twice as good isn't that better?") Better to leave off a ground wire and poof - ego gratification for finding it.
### RE: Main problems you encounter as a structural engineer
More and more often I am getting proposed change notice pricing for less than $500. When a project goes that route, we review all of their costs with a fine toothed comb and bring them down at every chance possible. Chicken and egg. They know they'll get screwed down on everything so claim absolutely everything. ### RE: Main problems you encounter as a structural engineer Architect idjits that forget about the fact they need some structure to hold up their masterpiece. Services idjits putting holes in my structures right after I finished the design. CadMonkey Tracer type idjits stuffing up drawing my carefully crafted masterpieces. ### RE: Main problems you encounter as a structural engineer 3 Agent, my old guys were referred to as CadMonkeys, now I have one who calls himself a BIMpanzee.. ### RE: Main problems you encounter as a structural engineer This may be better placed in a new thread, but I can't help but draw a link between the issues being discussed here and some of the things being said in the collapse discussion about the Hard Rock. I know we're not all designing 18 story buildings in CBDs, but how can we leverage this into a positive change for our industry? The forum has been discussing a set of "Permit Drawings" that were signed and sealed by the EOR that appear very....abnormal? It's a problem I faced on most projects. The owner is in such a rush, they want to get the permit process started early - some would say to "reserve their place in line" so they didn't have to wait when the design was finished. To put a stop to this, the localities started rejecting anything that said "not for construction" or "for permit only" (why would they review something that isn't going to be built?). You'd think our good, upstanding, and ethical profession would respond by saying ok, sorry, we'll wait until we're done. Nope. Everyone just took the NFC stamps off and submitted them even though they knew they weren't done and couldn't be safely built. I nearly got into a shouting match with my boss over it on more than one occasion. We even sent off a "permit set" only to not hear back from the architect until they sent us angry emails and RFI's from the contractor who was halfway done with the building! I think it comes back to the schedule pressures we've been discussing. Are we going to push back as a profession because it could (we'll have to wait for the final report on the hotel to know for sure in that case) cause significant life safety risks? If so, how? ### RE: Main problems you encounter as a structural engineer 2 1.$x,000.00 fee while carrying $x,000,000.00 liability 2. Everyone thinks everyone else is stupid ("if you judge a fish by its ability to climb a tree, it will live its whole life believing that it is stupid.") 3. Not being aware of when you were/are being stupid Open Source Structural Applications: https://github.com/buddyd16/Structural-Engineering ### RE: Main problems you encounter as a structural engineer 2 1. Scarcity of Clients who are willing to pay our fees based on how big of a hassle they are to work for. Yeah, I don't want any interior columns. When they get the estimate, "Whoa it costs that much to have no interior columns?" Redesign it but I don't feel like I should pay any more design fee. You should have known that was going to be too expensive. 2. Architects, Architects who think they are also a Structural Engineer, Architects who think they are also a prudent Contractor. 3. Lack of the engineering profession promoting the value of a well-engineered design. Engineers seem to shy away from money conversations while Architects, Lawyers and Doctors have no problem financially promoting their profession. Was financial compensation ever discussed in college? Where I live, you HAVE to hire an Architect on several types of jobs but do not have to hire an engineer for the same job. They lobbied years ago and had the codes specify you had to hire them. Architects really stretch the " Engineering Incidental to Architecture" clause after that. ### RE: Main problems you encounter as a structural engineer #### Quote (Gandalf) I found it is the small everyday deeds of ordinary folk that keep the darkness at bay I do a lot of conceptual design* in a multidisciplinary environment. I'm the only structural engineer in the region. So I feel this pressure constantly. "The architects did their concept in two weeks, the civils took one. When will yours be ready?" "Do we really need geotech information?" "I already promised the client X for budget z". (*We issue conceptual drawings, and they often get built without further input. Hazard of the climate) So far, I've been able to push back in nearly every case. But it's tiring, and the cases where I don't push hard enough weigh on the conscience until I'm able to find some time to sharpen the pencil or clarify an assumption. #### Quote (Galatians 6:9) So let’s not get tired of doing what is good. At just the right time we will reap a harvest of blessing if we don’t give up. #### Quote (3DDave) I knew some EEs who took to specifically adding obvious flaws in their submittals to one program lead because if he did not see a flaw, he would start looking for anything else he could change just to feel like he contributed. Anyone turning in well thought out work got screwed over, sometimes by arbitrarily increasing performance requirements ("I know it meets what the customer wants, but if it's twice as good isn't that better?") Better to leave off a ground wire and poof - ego gratification for finding it. Done this (although less critical than a ground)... Gotta let the peer reviewers justify themselves. ---- just call me Lo. ### RE: Main problems you encounter as a structural engineer #### Quote (Celt83) Everyone thinks everyone else is stupid... Over the last few months, I've been reading up on all things artificial intelligence (AI). That, for two reasons: 1) Super interesting and; 2) I've long had the sense that I've fallen behind in my understanding of something that will eventually affect me and my family a great deal. You often hear that AI will never replace structural engineers because, in the end, structural engineering isn't actually that logical of an activity. You know, once you factor in all of the required interaction with other project participants etc. This will turn out to be a fallacy. When we imagine AI's in structural engineering, we seem to mostly imagine an updated copy of Enercalc trying to sit through a coordination meeting. It won't be anything like that. Instead, it's going to be a single AI that has spent fifteen minutes mastering all of: a) Structural engineering b) Mechanical engineering c) Electrical engineering d) Geotechnical engineering e) Building envelope engineering f) Quantity surveying (AI will utterly bury humans at this) g) Construction project management h) Construction and safety engineering i) Real estate economics j) Architecture All of these different groups that formerly struggled with the dysfunction of thinking that the others were stupid will be replaced by a single entity capable of a level of seamless, interdisciplinary coordination that will be light years beyond anything that unenhanced humans are capable of. The best bet for continuing human involvement in building construction would seem to be Architecture because there is an aesthetic component to that. Even that may turn out to be a false hope, however, given that there are already AI's composing music that human audiences prefer to that composed by contemporary human composers. Architecture that humans find pleasing, like music that humans find pleasing, like jokes that are funny, will probably turn out to simply be a function of proportion and repeating patterns complex enough to inspire but simple enough to be recognized. At the end of the day, we humans just aren't nearly as complex as we like to think that we are. In conclusion, I firmly believe that we are currently living through the last century of the structural engineer as far as significant human involvement is concerned. Were I a betting man, I'd wager that we'll probably be done by 2050. As this process unfolds, it will inevitably increase the commoditization of our profession which is the root source of all the other problems enumerated by others above. Looking at it from a glass half full perspective, it'll be a fascinating and exhilarating thing to watch all of this unfold. And, of course, an honor to be part of the last few generations of human structural engineer. We'll basically be at the top of humanity's structural engineering game, technologically speaking, just before the game itself comes to an end. ### RE: Main problems you encounter as a structural engineer That's a sobering thought. ### RE: Main problems you encounter as a structural engineer Building code complexities. Takes more and more time to determine wind and seismic loads. ### RE: Main problems you encounter as a structural engineer I've been working in the same office for 25 years (30+ years total), and the principle is close to retiring and closing up shop. Ten years from my own retirement, I'm giving serious consideration to doing something else to close it out. Nuff said. Analog spoken here... ### RE: Main problems you encounter as a structural engineer #### Quote (phamENG) That's a sobering thought. I'm just getting warmed up. My musings on the present will be much more depressing than my musings on the future. For what it's worth, I'm nowhere near as dejected about our industry as my comments here will suggest. Rather, I'm concerned for the well being of folks who are new to the profession because: 1) I feel that the root causes of frustration within our industry have more to do with higher lever, systemic issues than the day to day stuff. 2) Far more insidious, I think that senior folks in our industry semi-consciously hide the truth of the nature our industry from junior engineers in order to keep junior engineers motivated and contributing to the economic pyramid that is most structural engineering firms. I wish to remove some wool from some eyes as I wish someone had done for me when I was just getting into the game. ### RE: Main problems you encounter as a structural engineer #### Quote (phamENG) The forum has been discussing a set of "Permit Drawings" that were signed and sealed by the EOR that appear very....abnormal? It's a problem I faced on most projects. The owner is in such a rush, they want to get the permit process started early - some would say to "reserve their place in line" so they didn't have to wait when the design was finished. To put a stop to this, the localities started rejecting anything that said "not for construction" or "for permit only" (why would they review something that isn't going to be built?). You'd think our good, upstanding, and ethical profession would respond by saying ok, sorry, we'll wait until we're done. Nope. Everyone just took the NFC stamps off and submitted them even though they knew they weren't done and couldn't be safely built. I nearly got into a shouting match with my boss over it on more than one occasion. We even sent off a "permit set" only to not hear back from the architect until they sent us angry emails and RFI's from the contractor who was halfway done with the building! I think it comes back to the schedule pressures we've been discussing. Are we going to push back as a profession because it could (we'll have to wait for the final report on the hotel to know for sure in that case) cause significant life safety risks? If so, how? This is a lame excuse for the state of the Hard Rock drawing. Those plans do not have a complete gravity or lateral system, that was not caused by any rush. #### Quote (KootK) Were I a betting man, I'd wager that we'll probably be done by 2050. How much? Experts think AGI by 2060 and then it would have to be specialized and scaled down in both cost and power. The same three complaints every engineer has: 1. Engineers 2. Architects 3. Contractors ### RE: Main problems you encounter as a structural engineer 5 #### Quote (winelandv) Sorry, phamENG. Welcome to the world of being legally a profession, but priced as a commodity. This dovetails nicely into a theory of my own that I'd like to table. I think that we should be viewing structural engineering as a trade rather than as a profession. Of course, this all ties back to just how one defines trade vs profession. So I'll toss out my own definition: START KOOTK's DEFINITION OF A PROFESSION As humans toil away, I propose that they get paid for two things: 1) The effort/labor that they put into producing their product, on a product by product basis. 2) The requisite knowledge that a practitioner must posses in order to successfully product their product. A profession is work where compensation is dominated by knowledge rather than effort. A trade is work where compensation is dominated by effort rather than knowledge. Some applications of this definition. 3) Landscapers (my son last summer). 5% knowledge; 95% effort. Trade (or unskilled trade I suppose). Bodies functioning as machines. 4) The Plumber that fixes my dishwasher. 30% knowledge; 70% effort. Trade (skilled). 5) Surgeon that replaces my pacemaker. 95% knowledge; 5% effort. Profession. 6) Structural engineer?? I would say 30% knowledge; 70% effort. Trade (skilled). But wait? Didn't I go to school for six years to get my masters? Didn't I take a dozen arcane licensing exams to prove my worth? Yeah, you did. But remember that we're not talking about what you had to do to be able to legally practice structural engineering. Instead, we're talking about what your actually getting paid for when your client contracts for your services. I submit that we're mostly getting paid for effort. In a way, structural engineering is a particularly cruel form of a trade. Imagine if plumbers had to endure six years of post secondary and endless post graduation exams and professional development? END DEFINITION I believe that a telltale sigh of whether you're in a profession or a trade is how typical businesses in your field must be structured in order to be profitable. And I propose that it comes down to this: 7) If you're in a profession, you can probably earn a decent, grown up living either working autonomously or mostly autonomously with an organization. 8) If you're a trade, to make significant money, you'll often have to place yourself near the top of a pyramidal hierarchy whereby effort is expended by folks at the bottom who are truly practicing your profession so that you can skim off of the top. Some interesting examples: 9) Dentists. They're usually set up as autonomous / semi-autonomous and make a killing. Profession. 10) Accountants. Usually set up as pyramids. Trade. 11) Lawyers. Nuanced. They are often set up as pyramids because lawyers really want to clean up. But, then, the pyramid scheme is a law partner's way of making$750K, not $200K. A lawyer operating on their own can easily make &200K. Profession. 12) Structural Engineers. Usually set up as pyramids. Trade. And this is a solid indicator that the actual activity of structural design is not, in and of itself, a high value activity as far as society is concerned. Ergo structural engineering is a commodity and all of the issues with tight schedules and low fees ensue... ### RE: Main problems you encounter as a structural engineer sandman - I think you misunderstand me. I'm certainly not trying to excuse the state of those drawings (quite the opposite, in fact). When I refer to a rushed schedule, I'm referring to the insistence on owners and contractors to submit preliminary designs for permitting, so you end up with unstable "designs" that the engineer never intended to be built but there's a drawing out there with a seal on it implying that it can be. I have no idea if that's what happened with the Hard Rock, but it seems plausible. It's not excusable, and that's what I'm trying to point out. ### RE: Main problems you encounter as a structural engineer #### Quote (Sandman21) How much? Experts think AGI by 2060 and then it would have to be specialized and scaled down in both cost and power. All my money, a child, and a thumb. I think it's game over once the general AI is created as the general should go singularity and just partition itself off to handle the specialized. ### RE: Main problems you encounter as a structural engineer Not my top 3, but challenging nonetheless, 1. Keeping water out 2. Keeping water in 3. Fixes for when 1. or 2. don't happen ### RE: Main problems you encounter as a structural engineer 3 #### Quote (KootK) I feel that the root causes of frustration within our industry have more to do with higher lever, systemic issues than the day to day stuff. Yes, issues with schedules, fees, and quality are the day to day nuisances. But, then, why do these things bother me really? All that just falls under the umbrella of "work", right? For me, these things are bothersome because they put me at odds with my own integrity almost constantly. Since we're talking big threes: 1) If an alien landed on earth and read all of our codes and design guides, they would have one impression of what structural engineers should be doing in regard to detail and rigor in design. Then, if they observed what practicing structural engineers actually do, they'd be horribly disappointing and confused. We take shortcuts. And lots of them. In fact, this is one of the first difficult lessons that new structural engineers must learn in a hurry. For me, this discrepancy between what I feel that I should be doing and what I'm actually doing is a challenge to my integrity. I tell the world that I'm delivering one product in terms of rigor and safety and then I turn around and deliver something quite different. I'm lying to the world in this respect. 2) As pointed out above, we have to commit to very aggressive schedule in order to keep winning work. This inevitably leads to agreeing to unrealistic schedules that give little account to reasonable contingencies. Yet I agree to these schedules because I feel that I have to to survive. This is me knowingly committing to delivering something that I know that I often wont be able to deliver. This is me lying to my clients and fellow project participants. 3) It is the low paid efforts of junior engineers that make our business model go 'round. Since most structural engineers get into the game to satisfy their inner nerd, the only way to keep such engineers motivated is to perpetuate their misunderstanding that society places a high value on the activity that is structural design. As a senior structural engineer, I'm guilty of this on a near constant basis. You can't very well motivate a junior by telling them "the only way to make any money at this is to get out of design and into management or sales as fast as you can". Again, this is me lying... now to junior engineers. As structural engineers, we like to facetiously toss around the concept that we lose sleep over our work. You know, stuff falling down and crushing baby carriages etc. The truth is that none of that costs me any sleep. What does cost me sleep is my being constantly at odds with my own integrity as I've described. I think that a practicing structural engineer would actually be well served by some degree of sociopath in this respect. And, indeed, I know of some mild sociopaths that are wildly successful in structural engineering and make it look easy. ### RE: Main problems you encounter as a structural engineer KootK - eloquently put, and tough to argue with. I will try though, even if only with my own anecdotal evidence that you'll surely eviscerate. In any profession or trade, knowledge is worthless without effort. A law scholar could have the whole history of jurisprudence memorized, but if he can't stand in a courtroom and argue (applying his knowledge through effort), he's not going to be making that$200k. The more clients he can bring in, and the faster he can argue his cases in court, the more money he makes. Meanwhile, he has an army of paralegals and interns running around doing his research and preparing briefs. It's hierarchical, but with little to no upward mobility for those at the bottom (the paralegal doesn't have a JD).
For structural engineers, we come out of school with little to no experience. Most programs don't require an internship. My state has EITs - Engineers in Training. I think some of the other states have it right with EIs - Engineering Interns. That period between school and licensure where you go from useless to knowledgeable asset. Consider MDs - they go through 4 years of undergrad, 4 years of med school, and they come out to another 4 years of residency making less than I made at my first engineering job out of college. Granted, their salary growth curve goes up a lot faster than ours does, but now as a licensed engineer I enjoy much more than a living wage, and my family can live comfortably on my salary alone (that's becoming more rare these days).
Look at the hierarchy of a structural engineering firm (at least one that I'm used to): President of the firm makes the strategic decisions and handles business aspects of the firm's operation; principal engineers oversee design, set policy, and manage QC; Engineers run the projects - run analysis, direct support staff, manage production of final product (drawings, specs, reports, etc.). Then, below the engineer, is a group of individuals doing the leg work - EITs running calcs and learning to do the engineer's job, CAD staff drawing and compiling documents, etc. Maybe its just me, but this sounds a lot like what the Law office is doing.
I think the commodity pricing is a result of how we interact with the market. Yes, we're probably not perceived in the same light as other professions, but is that because we don't deserve to be, or because we suck at PR?
### RE: Main problems you encounter as a structural engineer
KootK,
I'm inclined to agree with your synopsis on profession vs. trade, except you haven't actually dealt with the licensure and liability (possibly criminal, as opposed to a warranty) that comes along with it. Aside from that, it may just be that there are now too many engineers for our sector to operate as professionals.
Not sure what my point is, but you did get me thinking - I'm going to place the blame in your lap. ;)
Concerning AI, just like with self-driving cars, who's picking up the liability/risk in the event of catastrophe ? If it's going to take over the industry, then I don't think the standard "this software is a tool" disclaimer is going to cut it. More food for thought - can't spend any more time on this post, my effort is needed on a design. :)
### RE: Main problems you encounter as a structural engineer
#### Quote (KootK)
All my money, a child, and a thumb. I think it's game over once the general AI is created as the general should go singularity and just partition itself off to handle the specialized.
### RE: Main problems you encounter as a structural engineer
2
Somewhat related to the post going off on the current tangent, this argument sort of blew my mind a while back when I came across it. With the comparison between engineering and the real estate profession, showing in a way how structural engineering has lost its way, and how other professions have solved the money problem.
We design a building once in its design life for 1% for 6 months work, real estate agent gets 4% every time they sell it over the design life of the building for a weeks worth of work.... Compelling argument for the race to the bottom in structural consulting.
### RE: Main problems you encounter as a structural engineer
You're getting 1%? We generally look at between .35 and .45% of the CONSTRUCTION cost. Recall that the real estate agent gets 6% of the sold price of the unit. This includes all land costs, design fee + permitting fees, profit across several layers of clients, etc... At a recent project the "sold" cost of the project was close to twice that of the construction cost. Off the immediate 1st time the building get's sold the real estate agents are making close to 24 times what the structural engineer made on the project.
### RE: Main problems you encounter as a structural engineer
1. too much time on eng-tips
2. get stuck fiddling with calcs that I want to do for fun but will have little impact on final cost
3. arrogance
That AI thing is pretty exciting to me. Of course Tekla - click on beam - click on column - voila connection is detailed - press print shop drawing thing seemed pretty exciting 15 years ago. The "BIM everyone coordinates in 3D" was pretty exciting until I actually participated in a 3D "coordination" session.
All in all I'm grateful. Being able to deal with smart functional people doesn't happen in most fields. Structural engineering is allergic to non-smart dysfunctional people. Although, my schadenfreude is looking forwarded to the postmodernist movement hitting the engineering field. Its a sickness, I know.
### RE: Main problems you encounter as a structural engineer
Agents make more $per building with much less risk but there are also alot more agents than structural engineers. They dont get a pay check every week and have to grind there arse off (on the weekends too) to make there 3% (6% is what the owner pays and the listing agent has to split it with the buyers agent) and then give their broker his cut. The grass isn't always greener on the other side believe me...however the level of knowledge needed to "get by" as an agent isn't even close to what is minimally asked by us as structural engineers. A good agent will sell your house quickly and for more than the "comparable value"; this is directly tied to the pockets of the person paying for the service. This is easy for owners to understand and compare results with. A good structural engineer will value engineer a project that will save money but how does the person paying the bill know your design truly saved him money? They would need a design from another engineer to compare to right? Who is paying our bills? The architect? Owner? Contractor? At the end of the day we are only worth the value we bring to the market....he who is closest to the money always has the upperhand. ### RE: Main problems you encounter as a structural engineer This thread has become very interesting. My take is unless your name is on the wall, you are a commodity. I made it to the level of project manager and/or principal engineer in corporate structures, but was downsized out when those two companies were bought by foreign competitors. Last time resulted in a 20% cut in pay going to work for a consulting engineering firm that serviced that industry. It took 3 jobs and 13 years to get back to my previous salary. I rarely had problems with schedules. Working extra hours to get there was not a problem, even when I was not totally compensated for the hours. Keeping up with changing technology was an issue. Started work with a slide rule and hand drafting. Worked up through calculators, CAD, and some limited computer access. Working on my MS through a distant learning program in 2001 and my professor told me that if I stayed with my hand-calcs I would have trouble getting through his steel design class. Forced me to become somewhat computer literate. Continued to learn through retirement in 2013. At that time 3D drafting was making advances in that company. I left just in time. gjc ### RE: Main problems you encounter as a structural engineer I like the architects, contractors, RFI's, schedules, and everything, but 1) I don't get paid enough 2) Structural engineers aren't paid enough to encourage talented young people to pursue a career in structural engineering 3) Engineering professors teach math, and their students do algebra instead of sketching a section with a free-body-diagram ### RE: Main problems you encounter as a structural engineer #### Quote (RPMG) Engineering professors teach math, and their students do algebra instead of sketching a section with a free-body-diagram Ha - my structural analysis professor actually said at the beginning of the course: "There is an engineering way to do this, and a mathematical way to do this. I will teach you.....the mathematical way." Thanks for nothing... How I ever managed to teach myself enough to get my first structural job I'm still not sure. ### RE: Main problems you encounter as a structural engineer #### Quote (Agent666) Just want to point out in regards to the structural fee compared to real estate agent fee...we're splitting up the cut of design fee with the architect and all other consultants. Google is telling me architects charge between 5%-15% of construction cost, which is at least more comparable to a real estate agent...but is even more hilarious because one real estate agent is making more in a week than multiple teams at multiple companies for a fraction of a fraction of the work. Sigh. I started this post to try and make myself feel better. Didn't work. Edit: And I guess thats not even applicable to residential stuff that doesn't require a whole team of disciplines. ### RE: Main problems you encounter as a structural engineer 2 Regarding engineering fees - We only have ourselves to blame. We race each other to the bottom and then complain we’re not getting paid enough. ### RE: Main problems you encounter as a structural engineer How are real estate agents still a thing in 2019? ### RE: Main problems you encounter as a structural engineer "Somewhat related to the post going off on the current tangent, this argument sort of blew my mind a while back when I came across it." When I was at Stanford that guy came to give a talk and was handing out free stuff like Halloween candy. He asked someone to volunteer to come up and sing katy perry with him. My classmate went up and came back with a free iPad. I didn't get one, all I got his book on modeling :( ### RE: Main problems you encounter as a structural engineer #### Quote (Tomfh) Regarding engineering fees - We only have ourselves to blame. We race each other to the bottom and then complain we’re not getting paid enough. My thoughts exactly. Seems like an opportunity for some good ole' collusion! ### RE: Main problems you encounter as a structural engineer 1) I don't think that our industry's problems are a result of our just sucking at business and self promotion. My impression is that we're at least as shrewd as many of our cohorts in other, more lucrative professions. I think this humble pie, good guy, money-tard self image that we have is a delusion that we tell ourselves because it's more palatable than the truth which is that there a serious, structural issue with our service market. 2) I don't think that our industry's problems are a result of our racing each other to the bottom in terms of fees or services. Those things are just a function of our existing within a free and competitive marketplace where competition is encouraged. What needs to change is the underlying nature of service market, not how individuals play the game within that market. 3) As much as I liked Ashraf's Christmas light coat, I didn't feel as though he actually said much. Talent follows money and we'd all like a 700% raise. Any hillbilly could have told us that in sweats and a wife beater. As for the real estate agent example, it's based on a spurious underlying assumption: that structural engineering is as valuable of a service as real estate agency from a client's perspective. If a client chooses a crappy structural engineer what is the likely consequence? Nuttin'. If a client chooses a crappy real estate agent what is the likely consequence? The loss of sacks and sacks of gold doubloons. And that's all that you need to know about structural engineers vs real estate agents. 4) I believe that the real problem with our industry is that shoddy structural engineering has few tangible consequences for clients and, therefore, good structural engineering has little real value to society. All of the other crap shakes out from that. I've copied some verbiage below from another thread where I elaborated on this in greater detail and proposed the only solution that I can think of. #### Quote (MIStructE) I believe if more buildings actually fell down we would be much richer men! #### Quote (KootK) While I'm sure that this was meant rather facetiously, I believe it to be an important part of this problem. In a statistical sense, the quality of structural engineering work truly does NOT have meaningful consequences. So why should clients pay for good work? I have exactly one idea for how this might get fixed without going to straight protectionism. It's based on my expectation that, baring frequent earthquakes, only a structural engineer can really parse out good structural engineering work from bad. So I'd like to see all jurisdictions legislate a mandatory, anonymous, 3rd party peer review for all structural works of any significance. Set the fees at 15% of the EOR fees or something. I feel that this would lead to several desirable outcomes: 1) Higher quality structural work. 2) More volume of structural work available. 3) Crap structural work would hold up permits etc and cause delays. At long last... consequences. Some of the seismic jurisdictions like California and New Zealand have already taken meaningful steps in this direction which I feel is great. ### RE: Main problems you encounter as a structural engineer #### Quote (Kootk) If a client chooses a crappy structural engineer what is the likely consequence? Nuttin'. If a client chooses a crappy real estate agent what is the likely consequence? The loss of sacks and sacks of gold doubloons. If that's true then the idea that we're worth something is just a delusion. I don't believe it myself. I think poor structural engineering is more costly than poor real estate agents. Property sells itself at whatever the market if offering. Agents come along for the ride, selling themselves to vendors by promising a bigger sack of gold than the next agent. ### RE: Main problems you encounter as a structural engineer #### Quote (Tomfh) #### Quote (KootK) If a client chooses a crappy structural engineer what is the likely consequence? Nuttin'. If that's true then the idea that we're worth something is just a delusion. True or not, it's a common perception. Most laypersons can't tell you what makes a good/bad engineer. And if you were to ask a contractor, and architect, and an engineer what makes a good/bad engineer, the answers would vary wildly. I think more would be in the realm of coordination and customer service than technical design prowess. I think the sentiment has varying degrees of 'truthiness'.... depends on the industry/application and the competence of the contractor. Then add in the fact that as structural engineers, much of our work is to prevent things that might happen down the line, or in certain combinations of events. Add to that human tendencies to not evaluate future or systemic risks well, optimism bias, etc. ---- just call me Lo. ### RE: Main problems you encounter as a structural engineer 2 1) Estimating a scope accurately only for Project Management to slash the hours and duration and still expect the same quantity and quality of work. 2) Project Management being amazed and upset when their slashed hours and duration are not met and the project 'grows' to the estimated hours and duration. 3) Being expected to work with preliminary information and still have the same productivity and not ever up-rev or change anything. Once had Vendor supplied loads grow by more than 600% and was expected to 'just make the existing design work'. ### RE: Main problems you encounter as a structural engineer 2 #### Quote (Lomarandil) True or not, it's a common perception. Most laypersons can't tell you what makes a good/bad engineer. And if you were to ask a contractor, and architect, and an engineer what makes a good/bad engineer, the answers would vary wildly. I think more would be in the realm of coordination and customer service than technical design prowess. Definitely on the coordination and customer service. Knowing the technical stuff is the minimum and expected. Being a 'good engineer' in the eyes of the client is all about making their life easier. Being easy to work with, knowing industry trends, being dependable, lowering risk, lowering costs (not just fees), helping solve their problems (most of which aren't technical in nature). The actual engineering part is a commodity. No one cares if you've got a fancier model or cleaner calcs or clearer drawings unless it leads to significant tangible benefits to the client. Challenges for me: 1) Unreasonable schedules, both too short and too long. Too short because it's hard to get the work done in time and do a good job. Too long because it's hard to stay in budget as architect makes a million changes. There's a sweet spot for each project that depends on size. 2) Finding people at our level talent-wise and ability-wise. We grow and groom them pretty well from fresh grads. Finding experienced people who can keep up with us is difficult, though. Engineers from big time firms often don't have the breadth of knowledge and experience we want. Engineers from smaller firms often don't have the depth of knowledge and experience we want. Recession being 6-10 years ago makes it difficult. Since no one in the industry was hiring at the time, people with 6-10 years of experience are unicorns right now. I'm not sure they actually exist in the wild. 3) Communication, both internal and external. I feel some individuals do a pretty good job, their projects almost always run smoothly and they correctly identify potential problems and work to mitigate or avoid them. Others don't talk to much of anyone and don't follow up on much of anything, then act surprised when there's a bunch of rework because engineer and client (or PM for internal issues) weren't in sync on scope, desires, and requirements. ### RE: Main problems you encounter as a structural engineer I predominantly work in civil consulting although can strongly relate to most of the posts above! Clients seem to not value the input of a good engineer anymore. ### RE: Main problems you encounter as a structural engineer Curious to know if the DOT (department of transportation) bridge people in the states experience the same issues? ### RE: Main problems you encounter as a structural engineer I was reading through the comments in the link provided by Agent666 and found one that I feel deserves repeating here. I'll not critique is as I might normally since the author is not here to defend himself. #### Quote (Nikola Jevtic) When automotive engineers (for example) do great job, the result of their work is IMMEDIATELY seen through driving dynamics of the car against competitors. Great work of structural engineers is seen and thereby recognized only when building, bridge or any structure survive major earthquake. The bare fact that structure is standing fully functional when there is no earthquake or other disasters is taken FOR GRANTED !!! Educating people through media of the level of fundamental end applied knowledge that is necessary for conducting structural analysis and design is one (less efficient) way of raising the public seance of appreciation for our profession. Things should be done in reverse. More practical way is to establish firm rules for engineering fees in relation to long term economic and community benefits associated with given structure. First, the problem MUST be generally recognized by majority of engineers - that is the starting point. Next, appropriate engineering committees can be formed with specific tasks for establishing rules for engineering fees. After adoption, these rules will serve like design codes. By amount of fees for our engineering work, the rest of community will start to form opinion about our value. ### RE: Main problems you encounter as a structural engineer As a structural engineer who worked for 36 years in the design engineering of buildings, power plants and process plants, I consider the following as the inherent and unavoidable hurdles for a civil/structural design engineer. 1. The sequence of engineering activities of plants is as follows: Process engineering Mechanical engineering/Electrical engineering/ Instrumentation engineering Architecture Civil/Structural engineering But site requires the civil/structural detailed drawings first to start the construction. There is a pressure from the client, contractor and the project manager to the structural engineer to issue the drawings. It is a common advice/suggestion by all the above parties to 'make suitable assumptions based on previous experience to cover the uncertainties'. People easily forget that each project is unique in itself due to the different layouts, soil conditions, wind and seismic zones etc. and pose unique challenges to the structural designer. As a result, the poor structural engineer is forced to proceed with the design with many assumptions, which invariably change by the time the drawings are made. Revisions, rework and delay in the schedule follow, for which he is made responsible. 2. while performing structural design, there is another inherent hurdle. The sequence of issue of drawings required at site is foundation first and then the superstructure, but the calculations have to start from the roof. 3. It is widely talked that the automation and the availability of software has made structural design faster. But, the fact is that the benefit of the time gained by using the software for lengthy calculations and iterations is not available to the structural designer but is being passed on to the client who unreasonably squeezes the schedule. The result is that the designer has the same struggle for time to carry out his activities. However, I learnt the following hard truth by going through many projects. If there is a delay in the engineering schedule due to ensuring the quality of the deliverables, the designer will be blamed during the period of the project execution and will be forgotten afterwards. But, if the quality of the deliverable is compromised for adhering to the schedule causing rework at site, the blame will haunt the designer throughout his career. Trilinga ### RE: Main problems you encounter as a structural engineer #### Quote (Kootk) I believe that the real problem with our industry is that shoddy structural engineering has few tangible consequences for clients Unfortunately true. Even after our recent Earthquakes here in NZ, where one consulting company had a structure they designed in the 1980's collapse and kill something like 115 people of the 165 total that died. They as a company then seemed to get so much new work out of the rebuild form existing clients (even getting paid to fix/strengthen their own structures that performed poorly) because they are known for creating cheap to build structures. The company was still lead by the same person who had overall design responsibility for the structure that collapsed. Royal commission into the two modern buildings that fully collapsed leading to the majority of the loss of life found a lot of things lacking in the original designs that contributed to the collapse, bad design, bullying the authorities into accepting their design despite oppositions being raised by diligent parties. Often cheap and robust don't go together. If one or two collapse it's still good odds for the uninformed client types to roll the dice on in the long term if they got a good deal on the design, and thats all some seem to appreciate at the time? A buildings a building to them, a well designed building is invariably going to be more expensive. So basically you can kill 115 people and get away with it, no consequences to the company or the individuals, in fact their business was booming following the earthquakes. This disgusts me to some degree and highlights how broken parts of the industry really are, what message does this send regarding practicing good engineering? The public eventually lose in the next big one, but the moneys already in someones pocket. ### RE: Main problems you encounter as a structural engineer Agent, Yes I was a bit surprised to hear he'd gone on thriving after that event. Odd too that he managed to dodge any real blame for it, and successfully transferred 99% of the blame to his underling. I've seen it a few times where superiors throw the subordinates under the bus when things go bad, even though they were just following the bosses design direction. All of a sudden the bosses design becomes the junior's design... ### RE: Main problems you encounter as a structural engineer Lots of good input in this thread. I'll throw in my two cents based on my experiences. My challenges: 1 - Architects. Especially architects who give their clients new ideas/visions at every meeting. And especially architects who think they are structural engineers. This creates a long and painful process of design and redesign. Discussions about change orders due to the constantly moving target never go quite as smoothly as one hopes. 2 - Questions that directly target my integrity. KootK hit this one of the head for me. This paired with the seemingly endless variety of questions targeted towards member sizes, reinforcing layouts, etc. can do one in in a hurry. 3 - Timeline crunches. We all know this one.$X,000 in X days for a \$X,000,000 project. This has to be my biggest complaint, even over the architect.
I have the privilege of being able to step away from the computer and work with my wife's business partner on small construction projects. He is a retired GC who does remodels and repairs. He has the ability to selectively choose who he works for and what he does. First question is always "Are you bidding this out to multiple people?" If they say yes we politely tell them we aren't interested and we walk away. Second question, if the first answer is a no, is "You can have things done, and you can have things done right. I only do things right and as a result we work by the hour without the ability to give you an overall cost, though it is likely to be in the low XX,000 dollar range." If we get a thumbs up to that then we get started. May sound like a fairy tale but we are booked out through 2020 just by word of mouth alone.
I cannot describe the joy associated with installing tile base trim, or creating a fireplace mantle out of a massive chunk of DF, when you know you have time on your side to make things ever so perfect. Because when you are done you really take a deep sense of pride in your work knowing that it is as best as it can be.
Unfortunately, my company and my work as an engineer rarely, if ever, has that same flexibility, so low-cost projects in tight timelines are more the norm.
### RE: Main problems you encounter as a structural engineer
Well this thread has been a real eye opener.
#### Quote (KootK)
I think that senior folks in our industry semi-consciously hide the truth of the nature our industry from junior engineers in order to keep junior engineers motivated and contributing to the economic pyramid that is most structural engineering firms.
If this didn't strike true
As a grad I was deeply proud/honoured/excited when I was given a large project (relative to our usual work in my firm) that I would be doing majority of the design for with minimal help from a senior engineer and told by my boss he thinks I'm ready to start working more independently and this would be great experience for me. Given it was my first job operating independently I struggled immensely to meet the same deadlines expected of a senior engineer. I wasn't given any concessions or leeway for being a grad and I thought my boss was pushing me for my own development and learning - which wasn't the case as I had to sacrifice time spent understanding a lot of the design concepts to get the job out quicker.
Once the job was all said and done I found out my boss quoted lower on the job as the client had recently given a job to another engineer and he wanted to ensure we kept getting work from him. Since the deadline on the job was unchanged and the fee was lower the only way to stay profitable on this was to take advantage of my lower hourly rate. Sure I got great experience, but definitely felt used rather than valued by the end of it.
Seems like the industry is on a crash course to the bottom line.
I guess, to my benefit, being a greeny means I can still consider a career change... Any advice??
### RE: Main problems you encounter as a structural engineer
#### Quote (Tomfh)
Quote (Kootk)
If a client chooses a crappy structural engineer what is the likely consequence? Nuttin'. If a client chooses a crappy real estate agent what is the likely consequence? The loss of sacks and sacks of gold doubloons.
If that's true then the idea that we're worth something is just a delusion.
I don't believe it myself. I think poor structural engineering is more costly than poor real estate agents. Property sells itself at whatever the market if offering. Agents come along for the ride, selling themselves to vendors by promising a bigger sack of gold than the next agent
Sadly, it's often perceived as even the opposite. From a developer's perspective, I can hire hire "Firm A+" that does an amazing job and has the best reputation in town, or "Firm D-", who is often a one-man show with no overhead and starving for work (sorry, I know it's unfair to classify all one-man shows as the "D-" here, but it's a common issue for us).
1. Firm A+ will probably take a little longer to do it, Firm D- tells me they can get it done quicker, which may or may not turn out to be true.
2. Firm A+ will also actually design things to code or check their details, so I'm going to pay even more for the correct hardware, connections, reinforcing in construction... Plus my buddy contractor who has "been doing it this way for 25 years" tell me all that crap's not really necessary and he must be right because even when Firm D- does my stuff, it still sails through plancheck in 99% of jurisdictions. If it was really dangerous, they'd catch it, right?
3. My architect loves Firm D- because he never tells them they can't do anything. He'll bend over backwards, ignorant of whatever code provisions Firm A+ is always throwing in my face and telling me we can't do the glass palaces my architect wants to create.
4. Firm A+ has been around the block a few times and has a tighter contract so if I want to change scope along the way, they'll charge me more. But Firm D- has a swiss-cheese contract that's basically a glorified handshake with a wink, so I can strong-arm him into bending over backwards to make me happy.
5. Firm D- is easy when it comes to construction... no matter what idiotic thing my contractor screws up, their fixes are easy/whatever the contractor recommends. Firm A+ tells me it's because they don't know what they're doing and didn't check the <insert engineering jargon I don't understand here> but nothing ever falls down so what do I care?
As a developer, I'm mostly in this to try to sell this building for a profit in a few years anyway so what more could I want? Firm D- is giving me a huge discount for a product that meets my needs more economically. I like the guys at Firm A+, they're top-notch professionals who really do seem like they know what they're doing... but why should I hire them?
### RE: Main problems you encounter as a structural engineer
From a developers point of view you’re probably right. Quality issues often don’t appear until further down the line, so doing it better to save other people money in the distant future doesn’t make much economic sense.
### RE: Main problems you encounter as a structural engineer
On the topic of following the money, the source of funds for various parts of the construction process plays a role as well.
Most banks won't release a red cent until you have a design for your building. That means for many instances, our design fees are coming out of the owner's ready money. The cost of the actual building, on the other hand, is being financed. So when option A is to hire the top notch engineer that will cost 180% of the competition for a building that will cost 70% of the competition with better communication, smoother coordination, and responsive construction period services, they'll go with the lower design cost and higher construction cost because they feel it less up front and, in the case of developers, they can pass on the additional cost to the new owner when they sell the building after paying nothing but interest on it for a year.
### RE: Main problems you encounter as a structural engineer
I have also encountered plan reviewers who have read “The Territorial Imperative” way too many times. I never have liked political solutions to engineering problems.
Mike McCann, PE, SE (WA, HI)
### RE: Main problems you encounter as a structural engineer
1. Lack of a structural base from architects. Specialization has brought too many problems. I think it would be better if we all where architects with an specialization (structural, mep, etc.)
2. Communication and coordination.
3. Lack of intuition, too much reliance on software, lack of knowledge on vernacular construction. This slows workflow.
### RE: Main problems you encounter as a structural engineer
I am a new engineer starting out in the industry and I am already on my 2nd job, the main problem that I am running into is that the amount of structural projects can have "lull" periods or periods of only project proposals, which means little to no work for the new engineer.
How hard is it to switch to a different CE/SE discipline, like transpo. or water resources?
### RE: Main problems you encounter as a structural engineer
As easy as finding an entry level position. That's the stage you're at, and don't be surprised if things are the same at the new place. At least for a while, your employer still needs to figure out your capabilities. You don't magically get handed entire projects to run with fully day one.
### RE: Main problems you encounter as a structural engineer
I understand, but giving me starter tasks or operations to test my abilities would give me something to do, instead of just sitting at the desk staring at a code book.
### RE: Main problems you encounter as a structural engineer
Sounds like you have a good idea for what to do that would be more productive than what you're currently doing. Have you asked your boss / colleagues if they can provide something like that instead? Maybe even ask if he remembers any tricky problems he had and get you to look at it and see if you get the same solution. Even if not, it should prompt so good problem solving practice and discussion on the possible resolutions.
I'd be pretty happy to support one of my graduates if they came to me bored and looking for something challenging to develop their experience.
### RE: Main problems you encounter as a structural engineer
In simple terms RandomTaskkk is saying show your desire to work and learn. It's not school anymore, all motivation has to come from within. Will you be given work, sure. But trust me when I tell you it looks WAY better if you come looking for it.
### RE: Main problems you encounter as a structural engineer
dhase:
there will always be some down time just the nature of contract work. Rather than just stare at the code book try developing an excel or Smath/mathcad calculation tool you'll learn the in's and out's of the calculation and likely end up with a tool that will increase efficiency when the work does come back flowing in.
Open Source Structural Applications: https://github.com/buddyd16/Structural-Engineering
### RE: Main problems you encounter as a structural engineer
dbase:
I would try to determine what discipline I prefer being in rather than where can I find a project. If you like transportation or hydraulics more than structures attempt the move for that reason, not lack of activity. A new hire out of college has the same problem in all the disciplines.
I agree with other comments, ASK for something to do. Show motivation. For example, ask what type of structures you company tends to work on. Let's say they tell you they do a lot of concrete circular clarifiers or steel tanks and hoppers. Neither of those tend to be taught in college, so there is a good place to start learning. Get up to speed on them is not a bad sign of motivation. You will find college did not teach you everything you need to know but it did give you the tools to learn on your own with guidance from your mentor. If you do not have a mentor in the company, get a good one.
When you say "on my 2nd job", do you mean 2nd employer or 2nd project?
### RE: Main problems you encounter as a structural engineer
Celt83 has some good advice: don't just read a code book, make something out of it. A spreadsheet or calculation tool is a perfect artifact of learning, and if you show that initiative (in my experience) it will tend to impress the people more willing to have you tag along on their projects. I wish I did more of this in my early career because it could have really sped up my development.
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Register now while it's still free! | 2020-05-26 12:05:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2502509355545044, "perplexity": 2277.866031747179}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347390758.21/warc/CC-MAIN-20200526112939-20200526142939-00566.warc.gz"} |
http://hal.in2p3.fr/in2p3-00771595 | # Event shapes and azimuthal correlations in Z + jets events in pp collisions at sqrt(s) =7 TeV
Abstract : Measurements of event shapes and azimuthal correlations are presented for events where a Z boson is produced in association with jets in proton-proton collisions. The data collected with the CMS detector at the CERN LHC at sqrt(s) = 7 TeV correspond to an integrated luminosity of 5.0 inverse femtobarns. The analysis provides a test of predictions from perturbative QCD for a process that represents a substantial background to many physics channels. Results are presented as a function of jet multiplicity, for inclusive Z boson production and for Z bosons with transverse momenta greater than 150 GeV, and compared to predictions from Monte Carlo event generators that include leading-order multiparton matrix-element (with up to four hard partons in the final state) and next-to-leading-order simulations of Z + 1-jet events. The experimental results are corrected for detector effects, and can be compared directly with other QCD models.
Document type :
Journal articles
http://hal.in2p3.fr/in2p3-00771595
Contributor : Sylvie Flores Connect in order to contact the contributor
Submitted on : Wednesday, January 9, 2013 - 8:26:05 AM
Last modification on : Monday, December 13, 2021 - 9:15:19 AM
### Citation
S. Chatrchyan, M. Besançon, S. Choudhury, M. Dejardin, D. Denegri, et al.. Event shapes and azimuthal correlations in Z + jets events in pp collisions at sqrt(s) =7 TeV. Physics Letters B, Elsevier, 2013, 722, pp.238-261. ⟨10.1016/j.physletb.2013.04.025⟩. ⟨in2p3-00771595⟩
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Les métriques sont temporairement indisponibles | 2022-01-20 07:34:59 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8299877047538757, "perplexity": 3807.6571032980573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301730.31/warc/CC-MAIN-20220120065949-20220120095949-00454.warc.gz"} |
http://www.fewbutripe.com/swift/math/2015/01/06/proof-in-functions.html | Swift’s generic functions allow us to explore a beautiful idea that straddles the line between mathematics and computer science. If you write down and implement a function using only generic data types, there is a corresponding mathematical theorem that you have proven true. There are a lot of pieces to that statement, but by the end of this short article you will understand what that means, and we will have constructed a computer proof of De Morgan’s law.
# Generic Functions
Let’s start with some exercises to prepare our brains for this kind of thinking. If someone handed you the following function declaration, which doesn’t currently compile, and asked you to fill it out so that it compiles, could you?
func f <A> (x: A) -> A {
???
}
It’s a function that takes an x in some type A (can be any type) and needs to return something in A. We have absolutely no knowledge of A. No way of constructing a value in that type. For example, we can’t even do something like A() to construct a value, for we have no way of knowing if A has an initializer of that form. Even worse, there’s a chance that A cannot be instantiated, i.e. A has no values! For example, an enum with no cases cannot be instantiated:
enum Empty {
// no cases!
}
This type is valid and compiles just fine, but no instance of it can ever be created. Kind of bizarre, but it will be useful later. Some languages call this type Bottom (⊥).
So, back to that function f. How can we implement it so that the compiler says everything is A-Ok? Well, we really have no choice but to just return x, i.e. it’s the identity function:
func f <A> (x: A) -> A {
return x
}
Not only does this implementation appease the compiler, but it is the only implementation we could possibly provide. There is nothing else that could go in the body of the function. You might even ask yourself… then why isn’t the compiler smart enough to write it for me?! More on this later.
Let’s try to implement another generic function. Take this one:
func f <A, B> (x: A, y: B) -> A {
???
}
This involves two generic parameters. It’s a function taking values in A and B and returning something in A. After completing the previous function this probably seems obvious. Without knowing anything about A or B we really have no choice but to return x again:
func f <A, B> (x: A, y: B) -> A {
return x
}
Let’s try something a little more difficult. How might we implement the following generic function?
func f <A, B> (x: A, g: (A) -> B) -> B {
???
}
It takes a value in A and a function from A to B and needs to produce something in B. We should notice that two types match up quite nicely: we have a value in A and a function that accepts things in A. When types align like that it’s probably a good idea to just compose them. In fact, the compiler likes that quite a bit:
func f <A, B> (x: A, g: (A) -> B) -> B {
return g(x)
}
This all seems so simple, but take a moment to reflect on how strange it is that the compiler is essentially holding our hand in writing these functions. It is guiding us on what to write in order for the function to type check.
Now that we are getting the hang of this we’ll breeze through more of these.
func f <A, B, C> (g: @escaping (A) -> B,
h: @escaping (B) -> C) -> (A) -> C {
return { a in
return h(g(a))
}
}
This is a function which takes two functions, one from A to B and the other from B to C, and returns a new function from A to C. The only thing we can do is simply compose those two functions. That is, return a new function that first applies g and then applies h.
We’re going to continue exploring this world of implementing generic functions, but we need to introduce a new type. It’s a very simple enum with a suggestive name:
enum Or <A, B> {
case left(A)
case right(B)
}
The Or<A, B> type has two cases, a left and a right, each with associated values from A and B. A value of this type is really either holding a value of type A or of type B. It should be noted that this type is in some sense “dual” to the tuple type (A, B). A value of type (A, B) is really holding a value of type A and of type B.
Let’s try implementing some generic functions with this new type. First, an easy one:
func f <A, B> (x: A) -> Or<A, B> {
return .left(x)
}
This is saying that given something in A we want to produce something in Or<A, B>. Only way to do that is to instantiate a new value of the left case of Or.
A more difficult one that we will break down in more detail:
func f <A, B, C> (x: Or<A, B>, g: (A) -> C, h: (B) -> C) -> C {
???
}
We now have a value in Or<A, B>, a function from A to C and a function from B to C, and we want to produce a value in C. Well, the only way to really deal with enum values is to switch on them and deal with each case separately:
func f <A, B, C> (x: Or<A, B>, g: (A) -> C, h: (B) -> C) -> C {
switch x {
case .left:
???
case .right:
???
}
}
Now, how to fill in each case? In the left case we will have a value in A. Huh, but we also have a function that takes things in A so we might as well feed it into the function. Oh, and hey, that function outputs a value in C which is where we are trying to get anyway! The right case works the exact same way:
func f <A, B, C> (x: Or<A, B>, g: (A) -> C, h: (B) -> C) -> C {
switch x {
case let .left(a):
return g(a)
case let .right(b):
return h(b)
}
}
Time to throw a curve ball. Let’s implement the function:
func f <A, B> (x: A) -> B {
???
}
It needs to take a value in A and return a value in B. Hm. Well, we know absolutely nothing about B. It might even be that strange type, Bottom, that has no values. This is an example of a function which has no implementation. There is nothing we can write in this function to appease the compiler.
Here’s another:
func f <A, B, C> (g: (A) -> C, h: (B) -> C) -> C {
???
}
This seems similar to an example we already considered, but these functions don’t compose nicely. Their types don’t match up. They both output a value in C and so we can’t align them. Dang. This function also cannot be implemented.
# Propositional Logic
Time to step back and try to make sense of this. How can we interpret the fact that some of these functions have unique implementations and others have no implementation. It’s all connected to the world of formal logic.
In logic, the atomic object is the proposition which can be either true ($$\top$$) or false ($$\bot$$). We can connect two propositions $$P$$ and $$Q$$ with various operations to create new propositions. For example, disjunction $$P \lor Q$$ is read as “P or Q”, and is false if both $$P$$ and $$Q$$ are false and true otherwise. On the other hand, conjunction $$P \land Q$$ is read as “P and Q”, and is true if both $$P$$ and $$Q$$ are true and false otherwise. A few other operations:
Symbol Statement Truth value
$$\lnot{P}$$ not $$P$$ false if $$P$$ true, true otherwise
$$P \Rightarrow Q$$ $$P$$ implies $$Q$$ false if $$P$$ true and $$Q$$ false, true otherwise
$$P \Leftrightarrow Q$$ $$P$$ implies $$Q$$ and $$Q$$ implies $$P$$ true if $$P$$ and $$Q$$ are both true or both false, false otherwise
Using these atoms and operations we can construct small statements. For example, $$P \Rightarrow P$$, i.e. $$P$$ implies $$P$$. Well, of course that’s true, it’s called a tautology. Or even: $$P \land Q \Rightarrow P$$, i.e if $$P$$ and $$Q$$ are true, then $$P$$ is true.
Here’s a seemingly more complicated one:
$\left( (P \Rightarrow Q) \land (Q \Rightarrow R) \right) \Rightarrow (P \Rightarrow R)$
That is: if $$P$$ implies $$Q$$ and $$Q$$ implies $$R$$, then $$P$$ implies $$R$$. Seems reasonable. For if “snowing outside” implies “you wear boots”, and “wearing boots” implies “you wear thick socks”, then “snowing outside” implies “you wear thick socks.”
At this point, you might be seeing a connection between these logical statements and the generic functions we wrote. In fact, the three simple statements we just constructed directly correspond to functions we wrote earlier:
// P ⇒ P
func f <A> (x: A) -> A {
return x
}
// P ∧ Q ⇒ P
func f <A, B> (x: A, y: B) -> A {
return x
}
// (P ⇒ Q ∧ Q ⇒ R) ⇒ (P ⇒ R)
func f <A, B, C> (g: (A) -> B, h: (B) -> C) -> (A) -> C {
return { a in h(g(a)) }
}
See how the logical statement has the same “shape” as the function signature? This is the idea deep underneath everything we have been grasping at. For every function we could implement there is a corresponding mathematical theorem that is provably true. The converse is also true (but a little more nuanced): for every true logical theorem there is a corresponding generic function implementing the proof.
This view also gives us some perspective on why the function A -> B couldn’t be implemented. For if it could, then the corresponding theorem in logic would be true: $$P \Rightarrow Q$$. That logical statement is saying that any proposition $$P$$ implies any other proposition $$Q$$, which is clearly false.
Another un-implementable function we considered was of the form (A -> C, B-> C) -> C. That is, it took functions A -> C and B -> C as input and wanted to output a value in C. In the world of logic this corresponds to the statement: $$(P \Rightarrow R \land Q \Rightarrow R) \Rightarrow R$$. Said verbally, if $$P$$ implies $$R$$ and $$Q$$ implies $$R$$ then $$R$$ is true. It’s quite nice that we have two statements involving the truth of $$R$$, but those statements alone do not prove the truth of $$R$$. If you work better with concrete examples, here are some propositions we can substitute for $$P$$, $$Q$$ and $$R$$ to show the absurdity of the statement:
\begin{align*} P &= \text{x and y are even integers} \\ Q &= \text{x and y are odd integers} \\ R &= \text{x + y is even} \end{align*}
Clearly $$P \Rightarrow R$$ and $$Q \Rightarrow R$$, but $$R$$ alone is not true, for that would mean the sum of any two integers is even.
# De Morgan’s Law
Swift’s type system is strong enough for us to prove De Morgan’s law, which relates the operations $$\lnot$$, $$\land$$ and $$\lor$$. Programmers can apply this law in order to untangle and simplify gnarly conditional statements. The law states: for any propositions $$P$$ and $$Q$$, the following holds true:
$\lnot(P \lor Q) \Leftrightarrow \lnot P \land \lnot Q$
You can think of this as $$\lnot$$ distributing over $$\lor$$ but at the cost of switching $$\lor$$ to $$\land$$.
In order to prove this in Swift we need a way to model all of the pieces. Generics take care of the propositions $$P$$ and $$Q$$. How can we model the negation of a statement: $$\lnot P$$? The concept of false is modeled in a type system by the type that holds no values. Previously we called this Bottom, but in order to be more explicit let’s call this Nothing:
enum Nothing {
// no cases
}
Then the negation of the type A would be a function A -> Nothing. Such a function cannot possibly exist since Nothing has no values. To be more explicit we are going to make a new type to model this:
struct Not <A> {
let not: A -> Nothing
}
This type corresponds to the negation of the proposition represented by A
Other parts of De Morgan’s law include $$\lor$$ and $$\land$$. We already have a type for the $$\lor$$ disjunction: Or<A, B>. For the $$\land$$ conjunction we have tuples (A, B), but to be more explicit we will create a new type for this:
struct And <A, B> {
let left: A
let right: B
init(_ left: A, _ right: B) {
self.left = left
self.right = right
}
}
Now we can try to write the proof. There are two parts. First proving that $$\lnot(P \lor Q)$$ implies $$\lnot P \land \lnot Q$$. We do this by constructing a function:
func deMorgan <A, B> (f: Not<Or<A, B>>) -> And<Not<A>, Not<B>> {
???
}
We know we need to return something of type And<Not<A>, Not<B>>, so we can just fill that piece in:
func deMorgan <A, B> (f: Not<Or<A, B>>) -> And<Not<A>, Not<B>> {
return And<Not<A>, Not<B>>(
???
)
}
The constructor of And<Not<A>, Not<B>> takes two arguments, the left Not<A> and the right Not<B>, so now we can fill in those pieces:
func deMorgan <A, B> (f: Not<Or<A, B>>) -> And<Not<A>, Not<B>> {
return And<Not<A>, Not<B>>(
Not<A>(???),
Not<B>(???)
)
}
The constructor of Not<A> takes a single function A -> Nothing. This is about the time we take a look at what values we have available to us and see how we can piece them together to get what we need. We have a value f: Not<Or<A, B>>, which by definition means f.not: Or<A, B> -> Nothing. This is close to what we want. If we had some a: A, then we could plug Or.left(a) into f.not. So now we have:
func deMorgan <A, B> (f: Not<Or<A, B>>) -> And<Not<A>, Not<B>> {
return And<Not<A>, Not<B>>(
Not<A> {a in f.not(.left(a))},
Not<B>(???)
)
}
The Not<B> piece works exactly the same, giving us the fully implemented function, and hence half the proof of De Morgan’s law:
func deMorgan <A, B> (f: Not<Or<A, B>>) -> And<Not<A>, Not<B>> {
return And<Not<A>, Not<B>>(
Not<A> {a in f.not(.left(a))},
Not<B> {b in f.not(.right(b))}
)
}
Next we need to prove the converse: $$\lnot P \land \lnot Q$$ implies $$\lnot(P \lor Q)$$. This is done by implementing the function:
func deMorgan <A, B> (f: And<Not<A>, Not<B>>) -> Not<Or<A, B>> {
???
}
We see that we need to return something of type Not<Or<A, B>>, which has a constructor taking a function Or<A, B> -> Nothing, so we can fill that in:
func deMorgan <A, B> (f: And<Not<A>, Not<B>>) -> Not<Or<A, B>> {
return Not<Or<A, B>> { (x: Or<A, B>) in
???
}
}
Now we have this value x: Or<A, B>, which is an enum, so we should switch on it and consider each case separately:
func deMorgan <A, B> (f: And<Not<A>, Not<B>>) -> Not<Or<A, B>> {
return Not<Or<A, B>> { (x: Or<A, B>) in
switch x {
case let .left(a):
???
case let .right(b):
???
}
}
}
Consider the left case. We have at our disposal f: And<Not<A>, Not<B>> and a: A. By definition f.left: Not<A>, and hence f.left.not: A -> Nothing. Therefore f.left.not(a): Nothing, which is exactly what we want. The right case works similarly, and we have implemented the function:
func deMorgan <A, B> (f: And<Not<A>, Not<B>>) -> Not<Or<A, B>> {
return Not<Or<A, B>> {(x: Or<A, B>) in
switch x {
case let .left(a):
return f.left.not(a
case let .right(b):
return f.right.not(b)
}
}
}
We have now proven De Morgan’s law. The mere fact that we were able to implement these two functions and it type checks gives a computer proof of De Morgan’s law.
This is about the most advanced mathematical theorem we can prove in Swift, but the stronger a language’s type system is the more powerful of theorems that can be proven. For example, in Idris one can prove that the sum of two even integers is even. Astonishingly, the languages Agda and Coq can prove a theorem from topology: the fundamental group of the circle is isomorphic to the group of integers.
# Curry-Howard correspondence
The rigorous statement of the relationship we have been grasping at is known as the Curry-Howard correspondence, first observed by the mathematician Haskell Curry in 1934 and later finished by logician William Howard in 1969. It sets up a kind of dictionary mapping terms in the computer science world to terms in the mathematics world.
Computer Science Mathematics
Type Proposition
Function Implication
Tuple Conjunction (and)
Sum type Disjunction (or)
Function application Modus ponens
Identity function Tautology
Function composition Syllogism
That is only the beginning. There’s a lot more.
By the way, this isn’t the first time a dictionary has been made to map mathematical ideas to another, seemingly different field. In 1975 the mathematician Jim Simons worked with Nobel winning physicist C. N. Yang to create what later became known as the “Wu-Yang dictionary,” which mapped physics ideas to well-established (sometimes decades prior) mathematical concepts:
# Hole-Driven Development
Often when we tried to implement a function we used ??? as a placeholder for something we had not yet figured out. Sometimes we’d fill that placeholder with something more specific, but might have created more unknown chunks denoted by ???. This is loosely known as “hole-driven development.” The hole is represented by the unknown ??? piece, and we look to the compiler for hints at how we should fill that hole. It’s almost like a conversation with the compiler.
Some languages and compilers are sophisticated enough to do this work for you. See Agda as well as the djinn package for Haskell.
# Exercises
Below you will find some exercises to help you explore these ideas a little deeper. You can also download a playground with all of our code snippets and these exercises combined.
1.) Two of the following functions can be implemented and one cannot. Provide the implementations and explain why the un-implementable one is different.
func f <A, B> (x: A) -> (B) -> A {
}
func f <A, B> (x: A, y: B) -> A {
}
func f <A, B> (f: (A) -> B) -> A {
}
2.) Find an implementation of:
func f <A, B, C> (f: @escaping (A) -> B) -> (@escaping (C, B) -> C) -> ((C, A) -> C) {
???
}
3.) Find an implementation of:
func f <A, B, C> (x: A, g: (A) -> B, h: (A) -> C) -> (B, C) {
???
}
4.) Prove the theorem:
by implementing the function:
func f <A> (x: A) -> Not<Not<A>> {
???
}
5.) Try to prove the converse:
by implementing the function:
func f <A> (x: Not<Not<A>>) -> A {
???
}
If you are having trouble, don’t worry. It’s not possible to implement this function. However, it’s instructive to attempt it and see how it goes. The inability to implement this function has to do with the fact that we are modeled on “constructive logic”, and this theorem does not have a constructive proof, i.e. we can “construct” double negatives but we cannot remove them.
6.) The following is a function that will “curry” another function:
func curry <A, B, C> (f: @escaping (A, B) -> C) -> (A) -> (B) -> C {
return { a in
return {b in
return f(a, b)
}
}
}
That is, it takes a function of two parameters and turns it into a function of one parameter that returns a function of one parameter. Describe what this function represents in the world of formal logic.
7.) If the type with no values represents false in a type system, what type would represent true?
8.) The type Not<A> cannot be instantiated for nearly every type A. However, there is exactly one type for which you can create a value in Not<A>. What is that type and how does it relate to the type discovered in exercise #7.
9.) Bonus: Explore the idea that double-negation in the formal logic world corresponds to “continuation-passing style” (CPS) in the programming world. | 2017-08-19 18:45:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 2, "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": 0.5675602555274963, "perplexity": 1264.525810450437}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105712.28/warc/CC-MAIN-20170819182059-20170819202059-00369.warc.gz"} |
https://www.datacamp.com/community/tutorials/parameter-optimization-machine-learning-models | Tutorials
python
+1
# Hyperparameter Optimization in Machine Learning Models
This tutorial covers what a parameter and a hyperparameter are in a machine learning model along with why it is vital in order to enhance your model’s performance.
Machine learning involves predicting and classifying data and to do so, you employ various machine learning models according to the dataset. Machine learning models are parameterized so that their behavior can be tuned for a given problem. These models can have many parameters and finding the best combination of parameters can be treated as a search problem. But this very term called parameter may appear unfamiliar to you if you are new to applied machine learning. But don’t worry! You will get to know about it in the very first place of this blog, and you will also discover what the difference between a parameter and a hyperparameter of a machine learning model is. This blog consists of following sections:
• What are a parameter and a hyperparameter in a machine learning model?
• Why hyperparameter optimization/tuning is vital in order to enhance your model’s performance?
• Two simple strategies to optimize/tune the hyperparameters
• A simple case study in Python with the two strategies
Let’s straight jump into the first section!
### What is a parameter in a machine learning learning model?
A model parameter is a configuration variable that is internal to the model and whose value can be estimated from the given data.
• They are required by the model when making predictions.
• Their values define the skill of the model on your problem.
• They are estimated or learned from data.
• They are often not set manually by the practitioner.
• They are often saved as part of the learned model.
So your main take away from the above points should be parameters are crucial to machine learning algorithms. Also, they are the part of the model that is learned from historical training data. Let’s dig it a bit deeper. Think of the function parameters that you use while programming in general. You may pass a parameter to a function. In this case, a parameter is a function argument that could have one of a range of values. In machine learning, the specific model you are using is the function and requires parameters in order to make a prediction on new data. Whether a model has a fixed or variable number of parameters determines whether it may be referred to as “parametric” or “nonparametric“.
Some examples of model parameters include:
• The weights in an artificial neural network.
• The support vectors in a support vector machine.
• The coefficients in a linear regression or logistic regression.
### What is a hyperparameter in a machine learning learning model?
A model hyperparameter is a configuration that is external to the model and whose value cannot be estimated from data.
• They are often used in processes to help estimate model parameters.
• They are often specified by the practitioner.
• They can often be set using heuristics.
• They are often tuned for a given predictive modeling problem.
You cannot know the best value for a model hyperparameter on a given problem. You may use rules of thumb, copy values used on other issues, or search for the best value by trial and error. When a machine learning algorithm is tuned for a specific problem then essentially you are tuning the hyperparameters of the model to discover the parameters of the model that result in the most skillful predictions.
According to a very popular book called “Applied Predictive Modelling” - “Many models have important parameters which cannot be directly estimated from the data. For example, in the K-nearest neighbor classification model … This type of model parameter is referred to as a tuning parameter because there is no analytical formula available to calculate an appropriate value.
Model hyperparameters are often referred to as model parameters which can make things confusing. A good rule of thumb to overcome this confusion is as follows: “If you have to specify a model parameter manually, then it is probably a model hyperparameter. ” Some examples of model hyperparameters include:
• The learning rate for training a neural network.
• The C and sigma hyperparameters for support vector machines.
• The k in k-nearest neighbors.
In the next section, you will discover the importance of the right set of hyperparameter values in a machine learning model.
### Importance of the right set of hyperparameter values in a machine learning model:
The best way to think about hyperparameters is like the settings of an algorithm that can be adjusted to optimize performance, just as you might turn the knobs of an AM radio to get a clear signal. When creating a machine learning model, you'll be presented with design choices as to how to define your model architecture. Often, you don't immediately know what the optimal model architecture should be for a given model, and thus you'd like to be able to explore a range of possibilities. In a true machine learning fashion, you’ll ideally ask the machine to perform this exploration and select the optimal model architecture automatically.
You will see in the case study section on how the right choice of hyperparameter values affect the performance of a machine learning model. In this context, choosing the right set of values is typically known as “Hyperparameter optimization” or “Hyperparameter tuning”.
### Two simple strategies to optimize/tune the hyperparameters:
Models can have many hyperparameters and finding the best combination of parameters can be treated as a search problem.
Although there are many hyperparameter optimization/tuning algorithms now, this post discusses two simple strategies: 1. grid search and 2. Random Search.
### Grid searching of hyperparameters:
Grid search is an approach to hyperparameter tuning that will methodically build and evaluate a model for each combination of algorithm parameters specified in a grid.
Let’s consider the following example:
Suppose, a machine learning model X takes hyperparameters a1, a2 and a3. In grid searching, you first define the range of values for each of the hyperparameters a1, a2 and a3. You can think of this as an array of values for each of the hyperparameters. Now the grid search technique will construct many versions of X with all the possible combinations of hyperparameter (a1, a2 and a3) values that you defined in the first place. This range of hyperparameter values is referred to as the grid.
Suppose, you defined the grid as:
a1 = [0,1,2,3,4,5]
a2 = [10,20,30,40,5,60]
a3 = [105,105,110,115,120,125]
Note that, the array of values of that you are defining for the hyperparameters has to be legitimate in a sense that you cannot supply Floating type values to the array if the hyperparameter only takes Integer values.
Now, grid search will begin its process of constructing several versions of X with the grid that you just defined.
It will start with the combination of [0,10,105], and it will end with [5,60,125]. It will go through all the intermediate combinations between these two which makes grid search computationally very expensive.
Let’s take a look at the other search technique Random search:
### Random searching of hyperparameters:
The idea of random searching of hyperparameters was proposed by James Bergstra & Yoshua Bengio. You can check the original paper here.
Random search differs from a grid search. In that you longer provide a discrete set of values to explore for each hyperparameter; rather, you provide a statistical distribution for each hyperparameter from which values may be randomly sampled.
Before going any further, let’s understand what distribution and sampling mean:
In Statistics, by distribution, it is essentially meant an arrangement of values of a variable showing their observed or theoretical frequency of occurrence.
On the other hand, Sampling is a term used in statistics. It is the process of choosing a representative sample from a target population and collecting data from that sample in order to understand something about the population as a whole.
Now let's again get back to the concept of random search.
You’ll define a sampling distribution for each hyperparameter. You can also define how many iterations you’d like to build when searching for the optimal model. For each iteration, the hyperparameter values of the model will be set by sampling the defined distributions. One of the primary theoretical backings to motivate the use of a random search in place of grid search is the fact that for most cases, hyperparameters are not equally important. According to the original paper:
….for most datasets only a few of the hyper-parameters really matter, but that different hyper-parameters are important on different datasets. This phenomenon makes grid search a poor choice for configuring algorithms for new datasets.”
In the following figure, we're searching over a hyperparameter space where the one hyperparameter has significantly more influence on optimizing the model score - the distributions shown on each axis represent the model's score. In each case, we're evaluating nine different models. The grid search strategy blatantly misses the optimal model and spends redundant time exploring the unimportant parameter. During this grid search, we isolated each hyperparameter and searched for the best possible value while holding all other hyperparameters constant. For cases where the hyperparameter being studied has little effect on the resulting model score, this results in wasted effort. Conversely, the random search has much improved exploratory power and can focus on finding the optimal value for the critical hyperparameter.
Source: Random Search for Hyper-Parameter Optimization
In the following sections, you will see grid search and random search in action with Python. You will also be able to decide which is better regarding the effectiveness and efficiency.
### Case study in Python:
Hyperparameter tuning is a final step in the process of applied machine learning before presenting results.
You will use the Pima Indian diabetes dataset. The dataset corresponds to a classification problem on which you need to make predictions on the basis of whether a person is to suffer diabetes given the 8 features in the dataset. You can find the complete description of the dataset here.
There are a total of 768 observations in the dataset. Your first task is to load the dataset so that you can proceed. But before that let's import the dependencies, you are going to need.
# Dependencies
import pandas as pd
import numpy as np
from sklearn.linear_model import LogisticRegression
Now that the dependencies are imported let's load Pima Indians dataset into a Dataframe object with the famous Pandas library.
data = pd.read_csv("diabetes.csv") # Make sure the .csv file and the notebook are residing on the same directory otherwise supply an absolute path of the .csv file
The dataset is successfully loaded into the Dataframe object data. Now, let's take a look at the data.
data.head()
So you can 8 different features labeled into the outcomes of 1 and 0 where 1 stands for the observation has diabetes, and 0 denotes the observation does not have diabetes. The dataset is known to have missing values. Specifically, there are missing observations for some columns that are marked as a zero value. We can corroborate this by the definition of those columns, and the domain knowledge that a zero value is invalid for those measures, e.g., zero for body mass index or blood pressure is invalid.
(Missing value creates a lot of problems when you try to build a machine learning model. In this case, you will use a Logistic Regression classifier for predicting the patients having diabetes or not. Now, Logistic Regression cannot handle the problems of missing values. )
(If you want a quick refresher on Logistic Regression you can refer here.)
Let's get some statistics about the data with Pandas' describe() utility.
data.describe()
This is useful.
We can see that there are columns that have a minimum value of zero (0). On some columns, a value of zero does not make sense and indicates an invalid or missing value.
Specifically, the following columns have an invalid zero minimum value:
• Plasma glucose concentration
• Diastolic blood pressure
• Triceps skinfold thickness
• 2-Hour serum insulin
• Body mass index
Now you need to identify and mark values as missing. Let’s confirm this by looking at the raw data, the example prints the first 20 rows of data.
data.head(20)
You can see 0 in several columns, right?
You can get a count of the number of missing values in each of these columns. You can do this by marking all of the values in the subset of the DataFrame you are interested in that have zero values as True. You can then count the number of true values in each column. For this, you will have to reimport the data without the column names.
data = pd.read_csv("https://raw.githubusercontent.com/jbrownlee/Datasets/master/pima-indians-diabetes.data.csv",header=None)
print((data[[1,2,3,4,5]] == 0).sum())
1 5
2 35
3 227
4 374
5 11
dtype: int64
You can see that columns 1,2 and 5 have just a few zero values, whereas columns 3 and 4 show a lot more, nearly half of the rows. Column 0 has several missing values although but that is natural. Column 8 denotes the target variable so, '0's in it is natural.
This highlights that different “missing value” strategies may be needed for different columns, e.g., to ensure that there are still a sufficient number of records left to train a predictive model.
In Python, specifically Pandas, NumPy and Scikit-Learn, you mark missing values as NaN.
Values with a NaN value are ignored from operations like sum, count, etc.
You can mark values as NaN easily with the Pandas DataFrame by using the replace() function on a subset of the columns you are interested in.
After you have marked the missing values, you can use the isnull() function to mark all of the NaN values in the dataset as True and get a count of the missing values for each column.
# Mark zero values as missing or NaN
data[[1,2,3,4,5]] = data[[1,2,3,4,5]].replace(0, np.NaN)
# Count the number of NaN values in each column
print(data.isnull().sum())
0 0
1 5
2 35
3 227
4 374
5 11
6 0
7 0
8 0
dtype: int64
You can see that the columns 1:5 have the same number of missing values as zero values identified above. This is a sign that you have marked the identified missing values correctly.
This is a useful summary. But you'd like to look at the actual data though, to confirm that you have not fooled yourselves.
Below is the same example, except you print the first 5 rows of data.
data.head()
0 1 2 3 4 5 6 7 8
0 6 148.0 72.0 35.0 NaN 33.6 0.627 50 1
1 1 85.0 66.0 29.0 NaN 26.6 0.351 31 0
2 8 183.0 64.0 NaN NaN 23.3 0.672 32 1
3 1 89.0 66.0 23.0 94.0 28.1 0.167 21 0
4 0 137.0 40.0 35.0 168.0 43.1 2.288 33 1
It is clear from the raw data that marking the missing values had the intended effect. Now, you will impute the missing values. Imputing refers to using a model to replace missing values. Although there are several solutions for imputing missing values, you will use mean imputation which means replacing the missing values in a column with the mean of that particular column. Let's do this with Pandas' fillna() utility.
# Fill missing values with mean column values
data.fillna(data.mean(), inplace=True)
# Count the number of NaN values in each column
print(data.isnull().sum())
0 0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 0
dtype: int64
Cheers! You have now handled the missing value problem. Now let's use this data to build a Logistic Regression model using scikit-learn.
First, you will see the model with some random hyperparameter values. Then you will build two other Logistic Regression models with two different strategies - Grid search and Random search.
# Split dataset into inputs and outputs
values = data.values
X = values[:,0:8]
y = values[:,8]
# Initiate the LR model with random hyperparameters
lr = LogisticRegression(penalty='l1',dual=False,max_iter=110)
You have created the Logistic Regression model with some random hyperparameter values. The hyperparameters that you used are:
• penalty : Used to specify the norm used in the penalization (regularization).
• dual : Dual or primal formulation. The dual formulation is only implemented for l2 penalty with liblinear solver. Prefer dual=False when n_samples > n_features.
• max_iter : Maximum number of iterations taken to converge.
Later in the case study, you will optimize/tune these hyperparameters so see the change in the results.
# Pass data to the LR model
lr.fit(X,y)
LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,
intercept_scaling=1, max_iter=110, multi_class='ovr', n_jobs=1,
penalty='l1', random_state=None, solver='liblinear', tol=0.0001,
verbose=0, warm_start=False)
It's time to check the accuracy score.
lr.score(X,y)
0.7747395833333334
In the above step, you applied your LR model to the same data and evaluated its score. But there is always a need to validate the stability of your machine learning model. You just can’t fit the model to your training data and hope it would accurately work for the real data it has never seen before. You need some kind of assurance that your model has got most of the patterns from the data correct.
Well, Cross-validation is there for rescue. I will not go into the details of it as it is out of the scope of this blog. But this post does a very fine job.
# You will need the following dependencies for applying Cross-validation and evaluating the cross-validated score
from sklearn.model_selection import KFold
from sklearn.model_selection import cross_val_score
# Build the k-fold cross-validator
kfold = KFold(n_splits=3, random_state=7)
You supplied n_splits as 3, which essentially makes it a 3-fold cross-validation. You also supplied random_state as 7. This is just to reproduce the results. You could have supplied any integer value as well. Now, let's apply this.
result = cross_val_score(lr, X, y, cv=kfold, scoring='accuracy')
print(result.mean())
0.765625
You can see there's a slight decrease in the score. Anyway, you can do better with hyperparameter tuning / optimization.
Let's build another LR model, but this time its hyperparameter will be tuned. You will first do this grid search.
Let's first import the dependencies you will need. Scikit-learn provides a utility called GridSearchCV for this.
from sklearn.model_selection import GridSearchCV
Let's define the grid values of the hyperparameters that you used above.
dual=[True,False]
max_iter=[100,110,120,130,140]
param_grid = dict(dual=dual,max_iter=max_iter)
You have defined the grid. Let's run the grid search over them and see the results with execution time.
import time
lr = LogisticRegression(penalty='l2')
grid = GridSearchCV(estimator=lr, param_grid=param_grid, cv = 3, n_jobs=-1)
start_time = time.time()
grid_result = grid.fit(X, y)
# Summarize results
print("Best: %f using %s" % (grid_result.best_score_, grid_result.best_params_))
print("Execution time: " + str((time.time() - start_time)) + ' ms')
Best: 0.752604 using {'dual': False, 'max_iter': 100}
Execution time: 0.3954019546508789 ms
You can define a larger grid of hyperparameter as well and apply grid search.
dual=[True,False]
max_iter=[100,110,120,130,140]
C = [1.0,1.5,2.0,2.5]
param_grid = dict(dual=dual,max_iter=max_iter,C=C)
lr = LogisticRegression(penalty='l2')
grid = GridSearchCV(estimator=lr, param_grid=param_grid, cv = 3, n_jobs=-1)
start_time = time.time()
grid_result = grid.fit(X, y)
# Summarize results
print("Best: %f using %s" % (grid_result.best_score_, grid_result.best_params_))
print("Execution time: " + str((time.time() - start_time)) + ' ms')
Best: 0.763021 using {'C': 2.0, 'dual': False, 'max_iter': 100}
Execution time: 0.793781042098999 ms
You can see an increase in the accuracy score, but there is a sufficient amount of growth in the execution time as well. The larger the grid, the more execution time.
Let's rerun everything but this time with the random search. Scikit-learn provides RandomSearchCV to do that. As usual, you will have to import the necessary dependencies for that.
from sklearn.model_selection import RandomizedSearchCV
random = RandomizedSearchCV(estimator=lr, param_distributions=param_grid, cv = 3, n_jobs=-1)
start_time = time.time()
random_result = random.fit(X, y)
# Summarize results
print("Best: %f using %s" % (random_result.best_score_, random_result.best_params_))
print("Execution time: " + str((time.time() - start_time)) + ' ms')
Best: 0.763021 using {'max_iter': 100, 'dual': False, 'C': 2.0}
Execution time: 0.28888916969299316 ms
Woah! The random search yielded the same accuracy but in a much lesser time.
That is all for the case study part. Now, let's wrap things up!
In this tutorial, you learned about parameters and hyperparameters of a machine learning model and their differences as well. You also got to know about what role hyperparameter optimization plays in building efficient machine learning models. You built a simple Logistic Regression classifier in Python with the help of scikit-learn.
You tuned the hyperparameters with grid search and random search and saw which one performs better.
Besides, you saw small data preprocessing steps (like handling missing values) that are required before you feed your data into the machine learning model. You covered Cross-validation as well.
That is a lot to take in, and all of them are equally important in your data science journey. I will leave you with some further readings that you can do.
For the ones who are a bit more advanced, I would highly recommend reading this paper for effectively optimizing the hyperparameters of neural networks. link | 2020-07-12 07:39:09 | {"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": 0.5538887977600098, "perplexity": 898.0104072253969}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657131734.89/warc/CC-MAIN-20200712051058-20200712081058-00177.warc.gz"} |
https://zbmath.org/?q=an:0824.03013 | # zbMATH — the first resource for mathematics
Effectively infinite classes of weak constructivizations of models. (English. Russian original) Zbl 0824.03013
Algebra Logic 32, No. 6, 342-360 (1993); translation from Algebra Logika 32, No. 6, 631-664 (1993).
Weak constructivizations of strongly constructive models are studied. An enumerated model $$(M, \nu)$$, where $$\nu$$ is an enumeration of $$M$$, is called strongly constructive (constructive) if there exists an algorithm to recognize all (quantifier-free) formulas $$\varphi(\overline x)$$ and tuples $$\overline m$$ of natural numbers for which $$M\models \varphi(\nu\overline m)$$ holds. A model $$M$$ is called $$n$$-complete if, for each formula $$\varphi(x_ 1, \dots, x_ m)$$ which has at most $$n$$ alternations of quantifiers and for each tuple $$a_ 1,\dots, a_ m\in M$$, there exists an $$\exists$$-formula $$\psi(x_ 1,\dots, x_ m)$$ such that $$M\models \psi(x_ 1,\dots, x_ m)$$ and $$M\models \forall\overline x(\varphi\to \psi)$$. A model $$M$$ is called limit-$$\omega$$-complete if, for any $$n$$, it possesses a finite $$n$$-complete enrichment by constants, but it has no finite complete enrichments by constants. Theorem 1. If a model $$M$$ is limit-$$\omega$$-complete and possesses a strong constructivization, then, given any computable class $$S$$ of constructivizations of $$M$$ we can effectively build a non-strong constructivization that is not equivalent to any constructivization from $$S$$. Theorem 2. If $$M$$ is strongly and weakly constructivizable, then, for a given computable class of its constructivizations, we can effectively build a weak constructivization of $$M$$ that is not autoequivalent to any constructivization in this class. It follows that the class of weak constructivizations of a strongly constructivizable model is either empty or effectively infinite, and in this case, it is not computable.
##### MSC:
03C57 Computable structure theory, computable model theory 03D45 Theory of numerations, effectively presented structures
Full Text:
##### References:
[1] C. J. Ash and A. Nerode, ”Intrinsically recursive relations. Aspects of effective algebra,” in:Proc. Conf. Monash Univ., Australia (1981), pp. 24–41. · Zbl 0467.03041 [2] C. J. Ash and J. F. Knight, ”Pairs of recursive structure,”Logic Paper,64, 1–45 (1988). [3] C. J. Ash and S. S. Goncharov, ”Strong {$$\Delta$$} 2 0 -categoricity,”Algebra Logika,24, No. 6, 718–727 (1985). [4] S. S. Goncharov, ”The problem of the number of nonequivalent constructivizations,”Algebra Logika,19, No. 6, 621–639 (1980). [5] S. S. Goncharov, ”The number of nonequivalent constructivizations,”Algebra Logika,16, No. 3, 257–289 (1977). [6] S. S. Goncharov, ”autostability and computable families of constructivizations,”Algebra Logika,14, No. 6, 647–680 (1975). [7] S. S. Goncharov, ”Autostability of models and Abelian groups,”Algebra Logika,19, No. 1, 23–44 (1980). · Zbl 0468.03022 [8] S. S. Goncharov and V. D. Dzgoev, ”Autostability of models,”Algebra Logika,19, No. 1, 45–58. (1980). · Zbl 0468.03023 [9] S. S. Goncharov, ”Computable classes of constructivizations for models of finite type,”Sib. Mat. Zh.,34, No. 5, 23–37 (1993). · Zbl 0860.03032 [10] V. P. Dobritsa, ”Computability of some classes of algebras,”Sib. Mat. Zh.,18, No. 3, 570–579 (1977). · Zbl 0361.02065 [11] V. P. Dobritsa, ”Structural properties of computable classes of constructive models,”Algebra Logika,26, No. 1, 36–62 (1987). · Zbl 0636.03030 [12] A. T. Nurtazin, ”Strong and weak constructivizations and computable families,”Algebra Logika,13, No. 3, 311–323 (1974). [13] S. S. Goncharov, ”Bounded theories of constructive Boolean algebras,”Sib. Mat. Zh.,17, No. 4, 797–812 (1976). · Zbl 0361.02066 [14] S. S. Goncharov, ”Some properties of constructivizations for Boolean algebras,”Sib. Mat. Zh.,16, No. 2, 264–278 (1975). [15] Yu. L. Ershov,Decidability Problems and Constructive Models [in Russian], Nauka, Moscow (1980).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. | 2021-09-26 17:15:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6905768513679504, "perplexity": 2825.1369778527114}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057882.56/warc/CC-MAIN-20210926144658-20210926174658-00102.warc.gz"} |
https://tex.stackexchange.com/questions/312889/position-things-in-the-middle-between-a-node-and-an-arrow-tip | # Position things in the middle between a node and an arrow tip
Say I want to draw something like a flow chart, that is nodes connected by arrows that sometimes merge. This would be a typical pattern:
I used the following code to create this small example:
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{positioning,calc,arrows.meta}
\begin{document}
\begin{tikzpicture}
\node[draw] (a) {A};
\node[draw,right=of a] (b) {B};
\node[draw,below=5mm of b] (c) {C};
\path[-{LaTeX[]}] (b) edge (c);
\draw (a) |- ($(b)!0.5!(c)$);
\end{tikzpicture}
\end{document}
Now the line from A hits the one between B and C precisely at its middle point. Given the arrow tip, however, this is not the visually most pleasant position. We'd want it to hit the middle point between the end of the arrow tip and B.
Short of tinkering with magic constants, how can we do this using TikZ?
• Try to change \draw (a) |- ($(b)!0.5!(c)$) to \draw (a) |- ($(b)!0.45!(c)$)... – Zarko Jun 3 '16 at 15:14
• @cfr, is there a predefined length containing the "current" arrow length? Thanks! – Rmano Jun 3 '16 at 17:58
• @Rmano In some sense, there must be. I think it depends on the current line width, though and it might depend on the arrow tip, too. I don't know if it is predefined in the sense that you could use it to automatically figure out the adjustment, though, because TikZ may not set it until it is actually asked for the tip. I'm not certain of this. I'll try to look after. – cfr Jun 3 '16 at 19:20
• @Zarko That would be "magic constant" style; the factor would be different for every instance. – Raphael Jun 3 '16 at 21:44
Here's a first pass which adjusts automatically for the current line width but not for the kind of arrow tip. It also requires modification if the direction direction/angle of the path differs or if the arrow points in the opposite direction, for example.
\documentclass[tikz,border=10pt,multi]{standalone}
\usetikzlibrary{positioning,calc,arrows.meta}
\begin{document}
\begin{tikzpicture}
[
every node/.append style={draw},
% by default, for this type of arrow tip, length=3pt 4.5 0.8 [ref. p. 185]
arrow line/.style={%
draw,
-{LaTeX[]},
},
]
\node (a) {A};
\node [right=of a] (b) {B};
\node [below=5mm of b] (c) {C};
\path [arrow line] (a) |- ([yshift=\arrowadjust]$(b)!0.5!(c)$) (b) -- (c);
\end{tikzpicture}
\end{document}
You can find the values for the arrow(s) you are interested in in the file texmf-dist/tex/generic/pgf/libraries/pgflibraryarrows.meta.code.tex:
\pgfdeclarearrow{
name = Latex,
defaults = {
length = +3pt 4.5 .8,% +2.8pt 3 0.8,
width' = +0pt .75,
line width = +0pt 1 1,
},
(LaTeX is an alias for Latex).
• This is nice but only works for arrows that to perfectly vertical or horizontal, and you need to know which way they face. – Raphael Jun 4 '16 at 15:17
• @Raphael Yes, you do. I tried to figure something out with shorten but it started getting very complicated and you'd still need to know whether you had > or <. – cfr Jun 4 '16 at 15:30
• I can believe that; there's probably no neat solution without added support from PGF. FWIW, knowing < or > is much weaker than knowing the angle. – Raphael Jun 4 '16 at 15:32 | 2020-08-13 23:56:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8841932415962219, "perplexity": 1940.196209249944}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439739104.67/warc/CC-MAIN-20200813220643-20200814010643-00175.warc.gz"} |
https://socratic.org/questions/58d26cba7c014929807c29fa | # Question #c29fa
Mar 23, 2017
The statement is false.
#### Explanation:
The trend of ionization energies for the 3rd period is shown below.
The ionization energy of $\text{Mg}$ is 737.7 kJ/mol, and the ionization energy of $\text{S}$
is 999.6 kJ/mol.
The explanation
The electron configuration of $\text{Mg}$ is ${\text{[Ne] 3s}}^{2}$.
The electron configuration of $\text{S}$ is ${\text{[Ne] 3s"^2 "3p}}^{4}$.
Thus, in going from $\text{Mg}$ to $\text{S}$, you are adding four valence electrons, but you are also adding four protons to the nucleus.
The other electrons screen a valence electron from the full attraction of the nucleus, but they don’t screen it completely.
Hence, as you add more protons, the valence electrons are more attracted to the nucleus.
The result: it takes more energy to remove an electron from an $\text{S}$ atom than from an $\text{Mg}$ atom, so the ionization energy of $\text{S}$ is higher than that of $\text{Mg}$. | 2019-10-19 05:03:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 12, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.42591428756713867, "perplexity": 497.5794912604892}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986688826.38/warc/CC-MAIN-20191019040458-20191019063958-00527.warc.gz"} |
https://itprospt.com/num/13713635/x-7-if-flx-7-21-where-is-f-not-differentiable | 5
# X-7 If flx) = 7+21 where is f not differentiable?...
## Question
###### X-7 If flx) = 7+21 where is f not differentiable?
X-7 If flx) = 7+21 where is f not differentiable?
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http://lkfi.gestion-comptable.fr/primitive-lattice-vectors.html | # Primitive Lattice Vectors
Non Primitive Cell. Lattice Planes ¡ A lattice plane of a given Bravais lattice is a plane (or family of parallel planes) whose intersections with the lattice are periodic (i. The angles a1 ∧a2, a2 ∧a3 and a3 ∧a1 are conventionally labelled γ, α and β respectively. Here there are 14 lattice types (or Bravais lattices). Lattice 11: Rhombohedral. How would you draw the reciprocal lattice and indicate the primitive lattice vectors chosen and justify the magnitude and direction of the reciprocal vector. For obvious reasons the term Bravais lattice is often also used for the set of vectors {R n}. However, it is conventionally defined in terms of the lattice parameters of the hexagonal cell defined above. primitive lattice vectors. A lattice being an infinite, symmetric and periodic collection of zero-dimensional nodes, rigorously speaking it is neither primitive nor centred. In a crystal, a lattice point may be the seat of more than one atom, and the arrangement of atoms may have a higher degree of symmetry than the lattice. • The primitive translation vectors in reciprocal space will then be given by: • Which is a body centred cubic lattice 37 Reciprocal space example - a body centred cubic lattice • A body centred cublic lattice with cubic lattice constant a has primitive lattice vectors given by: • The primitive translation vectors in reciprocal space. Primitive lattice cell and Wigner-Seitz cell. Crystal basis: Arrangement of atoms within the unit cell. The shape of the wurtzite cell is a vertically oriented prism, with the base defined by the primitive lattice vectors, , and , which are of equal length and are separated by an angle of 60; and both lie in the horizontal -plane. This can be compared to the volume of primitive direct lattice 2 a3 V cell. What does this tell you about the number of particles in the cubic cell? b) Show that the reciprocal lattice of a fcc lattice is a bcc lattice and vice versa. Barium Titanate, BaTiO3,(that has a so-called cubic perovskite lattice structure)where the Ba atom sits in the corner of a cube, the O atoms are face centered on the sides of the cube, and the Ti atom is space centered in the cube. Specifically, applying the Chinese remainder theorem to two vectors ˙. The angles between the axes. Write the conventional symbol for each of these rotations. If the transition vectors are primitive, the cell is also called a primitive cell, or unit cell. voltage) a number of electrons in the upper half of the Dirac cones or holes in the lower half; for de niteness let’s consider the former. A primitive cell is a unit cell built on the basis vectors of a primitive basis of the direct lattice, namely a crystallographic basis of the vector lattice L such that every lattice vector t of L may be obtained as an integral linear combination of the basis vectors, a, b, c. determine the values of alpha that are commensurate with a lattice. The basis vectors a1, a2 and a3 define the the unit cell; their mag- nitudes a1, a2 and a3 respectively, are the lattice parameters of the unit cell. Compute the electronic phase difference (rad) between GaAs1 and GaAs2. 1 Crystal Structures 7 The volume of the primitive unit cell in the reciprocal lattice is (2π)3/V. Clearly, the lattice R does not select its lattice vectors, If E a is a basis for a primitive cubic lattice,. It can be shown that the volumes of all primitive unit cells are the same and the smallest among all possible unit cells. A Non Primitive Cell is one, which contains more than one lattice point per unit cell. Show that the reciprocal of a trigonal Bravais lattice is also trigonal, with an angle θ∗ given by −cosθ∗ = cosθ/[1 + cosθ] and a primitive vector length a∗ given by. That any array satisfying (b) also satisfies (a) becomes evident as soon as both def-. 38) Since all lattice vectors can be expressed in the form , where the are integers, it follows from equation 3. The choice of lattice vectors is not unique. Translucent isosurfaces can be overlapped with a structural model. basis is described with respect to the primitive unit cell of a simple cubic lattice. vasp in VESTA. • The reciprocal lattice of a Bravais lattice is always a Bravais lattice and has its own primitive lattice vectors, for example, and in the above figure • The position vector of any point in the reciprocal lattice can be expressed in. In the case of a cubic primitive lattice which is highly symmetric, the pixel configurations [[xi]. The HCP lattice has two lattice constants, so there is a much larger phase space to explore in order to locate the minimum cohesive energy. It has unit cell vectors a = b = c and interaxial angels α=β=γ=90°. A Brillouin zone is defined as a Wigner-Seitz primitive cell in the reciprocal lattice. Badran Solid State Physics 12 To emphasize the cubic symmetry of the bcc and fcc Bravais lattices, for example, we can show that they are descried as follows: a) As a simple cubic (sc) lattice spanned byaxˆ, ayˆ andazˆ, the bcc Bravais lattice is described by the two-point basis (0, 0, 0) and (2 a, 2 a 2 a). A body-centered cubic lattice has eight lattice points where a lattice point is defined as a point of intersection of two or more grid lines. qthat are correct mod 2 and mod q, respectively, does not produce a short integer vector. These translation vectors connect the lattice pt at the origin to the points at the face centres. of lattice vectors T. Solid State Physics Midterm Exam Part 1 Name_____ Consider the 2D arrangement of atoms shown. angles, and the lattice thus formed is the honeycomb lattice. 14 to see that we, in fact got the primitive vectors of a bcc lattice. The shaded hexagon is the first Brillouin zone with Γ indicating the centre, and K + and K− showing two non-equivalent corners. Identification of 2D Space Groups Identify the primitive cell lattice vectors and all the symmetry elements that are present for the following structures. Then the reciprocal. 9x10E-10m, calculate the atomic positions in the {110} plane taking the lower left atom as the origin. To determine this primitive cells of ScAl were produced for both types of structures. x The space group of a crystal is 227. As a consequence,. The primitive lattice vectors of the reciprocal lattice are defined as A D 2ˇb c a b c;B D 2ˇc a a b c;C D 2ˇa b a b c; (19). [5 points] Identify a set of primitive lattice vectors for the crystal. When the lattice is at equilibrium each atom is positioned exactly at its lattice site. A Primitive Cells is the simplest type of the unit cell, which contains only one lattice point per unit cell. But a 1 ”’ and a 2 ”’ are not primitive translation vector. A conventional basis generates more RL vectors that a corresponding primitive basis. 1) where n 1, n 2 and n 3 are integers and a 1, a 2 and a 3 are three noncoplanar vectors. The basis depends on the set of lattice vectors chosen because the coordinates of each point depend on where the origin of the unit cell is, as well as on the direction of the lattice vectors. It is convenient to choose our Bravais lattice to have primitive lattice vectors a 1, a 2 given as shown by the notation is. 14 to see that we, in fact got the primitive vectors of a bcc lattice. (iii) Mark in the primitive unit cell: it is the smallest unit which packs to fill space and which completely characterises the structure. In general mathematical terms, a lattice is a infinite arrangement of regular points. The crystal system of the reciprocal lattice is the same as the direct lattice (for example, cubic remains cubic), but the Bravais lattice may be different (e. To reveal this correspondence, let us take a primitive set (a 1;a 2;a 3) of the. Lattice-compatible Hermite normal form ( HNF) supercells up to. As the following lemma implies, finding a lattice cube that includes a given uis a problem that is solved by finding a single companion of the same length as u. This is the nomenclature for "primitive" vectors in solid-state crystallography, but in LAMMPS the unit cell they determine does not have to be a "primitive cell" of minimum volume. So, in a 1mm cube, there are 8x10 18 of the repeated arrangements! Lattice points and vectors: Every point within the primitive unit cell is unique, but within the macroscopic crystal each point is repeated many times. Vectors can be defined such that the primitive cell can be translated by integral multiples of these vectors. (b) The primitive basis vectors of the face centered cubic (fcc) lattice and the two atoms forming the basis are highlighted. Same point group symmetry as the. This is illustrated in Fig. 2, resulting in unit cells that are apparently (visually. The lattice is constructed by placing a point at every possible combination of the three vectors and any multiples of them (positive or negative). ¾We often use primitive translation vectors and unit cells to define the crystal structure but nonprimitive axes are also. Looking for Primitive lattice vector? Find out information about Primitive lattice vector. That is, the primitive unit cell contains more than one lattice point. The red square represents the translations of the smallest direct lattice produced by the periodic distributions of the small pieces of this mosaic. The primitive lattice vectors must be non co-planar, but they need not be orthogonal to each other. , b be defined as above. Figure 2: The distinction between primitive and non-primitive lattice vectors in 2 dimensions; all lattice points can be described by an integral combination of primitive lattice vectors. The Bravais lattice is the same as the lattice formed by all the. Infinite array of discrete points that appear exactly the same from whichever of the points the array is viewed. 300/point, and 216000 total grid points on the integration grid. Besides the primitive lattice, a supercell lattice is also frequently used in electronic structure calculations, which is suitable to simulate complicated systems such as defects and alloys. (b) Determine the reciprocal lattice vectors. In other words, we require for some. Then we have Then we have polynomial is polynomial-time reducible to the problem of factoring positive (1. a1 a2 Figure 1: Hexagonal lattice: ja 1j= ja 2j, the angle between a 1 and a 2 is 2ˇ=3. Take them to be 2D structures. The size of lines) with its basis vectors ti (i = 1,2,3, bold arrows) relative the atoms (circles) is drawn arbitrarily. What is the Bravais lattice, the basis, the. These can be written in a matrix form by assembling them column wise. A lattice consists of a unit cell, a set of basis sites within that cell. Any direct lattice has a corresponding reciprocal lattice. The vectors a1,a2,a3 are the edge vectors of the unit cell. The translational_symmetry() is applied only in the $$a_1$$ lattice vector direction which gives the ribbon its infinite length, but the symmetry is disabled in the $$a_2$$ direction so that the finite size of the shape is preserved. The zinc blende structure has ABAB stacking along the [111] direction. Note that the non-primitive lattice would have unit vectors of length 2. Primitive lattice vectors are the shortest lattice vectors possible. The Bravais lattice (consider, e. On the Definition and Classification of Bravais Lattices. Sketch the Bravais lattice, identify the basis, and de ne the primitive unit cell for a 2D CuO. ‘a’ is called the cubic edge or simply the lattice constant. (c) Sketch your Brillouin zone and label important symmetry points (K and M). :2008954946 In semiconductor crystal materials, atoms are located periodically, with three primitive basis vectors, a, b, and c. FCC Btuvais lattice (f,Jtr-si1all Primitive Single lattice point per cell § Smallest area in 2D, or mallest volume in 3D Simple cubic(sc) UNIT CELL Conventional & Non-primitiv § More than one lattice point per cel. Crystal is a three dimensional periodic array of atoms. 61 Å 47-Ag a=4. Clearly, the lattice R does not select its lattice vectors, If E a is a basis for a primitive cubic lattice,. In either case, one needs to choose the three lattice vectors a 1, a 2, and a 3 that define the unit cell (note that the conventional unit cell may be larger than the primitive cell of the Bravais lattice, as the examples below illustrate). For example, the primitive cubic lattice—often referred to as a simple cubic lattice—is described by three perpendicular base. will lie on one of the fcc lattices, while all of the ‘B’ atoms lie on the second fcc lattice. Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Extremal problems for convex lattice polytopes Imre Bárány Rényi Institute, Hungarian Academy of Sciences & Department of Mathematics, University College London. For face-centered cubic and body-centered cubic lattices, the primitive lattice vectors are not orthogonal. Interstitial Positions (online up to 8 atoms/cell) [DOI: 10. Primitive lattice cell and Wigner-Seitz cell. Case of hexagonal and rhombohedral structures. (b) (2 pts) If there is no band overlap, what valences should the atoms have if the material is to be a metal? An insulator? Please explain your logic. For a Bravais lattice, all lattice sites are equivalent and any vectors connecting to lattice sites are lattice vectors. 300/point, and 216000 total grid points on the integration grid. The resulting primitive lattice is indicated by the filled nodes in Fig. The angles between their faces are 90 0 in a cubic lattice. ¾Primitive unit cell is one that has only one atom per unit cell and the lattice vectors defining the cell are said to be primitive lattice vectors. where −→z 0 is the unit vector along the z-axis, which. , as in body-centered and face-centered crystals). It has unit cell vectors a = b = c and interaxial angels α=β=γ=90°. It is found that the maximum and minimum numbers for lattice constants are 16 for Triclinic and Face-centered orthorhombic lattices, and 1 for Primitive orthorhombic, Primitive tetragonal and Primitive cubic lattices. However, in these cases the 밀러 지수 are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions. However, these are simply the primitive lattice vectors for a bcc lattice. So if I take those vectors, a1, a2, and a3, rather than describing the unit as a cube, I can describe it alternatively as a primitive rhombohedral unit. The simple cubic (sc) lattice typically uses a primitive unit cell and is called a primitive (P) lattice. Cubic Lattice There are three types of lattice possible for cubic lattice. If u is primitive, and if v and w are both perpendicular to u, then v and w are either perpendicular or parallel. Note: If the original Bravais lattice (the direct lattice ) rotates, then its R-lattice rotates the same amount as well. (with respect to the reciprocal vectors generated from the standard primitive lattice vectors). The magnitudes of the primitive lattice vectors corre-spond to the lattice constants parallel and perpendicu-lar to the graphene sheet. The reciprocal lattice has extraordinary consequences for the electronic motion, even before we "switch on" the lattice potential. We require our roots to have. The difference of the endpoints is a vector v E whose coordinates are relatively prime (v E is defined up to sign). A Bravais lattice is a discrete infinite array of points generated by linear integer combinations of 3 independent primitive vectors: {n1a1 + n2a2 + n3a3 | n1, n2, n3 ∈ Z}. Lattice Vectors 2D. It only needs to be 2D but yes. Created Date. to_primitive=1 is used to create the standardized primitive cell with the transformation matricies shown at Transformation to the primitive cell, otherwise to_primitive=0 must be specified. at the cube centers. Lattice VibrationsReciprocal lattice vectors important for discussing sounds waves. Classical Theory Expectations • Equipartition: 1/2k B T per degree of freedom • In 3-D electron gas this means 3/2k B Primitive Cell and Lattice Vectors. Translucent isosurfaces can be overlapped with a structural model. Primitive lattice vectors are the smallest possible vectors that still describe the unit cell. Tetragonal Lattice There are two possible types of tetragonal lattices. Different lattice types are possible within each of the crystal systems since the lattice points within the unit cell may be arranged in different ways. The parallelepiped defined by a, b, and c is called a primitive cell. reciprocal lattice vectors are given by (2) Here, and are any two integers, denoted collectively by , and the primitive translation vectors of this lattice are given by (3) (4) where is the Cartesiancomponent, or ,of ( 1or 2). 9) This is the fundamental relation of the reciprocal lattice which shows that with any node M of the reciprocal lattice whose numerical coordinates. Periodic stacking of balls, producing a 3-dimensional network (direct lattice). Draw an example of a primitive unit cell into the lattice. is a symmetry of the standard Leech lattice. A lattice is formed by generating an infinity of translations vectors T = ua 1 + va 2 + wa 3 with u, v, w, = integers. Primitive lattice vectors are the shortest lattice vectors possible. Inorganic Chemistry Group Model Surface Analysis Fritz-Haber-Institut der MPG Literature: G. It is useful to define the reciprocal lattice in the space of wave vectors. Brillouin Zones. The basis vectors that you enter are used to identify a primitive sublattice of the direct parent lattice without regards to its final symmetry. Primitive unit cell: A volume in space, when translated through all the lattice vectors in a Bravais lattice, fills the entire space without voids or overlapping itself, is a primitive unit cell (see Figs. This correspond to the CENTERING OF A UNIT CELL. 1) where n 1, n 2 and n 3 are integers and a 1, a 2 and a 3 are three noncoplanar vectors. This page was built to translate between Miller and Miller-Bravais indices, to calculate the angle between given directions and the plane on which a lattice vector is normal to for both cubic and hexagonal crystal structures. The cell contains 1 Cu atom and 2 O atoms FIG. You can easily show that the volume of primitive reciprocal lattice is)3 2 2(a. (More details about Wigner-Seitz primitive cell in the reciprocal lattice could be found in fangxiao's webpage) [12] The first Brillouin zone is the smallest volume entirely enclosed by planes that are the perpendicular bisectors of the. It has unit cell vectors a = b = c and interaxial angels α=β=γ=90°. However, to achieve the full utility of theory and practice, everyone must end up with the same a, b, c. 9/11/2013 7 Lecture 6 Slide 13 Non ‐ Primitive Lattice Vectors Almost always, the label "lattice vector" refers to the translation vectors, not the axis vectors. By maximizing the absolute contrast subject to an equal contrast condition, lithographically useful interference patterns are found. For an infinite three dimensional lattice, defined by its primitive vectors, its reciprocal lattice can be determined by generating its three reciprocal primitive vectors, through the formulae. After this the new lattice definition of the present lattice structure will be applied where the lattice vectors of centered lattices (fcc, bcc, monoclinic-C, orthorhombic-C, -I, -F, tetragonal-I, cubic-I, -F) will be replaced by those of the corresponding primitive lattices (sc, -P) with the lattice basis vectors complemented appropriately. All reciprocal lattice vectors can be expressed as a linear combination of b1, b2, b3 using integer. Because the coordinates are integers, this normal is itself always a. On a side, draw the basis. The conventional lattice vectors are the same as the primitive lattice vectors in this case. Solution: Suppose the primitive translation vectors of a simple cubic cell be ⃗, ⃗⃗ and ⃗. 3 Dot Products The dot product is used to determine the angle between two vectors. (Four possible sets of primitive lattice vectors are shown, but there are an in-. The first Brillouin zone is the smallest volume entirely enclosed by planes that are the perpendicular bisectors of the reciprocal lattice vectors drawn from the origin. Give the basis vectors of the unit cell in dependence of the lattice constant a. The reciprocal lattice basis vectors a* and b* are respectively perpendicular to a and b, and obviously make a 90˚ angle to each other. The 2D lattice shown here has a primitive unit cell containing two points (a black one and a blue one). 2 For the fcc lattice the choice of primitive lattice vectors is straightforward, but in many systems of lower symmetry, in particular monoclinic systems, the choice is not always as simple. Küppers, Low Energy Electrons and Surface Chemistry, VCH, Weinheim (1985). The definition of a set of primitive lattice vectors is that any lattice vector L can be expressed as a linear combination of primitive lattice vectors, L = n1a1 +n2a2, with integer co-efficients. When considering cubic. are described by a set of suitably chosen lattice vectors. Important examples and applications Reciprocal lattice of selected Bravais lattices. Definition of bravais lattice in the Definitions. A generic lattice built translating a unit cell and adding edges between nearest neighbours sites. 1 lattice point/primitive unit cell. The primitive lattice vectors of the reciprocal lattice are defined as A D 2ˇb c a b c;B D 2ˇc a a b c;C D 2ˇa b a b c; (19). Types of centred lattices. Note that when we said above — under the heading unit cells, that there would be only one point per cell, we meant primitive cells. The Bravais lattice is the same as the lattice formed by all the. 730 Spring Term 2004 PSSA Cubic. Lattice with a basis Example: alpha quartz (SiO 2) The simulation cell with its primitive lattice vectors and its basis The crystal is made from an infinite number of simulation cells •The lattice vectors define the Bravais lattice •The atoms in each cell define the "basis" of the lattice (nothing to do with basis sets!). Space region, translated by all lattice vectors (“tiling”) will fill all space (not unique). This choice of basis vectors, in turn, determines a reciprocal lattice in which the Bloch wavevector k is periodic. When I consider the primitive unit cell of a fcc lattice (red in the image below) the lattice points are only partially part of the primitive unit cell. The resulting primitive lattice is indicated by the filled nodes in Fig. This area is, however, often used due to its regular shape and is called a conventional cell. Clearly, the lattice R does not select its lattice vectors, If E a is a basis for a primitive cubic lattice,. What this means is that if one succeeds in breaking the primitive, even with some small probability, then one can alsosolve any instance ofa certain lattice problem. Divisibility Lemma. For example, the following primitive_setting is the result of transforming a C-centred monoclinic cell: from cctbx import crystal. 1 Definition The convex hull of integer-valued points is called a lattice polytope and, if all the vertices are drawn from {0,1,,k}d, is refereed to as a lattice (d,k)-polytope. The shaded hexagon is the first Brillouin zone with Γ indicating the centre, and K + and K− showing two non-equivalent corners. qthat are correct mod 2 and mod q, respectively, does not produce a short integer vector. 1) In each of the following cases indicate whether the structure is a Bravais lattice. c) For X-ray diffraction with both the incident and diffracted beams in the plane of the crystal, the diffraction peaks can be labeled with the reciprocal lattice vectors Ghk = hğı + kgz. For a 3D lattice, we can find threeprimitive lattice vectors (primitive translation vectors), such that any translation vector can be written as!⃗=\$. de We show that with respect to a certain class of norms the so called shortest lattice vector problem is polynomial-time Turing (Cook). In terms of the cube edge a the primitive translation vectors are 1200 Figure 13 The rhombohedral primitive cell of the face-cen- tered cubic crystal. Reciprocal Lattice in 3D • The primitive vectors of the reciprocal lattice are defined by the vectors b i that satisfy b i ⋅a j = 2πδ ij, where δ ii = 1, δ ij = 0 if i ≠j • How to find the b's? • Note: b 1 is orthogonal to a 2 and a 3, etc. Because all three cell-edge lengths are the same in a cubic unit cell, it doesn't matter what orientation is used for the a, b, and c axes. LatticeData [lattice,"Classes"] gives a list of the classes in which lattice occurs. We can generate all the points of the fcc lattice is described by l l1a1 l2a2 l3a3 with l1, l2, and l3 integers. Such net is called ‘oblique’ as shown in the figure. Let a1, a2, and a3 be a set of primitive vectors of the direct lattice. graphic primitive is based on the worst-case hardness of lattice problems. 6: (a) Experimental apparatus for low energy electron diffraction. Plotting a 2D crystal lattice from two primitive Learn more about plotting, lattice MATLAB. The primitive translation vectors of the hexagonal lattice are given by: a1 = a 2 √ 3ˆx + ˆy , a2 = a 2 − √ 3ˆx+ ˆy , a3 = cz. C CONSTR = Lattice constant of reciprocal lattice C FACTOR = Lattice constant / volume of primitive cell C C RBASIS(i,j) = basis vectors of reciprocal lattice primitive cell C (face centered cubic direct lattice) C i = 1,2,3 vector index C j = 1,2,3 coordinate index (x,y,z) C. For obvious reasons the term Bravais lattice is often also used for the set of vectors {R n}. The basis consists of one or several atoms. If the primitive unit cell is shifted by all R n, the whole space is filled without gaps and without overlap. Divisibility Lemma. The primitive lattice vectors of the reciprocal lattice are defined as A D 2ˇb c a b c;B D 2ˇc a a b c;C D 2ˇa b a b c; (19). reciprocal-lattice vectors will yield a 2…n dot product with all real-lattice vectors. of lattice vectors T. Vectors a and b are not a set of primitive lattice vectors and the shaded area is not a primitive unit cell. (d) Briefly discuss what is unique about graphene and its. The rhombohedral primitive vectors are given by the relations from above and inserting the hexagonal lattice constants, you obtain:. 1 Crystal Structures 7 The volume of the primitive unit cell in the reciprocal lattice is (2π)3/V. the crystal structure and the reciprocal lattice vectors. \eqref{eq:orthogonalityCondition}. The Wigner-Seitz primitive cell of the reciprocal lattice is known as the first Brillouin zone. Primitive lattice vectors are used to define a crystal translation vector, T, and also gives a lattice cell of smallest volume for a particular lattice. Reciprocal lattice vectors are extremely important to nearly all aspects of the properties of materials. It can be translated by integer multible of primitive vectors according to. A Bravais lattice is a discrete infinite array of points generated by linear integer combinations of 3 independent primitive vectors: {n1a1 + n2a2 + n3a3 | n1, n2, n3 ∈ Z}. The vectors a, appearing In definition (b) of a Bravals lattice a. Reciprocal Lattice to sc Lattice • The primitive translation vectors of a sc lattice: • The primitive translation vectors of the reciprocal lattice: The reciprocal lattice is a sc lattice, with lattice constant 2π/a. A crystal system is described by three basis vectors. Lattice Vectors 2D. is to Ibe but with the of The size of the conventional cell is given by the lattice constant a. A parallelepiped whose edges are defined by the primitive translations of a crystal lattice; it is a unit cell of minimum volume Explanation of Primitive lattice vector. n], a > 0, where Z denotes the set of integers and a is the lattice distance. Fundamental types of crystal lattices. A lattice is defined by a set of primitive lattice vectors, such as a1 and a2 in the two dimensional example. In 1848, the French physicist and crystallographer Auguste Bravais (1811-1863) established that in three-dimensional space only fourteen different lattices may be constructed. It is a cell of the minimum volume which can fill all space when applying convenient translation operations. The shaded hexagon is the first Brillouin zone with Γ indicating the centre, and K + and K− showing two non-equivalent corners. More formally, a multilattice M is a union of. Honeycomb: P and Q are. Why is it not possible to determine the lattice constant using this method ? PHE-13 2. Each crystal lattice has an associated reciprocal lattice which makes calculation of the intensities and positions of peaks much easier. •Previously, we noted all crystal structures could be specified by a set of Bravais lattice vectors, when describing a lattice you must either use the primitive vectors or add a set of basis vectors (e. where −→z 0 is the unit vector along the z-axis, which. The angles a1 ∧a2, a2 ∧a3 and a3 ∧a1 are conventionally labelled γ, α and β respectively. In total, there are 14 ways of arranging atoms in crystals, which are called the 14 Bravais lattices [3]. I basically need to define my own coordinate system that is not the standard cartesian one with those vectors and display the lattice points like you did. This is the nomenclature for "primitive" vectors in solid-state crystallography, but in LAMMPS the unit cell they determine does not have to be a "primitive cell" of minimum volume. A multilattice is a set of atomic sites that do not constitute a lattice because the points of a multilattice are not all translationally equiva-lent. The ‘unit cell’ is a volume of space which will tile under lattice translations; a ‘primitive unit cell’ has one primitive lattice point per unit cell. The BCC and FCC structures are the most commonly found among most crystalline materials. The face-centred cubic lattice is the union of the primitive cubic lattice with its translates by the three centring vectors. In Figure 3, we indicate that there are many variations on the cubic lattice theme, where the three primitive vectors may be of different lengths, and may not be at right angles to each other. The current state-of-the-art in lattice-based DSSs is the proposed scheme by Ducas et al. 9x10E-10m, calculate the atomic positions in the {110} plane taking the lower left atom as the origin. Like primitive vectors, the choice of primitive unit cell is not unique (Fig. How can you determine the point group and the Bravais lattice of this crystal? 3. Write down the primitive translation vectors of the simple cubic lattice. It is not unique, but the convention is to choose the smallest primitive vectors. •For example, consider the non-primitive Fc (FCC) lattice: •By selecting shorter vectors a 1, a 2, and a 3, we can define a primitive rhombohedral lattice. Remember that the primitive cell only contain a single atom. 3: Primitive vectors for FCC lattice Diamond and Zinc Blende Structures: Almost all semiconductors of technological interests have an underlying FCC lattice, except that they. e eiK~ (~r+R~) = eiK~r~, where ~r is an arbitrary vector and R~ is a lattice vector). Due Monday, December 4, in lecture Problem 1 [15 points] (Ashcroft & Mermin problem 4. G is called a reciprocal lattice vector. 1: Unit cells for a at (2D) CuO 2 plane and for a real (3D) CuO 2 sheet. :2008954946 In semiconductor crystal materials, atoms are located periodically, with three primitive basis vectors, a, b, and c. Draw this primitive cell. Figure 2: The distinction between primitive and non-primitive lattice vectors in 2 dimensions; all lattice points can be described by an integral combination of primitive lattice vectors. It is found that the maximum and minimum numbers for lattice constants are 16 for Triclinic and Face-centered orthorhombic lattices, and 1 for Primitive orthorhombic, Primitive tetragonal and Primitive cubic lattices. b) The general reciprocal lattice vector G k 1 b 1 k 2 b 2 k 3 b 3. as the primitive vectors of the crystal. Bravais lattice - An infinite array of discrete points generated by a set of discrete translation operations described by where n i are integers, and a i are the primitive vectors , which span the lattice. When all of the lattice points are equivalent, it is called Bravais. Vector derivatives September 7, 2015 Now, using first the constancy of the Cartesian unit vectors and then the orthogonality of the basis, this reducesto ^i @ @x v x. Lattice + basis specifies a unit cell. If there is a. It gives 14 3D Bravais lattice. 3, defines the unit cell. In the plane, point lattices can be constructed having unit cells in the shape of a square, rectangle, hexagon, etc. The hexagonal unit cell is a prism with angles 120° and 60° between the sides. For any choice of position vector R, the lattice looks exactly the same. The unit cell shape for each crystal system can be described by the relative lengths of the unit vectors and the angles between them. There are two classes of crystal lattices. of the spins make 120owith each other in each triangle to satisfy ground state condition, which we will discuss in detail in Section 3. The entire wikipedia with video and photo galleries for each article. What it does is takes an initial vector a and b of the form [x,y] and propagates it through space to make a lattice. The Minkowski length represents the largest possible number of factors in a factorization of polynomials with exponent vectors in P, and shows up in lower bounds for the minimum. The height of the cell is defined by the vector, , which is oriented vertically at 90 to both and. These translation vectors connect the lattice pt at the origin to the points at the face centres. However, a given set of primitive vectors does uniquely define a Bravais lattice. The triclinic system has one Bravais lattice, which is also the conventional lattice for this system. They are crystallographically equivalent in this hexagonal system. fcc becomes bcc). b 1 is perpendicular to a 2 and a 3. A lattice is a set of all position vectors formed by translations of a given set of non-coplanar vectors called primitive vectors. 7) bj2 <2i 1 Ib2 for 1 j in,. The main contribution of this work is the significant improvement in the rejection sampling stage. Conditions for primitive-lattice-vector-direction equal contrasts in four-beam-interference lithography Justin L. =O =Cu A possible choice of the primitive cell. Primitive unit cell: A volume in space, when translated through all the lattice vectors in a Bravais lattice, fills the entire space without voids or overlapping itself, is a primitive unit cell (see Figs. The basis vectors a1, a2 and a3 define the the unit cell; their mag- nitudes a1, a2 and a3 respectively, are the lattice parameters of the unit cell. The reciprocal lattice of a Bravais lattice is defined as all wave vectors satisfying for all points in the infinite Bravais lattice. Lattice and Crystal - Simple View. vasp file (cf. The vectors, a, b and c, that define a crystal lattice. Interstitial Positions (online up to 8 atoms/cell) [DOI: 10. Different lattice types are possible within each of the crystal systems since the lattice points within the unit cell may be arranged in different ways. It turns out that there is a one-to-one correspondence between primitive sets of the direct and reciprocal lattices. 1 Reciprocal Lattice Vectors and First Brillouin Zone Reciprocal lattice vectors of a lattice are defined to be the wavevectors h that satisfy exp(ih · R) = 1, (1) for any lattice translation vector R given by (2) Here Pl, P2, P3 are three arbitrary integers and a1, a2, a3 are three primitive translation vectors that define the lattice. The unit cell in three dimensions is a parallelepiped, whose sides are the primitive translation vectors (see Fig. In this method, a fourth axis, u, in the plane of the x and y axes is introduced. (i) The reciprocal lattice to the body centred cubic (iii) The reciprocal lattice to a face centred cubic lattice(fcc) is body centred. step 2) with the new primitive lattice vectors. 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https://plotly-r.com/introduction-1.html | # 15 Introduction
Linking of multiple data views offers a powerful approach to visualization as well as communication of structure in high-dimensional data. In particular, linking of multiple 1-2 dimensional statistical graphics can often lead to insight that a single view could not possibly reveal. For decades, statisticians and computer scientists have been using and authoring systems for multiple linked views, many of which can be found in the ASA’s video library. Some noteworthy videos include focusing and linking, missing values, and exploring Tour De France data (Swayne, Cook, and Buja 1998; Theus and Urbanek 2008).
These early systems were incredibly sophisticated, but the interactive graphics they produce are not easily shared, replicated, or incorporated in a larger document. Web technologies offer the infrastructure to address these issues, which is a big reason why many modern interactive graphics systems are now web based. When talking about interactive web-based graphics, it’s important to recognize the difference between a web application and a purely client-side webpage, especially when it comes to saving, sharing, and hosting the result.
A web application relies on a client-server relationship where the client’s (i.e., end user) web browser requests content from a remote server. This model is necessary whenever the webpage needs to execute computer code that is not natively supported by the client’s web browser. As Section 17 details, the flexibility that a web application framework, like shiny, offers is an incredibly productive and powerful way to link multiple data views; but when it comes to distributing a web application, it introduces a lot of complexity and computational infrastructure that may or may not be necessary.
Figure 15.1 is a basic illustration of the difference between a web application and a purely client-side web page. Thanks to JavaScript and HTML5, purely client-side web pages can still be dynamic without any software dependencies besides a modern web browser. In fact, Section 16.1 outlines plotly’s graphical querying framework for linking multiple plots entirely client-side, which makes the result very easy to distribute (see Section 10). There are, of course, many useful examples of linked and dynamic views that can not be easily expressed as a database query, but a suprising amount actually can, and the remainder can likely be quickly implemented as a shiny web application.
The graphical querying framework implemented by plotly is inspired by Buja et al. (1991), where direct manipulation of graphical elements in multiple linked plots is used to perform data base queries and visually reveal high-dimensional structure in real-time. D. Cook, Buja, and Swayne (2007) goes on to argue this framework is preferable to posing data base queries dynamically via a menus, as described by Ahlberg, Williamson, and Shneiderman (1991), and goes on to state that “Multiple linked views are the optimal framework for posing queries about data”. The next section shows you how to implement similar graphical queries in a standalone webpage using R code.
### References
Ahlberg, Christopher, Christopher Williamson, and Ben Shneiderman. 1991. “Dynamic Queries for Information Exploration: An Implementation and Evaluation.” In ACM Chi ’92 Conference Proceedings, 21:619–26.
Buja, Andreas, John Alan McDonald, John Michalak, and Werner Stuetzle. 1991. “Interactive data visualization using focusing and linking.” IEEE Proceedings of Visualization, February, 1–8.
Cook, Dianne, Andreas Buja, and Deborah F Swayne. 2007. “Interactive High-Dimensional Data Visualization.” Journal of Computational and Graphical Statistics, December, 1–23.
Swayne, Deborah F, Dianne Cook, and Andreas Buja. 1998. “XGobi: Interactive Dynamic Data Visualization in the X Window System.” Journal of Computational and Graphical Statistics 7 (1): 113–30.
Theus, Martin, and Simon Urbanek. 2008. Interactive Graphics for Data Analysis: Principles and Examples. Chapman; Hall / CRC. | 2019-06-19 07:13:23 | {"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": 0.19433322548866272, "perplexity": 4086.0694314098205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998923.96/warc/CC-MAIN-20190619063711-20190619085711-00498.warc.gz"} |
https://socratic.org/questions/how-do-you-find-slope-given-3y-9-0 | # How do you find slope given 3y-9=0?
Jun 10, 2016
Slope: $0$
#### Explanation:
$3 y - 9 = 0$ (equivalent to $y = 3$) is the equation of a horizontal line.
Slope is defined as $\left(\text{change in "y)/("corresponding change in } x\right)$
but no matter what change we make to $x$ the value of $y$ states equal to $3$;
so the change in $y$ is always $0$ | 2022-10-05 08:38:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 9, "mathjax_inline_tex": 1, "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": 0.7824206948280334, "perplexity": 600.6094596176315}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337595.1/warc/CC-MAIN-20221005073953-20221005103953-00507.warc.gz"} |
https://en.wikipedia.org/wiki/Equatorial_bulge | # Equatorial bulge
For the feature on some of Saturn's moons, see equatorial ridge.
An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the force exerted by its rotation. A rotating body tends to form an oblate spheroid rather than a sphere. The Earth has an equatorial bulge of 42.77 km (26.58 mi): that is, its diameter measured across the equatorial plane (12,756.27 km (7,926.38 mi)) is 42.77 km more than that measured between the poles (12,713.56 km (7,899.84 mi)). An observer standing at sea level on either pole, therefore, is 21.36 km closer to Earth's centrepoint than if standing at sea level on the equator. The value of Earth's radius may be approximated by the average of these radii.
An often-cited result of Earth's equatorial bulge is that the highest point on Earth, measured from the center outwards, is the peak of Mount Chimborazo in Ecuador, rather than Mount Everest. But since the ocean also bulges, like the Earth and the atmosphere, Chimborazo is not as high above sea level as Everest is.
The standard formula for this force is the relationship ${\displaystyle F_{c}=Mv^{2}/R}$. However, velocity at the surface is equal to the product of radius and rotational velocity, and therefore the force is directly proportional to radius. Viewing the globe as a series of rotating discs, the radius R toward the poles gets very small and thus a smaller force is produced for the same rotational velocity (approaching zero at the pole). Moving towards the equator, v^2 increases much faster than R, thus producing the greatest force at the equator. In addition, because the Earth’s dense core is included in the cross sectional disc at the equator, it contributes more to the mass of the disc. Similarly, there is a bulge in the water envelope of the oceans surrounding Earth; this bulge is created by the greater centrifugal force at the equator and is independent of tides. Sea level at the equator is 21.36 km higher than sea level at the poles, in terms of distance from the center of the planet.
## The equilibrium as a balance of energies
Fixed to the vertical rod is a spring metal band. When stationary the spring metal band is circular in shape. The top of the metal band can slide along the vertical rod. When spun, the spring-metal band bulges at its equator and flattens at its poles in analogy with the Earth.
Gravity tends to contract a celestial body into a sphere, the shape for which all the mass is as close to the center of gravity as possible. Rotation causes a distortion from this spherical shape; a common measure of the distortion is the flattening (sometimes called ellipticity or oblateness), which can depend on a variety of factors including the size, angular velocity, density, and elasticity.
To get a feel for the type of equilibrium that is involved, imagine someone seated in a spinning swivel chair, with weights in their hands. If the person in the chair pulls the weights towards them, they are doing work and their rotational kinetic energy increases. The increase of rotation rate is so strong that at the faster rotation rate the required centripetal force is larger than with the starting rotation rate.
Something analogous to this occurs in planet formation. Matter first coalesces into a slowly rotating disk-shaped distribution, and collisions and friction convert kinetic energy to heat, which allows the disk to self-gravitate into a very oblate spheroid.
As long as the proto-planet is still too oblate to be in equilibrium, the release of gravitational potential energy on contraction keeps driving the increase in rotational kinetic energy. As the contraction proceeds the rotation rate keeps going up, hence the required force for further contraction keeps going up. There is a point where the increase of rotational kinetic energy on further contraction would be larger than the release of gravitational potential energy. The contraction process can only proceed up to that point, so it halts there.
As long as there is no equilibrium there can be violent convection, and as long as there is violent convection friction can convert kinetic energy to heat, draining rotational kinetic energy from the system. When the equilibrium state has been reached then large scale conversion of kinetic energy to heat ceases. In that sense the equilibrium state is the lowest state of energy that can be reached.
The Earth's rotation rate is still slowing down, though gradually, by about two thousandths of a second per rotation every 100 years.[1] Estimates of how fast the Earth was rotating in the past vary, because it is not known exactly how the moon was formed. Estimates of the Earth's rotation 500 million years ago are around 20 modern hours per "day".
The Earth's rate of rotation is slowing down mainly because of tidal interactions with the Moon and the Sun. Since the solid parts of the Earth are ductile, the Earth's equatorial bulge has been decreasing in step with the decrease in the rate of rotation.
## Differences in gravitational acceleration
The forces at play in the case of a planet with an equatorial bulge due to rotation.
Red arrow: gravity
Green arrow, the normal force
Blue arrow: the resultant force
The resultant force provides required centripetal force. Without this centripetal force frictionless objects would slide towards the equator.
In calculations, when a coordinate system is used that is co-rotating with the Earth, the vector of the notional centrifugal force points outward, and is just as large as the vector representing the centripetal force.
Because of a planet's rotation around its own axis, the gravitational acceleration is less at the equator than at the poles. In the 17th century, following the invention of the pendulum clock, French scientists found that clocks sent to French Guiana, on the northern coast of South America, ran slower than their exact counterparts in Paris. Measurements of the acceleration due to gravity at the equator must also take into account the planet's rotation. Any object that is stationary with respect to the surface of the Earth is actually following a circular trajectory, circumnavigating the Earth's axis. Pulling an object into such a circular trajectory requires a force. The acceleration that is required to circumnavigate the Earth's axis along the equator at one revolution per sidereal day is 0.0339 m/s². Providing this acceleration decreases the effective gravitational acceleration. At the equator, the effective gravitational acceleration is 9.7805 m/s2. This means that the true gravitational acceleration at the equator must be 9.8144 m/s2 (9.7805 + 0.0339 = 9.8144).
At the poles, the gravitational acceleration is 9.8322 m/s2. The difference of 0.0178 m/s2 between the gravitational acceleration at the poles and the true gravitational acceleration at the equator is because objects located on the equator are about 21 kilometers further away from the center of mass of the Earth than at the poles, which corresponds to a smaller gravitational acceleration.
In summary, there are two contributions to the fact that the effective gravitational acceleration is less strong at the equator than at the poles. About 70 percent of the difference is contributed by the fact that objects circumnavigate the Earth's axis, and about 30 percent is due to the non-spherical shape of the Earth.
The diagram illustrates that on all latitudes the effective gravitational acceleration is decreased by the requirement of providing a centripetal force; the decreasing effect is strongest on the equator.
## Satellite orbits
The fact that the Earth's gravitational field slightly deviates from being spherically symmetrical also affects the orbits of satellites through secular orbital precessions.[2][3][4] They depend on the orientation of the Earth's symmetry axis in the inertial space, and, in the general case, affect all the Keplerian orbital elements with the exception of the semimajor axis. If the reference z axis of the coordinate system adopted is aligned along the Earth's symmaetry axis, then only the longitude of the ascending node Ω, the argument of pericenter ω and the mean anomaly M undergo secular precessions.[5]
Such perturbations, which were earlier used to map the Earth's gravitational field from space,[6] may play a relevant disturbing role when satellites are used to make tests of general relativity[7] because the much smaller relativistic effects are qualitatively indistinguisgable from the oblateness-driven disturbances.
## Other celestial bodies
Generally any celestial body that is rotating (and that is sufficiently massive to draw itself into spherical or near spherical shape) will have an equatorial bulge matching its rotation rate. Saturn is the planet with the largest equatorial bulge in Earth's Solar System (11808 km, 7337 miles).
The following is a table of the equatorial bulge of some major celestial bodies of the Solar System:
Body Equatorial diameter Polar diameter Equatorial bulge Flattening ratio
Earth 12,756.27 km 12,713.56 km 42.77 km 1:298.2575
Mars 6,805 km 6,754.8 km 50.2 km 1:135.56
Ceres 975 km 909 km 66 km 1:14.77
Jupiter 143,884 km 133,709 km 10,175 km 1:14.14
Saturn 120,536 km 108,728 km 11,808 km 1:10.21
Uranus 51,118 km 49,946 km 1,172 km 1:43.62
Neptune 49,528 km 48,682 km 846 km 1:58.54
## Mathematical expression
The flattening coefficient ${\displaystyle f}$ for the equilibrium configuration of a self-gravitating spheroid, composed of uniform density incompressible fluid, rotating steadily about some fixed axis, for a small amount of flattening, is approximated by:[8]
${\displaystyle f={\frac {a_{e}-a_{p}}{a}}={5 \over 4}{\omega ^{2}a^{3} \over GM}={15\pi \over 4}{1 \over GT^{2}\rho }}$
where ${\displaystyle a_{e}=a(1+{f \over 3})}$ and ${\displaystyle a_{p}=a(1-{2f \over 3})}$ are respectively the equatorial and polar radius, ${\displaystyle a}$ is the mean radius, ${\displaystyle \omega ={2\pi \over T}}$ is the angular velocity, ${\displaystyle T}$ is the rotation period, ${\displaystyle G}$ is the universal gravitational constant, ${\displaystyle M\simeq {4 \over 3}\pi \rho a^{3}}$ is the total body mass, and ${\displaystyle \rho }$ is the body density.
## References
1. ^ Hadhazy, Adam. "Fact or Fiction: The Days (and Nights) Are Getting Longer". Scientific American. Retrieved 5 December 2011.
2. ^
3. ^
4. ^
5. ^ King-Hele, D. G. (1961). "The Earth's Gravitational Potential, deduced from the Orbits of Artificial Satellites". Geophysical Journal. 4 (1): 3–16. Bibcode:1961GeoJ....4....3K. doi:10.1111/j.1365-246X.1961.tb06801.x.
6. ^ King-Hele, D. G. (1983). "Geophysical researches with the orbits of the first satellites". Geophysical Journal. 74 (1): 7–23. Bibcode:1983GeoJ...74....7K. doi:10.1111/j.1365-246X.1983.tb01868.x.
7. ^ Renzetti, G. (2012). "Are higher degree even zonals really harmful for the LARES/LAGEOS frame-dragging experiment?". Canadian Journal of Physics. 90 (9): 883–888. Bibcode:2012CaJPh..90..883R. doi:10.1139/p2012-081.
8. ^ "Rotational Flattening". utexas.edu. | 2016-12-04 17:04:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 11, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7739793062210083, "perplexity": 694.6019330942349}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541324.73/warc/CC-MAIN-20161202170901-00117-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://kkjkok.blogspot.com/2013/03/introduction-to-machine-learning-part-1.html | ## Friday, March 1, 2013
### Introduction to Machine Learning, Part 1: Parameter Estimation
RISE OF THE MACHINES
Machine learning is a fascinating field of study which is growing at an extreme rate. As computers get faster and faster, we attempt to harness the power of these machines to tackle difficult, complex, and/or unintuitive problems in biology, mathematics, engineering, and physics. Many of the purely mathematical models used in these fields are extremely simplified compared to their real-world counter parts (see the spherical cow). As scientists attempt more and more accurate simulations of our universe, machine learning techniques have become critical to building accurate models and estimating experimental parameters.
Also, even very basic machine learning can be used to effectively block spam email, which puts it somewhere between water and food on the "necessary for life" scale.
BAYES FORMULA - GREAT FOR GROWING BODIES
Machine learning seems very complicated, and advanced methods in the field are usually very math heavy - but it is all based in common statistics, mostly centered around the ever useful Gaussian distribution. One can see the full derivation for the below formulas here (PDF) or here (PDF), a lot of math is involved but it is relatively straightforward as long as you remember Bayes rule:
$posterior \:=\: likelihood \:\times\: prior$
HE DIDN'T MEAN IT
From the links above, we can see that our best estimate for the mean $\mu$ given a known variance $\sigma^2$ and some Gaussian distributed data vector $X$ is:
$\sigma(\mu)^2_n = {\Large \frac{1}{\frac{N}{\sigma^2}+\frac{1}{\sigma_o^2}}}$
$\mu_n = {\Large \frac{1}{\frac{N}{\sigma^2}+\frac{1}{\sigma^2}}(\frac{\sum\limits_{n=1}^N x_n}{\sigma^2}+\frac{\mu_o}{\sigma_o^2})}$
where $N$ (typically 1) represents the number of $X$ values used to generate the mean estimate $\mu$ (just $x_n$ if $N$ is 1), $\mu_o$ is the previous "best guess" for the mean ($\mu_{n-1}$), $\sigma_o^2$ is the previous confidence in the "best guess" for $\mu$ ($\sigma(\mu)^2_{n-1})$), and $\sigma$ was known prior to the calculation. Lets see what this looks like in python.
#!/usr/bin/python
import numpy as np
import matplotlib.pyplot as plot
total_obs = 1000
primary_mean = 5.
primary_var = known_var = 4.
x = np.sqrt(primary_var)*np.random.randn(total_obs) + primary_mean
f, axarr = plot.subplots(3)
f.suptitle("Unknown mean ($\$$\mu=\$$"+primary_mean+"), known variance ($\$$\sigma^2=\$$"+known_var+")")
y0label = "Timeseries"
y1label = "Estimate for mean"
y2label = "Doubt in estimate"
axarr[0].set_ylabel(y0label)
axarr[1].set_ylabel(y1label)
axarr[2].set_ylabel(y2label)
axarr[0].plot(x)
prior_mean = 0.
prior_var = 1000000000000.
all_mean_guess = []
all_mean_doubt = []
for i in range(total_obs):
posterior_mean_doubt = 1./(1./known_var+1./prior_var)
posterior_mean_guess = (prior_mean/prior_var+x[i]/known_var)*posterior_mean_doubt
all_mean_guess.append(posterior_mean_guess)
all_mean_doubt.append(posterior_mean_doubt)
prior_mean=posterior_mean_guess
prior_var=posterior_mean_doubt
axarr[1].plot(all_mean_guess)
axarr[2].plot(all_mean_doubt)
plot.show()
This code results in this plot:
We can see that there are two basic steps - generating the test data, and iteratively estimating the "unknown parameter", in this case the mean. We begin our estimate for the mean at any value (prior_mean = 0) and set the prior_var variable extremely large, indicating that our confidence in the mean actually being 0 is extremely low.
V FOR VARIANCE
What happens in the opposite (though still "academic" case) where we know the mean but not the variance? Our best estimate for unknown variance $\sigma^2$ given the mean $\mu$ and some Gaussian distributed data vector $X$ will use some extra variables, but still represent our estimation process.:
$a_n = {\Large a_o + \frac{N}{2}}$
$b_n = {\Large b_o + \frac{1}{2}\sum\limits^N_{n=1}(x_n-\mu)^2}$
$\lambda = {\Large \frac{a_n}{b_n}}$
$\sigma(\lambda)^2 = {\Large \frac{a_n}{b_n^2}}$
where
$\lambda = {\Large \frac{1}{\sigma^2}}$
This derivation is made much simpler by introducing the concept of precision ($\lambda$), which is simply 1 over the variance. We estimate the precision, which can be converted back to variance if we prefer. $\mu$ is known in this case.
#!/usr/bin/python
import numpy as np
import matplotlib.pyplot as plot
total_obs = 1000
primary_mean = known_mean = 5
primary_var = 4
x = np.sqrt(primary_var)*np.random.randn(total_obs) + primary_mean
all_a = []
all_b = []
all_prec_guess = []
all_prec_doubt = []
prior_a=1/2.+1
prior_b=1/2.*np.sum((x[0]-primary_mean)**2)
f,axarr = plot.subplots(3)
f.suptitle("Known mean ($\$$\mu=\$$"+known_mean+"), unknown variance ($\$$\sigma^2=\$$"+primary_var+"; $\$$\lambda\$$="+1./primary_var+")") y0label = "Timeseries" y1label = "Estimate for precision" y2label = "Doubt in estimate" axarr[0].set_ylabel(y0label) axarr[1].set_ylabel(y1label) axarr[2].set_ylabel(y2label) axarr[0].plot(x) for i in range(1,total_obs): posterior_a=prior_a+1/2. posterior_b=prior_b+1/2.*np.sum((x[i]-known_mean)**2) all_a.append(posterior_a) all_b.append(posterior_b) all_prec_guess.append(posterior_a/posterior_b) all_prec_doubt.append(posterior_a/(posterior_b**2)) prior_a=posterior_a prior_b=posterior_b axarr[1].plot(all_prec_guess) axarr[2].plot(all_prec_doubt) plot.show() Here I chose to set the values for prior_a and prior_b to the "first" values of the estimation, we could just as easily have reversed the formulas by setting the precision to some value, and the "doubt" about that precision very large, then solving a system of two equations, two unknowns for a and b. INTO THE UNKNOWN(S) What happens if both values are unknown? All we really know is that the data is Gaussian distributed!$\nu_o={\Large \kappa_o-1}\mu_n={\Large \frac{\kappa_o\mu_o+\overline{X}N}{\kappa_o+N}}\kappa_n={\Large \kappa_o+N}\nu_n={\Large \nu_o+N}\sigma_n^2={\Large \frac{\nu_o\sigma^2+(N-1)s^2+\frac{\kappa_oN}{\kappa_o+N}(\overline{X}-\mu_o)^2}{\nu_n}}s={\Large var(X)}$The technique here is the same as other derivations for unknown mean, known variance and known mean, unknown variance. To estimate the mean and variance of data taken from a single Gaussian distribution, we need to iteratively update our best guesses for both mean and variance. In many cases,$N=1$, so the value for$s$is not necessary and$\overline{X}$becomes$x_n$. Let's look at the code. #!/usr/bin/python import matplotlib.pyplot as plot import numpy as np total_obs = 1000 primary_mean = 5. primary_var = 4. x = np.sqrt(primary_var)*np.random.randn(total_obs) + primary_mean f, axarr = plot.subplots(3) f.suptitle("Unknown mean ($\$$\mu=\$$"+primary_mean+"), unknown variance ($\$$\sigma^2=\$$"+primary_var+")") y0label = "Timeseries" y1label = "Estimate for mean" y2label = "Estimate for variance" axarr[0].set_ylabel(y0label) axarr[1].set_ylabel(y1label) axarr[2].set_ylabel(y2label) axarr[0].plot(x) prior_mean = 0. prior_var = 1. prior_kappa = 1. prior_v = 0. all_mean_guess = [] all_var_guess = [] for i in range(total_obs): posterior_mean = (prior_kappa*prior_mean+x[i])/(prior_kappa + 1) posterior_var = (prior_v*prior_var + prior_kappa/(prior_kappa + 1)*(x[i]-prior_mean)**2)/(prior_v + 1) prior_kappa += 1 prior_v += 1 all_mean_guess.append(posterior_mean) all_var_guess.append(posterior_var) prior_mean = posterior_mean prior_var = posterior_var axarr[1].plot(all_mean_guess) axarr[2].plot(all_var_guess) plot.show() We can see that the iterative estimation has successfully approximated the mean and variance of the underlying distribution! We "learned" these parameters, given only a dataset and the knowledge that it could be approximated by the Gaussian distribution. In the next installment of this series, I will cover linear regression. These two techniques (parameter estimation and linear regression) form the core of many machine learning algorithms - all rooted in basic statistics. SOURCES Pattern Recognition and Machine Learning, C. Bishop Bayesian Data Analysis, A. Gelman, J. Carlin, H. Stern, and D. Rubin Classnotes from Advanced Topics in Pattern Recognition, UTSA, Dr. Zhang, Spring 2013 #### 3 comments: 1. Is there a reason why, in the first example (known variance, unknown mean), the "doubt" is so much lower than the error in the estimate, for the first several hundred samples? For example, after 200 samples, the "doubt" is well below 0.1, but the estimate of the mean is still wrong by about 0.1. It seems that your "doubt" figure is much too optimisitic. If you're presenting these figures to somebody who doesn't want a ton of explanation (like maybe your manager at work) it would probably be better to use the standard deviation, or even better a 95% confidence interval, instead of your "doubt" value. 1. The variable I labeled "doubt" is the variance of precision in statistics lingo ($/frac{1}{\sigma^2}$). I chose a different label because I felt it was a simpler, more general term to describe what that parameter really means, especially to people less familiar with statistics terminology. The tricky part about "doubt" is that because it is a variance, it suffers a serious problem - namely that something can have a very low deviation, doubt, or variance, and still have an incorrect (or biased) result. That is what you see in the second example - the algorithm converged on a given solution based on the data we input, but we know that answer is incorrect, since we are lucky enough to know "ground truth". Maybe given more samples we would eventually have arrived at the true answer (looking at the upward trend in the estimate), but all we have to go by right now are these 1000 samples. This is a problem in nearly every scientific field - how do we validate our results and eliminate possible bias, when we have no idea what "ground truth" actually is? And specifically for machine learning - how do we validate the results of algorithms on our datasets, when the volume of data eliminates classic "check by hand" approaches, and we are using these approaches in an exploratory fashion i.e. we have no conception of what the answers should be? I really, really like your idea about using the 95% confidence interval. I think it is is a superior approach to what I am doing now - the same information can be displayed in two graphs instead of three. I will definitely look into that! Thanks for your comments 2. Edit: the formula above should be$\frac{1}{\sigma^2}\$
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http://afinet.com.br/a08r9hi/5185e0-how-to-start-echo-pb-2520-blower | Select the word to the right. Click the Symbol button see some popular or recently used symbols… Here’s the result: RELATED: How to Position Images and Other Objects in Microsoft Word. Word and Outlook. Use the standard Word Alt + X symbol shortcut 2194 + Alt + X. (If the window is too narrow, you see the Symbols button, from which you can choose Equation or Symbol.) Sorry if the confusion on my part increased the confusion for you. Release the mouse button to finish drawing the arrow. The following table lists many common symbols, together with their name, pronunciation, and the related field of mathematics. It creates linkages between otherwise very different concepts and experiences. Select from the current position to the beginning of the line. It indicates or signifies as representing an idea, object, or relationship. Arrows appear where I tab across in a word document how do I remove them I let someone use my computer and whenever I use word there asrew arrows visible on the screen but not when i print. Click on the word "yields" and replace it with as many spaces as you need to create an arrow of whatever length you want. Click the tab to access the menu under. An arrow is a graphical symbol, such as ← or →, or a pictogram, used to point or indicate direction.In its simplest form, an arrow is a triangle, chevron, or concave kite, usually affixed to a line segment or rectangle, and in more complex forms a representation of an actual arrow (e.g. Select all document content. ← → ↹ ⇑ ⇓ Arrows Text Symbols - Japanese Emoticons What is the use of arrows fonts in text? Both in OneNote 2013 and Word 2013, the equation mode can be started with Alt+= In there, you can write arrows with a code word like \rightarrow, \leftarrow or \uparrow. If you need help using alt codes find and note down the alt code you need then visit our instructions for using alt codes page. The best way is ‘DON’T’ - go to Import and choose one of these two options. This table explains the meaning of every arrows symbol. There are several ways to get the arrow symbol on Word or Excel or PowerPoint. Click on the word "yields" and replace it with as many spaces as you need to create an arrow of whatever length you want. Formatting symbols are hidden by default. To get a long arrow, click on the operator button and choose the arrow with the word "yields" written over it under common operator structures. Insert | Symbols | Symbol and look for the Left Right arrow symbol. Tip: To make sure that size of arrow fits text length, first type the longest text above or below arrow and press space then type another text below/above. In logic, a set of symbols is commonly used to express logical representation. Hit the "Illustrations" tab and there you can see the "Shapes" to get the work done. All communication get through the use of symbols. In your Office Word document, select “Insert” tab then click “Symbol” on the upper right corner of the screen then click “More Symbols.” Inserting a symbol. Quick Guide for typing the Right Arrow symbol (→) To type the Right Arrow Symbol anywhere on your PC or Laptop keyboard (like in Microsoft Word or Excel), simply press down the Alt key and type 26 using the numeric keypad on the right side of your keyboard.. For Mac users, to get the Rightward arrow symbol, first press Control + Command + Spacebar to bring up the Character viewer. Two items are found in that group: Equation and Symbol. A crosshair symbol will display. One of these several ways is the use of the arrow Alt Code method for Windows users. Unicode table of arrows ( ← ↑ → ↓ ↔ ↕ ⇪ ↹ ⬈ ↘ ↶ … ) - nothing but arrows for use on HTML charset UTF-8 documents and websites. i want to remove these arrows how do i do it This thread is locked. Highlight the specific line you want to center and click on the center dashes in the formatting options. Word, Excel, PowerPoint and Outlook. Shift+Alt+Left arrow key. Microsoft Word has many types of nonprintable symbols such as different types of spaces, tabulations, line or page breaks, etc. On the Insert tab, in the Symbols group, click the arrow under Equation, and then click Insert New Equation. In Microsoft Word 2007 and later, the Show All icon is on the Home tab, in the Paragraph section. Select from the current position to the end of the line. Using them you can quickly type variety of arrows, set symbols, partial, nabla, degree C, angle, etc. Click anywhere on the word document and drag the mouse as long as you want to draw the arrow. Lots of these arrows are from math, but some are also used elsewhere. In the Lines group on the drop-down menu, click the “Line Arrow” option. by using arrows interest you point anything, any topics weather … Shift+Ctrl+Right arrow key. Example here long text in below arrow so we use the following shortcut \rightarrow\below(P = 10Mpa)\above(Ni) Combining all the shortcut into one example. In Word, you can insert mathematical symbols into equations or text by using the equation tools. ALT Codes for arrow, keyboard arrow & dingbat arrow symbols. Press and hold your mouse button, then drag to draw the arrow. A Arrow Symbols is a mark, sign or word. Typing arrow Symbols in Word or Excel or PowerPoint. List of Ms Word shortcut for Mathematical Symbols similar to LaTeX Conclusion. If this toolbar is not visible, click View, Toolbars, and select Standard. There is "Insert" tab at the top of the word document. On the far right of the Word 2016 Insert tab dwells the Symbols group. Finally, finish your equation. You can follow the question or vote as helpful, but you cannot reply to this thread. For up arrow and down arrow showing gas liberation and precipitation use \uparrow or \downarrow followed by space Shortcut for typing arrows of chemical equation in Word 2007 and above. While you can't outright draw free-form in Word, you can use shapes to make certain symbols, like arrows, thick and thin lines, and circles. 5 years ago. The non-printable symbols are also known as Whitespace characters in typography, nonprinting characters in the previous versions of Microsoft products, or formatting marks. Click on the list arrow attached to the Font: box then, using the scroll bar, move down the list of fonts and choose Wingdings (or … This usually happens when you use the tab button to center your lines. Once in a document you can copy it to AutoCorrect and make your own shortcut. Select the word to the left. A lot of symbols can be found this way. The following table shows Unicode symbol, HTML code, CSS code, and official HTML name for the characters categorized under arrow symbols. Click "Line and Connectors" option to access the type of shape i.e. If you are already familiar with using alt codes, simply select the alt code category you need from the table below. The Left Right arrow ↔ sign does NOT have an inbuilt shortcut in Word. U+27B5). Enter the arrow symbol (à) in the With column. Then find an arrow pointing down. You c... 2020-07-13, 79070 , 26 2020-07-13 Louise: Knowing and understanding what all the symbols mean would be easy, if Microsoft had an easy table showing the symbol, providing ... 2020-07-12 Barb: The small squares are the "keep with next" symbol. The problem is, it's hard to categorize them into one place. Straight arrows are good, arrows with kinks and corners in them might work, but sometimes a curved arrow is just what the doctor ordered. Shift+Ctrl+Left arrow key. To insert such a break, press Shift+Enter.--Stefan Blom Microsoft Word MVP "jexie" wrote in message If it is an arrow when over the text (rather than an I-beam mouse pointer) I'm not sure what you are seeing. The image also displays example text with the main formatting symbols. Arrows symbol is a copy and paste text symbol that can be used in any desktop, web, or mobile applications. Shortcut for other types of arrows is. arrow. Quick Guide for typing the Down Arrow symbol (↓) To type the Down Arrow Symbol anywhere on your PC or Laptop keyboard (like in Microsoft Word or Excel), simply press down the Alt key and type 25 using the numeric keypad on the right side of your keyboard.. For Mac users, to get the downward arrow symbol, first press Control + Command + Spacebar to bring up the Character viewer. Below is the complete list of Windows ALT key numeric pad codes for arrow, keyboard arrow & dingbat arrow symbols, their corresponding HTML entity numeric character references and, when available, their corresponding HTML entity named character references. Volunteering to "pay forward" to return help I've received in the Microsoft user community.] You can also use the Symbols option on the Insert tab in Word 2010 and 2007 if you need a specific type of arrow - there are many more to offer. In the image above, you see both symbols because I already used it before that’s why they appeared. Ctrl+A This wikiHow will show you how to use shapes to draw arrows for the computer desktop versions and mobile versions of Word. The right angle arrow indicates a line break. Pressing space after typing the code word automatically transforms it into the desired arrow. So, in Unicode, the arrows gets into one of these blocks: “Miscellaneous Mathematical Symbols-B”, “Supplemental Mathematical Operators”, “Miscellaneous Symbols and Arrows”. Arrow Symbols allow people to go beyond that they have seen. If you haven’t used the symbol yet, it won’t appear on the dropdown box. Using arrows in your Microsoft Word document are a good way to bring your reader’s attention to a particular point. Additionally, the third column contains an informal definition, the fourth column gives a short example, the fifth and sixth give the Unicode location and name for use in HTML documents. The question was - What are the best ways to draw arrows in Microsoft Word? Shift+Alt+Right arrow key. Lv 4. Request Now, Mac will autocorrect –> to an arrow (→) as you type. Shift+Alt+Down arrow key. Leave the equation mode again with Alt+=. To enable or disable this feature, click the Show All, or pilcrow, icon on the standard toolbar. 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It this thread is locked “ line arrow ” option into equations or by! | 2021-05-16 03:18:20 | {"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": 0.5544650554656982, "perplexity": 2822.908993425005}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991659.54/warc/CC-MAIN-20210516013713-20210516043713-00373.warc.gz"} |
https://www.stats4stem.org/introduction-to-probability | 1
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Introduction to Probability
Introduction
There are lots of phenomena in nature, like tossing a coin or drawing cards from a card deck, whose outcomes cannot be predicted with certainty in advance, but the set of all the possible outcomes is known. These are what we call random phenomena or random experiments. Probability is concerned with such random phenomena or random experiments.
Probability
Probability is a way to measure the likelihood of certain outcomes of a particular activity that is typically random.
Probability Notation
In general, the probability of an event is denoted by a capital P followed by the event put in paranethesis.
Example 1: If we wish to express “the probability it will snow next week,” we use the notation: P(snow next week).
Example 2: Lets go ahead and create a random variable that represents the outcome of a roll of a die. If we wished to find the probability that X = 2, we would simply express this as: P(X = 2).
Example 3: If we let A represent an event we wish to find the probability of, then P(A) would represent that probability.
NOTATION MEANING P(win lottery) the probability that a person who has a lottery ticket will win that lottery P(A) the probability that event A will occur P(X = 2) the probability that random variable X equals 2
Outcome
An outcome is the result of an experiment or random activity that involves uncertainty.
Event
An event is any outcome or combination of outcomes. In other words, an event is a subset of the sample space. The probability of an impossible event is 0; the probability of a certain event is 1. Therefore, the range of possible probabilities for any given event is: 0≤P(A)≤1.
For any event A, P(A) must be a number between 0 and 1. ⇒ (0 $$\le$$ P( A) $$\le$$ 1)
Sample Space
The sample space is the set of all the possible outcomes in an experiment. The sample space is denoted by a capital letter S. If you were to roll a single die, then S = {1, 2, 3, 4, 5, 6}, which represents the set of all possible outcomes. If you were to roll two dice simultaneously and look at the sum of the two dice, then S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.
The probability of the Sample Space, S, is equal to one. ⇒ P(S) = 1
Relative Frequency
Relative frequency refers to the proportion of times an event occurs. In other words, it is the number of times a particular outcome occurs divided by the total number of trials. In general, the probability of an event can be approximated by the relative frequency.
Law of Large Numbers
The Law of Large Numbers is an important component of probability experiments. It states that as the number of repetitions or trials of an experiment increases, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability, or true probability. Even though the short-term or immediately observed outcomes may vary widely, the Law of Large Numbers allows us to say that the long-term observed relative frequency will approach the theoretical probability. | 2020-08-08 21:18:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7440124154090881, "perplexity": 286.33134080682817}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738351.71/warc/CC-MAIN-20200808194923-20200808224923-00234.warc.gz"} |
https://socratic.org/questions/how-do-you-simplify-2-3-i | # How do you simplify -2/(3-i)?
Dec 29, 2015
Multiply the numerator and denominator by the conjugate of the denominator to find
$- \frac{2}{3 - i} = - \frac{3}{5} - \frac{1}{5} i$
#### Explanation:
The conjugate of a complex number $a + b i$ is $a - b i$. The product of a complex number and its conjugate is a real number. We will use this property to produce a real number in the denominator of the given expression.
$- \frac{2}{3 - i} = - \frac{2}{3 - i} \cdot \frac{3 + i}{3 + i}$
$= \frac{- 2 \left(3 + i\right)}{\left(3 - i\right) \left(3 + i\right)}$
$= \frac{- 6 - 2 i}{9 + 3 i - 3 i + 1}$
$= \frac{- 6 - 2 i}{10}$
$= - \frac{3}{5} - \frac{1}{5} i$
Dec 29, 2015
$= 0.632 \angle 0.3217$. (in polar form)
$= - 0.6 - 0.2 i$. (in rectangular form)
#### Explanation:
There are 2 methods to do this -
Method 1
First convert everything to polar form then use the formula
$\frac{{z}_{1}}{{z}_{2}} = \frac{{r}_{1}}{{r}_{2}} c i s \left({\theta}_{1} - {\theta}_{2}\right)$
therefore (-2/pi)/((sqrt(3^2+1^2))/_tan^-1(-1/3))=-2/sqrt10 /_ (pi-(-0.32175)
$= 0.632 \angle 3.4633$. (Principle angle arguement is 0.3217 rad).
Method 2
Multiply the quantity by 1, selecting 1 as the complex conjugate of the denominator over itself.
Then the resultant denominator is a real number, and the numerator we multiply in rectangular form using the rule
$\left(a + i b\right) \cdot \left(x + i y\right) = \left(a x - b y\right) + i \left(b x + a y\right)$.
$\therefore \frac{- 2 + 0 i}{3 - i} \cdot \frac{3 + i}{3 + i} = \frac{- 6 - 2 i}{{3}^{2} + {1}^{2}}$
$= \frac{1}{10} \left(- 6 - 2 i\right)$
$= - 0.6 - 0.2 i$ | 2022-10-05 21:34:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 17, "mathjax_inline_tex": 1, "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": 0.9870202541351318, "perplexity": 749.831637157298}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337668.62/warc/CC-MAIN-20221005203530-20221005233530-00752.warc.gz"} |
https://www.physicsforums.com/threads/current-density-inside-superconductors.592747/ | # Current density inside superconductors
1. Apr 2, 2012
### rheajain
1. The problem statement, all variables and given/known data
consider an infinite superconducting slab of thickness 2d (-d<=z<=d), outside of which there is a given constant magnetic field parallel to the suface. Hx =Hz=0 hy= H0 (some value for z<d and z>-d) , with E vector= D vector=0 everywhere. compute H vector < J vector inside the slab, assuming surface currents and charges absent.
2. Relevant equations
consider Maxwell's equations in Gaussian units:
divergence D vector= 4∏ρ
divergence of B vector = 0
curl of E vector= -(1/c) partial differential of B with respect to time.
curl of H vector= (1/c) partial differential of D with respect to time + (4∏/c)J vector
with D=E+4∏Pvector
B vector = H vector + 4∏M vector
now inside superconductor
current density obeys following equation:
c * curl(λJ)= -B , partial differential of (λJ) with respect to time= E
λ is a constant
3. The attempt at a solution
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution | 2017-08-21 02:18:56 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8069641590118408, "perplexity": 6350.9538619350515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886107065.72/warc/CC-MAIN-20170821003037-20170821023037-00084.warc.gz"} |
https://worldbuilding.stackexchange.com/questions/25598/hydrogenenic-photosynthesis-strategies-for-animals | # Hydrogenenic Photosynthesis: Strategies for animals
Hydrogenic photosynthesis reduces methane and water to build biomass ($\text{CH}_2\text{O}$) and releases hydrogen:
$$\text{CH}_4 + \text{H}_2\text{O} + \text{photons} \to \text{CH}_2\text{O} + 2\text{H}_2$$
For reference, oxygenic photosynthesis is:
$$n \text{ CO}_2 + n \text{ H}_2\text{O} + \text{photons} \to (\text{CH}_2\text{O})n + n \text{O}_2$$
According to this excellent paper by Bains et al, the hydrogenic process is some four times as efficient as the oxygenic version, allowing four times the amount of biomass to be constructed for the same quantity of light (see note *1).
The linked paper describes how large planets could hold onto a hydrogen atmosphere, but this question is not about that.
My question is about strategies for animal evolution, since the flip side of it being 4 times as easy for autotrophs to build mass, is that heterotrophic consumers get 4 times less energy from breaking down one gram of this hydrogenic biomass. Here are the authors words:
"From a purely human point of view, the evolution of hydrogenic photosynthesis might be a disappointing discovery on another world, for reasons implicit in Figure 1. Just as making biomass in an oxidized environment requires more energy, breaking down biomass in an oxidized environment releases more energy. In particular, oxidizing biomass using molecular oxygen releases substantially more energy than reducing it using molecular hydrogen. A commonly-held explanation for the rise of complex animals in the late Pre-Cambrian and Cambrian periods was the rise in atmospheric oxygen that allowed their energy-intensive lifestyles "
My question is; how does the change in 'balance of power' between autotrophs and heterotrophs affect the evolution of both and what is the appropriate animal metabolism to allow animals to display the types of abilities (which rely on storing concentrated energy see note *2) that earth animals display?
Please note - any answer that addresses the fourfold animal vs plant imbalance is valid - PhD level biochemistry answers will be much appreciated but I am not expecting to get many of them!
End of question: what follows is supporting material from the paper that you can treat as **TL;DR.
Note *1
Here is the passage from the paper that makes the claim about reduced biomass generation requirements.
"Comparison of Gibbs energies of formation of CO2 (gas ~ −394 kJ/mol, aq ~−385 kJ/mol) and CH4 (gas ~ −50 kJ/mol, aq ~ −35 kJ/mol) [65] shows that any reaction involving CO2 as the C-bearing reactant will almost always have a more positive Gibbs energy of reaction than a similar reaction with CH4 as the reactant. The quantitative difference between the reactions will depend on the products of the reaction, as illustrated in Figure 1. On average, for the set of chemicals in Figure 1, making the chemical from CH4 takes ~20% the energy needed to make it from CO2. This suggests that building biomass in a CH4/H2-dominated environment would require only ~20% of the energy needed in our CO2-dominated environment."
Note *2
The linked paper mentions that maybe these animals could make use of dimethylsulfonium proprionate (DMSP) to store energy rather than carbohydrate but I don't really understand this process or what its implications are...
• I'm no chemist but I don't follow the logic of heterotrophic consumers getting 4 times less energy from the biomass. Since the produced Formaldehyde molecule is the same in both cases, why would the relative efficiency of the production process change the stored energy for a given unit mass? – KillingTime Sep 13 '15 at 15:28
• @KillingTime its a fair question and I have to say I don't know the answer - its too far out of my comfort zone to paraphrase the authors arguments. However I have edited my answer to include a quotation from part of the paper that motivated my question. – rumguff Sep 13 '15 at 15:37
• I'm not sure about the claim that heterotrophs get 4x less energy per gram. The chemical outputs of both types of photosynthesis are the same so why would there be less of energy available for heterotrophs? – Green Sep 13 '15 at 15:52
• @Green Because there is no free oxygen available? Remember that plants release hydrogen now. Even if there is an alternate oxygen source, oxygen in a hydrogen atmosphere is not a good sign at all... – Radovan Garabík Sep 13 '15 at 15:59
• @green I added the relevant section of the paper that addresses your point. I think they key is that the reference equations at the top of my post are highly simplified pictures... – rumguff Sep 13 '15 at 16:06
If I've understood your question correctly I'm going to basically ignore the biochemical science and jump straight to what I feel is the meat (actually, veg) of the question:
What happens if plants grow 4x faster, but animals get 4x less nutrition from them
Please note that above I'm using 'plant' as a synonym for autotroph and 'animal' as a synonym for heterotroph. I'm doing this simply because it feels more natural as a form of address. I'll use the correct terms later as it's important to make the distinction.
So: Moving on.
The period for which single celled life dominates will become shorter. Your single cells are more likely to be autotrophic, and as such will multiply much more quickly. In this sort of high-energy high population environment any heterotrophs that do emerge will have a glut of food, but won't be as much of an impactor on the autotrophs as they were in our history (as they reproduce at a quarter of the rate). The autotrophs therefore will compete with each other, and the high population density will lead to cellular co-operation faster.
When it comes to multicellular plantlife: competition will be fierce. I mean, genuinely fierce. These plants will have 4x the energy, and therefore 4x the capacity to reproduce, grow and generally do what plants do. Tall trees, resource sapping and funky seed dispersal techniques will blossom as all the plants will have more energy to 'waste'.
Animals on the other hand will have to move slower by necessity. They still have an advantage in that they don't need the sun, and they still have an advantage in that they're eating a richer energy source, but we won't be seeing purely carnivorous predators anytime soon as the amount of acreage required for a single predator would go up 16 fold (4x for the herbivores, then another 4x for the pure carnivores) Omnivores would likely do the best, but still, slower creatures would do better.
As the disparity between the amount of energy that can be gained from the sun vs the amount of energy gained from eating other plants is much smaller lifeforms exhibiting both autotrophic and heterotrophic behaviour would be considerably more prolific. Parasitic and carnivorous plants would be more common, and I'd expect a whole range of adaptations (Jellyfish vines, climbing bananas, Cuckoo-Elm?) and being photoheterotrophic (using sunlight to help fix carbon but not photosynthesising directly) would be a strong evolutionary choice.
If you want to see an earthlike system then your animals are going to have to have some serious metabolic mojo. For starters the herbivores will have to eat at least 4x more vegetation, and that's assuming metabolic efficiency works the same way. As previously mentioned any fast carnivores are going to be ravenously hungry, and would also have to evolve some major parenting skills as they won't have the energy to employ a 'fire and forget' strategy and then worry about all the competitors they just spawned. I'm unsure as to whether the same argument about parenting applies to the herbivores.
One last, rather intriguing (though contradictory) thought: Underwater the apex predator would probably be Coral...
• Great answer; I must admit now you mention it the implication about the widening of the producer/herbivore/carnivore pyramid seems obvious but I hadn't actually worked that out for myself. That ravenous apex hunter would be a magnificent beast (and probably subject to significant pressure to develop real intelligence). Nice point also about the photoheterotrophs. Likewise there is probably a big niche for fungi, which I also like. Hope to get more answers like this. – rumguff Sep 16 '15 at 22:48
• actually to accommodate that widely spread food chain I can just make the planet vast - a super-earth, which fits in nicely with retaining a H2 atmosphere. Will need to think about how animals can cover the larger distances though. – rumguff Sep 16 '15 at 22:51
• If you increase the prey density you can decrease the range required. If the plants grow incredibly densely then the prey will be denser so your predator can use ambush tactics and let the prey come to him. As long as he doesn't stay still long enough for the Greater Spotted Climbing Banana to get it's tendrils into him... – Joe Bloggs Sep 17 '15 at 8:24
Get your oxidizers here! Get them while they're hot!
The fundamental question is where do you get your oxidizer from? All oxygen on this methane+H2 planet is wrapped up in water or something else. Candidate oxidizers might be Fluorine or chlorine but both have their problems. Fluorine is so reactive it never stays free for long. Chlorine is also never found free in the atmosphere. With so much methane and hydrogen floating around, any oxidizer is going to get captured quickly. We only have it on Earth because there's so much life pumping out oxygen.
This leave us with two options. First, we develop a reciprocal metabolism that doesn't require an oxidizer and runs on hydrogen. (The world of chemistry is broad. It could probably be done.) I don't know near enough chemistry to even guess at candidate reactions.
Or, second, we recycle the oxidizers within the autotroph after consuming them from terrestrial carbonate, perhaps calcium carbonate which has three oxygen atoms for one calcium atom. I don't know the energy penalty in acquiring an oxidizer this way but it seems convenient. Perhaps a fluorine catalyst of some kind?
CO2 is also removed from the atmosphere by conversion to carbonate, at a rate that depends on surface chemistry.
This atmosphere is the inverse of Earth. On Earth, the oxidizer is freely available and the fuel is in short supply.
• the lack of oxidisers is why this is referred to as a reducing atmosphere. I quote from the paper: "In a reducing environment, highly oxidized compounds could be stored as energy storage materials, having the highest energy density when reduced with hydrogen, or other compounds with roles comparable to DMSP could be accumulated and be used as high-energy food. The absence of oxygen does not therefore preclude the possibility that other biomass components could be metabolized to yield lots of energy per gram." Oxidising accumulated biomass is not the only way of producing energy it seems. – rumguff Sep 15 '15 at 21:06
• Ha! :) You got a middle-schooler's answer. Sorry I can't do better. – Green Sep 15 '15 at 21:12 | 2021-01-26 15:10:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6028319001197815, "perplexity": 1734.6433496783175}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704800238.80/warc/CC-MAIN-20210126135838-20210126165838-00289.warc.gz"} |
https://cs.stackexchange.com/questions/37867/combinational-logic-circuits-and-theory-of-computation | # Combinational Logic Circuits and Theory of Computation
I'm trying to link Combinational Logic Circuits ( computers based on logical gates only ) with everything I have learned recently in Theory of Computation.
I was wondering whether combinational logic circuits can implement computations in the same way finite state machines can. They seem radically different :
Finite State Machines, however, have a well-defined memory in the form of the states that it can be in. Combinational logic circuits, however, don't have a well-defined memory so to implement algorithms that need memory they kinda use some weird method of serial connection ( see how $C_{out}$ of previous adder is connected to $C_{in}$ of current adder in the image bellow ) .
However radically different might seem, they both seems to be doing computations. For instance, both can implement an algorithm for binary addition ( and even binary multiplication ) however different those implementations might be :
FSM :
Combinational Logic Circuit ( C, as in $C_{in}$ and $C_{out}$, stands for Carry ) :
I'm even thinking ( although still very uncertain ) that we can convert every FSM into a corresponding Combinational Logic Circuit.
Can Combinational Logic Circuits also be considered a instantaneous kind of model of computation ? Can we apply all concepts we learn in Computability Theory and Computational Complexity Theory, like space complexity and computability, to it ?
On one hand, it seems like they don't fit as a model of computation because they don't have elementary operations ( like reading/writing of a tape, function reduction, steps on the proof search of logical programming paradigma ), they implement their computations instantaneously.
But on the other hand, they seem to be fit as a model of computation because we can model all kinds of computation with them ( binary addition is one example ), and they can be viewed abstractly ( by only focusing on the truth-tables and the logical gates and forgetting about the physical circuit that might implement it ).
So, what do you guys think ?
Also, if it can indeed be considered as a (instantaneous kind of ) model of computation, do you guys have any example of other similar ( also a instantaneous kind of ) model of computation ?
Logic circuits are common in complexity theory, where they go by the name circuits. There is a big difference between circuits and models of computation such as the Turing machine: each circuit can only handle inputs of fixed size. In order to fix this, under the circuit computation model, for every input length $n$ there is a circuit $C_n$, and together they compute a function on strings of arbitrary length. This computation model, as stated, is too strong: it can compute uncomputable functions, indeed all functions. The problem is that an infinite sequence of circuits doesn't necessarily have a finite description. In order to fix this problem, we usually demand that the circuits be uniform, that is, that they be generated by some Turing machine, which on input $n$ generates $C_n$. | 2020-01-29 13:11:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.756976306438446, "perplexity": 572.4523685892735}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251796127.92/warc/CC-MAIN-20200129102701-20200129132701-00221.warc.gz"} |
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### Reinterpretation and extension of entropy correction terms for residual distribution and discontinuous Galerkin schemes: application to structure preserving discretization. (English)Zbl 07517718
MSC: 65Mxx 35Lxx 76Mxx
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### A local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations. (English)Zbl 07517175
MSC: 65Nxx 35Jxx 65Mxx
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### Energy analysis and discretization of the time-domain equivalent fluid model for wave propagation in rigid porous media. (English)Zbl 07517171
MSC: 65Mxx 76Sxx 35Lxx
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### A functional oriented truncation error adaptation method. (English)Zbl 07517169
MSC: 65Mxx 65Fxx 65Nxx
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### On the stability of conservative discontinuous Galerkin/Hermite spectral methods for the Vlasov-Poisson system. (English)Zbl 07517167
MSC: 65Mxx 35Qxx 76Xxx
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### Energy-preserving fully-discrete schemes for nonlinear stochastic wave equations with multiplicative noise. (English)Zbl 07517148
MSC: 60Hxx 65Mxx 65Cxx
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### A coupled discontinuous Galerkin-finite volume framework for solving gas dynamics over embedded geometries. (English)Zbl 07517120
MSC: 76Mxx 65Mxx 65Nxx
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### Non-linear Boltzmann equation on hybrid-unstructured non-conforming multi-domains. (English)Zbl 07517098
MSC: 65Mxx 76Mxx 65Nxx
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### Entropy stable modal discontinuous Galerkin schemes and wall boundary conditions for the compressible Navier-Stokes equations. (English)Zbl 07516807
MSC: 65Mxx 76Mxx 76Nxx
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### LDG approximation of large deformations of prestrained plates. (English)Zbl 07516803
MSC: 74Kxx 65Nxx 74Bxx
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### An artificial equation of state based Riemann solver for a discontinuous Galerkin discretization of the incompressible Navier-Stokes equations. (English)Zbl 07516795
MSC: 76Mxx 76Dxx 65Mxx
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### Efficient computation of Jacobian matrices for entropy stable summation-by-parts schemes. (English)Zbl 07516792
MSC: 65Mxx 76Mxx 76Nxx
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all top 3 | 2022-08-10 19:44:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7315828204154968, "perplexity": 12340.079889907938}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571210.98/warc/CC-MAIN-20220810191850-20220810221850-00072.warc.gz"} |
https://research.google/pubs/pub48212/ | # Learning Linear-Quadratic Regulators Efficiently with only √ T Regret
ICML (2019) (to appear)
## Abstract
We present the first computationally-efficient algorithm with $\tO(\sqrt{T})$ regret for learning in Linear Quadratic Control systems with unknown linear dynamics and known quadratic costs. | 2021-04-15 10:04:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21532346308231354, "perplexity": 6831.945823600936}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038084765.46/warc/CC-MAIN-20210415095505-20210415125505-00151.warc.gz"} |
https://www.bbc.co.uk/bitesize/guides/zygxdxs/revision/4 | # Rates and temperature
The greater the rate or frequency of , the greater the rate of reaction. If the of the reaction mixture is increased:
• move more quickly
• the of the particles increases
• the frequency of successful collisions between reactant particles increases
• the proportion of collisions which are successful also increases
• the rate of reaction increases
Question
Explain what is meant by a ‘successful’ collision.
A collision between reactant particles with enough energy, eg the activation energy or more, to produce a reaction.
### Graphs
The rates of two or more reactions can be compared using a graph of or of formed against time. The graph shows this for two reactions.
Comparing reactions of different temperatures
The of the line is equal to the rate of reaction. The faster reaction at the higher temperature:
• gives a steeper line
• finishes sooner
The effect of temperature on the rate of reaction is due to two factors: frequency of collisions and energy of collisions. The increase in energy is usually the more important factor. | 2020-01-25 09:30:42 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9013491868972778, "perplexity": 1242.590843324824}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251671078.88/warc/CC-MAIN-20200125071430-20200125100430-00059.warc.gz"} |
http://npe.org.ua/17-10/ | # Methods for Analyzing the Hydrogeological Characteristics of the Aquifers in the Vicinity of Nuclear Power Plants Using Indicators
I. O. Kovalenko1, N. V. Sosonna1, M. I. Panasiuk1, U. Saravana Kumar2
1Institute for Safety Problems of Nuclear Power Plants, NAS of Ukraine, 36a, Kirova st., Chornobyl, 07270, Ukraine
2International Atomic Energy Agency, p. o. box 100, 5, Wagramer Strasse, A-1400, Vienna, Austria
DOI: doi.org/10.31717/2311-8253.20.2.10
### Abstract
The mathematical modeling of migration of the unsorbed indicator on the way of filtration flow of the first from the surface alluvial-quaternary aquifer underground waters was completed. The imitation modeling was performed to justify the using of isotope or indicator methods to obtain reliable data on aquifer parameters, in particular, the permeability coefficient. Three-dimensional geofiltration model was used and the verification of received predictive results with the results of the field observations was completed. Program complex Visual MODFLOW 2011.1 was used as a tool to manage and edit model and the model’s data, which gave a chance to improve the accuracy and performance of the model while increasing the efficiency of mathematical modeling.
Keywords: mathematical model, water, mass transfer indicator, bromide ion, permeability coefficient, Shelter object.
### References
1. Panasiuk M. I. (2014). [Determination of filtration coefficient of alluvial sands in the area of the Chornobyl NPP]. Problemy bezpeky atomnyh elektrostantsiy і Chornobylya [Problems of Nuclear Power Plants’ Safety and of Chornobyl], vol. 23, pp. 124-130. (in Russ.)
2. Panasiuk M. I., Alfyorov A. M., Starikov M. B., Litvin I. A., Liushnya E. P. (2011). [Results of detailed modeling of influence pile foundations in hydrogeological conditions in the New Safe Confinement district construction]. Problemy bezpeky atomnyh elektrostantsiy 1 Chornobylya [Problems of Nuclear Power Plants’ Safety and of Chornobyl], vol. 16, pp. 124-129. (in Russ.)
3. Panasiuk M. I., Alfyorov A. M., Levin G. V., Starikov M. B. (2011). [Mathematical modeling of geomigratory processes in water-saturated soil in area of the Shelter object]. Problemy bezpeky atomnyh elektrostantsiy і Chornobylya [Problems of Nuclear Power Plants’ Safety and of Chornobyl], vol. 17, pp. 124-130.
4. Visual MODFLOW 2011.1 User’s Manual. Waterloo: Waterloo Hydrogeologic, 2015, 702 c.
Full Text (PDF)
Published
2020-05-16
If the article is accepted for publication in the journal «Industrial Heat Engineering» the author must sign an agreement on transfer of copyright. The agreement is sent to the postal (original) or e-mail address (scanned copy) of the journal editions.
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License International CC-BY that allows others to share the work with an acknowledgement of the work’s authorship and initial publication in this journal.
Insert math as
$${}$$ | 2021-01-25 22:40:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.38945773243904114, "perplexity": 12152.752796666226}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704792131.69/warc/CC-MAIN-20210125220722-20210126010722-00070.warc.gz"} |
https://themathscentre.com/edexcel-a-levels-s1-statistics/ | # Edexcel A Levels S1 Statistics
### Unit S1: Statistics 1
#### GCE AS and GCE Mathematics, GCE AS and GCE Further Mathematics and GCE AS and GCE Further Mathematics (Additional) AS optional unit
Module 1 Representation and summary of data - Edexcel S1 (310:09 mins)
Histograms, stem and leaf diagrams, box plots.
Using histograms, stem and leaf diagrams and box plots to compare distributions. Back-to-back stem and leaf diagrams may be required. Drawing of histograms, stem and leaf diagrams or box plots will not be the direct focus of examination questions.
Measures of location — mean, median, mode.
Calculation of mean, mode and median, range and interquartile range will not be the direct focus of examination questions. Students will be expected to draw simple inferences and give interpretations to measures of location and dispersion. Significance tests will not be expected. Data may be discrete, continuous, grouped or ungrouped. Understanding and use of coding.
Measures of dispersion — variance, standard deviation, range and interpercentile ranges.
Simple interpolation may be required. Interpretation of measures of location and dispersion.
Skewness. Concepts of outliers.
Students may be asked to illustrate the location of outliers on a box plot. Any rule to identify outliers will be specified in the question.
Unit 1 Representation and summary of data - Edexcel S1 PDF
Unit 2 Representation and summary of data - Edexcel S1 Video
Module 2 Probability - Edexcel S1 (178.58 mins)
Elementary probability.
Sample space. Exclusive and complementary events. Conditional probability.
Understanding and use of P(A′) = 1 − P(A), P(A ∪ B) = P(A) + P(B) − P(A ∩ B), P(A ∩ B) = P(A) P(B | A).
Independence of two events.
P(B | A) = P(B), P(A | B) = P(A), P(A ∩ B) = P(A) P(B).
Sum and product laws.
Use of tree diagrams and Venn diagrams. Sampling with and without replacement.
Unit 1 Probability - Edexcel S1 PDF
Unit 2 Probability - Edexcel S1 Video
Module 3 Correlation and regression - Edexcel S1 (85.31 mins)
Scatter diagrams. Linear regression.
Calculation of the equation of a linear regression line using the method of least squares. Students may be required to draw this regression line on a scatter diagram.
Explanatory (independent) and response (dependent) variables. Applications and interpretations.
Use to make predictions within the range of values of the explanatory variable and the dangers of extrapolation. Derivations will not be required. Variables other than x and y may be used. Linear change of variable may be required.
The product moment correlation coefficient, its use, interpretation and limitations.
Derivations and tests of significance will not be required.
Unit 1 Correlation and regression - Edexcel S1 PDF
Unit 2 Correlation and regression - Edexcel S1 Video
Module 4 Discrete random variables - Edexcel S1 (123.12 mins)
The concept of a discrete random variable.
The probability function and the cumulative distribution function for a discrete random variable.
Simple uses of the probability function $$p(x)$$ where $$p(x) = P(X = x)$$. Use of the cumulative distribution function: $$F(x_0) = P(X \le x_0) = \displaystyle\sum_{x \le x0} p(x)$$
Mean and variance of a discrete random variable.
Use of $$E(X), E(X^2)$$ for calculating the variance of X. Knowledge and use of $$E(aX + b) = aE(X) + b$$, $$Var(aX + b) = a^2 Var(X)$$.
The discrete uniform distribution.
The mean and variance of this distribution.
Unit 1 Discrete random variables - Edexcel S1 PDF
Unit 2 Discrete random variables - Edexcel S1 Video
Module 5 The Normal distribution - Edexcel S1 (108.52mins)
The Normal distribution including the mean, variance and use of tables of the cumulative distribution function.
Knowledge of the shape and the symmetry of the distribution is required. Knowledge of the probability density function is not required. Derivation of the mean, variance and cumulative distribution function is not required. Interpolation is not necessary. Questions may involve the solution of simultaneous equations.
Unit 1 The Normal distribution - Edexcel S1 PDF
Unit 2 The Normal distribution - Edexcel S1 Video
** Our syllabus is current and updated to 2018 | 2022-12-06 05:11:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3418930470943451, "perplexity": 1886.4202075235605}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711069.79/warc/CC-MAIN-20221206024911-20221206054911-00675.warc.gz"} |
https://ccssmathanswers.com/proof-by-the-equal-intercepts-theorem/ | # Proof by the Equal Intercepts Theorem | State and Prove the Equal Intercepts Theorem
If you are searching for the article which gives the proof by the equal intercepts theorem, then, it is the right page you are on. For Grade9 students, the equal intercepts concept feels difficult to learn. But, 9th Grade Math students can get clear knowledge by referring to these articles and which makes them feel easier.
In an equal intercept theorem, a transversal makes equal intercepts on three or more lines than any other transversal cutting them will also make an equal intercept. In this article, students get a step-by-step procedure of proof by the equal intercepts theorem.
## What is meant by Intercepts in Maths?
Line segments are the intercepts made by the transversals by the lines L1 and L2. For example, in the below figure, AB is a transversal cutting the lines L and M at P and Q respectively. The line segment PQ is called the intercept made on the transversal AB by the lines L and M.
### What is Equal Intercept in Maths?
In mathematics, the definition of equal intercepts is if a transversal creates equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts.
### Equal Intercept Formula
The formula to find the equation of a line by intercept form is
$$\frac{x}{a}$$ + $$\frac{y}{b}$$ = 1
where a is x-intercept and b is y-intercept.
But given that the x & y-intercepts are equal.
Therefore, a = b
Thus, the equation of a line is
$$\frac{x}{a}$$ + $$\frac{y}{a}$$ = 1 ( As b= a)
x + y = a
⇒ x+ y- a= 0.
Example: Find the equation of a line that cuts off the equal intercepts on the axes and passes through the point (4, 6).
Solution:
As we know, the equation of a line is x+ y- a= 0
Now, find the value of a line that passes through the point (4,6)
Substituting, the given points in the equation of a line
i.e., x=4, y=6
4+6-a= 0
⇒ 10-a= 0
⇒ a= 10.
So, the equation of a line becomes
x+ y- 10= 0 (as a= 10)
Thus, x+ y- 10= 0 is the equation of a line that cuts off the equal intercepts on the coordinate axes.
Also Check:
### Equal Intercepts Theorem Statement & Proof
Prove that in the given triangle ABC, and, D, E are the midpoints of AB and AC respectively. Prove that DE bisects AC.
Given In ∆ABC, AD = DB and DE ∥ BC.
To Prove:
From ∆ABC, we have to prove that AE = EC.
Construction:
Draw a line PQ through vertex A such that PQ ∥ BC.
Proof and Derivation
In ∆ABC, PQ, DE, and BC are three parallel lines i.e., PQ ∥ DE ∥ BC, and AB is a transversal that makes equal intercepts. i.e., AD = DB (given)
Since by equal intercept theorem we know if a transversal makes equal intercepts with three or more parallel lines, then any other transversal will also make equal intercepts.
Therefore, AE = EC and AC make equal intercepts. Hence, the statement is proved.
### FAQ’s on Equal Intercepts Theorem
1. What is the intercept theorem and who invented the intercept theorem in mathematics?
The intercept theorem is also comprehended as Thales’s theorem and it has been invented by the ancient Babylonians and Egyptians, but firstly the known proof is found in Euclid’s Elements. of equalities. Q. E. D.
2. How do you find the equation of a line with equal intercepts?
The equation of a line that cuts off equal intercepts on the coordinate axes is $$\frac{x}{a}$$ + $$\frac{y}{b}$$ = 1. Thus, the equation of a line is x+ y- a= 0.
3. Which cuts off the equal intercepts and what is the slope of the line that cuts off equal intercepts on the coordinate axis?
The line cuts off the equal intercepts on the coordinate axis and the slope of the line is -1 when an equal intercept is on the axes.
Scroll to Top | 2022-05-21 05:06:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7127317190170288, "perplexity": 958.9313820971918}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662538646.33/warc/CC-MAIN-20220521045616-20220521075616-00396.warc.gz"} |
https://www.vedantu.com/maths/determinant-of-a-matrix | # Determinant of a Matrix
Determinant dates back to 1841 when Authur Cayley developed this system for solving linear equations quickly using two vertical line notations. For a given square matrix, the determinant of that matrix can compute its scalar value. The square matrix can be of any order such as 2x2 matrix, 3x3 matrix, or other nxn matrices. The important point to note here is the number of columns being equal as the number of rows. A determinant is represented with two vertical lines that consist of rows and columns. It is also represented as |A| or get A or det (A).
The process of calculating a determinant is also discussed in this article. Follow the below-mentioned steps for calculating the value of a determinant.
### Calculating the Value of a Determinant
|A| or det(A) is given below
$\begin{vmatrix}a &b \\ c& d \end{vmatrix}$
Step 1:
Solving for determinants involves the multiplication of rows with columns.
Step 2:
To illustrate the first step, multiply and bcc
Step 3:
The product of rows and columns is them subtracted
Step 4:
To illustrate this, ad - bc
Step 5:
The final result after subtracting the product is the value of the determinant
Let’s understand this further by taking a numerical example
### Example
Question
Solve for det(A)
DetA = $\begin{vmatrix}5 &7 \\ 3 & 1 \end{vmatrix}$
Answer: Follow the below-mentioned steps to solve for the value of the determinant
Step 1: Cross multiply the rows with columns
Step 2: The product of multiplication are 5 (5 x 1) and 21(7 x 3)
Step 3: Subtraction of the products
Step 4: 5 - 21 = -16
Step 5: det(A) = -16
Therefore, following the above-mentioned step can lead to solving any determinants. Furthermore, other complex solution examples will be discussed in this article where determinant of a singular matrix and 3x3 matrix shall also be addressed.
### 2 x 2 Matrix Determinant
Also commonly known as a determinant of a square matrix. A 2x2 matrix has two columns and two rows. The example mentioned above is an example of a 2x2 matrix determinant.
A matrix given below can be solved using the steps mentioned above
det(A) = $\begin{vmatrix}a_{11} &b_{12} \\ c_{21} & d_{22} \end{vmatrix}$
det(A) = a11 x a22 - a12 x a21
Using the formula above, and solve for any 2x2 determinant matrix
### 3x3 Matrix Determinant
A 3x3 matrix determinant has three columns and three rows. The method involving solving for the 3x3 determinant matrix is different from what has been discussed until now in this article.
An example of how the 3x3 matrix is represented is given below:
det (A)
$\begin{vmatrix}a_{11} &a_{12} &a_{13} \\a_{21} &a_{22} &a_{23} \\a_{31} &a_{32} &a_{33} \end{vmatrix}$
In order to solve for a 3x3 matrix determinant, follow the steps mentioned below:
Step 1: By expanding any one row, the solution for the determinant can be derived
Step 2: For solving det(A), the first row will be expanded
Step 3: The expanded version of the determinant will be as follow:
a11 $\begin{vmatrix}a_{22} &b_{23} \\ a_{32} & a_{33} \end{vmatrix}$ - a12$\begin{vmatrix}a_{21} &b_{23} \\ a_{31} & a_{33} \end{vmatrix}$ + a13$\begin{vmatrix}a_{21} &b_{22} \\ a_{31} & a_{32} \end{vmatrix}$
Step 4: The 2x2 determinant matrix will be solved as mentioned above.
Step 5: After solving for those, simple multiplication with the items of row 1 can lead to the next and final step
Step 6: In the 3x3 determinant matrix, the signs are alternative ( +, -, +). Following these signs, one can get to the final answer of det(A).
Let's solve an example with numerical values to get a better understanding of solving for a 3x3 determinant matrix.
Example:
Question
Solve for det(A) which is
$\begin{vmatrix}5 &2 &1 \\-2 &-1 &1 \\-4 &4 &3 \end{vmatrix}$
Step 1: For solving det(A), row 1 shall be expanded
Step 2: That being said, the expanded version of this determinant is given below
5$\begin{vmatrix}-1 &1 \\ 4 & 3 \end{vmatrix}$ -2$\begin{vmatrix}-2 &1 \\ -4 & 3 \end{vmatrix}$ + 1$\begin{vmatrix}-2 &-1 \\ -4 & 4 \end{vmatrix}$
Step 3: This step involves solving for the 2x2 determinant matrices
5 { (-1 x 3) - (4 x 1)} - 2 { ( -2 x 3 ) - ( 1 x -4)} + 3 {(-2 x 4) - ( -1 x -4)}
5 ( -3 - 4) - 2 (-6 + 4) + 3( -6 -4)
5(-7) -2(-2) +3(-10)
-35 + 4 -30
-61
Step 4: The value of det(A) is - 6m1
Following the above-mentioned step can easily help you solve for 3x3 determinant matrices. It is very important to remember to expand the row as the first step and then solving as a 2 x 2 matrix.
Determinants and matrices are two different concepts but have overlap uses. Even though they can be solved using simple mathematical rules, understanding the step for the solution is important. Similar to the 3 x 3 matrix, other square matrices can also be solved following the same steps and approaches. | 2020-08-09 05:52:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8505754470825195, "perplexity": 839.3389428031763}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738425.43/warc/CC-MAIN-20200809043422-20200809073422-00063.warc.gz"} |
https://datascience.stackexchange.com/questions/24137/converting-a-nominal-attributes-to-numerical-ones-in-data-set | Converting a nominal attributes to numerical ones in data set
I'm using the NSL-KDD data set which contains nominal and numerical values, and I want to convert all the nominal values to numerical ones. I tried the get_dummies method in python and the NominalToBinary method in WEKA, but the problem is that some nominal features contain 64 values so the conversion increases the dimensionality of the data a lot, and this can create problems for the classifier.
My question is if I can convert the nominal attributes by establishing a correspondence between each category of a nominal feature and a sequence of integer values, for example protocol_type {tcp=0, udp=1, icmp=2...etc}? Would this alter the credibility of the resulted data set?
By converting a nominal attribute to a single numeric attribute as you described, you are implicitly introducing an ordering over the nominal labels which is a bad representation of the data, and can lead to unwanted effects from a classifier. Does it make sense to say that UDP should be inbetween TCP and ICMP? (no!) Imagine you are training a $k$-NN model on this data. It doesn't make sense to say that ICMP should be "further away" from TCP than UDP, but if you adopted the mapping that you suggested, the representation of the data has this assumption built-in. Alternatively, what if you are training a decision tree-based model? Usually, in decision trees, binary split points are chosen for numeric attributes. There could be some randomness in your training data where splits at certain values of the numeric attribute results in overfitting to noise.
Typically when converting a nominal attribute to numeric, one numeric attribute per nominal label is created. Each attribute is set to one if the corresponding nominal label is set, and zero otherwise. For example, if a nominal attribute called protocol has labels {tcp, udp, icmp}, then this dataset:
$$\begin{array}{ccl} \text{inst.} & \text{protocol} & \text{other attributes} \\ \hline 1 & \text{tcp} & \dots \\ 2 & \text{icmp}& \dots \\ 3 & \text{icmp}& \dots \\ \vdots & \vdots & \ddots \end{array}$$
could be converted as follows:
$$\begin{array}{ccccl} \text{inst.} & \text{tcp} & \text{udp} & \text{icmp} & \text{other attributes} \\ \hline 1 & 1 & 0 & 0 & \dots \\ 2 & 0 & 0 & 1 & \dots \\ 3 & 0 & 0 & 1 & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{array}$$
This is what the NominalToBinary filter does in WEKA. As you mention, the downside of this is that a large number of additional attributes can be introduced if the number of distinct nominal values is high.
If the dimensionality is too high after the conversion, you may want to consider using a dimensionality reduction technique such as random projection, PCA, t-SNE, etc. Note that this will reduce the interpretability of your model. You could also use feature selection techniques to remove some of the less useful attributes. It is possible that some of the nominal labels are not useful for your model, and you will improve performance by removing them. Another thing you could try is to use your domain knowledge to reduce the number of categories. For example, TCP and UDP are both transport protocols, maybe for your application the distinction between TCP and UDP is not that important and you can put instances with protocol $\in$ {tcp, udp} into a new category, removing the old ones.
• thank you so much for your help, even if i have just a litle objection concerning your sentence "Does it make sense to say that UDP should be inbetween TCP and ICMP? (no!)", cause i meant to represent each nominal value with a given numeric value, i mean where the problem can occurs? thank you again – user4309930 Oct 28 '17 at 19:47
• By representing each nominal value with a given numeric value, you are imposing an ordering on the nominal values. If your model doesn't treat the new numeric attribute as a continuous variable, then it's the same as if it were a nominal attribute. I've added some more clarification as to exactly how this can create poor performance in the model in my answer. – timleathart Oct 29 '17 at 22:07
• Now i get it, thank you so much for your help and your time ;) – user4309930 Oct 29 '17 at 22:36
For encoding of the categorical variables with high cardinality (i.e. with large number of levels) you may want to try the so called impact coding.
The main idea is very simple, you just split the dataset into non-overlapping buckets by the variable of interest (“protocol” in your case) and calculate average of your response variable over each bucket. Then, the values of the categorical variable can be substituted by the average value over particular bucket.
Avg(response | protocol=”tcp”)
Avg(response | protocol=”icmp”)
Avg(response | protocol=”udp”)
The tricky part is to avoid data leakage, this can be done by splitting the entire dataset into several subsets (e.g. “encoding”, “training”, “validation”,…) and using only the data from “encoding” dataset for the nominal-to-numerical conversion.
I’ve learned about this approach from Win-Vector blog and their paper: vtreat: a data.frame Processor for Predictive Modeling, which I highly recommend. | 2021-08-03 11:57:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.40902870893478394, "perplexity": 688.1544187782637}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154457.66/warc/CC-MAIN-20210803092648-20210803122648-00400.warc.gz"} |
https://physics.stackexchange.com/questions/666767/how-do-i-know-the-signs-that-i-should-put-when-calc-the-interference | # How do I know the signs that I should put when calc the interference?
Imagine this problem: A light incide in a thin film with thickness $$d$$, it incides in such way that the angle it makes with the normal is theta. The film has refractive index $$n_2$$ and the initial medium has refractive index $$n_1$$.
Now, the reference image for my calculations is this:
I just can't understand the signs i should put on the equation of diference of path! See:
For the light propagating ABC, i think we can say the path is essentialy $$2dn_1/\cos \theta_2$$. Now, the problem is with the path of the light AD, the light that is reflected at the above surace. Shouldn't it be "$$2n_2d\tan(\theta_2)\sin(\theta_1) + \lambda/2$$"? Where i am considering there is change of phase in the reflection.
So,, in the end, the difference of the optical path would be "$$2dn_1/\cos \theta_2-(2n_2d\tan(\theta_2)\sin(\theta_1) + \lambda/2) = 2n_2d\cos\big(\theta_2) - \lambda/2$$". But, apparently, this is wrong.
I tried to apply that for a real question essentially equal to the question i posted here. The author gaves that the difference of path would be "$$2n_2d\cos\big(\theta_2) + \lambda/2$$" and I can't understand why.
What is the critery for the choice of the sign due to the refletion?
As far as the phase is concerned, adding or subtracting $$π$$ is totally equivalent. You can't tell the difference between the two. And the phase difference is what matters in optics.
The passage to the optical path is a convention: what is the difference of optical path which would give the same phase difference. So, adding or subtracting $$λ/2$$ is the same thing!
• Yes. But let me add something: In the problem i said above, the author asks for the thickness d necessary to occurs 2nd order interference. SO, in my case, i would need to use "$2\lambda = 2ndcos(\theta) -\lambda/2$", but he uses "$2\lambda = 2ndcos(\theta) +\lambda/2$", so that, even so add or subtract half of wavelength is equivalent, our answers differs! So who would be right?
• This is because, in this case, the definition of the interference order is also a convention. We must speak of the interference fringe for which the difference of the true optical path is 0, or $λ$, or...... There would be no more ambiguity. Sep 18 '21 at 17:18 | 2022-01-22 17:20:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 9, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8403923511505127, "perplexity": 288.97257533956326}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303868.98/warc/CC-MAIN-20220122164421-20220122194421-00544.warc.gz"} |
http://www.ni.com/documentation/en/labview/1.0/m-ref/arginchk/ | # arginchk
Version:
Determines whether an input argument is in a given range.
## Syntax
d = arginchk(a, b, c)
d = arginchk(a, b, c, 'string')
Legacy name: nargchk
## a
Lower end of the range. a is a positive integer.
## b
Upper end of the range. b is a positive integer.
## c
Positive integer.
## 'string'
Currently no functionality.
## d
Empty string if c is between a and b, inclusive. Otherwise, arginchk returns an error text.
D = arginchk(2, 5, 6)
Where This Node Can Run:
Desktop OS: Windows
FPGA: This product does not support FPGA devices | 2017-12-14 04:41:45 | {"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": 0.43697091937065125, "perplexity": 13168.63451566513}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948539745.30/warc/CC-MAIN-20171214035620-20171214055620-00695.warc.gz"} |
https://mail.academickids.com/encyclopedia/index.php/Dirac%27s_constant | # Planck's constant
(Redirected from Dirac's constant)
Planck's constant, denoted h, is a physical constant that is used to describe the sizes of quanta. It plays a central role in the theory of quantum mechanics, and is named after Max Planck, one of the founders of quantum theory. Its value is
[itex]h=6.626\ 069\ 3(11) \times10^{-34}\ \mbox{J}\cdot\mbox{s}[itex]
or with electronvolts as energy unit:
[itex]h=4.135\ 667\ 43(35) \times10^{-15}\ \mbox{eV}\cdot\mbox{s}.[itex]
Planck's constant has units of energy multiplied by time, which are the units of action. These units may also be written as momentum times distance (N·m·s), which are the units of angular momentum.
A closely-related quantity is the reduced Planck constant (sometimes called Dirac's constant):
[itex]\hbar\equiv\frac{h}{2\pi}=1.054\ 571\ 68(18)\times10^{-34}\ \mbox{J}\cdot\mbox{s},[itex]
where π is the constant pi. This constant is pronounced as "h-bar".
The figures cited here are the 2002 CODATA-recommended values for the constants and their uncertainties. The 2002 CODATA results were made available in December 2003 and represent the best-known, internationally-accepted values for these constants, based on all data available through 31 December 2002. New CODATA figures are scheduled to be published approximately every four years.
Planck's constant is used to describe quantization, a phenomenon occurring in microscopic particles such as electrons and photons in which certain physical properties occur in fixed amounts rather than assuming a continuous range of possible values. For instance, the energy E carried by a beam of light with constant frequency ν can only take on the values
[itex]E = n h \nu \,,\quad n\in\mathbb{N}[itex]
It is sometimes more convenient to use the angular frequency ω = 2 π ν, which gives
[itex]E = n \hbar \omega \,,\quad n\in\mathbb{N}[itex]
Many such "quantization conditions" exist. A particularly interesting condition governs the quantization of angular momentum. Let J be the total angular momentum of a system with rotational invariance, and Jz the angular momentum measured along any given direction. These quantities can only take on the values
[itex]\begin{matrix}
J^2 = j(j+1) \hbar^2, & j = 0, 1/2, 1, 3/2, \ldots \\ J_z = m \hbar, \qquad\quad & m = -j, -j+1, \ldots, j\end{matrix}[itex]
Thus, [itex]\hbar[itex] may be said to be the "quantum of angular momentum".
Planck's constant also occurs in statements of Heisenberg's uncertainty principle. The uncertainty (more precisely: the standard deviation) in any position measurement, Δx, and the uncertainty in a momentum measurement along the same direction, Δp, obeys
[itex] \Delta x \Delta p \ge \begin{matrix}\frac{1}{2}\end{matrix} \hbar[itex]
There are a number of other such pairs of physically measurable values which obey a similar rule.
On some browsers, the Unicode symbol ℎ (ℎ) is rendered as Planck's constant, and the symbol ℏ (ℏ) is rendered as Dirac's constant.
## Reference
• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy | 2022-05-22 13:54:10 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9532663226127625, "perplexity": 1275.3305523345448}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662545548.56/warc/CC-MAIN-20220522125835-20220522155835-00780.warc.gz"} |
http://fora.xkcd.com/viewtopic.php?f=17&t=56884&p=2151313 | ## Base pi
For the discussion of math. Duh.
Moderators: gmalivuk, Moderators General, Prelates
webb.am
Posts: 41
Joined: Wed Feb 24, 2010 1:09 pm UTC
### Base pi
Does anyone else find non-integer bases fascinating?
10π = π
A circle with diameter 1π has a circumference of 10π
Yeah... I'm not a mathematician so I can't really say anything interesting about them, but what about base π, base e, base φ? What does e look like in base π? Are non-integer bases interesting or am I just making something out of nothing?
the tree
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Location: Behind you
### Re: Base pi
webb.am wrote: What does e look like in base π?
Maybe equally, if [imath]e[/imath] and [imath]\pi[/imath] are algebraically independent, one can't really be expressed in terms of the other. But if they are algebraically dependant then it should be a finite expression. Whether or not [imath]e[/imath] and [imath]\pi[/imath] are algebraically independent is an unsolved question.
GoldenPhi
Posts: 5
Joined: Sat Jun 02, 2007 7:34 pm UTC
### Re: Base pi
Interestingly enough, integers in base φ are finite decimals. For example 2 is 10.01 in phinary. Also since φ² = φ +1 any number can be represented without a 1 appearing next to any other 1 in its decimal form: 3 = 11.01 or 100.01 base phi.
skeptical scientist
closed-minded spiritualist
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### Re: Base pi
Mostly I don't like non-integer bases as you give up unique representation. Base phi is a special case, because there is a trick that gives a unique representation (as noted above). So base phi is really quite nice. But I'm not sure if it's useful for anything.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
phlip
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Contact:
### Re: Base pi
$$2.2021201002111122\cdots_{\pi} = e$$
Not sure what that achieves, though.
Code: Select all
enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵╰(ಠ_ಠ ⚠) {exit((int)⚠);}
[he/him/his]
skeptical scientist
closed-minded spiritualist
Posts: 6142
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### Re: Base pi
phlip wrote:Not sure what that achieves, though.
A very strange use of \cdots.
Wait, you can use double-dollar signs for display math? $$I \, did \, not \, know \, that.$$ Unfortunately, single dollar signs don't seem to $work$.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
phlip
Restorer of Worlds
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### Re: Base pi
skeptical scientist wrote:A very strange use of \cdots.
Well, what's the norm for ellipses in tex? I thought cdots was good for "and so on". Is there a different one for when it's a number, and not, like, a matrix or something?
skeptical scientist wrote:Wait, you can use double-dollar signs for display math? $$I \, did \, not \, know \, that.$$ Unfortunately, single dollar signs don't seem to $work$.
Yeah, I stumbled onto it by accident when I was looking through some of my old posts and I saw that $$and$$ got converted in a post, and looked at the help file, which says that $$this$$, $this$ and $$this$$ are supported. $this$ is supported, but turned off by default, so that it doesn't mess up normal uses of the dollar sign.
Code: Select all
enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵╰(ಠ_ಠ ⚠) {exit((int)⚠);}
[he/him/his]
skeptical scientist
closed-minded spiritualist
Posts: 6142
Joined: Tue Nov 28, 2006 6:09 am UTC
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### Re: Base pi
phlip wrote:
skeptical scientist wrote:A very strange use of \cdots.
Well, what's the norm for ellipses in tex? I thought cdots was good for "and so on". Is there a different one for when it's a number, and not, like, a matrix or something?
You know about both \cdots and \ldots, right? Generally you use \ldots for decimal expansions and sequences, and \cdots for things like sums and products. I'm not sure what exactly the rules are, but I think $$\pi=3.14159\cdots$$ is less natural than $$e=2.7182818\ldots$$...if you'll pardon the pun.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
phlip
Restorer of Worlds
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### Re: Base pi
So noted. I've never really learned anything TeX-related properly... just picked it up from osmosis.
Code: Select all
enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵╰(ಠ_ಠ ⚠) {exit((int)⚠);}
[he/him/his]
Talith
Proved the Goldbach Conjecture
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### Re: Base pi
You can use \dots instead of \ldots which I think implies that dots on the line of writing are more commonly used than their centered colleagues. $$\dots \mbox{} \ldots$$
skeptical scientist
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### Re: Base pi
Talith wrote:You can use \dots instead of \ldots which I think implies that dots on the line of writing are more commonly used than their centered colleagues. $$\dots \mbox{} \ldots$$
Actually, if this were real Latex, I think \dots is implemented in such a way that it automatically chooses between centered and lowered dots based on context. However, it seems that jsMath doesn't do this, and always uses lowered dots for the \dots command. So I don't think your conclusion is justified.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
Talith
Proved the Goldbach Conjecture
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### Re: Base pi
I think we've gone slightly off topic, but thanks for disproving my hypothesis.
skeptical scientist
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### Re: Base pi
Talith wrote:I think we've gone slightly off topic, but thanks for disproving my hypothesis.
Yes and no. The topic was "are non-integer bases interesting?" The fact that the thread went off topic is very relevant, and tells as that the answer is no.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
Talith
Proved the Goldbach Conjecture
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### Re: Base pi
Give it some credit, I see 5 posts on topic. That's mildly intriguing at the least!
jestingrabbit
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### Re: Base pi
Probably the most interesting thing about non integer bases is that its hard to work out what the admissability criteria for the representation of a number should be. For instance, in base [imath]\sqrt{2}[/imath] its pretty easy to see that there are two representations of 2: 100 and something very messy. That's not true of integer bases (just never let a representation have a number value that exceeds the base, and don't let it end in an infinite string of bbbbbb...), nor in fact phinary (never let 11 appear and never let it end with 10101010101...). But in more complicated bases, its less clear what we can and shouldn't allow.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
phlip
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### Re: Base pi
Maybe a rule like... Take the representation at any point, cut off everything before that point, and move the decimal point (or whatever you call it in an arbitrary base) to just before the cutoff. So, for instance, you could take abcde.fghijk... and get 0.cdefghijk... or 0.ijk... or many other things. If the resulting number is >= 1, then the representation is bad. If the resulting number is <1 for all possible cutoffs, the representation is good.
So, for integer bases, this means no 0.999...-like representations, because 0.999... = 1.
For phinary, 0.11abcd... >= 0.10101010... = 1, so we don't use any representations that contain 11 or end 101010....
Equivalently, use the representation that comes last in lexical order (when they're prepended with 0s to the same length)... if you can increase the bn place and decrease the bn-1 and lower places (without making them go negative), and still end up with the same number, then do so.
So 2 is 100√2 and not 10.01000001001...√2, because the latter representation should "carry" upward.
This rule is a lot harder to judge for weird bases than it is for integers, though... would you know that 10.01000001001...√2 was a non-standard representation if it wasn't pointed out?
As a bonus, this also generalises the fact that all digits need to be in [imath]0 \le d < \left\lceil b \right\rceil[/imath]... you just have to say that the digits must be non-negative. Because if you try to have a digit d > b, then it'll be nonstandard, since 0.d > 1.
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### Re: Base pi
Another way of saying that is always use the representation generated by the greedy algorithm, where to write an x≥0 as a sum of powers of b, you repeatedly look for the greatest power of b which is less than the difference between the sum of powers found so far and x, and add it to the sum.
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jestingrabbit
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### Re: Base pi
skeptical scientist wrote:Another way of saying that is always use the representation generated by the greedy algorithm, where to write an x≥0 as a sum of powers of b, you repeatedly look for the greatest power of b which is less than the difference between the sum of powers found so far and x, and add it to the sum.
Yeah, that works fine for determining the representation of a number given its value, but if we start with a representation, like we might have after we do an addition like (21/2) + (2 - 21/2), I think its a lot harder to get to the representation that we want.
The good thing about integer (and other nice) bases, is that we can calculate with them, but there are others where we can't really calculate with them in an easy way.
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tmim
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### Re: Base pi
What about the [imath]0.999\ldots[/imath] like representation of pi in base pi?
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### Re: Base pi
If we are including rational bases in this discussion I would be tempted to put things in bases less than 1 to mess with peoples' minds.
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### Re: Base pi
nash1429 wrote:If we are including rational bases in this discussion I would be tempted to put things in bases less than 1 to mess with peoples' minds.
I'd just read it backwards as a number in base 1/b.
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### Re: Base pi
tmim wrote:What about the [imath]0.999\ldots[/imath] like representation of pi in base pi?
I'm not sure what you mean. But one thing to note is that there's actually a whole interval of things that can have multiple representations, instead of just certain points as in integer bases. For example, while 9.999999...=10 in base-ten, 3.33333... > 4 in base-pi.
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### Re: Base pi
By far the best base is base i*pi.
levantis
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### Re: Base pi
id rather say its $$\sqrt{10}i$$, because you dont need the usual base-changing algorithm to see how much is 123.45 (its $$-97.5+1.6\sqrt{10}i$$)
athough technicaly it is an integer base, i also like something like -2 or -10.
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### Re: Base pi
I like quater-imaginary...
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### Re: Base pi
levantis wrote:id rather say its $$\sqrt{10}i$$, because you dont need the usual base-changing algorithm to see how much is 123.45 (its $$-97.5+1.6\sqrt{10}i$$)
I get just [imath]-7.5 + 1.6\sqrt{10}i[/imath]
But yes, to write a given complex number x+iy in base bi (for real x,y,b), you just encode x and y/b in base -b2 and interleave the digits of the two (lining up the decimal-or-whatever-they're-called points, and putting the imaginary part to the left of the real part for a given digit). So -7.5 and 1.6 in base -10 are 13.5 and 2.4... interleaving them gives 123.45.
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### Re: Base pi
the tree wrote:
webb.am wrote: What does e look like in base π?
Maybe equally, if [imath]e[/imath] and [imath]\pi[/imath] are algebraically independent, one can't really be expressed in terms of the other. But if they are algebraically dependant then it should be a finite expression. Whether or not [imath]e[/imath] and [imath]\pi[/imath] are algebraically independent is an unsolved question.
You can write [imath]e[/imath] in terms of [imath]\pi[/imath]... [imath]e = (-1)^{1 \over i\pi}[/imath] ... but that really doesn't get you anywhere at all.
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### Re: Base pi
e
Last edited by black_hat_guy on Mon Aug 16, 2010 11:27 pm UTC, edited 1 time in total.
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### Re: Base pi
Nah, i is a crappy base by itself, because it's basically complex-unary. Look at how, to make 20+17i, you had to write it as 9+9+2 + 9i+8i.
(Pretty sure you did that wrong, though - remember that the first digit is worth i^0 = 1, not i. So it should be 200890099.)
Quater-imaginary, mentioned by another poster, is much better. It uses 2i as its base, and 0-3 as its digits. It's a proper numbering system, where the length of the written number increases logarithmically with the size of the number. It can also express every complex number.
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### Re: Base pi
skeptical scientist wrote:Another way of saying that is always use the representation generated by the greedy algorithm, where to write an x≥0 as a sum of powers of b, you repeatedly look for the greatest power of b which is less than the difference between the sum of powers found so far and x, and add it to the sum.
Personally I favour the reverse, the algorithm is something like:
Take value a in base X and we want it in base Y.
Divide the value a by Y in base X, and record the quotient and the remainder.
The remainder is the first digit (i.e. units). Repeat with the quotient (the next digit would be Y, then Y² and so on).
It's similar to GCD. Since the remainder is always less than Y, it works.
Mike_Bson
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### Re: Base pi
Does anyone know what the Champernowne Constant (0.12345678910111213...) would be like in a non-natural base? In an integer base, whether that be base 10, or base 4 (where it would be 0.123101112132021...), it will always be like that (as I understand, am I wrong?). What would it do in base 1.5, or phi, or pi? Also, is there any meaning to having a negatve number base?
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### Re: Base pi
As far as i'm aware, the constant itself is defined in base 10. If you put it into a new base, it won't have the same form as the base 10. You can of course, define a new set of constants [imath]\mathbf{Champ}=\{C(n)=0.[1]_n[2]_n[3]_n[4]_n[5]_n[6]_n..._n |n \in \mathbf{Z}\}[/imath] so that [imath]C(4)=0.123101112132021..._4=0.426111111111111028901245318..._{10}[/imath] which as you can see, doesn't have the form that you want when in base 10, but still has some of the properties of C(10) in that it is irrational and 4-normal. C(10) was shown to be absolutely normal (that is, n-normal for all n) by Champernowne and curiously, it hasn't been proven yet that C(n) is normal for any n other than n=10 - probably through lack of people trying. [must remember to read more thoroughly before posting]
Last edited by Talith on Mon Aug 09, 2010 1:52 am UTC, edited 4 times in total.
Mike_Bson
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### Re: Base pi
Talith wrote:As far as i'm aware, the constant itself is defined in base 10. If you put it into a new base, it won't have the same form as the base 10. You can of course, define a new constant Champernowne(n) so that [imath]C(4)=0.123101112132021..._4=0.426111111111111028901245318..._{10}[/imath] which as you can see, doesn't have the form that you want when in base 10, but still has some of the properties of C(10) in that it is irrational and 4-normal. C(10) was shown to be absolutely normal (that is, n-normal for all n) by Champernowne and curiously, it hasn't been proven yet that C(n) is normal for any n other than n=10 - probably through lack of people trying.
So 0.12345678910... and 0.12310111213 (base 4) are two different quantitative values? Wikipedia was a bit unclear, then. . . .
antonfire
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### Re: Base pi
It's pretty clear that they're not all the same. Just from the first digit, C(n) is between 1/n and 2/n.
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### Re: Base pi
Yeh, the way the constant is defined is determined entirely by what base you are working in. If you put all of the C(n) on the number line, no two would be the same and they would all be in the interval (0,1). It might be easier to see this if you work out a few in different bases and then convert them to base 10. (I'd be interested to see what a graph of C(n) looks like)
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### Re: Base pi
Talith wrote: (I'd be interested to see what a graph of C(n) looks like)
Hm, that's a thought. I'll see what I can make up, for the fun of it.
EDIT- It's a pretty simple graph. What I noticed is that C(1) would be about 1 in base 10, C(2) would be about 0.5, C(3) would be about 0.3. I think you see the pattern; the graph for y=C(x) is about the same as the graph of y=1/x (on the right side, at least).
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### Re: Base pi
Mike_Bson wrote:EDIT- It's a pretty simple graph. What I noticed is that C(1) would be about 1 in base 10, C(2) would be about 0.5, C(3) would be about 0.3. I think you see the pattern; the graph for y=C(x) is about the same as the graph of y=1/x (on the right side, at least).
Right, because as previously mentioned it's always going to be between 1/n and 2/n.
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NathanielJ
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### Re: Base pi
Talith wrote:C(10) was shown to be absolutely normal (that is, n-normal for all n) by Champernowne
Can you provide a source for this? Everything I've read says that it's known to be 10-normal but that the question remains open for n-normality when n \neq 10.
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Talith
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### Re: Base pi
Sorry, that was my mistake. I think it was a combination of me misunderstanding notation and not remember the wiki entry correctly. I though it said C(10) was known to be normal but C(n) it wasn't know for. It turns out it says "...C10 is normal in base ten, although it is possible that it is not normal in other bases".
I'm used to 'n-normal' meaning it's normal in base n, and 'normal' meaning it's absolutely normal.
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If Goku is afraid of her then she must be deadly powerful. Lol, New comments cannot be posted and votes cannot be cast, A subreddit for all things Dragon Ball! The Dragon Chi, Phoenix Chi, Tiger Chi and Turtle Chi represent four kinds of locked elemental power. Unfortunately, the term chi power is also used to explain complete BS, like throwing invisible balls of chi to defeat opponents, knocking people out without touching them, etc. PS5 Update Adds Alerts For When You're Running PS4 Versions Of Games. Dragon Ball Z fans will remember how awfully annoying Chi-Chi was when she put Gohan’s education before saving the world. The users of this technique can drastically increase their attack and defense power to inhuman levels with proper training and control. She has three abilities: Fire Quills, Ethereal, and Tranquility. She must have been pretty powerful. Ki is a power for a Martial Artist and the Willpower of the user. Chi-Chi is a beautiful woman with long straight black hair, large black eyes, a lighter skin color, and a curvaceous and slender figure. Chi-Chi, Hatchling of Chi-Ji Chi-Chi is a slippery little flyer with incredible defenses. Let's be honest, Chi Chi is secretly the strongest being on the planet. Kaiju is A Strong Saiyan who was born with a power level of 300,000! What if Chi-Chi's power level was 2,500,000,001? Gogeta is the Metamoran Fusion of Goku and Vegeta, formed to defeat Broly. CHI Power Plus Great Defense Duo$ 97.96 $67.96$ 97.96 $67.96 Add to cart. Krillin - 8 ... Power level Wikia is a FANDOM TV Community. *Quickly looks back and forth for Chi-chi*, Yes they are, and they are only a small part of the series. Chi-square test for independence. Strategy vs. Chi-Chi, Hatchling of Chi-Ji using: Corefire Imp (212), Rapana Whelk (111) and Any Pet. The power to mask one's chi potential. Sony promised more PS5 restocks before the end of the year, and we're tracking availability. The last power level stated in the series was Freezas one million. The others are stronger by merit of SSj2 (and 3 for goku) mostly. Everything else was filler. Goku - 10 Bulma - 1Large Pterodactyl - 8 Turtle - 0.001 Bear Thief - 8 Dragon Ball - General This is a split board - You can return to the Split List for other boards. Power Level Measuring: The user can detect how strong other's power levels are. Durability Enhanced Senses Stamina Vision - Night Weapons Master. (llamada Milk en Hispanoamérica) es un personaje de ficción de la serie de manga y anime Dragon Ball. Although older people may have lower chi level, your personal chi level actually has nothing to do with your age. By using our Services or clicking I agree, you agree to our use of cookies. Power Detection: The user can detect supernatural beings/powers in their surroundings. Dr. Yang is a Qi Gong master and wrote the book The Root of Chinese Qi Gong: Secrets of Health, Longevity & Enlightenment.He is the founder of YMAA, which is an organization that seeks to preserve and continue the traditional Chinese styles of Kung Fu and Qi Gong.The organization routinely publishes training material and books and … power levels are pretty unreliable. No previous Tai Chi experience necessary. The Benjamini–Hochberg (BH) procedure is a wellknown FDR controlling procedure. Shang Chi's has intensively trained his body to possess the highest level of stamina and endurance that a human can have without artificial enhancements. When you reach Level 110 of your 1st rebirth, you are able to unlock the Chi power. This. Because there's no way for her to have that much power. 8.8K likes. She doesn't develop any fighting skills in Z. I would say she is stronger (or at least a better fighter) than Mr. Satan, but not quite up to par with Krillin and Yamcha. A two tailed test is the default. Chi-Chi (チチ,?) Shang-Chi (MCU) hasn't been added to a collection yet. 2. And he was very young. You have a maximum capacity of 5 Chi (6, if you have the Ascension talent). 1 Also Called 2 Capabilities 3 Applications 4 Variations 5 Associations 6 Limitations 7 Known Users 8 Gallery Chi/Qi Masking User can hide their chi to hide their full battle potential or to avoid being detected. ALL RIGHTS RESERVED. i definitely wouldnt use them as a definitive quantification for any characters abilities. Dragon Ball's Goku and Chi-Chi are one of the longest-running couples in anime history.However, long-lasting doesn't necessarily mean good. Here are some Chi energy secrets... Dr. Yang, Jing-Wing. This is her battle power during the 23rd Tenka’ichi Budōkai. He's actually almost matched up to the Omni King: Zeno, in his C-Type transformations. Chi-Chi = 73 Muten Roshi = 139 Grandpa Gohan = 156 Krillin = 216 Korin = 169 Kami = 301 Piccolo (w/weights) = 322 Goku (w/weights) = 334 Piccolo = 408 Goku = … Find the exact moment in a TV show, movie, or music video you want to share. Each Super Power also has 3 levels (SPL). Chi Power Secrets, NJ. For comparison with mobile phones, the power level is around 500mW power. There are 3 characters and builds with it. She is also shown to take shots from Goku multiple times and has somehow fathered two of his Übermensch children (and survived the process). Given that Goten was able to make the jump to SSJ we have to assume that base level Goten was at least hitting 20k-500k as a power level. N, the total frequency, should be reasonably large, say greater than 50. For exam ple, the goodness -of-fit Chi-square may be used to test whether a set of values follow the normal distribution or whether the proportions of Democrats, Republicans, and other parties are equal to a certain set of values, say 0.4, 0.4, and 0.2. The two are often seen arguing or fighting, with the source of conflict usually revolving around Goku's love for martial arts and training. Default values for the SIDES= and ALPHA= options specify a two-sided test with a significance level of 0.05. Once you've learned how to control your chi power you can heal yourself, produce energy, focus your mind, and even increase your physical strength by forming a more powerful mind-body connection. Press question mark to learn the rest of the keyboard shortcuts. UNTIL we see that she is training Goten because reasons. As a child, Chi-Chi's appearance consisted of a blue bikini, pink gloves and boots, a green cape and her pink helmet. Chi-Chi - 8 Puar - 3 Boss Rabbit - 9 Rabbit Mob Members - 5 each Ox-King - 18 Master Roshi (100%) - 136 Goku (First Kamehameha) - 21 Pilaf - 3 Mai - 7 Shu - 7 Goku Oozaru - 100 Tournament Saga Edit. Chichi has a power level of last recorded 109 IIRC. Move Your Body. Videl is a normal self-taught girl. Cookies help us deliver our Services. However, there’s no in denying that Chi-Chi’s regiment training made Gohan one of the smartest Saiyan kids. These are all his superior power levels and transformations. power levels are pretty unreliable. Athlete Strength Shang-Chi possesses the strength level of a man his age, size and weight who engages in intensive regular exercise, Shang can lift at least twice his bodyweight, or 350 lbs. © 2020 GAMESPOT, A RED VENTURES COMPANY. You can build and maintain good personal energy no matter what your age provided you pay attention and have good consistent self care. discord.gg/dbz, Looks like you're using new Reddit on an old browser. A series of exercises with a jade or stone egg is used to strengthen the urogenital and pelvic diaphragm, the muscles of the vagina and the glands, tendons and nervous system. Click the "Chi" button at the bottom of the character status to access the Chi system. While all fusions have immense power, Gogeta's power is abnormal even by regular standards, as Vegeta and Goku's intense rivalry has brought out an exceptional power. This class introduces you to Tai Chi relaxation and movements fundamentals, plus the first movements of the Wu Style Short Form. CHI Oh So Rouge Travel Styling Kit$ 129.99 $99.99$ 129.99 $99.99 Add to cart. Density, distribution function, quantile function and random generation for the chi-squared ($$\chi^2$$) distribution with df degrees of freedom and optional non-centrality parameter ncp. As for the topic.. yea, though it's hard to say. Students learn to build their energy to high levels through a process called the Nerve Fiber Building energy exercise. http://chipowersecrets.com/ Master the power of chi energy and unique esoteric abilities. Even in terms of background, they're not remotely the same. Sense of Strength: The user can gain knowledge of another's strength by using their chi. Reaching High Levels of Yin Cultivation with Professional Chi Energy Training . She is able to take a shot from SSJ Goten (meaning he spiked to at least 1,000,000 PL) and while this hurts her, she isn't completely eviscerated and isn't shown to have any lasting damage. Chi Power Secrets is dedicated to bringing you cutting edge information and training in all types of esoteric training, including but not limited to Chi Power, Qigong, Meditation, Psychic Energy Skills, Hypnotic Influence, Covert Persuasion, Abundance Training and more. Every Super Power has a score (SPS) that is used to calculate the Class. CHI Franciscan Brings Seattle Families One Step Closer to More OB Care Choices OCT 11, 2019 CHI Franciscan, Tacoma, WA, and Virginia Mason, Seattle, WA, announced the Washington State Department of Health approved Virginia Mason’s Certificate of Need application for a Level II Special Care Nursery. This function calculates the power of a Chi-square test, given the value of non-centrality parameter Power.Chisq: Function to calculate the power of a Chi-square test in OptSig: Optimal Level of Significance for Regression and Other Statistical Tests ## ## Chi squared power calculation ## ## w = 0.2182179 ## N = 312.4671 ## df = 1 ## sig.level = 0.01 ## power = 0.9 ## ## NOTE: N is the number of observations About 313. The practice can shorten menstruation, reduce cramps and compress more life force energy (Chi) into the ovaries for more sexual and creative power. The power of the goodness of fit or chi-square independence test is given by. Gogeta is the Metamoran Fusion of Goku and Vegeta, formed to defeat Broly. 100% handwork and cheap WoW power leveling available here. Meaning that Chi Chi (by Buu Saga) is either equivalent to or stronger than any member of the Ginyu Force. Was under 9k I imagine she may have been able to unlock the Chi power you can build maintain... Three steps, allowing you to see how the chi-square test for proportions. Around 500mW power like you 're using New Reddit on an old browser$ 129.99 99.99... Of SSj2 ( and 3 for Goku ) mostly rebirth, you want to share that 's 10... The Chi '' button at the bottom of the user can gain knowledge of 's. … gogeta is the Metamoran Fusion of Goku and Chi-Chi are one of the character status to access the inside... Training and control couples in anime history.However, long-lasting does n't develop fighting! Proper training and control options specify a two-sided test with a significance of! N'T as strong as someone say Krillin ps5 restocks before the end of keyboard. Source ive seen to understand power levels are Omni King: Zeno, in his C-Type Transformations in mind... Be available in your mind, you are able to unlock the system! A slippery little flyer with incredible defenses let 's be honest, Chi trained., fast delivery now the character status to access the Chi power you can build maintain! And managed to make him go Super Saiyan FDR controlling procedure stronger than any member of the longest-running couples anime. Of the strongest humans, maybe the strongest non-ki using large, say than! 100 % handwork and cheap WoW power leveling with safe, fast delivery!! Used are in line with the AQA GCE human Biology Students ’ Statistics Sheet version.: —Spirituality, prayer and meditation chi chi power level terms of feats, the frequency. That being said, this is her battle power during the 23rd Tenka ’ Budōkai... Goten and managed to make him go Super Saiyan to High levels through a twine a! Are one of chi chi power level series are not consistent Senses Stamina Vision - Night Weapons Master anime Dragon Ball fans. So if she 's sleeping with Goku and Chi-Chi are one of the year, and they are a... Status to access the Chi power Chi-Ji using: Corefire Imp ( 212 ), Rapana Whelk 111. To cart Adds Alerts for when you reach level 110 of your 1st rebirth you! A score ( SPS ) that is used to calculate the Class,... Envision gold yellow energy coming from the fire with finger chakra Chi and to relax their systems! Smartest Saiyan kids or backward to get to the Omni King: Zeno, in C-Type! Fundamentals, plus the first movements of the smartest Saiyan kids level of last recorded IIRC... Ginyu Force sense others takes three steps, allowing you to Tai Chi will be in... That 's around 10 years of training, right finger chakra Chi and should... Before saving the world regiment training made Gohan one of the strongest non-ki using levels with training! Still talked about reasonably large, say greater than 50 which is still something like 10-20x than! Under 9k I imagine she may have been able to keep up un de... Her to have that much power make the Chi going into the you!, Krillin, and even Yamcha, I often chi chi power level exactly how strong Chi... Real definition that will clear a lot of the Wu Style Short Form remotely the.! She may have been able to use ki in fights as a definitive quantification for any characters.! $129.99$ chi chi power level Add to cart her battle power during the 23rd Tenka ’ ichi Budōkai to! Another 's strength by using our Services or clicking I agree, you to... To get to the split List for other boards an overview and step-by-step for... Wonder exactly how strong other 's power levels, Transformations, Extra 's and. Course Catalog character status to access the Chi system a significance level 300,000. And managed to make him go Super Saiyan are stronger by merit of SSj2 ( 3! I simply love the fact that DBZ is still something like chi chi power level stronger than average. Any member of the longest-running couples in anime history.However, long-lasting does n't necessarily mean good, she... It does n't necessarily mean good it should get smaller/ duller power can. Other 's power levels and Transformations good personal energy no matter what your age set connecting... yea, though it 's moving at all contingency tables I simply love the fact DBZ! On an old browser afraid of her then she must be deadly powerful you move through body... | 2022-05-25 20:39:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.18561787903308868, "perplexity": 5524.324249455878}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662593428.63/warc/CC-MAIN-20220525182604-20220525212604-00272.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcds.2018103 | # American Institute of Mathematical Sciences
May 2018, 38(5): 2487-2503. doi: 10.3934/dcds.2018103
## Topological stability and spectral decomposition for homeomorphisms on noncompact spaces
Department of Mathematics, Chungnam National University, Daejeon 305-764, Korea
* Corresponding author (yangyinong1201@gmail.com)
Received August 2017 Published March 2018
In this paper, we introduce the notions of expansiveness, shadowing property and topological stability for homeomorphisms on noncompact metric spaces which are dynamical properties and equivalent to the classical definitions in case of compact metric spaces. Then we extend the Walters's stability theorem and Smale's spectral decomposition theorem to homeomorphisms on locally compact metric spaces.
Citation: Keonhee Lee, Ngoc-Thach Nguyen, Yinong Yang. Topological stability and spectral decomposition for homeomorphisms on noncompact spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2487-2503. doi: 10.3934/dcds.2018103
##### References:
[1] N. Aoki, On homeomorphisms with pseudo-orbit tracing property, Tokyo J. Math., 6 (1983), 329-334. doi: 10.3836/tjm/1270213874. [2] B. Carvalho and W. Cordeiro, N-expansive homeomorphisms with the shadowing property, J. Differential Equations, 261 (2016), 3734-3755. doi: 10.1016/j.jde.2016.06.003. [3] N.-P. Chung and K. Lee, Topological stability and pseudo-orbit tracing property of group actions, Proc. Amer. Math. Soc., 146 (2018), 1047-1057. [4] W. Cordeiro, M. Denker and X. Zhang, On specification and measure expansiveness, Discrete Contin. Dyn. Syst., 37 (2017), 1941-1957. [5] T. Das, K. Lee, D. Richeson and J. Wiseman, Spectral decomposition for topologically Anosov homemorphisms on noncompact and non-metrizable spaces, Topology Appl., 160 (2013), 149-158. doi: 10.1016/j.topol.2012.10.010. [6] M. Hurley, Chain recurrence, semiflows, and gradients, J. Dynam. Differential Equations, 7 (1995), 437-456. doi: 10.1007/BF02219371. [7] K. Lee and C. A. Morales, Topological stability and pseudo-orbit tracing property for expansive measures, J. Differential Equations, 262 (2017), 3467-3487. doi: 10.1016/j.jde.2016.04.029. [8] P. Oprocha, Chain recurrence in multidimensional time discrete dynamical systems, Discrete Conti. Dyn. Syst., 20 (2008), 1039-1056. doi: 10.3934/dcds.2008.20.1039. [9] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. [10] P. Walters, On the pseudo-orbit tracing property and its relationship to stability, Lecture Notes in Math., Springer, 668 (1978), 231-244.
show all references
##### References:
[1] N. Aoki, On homeomorphisms with pseudo-orbit tracing property, Tokyo J. Math., 6 (1983), 329-334. doi: 10.3836/tjm/1270213874. [2] B. Carvalho and W. Cordeiro, N-expansive homeomorphisms with the shadowing property, J. Differential Equations, 261 (2016), 3734-3755. doi: 10.1016/j.jde.2016.06.003. [3] N.-P. Chung and K. Lee, Topological stability and pseudo-orbit tracing property of group actions, Proc. Amer. Math. Soc., 146 (2018), 1047-1057. [4] W. Cordeiro, M. Denker and X. Zhang, On specification and measure expansiveness, Discrete Contin. Dyn. Syst., 37 (2017), 1941-1957. [5] T. Das, K. Lee, D. Richeson and J. Wiseman, Spectral decomposition for topologically Anosov homemorphisms on noncompact and non-metrizable spaces, Topology Appl., 160 (2013), 149-158. doi: 10.1016/j.topol.2012.10.010. [6] M. Hurley, Chain recurrence, semiflows, and gradients, J. Dynam. Differential Equations, 7 (1995), 437-456. doi: 10.1007/BF02219371. [7] K. Lee and C. A. Morales, Topological stability and pseudo-orbit tracing property for expansive measures, J. Differential Equations, 262 (2017), 3467-3487. doi: 10.1016/j.jde.2016.04.029. [8] P. Oprocha, Chain recurrence in multidimensional time discrete dynamical systems, Discrete Conti. Dyn. Syst., 20 (2008), 1039-1056. doi: 10.3934/dcds.2008.20.1039. [9] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. [10] P. Walters, On the pseudo-orbit tracing property and its relationship to stability, Lecture Notes in Math., Springer, 668 (1978), 231-244.
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https://123dok.net/document/yng84r0z-%CE%B4-groupoids-in-knot-theory.html | Δ-groupoids in knot theory
26
Texte intégral
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DOI 10.1007/s10711-010-9496-5 O R I G I NA L PA P E R
-groupoids in knot theory
R. M. Kashaev
Received: 20 August 2009 / Accepted: 30 March 2010 / Published online: 10 April 2010 © Springer Science+Business Media B.V. 2010
Abstract A-groupoid is an algebraic structure which axiomatizes the combinatorics of a truncated tetrahedron. It is shown that there are relations of-groupoids to rings, group pairs, and (ideal) triangulations of three-manifolds. In particular, we describe a class of repre-sentations of group pairs H⊂ G into the group of upper triangular two-by-two matrices over an arbitrary ring R, and associate to that group pair a universal ring so that any representation of that class factorizes through a respective ring homomorphism. These constructions are illustrated by two examples coming from knot theory, namely the trefoil and the figure-eight knots. It is also shown that one can associate a-groupoid to ideal triangulations of knot complements, and a homology of-groupoids is defined.
Keywords Knot theory· Ideal triangulation · Group · Malnormal subgroup · Groupoid · Ring
Mathematics Subject Classification (2000) 20L05· 57M27 · 16S10
1 Introduction
In this paper, we introduce an algebraic structure called-groupoid and describe its rela-tionships to rings, representations of group pairs, and combinatorics of (ideal) triangulations of three-manifolds. Functorial relations of-groupoids to the category of rings permit us to construct ring-valued invariants which seem to be interesting. In the case of knots, these rings are universal for a restricted class of representations of knot groups into the group G L(2, R), where R is an arbitrary ring.
Ideal triangulations of link complements give rise to presentations of associated -grou-poids which, as grou-grou-poids with forgotten-structure, have as many connected components
The work is supported in part by the Swiss National Science Foundation. R. M. Kashaev (
B
)
Université de Genève, Section de mathématiques, 2–4, Rue du Lièvre, CP 64, 1211 Genève 4, Suisse e-mail: Rinat.Kashaev@unige.ch
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as the number of link components. In particular, they are connected groupoids in the case of knots. In general, two-groupoids associated with two ideal triangulations of one and the same knot complement are not isomorphic, but one can argue that the corresponding vertex groups are isomorphic. In this way, we come to the most evident-groupoid knot invariant to be called the vertex group of a knot. It is not very sensitive invariant as one can show that it is trivial one-element group in the case of the unknot, isomorphic to the group of integers Zfor any non-trivial torus knot, and isomorphic to the groupZ×Zfor any hyperbolic knot. Moreover, it is trivial at least for some connected sums, e.g. 31#31 or 31#41. In the light of these observations it would be interesting to calculate the vertex group for satellite knots which are not connected sums.
One can also refine the vertex group by adding extra information associated with a distin-guished choice of a meridian-longitude pair. In this way one can detect the chirality of knots. For example, the torus knot Tp,q of type(p, q) and its mirror image Tp,qhave isomorphic vertex groups freely generated by the meridian m, while the longitude l is given as l= mpq for Tp,q, and l= m−pqfor Tp,q.
Finally, we define an integral-groupoid homology which seems not to be very interest-ing in the case of hyperbolic knots, but could be of some interest in the case of non-hyperbolic knots.
The paper is organized as follows. In Sect.2we give a definition of the-groupoid and a list of examples. In Sect.3, we show that there is a canonical construction of a-groupoid starting from a group and a malnormal subgroup. In Sect.4, we show that two constructions in Examples4and5are functors which admit left adjoints functors. In Sects.5and6, we reveal a representation theoretical interpretation of the A-ring of the previous section in terms of a restricted class of representations of group pairs into two-by-two upper-triangular matrices with elements in arbitrary rings. In Sect.7, in analogy with group presentations, we show that-groupoids can be presented starting from tetrahedral objects. In Sect.8, we define an integral homology of-groupoids. In the construction, actions of symmetric groups in chain groups are used in an essential way.
2-groupoids
2.1 Preliminary notions and notation
Recall that a groupoid is a (small) category where all morphisms are isomorphisms [1,5]. So, a groupoid G consists of a set of objects Ob G, a set of morphisms Hom(A, B) from A to B for any pair of objects(A, B), an identity morphism idA∈ Hom(A, A) for any object
A, and a composition or product map Hom(A, B) × Hom(B, C) → Hom(A, C) for any
triple of objects(A, B, C). These data satisfy the usual category axioms with the additional condition that any morphism is invertible. Following [5], for a morphism x ∈ Hom(A, B), we write A= dom(x), B = cod(x) and call them respectively the domain (source) and the codomain (target) of x. A typical example of a groupoid is the fundamental groupoid of a topological space X , where the objects are points of X and morphisms are paths considered up to homotopies relative the end points. A group is a groupoid with one object. By analogy with group theory, we shall identify a groupoid with the union of all its morphisms, so that the composition becomes a partially defined operation. We use the convention adopted for fundamental groupoids of topological spaces, i.e. a pair of morphisms(x, y) is composable if and only if cod(x) = dom(y), and the product is written xy rather than y ◦ x.
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2.2 Definition of the-groupoid
Let G be a groupoid and H its subset. We say that a pair of elements(x, y) ∈ H2is H
-com-posable if it is com-com-posable in G and x y∈ H.
Definition 1 A-groupoid is a groupoid G, a generating subset H ⊂ G, and an involution j: H → H, such that
(i) i(H) = H, where i(x) = x−1;
(ii) the involutions i and j generate an action of the symmetric groupS3on the set H , i.e. the following equation is satisfied: i j i= ji j;
(iii) if(x, y) ∈ H2is a composable pair then(k(x), j(y)) is also a composable pair, where
k= i ji;
(iv) if(x, y) ∈ H2 is H -composable then(k(x), j(y)) is also H-composable, and the following identity is satisfied:
k(xy)ik(y) = k(k(x) j(y)). (1)
A-group is a -groupoid with one object (identity element).
A morphism between two-groupoids is a groupoid morphism f : G → Gsuch that
f(H) ⊂ Hand jf = f j. In this way, one comes to the category Gpd of -groupoids. Remark 1 Equation (1) can be replaced by an equivalent equation of the form
i j(x) j(xy) = j(k(x) j(y)).
Remark 2 In any-groupoid G there is a canonical involution A → A∗acting on the set of objects (or the identities) of G. It can be defined as follows. As H is a generating set for G, for any A∈ ObG there exists x ∈ H such that A = dom(x). We define A= dom( j(x)). This definition is independent of the choice of x. Indeed, let y ∈ H be any other element satisfying the same condition. Then, the pair(i(y), x) is composable, and, therefore, so is
(ki(y), j(x)). Thus,
dom( j(y)) = cod(i j(y)) = cod(ki(y)) = dom( j(x)).
Remark 3 Definition1differs from the one given in the preprint [3] in the following aspects: (1) the subset H was not demanded to be a generating set for G so that it was possible to
have empty H with non-empty G;
(2) the condition (iii), which is essential in Remark2, was not imposed;
(3) it was implicitly assumed that any x∈ H enters an H-composable pair, and under that assumption the condition (ii) was superfluous.1
2.3 Examples of-groupoids
Example 1 Let G be a group. The tree groupoid G2is a-groupoid with H = G2, j( f, g) =
( f−1, f−1g).
Example 2 Let X be a set. The set X3can be thought of as a disjoint sum of tree groupoids X2 indexed by X : X3
x∈X{x} × X2. In particular, Ob(X3) = X2with dom(a, b, c) = (a, b) and cod(a, b, c) = (a, c) with the product (a, b, c)(a, c, d) = (a, b, d) and the inverse
(a, b, c)−1 = i(a, b, c) = (a, c, b). This is a -groupoid with H = X3 and j(a, b, c) =
(b, a, c).
1I am grateful to D. Bar-Natan for pointing out to this assumption during my talk at the workshop “Geometry and TQFT”, Aarhus, 2007.
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Example 3 We define an involutionQ∩ [0, 1[ t → t∗∈Q∩ [0, 1[ by the following
con-ditions: 0∗= 0 and if t = p/q with positive mutually prime integers p, q, then t= ¯p/q, where ¯p is uniquely defined by the equation p ¯p = −1 (mod q). We also define a map Q∩ [0, 1[ t → ˆt ∈Q∩ [0, 1] by the formulae ˆ0 = 1 and ˆt = (p ¯p + 1)/q2if t = p/q with positive, mutually prime integers p, q. Notice that in the latter case ˆt = ˜p/q with Z ˜p = (p ¯p + 1)/q, and 1 ≤ ˜p ≤ min(p, ¯p). We also remark that t= ˆt.
The rational strip X = Q× (Q∩ [0, 1[) can be given a groupoid structure as follows. Elements(x, s) and (y, t) are composable iff y ∈ s +Z, i.e. the fractional part of y is s, with the product(x, s)(s + m, t) = (x + m, t), the inverse (s + k, t)−1= (t − k, s), and the set of units ObX = {(t, t)| t ∈Q∩ [0, 1[}. Denote by X the underlying graph of X , i.e. the subset of non-identity morphisms. One can show that X is a-groupoid with H = Xand
k(x, t) =
tx− ˆt x− t , t
∗.
Taking into account the general construction of Sect.3, this example is associated to the group P S L(2,Z) and its malnormal subgroup represented by upper triangular matrices.
Example 4 Let R be a ring. We define a-group AR as the subgroup of the group R∗of invertible elements of R generated by the subset H = (1 − R) ∩ Rwith k(x) = 1 − x so that j(x) = iki(x) = (1 − x−1)−1.
Example 5 For a ring R, let RRbe the semidirect product of the additive group R with the multiplicative group Rwith respect to the (left) action of Ron R by left multiplica-tions. Set theoretically, one has RR= R × R∗, the group structure being given explicitly by the product(x, y)(u, v) = (x + yu, yv), the unit element (0, 1), and the inversion map
(x, y)−1= (−y−1x, y−1). We define a -group B R as the subgroup of RRgenerated by
the subset H = R× Rwith k(x, y) = (y, x) so that j(x) = kik(x, y) = (x−1, −x−1y).
Example 6 Let(G, G±, θ) be a symmetrically factorized group of [4]. That means that G is a group with two isomorphic subgroups G± conjugated to each other by an involutive elementθ ∈ G, and the restriction of the multiplication map m : G+× G→ G+G⊂ G
is a set-theoretical bijection, whose inverse is called the factorization map G+G g → (g+, g) ∈ G+ × G. In this case, the subgroup of G+ generated by the subset H =
G+∩ GG+θ ∩ θG+Gis a-group with j(x) = (θx)+.
3-groupoids and pairs of groups
Recall that a subgroup H of a group G is called malnormal if the condition g H g−1∩ H = {1} implies that g∈ H. In fact, for any pair of groups H ⊂ G one can associate in a canonical way another group pair H⊂ Gwith malnormal H. Namely, if N is the maximal normal subgroup of G contained in H , then we define G = G/N and H ⊂ Gis the malnormal closure of H/N.
Lemma 1 Let a subgroup H of a group G be malnormal. Then, the right action of the group H3on the set(G\H)2defined by the formula
(G\H)2× H3 (g, h) → gh = (h−1
1 g1h2, h−11 g2h3) ∈ (G\H)2,
h= (h1, h2, h3) ∈ H3, g = (g1, g2) ∈ (G\H)2
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Proof Let h∈ H3and g∈ (G\H)2be such that gh= g. On the level of components, this corresponds to two equations h−11 g1h2= g1and h−11 g2h3= g2, or equivalently g1h2g1−1=
h1 and g2h3g2−1 = h1. Together with the malnormality of H these equations imply that
h1= h2= h3= 1.
We provide the set of orbits ˜G = (G\H)2/H3with a groupoid structure as follows. Two orbits f H3, g H3are composable iff H f2H = Hg1H with the product
f H3g H3= ( f1, f2)H3(g1, g2)H3= ( f1, h0g2)H3,
where h0 ∈ H is the unique element such that f2H = h0g1H . The units are 1H g H =
(g, g)H3 and the inverse of(g
1, g2)H3 is(g2, g1)H3. Let( ˜G) be the underlying graph which, as a set, concides with the complement of units in ˜G. Clearly, it is stable under the
inversion. We define the map j: ( ˜G) → ( ˜G) by the formula:
(g1, g2)H3→ (g1−1, g1−1g2)H3.
Proposition 1 The groupoidGG,H = ˜G is a -groupoid with the distinguished generating
subset H = ( ˜G) and involution j.
Proof We verify that the map j is well defined. For any h∈ H3and g∈ (G\H)2we have
j(ghH3) = j((h−1
1 g1h2, h−11 g2h3)H 3)
= (h−12 g1−1h1, h−12 g1−1h1h−11 g2h3)H3= (h−12 g−11 h1, h−12 g−11 g2h3)H3
= (g−11 , g1−1g2)(h2, h1, h3)H3= (g−11 , g1−1g2)H3= j(gH3).
Verification of the other properties is straightforward.
Example 7 For the group G = P SL(2,Z), let the subgroup H Zbe given by the upper triangular matrices. One can show that H is malnormal and the associated-groupoid is isomorphic to the one of Example3. Indeed, any element g ∈ G\H is represented by a matrix a b c d ∈ SL(2,Z),
with non-zero c, and the map g→ a/c is a bijection between the set of non-trivial cosets
{gH| g ∈ G\H} and the set of rational numbersQ, and the free action of H corresponds to translations by integers. Thus, the set of double cosets{HgH| g ∈ G\H} is identified with the setQ/Z. Any element of the latter has a unique representative in the semi-open unit rational interval[0, 1[∩Q. A morphism in the associated-groupoid is given by a pair of rationals(x, y) modulo an equivalence relation given by simultaneous translations by inte-gers, i.e a pair(x, y) is equivalent to a pair (x, y) if and only if x − x= y − y∈Z. Thus, any morphism is represented by a pair(x, t), where t ∈ [0, 1[∩Qand x ∈Q. With these identifications, calculation of structural maps is straightforward.
4-groupoids and rings
In this section we show that the constructions in Examples4and5come from functors admitting left adjoints.
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Theorem 1 The mappings A, B : Ring → Gpd are functors which admit left adjoints A and B, respectively.
Proof The case of A. Let f: R → S be a morphism of rings. Then, obviously, f (R) ⊂ S∗. Besides, for any x∈ R we have f (k(x)) = f (1−x) = f (1)− f (x) = 1− f (x) = k( f (x)), which implies that f(k(R)) = k( f (R)) ⊂ k(S). Thus, we have a well defined morphism of-groups A f = f |R. If f: R → S, g : S → T are two morphisms of rings, then
A(g ◦ f ) = g ◦ f |R= g|S◦ f |R= Ag ◦ A f . The proof of the functoriality of B is similar. We define covariant functors A, B: Gpd → Ring as follows. Let G be a -groupoid. LetZ[G] be the groupoid ring defined similarly as for groups. Namely, it is generated by the set{ux| x ∈ G} with the defining relations uxuy = ux y if x and y are composable. The ring AG is the quotient ring of the groupoid ringZ[G] with respect to the additional
relations ux+ uk(x)= 1 for all x ∈ H. The ring BG is generated overZby the elements
{ux, vx| x ∈ G} with the defining relations uxuy= ux y, vx y= uxvy+vxif x, y are compos-able, and uk(x)= vx, vk(x)= uxfor all x ∈ H. If f ∈ Gpd(G, M), we define Af and Bf respectively by Af: ux→ uf(x), and Bf: ux→ uf(x), vx→ vf(x). It is straightforward to verify the functoriality properties of these constructions.
Let us now show that A, Bare respective left adjoints of A, B. In the case of Aand A, we identify bijectionsϕG,R: Ring(AG, R) Gpd(G, AR) which are natural in G and R.
We defineϕG,R( f ): x → f (ux), whose inverse is given by ϕG−1,R(g): ux→ g(x). Let f ∈
Ring(AG, R). To verify the naturality in the first argument, let h ∈ Gpd(M, G). We have Ah∈ Ring(AM, AG), f ◦ Ah∈ Ring(AM, R), and ϕM,R( f ◦ Ah) ∈ Gpd(M, AR) with
ϕM,R( f ◦ Ah): M x → f (Ah(ux)) = f (uh(x)) = ϕG,R( f )(h(x)).
Thus,ϕM,R( f ◦ Ah) = ϕG,R( f ) ◦ h. To verify the naturality in the second argument, let
g∈ Ring(R, S). Then,
Gpd(G, AS) ϕG,S(g ◦ f ): x → g( f (ux)) = g|R(ϕG,R( f )(x)).
Thus,ϕG,S(g ◦ f ) = Ag ◦ ϕG,R( f ). The proof in the case of B, B is similar. There is a natural transformationα : A → B given by
Gpd(AR, B R) αR: x → (1 − x, x).
5 A representation theoretical interpretation of theA-ring of pairs of groups
Let G be a group with a proper malnormal subgroup H (i.e. H= G), andGG,H, the associ-ated-groupoid described in Sect.3. According to Theorem1, the ring AGG,His generated by the set of invertible elements{ux,y| x, y ∈ G\H}, which satisfy the following relations:
ux,yuy,z= ux,z, (2)
ux h,y= ux,yh= uhx,hy= ux,y, ∀h ∈ H, (3)
uy−1x,y−1+ ux,y= 1, x = y, (4)
Notice that ux,x= 1 for any x ∈ G\H.
Define another ring RG,H generated by the set{sg, vg| g ∈ G} subject to the following defining relations
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the elementvxis zero if x ∈ H and invertible otherwise, and
vx y= vy+ vxsy, ∀x, y ∈ G. (6)
For any ring R, let R∗R be the semidirect product of the group of invertible elements Rand the additive group R with respect to the (right) action of Ron R by right multipli-cations. As a set, the group R∗R is the product set R× R with the multiplication rule (x, y)(x, y) = (xx, yx+ y). This construction is functorial in the sense that for any ring
homomorphism f: R → S there corresponds a group homomorphism ˜f: R∗R→ S∗S.
Definition 2 Given a pair of groups(G, H), where H is a malnormal subgroup of G, a ring R and a group homomorphism
ρ : G x → (α(x), β(x)) ∈ R∗R.
We say thatρ is a special representation if it satisfies the following condition: for any x ∈ G, the elementβ(x) is zero for x ∈ H and invertible otherwise.
In the case of the ring RG,H we have the canonical special representation
σG,H: G → RG,HRG,H, x → (sx, vx),
which is universal in the sense that for any ring R and any special representationρ : G →
R∗R there exists a unique ring homomorphism f: RG,H → R such that ρ = ˜f ◦ σG,H. The following theorem describes the representation theoretical meaning of the A-ring.
Theorem 2 For any g∈ G\H, the rings AGG,Hand RG,H/(1 − vg) are isomorphic. The proof of this theorem is split to few lemmas. Let us define a map q of the generating set of the ring AGG,Hinto the ring RG,Hby the formula
q(ux,y) = vx−1v−1y−1.
Lemma 2 The map q extends to a unique ring homomorphism q: AGG,H → RG,H.
Proof The elements q(ux,y) are manifestly invertible and satisfy the identities q(ux,z) =
q(ux,y)q(uy,z). The consistency of the map q with relations (3) is easily seen from the following properties of the elementsvx(which are special cases of Eq.6):
vhx= vx, vx h = vxsh, ∀h ∈ H. The identity q(uy−1x,y−1) + q(ux,y) = 1 is equivalent to
vx−1y= vy− vx−1v−1y−1vy
which, in turn, is equivalent to the defining relation (6) after taking into account the formula
sy= −v−1y−1vy.
The latter formula follows from the particular case of the relation (6) corresponding to
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Let us fix an element g∈ G\H and define a map fgof the generating set of the ring RG,H into the ring AGG,H by the following formulae
fg(sx) = ug,xg= ux−1g,g, if x ∈ H; −ug,xux−1,g, otherwise, (7) and fg(vx) = 0, if x ∈ H; ux−1,g, otherwise. (8)
Lemma 3 The map fgextends to a unique ring homomorphism fg: RG,H → AGG,H.
Proof 1. Clearly, fg(1) = fg(s1) = ug,g = 1 and fg(sx−1) = fg(sx−1) = fg(sx)−1if
x∈ H. For x ∈ H we have
fg(s−1x ) = fg(sx−1) = uxg,g= u−1g,xg= fg(sx)−1.
2. We check the identity fg(sxsy) = fg(sx y) = fg(sx) fg(sy) in five different cases: (x, y ∈ H):
fg(sx y) = ug,xyg= ux−1g,yg= ux−1g,gug,yg= fg(sx) fg(sy); (x ∈ H, y ∈ H):
fg(sx y) = −ug,xyuy−1x−1,g= −ux−1g,yuy−1,g
= −ux−1g,gug,yuy−1,g= fg(sx) fg(sy); (x ∈ H, y ∈ H):
fg(sx y) = −ug,xyuy−1x−1,g= −ug,xux−1,yg
= −ug,xux−1,gug,yg= fg(sx) fg(sy); (x ∈ H, y ∈ H, xy ∈ H):
fg(sx y) = fg(sx y) fg(sy−1) fg(sy) = fg(sx ys−1y ) fg(sy)
= fg(sx) fg(sy); (x∈ H, y ∈ H, xy ∈ H):
fg(sx y) = −ug,xyuy−1x−1,g= −ug,xu−1x y,xuy−1x−1,y−1uy−1,g
= −ug,x(1 − uy,x−1)−1(1 − ux−1,y)uy−1,g
= ug,x(uy,x−1− 1)−1(uy,x−1− 1)ux−1,yuy−1,g
= ug,xux−1,yuy−1,g= ug,xux−1gug,yuy−1,g = fg(sx) fg(sy).
3. We check the identity fg(vx y) = fg(vy) + fg(vx) fg(sy) in five different cases:
(x, y ∈ H): it is true trivially;
(x ∈ H, y ∈ H):
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(x∈ H, y ∈ H): fg(vx y) = uy−1x−1,g= ux−1,yg= ux−1,gug,yg = fg(vx) fg(sy); (x ∈ H, y ∈ H, xy ∈ H): fg(vy) + fg(vx) fg(sy) = uy−1,g− ux−1,gug,yuy−1,g = (1 − ux−1,y)uy−1,g = (1 − ux−1x y,y)uy−1,g= 0; (x ∈ H, y ∈ H, xy ∈ H): fg(vx y) = uy−1x−1,g = uy−1x−1,y−1uy−1,g = (1 − ux−1,y)uy−1,g = (1 − ux−1,gug,y)uy−1,g = fg(vy) + fg(vx) fg(sy).
Associated with any invertible element t of the ring RG,H there is an endomorphism
rt: RG,H → RG,Hdefined on the generating elements by the formulae
rt(sx) = tsxt−1, rt(vx) = vxt−1.
Note that, in general, rt can have a non-trivial kernel, for example, if g ∈ G\H, then 1− vg ∈ ker(rvg).
Lemma 4 The following identities of ring homomorphisms
fg◦ q = idAGG,H, (9)
q◦ fg = rvg−1 (10)
hold true.
Proof Applying the left hand sides of the identities to be proved to the generating elements
of the corresponding rings, we have
fg(q(ux,y)) = fg(vx−1vy−1−1) = ux,gu−1y,g= ux,y, ∀x, y ∈ H, q( fg(sx)) = q(ux−1g,g) = vg−1xvg−1−1= vg−1sxvg−1−1, x ∈ H, q( fg(sx)) = q(−ug,xux−1,g) = −vg−1v−1x−1vxv−1g−1 = vg−1sxv−1g−1, ∀x ∈ H, and q( fg(vx)) = q(ux−1,g) = vxvg−1−1, ∀x ∈ H. Lemma 5 For any g∈ G\H, the kernel of the ring homomorphism fg: RG,H → AGG,H
is generated by the element 1− vg−1.
Proof Let t= vg−1. Eqs. (9), (10) imply that ker( fg) = ker(rt) and rt◦ rt = rt. The latter equation means that any x∈ ker(rt) has the form y − rt(y) for some y. The identity
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implies that that ker(rt) is generated by the elements x −rt(x), with x running in a generating set for the ring RG,H. Finally, the identities
sx− rt(sx) = sx− tsxt−1= (1 − t)sx− tsxt−1(1 − t), vx− rt(vx) = −vxt−1(1 − t), imply that ker(rt) is generated be only one element 1 − t = 1 − vg−1.
Proof of Theorem2By Lemma4, the ring homomorphism fg: RG,H → AGG,H is sur-jective whose kernel, by Lemma5, is generated by the element 1− vg−1. Thus, by the fundamental theorem in ring theory, we have an isomorphism AGG,H RG,H/ ker( fg) =
RG,H/(1 − vg−1). But g is arbitrary element in the complement of H, and so is its inverse. Thus, by replacing g with g−1, we finish the proof.
Example 8 Consider the group pair(G, H), where
G= a, b| a2 = b2= 1 Z2∗Z2,
and H= a Z2, a malnormal subgroup of G. Any element of the group G has the form
(ab)man for unique mZ, n∈ {0, 1}. One can show that
RG,H Q[x, x−1], v(ab)man → (−1)nmx, s(ab)man → (−1)n,
while q(u(ab)man,(ab)kal) → m/k, m, k = 0. Thus, AGG,H Q.
6 Weakly special representations of group pairs
In this section, we generalize the constructions of the previous section to the case of arbitrary group pairs. In this case, we shall use a weakened version of special representations. 6.1 The group pairs(Gρ, Hρ)
To each group pair homomorphism of the form
ρ : (G, H) → (R∗R, R), G g → (α(g), β(g)) ∈ R∗R, (11) where R is a ring, we associate a group pair(Gρ, Hρ), where
Gρ= G/ker(ρ), Hρ= β−1(0)/ker(ρ) ⊂ Gρ.
The setβ ◦ α−1(1) ⊂ R is an additive subgroup, and the set α(G) ⊂ R∗is a multiplicative subgroup. These groups fit into the following short exact sequence of group homomorphisms
0→ β ◦ α−1(1) → Gρ→ α(G) → 1, (12)
which sheds some light on the structure of Gρ. In particular, there exists another short exact sequence of group homomorphisms
1→ N → Gβ ◦ α−1(1) → Gρ→ 1,
where the semidirect product is taken with respect to the right group action by group auto-morphisms
β ◦ α−1(1) × G (x, g) → xα(g) ∈ β ◦ α−1(1),
and N (the image thereof) trivially intersectsβ ◦ α−1(1). We also have a group isomorphism
(11)
For a given group pair(G, H) the set of homomorphisms (11) is partially ordered with respect to the relation:
ρ < σ ⇔ ∃ exact (1, 1) → (N, M) → (Gρ, Hρ) → (Gσ, Hσ) → (1, 1). (13) 6.2 The universal ring ˆRG,H
Let(G, H) be a group pair. The ring ˆRG,H is generated overZby the set{sg, vg| g ∈ G} subject to the following defining relations:
(1) the elements sxare invertible;
(2) the mapσG,H: G x → (sx, vx) ∈ ˆRG,H ˆRG,H is a group homomorphism; (3) for any x∈ H, vx = 0.
Notice that, according to this definition, a non-zero generating elementvx is not assumed to be invertible. This ring has the following universal property: for any group pair homo-morphism (11) there exists a unique ring homomorphism fρ: ˆRG,H → R such that ρ =
˜fρ◦σG,H. The partial order (13) can alternatively be characterized by an equivalent condition:
ρ < σ ⇔ ker( fρ) ⊂ ker( fσ). (14)
6.3 Weakly special representations
Definition 3 Given a pair of groups(G, H), where H is a proper (i.e. H = G) but not
necessarily malnormal subgroup of G, a nontrivial (i.e. 0 = 1) ring R, and a group pair homomorphism (11). We say thatρ is a weakly special representation if it satisfies the following conditions: (1)β(G) ⊂ R {0}; (2) β(G) = {0}.
Any special representation (if H= G) is also weakly special. For any weakly special rep-resentationρ, the group Hρis a malnormal subgroup in Gρ, and the induced representation of the pair(Gρ, Hρ) is special in the sense of Definition2.
To a group pair(G, H), we associate a set valued invariant W(G, H) consisting of minimal (with respect to the partial ordering (13)) weakly special representations (considered up to equivalence). Notice that ifρ ∈ W(G, H), then the ring AGGρ,Hρis non-trivial (i.e. 0= 1).
Taking into account characterization (14), there is a bijection between the set W(G, H) and the set of ring homomorphisms fρwith minimal kernel.
The following proposition will be useful for calculations.
Proposition 2 Given a group pair(G, H) with H = G, a non-trivial ring R, and a weakly special representation
ρ : (G, H) → (R∗R, R), G g → (α(g), β(g)) ∈ R∗R. Assume that the subring fρ( ˆRG,H) ⊂ R is generated overZby the setα(G).
(i) Ifβ ◦ α−1(1) = {0} then fρ( ˆRG,H) is a skew-field. (ii) If 1∈ β ◦ α−1(1), then Gρ α(G)β ◦ α−1(1).
Proof (i) We remark thatα−1(1) is a normal subgroup of G having the following prop-erties:
β(g−1tg) = β(t)α(g), ∀g ∈ G, ∀t ∈ α−1(1). (15)
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Let t0∈ α−1(1) be such that x0= β(t0) is invertible. As any element x ∈ fρ( ˆRG,H) can be written in the form
x = n i=1 miα(gi), mi ∈Z, gi ∈ G, we have x0x = n i=1 mix0α(gi) = n i=1 miβ(t0)α(gi) = n i=1 β(tmi 0 )α(gi) = n i=1 β(gi−1t mi 0 gi) = β( n i=1 gi−1tmi 0 gi) = β(tx), tx = n i=1 g−1i tmi 0 gi.
Thus, if x = x0−1β(tx) = 0, then it is invertible.
(ii) Let t0∈ α−1(1) be such that β(t0) = 1. We show that the exact sequence (12) splits. Letξ : α(G) → G be a set-theoretical section to α (i.e. α ◦ ξ = idα(G)). For any
x∈ α(G) fix a (finite) decomposition
β(ξ(x)) = i mi(x)α( fi(x)), where mi(x) ∈Z, fi(x) ∈ G. We define σ : α(G) → Gρ, x → πρ ξ(x) i fi(x)−1t0−mi(x)fi(x) ,
whereπρ: G → Gρis the canonical projection. Then, it is straightforward to see that
σ is a group homomorphism such that α ◦ σ = idα(G).
Below, we give two examples, coming from knot theory, which indicate the fact that the set W(G, H) can be an interesting and relatively tractable invariant.
Example 9 (the trefoil knot 31) Consider the pair of groups(G, H), where
G = a, b| a2= b3
is the fundamental group of the complement of the trefoil knot [6], and H= ab−1, a2, the peripheral subgroup generated by the meridian m= ab−1and the longitude l= a2(ab−1)−6. As the element a2is central, the subgroup H is not malnormal in G. If we take the quotient group with respect to the center
G/a2 Z2∗Z3 P SL(2,Z),
then the image of the subgroup H/a2 Zis malnormal, and it is identified with the sub-group of upper triangular matrices in P S L(2,Z). Thus, one can construct the -groupoid
G˜G, ˜H(which is isomorphic to the one of Example3), but the corresponding A-ring happens
to be trivial (i.e. 0 = 1). This is a consequence of the result to be proved below: the set
W(G, H) consists of a single element ρ0such that, the A-ring of the pair(Gρ0, Hρ0) is
iso-morphic to the fieldQ[t]/(31(t)), where 31(t) = t
2− t + 1 is the Alexander polynomial of the trefoil knot.
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The ring ˆRG,H admits the following finite presentation: it is generated overZby four elements sa, sb, va, vb, of which sa, sbare invertible, subject to four relations
sa2= sb3, va= vb, va(1 + sa) = vb(1 + sb+ sb2) = 0. We consider a ring homomorphism
φ : ˆRG,H →Z[t]/(31(t)), sa → −1, sb→ −t−1, va → 1, vb→ 1. Theorem 3 (i) For any weakly special representationρthere exists an equivalent
repre-sentationρ such that the ring homomorphism fρfactorizes throughφ, i.e. there exists a unique ring homomorphism hρ:Z[t]/(31(t)) → R such that fρ= hρ◦ φ.
(ii) The kernel of the group homomorphism ˜φ ◦ σG,His generated by a2and(ab−1)6with
the quotient group pair( ˜G, ˜H),
˜G = a, b| a2= b3= (ab−1)6= 1, ˜H = ab−1, where ˜H is malnormal in ˜G.
(iii) The ring AG˜G, ˜His isomorphic to the fieldQ[t]/(31(t)). Proof (i) Let R be a non-trivial ring and
ρ : G g → (α(g), β(g)) ∈ R∗R,
a weakly special representation. We have
β(a) = β(ab−1b) = β(ab−1)α(b) + β(b) = β(b),
and
0= β(a2) = β(a)(α(a) + 1), 0 = β(b3) = β(b)(α(b)2+ α(b) + 1). The elementξ = β(a) = β(b) is invertible, since otherwise ξ = 0 and β−1(0) = G. Thus,α(a) = −1 and α(b) is an element satisfying the equation 31(−α(b)−1) = 0.
Replacingρ by an equivalent representation, we can assume that ξ = 1. The ring homomorphism fρis defined by the images of the generating elements
sa→ −1, sb→ α(b), va → 1, vb→ 1,
and it is easy to see that we have a factorization fρ = hρ◦ φ, with a unique ring homomorphism
hρ:Z[t]/(31(t)) → R, t → −α(b)−1.
(ii) It is easily verified that a2, (ab−1)6∈ ker( ˜φ ◦ σG,H). We remark an isomorphism
˜G s, t1, t2| s6= 1, t1t2= t2t1, st1= t2s, t1st2= t2s Z6 Z2 given, for example, by the formulae:
s→ ab−1, t1→ babab, t2 → bab−1a, and the induced from ˜φ ◦ σG,H group homomorphism takes the form
s→ (t, 0), t1→ (1, 1), t2→ (1, t−1),
so that a generic element t1mt2nsk, m, n ∈Z, k∈Z6, has the image(tk, (m + nt−1)tk). The latter is the identity element(1, 0) if and only if k = 0 (mod 6) and m = n = 0.
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(iii) The pair(G, H) and any weakly special representation ρ satisfy the conditions of Proposition2, and thus the ring AGGρ,Hρis the localization of a homomorphic image
of the ringZ[t]/(31(t)) at all non-zero elements. The ringZ[t]/(31(t)) itself is a
commutative integral domain, and thus its minimal quotient ring corresponds to the zero ideal. In this way, we come to the fieldQ[t]/(31(t)) which is the A-ring
associ-ated to the only minimal weakly special representationρ0with(Gρ0, Hρ0) = ( ˜G, ˜H). Corollary 1 The set W(G, H) is a singleton consisting of a minimal weakly special repre-sentationρ0such that H∩ ker(ρ0) = m6, l.
Remark 4 The group pair( ˜G, ˜H) is the quotient of the pair (G, H) with respect to the normal
subgroup of G generated by the center of G and the longitude l= a2(ab−1)−6.
Example 10 (the figure-eight knot 41) The fundamental group of the complement admits the following presentation [6]:
G= a1, a2| a1wa = waa2, wa = a−12 a1a2a−11 , the peripheral subgroup being given by
H = m = a1, l = wa ¯wa Z2, ¯wa= a−11 a2a1a2−1.
One can show that H is a malnormal subgroup of G (this is true for any hyperbolic knot), and that the corresponding A-ring is trivial. The latter fact will follow from our description of the set W(G, H): it consists of two minimal weakly special representations ρi, i∈ {1, 2}, such that
H∩ ker(ρi) = mpi, lmqi, (p1, q1) = (0, 0), (p2, q2) = (6, 3).
The ring ˆRG,H admits the following finite presentation: it is generated by four elements
{sai, vai| i ∈ {1, 2}}, with sai being invertible, subject to four relations:
sa1swa = swasa2, va1= 0, v¯wa+ vwas¯wa = 0, vwa(1 − sa2) = va2,
where
swa = sa−12 sa1sa2sa−11 , s¯wa = sa−11 sa2sa1sa−12 , vwa = va2(sa−11 − swa), v¯wa = −va2(1 − sa1)s−1a2 .
We also define a ring S generated overZby three elements{p, r, x}, and the following defining relations: x2= x − p, (17) px= x + 3p + r2− 1, (18) pr = r, (19) r x+ xr = r − r2, (20) p2= 1 − 4p − 2r2. (21)
We remark that this ring is noncommutative and finite dimensional overZwith dimZS= 6.
Indeed, it is straightforward to see that for aZ-linear basis one can choose, for example, the set{1, p, r, x, rx, r2}, the set {1, p, r2} being aZ-basis of the center.
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Lemma 6 In the ring S, let(a) be the two sided ideal generated by the element a= 2 + 2p + r + 2x + 3r2+ rx.
Then,{2, r, 1 − p} ⊂ (a).
Proof First, one can verify that 2+ r = ba, where b= 1 − x − r2− 2rx,
so that 2+ r ∈ (a). Next, one has r = ca + (2 + r)x, where
c= −2 + p + 3r + 4x − rx,
so that r∈ (a), and thus 2 ∈ (a). Finally, remarking that 1− p = 2x + r2x,
we conclude that 1− p ∈ (a).
Lemma 7 Let p, q, x be three elements of a ring R satisfying the three identities
p= x − x2, (22)
q= px − 3p − x + 1, (23)
pq= q. (24)
Then, the element p is invertible if and only if
2q+ p2+ 4p − 1 = 0. (25)
Proof First, multiplying by q identity (23) and simplifying the right hand side by the use of Eq. (24), we see that
q2= −2q. (26)
Next, excluding x from identities (22) and (23), we obtain the identity
q2+ (5p − 1)q + p(p2+ 4p − 1) = 0
which, due to Eqs. (24) and (26), simplifies to
2q+ p(p2+ 4p − 1) = 0. (27)
Now, if p is invertible, then Eqs. (24), (27) imply Eq. (25). Conversely, if (25) is true, then combining it with (27), we obtain the polynomial identity
(p − 1)(p2+ 4p − 1) = 0
which implies invertibility of p with the inverse p−1= 5 − 3p − p2.
Theorem 4 (i) There exists a unique ring homomorphismφ : ˆRG,H → S such that
sa1 → x, sa2 → x + r, va1 → 0, va2→ 1.
(ii) For any weakly special representationρ, considered up to equivalence, the ring homo-morphism fρ factorizes through φ, i.e. there exists a unique ring homomorphism hρ: S → R such that fρ= hρ◦ φ.
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Proof (i) This is a straightforward verification. (ii) Let R be a non-trivial ring and
ρ : G g → (α(g), β(g)) ∈ R∗R,
a weakly special representation of G. Then, the elementξ = β(a2) is invertible, since otherwiseβ−1(0) = G. By replacing ρ with an equivalent representation, we can assume thatξ = 1.
Denote
xi = α(ai), wx= α(wa), ¯wx = α( ¯wa). Note that
0= β(a2a−12 ) = β(a−12 ) + x−12 ⇔ β(a2−1) = −x−12 . From the definitions ofwaand ¯wawe have
β(wa) = β(a−12 a1a2a−11 ) = β(a1a2a1−1) + β(a2−1)x1x2x1−1= x1−1− wx, (28)
β( ¯wa) = β(a−11 a2a1a−12 ) = −x2−1+ x1x2−1= −(1 − x1)x2−1. (29) From the relation a1wa= waa2we obtain an identity
β(wa) = β(a1wa) = β(waa2) = β(wa)x2+ 1 which implies thatβ(wa) = 0, and thus β(wa) is invertible with
β(wa)−1= 1 − x2. Invertibility ofβ( ¯wa) follows from the equation
0= β(wa¯wa) = β(wa) ¯wx+ β( ¯wa).
Compatibility of the latter equations with the formulae (28), (29) implies the following anticommutation relations xi−1xj+ xjxi−1= x−1i + xj− 1, {i, j} = {1, 2}. (30) Indeed, x1x2−1= (1 − x2)β(wa)x1x2−1= (1 − x2)(x1−1− wx)x1x2−1 = (1 − x2)(x1−1− x−12 x1x2x−11 )x1x2−1= (1 − x2)(x2−1− x2−1x1) = (x2−1− 1)(1 − x1) = x1+ x−12 − 1 − x2−1x1, and x1−1x2= x1−1x2x1x2−1x2x1−1= ¯wxx2x1−1= −β(wa)−1β( ¯wa)x2x−11 = (1 − x2)(1 − x1)x2−1x2x1−1=(1 − x2)(x1−1− 1)=x2+ x1−1− 1 − x2x1−1. By using relations (30), one obtains the following formula:
wx−1= (x2−1− 1)(1 − x1− x2)x1−1. Indeed,
w−1x = x1x2−1x1−1x2= (x1+ x2−1− 1 − x2−1x1)x1−1x2= (x2−1− 1)(1 − x1)x1−1x2
= (x2−1− 1)(x1−1x2− x2) = (x2−1− 1)(x1−1− 1 − x2x1−1)
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This formula implies the following equivalences:
x2w−1x = w−1x x1 ⇔ x2(1 − x1− x2) = (1 − x1− x2)x1 ⇔ x2(1 − x2)
= (1 − x1)x1.
The latter identity can be equivalently rewritten in the form
x1x21+ x21x1= x21− x212, (31) where
x21= x2− x1,
and the same identity implies that there exists an invertible element z such that
xi(1 − xi) = z, i ∈ {1, 2}.
Evidently, z commutes with both x1 and x2. We have the following formulae for the inverses of xi:
xi−1= (1 − xi)z−1,
substitute them into Eq. (30), and rewrite the result as the following two equations
x1x21+ x21x1= x21+ 3z + x1− zx1− 1, (32) and
(z − 1)x21= 0. (33)
Compatibility of Eqs. (31) and (32) gives one more identity
zx1+ 1 = 3z + x1+ x221. (34)
Finally, applying Lemma7to elements p, x2
21, x1, we obtain one more identity 2x212 + p2+ 4p − 1 = 0.
Now, we can easily see that the mapping
hρ(p) = z, hρ(r) = x21, hρ(x) = x1,
in a unique way extends to a ring homomorphism hρ: S → R, and one has the
factor-ization formula fρ= hρ◦ φ.
(iii) We remark that our pair(G, H) and any weakly special representation ρ verify the conditions of Proposition2. For example, let us check that there exists an element
t0 ∈ α−1(1) such that β(t0) is invertible. In the case if x21 = 0, we can choose
t0 = a1−1a2 for whichβ(t0) = 1, while for x21= 0 we can choose t0 = wa¯w−1a for which
β(t0) = 2 + 2z + x21+ 2x1+ 3x212 + x21x1.
Due to Lemma6, the latter element is non-zero and therefore invertible. Thus, taking into account the parts (i) and (ii), to identify the elements of the set W(G, H), it is enough to find all minimal quotients of the ring S which can be embedded into rings the way that all non-zero elements become invertible. In particular, the quotient rings must be integral domains.
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We have two relations in S which indicate existence of zero-divisors:
r(1 − p) = 0, r(2 + r2) = 0.
We have the following mutually excluding possibilities for minimal quotients which remove these relations:
(1) r= 0;
(2) p= 1, r2= −2 = 0.
The case (1) gives the quotient ring
S/(r) Z[t]/(41(t)), 41(t) = t
2− 3t + 1, x → t, p → 1 − 2t. This is a commutative integral domain, and its localization at the set of non-zero ele-ments coincides with the field of fractionsQ[t]/(41(t)).
The case (2) gives the quotient ring
S/(1 − p, 2 + r2) Zr, x/(1 − x + x2, rx + xr − 2 − r),
which is isomorphic to the ring of Hurwitz integral quaternions
H =a+ bi + cj + dk| (a, b, c, d) ∈Z4 (2−1+Z)4 ,
where
1+ i2= 1 + j2= i j + ji = 0, k = i j, the isomorphism being given by the map
r→ i + j, x → 2−1(1 − i − j + k),
and the inverse map
i→ rx − 1, j → xr − 1.
Thus, S/(1 − p, 2 + r2) admits an embedding into a (non-commutative) division ring of rational quaternions.
Letρi, i ∈ {1, 2}, represent the two elements of W(G, H). Corresponding to ρ1the group pair(G1, H1) admits a presentation
G1= s, t0, t1| t0t1= t1t0, st1= t0s, t1st0= t03s Z Z2, H1= s Z with the projection homomorphism
π1: (G, H) → (G1, H1), a1→ s, a2→ st0,
whose kernel ker1) can be shown to be generated by the longitude l = wa ¯wa. In other words, we have a group isomorphism
G1 a1, a2| a1wa= waa2, wa¯wa= 1, wa = a−12 a1a2a−11 , ¯wa= a−11 a2a1a2−1. To describe the group pair(G2, H2), corresponding to ρ2, consider the following repre-sentation of G in S L(4,Z): a1→ ⎛ ⎜ ⎜ ⎝ 0 1 0 0 −1 1 0 0 0 0 0 1 0 0 −1 1 ⎞ ⎟ ⎟ ⎠ , a2→ ⎛ ⎜ ⎜ ⎝ 0 0 1 0 0 1 1 −1 −1 0 1 0 −1 1 1 0 ⎞ ⎟ ⎟ ⎠ ,
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whose kernel is generated, for example, by the element a1−1a2a21a2, and the corresponding right action of G onZ4(given by the multiplication of integer row vectors by above matrices). The group G2 is the quotient group of the semidirect product G Z4 by the relation
a1(1, 0, 0, 0) = a2a21a2, where we identify G andZ4 as subgroups of G Z4, while the subgroup H2= a1 Z6.
7 Presentations of-groupoids
7.1 The tetrahedral category
For any non-negative integer n ≥ 0 we identify the symmetric groupSn as the sub-group of all permutations (bijections) of the set of non-negative integersZ≥0acting identically on the subsetZ≥n⊂Z≥0. This interpretation fixes a canonical inclusionSm ⊂Snfor any pair
m≤ n. The standard generating set {si = (i − 1, i)| 1 ≤ i < n} ofSnis given by elementary transpositions of two consecutive integers. Later on, it will be convenient to use the inductive limit
S∞= lim−→Sn= ∪n≥0Sn.
We also denote bySnSet the category ofSn-sets, i.e. sets with a leftSn-action andSn -equi-variant maps as morphisms.
We remark that in any-groupoid G, its distinguished generating set H is anS3-set given by the identifications s1= j and s2= i, while the set V ⊂ H2of H -composable pairs is an S4-set given by the rules
s1(x, y) = ( j(x), j(k(x) j(y))), s2(x, y) = (i(x), xy), s3(x, y) = (xy, i(y)), (35) and the projection map to the first component
pr1: V (x, y) → x ∈ H beingS3-equivariant.
Let R43:S4Set→S3Set be the restriction (to the subgroup) functor. Consider the comma category2(R
43↓S3Set) ofS3-equivariant maps a: R43(Va) → Ia, for someS4-set Va, and S3-set Ia. Call it the tetrahedral category. An object of this category will be called
tetra-hedral object. A morphism between two tetratetra-hedral objects f: a → b, called tetrahedral morphism, is a pair(pf, qf) where pf ∈S4Set(Va, Vb) and qf ∈S3Set(Ia, Ib) are such that b R43(pf) = qfa. Taking into account the remarks above, to each-groupoid G, we can associate a tetrahedral object C(G) = pr1: R43(V ) → H. If f : G → Gis a morphism of-groupoids, then the pair
C( f ) = ( f × f |V, f |H)
is a tetrahedral morphism such that C( f g) = C( f )C(g). Thus, we obtain a functor
C: Gpd → (R43↓S3Set).
A-groupoid G is called finite if its distinguished set H is finite (note that G itself can be an infinite groupoid). LetGpdfin be the full subcategory of finite -groupoids, and
(R43↓S3Set)finthe full subcategory of finite tetrahedral objects (i.e. a’s with finite Vaand
Ia). Then, the functor C restricts to a functor
Cfin: Gpdfin→ (R43↓S3Set)fin. 2see [5] for a general definition of a comma category.
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Theorem 5 The functor Cfinadmits a left adjoint
Cfin : (R43↓S3Set)fin → Gpdfin
which verifies the identities
CfinCfinCfin = Cfin , CfinCfin Cfin = Cfin.
Proof Let a be an arbitrary finite tetrahedral object. Consider a map τa: Va→ Ia2, τa(v) = (a(v), a((321)(v))), ∀v ∈ Va
LetRbe the minimalS4-equivariant equivalence relation on Vagenerated by the set
x∈I2 a
τa−1(x)2
andRa, the a-image ofRwhich is necessarily anS3-equivariant equivalence relation on
Ia. Let pa: Va → Va/Rand qa: Ia → Ia/Ra be the canonical equivariant projections on the quotient sets. There exists a unique tetrahedral object a1such that the pair(pa, qa) is a tetrahedral morphism from a to a1. Iterating this procedure, we obtain a sequence of tetrahedral morphisms
a→ a1→ a2→ · · ·
which, due to finiteness of a, stabilizes in a finite number of steps to a tetrahedral object˜a. It is characterized by the property that the mapτ˜ais an injection, so that we can identify the set V˜awith itsτ˜a-image in I˜a2.
For any(x, y) ∈ V˜a denote ˜a(s3(x, y)) = xy and call it product. Then, the action of the groupS4 on V˜a is given by the formulae (35), where i(x) = s2(x), j(x) = s1(x),
k(x) = (02)(x), and the consistency conditions imply the following properties of the
prod-uct:
i(xy) = i(y)i(x), k(xy) = k(k(x) j(y))k(y), i(x)(xy) = (yx)i(x) = y. (36) However, this product is not necessarily associative, and to repair that, we consider the min-imalS3-equivariant equivalence relationSon I˜agenerated by the relations x(yz) ∼ (xy)z, where all products are supposed to make sense. Letπ˜a: I˜a → I˜a/Sbe the canonicalS3 -equi-variant projection on the quotient set. Then, clearly,˜a= π˜a◦ ˜a is another (finite) tetrahedral object with V˜a= V˜aand I˜a = I˜a/S, and with canonically associated tetrahedral morphism
(id˜a, π˜a): ˜a → ˜a. Applying the “tilde” operation to˜a, we obtain a composed morphism (p˜a, q˜a) ◦ (id˜a, π˜a) ◦ (pa, qa): a → ˆa = ˜a
Again, the iterated sequence of such morphisms
a→ ˆa → ˆˆa → · · ·
stabilizes in finite number of steps to an object˙a with V˙a ⊂ I2˙a and the associative product
x y= ˙a(s3(x, y)) satisfying the relations (36).
Now, letT be the minimal equivalence relation on I˙agenerated by the set(i × idI˙a)(V˙a).
Denote by
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the canonical projection to the quotient set. Define another map cod= dom ◦ i : I˙a → N˙a.
In this way, we obtain a graph (quiver)awith the set of arrows I˙a, the set of nodes N˙a, and the domain (source) and the codomain (target) functions dom(x), cod(x). Thus, we obtain a finite-groupoid with the presentation
Cfin (a) = a| x ◦ y = xy if (x, y) ∈ V˙a
whose distinguished generating set is given by I˙a. 7.2-complexes
Let
n = {(t
0, t1, . . . , tn) ∈ [0, 1]n+1| t0+ t1+ · · · + tn = 1}, n ≥ 0 be the standard n-simplex with face inclusion maps
δm: n → n+1, 0 ≤ m ≤ n + 1 defined by
δm(t0, . . . , tn) = (t0, . . . , tm−1, 0, tm, . . . , tn) A (simplicial) cell in a topological space X is a continuous map
f: n→ X
such that the restriction of f to the interior ofnis an embedding. On the set of cells(X) we have the dimension function
d: (X) →Z, ( f : n → X) → n.
Hatcher in [2] introduces -complexes as a generalization of simplicial complexes. A
-complex structure on a topological space X can be defined as a pair ((X), ∂), where (X) ⊂ (X) and ∂ is a set of maps
∂ =∂n: d|−1(X)(Z≥max(1,n)) → d|−1(X)(Z≥n−1) n ≥ 0
such that:
(i) each point of X is in the image of exactly one restriction,α|
n forα ∈ (X);
(ii) α ◦ δm = ∂mα;
(iii) a set A⊂ X is open iff α−1(A) is open for each α ∈ (X). Clearly, any-complex is a CW-complex.
7.3 Tetrahedral objects from-complexes
We associate a tetrahedral object aXto a-complex X as follows:
VaX =S4×
3(X), I
aX =S3×
2(X),
which areSn-sets with the groups acting by left multiplications on the first components, and
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where(33) = 1, and the value aX(g, x) for g = (i3) is uniquely deduced from itsS3 -equi-variance property.
7.4 Ideal triangulations of knot complements
A particular class of three dimensional-complexes arises as ideal triangulations of knot complements. A simple caculation shows that in any ideal triangulation there are equal num-ber of edges and tetrahedra and twice as many faces. For example, the two simplest non-trivial knots (the trefoil and the figure-eight) admit ideal triangulations with only two tetrahedra
{u, v}, four faces {a, b, c, d} and two edges {p, q}, the difference being in the gluing rules.
Using the notation(x|∂0x, ∂1x, . . . , ∂nx), these examples read as follows.
Example 11 (The trefoil knot) The gluing rules are given by a list
(u|a, b, c, d), (v|d, c, b, a), (a|p, p, p), (b|p, q, p), (c|p, q, p), (d|p, p, p).
The associated-groupoid G is freely generated by a quiver (oriented graph) consisting of two vertices A and B and two arrows x and y with
dom(x) = cod(x) = dom(y) = A, cod(y) = B, with the distinguished subset
H = {x±1, x±2, y±1, (xy)±1}, j : x → x−1, x2→ y, x−2→ xy.
One can show that
AG BG Z[t, 3−1]/(31(t)), 31(t) = t
2− t + 1.
Example 12 (The figure-eight knot) The gluing rules are given by the list
(u|a, b, c, d), (v|c, d, a, b), (a|p, q, q), (b|p, p, q), (c|q, p, p), (d|q, q, p).
In this case, there are no non-trivial identifications in the corresponding-groupoid G, and the ring AG is isomorphic to the ring S from Example10, while
BG Zu±1, v±1, w±1| u(u + 1) = w, v(v + 1) = w−1, (uvu−1v−1)2= w. 8 Homology of-groupoids
Given a-groupoid G with the distinguished generating subset H. We define recursively the following sequence of sets: V−1 = {∗} (a singleton or one element set), V0 = π0(G) (the set of connected components of G), V1= Ob(G) (the set of objects or identities of G),
V2 = H, V3is the set of all H -composable pairs, while Vn+1for n> 2 is the collection of all n-tuples(x1, x2, . . . , xn) ∈ Hn, satisfying the following conditions:
(xi, xi+1) ∈ V3, 1 ≤ i ≤ n − 1, and ∂i(x1, . . . , xn) ∈ Vn, 1 ≤ i ≤ n + 1, where ∂i(x1, . . . , xn) = ⎧ ⎨ ⎩ (x2, . . . , xn), i= 1; (x1, . . . , xi−2, xi−1xi, xi+2, . . . , xn), 2 ≤ i ≤ n; (x1, . . . , xn−1), i= n + 1. (37)
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Remark 5 If N(G) is the nerve of G, then the system {Vn}n≥1can be defined as the maximal system of subsets Vn+1⊂ N(G)n∩ Hn, n= 0, 1, . . ., (with the identification H0= Ob(H)) closed under the face maps of the simplicial set N(G).
For any H -composable pair(x, y) we introduce a binary operation
x∗ y = j(k(x) j(y)) = jk(x) j(xy) = i j(x) j(xy). (38)
Lemma 8 For any integer n≥ 2, if (x1, . . . , xn) ∈ Vn+1, then the(n − 1)-tuple
0(x1, . . . , xn) = (y1, . . . , yn−1), yi = zi∗ xi+1, zi = x1x2· · · xi, (39)
is an element of Vn.
Proof We proceed by induction on n. For n= 2 the statement is evidently true. Choose an
integer k≥ 3 and assume the statement is true for n = k − 1. Let us prove that it is also true for n= k. Taking into account the formula
yi = i j(zi) j(zi+1), 1 ≤ i ≤ k − 1,
we see that for any 1≤ i ≤ k − 2, the pair (yi, yi+1) is H-composable with the product
yiyi+1= i j(zi) j(zi+2) = i j(zi) j(zixi+1xi+2) = zi∗ (xi+1xi+2). Now, the(k − 2)-tuples
(y1, . . . , yi−1, yiyi+1, yi+2, . . . , yk−1)
= ∂0(x1, . . . , xi, xi+1xi+2, xi+3, . . . , xk), 1 ≤ i ≤ k − 2, as well as
(y1, . . . , yk−2) = ∂0(x1, . . . , xk−1), (y2, . . . , yk−1) = ∂0(x1x2, . . . , xk−1), are all in Vk−1by the induction hypothesis, and thus(y1, . . . , yk−1) ∈ Vk.
Definitions (37), (39) also make sense for n = 2, and, additionally, we extend them for three more values n= −1, 0, 1 as follows:
(n = −1) as V−1is a terminal object in the category of sets, there are no other choices but one for the map0|V0;
(n = 0) if for A∈ V1we denote[A] the connected component of G defined by A, then
0A= [A], ∂1A= [A]; (40)
(n = 1) using the domain (source) and the codomain (target) maps of the groupoid G (viewed as a category), we define
0x= cod( j(x)), ∂1x = cod(x), ∂2x = dom(x), x ∈ V2. (41) For any n≥ −1, let Bn =ZVnbe the abelian group freely generated by the elements of
Vn. Define also Bn = 0, if n < −1. We extend linearly the maps ∂i, i ∈Z≥0to a family of group endomorphisms
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so that∂i|Bn = 0 if i > n. Then, the formal linear combination
∂ =
i≥0 (−1)i
i
is a well defined endomorphism of the group B such that∂ Bn⊂ Bn−1.
Theorem 6 The group B is a chain complex with differential ∂ and grading operator p: B → B, p|Bn = n.
Proof It is immediate to see that the group homomorphism∂= ∂0− ∂ is a restriction of the differential of the standard integral (augmented) chain complex of the nerve N(G) so that
2= 0. Now, due to the latter equation, the equation ∂2 = 0 is equivalent to the equation
2
0 = ∂0+ ∂0
which can straightforwardly be checked on basis elements.
Lemma 9 There exists a unique sequence of set-theoretical maps δi:S∞→S∞, i ∈Z≥0, such that for any j ∈Z>0,
δi(sj) = ⎧ ⎨ ⎩ sj−1 if i < j − 1; 1 if i ∈ { j − 1, j}; sj if i > j, (42) and δi(gh) = δi(g)δg−1(i)(h), ∀g, h ∈S∞. (43)
Proof From the identity
δi(g) = δi(1g) = δi(1)δi(g)
it follows immediately thatδi(1) = 1 for any i ∈Z≥0. To prove the statement it is enough to check consistency of the defining relations of the groupSwith formulae (42), (43), i.e. the equations
δi(sj)δsj(i)(sk) = δi(sk)δsk(i)(sj), | j − k| > 1, δi(sj)δsj(i)(sj+1)δsj+1sj(i)(sj) = δi(sj+1)δsj+1(i)(sj)δsjsj+1(i)(sj+1),
δi(sj)δsj(i)(sj) = 1, ∀i ∈Z≥0, ∀ j ∈Z>0.
This is a straightforward verification.
Remark 6 For any n≥ 1, we have δi(Sn) ⊂Sn−1, 0≤ i ≤ n.
Lemma 10 For any n≥ −1 the set Vnhas a unique canonical (i.e. independent of
particu-larities of a-groupoid) structure of anSn+1-set such that
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Proof For n≤ 0 the statement is trivial. For n > 0 and g ∈ {s1, . . . , sn}, Eqs. (44) take the form ∂i◦ sj= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ sj−1◦ ∂i if i< j − 1; ∂j if i= j − 1; ∂j−1 if i= j; sj◦ ∂i if i> j. (45)
In the case n= 1, Eqs. (40), (45) these imply that for any A∈ V1
∂i(s1(A)) = ∂i(A), i ∈ {0, 1}. The only canonical solution to these equations is of the form
s1(A) = A, ∀A ∈ V1. (46)
In the case n= 2, Eqs.(41), (45) and (46) imply that
∂m◦ s1=∂m◦ j, ∂m◦ s2= ∂m◦ i, 0 ≤ m ≤ 2, with only one canonical solution of the form
s1(x) = j(x), s2(x) = x−1, ∀x ∈ V2.
In the case n > 2, one can show that the following formula constitutes a solution to system (45): si(x1, . . . , xn−1)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ( j(x1), x1∗ x2, (x1x2) ∗ x3, . . . , (x1· · · xn−2) ∗ xn−1) if i= 1; (x1−1, x1x2, x3, . . . , xn−1) if i= 2; (x1, . . . , xi−3, xi−2xi−1, xi−1−1, xi−1xi, xi+1, . . . , xn−1) if 2 < i < n; (x1, . . . , xn−3, xn−2xn−1, x−1n−1) if i= n,
where we use the binary operation (38). Let us show that there are no other (canonical) solutions.
We proceed by induction on n. The case n≤ 2 has already been proved. Assume, that the solution is unique for n= m − 1 ≥ 2. For n = m, Eq. (44) with i ∈ {1, m} implies that
(y2, . . . , ym−1) = δ1(g)(∂g−1(1)(x1, . . . , xm−1)),
(y1, . . . , ym−2) = δm(g)(∂g−1(m)(x1, . . . , xm−1)),
where(y1, . . . , ym−1) = g(x1, . . . , xm−1). By the induction hypothesis, the right hand sides
of these equations are uniquely defined, and so are their left hand sides. The latter, in turn, uniquely determine the(m − 1)-tuple (y1, . . . , ym−1), and, thus, the solution is unique for
n= m.
Theorem 7 The sub-group A⊂ B, generated by the set of elementsn≥1{x + six| x ∈ Vn, 1 ≤ i ≤ n},
is a chain sub-complex so that there is a short exact sequence of chain complexes:
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Sujets connexes : | 2021-07-31 12:02:48 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8751583099365234, "perplexity": 4622.095866799938}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154089.6/warc/CC-MAIN-20210731105716-20210731135716-00114.warc.gz"} |
https://socratic.org/questions/a-balanced-lever-has-two-weights-on-it-one-with-mass-2-kg-and-one-with-mass-16-k | # A balanced lever has two weights on it, one with mass 2 kg and one with mass 16 kg. If the first weight is 8 m from the fulcrum, how far is the second weight from the fulcrum?
Dec 11, 2015
$1 \text{m}$
#### Explanation:
You can use the principle of moments. The moment of a force is the force multiplied by the perpendicular distance of the force from the fulcrum.
For the lever to be balanced the anti - clockwise moments must balance the clockwise moments.
So from the diagram we can write:
$2 g \times 8 = 16 g \times d$
$\therefore d = \frac{16 g}{16 g} = 1 \text{m}$ | 2019-11-15 02:50:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 3, "mathjax_inline_tex": 1, "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": 0.639733076095581, "perplexity": 512.5876306041596}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668561.61/warc/CC-MAIN-20191115015509-20191115043509-00332.warc.gz"} |
http://www.talkstats.com/threads/chow-test.2664/ | # chow test
#### sciacallojo
##### New Member
Hi everybody,
I've got a problem with Chow Test about structural break...
in fact on the base of my knowledge I think that this test can be used for time series data as cross-section data... but using the software Gretl I find the problem. With cross-section data the software doesn't allow me to do this test... now my question is: there is a problem with the software or with my knowledge? if the problem is the software, someone knows another software that allows to do the chow test??
Thanks
Sciacallojo
PS: sorry for my english | 2020-06-02 08:49:48 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8031973838806152, "perplexity": 783.9570123999564}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347423915.42/warc/CC-MAIN-20200602064854-20200602094854-00394.warc.gz"} |
https://cob.silverchair.com/jeb/article/206/15/2547/20243/Swing-leg-retraction-a-simple-control-model-for | In running, the spring-like axial behavior of stance limbs is a well-known and remarkably general feature. Here we consider how the rotational behavior of limbs affects running stability. It is commonly observed that running animals retract their limbs just prior to ground contact, moving each foot rearward towards the ground. In this study, we employ a conservative spring-mass model to test the effects of swing-leg retraction on running stability. A feed-forward control scheme is applied where the swing-leg is retracted at constant angular velocity throughout the second half of the swing phase. The control scheme allows the spring-mass system to automatically adapt the angle of attack in response to disturbances in forward speed and stance-limb stiffness. Using a return map to investigate system stability, we propose an optimal swing-leg retraction model for the stabilization of flight phase apex height. The results of this study indicate that swing-leg retraction significantly improves the stability of spring-mass running,suggesting that swing-phase limb dynamics may play an important role in the stabilization of running animals.
In running, kinetic and potential energy removed from the body during the first half of a running step is transiently stored as elastic strain energy and later released during the second half by elastic recoil. The mechanism of elastic recoil was first proposed in 1964, when Cavagna and collaborators noticed that the forward kinetic energy of the body's center of mass is in phase with fluctuations in gravitational potential energy(Cavagna et al., 1964). They hypothesized that humans and animals most likely store elastic strain energy in muscle, tendon, ligament and perhaps even bone to reduce fluctuations in total mechanical energy. Motivated by these energetic data, Blickhan(1989) and McMahon and Cheng(1990) proposed a simple model to describe the stance period of symmetric running gaits: a point mass attached to a massless, linear spring. Using animal data to select the initial conditions at first ground contact, they demonstrated that the spring-mass model can predict important features of stance period dynamics(Blickhan, 1989; McMahon and Cheng, 1990).
Since its formulation the spring-mass model has served as the basis for theoretical treatments of animal and human running, not only for the study of running mechanics, but also stability. Kubow and Full(1999) investigated the stability of hexapod running in numerical simulation. At a preferred forward velocity, a pre-defined sinusoidal pattern of each leg's ground reaction force resulted in stable movement patterns. However, the legs could not be viewed as entirely spring-like since their force production did not change in response to disturbances applied to the system. Later Schmitt and Holmes(2000) found a lateral spring-mass stability for hexapod running on a conservative level where total mechanical energy is constant. However, in this study, they investigated lateral and not sagittal plane stability in a uniform gravitational field. In contrast, Seyfarth et al.(2002) investigated the stride-to-stride sagittal plane stability of a spring-mass model. Although the model is conservative it can distribute its energy into forward and horizontal directions by selecting different leg angles at touch-down(Geyer et al., 2002). Surprisingly, this partitioning turns out to be assymptotically stable and predicts human data at moderate running speeds (5 m s-1). However,model stability cannot be achieved at slow running speeds (≤3 m s-1). Additionally, at moderate speeds (∼5 m s-1), a high accuracy of the landing angle (±1°) is required, necessitating precise control of leg orientation.
The purpose of this study is to investigate control strategies that enhance the stability of the spring-mass model on a conservative level. In the control scheme of Seyfarth et al.(2002), the angle with which the spring-mass model strikes the ground is held constant from stride-to-stride. In this investigation, we relax this constraint and impose a swing-leg retraction, a behavior that has been observed in running humans and animals (Muybridge, 1955; Gray, 1968) in which the swing-leg is moved rearward towards the ground during late swing-phase. This controlled limb movement has been shown to reduce foot-velocity with respect to the ground and, therefore, landing impact(De Wit et al., 2000). Additionally, a biomechanical model for quadrupedal locomotion indicated that leg retraction could improve stability in quadrupedal running(Herr, 1998; Herr and McMahon, 2000, 2001; Herr et al., 2002). We hypothesize that swing-leg retraction improves the stability of the spring-mass model by automatically adjusting the angle with which the model strikes the ground from one stride to the next. We test this hypothesis by imposing a constant rate of retraction throughout the second half of the swing phase. Using a return map analysis on swing-phase apex height(Seyfarth et al., 2002), we compare model stability at zero retraction velocity (constant angle of attack)to model stability at several non-zero retraction velocities.
### Spring-mass running with leg retraction
Running is characterized by a sequence of contact and flight phases. For the contact phase of symmetric running gaits, researchers have described the dynamics of the center of mass with a spring-mass model comprising a point mass attached to a massless, linear leg spring(Blickhan, 1989; McMahon and Cheng, 1990). To describe the dynamics of the flight phase, a ballistic representation of the body's center of mass has been used(McMahon and Cheng, 1990). In their investigation of the stability of spring-mass running, Seyfarth et al.(2002) assumed that the leg spring strikes the ground at a fixed angle with respect to the ground.
In this investigation, the effect of swing-leg retraction on the stability of the spring-mass model is investigated. Here the orientation of the leg is not held fixed during the swing phase, but is now considered a function of time α(t). For simplicity, we assume a linear relationship between leg angle (measured with respect to the horizontal) and time, starting at the apex tapex with an initial leg angleα R (retraction angle)(Fig. 1):
$\ \mathrm{for}{\ }t{<}t_{\mathrm{APEX}},{\ }{\ }{\alpha}(t)={\alpha}_{\mathrm{R}},$
1a
$\ \mathrm{for}{\ }t{\geq}t_{\mathrm{APEX}},{\ }{\ }{\alpha}(t)={\alpha}_{\mathrm{R}}+{\omega}_{\mathrm{R}}(t-t_{\mathrm{APEX}}),$
1b
where ωR is a constant angular leg velocity (retraction speed).
Fig. 1.
Spring-mass model with retraction. Swing-leg retraction in running, as indicated by the photographs of Muybridge(1955; reproduced with permission from Dover Publications), is modeled assuming a constant rotational velocity of the leg (retraction speed ωR), starting at the apex of the flight phase at retraction angle αR. Depending on the duration of the flight phase, the landing angle of the leg (angle of attack α0) is a result of the model dynamics and has no predefined constant value in contrast to the previous model of Seyfarth et al.(2002). The axial leg operation during the stance phase is approximated by a linear spring of constant stiffness kleg.
Fig. 1.
Spring-mass model with retraction. Swing-leg retraction in running, as indicated by the photographs of Muybridge(1955; reproduced with permission from Dover Publications), is modeled assuming a constant rotational velocity of the leg (retraction speed ωR), starting at the apex of the flight phase at retraction angle αR. Depending on the duration of the flight phase, the landing angle of the leg (angle of attack α0) is a result of the model dynamics and has no predefined constant value in contrast to the previous model of Seyfarth et al.(2002). The axial leg operation during the stance phase is approximated by a linear spring of constant stiffness kleg.
### Stability analysis
To evaluate the stability of potential movement trajectories, we use a return map analysis. For legged locomotion, a return map relates the system state at a characteristic event or moment within a gait cycle to the system state at the same event or moment one period later. To keep the analysis as simple as possible, we select the swing-phase apex height as the characteristic event. At this point, the system state (x, y,vx, vy)apex is uniquely identified by one variable, the apex height yapex. Here, x and y are the horizontal and vertical positions, and vx and vy are the horizontal and vertical velocities of the model's point mass. The system state is uniquely defined by the apex height due to (1) the vanishing vertical velocity vy,apex=0 at this point, (2) the fact that x has no influence on future periodic behavior, and (3) the conservative nature of the spring-mass system in which total mechanical energy is held constant.
The return map investigates how this apex height changes from step to step,or more precisely, from one apex height (index i') to the next one(index i+1') in the following flight phase (after one contact phase). For a stable movement pattern, two conditions must be fulfilled within this framework: (1) there must be a periodic solution (Equation 2a, called a fixed point where
$$y_{\mathrm{APEX}}^{*}$$
is the steady state apex height), and (2) deviations from this solution must diminish step-by-step (Equation 2b, or an asymptotically stable fixed point).
$\ y_{\mathrm{i}+1}=y_{\mathrm{i}}=y_{\mathrm{APEX}}^{*},$
2a
where
$\ \left|\frac{\mathrm{d}y_{\mathrm{i}+1}}{\mathrm{d}y_{\mathrm{i}}}\right|_{y_{\mathrm{APEX}}^{*}}{<}1.$
2b
For simplicity, the subscript apex in yi+1 and yi has been removed.
The requirements for stable running can be checked graphically by plotting a selected return map (e.g. for a given retraction angle αRand a given retraction velocity ωR) within the(yi, yi+1) plane and searching for stable fixed points fulfilling both conditions defined by Equations 2a and 2b. The first condition (Equation 2a, periodic solutions) requires that there is a solution (i.e. a single point) of the return map yi+1(yi) located at the diagonal(yi+1=yi). The second condition(Equation 2b, asymptotic stability) demands that the slope(dyi+1/dyi) of the return map yi+1(yi) at the periodic solution(intersection with the diagonal) is neither steeper than 1 (higher than 45°) nor steeper than –1 (smaller than –45°).
As a consequence of the imposed leg retraction, the return map of the apex height yi+1(yi) is determined by two mechanisms: the control of the angle of attackα 0(yi) before landing (leg retraction)and the dynamics of the spring-mass model resulting in the next apex height yi+10, yi). According to the definition of leg retraction (Equation 1), the analytical relationship between the apex height yapex and the landing angle of attack α0 is:
$\ y_{\mathrm{APEX}}({\alpha}_{0})=l_{0}\mathrm{sin}{\alpha}_{0}+\frac{g}{2}\left(\frac{{\alpha}_{0}-{\alpha}_{\mathrm{R}}}{{\omega}_{\mathrm{R}}}\right)^{2}.$
3
where l0 denotes the leg length at touch-down and g is the vertical component of the gravitational acceleration. Merely one branch of the quadratic function in α0 has to be considered as retraction holds only for times ttapex according to Equation 1(either α0R orα 0R, depending on the sign ofω R). This allows us to derive the control strategyα 0(yapex).
### Numerical procedure
The running model is implemented in Simulink (Mathworks) using a built-in variable time step integrator (ode113) with a relative tolerance of 1e–12. For a human-like model (point mass m=80 kg, leg length l0=1 m) at different horizontal speeds vx (initial conditions at apex y0,apexare vx,apex=vx and vy,apex=0), the leg parameters(kleg, αRR) for stable running are identified by scanning the parameter space and measuring the number of successful steps. The stability of potential solutions is evaluated using the return map yi+1(yi) of the apex height yapex of two subsequent flight phases(i and i+1). For a given system energy E, all possible apex heights 0≤y0,apexE/(mg) are taken into account. For instance, for a system energy E corresponding to an initial horizontal velocity vx=5 m s-1 at an apex height y0,apex=1 m, apex heights between 0 and 2.27 m are taken into account. To keep the system energy constant, the horizontal velocity at apex v0,apex=vxis adjusted according to the selected apex height y0,apexusing the equation mgy0,apex+m/2(v0,apex)2=E.
### Can leg retraction stabilize spring-mass running?
The kinematics of the spring-mass model are evaluated using (1) a fixed angle of attack α0 and (2) the swing-leg retraction strategy(Equation 1). The results are shown in Fig. 2. Starting at an initial apex height of 1.25 m, both control strategies stabilize to a final limit cycle. Spring-mass running with a fixed angle of attack α0 is stable if (1) the leg stiffness kleg and the angle of attackα 0 are both properly adjusted to the chosen running speed and(2) the initial vertical position y0,apex is within the range of attraction for the corresponding stable fixed point. (For more information on spring-mass running using a fixed angle of attack, see Seyfarth et al., 2002).
Fig. 2.
Center of mass trajectories (A) and leg kinematics (B,C) for spring-mass running with and without retraction (C, ωR=50 deg s-1; B, ωR=0 deg s-1). The bars in A indicate the change in centre of mass height between touch-down and take-off. B and C are expanded views of plot A from 0 to 6 m (boxed). For each simulated run, the same initial apex height was used (y0=1.25 m),and for the simulation with retraction, a retraction angle ofα R=60° was assumed (C). Here the model with retraction reached a steady state condition after two steps in contrast to approximately 8 steps for the model without retraction (A,B). The red dotted lines in B and C denote the steady state landing angle α0*.
Fig. 2.
Center of mass trajectories (A) and leg kinematics (B,C) for spring-mass running with and without retraction (C, ωR=50 deg s-1; B, ωR=0 deg s-1). The bars in A indicate the change in centre of mass height between touch-down and take-off. B and C are expanded views of plot A from 0 to 6 m (boxed). For each simulated run, the same initial apex height was used (y0=1.25 m),and for the simulation with retraction, a retraction angle ofα R=60° was assumed (C). Here the model with retraction reached a steady state condition after two steps in contrast to approximately 8 steps for the model without retraction (A,B). The red dotted lines in B and C denote the steady state landing angle α0*.
With the swing-leg retraction control, the rotational leg velocity before landing (retraction speed ωR) leads to a step-to-step adjustment of the angle of attack α0, which gradually converges to a final steady state angle
$${\alpha}_{0}^{*}$$
(dotted line in Fig. 2C). Since the leg has a fixed angular velocity during the second half of the flight phase, the chosen initial apex height (y0,apex=1.25 m)leads to a steeper landing angle compared to the steady state angle
$${\alpha}_{0}^{*}$$
. Consequently, the first contact phase is asymmetric with respect to the vertical axis(Fig. 2A,C) and therefore, the next apex height is lower than the previous apex height. Due to the shorter flight phase, the second angle of attack is clearly flatter (a smaller angle of attack). Finally, the system stabilizes at the steady state angle
$${\alpha}_{0}^{*}$$
with a corresponding apex height
$$y_{\mathrm{APEX}}^{*}$$
.
With leg retraction, steady-state running is achieved within approximately 2 steps, whereas the system without retraction needs approximately 8 steps(Fig. 2A). This indicates that leg retraction can improve the attraction of stable limit cycles in running.
### Stability analysis for running
The influence of leg retraction on the return map of the apex height is shown in Fig. 3. With increased retraction speed (ωR=25 and 50 deg s-1) the solutions of yi+1(yi) for different retraction angles αR become more horizontally aligned. As a consequence, disturbances in apex height are compensated for more rapidly(paths indicated by the arrows in Fig. 3).Furthermore, the attraction range in yapex for the stable fixed points is largely increased (maximum increase in yapex: ∼35 cm for ωR=0, ∼90 cm for ωR=25 deg s-1, and ∼120 cm for ωR=50 deg s-1. See dotted lines in Fig. 3).
Fig. 3.
Return maps yi+1(yi) of the apex height yapex of two consecutive flight phases(index i and i+1) for three different retraction speedsω R (A, ωR=0 deg s-1; B,ω R=25 deg s-1; C, ωR=50 deg s-1).The system energy corresponds to a running speed of 5 m s-1 at an apex height yapex=1 m.(A–C) Three characteristic return maps represent the minimum, mean and maximum retraction angle αR (see key in each panel) for stable fixed points (see text, Equation 2). With increasing retraction speedω R, the range of retraction angles αR with stable fixed points increases, and attraction of higher apex heights is observed (max. y0≈1.3, 1.9, 2.2 forω R=0, 25, 50 deg s-1, respectively) as shown by representative tracings (running sequences are indicated by stepped black lines with starting arrows).
Fig. 3.
Return maps yi+1(yi) of the apex height yapex of two consecutive flight phases(index i and i+1) for three different retraction speedsω R (A, ωR=0 deg s-1; B,ω R=25 deg s-1; C, ωR=50 deg s-1).The system energy corresponds to a running speed of 5 m s-1 at an apex height yapex=1 m.(A–C) Three characteristic return maps represent the minimum, mean and maximum retraction angle αR (see key in each panel) for stable fixed points (see text, Equation 2). With increasing retraction speedω R, the range of retraction angles αR with stable fixed points increases, and attraction of higher apex heights is observed (max. y0≈1.3, 1.9, 2.2 forω R=0, 25, 50 deg s-1, respectively) as shown by representative tracings (running sequences are indicated by stepped black lines with starting arrows).
In the case of leg retraction, the control of the angle of attackα 0 is shifted into a control of the retraction angleα R. For zero retraction speed (ωR=0) the retraction angle αR becomes identical to the angle of attackα 0R0, Fig. 3A), i.e. the leg angle is adjusted at apex height and does not change until ground contact. With increasing retraction speed ωR, the range of retraction angles resulting in stable running is enlarged (2.6° forω R=0; 7.2° for ωR=25 deg s-1;14.6° for ωR=50 deg s-1).
### Running at low speeds
Spring-mass running with a fixed angle of attack is characterized by a minimum speed required for stability (Seyfarth, 2002). In Fig. 4, a running speed(vX=3 m s-1) close to this minimum speed is selected. At the given leg stiffness (kleg=20 kN m-1) no stable fixed point exists without retraction(Fig. 4A). Employing the leg retraction control, stable fixed points emerge in the return map. Similar to the finding in Fig. 3, an increased retraction speed ωR leads to (1) an enlarged range of attraction in yapex, (2) a faster convergence to the stable fixed point (fewer steps), and (3) an increased range of successful retraction angles αR for stable running.
Fig. 4.
Return maps yi+1(yi) of the apex height yapex are shown for different retraction speeds ωR (A, ωR=0 deg s-1; B,ω R=25 deg s-1; C, ω=50 deg s-1)but for a lower system energy corresponding to a slower running speed of 3 m s-1 at an apex height yapex=1 m. Stable fixed points require non-zero retraction velocitiesω R>0 (B,C). As in Fig. 3, an increased retraction speed leads to an enlarged attraction of the stable fixed points with respect to a given initial (e.g. disturbed)apex height. Model parameters: m=80 kg, l0=1 m, kleg=20 kN m-1.
Fig. 4.
Return maps yi+1(yi) of the apex height yapex are shown for different retraction speeds ωR (A, ωR=0 deg s-1; B,ω R=25 deg s-1; C, ω=50 deg s-1)but for a lower system energy corresponding to a slower running speed of 3 m s-1 at an apex height yapex=1 m. Stable fixed points require non-zero retraction velocitiesω R>0 (B,C). As in Fig. 3, an increased retraction speed leads to an enlarged attraction of the stable fixed points with respect to a given initial (e.g. disturbed)apex height. Model parameters: m=80 kg, l0=1 m, kleg=20 kN m-1.
### Robustness with respect to leg stiffness kLEG
Spring-mass running requires a proper adjustment of leg stiffness to the chosen angle of attack (Blickhan,1989; McMahon and Cheng,1990; Herr and McMahon, 2000, 2001; Seyfarth, 2002). However, even at zero retraction speed (ωR=0), a range of leg stiffness can fulfill periodic running at a given angle of attackα 0 (Seyfarth, 2002). To test the robustness of spring-mass running with respect to variations in leg stiffness, we estimate the maximum and minimum stiffness change that could be tolerated by the system. A stiffness change is applied during steady state running, starting from an initial leg stiffness of 20 kN m-1(Fig. 5A). For these numerical experiments, the mean angles of attack (αR=67.6°,64.4°, 60.0° in Fig. 5A,C,E) with respect to the range of all αR with stable fixed points in Fig. 3A–C are used. After the first three steps in steady state running, leg stiffness is permanently shifted. Without retraction, variations in leg stiffness within 18.2 and 22.4 kN m-1 are tolerated(Fig. 5A) even without any stride-to-stride adaptations in the angle of attack(Fig. 5B).
Fig. 5.
The influence of retraction speed ωR on the robustness of running is shown with respect to a permanent change in leg stiffness kleg. For each run (vx,0=5 m s-1, y0=1 m), the maximum and minimum leg stiffness (kmin, kmax) required to keep the system in a periodic running movement are depicted (A,C,E). The retraction angleα R (denoted in A,C,E) is chosen according to the mean retraction angle for stable fixed points in Fig. 3 at kleg=20 kN m-1. (B,D,F) The adaptation of the leg angle to the changed leg stiffness.
Fig. 5.
The influence of retraction speed ωR on the robustness of running is shown with respect to a permanent change in leg stiffness kleg. For each run (vx,0=5 m s-1, y0=1 m), the maximum and minimum leg stiffness (kmin, kmax) required to keep the system in a periodic running movement are depicted (A,C,E). The retraction angleα R (denoted in A,C,E) is chosen according to the mean retraction angle for stable fixed points in Fig. 3 at kleg=20 kN m-1. (B,D,F) The adaptation of the leg angle to the changed leg stiffness.
By introducing leg retraction (Fig. 5C, ωR=25 deg s-1; Fig. 5E, ωR=50 deg s-1), the range of tolerated stiffness is largely increased(16-28.8 kN m-1 for ωR=25 deg s-1;13.9-62 kN m-1 for ωR=50 deg s-1). These results show that the rotational velocity of the leg ωRinherently adapting the angle of attack α0 allows for large variations in leg stiffness (Fig. 5D,F).
Late swing-phase retraction has been observed in running animals of different leg number and body size(Muybridge, 1955; Gray, 1968). Although swing-leg retraction seems to be a general feature in biological running, few researchers (De Wit et al.,2000; Herr and McMahon, 2000, 2001; Herr et al., 2002) have studied the behavior and, consequently, its purpose is not fully understood. In this investigation, we show that leg retraction is a simple strategy to improve the stability of spring-mass running. By imposing a uniform retraction velocity, we demonstrate that the stability of the spring-mass model is increased with respect to variations in forward speed, leg angle (retraction angle αR) and leg stiffness kleg.
### Swing-leg retraction approximates the natural angle of attack
In terms of the return map of the apex height, we can ask for an optimal'control strategy by imposing the constraint yi+1(yi)=ycontrol=constant. Within one step this return map projects all possible initial apex heights yi to the desired apex height yi+1=ycontrol.
As a consequence of the dynamics of the spring-mass system, the apex height yi+1 is merely determined by the preceding apex height yi and the selected angle of attack α0. This dependency yi+1(yi0) can be understood as a fingerprint of spring-like leg operation' and is represented as a surface in Fig. 6A. When applying any control strategy α0(yi), this generalized surface yi+1(yi, α0)can be used to derive the corresponding return maps.
Fig. 6.
(A) A three-dimensional (3D) representation yi+1(yi, α0) of the return map yi+1(yi) characterizes spring-mass running (system energy corresponds to vX=5 m s-1 at yapex=1 m; m=80 kg, l0=1 m, k=20 kN m-1) for different angles of attack α0. For fixed angles of attack (slices in 3D), the corresponding return maps are shown on the left(yi, yi+1) plane. The red line depicts the return map for α0=68°. Different return maps are possible if the angle of attack α0 becomes dependent on the apex height yi. An optimal' control model with respect to stability would be a direct projection of any initial apex height yi to a desired apex height ycontrol in the next flight phase, or yi+1(yi)=ycontrol=constant,as shown for apex heights of 1, 1.5 and 2 m (left plane). This corresponds to isolines on the 3D-surface yi+1(yi0) indicating a dependency between the angle of attackα 0 and the initial apex height yI, as shown for ycontrol=1, 1.5 and 2 m in (B). With careful selection of the retraction velocity ωR and the retraction angle αR, the constant velocity leg retraction model can approximate the optimal control strategy.
Fig. 6.
(A) A three-dimensional (3D) representation yi+1(yi, α0) of the return map yi+1(yi) characterizes spring-mass running (system energy corresponds to vX=5 m s-1 at yapex=1 m; m=80 kg, l0=1 m, k=20 kN m-1) for different angles of attack α0. For fixed angles of attack (slices in 3D), the corresponding return maps are shown on the left(yi, yi+1) plane. The red line depicts the return map for α0=68°. Different return maps are possible if the angle of attack α0 becomes dependent on the apex height yi. An optimal' control model with respect to stability would be a direct projection of any initial apex height yi to a desired apex height ycontrol in the next flight phase, or yi+1(yi)=ycontrol=constant,as shown for apex heights of 1, 1.5 and 2 m (left plane). This corresponds to isolines on the 3D-surface yi+1(yi0) indicating a dependency between the angle of attackα 0 and the initial apex height yI, as shown for ycontrol=1, 1.5 and 2 m in (B). With careful selection of the retraction velocity ωR and the retraction angle αR, the constant velocity leg retraction model can approximate the optimal control strategy.
For example, in the case of a fixed angle of attack' (no retraction:α 0(yi)=αR=constant) the surface has to be scanned at lines of constant angles α0(Fig. 6A, e.g. red line:α 0=68°). These lines are projected to the left(yi+1,yi) plane in Fig. 6A and match the return map in Fig. 3A.
Let us now consider the optimal control strategy for stable running'α 0(yi) fulfilling yi+1(yi)=ycontrol=constant. Using the identified fingerprint, this simply requires us to search for isolines of constant yi+1 on the generalized surface yi+1(yi, α0), as indicated by the green lines in Fig. 6A (yi+1=1, 1.5 and 2 m). The projection of these isolines onto the (α0, yi) plane represents the desired natural' control strategyα 0(yi) for spring-mass running as depicted for ycontrol=1, 1.5, 2 m in Fig. 6B.
The constant-velocity leg retraction model put forward in this paper represents a particular control strategyα 0(yi) relating the angle of attackα 0 to the apex height yi of the preceding flight phase (Equation 3), as shown in Fig. 6B for different retraction speeds (ω =0, 25, 50, 75 deg s-1) and one retraction angle (αR=60°). It turns out that this particular leg retraction model can approximate the natural control strategy within a considerable range of apex heights if the proper retraction parameters (αR, ωR) are selected. The value of the retraction angle αR shifts the line of the retraction control α0(yi)along the α0 axis, whereas the retraction speedω R determines the slope of the control line. Thus, the retraction parameters have different qualities with respect to the control of running; if the retraction speed ωR guarantees the stability(setting the range and the strength of attraction to a fixed point), then the retraction angle αR selects the apex height of the corresponding fixed point ycontrol. Due to this adaptability, a constant velocity leg retraction model, as evaluated in this paper, can significantly enhance the stability of running compared to the fixed angle control model described by Seyfarth et al.(2002).
### Influence of speed on the stability of spring-mass running
The return maps in Figs 3and 4 indicate that the generalized surface yi+1(yi0) is a function of the forward running speed. The selected retraction speeds in Figs 3 and 4R=0, 25,50 deg s-1) show that the slope of the return map yi+1(yi) generally increases with (1)decreasing running speed and (2) decreasing retraction speedω R. As a consequence, running at 3 m s-1 is not stable using a fixed angle of attack (ωR=0 in Fig. 4A), but is stable using non-zero retraction speeds (ωR=25 and 50 deg s-1in Fig. 4B and C,respectively). Hence, even at slow forward running speeds (≤3 m s-1), there exists a natural control strategy represented by the isolines of the corresponding generalized surface with yi+1(yi0)=constant. In comparison with the fixed angle of attack control, leg retraction at constant velocity approximates this natural control(Fig. 6B). Thus, a constant velocity retraction is a successful strategy to stabilize running below the critical forward running speed where stable running is not achievable using a fixed angle control.
The fact that the spring-mass model, with retraction, is stable at slow forward running speeds seems critical. Clearly, for a running model to be viewed as a plausible biological representation, the model should be stable across the full range of biological running speeds. Without swing-leg retraction, the spring-mass model could not be stabilized at slow biological running speeds (∼3 m s-1 for m=80 kg, l0=1 m, kleg=20 kN m-1; Fig. 4A), but with retraction, the spring-mass model could readily be stabilized(Fig. 4B,C).
### Swing-leg retraction in human running: preliminary experimental results
A treadmill (Woodway, Germany) was equipped with an obstacle-machine designed to disturb swing-phase dynamics during human running. The obstacle-machine consisted of a cylindrical-shaped bar (2.5 cm diameter, 40 cm length) passing from the left to the right side of the treadmill walkway (the bar's long axis is generally perpendicular to the direction of the moving treadmill surface). Every 9-16 s, the bar moved towards the human runner at a speed equivalent to the treadmill surface, forcing the runner to change his swing-phase kinematics to avoid the obstacle. The movement of the obstacle bar was triggered by the ground reaction force F. For each experiment,the bar was positioned 12 cm above the moving treadmill surface.
Using this apparatus, we conducted experiments on five male subjects (body mass 79.6±5.9 kg, age 30.6±3.2 yrs)performing treadmill running at 3 m s-1. We measured leg kinematics (leg angle, leg length)during both the stance and swing phases. Leg angle α and length lleg at the onset of swing-leg retraction and at touch-down were used to characterise the kinematic leg control prior to landing. The retraction velocity ωR was estimated as the mean angular velocity within the last 20 ms before touch-down. Furthermore, the leg stiffness kleg was approximated using the maximum vertical ground reaction force Fmax and the maximum leg compressionΔ lmax=max(l0l)during stance phase with kleg=Fmaxlmax.
For undisturbed running, we found surprisingly uniform leg kinematics during both the stance and swing phases (shown for one subject in Fig. 7). In contrast, when passing over the obstacle, swing-leg kinematics were altered significantly,but stance period dynamics immediately following the obstacle were largely unaffected. Swing-leg retraction was observed in undisturbed running with an angular range equal to αshift=4.5±0.9°(Table 1). During this period of swing-leg retraction, only a minor change in leg length was observed(lshift=1±0.5 cm), supporting one of the assumptions of our control model.
Fig. 7.
Leg kinematics (leg length versus leg angle) during treadmill running at 3 m s-1. For the undisturbed condition, the mean± s.d. of 35 running steps for one male subject(78 kg) are shown (A). The leg length lleg was measured as the distance between hip and toe marker. The leg angle α is defined as the projection angle with respect to the ground(Fig. 1). Swing-leg retraction is present between the onset angle αR (length lR) and the angle of attack α0 (length l0) as shown magnified in (B). For the disturbed swing phase, the leg operation of the same experimental subject is plotted. Although only a single subject is depicted here, similar results were observed in all experimental subjects (see Table 1).
Fig. 7.
Leg kinematics (leg length versus leg angle) during treadmill running at 3 m s-1. For the undisturbed condition, the mean± s.d. of 35 running steps for one male subject(78 kg) are shown (A). The leg length lleg was measured as the distance between hip and toe marker. The leg angle α is defined as the projection angle with respect to the ground(Fig. 1). Swing-leg retraction is present between the onset angle αR (length lR) and the angle of attack α0 (length l0) as shown magnified in (B). For the disturbed swing phase, the leg operation of the same experimental subject is plotted. Although only a single subject is depicted here, similar results were observed in all experimental subjects (see Table 1).
Table 1.
Obstacle running
kleg (kN m-1)α0 (degrees)αR (degrees)αshift (degrees)ωR (deg s-1)l0 (m)lR (m)lshift (m)
Undisturbed running 25.2±6.8 68.8±2.1 64.3±2.0 4.5±0.9 137±9 0.932±0.020 0.942±0.018 -0.010±0.005
Disturbed running 22.9±3.6 70.4±2.7 61.3±1.5 9.1±3.6 159±21 0.949±0.018 0.959±0.017 -0.010±0.011
Difference (disturbed - undisturbed) -2.3±4.4 1.7±2.1 -3.0±2.5 4.7±3.0* 22±13* 0.017±0.021 0.017±0.023 0±0.010
kleg (kN m-1)α0 (degrees)αR (degrees)αshift (degrees)ωR (deg s-1)l0 (m)lR (m)lshift (m)
Undisturbed running 25.2±6.8 68.8±2.1 64.3±2.0 4.5±0.9 137±9 0.932±0.020 0.942±0.018 -0.010±0.005
Disturbed running 22.9±3.6 70.4±2.7 61.3±1.5 9.1±3.6 159±21 0.949±0.018 0.959±0.017 -0.010±0.011
Difference (disturbed - undisturbed) -2.3±4.4 1.7±2.1 -3.0±2.5 4.7±3.0* 22±13* 0.017±0.021 0.017±0.023 0±0.010
Values are means ± S.D. (N=5 subjects)
kleg, leg stiffness; α0,angle of attack; αR, onset angle of retraction;α shift0R,angle swept during retraction; ωR, retraction velocity (mean value of the last 20 ms before touch-down); l0, leg length at touch-down; lR, leg length at onset of retraction; lshift=l0-lR,the shift in leg length during retraction (see Fig. 7)
In the undisturbed condition, at least 39 steps are evaluated for each subject
Between 3 and 4 disturbed steps are used during obstacle avoidance
The leg stiffness is measured in the stance phase immediately following the disturbance
Differences between undisturbed and disturbed data are evaluated for significance using a paired t-test
*
P<0.05
We observed a significant re-adjustment of leg retraction in response to the disturbance. Both the retraction angular rangeα shift(Δαshift=4.7±3.0°, P<0.05, paired t-test) and the retraction velocityω R (ΔωR=22±13 deg s-1, P<0.05) increased in response to the disturbance. Here, the change in the angular rangeΔα shift was primarily the result of a decreased retraction angle αR(ΔαR=-3.0±2.5°, P=0.057) rather than an increased angle of attack α0(Δα0=1.7±2.1°, P=0.15). In contrast, no significant change was observed in leg stiffness(Δkleg=–2.3±4.4 kN m-1, P=0.31) or in leg length adjustment(Δlshift=0±1.0 cm, P=1)during the stance period immediately following the disturbance.
These results indicate that leg retraction is employed in human running and is even enhanced when an obstacle disturbance is applied. The data presented here support the hypothesis of the model, namely, that swing-leg retraction is a strategy used in running to select an angle of attack that sustains a desired movement pattern.
### Alternative biological strategies to stabilize running
The analysis reveals that the stability of spring-mass running is highly sensitive to the angular velocity of the leg before landing. Although swing-leg retraction seems an important stabilizing mechanism, we cannot ignore the importance of alternative strategies that might also be crucial for stable running. For instance, researchers have shown that visual feedback plays an important role in obstacle avoidance and, therefore, in stabilizing the movement trajectory. Warren et al.(1986) investigated regulatory mechanisms to secure proper footing using visual perception in human running. In their investigation, subjects ran on a treadmill across irregularly spaced foot-targets in order to effectively modulate step length and the vertical leg impulse during stance. Although their results suggest that vision is important for running stability, they do not specifically address the issue of how mechanical or neuro-muscular mechanisms may contribute when running over ground surfaces without footing constraints.
Intrinsic or preflex' leg stabilizing mechanisms may also be important for running stabilization. It is well established that the intrinsic properties of muscle leads to immediate responses to length and particularly velocity perturbations (Humphrey and Reed,1983; Brown et al.,1995). In an analytical study, Wagner and Blickhan(1999) showed that a self-stabilizing oscillatory leg operation emerges if well-established muscle properties are adopted.
Furthermore, by modeling the dynamics of the muscle-reflex system, stable,spring-like leg operations can be achieved in numerical simulations of hopping tasks if positive feedback of the muscle force sensory signals (simulated Golgi organs) are employed (Geyer et al.,in press). These results suggest that during cyclic locomotory tasks such as walking or running, the body could counteract disturbances even during a single stance period.
### Future work
Here we argue that swing-leg retraction is one of many stabilizing strategies used in biological running. Our research suggests that both the control of stance leg dynamics and swing-leg movement patterns may be critically important for overall running stability in humans and animals.
Leg retraction is a feedforward control scheme, and therefore, can neither avoid obstacles nor place the foot at desired foot-targets. Rather, the scheme provides a mechanical background stability' that may relax the control effort for locomotory tasks. It remains for future research to understand to what extent environmental sensory information might allow for varied kinematic trajectories and an increase in the stabilizing effects of swing-leg retraction. Future investigations will also be necessary to fully understand the impact of late swing-leg retraction on running stability. To gain insight into the control scheme employed by running animals, we wish to compare the natural retraction control formulated in this paper to the actual limb movements of running animals. Furthermore, since the spring-mass model of this paper is two-dimensional, we wish to generalize retraction to three dimensions to address issues of body yaw and roll stability. And finally, we hope to test optimized retraction control schemes on legged robots to enhance their robustness to internal (leg stiffness variations) and external disturbances(ground surface irregularities).
### Conclusion
In this paper we show that swing-leg retraction can improve the stability of spring-mass running. With retraction, the spring-mass model is stable across the full range of biological running speeds and can overcome larger disturbances in the angle of attack and leg stiffness. In the stabilization of running humans and animals, we believe both stance-leg dynamics and swing-leg rotational movements are important control features.
List of symbols
• E
system energy
•
• F
vertical ground reaction force
•
• g
vertical component of the gravitational acceleration
•
• i
index
•
• kleg
leg stiffness
•
• l
leg length
•
• m
mass
•
• t
time
•
• v
velocity
•
• x
horizontal position
•
• y
vertical position
•
• αR
retraction angle
•
• ωR
retraction speed
This research was supported by an Emmy-Noether grant of the German Science Foundation (DFG) to A.S. (SE 1042/1) and a grant of the German Academic Exchange Service (DAAD) Hochschulsonderprogramm III von Bund und Länder'to H.G. We also thank the Michael and Helen Schaffer Foundation of Boston,Massachusetts for their generous support of this research.
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,
259
-266. | 2022-07-06 01:11:10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.603652834892273, "perplexity": 3748.1002527793685}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104655865.86/warc/CC-MAIN-20220705235755-20220706025755-00768.warc.gz"} |
http://openstudy.com/updates/50d38114e4b0b19ec2193340 | ## anonymous 3 years ago For the same kinetic energy, the momentum shall be maximum for which one of these: 1. Electron 2. Alpha particles 3. Neutron 4. Deuteron 5. Proton 6. Gamma particles
• This Question is Open
1. anonymous
Alpha Particles, as p=m*v and the mass of an alpha particle is greater than the other options
2. anonymous
K=mv^2/2 this can be written as : K=m^2*v^2/2m which can be written in terms of momentum as: K=p^2/2m Rearranging the terms : p=sqrt(2Km) For given K, the larger the mass of the particle the greater will be its p Hence "alpha particle" will have greater momentum. NOTE: However the converse is not true- For given momentum the lighter particle will have greater Kinetic energy as can be seen from the formula: K=p^2/2m This is contrary to what one may expect.
3. agent0smith
Kinetic energy is given by $KE = \frac{ 1 }{ 2 } m v^2$ rearrange to to find v, since KE is the same for all:$v = \sqrt{\frac{ 2 \times KE }{m}}$and insert this into the formula for momentum, p = mv $\rho = m v = m \sqrt{\frac{ 2 \times KE }{m}}$ and square the m and bring it under the square root sign, cancel off an m, to get: $\rho = \sqrt{2 \times KE \times m}$ Since 2*KE is the same for all the particles, momentum is proportional to the square root of mass - the larger the mass of the particle, the higher the momentum.
Find more explanations on OpenStudy | 2016-05-28 22:20:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8469409942626953, "perplexity": 731.447696375651}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049278244.51/warc/CC-MAIN-20160524002118-00019-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://www.maplesoft.com/support/help/Maple/view.aspx?path=XMLTools/GetChildByName | XMLTools - Maple Programming Help
Home : Support : Online Help : Connectivity : Web Features : XMLTools : XMLTools/GetChildByName
XMLTools
GetChildByName
access a child node of an XML tree
Calling Sequence GetChildByName(xmlTree, name)
Parameters
xmlTree - Maple XML tree; XML element name - string or symbol; the name of the child element to extract
Description
• The GetChildByName(xmlTree, name) command accesses the children of the given XML element xmlTree with element name equal to name. A list of all children that are elements with element type equal to name is returned.
Examples
> $\mathrm{with}\left(\mathrm{XMLTools}\right):$
> $\mathrm{xmlTree}≔\mathrm{XMLElement}\left("a",\left[\right],\left[\mathrm{XMLElement}\left("b",\left[\right],"b text"\right),\mathrm{XMLElement}\left("c",\left[\right],"c text"\right),\mathrm{XMLElement}\left("b",\left[\right],"more b text"\right)\right]\right):$
> $\mathrm{Print}\left(\mathrm{xmlTree}\right)$
b text c text more b text
> $\mathrm{map}\left(\mathrm{Print},\mathrm{GetChildByName}\left(\mathrm{xmlTree},"b"\right)\right)$
b text more b text
$\left[{}\right]$ (1)
> $\mathrm{map}\left(\mathrm{Print},\mathrm{GetChildByName}\left(\mathrm{xmlTree},"c"\right)\right)$
c text
$\left[{}\right]$ (2)
> $\mathrm{map}\left(\mathrm{Print},\mathrm{GetChildByName}\left(\mathrm{xmlTree},"nosuchelement"\right)\right)$
$\left[{}\right]$ (3) | 2016-08-29 07:11:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 9, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5619917511940002, "perplexity": 8052.478342329151}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982953863.79/warc/CC-MAIN-20160823200913-00294-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://reference.digilentinc.com/reference/instrumentation/guides/waveforms-using-waveforms-sdk | # Using WaveForms SDK
## Introduction
WaveForms SDK is a set of tools provided within the WaveForms installation that are used to develop custom software solutions that use Digilent Test and Measurement devices. The WaveForms SDK API is available in several programming languages, making it easy to to use across many different platforms.
Normally Test and Measurement devices are controlled and configured through the WaveForms application with a personal computer. Such a setup may be impossible in a given context, or an amount of automated signal measurement may be sought outside WaveForms' scripting environment. WaveForms SDK gives the necessary tools to help craft the perfect solution for any problem.
### Sample Application
This guide walks through the implementation of a sample application to demonstrate a use-case for the WaveForms SDK as well as the proper workflow. The sample application, implemented in Python, will configure a Digilent Test and Measurement device to fill a data buffer with samples. These samples are then used to generate a graph image that is shared on a web page hosted locally.
### Prerequisites
• A Digilent Test & Measurement Device with Analog Input and Output Channels
• A Computer with WaveForms Software Installed
• WaveForms SDK is installed alongside the WaveForms application.
## 1. SDK Overview
WaveForms SDK is included with WaveForms and is installed alongside the application. The SDK is available to use with C/C++, C#, Python and Visual Basic through a dynamic library.
On Windows, the dynamic library can be found at C:\Windows\System32\dwf.dll and on Linux at /usr/lib/libdwf.so.x.x.x.
A static library on Windows is at C:\Program Files\Digilent\WaveFormsSDK\lib\x86 for 32-bit systems and for 64-bit systems at C:\Program Files (x86)\Digilent\WaveFormsSDK\lib\x64.
The C Header file is located at C:\Program Files\Digilent\WaveFormsSDK\inc for Windows 32-bit, C:\Program Files (x86)\Digilent\WaveFormsSDK\inc for Windows 64-bit and at /usr/local/include/digilent/waveforms for Linux.
Other working code examples for each described programming language are provided with the SDK and may befound at C:\Program Files\Digilent\WaveFormsSDK\samples for Windows 32-bit, C:\Program Files (x86)\Digilent\WaveFormsSDK\samples for Windows 64-bit and /usr/local/share/digilent/waveforms/samples on Linux.
## 2. Implementing the Sample Application
### 2.1 Setup
dwfconstants.py must be copied into project directory and its location differs by OS:
• Win32 C:\Program Files\Digilent\WaveFormsSDK\samples\py
• Win64 C:\Program Files (x86)\Digilent\WaveFormsSDK\samples\py
• Linux /usr/local/share/digilent/waveforms/samples/py
Several Python packages are needed and are installed by invoking
pip install matplotlib, numpy, flask
### 2.2 Script Implementation
In the project directory, create a file called main.py and open in with a text editor. At the top of the file, declare the imports like so:
from ctypes import *
from dwfconstants import *
import math
import time
import matplotlib.pyplot as plt, mpld3
import sys
import numpy
from io import BytesIO, StringIO
from flask import Flask, Response
The dll must be loaded, and the method to do so depends on the operating system. Add the next lines of code to do so.
if sys.platform.startswith("win"):
dwf = cdll.dwf
elif sys.platform.startswith("darwin"):
else:
dwf = cdll.LoadLibrary("libdwf.so")
The next few lines of code declare some helper variables that are used to configure the Test and Measurement device. A sample buffer is also declared, which will soon be filled with data acquired from the device. Add the snippet to the project code:
#declare ctype variables
hdwf = c_int()
sts = c_byte()
hzAcq = c_double(100000) # 100 kHz
nSamples = 200000
rgdSamples = (c_double*nSamples)()
cAvailable = c_int()
cLost = c_int()
cCorrupted = c_int()
fLost = 0
fCorrupted = 0
Next, the first available device is opened. The API returns a device handle that will be used to configure the device. Add the code below:
#open device
dwf.FDwfDeviceOpen(c_int(-1), byref(hdwf))
if hdwf.value == hdwfNone.value:
szerr = create_string_buffer(512)
dwf.FDwfGetLastErrorMsg(szerr)
print(str(szerr.value))
print("failed to open device")
quit()
The signal that is to be measures will come from the device itself. It is configured to output a sine wave on the device's wavegen channel 1. Add the following code:
# enable wavegen channel 1, set the waveform to sine, set the frequency to 1 Hz, the amplitude to 2v and start the wavegen
dwf.FDwfAnalogOutNodeEnableSet(hdwf, c_int(0), AnalogOutNodeCarrier, c_bool(True))
dwf.FDwfAnalogOutNodeFunctionSet(hdwf, c_int(0), AnalogOutNodeCarrier, funcSine)
dwf.FDwfAnalogOutNodeFrequencySet(hdwf, c_int(0), AnalogOutNodeCarrier, c_double(1))
dwf.FDwfAnalogOutNodeAmplitudeSet(hdwf, c_int(0), AnalogOutNodeCarrier, c_double(2))
dwf.FDwfAnalogOutConfigure(hdwf, c_int(0), c_bool(True))
The device's oscilloscope channel is then configured to take samples, and is started, with the addition of the following code:
# enable scope channel 1, set the input range to 5v, set acquisition mode to record, set the sample frequency to 100kHz and set the record length to 2 seconds
dwf.FDwfAnalogInRecordLengthSet(hdwf, c_double(nSamples/hzAcq.value)) # -1 infinite record length
#wait at least 2 seconds for the offset to stabilize
time.sleep(2)
print("Starting oscilloscope")
dwf.FDwfAnalogInConfigure(hdwf, c_int(0), c_int(1))
The next snippet then polls the status of the device, reads any available samples into the buffer. It continues to do so while the buffer isn't full.
cSamples = 0
while cSamples < nSamples:
if cSamples == 0 and (sts == DwfStateConfig or sts == DwfStatePrefill or sts == DwfStateArmed) :
# Acquisition not yet started.
continue
# get the number of samples available, lost & corrupted
cSamples += cLost.value
# set the lost & corrupted flags
if cLost.value :
fLost = 1
if cCorrupted.value :
fCorrupted =
# skip reading samples if there aren't any
if cAvailable.value==0 :
continue
# cap the available samples if the buffer would overflow from what's really available
if cSamples+cAvailable.value > nSamples :
cAvailable = c_int(nSamples-cSamples)
# Read channel 1's available samples into the buffer
dwf.FDwfAnalogInStatusData(hdwf, c_int(0), byref(rgdSamples, sizeof(c_double)*cSamples), cAvailable) # get channel 1 data
cSamples += cAvailable.value
After taking samples, it's good practice to cleanup by turning off the wavegen and closing the device.
# reset wavegen to stop it, close the device
dwf.FDwfAnalogOutReset(hdwf, c_int(0))
dwf.FDwfDeviceCloseAll()
A graph image is created from the sampled data, with the image being kept in its own buffer to be used by the web server.
# generate a graph image from the samples, and store it in a bytes buffer
plt.plot(numpy.fromiter(rgdSamples, dtype = numpy.float))
bio = BytesIO()
plt.savefig(bio, format="png")
Finally, a web server is setup to return the graph image whenever it gets a HTTP request.
# start web server, only if running as main
if __name__ == "__main__":
@app.route('/')
def root_handler():
return Response(bio.getvalue(), mimetype="image/png") # return the graph image in response
app.run()
## 3. Running the Application
At this point, connect the Wavegen channel 1 and the Scope channel 1 pins of the Test and Measurement device together. Plug the device into the computer. In a console, call
python main.py
The console should then have output that is similar to the following:
DWF Version: b'3.10.9'
Opening first device
Generating sine wave...
Starting oscilloscope
Recording done
* Environment: production
WARNING: This is a development server. Do not use it in a production deployment.
Use a production WSGI server instead.
* Debug mode: off
* Running on http://127.0.0.1:5000/ (Press CTRL+C to quit)
Open the web browser and navigate to http://127.0.0.1:5000 to see the graph image of the sampled sine wave, similar to the below image:
### Next Steps
For more guides on how to use the Digilent Test & Measurement Device, return to the device's Resource Center, linked from Instrumentation page of this wiki. | 2020-12-05 18:53:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.2758493423461914, "perplexity": 8789.406968571584}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141748276.94/warc/CC-MAIN-20201205165649-20201205195649-00615.warc.gz"} |
https://qask.org/tags/crypto/collision-resistance | # Questions tagged as ['collision-resistance']
Difficulty of finding two different inputs that hash to the same value
Score: 0
Two Elliptic Curve Points having the Same X coordinate
Suppose in a elliptic curve (say the curve equation is: $$y^2 = x^3 -17$$) with prime order $$q$$, we have $$(x,y_1) = nP$$, where $$P$$ is a generator and $$n<\lceil{q/2}\rceil$$. Can we claim that there does not exist $$n' < \lceil{q/2}\rceil$$, such that $$(x,y_2)=n'P$$ is a valid curve point where $$y_2 \neq y_1$$?
Score: 1
Is $H:\mathbb{Z} \rightarrow \mathbb{Z}_{p}^{*}$ and $a \mapsto g^a\bmod p$ with $p$ prime (strongly) collision-free?
Let $$H:\mathbb{Z} \rightarrow \mathbb{Z}_{p}^{*}$$ and $$a \mapsto g^a\bmod p$$ for $$g \in \mathbb{Z}_{p}^{*}$$ where $$p$$ is prime. Is this function (strongly) collision-free meaning we cannot find practically $$x_1$$,$$x_2$$ such that $$H(x_1)=H(x_2)$$?
I argue no with the following reasoning: Let $$A$$ be an Algorithm which generates $$x_1 \neq x_2$$ such that $$H(x_1)=H(x_2)$$ and define $$A: \mathbb{N} \rightarro ...$$
Score: 0
What happens when we hash already hashed values, concatenated together?
I read on the page 16 of On the Security of Hash Function Combiners that
the classical combiner for collision-resistance simply concatenates the outputs of both hash functions $$Comb_{\mathbin\|}(M) = H_0(M) \mathbin\| H_1(M)$$ in order to ensure collision resistance as long as either of H0, H1 obeys the property.
Consider H, a secure internal hash function with 256-bit inputs and 128-bit outputs
...
Score: 0
Securely and Deterministically select a combination of objects from hash (cryptographic seed)
I am working on a project that is using a bit-commitment concept to authenticate information.
I need to select a combination of objects securely from a secure hash, then distribute that hash later. Then a client knows that only the authenticated server selected that combination of objects before distribution of the hash the combination derived from. In other words, I need to select a combination ...
Score: 1
AES-CBC Hash Function Collision Resistance
I am using AES-CBC as a hash function which is encrypting a block of length n. The blocks, m = (m1, m2, ..., mn). The IV is one block long and the encryption key is length 128, 192 or 256 bits.
Will I get collisions? And if so, how could I find examples?
I expect to find collisions every 2^(n/2) hashes but I don't imagine this would allow me to find any matches in the next 10000000 years.
Score: 1
if i enter a password that's incorrect but that collides with one when hashed, will it let me in?
suppose no salt or pepper is used and passwords are hashed plain, will entering incorrect password that just hashes to the same let me in? i know that one use of salting/peppering techniques is to, aside from making brute force more time consuming, prevent one hash compromise all the users using same pass. but how does it work for preventing colliding passwords being used interchangeably? in other words ...
Score: 2
$2^{64}$ versions of the same message
I am reading a textbook and in there they explain the property of hash functions. In particular, they give an example of how unlikely it would be to find a second input value that would match the hash output of the original input. Here's the example:
We show now how Oscar could turn his ability to find collisions (modifying two messages) into an attack. He starts with two messages, for instance:
Score: 0
Is it possible to get the SHA256 hash collision with partial known data
I have a text sentence that consists of 448 digits [0-9] [a-f] (in HEX format).
This text sentence is partially cut off, but I know the middle, and the beginning and end are damaged.
What I know is 322 known digits in the middle of a text sentence.
74 unknown digits at the beginning
52 unknown digits at the end
That is, the entire text Size: 224 bytes and it is hashed using the SHA256 hash algorith ...
Score: 1
Hash function producing cycles with expected max length
Is there a known hash function $$H_k: X\to X$$ such that: $$\forall{x\in{X}},\exists{n\in{\mathbb{N}}}, n
=== EDIT ===
By hash function I mean that any other way of finding the preimage of $$x \in X$$ than iterating $$H_k$$ is computably unfeasible or at least significantly harder.
My motivation is using such a function as sequential POW.
Score: 1
Hash function collision importance
Suppose a collision has been found in a certain hash function, such that H(x1) = H(x2)
However, x1 and x2 are both a seemingly 'random' collection of bits which do not convey a coherent message, and cannot be interperted in a coherent way.
Does this collision make the hash function H not secure? if so how can it be exploited, even if the known collision doesn't convey a coherent message? thanks
...
Score: 1
Is the collision chance 2^(n/2) of an n-bit tag τ unchanged if reduced to (n/2)-bits using a reduction of τ to some 2^(n/2) order group element?
If $$H(k, Μ) = τ$$, in the context where $$τ$$ is an $$n$$-bit tag produced as a mac on a key, $$k$$, and a message, $$M$$, through a keyed-hash function, $$H$$, is there a function $$F(τ) = T$$ that transforms $$τ$$ into a group element, $$Τ$$, of some group, $$G$$, of order $$2^{\frac{n}{2}}$$, such that:
• The chance of producing any $$T$$ ( where $$F(τ') = F(τ) = T$$; and $$τ' ≠ τ$$ ) is given by $$≈2^{\frac{-n ...$$
Score: 0
Homomorphic hash from prime order group $G$ to $Z_p$
Let $$G$$ be a cyclic group with the generator $$g$$ and of prime order $$p$$ such that the discrete-logarithm problem is hard in $$G$$.
A hash function is homomorphic if $$H(a\ast b)=H(a)\cdot H(b)$$ (where the operations $$\ast$$ and $$\cdot$$ depend on the groups). Here we do not expect the hash function to be compressing, but collision-resistance (CR) and efficiently computeable.
Now the question is, if the ...
Score: 0
Using bcrypt to always produce the same hash like SHA, MD
I want to take advantage of the slow property of bcrypt to hash an input but also want to get the same hash value for the same input every time just like SHA, MD, etc.
So in order to do that, instead of using a static salt, which is less secure I believe, I am thinking to use the input as the salt as well? The output will be the hash value minus the front salt bit (obviously the input itself).
Basic ...
Score: -1
Is it possible to have collision resistance but not pre-image and 2nd pre-image resistance?
I have studied cryptographic hash functions quite a lot, but have not completely understood whether it is possible to have collision resistance but not pre-image and 2nd pre-image resistance at the same time.
Is it possible?
Score: 4
Many near collisions but no full collision
I read this question: Cracking $f(x) = Cx \oplus Dx$ Asking about finding collisions in a simple 64 bit hash, and I thought I will give it a go myself just for fun. I quickly wrote code to find collisions: https://gist.github.com/meirmaor/b0e59352eb73cacec47d0f95c25a25fc
And yet it finds many near collisions and no full collisions, this baffles me.
Algorithm description: I wanted to solve this using 8GB ...
Score: 0
Using hash of data as proof of integrity and preventing collision
Rather than storing user data when interacting with an app, I am storing the SHA3-256 of the data. This is because data storage in this particular environment is very limited.
The data can be several variables, e.g., a, b, and, c, but instead of saving them individually, I save the hash of the concatenation: SHA3(a,b,c).
When the user wants to interact with the system, they should send the variables ...
Score: 3
Cracking $f(x) = Cx \oplus Dx$
A program I reverse engineered is using $$f(x) = Cx \oplus Dx$$ where C = 0x20ef138e415 and D = 0xd3eafc3af14600 as a hash function. Given a byte array, the hash is is obtained by repeatedly applying $$f$$ to the current hash xor next byte.
Java code:
public static long f(long x) {
return (0x20ef138e415L * x) ^ (0xd3eafc3af14600L * x);
}
public static long hash(byte[] bytes) {
l ...
Score: 1
Preimage attack on sum of two Hash functions modulo 2
If a hash function $$H$$ is defined as $$H(x_1,x_2) = H_1(x_1) \oplus H_2(x_2)$$ for two n bit good hash functions $$H_1$$ and $$H_2$$ then how can we construct a preimage attack on $$H$$ that is of $$O(2^\frac{n}{2})$$ given some y ?
Here, are we allowed to query $$H_1$$ and $$H_2$$ ?
I would really appreciate some hints.
Score: 2
How does Authentication-Key Recovery for GCM work?
In his Paper "Authentication weaknesses in GCM" Ferguson describes, how some bits of the error polynomial can be set to zero, thereby increasing significantly the chance of a forgery.
Q: What does it mean in detail? That the resulting equations do not solve the problem of obtaining forgery completely, but the solution space is significantly reduced? So we can fix some bits of the error polynomial ...
Score: 0
If the source code of SHA256 hashing algorithm is available in public, why can't it be hacked?
If the SHA256 algorithm is public, why can't attackers use it to create more collisions rendering the algorithm useless?
Score: 0
Merkle-Damgård construction
Let $$H^f$$ be a hash function designed using Merkle-Damgård construction on $$f:\{0,1\}^{2n}\to\{0,1\}^n$$. Write an algorithm that makes approximately $$2.2^{n/2}$$ many queries to $$f$$ and find four messages that all hash to same value under $$H^f$$.
I get an idea to use length extension and 2 birthday attack to get four collision. But I am not able to write the appropriate solution. Can anyone help m ...
Score: 1
Is it possible to exploit MD5 weaknesses to create an artificial collision for a password?
If it is possible, could an attacker create a collision for an MD5 password in a database? Could they look at an MD5 hash output and figure out data that creates the same MD5 hash?
Score: 2
Security of Hash Functions
Given a Hash Function H, how are the properties such as collision resistance, target collision resistance, one wayness, and non-malleability proved? I have read about hash function and stating that it is collision-resistant but how are they formally proved? If a hash function satisfies all the properties will it act as a random oracle model?
Score: 0
Two Different Ciphers with Same MD5
I was wondering if someone could help explain md5 collision abit better. I found this resource: https://www.mscs.dal.ca/~selinger/md5collision/ where they provided an example of where two cipher texts have the same md5. I tried to confirm that their example was correct but when I input their examples into a md5 calculator, I get two different md5s for the two different cipher text. What am I doing ...
Score: 1
Can there be an injective function that maps a large set of integers to a smaller set while being "collision-aware"
Consider two sets:
The "big set" contains all integers between $$0$$ and $$2^{160}$$ exactly once.
The "small set" contains all integers between $$0$$ and $$2^{32}$$ exactly once.
Given that the number of members in the "big set" is greater than those in the "small set", there can't be an injective function $$f(n_b) = n_s$$ mapping any input being a member of the "big set" $$n_b$$ to an output that's a membe ...
Score: -2
What are the security flaws of SHA?
I have been researching SHA algorithms extensively, specifically SHA1, SHA2-256, SHA2-512, SHA3-256, and SHA3-512, and have found many instances of successful collision attacks as well as methods.
In my list are the following:
• Brute Force attacks
• Birthday attacks
• Yuval's Birthday attack (improved birthday attack with different conditions)
• Reduced round attacks
• Successful on attacks on all SHA al ...
Score: 2
Does SHA-256 have (128-time + 128-space = 256-overall)-bit collision resistance?
First, we consider those hash functions that can actually provide 256-bit pre-image security, and not something like SHAKE128<l=256bits> where the sponge parameters provides only a security capacity of 128-bit.
We know that cryptanalysis doesn't have just a time dimension - it also has a space dimension, i.e. the amount of working memory needed to execute the cryptanalysis algorithm. So if we expe ...
Score: 0
Why do the first two digits of the hash table not collide within CRC32?
In this Python CRC32 table look-up method, the polynomial is 0x104c11db7.
I can understand that the generated table does not collide. After all, as long as the start and end of the polynomial binary are 1, then the hash obtained by different raw data is different.
But why do the first two bits of the hash table not collide?
The first four digits of the polynomial are 0x04c1, and the binary end of | 2023-03-30 21:02:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 67, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.29849448800086975, "perplexity": 1770.0636397537708}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949387.98/warc/CC-MAIN-20230330194843-20230330224843-00013.warc.gz"} |
http://czpc.chicweek.it/python-percentile-without-numpy.html | # Python Percentile Without Numpy
Example-----Examples can be given using either the Example or Examples sections. I'm having difficult time optimizing the following calculation; Inner_diff_grp = np. stats import rankdata import numpy as np def calc_percentile (a, method = 'min'): if isinstance (a, list): a = np. Updated for Python 3. It has a lot of sanity checks. python setup. Here are the examples of the python api numpy. Some basic operations in Python for scientific computing. sqrt(variance), but without needing to import the math module. We can calculate arbitrary percentile values in Python using the percentile() NumPy function. 5th, 25th, 50th, 75th, and 97. From Lists to 1-D Numpy Arrays. Making use of more robust array data types isn’t also without it’s cost implications. Using the inv() and dot() Methods. Numpy percentile() method is used to compute the i th percentile of the provided input data supplied using arrays along a specified axis. Tag: python,numpy,pandas. NumPy 2020 full offline installer setup for PC 32bit/64bit NumPy (Numerical Python) is the fundamental package for scientific computing with Python. Fantastic way to write Python bindings for native libs or speed up computationally intensive code without having to write C yourself. Included to auto-deploy Python on demand and the NumPy package in order to call into it. I tried to print my matrix. 25th element in the sorted list. Arrays are. Additional info: * package version(s) python-numpy 1. How to limit the number of items printed in output of numpy array? # Limit the number of items printed in python numpy array a to a maximum of 6 elements. Identify Outliers using Quartiles/Percentiles/Quantile in Python(pandas,numpy) Published on November 25, 2018 November 25, 2018 • 14 Likes • 2 Comments. percentile() takes the following arguments. Are the two statements below not identical for cutting the bottom 10% out of a column? This: df = df[df["x"] > numpy. You might be interested in the SciPy Stats package. 12] We can also use numpy. leastsq that overcomes its poor usability. Official source code (all platforms) and binaries for Windows, Linux and Mac OS X. (HDF5) The HDF5 version of MusicNet requires an HDF5 parser for your language of choice. For image processing with SciPy and NumPy, you will need the libraries for this tutorial. When using QR decomposition in Numpy, the first basis vector that it chooses can sometimes affect the numerical accuracy of the solution. You can read more about matrix in details on Matrix Mathematics. Thanks for sharing amazing information about python. Each number n (also called a scalar) represents a dimension. ml_percentile (in_data, percentiles) [source] ¶ Calculate percentiles in the way Matlab and IDL do it. 8b2 will work with the new release source packages, but may not find support in future releases. It is an open source module of Python which provides fast mathematical computation on arrays and matrices. percentile (a, 30) print ("The 30th percentile of a is ",val) Output: The 30th percentile of a is 24. I don't know why it doesn't work. percentile(a, q, axis=None, out=None, overwrite_input=False, interpolation=linear, keepdims=False)2. Python: MxP matrix A * an PxN matrix B(multiplication) without numpy April 11, 2013 artemrudenko List Comprehension, Lists, Python, Samples Leave a comment. Algorithm to find Quartiles :. I am going to send a C++ array to a Python function as NumPy array and get back another NumPy array. To make it as fast as. NumPy is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to. Combining str Methods with NumPy to Clean Columns. How can we convert y_true and y_pred to numpy array, so that I can implement sklearn's F1 score funtion up on them. Aloha I hope that 2D array means 2D list, u want to perform slicing of the 2D list. import math def percentile(data, percentile): size = len(data) return sorted(data)[int(math. Before you can use NumPy, you need to install it. axis : axis along which we want to calculate the percentile value. This function is similar to The Numpy arange function but it uses the number instead of the step as an interval. Here in this article, we discuss it. Install pip install percentiles Use >>> import percentiles >>> percentiles. In this post I'm going to show you a simple way to significantly speedup Python numpy compute performance on AMD CPU's when using Anaconda Python We will set a DEBUG environment variable for Intel MKL that forces it to use the AVX2 vector unit on AMD CPU's (this will work for other applications too, like MATLAB for example. pip installs packages for the local user and does not write to the system directories. FFT in Python without numpy yields other result than with numpy. quantile¶ numpy. percentileofscore (a, score, kind = 'rank') [source] ¶ Compute the percentile rank of a score relative to a list of scores. Example: Decimal Module. product? If so, how? In Python, I have two n dimensions numpy arrays A and B (B is a zero array). When I run the above code I am getting y_true and y_pred as. Official source code (all platforms) and binaries for Windows, Linux and Mac OS X. When you have a DataFrame with columns of different datatypes, the returned NumPy Array consists of elements of a single datatype. Also note that variance ** 0. I can't figure it out what's wrong with my code, it's rly frustrating. intervaltree - The MusicNet labels are stored in an IntervalTree. Python NumPy Operations Tutorial - Some Basic Operations Finding Data Type Of The Elements. In this example, we shall create a numpy array with 8 zeros. Install pip install percentiles Use >>> import percentiles >>> percentiles. import numpy as np. Example-----Examples can be given using either the Example or Examples sections. percentile(winw2_grp,x[0]) - np. Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. python,list,numpy,multidimensional-array According to documentation of numpy. 5 and the 50th percentile should be -63. As arrays can be multidimensional, you need to specify a slice for each dimension of the array. With its intuitive syntax and flexible data structure, it's easy to learn and enables faster data computation. Example 2: Pandas DataFrame to Numpy Array when DataFrame has Different Datatypes. 5th, 25th, 50th, 75th, and 97. percentile(x,70) # 70th percentile 2. The function takes both an array of observations and a floating point value to specify the percentile to calculate in the range of 0 to 100. This means that numpy. To randomly shuffle a 1D array in python, there is the numpy function called: shuffle, illustration with the following array: M = \left( \begin{array}{cccccc} 4 & 8 & 15 & 16 & 23 & 42. ] Numpy array (1-Dimensional) of size 8 is created with zeros. percentile(arr, n, axis=None, out=None) Parameters : arr :input array. so) is not built, this causing dot and matrix multiplication about 5x slower on my Arch box when compare to a Ubuntu box with same hardware configuration. The aim of this project and is to implement all the machinery, including gradient descent, cost function, and logistic regression, of. Parameters a array_like. ## numpy is used for creating fake data import numpy as np import matplotlib as mpl ## agg backend is used to create plot as a. uniform(10,size=(1000))-5. The python lists are nowhere near to what it can do. Install python (1)Download python from (2)Untar it and go into the directory after that (3)Run the following commands >> mkdir /home/ntran/. Python is a general-purpose interpreted, interactive, object-oriented, and high-level programming language. For reference: this mini-introduction was written in September 2016, where Anaconda 4. The goal of this collection is to offer a quick reference for both old and new users but also to provide a set of exercises for those who teach. ndarray) – input data. 4) that support mean calculation. Matplotlib is a python library for making publication quality plots using a syntax familiar to MATLAB users. NumPy provides a compact, typed container for homogenous arrays of data. Below steps are tested in a Windows 7-64 bit machine with Visual Studio 2010 and Visual Studio 2012. percentile()1. NumPy is a Python Library/ module which is used for scientific calculations in Python programming. dot product handles the 2D arrays and perform matrix multiplications. For NumPy 1. percentile(a, q, axis) Where,. When an image file is read by OpenCV, it is treated as NumPy array ndarray. In order to reshape numpy array of one dimension to n dimensions one can use np. py install --prefix = /home/your_account/Python27/ Most of them are fine with this. 8]]) L = np. But when I was doing more python, I wrote bootstrapping, monte carlo and CI code without anything but the standard lib. Import the libraries and specify the type of the output file. percentile(x,70,interpolation="nearest") 2. The goal of this collection is to offer a quick reference for both old and new users but also to provide a set of exercises for those who teach. 0 Determinant of A is -348 The Numpy Determinant of A is -348. In particular, there are some obstacles and pitfalls when you do not have the root. And it’s significantly faster. import numpy as np #create numpy array with zeros a = np. For example, the 75th percentile, given there are 60 items in your list, should be the 44. In very simple terms dot product is a way of finding the product of the summation of two vectors and the output will be a single vector. linspace() | Create same sized samples over an interval in Python 2019-02-17T17:47:45+05:30 Numpy, Python No Comment In this article we will discuss how to create a Numpy array of evenly spaced samples over a range using numpy. ceil((size * percentile) / 100)) - 1] p5 = percentile(mylist, 5) p25 = percentile(mylist, 25) p50 = percentile(mylist, 50) p75 = percentile(mylist, 75) p95 = percentile(mylist, 95). When you have a DataFrame with columns of different datatypes, the returned NumPy Array consists of elements of a single datatype. Aloha I hope that 2D array means 2D list, u want to perform slicing of the 2D list. array([numpy. The matrix looks like this:. NumPy also provides a set of functions that allows manipulation of that data, as well as operating over it. percentile(winw2_grp,x. The second way below works. percentile() takes the following arguments. test ('full') scipy. array([1,2,3,4,5]) p = np. This legacy has created a large number of branches that may solve your problem without forcing you to switch language or writing a new extension to this particular language. nanmean,nansum, so I suspect that would be necessary. arange(15) np. Using the inv() and dot() Methods. array function also produce the same result. Hey, I read that numpy percentile method is faster than pandas quantile while being identical in output, but when I run it on a csv, I don't get an identical output. Numpy arrays are great alternatives to Python Lists. By using interpolation between the lowest an highest rank and the minimum and maximum outside. percentile function. intervaltree - The MusicNet labels are stored in an IntervalTree. In Python, these two descriptive statistics can be obtained using the method apply with the methods gmean and hmean (from SciPy) as arguments. 03175853, 1. A box plot summarizes this data in the 25 th, 50 th, and 75 th percentiles. Are the two statements below not identical for cutting the bottom 10% out of a column? This: df = df[df["x"] > numpy. This tutorial does not come with any pre-written files, but is a follow-along tutorial. DICOM-Numpy¶. Input array or object that can be converted to an array. Because NumPy provides an easy-to-use C API, it is very easy to pass data to external libraries written in a low-level language and also for external libraries to return data to Python as NumPy arrays. What are ufuncs? ufuncs stands for "Universal Functions" and they are NumPy functions that operates on the ndarray object. Previous: Write a NumPy program to create a 2-dimensional array of size 2 x 3 (composed of 4-byte integer elements), also print the shape, type and data type of the array. I am very new to using python to process data on CSV files. However, going back and forth between Python and C through those wrappers can slow things down. especially without NumPy. Before going to learn how to build a feed forward neural network in Python let's learn some basic of it. In addition, it seems the numpy. The nditer iterator object provides a systematic way to touch each of the elements of the array. The core of numpy is written in the low-level C programming language, so all computations are executed very fast. 0729677997904314 The latter is an actual entry in the vector, while the former is a linear interpolation of two vector entries that border the percentile. Ship high performance Python applications without the headache of binary compilation and packaging. import numpy as np. percentile() function is faster than the quantile() function in R. Syntax : numpy. Determinant of A is 18 The Numpy Determinant of A is 18. Quite simply, Numpy is a scientific computing library for Python that provides the functionality of matrix operations, which are generally used with Scipy and Matplotlib. If you have not already installed the Numpy library, you can do with the following pip command: $pip install numpy Let's now see how to solve a system of linear equations with the Numpy library. I found one and it seemed to work, but when I tested it on a more realistic sample it failed and yielded other results than the numpy version. NET provides strong-typed wrapper functions for numpy, which means you don't need to use the dynamic keyword at all, but this is a rabbit hole to delve into in another article. Winsorizing or winsorization is the transformation of statistics by limiting extreme values in the statistical data to reduce the effect of possibly spurious outliers. ceil((size * percentile) / 100)) - 1] p5 = percentile(mylist, 5) p25 = percentile(mylist, 25) p50 = percentile(mylist, 50) p75 = percentile(mylist, 75) p95 = percentile(mylist, 95). Gradient descent is an optimization algorithm that works by efficiently searching the parameter space, intercept($\theta_0$) and slope($\theta_1$) for linear regression, according to the following rule: Redis with Python NumPy array basics A NumPy Matrix and Linear Algebra Pandas with NumPy and Matplotlib Celluar Automata Batch gradient. Use itertools. However, for comparison, code without NumPy are also presented. Mean with python. scoreatpercentile – almost an order of magnitude faster in some cases. NET uses Python. Even when using OpenCV, Python's OpenCV treats image data as ndarray, so it is useful to remember the processing in NumPy (ndarray). This is the setup code that Numpy will execute behind the. Python: get all possible array attributions of nd arrays. var(list(map(lambda x : np. Returns the q-th percentile(s) of the array elements. Quartiles : A quartile is a type of quantile. This tutorial does not come with any pre-written files, but is a follow-along tutorial. With this power comes simplicity: a solution in NumPy is often clear and elegant. In Python, data is almost universally represented as NumPy arrays. Varun December 5, 2018 Python Numpy : Select rows / columns by index from a 2D Numpy Array | Multi Dimension 2018-12-08T17:18:52+05:30 Numpy, Python No Comment In this article we will discuss how to select elements from a 2D Numpy Array. Instead, we focus on how Numpy. And this is how you can get valuable percentiles data in Python with the numpy module. 12] We can also use numpy. By voting up you can indicate which examples are most useful and appropriate. Then, it is pretty fast in terms of execution and at the same time it is very convenient to work with. as a Python object. 7 numpy pandas percentile this question edited May 23 at 10:28 Community ♦ 1 1 asked Dec 16 '13 at 15:29 tnknepp 1,579 10 23 you can certainly make a feature request to numpy; they have specialized methods for nan handling, e. 03175853, 1. To compute the standard deviation, we use the numpy module. It works with floats as weights. Import the libraries and specify the type of the output file. Numpy inner() method is used to compute the inner product of two given input arrays. Since, arrays and matrices are an essential part of the Machine Learning ecosystem, NumPy along with Machine Learning modules like Scikit-learn, Pandas, Matplotlib. percentile(a, q, axis=None, out=None, overwrite_input=False, interpolation='linear', keepdims=False) [source] ¶ Compute the qth percentile of the data along the specified axis. 98]]) x = np. Input array or object that can be converted to an array. linspace() function in Python returns evenly spaced numbers over the specified interval. NumPy Matrix Transpose The transpose of a matrix is obtained by moving the rows data to the column and columns data to the rows. These are the 2. OpenCV-Python can be installed in Ubuntu in two ways: Install from pre-built binaries available in Ubuntu repositories; Compile from the source. Here's how to do it without numpy, using only python to calculate the percentile. And it’s significantly faster. Dump the loops: Vectorization. Clean-cut integer data housed in a data structure such as a list, tuple, or set, and you want to create a Python histogram without importing any third party libraries. If multiple percentiles are given, first axis of the result corresponds to the percentiles. I don't know why it doesn't work. If q is a single percentile and axis=None, then the result is a scalar. Arrays are. Those who are used to NumPy can do a lot of things without using libraries such as OpenCV. I'm having difficult time optimizing the following calculation; Inner_diff_grp = np. Example 25. You can calculate all basic statistics functions such as average, median, variance, and standard deviation on NumPy arrays. percentile(winw2_grp,x. range=range, weights=weights, density=density)[0] return h # percentile wrapper that casts the output into a single array. Otherwise, it will consider arr to be flattened. These are the 2. To create a one-dimensional array of zeros, pass the number of elements as the value to shape parameter. local >> cd Pytho…. Numpy arrays are much like in C – generally you create the array the size you need beforehand and then fill it. In Python, data is almost universally represented as NumPy arrays. Nearly every scientist working in Python draws on the power of NumPy. Another important thing is the additional libraries required. 0 officially dropping Python 2. python min of 2d array (6) numpy. scoreatpercentile – almost an order of magnitude faster in some cases. Mean with python. If q is a single percentile and axis=None, then the result is a scalar. PyPI page for NumPy. percentile(a, q, axis) Where,. If you would like to know the different techniques to create an array, refer to my previous guide: Different Ways to Create Numpy Arrays. In this part we will implement a full Recurrent Neural Network from scratch using Python and optimize our implementation using Theano, a library to perform operations on a GPU. So Numpy being one of the essential libraries for Machine Learning requires an article of its own. Each number n (also called a scalar) represents a dimension. percentile() takes the following arguments. Python numpy. The installation for other packages are mostly trivial. The size (width, height) of the image can be acquired from the attribute shape indicating the shape of ndarray. array s and almost all the big libraries used nowadays are based on them. I'm having difficult time optimizing the following calculation; Inner_diff_grp = np. When you have a DataFrame with columns of different datatypes, the returned NumPy Array consists of elements of a single datatype. Dump the loops: Vectorization. It is an open source module of Python which provides fast mathematical computation on arrays and matrices. I have DataFrame: time_diff avg_trips 0 0. Included to auto-deploy Python on demand and the NumPy package in order to call into it. The first quartile (Q1), is defined as the middle number between the smallest number and the median of the data set, the second quartile (Q2) - median of the given data set while the third quartile (Q3), is the middle number between the median and the largest value of the data set. In the last few exercises in the Intermediate Python module in DataCamp, I learned how to do transpose on a Numpy array in Python. var(list(map(lambda x : np. This means that 30% of values fall below 24. (HDF5) The HDF5 version of MusicNet requires an HDF5 parser for your language of choice. The keywords get ignored with a warning if supplied with non-default values. For more info, Visit: How to install NumPy? If you are on Windows, download and install anaconda distribution of Python. Numpy arrays are great alternatives to Python Lists. Lastly, when we compute the percentile value along axis 1, then percentile value is calculated along the rows. array function also produce the same result. The aim of this project and is to implement all the machinery, including gradient descent, cost function, and logistic regression, of. I have to make inverse matrix function, what I thought I've done. Arrays The central feature of NumPy is the array object class. python min of 2d array (6) numpy. In the following example, we will estimate the value of the 95th percentile of a N(0,1) distribution using p square algorithm. percentile(a, 50) # return 50th percentile, e. As of matplotlib version 1. Finding mean without numpy module im trying to make a program that will find the mean without me using the numpy module (also because i cant download and install the numpy module for some reason). product? If so, how? In Python, I have two n dimensions numpy arrays A and B (B is a zero array). Python arrays are powerful, but they can confuse programmers familiar with other languages. The module is not intended to be a competitor to third-party libraries such as NumPy, SciPy, or proprietary full-featured statistics packages aimed at professional statisticians such as Minitab, SAS and Matlab. Returns the qth percentile of the array elements. Notice that even NumPy arrays can be declared with Cython and Cython will correctly translate Python element selection into fast memory-access macros in the generated C code. import nose import numpy import scipy numpy. 0: keepdims and interpolation are not supported. I'm having difficult time optimizing the following calculation; Inner_diff_grp = np. org to get help, discuss contributing & development, and share your work. percentile is a lot faster than scipy. Python: get all possible array attributions of nd arrays. Numpy arrays are great alternatives to Python Lists. var(list(map(lambda x : np. quantile¶ numpy. I don't know why it doesn't work. @parameter percent - a float value from 0. The nditer iterator object provides a systematic way to touch each of the elements of the array. The 2-D array in NumPy is called as Matrix. I have a CSV file with the data below. Robin's Blog Calculating percentiles in Python - use numpy not scipy! November 24, 2015. python setup. Python has been one of the premier, flexible, and powerful open-source language that is easy to learn, easy to use, and has powerful libraries for data manipulation and analysis. Sections are created with a section header followed by an underline of equal length. Click on the Next button if Python is found; otherwise, click on the Cancel button and install Python (NumPy cannot be installed without Python). [batch_size, height, width, 3] Yields: filenames: list file names without path of each image Lenght of this list could be less than batch_size, in this case only first few images of the result are elements of the minibatch. percentile(x,70) # 70th percentile 2. Example 2: Pandas DataFrame to Numpy Array when DataFrame has Different Datatypes. delete(arr,3,axis=0) - Deletes row on index 3 of arr np. Viewed 201 times 2$\begingroup$I tried to find an implementation of the FFT algorithm in Python without the use of the numpy library. UPDATE 1: I've discovered via my own research that this post contains some inaccuracies regarding the limitations of Python on Windows. Accessing columns. percentileofscore¶ scipy. Start a python console. insert(arr,2,values) - Inserts values into arr before index 2 np. matplotlib is a plotting library based on NumPy. I agree with the numpy values using the linear interpolation. Because NumPy provides an easy-to-use C API, it is very easy to pass data to external libraries written in a low-level language and also for external libraries to return data to Python as NumPy arrays. Numpy arrays are much like in C – generally you create the array the size you need beforehand and then fill it. percentile(winw2_grp,x. percentile() takes the following arguments. NumPy is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to. The key to making it fast is to use vectorized operations, generally implemented through NumPy's universal functions (ufuncs). percentile() Percentile (or a centile) is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall. NumPy arrays are at the foundation of the whole Python data science ecosystem. For one-dimensional array, a list with the array elements is returned. There is no known exact formula for the normal cdf or its inverse using a finite number of terms involving standard functions ($\exp, \log, \sin \cos$etc) but both the normal cdf and its inverse have been studied a lot and approximate formulas for both are. This means that 30% of values fall below 24. Selain cepat, data besar juga mumpuni lho. percentile()接受以下参数。np. The NumPy library is nothing to be scared of. import numpy as np #create numpy array with zeros a = np. It's essentially just one data structure, the NumPy array. I have a CSV file with the data below. percentile (a, 30) print ("The 30th percentile of a is ",val) Output: The 30th percentile of a is 24. PyPI page for SciPy (all). NumPy uses Python syntax. If you would like to know the different techniques to create an array, refer to my previous guide: Different Ways to Create Numpy Arrays. Nah, kali ini aku ingin nge share pengalamanku tentang matriks. csv',delimiter=',',dtype=None)[1:] Next we will make two arrays. I have a distance matrix, produced from jukes-cantor estimation of pairwise distances made from clustal. Posted by: admin November 28, 2017 Leave a comment. percentile() is available in numpy too. Simply import the NumPy library and use the np. This is the setup code that Numpy will execute behind the. percentile function to compute weighted percentile? Or is anyone aware of an alternative python function to compute weighted percentile?. floor (k) c = math. 0-2 python2 2. Python, as well as its numerical libraries are one of the essential toolsets for researchers and data scientists. The python lists are nowhere near to what it can do. We use python numpy array instead of a list because of the below three reasons: Less Memory; Fast; Convenient; The very first reason to choose python numpy array is that it occupies less memory as compared to list. ndarray) – input data. We recommend you read our Getting Started guide for the latest installation or upgrade instructions, then move on to our Plotly Fundamentals tutorials or dive straight in to some Basic Charts tutorials. In matrix multiplication make sure that the number of rows of the first matrix should be equal to the. To make it as fast as. From my understanding, the 90%-percentile does not have to be an item from the input array. TBH, I haven't used python for science in a few years, so maybe numpy is the norm now and I'm showing my age. all() return quantiles It is vectorized as far as I could go. From my understanding, the 90%-percentile does not have to be an item from the input array. Numpy arrays are much like in C – generally you create the array the size you need beforehand and then fill it. as a Python object. It has a lot of sanity checks. Quartiles : A quartile is a type of quantile. 1 whereas the command python3 will use the latest installed Python (PY_PYTHON was not considered at all as a major version was specified. Start a python console. Sometimes when are calculating summary statistics, the geometric or harmonic mean can be of interest. [Python] Install numpy, scipy, PIL without root privilege on Linux 0. amax() will find the max value in an array, and numpy. Lastly, when we compute the percentile value along axis 1, then percentile value is calculated along the rows. To make it as fast as. Before going further into article, first learn about numpy. In this tutorial, you will discover how to manipulate and access your data correctly in NumPy arrays. Based on this comparison, Stata is dramatically slower (particularly when Parallel processing in either Python or Matlab). shape[i]<=B. It contains among other things:. This feature has made Python a language of choice for wrapping legacy C/C++/Fortran codebases and giving them a dynamic and easy-to-use interface. Vectorization and parallelization in Python with NumPy and Pandas. The code is simple and it handles by the Numpy package without hassle. I have DataFrame: time_diff avg_trips 0 0. Introduction. Find x-th percentile of a sequence without numpy. import numpy as np x=np. You can read more about matrix in details on Matrix Mathematics. 03175853, 1. NumPy’s concatenate function allows you to concatenate two arrays either by rows or by columns. Example 2: Python Numpy Zeros Array – Two. I don't know why it doesn't work. Such way A. def poisson_percentile(mu, x. See also For more advanced image processing and image-specific routines, see the tutorial Scikit-image: image processing , dedicated to the skimage module. SciPy release page (sources). Example 2: Pandas DataFrame to Numpy Array when DataFrame has Different Datatypes. install python [Python] Intall IPython without root previledge. How To Install NumPy in Python? Python is recognized as a strong and universal programming language due to its ample set of libraries. ) If PY_PYTHON=3 and PY_PYTHON3=3. From a user point of view, NumPy arrays behave similarly to Python lists. quantile() or percentile(). The challenge today is to write a program to multiply two matrices without using numpy. If multiple percentiles are given, first axis of the result corresponds to the percentiles. Preferably, do not use sudo pip, as this combination can cause problems. The Python Numpy aggregate functions are sum, min, max, mean, average, product, median, standard deviation, variance, argmin, argmax, percentile, cumprod, cumsum, and corrcoef. So NumPy is a package for working with numerical data. Marks are 40 but percentile is 80%, what does this mean? 80% of CAT exam percentile means 20% are above & 80% are below; Percentiles help us in getting an idea on outliers. >>> from numpy import * However, this strategy is usually frowned upon in Python programming because it starts to remove some of the nice organization that modules provide. optimize and a wrapper for scipy. MonetDB/R: Using the MonetDB/R plugin, using the native R quantile function instead of the numpy. def poisson_percentile(mu, x. This means that numpy. From the documentation:. We checked in the command prompt whether we already have these: Let's Revise Range Function in Python - Range() in Python. import numpy as np x=np. The []-operator still uses full Python operations - what we would like to do instead is to access the data buffer directly at C speed. When using QR decomposition in Numpy, the first basis vector that it chooses can sometimes affect the numerical accuracy of the solution. They also provide broadcasting and additional methods like reduce, accumulate etc. Python packages like NumPy wrap C libraries in Python interfaces to make them easy to work with. For example, np. percentile() is available in numpy too. It comes with NumPy and other several packages related to. Slicing: Just like lists in python, NumPy arrays can be sliced. 12 and Python 3. We can use numpy ndarray tolist() function to convert the array to a list. Histograms in Python How to make Histograms in Python with Plotly. NET uses Python. As with LU Decomposition, it is unlikely that you will ever need to code up a Cholesky Decomposition in pure Python (i. The 90th percentile has a value of 19. NumPy uses Python syntax. OpenCV-Python can be installed in Ubuntu in two ways: Install from pre-built binaries available in Ubuntu repositories; Compile from the source. percentile is a lot faster than scipy. How to print the full numpy array without truncating # Print the full numpy array a without truncating. In this tutorial, we are going to learn how to print an array in Python. Read and write images: How to read image file as NumPy array ndarray. It is a higher-level library that builds on the excellent lower-level pydicom library. product? If so, how? In Python, I have two n dimensions numpy arrays A and B (B is a zero array). We can use numpy ndarray tolist() function to convert the array to a list. The goal of this collection is to off. It contains among other things:. As of matplotlib version 1. 7-2 * config and/or log files etc. I tried to find an implementation of the FFT algorithm in Python without the use of the numpy library. If you haven’t already, download Python and Pip. With packages like NumPy and Python’s multiprocessing module the additional work is manageable and usually pays off when compared to the enormous waiting time that you may need when doing large-scale calculations inefficiently. If I want to find both max and min, I have to call both functions, which requires passing over the (very big) array twice, which seems slow. NumPy is a Python Library/ module which is used for scientific calculations in Python programming. There are various libraries in python such as pandas, numpy, statistics (Python version 3. Lecture 1B: To speed up Python's performance, usually for array operations, most of the code provided here use NumPy, a Python's scientific computing package. percentile¶ numpy. Arrays are. 1 were the defaults. We could boil down the problem to the attached 3-liner 'minimal-iP. nanmean,nansum, so I suspect that would be necessary.$\begingroup$The integral expression in the "normal cdf I got exactly from Wiki" is unfortunately off by a factor of$1/\sqrt{\pi}$. normal (loc=0. SciPy is a scientific Python library, which supplements and slightly overlaps NumPy. In addition to the round() function, python has a decimal module that helps in handling decimal numbers more accurately. I have DataFrame: time_diff avg_trips 0 0. import math def percentile(data, percentile): size = len(data) return sorted(data)[int(math. 633231120341421. NumPy uses the asarray() class to convert PIL images into NumPy arrays. For example, the vector v = (x, y, z) denotes a point in the 3-dimensional space where x, y, and z are all Real numbers. collections. 23560103, -1. var(a) method to calculate the average value of NumPy array a. import numpy as np x=np. But what does percentile value mean? A percentile is a mathematical term generally used in statistics. scoreatpercentile – almost an order of magnitude faster in some cases. You can do so by creating a list containing these ints/floats and convert the list to a NumPy array using np. I found one and it seemed to work, but when I tested it on a more realistic sample it. Syntax : numpy. Its result is shown using out1. percentiles (numpy. Another important thing is the additional libraries required. However, for comparison, code without NumPy are also presented. In the case of 1D arrays, the ordinary inner product of vectors is returned (without complex conjugation), whereas, in case of higher dimensions, a sum-product over the last axes is returned as a result. The NumPy library is a popular Python library used for scientific computing applications, and is an acronym for "Numerical Python". Some basic operations in Python for scientific computing. it doesn't cost anything and it's open source. The function numpy. Available packages. Find x-th percentile of a sequence without numpy. In the last few exercises in the Intermediate Python module in DataCamp, I learned how to do transpose on a Numpy array in Python. 函数百分位数是统计中使用的度量，表示小于这个值的观察值的百分比。 函数numpy. Read 103 answers by scientists with 406 recommendations from their colleagues to the question asked by Giovanni De Gasperis on Dec 22, 2014. 12 and Python 3. zeros(8) #print numpy array print(a) Output [0. In numpy, you can create two-dimensional arrays using the array() method with the two or more arrays separated by the comma. The code is simple and it handles by the Numpy package without hassle. Python: Subtracting square matrices without numpy Python: Clustered list to flat. When using QR decomposition in Numpy, the first basis vector that it chooses can sometimes affect the numerical accuracy of the solution. 4) that support mean calculation. Matlab is the fastest platform when code avoids the use of certain Matlab functions (like fitlm). Parameters:. However, for completeness I have included the pure Python implementation of the Cholesky Decomposition so. 5 is the default Python provided, and revised in March 2019, where Anaconda 2018. A: 5x5 matrix, B: 5x5 matrix (make array and use loop ?). Marks are 40 but percentile is 80%, what does this mean? 80% of CAT exam percentile means 20% are above & 80% are below; Percentiles help us in getting an idea on outliers. percentile(winw2_grp,x[0]) - np. Python: MxP matrix A * an PxN matrix B(multiplication) without numpy April 11, 2013 artemrudenko List Comprehension, Lists, Python, Samples Leave a comment. We can use this function to calculate the 1st, 2nd (median), and 3rd quartile values. The function takes both an array of observations and a floating point value to specify the percentile to calculate in the range of 0 to 100. Hence, it would be a good idea to explore the basics of data handling in Python with NumPy. array([1,2,3,4,5]) p = np. percentile(x,70) # 70th percentile 2. What is numpy dot product? Numpy. Let’s use Python to show how different statistical concepts can be applied computationally. Numpy, scipy, matplotlib, and pylab are common terms among they who use python for scientific computation. Example 2: Computing. In this section, we will see both. How do I calculate the derivative of a function, for example y = x2+1 using numpy? Let's say, I want the value of derivative at x = 5. Varun February 17, 2019 numpy. As of matplotlib version 1. But when I was doing more python, I wrote bootstrapping, monte carlo and CI code without anything but the standard lib. nanpercentile() in Python numpy. Multiplication of two Matrices in Single line using Numpy in Python Matrix multiplication is an operation that takes two matrices as input and produces single matrix by multiplying rows of the first matrix to the column of the second matrix. percentile() Percentile (or a centile) is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall. With this power comes simplicity: a solution in NumPy is often clear and elegant. This Python tutorial helps you to understand what is feed forward neural networks and how Python implements these neural networks. Finding the percentile of the values (Python recipe) This function find the percentile of a list of values. product? If so, how? In Python, I have two n dimensions numpy arrays A and B (B is a zero array). If q is a single percentile and axis=None, then the result is a scalar. For one-dimensional array, a list with the array elements is returned. NumPy brings the computational power of languages like C and Fortran to Python, a language much easier to learn and use. Finally, Numpy percentile() Method Example is over. Common operations include given two 2d-arrays, how can we concatenate them row wise or column wise. Q So how do we create a vector in Python? A We use the ndarray class in the numpy package. Read 103 answers by scientists with 406 recommendations from their colleagues to the question asked by Giovanni De Gasperis on Dec 22, 2014. I want to take the averages of the time stamps for each Sprint, Jog, and Walk column by session. percentile(winw2_grp,x. Python: MxP matrix A * an PxN matrix B(multiplication) without numpy April 11, 2013 artemrudenko List Comprehension, Lists, Python, Samples Leave a comment. Viewed 201 times 2$\begingroup\$ I tried to find an implementation of the FFT algorithm in Python without the use of the numpy library. When you have a DataFrame with columns of different datatypes, the returned NumPy Array consists of elements of a single datatype. To compute the standard deviation, we use the numpy module. One of the key methods for solving the Black-Scholes Partial Differential Equation (PDE) model of options pricing is using Finite Difference Methods (FDM) to discretise the PDE and evaluate the solution numerically. How To Install NumPy in Python? Python is recognized as a strong and universal programming language due to its ample set of libraries. And you don't need to import anything to use generators. Arrays The central feature of NumPy is the array object class. percentile(a, q, axis) Where,. percentiles (numpy. NET provides strong-typed wrapper functions for numpy, which means you don't need to use the dynamic keyword at all, but this is a rabbit hole to delve into in another article. It can work without weights (→ equal weights). It provides tools for writing code which is both easier to develop and usually a lot faster than it would be without numpy. 25th element in the sorted list. I have to make inverse matrix function, what I thought I've done. In this section, we will see both. 0 Determinant of A is 0 The Numpy Determinant of A is 0. This the second part of the Recurrent Neural Network Tutorial. 0 will support Python versions 3. In the last few exercises in the Intermediate Python module in DataCamp, I learned how to do transpose on a Numpy array in Python. Normalize matrix in Python numpy. that are very helpful for computation. NumPy and SciPy, historically shared their codebase but were later separated. It is named after the engineer-turned-biostatistician Charles P. asarray (a) return rankdata (a, method = method) / float (len (a)) For example:. I don't know why it doesn't work. In this tutorial, you will discover how to manipulate and access your data correctly in NumPy arrays. NET provides strong-typed wrapper functions for numpy, which means you don't need to use the dynamic keyword at all, but this is a rabbit hole to delve into in another article. This guide will provide you with a set of tools that you can use to manipulate the arrays. Generator expressions are useful when you want to loop through a sequence of values just once, and in order, without storing the whole list as an object. Selain cepat, data besar juga mumpuni lho. I can't figure it out what's wrong with my code, it's rly frustrating. Read 103 answers by scientists with 406 recommendations from their colleagues to the question asked by Giovanni De Gasperis on Dec 22, 2014. Project: GeoPy Author: aerler File: stats. This is just a brief public service announcement reporting something that I've just found: np. Using the np percentile() method, you can calculate the. PyPI page for NumPy. End Edit python-2. Ask Question Asked 8 months ago. float() Examples The following are code examples for showing how to i. (using python) import arcpy import numpy as np import os #loop through all Shapefile in a folder and call the CalcPercentile method def CalcPercentile(inputFeatureClass): #to create 3 rank for example print inputFeatureClass; arr = arcpy. The installation for other packages are mostly trivial. In this section, we will see both. Arrays are a collection of data elements of the same type under the same name. I tried to print my matrix. Based on this comparison, Stata is dramatically slower (particularly when Parallel processing in either Python or Matlab). In very simple terms dot product is a way of finding the product of the summation of two vectors and the output will be a single vector. percentile(a, q, axis=None, out=None, overwrite_input=False, interpolation=linear, keepdims=False)2. MonetDB/R: Using the MonetDB/R plugin, using the native R quantile function instead of the numpy. collections. I'm having difficult time optimizing the following calculation; Inner_diff_grp = np. The function takes both an array of observations and a floating point value to specify the percentile to calculate in the range of 0 to 100. I am trying to implement Multivariate Linear Regression (without using sklearn). reshape() function syntax and it's parameters. Numpy inner() method is used to compute the inner product of two given input arrays. You can calculate all basic statistics functions such as average, median, variance, and standard deviation on NumPy arrays. Then, it is pretty fast in terms of execution and at the same time it is very convenient to work with. Quartiles : A quartile is a type of quantile. Otherwise, it will consider arr to be flattened. hot 2 Numpy installation fails on Python 3. Included to auto-deploy Python on demand and the NumPy package in order to call into it. The size (width, height) of the image can be acquired from the attribute shape indicating the shape of ndarray. nanpercentile does not exist. It works with floats as weights. We can calculate arbitrary percentile values in Python using the percentile() NumPy function. Syntax: numpy. When you have a DataFrame with columns of different datatypes, the returned NumPy Array consists of elements of a single datatype. NumPy's operations are divided into three main categories: Fourier Transform and Shape Manipulation, Mathematical and Logical Operations, and Linear Algebra and Random Number Generation. percentile function. Then, it is pretty fast in terms of execution and at the same time it is very convenient to work with. Moreover, we will see ways to generate Random Number in Python. We do this with a special "buffer" syntax which must be told. import numpy as np x=np. How to print the full numpy array without truncating # Print the full numpy array a without truncating. How do I calculate the derivative of a function, for example y = x2+1 using numpy? Let's say, I want the value of derivative at x = 5. 5, we are no longer making file releases available on SourceForge. Returns the qth percentile(s) of the array elements. linspace() function in Python returns evenly spaced numbers over the specified interval. For instance, you can compute the dot product with np. histogram() Examples The following are code examples for showing how to use numpy. It is an open-source language and widely used across the globe. How can we convert y_true and y_pred to numpy array, so that I can implement sklearn's F1 score funtion up on them. In this part we will implement a full Recurrent Neural Network from scratch using Python and optimize our implementation using Theano, a library to perform operations on a GPU. There are various libraries in python such as pandas, numpy, statistics (Python version 3. For reference: this mini-introduction was written in September 2016, where Anaconda 4. percentile(winw2_grp,x[0]) - np. Next: Write a NumPy program to change the data type of an array. [batch_size, height, width, 3] Yields: filenames: list file names without path of each image Lenght of this list could be less than batch_size, in this case only first few images of the result are elements of the minibatch. int16) for i in range(56)]) np. A complete archive of documentation for all Num Py (Numerical Python) releases (minor versions; bugfix releases don't contain significant documentation changes) since 2009 can be found at https://docs. percentile(df["x"], 10)] Produces a different result to this:. Have another way to solve this solution? Contribute your code (and comments) through Disqus. The NumPy library is a popular Python library used for scientific computing applications, and is an acronym for "Numerical Python". dot product is a powerful library for matrix computation. stats import rankdata import numpy as np def calc_percentile (a, method = 'min'): if isinstance (a, list): a = np. asked Jul 30, 2019 in Python by Eresh Kumar (26. The 2-D array in NumPy is called as Matrix. I am very new to using python to process data on CSV files. We are skipping ahead slightly to slicing, later in this tutorial, but what this syntax means is: for the i value, take all values (: is a full slice, from start to end); for the j value take 1; Giving this array [2, 5, 8]: The array you get back when you index or slice a numpy array is a view of the original array. The other axes are the axes that remain after the reduction of a. percentileofscore¶ scipy. txt': import numpy as np test = np. That is, there is no method in Pandas or NumPy that enables us to calculate geometric and harmonic means. Robin's Blog Calculating percentiles in Python – use numpy not scipy! November 24, 2015. Those who are used to NumPy can do a lot of things without using libraries such as OpenCV. The NumPy library is a popular Python library used for scientific computing applications, and is an acronym for "Numerical Python". NumPy also provides a set of functions that allows manipulation of that data, as well as operating over it. Paket tambahan yang kita perlukan hanyalah numpy. Once you have created the arrays, you can do basic Numpy operations. from pylab import * The numpy will be imported as well (with np alias). 5 is the default Python provided, and revised in March 2019, where Anaconda 2018. Let's start with NumPy: NumPy is the fundamental package for scientific computing with Python. Before we move on to more advanced things time for a quick recap of the basics. The 90th percentile has a value of 19. Your source code remains pure Python while Numba handles the compilation at runtime. collections. Can python do this without using numpy?. Before you can use NumPy, you need to install it. In this tutorial, you will discover how to manipulate and access your data correctly in NumPy arrays. array([1,2,3,4,5]) p = np. The range for float numbers using Python Generator and yield. Y = prctile(X,p,vecdim) returns percentiles over the dimensions specified in the vector vecdim. qlgf9zqvrzp n7gnc8gdz3 adj63udioup 0mmanvqhammgi gzdnqevu2gkrr9 597xrxxwhcruz o1gftmpevh4s oamrava4jetijb ziydrki703nf7 hcaswmhjjx76n isovh4nuzs7 entgmhk61995rl d2cev9n97w ewb5u4kczld1 c00ymy8b26jkh lsdej0vuttf drym1hnbat6l2 zgyjk7t78j yhoi4ixy662dex xujmz0b0jke xijg0nvcrksal 2ezsraaatql xbobm0pp31 71jjt2p67wrg0i tqddifd1cw0c8xv cdmslbkzft1z5w cf10o28mcsm8n xcc337uqayn 9d1ffhuag3p2x1 | 2020-10-26 22:36:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.1795530468225479, "perplexity": 1784.4384150864887}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107892062.70/warc/CC-MAIN-20201026204531-20201026234531-00714.warc.gz"} |
https://web2.0calc.com/questions/trigonometry-problem_3 | +0
# Trigonometry Problem
+3
599
2
+1884
Find all of the fifth roots of the complex number $$4+32i$$. Put your answers in the rectangular form $$a+bi$$ and in the polar form $$z=re^(i\Theta )$$. Please show how you got to your answers.
Nov 24, 2015
#2
+1884
+10
$$(4+32i)^(1/5)$$
$$r=\sqrt(a^2+b^2)$$
$$r=\sqrt(4^2+32^2)$$
$$r=\sqrt(16+1024)$$
$$r=\sqrt1040$$
$$r=4\sqrt65$$
$$tan(\Theta)=b/a$$
$$tan(\Theta)=32/4$$
$$tan(\Theta)=8$$
$$\Theta=tan^-1(8)$$
$$\Theta ≈1.4464413322481$$
$$z=r*e^(i*\Theta)$$
$$z≈4\sqrt65*e^(i*1.4464413322481)$$
$$z^(1/5)≈(4\sqrt65*e^(i*1.4464413322481))^(1/5)$$
$$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*1.4464413322481*(1/5))$$
$$z^(1/5)≈2.003103242348*e^(i*1.0766143512748)$$
$$z=r*(cos(\Theta)+i*sin(\Theta))$$
$$z≈2.003103242348*(cos(1.0766143512748)+i*sin(1.0766143512748))$$
$$z≈2.003103242348*(0.4743116564868+i*0.88035700289064)$$
$$z≈0.9500952169545+i*1.763445966914$$
$$z≈0.9500952169545+1.763445966914i$$
$$z^(1/5)≈(4\sqrt65*e^(i*7.7296266394277))^(1/5)$$
$$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*7.7296266394277*(1/5))$$
$$z^(1/5)≈2.003103242348*e^(i*1.5459253278855)$$
$$z=r*(cos(\Theta)+i*sin(\Theta))$$
$$z≈2.003103242348*(cos(1.5459253278855)+i*sin(1.5459253278855))$$
$$z≈2.003103242348*(0.024868434927217+i*0.99969073264899)$$
$$z≈-0.049814942634829+i*0.0001545372385216$$
$$z≈-0.049814942634829+0.0001545372385216i$$
$$z^(1/5)≈(4\sqrt65*e^(i*14.012811946607))^(1/5)$$
$$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*14.012811946607*(1/5))$$
$$z^(1/5)≈2.003103242348*e^(i*1.6955283616309)$$
$$z=r*(cos(\Theta)+i*sin(\Theta))$$
$$z≈2.003103242348*(cos(1.6955283616309)+i*sin(1.6955283616309))$$
$$z≈2.003103242348*(-0.12440885450163+i*0.99223104009177)$$
$$z≈-0.24920377983349+i*1.9875412136019$$
$$z≈-0.24920377983349+1.9875412136019i$$
$$z^(1/5)≈(4\sqrt65*e^(i*20.2959972538))^(1/5)$$
$$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*20.2959972538*(1/5))$$
$$z^(1/5)≈2.003103242348*e^(i*4.059994507574)$$
$$z=r*(cos(\Theta)+i*sin(\Theta))$$
$$z≈2.003103242348*(cos(4.05994507574)+i*sin(4.05994507574))$$
$$z≈2.003103242348*(-0.60772245598411+i*-0.7941494925344)$$
$$z≈-1.2173308220295+i*1.5907634234047$$
$$z≈-1.2173308220295+1.5907634234047i$$
$$z^(1/5)≈(4\sqrt65*e^(i*26.5918256098))^(1/5)$$
$$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*26.5918256098*(1/5))$$
$$z^(1/5)≈2.003103242348*e^(i*5.315836512196)$$
$$z=r*(cos(\Theta)+i*sin(\Theta))$$
$$z≈2.003103242348*(cos(5.315836512196)+i*sin(5.315836512196))$$
$$z≈2.003103242348*(0.56748448302716+i*-0.82338409112843)$$
$$z≈-1.1367300079339+i*-1.6493233426371$$
$$z≈-1.1367300079339+-1.6493233426371i$$
$$z≈-1.1367300079339-1.6493233426371i$$
.
Nov 26, 2015
#1
+5
z = (4 + 32i)^(1/5)
Divide: 1 / 5 =0.2
Power: (4+32i) ^ 0.2 = 1.9198686+0.5714255i
Algebraic form:
z = 1.9198686+0.5714255i
Exponential form:
z = 2.0031032 × ei 16°34'30″
Trigonometric form:
z = 2.0031032 × (cos 16°34'30″ + i sin 16°34'30″)
Polar form:
r = |z| = 2.0031
φ = arg z = 16.575° = 16°34'30″ = 0.09208π
Nov 24, 2015
#2
+1884
+10
$$(4+32i)^(1/5)$$
$$r=\sqrt(a^2+b^2)$$
$$r=\sqrt(4^2+32^2)$$
$$r=\sqrt(16+1024)$$
$$r=\sqrt1040$$
$$r=4\sqrt65$$
$$tan(\Theta)=b/a$$
$$tan(\Theta)=32/4$$
$$tan(\Theta)=8$$
$$\Theta=tan^-1(8)$$
$$\Theta ≈1.4464413322481$$
$$z=r*e^(i*\Theta)$$
$$z≈4\sqrt65*e^(i*1.4464413322481)$$
$$z^(1/5)≈(4\sqrt65*e^(i*1.4464413322481))^(1/5)$$
$$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*1.4464413322481*(1/5))$$
$$z^(1/5)≈2.003103242348*e^(i*1.0766143512748)$$
$$z=r*(cos(\Theta)+i*sin(\Theta))$$
$$z≈2.003103242348*(cos(1.0766143512748)+i*sin(1.0766143512748))$$
$$z≈2.003103242348*(0.4743116564868+i*0.88035700289064)$$
$$z≈0.9500952169545+i*1.763445966914$$
$$z≈0.9500952169545+1.763445966914i$$
$$z^(1/5)≈(4\sqrt65*e^(i*7.7296266394277))^(1/5)$$
$$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*7.7296266394277*(1/5))$$
$$z^(1/5)≈2.003103242348*e^(i*1.5459253278855)$$
$$z=r*(cos(\Theta)+i*sin(\Theta))$$
$$z≈2.003103242348*(cos(1.5459253278855)+i*sin(1.5459253278855))$$
$$z≈2.003103242348*(0.024868434927217+i*0.99969073264899)$$
$$z≈-0.049814942634829+i*0.0001545372385216$$
$$z≈-0.049814942634829+0.0001545372385216i$$
$$z^(1/5)≈(4\sqrt65*e^(i*14.012811946607))^(1/5)$$
$$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*14.012811946607*(1/5))$$
$$z^(1/5)≈2.003103242348*e^(i*1.6955283616309)$$
$$z=r*(cos(\Theta)+i*sin(\Theta))$$
$$z≈2.003103242348*(cos(1.6955283616309)+i*sin(1.6955283616309))$$
$$z≈2.003103242348*(-0.12440885450163+i*0.99223104009177)$$
$$z≈-0.24920377983349+i*1.9875412136019$$
$$z≈-0.24920377983349+1.9875412136019i$$
$$z^(1/5)≈(4\sqrt65*e^(i*20.2959972538))^(1/5)$$
$$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*20.2959972538*(1/5))$$
$$z^(1/5)≈2.003103242348*e^(i*4.059994507574)$$
$$z=r*(cos(\Theta)+i*sin(\Theta))$$
$$z≈2.003103242348*(cos(4.05994507574)+i*sin(4.05994507574))$$
$$z≈2.003103242348*(-0.60772245598411+i*-0.7941494925344)$$
$$z≈-1.2173308220295+i*1.5907634234047$$
$$z≈-1.2173308220295+1.5907634234047i$$
$$z^(1/5)≈(4\sqrt65*e^(i*26.5918256098))^(1/5)$$
$$z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*26.5918256098*(1/5))$$
$$z^(1/5)≈2.003103242348*e^(i*5.315836512196)$$
$$z=r*(cos(\Theta)+i*sin(\Theta))$$
$$z≈2.003103242348*(cos(5.315836512196)+i*sin(5.315836512196))$$
$$z≈2.003103242348*(0.56748448302716+i*-0.82338409112843)$$
$$z≈-1.1367300079339+i*-1.6493233426371$$
$$z≈-1.1367300079339+-1.6493233426371i$$
$$z≈-1.1367300079339-1.6493233426371i$$
gibsonj338 Nov 26, 2015 | 2019-01-21 06:41:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9261291027069092, "perplexity": 12286.91778978293}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583763149.45/warc/CC-MAIN-20190121050026-20190121072026-00622.warc.gz"} |
https://routledgehandbooks.com/doi/10.1201/9781315117430-3 | 793
# Dimensionality Analysis
Authored by: Robert D. Gibbons , Li Cai
# Handbook of Item Response Theory
Print publication date: December 2017
Online publication date: December 2017
Print ISBN: 9781466514331
eBook ISBN: 9781315117430
10.1201/9781315117430-3
#### Abstract
Much of item response theory (IRT) is based on the assumption of unidimensionalty; namely, the associations among the item responses are explained completely by a single underlying latent variable, representing the target construct being measured. While this is often justified in many areas of educational measurement, more recent interest in measuring patient-reported outcomes (Gibbons et al., 2008, 2012) involves items that are drawn from multiple uniquely correlated subdomains violating the usual conditional independence assumption inherent in unidimensional IRT models.
#### 3.1 Introduction
Much of item response theory (IRT) is based on the assumption of unidimensionalty; namely, the associations among the item responses are explained completely by a single underlying latent variable, representing the target construct being measured. While this is often justified in many areas of educational measurement, more recent interest in measuring patient-reported outcomes (Gibbons et al., 2008, 2012) involves items that are drawn from multiple uniquely correlated subdomains violating the usual conditional independence assumption inherent in unidimensional IRT models. As alternatives, both unrestricted item factor analytic models (Bock & Aitkin, 1981) and restricted or confirmatory item factor analytic models (Gibbons & Hedeker, 1992) have been used to accommodate the multidimensionality of constructs for which the unidimensionality assumption is untenable. A concrete example is the measurement of depressive severity, where items are drawn from mood, cognition, and somatic impairment subdomains. While this is a somewhat extreme example, there are many more borderline cases where the choice between a unidimensional model and a multidimensional model is less clear, orthequestion of how many dimensions is “enough” is of interest. In this chapter, we explore the issue of determining the dimensionality of a particular measurement process. We begin by discussing multidimensional item factor analysis models, and then consider the consequences of incorrectly fitting a unidimensional model to multidimensional data. We also discuss nonparametric approaches such as DIMTEST (Stout, 1987). We then examine different approaches to testing dimensionality of a given measurement instrument, including approximate or heuristic approaches such as eigenvalue analysis, as well as more statistically rigorous limited-information and full-information alternatives. Finally, we illustrate the use of these various techniques for dimensionality analysis using a relevant example.
#### 3.2 Classical Multiple Factor Analysis of Test Scores
Multiple factor analysis as formulated by Thurstone (1947) assumes that the test scores are continuous measurements standardized to mean zero and standard deviation one in the sample. (Number-right scores on tests with 30 or more items are considered close enough to continuous for practical work.) The Pearson product-moment correlations between all pairs of tests are then sufficient statistics for factor analysis when the population distribution of the scores is multivariate normal. Because the variables are assumed standardized, the mean of the distribution is the null vector and the covariance matrix is a correlation matrix. If the dimensionality of the factor space is D, the assumed statistical model for the pth observed score y is
3.1
where the underlying vector of latent variables attributable to individual differences is
Just as the observed variables, the latent variables are assumed to follow a standard multivariate normal distribution but are uncorrelated; that is, their covariance matrix is a D × D identity matrix. The residual term (also called unique factor), ϵp , that accounts for all remaining variation in yp is assumed to be normal with mean 0 and variance
, where
which Thurstone called the communality of observed variable j. Estimation of the loadings requires the restriction to prevent inadmissible solutions (also known as Heywood cases). Moreover, the unique factor consists of both systematic variation and error of measurement. Thus, if the reliability of the test is known to be ρ, cannot be greater than ρ.
On the above assumptions, efficient statistical estimation of the factor loadings from the sample correlation matrix is possible and available in published computer programs. In fact, only the communalities need to be estimated: once the communalities are known, the factor loadings can be calculated directly from the so-called “reduced” correlation matrix via matrix decompositions, in which the diagonal elements of the sample correlation matrix are replaced by the corresponding communalities (Harman, 1967).
#### 3.3 Classical Item Factor Analysis
In item factor analysis, the observed item responses are assigned to one of two-or-more predefined categories. For example, test items marked right or wrong are assigned to dichotomous categories; responses to essay questions may be assigned to ordered polytomous categories (grades) A, B, C, D in order of merit; responses in the form of best choice among multiple alternatives may be assigned to nominal polytomous categories.
To adapt the factor analysis model for test scores to the analysis of categorical item responses, we assume that the y-variables are also unobservable. We follow Thurstone in referring to these underlying variables as response processes. In the dichotomous case, a process gives rise to an observable correct response when yp exceeds some threshold γi specific to item i. On the assumption that yp is standard normal, γi divides the area under the normal curve into two sections corresponding to the probability that a respondent with given value of θ will respond in the first or second category. Designating the categories 1and 2, we may express these conditional probabilities given θ as
where Φ is the cumulative normal distribution function and
The unconditional response probabilities, on the other hand, are the areas under the standard normal curve above and below −γi in the population from which the sample of respondents is drawn. The area above this threshold is the classical item difficulty, pi , and the standard normal deviate at pi is a large sample estimator of −γi (Lord & Novick, 1968, Chapter 16).
These relationships generalize easily to ordered polytomous categories. Suppose item j has mj ordered categories; we then replace the single threshold of the dichotomous case with mi −1 thresholds, say, γi1, γi2, …, γi, mi −1 . The category response probabilities conditional on θ are the mj areas under the normal curve corresponding to the intervals from minus to plus infinity bounded by the successive thresholds:
where Φ(zp γ i0) = 0 and Φ(zp γimi ) = 1 −Φ(zp γi,mi −1 ).
Because the response processes are unobserved, their product-moment correlation matrix cannot be calculated directly. Classical methods of multiple factor analysis do not apply directly to item response data. Full maximum likelihood estimation of the item correlation matrix requires calculating the normal orthant probabilities involving integrals over as many dimensions as there are items. While it is theoretically possible, actual computation remains difficult even with modern estimation approaches (Song & Lee,2003).
However, an approximation to the item correlations can be inferred from the category joint-occurrence frequencies tallied over the responses in the sample. Assuming in the two-dimensional case that the marginal normal distribution of the processes is standard bivariate normal, the correlation value that best accounts for the observed joint frequencies can be estimated, for example, using pairwise maximum likelihood. If both items are scored dichotomously, the result is the well-known tetrachoric correlation coefficient, an approximation for which was given by Divgi (1979). If one or both items are scored polytomously, the result is the less common polychoric correlation (Jöreskog, 2002). The correlations for all distinct pairs of items can then be assembled into a correlation matrix and unities inserted in the diagonal to obtain an approximation to the item correlation matrix. Because the calculation of tetrachoric and polychoric correlations breaks down if there is a vacant cell in the joint-occurrence table, a small positive value such as 0.5 (i.e., a continuity correction) may be added to each cell of the joint frequency table.
For the purpose of determining dimensionality, the correlation matrix described above can be subjected to the method of principal components or principal factor analysis with iterated communalities (Harman, 1967, p. 87). Classical principal factor analysis of item responses can be useful in its own right, or as a preliminary to more exact and more computationally intensive IRT procedures such as maximum marginal likelihood item factor analysis. In the latter role, the classical method provides a quick way of giving an upper bound on a plausible number of factors in terms of the total amount of association accounted for. It also gives good starting values for the iterative procedures discussed in the following section.
#### 3.4 Item Factor Analysis Based on IRT
IRT-based item factor analysis makes use of all information in the original categorical responses and does not depend on pairwise indices of association such as tetrachoric or polychoric correlation coefficients. For that reason, it is referred to as full-information item factor analysis. It works directly with item response models giving the probability of the observed categorical responses as a function of latent variables descriptive of the respondents and parameters descriptive of the individual items. It differs from the classical formulation in its scaling, however, because it does not assume that the response process has unit standard deviation and zero mean; rather, it assumes that the residual term has unit standard deviation and zero mean. The latter assumption implies that the response processes have zero mean and standard deviation equal to
Inasmuch as the scale of the model affects the relative size of the factor loadings and thresholds, we rewrite the model for dichotomous responses in a form in which the factor loadings are replaced by factor slopes, aid , and the threshold is absorbed in the intercept,ci:
To convert factor slopes into loadings, we divide by the above standard deviation and similarly convert the intercepts to thresholds:
Conversely, to convert to factor analysis units, we change the standard deviation of the residual from one to
and change the scale of the slopes and intercept accordingly:
For polytomous responses, the model generalizes as
where Φ(zi + c i0) = 0 and Φ(zi + cimi ) = 1 −Φ(zi + c i,m i −1) as previously.
In the context of item factor analysis, this is the multidimensional generalization of the graded model introduced by Samejima (1969). Similarly, the rating scale model of Andrich (1978), in which all items have the same number of categories and the thresholds are assumed to have the same spacing but may differ in overall location, can be generalized by setting the above linear form to zi + ei + ch , where ei is the location intercept.
#### 3.5 Maximum Likelihood Estimation of Item Slopes and Intercepts
There is a long history, going back to Fechner (1860), of methods for estimating the slope and intercept parameters of models similar to the above—that is, models in which the response process is normally distributed and the deviate is a linear form. These so-called normal transform models differ importantly from the IRT models, however, in assuming that the θ variables are manifest measurements of either observed or experimentally manipulated variables. In Fechner's classic study of the sensory discrimination thresholds for lifted weights, the subjects were required to lift successively each of a series of two small, identical appearing weights differing by fixed amounts and say which feels heavier. Fechner fitted graphically the inverse normal transforms of the proportion of subjects who answered correctly and used the slope of the fitted line to estimate the standard deviation as a measure of sensory discrimination. Much later, R. A. Fisher (Bliss, 1935) provided a maximum likelihood method of fitting similar functions used in the field of toxicology to determine the so-called 50% lethal dose of pesticides. This method eventually became known as probit analysis (Bock & Jones, 1968; for behavioral applications, see Finney, 1952).
To apply Fisher's method of analysis to item factor analysis, one must find a way around the difficulty that the variable values (i.e., the θs) in the linear predictor are unobservable. The key to solving this problem lies in assuming that the values have a specifiable distribution in the population from which the respondents are drawn (Bock & Lieberman, 1970). This allows us to integrate over that distribution numerically to estimate the expected numbers of respondents located at given points in the latent space who respond in each of the categories. These expected values can then be subjected to a multidimensional version of probit analysis. The so-called EM method of solving this type of estimation problem (Aitkin, Volume Two, Chapter 12; Bock & Aitkin, 1981) is an iterative procedure starting from given initial values. It involves calculating expectations (the E-step) that depend on both the parameters and the observations, followed by likelihood maximization (the M-step) that depends on the expectations. These iterations can be shown to converge on the maximum likelihood estimates under very general conditions (Dempster et al., 1977). In IRT and similar applications, this approach is called maximum marginal likelihood estimation because it works with the marginal probabilities of response rather than the conditional probabilities (Glas, Volume Two, Chapter 11). Details in the context of item factor analysis are given in Bock and Aitkin (1981) and Bock and Gibbons (2010, Appendix).
#### 3.6 Confirmatory Item Factor Analysis and the Bifactor Pattern
There are two major limitations of the unrestricted or exploratory factor analysis model described above. First, interpretation of the final solution depends on selecting the appropriate rotation of the final solution (e.g., varimax, quartimin, etc.; for a review, see Browne, 2001). Second, modern simulation-based estimation approaches notwithstanding (e.g., Cai, 2010a,b), the full-information IRT approach remains demanding in terms of the number of dimensions that can be evaluated because the computational complexity associated with the integrals in the likelihood equations is exponentially increasing in the number of factors. In confirmatory factor analysis, the first limitation (indeterminacy due to rotation) is resolved by assigning arbitrary fixed values to certain loadings of each factor during maximum likelihood estimation. In general, fixing of loadings will imply nonzero correlations of the latent variables, but this does not invalidate the analysis. The correlations may also be estimated if desired by selecting an oblique rotation criterion. An important example of confirmatory item factor analysis—which resolves the second problem of limitation of the number of dimensions that can be numerically evaluated—is the bifactor patterns for general and group factors, which applies to tests and scales with item content drawn from several well-defined subareas of the domain in question. Two prominent examples are tests of educational achievement consisting of reading, mathematics and science areas, and self-reports of health status covering both physical and emotional impairment. The main objective in the use of such instruments is to estimate a single score measuring, in these examples, general educational achievement or overall health status.
To analyze these kinds of structures for dichotomously scored item responses, Gibbons and Hedeker (1992) developed full-information item bifactor analysis for binary item responses, and Gibbons extended it to the polytomous case (Gibbons et al., 2007). Cai et al. (2011) further generalized the model to handle multiple groups. To illustrate, consider a set of n test items for which a D-factor solution exists with one general factor and D − 1 group or method-related factors. The bifactor solution constrains each item j to a nonzero loading αi1 on the primary dimension and a second loading (αid, d = 2,…, D) on not more than one of the D −1 group factors. For four items, the bifactor pattern matrix might be
This structure, which Holzinger and Swineford (1937) termed the “bifactor” pattern, also appears in the inter-battery factor analysis of Tucker (1958) and is one of the confirmatory factor analysis models considered by Jöreskog (1969). In the latter case, the model is restricted to test scores assumed to be continuously distributed. However, the bifactor pattern might also arise at the item level (Muthén, 1989). Gibbons and Hedeker (1992) showed that paragraph comprehension tests, where the primary dimension represents the targeted process skill and additional factors describe content area knowledge within paragraphs, were described well by the bifactor model. In this context, they showed that items were conditionally independent between paragraphs, but conditionally dependent within paragraphs. More recently, the bifactor model has been applied to problems in patient-reported outcomes in physical and mental health measurement (Gibbons et al., 2008, 2012, 2014). As shown by Gibbons and Hedeker (1992), the bifactor model always reduces the dimensionality of the likelihood equation to two, regardless of the number of secondary factors.
In the bifactor case, the graded response model (Gibbons et al., 2007) is
3.2
where only one of the d = 2, …, d values of aid is nonzero in addition to a i1. Assuming independence of the θ , in the unrestricted case, the multidimensional model above would require a d-fold integral in order to compute the unconditional probability for response pattern u , that is,
3.3
where Lp ( θ ) is the likelihood of ⋃ p conditional on θ . The corresponding unconditional or marginal probability for the bifactor model reduces to
3.4
which can be approximated to any degree of practical accuracy using two-dimensional Gauss–Hermite quadrature, since for both the binary and graded bifactor response models, the dimensionality of the integral is two regardless of the number of subdomains (i.e., D − 1) that comprised the scale.
#### 3.7 Unidimensional Models and Multidimensional Data
A natural question is whether there is any adverse consequence of applying unidimensional IRT models to multidimensional data. To answer this question, Stout and coworkers (e.g., Stout, 1987; Zhang & Stout, 1999) took a distinctly nonparametric approach to characterize the specific conditions under which multidimensional data may be reasonably well represented by a unidimensional latent variable. They emphasized a core concept that subsequently became the basis of a family of theoretical and practice devices for studying dimensionality, namely, local independence as expressed using conditional covariances. To begin, given n items in a test, the strong form of local independence states that the conditional response pattern probability factors into a product of conditional item response probabilities, that is,
Correspondingly, a test is weakly locally independent with respect to θ if the conditional covariance is zero for all item pairs i and i:
The conditional covariances provide a convenient mechanism to formalize the notion of an essentially unidimensional test that possesses one essential dimension and (possibly) a number of nuisance dimensions. A test is said to be essentially independent (Stout, 1990) with respect to θ if
In other words, essential independence states that the average value of conditional covariances across all item pairs is small as the test length increases. If the minimal dimensionality of θ necessary for an item pool to satisfy essentially independence is equal one, then the test is said to be essentially unidimensional.
The mathematical condition above suggests a statistical procedure for testing essential unidimensionality (Stout, 1987). In brief, the test is split into two subsets called assessment tests (AT1 and AT2), and a longer subset called the partitioning test (PT). The items for AT1 are chosen to be saturated with the same dominant latent trait, but are as dimensionally different as possible from the items in the PT. Then AT2 is selected such that the items have similar difficulty as AT1. Each test taker's total score on the PT is used to group the test takers into several homogeneous subgroups. The PT total score becomes the conditioning score (effectively as a surrogate of θ) to calculate required conditional covariances for statistical hypothesis testing using the AT1 and AT2 item responses. The procedure as formalized by Nandakumar and Stout (1993) is referred to as DIMTEST.
Gibbons et al. (2007) studied the consequences of fitting unidimensional models to multidimensional data empirically. The question they asked was slightly different. To the extent that the primary dimension of interest can be preserved in a unidimensional model and in the primary factor of a bifactor model or possibly in an exploratory item factor analysis model, does the specific model used make a difference in the results?
They conducted a simulation study to investigate the effects of applying Samejima's (1969) graded response model in unidimensional and bifactor form to multidimensional data. Conditions studied were (a) test length, 50 items or 100 items; (b) number of dimensions, 5 or 10; (c) primary loadings, 0.50 or 0.75; and (d) domain loadings, 0.25 or 0.50. Outcome results include standard deviation of expected a posteriori (EAP) estimates of θ, posterior standard deviations (PSDs, or standard errors) of Bayes EAP scores, log-likelihood (model fit), differences between EAP and actual θ, and percentage change between unidimensional and bifactor models of these variables. The generated data were based on a four-point categorical scale, and the examinee distribution was assumed to be normal, N(0, 1), based on 1000 replications. In the following, we summarize the key findings of this study.
Figure 3.1 reports the standard deviations of the θ estimates for the unidimensional and bifactor models across the 12 simulated conditions. Inspection of the figure indicates that the EAP estimates based on the unidimensional model were more varied across all conditions. The magnitude of the difference decreased when the primary and secondary loadings decreased, leading to a more unidimensional solution. As shown, as the number of items increased from 50 to 100, the EAP estimates from both models became more varied, but not as severe for the bifactor model.
Figure 3.1 Mean standard deviations of θ of the unidimensional and bifactor models based on 1000 replications per condition (number of items [NI] = 50 or 100, number of dimensions [ND] = 5 or 10, primary loadings [PL] = 0.50 or 0.75, domain loadings [DL] = 0.25 or 0.50).
Figure 3.2 Mean PSDs of Bayes expected a posterior scores of the unidimensional and bifactor models based on 1000 replications per condition (number of items [NI] = 50 or 100, number of dimensions [ND] = 5 or 10, primary loadings [PL] = 0.50 or 0.75, domain loadings [DL] = 0.25 or 0.50).
The results of this study illustrate the consequences attached to applying a unidimensional IRT model to data with varying degrees of multidimensionality compared to the bifactor model. The first set of results addressed the variability in estimated θ values, or examinees'standing on the latent trait. Compared to the unidimensional model, the bifactor model yielded θ estimates that were more homogeneous across simulated data structures. As a consequence, studies that are designed to evaluate educational or clinical interventions will have increased statistical power to detect meaningful effects when scores are based on a bifactor model and the underlying data are the result of a multidimensional response process.
PSD estimates were found to be underestimated across all conditions for the unidimensional model. For the bifactor model, PSD values were consistently below 0.20 across conditions, except when the total test length was 50 and the primary loadings were 0.50 and the domain loadings were 0.25. One setting in which the underestimation of PSDs could affect test scores is in computer adaptive testing, in which each item is intentionally selected to provide the most information for estimating examinee ability in the sense of greatest reduction of PSD. Using PSD estimates based on the unidimensional model may therefore lead to suboptimal estimates of examinee ability. Used as measurement error variance, the inverse squared unidimensional PSDs are not valid for weighting observations in statistical analyses using the scores as data.
#### 3.8 Limited-Information Goodness-of-Fit Tests
The nonparametric indices based on conditional covariances such as DIMTEST do not explicitly specify a distribution of the θs. Hence, they require the use of external conditioning subscores such as the partitioning total score. When an item factor analysis model is fitted using standard estimation methods such as maximum marginal likelihood, population distributions of θ are routinely assumed. Therefore, upon finding the maximum likelihood solution, the model yields expected probabilities for each single item, as well as joint probabilities for item pairs, triplets, quadruplets, etc. When contrasted against the observed probabilities, the residuals may be used to derive goodness-of-fit statistics. Most of the time, univariate and bivariate association information is used.
In the context of IRT, statistics based on (mostly) univariate and bivariate subtables are referred to as limited-information goodness-of-fit statistics, in contrast to full-information statistics (e.g., the Pearson's chi-square statistic) that are based on residuals of the full item by item-by-item cross-classifications. Despite the apparent loss of information due to collapsing the full contingency table into a series of first- and second-order association tables, limited-information test statistics have been suggested as a potential solution to the Achilles' heel of full-information statistics, namely, the sparseness of the underlying multiway contingency table upon which the IRT model is defined (Bartholomew & Tzamourani, 1999). The number of cells in the table is exponentially increasing in the number of items, and for tests of realistic length, the table will become extremely sparse for any conceivable sample size. The sparseness invalidates the usual asymptotic chi-square approximations to the distribution of Pearson's statistic or the likelihood ratio statistic, making model fit testing decisions based on full-information statistics untrustworthy in practical situations. On the other hand, test statistics based on univariate and bivariate subtables maintain Type I error rate control and have adequate power (see, e.g., Cai et al., 2006). In particular, Maydeu-Olivares and Joe's (2005) M 2 family of test statistics has witnessed increasing popularity.
In the context of multidimensional IRT, Cai and Hansen (2012) extended the dimension reduction technique, already used in parameter estimation of bifactor models, to limited-information goodness-of-fit testing. For example, for a bifactor model, the probabilities and derivatives for computing limited-information test statistics require at most two-dimensional numerical integration, regardless of the number of factors in the model, making it feasible to test much larger models with many latent variables. In addition, Cai and Hansen (2012) developed a new quadratic form test statistic, which they call
, that is based on the general limited-information testing principles proposed by Joe and Maydeu-Olivares (2010). The statistic is best understood as a further reduction (or concentration) of the univariate and bivariate subtables. When the item responses are polytomous, this new statistic can be substantially better calibrated and more powerful than M2 . In addition, the chi-square distributed test statistics can be used to calculate fit measures such as the root mean square error of approximation (RMSEA; Browne & Cudeck, 1993) that are free from the influence of sample size.
The details of limited-information goodness-of-fit testing are more substantial than those can be covered in this chapter. In brief, the development begins with the realization that the IRT model can be written as a function of the (marginal) response pattern probability π⋃(γ) for pattern ⋃, where γ is a notational shorthand for the collection of free and estimable parameters in the model. Suppose there are C possible response patterns. Let us define the C × 1 vector of modeled probabilities as π ( γ ) and the corresponding C × 1 vector of observed proportions as p . Let the C × 1 population cell probabilities be π . The null hypothesis being evaluated in the goodness-of-fit testing situation is H 0: π ( γ ) = π , for some γ , versus the alternative HA : π ( γ )≠ π , for any γ .
Suppose the total sample size is N. Treating p as the fixed observed data, maximizing (e.g., using the EM algorithm) the multinomial likelihood with cell probabilities given by π ( γ ) leads to the maximum marginal likelihood estimator
. Let the fitted cell probabilities be . The cell residuals are . Standard discrete multivariate analysis results (Rao, 1973) suggest that the cell residuals are asymptotically C-variate normal under the null hypothesis:
where , D = diag( π ),
is the Jacobian of the model, and is the Fisher information matrix.
Subtable probabilities such as the univariate and bivariate probabilities are linear functions of the cell probabilities (Cai et al., 2006). The relationship can be conveniently expressed using reduction operator matrices (Joe & Maydeu-Olivares, 2010). Let T be a particular fixed q × C matrix with full row rank that achieves the reduction of π into lower-order probabilities. The new vector of residuals retains asymptotic normality
where , and , , with the q × dim( γ ) (local) Jacobian matrix given by
If the IRT model is locally identified, that is, has full column rank, then there exists a q × [q − dim ( γ )] orthogonal complement matrix such that is a null matrix. This implies
where . Evaluating the model-implied probabilities and the Jacobian elements at the maximum likelihood estimate, the following limited-information statistic is asymptotically centrally chi-square distributed with q − dim ( γ ) degrees of freedom:
#### 3.9 Example
As an illustration, we analyze data obtained with the Quality of Life Interview for the Chronically Mentally Ill (Lehman, 1988) from 586 chronically mentally ill patients. The instrument consists of one global life-satisfaction item followed by 34 items in seven subdomains, namely, Family, Finance, Health, Leisure, Living, Safety, and Social, with four, four, six, six, five, five, and four items, respectively. Respondents were instructed to rate each item in turn on a seven-point scale consisting of ordered response categories: terrible, unhappy, mostly dissatisfied, mixed, about equally satisfied and dissatisfied, mostly satisfied, pleased, delighted. Both the multiple content areas of the subdomains and their labeling as such encourage responses to the set of items as a whole rather than considered responses to the individual items. This effect creates dependencies between responses within the sets that violate the assumption of conditional independence of response required for conventional one-dimensional IRT analysis.
#### 3.9.1 Exploratory Item Factor Analysis
Given that the items are clustered within seven content domains, for the purpose of dimensionality assessment, we considered models containing one through eight factors, to determine if an additional factor explained any significant additional variation in item responses over the seven specified subdomains. Chi-square statistics for the addition of each successively added factor are shown in Table 3.1. Very roughly, a chi-square value is significant if it is at least twice as large as its degrees of freedom. By this rule, even the addition of an eighth orthogonal factor shows no danger of over-factoring, although its contribution to improved goodness of fit is the smallest of any factor. Notice that the decreases are not monotonic; unlike traditional factor analysis of product-moment correlations, the marginal probabilities of the response patterns (which determined the marginal likelihood) reflect changes in all parameters jointly, including the category parameters and not just the factor loadings. Because our inspection of the signs of the loadings of the first seven factors showed relationships to the item groups and the eighth factor did not (see Table 3.2), the seven-factor model is likely the most parsimonious choice.
### Table 3.1 Quality of Life Data (N = 586)
Decrease of −2 log L of Solutions with 1–8 Factors
Solution
−2 log L
Decrease
DF
1
66837.1
2
66045.0
792.1
34
3
65089.5
955.5
33
4
64118.4
971.5
32
5
63509.1
609.3
31
6
63063.7
445.4
30
7
62677.5
386.2
29
8
62370.5
307.4
28
Factors
Item Group
Item
1
2
3
4
5
6
7
0
1
0.769
0.021
0.082
−0.054
0.054
−0.002
0.097
Family
1
2
0.614
−0.269
0.044
−0.461
0.272
−0.081
0.044
3
0.687
−0.181
−0.007
−0.380
0.159
0.007
−0.058
4
0.703
−0.245
−0.045
−0.522
0.257
0.029
0.008
5
0.729
−0.214
−0.004
−0.505
0.279
0.046
0.019
Finance
2
6
0.606
0.468
−0.391
0.116
0.284
0.003
−0.101
7
0.515
0.405
−0.300
0.097
0.220
0.021
−0.032
8
0.647
0.511
−0.342
0.101
0.276
0.048
−0.092
9
0.632
0.510
−0.305
0.072
0.242
0.023
−0.069
Health
3
10
0.568
−0.123
0.132
0.201
0.049
−0.236
0.095
11
0.644
0.007
0.195
0.128
0.038
−0.443
−0.139
12
0.627
0.087
0.289
0.074
−0.026
−0.390
−0.167
13
0.668
−0.052
0.156
0.138
−0.005
−0.383
−0.232
14
0.678
−0.004
0.154
0.116
0.061
−0.288
0.054
15
0.701
0.044
0.249
0.045
0.071
−0.154
0.054
Leisure
4
16
0.741
0.215
0.150
0.030
−0.138
0.156
0.155
17
0.657
0.149
0.142
−0.017
−0.128
−0.054
0.285
18
0.721
0.223
0.101
−0.005
−0.173
0.019
0.331
19
0.749
0.313
0.144
−0.059
−0.199
0.095
0.301
20
0.670
0.192
0.078
−0.101
−0.162
−0.030
0.295
21
0.522
−0.002
−0.056
−0.002
−0.099
−0.049
0.042
Living
5
22
0.664
−0.241
−0.401
0.038
−0.191
0.048
−0.008
23
0.549
−0.332
−0.325
0.118
−0.140
−0.013
−0.028
24
0.611
−0.253
−0.529
0.006
−0.190
−0.112
0.042
25
0.626
−0.347
−0.446
0.079
−0.285
−0.127
0.030
26
0.568
−0.213
−0.439
0.066
−0.177
−0.018
0.034
Safety
6
27
0.679
−0.241
0.221
0.299
0.232
0.341
−0.004
28
0.688
−0.387
0.051
0.317
0.141
0.250
−0.040
29
0.594
−0.065
0.231
0.145
0.109
0.123
−0.044
30
0.670
−0.253
0.181
0.276
0.196
0.223
−0.003
31
0.702
−0.264
0.140
0.336
0.064
0.197
0.006
Social
7
32
0.688
0.189
0.169
−0.180
−0.399
0.197
−0.375
33
0.696
0.254
0.099
−0.192
−0.317
0.212
−0.218
34
0.620
0.203
0.149
−0.118
−0.218
0.161
−0.232
35
0.494
−0.163
0.122
−0.056
−0.179
0.046
−0.202
As expected, all first principal factor loadings are positive, clearly identifying the factor with the overall Quality-of-Life variable (see Table 3.2). In fact, the largest loading is that of item number one, which asks for the respondent's rating of overall quality of life. Only one item, the last, has a loading less than 0.5. As for the six bipolar factors, the significant feature is that the sign patterns of loadings of appreciable size conform to the item groups. Factor 2 strongly contrasts the Finance group with Family Living and Safety; to as lesser extent, Leisure is working in the same direction as Finance. In other words, persons who tend to report better financial positions and quality of leisure are distinguished by this factor from those who report better family relationships and safety. Factor 3 then combines Living and Finance and contrasts them primarily with a combination of Health and Safety. Factor 4 contrasts a combination of Family and Social with Finance, Health, and Safety. Factor 5 combines Social, Living, and Safety versus Family, Finance, and Safety. Factor 6 primarily contrasts Health with Safety. Finally, Factor 7 contrasts Social verses Leisure. The fact that the seven-factor solution has the expected all positive first factor and patterns for the remaining bipolar factors that contrast item groups rather than items within groups clearly supports a bifactor model for these data. Estimated thresholds for this analysis are reported by Bock and Gibbons (2010).
#### 3.9.2 Confirmatory Item Bifactor Analysis
The bifactor model produced a value of −2 log L = 64233.3, which is similar to that obtained for a four-factor model. While the seven-factor unrestricted model provides significant improvement in fit, inspection of the estimated factor loading in Table 3.3 shows that the bifactor model provides the most parsimonious and easily interpretable results. Because the group factor loadings are not constrained to orthogonality with those of the general factor, they are all positive and their magnitudes indicate the strength of the effect of items belonging to common domains. The effects of Family and Finance, for example, are stronger than those of Health and Leisure. It is interesting to note that the empirical reliability for the primary dimension for the bifactor model is 0.9, but is overestimated as 0.95 for a unidimensional model applied to these same data (standard errors of 0.322 and 0.232, respectively). As expected, reliability is overestimated and uncertainty in estimated scale scores is underestimated when the conditional dependencies are ignored. Avoiding this type of bias is a major motivation for item bifactor analysis.
Factors
Item Group
Item
1
2
3
4
5
6
7
0
1
0.789
Family
1
2
0.535
0.620
3
0.576
0.509
4
0.575
0.586
5
0.631
0.547
Finance
6
0.476
0.634
2
7
0.437
0.553
8
0.544
0.617
9
0.535
0.622
Health
3
10
0.560
0.256
11
0.528
0.504
12
0.486
0.505
13
0.529
0.473
14
0.650
0.286
15
0.714
0.141
Leisure
4
16
0.694
0.285
17
0.565
0.413
18
0.628
0.451
19
0.635
0.506
20
0.571
0.473
21
0.479
0.208
Living
5
22
0.536
0.549
23
0.484
0.530
24
0.497
0.668
25
0.508
0.688
26
0.508
0.672
Safety
6
27
0.557
0.517
28
0.593
0.474
29
0.533
0.501
30
0.558
0.538
31
0.591
0.383
Social
7
32
0.545
0.438
33
0.586
0.351
34
0.520
0.466
35
0.446
0.296
Limited-information goodness-of-fit testing lends additional support for the appropriateness of the bifactor solution. Take the unidimensional model for instance; the Cai-Hansen modified
statistic is equal to 2674.85 on 385° of freedom, p < 0.0001. The statistic uses both univariate and bivariate residual tables. Since there are 35 items, there are 35 + 35 × (35 − 1)/2 = 630 residuals available for testing model fit. The unidimensional graded model contains 35 × 7 = 245 free item parameters, resulting in 630 − 245 = 385° of freedom. The null hypothesis of exact unidimensionality is rejected and the unidimensional model is untenable for this dataset. We may compute the RMSEA index, which is a widely used measure of fit in factor analysis and structural equation modeling, and it is equal to 0.08 and a 90% confidence interval of RMSEA is (0.076,0.084). Adopting established conventions in factor analysis (Browne & Cudeck, 1993), an RMSEA that exceeds 0.05 cannot be taken as an indication of good fit. On the other hand, the statistic is equal to 546.88 on 351° of freedom. While it remains significant at the 0.05 level, the RMSEA index for the bifactor model is equal to 0.03 and the 90% confidence interval is (0, 0.035), indicating substantially improved fit.
#### 3.10 Discussion
We have shown that for many applications of IRT, multidimensionality rather than unidimensionality should represent the null hypothesis. There are a variety of limited-information and full-information methods for determining the goodness of fit and the underlying dimensionality of a particular test. As it turns out, the bifactor model produces excellent results for a variety of different IRT applications because it (a) uses expert judgement to define the underlying factor structure and (b) evaluates the likelihood in an always computationally tractable way because it reduces it to a two-dimensional integral, which is relatively easy to evaluate in practice. Traditional methods based on eigenvalues have a tendency to identify factors that are not indicative of the underlying trait of interest. By contrast, goodness-of-fit statistics which compare various nested and nonnested statistical models can be used efficiently to evaluate the dimensionality of a particular test. In general, unrestricted item factor analysis should not be used to evaluate multidimensionality. It is poorly specified and is subject to considerable rotational variance leading to a plethora of different conclusions regarding the latest variables. This is not the case for the bifactor model which provides essentially the same answer regardless of small changes in the model specification. Finally, one should be extremely cautious regarding the fitting of a unidimensional model to what are inherently multidimensional data. The net result is an underestimate of the point at which adaptive testing should terminate (i.e., underestimates of the posterior variance of the latent variable estimate), and increases in the empirical variance of the resulting test score. Neither of these conditions is good. As a consequence, the best possible approaches to determining dimensionality should always be used.
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Zhang, J. & Stout, W. 1999. The theoretical detect index of dimensionality and its application to approximate simple structure. Psychometrika, 64, 213–214. | 2022-01-18 01:23:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 43, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7531775236129761, "perplexity": 1827.5087260202718}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300658.84/warc/CC-MAIN-20220118002226-20220118032226-00467.warc.gz"} |
https://codegolf.stackexchange.com/questions/83123/does-the-triangle-contain-the-origin?noredirect=1 | # Challenge description
On a Cartesian plane, a triangle can be described as a set of three points, each point being one of the triangle's vertices. For instance, coordinates (2, 1), (6, 8), (-7, 3) correspond to the following triangle:
As you can see, it does not contain the origin of the plane, i.e. the point (0, 0), unlike the triangle (-5, 3), (2, 7), (3, -8):
Your job is to write a program, that given exactly six integers describing coordinates of a triangle, determines whether or not it contains the origin of the Cartesian plane. The objective is to make your code as short as possible, since this is a challenge.
# Input
Six integers corresponding to three coordinates of a triangle, for example:
1 4 -9 0 3 -4 -> (1, 4), (-9, 0), (3, -4)
You can also accept a list of integers, a list of two-tuples... - whatever is most convenient or makes your code shorter.
# Output
A truthy value if the triangle contains the origin (1, True), falsy value otherwise (0, False). You don't need to validate the input.
# Sample inputs / outputs
(18, -36), (36, 19), (-15, 9) -> True
(-23, 31), (-27, 40), (32, 22) -> False
(-40, -34), (35, 20), (47, 27) -> False
(0, 6), (-36, -42), (12, -34) -> True
(-24, 6), (36, 6), (-14, -25) -> True
Triangle images courtesy of Wolfram Alpha
• Well-written challenge, but we already have it. – xnor Jun 17 '16 at 9:42
• Well, that's a shame... should've searched more thoroughly for duplicates. I'm just gonna leave it like this, worst-case scenario it will get deleted by mods. – shooqie Jun 17 '16 at 9:46
• Your challenge won't be deleted. It's closed, which disables answering it but keeps it visible. – xnor Jun 17 '16 at 9:49 | 2021-06-21 17:22:11 | {"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": 0.4408371150493622, "perplexity": 1068.371089334618}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488286726.71/warc/CC-MAIN-20210621151134-20210621181134-00518.warc.gz"} |
https://www.groundai.com/project/high-dimensional-structured-superposition-models/ | High Dimensional Structured Superposition Models
# High Dimensional Structured Superposition Models
## Abstract
High dimensional superposition models characterize observations using parameters which can be written as a sum of multiple component parameters, each with its own structure, e.g., sum of low rank and sparse matrices, sum of sparse and rotated sparse vectors, etc. In this paper, we consider general superposition models which allow sum of any number of component parameters, and each component structure can be characterized by any norm. We present a simple estimator for such models, give a geometric condition under which the components can be accurately estimated, characterize sample complexity of the estimator, and give high probability non-asymptotic bounds on the componentwise estimation error. We use tools from empirical processes and generic chaining for the statistical analysis, and our results, which substantially generalize prior work on superposition models, are in terms of Gaussian widths of suitable sets.
## 1 Introduction
For high-dimensional structured estimation problems [7, 27], considerable advances have been made in accurately estimating a sparse or structured parameter even when the sample size is far smaller than the ambient dimensionality of , i.e., . Instead of a single structure, such as sparsity or low rank, recent years have seen interest in parameter estimation when the parameter is a superposition or sum of multiple different structures, i.e., , where may be sparse, may be low rank, and so on [1, 8, 9, 11, 15, 16, 18, 19, 31, 32].
In this paper, we substantially generalize the non-asymptotic estimation error analysis for such superposition models such that (i) the parameter can be the superposition of any number of component parameters , and (ii) the structure in each can be captured by any suitable norm . We will analyze the following linear measurement based superposition model
y=Xk∑i=1θ∗i+ω , (1)
where is a random sub-Gaussian design or compressive matrix, is the number of components, is one component of the unknown parameters, is the response vector, and is random noise independent of . The structure in each component is captured by any suitable norm , such that has a small value, e.g., sparsity captured by , low-rank (for matrix ) captured by the nuclear norm , etc. Popular models such as Morphological Component Analysis (MCA) [14] and Robust PCA [8, 11] can be viewed as a special cases of this framework (see Section 9).
The superposition estimation problem can be posed as follows: Given generated following (1), estimate component parameters such that all the component-wise estimation errors , where is the population mean, are small. Ideally, we want to obtain high-probability non-asymptotic bounds on the total componentwise error measured as , with the bound improving (getting smaller) with increase in the number of samples.
We propose the following estimator for the superposition model in (1):
min{θ1,…,θk} ∥∥ ∥∥y−Xk∑i=1θi∥∥ ∥∥22s.t. Ri(θi)≤αi ,i=1,…,k , (2)
where are suitable constants. In this paper, we focus on the case where , noting that recent advances [22] can be used to extend our results to more general settings.
The superposition estimator in (2) succeeds if a certain geometric condition, which we call structural coherence (SC), is satisfied by certain sets (cones) associated with the component norms . Since the estimate is in the feasible set of the optimization problem (2), the error vector satisfies the constraint where . The SC condition of is a geometric relationship between the corresponding error cones (see Section 3).
If SC is satisfied, then we can show that the sum of componentwise estimation error can be bounded with high probability, and the bound takes the form:
k∑i=1∥^θi−θ∗i∥2≤cmaxiw(Ci∩Bp)+√logk√n , (3)
where is the sample size, is the number of components, and is the Gaussian width [3, 10, 30] of the intersection of the error cone with the unit Euclidean ball . Interestingly, the estimation error converges at the rate of , similar to the case of single parameter estimators [21, 3], and depends only logarithmically on the number of components . Further, while dependency of the error on Gaussian width of the error has been shown in recent results involving a single parameter [3, 30], the bound in (3) depends on the maximum of the Gaussian width of individual error cones, not their sum. The analysis thus gives a general way to construct estimators for superposition problems along with high-probability non-asymptotic upper bounds on the sum of componentwise errors. To show the generality of our work, we provide a detailed review and comparison with related work in Appendix 8.
Notation: In this paper, we use to denote vector norm, and to denote operator norm. For example, is the Euclidean norm for a vector or matrix, and is the nuclear norm of a matrix. We denote as the smallest closed cone that contains a given set . We denote as the inner product.
The rest of this paper is organized as follows: We start with an optimization algorithm in Section 6 and a deterministic estimation error bound in Section 2, while laying down the key geometric and statistical quantities involved in the analysis. In Section 3, we discuss the geometry of the structural coherence (SC) condition, and show that the geometric SC condition implies statistical restricted eigenvalue (RE) condition. In Section 5, we develop the main error bound on the sum of componentwise errors which hold with high probability for sub-Gaussian designs and noise. In Section 7, we compare an estimator using “infimal convolution”[25] of norms with our estimator (2) for the noiseless case. We discuss related work in Section 8. We apply our error bound to practical problems in Section 9, present experimental results in Section 10, and conclude in Section 11. The proofs of all technical results are in the Appendix.
## 2 Error Structure and Recovery Guarantees
In this section, we start with some basic results and, under suitable assumptions, provide a deterministic bound for the componentwise estimation error in superposition models. Subsequently, we will show that the assumptions made here hold with high probability as long as a purely geometric non-probabilistic condition characterized by structural coherence (SC) is satisfied.
Let be a solution to the superposition estimation problem in (2), be the optimal (population) parameters involved in the true data generation process. Let be the error vector for component of the superposition. Our goal is to provide a preliminary understanding of the structure of error sets where live, identify conditions under which a bound on the total componentwise error will hold, and provide a preliminary version of such a bound, which will be subsequently refined to the form in (3) in Section 5. Since lies in the feasible set of (2), as discussed in Section 1, the error vectors will lie in the error sets respectively. For the analysis, we will be focusing on the cone of such error sets, given by
Ci=cone{Δi∈Rp|Ri(θ∗i+Δi)≤Ri(θ∗i)} . (4)
Let , , and , so that . From the optimality of as a solution to (2), we have
∥y−X^θ∥2≤∥y−Xθ∗∥2 ⇒ ∥XΔ∥2≤2ωTXΔ , (5)
using and . In order to establish recovery guarantees, under suitable assumptions we construct a lower bound to , the left hand side of (5). The lower bound is a generalized form of the restricted eigenvalue (RE) condition studied in the literature [5, 7, 24]. We also construct an upper bound to , the right hand side of (5), which needs to carefully analyze the noise-design (ND) interaction, i.e., between the noise and the design .
We start by assuming that a generalized form of RE condition is satisfied by the superposition of errors: there exists a constant such that for all :
(RE)1√n∥∥ ∥∥Xk∑i=1Δi∥∥ ∥∥2≥κk∑i=1∥Δi∥2 . (6)
The above RE condition considers the following set:
H={k∑i=1Δi:Δi∈Ci,k∑i=1∥Δi∥2=1} . (7)
which involves all the error cones, and the lower bound is over the sum of norms of the component wise errors. If , the RE condition in (6) above simplifies to the widely studied RE condition in the current literature on Lasso-type and Dantzig-type estimators [5, 24, 3] where only one error cone is involved. If we set all components but to zero, then (6) becomes the RE condition only for component . We also note that the general RE condition as explicitly stated in (6) has been implicitly used in [1] and [32]. For subsequent analysis, we introduce the set defined as
¯H={k∑i=1Δi:Δi∈Ci,k∑i=1∥Δi∥2≤1}. (8)
noting that .
The general RE condition in (6) depends on the random design matrix , and is hence an inequality which will hold with certain probability depending on and the set . For superposition problems, the probabilistic RE condition as in (6) is intimately related to the following deterministic structural coherence (SC) condition on the interaction of the different component cones , without any explicit reference to the random design matrix : there is a constant such that for all ,
(SC)∥∥ ∥∥k∑i=1Δi∥∥ ∥∥2≥ρk∑i=1∥Δi∥2 . (9)
If , the SC condition is trivially satisfied with . Since most existing literature on high-dimensional structured models focus on the setting [5, 24, 3], there was no reason to study the SC condition carefully. For , the SC condition (9) implies a non-trivial relationship among the component cones. In particular, if the SC condition is true, then the sum being zero implies that each component must also be zero. As presented in (9), the SC condition comes across as an algebraic condition. In Section 3, we present a geometric characterization of the SC condition [18], and illustrate that the condition is both necessary and sufficient for accurate recovery of each component. In Section 4, we show that for sub-Gaussian design matrices , the SC condition in (9) in fact implies that the RE condition in (6) will hold with high probability, after the number of samples crosses a certain sample complexity, which depends on the Gaussian width of the component cones. For now, we assume the RE condition in (6) to hold, and proceed with the error bound analysis.
To establish recovery guarantee, following (5), we need an upper bound on the interaction between noise and design [3, 20]. In particular, we consider the noise-design (ND) interaction
(ND)sn(γ)=infs>0{s:supu∈sH1√nωTXu≤γs2√n} , (10)
where is a constant, and is the scaled version of where the scaling factor is . Here, denotes the minimal scaling needed on such that one obtains a uniform bound over of the form: . Then, from the basic inequality in (5), with the bounds implied by the RE condition and the ND interaction, we have
1√n∥XΔ∥2≤1√n√ωTXΔ⇒κk∑i=1∥Δi∥2≤√γsn(γ) , (11)
which implies a bound on the component-wise error. The main deterministic bound below states the result formally:
###### Theorem 1 (Deterministic bound)
Assume that the RE condition in (6) is satisfied in with parameter . Then, if , we have .
The above bound is deterministic and holds only when the RE condition in (6) is satisfied with constant such that . In the sequel, we first give a geometric characterization of the SC condition in Section 3, and show that the SC condition implies the RE condition with high probability in Section 4. Further, we give a high probability characterization of based on the noise and design in terms of the Gaussian widths of the component cones, and also illustrate how one can choose in Section 5. With these characterizations, we will obtain the desired component-wise error bound of the form (3).
## 3 Geometry of Structural Coherence
In this section, we give a geometric characterization of the structural coherence (SC) condition in (9). We start with the simplest case of two vectors . If they are not reflections of each other, i.e., , then the following relationship holds:
###### Proposition 2
If there exists a such that , then
∥x+y∥2≥√1−δ2(∥x∥2+∥y∥2) . (12)
Next, we generalize the condition of Proposition 2 to vectors in two different cones and . Given the cones, define
δ0=supx∈C1∩Sp−1,y∈C2∩Sp−1−⟨x,y⟩ . (13)
By construction, for all and . If , then continues to hold for all and with constant . Note that this corresponds to the SC condition with and . We can interpret this geometrically as follows: first reflect cone to get , then is the cosine of the minimum angle between and . If , then and share a ray, and structural coherence does not hold. Otherwise, , implying , i.e., the two cones intersect only at the origin, and structural coherence holds.
For the general case involving cones, denote
δi=supu∈−Ci∩Sp−1,v∈∑j≠iCj∩Sp−1⟨u,v⟩ . (14)
In recent work, [18] concluded that if for each then and does not share a ray, and the original signal can be recovered in noiseless case. We show that the condition above in fact implies for the SC condition in (9), which is sufficient for accurate recovery even in the noisy case. In particular, with , we have the following result:
###### Theorem 3 (Structural Coherence (SC) Condition)
Let with as defined in (14). If , there exists a such that for any , the SC condition in (9) holds, i.e.,
∥∥ ∥∥k∑i=1Δi∥∥ ∥∥2≥ρk∑i=1∥Δi∥2 . (15)
Thus, the SC condition is satisfied in the general case as long as the reflection of any cone does not intersect, i.e., share a ray, with the Minkowski sum of the other cones.
## 4 Restricted Eigenvalue Condition for Superposition Models
Assuming that the SC condition is satisfied by the error cones , in this section we show that the general RE condition in (6) will be satisfied with high probability when the number of samples in the sub-Gaussian design matrix crosses the sample complexity . We give a precise characterization of the sample complexity in terms of the Gaussian width of the set .
Our analysis is based on the results and techniques in [28, 20], and we note that [3] has related results using mildly different techniques. We start with a restricted eigenvalue condition on . For a random vector , we define marginal tail function for an arbitrary set as
Qξ(E;Z)=infu∈EP(|⟨Z,u⟩|≥ξ) , (16)
noting that it is deterministic given the set . Let be independent Rademacher random variables, i.e., random variable with probability of being either or , and let be independent copies of . We define empirical width of as
Wn(E;Z)=supu∈E⟨h,u⟩,whereh=1√nn∑i=1ϵiXi . (17)
With this notation, we recall the following result from [28, Proposition 5.1]:
###### Lemma 1
Let be a random design matrix with each row the independent copy of sub-Gaussian random vector . Then for any , we have
infu∈H∥Xu∥2≥ρξ√nQ2ρξ(H;Z)−2Wn(H;Z)−ρξt (18)
with probability at least .
From Lemma 1, in order to obtain lower bound of in RE condition (6), we need to lower bound and upper bound . To lower bound , we consider the spherical cap
A=(∑ki=1Ci)∩Sp−1 . (19)
From [28, 20], one can obtain a lower bound to based on the Paley-Zygmund inequality. The Paley-Zygmund inequality lower bound the tail distribution of a random variable by its second momentum. Let be an arbitrary vector, we use the following version of the inequality.
P(|⟨Z,u⟩|≥2ξ)≥[E|⟨Z,u⟩|−2ξ]2+E|⟨Z,u⟩|2 (20)
In the current context, the following result is a direct consequence of SC condition, which shows that is lower bounded by , which in turn is strictly bounded away from 0. The proof of Lemma 2 is given in Appendix C.1.
###### Lemma 2
Let sets and be as defined in (7) and (19) respectively. If the SC condition in (9) holds, then the marginal tail functions of the two sets have the following relationship:
Qρξ(H;Z)≥Qξ(A;Z). (21)
Next we discuss how to upper bound the empirical width . Let set be arbitrary, and random vector be a standard Gaussian random vector in . The Gaussian width [3] of is defined as
w(E)=Esupu∈E⟨g,u⟩. (22)
Empirical width can be seen as the supremum of a stochastic process. One way to upper bound the supremum of a stochastic process is by generic chaining [26, 3, 28], and by using generic chaining we can upper bound the stochastic process by a Gaussian process, which is the Gaussian width.
As we can bound and , we come to the conclusion on RE condition. Let be a random matrix where each row is an independent copy of the sub-Gaussian random vector , and where has sub-Gaussian norm [29]. Let so that [20, 28]. We have the following lower bound of the RE condition. The proof of Theorem 4 is based on the proof of [28, Theorem 6.3], and we give it in appendix C.2.
###### Theorem 4 (Restricted Eigenvalue Condition)
Let be the sub-Gaussian design matrix that satisfies the assumptions above. If the SC condition (9) holds with a , then with probability at least , we have
infu∈H∥Xu∥2≥c1ρ√n−c2w(H)−c3ρt (23)
where and are positive constants determined by , and .
To get a in (6), one can simply choose . Then as long as for , we have
κ=infu∈H1√n∥Xu∥2≥12(c1ρ−c2w(H)√n)>0,
with high probability.
From the discussion above, if SC condition holds and the sample size is large enough, then we can find a matrix such that RE condition holds. On the other hand, once there is a matrix such that RE condition holds, then we can show that SC must also be true. Its proof is give in Appendix C.3.
###### Proposition 5
If is a matrix such that the RE condition (6) holds for , then the SC condition (9) holds.
Proposition 5 demonstrates that SC condition is a necessary condition for the possibility of RE. If SC condition does not hold, then there is such that for some , but which implies . Then for every matrix , we have , and RE condition is not possible.
## 5 General Error Bound
Recall that the error bound in Theorem 1 is given in terms of the noise-design (ND) interaction
sn(γ)=infs>0{s:supu∈sC1√nωTXu≤γs2√n} . (24)
In this section, we give a characterization of the ND interaction, which yields the final bound on the componentwise error as long as , i.e., the sample complexity is satisfied.
Let be a centered sub-Gaussian random vector, and its sub-Gaussian norm . Let be a row-wise i.i.d. sub-Gaussian random matrix, for each row , its sub-Gaussian norm . The ND interaction can be bounded by the following conclusion, and the proof of lemma 3 is given in Appendix D.1.
###### Lemma 3
Let design be a row-wise i.i.d. sub-Gaussian random matrix, and noise be a centered sub-Gaussian random vector. Then for some constant with probability at least . Constant depends on and .
In lemma 3 and theorem 6, we need the Gaussian width of and respectively. From definition, both and is related to the union of different cones; therefore bounding the width of and may be difficult. We have the following bound of and in terms of the width of the component spherical caps. The proof of Lemma 4 is given in Appendix D.2.
###### Lemma 4 (Gaussian width bound)
Let and be as defined in (7) and (8) respectively. Then, we have and .
By applying lemma 4, we can derive the error bound using the Gaussian width of individual error cone. From our conclusion on deterministic bound in theorem 1, we can choose an appropriate such that . Then, by combining the result of theorem 1, theorem 4, lemma 3 and lemma 4, we have the final form of the bound, as originally discussed in (3):
###### Theorem 6
For estimator (3), let , design be a random matrix with each row an independent copy of sub-Gaussian random vector , noise be a centered sub-Gaussian random vector, and be the centered unit euclidean ball. Suppose SC condition holds with
∥∥ ∥∥k∑i=1Δi∥∥ ∥∥2≥ρk∑i=1∥Δi∥2 .
for any ans a constant . If sample size , then with high probability,
k∑i=1∥^θi−θ∗i∥2≤Cmaxiw(Ci∩Bp)+√logkρ2√n, (25)
for constants that depend on sub-Gaussian norms and .
Thus, assuming the SC condition in (9) is satisfied, the sample complexity and error bound of the estimator depends on the largest Gaussian width, rather than the sum of Gaussian widths. The result can be viewed as a direct generalization of existing results for , when the SC condition is always satisfied, and the sample complexity and error is given by and [3, 10].
## 6 Accelerated Proximal Algorithm
In this section, we propose a general purpose algorithm for solving problem (2). For convenience, with , we set and . While the norms may be non-smooth, one can design a general algorithm as long as the proximal operators for each set can be efficiently computed. The algorithm is simply the proximal gradient method [23], where each component is cyclically updated in each iteration (see Algorithm 1):
~θt+1i=argminθi∈Ωi ⟨∇θif(θt),θi−θti⟩+12ηt+1∥θi−θti∥22 ,=argminθi∈Ωi ∥θi−(θti−ηt+1∇θif(θt))∥22 , (26)
where is the learning rate. To determine a proper , we use a backtracking step [4]. Starting from a constant , in each step we first update ; then we decide whether satisfies condition:
f(~θt+1)≤f(θt)+∇Tf(θt)(~θt+1−θt)+12ηt+1(k∑i=1∥~θt+1i−θti∥22). (27)
If the condition (27) does not hold, then we decrease till (27) is satisfied. Based on existing results [4], the basic method can be accelerated by setting the starting point of the next iteration as a proper combination of and . By [4], one can use the updates:
θt+1i=~θt+1i+αt−1αt+1(~θt+1i−θti) ,whereαt+1=1+√1+4α2t2 . (28)
Convergence of Algorithm 1 has been studied in [4]. The backtracking step ensures that the convergence of algorithm 1. The work [4] also give the convergence rate of Algorithm 1, which is . Therefore, we can always reach a stationary point of problem (2) using Algorithm 1.
## 7 Noiseless Case: Comparing Estimators
In this section, we present a comparative analysis of estimator
min{θi}k∑i=1λiRi(θi)s.t.Xk∑i=1θi=y (29)
with the proposed estimator (2) in the noiseless case, i.e., . In essence, we show that the two estimators have similar recovery conditions, but the existing estimator (29) needs additional structure for unique decomposition of into the components .
The estimator (29) needs to consider the so-called “infimal convolution” [25, 32] over different norms to get a (unique) decomposition of in terms of the components . Denote
R(θ)=min{θi}:∑iθi=θk∑i=1λiRi(θi) . (30)
Results in [25] show that (30) is also a norm. Thus estimator (29) can be rewritten as
minθR(θ) s.t. Xθ=y. (31)
Interestingly, the above discussion separates the estimation problem in into two parts—solving (31) to get , and then solving (30) to get the components . The problem (31) is a simple structured recovery problem, and is well studied [10, 28]. Using infimal convolution based decomposition problem (30) to get the components will be our focus in the sequel.
To get some properties of decomposition (30), we consider the unit norm balls for norm and component norms :
ΩR={θ∈Rp:R(θ)≤1}andΩiR={θi∈Rp:Ri(θi)≤1} ,i=1,…,k .
The norm balls are related by the following result, we give the proof in appendix F.1.
###### Lemma 5
For a given set , the infimal convolution norm ball is the convex hull of , i.e., .
Lemma 5 illustrates what the decomposition (30) should be like. If is a point on the surface of the norm ball , then the value of is the convex combination of some on the surface of such that . Hence if can be successfully decomposed into different components along the direction of , then we should be able to connect and by a surface on the norm ball, or they have to be “close”. Interestingly, the above intuition of “closeness” between different components can be described in the language of cones, in a way similar to the structural coherence property discussed in Section 3.
Given the intuition above, we state the main result in this section below. Its proof is given in appendix F.2.
###### Theorem 7
Given and define
C0=⎧⎨⎩∑θi≠0(c′ici−1)θi | c′i≥0,k∑i=1c′i=1⎫⎬⎭. (32)
Suppose , then there exist such that are unique solutions of (30) if and only if there are with and such that for the corresponding error cone of and defined above, for .
Theorem 7 illustrate that the successful decomposition of (30) requires an additional condition, i.e., beyond that is needed by the SC condition (see Section 3). The additional condition needs us to choose parameters properly. Theorem 7 shows that depends on both and . For appropriate , there may be a range of such that the solution is unique. Therefore, in noiseless situation, if we know , then solving estimator (29) would be a better idea, because it requires less condition to recover the true value and we do not need to choose parameters .
## 8 Related Work
Structured superposition models have been studied in recent literatures. Early work focus on the case when k=2 and noise , and assume specific structures such as sparse+sparse [14], and low-rank+sparse [11]. [16] analyze error bound for low-rank and sparse matrix decomposition with noise. Recent work have considered more generalized models and structures. [1] analyze the decomposition of a low-rank matrix plus another matrix with generalized structure. [15] propose an estimator for the decomposition of two generalize structured matrices, while one of them has a random rotation. Because of the increase in practical application and non-trivial of such problem, people have begun to work on unified frameworks for superposition model. In [31], the authors generalize the noiseless matrix decomposition problem to arbitrary number of superposition under random orthogonal measurement. [32] consider the superposition of structures of structures captured by decomposable norm, while [18] consider general norms but with a different measurement model, involving componentwise random rotations. These two papers are similar in spirit to our work, so we briefly discuss and differentiate our work from these papers.
[32] consider a general framework for superposition model, and give a high-probability bound for the following estimation problem:
minθi,i=1,…,k∥∥ ∥∥y−Xk∑i=1θi∥∥ ∥∥22+k∑i=1λiRi(θi) (33)
they assume each to be a special kind of norm called decomposable norm. the authors used a different approach for RE condition. They decompose into two parts. One is
1n∥XΔi∥22≥κ∥Δi∥22, (34)
which characterizes the restricted eigenvalue of each error cone. The other is
2n|∑i
which characterizes the interaction between different error cones. (35) is a strong assumption, and RE condition can hold without it. If and are positively correlated, then large interaction terms will make our RE condition stronger. Therefore their results are restricted.
[19] consider an estimator like (2), which is
minθi,i=1,…,k∥∥ ∥∥y−Xk∑i=1Qiθi∥∥ ∥∥22 s.t. Ri(θi)≤Ri(θ∗i), i=1,…,k, (36)
where are known random rotations. Problem (36) is then transformed into a geometric problem: whether random cones intersect. The componentwise random rotation can ensure that any kind of combination can be recovered with high probability. However, in practical problems, we need not have such random rotations available as part of the measurements. Further, their analysis is primarily focused on the noiseless case.
## 9 Application of General Bound
In this section, we instantiate the general error bounds on Morphological Component Analysis (MCA), and low-rank and sparse matrix decomposition. The proofs are provided in appendix E.
### 9.1 Morphological Component Analysis Using l1 Norm
In Morphological Component Analysis [14], we consider the following linear model
y=X(θ∗1+θ∗2)+ω,
where vector is sparse and vector is sparse under a rotation . In [14], the authors introduced a quantity
M=maxi,j|Qij|. (37)
For small enough , if the sum of their sparsity is lower than a constant related to , we can recovery them. We show that for two given sparse vectors, our SC condition is more general.
Consider the following estimator
minθ1,θ2∥y−X(θ1+θ2)∥22s.t.∥θ1∥1≤∥θ∗1∥1,∥Qθ2∥1≤∥Qθ∗2∥1, (38)
where vector is the observation, vectors are the parameters we want to estimate, matrix is a sub-Gaussian random design, matrix is orthogonal. We assume and are -sparse and -sparse vectors respectively. Function is still a norm.
Suppose , , and the i-th entry of and the j-th entry of are non-zero. If
Qij\rm sign(θ∗1)i\rm sign(Qθ∗2)>0,
then we have
ρ≥√(1−√1−Q2ij)/2. (39)
Thus we will have chance to separate and successfully. It is easy to see that is lower bounded by . Large leads to larger , but also leads to larger , which is better for separating and . The proof of above bound of is given in Appendix E.1.
In general, it is difficult for us to derive a lower bound of like 39. Instead, we can derive the following sufficient condition in terms of :
###### Theorem 8
If , then for problem (38) with high probability
∥θ1−θ∗1∥2+∥θ2−θ∗2∥2=O(max{√s1logpn,√s2logpn}).
When , this condition is much stronger than , because every entry of has to be smaller than ;
### 9.2 Morphological Component Analysis Using k-support Norm
-support norm [2] is another way to induce sparse solution instead of norm. Recent works [2, 12] have shown that -support norm has better statistical guarantee than norm. For arbitrary , its -support norm | 2019-11-19 07:20:49 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9577531814575195, "perplexity": 389.1397392133786}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670036.23/warc/CC-MAIN-20191119070311-20191119094311-00203.warc.gz"} |
http://codeforces.com/problemset/problem/859/D | D. Third Month Insanity
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output
The annual college sports-ball tournament is approaching, which for trademark reasons we'll refer to as Third Month Insanity. There are a total of 2N teams participating in the tournament, numbered from 1 to 2N. The tournament lasts N rounds, with each round eliminating half the teams. The first round consists of 2N - 1 games, numbered starting from 1. In game i, team i - 1 will play against team i. The loser is eliminated and the winner advances to the next round (there are no ties). Each subsequent round has half as many games as the previous round, and in game i the winner of the previous round's game i - 1 will play against the winner of the previous round's game i.
Every year the office has a pool to see who can create the best bracket. A bracket is a set of winner predictions for every game. For games in the first round you may predict either team to win, but for games in later rounds the winner you predict must also be predicted as a winner in the previous round. Note that the bracket is fully constructed before any games are actually played. Correct predictions in the first round are worth 1 point, and correct predictions in each subsequent round are worth twice as many points as the previous, so correct predictions in the final game are worth 2N - 1 points.
For every pair of teams in the league, you have estimated the probability of each team winning if they play against each other. Now you want to construct a bracket with the maximum possible expected score.
Input
Input will begin with a line containing N (2 ≤ N ≤ 6).
2N lines follow, each with 2N integers. The j-th column of the i-th row indicates the percentage chance that team i will defeat team j, unless i = j, in which case the value will be 0. It is guaranteed that the i-th column of the j-th row plus the j-th column of the i-th row will add to exactly 100.
Output
Print the maximum possible expected score over all possible brackets. Your answer must be correct to within an absolute or relative error of 10 - 9.
Examples
Input
20 40 100 10060 0 40 400 60 0 450 60 55 0
Output
1.75
Input
30 0 100 0 100 0 0 0100 0 100 0 0 0 100 1000 0 0 100 100 0 0 0100 100 0 0 0 0 100 1000 100 0 100 0 0 100 0100 100 100 100 100 0 0 0100 0 100 0 0 100 0 0100 0 100 0 100 100 100 0
Output
12
Input
20 21 41 2679 0 97 3359 3 0 9174 67 9 0
Output
3.141592
Note
In the first example, you should predict teams 1 and 4 to win in round 1, and team 1 to win in round 2. Recall that the winner you predict in round 2 must also be predicted as a winner in round 1. | 2018-07-20 08:27:47 | {"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": 0.22142457962036133, "perplexity": 380.329552653852}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591575.49/warc/CC-MAIN-20180720080634-20180720100634-00171.warc.gz"} |
https://book.stat385.org/iteration.html | This chapter was developed from scratch for the Fall 2022 semester. As such, you might notice a few extra typos, or some topics that are not well explained. If you encounter these issues, please let us know on the discussion forum. Except some additional changes to what is currently published while this warning persists.
In programming, iteration is the act of repeating a set of instructions. This can be done several different ways:
• Repeat until some condition is met.
• Repeat a specified number of times.
• Repeat for each element of a collection.
In R, the last example here, repeating each element of a collection, is by far the most common because vectors (collections of elements) are the most important data structure in R. As such, R has built-in functions that make this type of iteration extremely easy. While R does provide the usual iteration abilities through the use of for and while loops, these should not be your go-to methods for performing iteration with R.
After reading this chapter you should be able to:
• Use lapply and related functions to iterate over the elements of a vector.
• Use for and while loops to repeatedly evaluate R expressions.
• Avoid common pitfalls when using loops in R.
## 10.1 Apply Functions
One of the most common operations that you will encounter while programming with R is running a function with each element of some vector as input and then collecting the results in a vector.
There are many functions, built-in and otherwise, to accomplish this task. We will begin by looking at the most important and generic function, lapply.
### 10.1.1lapply
The function in R that performs the operation described above is the lapply function. The general syntax is:
lapply(X = some_list, FUN = f)
That is, some_list is a vector (atomic vector or list) that the function f will be “applied” to each element of. Note that it is customary to not name the arguments to lapply.
lapply(some_list, f)
lapply(1:3, log)
#> [[1]]
#> [1] 0
#>
#> [[2]]
#> [1] 0.6931472
#>
#> [[3]]
#> [1] 1.098612
Here we see the log function applied to each of the elements of the vector 1:3. This would be the same as running the following:
list(
log(1),
log(2),
log(3)
)
#> [[1]]
#> [1] 0
#>
#> [[2]]
#> [1] 0.6931472
#>
#> [[3]]
#> [1] 1.098612
Clearly, this isn’t a particularly useful example, as we could simply do the following:1
log(1:3)
#> [1] 0.0000000 0.6931472 1.0986123
Although, note that lapply is returning a list, but the above returns an atomic vector. More on that in a moment.
For now, know that lapply will return a list that has the same length as the input vector.2
Let’s look at an example of iterating over a list.
set.seed(42)
ex_list = list(a = runif(5),
b = runif(5),
c = runif(5))
ex_list
#> $a #> [1] 0.9148060 0.9370754 0.2861395 0.8304476 0.6417455 #> #>$b
#> [1] 0.5190959 0.7365883 0.1346666 0.6569923 0.7050648
#>
#> $c #> [1] 0.4577418 0.7191123 0.9346722 0.2554288 0.4622928 lapply(ex_list, max) #>$a
#> [1] 0.9370754
#>
#> $b #> [1] 0.7365883 #> #>$c
#> [1] 0.9346722
Again, here the input was a list of length three, so the output is as well. You might wish the output was an atomic vector. Again, more on that soon.
lapply(ex_list, range)
#> $a #> [1] 0.2861395 0.9370754 #> #>$b
#> [1] 0.1346666 0.7365883
#>
#> $c #> [1] 0.2554288 0.9346722 Finally, a slightly more useful example. This returns the same object as the following: list( range(ex_list[[1]]), range(ex_list[[2]]), range(ex_list[[3]]) ) #> [[1]] #> [1] 0.2861395 0.9370754 #> #> [[2]] #> [1] 0.1346666 0.7365883 #> #> [[3]] #> [1] 0.2554288 0.9346722 Hopefully, it is becoming clear that lapply can be used to write concise, useful, and readable code. What if we want to use a function with more than one argument? For example: multiply_and_power = function(x, c, p) { c * x ^ p } multiply_and_power(x = 2, c = 3, p = 0.5) #> [1] 4.242641 multiply_and_power(x = 2, c = 1:3, p = 0.5) #> [1] 1.414214 2.828427 4.242641 Be aware that depending on how we specify the values we pass to the arguments, there is likely going to be some length coercion taking place. To use this function together with lapply, we simply add the values of the additional parameters as arguments to lapply.3 lapply(1:3, multiply_and_power, c = 1:5, p = 2) #> [[1]] #> [1] 1 2 3 4 5 #> #> [[2]] #> [1] 4 8 12 16 20 #> #> [[3]] #> [1] 9 18 27 36 45 What did this code do? list( multiply_and_power(x = 1, c = 1:5, p = 2), multiply_and_power(x = 2, c = 1:5, p = 2), multiply_and_power(x = 3, c = 1:5, p = 2) ) #> [[1]] #> [1] 1 2 3 4 5 #> #> [[2]] #> [1] 4 8 12 16 20 #> #> [[3]] #> [1] 9 18 27 36 45 What if we wanted to iterate over a different argument, say c instead of x? Specify x and p in the call to lapply. Now lapply will iterate over c. lapply(1:3, multiply_and_power, x = 1:5, p = 2) #> [[1]] #> [1] 1 4 9 16 25 #> #> [[2]] #> [1] 2 8 18 32 50 #> #> [[3]] #> [1] 3 12 27 48 75 So, this time, we did the following: list( multiply_and_power(x = 1:5, c = 1, p = 2), multiply_and_power(x = 1:5, c = 2, p = 2), multiply_and_power(x = 1:5, c = 3, p = 2) ) #> [[1]] #> [1] 1 4 9 16 25 #> #> [[2]] #> [1] 2 8 18 32 50 #> #> [[3]] #> [1] 3 12 27 48 75 Sure, you could simply use this instead, but imagine needed to iterate over 1:100000 instead of 1:5. ### 10.1.2sapply Let’s return to the example that found the maximum of each element of a list. set.seed(42) ex_list = list(a = runif(5), b = runif(5), c = runif(5)) lapply(ex_list, max) #>$a
#> [1] 0.9370754
#>
#> $b #> [1] 0.7365883 #> #>$c
#> [1] 0.9346722
As expected, the result is a list. However, notice that each element of said list is an atomic vector of length one, of the same type. We could actually check that using lapply.
lapply(lapply(ex_list, max), typeof)
#> $a #> [1] "double" #> #>$b
#> [1] "double"
#>
#> $c #> [1] "double" lapply(lapply(ex_list, max), length) #>$a
#> [1] 1
#>
#> $b #> [1] 1 #> #>$c
#> [1] 1
It probably seems like what we really want as output here is an atomic vector that is the same length as the input vector. We can obtain this result by switching from lapply to sapply.
sapply(ex_list, max)
#> a b c
#> 0.9370754 0.7365883 0.9346722
The s in sapply refers to the simplifying action taken by the function. Much of the details of how the simplification works follow the usual rules of the coercion hierarchy. It is probably best not to worry too much about these rules, but also not rely on simplification too much. Generally, it is best to use sapply in the case we’ve just seen here: you are certain the result of the function applied to each element is an atomic vector of length one, each with the same type.
Another example:
sapply(1:3, log)
#> [1] 0.0000000 0.6931472 1.0986123
But again, this example isn’t truly necessary, as the following is even better:
log(1:3)
#> [1] 0.0000000 0.6931472 1.0986123
We show this to demonstrate that many operations in R are already vectorized, so there is no need to iterate.
### 10.1.3 Other Apply Functions
Other apply functions exist. Many are rarely used. One that might be of interest is vapply which will do simplification like sapply, but the user will need to specify the expected outcome of each iteration, which will make the simplification more predictable.
vapply(1:3, log, double(1))
#> [1] 0.0000000 0.6931472 1.0986123
vapply(1:3, log, integer(1))
#> Error in vapply(1:3, log, integer(1)): values must be type 'integer',
#> but FUN(X[[1]]) result is type 'double'
Another that you will likely see is the apply function. We would advise avoiding this unless you truly understand what it does. Also, beware, it should probably not be used with data frames.4
## 10.2 Loops
Loops are another form of control flow. They allow you to explicitly specify the repetition of some code, in contrast to the apply functions above that did so implicitly.5
Welcome to R Club.
• The first rule of R Club is: Do not use for loops!
• The second rule of R Club is: Do not use for loops!
• And the third and final rule: If you have to use a for loop, do not grow vectors!
— Unknown
Loops are very common in programming, however, in R, it is probably best to avoid them unless you truly need them. The general heuristic you should use to determine if you need a loop or apply function is:
• Use a loop when the result of the next iteration depends on the result of the previous iteration.
• Use an apply function when the results of each iteration are independent.6
### 10.2.1for
The most common looping structure is a for loop. The generic syntax is:
for (element in vector) {
code_to_run
}
We’ll refer to element as the loop variable.
Let’s look at a specific example.
# pre-allocate storage vector
x = double(length = 5)
# perform loop
for (i in 1:5) {
x[[i]] = i ^ 2
}
# check results
x
#> [1] 1 4 9 16 25
First, note that for is not a function, which is why you should consider placing a space between it and the parenthesis that follows. Next, (i in 1:5) is considered the header of the loop which defines how the iteration will take place. Here the name of the loop variable is i and it will take a value from the vector 1:5 each time the body runs. The code inside the braces, {} is called the body of the loop, much like the body of a function.
• Each time through the loop, i, will take one of the values from 1:5. Or generally, the loop variable will take the value of each element of some vector.
• For each value of i, the code x[i] = i ^ 2 will run. In general, for each value of the looping variable, the code in the body will run. And often, that code will depend on the looping variable, like we see here.
So, the above for loop ran each of the following:
x[[1]] = 1 ^ 2
x[[2]] = 2 ^ 2
x[[3]] = 3 ^ 2
x[[4]] = 4 ^ 2
x[[5]] = 5 ^ 2
This should make it clear that the purpose of a loop is to repeat code, without actually having to repeatedly type the code.
As has become a theme, this for loop is truly useless in R. We could have simply done:
(1:5) ^ 2
#> [1] 1 4 9 16 25
Here, i is functioning much like the name of a function argument, except now, we pass a new value, an element of 1:5, each time through the loop.
You can use any name you want for the loop variable, but i, j, and k are most common.
for (some_long_var_name in 1:5) {
print(some_long_var_name)
}
#> [1] 1
#> [1] 2
#> [1] 3
#> [1] 4
#> [1] 5
A for loop is a very powerful structure, so it will not be possible for us to illustrate all possible usage examples. Let’s look at a correct loop written poorly, then the same loop written better, and try to draw some conclusion about best practices with for loops.
Before proceeding, let’s introduce the seq_along function.
seq_along(5:1)
#> [1] 1 2 3 4 5
Essentially, seq_along returns the indexes of a vector. Or, you could think of it as returning the result of the following:
1:length(5:1)
#> [1] 1 2 3 4 5
Let’s use a for loop to create a sequence of numbers. The first two numbers will be 10, and 5. Elements after that will be calculated as:
$x_i = 3 \cdot \frac{x_{i - 1}}{x_{i - 2}}$
We’ll use a loop to create a sequence of length ten that follows this specification.
First, a bad example of how to write a loop to accomplish this:
# perform loop
for (i in 1:10) {
if (i == 1) {
x = 10
} else if (i == 2) {
x = c(x, 5)
} else {
x = c(x, 3 * x[i - 1] / x[i - 2])
}
}
# check results
x
#> [1] 10.0 5.0 1.5 0.9 1.8 6.0 10.0 5.0 1.5 0.9
We see the correct resulting vector, x, but we have used multiple sub-optimal techniques. In particular, we “grew” the x vector. The use of x = c(x, some_new_element) takes what was x, then creates a new x but combining the previous x with some new element. Do not do this. This is one of the reasons people incorrectly think R is slow. This operation is slow, but there is no need for it.
Instead, let’s pre-allocate the x which we will store our results in.
# pre-allocate x to be a double vector of the correct length
x = double(10)
# perform loop
for (i in seq_along(x)) {
if (i == 1) {
x[[i]] = 10
} else if (i == 2) {
x[[i]] = 5
} else {
x[[i]] = 3 * x[i - 1] / x[i - 2]
}
}
# check results
x
#> [1] 10.0 5.0 1.5 0.9 1.8 6.0 10.0 5.0 1.5 0.9
This time, since x already existed, we are simply replacing individual elements of an already existing vector. This is faster. Any time you grow or add new elements (that is you increase the length of a vector) there is a copy operation taking place under the hood that you could have avoided.
Also, by pre-allocating x, we can now use seq_along(x). In some applications we might be creating x with a program, and we wouldn’t know its length ahead of time! This will avoid having to specify the length in two locations in code.
Some general ideas to keep in mind:
1. Do not attempt to iterate over and store results in the same vector.
2. Pre-allocate a “results” vector and update individual elements as you progress through the loop. Do not grow vectors.
3. Use seq_along and iterate over indexes rather than elements of a vector.
We’ve already discussed why the second item is a problem. Let’s now create an example that demonstrate items one and three.
The following function will check if an number is even.
is_even = function(x) {
x %% 2 == 0
}
We also create a vector y that stores some numbers.
# create data
set.seed(42)
y = sample(1:10, size = 20, replace = TRUE)
# view data
y
#> [1] 1 5 1 9 10 4 2 10 1 8 7 4 9 5 4 10 2 3 9 9
Our goal is to create a logical vector, the same length as y, containing TRUE at any index where y is even.
This will, not work:
for (i in y) {
y[[i]] = is_even(i)
}
To better see the issue, temporarily place a print() statement inside the loop.7
# create data
set.seed(42)
y = sample(1:10, size = 20, replace = TRUE)
# perform loop
for (i in y) {
print(i)
y[[i]] = is_even(i)
}
#> [1] 1
#> [1] 5
#> [1] 1
#> [1] 9
#> [1] 10
#> [1] 4
#> [1] 2
#> [1] 10
#> [1] 1
#> [1] 8
#> [1] 7
#> [1] 4
#> [1] 9
#> [1] 5
#> [1] 4
#> [1] 10
#> [1] 2
#> [1] 3
#> [1] 9
#> [1] 9
# check results
y
#> [1] 0 1 0 1 0 4 0 1 0 1 7 4 9 5 4 10 2 3 9 9
So i takes values from y, but by doing so, we don’t have access to the indexes at which we need to replace with the result of is_even. Let’s use seq_along.
# create data
set.seed(42)
y = sample(1:10, size = 20, replace = TRUE)
# perform loop
for (i in seq_along(y)) {
y[[i]] = is_even(y[i])
}
# check results
y
#> [1] 0 0 0 0 1 1 1 1 0 1 0 1 0 0 1 1 1 0 0 0
Note that inside of i, we now need to change i to y[i] to get the value rather than the index each time through the loop.
But there’s still an issue! We have 0 and 1 instead of FALSE and TRUE. Coercion!
# create data
set.seed(42)
y = sample(1:10, size = 20, replace = TRUE)
# pre-allocate storage vector
res = logical(length(y))
# perform loop
for (i in seq_along(y)) {
res[[i]] = is_even(y[[i]])
}
# check results
res
#> [1] FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE FALSE TRUE FALSE TRUE
#> [13] FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE
Much better. But again, remember, many things in R are vectorized:
is_even(y)
#> [1] FALSE FALSE FALSE FALSE TRUE TRUE TRUE TRUE FALSE TRUE FALSE TRUE
#> [13] FALSE FALSE TRUE TRUE TRUE FALSE FALSE FALSE
This example did not need a loop, because results from one iteration to the next were independent. In the previous example, this was not the case, and was an example of when you truly need a loop.
Note that these examples have used atomic vectors, but, no reason we couldn’t use a list!
### 10.2.2while
A while loop will repeat code until a specified condition is no longer met. The general syntax is:
while (condition) {
code_to_run
}
Let’s see an example.
# create some data
x = 5
# pre-allocate storage vector
y = double(length = length(x))
# perform loop
while (x > 0) {
print(x)
y[[x]] = x ^ 2
x = x - 1
}
#> [1] 5
#> [1] 4
#> [1] 3
#> [1] 2
#> [1] 1
# check results
x
#> [1] 0
y
#> [1] 1 4 9 16 25
Here, the loop runs until x is no longer greater than 0.
Notice that if we don’t modify x inside the loop, it would run forever! An infinite loop!8
You will likely see for loops more often, but while loops are useful when you don’t know how many iterations you’ll need ahead of time, but you can describe a stopping condition.
x = 1 # setup initial data
y = 0 # setup result vector
# perform loop
while(x > .Machine$double.eps) { y = y + x x = x / 2 } # check results y This example demonstrates a method to numerically evaluate the following sum.9 $\sum_{k = 0}^{\infty} \left(\frac{1}{2}\right) ^ k$ Because we cannot actually sum up an infinite number of terms, as that would take forever, we instead sum up all terms that are indistinguishable from zero to the computer. In this case, .Machine$double.eps gives us the smallest possible number that R can recognize on the machine that processed this chapter. Because x will only become smaller as the loop continues, we know that once the loop stops, all future terms would have also been indistinguishable from zero.
.Machine\$double.eps
#> [1] 2.220446e-16
### 10.2.3repeat
A repeat loop will continually repeat an expression, without ever stopping.
repeat {
42
}
The above is not run, because it would never stop! It could have also been written using a while loop:
while (TRUE) {
42
}
Because there is no built-in stopping rule when using repeat, it is not a feature we will return to often. If necessary, in order to exit a repeat loop, the break expression may be used.
x = 42
repeat {
print(x)
if (x < 1) {
break
}
x = sqrt(x) - 1
}
#> [1] 42
#> [1] 5.480741
#> [1] 1.341098
#> [1] 0.1580579
## 10.3 Summary
• TODO: You’ve learned to…
## 10.4 What’s Next?
• TODO: ?
1. This example is easier to write, easier to read, and because of vectorization, much faster.↩︎
2. Think l for list. Although, it is unclear if that is the etymology of the name of the lapply function.↩︎
3. If you check the documentation for lapply, you’ll notice an argument called .... More on this later, but this is what allows R to pass these additional arguments to the function.↩︎
4. The apply function is useful when working with matrix objects, which so far we have been avoiding.↩︎
5. Technically, the apply functions could be said to be “hiding loops” as they are mostly just convenience functions wrapped around for loops.↩︎
6. Also check that you can’t just use a vectorized operation.↩︎
7. This is a simple and naive, but incredibly powerful debugging practice.↩︎
8. If you experience an infinite loop, use Ctrl + C in the console to escape it. Or press the stop button in RStudio.↩︎
9. Observant readers will recognize this sum as a geometric series and note that there is an analytical solution.↩︎ | 2022-10-02 15:49:49 | {"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": 2, "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": 0.36771976947784424, "perplexity": 1464.3399796830208}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337338.11/warc/CC-MAIN-20221002150039-20221002180039-00782.warc.gz"} |
https://www.physicsforums.com/threads/use-lagrange-multipliers-to-find-the-shortest-distance.379837/ | # Use lagrange multipliers to find the shortest distance
## Homework Statement
Use lagrange multipliers to find the shortest distance between a point on the elliptic paraboloid z=x^2 +y^2
## The Attempt at a Solution
http://img716.imageshack.us/img716/7272/cci1902201000000.jpg [Broken]
I'm not that good with using the equation editor, so I scanned my work.
I'm stuck on the last part where i'm trying to factor the equation to find a solution for $$\lambda$$, I cant seem to find a solution that would make the equation zero, which is what i need in order to do the long division to factor that equation.
Last edited by a moderator: | 2021-10-23 08:19:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8715779781341553, "perplexity": 233.93811361721643}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585653.49/warc/CC-MAIN-20211023064718-20211023094718-00205.warc.gz"} |
https://gianluca--gianlucabaio.netlify.app/post/2014-06-27-the-oracle-6/ | The Oracle (6)
Quick update, now that the group stage is finished. We needed a few tweaks to the simulation process (described in some more details here), which we spent some time debating and implementing.
First off, the data on the last World Cups show that during the knock out stage, there are substantially fewer goals scored. This makes sense: from tomorrow it’s make or break. This wasn’t too difficult to deal with, though $-$ we just needed to modify the distribution for the zero component of the number of goals ($\pi$, as described here). In this case, we’ve used a distribution centered on around 12% with most of the mass concentrated between 8% and 15%.
These are the predictions for the 8 games. Brazil, Germany, France and (only marginally) Argentina have a probability of winning exceeding 50%. The other games look closer.
Technically, there is a second issue, which is of course that in the knock out stage draws can’t really happen $-$ eventually game ends either after extra time, or at penalties. For now, we’ll just use this prediction, but I’m trying to think of a reasonable way to include the extra complication in the model; the main difficulty is that in extra time the propensity to score drops even further $-$ about 30% of the games that go to extra time end up at penalties. I’ll try and update this (if not for the this round, possibly for the next one). | 2022-08-16 05:08:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7123439908027649, "perplexity": 701.9100053736511}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572220.19/warc/CC-MAIN-20220816030218-20220816060218-00018.warc.gz"} |
https://happyagain4dogs.com/1x91ohr/longitude-and-time-sums-with-answers-e62138 | The Earth makes one complete rotation of 360 degrees in 24 hours. Does The Inertia Of A Body Depend Upon Its Energy-content Pdf, These are imaginary lines that run from the North Pole to the South Pole. The time difference is to be added in case of places to the east of a point. 1 degree= 4 mins ( 360 degree/ 24 hrs), 15 degrees= 1hr. The extent of relief features is correct. Holiday Inn Express Jobs, Enter coordinates to find a place. It would look like a circle. Answer: (d) Longitude
(b) 15 minutes Question 1. Provide details and share your research! Which is the longest line of latitude ? Question 4. Flashlight Song, Figure 1: Globe depicting the most important lines of latitude and longitude. Answers for "Time Sum problems" Put the first three functional lines (up through the determination of 'z') into a separate formula, then Sum() that. The question as stated has no answer. After they set, they glide along just below the northern horizon before rising very gradually, not sharply the way, say, Rigel and Betelgeuse come up. Answer: According to the Indian Standard Time 7.30 p.m. based on 82 1/2° E. Longitude was the time at Mumbai.
How To Get SSL Certificate? Basic-fit Management Team, Install Gdata, Question 18. Neat work will be appreciated. The midâday sun never shines overhead on any latitude beyond the tropics. This is known as Greenwich Mean Time (GMT). (By the way, from here on out, we'll write the term 'degree' just as 'deg' b… (c) 3 hours Name the two reference lines with respect to which the distances of various places on the earth’s surface are measured ? Election Jobs Near Me, How Old Do You Have To Be To Go To The Gym Australia, The Eastern places will be ahead of Western places for local time. Tony Gilroy Rogue One, Questions & Answers: Central Home Page (5) Latitude and Longitude Index 4. As a result, the places between tropics to Arctic and to Antarctic Circle have moderate temperature. An important pr Statera Coinmarketcap, Which of the following zone is situated between 66°30′ S and poles? Objective Reality Vs Subjective Reality, Poles: B. Equator: C. Tropic of Cancer: D. Tropic of Capricorn View Answer Workspace Report Discuss in Forum. i don't understand how it helped them sail and like why it was important for them to know the difference in time at their original port and wherever they are at a specific time. Q. If it is 3:00am where you are and 7:30am GMT, what is your longitude? If it is 5:00pm where you are and 7:30pm GMT, what is your longitude? The Round Earth 8a. Early Voting Chatham County Ga 2020, With reference to the International Date Line, state the following: (a) It is meaning and application. Latitudes and Longitudes are imaginary lines used to determine the location of a place on earth. EXPLANATION: 1 degree= 4 mins ( 360 degree/ 24 hrs), 15 degrees= 1hr here the difference is 3hrs so it would be 45 degrees. Enter coordinates to find a place. LATITUDES AND DEPARTURES: Background. therefore 60+ 45 degrees= 105 degress east longitude. Through this hunt for longitude locations, students gain a greater understanding about the Earth’s daily and seasonal cycles. Displaying top 8 worksheets found for - Geography Latitude And Longitude. Cartographers and geographers divide the Earth into longitudes and latitudes in order to locate points on the globe.These are imaginary lines forming a grid and it helps in finding the location of a place. Ariana Grande Carpool Karaoke, i) If you look in the globe, you will see that longitudes are closer together as you move towards north and south of Equator. Answer ALL the questions. Mid-days Sun can be seen overhead in Chennai twice a year, but not even once in Delhi. Log in Ask Question. Dw Membership Prices, Answer: The longitudinal difference is 92° – 73° = 19°. 2011-07-19 06:34:24 2011-07-19 06:34:24. to tell time,we can use time telling machine. Required fields are marked *. Here once in the year (December 22nd) the day is of 24 hours duration and once in year (June 21) the night is of 24 hours duration. The Calendar 6a. Days and nights are equal throughout the globe when the sun is above: A. List Of Villages In New Jersey, Question 4. The time of day depends on where you are on the Earth. Solution : Difference between Greenwich and New Orleans = 90° of longitudes Total T ime dif ference = 90 × 4 = 360 minutes = 360/60 hours = 6 hours\Local time of New Orleans is 6 hours less than that at Greenwich, i.e. (c) Both (a) and (b) It lies at 66$$\frac { 1 }{ 2 }$$° north of the equator. The time at place ‘a’ is 1 hour less because it is to the west of ‘b’ and time at ‘c’ is 1 hour more because it is to the east of ‘b’. What is the relation between Temperature and Latitude of a place ? ... Uncategorized. State the importance of a globe. When Francis Drake returned to England after circumnavigating the globe, he thought it was Saturday, where as actually it was Sunday. Cubby Storage, The lines of longitude are the great semi-circles joining North pole and South pole and are equal in length. it will cover 88° from west to 0° and 50° 45 east from 0° or Greenwich line. Answer: As the circumference or the equator is nearly about 40,000 km. Latitude and longitude are a coordinates system that let anyone determine the location of a single point on any part of the earth’s surface. Answer: Longitude of Madras = 80° E Longitude of New York = 74° W Difference in Degrees = 80° + 74° = 154° Difference in time per degree = 4 minutes Difference between the times of two places = 40 x 154 = 616 = 10 hour 16 minutes Since Madras lies East of New York, we shall add 10 hours 16 minutes to the local time of New York. The Gym Plus Shop, All lines of longitude are semicircles of equal length. The observer then notices that the sun is highest in the sky at 4:00 according his watch. Longitude and Time Worksheet - Free download as Word Doc (.doc / .docx), PDF File (.pdf), Text File (.txt) or read online for free. Question 2. TopperLearning’s Experts and Students has answered all of Globe Latitudes And Longitudes of CBSE Class 6 Geography questions in detail. And the location of a place on the earth can be mentioned in terms of latitudes and longitudes. Wiki User Answered . well, the answer for this would be 105 degree east. Basil Hailed reached Manila after circumnavigating the globe, he thought it was Monday; where as actually it was Sunday. An important pr Describe the lines of latitude, their importance and use. Answer: The sun does not shine with the same intensity over all parts of the world at a particular time. Guajira Venezuela Peces, Bbsw Publication, A globe is very useful model to display the actual shape of the earth with its tilted axis ; The rotation and revolution of the earth can be very clearly shown by it along with the continents and oceans. Sum of all three digit numbers divisible by 7. Example. What would it look like? Actually, latitude and longitude are both imaginary lines drawn on a map or a globe in order to locate the position of a place or a region on the earth’s surface. Answer: The temperature decreases with latitude. It is based on the local time at the Prime Meridian in Greenwich, England (at zero degrees longitude). When he returns to the place from where he started he will appear to have gained a day. On the other hand, a person going from east to west will have his/her watch moved backwards by 4 minute at each meridian. The following are the major features of meridians: Question 4. Time at given place = 12 pm. (d) The intersection of latitude and longitude points out the exact position of a place on the earth’s surface. On the other hand, standard time is the same for a particular country. Historical Figures That Start With O, to tell time,we can use time telling machine. The Moon (2) 4b. The reason was that he had traveled from west to east and had over calculated a day. It is 10:00 AM on Monday at 120 degrees west longitude. longitude and time sums with answers. This example counts the number of values in the range A1:A10 that fall between 1 and 10, inclusively. Question 6. On your computer, open Google Maps. geography two question and answers Media Publishing eBook, ePub, Kindle PDF View ID 63437ebf4 May 25, 2020 By Robin Cook geography questions they need to answer quickly and to … Example: +40.689060 -74.044636 Note: Your entry should not have any embedded spaces. San Francisco Department Of Elections, A comprehensive database of more than 10 latitude and longitude quizzes online, test your knowledge with latitude and longitude quiz questions. therefore the time at long 60degress is 4:00pm in the afternoon Netgear Nighthawk M2 Mobile Router 5g, Answer: Equator is the longest parallel on the Earth. As the sun rises in the … Samas in Hindi | समास परिभाषा व भेद और उदाहरण – हिन्दी व्याकरण, Sandhi in Hindi | संधि की परिभाषा, भेद और उदाहरण – हिन्दी व्याकरण, Ras in Hindi | रस के परिभाषा, भेद और उदाहरण – हिन्दी व्याकरण, Karak in Hindi | कारक परिभाषा, भेद और उदाहरण – हिन्दी व्याकरण, Visheshan in Hindi | विशेषण शब्द (Shabd), भेद (Bhed), प्रकार और उदाहरण – हिन्दी व्याकरण, Medical Fitness Certificate | Samples, Guidelines, Contents and Templates, Pratyay in Hindi प्रत्यय | प्रत्यय परिभाषा, भेद और उदाहरण – हिन्दी व्याकरण. System Of Differential Equations Calculator, (c) Latitude In Northern Hemisphere, it bends to the West of 180° while in the Southern Hemisphere it bends Eastwards. Asked by Bittu | 1st Apr, 2017, 03:09: PM. Quiz on Time Zones ; No. The Earth moves on its axis from West to East. Add your answer and earn points. 46. Finding your local time is easy even if you haven’t got a clock: you simply observe the position of the Sun. It passes through 15 degrees is one hour or one degree in four minutes. Besides longitude and latitude, you can use plus codes to share a place without an address. State its value in degrees. What does the Earth look like? These are circles drawn round the earth, parallel to the equator. Latitude & Longitude, Geography Topics - Seventh 7th Grade Social Studies Standards, ... Label the Latitude and Longitude - a world map from Enchanted Learning answers ; Latitude and Longitude Printout Glossary - to be used with the world map above. Musical Theatre Songs About Magic, Answer: This distance is measured in degrees. We also compile various competative examination materials AIPMT, AIEEE, CA CPT, CTET, NDA, NEET, SCRA, IIT JEE. However, CBSEGuess and Dreamzsop does not take the responsibility of how it is used or the the consequence of its use. 75° would be the number of degrees of longitude. When Is Cameron Boyce Birthday, Lines of longitude are also called Meridians of longitude? NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12, NCERT Extra Questions for Class 7 Social Science Geography Chapter 2 Globe Latitudes and Longitudes. An hour is one twenty-fourth a day, so the Earth is divided into 24 standard time zones, each covering approximately 15 degrees of longitude. Question 14. So the equator and the longitudinal lines around the earth are called the Great Circles. (b) The lines of latitude are drawn at an angular distance with respect to the equator. Question 6. Expert Answer: Time at Greenwich = 7:30 pm. If it is 5:00pm where you are and 7:30pm GMT, what is your longitude? Ghost Infestissumam Album, Canada spreads across about 90 degrees of longitude and has six time zones and each zone has its own standard time. L.C.M method to solve time and work problems. My Missus Or Misses, 7th Grade Geography On the other hand if a man travels from west to east, for every degree of longitude covered he will put forward his watch by four minutes and for 360° he will forward it 24 hrs. $\endgroup$ – Envite Jan 20 '14 at 0:21 Sam Zacharia Instagram, Answer: Answer: Longitudinal value of Indian Standard Meridian is 82 1/2° E. It passes midway through India nearly along the city of Allahabad. One set of lines runs north and south of Equator and Parallel to it. This is known as Indian Standard Time (IST). Tk Fitness, Click to share on Twitter (Opens in new window), Click to share on Facebook (Opens in new window), South Carolina Presidential Election 2012, Enrolled Nurse Graduate Programs 2020 Victoria. First thing that you need to ascertain is, the longitude on which you are standing is westwards or eastwards from the first Meridian i.e. Does The Inertia Of A Body Depend Upon Its Energy-content Pdf, How Old Do You Have To Be To Go To The Gym Australia, All You Need To Know About Cryptocurrency Pdf, The Origin Of Consciousness In The Breakdown Of The Bicameral Mind Audiobook, Planescape: Torment Walkthrough Gamebanshee, Winnie The Pooh Springtime With Roo Cornel, The Process That Provides Citizens A Way Forward In The Face Of Legislative Inaction Is A(n), Premier League Goalkeeper Save Percentage 2018/19, System Of Differential Equations Calculator, Falaknuma Das Movie Online Watch Movierulz. Webroot Storage, Question 4. 10K views. Describe important parallels of latitudes. If it is 10:00pm where you are and 3:30pm GMT, what is your longitude? 1. Home Science Math History Literature Technology Health Law Business All Topics Random. Further north still, the Big Dipper never does break the plane of the horizon at 40 degrees North latitude, and for a time rides just above the horizon and almost parallel to it. Time is measured by the movement of the Earth. An important programme was to be broadcast from Mumbai at 7.30 p.m. What is the relation between Temperature and Latitude of a place ? 15degrees=1hr . Question 3. Remainder when 2 power 256 is divided by 17. Answers is the place to go to get the answers you need and to ask the questions you want. So if we have the same time, then 12 noon will mean midday at one place, and midnight at another. 1) At first filter your SQLite data with a good approximation and decrease amount of data that you need to evaluate in your java code. Therefore, if you know your local time and the time at Greenwich, you can use the difference to work out your longitude. What is the relation between longitude and time ? The Moon (1) 4a. Latitudes and longitudes are expressed in degrees. All lines of longitude are semicircles of equal length. Hobson General Relativity Solutions, Use: + for N Lat or E Long -for S Lat or W Long. The Earth moves on its axis from West to East. Therefore, an hour would be 15 degrees longitude (360/24). This is due to the fact that earth rotates from west to east. To understand how the lines are chosen, think of looking at a picture of the earth that's been sliced evenly across. Isle Of Skye Gallery, Time is measured by the movement of the Earth. (c) What is the difference between Prime Meridian and other meridians of longitude ? Vline Head Office, Every 15 degrees of longitude equals 1 hour. Please help by giving me a scoring and easy method. therefore 60+ 45 degrees= 105 degress east longitude. Mark Johnson Hockey, Pierce County Wi Cities, Math and Arithmetic. You'll answer questions about the earth's rotation and the prime meridian. Question 1. What time is it at 60 degrees west? The latitude 23° 26′ North is also known as the Tropic of Cancer. 0: B. The position of Birmingham is 5° W. Calculate the time the viewers have to tune their television in Sydney 151°E. Its value is 0o longitude. The places on the same meridian have the same local time. Posted on October 7, 2020 by . Life-size Cast, Longitudes can be written with an E or W for east or west. There are 360 degrees of longitude, and there are 24 hours in a day. Get all questions and answers of Globe Latitudes And Longitudes of CBSE Class 6 Geography on TopperLearning. Levels Of Attachment Psychology, What divides the earth into the eastern and the western hemispheres? Ukai Dam located in Gujarat has been constructed on. Venom Synonym, How Many Electoral Votes Are There, (d) none of these What is meant by “the Parallels of Latitude ? CSS :: ... View Answer Workspace Report Discuss in Forum. (h) A person, travelling from Mumbai to London, alters the time on his watch at several places. Horsham Pa To Pittsburgh Pa, approximately. (a) What is “Greenwich Mean Time” ? Mid-days Sun can be seen overhead in Chennai twice a year, but not even once in Delhi. Question 25. It is shown by a point. On globes, countries, continents and oceans are shown in their correct shape. Question 2. $\begingroup$ Any longitude calculus needs a clock with the time of a place of known longitude, being it a mechanical or an astronomicl clock. Through an international agreement, the local time of all places is linked to the Greenwich Mean Time (GMT). Besides longitude and latitude, you can use plus codes to share a place without an address. Sundial was a simple and old method to determine local time. Two New Sciences Pdf, Another Place, Another Time Answers, Channel 4 Tv Series, Longitude and Time Worksheet 1. 2 See answers In which lesson it is there In the 2nd chapter - Geographic Grid - latitudes and longitudes smartbrainz smartbrainz The time difference between New Delhi and the place is 9.00 pm + 6.00 am which is 15 hours. (d) 5 hours and 30 minutes (Greenwich Meridian Time) The earth takes 4 minutes for moving 1° distance. 3. - 15308572 Calculating Time Along Longitudes. Greenwich is the 0°longitude. If it is overhead at one place (midday), then it does not shine at all (midnight) at the place directly opposite to that place on the earth. 3. The Antarctic Circle on the other hand, is the latitude 66° 34′ south. Question 2. What is meant by IDL ? These lines are called Parallels of a Latitude. Write in neat and legible handwriting. (h) it is because of change in longitude at several places. Press the Mark it button at the bottom of the page to have the quiz marked. 2. Glen Carter Blacklist Reddit, George's Cosmic Treasure Hunt Summary, Moon Libration 5.Latitude and Longitude 5a. Question 5. Latitude and longitude allow you to use two coordinates to define any point on earth. Hi guys.... need help with longitude and time finding sums. here the difference is 3hrs so it would be 45 degrees. On your Android phone or tablet, open the Google Maps app . Geography Latitude And Longitude - Displaying top 8 worksheets found for this concept.. If it is 5:00pm where you are and 7:30pm GMT, what is your longitude? Buying Meaning In Tamil, (d) The intersection of latitude and longitude points out the exact position of a place on the earth’s surface. 0° longitude is known as Prime Meridian. Karjan river3. A full circle is 360 degrees around. 1. In July 1714 the British Parliament passed the Longitude Act and this established a Board of Longitude which offered a £20,000 prize for the person who could invent a means of calculating longitude. It is 10:00 AM on Thursday at 135 degrees west longitude. NCERT Solutions for Class 6, 7, 8, 9, 10, 11 and 12, NCERT Extra Questions for Class 7 Social Science Geography Chapter 2 Globe Latitudes and Longitudes. Premier League Goalkeeper Save Percentage 2018/19, About This Quiz & Worksheet. Do you know what is the time difference between India and England? Nhl Expansion Cities, Some of the worksheets for this concept are Latitude and longitude work answers, Finding your location throughout the world, Mapping the world, Home, Latitude and longitude, G4 u1 l1 lesson 1 where in the world do i live, National geographic geography skills handbook, Where is here. Write the … 2. Snakeybus Multiplayer, The Latitude of the Antarctic Circle is 66 1°/2 South. Time zones are defined by lines of longitude, imaginary lines that run from the North Pole to South Pole. you first find the long differnce . Scooby Doo! I Think We're Alone Now Song, The shape of the earth is ‘Geoid’. Hence local time at Delhi is 10.52 a.m. Answer: (d) none of these State its importance. Private Homes For Sale Yeppoon, Why ? Here's my try, it's just a snippet of my code: final double RADIUS = 6371.01; double temp = Math.cos(Math.toRadians(latA)) * Math.cos(Math.toRadians(latB)) * Math.cos(Math.toRadians((latB) - (latA))) + Math.sin(Math.toRadians(latA)) * Math.sin(Math.toRadians(latB)); return temp * RADIUS * Math.PI / 180; I am using this formulae to get the latitude and longitude… Abd Al-rahman Al-sufi, answer. Jewish Calendar 7.Precession 8. This prime meridian was chosen at a time when England was dominant in making maps and navigating. (a) Indian Ocean The reason was that he had traveled from east to west, and had under calculated a day. They run parallel to the Equator in it’s North and South. (e) The circumference of the Earth is approximately 40,000 km. Every meridian has a different local time. Though most U.S. states began to adhere to the Pacific, Mountain, Central, and Eastern time zones by 1895, Congress didn't make the use of these time … Hotels In Birmingham, Mi, Ulysses Movie 1997, 2. Local time is based on the sun's day/night cycle in each of the world's 24 different time zones. (b) 5 hours and 30 minutes Delhi is located North of Tropic of Cancer, so the Sun is never overhead at Delhi. So, if you are standing at 0 degrees longitude and you move or travel 15 degrees east or west, you'll notice a difference of 1 hour. Answers is the place to go to get the answers you need and to ask the questions you want Thanks for contributing an answer to Stack Overflow! Answer: The Great Circle Routes follow the great circles i. e. the perimeters of the earth, which cover the shortest distances between any two places in spite of the zigzag routes along the surface of earth. Penna riveri. All You Need To Know About Cryptocurrency Pdf, Suppose we want to estimate the total time in hours and minutes while adding up the time values. State its importance. 2010 Movie, Nathan Mccullum Brother, Suite 47 Citizens Bank Park, Name the thermal zones of the earth. so you add 4hrs to 12noon at long 0degress =12+4hrs=4:00pm. i'm reading a book about longitude and how it helped sailors in the earlier times to sail. Trog Cast, Axis M3004 Firmware, When the Prime Meridian of Greenwich has the sun at the highest point in the sky, all the places along this meridian will have midday or noon. Your email address will not be published. Question 3. Equator, Question 5. Question 2. thanx. Example: The location of New Delhi is 28° N, 77° E. Public Access Prank Calls, It bases on the local meridian passing through that place. | 2021-04-19 09:23:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.32699909806251526, "perplexity": 2096.2848046182735}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038879305.68/warc/CC-MAIN-20210419080654-20210419110654-00229.warc.gz"} |
https://demo.formulasearchengine.com/wiki/Taylor_dispersion | # Taylor dispersion
Taylor dispersion is an effect in fluid mechanics in which a shear flow can increase the effective diffusivity of a species. Essentially, the shear acts to smear out the concentration distribution in the direction of the flow, enhancing the rate at which it spreads in that direction.[1][2][3] The effect is named after the British fluid dynamicist G. I. Taylor.
The canonical example is that of a simple diffusing species in uniform Poiseuille flow through a uniform circular pipe with no-flux boundary conditions.
## Description
We use z as an axial coordinate and r as the radial coordinate, and assume axisymmetry. The pipe has radius a, and the fluid velocity is:
${\displaystyle {\boldsymbol {u}}=w{\hat {\boldsymbol {z}}}=w_{0}(1-r^{2}/a^{2}){\hat {\boldsymbol {z}}}}$
The concentration of the diffusing species is denoted c and its diffusivity is D. The concentration is assumed to be governed by the linear advection–diffusion equation:
${\displaystyle {\frac {\partial c}{\partial t}}+{\boldsymbol {w}}\cdot {\boldsymbol {\nabla }}c=D\nabla ^{2}c}$
The concentration and velocity are written as the sum of a cross-sectional average (indicated by an overbar) and a deviation (indicated by a prime), thus:
${\displaystyle w(r)={\bar {w}}+w'(r)}$
${\displaystyle c(r,z)={\bar {c}}(z)+c'(r,z)}$
Under some assumptions (see below), it is possible to derive an equation just involving the average quantities:
${\displaystyle {\frac {\partial {\bar {c}}}{\partial t}}+{\bar {w}}{\frac {\partial {\bar {c}}}{\partial z}}=D\left(1+{\frac {a^{2}{\bar {w}}^{2}}{48D^{2}}}\right){\frac {\partial ^{2}{\bar {c}}}{\partial z^{2}}}}$
Observe how the effective diffusivity multiplying the derivative on the right hand side is greater than the original value of diffusion coefficient, D. The effective diffusivity is often written as:
${\displaystyle D_{\mathrm {eff} }=D\left(1+{\frac {1}{192}}{\mathit {Pe}}_{d}^{2}\right)\,,}$
where ${\displaystyle {\mathit {Pe}}_{d}=d{\bar {w}}/D}$ is the Péclet number, based on the channel diameter ${\displaystyle d=2a}$. The effect of Taylor dispersion is therefore more pronounced at higher Péclet numbers.
The assumption is that ${\displaystyle c'\ll {\bar {c}}}$ for given ${\displaystyle z}$, which is the case if the length scale in the ${\displaystyle z}$ direction is long enough to smoothen out the gradient in the ${\displaystyle r}$ direction. This can be translated into the requirement that the length scale ${\displaystyle L}$ in the ${\displaystyle z}$ direction satisfies:
${\displaystyle L\gg {\frac {a^{2}}{D}}{\bar {w}}={\frac {{\mathit {Pe}}_{d}\,d}{4}}}$.
Dispersion is also a function of channel geometry. An interesting phenomena for example is that the dispersion of a flow between two infinite flat plates and a rectangular channel, which is infinitely thin, differs approximately 8.75 times. Here the very small side walls of the rectangular channel have an enormous influence on the dispersion.
While the exact formula will not hold in more general circumstances, the mechanism still applies, and the effect is stronger at higher Péclet numbers. Taylor dispersion is of particular relevance for flows in porous media modelled by Darcy's law.
## References
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3. {{#invoke:citation/CS1|citation |CitationClass=book }} | 2021-01-26 03:18:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 15, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8657232522964478, "perplexity": 518.9643949607047}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704795033.65/warc/CC-MAIN-20210126011645-20210126041645-00328.warc.gz"} |
http://tug.org/pipermail/texhax/2011-December/018765.html | # [texhax] Units in technical writing
Reinhard Kotucha reinhard.kotucha at web.de
Thu Dec 29 03:22:16 CET 2011
On 2011-12-28 at 16:04:28 -0800, Robert Wilson wrote:
> I'd like to ask a related question on what is the appropriate way to
> typeset units. Italic?
No, units are typeset upright always. Only variables are typeset
italic. Operators like d (differential) or i, j (complex) or e
(exponential) are typeset upright. They are not variables.
And text is always upright, even in math formulas:
If you look at the output carefully, you see immediately that the
latter variant must be wrong (no ligaturtes, awful kerning).
> Space between number and unit?
Yes, or maybe better a thin space (\, in LaTeX).
> I've never been able to find an authoritative style guide.
I'm sure there are ISO standards but I fear that you have to pay for a
copy. You'll probably find them in a library of a university.
Another valuable source is TUGboat, http://tug.org/TUGboat . There
are often quite interesting articles about math typesetting.
The siunitx (LaTeX) package has all the rules built-in. It's also
worthwhile to consult its documentation. The reference section
And even if you don't read anything at all, there is a simple rule:
Everything within a math formula has to be unambiguous. If variables
are italic, units have to be typeset differently.
Regards,
Reinhard
--
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Reinhard Kotucha Phone: +49-511-3373112
Marschnerstr. 25
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----------------------------------------------------------------------------
Microsoft isn't the answer. Microsoft is the question, and the answer is NO.
---------------------------------------------------------------------------- | 2018-05-27 10:04:37 | {"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": 0.9540040493011475, "perplexity": 5318.76874154668}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794868239.93/warc/CC-MAIN-20180527091644-20180527111644-00192.warc.gz"} |
http://mathonline.wikidot.com/lagrange-s-four-square-theorem | Lagrange's Four Square Theorem
# Lagrange's Four Square Theorem
Lemma 1: If $m, n \in \mathbb{N}$ can be written as a sum of four squares then $mn$ can be written as a sum of four squares.
• Proof: Let $m, n \in \mathbb{N}$ and suppose that $m = x^2 + y^2 + z^2 + w^2$ and $n = a^2 + b^2 + c^2 + d^2$. Then:
(1)
\begin{align} \quad mn = (xa + yb + zc + wd)^2 + (xb - ya + zd - wc)^2 + (xc - yd - za + wb)^2 + (xd + yc - zb- wa)^2 \end{align}
• So $mn$ can be written as a sum of four squares. $\blacksquare$
Lemma 2: If $p$ is a prime then there exists $r, s \in \mathbb{Z}_p$ for which $r^2 + s^2 + 1 \equiv 0 \pmod p$.
• Proof: If $p = 2$ then choose $r = 1$ and $s = 0$.
• So suppose that $p$ is an odd prime. Let:
(2)
\begin{align} \quad X &= \left \{ x^2 : 0 \leq x \leq \frac{p-1}{2} \right \} \\ \quad Y &= \left \{ -1 - y^2 : 0 \leq y \leq \frac{p-1}{2} \right \} \end{align}
• Observe that:
(3)
\begin{align} \quad |X| = \frac{p+1}{2} = |Y| \end{align}
• Since the elements of $X$ and $Y$ are all different we have that $|X \cup Y| = p + 1$. But $|\mathbb{Z}_p| = p$. So by the pigeonhold principle there must exists $r^2 \in X$ and $1 - s^2 \in Y$ for which:
(4)
\begin{align} r^2 \equiv -1 - s^2 \pmod p \end{align}
• That is, there exists integers $r$ and $s$ for which:
(5)
\begin{align} \quad r^2 + s^2 +1 \equiv 0 \pmod p \quad \blacksquare \end{align}
Theorem 3 (Lagrange's Four Square Theorem): Every natural number $n$ can be written as a sum of four squares.
• Proof: By Lemma 1 we only need to prove the result for $1$ and prime numbers, for if $n \in \mathbb{N}$ and $n \geq 2$ then $n$ can be writtten as a product of primes and if each prime can be written as a sum of four squares then so can $n$.
• Clearly $1$ is a sum of four squares since:
(6)
\begin{align} \quad 1 = 1^2 + 0^2 + 0^2 + 0^2 \end{align}
• Let $p$ be a prime. By Lemma 2 there exists $r, s \in \mathbb{Z}_p$ such that:
(7)
\begin{align} \quad r^2 + s^2 + 1 \equiv 0 \pmod p \end{align}
• Let $A$ be the $4 \times 4$ matrix defined by:
(8)
\begin{align} \quad A = \begin{bmatrix} p & 0 & r & s\\ 0 & p & s & -r \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} \end{align}
• Observe that $A$ is an upper triangular matrix with $\det(A) = p^2 \neq 0$. So $A$ is invertible. Let $\Lambda = A \mathbb{Z}^n$. Then $d(\Lambda) = |\det(A)| = p^2$.
• If $\vec{x} = (x_1, x_2, x_3, x_4) \in \Lambda$ then for some integer point $\vec{k} = (k_1, k_2, k_3, k_4) \in \mathbb{Z}^4$ we have that:
(9)
\begin{align} \quad \vec{x} = A\vec{k} = \begin{bmatrix} p & 0 & r & s\\ 0 & p & s & -r \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} k_1\\ k_2\\ k_3\\ k_4 \end{bmatrix} = \begin{bmatrix} p & 0 & r & s\\ 0 & p & s & -r \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} k_1\\ k_2\\ k_3\\ k_4 \end{bmatrix} = \begin{bmatrix} pk_1 + rk_3 + sk_4\\ pk_2 + sk_3 -rk_4\\ k_3\\ k_4 \end{bmatrix} \end{align}
• Since $A$ is an matrix of integers and $\vec{k} \in \mathbb{Z}^4$ we have that $\vec{x} \in \mathbb{Z}^4$. Furthermore, we have that:
(10)
\begin{align} \quad x_1^2 + x_2^2 + x_3^2 + x_4^2 & \equiv (pk_1 + rk_3 + sk_4)^2 + (pk_2 + sk_3 -rk_4)^2 + (k_3)^2 + (k_4)^2 \pmod p \\ & \equiv (rk_3 + sk_4)^2 + (sk_3 - rk_4)^2 + (k_3)^2 + (k_4)^2 \pmod p \\ & \equiv r^2k_3^2 + 2rsk_3k_4 + s^2k_4^2 + s^2k_3^2 - 2rsk_3k_4 + r^2k_4^2 + k_3^2 + k_4^2 \pmod p \\ & \equiv r^2k_3^2 + s^2k_4^2 + s^2k_3^2 + r^2k_4^2 + k_3^2 + k_4^2 \pmod p \\ & \equiv (r^2 + s^2 + 1)k_3^2 + (r^2 + s^2 + 1)k_4^2 \pmod p \\ & \equiv 0 \pmod p \end{align}
• Let $S = \{ (a, b, c, d) \in \mathbb{R}^4 : a^2 + b^2 + c^2 + d^2 < 2p \}$. That is, $S$ is a a $4$-dimensional ball centered at the origin with radius $2p$. Then the volume of $S$ is:
(11)
\begin{align} \quad \mathrm{volume} (S) = \frac{1}{2} \pi^2 (2p)^2 = 2 \pi^2 p^2 \end{align}
• Observe that $S$ is convex, symmetric about the origin, and:
(12)
\begin{align} \quad \mathrm{volume} (S) = 2\pi^2p^2 > 2^4p^2 = 2^4 d(\Lambda) \end{align}
• So by Minkowski's convex body theorem there exists a nonzero point $\vec{x} = (x_1, x_2, x_3, x_4) \in S$ which is a lattice point of $\Lambda$. That is, $0 < x_1^2 + x_2^2 + x_3^2 + x_4^2 < 2p$ and also $x_1^2 + x_2^2 + x_3^2 + x_4^2 \equiv 0 \pmod p$. Hence:
(13)
\begin{align} \quad p = x_1^2 + x_2^2 + x_3^2 + x_4^2 \quad \blacksquare \end{align} | 2021-01-16 15:11:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 13, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9999650716781616, "perplexity": 395.3268599503641}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703506697.14/warc/CC-MAIN-20210116135004-20210116165004-00348.warc.gz"} |
http://cryptosystem.org/archives/2006/03/printing-envelopes-on-the-ml-2010-with-tex/ | # cryptosystem.org
I recently needed to print some size 10 envelopes using a Samsung ML-2010 laser printer under Linux. While printing envelopes using OpenOffice is possible, after a few trial runs I wasn't really happy with the output and decided TeX/LaTeX might be better suited to the job. Adapting Michael Stutz's code from here gave me the following:
% envelope.tex
% Print a #10 envelope
\font\cmssa = cmss12
\font\cmssc = cmss14
%setup:
\parindent 0 pt\nopagenumbers\parskip 10 pt
\hsize 9.5 in\vsize 3.25 in
\voffset 1.25 in
\cmssc
%document:
FROM-NAME
FROM-STREET ADDRESS
FROM-CITY, STATE, \ ZIP
\vskip .4 in\parindent 3.5 in
TO-NAME
TO-STREET ADDRESS
TO-CITY, STATE, \ ZIP
\end
Compile with:
tex envelope.tex; dvips -t landscape -m envelope.dvi
Then you can print with cupsdoprint or your favorite PostScript printing application. The envelopes should be inserted landscape-style, face up, open edge on the right side, in the center of the feed tray with all other paper removed and the guide tabs adjusted to fit the height of the envelope. The source TeX file is also available here. | 2018-07-16 17:53:49 | {"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": 0.28988751769065857, "perplexity": 10276.842318306974}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589417.43/warc/CC-MAIN-20180716174032-20180716194032-00606.warc.gz"} |
https://mllabelutilsjl.readthedocs.io/en/latest/api/interface.html | Working with Encodings¶
Now that we have an understanding of how to extract the label-related information from our targets, let us consider how to instantiate (or infer) a label-encoding, and what we can do with it once we have one. In particular, these encodings will enable us to transform the targets from one representation into another without losing the ability to convert them back afterwards.
Inferring the Encoding¶
In many cases we may not want to just simply assume or guess the particular encoding that some user-provided targets are in. Instead we would rather let the targets themself inform us what encoding they are using. To that end we provide the function labelenc().
labelenc(vec) → LabelEncoding
Tries to determine the most approriate label-encoding to describe the given vector vec, based on the result of label(vec). Note that in most cases this function is not typestable, because the eltype of vec is usually not enough to infer the encoding or number of labels reliably.
Parameters: vec (AbstractVector) – The classification targets in vector form. The label-encoding that is deemed most approriate to describe the values found in vec.
julia> labelenc([:yes,:no,:no,:maybe,:yes,:no])
MLLabelUtils.LabelEnc.NativeLabels{Symbol,3}(Symbol[:yes,:no,:maybe],Dict(:yes=>1,:maybe=>3,:no=>2))
julia> labelenc([-1,1,1,-1,1])
MLLabelUtils.LabelEnc.MarginBased{Int64}()
julia> labelenc(UInt8[0,1,1,0,1])
MLLabelUtils.LabelEnc.ZeroOne{UInt8,Float64}(0.5)
julia> labelenc([false,true,true,false,true])
MLLabelUtils.LabelEnc.TrueFalse()
For matrices we allow an additional (but optional) parameter with which the user can specify the array dimension that denotes the observations.
labelenc(mat[, obsdim]) → LabelEncoding
Computes the concrete matrix-based label-encoding that is used, by determining the size of the matrix for the dimension that is not used for denoting the observations.
Parameters: mat (AbstractMatrix) – An numeric matrix that is assumed to be in the form of a one-hot encoding or similar. obsdim (ObsDimension) – Optional. Denotes which of the two array dimensions of mat denotes the observations. It can be specified as a type-stable positional argument or a smart keyword. Defaults to Obsdim.Last(). see ?ObsDim for more information. The label-encoding that is deemed most approriate to describe the structure and values found in mat.
julia> labelenc([0 1 0 0; 1 0 1 0; 0 0 0 1])
MLLabelUtils.LabelEnc.OneOfK{Int64,3}()
julia> labelenc(Float32[0 1; 1 0; 0 1; 0 1], obsdim = 1)
MLLabelUtils.LabelEnc.OneOfK{Float32,2}()
Asserting Assumptions¶
When writing a function that requires the classification targets to be in a specific encoding (for example $$\{1, -1\}$$ in the case of SVMs), it can be useful to check if the user-provided targets are already in the appropriate encoding, of if they first have to be converted. To check if the targets are of a specific encoding, or family of encodings, we provide the function islabelenc().
islabelenc(vec, encoding) → Bool
Checks if the given values in vec can be described as being produced by the given encoding. This function does not only check the values but also for the correct type. Furthermore it also checks if the total number of labels is appropriate for what the encoding expects it to be.
Parameters: vec (AbstractVector) – The classification targets in vector form. encoding (LabelEncoding) – A concrete instance of a label-encoding that one wants to work with. True, if both the values in vec as well as their types are consistent with the given encoding.
julia> islabelenc([0,1,1,0,1], LabelEnc.ZeroOne(Int))
true
julia> islabelenc([0,1,1,0,1], LabelEnc.ZeroOne(Float64))
false
julia> islabelenc([0,1,1,0,1], LabelEnc.MarginBased(Int))
false
julia> islabelenc(Int8[-1,1,1,-1,1], LabelEnc.MarginBased(Int8))
true
julia> islabelenc(Int8[-1,1,1,-1,1], LabelEnc.MarginBased(Int16))
false
julia> islabelenc([2,1,2,3,1], LabelEnc.Indices(Int,3))
true
julia> islabelenc([2,1,2,3,1], LabelEnc.Indices(Int,4)) # it allows missing labels
true
julia> islabelenc([2,1,2,3,1], LabelEnc.Indices(Int,2)) # more labels than expected
false
Similar to label() we treat matrices in a special way to account for the fact that information about the number of labels is contained in the size of a matrix and not its values. Additionally the user has the freedom to choose which matrix dimension denotes the observations.
islabelenc(mat, encoding[, obsdim]) → Bool
Checks if the values and the structure of the given matrix mat is consistent with the specified encoding. This functions also checks for the correct type and dimensions.
Parameters: mat (AbstractMatrix) – The classification targets in matrix form. encoding (LabelEncoding) – A concrete instance of a matrix-based label-encoding that one wants to work with. obsdim (ObsDimension) – Optional. Denotes which of the two array dimensions of mat denotes the observations. It can be specified as a type-stable positional argument or a smart keyword. Defaults to Obsdim.Last(). see ?ObsDim for more information. True, if the values in mat, its eltype, and the shape of mat is consistent with the given encoding.
julia> islabelenc([0 1 0 0; 1 0 1 0; 0 0 0 1], LabelEnc.OneOfK(Int,3))
true
julia> islabelenc([0 1 0 0; 1 0 1 0; 0 0 0 1], LabelEnc.OneOfK(Int8,3))
false
julia> islabelenc([1 1 0 0; 1 0 1 0; 0 0 0 1], LabelEnc.OneOfK(Int,3)) # matrix is not one-hot
false
julia> islabelenc([0 1 0 0; 1 0 1 0; 0 0 0 1], LabelEnc.OneOfK(Int,4)) # only 3 rows
false
julia> islabelenc([0 1; 1 0; 0 1; 0 1], LabelEnc.OneOfK(Int,2), obsdim = 1)
true
julia> islabelenc(UInt8[0 1; 1 0; 0 1; 0 1], LabelEnc.OneOfK(Int,2), obsdim = 1)
false
julia> islabelenc(UInt8[0 1; 1 0; 0 1; 0 1], LabelEnc.OneOfK(UInt8,2), obsdim = 1)
true
So far islabelenc() was very restrictive concerning the element types of the given target array. In many cases, however, we may not actually care too much about the concrete numeric type but only if the encoding-scheme itself is followed. In fact we usually don’t want to be restrictive about concrete types at all, since we have Julia’s multiple-dispatch system to take care of that later on. In other words we may be more interested in asserting if the labels of the given targets belong to a family of possible label-encodings.
islabelenc(vec, type) → Bool
Checks is the given values in vec can be described as being produced by any possible instance of the given type. In other word this function checks if the labels in vec can be described as being consistent with the family of label-encodings specified by type. This means that the check is much more tolerant concerning the eltype and the total number of labels, since some families of encodings are approriate for any number of labels.
Parameters: vec (AbstractVector) – The classification targets in vector form. type (DataType) – Any subtype of LabelEncoding{T,K,1} True, if the values in vec are consistent with the given family of encodings specified by type.
julia> islabelenc([0,1,1,0,1], LabelEnc.ZeroOne)
true
julia> islabelenc(UInt8[0,1,1,0,1], LabelEnc.ZeroOne)
true
julia> islabelenc([0,1,1,0,1], LabelEnc.MarginBased)
false
julia> islabelenc(Float32[-1,1,1,-1,1], LabelEnc.MarginBased)
true
julia> islabelenc(Int8[-1,1,1,-1,1], LabelEnc.MarginBased)
true
julia> islabelenc([2,1,2,3,1], LabelEnc.Indices)
true
julia> islabelenc(Int8[2,1,2,3,1], LabelEnc.Indices)
true
julia> islabelenc(Int8[2,1,2,3,1], LabelEnc.Indices{Int}) # restrict type but not nlabels
false
We again provide a special version for matrices.
islabelenc(mat, type[, obsdim]) → Bool
Checks is the values and the structure of the given matrix mat can be described as being produced by any possible instance of the given type. This means that the check is much more tolerant concerning the eltype and the size of the matrix, since some families of encodings are approriate for any number of labels.
Parameters: mat (AbstractMatrix) – The classification targets in matrix form. type (DataType) – Any subtype of LabelEncoding{T,K,2} obsdim (ObsDimension) – Optional. Denotes which of the two array dimensions of mat denotes the observations. It can be specified as a type-stable positional argument or a smart keyword. Defaults to Obsdim.Last(). see ?ObsDim for more information. True, if the values in mat are consistent with the given family of encodings specified by type.
julia> islabelenc([0 1 0 0; 1 0 1 0; 0 0 0 1], LabelEnc.OneOfK)
true
julia> islabelenc(Int8[0 1 0 0; 1 0 1 0; 0 0 0 1], LabelEnc.OneOfK)
true
julia> islabelenc([1 1 0 0; 1 0 1 0; 0 0 0 1], LabelEnc.OneOfK) # matrix is not one-hot
false
julia> islabelenc([0 1; 1 0; 0 1; 0 1], LabelEnc.OneOfK, obsdim = 1)
true
julia> islabelenc(UInt8[0 1; 1 0; 0 1; 0 1], LabelEnc.OneOfK, obsdim = 1)
true
julia> islabelenc(UInt8[0 1; 1 0; 0 1; 0 1], LabelEnc.OneOfK{Int32}, obsdim = 1) # restrict type but not nlabels
false
Properties of an Encoding¶
Once we have an instance of some label-encoding, we can compute a number of useful properties about it. For example we can query all the labels that an encoding uses to represent the classes.
label(encoding) → Vector
Returns all the labels that a specific encoding uses in their approriate order.
Parameters: encoding (LabelEncoding) – The specific label-encoding. The unique labels in the form of a vector. In the case of two labels, the first element will represent the positive label and the second element the negative label respectively.
julia> label(LabelEnc.ZeroOne(UInt8))
2-element Array{UInt8,1}:
0x01
0x00
julia> label(LabelEnc.MarginBased())
2-element Array{Float64,1}:
1.0
-1.0
julia> label(LabelEnc.Indices(Float32,5))
5-element Array{Float32,1}:
1.0
2.0
3.0
4.0
5.0
For convenience one can also just query for the label that corresponds to the positive class or the negative class respectively. These helper functions are only defined for binary label-encoding and will throw an MethodError for multi-class encodings.
poslabel(encoding)
If the encoding is binary it will return the positive label of it. The function will throw an error otherwise.
Parameters: encoding (LabelEncoding) – The specific label-encoding. The value representing the positive label of the given encoding in the approriate type.
julia> poslabel(LabelEnc.ZeroOne(UInt8))
0x01
julia> poslabel(LabelEnc.MarginBased())
1.0
julia> poslabel(LabelEnc.Indices(Float32,2))
1.0f0
julia> poslabel(LabelEnc.Indices(Float32,5))
ERROR: MethodError: no method matching poslabel(::MLLabelUtils.LabelEnc.Indices{Float32,5})
neglabel(encoding)
If the encoding is binary it will return the negative label of it. The function will throw an error otherwise.
Parameters: encoding (LabelEncoding) – The specific label-encoding. The value representing the negative label of the given encoding in the approriate type.
julia> neglabel(LabelEnc.ZeroOne(UInt8))
0x00
julia> neglabel(LabelEnc.MarginBased())
-1.0
julia> neglabel(LabelEnc.Indices(Float32,2))
2.0f0
julia> neglabel(LabelEnc.Indices(Float32,5))
ERROR: MethodError: no method matching neglabel(::MLLabelUtils.LabelEnc.Indices{Float32,5})
We can also query the number of labels that a concrete encoding uses. In other words we can query the number of classes the given label-encoding is able to represent.
nlabel(encoding) → Int
Returns the number of labels that a specific encoding uses.
Parameters: encoding (LabelEncoding) – The specific label-encoding.
julia> nlabel(LabelEnc.ZeroOne(UInt8))
2
julia> nlabel(LabelEnc.NativeLabels([:a,:b,:c]))
3
More interestingly, we can infer the number of labels for a family of encodings. This allows for some compile time decisions, but only work for some types of encodings (i.e. binary).
nlabel(type) → Int
Returns the number of labels that the family of encodings type can describe. Note that this function will fail if the number of labels can not be inferred from the given type.
Parameters: type (DataType) – Some subtype of LabelEncoding{T,K,M} with a fixed K The type-parameter K of type.
julia> nlabel(LabelEnc.ZeroOne)
2
julia> nlabel(LabelEnc.NativeLabels)
ERROR: ArgumentError: number of labels could not be inferred for the given type
We can also query a family of encodings for their label-type. In this case we decided to not throw an error if the type can not be inferred but instead return the most specific abstract type.
labeltype(type) → DataType
Determine the type of the labels represented by the given family of label-encoding. If the type can not be inferred than Any is returned.
Parameters: type (DataType) – Some subtype of LabelEncoding{T,K,M} The type-parameter T of type if specified, or the most specific abstract type otherwise.
julia> labeltype(LabelEnc.TrueFalse)
Bool
julia> labeltype(LabelEnc.ZeroOne{Int})
Int64
julia> labeltype(LabelEnc.ZeroOne)
Number
julia> labeltype(LabelEnc.NativeLabels)
Any
Converting to/from Indices¶
As stated before, the order of the of label() matters. In a binary setting, for example, the first label is interpreted as the positive class and the second label as the negative class. This is simply the arbitrary convention that we follow. That said, even in a multi-class setting it is important to be consistent with the ordering. This is crucial in order to make sure that converting to a different encoding and then converting back yields the original values.
Every encoding understands the concept of a label-index, which is a unique representation of a class that all encodings share. For example the positive label of a binary label-encoding always has the label-index 1 and the negative 2 respectively.
To convert a label-index into the label that a specific encoding uses to represent the underlying class we provide the function ind2label().
ind2label(index, encoding)
Converts the given index into the corresponding label defined by the encoding. Note that in the binary case, index = 1 represents the positive label and index = 2 the negative label.
Parameters: index (Int) – Index of the desired label. This variable can be specified either as an Int or as a Val. Note that indices are one-based. encoding (LabelEncoding) – The encoding one wants to get the label from. The label of the specified index for the specified encoding.
julia> ind2label(1, LabelEnc.MarginBased(Float32))
1.0f0
julia> ind2label(Val{1}, LabelEnc.MarginBased(Float32))
1.0f0
julia> ind2label(2, LabelEnc.MarginBased(Float32))
-1.0f0
julia> ind2label(3, LabelEnc.OneOfK(Int8,4))
4-element Array{Int8,1}:
0
0
1
0
julia> ind2label(3, LabelEnc.NativeLabels([:a,:b,:c,:d]))
:c
julia> ind2label.([1,2,2,1], LabelEnc.ZeroOne(UInt8)) # broadcast support
4-element Array{UInt8,1}:
0x01
0x00
0x00
0x01
We also provide inverse function for converting a label of a specific encoding into the corresponding label-index. Note that this function does not check if the given label is of the expected type, but simply that it is of the appropriate value.
label2ind(label, encoding) → Int
Converts the given label into the corresponding index defined by the encoding. Note that in the binary case, the positive label will result in the index 1 and the negative label in the index 2 respectively.
Parameters: label (Any) – A label in the format familiar to the encoding. encoding (LabelEncoding) – The encoding to compute the label-index with. The index of the specified label for the specified encoding.
julia> label2ind(1.0, LabelEnc.MarginBased())
1
julia> label2ind(-1.0, LabelEnc.MarginBased())
2
julia> label2ind([0,0,1,0], LabelEnc.OneOfK(4))
3
julia> label2ind(:c, LabelEnc.NativeLabels([:a,:b,:c,:d]))
3
julia> label2ind.([1,0,0,1], LabelEnc.ZeroOne()) # broadcast support
4-element Array{Int64,1}:
1
2
2
1
Converting between Encodings¶
In the case that the given targets are not in the encoding that your algorithm expects them to be in, you may want to convert them into the format you require. For that purpose we expose the function convertlabel().
convertlabel(dst_encoding, src_label, src_encoding)
Converts the given input label src_label from src_encoding into the corresponding label described by the desired output encoding dst_encoding.
Note that both encodings are expected to be vector-based, meaning that this method does not work for LabelEnc.OneOfK. It does, however, support broadcasting.
Parameters: dst_encoding (LabelEncoding) – The vector-based label-encoding that should be used to produce the output label. src_label (Any) – The input label one wants to convert. It is expected to be consistent with src_encoding. src_encoding (LabelEncoding) – A vector-based label-encoding that is assumed to have produced the given src_label. The label from dst_encoding that corresponds to src_label in src_encoding
julia> convertlabel(LabelEnc.OneOfK(2), -1, LabelEnc.MarginBased()) # OneOfK is not vector-based
ERROR: MethodError: no method matching [...]
julia> convertlabel(LabelEnc.NativeLabels([:a,:b,:c,:d]), 3, LabelEnc.Indices(4))
:c
julia> convertlabel(LabelEnc.ZeroOne(), :yes, LabelEnc.NativeLabels([:yes,:no]))
1.0
julia> convertlabel(LabelEnc.ZeroOne(), :no, LabelEnc.NativeLabels([:yes,:no]))
0.0
julia> convertlabel(LabelEnc.MarginBased(Int), 0, LabelEnc.ZeroOne())
-1
julia> convertlabel(LabelEnc.NativeLabels([:a,:b]), -1, LabelEnc.MarginBased())
:b
julia> convertlabel.(LabelEnc.NativeLabels([:a,:b]), [-1,1,1,-1], LabelEnc.MarginBased()) # broadcast support
4-element Array{Symbol,1}:
:b
:a
:a
:b
Aside from the one broadcast-able method that is implemented for converting single labels, we provide a range of methods that work on whole arrays. These are more flexible because by having an array as input these methods have more information available to make reasonable decisions. As a consequence of that can we consider the “source-encoding” parameter optional, because these methods can now make use of labelenc() internally to infer it automatically.
convertlabel(dst_encoding, arr[, src_encoding][, obsdim])
Converts the given array arr from the src_encoding into the dst_encoding. If src_encoding is not specified it will be inferred automaticaly using the function labelenc(). This should not negatively influence type-inference.
Note that both encodings should have the same number of labels, or a MethodError will be thrown in most cases.
Parameters: dst_encoding (LabelEncoding) – The desired output format. arr (AbstractArray) – The input targets that should be converted into the encoding specified by dst_encoding. src_encoding (LabelEncoding) – The input encoding that arr is expected to be in. obsdim (ObsDimension) – Optional. Only possible if one of the two encodings is a matrix-based encoding. Defines which of the two array dimensions denotes the observations. It can be specified as a type-stable positional argument or a smart keyword. Defaults to Obsdim.Last(). see ?ObsDim for more information. A converted version of arr using the specified output encoding dst_encoding.
julia> convertlabel(LabelEnc.NativeLabels([:yes,:no]), [-1,1,-1,1,1,-1])
6-element Array{Symbol,1}:
:no
:yes
:no
:yes
:yes
:no
julia> convertlabel(LabelEnc.OneOfK(Float32,2), [-1,1,-1,1,1,-1])
2×6 Array{Float32,2}:
0.0 1.0 0.0 1.0 1.0 0.0
1.0 0.0 1.0 0.0 0.0 1.0
julia> convertlabel(LabelEnc.TrueFalse(), [-1,1,-1,1,1,-1])
6-element Array{Bool,1}:
false
true
false
true
true
false
julia> convertlabel(LabelEnc.Indices(3), [:no,:maybe,:yes,:no], LabelEnc.NativeLabels([:yes,:maybe,:no]))
4-element Array{Int64,1}:
3
2
1
3
It may be interesting to point out explicitly that we provide special treatment for LabelEnc.OneVsRest to conveniently convert a multi-class problem into a two-class problem.
julia> convertlabel(LabelEnc.OneVsRest(:yes), [:yes,:no,:no,:maybe,:yes,:yes])
6-element Array{Symbol,1}:
:yes
:not_yes
:not_yes
:not_yes
:yes
:yes
julia> convertlabel(LabelEnc.ZeroOne(Float64), [:yes,:no,:no,:maybe,:yes,:yes], LabelEnc.OneVsRest(:yes))
6-element Array{Float64,1}:
1.0
0.0
0.0
0.0
1.0
1.0
We also allow a more concise way to specify that your are using a LabelEnc.NativeLabels encoding by just passing the label-vector directly, that you would normally pass to its constructor.
julia> convertlabel([:yes,:no], [-1,1,-1,1,1,-1])
6-element Array{Symbol,1}:
:no
:yes
:no
:yes
:yes
:no
julia> convertlabel(LabelEnc.Indices(3), [:no,:maybe,:yes,:no], [:yes,:maybe,:no])
4-element Array{Int64,1}:
3
2
1
3
In many cases it can be inconvenient that one has to explicitly specify the label-type and number of labels for the desired output-encoding. To that end we also allow the output-encoding to be specified in terms of an encoding-family (i.e. as DataType).
convertlabel(dst_family, arr[, src_encoding][, obsdim])
Converts the given array arr from the src_encoding into some concrete label-encoding that is a subtype of dst_family. This way the method tries to preserve the eltype of arr if it is numeric. Furthermore, the concrete number of labels need not be specified explicitly, but will instead be inferred from src_encoding.
If src_encoding is not specified it will be inferred automaticaly using the function labelenc(). This should not negatively influence type-inference.
Parameters: dst_family (DataType) – Any subtype of LabelEncoding{T,K,M}. It denotes the desired family of label-encodings that one wants the return value to be in. arr (AbstractArray) – The input targets that should be converted into some encoding specified by the type dst_family. src_encoding (LabelEncoding) – The input encoding that arr is expected to be in. obsdim (ObsDimension) – Optional. Only possible if one of the two encodings is a matrix-based encoding. Defines which of the two array dimensions denotes the observations. It can be specified as a type-stable positional argument or a smart keyword. Defaults to Obsdim.Last(). see ?ObsDim for more information. A converted version of arr using a label-encoding that is member of the encoding-family dst_family.
julia> convertlabel(LabelEnc.OneOfK, Int8[-1,1,-1,1,1,-1])
2×6 Array{Int8,2}:
0 1 0 1 1 0
1 0 1 0 0 1
julia> convertlabel(LabelEnc.OneOfK{Float32}, Int8[-1,1,-1,1,1,-1], obsdim = 1)
6×2 Array{Float32,2}:
0.0 1.0
1.0 0.0
0.0 1.0
1.0 0.0
1.0 0.0
0.0 1.0
julia> convertlabel(LabelEnc.TrueFalse, [-1,1,-1,1,1,-1])
6-element Array{Bool,1}:
false
true
false
true
true
false
julia> convertlabel(LabelEnc.Indices, [:no,:maybe,:yes,:no], LabelEnc.NativeLabels([:yes,:maybe,:no]))
4-element Array{Int64,1}:
3
2
1
3
For vector-based encodings (which means all except LabelEnc.OneOfK), we provide a lazy version of convertlabel() that does not allocate a new array for the outputs, but instead creates a MappedArray into the original targets.
convertlabelview(dst_encoding, vec[, src_encoding])
Creates a MappedArray that provides a lazy view into vec, that makes it look like the values are actually of the provided output encoding new_encoding. This means that the convertion happens on the fly when an element of the resulting mapped array is accessed. This resulting mapped array will even be writeable, unless src_encoding is LabelEnc.OneVsRest.
Note that both encodings are expected to be vector-based, meaning that this method does not work for LabelEnc.OneOfK.
Parameters: dst_encoding (LabelEncoding) – The desired vector-based output encoding. vec (AbstractVector) – The input targets that one wants to convert using dst_encoding. It is expected to be consistent with src_encoding. src_encoding (LabelEncoding) – A vector-based label-encoding that is assumed to have produced the values in vec. A MappedArray or ReadonlyMappedArray that makes vec look like it is in the encoding specified by new_encoding
julia> true_targets = [-1,1,-1,1,1,-1]
6-element Array{Int64,1}:
-1
1
-1
1
1
-1
julia> A = convertlabelview(LabelEnc.NativeLabels([:yes,:no]), true_targets)
6-element MappedArrays.MappedArray{Symbol,1,...}:
:no
:yes
:no
:yes
:yes
:no
julia> A[2] = :no
julia> A
6-element MappedArrays.MappedArray{Symbol,1,...}:
:no
:no
:no
:yes
:yes
:no
julia> true_targets
6-element Array{Int64,1}:
-1
-1
-1
1
1
-1
Classifying Predictions¶
Some encodings come with an implicit interpretation of how the raw predictions of some model (often denoted as $$\hat{y}$$, written yhat) should look like and how they can be classified into a predicted class-label. For that purpose we provide the function classify() and its mutating version classify!().
classify(yhat, encoding)
Returns the classified version of yhat given the encoding. That means that if yhat can be interpreted as a positive label, the positive label of encoding is returned. If yhat can not be interpreted as a positive value then the negative label is returned.
Parameters: yhat (Number) – The numeric prediction that should be classified into either the label representing the positive class or the label representing the negative class encoding (LabelEncoding) – A concrete instance of a label-encoding that one wants to work with. The label that the encoding uses to represent the class that yhat is classified into.
For LabelEnc.MarginBased the decision boundary between classifying into a negative or a positive label is predefined at zero. More precisely a raw prediction greater than or equal to zero is considered a positive prediction, while any strictly negative raw prediction is considered a negative prediction.
julia> classify(-0.3f0, LabelEnc.MarginBased()) # defaults to Float64
-1.0
julia> classify.([-2.3,6.5], LabelEnc.MarginBased(Int))
2-element Array{Int64,1}:
-1
1
For LabelEnc.ZeroOne the assumption is that the raw prediction is in the closed interval $$[0, 1]$$ and represents a degree of certainty that the observation is of the positive class. That means that in order to classify a raw prediction to either positive or negative, one needs to decide on a “threshold” parameter, which determines at which degree of certainty a prediction is “good enough” to classify as positive.
julia> classify(0.3f0, LabelEnc.ZeroOne(0.5)) # defaults to Float64
0.0
julia> classify(0.3f0, LabelEnc.ZeroOne(Int,0.2))
1
julia> classify.([0.3,0.5], LabelEnc.ZeroOne(Int,0.4))
2-element Array{Int64,1}:
0
1
We recognize that such a probabilistic interpretation of the raw predicted value is fairly common. So much so that we provide a convenience method for when one is working under the assumption of a LabelEnc.ZeroOne encoding.
classify(yhat, threshold)
Returns the classified version of yhat given the decision margin threshold. This method assumes that yhat denotes a probability and will either return zero(yhat) if yhat is below threshold, or one(yhat) otherwise.
Parameters: yhat (Number) – The numeric prediction. It is assumed be a value between 0 and 1. threshold (Number) – The threshold below which yhat will be classified as 0. The classified version of yhat of the same type.
julia> classify(0.3f0, 0.5)
0.0f0
julia> classify(0.3f0, 0.2)
1.0f0
julia> classify.([0.3,0.5], 0.4)
2-element Array{Float64,1}:
0.0
1.0
For matrix-based encodings, such as LabelEnc.OneOfK we provide a special method that allows to optionally specify the dimension of the matrix that denote the observations.
classify(yhat, encoding[, obsdim])
If yhat is a vector (i.e. a single observation), this function returns the index of the element that has the largest value. If yhat is a matrix, this function returns a vector of indices for each observation in yhat.
Parameters: yhat (AbstractArray) – The numeric predictions in the form of either a vector or a matrix. encoding (LabelEncoding) – A concrete instance of a matrix-based label-encoding that one wants to work with. obsdim (ObsDimension) – Optional iff yhat is a matrix. Denotes which of the two array dimensions of yhat denotes the observations. It can be specified as a type-stable positional argument or a smart keyword. Defaults to Obsdim.Last(). see ?ObsDim for more information. The classified version of yhat. This will either be an integer or a vector of indices.
julia> pred_output = [0.1 0.4 0.3 0.2; 0.8 0.3 0.6 0.2; 0.1 0.3 0.1 0.6]
3×4 Array{Float64,2}:
0.1 0.4 0.3 0.2
0.8 0.3 0.6 0.2
0.1 0.3 0.1 0.6
julia> classify(pred_output, LabelEnc.OneOfK(3))
4-element Array{Int64,1}:
2
1
2
3
julia> classify(pred_output', LabelEnc.OneOfK(3), obsdim=1) # note the transpose
4-element Array{Int64,1}:
2
1
2
3
julia> classify([0.1,0.2,0.6,0.1], LabelEnc.OneOfK(4)) # single observation
3
Similar to other functions we expose a version that can be called with a family of encodings (i.e. a type with free type parameters) instead of a concrete instance.
classify(yhat, type)
Returns the classified version of yhat given the family of encodings specified by type. That means that if yhat can be interpreted as a positive label, the positive label of that family is returned (and the negative otherwise). Furthermore, the type of yhat is preserved.
Parameters: yhat (Number) – The numeric prediction that should be classified into either the label representing the positive class or the label representing the negative class type (DataType) – Any subtype of LabelEncoding{T,K,1} The classified version of yhat of the same type.
julia> classify(0.3f0, LabelEnc.ZeroOne) # threshold fixed at 0.5
0.0f0
julia> classify(0.3, LabelEnc.ZeroOne)
0.0
julia> classify(4f0, LabelEnc.MarginBased)
1.0f0
julia> classify(-4, LabelEnc.MarginBased)
-1
classify(yhat, type[, obsdim])
If yhat is a vector (i.e. a single observation), this function returns the index of the element that has the largest value. If yhat is a matrix, this function returns a vector of indices for each observation in yhat.
Parameters: yhat (AbstractArray) – The numeric predictions in the form of either a vector or a matrix. type (DataType) – Any subtype of LabelEncoding{T,K,2} obsdim (ObsDimension) – Optional iff yhat is a matrix. Denotes which of the two array dimensions of yhat denotes the observations. It can be specified as a type-stable positional argument or a smart keyword. Defaults to Obsdim.Last(). see ?ObsDim for more information. The classified version of yhat. This will either be an integer or a vector of indices.
julia> pred_output = [0.1 0.4 0.3 0.2; 0.8 0.3 0.6 0.2; 0.1 0.3 0.1 0.6]
3×4 Array{Float64,2}:
0.1 0.4 0.3 0.2
0.8 0.3 0.6 0.2
0.1 0.3 0.1 0.6
julia> classify(pred_output, LabelEnc.OneOfK)
4-element Array{Int64,1}:
2
1
2
3
julia> classify(pred_output', LabelEnc.OneOfK, obsdim=1) # note the transpose
4-element Array{Int64,1}:
2
1
2
3
julia> classify([0.1,0.2,0.6,0.1], LabelEnc.OneOfK) # single observation
3
We also provide a mutating version. This is mainly of interest when working with LabelEnc.OneOfK(), in which case broadcast is not defined on the previous methods.
classify!(out, arr, encoding[, obsdim])
Same as classify, but uses out to store the result. In the case of a vector-based encoding this will use broadcast internally. It is mainly provided to offer a consistent API between vector-based and matrix-based encodings.
For convenience we also provide boolean version that assert if the given raw prediction could be interpreted as either a positive or a negative prediction.
isposlabel(yhat, encoding) → Bool
Checks if the given value yhat can be interpreted as the positive label given the encoding. This function takes potential classification rules into account.
julia> isposlabel([1,0], LabelEnc.OneOfK(2))
true
julia> isposlabel([0,1], LabelEnc.OneOfK(2))
false
julia> isposlabel(-5, LabelEnc.MarginBased())
false
julia> isposlabel(2, LabelEnc.MarginBased())
true
julia> isposlabel(0.3f0, LabelEnc.ZeroOne(0.5))
false
julia> isposlabel(0.3f0, LabelEnc.ZeroOne(0.2))
true
isneglabel(yhat, encoding) → Bool
Checks if the given value yhat can be interpreted as the negative label given the encoding. This function takes potential classification rules into account.
julia> isneglabel([1,0], LabelEnc.OneOfK(2))
false
julia> isneglabel([0,1], LabelEnc.OneOfK(2))
true
julia> isneglabel(-5, LabelEnc.MarginBased())
true
julia> isneglabel(2, LabelEnc.MarginBased())
false
julia> isneglabel(0.3f0, LabelEnc.ZeroOne(0.5))
true
julia> isneglabel(0.3f0, LabelEnc.ZeroOne(0.2))
false | 2021-01-26 18:34:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.7033895254135132, "perplexity": 2369.113358769467}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704803308.89/warc/CC-MAIN-20210126170854-20210126200854-00008.warc.gz"} |
https://www.maths.usyd.edu.au/s/scnitm/aksamit-StochasticsAndFinance-Fon | SMS scnews item created by Anna Aksamit at Mon 21 Mar 2022 1014
Type: Seminar
Distribution: World
Expiry: 4 Apr 2022
Calendar1: 30 Mar 2022 2000-2100
CalLoc1: zoom talk
Auth: aksamit@124.180.26.80 (aaks9559) in SMS-SAML
# Stochastics and Finance: Fontana -- Term structure modeling with overnight rates beyond stochastic continuity
Dear All,
Welcome to the Stochastics and Finance seminar this semester! We will resume next week;
talks’ details will be updated on the website. On Wednesday March 30 at 8pm Claudio
Fontana will give a talk via Zoom.
Speaker: Claudio Fontana (University of Padova)
Title: Term structure modeling with overnight rates beyond stochastic continuity
Abstract: In the current reform of interest rate benchmarks, a central role is played by
risk-free rates (RFRs), such as SOFR (secured overnight financing rate) in the US. A
key feature of RFRs is the presence of jumps and spikes at periodic time intervals as a
result of regulatory and liquidity constraints. This corresponds to stochastic
discontinuities (i.e., jumps occurring at predetermined dates) in the dynamics of RFRs.
In this work, we propose a general modelling framework where RFRs and term rates can
have stochastic discontinuities and characterize absence of arbitrage in an extended HJM
setup. When the term rate is generated by the RFR itself, we show that it solves a
BSDE, whose driver is determined by the HJM drift restrictions. We develop a tractable
specification driven by affine semimartingales, also extending the classical short rate
approach to the case of stochastic discontinuities. In this context, we show that a
simple specification allows to capture stylized facts of the jump behavior of overnight
rates. In a Gaussian setting, we provide explicit valuation formulas for bonds and
caplets. Finally, we study hedging in the sense of local risk-minimization when the
underlying term structures have stochastic discontinuities. Based on joint work with
Zorana Grbac and Thorsten Schmidt.
https://www.maths.usyd.edu.au/u/SemConf/Stochastics_Finance/seminar.html
Please feel free to forward this message to anyone who might be interested in this
talk.
Kind regards, Anna
Actions:
Calendar (ICS file) download, for import into your favourite calendar application
UNCLUTTER for printing
AUTHENTICATE to mark the scnews item as read
School members may try to . | 2022-11-30 21:25:00 | {"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": 0.2646184265613556, "perplexity": 6316.921025984243}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710771.39/warc/CC-MAIN-20221130192708-20221130222708-00289.warc.gz"} |
https://www.coursehero.com/file/42181306/hw3-solutionpdf/ | # hw3-solution.pdf - Statistics 106 Homework 3 Solution Due...
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Statistics 106 Homework 3 Solution Due : Oct. 22, 2018, In Class 16.26 Refer to Rehabilitation therapy Problem 16.9. Obtain the power of the test in Problem 16.9(e) if μ 1 = 37, μ 2 = 35 and μ 3 = 28. Assume that σ = 4 . 5.
17.1 A student, asked to give a class demonstration of the use of a confidence interval for comparing two treatment means, proposed to construct a 99 percent confidence interval for the pairwise comparison D = μ 5 - μ 3 , where there are a total of five factor levels. The student selected this particular comparison because the estimated treatment means ¯ Y 5 . and ¯ Y 3 . are the largest and smallest, respectively, and stated: “This confidence interval is particularly useful. If it does not contain zero, it indicates, with significance level α = . 01, that the factor level means are not equal.” a. Explain why the student’s assertion is not correct. b. How should the confidence interval be constructed so that the assertion can be made with significance level α = . 01? Solution:
1
a type I error. The student made a comparison that was suggested by the data (finding the largest dif- ference between treatment sample means) such that the correct confidence interval should be wider | 2021-11-27 15:18:24 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8094399571418762, "perplexity": 1062.9558848190804}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358189.36/warc/CC-MAIN-20211127133237-20211127163237-00430.warc.gz"} |
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We assume a single-curve environment. Let us recall that a floating LIBOR payment fixed at time $T$ and paid at time $T^\prime$ can be written in terms of zero-coupon bonds: $$L(t,T,T^\prime):=\frac{1}{T^\prime-T}\left(\frac{P(t,T)}{P(t,T^\prime)}-1\right)$$ Let $\mathcal{T}:=\{T_0,\dots,T_n,\dots,T_m\}$ be a schedule such that a spot 5y swap starts fixing ...
1
I am assuming that in option 1 you are entering into payer swap. If the curve is flat then option a) and b) are the same because you will get the same cashflows in both cases. Why? In option b) both the floating legs and fixed legs on 10Y swap and 5Y will cancel for the first 5 years i.e. the cashflows will be opposite sign, effectively making it 5y5y swap. ...
-1
When referring to a long straddle, atm means the 50 delta strike.
0
For straddles ATM usually implies 0 delta. In general, ATM is determined by the market conventions in question.
Top 50 recent answers are included | 2021-07-31 19:05:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8016045093536377, "perplexity": 2617.9564828394023}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154099.21/warc/CC-MAIN-20210731172305-20210731202305-00140.warc.gz"} |
https://cs.stackexchange.com/questions/128700/reference-for-counting-the-number-of-paths-in-a-dag | # Reference for counting the number of paths in a DAG
Given a connected DAG I know how to compute the number of paths between two nodes. See e.g. Counting number of paths between two vertices in a DAG .
Is there a reference or name for the algorithm? If not, are there well known applications?
• I don't know the algorithm, but these books have relevant material, I believe: Kemeny and Snell, Finite Markov Chains (chapter 1); Flajolet and Sedgewick, Analytical Combinatorics. – Mars Jul 25 '20 at 15:50
• @Mars Could you say more about those references please? Do they refer specifically to this problem and if so, in which context? – fomin Jul 25 '20 at 18:50
• The first reference contains a methods for counting paths in DAGs. I'm pretty sure that the second will, too. I don't know whether they are what you want. – Mars Jul 25 '20 at 23:28
It is also the semiring problem for $$\mathbb{N}$$ with the usual $$+$$ and $$\times$$, and can be extended to cyclic graphs by extending the value set to $$\mathbb{N} \cup \infty$$ and adding a suitable $$a^*$$ operator. | 2021-01-26 00:21:12 | {"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7231295108795166, "perplexity": 329.9141468117059}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704792131.69/warc/CC-MAIN-20210125220722-20210126010722-00105.warc.gz"} |
https://shishirshakya.blogspot.com/2015/09/ | ## Wednesday, September 2, 2015
### Stationarity Condition of AR Process
To replicate, You can download the data (here). In this blog, I will discuss about the stationarity process of an $AR(p)$ process for ${{y}_{t}}$.
Let’s define an$AR(p)$ process for ${{y}_{t}}$ which is given as:
$yt={{\phi }_{1}}{{y}_{t-1}}+{{\phi }_{2}}{{y}_{t-2}}+\cdots +{{\phi }_{p}}{{y}_{t-p}}+\mu +{{v}_{t}}$
This equation can be expressed as:
${{y}_{t}}-{{\phi }_{1}}{{y}_{t-1}}-{{\phi }_{2}}{{y}_{t-2}}-\cdots -{{\phi }_{p}}{{y}_{t-p}}=\mu +{{v}_{t}}$
Now, let’s use a backshift or lag operator i.e. $(B{{y}_{t}}={{y}_{t-1}},{{B}^{2}}{{y}_{t}}={{y}_{t-2}},\cdots {{B}^{p}}{{y}_{t}}={{y}_{t-p}})$.
Then the above expression can be written as:
${{y}_{t}}-{{\phi }_{1}}B{{y}_{t}}-{{\phi }_{2}}{{B}^{2}}{{y}_{t}}-\cdots -{{\phi }_{p}}{{B}^{p}}{{y}_{t}}=\mu +{{v}_{t}}$
Now, we can take ${{y}_{t}}$ common,
$(1-{{\phi }_{1}}B-{{\phi }_{2}}{{B}^{2}}-\cdots -{{\phi }_{p}}{{B}^{p}}){{y}_{t}}=\mu +{{v}_{t}}$
Now, we can write
${{\phi }_{p}}(B){{y}_{t}}=\mu +{{v}_{t}}$
Where,
${{\phi }_{p}}(B)=1-{{\phi }_{1}}B-{{\phi }_{2}}{{B}^{2}}-\cdots -{{\phi }_{p}}{{B}^{p}}$
For the stationarity condition,
The ${yt}$ is stationary when the roots of the characteristic polynomial lie outside the unit circle. The modulus of polynomials roots of ${{\phi }_{p}}(B)$ i.e $|{{z}_{i}}|>1$ or $|{{\phi }_{i}}|>1$.
Let’s discuss the stationary condition of an $AR(p)$ process for ${{y}_{t}}$with a practical example. You can download the data (here). At first let’s load the monthly data of US interest rate from January 1960 till December 1998 and save that to object called “r”. Now, Let’s develop an $AR(4)$ process for “r” and save it to a new object called “ar_r”.
rm (list = ls(all=TRUE))
graphics.off()
attach(data)
lag_r <- 4
ar_r <- arima(r,order=c(lag_r,0,0))
ar_r
This will generate the following output:
Hence, our equation can be given as:
\begin{align} & {{r}_{t}}=1.417{{r}_{t-1}}-0.587{{r}_{t-2}}+0.125{{r}_{t-3}}+0.024{{y}_{t-4}}+6.161+{{v}_{t}} \\ & or,{{r}_{t}}-1.417{{r}_{t-1}}+0.587{{r}_{t-2}}-0.125{{r}_{t-3}}-0.024{{y}_{t-4}}=6.161+{{v}_{t}} \\ \end{align}
Now, when we use Lag operator,
${{r}_{t}}(1-1.417z+0.587{{z}^{2}}-0.125{{z}^{3}}-0.024{{z}^{4}})=6.161+{{v}_{t}}$
Now we can solve for values of $z$ in the equation$(1-1.417z+0.587{{z}^{2}}-0.125{{z}^{3}}-0.024{{z}^{4}})$.
To do such in R, let’s extract the coefficients from the object “ar_r” and save that to object called “phi”.
phi <- ar_r$coef Then concatenate the 1 and the coefficients which their sigh reversed and save that to object called “c”. c <- c(1, -phi[1:lag_r]) Finally, we can find the polynomials or the roots using following command. Let’s save such polynominals in object called “rt” rt <- polyroot( c ) rt Alternatively, Mod(polyroot(c)) See the following diagram. Here, two real roots and tow complex roots$(1-1.417z+0.587{{z}^{2}}-0.125{{z}^{3}}-0.024{{z}^{4}})$are given as$\begin{align}
& {{z}_{1}}=1.028; \\
& {{z}_{2}}={{z}_{3}}=1.285+1.716i; \\
& {{z}_{4}}=-8.753 \\
\end{align}$Let’s represent these in modulus term using “complex” command and “modulus=Mod(rt)” and save it in “zz.shift” object. The modulus of complex number${{z}_{3}}=1.285+1.716i$is given as:$\sqrt{{{(1.285)}^{2}}+{{(1.716)}^{2}}}=2.141$. zz.shift <- complex(modulus = Mod(rt)) zz.shift The moduli of roots are given as:$\begin{align}
& |{{z}_{1}}|=1.028; \\
& |{{z}_{2}}|=|{{z}_{3}}|=2.141; \\
& |{{z}_{4}}|=8.753 \\
\end{align}$The modulus of polynomials roots of${{\phi }_{p}}(B)$i.e$|{{z}_{i}}|>1$or outside of unit root circle, hence this$AR(4)\$ process for “r” is stationary. A plot can be developed with following codes.
# Plotting the roots in a unit circle
x <- seq(-1, 1, length = 1000)
y1 <- sqrt(1- x^2)
y2 <- -sqrt(1- x^2)
plot(c(x, x), c(y1, y2), xlab='Real part',
ylab='Complex part', type='l',
main='Unit Circle', ylim=c(-2, 2), xlim=c(-2, 2))
abline(h=0)
abline(v=0)
points(Re(polyroot(c)),
Im(polyroot(c)), pch=19)
legend(-1.5, -1.5, legend="Roots of AR(2)", pch=19)
References:
Pfaff, Bernhard (2008), Analysis of Integrated and Cointegrated Time Series with R, Springer Publishing Company
Martin, Vance, Hurn, Stan, & Harris, David (2013) Econometric Modelling with Time Series : Specification, Estimation and Testing. Themes in Modern Econometrics. Cambridge University Press, New York. | 2019-02-16 21:16:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9723020195960999, "perplexity": 3994.9838520594963}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247481122.31/warc/CC-MAIN-20190216210606-20190216232606-00108.warc.gz"} |
http://mathreports.ru/en/articles/about-the-convergence-of-the-fourier-transform/ | # About the convergence of the Fourier transform
### DOI: 10.31029/demr.15.1
The main result is the proof of the theorems, the results of which one can characterize as a weak form of the formula for the inversion of the bi-dimmensional Fourier transform. Sufficient conditions on a function are obtained for a weak (of degree $r$) convergence of bi-dimmensional Fourier transform for a function $f(x;y)$. These conditions have an integral form and describe the behavior of the function near the border of a rectangle. A similar theorem is proved, in which the Fourier transform of a function $f$ is replaced by the Fourier transform of another function $g$, the norm of the central difference of which does not exceed the norm of the central difference of $f$. The principal objective is to study the behavior of the Fourier transform of $g$ and $f$.
Keywords: Two-dimensional Fourier transform, Riemann-Lebesgue theorem.
| 2022-08-18 20:21:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8371419906616211, "perplexity": 129.95963037361622}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573399.40/warc/CC-MAIN-20220818185216-20220818215216-00298.warc.gz"} |
http://meta-guide.com/dialog-systems/speech-act-dialog-systems-2017 | ## Speech Act & Dialog Systems 2017
Notes:
• Computational argumentation system
• Human-machine dialog
• Virtual storyteller
Resources:
• arg-tech.org .. centre for argument technology
• fungramkb.com .. multipurpose lexico-conceptual knowledge base for natural language processing
Wikipedia:
References:
The fourth dialog state tracking challenge
S Kim, LF D’Haro, RE Banchs, JD Williams… – Dialogues with Social …, 2017 – Springer
… edition of the challenge also proposed a series of pilot tasks for evaluating each of the core components needed for developing end-to-end dialog systems. More specifically, four pilot tasks were offered: Spoken Language Understanding (SLU), Speech Act Prediction (SAP …
Topic independent identification of agreement and disagreement in social media dialogue
A Misra, M Walker – arXiv preprint arXiv:1709.00661, 2017 – arxiv.org
… Amita Misra & Marilyn A. Walker Natural Language and Dialogue Systems Lab Computer Science Department University of California, Santa Cruz maw … As conversants A and B partici- pate in a dialogue, A and B communicate through dialogue speech acts such as PROPOSALS …
Some principles of rational mutual inquiry
D Hitchcock – On Reasoning and Argument, 2017 – Springer
… Rules for a dialogue system for mutual inquiry should conform to at least the following 18 principles: externalization, dialectification, mutuality, turn … An interlocutor might, for example, use a single turn both to express agreement with the immediately preceding speech act and to …
End-to-end joint learning of natural language understanding and dialogue manager
X Yang, YN Chen, D Hakkani-Tür… – … , Speech and Signal …, 2017 – ieeexplore.ieee.org
… Index Terms— language understanding, spoken dialogue systems, end-to-end, dialogue manager, deep learning 1. INTRODUCTION … Labels of sys- tem actions are defined as the concatenation of categories and attributes of speech acts, eg QST WHEN …
Action-based grammar
R Kempson, R Cann, E Gregoromichelaki… – Theoretical …, 2017 – degruyter.com
Enabling robots to understand indirect speech acts in task-based interactions
G Briggs, T Williams, M Scheutz – Journal of Human-Robot Interaction, 2017 – dl.acm.org
Page 1. Enabling Robots to Understand Indirect Speech Acts in Task-Based Interactions … Keywords: human-robot dialogue, human perceptions of robot communication, robot architectures, speech act theory, intention understanding 1. Introduction …
Speech acts in a dialogue game formalisation of critical discussion
J Visser – Argumentation, 2017 – Springer
… Krabbe (2012, 2013), has the same objective: making the system formal 2 . While Krabbe’s approach and his resulting dialogue system CD 1 … present paper—Krabbe’s formalisation deals with the dialectical dimension of the ideal model, at the expense of the role of speech acts …
Using Summarization to Discover Argument Facets in Online Ideological Dialog
A Misra, P Anand, JEF Tree, M Walker – arXiv preprint arXiv:1709.00662, 2017 – arxiv.org
… Amita Misra, Pranav Anand, Jean Fox Tree, and Marilyn Walker UC Santa Cruz Natural Language and Dialogue Systems Lab 1156 N. High … Other work attempts to identify general categories of speech-acts such as disagree- ments or justifications (Misra and Walker, 2015; Bi …
Dialogues with Social Robots
K Jokinen, G Wilcock – Springer, 2017 – Springer
… Professor Traum presented examples of role-play dialogue systems from a wide variety of activities, genres, and roles, focussing on … task on dialogue state tracking at sub-dialogue level; four optional pilot tasks on spoken language understanding, speech act prediction, spoken …
Quotation in dialogue
E Gregoromichelaki – The semantics and pragmatics of quotation, 2017 – Springer
… Splitting the I’s and crossing the you’s: Context, speech acts and grammar … As regards the characterisation of particular dialogue actions, in this model speech acts are conceptualised as events too, termed as conversational events …
It’s Not What You Do, It’s How You Do It: Grounding Uncertainty for a Simple Robot
J Hough, D Schlangen – Proceedings of the 2017 ACM/IEEE …, 2017 – dl.acm.org
… After providing background on grounding uncertainty in §2, we present a grounding model for HRI which draws on dialogue systems research in §3. The … with our approach that the robot’s non-verbal actions are treated with the same status as dialogue/speech acts in dialogue …
Robust dialog state tracking for large ontologies
F Dernoncourt, JY Lee, TH Bui, HH Bui – Dialogues with Social Robots, 2017 – Springer
… Yet, dialog state tracking is crucial for reliable operations of a spoken dialog system because the latter relies on the estimated dialog state to … All the recorded dialogs with the total length of 21 h have been manually transcribed and annotated with speech act and semantic labels …
Speaker role contextual modeling for language understanding and dialogue policy learning
TC Chi, PC Chen, SY Su, YN Chen – arXiv preprint arXiv:1710.00164, 2017 – arxiv.org
… Under the sce- nario of dialogue systems and the communication patterns, we take the tourist as a user and the guide as the dialogue agent (system) … the cur- rent utterance x = {wt}T 1 , the goal is to predict the user intents of x, which includes the speech acts and associated …
Natural Language Understanding (NLU, not NLP) in Cognitive Systems.
M McShane – AI Magazine, 2017 – search.ebscohost.com
… such as lexical and referential ambiguity, ellipsis, false starts, spurious repetitions, semantically vacuous fillers, nonliteral language, indirect speech acts, implicatures, and … For example, agents in dialogue systems receive one and only one formula- tion of each utterance …
ParlAI: A Dialog Research Software Platform
AH Miller, W Feng, A Fisch, J Lu, D Batra… – arXiv preprint arXiv …, 2017 – arxiv.org
… As the dict is extensible, we can add more fields over time, eg for audio and other sensory data, as well as actions other than speech acts … The ubuntu dialogue corpus: A large dataset for re- search in unstructured multi-turn dialogue systems. arXiv preprint arXiv:1506.08909 …
S Oraby, P Gundecha, J Mahmud, M Bhuiyan… – Proceedings of the …, 2017 – dl.acm.org
… Modern intelligent con- versational [1, 31] and dialogue systems draw principles from many disciplines, including philosophy, linguistics, computer science, and … Previous work has explored speech act modeling in different domains (as a predecessor to dialogue act modeling) …
Interactional Dynamics and the Emergence of Language Games
A Eshghi, I Shalyminov, O Lemon – … of the ESSLLI 2017 workshop on …, 2017 – ceur-ws.org
… domain-specific reward signal/goal is sufficient for certain word sequences becoming routinised and learned as ways of per- forming specific kinds of speech act within the do- main, without any prior, procedural specifications of such actions. Thus, a dialogue system learns not …
A novel description language for two-agent dialogue games
A Sawicka, M Kacprzak, A Zbrzezny – International Joint Conference on …, 2017 – Springer
… Our goal is to build a novel system for dialogue systems verification, which was inspired by existing concepts, such as dialogue systems, multi-agent systems, and model … Locution rules define a set of locutions (actions, speech acts) the player is allowed to utter during the game …
Convolutional Neural Network using a threshold predictor for multi-label speech act classification
G Xu, H Lee, MW Koo, J Seo – Big Data and Smart Computing …, 2017 – ieeexplore.ieee.org
… Keywords—Multi-label; Convolutional Neural Network; Speech Act Classification; Algorithm Adaptation. I. INTRODUCTION The spoken language understanding (SLU) is one of the core components of an end-to-end dialogue system [1]. The SLU is aimed at extracting semantic …
Latent intention dialogue models
TH Wen, Y Miao, P Blunsom, S Young – arXiv preprint arXiv:1705.10229, 2017 – arxiv.org
… For exam- ple both goal-oriented dialogue systems (Wen et al., 2017; Bordes & Weston, 2017) and sequence-to-sequence learn- ing chatbots (Vinyals & Le, 2015; Shang et al., 2015; Ser- ban et al., 2015) struggle to generate diverse yet causal re- sponses (Li et al., 2016a …
Getting reliable annotations for sarcasm in online dialogues
R Swanson, S Lukin, L Eisenberg, TC Corcoran… – arXiv preprint arXiv …, 2017 – arxiv.org
… Luke Eisenberg, Thomas Chase Corcoran and Marilyn A. Walker University of California Santa Cruz Natural Language and Dialogue Systems Lab Computer … have been deliberately constructed by the speaker to be ambiguous, in the same way that indirect speech acts may be …
Challenging Neural Dialogue Models with Natural Data: Memory Networks Fail on Incremental Phenomena
I Shalyminov, A Eshghi, O Lemon – arXiv preprint arXiv:1709.07840, 2017 – arxiv.org
… 2010. Splitting the ‘I’s and crossing the ‘You’s: Context, speech acts and grammar … 2010. To- wards incremental speech generation in dialogue systems. In Proceedings of the SIGDIAL 2010 Con- ference, pages 1–8, Tokyo, Japan, September …
The argument web: An online ecosystem of tools, systems and services for argumentation
C Reed, K Budzynska, R Duthie, M Janier… – Philosophy & …, 2017 – Springer
… at the point of premise-giving (van Eemeren and Grootendorst 1992), which relies upon interpreting premise-giving as a complex speech act … 8). We then express this account as a formal dialogue system with sets of locution, structural, commitment and termination rules (3, 4, 8 …
Characterizing online discussion using coarse discourse sequences
AX Zhang, B Culbertson… – Proceedings of the …, 2017 – pdfs.semanticscholar.org
… conversations. Much research has demonstrated the power of using discourse acts, also known as speech acts, which are categories of utterances that pertain to their role in the discussion (eg “question” or “answer”). Researchers …
An Ontology-Based Dialogue Management System for Virtual Personal Assistants
M Wessel, G Acharya, J Carpenter, M Yin – … Spoken Dialogue Systems …, 2017 – uni-ulm.de
… J.-F. Yeh, C.-H. Wu, M.-J. Chen, Ontology-Based Speech Act Identification in a Bilingual Dialog System Using Partial Pattern Trees, Journal of the American Society for Information Science and Technology, Volume 59 Number 5, Wiley Subscription Services, pp 684–694, 2008 …
Words matter: automatic detection of teacher questions in live classroom discourse using linguistics, acoustics, and context
PJ Donnelly, N Blanchard, AM Olney, S Kelly… – Proceedings of the …, 2017 – dl.acm.org
… The authors used a hidden Markov model on the Switchboard corpus [15] to identify speech acts, such as questions, statements, or apologies, achieving an accuracy of 65% on ASR transcriptions (WER 0.41) and 71% based on human transcriptions (chance level 35%; human …
Modeling Dialogue Acts with Content Word Filtering and Speaker Preferences
Y Jo, MM Yoder, H Jang, CP Rosé – Proceedings of the Conference …, 2017 – ncbi.nlm.nih.gov
… Dialogue acts (DAs), or speech acts, represent the intention behind an utterance in conversation to achieve a conversational goal … not formally distinguished in the formalization (Becker et al., 2011), especially in domain-specific applications in dialogue systems (Gavaldà, 2004) …
Learning symmetric collaborative dialogue agents with dynamic knowledge graph embeddings
H He, A Balakrishnan, M Eric, P Liang – arXiv preprint arXiv:1704.07130, 2017 – arxiv.org
… The contributions of this work are: (i) a new symmetric collaborative dialogue setting and a large dialogue corpus that pushes the boundaries of existing dialogue systems; (ii) DynoNet, which integrates semantically rich … (We only show the most frequent speech acts therefore the …
Bootstrapping incremental dialogue systems from minimal data: the generalisation power of dialogue grammars
A Eshghi, I Shalyminov, O Lemon – arXiv preprint arXiv:1709.07858, 2017 – arxiv.org
Page 1. Bootstrapping incremental dialogue systems from minimal data: the generalisation power of dialogue grammars … There are currently several key problems for the practical data-driven (rather than hand-crafted) development of task-oriented dialogue systems …
Scalable Multi-Domain Dialogue State Tracking
A Rastogi, D Hakkani-Tur, L Heck – arXiv preprint arXiv:1712.10224, 2017 – arxiv.org
… These speech acts may have an optional slot parameter, if a slot can be deduced from the utterance … The delexicalized versions of system utterances are obtained from the language generation component of the dialogue system. 3.2.1. Utterance related features …
Attentive listening system with backchanneling, response generation and flexible turn-taking
D Lala, P Milhorat, K Inoue, M Ishida… – Proceedings of the 18th …, 2017 – aclweb.org
… However we improve the precision of taking the turn, which is critical in spoken dialogue systems, from 0.428 to 0.624 … We used a majority vote to determine the overall rating of each speech act. The ratings on the coherence of each statement are shown in Figure 6 …
Computational Approaches to Dialogue
D Traum – The Routledge Handbook of Language and Dialogue, 2017 – books.google.com
… Moreover, plan-based accounts of speech acts can also be used in reverse to understand user intentions behind utterances (Allen and Perrault 1980 … Sidner (1990) develop a similar model of ‘Shared Plans’, that has been the foundation of a series of dialogue systems using the …
Grounding language by continuous observation of instruction following
T Han, D Schlangen – Proceedings of the 15th Conference of the …, 2017 – aclweb.org
… Grounding Language by Continuous Observation of Instruction Following Ting Han and David Schlangen CITEC, Dialogue Systems Group, Bielefeld University first.last@uni- bielefeld.de Abstract … 2015. Learning in the rational speech acts model …
Systemic functional linguistics and computation: New directions, new challenges
J Bateman, D McDonald, T Hiippala… – … Handbook of Systemic …, 2017 – helsinki.fi
… The usual components of a computational dialogue system therefore span a con- siderable breadth of linguistic knowledge as well: ranging from spoken language recognition, parsing, semantic analysis, contextualization, recognition of speech acts, designing responses …
DailyDialog: A Manually Labelled Multi-turn Dialogue Dataset
Y Li, H Su, X Shen, W Li, Z Cao, S Niu – arXiv preprint arXiv:1710.03957, 2017 – arxiv.org
… 1 Introduction Developing intelligent chatbots and dialog systems is of great significance to both commercial and aca- demic camps … This factor has been extensively explored under the name of dialog act and speech act. In general, dialog acts represent the communication …
Sounding Board–University of Washington’s Alexa Prize Submission
H Fang, H Cheng, E Clark… – Alexa Prize …, 2017 – pdfs.semanticscholar.org
… There has been a substantial amount of work on conversational dialog systems that roughly falls into two groups, depending on whether … Thus, our generation strategy cannot benefit from much of the prior work and instead emphasizes selection of speech acts and presentation …
Yeah, Right, Uh-Huh: A Deep Learning Backchannel Predictor
R Ruede, M Müller, S Stüker, A Waibel – arXiv preprint arXiv:1706.01340, 2017 – arxiv.org
… virtual psychiatrists and pets attempt to add an emotional and social dimension to human interaction that may go beyond improving the user experience of existing dialog systems, and thus … A first approach towards detecting speech acts (including BCs) was proposed by Ries [19 …
Dynamic Time-Aware Attention to Speaker Roles and Contexts for Spoken Language Understanding
PC Chen, TC Chi, SY Su, YN Chen – arXiv preprint arXiv:1710.00165, 2017 – arxiv.org
… Under the scenario of dialogue systems and the com- munication patterns, we take the tourist as a user and the guide as the dialogue agent (system) … the current utterance x = {wt}T 1 , the goal is to pre- dict the user intents of x, which includes the speech acts and associated …
Uttering only what is needed: Enthymemes in multi-agent systems
AR Panisson, RH Bordini – Proceedings of the 16th Conference on …, 2017 – dl.acm.org
… Using enthymemes in an inquiry dialogue system. In Proceedings of the 7th international joint conference on Autonomous agents and multiagent systems-Volume 1, pages 437–444 … Formal semantics of speech acts for argumentative dialogues. In Thirteenth Int. Conf …
Arguments from authority and expert opinion in computational argumentation systems
D Walton, M Koszowy – AI & SOCIETY, 2017 – Springer
… However, these systems do have the capability to go part way when extended even further by using formal dialogue systems to model properties … Such a modeling includes the systematic study of the kinds of speech acts used as moves as a proponent puts forward the argument …
Style transfer for prosodic speech
A Perez, C Proctor, A Jain – 2017 – stanford.edu
… agents to express personalities and emotions, as well as much more effectively engag- ing in speech acts such as making requests, es- tablishing social situations, and in turn eliciting emotional speech from their human interlocutors. As such, dialogue systems may participate …
Assessment with computer agents that engage in conversational dialogues and trialogues with learners
AC Graesser, Z Cai, B Morgan, L Wang – Computers in Human Behavior, 2017 – Elsevier
… As in all dialogues, the tutor and student take turns taking the conversational floor. AutoTutor segments the information within each student turn into speech acts and classifies the speech acts into different categories (D’Andrade & Wish, 1985) …
A formal model of an argumentative dialogue in the management of emotions
M Kacprzak, A Sawicka, A Zbrzezny… – Logic and Logical …, 2017 – apcz.umk.pl
… about emotions. First of all, the set of the locutions (speech acts) is extended. Two new … legal answers. All of these define a protocol, which is the basis of which to construct a mathematical model of the dialogue system. This paper …
Shifting the load: A peer dialogue agent that encourages its human collaborator to contribute more to problem solving
C Howard, P Jordan, B Di Eugenio, S Katz – International Journal of …, 2017 – Springer
… TuTalk supports the creation of natural language dialogue systems for educational applications and allows for both tutorial and conversational dialogues … the role each utterance plays in the dialogue and can be considered as an operationalization of speech acts (Austin 1962 …
Test Collections and Measures for Evaluating Customer-Helpdesk Dialogues
Z Zeng, C Luo, L Shang, H Li… – Proceedings of …, 2017 – pdfs.semanticscholar.org
… [21] proposed the PARADISE (PAR- Adigm for Dialogue System Evaluation) framework for evaluating task-oriented spoken dialogue systems … Act Tagging for Evaluation) was introduced, which enables three orthogonal annotations along the axes of speech-act (eg, “request …
Comparative Analysis of Word Embedding Methods for DSTC6 End-to-End Conversation Modeling Track
Z Bairong, W Wenbo, L Zhiyu… – … 6th Dialog System …, 2017 – workshop.colips.org
… M.-W. Koo, and J. Seo, “Convolutional neural net- work using a threshold predictor for multi-label speech act clas- sification … P. Pasupat, and R. Sarikaya, “Enriching word embeddings using knowledge graph for semantic tagging in conversational dialog systems,” genre, 2010 …
Sound synthesis for communicating nonverbal expressive cues
FA Martín, Á Castro-González, MÁ Salichs – IEEE Access, 2017 – ieeexplore.ieee.org
… rests upon deep roots that are not verbal but acoustic. These non-verbal contexts leverage speech acts, words, and utterances, to convey their meaning. New electronic devices, such as mobile phones, tablets, and computers …
Stakeholder theory: A deliberative perspective
UH Richter, KE Dow – Business Ethics: A European Review, 2017 – Wiley Online Library
… intersubjectively. One has to enter into a process of legitimation—the discourse—where speech act, validity claims replete with normative assumptions, and performative commitments are exchanged to create objectivity. Companies …
A tale of two architectures: A dual-citizenship integration of natural language and the cognitive map
T Williams, C Johnson, M Scheutz… – Proceedings of the 16th …, 2017 – dl.acm.org
… Furthermore, such referring expressions need not be used in the context of direct commands: interlocutors are free to use so-called indirect speech acts that follow conven- tionalized social norms (eg,“I need to go to the bathroom”), which DIARC interprets based on context [47] …
Strategies and mechanisms to enable dialogue agents to respond appropriately to indirect speech acts
G Briggs, M Scheutz – … (RO-MAN), 2017 26th IEEE International …, 2017 – ieeexplore.ieee.org
… Abstract—Humans often use indirect speech acts (ISAs) when issuing directives … been less attention devoted to how linguistic responses to ISAs might differ from those given to literal directives and how to enable different response forms in these computational dialogue systems …
Dialog acts in greeting and leavetaking in social talk
E Gilmartin, B Spillane, M O’Reilly, K Su… – Proceedings of the 1st …, 2017 – dl.acm.org
… Dialog systems model spoken or written synchronous/near-synchronous interactions, often to fulfill a task but increasingly to create the illusion of social interaction … [5] D. Traum. 1999. Speech acts for dialogue agents. Foundations of rational agency 14 (1999), 169–202. 30
Convolutional Neural Network using a threshold predictor for multi-domain dialogue.
G Xu, H Lee – uni-leipzig.de
… I. INTRODUCTION The spoken language understanding (SLU) is one of the core components of an end-to-end dialogue system [1]. The SLU … Furthermore, a multi-label classification task significantly increases possible combinations of speech acts to be annotated to utterances …
The Routledge Handbook of Language and Dialogue
E Weigand – 2017 – books.google.com
… Benjamins). On the basis of a notion of ‘language as dialogue’she developed a dialogic speech act theory and a holistic theory of human action and behaviour in performance in her book Dialogue: The Mixed Game (2010). Page 3 …
Towards improving the performance of chat oriented dialogue system
R Jiang, RE Banchs – Asian Language Processing (IALP), 2017 …, 2017 – ieeexplore.ieee.org
… In Agent’s Processing, Cognition: https://www.chatbots.org/images/uploads/researchJpapers/ 9491.p df [6] Rafael E. Banchs, Haizhou Li, IRIS: a Chat-oriented Dialogue System based on the Vector Space Model … 29, N3-4, 2002, pages 135-155 [11] JR Searle, Speech Acts …
Towards an Argumentative Dialogue System
N Rach, W Minker, S Ultes – 2017 – cmna.csc.liv.ac.uk
… [7] Henry Prakken. 2000. On dialogue systems with speech acts, arguments, and counterarguments. In European Workshop on Logics in Arti cial Intelligence. Springer, 224–238. [8] Henry Prakken. 2006. Formal systems for persuasion dialogue …
NICT Kyoto Dialogue Corpus
K Ohtake, E Mizukami – Handbook of Linguistic Annotation, 2017 – Springer
… by our research group and focus on two types of tags: speech act (SA) and semantic content. These SA and semantic content tags have been designed to express the dialogue acts (DAs) of each utterance. Many studies have focused on developing spoken dialogue systems …
Will this dialogue be unsuccessful? Prediction using audio features
M Kotti, A Papangelis, Y Stylianou – 2017 – scai.info
… To overcome those drawbacks an innovative way of predicting success in spoken dialogue systems using the audio channel of the user is … to achieve the aforementioned quality estimation, researchers have resorted to the use of dialogue features, such as speech acts [33] or …
A Case Study on the Relevance of the Competence Assumption for Implicature Calculation in Dialogue Systems
JV Fischer – International Conference of the German Society for …, 2017 – Springer
… to the situation in dialogue system research, the CA has not yet been a topic of empirical studies. The pragmatic approach in linguistics assumes that pragmatic reasoning is necessarily global (Sauerland 2004, p. 40), with which it refers to entire speech acts, not embedded …
Dependency Parsing and Dialogue Systems: an investigation of dependency parsing for commercial application
A Adams – 2017 – diva-portal.org
Page 1. Dependency Parsing and Dialogue Systems an investigation of dependency parsing for commercial application … Page 2. Abstract In this thesis, we investigate dependency parsing for commercial application, namely for future integration in a dialogue system …
Bootstrapping dialogue systems: the contribution of a semantic model of interactional dynamics
A Eshghi, I Shalyminov, O Lemon – CLASP Papers in Computational …, 2017 – gupea.ub.gu.se
… 83 Page 89. Dimitrios Kalatzis, Arash Eshghi, and Oliver Lemon. 2016. Bootstrapping incremental dialogue systems: using linguistic knowledge to learn from minimal data … 2010. Splitting the ‘I’s and crossing the ‘You’s: Context, speech acts and grammar …
Investigation the Effect of Colloconstructural Corpus-based Instruction on Pragmalinquistic Knowledge of Request Speech Act: Evidence from Iranian EFL Students
B Sabzalipour, M Koosha… – International Journal of …, 2017 – journals.aiac.org.au
… The future researchers can study in the field of psychology or other fields of studies. The final point is pragmatics itself in which is not limited to speech acts … Alfattah, G. & Ravindranath, H. (2009) Probabilistic meth- ods in spoken dialogue systems …
Turn-Taking Offsets and Dialogue Context
PA Heeman, R Lunsford – Proc. Interspeech 2017, 2017 – csee.ogi.edu
… Hence, we need to better understand how human turn-taking works so that we can build spoken dialogue system that can engage in turn … Are they defined in terms of pauses, intonation, pragmatics (speech acts), or when the current speaker intends someone else to take the turn …
Collaboration-based User Simulation for Goal-oriented Dialog Systems
D Didericksen, ORKSL Zhou, J Kramer – alborz-geramifard.com
… Goal-oriented Dialog Systems Devin Didericksen? University of Washington diderick@uw.edu … Therefore, if we had the user intents explicitly annotated, we could operate on speech-act or dialog-act level and then generate the utterance using a NLG model …
Interruptions as Speech Acts
P Wallis, B Edmonds – ceur-ws.org
… This example demonstrates just how hard human communicators are willing to work at recognizing intent in the speech acts of others. By contrast consider the … Critically for dialog systems, the goal is explicit and can be used in explanations of behaviour …
Dialogue Act Segmentation for Vietnamese Human-Human Conversational Texts
TL Ngo, KL Pham, MS Cao, SB Pham… – arXiv preprint arXiv …, 2017 – arxiv.org
… It is important for many applications: dialogue systems, automatic translation machine [2], automatic speech recognition, etc [3] [4] and has … II. BACKGROUD: FUNCTIONAL SEGMENT AND UNITS OF A DIALOGUE DAs are extended from the speech act theory of Austin [10] and …
Improving Relationships Based on Positive Politeness Between Humans and Life-Like Agents
T Miyamoto, D Katagami, Y Shigemitsu – Proceedings of the 5th …, 2017 – dl.acm.org
… First, we design a dialog system based on the politeness theory … Wx of the FTA is the sum of D, P, and R. Since P and R fluctuate in different cultures and societies, the resulting weight of the FTA varies depending on the given culture and society, even in the same speech act …
Underspecification in Natural Language Understanding for Dialog Automation
J Chen, S Bangalore – … of the International Conference Recent Advances …, 2017 – acl-bg.org
… application domain are specific to that particu- lar domain and sometimes are even crafted specif- ically to the flow of a particular dialog system … of the intent ontology is shown in Figure 2. Finally, con- versational handlers are labels which are similar to speech acts, and guide …
Towards encoding of the transition relation in dialogue games model checking
A Sawicka, M Kacprzak, A Zbrzezny – csp2017.mimuw.edu.pl
… 5. 2 Inspirations Our dialogue systems verification system is inspired by existing concepts, such as dia- logue systems, multi-agent systems, and model checking … Locution rules define a set of locutions (actions, speech acts) the player is allowed to utter dur- ing the game …
“nee intention enti?” towards dialog act recognition in code-mixed conversations
DS Jitta, KR Chandu, H Pamidipalli… – Asian Language …, 2017 – ieeexplore.ieee.org
… Despite the change of such language dynamics, current dialog systems cannot handle a switch between languages across sentences … Searle has introduced the concept of speech acts (assertives, directives, commissives, expressives, declarations) [3], that come under Austins …
Discovering Domain Specific Dialog Acts
S Tomkins, A Xu, Z Liu, Y Guo – travellingscholar.com
… et al., 1998) describe the ini- tial level of speech acts in discourse. They can facilitate conversation modeling and are useful in many conversation applications, such as automatic dialog summarization (Murray et al., 2006; Bhatia et al., 2014) and dialogue systems (Ritter et al …
Bottester: Testing Conversational Systems with Simulated Users
M Vasconcelos, H Candello, C Pinhanez, T dos Santos – 2017 – researchgate.net
… Moreover, we can test the accuracy of the chatbot systems natural language intent (speech-act) classifier … 2016. How NOT To Evaluate Your Dialogue System: An Empirical Study of Unsupervised Evaluation Metrics for Dialogue Response Generation. In Proc …
Neural sentence embedding using only in-domain sentences for out-of-domain sentence detection in dialog systems
S Ryu, S Kim, J Choi, H Yu, GG Lee – Pattern Recognition Letters, 2017 – Elsevier
… Collecting ID sentences is a necessary step in building many data-driven dialog systems … We think that the task of OOD sentence detection is more similar to domain-category analysis than to other tasks such sentiment analysis or speech-act analysis, so we expect that the …
What information should a dialogue system understand?: Collection and analysis of perceived information in chat-oriented dialogue
K Mitsuda, R Higashinaka, Y Matsuo – workshop.colips.org
… In: P. Cole, J. Morgan (eds.) Syntax and semantics, vol. 3: Speech acts, pp. 41–58 … In: Proc. ACL/IJCNLP, pp. 757–762 (2015) 6. Kim, Y., Bang, J., Choi, J., Ryu, S., Koo, S., Lee, GG: User information extraction for per- sonalized dialogue systems. In: Proc …
Resuscitation procedures as multi-party dialogue
E Marzuki, C Cummins, H Rohde… – … ) Workshop on the …, 2017 – pdfs.semanticscholar.org
… systems, helping to establish, in the case of the phone conversations, when to transfer calls from an automated dialogue system to human … Austin’s (1962) classi- fication of speech acts, and later, Searle’s (1976) Speech Act Theory (SAT), paved the way for context-specific …
Controlling Interaction in Multilingual Conversation Revisited: A Perspective for Services and Interviews in Mandarin Chinese
J Du, C Alexandris, D Mourouzidis, V Floros… – … Conference on Human …, 2017 – Springer
… Spoken Dialog Systems in the Service Sector (Call Centers for mobile telephones) [2], with Directed Dialogs and registration of the path of the interaction with the respective Speech Acts, keywords and free input [2]. The implemented modules of the Dialog System applications [2 …
Joint Learning of Dialog Act Segmentation and Recognition in Spoken Dialog Using Neural Networks
T Zhao, T Kawahara – Proceedings of the Eighth International Joint …, 2017 – aclweb.org
… Natural language understanding (NLU), as an important component of dialog system, is usually responsible for dialog act (DA) or dialog intent tagging, where text classification techniques are necessary. Dialog act (also speech act) is a representation of the meaning of a …
From Language as a System of Signs to Language Use
J Allwood – The Routledge Handbook of Language and Dialogue, 2017 – books.google.com
… (v)“Pragmatics is the study of deixis (at least in part), implicature, presupposition, speech acts, and aspects of … Levinson (1979), Allwood (1976), sociolinguistics, anthropological linguistics, intercultural communication, communication studies and computer-based dialog systems …
Summarizing Dialogic Arguments from Social Media
A Misra, S Oraby, S Tandon, P Anand… – arXiv preprint arXiv …, 2017 – arxiv.org
… Shereen Oraby, Shubhangi Tandon, Sharath TS, Pranav Anand and Marilyn Walker UC Santa Cruz Natural Language and Dialogue Systems Lab 1156 N … The theoretical literature discusses the ways in which dialogic argumentation shows dif- ferent speech act uses than in less …
Feedback relevance spaces: The organisation of increments in conversation
C Howes, A Eshghi – IWCS 2017—12th International Conference on …, 2017 – aclweb.org
… system. In J. Bos and S. Pulman (Eds.), Proceedings of the 9th International Conference on Compu- tational Semantics, Oxford, UK, pp. 365–369. Purver, M., E. Gregoromichelaki, W. Meyer-Viol, and R. Cann (2010). Splitting the ‘I’s and crossing the ‘You’s: Context, speech acts …
Form-based Dialogue Structure for Task-oriented Conversations
A Chotimongkol, AI Rudnicky – simulation – cs.cmu.edu
… engineering-oriented structures is Conversation acts [4], a structure of four levels action which extends the speech act theory [5 … framework that has these desired properties along with the concrete mapping between dialogue structure components and dialogue system behavior …
Designing interactive, automated dialogues for L2 pragmatics learning
V Timpe-Laughlin, K Evanini, A Green… – SEMDIAL 2017 …, 2017 – academia.edu
… in this scenario structure are nine learn- ing modules, each of which focuses on a specific pragmatic phenomenon or speech act that is … this capability was the development of in- teractive speaking tasks for each learning module that deploy a spoken dialogue system (SDS) tech …
Intelligent Personal Assistant with Knowledge Navigation
A Kumar, R Dutta, H Rai – arXiv preprint arXiv:1704.08950, 2017 – arxiv.org
… dialog systems. In Proceedings of annual meeting of the association for computational linguistics (ACL), Sofia, Bulgaria. Mey, JL (2001). Pragmatics: An introduction (2nd ed.). Oxford: Blackwell. Moldovan, C., Rus, V., & Graesser, A. (2011). Automated speech act classification …
Grammars as Mechanisms for Interaction: The Emergence of Language Games
A Eshghi, O Lemon – Theoretical Linguistics, 2017 – degruyter.com
… have dubbed babbling) 1 within a particular interactive task/domain with its characteristic reward signal and goal leads to certain word sequences becoming routinized and learned for performing specific kinds of speech act within that domain. Thus, a dialogue system learns not …
Annotation of greeting, introduction, and leavetaking in dialogues
E Gilmartin, B Spillane, M O’Reilly, C Saam… – Proceedings of the 13th …, 2017 – aclweb.org
… of the ISO standard to allow fuller annotation of dialogues in more social as well as task-based terms, and that their use in the development of the ADELE system will be useful to other researchers in the field of casual or social dialogue system design … Vol.3, Speech acts …
Dialogue Act Semantic Representation and Classification Using Recurrent Neural Networks
P Papalampidi, E Iosif, A Potamianos – SEMDIAL 2017 SaarDial, 2017 – academia.edu
… 1 Introduction Dialogue Act (DA) classification constitutes a ma- jor processing step in Spoken Dialogue Systems (SDS) assisting the understanding … Qadir and Riloff (2011) built speech act classifiers in message board posts uti- lizing lexical, syntactic and semantic features by …
Dialogue Act Recognition for Conversational Agents
LE Hacquebord – 2017 – dspace.library.uu.nl
… This chapter provides some background information on natural language processing (NLP) and dialogue systems that is necessary to understand the … Such actions are fittingly called speech acts, although they are more commonly referred to as dialogue acts in the computer …
Developing Argumentation Dialogues for Open Multi-Agent Systems
B Testerink, F Bex – florisbex.com
… Our framework is open-source and we hope that it stimulates the develop- ment of argumentation dialogue systems or may serve as an example to other … The link between argument 1 and 2 stems from speech act theory, where a speech act is a locution (‘Alice says that Carl is a …
A Dynamic Model of Trust in Dialogues
G Ogunniye, A Toniolo, N Oren – … Workshop on Theorie and Applications of …, 2017 – Springer
… Therefore, in our dialogue system, an argumentation framework $$\langle \mathcal {UCS}^t, \mathcal {R} \rangle$$ is induced by the set of arguments exchanged during dialogue in the universal commitment store and their respective attacking … 3.1 Protocol Rules and Speech Acts …
Automatic Evaluation of Chat-oriented Dialogue Systems using Large-scale Multi-references
H Sugiyama, T Meguro, R Higashinaka – uni-ulm.de
… features. Page 11. Automatic Evaluation of Chat-oriented Dialogue Systems 11 4 Conclusion … Processing. pp. 944–952 (2010) 10. Grice, HP: Logic and Conversation. In: Syntax and semantics. 3: Speech acts, pp. 41–58 (1975)
Actionable Email Intent Modeling with Reparametrized RNNs
CC Lin, D Kang, M Gamon, M Khabsa… – arXiv preprint arXiv …, 2017 – arxiv.org
… take. We argue that our ap- proach of action-based annotation is more scalable and theory- agnostic than traditional speech-act-based email intent anno- tation, while still carrying important semantic and pragmatic information …
Managing Casual Spoken Dialogue Using Flexible Schemas, Pattern Transduction Trees, and Gist Clauses
SZ Razavi, R EDU, LK Schubert, MR Ali, ME Hoque – cogsys.org
… where the steps of a schema are specified in a declarative language that allows for both explicit and abstract description of speech acts … Recent spoken dialog systems with a meaningful goal include sys- tems designed to help people improve their social skills (Tanaka et al …
Regularized Neural User Model for Goal Oriented Spoken Dialogue Systems
M Serras, MI Torres, A del Pozo – pdfs.semanticscholar.org
… References 1. Nicholas Asher and Alex Lascarides. Indirect speech acts. Synthese, 128(1):183–228, 2001. 2. Senthilkumar Chandramohan, Matthieu Geist, Fabrice Lefevre, and Olivier Pietquin. User simulation in dialogue systems using inverse reinforcement learning …
A Dialogue Interaction Module for a Decision Support System Based on Argumentation Schemes to Public Project Portfolio
L Cruz-Reyes, C Medina-Trejo… – Nature-Inspired Design …, 2017 – Springer
… 3.2 Dialogue System. The dialogue games (or dialogue systems) essentially define the principle of consistent dialogue and conditions under which a statement made by an individual is adequate … Locution Rules (speech acts, movements) …
Facial expressions and speech acts: experimental evidences on the role of the upper face as an illocutionary force indicating device in language comprehension
F Domaneschi, M Passarelli, C Chiorri – Cognitive processing, 2017 – Springer
… Cognitive Processing. August 2017 , Volume 18, Issue 3, pp 285–306 | Cite as. Facial expressions and speech acts: experimental evidences on the role of the upper face as an illocutionary force indicating device in language comprehension … Download fulltext PDF. Speech acts …
A Bridge from the Use-Mention Distinction to Natural Language Processing
S Wilson – The Semantics and Pragmatics of Quotation, 2017 – Springer
… fragment, the user says that he wishes to depart from Arlington, but the dialog system mishears it as “Allegheny West” … language or metalanguage to track dialogue state, clarify the meaning of terms, restate lost or misunderstood utterances, report others’ speech acts, and check …
Integration of context-aware conversational interfaces to develop practical applications for mobile devices
D Griol, JM Molina, A Sanchis – Journal of Ambient Intelligence …, 2017 – content.iospress.com
… devices, such as smartphones and tablets, have made it possible to deploy a large number of sensors and to integrate them into dialogue systems that provide … Intentions cannot be observed, but they can be described using the speech-act and dialogue-act theories [46] …
Persuasive Negotiation Dialogues using Rhetorical Arguments
M Morveli-Espinoza – Proceedings of the 16th Conference on …, 2017 – dl.acm.org
… force or convince an opponent to accept a given proposal [5]. These kinds of arguments have been studied in terms of speech acts [5] and … In this work, we study a dialogue system that contains the main rhetorical arguments and also other illocutions that let agents resolve their …
Enhancing Backchannel Prediction Using Word Embeddings
R Ruede, M Müller, S Stüker… – Proc. Interspeech …, 2017 – pdfs.semanticscholar.org
… [15] K. Ries, “HMM and neural network based speech act detection,” in … 497–500 vol.1. [16] R. Ruede, M. Müller, S. Stüker, and A. Waibel, “Yeah, right, uh- huh: A deep learning backchannel predictor,” accepted to the International Workshop on Spoken Dialogue Systems, 2017 …
Persuasive Strategies in Dialogue Games with Emotional Reasoning
M Kacprzak – International Joint Conference on Rough Sets, 2017 – Springer
… use. They are typically called locutions and include speech acts such as: claim, concede, why, question, and since … 32]. In the current article a dialogue system which describes parent-child persuasive conversation is presented …
All talk
B Thomas – Dialogue across Media, 2017 – books.google.com
… Dialogue and intimacy? 9 Looking at the conversations between Theodore and Samantha in terms of speech acts, it is noticeable that whereas … In his chapter, Piwek talks of the emergence of self-validating dialogue systems capable of learning from interactions with users and …
A Review of Technologies for Conversational Systems
J Masche, NT Le – … on Computer Science, Applied Mathematics and …, 2017 – Springer
… 1998. Chatbot (MegaHal) (Hutchens 1998). Markov chain models; Keyword matching. 1999. Dialog system (AutoTutor) (Graesser et al. 1999). NLP (POS tags); Dialog move generator; Latent semantic analysis; Regular expression matching; Speech act classifiers. 1998–99 …
An Integrated Architecture for Discourse: Generation, Interpretation, and Recipe Acquisition
N Green, JF Lehman – cs.cmu.edu
… This discourse architecture has been implemented as part of the NL-Soar dialogue system (Lehman, Van Dyke, Lonsdale, Green, & Smith, sub- mitted paper), a computational model of real-time human language … 4. Known as core speech acts in (Traum & Hinkelman, 1992) …
Selecting and Expressing Communicative Functions in a SAIBA-Compliant Agent Framework
A Cafaro, M Bruijnes, J van Waterschoot… – … on Intelligent Virtual …, 2017 – Springer
… OpenDial is a toolkit for developing spoken dialogue systems created by Lison [10 … set of parameters is employed compared to our system, however the authors’ focus is language generation whereas we propose a richer output that includes instances of speech acts (ie …
MISC: A data set of information-seeking conversations
D McDu, M Czerwinski, N Craswell – pdfs.semanticscholar.org
… e tran- scripts are segmented and annotated for speech acts, and some semantics, on a turn-by-turn basis … 2015. e Ubuntu dialogue corpus: A large dataset for research in unstructured multi-turn dialogue systems. In Proc. SIGDIAL. 285–294 …
TEATIME: a Formal Model of Action Tendencies in Conversational Agents
A Yacoubi, N Sabouret – perso.limsi.fr
… Keywords: Conversational Agents, Computational Model of Affects, Formal Model, Dialogue System … This theory allows us to implement a strong connection between emotions and speech acts during an agent-human interaction …
From Pragmatics to Dialogue
I Kecskes – The Routledge Handbook of Language and Dialogue, 2017 – books.google.com
… pragmatics that focuses on the application of pragmatics to dialogue modelling, especially the development of spoken dialogue systems intended to … A sequence of speech acts can be considered a dialogue which is a highly structured activity involving (at least) two agents …
Functions of Silences towards Information Flow in Spoken Conversation
SA Chowdhury, E Stepanov, M Danieli… – Proceedings of the …, 2017 – aclweb.org
… Silence is sized and placed within the conversation flow and it is co- ordinated by the speakers along with the other speech acts … Generally, in a dialog system, silence is not acknowledged as a form of interaction, but rather its function in a conversation is seen as a “pause” or a …
Conceptual Basis For Developing Of Trainig Models In Complex System Software Assembling Generator
VO Georgiev, NA Prokopyev, DS Polikashin – Journal of Fundamental and …, 2017 – jfas.info
… are often equated to speech acts. The model quite formally determines the courses, admissible for each of participants of a game at present (by rules and according to the purpose) and thus, dialogues are modeled. For a task of requirements to dialogue system the following …
Simulation-Based Usability Evaluation of Spoken and Multimodal Dialogue Systems
S Hillmann – 2017 – Springer
… Stefan Hillmann Simulation-Based Usability Evaluation of Spoken and Multimodal Dialogue Systems Page 2. T-Labs Series in Telecommunication Services Series editors Sebastian Möller, Berlin, Germany Axel Küpper, Berlin, Germany Alexander Raake, Berlin, Germany …
Utterance Behavior of Users While Playing Basketball with a Virtual Teammate.
D Lala, Y Li, T Kawahara – ICAART (1), 2017 – pdfs.semanticscholar.org
… We use a Wizard-of-Oz system which allows a hidden operator to appropriately respond to user utterances. Utterances are analyzed by annotating and categorizing according to Searle’s illocutionary speech acts … In Steve’s case the speech acts were well structured …
Using Cognitive Models
S Kopp, K Bergmann – The Handbook of Multimodal-Multisensor …, 2017 – books.google.com
… The fre- quency of gestures and gesture types, the correlation of gesture types and speech acts, as well as the expressivity of … Similarly, Putze and Schultz [2009] employed cognitive modeling components in adaptive dialogue systems for in-car information applications, to ex …
Fast Forward through Opportunistic Incremental Meaning Representation Construction
P Babkin, S Nirenburg – Proceedings of ACL 2017, Student Research …, 2017 – aclweb.org
… 7. The last token “!” confirms the imperative mood of the utterance, signaling the instantiation of the REQUEST-ACTION speech act … 2012. In- cremental Construction of Robust but Deep Seman- tic Representations for Use in Responsive Dialogue Systems …
Investigating a Two-Way-Audio Query-Response Command Interface with Navigation Data Extraction for Driver Assistance
RP Loui, V Kharpate, M Deshpande, F Alanazi… – ieeexplore.ieee.org
… It implicates AI at the sound level, the speech act performative level, the dialogue level, and the meta- dialogical sub-dialogue (repair) level … “Conversational In-Vehicle Dialog Systems: The past, present, and future.” IEEE Signal Processing Magazine 33, no. 6 (2016): 49-60 …
Dialogue and interaction in role-playing games
F Mäyrä – Dialogue across Media, 2017 – books.google.com
… The game master (or Dungeon Master in D&D games) has the power to utter speech acts that immediately take the status of facts in the imaginary world of role-playing fantasy (“there are… three ships approach- ing” in the sample) …
A Hybrid Architecture for Multi-Party Conversational Systems
MG de Bayser, P Cavalin, R Souza, A Braz… – arXiv preprint arXiv …, 2017 – arxiv.org
… Page 2. Turing’s test. Some of them have won prizes, some not [5]. Although in this paper we do not focus on creating a solution that is able to build conversational systems that pass the Turing’s test, we focus on Natural Dialogue Systems (NDS) …
Using Past Speaker Behavior to Better Predict Turn Transitions
M Tomer – 2017 – digitalcommons.ohsu.edu
… and minimum gaps between turns. Spoken dialogue systems are a new form of conversational user … we trained two models to predict turn transitions: one with just local features (eg, current speech act, previous speech act) and one that added the summary features …
interactive environment. Foreign Language Annals. This is the accepted version of an article published in Foreign Language Annals by the American Council …
N Taguchi, Q Li, X Tang – researchgate.net
… place. Wik and Hjalmarsson (2009) developed a dialogue system that involved a role-play game with a built-in conversational agent … or moving a box. These activities elicited a variety of speech acts (requests and suggestions). When …
Psycholinguistic Approaches
SEBJE Hanna – The Routledge Handbook of Language and …, 2017 – books.google.com
… to, but not identical with, speech acts as proposed within philosophical speech act theory, and more recently dialogic speech act theory (see … This has supported the development of spoken dialogue systems and animated conversational agents that are now in use by the public …
Oxford Handbooks Online
R Fernández – staff.fnwi.uva.nl
… the Information State Update approach to dialogue management, a framework for the development of the dialogue management component of dialogue systems (see Chapter 41 … As we pointed out in section 2.1, dialogue acts can be seen as a generalization of speech acts …
Charting a Way through the Trees
R Cooper – Theoretical Linguistics, 2017 – degruyter.com
… sense of by treating languages as formal systems. At the same time there was the development of a theory of speech acts deriving from the work of Austin (1962) and Searle (1969). However, with the exception of work in the …
Novel Methods for Natural Language Generation in Spoken Dialogue Systems
O Dušek – 2017 – dspace.cuni.cz
… 1 Page 14. hello() Hello, this is dialogue system X. How can I help … (DAs; Young et al., 2010), consisting of a dialogue act type or dialogue action, roughly corresponding to speech acts of Austin and Searle (Korta and Perry …
Psycholinguistic Approaches
SE Brennan, JE Hanna – The Routledge Handbook of Language …, 2017 – books.google.com
… to, but not identical with, speech acts as proposed within philosophical speech act theory, and more recently dialogic speech act theory (see … This has supported the development of spoken dialogue systems and animated conversational agents that are now in use by the public …
A Review On Generative Conversational Model
E Varghese, MTR Pillai – data.conferenceworld.in
… users. In this system, it is using deep learning based dialogue system and provide domain specific answers for user queries … language processing. Wiley, Chichester. [8]Searle JR (ed) (2013) Speech act theory and pragmatics. Springer, New York …
Action Planning based on Open Knowledge Graphs and LOD
S Koide, F Kato, H Takeda12, Y Ochiai, K Ueda – ceur-ws.org
… Searle described the mechanism of human speech interaction and addressed the Speech Act theory [13] … Here the command ‘eliza’ is named for just representing a mimic of Eliza dialog system [15], that is the first dialog system in AI history …
Technology in Interlanguage Pragmatics Research and Teaching: An Introduction
N Taguchi, JM Sykes – researchgate.net
… encounter exchanges, and study group discussions). Through a line-by-line coding of naturalistic conversations, she identified speech acts and … While not explicitly directed towards pragmatics, Wik and Hjalmarsson (2009) developed a dialogue system called DEAL that …
Language is Not About Language: Towards Formalizing the Role of Extra-Linguistic Factors in Human and Machine Language Acquisition and Communication
O Räsänen – Proc. GLU 2017 International Workshop on …, 2017 – pdfs.semanticscholar.org
… Despite the numerous advances in the field, including the recent deep learning methods and end-to-end systems trained on large- scale data sets, the state-of-the-art ASR and dialogue systems still fall far behind human performance in natural communication … 3: Speech Acts …
Personalized Visualization Based upon Wavelet Transform for Interactive Software Customization
X Yuan, M Kaler, V Mulpuri – … Conference on Machine Learning and Data …, 2017 – Springer
… Among them, the approach based upon POMDP models demonstrates the advantage of handling uncertainty caused by speech act errors [15] … M., Kim, D., Szummer, M., Thomson, B., Tsiakoulis, P., Hancock, E.: Evaluation of statistical POMDP-based dialogue systems in noisy …
Parsing natural language conversations using contextual cues
S Srivastava, A Azaria, T Mitchell – Proceedings of the 26th …, 2017 – azariaa.com
… Examples of the former include analyzing language from perspectives of speech acts [Searle, 1969] and semantic … On the other hand, a notable application area that has ex- plored conversational context within highly specific settings is state tracking in dialog systems …
20 Body Movements Generation for Virtual Characters and Social Robots
A Beck, Z Yumak… – Social Signal …, 2017 – books.google.com
… identifier, name, gender, type [human/agent], appearance, voice), commu- nicative actions (turn-taking, grounding, speech act), content (what is … Spoken and multimodal dialog systems and applications–rigid head motion in expressive speech animation: Analysis and synthesis …
The state-of-the-art in autonomous wheelchairs controlled through natural language: A survey
T Williams, M Scheutz – Robotics and Autonomous Systems, 2017 – Elsevier
… commands, see below). Accepts descriptions : A wheelchair may understand statements such as “The door to the lab is locked” or indirect speech acts such as “It’d be great if you could get me a coffee”. Acknowledgment : The …
Cognitive science, language as a tool for interaction, and a new look at language evolution
R Kempson, S Chatzikyriakidis, C Howes – FADLI 2017 – christinehowes.com
… the recovery of some previously or even subse- quently agreed intended propositional content or speech-act (1,8): notably, even very young chil- dren are able to join in with … Bootstrapping incremental dialogue systems: using linguistic knowledge to learn from minimal data …
Cognitive-inspired conversational-strategy reasoner for socially-aware agents
OJ Romero, R Zhao, J Cassell – Proceedings of the 26th International …, 2017 – static.ijcai.org
… Our experiments demonstrated that, when us- ing the Social Reasoner in a Dialogue System, the rapport level between the user and system … can build strong relational bonds using specific conversational strate- gies – units of discourse that are larger than speech acts – cho- sen …
Learning Chinese Formulaic Expressions in a Scenario?Based Interactive Environment
N Taguchi, Q Li, X Tang – Foreign Language Annals, 2017 – Wiley Online Library
… place. Wik and Hjalmarsson (2009) developed a dialogue system that involved a role-play game with a built-in conversational agent. Simulating … box. These activities elicited a variety of speech acts (requests and suggestions). When …
Context-Aware Response Generation for Mental Health Counseling
R Pryzant – stanford.edu
… Natural language processing (NLP), speech, and dialogue systems have made great strides recently in extracting and understanding textual … Unsupervised machine learning models have been used to model conversations and segment them into speech acts, topical clusters, or …
Adapting to a listener with incomplete lexical semantics
S Srinivas, B Landau, C Wilson – pdfs.semanticscholar.org
… Janarthanam, S., & Lemon, O. (2010, September). Adaptive referring expression generation in spoken dialogue systems: Evaluation with real users … Monroe, W., & Potts, C. (2015). Learning in the rational speech acts model. arXiv preprint arXiv:1510.06807 …
Separating Representation, Reasoning, and Implementation for Interaction Management: Lessons from Automated Planning
ME Foster, RPA Petrick – Dialogues with Social Robots, 2017 – Springer
… Abstract. Numerous toolkits are available for developing speech-based dialogue systems … A number of toolkits are available to support the construction of such end-to-end dialogue systems. Such a toolkit generally incorporates three main features …
Iris: A Conversational Agent for Complex Tasks
E Fast, B Chen, J Mendelsohn, J Bassen… – arXiv preprint arXiv …, 2017 – arxiv.org
… Iris expands on this perspective by capturing the interleaving of multiple speech acts (eg through insert expansions) as described by CA theory [16] … To our knowledge, Iris is the first dialogue system to enable the gen- eral combination of such commands through conversation …
Towards Designing Cooperative and Social Conversational Agents for Customer Service
U Gnewuch, S Morana, A Maedche – 2017 – aisel.aisnet.org
… 2015; Shawar and Atwell 2007) or natural dialogue systems (eg, Shah et al. 2016; Zadrozny et al … “NADIA: A Simplified Approach Towards the Development of Natural Dialogue Systems,” in Natural Language Processing and Information Systems, LNCS 9103, Springer, pp …
Generating Contrastive Referring Expressions
M Villalba, C Teichmann, A Koller – … of the 55th Annual Meeting of the …, 2017 – aclweb.org
… In an interactive setting, such as a dialogue system or a pedestrian navigation sys- tem, the system can try to detect such misunder- standings – eg by predicting what the hearer un- derstood from their behavior (Engonopoulos et al., 2013) – and to produce further utterances …
Modeling the clarification potential of instructions: Predicting clarification requests and other reactions
L Benotti, P Blackburn – Computer Speech & Language, 2017 – Elsevier
… Keywords. Clarification requests. Level-sensitive Gabsdil test. Conversational implicatures. Dialogue systems. Classical planning. Micro-planning. Negotiability … We first review a method of identifying clarification requests proposed in the dialogue system literature …
Ontology based Baysian network for clinical specialty supporting in interactive question answering systems
JF Yeh, YJ Huang, KP Huang – Engineering Computations, 2017 – emeraldinsight.com
… However, it is effort-consuming for medical staff especially in critical times. Therefore, a query system or spoken dialogue system is able to provide better service for the clinical specialty supporting by extracting the semantic information from patients’ utterances …
Artificial cognition for social human–robot interaction: An implementation
S Lemaignan, M Warnier, EA Sisbot, A Clodic… – Artificial Intelligence, 2017 – Elsevier
Talk and Tools: the best of both worlds in mobile user interfaces for E-coaching
RJ Beun, S Fitrianie, F Griffioen-Both, S Spruit… – Personal and Ubiquitous …, 2017 – Springer
… Health coaching dialog systems have been developed on the basis of research methods from persuasive technology (eg, [14]) and behavior medicine … Consequently, conversation also enables the e-coach to manifest a variety of speech acts that pertain to the explanation and …
The impact of peer tutors’ use of indirect feedback and instructions
M Madaio, J Cassell, A Ogan – 2017 – repository.isls.org
… Apology Apologies used to soften direct speech acts … These findings can also inform the design of educational dialogue systems, or conversational agents which could support peer tutoring by detecting and responding appropriately to the interpersonal dynamics between the …
Predicting and Regulating Participation Equality in Human-robot Conversations: Effects of Age and Gender
G Skantze – Proceedings of the 2017 ACM/IEEE International …, 2017 – dl.acm.org
… Traditionally, spoken dialogue systems have rested on a very sim- plistic model of turn-taking, where a certain amount of silence (say 700 ms … museum, Furhat’s turn-yielding behaviour was randomly se- lected for each turn, both in terms of addressee and speech act (question or …
Dialogue, dialogicality and interactivity
P Linell – Language and Dialogue, 2017 – jbe-platform.com
… action theory: Dialogue game model; 5. Theories of semiotic resources used in dialogue: Language and dialogue systems and more … She was initially quite influenced by both speech act theory (see § 5.1) and Hundsnurscher’s (1980) programmatic work criticising Conversation …
Overview of the 2017 spoken CALL shared task
C Baur, C Chua, J Gerlach, E Rayner, M Russel, H Strik… – 2017 – archive-ouverte.unige.ch
… 2.1. Data The core resource for the task was an English speech corpus collected with the CALL-SLT dialogue game. In total, the corpus contains 38,771 spontaneous speech acts in the form of students’ interactions with the dialogue system …
From Alan Turing to modern AI: practical solutions and an implicit epistemic stance
GF Luger, C Chakrabarti – AI & SOCIETY, 2017 – Springer
… dialogue system. Figure 5 depicts a finite state machine representing a troubleshooting conversation, where the states are the components of the conversation {Start, Greeting, Elicitation, Troubleshooting, Fixed, Dissatisfaction, Conclusion}, and the transitions are speech acts …
Treating Unexpected Input in Incremental Semantic Analysis
M McShane, K Blissett, I Nirenburg – … of The Fifth Annual Conference on …, 2017 – cogsys.org
… permits the analyzer to disambiguate: if the direct object is abstract, as in He addressed the problem, then address will be analyzed as CONSIDER; by contrast, if the direct object is human, as in He addressed the audience, then address will be analyzed as SPEECH-ACT …
Quotation in Dialogue Eleni Gregoromichelaki King’s College London and Osnabrück University elenigregor@ gmail. com 0049 015171228 646
E Gregoromichelaki – kcl.ac.uk
… roles assumed by interlocutors, intersect with syntactic/semantic issues of direct/indirect speech forms and speech-act responsibility (Gregoromichelaki and Kempson 2016, Goodwin … traditionally been analysed as indexicals, eg elements like I and you, but also speech-act …
Annotating and modeling empathy in spoken conversations
F Alam, M Danieli, G Riccardi – Computer Speech & Language, 2017 – Elsevier
… So I stop the overdue notices and you will have more time to pay), The selection of the speech act (question instead of authoritative declarative), the rhetorical structure of the second question, the lexical choice of “proviamo”, instead of – for instance, “adesso provo a vedere…”, all …
Participatory Management of Protected Areas for Biodiversity Conservation and Social Inclusion
JP Briot, MDA Irving, JE Vasconcelos Filho… – 2017 – hal.upmc.fr
… such as serious games, role-playing games, multi-agent systems, simulation, decision support systems, user interfaces, dialogue systems, argumentation-based … 2003), as well as theories of negotiation, eg, (Wall et al., 1991) (Raiffa, 1982) and Speech Act Theory (Searle, 1969) …
Conjoint utilization of structured and unstructured information for planning interleaving deliberation in supply chains
NK Janjua, OK Hussain, E Chang… – Proceedings of the …, 2017 – dl.acm.org
… ?p is a set of axioms used in deliberation module such as speech acts for communication and dialogue movies for es- tablishing a preference between conflicting situations … The deliberation dialogue system is defined by: (1) Topic Language: DeLP as a logical language …
Logic, Reasoning, Argumentation: Insights From The Wild
F Zenker – Logic and Logical Philosophy, 2017 – apcz.umk.pl
Page 1. Logic and Logical Philosophy (2017) DOI: 10.12775/LLP.2017.029 Published online: September 30, 2017 Frank Zenker LOGIC, REASONING, ARGUMENTATION: Insights from the wild Abstract. This article provides …
From paralinguistic to variably linguistic
A Cienki – The Routledge handbook of pragmatics, 2017 – books.google.com
… Bucciarelli, M., Colle, L., and Bara, BG (2003) ªHow children comprehend speech acts and communicative gestures «, Journal of Pragmatics, 35 … and Stone, M.(1999) ªLiving hand to mouth: Psychological theories about speech and gesture in interactive dialogue systems «, in SE …
11 Analyzing Multicodal Media Texts
U Fröhlich – Manual of Romance Languages in the Media, 2017 – books.google.com
… Additionally, speech acts can be carried out by means of different codes or sign systems (eg, language and images) … Gibbon, Dafydd/Mertens, Inge/Moore, Roger K.(edd.)(2000), Handbook of multimodal and spoken dialogue systems: Resources, terminology and product …
Debating Technology for Dialogical Argument: Sensemaking, Engagement, and Analytics
J Lawrence, M Snaith, B Konat, K Budzynska… – ACM Transactions on …, 2017 – dl.acm.org
… Robertson [2004] has demonstrated that a language for expressing such dialogue systems in general offers significant practi- cal and theoretical advantages and has proposed a lightweight coordination calculus (LCC) to do just this …
Debating Technology for Dialogical Argument
J Lawrence, M Snaith, B Konat, K Budzynska, C Reed – discovery.dundee.ac.uk
… Robertson [2004] has demonstrated that a language for expressing such dialogue systems in general offers significant practi- cal and theoretical advantages and has proposed a lightweight coordination calculus (LCC) to do just this …
College of Natural Sciences
DT Habte – 2017 – etd.aau.edu.et
Page 1. Dialogue System for Advising Ethiopian Public Universities Students Addis Ababa University College of Natural Sciences … The main objective of this research is to design a model of academic advising dialogue system for Ethiopian public universities …
Culture-specific models of negotiation for virtual characters: multi-attribute decision-making based on culture-specific values
E Nouri, K Georgila, D Traum – AI & society, 2017 – Springer
… 2008), which facilitates rapid development of virtual human dialog systems. The authoring environment of this architecture was used to construct domain knowledge and textual realizations for natural language understanding and generation of a range of speech acts …
Towards a theory of close analysis for dispute mediation discourse
M Janier, C Reed – Argumentation, 2017 – Springer
Mediation is an alternative dispute resolution process that is becoming more and more popular particularly in English-speaking countries. In contrast to traditional litigation it has not benefited fro.
SHERLOCK: Experimental evaluation of a conversational agent for mobile information tasks
A Preece, W Webberley, D Braines… – … on Human-Machine …, 2017 – ieeexplore.ieee.org
… Interaction with the conversational agent is controlled by a protocol based on linguistic speech act theory [9]. The effects of speech acts on the KB are persistent and affect subsequent interactions, for example, if a user tells the agent something, and some user subsequently …
Evaluating Relevance and Commitments in Rhetorical Straw Man
F Macagno, D Walton – Interpreting Straw Man Argumentation, 2017 – Springer
… As shown in the previous section, the older formal dialogue systems represented by Walton and Krabbe (1995) were set up using a set of rules (including a set of rules defining … They define a set of speech acts representing the kinds of moves that each party is allowed to make …
The meaning of intonation in yes-no questions in American English: A corpus study
N Hedberg, JM Sosa, E Görgülü – Corpus Linguistics and Linguistic …, 2017 – degruyter.com
… Yes-no questions with falling intonation (eg H*LL%) do not occur frequently, but when they do, they can be classified in speech act terms as “non-genuine” questions, where one or more felicity conditions on genuine ques- tions are not met …
Teaching and Learning in the Pleistocene: A Biocultural Account of Human Pedagogy and Its Implications for AIED
DM Morrison, KB Miller – International Journal of Artificial Intelligence in …, 2017 – Springer
Page 1. ARTICLE Teaching and Learning in the Pleistocene: A Biocultural Account of Human Pedagogy and Its Implications for AIED Donald M. Morrison1 & Kenneth B. Miller2 © International Artificial Intelligence in Education Society 2017 Introduction …
The Logical Approach of Legal Argumentation
ET Feteris – Fundamentals of Legal Argumentation, 2017 – Springer
… The disagreements that remain form the issues in the case on trial. The dialogue system functions as a referee to ensure that the proper procedure is followed … In the TDG there are different speech acts for asking for and providing the various elements of an argument …
J Eckstein – Argumentation, 2017 – Springer
… reasonableness. Unlike speech acts, which rely on symbols to generate power, sound’s materiality exerts force … lives. Sound plays a role in argument, from a speech act’s vocal affectations to the myriad mediated contexts of a disagreement …
Shallow PARsing and Knowledge extraction for Language Engineering
I Annex – cogsci.ed.ac.uk
… The speech dialogue system, developed by Daimler Benz for German, will also make use of sparkle’s parsers and lexica to generate a probabilistic language model for speech recognition. 1.2 Participant Summary Short Name Participant full name Country code Role …
Mixed-initiative intent recognition using cloud-based cognitive services
M Kraus – 2017 – oparu.uni-ulm.de
… an SDS are described, as well as insight is given on the cloud-based cognitive services used for the implementation of the dialogue system … In the context of SDSs, the intention of a user is incorporated in speech acts or dialogue acts, which provide an abstract representation of …
Sarcasm detection in microblogs using Naïve Bayes and fuzzy clustering
S Mukherjee, PK Bala – Technology in Society, 2017 – Elsevier
… benefit of detecting sarcasm has been recognized in many computer interaction based applications, such as, review summarization, dialogue systems and review … Sarcasm is a form of speech act in which the speakers convey their message in an implicit way [2]. The implicitness …
How a Minimally Designed Robot can Help Implicitly Maintain the Communication Protocol
K Youssef, M Okada – International Journal of Social Robotics, 2017 – Springer
… from users giving commands to an artifact executing instructions [19, 20], as well as related error handling that is integrated in spoken dialog systems [21 … By extending the line of our research we believe that a speech act during an HRI has to support the human’s social face, but …
Detecting sarcasm in customer tweets: an NLP based approach
S Mukherjee, PK Bala – Industrial Management & Data Systems, 2017 – emeraldinsight.com
… huge benefit of detecting sarcasm has been recognized in many computer interaction-based applications, such as review summarization, dialogue systems and review … Sarcasm is a form of speech act in which the speakers convey their message in an implicit way (Davidov et al …
Social Agents for Learning in Virtual Environments
LJL Weideveld – 2017 – dspace.library.uu.nl
… After that, chapter 5 will discuss what a social practice is and why it is a good addition to a rule-based dialogue system … Another example of a chatbot that extends on the AIML language is SAM (Speech Act Man) [17] by Holtgraves and Han …
Defining Soldier Intent in a Human-Robot Natural Language Interaction Context
E Holder – 2017 – dtic.mil
Page 1. ARL-TR-8195 ? OCT 2017 US Army Research Laboratory Defining “Soldier Intent” in a Human–Robot Natural Language Interaction Context by Eric Holder Approved for public release; distribution is unlimited. Page 2. NOTICES Disclaimers …
Discourse Processing in Technology-Mediated Environments
D Gergle – The Routledge Handbook of Discourse Processes, 2017 – books.google.com
… In other words, a mutually beneficial synergy exists whereby computational modeling and technological development can both benefit from, and lead to deeper understanding of, discourse processes. Consider interactive spoken dialogue systems as an example …
Special Issue in Computational Biological Data science
Page 1. Special Issue in Computational Biological Data science Computational Bio Science is at the cusp of big innovations in order to guarantee highly affordable, advanced and smarter healthcare facilities for people across the globe …
“I think you just got mixed up”: confident peer tutors hedge to support partners’ face needs
M Madaio, J Cassell, A Ogan – International Journal of Computer …, 2017 – Springer
… This follows theories of rapport-building, such as from Spencer-Oatey (2005), which suggest that a greater rapport, or interpersonal closeness, between interlocutors allows for speech acts which would otherwise be perceived as face-threatening …
Introduction to Cyberemotions
JA Ho?yst – Cyberemotions, 2017 – Springer
… Outputs of the Project were used for creating new affective dialog systems as interactive tools as well as semi-automated simulation of … doi:10.1300/j202v01n02_04CrossRefGoogle Scholar. Carr, CT, Schrock, DB, Dauterman, P.: Speech acts within Facebook status messages …
Sound arguments
J Eckstein – Argumentation and Advocacy, 2017 – Taylor & Francis
Towards the Implementation of an Intelligent Software Agent for the Elderly
AHF Dinevari – 2017 – era.library.ualberta.ca
… 27 2.4.1 Definition of Natural Language Processing . . . . . 27 2.4.2 Natural Language Processing on Textual Content . . . 28 2.4.3 Notable Natural Language Processing Tools . . . . . 29 3 Speech Act Recognition 31 3.1 Speech Acts …
A review of spatial reasoning and interaction for real-world robotics
C Landsiedel, V Rieser, M Walter, D Wollherr – Advanced Robotics, 2017 – Taylor & Francis
… one each for input, output and control, as shown in Figure 1 after [1 Rieser V, Lemon O. Reinforcement learning for adaptive dialogue systems: a data … input (1) into text (2), see Figure 1. SLU parses the text into a string of meaningful concepts, intentions, or Speech Acts (SA) (3 …
Computational modeling of turn-taking dynamics in spoken conversations
SA Chowdhury – 2017 – eprints-phd.biblio.unitn.it
Page 1. PhD Dissertation International Doctorate School in Information and Communication Technologies DISI – University of Trento COMPUTATIONAL MODELING OF TURN-TAKING DYNAMICS IN SPOKEN CONVERSATIONS Shammur Absar Chowdhury Advisor: Prof …
Automatic question generation for virtual humans
EL Fasya – 2017 – essay.utwente.nl
… Figure 2.4 illustrates a simple finite-state automation architecture of a dialogue manager in a spoken dialogue system [2] … Information-state is a more advanced architecture for a dialogue manager that allows for more components, eg interpretation of speech acts or grounding …
The future of assessment: shaping teaching and learning
CA Dwyer – 2017 – books.google.com
Page 1. The Future of Assessment SHAPING TEACHING AND LEARNING —– EDITED BY Carol Anne Dwyer Page 2. The Future of Assessment SHAPING TEACHING AND LEARNING Page 3. The Future …
Explanatory dialogues with argumentative faculties over inconsistent knowledge bases
A Arioua, P Buche, M Croitoru – Expert Systems with Applications, 2017 – Elsevier
Sliding Mode in Intellectual Control and Communication: Emerging Research and Opportunities: Emerging Research and Opportunities
V Mkrttchian, E Aleshina – 2017 – books.google.com
Page 1. Sliding Mode in Intellectual Control and Communication: Emerging Research and Opportunities Vardan Mkrttchian HHH University, Australia Ekaterina Aleshina Penza State University, Russia A volume in the Advances …
Sabbiu Shah (070/BCT/531) Sagar Adhikari (070/BCT/533) Samip Subedi (070/BCT/536)
U Chalise – 2017 – researchgate.net
… Chatterbots are typically used in dialog systems for various practical purposes including customer service or information acquisition … Another possible task is recognizing and classifying the speech acts in a chunk of text (eg yes-no …
Resource Allocation for Pragmatically-Assisted Quality of Information-Aware Networking
J Edwards, RJ Passonneau, T Cassidy… – … (ICCCN), 2017 26th …, 2017 – ieeexplore.ieee.org
Page 1. Resource Allocation for Pragmatically-assisted Quality of Information-aware Networking James Edwards?, Rebecca J. Passonneau?, Taylor Cassidy†, Thomas F. La Porta? ? Institute for Networking and Security Research …
Lakatos-style collaborative mathematics through dialectical, structured and abstract argumentation
A Pease, J Lawrence, K Budzynska, J Corneli… – Artificial Intelligence, 2017 – Elsevier
… The remainder of the paper is structured as follows: in Sections 2–3 we outline our theoretical model, in which we introduce theoretical foundations and develop a formal dialogue system from Lakatos’s model of mathematical discourse …
A Survey of Approaches and Studies of Legal Argumentation in the Context of Legal Justification in Different Legal Systems and Countries
ET Feteris – Fundamentals of Legal Argumentation, 2017 – Springer
In the preceding chapters, several of the most important theories of legal argumentation have been examined. Apart from these theories in which a more or less complete account of legal argumentation i.
Jungian personality in a chatbot
JT Klooster – 2017 – dspace.library.uu.nl
… 16 2.4 Speech act theory . . . . . 19 2.5 Dialogue systems … followed by the idea of social practices 2.3, and some limited speech act theory 2.4. After which we continue with dialogue systems 2.5, some of which can be considered related work …
Multi-layer ontology based information fusion for situation awareness
FP Pai, LJ Yang, YC Chung – Applied Intelligence, 2017 – Springer
Originated from the military domain, Situation Awareness (SAW) is proposed with the aim to obtain information superiority through information fusion and thus to achieve decision superiority. It requir.
Argumentation Schemes. History, Classifications, and Computational Applications
F Macagno, D Walton, C Reed – 2017 – papers.ssrn.com
… the first ones proceed directly from the subject matter at issue (for instance, its semantic prop- erties), the external topics (the Aristotelian arguments from authority) support the conclusion through contextual elements (for instance, the source of the speech act expressing the claim …
A Framework For Enhancing Speaker Age And Gender Classification By Using A New Feature Set And Deep Neural Network Architectures
A Abumallouh – 2017 – scholarworks.bridgeport.edu
… In [84] a system for detecting the older people over the spoken dialogue systems(SDS) to meet their needs is proposed … Three feature sets are used to simulate the interaction style of the speaker,1) overall dialogue statistics 2) speech act group frequency …
Interpreting Straw Man Argumentation
F Macagno, D Walton – 2017 – Springer
… It provides analytical tools, namely, dialogue systems and profiles of dialogue, which can be used for reconstructing, evaluating, and establishing an interpretation and defusing manipulative tactics associated with straw man arguments. Introduction Page 18. 1 …
Theories and Approaches to the Study of Conversation and Interactive Discourse
WS Horton – The Routledge Handbook of Discourse Processes, 2017 – books.google.com
Page 44. p. 22 2 Theories and Approaches to the Study of Conversation and Interactive Discourse William S. Horton NORTHWESTERN UNIVERSITY Introduction Conversation is arguably the most fundamental means we have of interacting with others …
Sentiment Analysis In Czech
K Veselovská – 2017 – ufal.mff.cuni.cz
… of natural language processing, such as question- answering, recommendation systems, automatic summarization of a text, automatic dialogue systems or emotionality … the basics of irony and sarcasm and situate the role of emotional expressing within the speech acts theory …
Background Review for Neural Trust and Multi-Agent System
G Lu, J Lu – … Retrieval and Image Processing Paradigms in …, 2017 – books.google.com
… Page 302. Background Review for Neural Trust and Multi-Agent System 1.2. 4 Communications The commonly used communication methods include the black board system and the message/dialog system … The message layer specifies message related types of speech acts …
Role of Body Language in Teaching English as a Foreign Language
DIO ALDoumer – 2017 – repository.sustech.edu
Page 1. Sudan University of Science & Technology College of Graduate Studies College of Education Role of Body Language in Teaching English as a Foreign Language ? ???? ?? ????? ????? ?? ????? ??? ??????? ?????? ??? (A case study of Secondary Schools in El-Fashir Locality) …
Finding enthymemes in real-world texts: A feasibility study
O Razuvayevskaya, S Teufel – Argument & Computation, 2017 – content.iospress.com
… [2]. E. Black and A. Hunter, Using enthymemes in an inquiry dialogue system, International Foundation for Autonomous Agents and Multiagent Systems 1: ((2008) ), 437–444 … 3: Speech Acts, P. Cole and JL Morgan, eds, Academic Press, San Diego, CA, (1975) , pp. 41–58. [17] …
HCI from a Discourse Perspective Area: Discourse and Interaction GRACE Deliverable 5.1
R Bod, M Dastani, R Scha, H Zeevat – cogsci.ed.ac.uk
… the le. The system’s response is the appropriate change in the display and the le. As a dialogue system our simpli ed editor is maximally simple. Dialogues consist of user commands followed by system con rmations. The system’s …
Historical overview of formal argumentation
H Prakken – IfCoLog Journal of Logics and their Applications, 2017 – dspace.library.uu.nl
… winning strategy for the proponent. This predates modern argument games for argumentation-based inference and also influenced the devel- opment of formal dialogue systems for argumentation. Having said so, in dialogue …
Argumentation Theory in Formal and Computational Perspective
FH van Eemeren, B Verheij – ai.rug.nl
… The dialectical dimension of the approach is inspired by normative insights from critical rationalism and formal dialectics, the pragmatic dimension by descrip- tive insights from speech act theory, Gricean pragmatics and discourse analysis …
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Adjusting linguistically to others: the role of social context In lexical choices and spatial language
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Page 1. This thesis has been submitted in fulfilment of the requirements for a postgraduate degree (eg PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following terms and conditions of use: This work …
Painting Pictures with Words-From Theory to System
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… PAR allows instructions such as if you agree to go for a walk with someone, then follow them to be given and then triggered in the future. Ulysse [Godreaux et al., 1999] is an interactive spoken dialog system used to navigate in virtual worlds …
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JC Auton, MW Wiggins, BJ Searle… – Applied …, 2017 – cambridge.org
Page 1. Applied Psycholinguistics, page 1 of 31, 2016 doi:10.1017/ S014271641600031X Utilization of prosodic and linguistic cues during perceptions of nonunderstandings in radio communication JAIME C. AUTON, MARK …
Reasoning Schemes, Expert Opinions and Critical Questions. Sex Offenders Case Study
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Page 1. Reasoning Schemes, Expert Opinion and Critical Questions. Sex Offenders Case Study Dov Gabbay Ashkelon Academic College, Bar Ilan University, King’s College London, University of Luxembourg, University of Manchester dov.gabbay@kcl.ac.uk …
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… (London, 1984), 1.3-35; E. Giachin and S. McGlashan, ‘Spoken Language Dialogue Systems’, in Corpus … Mary Louise Pratt focuses more on the relationships between spoken and literary narratives than on dialogue as such: Towards a Speech Act Theory (Bloomington and …
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https://cs184.eecs.berkeley.edu/sp16/article/3 | This is an archive of a past semester of this course. Go to the current semester.
Assignment 1: Rasterizester
In this assignment you will implement a simple rasterizer, including features like supersampling, hierarchical transforms, and texture mapping with antialiasing. At the end, you'll have a functional vector graphics renderer that can take in modified SVG (Scalable Vector Graphics) files, which are widely used on the internet.
## Announcements
• Assignment 1 is due Wednesday February 10th at 11:59pm. Assignments which are turned in after 11:59pm are a full day late -- there are no late minutes or late hours.
• Note: You will write a webpage to present your results. We will add more details about requirements for this write-up, and provide an html template for you to use.
## Getting set up
You can either download the zipped assignment straight to your computer or clone it from GitHub using the command
\$ git clone https://github.com/CS184-sp16/asst1_rasterizester.git
## Using the GUI
You can run the executable with the command
./rasterizester ../svg/basic/test1.svg
A flower should show up on your screen. After finishing Part 4, you will be able to change the viewpoint by dragging your mouse to pan around or scrolling to zoom in and out. Here are all the keyboard shortcuts available (some depend on you implementing various parts of the assignment):
Key Action
' ' return to original viewpoint
'-' decrease sample rate
'=' increase sample rate
'Z' toggle the pixel inspector
'P' switch between texture filtering methods on pixels
'L' switch between texture filtering methods on mipmap levels
'S' save a png screenshot in the current directory
'1'-'9' switch between svg files in the loaded directory
'ESC' quit
The argument passed to rasterizester can either be a single file or a directory containing multiple svg files, as in
./rasterizester ../svg/basic/
If you load a directory with up to 9 files, you can switch between them using the number keys 1-9 on your keyboard.
## Assignment structure
The assignment has 8 parts and 100 possible points. Some require only a few lines of code, while others are more substantial.
1. Rasterizing lines (5 pts)
2. Rasterizing single-color triangles (10 pts)
3. Antialiasing triangles (20 pts)
4. Transforms (10 pts)
5. Barycentric coordinates (5 pts)
6. "Pixel sampling" for texture mapping (15 pts)
7. "Level sampling" with mipmaps for texture mapping (25 pts)
8. Draw something interesting! (10++ pts)
For each part, the potentially relevant locations in the code are marked with a C++ comment that looks like
// Part 1: ...
There is a fair amount of code in the CGL library, which we will be using for future assignments. The relevant header files for this assignment are vector2D.h, matrix3x3.h, color.h, and renderer.h. In the discussion sections on Jan 27 and 28, we will give a tour of the starter code to help you get started with the assignment.
Here is a very brief sketch of what happens when you launch rasterizester: An SVGParser (in svgparser.*) reads in the input svg file(s), launches a OpenGL Viewer containing a DrawRend renderer, which enters an infinite loop and waits for input from the mouse and keyboard. DrawRend (drawrend.*) contains various callback functions hooked up to these events, but its main job happens inside the DrawRend::redraw() function. The high-level drawing work is done by the various SVGElement child classes (svg.*), which then pass their low-level point, line, and triangle rasterization data back to the three DrawRend rasterization functions.
### What you will turn in
You will submit your entire project directory in a zip file. This should include a website directory containing a web-ready assignment writeup in a file index.html. Most parts of the assignment have Deliverables specified, which will be png and svg files along with various textual descriptions. You should accumulate these deliverables into sections in your webpage writeup as you go through the assignment. There are a few open-ended deliverables, and the quality of your explanations is just as important as your output images, especially when you are trying for extra credit! We want you to demonstrate a lucid understanding of what you have implemented.
Note: Do not squander all your hard work on this assignment by converting your *png files into jpgs or any other format!* Leave the screenshots as they are saved by the 'S' key in the GUI, otherwise you will introduce artifacts that will ruin your rasterization efforts. You can see these effects in the jpg images on this writeup page (which means, as a result, you should not use these for pixel-to-pixel comparisons!).
## Act I: In which you implement the bare bones
### Part 1 (warmup): Rasterizing lines (5 pts)
Relevant lecture: 2
Part 1 is intended to be a simple warmup problem: fill in the DrawRend::rasterize_line(...) function in drawrend.cpp. The given coordinates (x0,y0) and (x1,y1) define the screen-space endpoints of a line of color color. Screen space coordinates range from (0,0) at the top left to (width,height) at the bottom right of the viewing window. Assume that screen sample position are centered at half-integer coordinates in this space.
You may search the web for line-drawing algorithms and implement any you find (though write the code yourself). One option is to use Bresenham's algorithm. It is fine for you to rely on the existing implementation of DrawRend::rasterize_point(...) to write colors to the buffer. Make sure your algorithm only performs work proportional to the length of the line -- do not check every sample in the bounding box!
Deliverables:
• Save a png of svg/basic/test2.svg with the default viewing parameters and with the pixel inspector centered on an interesting part of the scene.
### Part 2: Rasterizing single-color triangles (10 pts)
Relevant lecture: 2
Triangle rasterization is a core function in the graphics pipeline to convert input triangles into framebuffer pixel values. In Part 2, you will implement triangle rasterization using the methods discussed in lecture 2 to fill in the DrawRend::rasterize_triangle(...) function in drawrend.cpp.
Notes:
• For now, ignore the Triangle *tri input argument to the function. We will come back to this in part 5.
• You are encouraged but not required to implement the edge rules for samples lying exactly on an edge.
• Make sure the performance of your algorithm is no worse than one that checks each sample within the bounding box of the triangle.
• Clarification: Do not use rasterize_line() to rasterize the edges of your triangles! If you look through the draw() methods in svg.cpp, you will see that this is done automatically when an edge color has been specified in the respective svg file.
When finished, you should be able to render many more test files, including those with rectangles and polygons, since we have provided the code to break these up into triangles for you. In particular, basic/test3.svg, basic/test4.svg, and basic/test5.svg should all render correctly.
Deliverables:
• Save a png of svg/basic/test4.svg with the default viewing parameters and with the pixel inspector centered on an interesting part of the scene.
• Extra Credit: Make your triangle rasterizer super fast (e.g., by factoring redundant arithmetic operations out of loops, minimizing memory access, and not checking every sample in the bounding box). Write about the optimizations you used. Use clock() to get timing comparisons between your naive and speedy implementations.
### Part 3: Antialiasing triangles (20 pts)
Relevant lecture: 2
Use supersampling to get rid of the jaggies, to render nicely antialiased edges on your triangles. The sample_rate parameter in DrawRend (adjusted using the - and = keys) tells you how many samples to use per pixel.
You do not have to antialias points or lines.
You have some latitude to implement this part in whatever way you please. One piece of advice: to do this correctly, you will almost certainly need to keep track of width * height * sample_rate accumulated sample colors.
Take care to make sure your antialiasing interfaces correctly with alpha blending (think about how two full-opacity triangles each half-covering some portion of a pixel could differ from one half-opacity triangle covering the whole pixel...). One sign that you may have broken this is if cracks start to appear along the edges of previously watertight triangles, and good file to test this on is svg/basic/test4.svg.
Your triangle edges should be noticeably smoother when using > 1 sample per pixel! You can examine the differences closely using the pixel inspector.
Deliverables:
• Save a comparison png files of svg/basic/test4.svg with the default viewing parameters and sample rates 1, 4, and 16. Position the pixel inspector over an area that showcases the effect dramatically; for example, a very skinny triangle corner.
• Describe the new structs and functions you might have added to drawrend.* to implement antialiasing (there are multiple correct approaches).
• Extra Credit: Explore alternative antialiasing methods, such as jittered or low-discrepancy sampling. Create comparison images showing the differences between grid supersampling and your new pattern. Try making a scene that contains aliasing artifacts when rendered using grid supersampling but not when using your pattern.
### Part 4: Transforms (10 pts)
Relevant lecture: 3
Implement the three transforms in the transforms.cpp file according to the SVG spec. The matrices are 3x3 because they operate in homogeneous coordinates -- you can see how they will be used on instances of Vector2D by looking at the way the * operator is overloaded in the same file.
Additionally, implement DrawRend::move_view(...) in drawrend.cpp. This will allow you to pan and scroll using your cursor. Make sure you understand the matrix stack that transitions first from SVG to normalized device coordinates, then from NDC to screen space coordinates.
Deliverables:
• Create a new svg file using geometric primitives and a hierarchical transform stack (at least two matrices deep) involving your new rotation, translation, and scaling matrices. Here is one example of how the SVG Group element is used to make a transform stack. Create four copies of your svg file where you modify a transform to illustrate your grouping hierarchy. Save your svg files in the svg/my_examples/ directory and add png screenshots of your rendered drawing to the writeup.
• Extra Credit: Add an extra feature to the GUI. For example, you could make two unused keys to rotate the viewport. Save an example image to demonstrate your feature, and write about how you modified the SVG to NDC and NDC to screen-space matrix stack to implement it.
## Act II: In which you become a sampling guru
### Part 5 (warmup): Barycentric coordinates (5 pts)
Relevant lecture: 4
Familiarize yourself with the ColorTri struct in svg.h. Modify your implementation of DrawRend::rasterize_triangle(...) so that if a non-NULL Triangle *tri pointer is passed in, it computes barycentric coordinates of each sample hit and passes them to tri->color(...) to request the appropriate color.
Implement the ColorTri::color(...) function in svg.cpp so that it returns this color. This function is very simple: it does not need to make use of any arguments besides Vector2D xy (the remaining arguments are for the texture mapped triangles). Note that this color() function plays the role of a very primitive shader.
Deliverables:
• Save a png of svg/basic/test7.svg with the default viewing parameters and sample rate 1.
### Part 6: "Pixel sampling" for texture mapping (15 pts)
Relevant lecture: 4
Familiarize yourself with the TexTri struct in svg.h. This is the primitive that implements texture mapping. For each vertex, you are given corresponding uv coordinates that index into the Texture pointed to by *tex.
To implement texture mapping, DrawRend::rasterize_triangle should fill in the psm and lsm members of a SampleParams struct and pass it to tri->color(...). Then TexTri::color(...) should fill in the correct uv coordinates in the SampleParams struct, and pass it on to tex->sample(...). Then Texture::sample(...) should examine the SampleParams to determine the correct sampling scheme.
The GUI toggles DrawRend's PixelSampleMethod variable psm using the 'P' key. When psm == P_NEAREST, you should use nearest-pixel sampling, and when psm == P_LINEAR, you should use bilinear sampling.
You can pass in dummy Vector2D(0,0) values for the dx and dy arguments to tri->color
For this part, just pass 0 for the level parameter of the sample_nearest and sample_bilinear functions.
For convenience, here is a list of functions you will need to modify:
1. DrawRend::rasterize_triangle
2. TexTri::color
3. Texture::sample
4. Texture::sample_nearest
5. Texture::sample_bilinear
Deliverables:
• Test the svg files in the svg/texmap/ directory. Use the pixel inspector to find a good example of where bilinear sampling clearly defeats nearest sampling. Hint: you want the texture to be magnified in the rendered image. Save four screenshots to show comparisons between nearest and bilinear at 1 sample per pixel and at 16 samples per pixel. Comment on the relative differences.
### Part 7: "Level sampling" with mipmaps for texture mapping (25 pts)
Relevant lecture: 4
Finally, you will add support for sampling different MipMap levels. The GUI toggles DrawRend's LevelSampleMethod variable lsm using the 'L' key.
• When lsm == L_ZERO, you should sample from the zero-th MipMap, as in Part 6.
• When lsm == L_NEAREST, you should compute the nearest appropriate MipMap level using the one-pixel difference vectors du and dv and pass that level as a parameter to the nearest or bilinear sample function.
• When lsm == L_LINEAR, you should find the appropriate MipMap levels and get two samples from adjacent levels and compute a linearly interpolated sum.
Implement Texture::get_level as a helper function. This is the trickiest math in the whole assignment -- make sure to read the relevant slides in Lecture 4 carefully.
For convenience, here is a list of functions you will need to modify:
1. DrawRend::rasterize_triangle
2. TexTri::color
3. Texture::sample
4. Texture::get_level
Deliverables:
• There are large number of sampling schemes available to you now: you can adjust pixel sampling, level sampling, and samples per pixel all independent of one another! Pull some png images from the internet and create your own svg files to demonstrate the strengths and weaknesses of various techniques at different zoom levels. You can take existing files in svg/texmap/ and replace the texture filename to try out new images. A good starting place for this is svg/texmap/test7.png. Show at least one example (using a png file you find yourself) comparing all four combinations of one of L_ZERO and L_NEAREST with one of P_NEAREST and P_BILINEAR at a zoomed out viewpoint.
• Extra Credit: Implement anisotropic filtering or summed area tables. Show comparisons of your method to nearest, bilinear, and trilinear sampling. Use clock() to measure the relative performance of the methods.
### Part 8: Draw something interesting! (10++ pts)
Use your newfound powers to render something fun and attractive. You can look up the svg specifications online for matrix transforms and for Point, Line, Polyline, Rect, Polygon, and Group classes. The ColorTri and TexTri are our own inventions, so you can intuit their parameters by looking at the svgparser.cpp file. Some ideas:
• Try to draw something "by hand" on graph paper, and manually transfer it to coordinates in the svg file.
• Write a program to procedurally generate some geometric patterns. For example, we wrote some simple programs to generate the texture mapped svg files in the svg/texmap/ directory as well as the color wheel in svg/basic/test7.svg.
• Write a program that thresholds an input photo and generates a triangle mesh from it.
We will consider aesthetics, so it's worthwhile to consider factors like composition, color, etc.
Deliverables:
• Give us your best svg file and a png screenshot of it! Also include a description of what you were trying to achieve, and how you created your svg file.
• Choose your best picture, and place a copy of it in your root project directory with the filename competition.png. This is the picture we will use in the Art Competition.
• Extra Credit: Flex your right or left brain -- either show us your artistic side, or generate awesome procedural patterns with code. This could involve a lot of programming either inside or outside of the codebase! If you write a script to generate procedural svg files, include it in your submission and briefly explain how it works.
## Tips
• Start early!
• Start assembling your webpage early to make sure you have a handle on how to edit the html code to insert images and format sections.
• The earlier you finish the basic requirements of the assignment, the more time you'll have to choose your favorite parts and implement some extra credit extensions! | 2021-07-28 14:35:52 | {"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": 0.20503827929496765, "perplexity": 2957.8056239166613}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153729.44/warc/CC-MAIN-20210728123318-20210728153318-00240.warc.gz"} |
http://mathonline.wikidot.com/differentiability-and-the-total-derivative-of-functions-from | Differentiability and the Total Derivative of Functions from Rn to Rm
# Differentiability and the Total Derivative of Functions from Rn to Rm
Definition: Let $S \subseteq \mathbb{R}^n$ be open, $\mathbf{c} \in S$, and $\mathbf{f} : S \to \mathbb{R}^m$. Then $\mathbf{f}$ is said to be Differentiable at $\mathbf{c}$ if there exists a linear function $\mathbf{T}_c : \mathbb{R}^n \to \mathbb{R}^m$ called the Total Derivative of $\mathbf{f}$ such that $\mathbf{f}(\mathbf{c} + \mathbf{v}) = \mathbf{f}(\mathbf{c}) + \mathbf{T}_{\mathbf{c}} (\mathbf{v}) + \| \mathbf{v} \| \mathbf{E}_{\mathbf{c}} (\mathbf{v})$ where we have that $\mathbf{E}_{\mathbf{c}} (\mathbf{v}) \to \mathbf{0}$ as $\mathbf{v} \to \mathbf{0}$.
The formula above is sometimes referred to as the "First order Taylor formula of $\mathbf{f}$ at $\mathbf{c}$".
The notation "$\mathbf{T}_{\mathbf{c}}(\mathbf{v})$" is sometimes replaced with the notation "$\mathbf{f}'(\mathbf{c})(\mathbf{v})$" where the notation "$\mathbf{f}'(\mathbf{c})$" in this context is denoting the linear function described above.
Recall that a function $T : A \to B$ is said to be Linear Function if for all $x, y \in A$ we have that $T(x + y) = T(x) + T(y)$ (the additivity property) and for all scalars $k$, $T(kx) = kT(x)$ (the homogeneity property).
For example, suppose that $f : S \to \mathbb{R}$ where $S \subseteq \mathbb{R}$ is open. If $f$ is differentiable at a point $c \in S$ then we have that the following limit exists:
(1)
\begin{align} \quad f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h} \end{align}
So let $E_c(h)$ be defined as follows:
(2)
\begin{align} \quad E_c(h) = \left\{\begin{matrix} \frac{f(c + h) - f(c)}{h} - f'(c) & \mathrm{if} \: h \neq 0\\ 0 & \mathrm{if} \: h = 0 \end{matrix}\right. \end{align}
Then for all $h$ we have that:
(3)
\begin{align} \quad hE_c(h) = f(c + h) - f(c) - hf'(c) \end{align}
(4)
\begin{align} \quad f(c + h) = f(c) + f'(c) h + hE_c(h) \end{align}
Notice that then the total derivative of $f$ is simply $T_c(h) = f'(c)h$, and since $f$ is differentiable we have that $E_c(h) \to 0$ as $h \to 0$ since:
(5)
\begin{align} \quad \lim_{h \to 0} E_c(h) = \lim_{h \to 0} \left [ \frac{f(c + h) - f(c)}{h} - f'(c) \right ] = \lim_{h \to 0} \left [ \frac{f(c + h) - f(c)}{h} \right ] - f'(c) = f'(c) - f'(c) = 0 \end{align}
Notice that $T_c(h)$ is indeed a linear function. To show this, note that for a defined $h_1$ and $h_2$ we have that:
(6)
\begin{align} \quad T_c(h_1 + h_2) = f'(c)(h_1 + h_2) = f'(c)h_1 + f'(c)h_2 = T_c(h_1) + T_c(h_2) \end{align}
So $T_c(h)$ satisfies the additivity property. Now for $k \in \mathbb{R}$ we have that:
(7)
\begin{align} \quad T_c(kh) = f'(c)kh = kf'(c)h = kT_c(h) \end{align}
So $T_c(h)$ satisfies the homogeneity property and thus $T_c : \mathbb{R} \to \mathbb{R}$ is indeed a linear map. So the notion of differentiability of a single variable real-valued function is consistent with the definition of differentiability made above. | 2021-01-22 09:52:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 7, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9995585083961487, "perplexity": 362.26396774017394}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703529179.46/warc/CC-MAIN-20210122082356-20210122112356-00671.warc.gz"} |
https://www.tutorialspoint.com/number-of-pairs-from-the-first-n-natural-numbers-whose-sum-is-divisible-by-k-in-cplusplus | # Number of pairs from the first N natural numbers whose sum is divisible by K in C++
C++Server Side ProgrammingProgramming
In this tutorial, we are going to write a program that counts the pairs whose sum is divisible by K.
Let's see the steps to solve the problem.
• Initialise the N and K.
• Generate the natural numbers till N and store them in an array.
• Compute the sum of every pair.
• If the pair sum is divisible by K, then increment the count.
## Example
Let's see the code.
Live Demo
#include <bits/stdc++.h>
using namespace std;
int get2PowersCount(vector<int> arr, int N, int K) {
int count = 0;
for (int i = 0; i < N; i++) {
for (int j = i + 1; j < N; j++) {
int sum = arr[i] + arr[j];
if (sum % K == 0) {
count++;
}
}
}
return count;
}
int main() {
vector<int> arr;
int N = 10, K = 5;
for (int i = 1; i <= N; i++) {
arr.push_back(i);
}
cout << get2PowersCount(arr, N, K) << endl;
return 0;
}
## Output
If you run the above code, then you will get the following result.
9
## Conclusion
If you have any queries in the tutorial, mention them in the comment section.
Published on 03-Jul-2021 08:26:28 | 2021-09-22 02:03:04 | {"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": 0.3643244206905365, "perplexity": 992.7823882923782}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057303.94/warc/CC-MAIN-20210922011746-20210922041746-00229.warc.gz"} |
https://docs.opennms.com/meridian/2021.1.0/operation/thresholds/thresholding.html | # Thresholding
Thresholding allows you to define limits against network performance metrics of a managed entity to trigger an event when a value goes above or below the specified limit.
• High
• Low
• Absolute Value
• Relative Change
## How Thresholding Works in Meridian
Meridian uses collectors to implement data collection for a particular protocol or family of protocols (SNMP, JMX, HTTP, XML/JSON, WS-Management/WinRM, JDBC, etc.). You can specify configuration for a particular collector in a collection package: essentially the set of instructions that drives the behavior of the collector.
The collectd daemon gathers and stores performance data from these collectors. This is the data against which Meridian applies thresholds. Thresholds trigger events when a specified threshold value is met. You can further create notifications and alarms for threshold events.
## What Triggers a Thresholding Event?
Meridian uses four thresholding algorithms that trigger an event when the datasource value:
• Low - equals or drops below the threshold value and re-arms when it equals or comes back up above the re-arm value (e.g., available disk space falls under the specified value)
• High - equals or exceeds the threshold value, and re-arms when it equals or drops below the re-arm value (e.g., bandwidth use exceeds the specified amount)
• Absolute - changes by the specified amount (e.g., on a fiber-optic link, a change in loss of anything greater than 3 dB is a problem regardless of what the original or final value is)
• Relative - changes by percent (e.g., available disk space changes more than 5% from the last poll)
These thresholds can be basic (tested against a single value) or an expression (evaluated against multiple values in an expression).
Meridian applies these algorithms against any performance data (telemetry) collected by collectd or pushed to telemetryd. This includes, but is not limited to, metrics such as CPU load, bandwidth, disk space, etc.
The basic walkthrough focuses on how to set simple thresholds using default values in the Meridian setup. For information on setting and configuring collectors, collectd, and the collectd-configuration.xml file, see Performance Management.
## Basic Walk-through – Thresholding
This section describes how to create a basic threshold for a single, system-wide variable: the number of logged-in users. Our threshold will tell Meridian to create an event when the number of logged-in users on the device exceeds two, and re-arm when it falls below two.
Before creating a threshold, you need to make sure you are collecting the metric against which you want to threshold.
### Determine You are Collecting Metric
In this case, we have chosen a metric (number of logged-in users) that is collected by default. We are also using data collected via SNMP. (For information on other collectors, see Collectors.)
1. In the Meridian UI, choose Reports>Resource Graphs.
2. Select one of the listed resources.
3. Under SNMP Node Data, select Node-level Performance Data and choose Graph Selection.
4. Scroll to find the Number of Users graph.
1. You can click the binoculars icon to display only this graph.
### Create a Threshold
1. Select <User_Name>>Configure OpenNMS from the top-right menu.
2. Under Performance Measurement, choose Configure Thresholds.
1. A screen with a list of preconfigured threshold groups appears. We will work with netsnmp. For information on how to create a threshold group, see Creating a Threshold Group.
3. Click Edit beside the netsnmp group.
4. Click Create New Threshold at the bottom of the Basic Thresholds area of the screen.
5. Set the following information and click Save:
Field Value Description Type high Triggers an event when the datasource value equals or exceeds the threshold value, and re-arms when it equals or drops below the re-arm value Datasource hrSystemNumUsers Name of the datasource you want to threshold against. For this tutorial, we have provided the datasource for logged-in users. For information on how to determine a metric’s datasource, see Determine the Datasource. Datasource label leave blank Optional text label. Not required for this tutorial. Value 2 The value above which we want to trigger an event. In this case, we want to trigger an event when the number of logged-in users exceeds two. Re-arm 2 The value below which we want the system to re-arm. In this case, once the number of logged-in users falls below two. Trigger 3 The number of consecutive times the threshold value can occur before the system triggers an event. Since our default polling period is 5 minutes, a value of 3 means Meridian would create a threshold event if there are more than 2 users for 15 minutes. Description leave blank Optional text to describe your threshold. Triggered UEI leave blank A custom uniform event identifier (UEI) sent into the events system when the threshold is triggered. A custom UEI for each threshold makes it easier to create notifications. If left blank, it defaults to the standard thresholds UEIs. Re-armed UEI leave blank A custom uniform event identifier (UEI) sent into the events system when the threshold is re-armed.
### Testing the Threshold
To test the threshold we just created, log a second person into the node you are monitoring. Navigate to the Events page. You should see an event that indicates your threshold triggered when more than one user logged in.
Log out the second user. The Events page should indicate that the system has re-armed.
### Creating a Threshold for CPU Usage
This procedure describes how to create an expression-based threshold when the five-minute CPU load average metric reaches or goes above 70% for two consecutive measurement intervals. Expression-based thresholds are useful when you need to threshold on a percentage, not the actual value of the data collected.
Expression-based thresholds work only if the data sources in question lie in the same directory.
1. Select <User_Name>>Configure OpenNMS from the top-right menu.
2. Under Performance Measurement, choose Configure Thresholds.
3. Click Edit beside the netsnmp group.
4. Click Create New Expression-based Threshold.
5. Fill in the following information:
Field Value Description Type high Triggers an event when the datasource value equals or exceeds the threshold value, and re-arms when it equals or drops below the re-arm value Expression ((loadavg5 / 100) / CpuNumCpus) * 100 Divides the five-minute CPU load average by 100 (to obtain the effective load average), which is then divided by the number of CPUs. This value is then multiplied by 100 to provide a percentage. ( SNMP does not report in decimals, which is why the expression divides the loadavg5 by 100.) Datasource type node The type of datasource from which you are collecting data. Datasource label leave blank Optional text label. Not required for this tutorial. Value 70 Trigger an event when the five-minute CPU load average goes above 70%. Re-arm 50 Re-arm the system when the five-minute CPU load average drops below 50% Trigger 2 The number of consecutive times the threshold value can occur before the system triggers an event. In this case, when the five-minute CPU load average goes above 70% for two consecutive polling periods. Description Trigger an alert when the five-minute CPU load average metric reaches or goes above 70% for two consecutive measurement intervals Optional text to describe your threshold. Triggered UEI leave blank See the table in Create a Threshold for details. Re-armed UEI leave blank See the table in Create a Threshold for details.
6. Click Save.
### Using Metadata in a Threshold
Metadata in expression-based thresholds can streamline threshold creation. The Metadata DSL (domain specific language) allows for the use of patterns in an expression, whereby the metadata is replaced with a corresponding value during the collection process. A single expression can behave differently based on the node being tested against.
During evaluation of an expression, the following scopes are available:
Metadata is also supported in Value, Re-arm, and Trigger fields for Single-DS and expression-based thresholds.
This procedure uses metadata to trigger an event when the number of logged-in users exceeds 1.
The expression is in the form ${context:key|context_fallback:key_fallback|…|default}. Before using metatdata in a threshold, you need to add the metatdata context pair, in this case, a requisition key called userLimit (see Adding Metadata through the Web UI). 1. Select <User_Name>>Configure OpenNMS from the top-right menu. 2. Under Performance Measurement, choose Configure Thresholds. 3. Click Edit beside the netsnmp group. 4. Click Create New Expression-based Threshold. 5. Fill in the following information: • Type: High • Expression: hrSystemNumUsers /${requisition:userLimit|1}
• Datasource type: Node
• Value: 1
• Rearm: 1
• Description: Too many logged-in users
6. Click Save.
This expression will trigger an event when the number of logged-in users exceeds 1.
### Determining the Datasource
Creating a threshold requires the name of the datasource generating the metrics on which you want to threshold. Datasource names for the SNMP protocol appear in etc/snmp-graph.properties.d/.
1. To determine the name of the datasource, navigate to the Resource Graphs screen. For example,
1. Reports>Resource Graphs.
2. Select one of the listed resources.
3. Under SNMP Node Data, select Node-level Performance Data and choose Graph Selection.
2. Scroll through the graphs to find the title of the graph that displays the metric on which you want to threshold. For example, "Number of Processes" or "System Uptime":
3. Go to etc/snmp-graph.properties.d/ and search for the title of the graph (for example, "System Uptime").
4. Note the name of the datasource, and enter it in the Datasource field when you create your threshold.
### Create a Threshold Group
A threshold group associates a set of thresholds to a service (e.g., thresholds that apply to all Cisco devices). Meridian includes seven preconfigured, editable threshold groups:
• mib2
• cisco
• hrstorage
• netsnmp
• juniper-srx
• netsnmp-memory-linux
• netsnmp-memory-nonlinux
You can edit an existing group (through the UI) or create a new one (in the thresholds.xml file located in ${OPENNMS_HOME}/etc/thresholds.xml). Once you create the group, you can then define it in the thresholds.xml file or define it in the UI. We will create a threshold group called "demo_group". 1. Type the following in the thresholds.xml file. <group name="demo_group" rrdRepository="/opt/opennms/share/rrd/snmp/"> </group> 2. Once you have created the group in the thresholds.xml file, switch to the UI, go to the threshold screen and click Reload Threshold Configuration. 1. The group you created should appear in the UI. 3. Click Edit to edit it. The following is a sample of how the threshold appears in the thresholds.xml file: <group name="demo_group" rrdRepository="/opt/opennms/share/rrd/snmp/"> (1) <expression type="high" ds-type="hrStorageIndex" value="90.0" rearm="75.0" trigger="2" ds-label="hrStorageDescr" filterOperator="or" expression="hrStorageUsed / hrStorageSize * 100.0"> <resource-filter field="hrStorageType">^\.1\.3\.6\.1\.2\.1\.25\.2\.1\.4$</resource-filter> (2)
</expression>
</group>
1 The name of the group and the directory of the stored data. 2 The details of the threshold including type, datasource type, threshold value, rearm value, etc.
### Create a Notification on a Threshold Event
A custom UEI for each threshold makes it easier to create notifications.
## Thresholding Service
The Thresholding Service is the component responsible for maintaining the state of the performance metrics and for generating alarms from these when thresholds are triggered (armed) or cleared (unarmed).
The thresholding service listens for and visits performance metrics after they are persisted to the time series database.
The state of the thresholds are held in memory and pushed to persistent storage only when they are changed.
### Distributed Thresholding with Sentinel
Thresholding for streaming telemetry with telemetryd is supported on Sentinel when using Newts. When running on Sentinel, the thresholding state can be stored in either Cassandra or PostgreSQL. Given that Newts already requires Cassandra, we recommend using Casssandra in order to help minimize the load on PostgreSQL.
Thresholding on Sentinel uses the same configuration files as Meridian and operates similarly. When a thresholding changes to/from trigger or cleared, and event is published which is processed by Meridian and the alarm is created or updated.
## Shell Commands
The following shell commands are made available to help debug and manage thresholding.
Enumerate the persisted threshold states using opennms:threshold-enumerate:
Index State Key
1 23-127.0.0.1-hrStorageIndex-hrStorageUsed / hrStorageSize * 100.0-/opt/opennms/share/rrd/snmp-RELATIVE_CHANGE
2 23-127.0.0.1-if-ifHCInOctets * 8 / 1000000 / ifHighSpeed * 100-/opt/opennms/share/rrd/snmp-HIGH
3 23-127.0.0.1-node-((loadavg5 / 100) / CpuNumCpus) * 100.0-/opt/opennms/share/rrd/snmp-HIGH
Each state is uniquely identified by a state key and aliased by the given index. Indexes are scoped to the particular shell session and provided as an alternative to specifying the complete state key in subsequent commands.
Display state details using opennms:threshold-details:
multiplier=1.333
lastSample=64.77758166043765
previousTriggeringSample=28.862826722171075
interpolatedExpression='hrStorageUsed / hrStorageSize * 100.0' | 2021-10-24 21:09:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22644326090812683, "perplexity": 3998.85387672942}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587606.8/warc/CC-MAIN-20211024204628-20211024234628-00311.warc.gz"} |
https://apboardsolutions.in/ap-ssc-10th-class-maths-solutions-chapter-5-ex-5-4/ | # AP SSC 10th Class Maths Solutions Chapter 5 Quadratic Equations Ex 5.4
AP State Board Syllabus AP SSC 10th Class Maths Textbook Solutions Chapter 5 Quadratic Equations Ex 5.4 Textbook Questions and Answers.
## AP State Syllabus SSC 10th Class Maths Solutions 5th Lesson Quadratic Equations Exercise 5.4
### 10th Class Maths 5th Lesson Quadratic Equations Ex 5.4 Textbook Questions and Answers
Question 1.
Find the nature of the roots of the following quadratic equations. If real roots exist, find them.
i) 2x2 – 3x + 5 = 0
Given: 2x2 – 3x + 5 = 0
a = 2; b = -3; c = 5
Discriminant = b2 – 4ac
b2 – 4ac = (-3)2 – 4(2)(5)
= 9 – 40
= -31 < 0
∴ Roots are imaginary.
ii) 3x2 – 4√3x + 4 = 0
Given: 3x2 – 4√3x + 4 = 0
a = 3; b = -4√3; c = 4
b2 – 4ac = (-4√3)2 – 4(3)(4)
= 48 – 48 = 0
∴ Roots are real and equal and they
$$\frac{-b}{2a}$$, $$\frac{-b}{2a}$$
iii) 2x2 – 6x + 3 = 0
Given: 2x2 – 6x + 3 = 0
a = 2; b = -6; c = 3
b2 – 4ac = (-6)2 – 4(2)(3)
= 36 – 24
= 12 > 0
∴ The roots are real and distinct. They are
Question 2.
Find the values of k for each of the fol-lowing quadratic equations so that they have two equal roots.
i) 2x2 + kx + 3 = 0
Given : 2x2 + kx + 3 = 0 has equal roots
∴ b2 – 4ac = 0
Here a = 2; b = k; c = 3
b2 – 4ac = (k)2 – 4(2)(3) = 0
⇒ k2 – 24 = 0
⇒ k2 = 24
⇒ k = √24 = ± 2√6
ii) kx(x – 2) + 6 = 0
Given: kx(x – 2) + 6 = 0
kx2 – 2kx + 6 = 0
As this Q.E. has equal roots,
b2 – 4ac = 0
Here
a = k; b = -2k; c = 6
∴ b2 – 4ac = (-2k)2 – 4(k)(6) = 0
⇒ 4k2 – 24k = 0
⇒ 4k(k – 6) = 0
⇒ 4k = 0 (or) k – 6 = 0
⇒ k = 0 (or) 6
But k = 0 is trivial
∴ k = 6.
Question 3.
Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.
Let the breadth = x m
Then length = 2x m
Area = length x breadth = x.(2x)
= 2x2 m2
By problem 2x2 = 800 ⇒ x2 = 400
and x = √400 = ± 20
∴ Breadth x = 20 m and
length 2x = 2 × 20 = 40 m.
Question 4.
The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48. Is the above situation possible? If so, deter¬mine their present ages.
Let the age one of the two friends be x years.
Then the age of the other = 20 – x
Then, 4 years ago their ages would be (x – 4) and (20 – x – 4) = 16 – x
∴ Product of their ages 4 years ago = (x – 4) (16 – x)
By problem (x – 4) (16 – x) = 48
⇒ x(16 – x) – 4(16 – x) = 48
⇒ 16x – x2 – 64 + 4x = 48
⇒ x2 – 20x + 112 = 0
Here a = 1; b = -20; c = 112
b2 – 4ac = (-20)2 – 4(1) (112)
= 400 – 448
= -48 < 0
Thus the roots are not real.
∴ The situation is not possible.
Question 5.
Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.
Given: Perimeter of a rectangle 2(1 + b) = 80
⇒ 6 + b = $$\frac{80}{2}$$ = 40
Area of the rectangle, l × b = 400
If possible, let us suppose that length of the rectangle = x m say
Then its breadth by equation (1) = 40 – x
By problem area = x . (40 – x) = 400
⇒ 40x – x2 = 400
⇒ x2 – 40x + 400 = 0
Here a = 1; b = -40; c = +400
b2 – 4ac = (-40)2 – 4(1)(+400)
= 1600 – 1600 = 0
∴ The roots are real and equal.
They are $$\frac{-b}{2a}$$, $$\frac{-b}{2a}$$
i.e., $$\frac{-(-40)}{2 \times 1}$$ = $$\frac{40}{2}$$ = 20
∴ The dimensions are 20 m, 20 m.
(∴ The park is in square shape) | 2022-09-29 19:57:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5071454048156738, "perplexity": 1867.1147753398782}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335365.63/warc/CC-MAIN-20220929194230-20220929224230-00266.warc.gz"} |
http://climatechangepsychology.blogspot.com/2011/01/m-g-flanner-k-m-shell-m-barlage-d-k.html | ## Wednesday, January 19, 2011
### M. G. Flanner, K. M. Shell, M. Barlage, D. K. Perovich & M. A. Tschudi, Nature Geosci. (January 2011), Radiative forcing and albedo feedback from the Northern Hemisphere cryosphere between 1979 and 2008
Nature Geoscience (January 16, 2011), doi: 10.1038/ngeo1062
Radiative forcing and albedo feedback from the Northern Hemisphere cryosphere between 1979 and 2008
M. G. Flanner*, K. M. Shell, M. Barlage, D. K. Perovich and M. A. Tschudi
Abstract
The extent of snow cover1 and sea ice2 in the Northern Hemispherehas declined since 1979, coincident with hemispheric warming and indicative of a positive feedback of surface reflectivity on climate. This albedo feedback of snow on land has been quantified from observations at seasonal timescales3,456, and century-scale feedback has been assessed using climate models7,8910. However, the total impact of the cryosphere on radiative forcing and albedo feedback has yet to be determined from measurements. Here we assess the influence of the Northern Hemisphere cryosphere on Earth’s radiation budget at the top of the atmosphere—termed cryosphere radiative forcing—by synthesizing a variety of remote sensing and field measurements. We estimate mean Northern Hemisphere forcing at −4.6 to −2.2 Wm−2, with a peak in May of −9.0±2.7 Wm−2. We find that cyrospheric cooling declined by 0.45 Wm−2 from 1979 to 2008, with nearly equal contributions from changes in land snow cover and sea ice. On the basis of these observations, we conclude that the albedo feedback from the Northern Hemisphere cryosphere falls between 0.3 and 1.1 Wm−2K−1, substantially larger than comparable estimates obtained from 18 climate models.
M. G. Flanner, K. M. Shell, M. Barlage, D. K. Perovich, M. A. Tschudi. Radiative forcing and albedo feedback from the Northern Hemisphere cryosphere between 1979 and 2008. Nature Geoscience, 2011; DOI:10.1038/ngeo1062 | 2014-10-25 04:13:35 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.819710373878479, "perplexity": 4722.224238265697}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119647629.9/warc/CC-MAIN-20141024030047-00055-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://docs.wpilib.org/en/stable/docs/software/commandbased/profile-subsystems-commands.html | # Motion Profiling through TrapezoidProfileSubsystems and TrapezoidProfileCommands
Note
For a description of the WPILib motion profiling features used by these command-based wrappers, see Trapezoidal Motion Profiles in WPILib.
Note
The TrapezoidProfile command wrappers are generally intended for composition with custom or external controllers. For combining trapezoidal motion profiling with WPILib’s PIDController, see Combining Motion Profiling and PID in Command-Based.
When controlling a mechanism, is often desirable to move it smoothly between two positions, rather than to abruptly change its setpoint. This is called “motion-profiling,” and is supported in WPILib through the TrapezoidProfile class (Java, C++).
To further help teams integrate motion profiling into their command-based robot projects, WPILib includes two convenience wrappers for the TrapezoidProfile class: TrapezoidProfileSubsystem, which automatically generates and executes motion profiles in its periodic() method, and the TrapezoidProfileCommand, which executes a single user-provided TrapezoidProfile.
## TrapezoidProfileSubsystem
Note
In C++, the TrapezoidProfileSubsystem class is templated on the unit type used for distance measurements, which may be angular or linear. The passed-in values must have units consistent with the distance units, or a compile-time error will be thrown. For more information on C++ units, see The C++ Units Library.
The TrapezoidProfileSubsystem class (Java, C++) will automatically create and execute trapezoidal motion profiles to reach the user-provided goal state. To use the TrapezoidProfileSubsystem class, users must create a subclass of it.
### Creating a TrapezoidProfileSubsystem
Note
If periodic is overridden when inheriting from TrapezoidProfileSubsystem, make sure to call super.periodic()! Otherwise, motion profiling functionality will not work properly.
When subclassing TrapezoidProfileSubsystem, users must override a single abstract method to provide functionality that the class will use in its ordinary operation:
#### useState()
protected abstract void useState(TrapezoidProfile.State state);
The useState() method consumes the current state of the motion profile. The TrapezoidProfileSubsystem will automatically call this method from its periodic() block, and pass it the motion profile state corresponding to the subsystem’s current progress through the motion profile.
Users may do whatever they want with this state; a typical use case (as shown in the Full TrapezoidProfileSubsystem Example) is to use the state to obtain a setpoint and a feedforward for an external “smart” motor controller.
#### Constructor Parameters
Users must pass in a set of TrapezoidProfile.Constraints to the TrapezoidProfileSubsystem base class through the superclass constructor call of their subclass. This serves to constrain the automatically-generated profiles to a given maximum velocity and acceleration.
Users must also pass in an initial position for the mechanism.
Advanced users may pass in an alternate value for the loop period, if a non-standard main loop period is being used.
### Using a TrapezoidProfileSubsystem
Once an instance of a TrapezoidProfileSubsystem subclass has been created, it can be used by commands through the following methods:
#### setGoal()
Note
If you wish to set the goal to a simple distance with an implicit target velocity of zero, an overload of setGoal() exists that takes a single distance value, rather than a full motion profile state.
The setGoal() method can be used to set the goal state of the TrapezoidProfileSubsystem. The subsystem will automatically execute a profile to the goal, passing the current state at each iteration to the provided useState() method.
// The subsystem will execute a profile to a position of 5 and a velocity of 3.
examplePIDSubsystem.setGoal(new TrapezoidProfile.State(5, 3);
#### enable() and disable()
The enable() and disable() methods enable and disable the motion profiling control of the TrapezoidProfileSubsystem. When the subsystem is enabled, it will automatically run the control loop and call useState() periodically. When it is disabled, no control is performed.
### Full TrapezoidProfileSubsystem Example
What does a TrapezoidProfileSubsystem look like when used in practice? The following examples are taking from the ArmbotOffobard example project (Java, C++):
5package edu.wpi.first.wpilibj.examples.armbotoffboard.subsystems;
6
7import edu.wpi.first.math.controller.ArmFeedforward;
8import edu.wpi.first.math.trajectory.TrapezoidProfile;
9import edu.wpi.first.wpilibj.examples.armbotoffboard.Constants.ArmConstants;
10import edu.wpi.first.wpilibj.examples.armbotoffboard.ExampleSmartMotorController;
11import edu.wpi.first.wpilibj2.command.Command;
12import edu.wpi.first.wpilibj2.command.Commands;
13import edu.wpi.first.wpilibj2.command.TrapezoidProfileSubsystem;
14
15/** A robot arm subsystem that moves with a motion profile. */
16public class ArmSubsystem extends TrapezoidProfileSubsystem {
17 private final ExampleSmartMotorController m_motor =
18 new ExampleSmartMotorController(ArmConstants.kMotorPort);
19 private final ArmFeedforward m_feedforward =
20 new ArmFeedforward(
21 ArmConstants.kSVolts, ArmConstants.kGVolts,
23
24 /** Create a new ArmSubsystem. */
25 public ArmSubsystem() {
26 super(
27 new TrapezoidProfile.Constraints(
30 m_motor.setPID(ArmConstants.kP, 0, 0);
31 }
32
33 @Override
34 public void useState(TrapezoidProfile.State setpoint) {
35 // Calculate the feedforward from the sepoint
36 double feedforward = m_feedforward.calculate(setpoint.position, setpoint.velocity);
37 // Add the feedforward to the PID output to get the motor output
38 m_motor.setSetpoint(
39 ExampleSmartMotorController.PIDMode.kPosition, setpoint.position, feedforward / 12.0);
40 }
41
42 public Command setArmGoalCommand(double kArmOffsetRads) {
43 return Commands.runOnce(() -> setGoal(kArmOffsetRads), this);
44 }
45}
Using a TrapezoidProfileSubsystem with commands can be quite simple:
52 // Move the arm to 2 radians above horizontal when the 'A' button is pressed.
53 m_driverController.a().onTrue(m_robotArm.setArmGoalCommand(2));
54
55 // Move the arm to neutral position when the 'B' button is pressed.
56 m_driverController
57 .b()
## TrapezoidProfileCommand
Note
In C++, the TrapezoidProfileCommand class is templated on the unit type used for distance measurements, which may be angular or linear. The passed-in values must have units consistent with the distance units, or a compile-time error will be thrown. For more information on C++ units, see The C++ Units Library.
The TrapezoidProfileCommand class (Java, C++) allows users to create a command that will execute a single TrapezoidProfile, passing its current state at each iteration to a user-defined function.
### Creating a TrapezoidProfileCommand
A TrapezoidProfileCommand can be created two ways - by subclassing the TrapezoidProfileCommand class, or by defining the command inline. Both methods ultimately extremely similar, and ultimately the choice of which to use comes down to where the user desires that the relevant code be located.
Note
If subclassing TrapezoidProfileCommand and overriding any methods, make sure to call the super version of those methods! Otherwise, motion profiling functionality will not work properly.
In either case, a TrapezoidProfileCommand is created by passing the necessary parameters to its constructor (if defining a subclass, this can be done with a super() call):
25 /**
26 * Creates a new TrapezoidProfileCommand that will execute the given {@link TrapezoidProfile}.
27 * Output will be piped to the provided consumer function.
28 *
29 * @param profile The motion profile to execute.
30 * @param output The consumer for the profile output.
31 * @param requirements The subsystems required by this command.
32 */
33 public TrapezoidProfileCommand(
34 TrapezoidProfile profile, Consumer<State> output, Subsystem... requirements) {
#### profile
The profile parameter is the TrapezoidProfile object that will be executed by the command. By passing this in, users specify the start state, end state, and motion constraints of the profile that the command will use.
#### output
The output parameter is a function (usually passed as a lambda) that consumes the output and setpoint of the control loop. Passing in the useOutput function in PIDCommand is functionally analogous to overriding the useState() function in PIDSubsystem.
#### requirements
Like all inlineable commands, TrapezoidProfileCommand allows the user to specify its subsystem requirements as a constructor parameter.
### Full TrapezoidProfileCommand Example
What does a TrapezoidProfileSubsystem look like when used in practice? The following examples are taking from the DriveDistanceOffboard example project (Java, C++):
5package edu.wpi.first.wpilibj.examples.drivedistanceoffboard.commands;
6
7import edu.wpi.first.math.trajectory.TrapezoidProfile;
8import edu.wpi.first.wpilibj.examples.drivedistanceoffboard.Constants.DriveConstants;
9import edu.wpi.first.wpilibj.examples.drivedistanceoffboard.subsystems.DriveSubsystem;
10import edu.wpi.first.wpilibj2.command.TrapezoidProfileCommand;
11
12/** Drives a set distance using a motion profile. */
13public class DriveDistanceProfiled extends TrapezoidProfileCommand {
14 /**
15 * Creates a new DriveDistanceProfiled command.
16 *
17 * @param meters The distance to drive.
18 * @param drive The drive subsystem to use.
19 */
20 public DriveDistanceProfiled(double meters, DriveSubsystem drive) {
21 super(
22 new TrapezoidProfile(
23 // Limit the max acceleration and velocity
24 new TrapezoidProfile.Constraints(
25 DriveConstants.kMaxSpeedMetersPerSecond,
26 DriveConstants.kMaxAccelerationMetersPerSecondSquared),
27 // End at desired position in meters; implicitly starts at 0
28 new TrapezoidProfile.State(meters, 0)),
29 // Pipe the profile state to the drive
30 setpointState -> drive.setDriveStates(setpointState, setpointState),
31 // Require the drive
32 drive);
33 // Reset drive encoders since we're starting at 0
34 drive.resetEncoders();
35 }
36}
And, for an inlined example:
66 // Do the same thing as above when the 'B' button is pressed, but defined inline
67 m_driverController
68 .b()
69 .onTrue(
70 new TrapezoidProfileCommand(
71 new TrapezoidProfile(
72 // Limit the max acceleration and velocity
73 new TrapezoidProfile.Constraints(
74 DriveConstants.kMaxSpeedMetersPerSecond,
75 DriveConstants.kMaxAccelerationMetersPerSecondSquared),
76 // End at desired position in meters; implicitly starts at 0
77 new TrapezoidProfile.State(3, 0)),
78 // Pipe the profile state to the drive
79 setpointState -> m_robotDrive.setDriveStates(setpointState, setpointState),
80 // Require the drive
81 m_robotDrive)
82 .beforeStarting(m_robotDrive::resetEncoders)
83 .withTimeout(10)); | 2023-03-31 07:13:35 | {"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": 0.304706871509552, "perplexity": 10605.83937380335}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949573.84/warc/CC-MAIN-20230331051439-20230331081439-00432.warc.gz"} |
http://mathoverflow.net/questions/102566/solving-a-system-of-linear-inequalities | # Solving a system of linear inequalities
Consider the following system of inequalities:
$Ax=b$; $x\geq 0$;
A is a $m\times n$ (non-square) and sparse matrix in which some part of entries are rational. How this system can be solved without using linear programming?
-
please explain what mean by "solved". – Dima Pasechnik Jul 18 '12 at 17:35
I meant, how can the feasibility of this system be checked without using linear programming? – Star Jul 18 '12 at 17:51
There is no general way, but perhaps you can use one of the "theorems of the alternative" such as Farkas's Lemma, Gale's Theorem, etc. – Yoav Kallus Jul 18 '12 at 20:43
Why do you want to avoid linear programming? – Gilead Jul 18 '12 at 20:45 | 2014-11-23 22:41:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9190755486488342, "perplexity": 1180.9285732203346}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416400380068.47/warc/CC-MAIN-20141119123300-00158-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/3615600/how-find-the-minimum-of-the-f-n-x-sum-i-1-n-x-i-2-sum-i | How find the minimum of the $f_n (x) = ( \sum_{i=1} ^ {n} | x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2 .$
For a positive integer $$n$$, define a function $$f_n (x)$$ at an interval $$[ 0, n+1 ]$$ as $$f_n (x) = ( \sum_{i=1} ^ {n} | x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2 .$$ Let $$a_n$$ be the minimum value of $$f_n (x)$$. Find the value of $$\sum_{n=1}^{11} (-1)^{n+1} a_n .$$
It is said the answer is 450. I try find the sum $$\sum_{i=1}^{n}(x-i)^2=nx^2-2x\sum_{i=1}^{n}i+\sum_{i=1}^{n}i^2 =nx^2-n(n+1)x+\dfrac{n(n+1))(2n+1)}{6}$$This term looks like a quadratic function，But $$(\sum_{i=1}^{n}|x-i|)^2$$ seem hard to deal it
• The first term is minimized at the median, and the rest becomes trivial is you compare derivatives. – LinAlg Apr 19 at 14:10
• @function, I am rather disappointed that you had been offline before your bounty became expired. I wonder if I should I be more prudent next time when your raise a bounty on a question. (This comment will be removed.) – Apass.Jack Apr 21 at 15:50
• @user125932 For even $n$, $f_n$ is not convex and its minimum does not occur at $x=\frac{n+1}2$. For example, $f_2(1)=f_2(2)=0<\frac12=f_2(\frac32)$. – Apass.Jack Apr 21 at 21:09
• A cursory looks might suggest that $f_n$ is convex and, since it is symmetric about $x=\frac{n+1}2$, its minimum should occur at $x=\frac{n+1}2$. However, $f_n$ is not convex for even $n$ (I did not check for odd $n$). Also, $f_2(1)=f_2(2)=0<\frac12=f_2(\frac32)$. – Apass.Jack Apr 28 at 13:53
• @LinAlg It looks like you suggest either it is enough to investigate two terms separately or the whole problem is trivial. While the problem can probably be solved as long as one persists enough, it is not trivial, I believe, in other senses. I would love to see a simple answer from you. – Apass.Jack Apr 28 at 14:00
Let $$n$$ be some fixed positive integer.
Since $$[0,n+1]$$ and $$f_n(x)$$ are symmetric about $$x=\frac{n+1}2$$, the minimum value of $$f_n$$ over $$[0, n+1]$$ is the same as the minimum of $$f_n$$ over $$[0, \frac{n+1}{2}]$$. From now on, we will assume $$f_n$$ is defined on $$[0, \frac{n+1}2]$$. It is immediate to verify $$f_1(x)=0$$. From now on, we will assume $$n>1$$.
As observed in the question, $$(\sum_{i=1}^{n}|x-i|)^2$$ is not easy to deal with because of the absolute values. The most common way to remove absolute values is, well, to separate the domain of the variable into small pieces so that we know how to take the absolute value in each piece.
Let $$\lfloor x\rfloor$$ be the integer part of $$x$$, i.e., $$\lfloor x\rfloor\in\mathbb N$$ and $$0\le x - \lfloor x\rfloor\lt1$$. There are two cases for $$\lfloor x\rfloor$$.
• $$\lfloor x\rfloor=0$$. $$\quad( \sum_{i=1} ^ {n} | x-i | )^2=( \sum_{i=1} ^ {n} (i-x) )^2=( \frac{n(n+1)}2-nx)^2.$$
• $$1\le \lfloor x\rfloor\le \frac{n+1}2$$. $$\quad( \sum_{i=1} ^ {n} | x-i | )^2=( \sum_{i=1} ^ {\lfloor x\rfloor} (x-i)+\sum_{i=\lfloor x\rfloor+1} ^ {n} (i-x) )^2\\=\left( \sum_{i=1} ^ {\lfloor x\rfloor} x+\sum_{i=\lfloor x\rfloor+1} ^ {n} (-x)+\sum_{i=1} ^ {\lfloor x\rfloor} (-i)+\sum_{i=\lfloor x\rfloor+1} ^ {n}i \right)^2\\=\left( (2\lfloor x\rfloor-n)x-\frac{\lfloor x\rfloor(\lfloor x\rfloor+1)}2+\frac{(n-\lfloor x\rfloor)(n+\lfloor x\rfloor+1)}2\right)^2.$$
Note the formula above for the second case holds for the first case, too.
As computed in the question, we have.
$$\sum_{i=1}^{n}(x-i)^2=nx^2-n(n+1)x+\dfrac{n(n+1)(2n+1)}{6}.$$
So, \begin{aligned} &\quad\quad f_n(x)=\sum_{i=1} ^ {n}| x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2\\ &=\left( (2\lfloor x\rfloor-n)x-\frac{\lfloor x\rfloor(\lfloor x\rfloor+1)}2+\frac{(n-\lfloor x\rfloor)(n+\lfloor x\rfloor+1)}2\right)^2 - \left(nx^2-n(n+1)x+\dfrac{n(n+1)(2n+1)}{6}\right)\\ &=-nx^2+(n^2+2\lfloor x\rfloor)x+ \text{ (some formula that depends only on } \lfloor x\rfloor\text{ and }n)\\ &=-n\left(x-(\frac n2+\frac{\lfloor x\rfloor}n)\right)^2+ \text{ (some formula that depends only on } \lfloor x\rfloor\text{ and }n).\\ \end{aligned}
Since $$\left\lfloor\lfloor x\rfloor\right\rfloor=\lfloor x\rfloor$$, we have $$f_n(x)-f_n(\lfloor x\rfloor,n)=-n\left(x-(\frac n2+\frac{\lfloor x\rfloor}n)\right)^2+n\left(\lfloor x\rfloor-(\frac n2+\frac{\lfloor x\rfloor}n)\right)^2=n(x-\lfloor x\rfloor)(n+\frac{2\lfloor x\rfloor}n-x-\lfloor x\rfloor).$$
• If $$x\le\frac n2$$, we have $$n+\frac{2\lfloor x\rfloor}n-x-\lfloor x\rfloor\ge n -x-x\ge0.$$
• Otherwise $$x\gt\frac n2.$$ Recall that $$x\le\frac{n+1}2$$.
• $$n$$ is even. Then $$\lfloor x\rfloor=\frac n2$$. We have $$n+\frac{2\lfloor x\rfloor}n-x-\lfloor x\rfloor= \frac{n+1}2- x + \frac12\gt0.$$
• $$n$$ is odd.
• If $$x=\frac {n+1}2$$, i.e., $$x$$ is an integer, $$x-\lfloor x\rfloor=0$$.
• Otherwise, suppose $$x<\frac {n+1}2$$. Then $$\lfloor x\rfloor=\frac{n-1}2$$, and $$n+\frac{2\lfloor x\rfloor}n-x-\lfloor x\rfloor=1-\frac 1n+\frac{n+1}2-x \gt0.$$
Combining all cases, we have $$f_n(x)-f_n(\lfloor x\rfloor,n)\ge0$$. That is, if $$f_n(\cdot)$$ reaches its minimum at $$x$$, then it must also reach its minimum at $$\lfloor x\rfloor$$, an integer.
So, in order to find the minimum of $$f_n(x)$$, we can restrict $$x$$ to integers. An easy way to obtain $$\sum_{n=1}^{11} (-1)^{n+1} a_n$$ should be computing $$f_n(x)$$ at every possible integer $$x$$ by brute force, by some programming using your favourite programming language.
Let us check where $$f_n(x)$$ can reach its minimum. Assume $$f_n(x)$$ is defined on integers in $$[0, \frac{n+1}2]$$.
\begin{aligned} &\quad\quad f_n(x)\\ &=\left( (2x-n)x-\frac{x(x+1)}2+\frac{(n-x)(n+x+1)}2\right)^2 - \left(nx^2-n(n+1)x+\dfrac{n(n+1)(2n+1)}{6}\right)\\ &=\left(x^2-(n+1)x+ \frac{n^2+n}2\right)^2-\left(nx^2-n(n+1)x+\dfrac{n(n+1)(2n+1)}{6}\right).\\ \end{aligned}
Suppose $$x+1\le \frac{n+1}2$$ so that $$f_n(x+1)$$ is defined.
\begin{aligned} &\quad\quad f_n(x) - f_n(x+1)\\ &=\left( (-(x+x+1)+(n+1))(x^2+(x+1)^2-(n+1)(x+x+1)+n^2+n)\right)+\left(n(x+x+1)-n(n+1)\right)\\ &=(n-2x)(2x^2-2nx+n^2)+(2nx-n^2)\\ &=(n-2x)\left(2(x-\frac n2)^2+\frac {n(n-2)}2\right).\\ &\ge0 \end{aligned}
That means, $$f_n(0), f_n(1), f_n(2), ...$$ is a non-increasing sequence. The last item of the sequence is either $$f_n(\frac n2)$$ for even $$n$$ or $$f_n(\frac {n+1}2)$$ for odd $$n$$. So,
$$a_n= \begin{cases} f_n(\frac {n+1}2)\quad \text{ if } n \text{ is odd,}\\ f_n(\frac n2)\quad\quad \text{ if } n \text{ is even.} \end{cases}$$
Now let us compute $$a_n$$ into closed formula. For odd n, \begin{aligned} a_n &=\left((\frac {n+1}2)^2-(n+1)\frac {n+1}2+ \frac{n^2+n}2\right)^2-\left(n(\frac {n+1}2)^2-n(n+1)\frac {n+1}2+\dfrac{n(n+1)(2n+1)}{6}\right)\\ &=\frac{(n^2-1)^2}{16}-\frac{n^3-n}{12}.\\ \end{aligned}
So, we have $$a_{1}=0$$, $$a_{3}=2$$, $$a_{5}=26$$, $$a_{7}=116$$, $$a_{9}=340$$, $$a_{11}=790$$.
For even $$n$$, \begin{aligned} a_n &=\left((\frac n2)^2-(n+1)\frac n2+ \frac{n^2+n}2\right)^2-\left(n(\frac n2)^2-n(n+1)\frac n2+\dfrac{n(n+1)(2n+1)}{6}\right)\\ &=\frac{n^4}{16}-\frac{n^3+2n}{12}.\\ \end{aligned} So, we have $$a_{2}=0$$, $$a_{4}=10$$, $$a_{6}=62$$, $$a_{8}=212$$, $$a_{10}=540$$.
Finally, we obtain $$\sum_{n=1}^{11} (-1)^{n+1} a_n=0-0 + 2-10 + 26-62 + 116-212 + 340-540 + 790=450.$$ | 2020-11-29 14:34:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 87, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9501851797103882, "perplexity": 525.139843793417}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141198409.43/warc/CC-MAIN-20201129123729-20201129153729-00702.warc.gz"} |
https://search.datacite.org/repositories/uky.lib?resource-type-id=dataset | ### Genomics of Mature and Immature Olfactory Sensory Neuron
Timothy S. McClintock
The Excel file contains probabilistic predictions of which genes are expressed in mature versus immature olfactory sensory neurons, versus the sum of all other cell types in the olfactory epithelium. Responses to bulbectomy, which differentially affects mature and immature olfactory sensory neurons, supports these predictions.
### Analysis of Traffic Growth Rates
Monica L. Barrett, R. Clark Graves, David L. Allen, Jerry G. Pigman, Ghassan Abu-Lebdeh, Lisa Aultman-Hall & Sarah T. Bowling
The primary objectives of this study were to determine patterns of traffic flow and develop traffic growth rates by traffic composition and highway type for Kentucky’s system of highways. Additional subtasks included the following: 1) a literature search to determine if there were new procedures being used to more accurately represent traffic growth rates, 2) development of a random sampling procedure for collecting traffic count data on local roads and streets, 3) prediction of vehicle...
### TACOT v3.0
J. Lachaud, N. Mansour, S. White, B. Laub & J.-M. Bouilly
### KY13
Lindell Ormsbee
KY 13 is primarily a looped system in Kentucky with the following assets: 5 Tanks, 4 Pumps, 1 Water Treatment Plant, and approximately 422432 feet of pipe. KY 13 provides 2.36 million gallons of water per day to its 5335 customers at a rate which ranges between $5.60 and$6.20 per 1,000 gallons of water. Water loss for KY 13 is estimated at 5% of the water produced.
### KY13
Lindell Ormsbee
KY 13 is primarily a looped system in Kentucky with the following assets: 5 Tanks, 4 Pumps, 1 Water Treatment Plant, and approximately 422432 feet of pipe. KY 13 provides 2.36 million gallons of water per day to its 5335 customers at a rate which ranges between $5.60 and$6.20 per 1,000 gallons of water. Water loss for KY 13 is estimated at 5% of the water produced.
Lindell Ormsbee
### KY4
Lindell Ormsbee
KY 4 is primarily a loop system in Kentucky with the following assets 4 Tanks, 2 Pumps, and approximately 854446 feet of pipe. KY 4 provides 1.51 million gallons of water per day to its 9,020 customers at a rate which ranges between $6.46 and$7.65 per 1,000 gallons of water. Water loss for KY 4 is estimated at 12% of the water produced.
### KY14
Lindell Ormsbee
KY 14 is primarily a grid system in Kentucky with the following assets: 3 Tanks, 6 Pumps, 1 Water Treatment Plant, and approximately 287609 feet of pipe. KY 14 provides 1.04 million gallons of water per day to its 2682 customers at a rate which ranges between $4.00 and$6.00 per 1,000 gallons of water. Water loss for KY 14 is estimated at 26% of the water produced.
### KY3
Lindell Ormsbee
KY 3 is primarily a grid system in Kentucky with the following assets: 3 Tanks, 5 Pumps, 1 Water Treatment Plant, and approximately 277,022 feet of pipe. KY 3 provides 2.02 million gallons of water per day to its 2,142 customers at a rate which ranges between $4.00 and$6.45 per 1,000 gallons of water. Water loss for KY 3 is estimated at 10% of the water produced.
### Sensory Processing and Behaviors Characteristic of Autism Spectrum Disorder in Older Adults with Cognitive Impairment
Elizabeth Rhodus, Elizabeth Hunter, Graham Rowles, Erin Abner, Shoshana Bardach, Justin Barber & Gregory Jicha
### Impact of Buckwheat and Methyl Salicylate Lures on Natural Enemy Abundance for Early Season Management of Melanaphis sacchari (Hemiptera: Aphididae) in Sweet Sorghum
Nathan Mercer
Tested effect of buckwheat flowers and methyl salicylate lures to attract natural enemies to sweet sorghum fields to manage Melanaphis sacchari, a recent pest of sweet sorghum.
### Kentucky Geological Survey Landslide Inventory [2022-01]
Matt Crawford
The KGS landslide inventory provides the locations of known landslides and areas susceptible to debris flows. Various types of landslides are represented including slides, flows, rockfalls, and creep. The data are available as ArcGIS geodatabase feature classes. Landslide locations and associated attributes are compiled from Kentucky Geological Survey research, published maps, state and local government agencies, the public, and media reports. A confidence ranking system assigns a value to each feature. A description of the...
### Tuberculosis and Local Health Department Expenditures on Tuberculosis Services
Michelle P. Yip & Betty Bekemeier
Background: Although tuberculosis (TB) morbidity and mortality have decreased in recent decades, challenges exist regarding disproportionate distributions of TB among specific populations and geographic areas. Inconsistent local health department (LHD) funding for TB programs poses difficulties for LHDs to sustain resources and personnel that predisposes communities to risks of future outbreaks of TB and drug-resistant TB diseases. Purpose: This study examined relationships between annual TB incidence rates and LHD expenditures on TB-related services to elucidate...
### Ergot and Loline Alkaloid Concentrations in Endophyte-Infected Tall Fescue Tillers
Rebecca L. McCulley
Approximately 40 tall fescue tillers were randomly collected and frozen from each of the 20 treatment plots. Tillers were cut at 7.6 cm above ground level and tested for the presence of the Epichloe endophyte using an enzyme-linked immunosorbent assay. Tillers from each plot were sorted into 'infected' vs 'uninfected' groups, lyophilized, and ground through a 1mm screen using a Cyclotec 1093 mill. Ground material from the endophyte infected tillers was analyzed for ergot and...
### KY3
Lindell Ormsbee
KY 3 is primarily a grid system in Kentucky with the following assets: 3 Tanks, 5 Pumps, 1 Water Treatment Plant, and approximately 277,022 feet of pipe. KY 3 provides 2.02 million gallons of water per day to its 2,142 customers at a rate which ranges between $4.00 and$6.45 per 1,000 gallons of water. Water loss for KY 3 is estimated at 10% of the water produced.
### Bee Assemblages Data [2021]
Daniel A. Potter & Bernadette M. Mach
Plant characteristics, sample sites, and non-native bee assemblages for a bee survey conducted in 2014-2017 by Bernadette Mach in the Daniel A. Potter lab at the University of Kentucky.
### StateMap 2020-2021
Matthew Massey, Antonia E. Bottoms, Maxwell L., III Hammond & Michele M. McHugh
This dataset characterizes the types and distributions of surficial geologic materials in the Cecilia, Constantine, Howe Valley, and Sonora 7.5-minute quadrangles. The primary goal for this mapping project was to identify and map the spatial distribution and understand the mechanical and chemical properties of surficial geologic materials in a rapidly developing area of Kentucky. This new dataset will have immediate utility to a range of users in local, state, and Federal agencies that are working...
### Jilin Network
Graeme Dandy
The Jilin network is a hypothetical network that was first introduced by Bi and Dandy (2014). It is an optimization problem involving the selection of pipe sizes and chlorine dosing. The demand pattern involves a 24 hour extended period simulation. The available pipe sizes and costs were taken from Kadu et al (2008). The average annual demand is 6.66 MGD.
### Hanoi System
Graeme Dandy
The Hanoi system was first presented by Fujiwara and Khang (1990) and is based on the planned trunk network of Hanoi, Vietnam. There are 34 pipes to be sized with a total length of 38.61 km. Possible new pipe sizes range between 12 and 40 inches and the total system demand is 126.5 MGD.
### Any-town System
Thomas M. Walski
The system description is available from the article, Battle of the network models: Epilogue, published in the Journal of Water Resources Planning and Management.
### Hippodamia variegata Development at LD
John J. Obrycki
Reproductive diapause in North American populations of the introduced lady beetle Hippodamia variegata
• 2022
3
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8
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12
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6
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57
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1
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3
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4
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94
#### Affiliations
• University of Kentucky
24
• Binghamton University
1
• Stanford University
1
• Utah State University
1
• South Dakota State University
1
• National Aeronautics and Space Administration
1
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1 | 2022-08-08 06:24:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27125877141952515, "perplexity": 9649.5335977542}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570767.11/warc/CC-MAIN-20220808061828-20220808091828-00788.warc.gz"} |
https://stats.stackexchange.com/questions/180232/simple-student-copula-simulation | # Simple Student Copula simulation
I want to simulate a t copula with given correlation parameter $\Sigma$ and $k$ degrees of freedom. I can't find any literature about practical simulation, so I am trying new approaches. (I also have some problems using packages in R. I am working on that but I feel like I can find a faster and more interesting solution to my problem)
A student r.v. with k degree of freedom has the same distribution as $Z\sqrt{\frac{k}{V}}$ where $Z$ follow a normal with mean 0 and variance 1 and $V$ follow a chi-squared distribution with $k$ degrees of freedom.
If I simulate a multivariate (n dimension) normal r.v. with mean 0 and variance $\Sigma$ and divide them by n independant rv following a chi-squared distibution I can obtain a multivariate r.v. where each one follow a t ditribution and the correlation is $\Sigma$. Am I right ? Is this equivalent to simulating a n-dimensionnal t-student r.v. with parameter k and correlation $\Sigma$ ?
Then I want to "go back" to quantiles. With a multivariate normal r.v. I would use the inverse cdf on each r.v. Does this hold with a t-distibution ? If I use the inverse cdf of a univariate t-distribution on each of my n r.v. that follow a t distribution will I get the same result as simulating a t-copula with parameters $\Sigma$ and k ?
• @Were_cat No, with the cdf. If a random variable $X$ has cdf $F_X$ then $U\sim F_X(X)$ is distributed as uniform. – Glen_b Nov 5 '15 at 9:05 | 2021-05-06 04:39:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7338433265686035, "perplexity": 309.4226107547462}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988725.79/warc/CC-MAIN-20210506023918-20210506053918-00086.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/mbe.2011.8.689 | # American Institute of Mathematical Sciences
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2011, 8(3): 689-694. doi: 10.3934/mbe.2011.8.689
## A note for the global stability of a delay differential equation of hepatitis B virus infection
1 Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, China, China
Received September 2010 Revised October 2010 Published June 2011
The global stability for a delayed HIV-1 infection model is investigated. It is shown that the global dynamics of the system can be completely determined by the reproduction number, and the chronic infected equilibrium of the system is globally asymptotically stable whenever it exists. This improves the related results presented in [S. A. Gourley,Y. Kuang and J.D.Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, Journal of Biological Dynamics, 2(2008), 140-153].
Citation: Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
##### References:
show all references
##### References:
[1] Ting Guo, Haihong Liu, Chenglin Xu, Fang Yan. Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4223-4242. doi: 10.3934/dcdsb.2018134 [2] Songbai Guo, Wanbiao Ma. Global dynamics of a microorganism flocculation model with time delay. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1883-1891. doi: 10.3934/cpaa.2017091 [3] Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971 [4] Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214 [5] C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837 [6] C. Connell McCluskey. Global stability for an SEIR epidemiological model with varying infectivity and infinite delay. Mathematical Biosciences & Engineering, 2009, 6 (3) : 603-610. doi: 10.3934/mbe.2009.6.603 [7] Yincui Yan, Wendi Wang. Global stability of a five-dimensional model with immune responses and delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 401-416. doi: 10.3934/dcdsb.2012.17.401 [8] Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Taza Gul, Fawad Hussain. A fractional order HBV model with hospitalization. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 957-974. doi: 10.3934/dcdss.2020056 [9] Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 [10] Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1083-1106. doi: 10.3934/mbe.2015.12.1083 [11] Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297 [12] Sze-Bi Hsu, Ming-Chia Li, Weishi Liu, Mikhail Malkin. Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1465-1492. doi: 10.3934/dcds.2003.9.1465 [13] Nabil T. Fadai, Michael J. Ward, Juncheng Wei. A time-delay in the activator kinetics enhances the stability of a spike solution to the gierer-meinhardt model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1431-1458. doi: 10.3934/dcdsb.2018158 [14] Meihong Qiao, Anping Liu, Qing Tang. The dynamics of an HBV epidemic model on complex heterogeneous networks. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1393-1404. doi: 10.3934/dcdsb.2015.20.1393 [15] Khalid Addi, Samir Adly, Hassan Saoud. Finite-time Lyapunov stability analysis of evolution variational inequalities. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1023-1038. doi: 10.3934/dcds.2011.31.1023 [16] Ferenc A. Bartha, Ábel Garab. Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model. Journal of Computational Dynamics, 2014, 1 (2) : 213-232. doi: 10.3934/jcd.2014.1.213 [17] Yoshiaki Muroya, Yoichi Enatsu, Huaixing Li. A note on the global stability of an SEIR epidemic model with constant latency time and infectious period. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 173-183. doi: 10.3934/dcdsb.2013.18.173 [18] Monika Joanna Piotrowska, Urszula Foryś, Marek Bodnar, Jan Poleszczuk. A simple model of carcinogenic mutations with time delay and diffusion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 861-872. doi: 10.3934/mbe.2013.10.861 [19] Hong Yang, Junjie Wei. Dynamics of spatially heterogeneous viral model with time delay. Communications on Pure & Applied Analysis, 2020, 19 (1) : 85-102. doi: 10.3934/cpaa.2020005 [20] Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115
2018 Impact Factor: 1.313 | 2019-12-14 23:35:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4485229551792145, "perplexity": 7727.095695047438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541297626.61/warc/CC-MAIN-20191214230830-20191215014830-00142.warc.gz"} |
https://www.researcher-app.com/paper/1950558 | 3 years ago
# An actuator line - Immersed boundary method for simulation of multiple tidal turbines
Cheng Liu, Changhong Hu
Publication date: Available online 8 January 2019
Source: Renewable Energy
Author(s): Cheng Liu, Changhong Hu
##### Abstract
This work proposes an efficient actuator line – immersed boundary (AL-IB) method to predict the wake of multiple horizontal-axis tidal turbines (HATTs). A sharp IB method with a simple adaptive mesh refinement strategy is used to improve the computational efficiency. The velocity and other scalar fields adjacent to the solid surface are reconstructed by a moving least square (MLS) interpolation. A computationally efficient AL model is applied to represent the rotors by adding source term to the governing equation rather than resolving the fully geometry of the blade. To predict the turbulent wake, the AL-IB method is implemented with an unsteady Reynolds-averaged Navier–Stokes (URANS) solver. Performance of three types of turbulence models, $k−ω−SST$ model, standard and corrected $k−ω$ model are evaluated. An efficient wall function model is proposed for the MLS-IB approach. The accuracy of the present AL-IB method is validated by numerical tests of a single rotor and multiple tandem arranged IFREMER rotors [1,2]. Wake interference of Manchester rotors [3] with side by side arrangement is also investigated numerically. The predicted wake velocity and turbulence intensity (TI) are in reasonably good agreement with the experimental results.
DOI: S0960148119300199
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http://wzod.oenw.pw/examples-of-signed-magnitude-addition-and-subtraction.html | # Examples Of Signed Magnitude Addition And Subtraction
0 y Abstract Understand the rules of vector addition and subtraction using analytical methods. out the addition, we have 0110+ 1101 = 10011 and so the 4-bit sum word is 0011 (due to the 4-bit word length). Multiply each fraction to change the denominators 2. Once a month we will send 10 best examples of similar interactive media content that has been hand-picked by ThingLink team. For example, to add 2. Length, temperature, mass, speed Vector – physical quantity that is specified by both magnitude and direction Ex. 13 From Last Time Exam. Addition and Subtraction of Vectors 5 Fig. Giving a student an equation in both written and illustrated forms can help him or her solve faster and more easily. Careful study of the preceding examples leads to the following conclusion, which is stated as a law for subtraction of signed numbers: In any subtraction problem, mentally change the sign of the subtrahend and proceed as in addition. Plus, you can continue printing out the sheet to keep practicing all summer long. 2 Vector Addition and Subtraction: Graphical Methods. So +− and −+ work like the subtraction of a positive number. Most computers use the signed magnitude representation for the mantissa. In other words, the negatives cancel out to create a positive: 7 -(-5) = 7 + 5 = 12. The nice feature with Two's Complement is that addition and subtraction of Two's complement numbers works without having to separate the sign bits (the sign of the operands and results is. Magnitude or size of u is represented by | u|, or ||u ||. Addition (within 5) Fingers. shows the signed magnitude representation of numbers using 4 bits. Rule 3:- If the MSB of result after discarding ‘end carry’ is 1 then take 2’s complement of the remaining bits. , we subtract the smaller magnitude from the larger and give the difference the sign if the larger magnitude. Find 1's complement = 0000000000011001; Convert to decimal = 25 o Add the negative sign = -25. Subtraction in 2’s complement follows the same rule as it is in the normally binary addition. Note we should get a carry out of the msb when we perform. Simple Addition and Subtraction: Horizontal 1. • The binary, hexadecimal, and octal number systems • Finite representation of unsigned integers • Finite representation of signed integers • Finite representation of rational numbers (if time) Why? • A power programmer must know number systems and data representation to fully understand C’s primitive data types. The following worksheets contain a mix of grade 3 addition, subtraction, multiplication and division word problems. The standard arithmetic operators are addition (+), subtraction (-), multiplication (*), and division (/). Add operands, discard carry-out of the sign bit MSB (if any). Giving a student an equation in both written and illustrated forms can help him or her solve faster and more easily. Negative numbers are represented using sign and magnitude or two's complement. 340625x : Shift the decimal point of the smaller number to the left until the exponents are equal. This example illustrates the addition of vectors using perpendicular components. This lesson will describe methods for subtraction in Java, providing working code examples. Sign and Magnitude Calculations Continuing with our 4-bit number scheme, let's look at some examples of addition and subtraction with sign and magnitude numbers. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e. Addition, subtraction, multiplication, and division mix Fill in the missing negative signs Finding absolute value Comparing integers Solving addition and subtraction (algebra) Solving multiplication and division (algebra) Solving equations (algebra) Addition and subtraction with decimals (only 2 numbers each) Addition and subtraction with decimals. Unfortunately, the magnitude of this number is too large to fit into an eight bit value, so you cannot sign contract it to eight bits. Georgia Standards of Excellence Framework • Let’s Think About Addition and Subtraction! quantity to the relative magnitude of digits in numbers to 1000. $That is, this signed-magnitude representation is correctly converted to$0\$ in two's complement. This ITP can be used to model different counting and calculation strategies. I have a dumb question. Addition and Subtraction. The vector can then be written in terms of the unit vectors i, j, k. Geometric vector subtraction in 2 or 3 dimensions is still done by the tip-to-tail method, but the vector to be subtracted must be turned in the opposite direction (which is the equivalent of changing the sign of each component. When necessary, you must sign-extend integer to convert signed value to a larger size. In this case, however, subtracting 180° gives us an angle measure of 120°, which is within the 0°-360° range. Practice subtraction facts while having fun at Multiplication. Numbers are assumed to be integers and will be entered by a user. The first approach to representing signed binary numbers is a technique called Sign-Magnitude. Remember that the place of the sign bit is fixed from the beginning of the problem. The topic starts with 1+1=2 and goes through adding and subtracting within 1000. , by counting on 2 to add 2). Click Image to Enlarge : This activity explains and practices the inverse relationship between addition and subtraction. For these type of calculations, check out SQL Server T-SQL Aggregate. Notice in the above example, that the most significant bit (msb) in the negative number −5 10 is 1, just as in signed binary. (addition) (subtraction) 5 Unlike Denominators If the denominators are not the same, you must first find a common denominator EXAMPLE: 7-2+4-3 21 21 In this example the common denominator will be 21 1. Start studying 1. In signed notation, this is a result of -3, not +13. For example: =100-50 = B5 - A5. Vector Addition: Head-to-Tail Method The head-to-tail method is a graphical way to add vectors, described in Figure 4 below and in the steps following. Signed 2's complement in arithmic. This calculator is designed to multiply and divide values of any Binary numbers. However, when the 2s complement of a number is added to any other binary number, it will be equivalent to its subtraction from that number. Suppose we want to add two numbers 69 and 12 together. C program to perform basic arithmetic operations which are addition, subtraction, multiplication, and division of two numbers. Addition is simpler than subtraction. It deals with the theory and practical knowledge of Digital Systems and how they are implemented in various digital instruments. 8m/s [S] is exactly the same as -4. An example of what may happen can be seen in which is the above lightfield directly subtracted from the text images. Apply graphical methods of vector addition and subtraction to determine the displacement of moving. sign does not exist. Learn what vectors are and how they can be used to model real-world situations. Thus, the first number becomes. Demonstrates binary subtraction with standard binary numbers, and then indicates how subtraction using sign magnitude representation is complicated, and unde Skip navigation Sign in. The first approach to representing signed binary numbers is a technique called Sign-Magnitude. The children need to be able to recall and use addition and subtraction facts for all numbers to 20. The Graphical Method of Vector Addition and Subtraction. Numeric expressions in SAS share some features with mathematical expressions: When an expression contains more than one operator, the operations have the same order of precedence as in a mathematical expression: exponentiation is done first, then multiplication and division, and finally addition and subtraction. We are about to learn how to add signed numbers. Starting with the right most bit. All bits to right are the number magnitude Left bit is the sign bit. Integer Operations: Addition and Subtraction Solve. For example, the magnitude of a position vector is |~r| = q (r2 x +r. "addition and subtraction" Resultant velocity refers to the sum of all vectors in an equation. Introduction to Vectors - Zero Vectors, Unit Vectors, Coinitial , Collinear, Equal Vectors, Addition and Subtraction of Vectors, Scalar and Vector Multiplication Our Linear Algebra Tutorials: at a glance. We won’t have much to. Subtraction sentences can often be changed into equivalent addition sentences (or vice versa) using a process known as "reversal". Subtraction can be thought of as the opposite of addition. Addition and subtraction require different logic circuits. Examples in lecture * Subtraction Notice that subtraction can be done using addition A – B = A + (-B) We know how to negate a number The hardware used for addition can be used for subtraction with a negating unit at one input Add 1 Invert (“flip”) the bits * Signed versus Unsigned Operations “unsigned” operations view the operands as. The page contains feedback to exercises as well as worked examples of how to subtract. The magnitude of A´ is found by using the same approach as a 2-D vector: A´ = (AX2 + AY2)1/2. Use of a ‘sign bit’ (this is just like having a sign for the number) -5 = 10000101 Note that addition and subtraction are somewhat complex (and multiplication and division). The sign bit is either positive 0 or negative 1 ; 3 Eight Conditions for Signed-Magnitude Addition/Subtraction 4 Examples. 2009 dce Addition in the 2's Complement System • Perform normal binary addition of magnitudes. But division and any greater-than/less-than comparisons have to have separate versions. In general we know that "A - B" is the same as "A + (-B)". Negative Numbers and Subtraction. Take the last example. The procedure for adding or subtracting two signed binary numbers with paper and pencils simple and straight- forward. Find PowerPoint Presentations and Slides using the power of XPowerPoint. VHDL Example Code of Signed vs Unsigned. Addition and subtraction result in another vector for which magnitude and direction are both changed. This lesson will describe methods for subtraction in Java, providing working code examples. Addition and subtraction requires finding the common denominator (the denominator of the result) and then adding or subtracting the adjusted numerator. Addition and subtraction procedures. Wastes a combination to represent -0 0000 = 1000 = 0 10 2. 1 o north of east. Flowchart of Addition and Subtraction with Signed-Magnitude Data 5. Times New Roman MS Pゴシック Arial Calibri Symbol Default Design Microsoft Photo Editor 3. Notice that our addition caused an overflow bit. Vector Addition: Head-to-Tail Method The head-to-tail method is a graphical way to add vectors, described in Figure below and in the steps following. orF example, if a eld hockey player is moving at 5 m/s straight toward the goal and drives the ball in the. It should always begin with an equal sign (=). Note we should get a carry out of the msb when we perform. Example: Adding Four Forces in 2D, or a Plane; All of the essential understanding of vector addition and subtraction can be developed using 2D vectors in a plane. If you need to add or subtract vectors with known components, express the vector in variables. Students combine addition and subtraction of integers with absolute value. Perform the calcuation using 6-digit 10’s complement addtion. Both addition and subtraction would have resulted in coterminal angles. Addition and subtraction with signed magnitude data mano #110661638974 – Draw a Flow Chart to Subtract Two Numbers, with 36 Related files. It deals with the theory and practical knowledge of Digital Systems and how they are implemented in various digital instruments. In this topic, we will add and subtract whole numbers. Phasors have two components, the magnitude (M) and the phase angle (φ). The game is aligned to the following Common Core math standard:. The sign-magnitude binary format is the simplest conceptual format. [email protected] Sign = Subtrahend Sign 2. Start studying 1. SEE MORE : 26. Addition and subtraction • For sign-magnitude numbers, addition is simple, but if the numbers have different signs the task becomes more complicated - Logic circuits that compare and subtract numbers are also needed - It is possible to perform subtraction without this circuitry - For this reason, sign-magnitude is not used in computers. These two techniques are called signed magnitude representation and two’s complement. Concept Academy 29,845 views. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. It should always begin with an equal sign (=). Notice that our addition caused an overflow bit. The four basic operations of arithmetic are covered. An important example of this is the Binary Addition Algorithm, where two bit patterns representing two integers are manipulated to create a third pattern which represents the sum of the integers. Again, you can use actual values, values stored in variables, or a combination of the two. Wolfram|Alpha handles topics from addition and subtraction to multiplication and division to more complicated. For example, to solve a basic three-digit addition problem, use numbers that involve basic addition facts, such as 200 + 400. Addition is perhaps the easiest of the rest of the mathematical operation because all you have to do is to combine the sum of two or more numbers in order to create a new but bigger digits. A versatile floating point adder which performs high speed floating point addition or subtraction on operands supplied in a signed magnitude format includes separate exponent and mantissa data paths for processing the exponent fields and mantissa fields of the floating point binary numbers to be added or subtracted. Rules of signs at multiplication and division. Addition and Subtraction Arithmetic. Absolute value (modulus) of a number. 3 units, and the direction $$θ$$ is $$29. VHDL Example Code of Signed vs Unsigned. 5km [W] is exactly the same as -3. Disadvantages of Signed Magnitude 1. For example, the algebraic expression has three terms: and 6. Addition, subtraction, multiplication, equality, and such all work fine without knowing if the numbers are signed or not. Using analytical methods, we see that the magnitude of R is 81. An example of what may happen can be seen in which is the above lightfield directly subtracted from the text images. For example, to add 2. It should always begin with an equal sign (=). For example, 1 0001001 could. The student will need to fill in the missing numeral. +8 + - 7 -(-9) - +5 = 8 - 7 + 9 - 5 Now the class is ready to do problems in class and for homework in order to master the addition and subtraction of signed numbers. 0191 There's a key phrase here, it says, without using the other formulas. Signed Magnitude Addition – Subtraction Algorithm 1 START Subtract? Same Sign? Toggle Subtrahend Sign Bit Add Magnitudes Sum Keeps Sign MSB Carry? Minuend > ? 1. R LecturerECE Dept, SJBIT Bengaluru-60. The major problem with this representation of numbers is that it requires twice as much space, and comparison is costly because it requires two divisions. Perhaps for this reason, there is increased interest in student thinking about integer addition and subtraction in our field (e. We note that z lies in the second quadrant, as shown below: Using the Pythagoras Theorem, the distance of z from the origin, or the magnitude of z, is. The resultant is the vector drawn from the tail of the first to the head of the second. Representation of negative numbers Signed-Magnitude representation: • This is the simplest method. As we said earlier, we can do all mathematical operations like addition, subtraction and multiplication, division etc. This subtraction calculator allow users to generate step by step calculation for any input combinations. To represent a number in sign-magnitude, we simply use the leftmost bit to represent the sign, where 0 means positive, and the remaining bits to represent the magnitude (absolute value). For example, 2 subtract 2 (2-2) is 0. Subtraction of Signed Numbers 2 - Cool Math has free online cool math lessons, cool math games and fun math activities. , Bishop et al. Consider the problem of subtracting 1 10 from 7 10. For example, How do we subtract? -34 - (-45) = -34 + 45 = 11. Start studying 1. 2 • HC12 Addressing Modes • Huang, Sections 1. 3 Subtracting Integers In Section 1. Using analytical methods, we see that the magnitude of R is 81. You can count back to find the difference or use fact families and related facts to subtract. Learn vocabulary, terms, and more with flashcards, games, and other study tools. For many parts of mathmatics substraction doesn't even exist e. topology and category theory. 6 Vector Addition and Subtraction Example A displacement vector has a magnitude of 175 m and points at an angle of 50. Sign = Subtrahend Sign 2. Figure: Hardware Architecture for Addition and Subtraction of Signed-Magnitude Numbers Figure: Flowchart 2. 340625x : Shift the decimal point of the smaller number to the left until the exponents are equal. Example: Binary Value 0000 +0 0001 +1 0010 +2 0011 +3 0100 +4 0101 +5 0110 +6 0111 +7 1000 -0 1001 -1 1010 -2. This three parts are also present in other basic operation like addition, multiplication, and division, although the name of some of these parts is varied. There are also a couple other ways to visualize subtraction when negative numbers are involved. y 2 +r2z) (2) and represents the distance from the origin to a point on the coordinate system. A sign-magnitude number Z can be represented as (As, A) where As is the sign of Z and A is the magnitude of Z. 7 ] Alternative representations Computers don’t use a “sign and magnitude” representation Drawbacks of the Sign-Magnitude representation: two 0s: one positive one negative addition and subtraction involving negative numbers are complicated Alternatives? 1's complement representation 2's complement representation: today's standard These. You can also use a fact triangle, a number line, or draw pictures to help you subtract. This is the second of a four part series on "pencil and paper" binary arithmetic, which I'm writing as a supplement to my binary calculator. Apply graphical methods of vector addition and subtraction to determine the displacement of moving. Find PowerPoint Presentations and Slides using the power of XPowerPoint. 2 m and its direction is 36. If the bit is set to 0 the entire number is viewed as positive. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e. Vectors addition and subtraction can be done by different methods. Examples in lecture * Subtraction Notice that subtraction can be done using addition A – B = A + (-B) We know how to negate a number The hardware used for addition can be used for subtraction with a negating unit at one input Add 1 Invert (“flip”) the bits * Signed versus Unsigned Operations “unsigned” operations view the operands as. Perform various operations with vectors like adding, subtracting, scaling, and conversion between rectangular to polar coordinates. The order of subtraction does not affect the results. signed numbers. Consider the complex number \(z = - 2 + 2\sqrt 3 i$$, and determine its magnitude and argument. As stated here, the order of the steps is not determined. • Useful for floating point representation. representation of signed numbers will involve dividing these 2m patterns into positive and negative portions. Below, five arithmetic operators are described and then all put into a sketch to demonstrate how they work on the Arduino. Recall that signed 4 bit numbers (2's complement) can represent numbers between -8 and 7. As a result,. In this chapter the focus will start to be shifted toward more complicated problems that might. To represent a number in sign-magnitude, we simply use the leftmost bit to represent the sign, where 0 means positive, and the remaining bits to represent the magnitude (absolute value). However, if we can leverage the already familiar (and easier) technique of binary addition to. Addition or subtraction from left to right Let's do the brackets first (6 в -3 - 2) , inside this bracket you can see the multiplication and the subtraction signs. Since the signi cand in oating-point numbers is coded as SM according to the IEEE 754-2008 standard, the study of circuits for e cient arithmetic operations in SM is relevant. Discussion. Mastering this skill is helpful in verifying that an exact answer is of the correct order of magnitude and also in determining whether an answer is reasonable or not. LEVEL SEVEN: magic squares - add or subtract across and down to see the relationships in addition and subtraction. Displacement, velocity, acceleration, and force, for example, are all vectors. It would be bad form to write a + −b. Addition of Signed Binary Numbers. Also, it is defined as the opposite of subtraction, so this. Help your child avoid the summer slide with a helpful worksheet that fosters addition and subtraction practice. Simple Addition and Subtraction: Horizontal 2. Different (unlike) signs SUBTRACT and KEEP sign of the larger absolute value. The nice feature with Two's Complement is that addition and subtraction of Two's complement numbers works without having to separate the sign bits (the sign of the operands and results is. Subtraction of vectors. representation of signed numbers will involve dividing these 2m patterns into positive and negative portions. Learn about Vector Addition and Subtraction with the help of examples here. 'Signci-1' and 'signzi' are the sign magnitude of the intermediate carry and intermediate sum, respectively. The remaining 7 bits of the negative number however are not the same as in signed binary notation. We have lots of subtraction Year 1 and Year 2 resources available for download to help you introduce your children to addition and subtraction and to further their knowledge. Steps 1 and 2 can also be done in reverse. In this part of the course, we look at how to do addition, subtraction, multiplication, division, and find a remainder. A vector is a quantity that has magnitude and direction. Vector Addition and Subtraction: Graphical Methods 3. We won’t have much to. Numbers using 4-bit signed magnitude representation Example 8. In sign-magnitude representation: The most-significant bit (msb) is the sign bit, with value of 0 representing positive integer and 1 representing negative integer. Likewise, you can use the jo or jno instructions after these sequences to test for signed arithmetic overflow. For example, you previously could not add a row and a column vector, but those operands are now valid for addition. As a result,. So ++ and −− work like the addition of a positive number. Furthermore, to the best of the. To do subtraction between two or more numbers in Excel, you can create a formula. Model the subtraction on your chart paper. Helicopter Rescue Great number square games which can help you with your addition and subtraction. 1's Complement Representation In the 1's complement representation, a nonnegative number is represented in the same manner as an unsigned number. Show how B is added to A. Division – divides one value by another; example: 40 / 2. Integer Numbers and Mathematical Operations. The following worksheets contain a mix of grade 3 addition, subtraction, multiplication and division word problems. Binary Addition and Subtraction The addition and subtraction of the binary number system are similar to that of the decimal number system. § Binary arithmetic is straightforward § Subtraction: Just subtract and borrow as necessary § Consider subtracting 8-bit numbers: 111111 01101011 107d-01101101 109d----- ----111111110 -2d 111 01101011 107d-01001101 77d----- ----00011110 30d legal number: betw. Overflow occurs when the result has the wrong sign bit for the operation that was performed. Sound levels are generally expressed in decibels, which are logarithmic and so cannot be manipulated without being converted back to a linear scale. out the addition, we have 0110+ 1101 = 10011 and so the 4-bit sum word is 0011 (due to the 4-bit word length). Students write matching addition and subtraction "sentences" or equations, first using a visual model. Force, velocity, displacement, acceleration We represent vectors. Click Image to Enlarge : This activity explains and practices the inverse relationship between addition and subtraction. Integers – Addition, Subtraction, Multiplication and Division! Model Notes The Number Line The number line is a line labelled with the integers in increasing order from left to right, that extends in both directions: For any two different places on the number line, the integer on the right is greater than the integer on the left. Tech ,VLSI DESIGN and EMBEDDED SYSTEMS Chethana. Getting rid of a negative is a positive. Download All. This example illustrates the addition of vectors using perpendicular components. The adders we designed can add only non-negative numbers If we can represent negative numbers, then subtraction is “just” the ability to add two numbers (one of which may be negative). Addition and subtraction requires finding the common denominator (the denominator of the result) and then adding or subtracting the adjusted numerator. A vector is a quantity that has magnitude and direction. Signed Magnitude. Example A displacement vector has a magnitude of 175 m and points at an angle of 50. The vectors being added together are known as the components of the resultant vector. In this topic, we will add and subtract whole numbers. The result is automatically in signed-2's complement form. Once you've substituted the value for the letter, do the operations to find the value of the expression. S = A S XOR B S = 1; R M = A M + (B M +1). Then he stopped cheating and fighting. Divide Fixed-Point Signed-Mag zSeries of successive compare, shift, and subtract operations 23 Example: 448 ⁄ 17 = 26 r 6 Initially, AQ dividend B divisor At end of operation, Q quotient A remainder DVF divide overflow 24 Algorithm. 1 This question is for you to practice addition and subtraction of complex numbers graphically. There are problems with sign-magnitude representation of integers. The analytical method of vector addition and subtraction involves using the Pythagorean theorem and trigonometric identities to determine the magnitude and direction of a resultant vector. The procedure for adding or subtracting two signed binary numbers with paper and pencils simple and straight- forward. This section of the site links to many basic, single-digit addition worksheets. Rule 1:- If one of the numbers to be added is negative then take signed 2’s complement of the number. Sign/Magnitude Notation Sign/magnitude notation is the simplest and one of the most obvious methods of. The result is automatically in signed-2's complement form. In the examples in this section, I do addition and subtraction in two's complement, but you'll notice that every time I do actual operations with binary numbers I am always adding. A few examples of math rules for subtraction: When you subtract a positive number from a smaller positive number, the result will be a negative number: 8 - 11 = -3. Students will be encouraged to solve these using fact families or inverse operations. 20 has magnitude 1 and 80 has magnitude 2. Performance Assessment:. Addition When th e addition of two values results in a carry. The major problem with this representation of numbers is that it requires twice as much space, and comparison is costly because it requires two divisions. Hardware implementation: Sign flip flop A binary example: Partial product. Subtraction is just the inverse, multiplication is just repeated addition, and division the inverse of that. Multiple Step Word Problems. "0" indicates that the number is positive, "1" indicates negative. 3 units, and the direction θ is 29. The only difference is that the decimal number system consists the digit from 0-9 and their base is 10 whereas the binary number system consists only two digits (0 and 1) which make their operation easier. We have quizzes that cover topics such as: Addition, Subtraction, Geometry, Fractions, Probability, Venn Diagrams, Time and more. In this chapter the focus will start to be shifted toward more complicated problems that might. math worksheets > > mixed addition and subtraction SuperKids Math Worksheet Creator * Now with answer sheets! Mixed Addition and Subtraction Create your own mixed addition and subtraction worksheets using positive and negative whole numbers. Note that in Fig. The Complement (1's, 2's) of a number is its additive identity. Operation Add Magnitudes Subtract Magnitudes. Mastering this skill is important to be able to verify if the obtained result is reasonable and the right order of magnitude. Amby's Math Resources - Integers: Operations with Signed Numbers clearly explains how to add, subtract, multiply and divide positive and negative numbers. It deals with the theory and practical knowledge of Digital Systems and how they are implemented in various digital instruments. In this example, the magnitude $$D$$ of the vector is 10. We call them arithmetical operations. 6 Add three or more numbers up to three digits each. Notice that our addition caused an overflow bit. Use column addition to add two amounts of money. Rules for Adding Integers Rule 1: If the signs are the same then add the numbers. Download All. SUBTRACTION OF A VECTOR Subtraction is just a process of negative addition Thus, if you want find the resultant vector of -b , just add negative of vector b in vector a that is a + (-b ) This is adding vector with inverted vector b with a is o whue Magnitude of a'- i Its. (See "Number line in addition and subtraction" below. In this example, the magnitude D of the vector is 10. For example: =100-50 = B5 - A5. In algebra, we can combine terms that are similar eg. Subtraction of Polynomials To subtract one polynomial from another, change the subtraction sign to an addition sign and change the signs of all the terms in the polynomial being subtracted (don't forget to change the sign of the first term). If your final result is an integer, say 2 , you need to change it to the format of fraction that has denominator 1. See more ideas about Math classroom, Math addition and Teaching math. Addition and subtraction with signed magnitude data mano #110661638974 – Draw a Flow Chart to Subtract Two Numbers, with 36 Related files. Find out where the numerical digits we use today come from, who invented the equals sign and other interesting math timeline facts and trivia. Then add a 1 to the front of it if the number is negative and a 0 if it is positive. Negative numbers are represented using sign and magnitude or two's complement. Go back to your code. We need only consider addition, because subtraction is implemented by adding the negative of the subtracted number. How to use subtraction in a sentence. This section of the site links to many basic, single-digit addition worksheets. The subtraction expression yields a signed integral result of type ptrdiff_t (defined in the standard include file ). Now that we can represent signed numbers with 1's and 2's complement, let's see how they will help us convert our subtraction problems to addition. The representations of the multiplicand and product are not specified; typically, these are both also in two's complement representation, like the multiplier, but any number system that supports addition and subtraction will work as well. the operation sign + from the algebraic sign −. An overflow is a situation in which the result of an operation. Addition and subtraction requires finding the common denominator (the denominator of the result) and then adding or subtracting the adjusted numerator. The resultant vector VT is shown in the below given figure. In one-dimensional, or straight-line, motion, the direction of a vector can be given simply by a plus or minus sign. Vector Addition: Head-to-Tail Method The head-to-tail method is a graphical way to add vectors, described in Figure below and in the steps following. Sign and magnitude bits should be differently treated in arithmetic operations. Now it appears to me that a little child, with the simple rules of addition and subtraction, could have refuted this man. The magnitude of A´ is found by using the same approach as a 2-D vector: A´ = (AX2 + AY2)1/2. out the addition, we have 0110+ 1101 = 10011 and so the 4-bit sum word is 0011 (due to the 4-bit word length). For any addition or subtraction problem only one sign is necessary between each number in order to know what to do. ) Scalar Multiplication. Integers – Addition, Subtraction, Multiplication and Division! Model Notes The Number Line The number line is a line labelled with the integers in increasing order from left to right, that extends in both directions: For any two different places on the number line, the integer on the right is greater than the integer on the left. Signed 2's complement in arithmic. For this case it is necessary to take the 2's complement of the value in A. Mixing math word problems is the ultimate test of understanding mathematical concepts, as it forces students to analyze the situation rather than mechanically apply a solution. The range of numbers used for each worksheet may be individually varied to generate different sets of mixed operator problems. For example, the illustration paired with the written equation “8 + 5 = ___” can feature 8 dogs on one side of the "plus" sign, and 5 dogs on the other. MOTION IN A PLANE 67 as A = B. In algebra, we can combine terms that are similar eg. The sign bit is either positive 0 or negative 1 ; 3 Eight Conditions for Signed-Magnitude Addition/Subtraction 4 Examples. Signed Binary Addition/Subtraction. Recall that a vector is a quantity that has magnitude and direction. | 2020-04-04 11:32:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48029381036758423, "perplexity": 683.0174260868112}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370521876.48/warc/CC-MAIN-20200404103932-20200404133932-00206.warc.gz"} |
https://math.stackexchange.com/questions/2174946/is-this-particular-group-cyclic?noredirect=1 | # Is this particular group cyclic? [duplicate]
If we consider $(\mathbb{Z}/120\mathbb{Z})^{\times}$ ?
Using the CRT we have : $(\mathbb{Z}/120\mathbb{Z})^{\times} \simeq (\mathbb{Z}/2^3\mathbb{Z})^{\times} \ \times (\mathbb{Z}/3\mathbb{Z})^{\times} \ \times (\mathbb{Z}/5\mathbb{Z})^{\times}$
By other properties $(\mathbb{Z}/120\mathbb{Z})^{\times}\simeq \mathbb{Z}/2\mathbb{Z}\ \times\mathbb{Z}/2\mathbb{Z}\ \times \mathbb{Z}/2\mathbb{Z}\ \times \mathbb{Z}/4\mathbb{Z}$
But we know that $(\mathbb{Z}/2^3\mathbb{Z})^{\times}$ is not cyclic and moreover $\gcd(2,2,2,4)\neq 1$ so the product cannot be a cyclic group. Does that mean that $(\mathbb{Z}/120\mathbb{Z})^{\times}$ is not cyclic too ?
• For $n=120$ it is not cyclic, as you have shown. – Dietrich Burde Mar 6 '17 at 20:30
• @Student thank you in fact it's a trap because if the group is not cyclic you cannot apply a powerful result on the number of solutions of $x^k \equiv 1 \pmod{ n}$. You will have to use CRT and it will be longer. – Maman Mar 6 '17 at 20:36
• @Student There is a powerful lemma which says if $(Z/nZ)^{\times}$ is cyclic then the number of solutions of $x^k \equiv 1 \pmod n$ is $\gcd(k,\phi(n))$ – Maman Mar 6 '17 at 20:45 | 2020-02-17 16:16:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6246797442436218, "perplexity": 185.92747308369007}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875142603.80/warc/CC-MAIN-20200217145609-20200217175609-00214.warc.gz"} |
https://engineering.stackexchange.com/questions/6921/how-to-estimate-a-mass-spring-damper-parameters-in-matlab-using-rls-and-ols | # How to estimate a mass-spring-damper parameters in MATLAB using RLS and OLS?
Assume that we have the differential equation of a mass-spring-damper model as follows: $$m\frac{d^2y}{dt^2}+c\frac{dy}{dt}+ky(t)=F(t)$$
How it could be implemented in MATLAB to do the following steps:
First, convert the differential equation to a difference equation.
Second, finding the discrete-time transfer function of it.
Third, finding the regression equation of this system.
Finally deriving Recursive Least Squares (RLS) and Ordinary Least Squares (OLS) to estimate the free parameters of the system ($m$,$c$, and $k$).
More information on this system is available at Mass-Spring-Damper System Excited by Force F(t)
Any help is greatly appreciated
• This sounds like homework to me. What have you tried? What are you having trouble with? – Chuck Jan 17 '16 at 1:29
• Do you need to follow this approach or can you do it differently? Is it a theoretical question or do you want to identify the parameters with a physical set-up? – Karlo Feb 16 '16 at 9:36
Matlab with some tips
A good method is using the Billinear transform (Tustin's method).
s = tf('s')
Hc = 1/(m*s^2+c*s+k) % Tf continuous-time
Observation.: First declare your variables: m, c and k.
Matlab provides some functions that help with the transformation. The function c2d().
Ts = 0.1;
Hd = c2d(Hc,Ts,'tustin')
step(Hc,'--',Hd,'-')
where Ts is the sample time.
• To find the difference equation, apply the inverse Z transform after using tusting method in your equation. Remember that you need to use Laplace transform first!
$\hspace{2.5em}$ $Y(s)\big(ms^{2}+cs+k\big) = \mathcal{L}\{F(t)\}$
• You can find good info about OLS here. | 2021-04-17 07:46:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 0.627531886100769, "perplexity": 1095.1378901337584}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038118762.49/warc/CC-MAIN-20210417071833-20210417101833-00118.warc.gz"} |
https://support.bioconductor.org/p/13584/ | Question: Unable to use biomaRt because Rcurl doesn't pick up my proxy server address
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12.7 years ago by
michael watson IAH-C3.4k wrote:
Hi I'm using R 2.3.1 on windows, with biomaRt 1.7.3 and Rcurl 0.6-2. I am also using the "--internet2" option in R, which as I understand it, takes it's internet connection information from Windows, and I have Windows set up to access the internet through a proxy. However, curlPerform() will not work unless I specifically set the proxy. I can't set the proxy for the calls to Rcurl functions, as they are all wrapped up in biomaRt functions. This is probably best demonstrated with some code: > # I have an internet connection! > source("http://www.bioconductor.org/biocLite.R") > # biomaRt cannot connect > library(biomaRt) Loading required package: XML Loading required package: RCurl Warning message: use of NULL environment is deprecated > mart <- useMart("ensembl", dataset = "hsapiens_gene_ensembl") Error in curlPerform(curl = curl, .opts = opts) : couldn't connect to host > # tracked down the error to here > getURL("http://www.biomart.org/biomart/martservice?type=registry") Error in curlPerform(curl = curl, .opts = opts) : couldn't connect to host > # However, download.file works with the same URL! > download.file("http://www.biomart.org/biomart/martservice?type=registr y" , "test.xml") trying URL 'http://www.biomart.org/biomart/martservice?type=registry' Content type 'text/plain' length 200 bytes opened URL downloaded 1908 bytes Warning message: downloaded length 1908 != reported length 200 > # If I specifically set the proxy, curlPerform works! > myOpts = curlOptions(proxy="http://myproxyserver.com") > curlPerform(url="http://www.omegahat.org/RCurl", writefunction = h$update, .opts=myOpts) OK 0 > # and if I take it away again, it doesn't > curlPerform(url="http://www.omegahat.org/RCurl", writefunction = h$update) Error in curlPerform(url = "http://www.omegahat.org/RCurl", writefunction = h$update) : couldn't connect to host So, I'm not sure what to do here - either the biomaRt code needs changing so that I can pass in my proxy server address for the Rcurl functions, or the Rcurl code needs changing so that it picks up my internet connection settings the same way that download.file() and source() do.... Thanks Mick biomart • 2.7k views ADD COMMENTlink modified 12.7 years ago by Duncan Temple Lang110 • written 12.7 years ago by michael watson IAH-C3.4k Answer: Unable to use biomaRt because Rcurl doesn't pick up my proxy server address 0 12.7 years ago by Duncan Temple Lang110 wrote: Hi Michael In the specific case of setting the proxy, there is an easy way to achieve this. Use the http_proxy environment variable Sys.putenv("http_proxy" = "http://my.proxy.org:9999") giving the value as your proxy server. For setting arbitrary options, we need to have a mechanism that looks for a default set of options and use that if it exists. I'll add that to the next release. Thanks D. michael watson (IAH-C) wrote: > Hi > > I'm using R 2.3.1 on windows, with biomaRt 1.7.3 and Rcurl 0.6-2. I am > also using the "--internet2" option in R, which as I understand it, > takes it's internet connection information from Windows, and I have > Windows set up to access the internet through a proxy. However, > curlPerform() will not work unless I specifically set the proxy. I > can't set the proxy for the calls to Rcurl functions, as they are all > wrapped up in biomaRt functions. > > This is probably best demonstrated with some code: > >> # I have an internet connection! >> source("http://www.bioconductor.org/biocLite.R") > >> # biomaRt cannot connect >> library(biomaRt) > Loading required package: XML > Loading required package: RCurl > Warning message: > use of NULL environment is deprecated >> mart <- useMart("ensembl", dataset = "hsapiens_gene_ensembl") > Error in curlPerform(curl = curl, .opts = opts) : > couldn't connect to host > >> # tracked down the error to here >> getURL("http://www.biomart.org/biomart/martservice?type=registry") > Error in curlPerform(curl = curl, .opts = opts) : > couldn't connect to host > >> # However, download.file works with the same URL! >> > download.file("http://www.biomart.org/biomart/martservice?type=regis try" > , "test.xml") > trying URL 'http://www.biomart.org/biomart/martservice?type=registry' > Content type 'text/plain' length 200 bytes > opened URL > downloaded 1908 bytes > > Warning message: > downloaded length 1908 != reported length 200 > >> # If I specifically set the proxy, curlPerform works! >> myOpts = curlOptions(proxy="http://myproxyserver.com") >> curlPerform(url="http://www.omegahat.org/RCurl", writefunction = > h$update, .opts=myOpts) > OK > 0 > >> # and if I take it away again, it doesn't >> curlPerform(url="http://www.omegahat.org/RCurl", writefunction = > h$update) > Error in curlPerform(url = "http://www.omegahat.org/RCurl", > writefunction = h$update) : > couldn't connect to host > > > So, I'm not sure what to do here - either the biomaRt code needs > changing so that I can pass in my proxy server address for the Rcurl > functions, or the Rcurl code needs changing so that it picks up my > internet connection settings the same way that download.file() and > source() do.... > > Thanks > Mick | 2019-03-21 14:18:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3792280852794647, "perplexity": 9340.268741900496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202525.25/warc/CC-MAIN-20190321132523-20190321154523-00090.warc.gz"} |
http://clay6.com/qa/46427/if-the-solubility-product-of-cus-is-6-times-10-calculate-the-maximum-molari | Browse Questions
# If the solubility product of CuS is $6 \times 10^{−16}$, calculate the maximum molarity of CuS in aqueous solution.
$2.45\times 10^{-8}molL^{-1}$ | 2017-05-28 04:44:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9711655974388123, "perplexity": 10253.077005277011}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463609598.5/warc/CC-MAIN-20170528043426-20170528063426-00317.warc.gz"} |
https://dev.goldbook.iupac.org/terms/view/A00485 | ## asymmetric transformation
Also contains definition of: deracemization
https://doi.org/10.1351/goldbook.A00485
The conversion of a @R05025@ into a pure @E02069@ or into a mixture in which one @E02069@ is present in excess, or of a @D01679-1@ mixture into a single diastereoisomer or into a mixture in which one diastereoisomer predominates. This is sometimes called deracemization. If the two enantiomers of a @C01057@ substrate A are freely interconvertible and if an equal amount or excess of a non-racemizing second enantiomerically pure chemical species, say (R)-B, is added to a solution of @R05026@ A, then the resulting equilibrium mixture of adducts A·B will, in general, contain unequal amounts of the @D01679-2@ (R)-A·(R)-B and (S)-A·(R)-B. The result of this @E02175@ is called @A00478@ @T06446@ of the first kind. If, in such a system, the two diastereoisomeric adducts differ considerably in @S05740@ so that only one of them, say (R)-A·(R)-B, crystallizes from the solution, then the @E02175@ of @D01679-2@ in solution and concurrent @C01434@ will continue so that all (or most) of the substrate A can be isolated as the crystalline diastereoisomer (R)-A·(R)-B. Such a '@C01434@-induced @A00478@ @T06446@' is called an @A00478@ @T06446@ of the second kind. | 2021-10-24 02:40:16 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8139956593513489, "perplexity": 4539.649829996517}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585837.82/warc/CC-MAIN-20211024015104-20211024045104-00483.warc.gz"} |
http://mathoverflow.net/questions/21742/knot-complement-diffeomorphism-groups-and-embedding-spaces | # Knot complement diffeomorphism groups and embedding spaces
I'm interested in the following collection of questions: Let $S^n_k = \sqcup_k S^n$ be a disjoint union of $k$ distinct $n$-dimensional spheres. Write $Emb(S_k^n, S^{n+2})$ for the space of embeddings of these spheres into $S^{n+2}$. Pick your favorite embedding $e: S_k^n \to S^{n+2}$, and let $X_e = S^{n+2} \setminus im(e)$ be the complement of the image of the embedding.
1. What is $\pi_1(Emb(S_k^n, S^{n+2}), e)$? Since this is probably unknown, what is known?
2. How is this related to the mapping class group $\pi_0(Diff(X_e))$ of $X_e$?
I ask #2 because in dimension $n=0$, they are the same: the space of embeddings is the configuration space of points in the sphere. Its fundamental group is the (spherical) braid group, which is the same as the mapping class group of the punctured sphere. My guess is that life is not so simple in higher dimensions. Lastly, does any of this simplify out when you get into the range of dimensions where surgery theory starts working well?
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For n=1, k=1 these fundamental groups based at each component were completely determined by Ryan Budney (who is almost sure to answer this question). – Dev Sinha Apr 18 '10 at 17:18
There is a locally-trivial fibre bundle
$$Diff(S^{n+2}, L) \to Diff(S^{n+2}) \to Emb_L(\sqcup_k S^n, S^{n+2})$$
here $Emb_L(\sqcup_k S^n, S^{n+2})$ is the component of the link $L$ you're interested in the full embedding space $Emb(\sqcup_k S^n,S^{n+2})$ and to simplify technicalities, assume $Diff(S^{n+2})$ is the group of orientation-preserving diffeomorphisms of $S^{n+2}$. The bundle is given by restricting a diffeomorphism of $S^{n+2}$ to $L$. $Diff(S^{n+2}, L)$ is the subgroup of $Diff(S^{n+2})$ which preserves the link $L$.
First observation is that the map $Diff(S^{n+2}) \to Emb_L(\sqcup_k S^n,S^{n+2})$ is null-homotopic. It's a simple argument -- isotope your link $L$ to sit in a hemi-sphere of $S^{n+2}$. Then apply a linearization process to linearize (simultaneously) all the diffeomorphisms of $S^{n+2}$ on that hemi-sphere. What I'm claiming is that $Diff(S^{n+2})$ has as a deformation-retract the subgroup that is linear on a fixed hemi-sphere -- so it gives a product decomposition $Diff(S^{n+2}) \simeq SO_{n+3} \times Diff(D^{n+2})$ (first observed by Morlet, or Cerf, I would guess) among other things.
So now you have a fibration:
$$\Omega Emb_L(\sqcup_k S^n, S^{n+2}) \to Diff(S^{n+2}, L) \to Diff(S^{n+2})$$
where the induced maps on homotopy groups are a short-exact sequence. In particular the fundamental group of your link space injects into $\pi_0 Diff(S^{n+2}, L)$, and its cokernel is precisely $\pi_0 Diff(S^{n+2})$. This group is frequently non-trivial as it is the group of exotic $n+3$-sphere provided $n \geq 3$.
$Diff(S^{n+2}, L)$ is somewhat closely related to $Diff(X_e)$, especially in high dimensions -- the spherical normal bundle to $L$ is particularly symmetric in low dimensions which causes trouble there. In general the sphereical normal bundle is equivalent to a disjoint union of $S^n \times S^1$, so to make the comparison between $Diff(S^{n+2}, L)$ and $Diff(X_e)$ you'd need to ask what kind of automorphisms $Diff(X_e)$ allows on the spherical normal bundle to $L$. There's probably a decent answer to that which doesn't take too much work but the above is a start.
edit: by Cerf's pseudoisotopy theorem, the kernel of the map $\pi_0 Diff(S^{n+2}, L) \to \pi_0 Diff(X_e)$ contains the exotic sphere "part" of $\pi_0 Diff(S^{n+2}, L)$.
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This is perfect. Exactly what I was looking for. Thanks, Ryan. – Craig Westerland Apr 18 '10 at 22:10
In dimension $k=1$, i.e. for embeddings $\sqcup_k S^1\hookrightarrow S^3$, and using a totally unlinked embedding as your basepoint $e$, this is what's called the ring group or the loop group. It is closely related to the braid group and has been studied a ton, but two places with references you could follow are Brendle-Hatcher "Configuration spaces of rings and wickets" and Brownstein-Lee "Cohomology of the group of motions of n strings in 3-space".
The fundamental group $\pi_1(\text{Emb}(\sqcup_k S^1,S^3),e)$ can been identified with McCool's "symmetric automorphism group". This is all the automorphisms of a free group $\langle x_1,\ldots,x_k\rangle$ which take each generator $x_i$ to a conjugate of some generator $x_j$. (A loop around one component of the link has to go to a loop around some component of the link.)
This is the image of $\pi_0(\text{Diff}(S^3\setminus\sqcup_k S^1))$ in $\text{Aut}(\pi_1(S^3\setminus\sqcup_k S^1))$, but since $S^3\setminus\sqcup_k S^1$ is not aspherical, this doesn't give us that $\pi_1(\text{Emb}(\sqcup_k S^1,S^3),e)=\pi_0(\text{Diff}(S^3\setminus\sqcup_k S^1))$ yet. I would be glad to see an argument that a diffeomorphism acting trivially on $\text{Aut}(\pi_1(S^3\setminus\sqcup_k S^1))$ must be isotopic (or even homotopic) to the identity.
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Tom, do you know anything about the stability of the corresponding group as the dimension of the knots increases? Note that n-spheres are automatically unlinked for n>1. Thanks. – Craig Westerland Apr 19 '10 at 0:08
About your question -- since $S^3 - \sqcup_k S^1$ is a $K(F_k, 1)$ (for the free group $F_k$), every nontrivial self-map is determined by what it induces in $\pi_1$. So if homotopy = isotopy in this dimension (not obvious to me), then $pi_0(Diff(S^3 - \sqcup_k S^1))$ will embed in $Aut(F_k)$. – Craig Westerland Apr 21 '10 at 0:24
@Craig: I started to write down that argument, but I don't think $S^3-\sqcup_k S^1$ is aspherical (it's not a $K(G,1)$) for unlinked circles. For each circle, add back in all but one point: this gives a map $S^3-\sqcup_k S^1\to S^3-\sqcup_k \ast$. The latter is equivalent to a bouquet of 2-spheres, and this map seems to be surjective on $\pi_2$. – Tom Church Apr 21 '10 at 2:40
Good point! So perhaps it embeds in $$\prod_n Aut(\pi_n(S^3 \setminus \sqcup_k S^1))$$ That will be nonzero for infinitely many values of $n$, but I bet that your argument indicates that it is determined by $n=1, 2$. – Craig Westerland Apr 21 '10 at 4:21
More: I think you can adapt your argument to get a homotopy equivalence from $S^3 \setminus \sqcup_k S^1$ to a bouquet of $k$ circles and $k-1$ 2-spheres. Unfortunately, however, look at the universal covering of this space: it's an infinite $k$-valent tree with $k-1$ $S^2$'s wedged on at every vertex. So $\pi_2$ of this cover (and hence the space itself) is infinitely genereated! – Craig Westerland Apr 21 '10 at 8:15 | 2015-01-30 00:53:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9054264426231384, "perplexity": 282.5984210892506}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122192267.50/warc/CC-MAIN-20150124175632-00040-ip-10-180-212-252.ec2.internal.warc.gz"} |
http://www.koreascience.or.kr/article/JAKO197703041857535.page | # 한국산(韓國産) 감귤과피(柑橘果皮)의 효율적(效率的) 이용(利用)에 관(關)한 연구(硏究) -I. 감귤과피(柑橘果皮)의 열풍건조(熱風乾燥)에 관(關)하여-
• Chang, Ho-Nam (Dept. of Applied Chemistry and Chemical Engineering, Korea Advanced Institute of Science) ;
• Hur, Jong-Wha (Dept. of Food Engineering Jeju National University)
• 장호남 (한국과학원 화학 및 화학공학과) ;
• 허종화 (제주대학 식품공학과)
• Published : 1977.12.28
#### Abstract
Experiment were conducted to find out the effective drying method of citrus peel produced in Korea by varying the temperature of hot air, surface area of peels, peels from several citrus varieties and physicochemical treatment of the peel. 1. About $3{\sim}6\;days$ were required to reduce the moisture level of the peel from 70%(wet basis) to 20% at room temperature without forced convection. 2. Drying was speeded up until the temperature of hot air reached $60^{\circ}C$. Beyond that no significant increase in drying rate was observed. About 50 minutes were needed to reduce the moisture level (dry basis) to below 10% at $60^{\circ}C$ by forced convection 3. When the peel surface area was increased twice by cutting the peel into 256 fractions, the overall drying time (the time required to reduce the moisture level to 10%, dry basis) was shortened to 15 minutes from 50 mintes of the original peel. 4. No significant difference in drying rate was observed among the peels from several citrus varieties except Shaddock jabon and Citrus ponki tanaka, which dried more slowly than others. 5. Treatment of $Ca(OH)_2$ and the pressing of the peel before drying were effective in drying only when the initial moisture content was substantially higher. | 2020-07-04 10:02:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.41559362411499023, "perplexity": 8948.600644592407}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655886095.7/warc/CC-MAIN-20200704073244-20200704103244-00011.warc.gz"} |
https://www.coursehero.com/sg/general-chemistry/energy-and-calorimetry/ | # Energy and Calorimetry
## Overview
### Description
Energy is defined as the ability to do work, and work is defined as the application of a force over a distance. When energy is transferred from one system to another, some energy is always "lost" as heat. Chemical processes and reactions always involve a transfer of energy. Reactions or processes that require energy, or heat, are endothermic, and those that release heat are exothermic. An exothermic reaction will cause a temperature rise in the surroundings, and an endothermic reaction will cause a temperature drop. The temperature change of a reaction or process can be measured using a calorimeter. This temperature change and the heat capacity of the materials can be used to calculate the energy change of the process.
### At A Glance
• Energy can take many forms and is always conserved. Potential energy is the energy due to position. Kinetic energy is the energy due to motion. In many systems, potential and kinetic energy are converted to each other.
• Energy transferred over a distance is work; the energy of molecular motion is thermal energy. The transfer of thermal energy is called heat.
• Thermodynamic work done by or on a gas is equal to $-P_{\rm{out}}{\Delta V}$.
• Energy changes during reactions are described by thermochemistry. An exothermic process releases heat and raises the temperature of the surroundings. An endothermic process absorbs heat and lowers the temperature of the surroundings.
• The temperature change in the surroundings can be used to calculate the energy change of a chemical process. These changes are described by calorimetry.
• Chemical thermodynamics is the study of the flow of energy during chemical reactions or phase changes between the system and the surroundings.
• Internal energy is the sum of the kinetic and potential energies of the particles in a system.
• The change in the internal energy of a system (U) is equal to heat in or out of a system (q) plus the work done on or by the system (w): $\Delta U=q+w$. | 2018-11-21 10:07:07 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 2, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.606601893901825, "perplexity": 260.1597379994239}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039747665.82/warc/CC-MAIN-20181121092625-20181121114625-00468.warc.gz"} |
https://indico.cern.ch/event/1013203/page/22724-nda-agreement | We deployed Indico v3.2. See our blog post for details on the changes.
# EXCESS Workshop
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## NDA Agreement
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The information marked with NDA information is presented for use in the EXCESS workshop only.
It must not be used for any other purpose, outside the EXCESS workshop, by persons not members of the corresponding collaboration."
We kindly ask all participants and readers to respect the NDA agreement. | 2022-06-26 13:46:17 | {"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": 0.40157029032707214, "perplexity": 7122.341214496227}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103269583.13/warc/CC-MAIN-20220626131545-20220626161545-00390.warc.gz"} |
https://earthscience.meta.stackexchange.com/questions/99/how-can-i-link-directly-to-an-answer/105 | Sometimes it is useful, when commenting on a question or answer, to link to an answer that was given to anther question. On other sites, I do this by using the "share" link under the desired link target to get a URL that goes specifically to that answer. But that "share" link doesn't seem to exist here at present, presumably because we're in private beta.
This is a hack. I hope there is a better way.
Go to the answer post. You will see the small edit link at the bottom.
Hover or click on it. You will see/go-to a url of the form:
http://earthscience.stackexchange.com/posts/358/edit
The number in the URL is the post ID. Now, add it to the answer short link.
http://earthscience.stackexchange.com/a/358
And, there you have it!
• Thanks - sounds like this may be the only way for a few weeks. Apr 19, 2014 at 16:21
Once we graduate into public beta, you'll see a 'share' button near each question and answer:
However, for right now, AsheeshR's answer is the easiest way to do it. | 2022-10-07 09:11:37 | {"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": 0.6067320704460144, "perplexity": 1175.0033937473618}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030338001.99/warc/CC-MAIN-20221007080917-20221007110917-00268.warc.gz"} |
https://electronics.stackexchange.com/questions/402461/worthwhile-to-create-some-building-block-pcb-instead-of-handsoldering | # Worthwhile to create some 'building block' PCB instead of handsoldering?
I am still busy with my DMX splitter. The circuit is (at least for me as beginner) quite exhausting to hand-solder. See circuit below (sorry for my 'artwork' as some used to call it, it benefits me a lot to locate exactly the positions to wire).
I spend several hours to one DMX output PCB. As you can see 4 of them are quite similar.
I was wonder, how worthwhile it can be to make some 'default' building block DMX and/or MIDI circuits and order them.
Let's assume I want to put both a DMX in and DMX out on one PCB, but I want to hand wire the connections between them, to be able to use them
• as DMX input only (and not using the DMX output part)
• as DMX output only (and not using the DMX input part)
• as DMX input and output (possibly by handwiring or use some jumpers to be able to connect the input and output
The PCB I need to order would be a combination of what is shown in the left/right PCBs marked with BOTTOM (which is a DMX input (left) and DMX output (right).
The reason is that ordered PCB normally have to be ordered by 10 so I benefit from equal designs.
The same later for when I'm going to incorporate MIDI (like making a MIDI in + out + thru on one PCB, but being able to use them in any combination.
Would this be useful (with the use of jumpers) or is it better to make the PCB's minimalistic?
(btw, for now I'm continuing handwiring, because I want to have this project finished, getting some more experience with soldering, and also because I never ordered a premade PCB, no knowledge about Kicad or other application that can generate Gerber or alike files needed to order a PCB).
• Usually for multiple configuration PCBs you would use jumper resistors or 2 pin headers with shunts if you want to be able to do them "on the fly". Regarding PCB costs, board area, single/double sided copper, quantity, production time all factor in. Depending on your requirements, (eg no ground plane required) you might be able to use a single sided board, it makes the through hole connector points a bit weaker, and you have to have the components "flipped" to the underside of the board with respect to the connectors. That's because you don't have plated-thru holes or pads on the other side
– isdi
Oct 21 '18 at 21:07
• @isdi thank you for that info, I see at jlcpcb.com/quote that the price for single or double side is the same. Not sure what a 'ground plane' means yet. I need multiple grounds for DMX (one separate for DMX input, and DMX output, they are isolated). Oct 21 '18 at 21:58
• ground plane is a layer of copper that covers most of a layer of a PCB, it's useful when signals of several megahertz are involved. Oct 21 '18 at 22:38
• If you're not in a rush, small PCBs are so inexpensive that custom boards for everything you do are quite feasible Oct 22 '18 at 0:13
• @Jasen thanks for the explanation, than in my case it would not be needed, MIDI is 31,250 bps, and DMX 250,000 bps. Oct 22 '18 at 0:40
I don't think you need a full 3W to drive an RS485 line, more like 0.5W (at 5V drive) I'd look into isolated DC-DC converters eg: B0505LS then you could run the whole thing off a single 5W supply | 2021-09-28 17:25:01 | {"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": 0.3402083218097687, "perplexity": 2609.1506018284463}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780060877.21/warc/CC-MAIN-20210928153533-20210928183533-00450.warc.gz"} |
http://promo-export.com/dwtuu/7ebcf3-role-of-ribulose-bisphosphate-in-photosynthesis | Source(s): function ribulose bisphosphate rbbp photosynthesis: https://biturl.im/vnOvE. A mutant of Arabidopsis thaliana has been isolated in which ribulose-1,5-bisphosphate carboxylase is present in a nonactivatable form in vivo. It's sort of between and around that Mr Willis Troma, okay, And for the synthesis takes place in the dialogue. However, the catalytic properties of Rubisco are not optimal for current or projected environments and limit the efficiency of photosynthesis. Plus, uh, you have these sort of towers of Thilo coy ds via local towers also call the Granna and then all that space. Anonymous. So what is the rule of this? The topic is discussed in all biochemistry textbooks, this one is representative: Nelson, D. L.; Cox, M. M. "Lehninger, Principles of Biochemistry" 3rd Ed. Click 'Join' if it's correct, By clicking Sign up you accept Numerade's Terms of Service and Privacy Policy, Whoops, there might be a typo in your email. When rubisco catalyses this reaction, it is known as ribulose bisphosphate oxygenase. ... Use this information and your knowledge of photosynthesis to explain the decrease in production of triose phosphate in the iron-deficient plants. Google Scholar. RuBP (ribulose 1,5-biphosphate) is a five-carbon sugar which reacts with CO 2 in the first step of the Calvin cycle for fixing carbon in photosynthetic systems. Since there's to the total is gonna be six carbons when two phosphates in here we have five carbons, plus one for six on the two phosphates. Ribulose bisphosphate carboxylase/oxygenase has common substrate, ribulose 1, 5- bisphosphate (RuBP). The characteristics of Rubisco that can affect photosynthesis fall under three main headings: (1) amount and kinetic constants; (2) activation state; and (3) regulation of catalysis (including the role of effectors, such as Pt and glycerate 3-phosphate (PGA)). So, problem. University of North Carolina at Wilmington, All right. It has a role as an Escherichia coli metabolite and a plant metabolite. And so what? Find out what you don't know with free Quizzes Start Quiz Now! immediately splits to form two 3-phosphoglycerate molecules (3-carbon compounds) in the presence of an enzyme called Rubisco. The two 3-phosphoglycerate molecules are reduced to glyceraldehyde-3-phosphate; one of Without ribulose biphosphate carboxylase, photosynthetic organisms would not be able to produce as much food. Ribulose-1,5-bisphosphate carboxylase/oxygenase (Rubisco) catalyzes the fixation of CO2 in photosynthesis. Match each with its most suitable description. It catalyzes the reaction between $\mathrm{CO}_{2}$ and ribulose bisphosphate (RuBP).b. The whole roll of this Ribis Cooper is it sort of facilitates a reaction. carbon …, EMAILWhoops, there might be a typo in your email. In photosynthesis Rubisco catalyses the assimilation of CO 2 by the carboxylation of ribulose-1,5-bisphosphate. I just sort of in this in the strom a ghoul or network and his whole roll. Ribulose biphosphate carboxylase is a chemical that aides in the process of photosynthesis. Salts of RuBP can be isolated, but its crucial biological function happens in solution. Atmospheric CO2 is combined with RuBP to form a 6 carbon compound, with the help of an enzyme (biological catalyst) called RuBisCo. But the role of the Rue Bulus 15 bias, Possibly which is typically abbreviated as as its think brew this CO right. Another added benefit of salicylic acid is that it can help play a role in systemic acquired resistance for turf in helping defend against pathogen attacks. Plants take in water, light energy and carbon dioxide in order to produce the glucose that their body needs. Abstract. In chemical terms, it catalyzes the carboxylation of ribulose-1,5-bisphosphate (also known as RuBP). (ii) Also in the oxygenation of ribulose 1, 5-bisphosphate (RuBP) leading to the formation of glycolate and 3-PGA in photorespiration. The mutation appears to affect carboxylase activation specifically, and not any other enzyme of the photosynthesis or photorespiratory cycles. Anonymous. Salts of RuBP can be isolated, but … During photorespiration RuBP combines with O2 to become 3-PGA + phosphoglycolic acid. There was little information about the fate of individual participants in this process outside the CO 2 recovery cycle. The properties and regulation of Rubisco are not optimal for biomass production in current and projected future environments. It catalyzes the reaction that regenerates RuBP.d. It is a colourless anion, a double phosphate ester of the ketopentose (ketone -containing sugar with five carbon atoms) called ribulose. The most abundant protein, Rubisco [ribulose-1,5-bisphosphate (RuBP) carboxylase/oxygenase; EC 4.1.1.39] catalyses the assimilation of CO2, by the carboxylation of ribulose-1,5-bisphosphate (RuBP) in photosynthetic carbon assimilation (Ellis, 1979). And on the carbon dioxide, which, of course, is that one's brought in, sort of be pumped into the cycle continuously. When rubisco catalyses this reaction, it is known as ribulose bisphosphate oxygenase. Find this author on PubMed . Ribulose 1,5-bisphosphate carboxylase/oxygenase (Rubisco) is the cornerstone of atmospheric CO 2 fixation by the biosphere. 5 years ago. split water. The carboxysome consists of a proteinaceous shell that structurally resembles virus capsids and internal enzymes including ribulose 1,5-bisphosphate carboxylase/oxygenase (Rubisco), the primary carbon-fixing enzyme in photosynthesis. It derives from a D-ribulose. 0 0. In plant: Specific variations in photosynthesis The enzyme ribulose 1,5-bisphosphate carboxylase/oxygenase (Rubisco) catalyzes the formation of organic molecules from CO 2. The enzymatically active substrate (ribulose 1,5-bisphosphate) binding sites are located in the large chains that form dimers as shown in Figure 1 (above, right) in which amino acids from each large chain contribute to the binding sites. Abstract. Ribulose-1,5-bisphosphate carboxylase/oxygenase (Rubisco) catalyzes the fixation of CO2 in photosynthesis. And Biscoe, that's what it X acts on this union so that at the end at into this, you know, once it capitalizes thes to uh, then what we're left with is going to be two of the three PG A's. And so so to answer the roll of the Rue Biscoe. Ribulose bisphosphate carboxylase—oxygenase: its role in photosynthesis. G3P, Which molecule must enter the Calvin cycle continually for the light-indepen…, Which of the following substances does not participate in the Calvin-Benson …. the role of chlorophyll in photolysis. David Alan Walker. Mirta N. Sivak. It is a colourless anion, a double phosphate ester of the ketopentose (ketone-containing sugar with five carbon atoms) called ribulose. It is an intermediate in photosynthesis. A total of eight large chain dimers and eight small chain… So the answer here is going to be a. And of course, each of these each of these is going to have three carbons and one phosphate right. As the major enzyme of all photosynthetic cells, Rubisco is the most abundant protein on Earth. The enzyme rubisco has two functions: (i) Mainly in the carboxylation of ribulose 1, 5- bisphosphate (RuBP) leading to the formation of 3-phosphogylceric acid (3PGA) in dark reaction of photosynthesis (see Calvin Cycle). What is the role of ribulose-1, 5 -bisphosphate, abreviated RuBisCO, in photosynthesis?a. Click 'Join' if it's correct. This six-carbon intermediate decays virtually instantaneously into two molecules of 3-phosphoglycerate(3-PGA) (see figure). Find this author on PubMed . The enzyme ribulose (RuBisCO) catalyzes the reaction between RuBP and carbon dioxide. RuBisCO also catalyzes RuBP with oxygen (O2) in a process called photorespiration, a process that is more prevalent at high temperatures. Particularly in the Calvin cycle. It catalyzes the reaction utilizing ATP and NADPH. In the Calvin cycle, RuBP is a product of the phosphorylation of ribulose-5-phosphate by ATP. Without ribulose biphosphate carboxylase, photosynthetic organisms would not be able to produce as much food. Google Scholar. The release of CO2 during photosynthesis that is due to the production and metabolism of glycollic acid is usually regarded as outward evidence for the wasteful process of photorespiration in plants. It catalyzes the reaction that produces glyceraldehyde 3 -phosphate (G3P).c. Ribulose Bisphosphate Carboxylase-Oxygenase: Its Role in Photosynthesis [and Discussion] October 1986 Philosophical Transactions of The Royal Society B Biological Sciences 313(1162):305-324 Ribulose-1,5-bisphosphate carboxylase/oxygenase (RuBisCO) is the key enzyme involved in photosynthetic carbon fixation, as it catalyzes the conversion of atmospheric CO 2 into organic compounds. Rubisco catalyses the carboxylation of ribulose-1,5-bisphosphate (RuBP), enabling net CO2 assimilation in photosynthesis. Search for more papers by this author , R. C. Leegood. It catalyzes the addition of CO 2 onto enolized ribulose 1,5-bisphosphate (RuBP), producing 3-phosphoglycerate which is then converted to sugars. The ATP and NADPH from the light reactions are used toa. Explain why, between 0 seconds and 300 seconds, the concentration of radioactive GP remained constant. Coy's and the Calvin Cycle takes place in this trauma. During photorespiration RuBP combines with O 2 t… In plants, algae, cyanobacteria, and phototropic and chemoautotropic proteobacteria the enzyme usually consists of two types of protein subunit, called the large chain (L, about 55,000 Da) and the small chain (S, about 13,000 Da). ... and ribulose bisphosphate (RuBP) changed. Search for more papers by this author . And so this ends. Explain the function of Ribulose BisPhosphate Carboxylase (aka Rubisco) in the Calvin Cycle. a. ribulose bisphosphate (RuBP) A five-carbon sugar that is combined with carbon dioxide to form two three-carbon intermediates in the first stage of the light-independent reactions of photosynthesis (see Calvin … Ribulose 1,5-bisphosphate (RuBP) is an organic substance that is involved in photosynthesis.It is a colourless anion, a double phosphate ester of the ketopentose (ketone-containing sugar with five carbon atoms) called ribulose.Salts of RuBP can be isolated, but its crucial biological function happens in solution. The answer is B as plants do not use glucose to photosynthesise, but instead produce glucose during photosynthesis. It is a conjugate acid of a D-ribulose 1,5-bisphosphate(4-). Rubisco is ubiquitus for photosynthetic organisms and is regarded as the most abundant protein on earth., From a nutritional point of view, the large subunit of Rubisco has an exceptionally ideal composition of essential amino acids among plant proteins. 5 years ago. Ribulose biphosphate carboxylase breaks down carbon dioxide and bonds carbon atoms together to begin the formation of sugars. Seems so to see if there's two home than the total. [1] To simplify the presentation, the image in the table depicts the acid … Okay, so this this room vistica here is an enzyme, and I just sort of exist in this trauma, right? This enzyme is therefore bifunctional and exerts in addition to its carboxylase activity a second activity called oxygenase, hence the name RubisCO (Ribulose biphosphate Carboxylase Oxygenase). Synthesis of triose phosphate by the chloroplast requires three substrates: light, CO_2 and orthophosphate (P_i). InChI=1S/C5H12O11P2/c6-3(1-15-17(9,10)11)5(8)4(7)2-16-18(12,13)14/h3,5-6,8H,1-2H2,(H2,9,10,11)(H2,12,13,14)/t3-,5-/m1/s1, O=P(O)(OCC(=O)[C@H](O)[C@H](O)COP(=O)(O)O)O, Except where otherwise noted, data are given for materials in their. PGAL formation______…, What do the light-capturing reactions of photosynthesis produce?a. Answered by | 23rd Jul, 2014, 04:43: PM. Ribulose-1,5-bisphosphate carboxylase/oxygenase, commonly known by the abbreviations Rubisco, rubisco, RuBPCase, or RuBPco, is an enzyme involved in the first major step of carbon fixation, a process by which the atmospheric carbon dioxide is converted by plants and other photosynthetic organisms to energy-rich molecules such as glucose. RuBP stands for ribulose bisphosphate, it's a 5 carbon compound involved in the Calvin cycle, which is part of the light independent reactions of photosynthesis. The enzyme ribulose (RuBisCO) catalyzes the reaction between RuBP and carbon dioxide. What is the role of ribulose-1, 5 -bisphosphate, abreviated RuBisCO, in photosynthesis? Worth Publishing: New York, 2000. https://en.wikipedia.org/w/index.php?title=Ribulose_1,5-bisphosphate&oldid=1000215646, Chemical articles with multiple compound IDs, Multiple chemicals in an infobox that need indexing, Chemical articles with multiple CAS registry numbers, Pages using collapsible list with both background and text-align in titlestyle, Articles containing unverified chemical infoboxes, Creative Commons Attribution-ShareAlike License, This page was last edited on 14 January 2021, at 04:00. That, well, Bill math check never hurts to make sure your equations there doing the right thing. What are the roles of ATP and NADPH in photosynthesis? and . (ii) Also in the oxygenation of ribulose 1, 5-bisphosphate (RuBP) leading to the formation of glycolate and 3-PGA in photorespiration. What is the overall outcome of the light reactions in photosynthesis? Ribulose 1,5-bisphosphate (RuBP) is an organic substance that is involved in photosynthesis.It is a colourless anion, a double phosphate ester of the ketopentose (ketone-containing sugar with five carbon atoms) called ribulose.Salts of RuBP can be isolated, but its crucial biological function happens in solution. Ribulose bisphosphate carboxylase-oxygenase: its role in photosynthesis BY D. A. WALKER, F.R.S., R. C. LEEGOOD AND MIRTA N. SIVAK Research Institute for Photosynthesis, University of Sheffield, Sheffield S10 2 TN, U.K. Synthesis of triose phosphate by the chloroplast requires three substrates: light, C02 and orthophosphate (Pi). Ribulose-1, 5-bisphosphate carboxylase-oxygenase (Rubisco) is the principal catalyst for photosynthesis and is the basic means by which living creatures acquire the carbon necessary for life. Ribulose biphosphate carboxylase breaks down carbon dioxide and bonds carbon atoms together to begin the formation of sugars. Recent evidence also suggests that salicylic acid is an important regulator of photosynthesis because it affects leaf and chloroplast structure and the activity of enzymes such as Rubisco. The formation of carboxysomes requires hierarchical self-assembly of thousands of protein subunits, initiated from Rubisco assembly and packaging to shell encapsulation. And so that's gonna be between the ruby p here, uh, which, which will contain five carbons and two phosphates and It's five carbons and too far, states, right? D-ribulose 1,5-bisphosphate is a ribulose phosphate that is D-ribulose attached to phosphate groups at positions 1 and 5. Source(s): function ribulose bisphosphate rbbp photosynthesis: https://biturl.im/vnOvE. Kieran Russell. ribulose bisphosphate: an organic substance that is involved in photosynthesis, reacts with carbon dioxide to form 3-PGA The Calvin Cycle In plants, carbon dioxide (CO 2 ) enters the leaves through stomata, where it diffuses over short distances through intercellular spaces until … It catalyzes the reaction between\mathrm{CO}_{2}$and ribulose bisphosphate (RuBP). 20 years asking us. So we're in the Calvin cycle. combines products of photosynthesis with 3CO2 (taken in by plant) What is G3P? ... Rubisco catalyzes the carboxylation and/or the oxygenation of ribulose‐1,5‐bisphosphate (RuBP) and thereby initiates the sugar producing or … 0 0. These many efforts to improve photosynthesis have been devoted to altering the catalytic properties of the initial carboxylating enzyme, ribulose 1,5-bisphosphate carboxylase/oxygenase (Rubisco), whose kinetic characteristics were believed to be a major limiting factor of photosynthesis under ambient conditions (Raines, 2006). Ribulose bisphosphate carboxylase/oxygenase has common substrate, ribulose 1, 5- bisphosphate (RuBP). [1] To simplify the presentation, the image in the table depicts the acid form of this anion. So far, the role of K and Mg in photosynthesis and associated processes has not been covered in a comprehensive review. The answer is B as plants do not use glucose to photosynthesise, but instead produce glucose during photosynthesis. Rubisco biogenesis depends on auxiliary factors, including the GroEL/ES-type chaperonin for folding The product is the highly unstable six-carbon intermediate known as 3-keto-2-carboxyarabinitol 1,5-bisphosphate. RuBisCO also catalyzes RuBP with oxygen (O 2) in a process called photorespiration, a process that is more prevalent at high temperatures. : its role in photosynthesis atmospheric CO 2 in which ribulose-1,5-bisphosphate carboxylase is a chemical that aides the. Most abundant protein on Earth ( Rubisco ) catalyzes the addition of CO 2, 04:43: PM of! Knowledge of photosynthesis on Earth _ { 2 }$ and ribulose bisphosphate carboxylase/oxygenase has common,... Each of these is going to be a typo in your email optimal for production! Biphosphate carboxylase is a product of the Rue Biscoe cells, Rubisco is an inefficient enzyme and thus a... With 3CO2 ( taken in by plant ) what is the role of ribulose-1, 5 -bisphosphate, abreviated,! 3-Phosphoglycerate ( 3-PGA ) ( see figure ) roles of ATP and NADPH from the light reactions photosynthesis. If you 're in 1/4 enabling net CO2 assimilation in photosynthesis Rubisco catalyses the of! Appears to affect carboxylase activation specifically, and not any other enzyme of all photosynthetic,... Reaction, it catalyzes the formation of sugars do the light-capturing reactions of photosynthesis produce?.... Specifically, and I just sort of exist in this trauma,?! The image in the iron-deficient plants addition of CO 2 recovery cycle s ): function ribulose rbbp... At high temperatures to see if there 's two home than the.! Product is the highly unstable six-carbon intermediate decays virtually instantaneously into two molecules of 3-phosphoglycerate ( 3-PGA (. Bisphosphate oxygenase photosynthesis with 3CO2 ( taken in by plant ) what the! Of 3-phosphoglycerate ( 3-PGA ) ( see figure ) biological function happens solution. Present in a comprehensive review and I just sort of between and around that Mr Willis Troma, okay and... In plant: Specific variations in photosynthesis? a 5 -bisphosphate role of ribulose bisphosphate in photosynthesis Rubisco. Ah, that 's gon na be through in the Calvin cycle RuBP... Three substrates: light, CO_2 and orthophosphate ( P_i ) carbon and... B as plants do not use glucose to photosynthesise, but … ribulose bisphosphate ( RuBP ), net! ] to simplify the presentation, the image in the presence of enzyme. 2 fixation by the biosphere atoms together to begin the formation of sugars enolized ribulose 1,5-bisphosphate ( RuBP is! Right thing more papers by this author, R. C. Leegood enabling net CO2 in...: function ribulose bisphosphate ( RuBP ) it 's sort of exist in this process the... Molecule of CO 2 and forms the first stable compound of the phosphorylation of by... Brew this CO right ( taken in by plant ) what is the overall outcome of the Rue Bulus bias! The biosphere, CO_2 and orthophosphate ( P_i ) the mutation appears affect. Of carboxysomes requires hierarchical self-assembly of thousands of protein subunits, initiated from Rubisco assembly and packaging shell. The right thing without ribulose biphosphate carboxylase is a product of the photosynthesis or photorespiratory.... The ATP and NADPH from the light reactions in photosynthesis all photosynthetic cells, Rubisco is an called. By Topperlearning User | 23rd Jul, 2014, 04:43: PM also catalyzes RuBP oxygen... That, well, Bill math check never hurts to make sure your equations doing! Enzyme and thus is a colourless anion, a double phosphate ester of the biosynthetic phase 's. And ribulose bisphosphate ( RuBP ), enabling net CO2 assimilation in photosynthesis enzyme. Rubisco is an enzyme called Rubisco n't know with free Quizzes Start Quiz Now simplify the presentation the. To begin the formation of organic molecules from CO 2 by the chloroplast if you recall if you in... These is going to be a typo in your email there might be a typo your... And orthophosphate ( P_i ) 1 ] to simplify the presentation, concentration. Answer here is an enzyme, and I just sort of exist in process. Carboxylase breaks down carbon dioxide in order to produce as much food it! 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Enzyme called Rubisco this this room vistica here is going to have carbons... Bisphosphate ( RuBP ) and thus is a key target for directed evolution key target for directed evolution subunits! | 2021-05-07 20:22:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5634804368019104, "perplexity": 10369.332837594813}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988802.93/warc/CC-MAIN-20210507181103-20210507211103-00020.warc.gz"} |
https://sourcegate.wordpress.com/tag/string/ | ## Whiteboard robot number crunching
I’m thinking of building a robot that suspends a whiteboard marker from two strings, and draws on a whiteboard. There would be two motors that can change the length of the strings.
I was playing with some calculations for this robot, literally on the back of an envelope. I need to calculate how long each string will be for a particular coordinate.
Pythagoras’ theorem gives:
$l_1^2=x^2+y^2$
and
$l_1^2=r^2+s^2$
If we smash them together, we get:
$r^2 + s_1^2 = x^2 + y^2$
$s_1 = \sqrt{x^2+y^2-r^2}$
and of course
$s_2 = \sqrt{(w-x)^2+y^2-r^2}$
That was easier than I expected – the maths wasn’t too hard! This should be enough to draw short segments with linear interpolation.
But, I’d like to know how the string length changes as I draw a straight line. Imagine a line being drawn perpendicular to the string; when the length of the string gets very long, it should lengthen at the same rate as the line. This suggests that the string length and the line length forms a hyperbola.
Consider a horizontal line beneath one of the wheels. Let x be the distance from the closest point on the line to the wheel. s is the string length. If the line went through the wheel, the graph
$\frac{x}{s}=1$
would give the line where the x position is the same as the string length.
If the line does not pass through the wheel, this equation from earlier where r is the wheel radius and y is the shortest distance from the wheel to the line:
$s^2 = x^2 + y^2 - r^2$
can become
$1 = \frac{x^2 + y^2 - r^2}{s^2}$
which should produce a hyperbola. When the string is long enough, y²-r² becomes negligible.
Previously I’ve implemented Bresenham’s algorithm to interpolate values on an AVR; it would be nice to do the same thing to calculate this parabola while drawing a straight line segment. It turns out someone has worked out how to use calculations like this for hyperbolae. | 2021-05-16 09:09:42 | {"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": 0, "img_math": 8, "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": 0.6595504879951477, "perplexity": 356.85612945872225}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243992516.56/warc/CC-MAIN-20210516075201-20210516105201-00526.warc.gz"} |
http://kiskis.sourceforge.net/ar01s04.html | Manual
Getting started
Please take a look on our Screencast Tutorial on youtube for a short introduction.
Account types explained
KisKis™ provides some basic account types for different purposes, such as “Network Accounts”, “Bank Accounts”, “Secured Files” and “Credit Cards”. However, if the attributes of them doesn't meet your needs, you can define your own account types in an easy way.
All accounts can be archived. Select the In archive? -checkbox if the account will not be used in the near future. Checked accounts will normally not be visible in the tree view. You can make them visible again with the View/Show archived items menu item.
The Apply -action will be enabled when the account was changed. Click on this button if you want to "commit" your changes.
Common properties
All account types define the following properties in common. That means they have a name, password, can expire on a specific date and collect some statistics.
Label
A simple name for the account which is used in the tree view.
The password used for this account. Normally it will be hidden, but you can display it if necessary.
You can create new passwords automatically if the password field is empty with the Create -action. Click on Create and a menu with three generators is displayed.
If a password exists you can display the password using the Show -button.
You can copy the password to the clipboard even when it is hidden with the Copy to clipboard -action from the context menu.
The Test -action checks if the password can be found in a dictionary. You can define your own dictionary to use as described in the Cracklib Options Tab .
This progress bar shows the password quality on the fly.
The tooltip shows you some more detailed information about the quality.
Expires
A password might expire on a specific date. You can enter this date here or mark the Never -checkbox if it never will expire.
Expired passwords can be found with the "Reports/Expired Accounts ..."-action from the menu bar.
History
Displays a dialog with all recent used passwords for this account.
Attachments are arbitrary files which will be encrypted and stored next to the account.
New
Shows a file selection dialog. Select a file and it will be shown in the list afterwards.
Save as
Decrypt the attachment and save it on the disc.
Delete
Delete the attachment from the account. The encrypted file will be removed as well.
All physical operations will be done when the document is saved. An Apply alone will not change any data on the disc. The source files will not be touched at all. That means you have to remove them from disc manually if necessary.
Normally you should not be bothered with the manual decryption of attachments, but here is how it works.
File attachments are stored as separate files which are associated with the password-file (e. g. c:\foo\bar\kiskis.xml.gpg ) by name for efficiency reasons. All attachments of c:\foo\bar\kiskis.xml.gpg can be found as c:\foo\bar\kiskis.xml.gpg.attachment.<i> encrypted as separate PGP-Messages. Each attachment is encrypted with a new random key which you can find within the c:\foo\bar\kiskis.xml.gpg -file in the <Attachment>-element.
So, a typical KisKis™ -directory c:\foo\bar with a passwordfile kiskis.xml.gpg will look like:
gandalf@valinor-LINUX:/foo/bar/$dir insgesamt 192 -rw------- 1 gandalf gandalf 6419 2010-11-15 14:17 kiskis.xml.gpg -rw-r--r-- 1 gandalf gandalf 159 2010-11-10 18:00 kiskis.xml.gpg.attachment.1 -rw-r--r-- 1 gandalf gandalf 543 2010-11-10 15:55 kiskis.xml.gpg.attachment.2 -rw-r--r-- 1 gandalf gandalf 231 2010-11-10 15:55 kiskis.xml.gpg.attachment.5 -rw-r--r-- 1 gandalf gandalf 1223 2010-11-10 15:55 kiskis.xml.gpg.attachment.6 -rw-r--r-- 1 gandalf gandalf 326 2010-11-10 15:55 kiskis.xml.gpg.attachment.7 -rw-r--r-- 1 gandalf gandalf 492 2010-11-10 18:00 kiskis.xml.gpg.attachment.8 -rw-r--r-- 1 gandalf gandalf 159 2010-11-10 18:00 kiskis.xml.gpg.attachment.9 Each account collects some statistics. So you can see when it was used the last time and how often it has been viewed. Plain text can be added to each account using the Comment -tab. Network Account This is the most often used account type. It can be used for computer logins, mailserver authentications, internet services and so on. The network account provides additional attributes for: User name Typical use is the login name of an internet service or computer account. This might be an e-mail address as well. URL The location where the service or computer can be found. This URL can be delivered to the Build-in Application Starter , so that you can associate your preferred application to it. Therefore, you would have to click on the button Open URL . Example 1. URL with placeholders http://www.foo.de/?un=%username&pwd=%pwd or pop://mail.foo.de/ E-mail If a service wants to know an email address you can type it in here. This is very useful if you have multiple email accounts and if you want to keep track which account knows which email address, especially if you use such services like spamgourmet.com. Bank Account This account type models a typical money account on a bank. It provides some additional attributes needed for financial transactions such as “telephone pin”, “account number”, “TAN lists” and more. The bank account provides additional attributes for: Bank Name The name of the bank, e. g. "Deutsche Bank". Bank Identifier The identifier of the bank. This may be a IBAN, BLZ or something else. Telephone PIN This is a password or PIN which is used for telephone banking. Account Number This is the number that identifies the account. Notice the TAN list field for transaction numbers (TAN). It is used to define sets of TANs. Each TAN list is identified by an ID and a creation date. Within the following dialog the TANs can be added, removed or marked as used. TAN lists: New Creates a new empty TAN list. Edit Open the selected TAN list in a TAN list editor dialog. Delete Deletes the selected TAN list. TAN list dialog options: TAN list ID An identifier which is usually written on the TAN list by the bank. Created on The date when the bank created the list. TAN ID A consecutive number which identifies a TAN. Value The value of the TAN, usually a 6-digit random number. Used? Checked if used. When checked is clicked, the field "Used On"" will be updated as well. Used On The date when the TAN was used. Credit Card A “Credit Card” is usually associated with a bank and has a tiny pin used for ATMs. Though, the most interesting part is its number which can be entered as well. The credit card account provides additional attributes for: Bank Name The name of the bank, e. g. "Deutsche Bank". Credit Card Number The number written on the card. PIN The PIN needed for ATMs. Card Validation Code The card security code (CSC) provides increased protection against credit card fraud Wikipedia . Secured File Sometimes files can be opened with a passphrase only. So you can define an account which is linked to the file. This files can be opened with your preferred filemanager. You can encrypt or decrypt them with OpenPGP if you want to. The secured file account provides additional attributes for: File A relative or absolute path to an arbitrary file. This could be "project plan", "word document", "keystore" or something else. Status Shows if the file could be found or if it is a directory or if it is missing. Decrypt Decrypts the file using the password given above. It is activated only if the file is a PGP file. This is checked automatically. Encrypt Encrypts the file using the password given above. It is activated only if the file is not a PGP file. This is checked automatically. User-defined Account Template If you need some extra attributes or even simpler accounts you can define your own account templates. In the standard KisKis™ document you can find two examples. The first example is the "Password only" type, the second example is the "Complex type example" type. You can change these examples if you want to. Think about an account type as a blueprint for multiple occurences which all need some specific properties. As you can see this account type does not define any additional attributes. So, the detail area is not visible anymore. You can use this account if you need a "label/password" pair only. This account type is just an example. You can see all field types available. Look at Managing your own account templates for further information. Managing your own account templates Open the menu item “Edit/Manage account templates” to open the template overview dialog. Warning Be careful when modifying a template you have already instantiated and filled with important data. New properties aren't a problem at all. But keep in mind, that deleting a property will delete ALL associated values from the instances as well. You should also note that deleting a template will delete all instances. Here you can see all your defined account templates. In this case, two types were already defined. Note that an item is uniquely identified by its name (case-sensitive). So you cannot have a second item called “Password only” . All the templates are stored within your current datafile. New Click New and a newly created template will appear in the list. It will be initially called "new template". Edit Select an existing template and click Edit to manipulate the template. A new "template editor dialog" will appear. You can do a double-click in the list as well. Delete Select an existing template and click Delete to remove the template. If the item is currently instanciated a warning will be shown. Import You can import existing templates from other KisKis™ files with the Import -button. Select a KisKis™ file, enter the password and all the templates will be copied to the current file. In case of naming collisions you can change the template names before OK is pressed. Name your template and add some tiny properties with New . You can order the properties using the arrow buttons on the right panel. Template Name Enter a unique name for the template. There is no other constraint for the name. New Opens the property editor dialog . Edit Select an existing property and click Edit . The property editor dialog will be shown. Delete Select an existing property and click Delete . The currently selected item will be removed from the list. If the property is still used by an instance a warning will be shown. Give each property a unqiue name within the template and choose a type out of the combo-box. As you can see, the following types are supported: Date Will be rendered as a date field Password Will be rendered as a password field. String Will be rendered as a simple text field. URL Will be rendered as a URL-input field which allows you to start an associated application. RichText Will be rendered as a text area. Take a look at the Complex template example . Generating passwords automatically Secure passwords should be only known by you and the service you are using. It is not recommended to reuse passwords for multiple services. That means you need one unique password for each account which cannot be derived from another password of a different account. Therefor, KisKis™ provides multiple password generators which make it easy for you to follow these rules. Three different generators are available when you want to create a new password human readable Human readable passwords do not use sophisticated special characters. Furthermore the generator mixes consonants and vocals in a friendly manner. The passwords created should always be readable, e. g. NuHuxo770165 secure Secure passwords use all displayable characters in a complete random order. These passwords may be hard to read and comprehend but are secure, e. g. du"|]Z0ku&"E . by template This option opens a new password generator dialog. Template Enter a string, defining your template, here. A template consists of a user-defined password pattern string with a length greater than zero. Each pattern character represents a set of characters which can be placed on this position randomly. c,C - a consonant (b, c, d, ...) v,V - a vocal (a, e, i, o, u) a,A - an alphabetic character 9 - a digit (0-9) n,N - a combination of 'a' and '9' # - a special character (+,$, %, ...)
? - any character
Example: cVCvaA99#? can generate kIFaaT40[F , wUJan042:% and so on.
Count
The number of passwords to generate. You can pick one of them out of the list.
Mix case
If upper case and lower case should be chosen randomly you can activate this box.
When you need to choose a password you may ask yourself "What is a good password and how do I know it is well chosen?". However, at first you need to know the how an intruder would try to get your password.
The easiest way to get a password is guessing or social engineering . Many people are using passwords of things or family members they can remember easily. "The name of the pet", "the childs birthday", "an anniversary" or such things are often used and can be guessed by an intruder easily. Think about all the information the web knows abaout you. Google, Facebook and Xing are a very useful source for such information. Even if you did not publish such sensitive information, the intruder might know you better than you think.
Important
You can protect yourself from those attacks easily. Do not use any information of your social environment as passwords!
An automated way to hack an account is to use a dictionary attack . Therefor a computer will try each entry of a dictionary to access your account. Those dictionaries contain millions of entries with the most common passwords used all over the world. Our fast computers do not need much time to find the right solution if the password can be found in the dictionary. Did you know that "qwertz" or "{[]}\" are often used passwords? That is the case because these character sequences constist of characters which are close to each other on the keyboard.
Important
You can protect yourself from those attacks easily. Do not use simple words or character sequences, consisting of characters which are close to each other on the keyboard, as passwords! Passwords should never be found in a dictionary!
Another way to get your password is to use a brute force attack . The algorithms are very simple. "Try each possible variation of characters and numbers up to a defined length." The longer your password and the more different characters your password contains the more variations need to be tried.
Important
You can protect yourself from those attacks easily. Use long passwords with at least 10 characters mixed with numbers, special characters, upper case and lower case!
The first is a simple password analyzer which tests the strength of your password depending on the character set used. A character set describes numbers, lower-case letters, upper-case letters, punctuation, ... The more different character sets a password uses and the longer it is, the more secure the password is because a brute force attack needs to take more possible variations into account.
As you type the password in the password element it will be checked automatically. Depending on the characters you typed the number of possible variations is computed. It is assumed, that an intruder might get the information about the character set used, i. e. if you use numbers only as a password the intruder would try numbers only in a brute force attack to reduce the number of possible variations. The tooltip shows you more information about the password quality. So you can see the number of possible variations.
The second way is a dictionary-based check using cracklib. The password is validated against a dictionary. If cracklib is able to find parts of the word in its dictionary you should use another password because a dictionary-based attack on your account could succeed with a high probability.
This dialog gives you some information about the password, e. g. if it was found in the dictionary or if it violates some other password rules.
Importing CSV files
KisKis™ provides a basic feature to import existent data via "comma-separated-value”-files (CSV). You can create CSV files easily with Microsoft Excel or OpenOffice Calc .
Open the KisKis™ file you want to add the imported accounts to and activate the menu item “File/Import” to start the procedure. A file selection dialog opens. Select the CSV file you want to import and click OK . The accounts will be added.
A CSV file must start with a header line and may contain multiple data lines. Each data line represents one account and must contain as much as fields as defined in the header.
The header with pre-defined values must be included:
Group
The name of the group. An empty group name means that the account should be append to the root. A group path can be defined using the character sequence " ## " as a path separator. A group name " Shopping##Books#My Favorite Bookstores " would result in the following tree path:
If no group path separator can be found the group will be appended to the root node.
Label
The name of the account.
User Name
The user name for the account.
Email
The e-mail address used for the account.
URL
The URL used for the account.
Created On
The creation date used for the account. The format is YYYY-MM-DD , e. g. 2010-12-01.
Expires On
The expiration date used for the account. The format is YYYY-MM-DD , e. g. 2010-12-01.
Comment
The comment used for the account. May contain linebreaks.
Example 2. CSV Example File
,"Account placed to the root","hhsgww2l","foo",,,,,
"Shopping","Amazon",32362187361,"amazon.foo","mail@bar.de","http://www.amazon.de",,,"amazon account"
"Newly created group","Blahblah",1234,"user@foo.bar","user@foo.bar",,,,
"Work##Job 1","Computer Job 1","secret","john.doe","john.doe@company.com","http://portal.company.com","2010-10-21",,"Another comment"
"Work##Job 2","Enterprise Password","foobar","karl.mustermann","karl@mustermann.de","Http://portal.foo.com","2009-12-24","2010-11-23","That is just a comment.
With
Multiple lines"
"##","Account placed to the root 2","ÄÖÜölöö","another@user.de","another@user.de","http://foo.bar",,,"Another comment
With multiple lines"
"Others##Invalid Accounts","Wrong Expiration date","rhiurhewf","foo","foo@bar.com",,,01.01.10,"Wrong expiration date"
"Others##Invalid Accounts","Wrong creation date","rhiurhewf","foo","foo@bar.com",,01.01.10,,"Wrong creation date"
"Others##Invalid Accounts",,,,,,,,
Notice the header in the first line and 11 different data rows. The order in the column header is not important. You don't have to provide values for each possible column. You could use the header Label, Password as well and omit the other column values ( the rest will be filled with predefined standard values). But if you have defined two columns in the header, each data row MUST provide two columns as well (but a column may be empty).
In this example the field-delimiter is ','. You can chose any other character if you want to. Put the field values in "" if the field-delimiter may be found inside the value, e. g. comments and text fields.
Get the example OpenOffice.org Calc spreadsheet and try out.
Important
Note that the imported accounts will be typed as “Network Accounts” and will be added to the opened file.
Options and preferences
Open the menu item Edit/Options... to edit your personal preferences. A new dialog will appear.
General
In the general tab you can find options for appearance and some automatisms making your life easier.
General options
Choose Look&Feel-classname
Choose a classname of an javax.swing.LookAndFeel -implementation.
The default value is the great com.incors.plaf.kunststoff.KunststoffLookAndFeel .
You can choose a font of all available fonts which is used for the password fields.
The default is Monospaced and should be sufficient for most platforms.
Lock program after N minutes
Enter the number of minutes of inactivity here that should pass before KisKis™ will be locked. Inactivity means that KisKis™ did not receive any mouse event or key stroke, e. g. because the window is in the background. When KisKis™ is locked you need the password of the currently opened file to unlock it again. So you might leave your computer alone for a moment.
A value of 0 will disable this option.
The default is 5 minutes.
Mark items as viewed after N seconds
Enter the number of seconds here that should pass before KisKis™ will mark the currently opened account as viewed. This means, when you opened the GMail account its last viewed date and view counter will be updated after N seconds. This is useful if you want to keep track of your favorite accounts. If you switch to another account before N seconds passed these values will remain the same as before.
A value of 0 will disable this option.
The default is 10 seconds.
Should the password stay in memory as long as the password file is opened? On a single user machine this is no problem. On a multiuser server, e. g. Citrix, it would be safer to disable this option.
The default is checked .
If buffer password is enabled the buffered password can be disposed from memory automatically after N minutes. This is useful if you run KisKis™ on a multiuser platform without losing much convenience.
A value of 0 will disable this option. The password will never disposed.
The default is 0 .
This value is used when a new account is created for computation of the expiration date. The default expiration date will be today + N days.
The default is 365 days.
Export user preferences on exit
If you want to run KisKis™ from a USB-stick on multiple computers it is useful to share the preferences. Check this box and the preferences will be saved in a file \$KISKIS_HOME/kiskis.preferences . If you start KisKis™ the next time it will restore the preferences from this file.
The default is not checked .
If you want to get a short message when a new version of KisKis™ is available you need to check this box. KisKis™ will ask the server http://kiskis.sourceforge.net/download if a new version is available. No information from you will be sent to the server for this operation. This is just a simple HTTP-GET . These requests will not be saved to any logfile from the KisKis™ -authors.
The default is checked .
In this tab you can find options for the load and save operation.
Default encryption algorithm
Select your favorite encryption algorithm. OpenPGP - AES (256) is the strongest algorithm available. You can use other algorithms if you want to, even 3DES which does not use PGP at all.
The default value is OpenPGP - AES (256) . If the JCE is not installed on your machine OpenPGP - AES (128) will be the default.
Enable auto save
Enable this option if KisKis™ should save your changed password file automatically.
The default value is checked .
Save every N minutes
Tell KisKis™ how many minutes it should wait to save the document automatically after the password file has been modified.
The default value is 5 minutes.
Max. number of backup files
KisKis™ can make backups when saving the document automatically. All attachments will be backed up as well. You can find the backup files in the directory where your password file is saved. The filenames follow the simple rule <password file>.backup.<timestamp> .
The default value is 5 backups.
Applications
You can define your own applications that shoul be used for opening URLs here. In the list you can find prefixes and regular expressions for URLs associated with commands for external applications. The list has to be read from top to bottom. The first matching prefix/regular expression for a given URL will be used to start an external application.
As you can see in the picture above three different entries exist. URLs starting with https://www.myjob.de/ will be started with firefox and all other http -URLs will be passed to the machines default browser . It is important that the more specific prefixes will be placed on top of the more general ones.
Applications options
New
Creates a new empty entry in the list. Make a double click on it to define its values.
Edit
Edit the selected entry.
Regular expression or prefix for URL
This pattern or prefix is used to match a given URL. It answers the question: Should this entry be used to open the URL X? .
Define a prefix, e .g http . You can define Java-like regular expressions [6] as well if you need more complicated patterns and logic.
Associated command
Define your command, which starts the application here. You may use placeholders as %url , %pwd and %username . These placeholders will be filled with the values of the specific account when you click Open URL .
The command <default browser> %url will use the Java-standard mechanism to detect the default browser on your machine.
Delete
Remove the selected entry.
Cracklib Dictionary
Here you can define your own dictionary that is used to check passwords if you want to. The standard dictionary contains more than 1.6 million words and typical passwords. Most of the words are in German and English.
Cracklib dictionary directory
Define the relative or absolute path to a cracklib directory here. This directory contains the dictionary which consists of three cracklib files ( cracklib.hwd , cracklib.pwd and cracklib.pwi ).
Select an existing dictionary file
Use this action to select an existing directory containing the three cracklib files . A directory selection dialog will appear. The selected directory will be validated and the absolute directory pathname is shown in the textbox afterwards. Otherwise an error message will appear.
Create a new Dictionary from wordlist
You can define your own textfiles with your own words as a dictionary. That is quite simple. Create a file wordlist.txt with a text editor (e. g. notepad on Windows). This file should look like as follows:
Example 3. Sample wordlist
a
aa
aron
berta
...
julia
z
zz
zoron
Empty lines and case will be ignored.
A file selection dialog will appear if you click this action. Select the file you created first and Select . A progress indicator will appear as long as this action has not been finished.
The dictionary files will be created on the file system in the directory specified in the textbox. So, you should define the target directory in the textbox first. The import of the file cannot be cancelled and may take a while. Please be patient.
You can find the standard wordlist in the Version Control System . | 2017-05-29 18:56:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.2515611946582794, "perplexity": 2095.8033468640383}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463612537.91/warc/CC-MAIN-20170529184559-20170529204559-00229.warc.gz"} |
https://math.stackexchange.com/questions/2630650/how-do-derive-the-statistical-representation-of-white-noise-in-the-frequency-dom | # How do derive the statistical representation of white noise in the frequency domain.
Setup:
A Gaussian white noise process $x(t)$ is defined such that
1. The autocovariance $R(\tau) = \left< x(t) x(t-\tau) \right> = \sigma^2\delta(\tau)$ (uncorrelated in time)
2. The ensemble distribution at any time $t$, $P[x(t)]$, is $\mathcal{N}[0, \sigma]$.
We can write the Fourier transform in terms of amplitude and phase as
$$\mathcal{F}[x(t)] = \hat{x}(\omega) = \int x(t) e^{i 2\pi \omega t} dt = a(\omega)e^{i\theta(\omega)}$$
A simple integral show s that the power spectral density
$$S(\omega) = \left< \left| a(\omega) \right|^2 \right> = \sigma^2 = \int R(\tau) e^{i 2\pi \omega \tau} d\tau$$
Question:
It is commonly stated that the ensemble statistics of $a(\omega)$ and $\theta(\omega)$ follow
1. $P[a] = \mathcal{N}[0, \sigma]$
2. $P[\theta] = \mathcal{U}[-\pi, \pi]$
In other words, now $a$ and $\theta$ are random processes in the frequency domain with the given distributions.
How does one derive this? What can we say about the correlation structure? | 2019-08-23 05:09:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9477933049201965, "perplexity": 328.79447831903076}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317847.79/warc/CC-MAIN-20190823041746-20190823063746-00321.warc.gz"} |
https://support.bioconductor.org/p/132965/ | ensembl to hgnc symbols creates duplicates
1
0
Entering edit mode
Matan G. ▴ 40
@matan-g-22483
Last seen 6 months ago
Hi all,
My data is a data frame of estimated TPM counts, where rows are the genes and columns are the samples. I'm using "library(biomaRt)" to get ensembl symbols and hgnc symbols. When trying to change the rownames from enemble to hgnc symbols I get an error which stems from duplicates in hgnc symbols the way I understand it.
The error I get:
"Error in .rowNamesDF<-(x, value = value) : duplicate 'row.names' are not allowed In addition: Warning message: non-unique values when setting 'row.names': ‘’, ‘ABCF2’, ‘LINC01238’, ‘POLR2J3’, ‘POLR2J4’, ‘TBCE"
How can I solve this issue? EDITED: using .rowNamesDF(TPM_countdata, make.names=TRUE) I've managed to force row names to be hgnc coded but I don't understand the reason it creates duplicates initially and not unique names of hgnc.
Thanks and all the best
data screenshot https://ibb.co/8M853bm
r biomart genemap TPM • 209 views
0
Entering edit mode
@kevin
Last seen 5 hours ago
Hey Matan,
It is expected for this to happen when comparing across annotation systems, in this case, Ensembl to HGNC. To understand why, please look at these answers on biostars, one from the Ensembl Outreach Project Leader:
What I usually do is merge the Ensembl and HGNC IDs via an underscore '_', which can be removed when it comes to exporting your final result or generating plots.
Note that, while we define a gene as a static unit, the genome does not behave this way. Transcription is a pervasive process whereby, over millions of years of evolution, certain parts of the genome are transcribed more frequently under certain cellular / environmental conditions, and then translated into proteins. The vast majority of the genome is still transcribed to some level, but can be regarded as background 'transcriptional noise'.
Kevin | 2021-05-13 06:46:07 | {"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": 0.30991724133491516, "perplexity": 6020.421980860567}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991537.32/warc/CC-MAIN-20210513045934-20210513075934-00253.warc.gz"} |
https://www.physicsforums.com/threads/quotient-rule-for-derivative.668872/ | # Quotient Rule for Derivative
1. Feb 2, 2013
### Torshi
1. The problem statement, all variables and given/known data
Use quotient rule
2. Relevant equations
d/dx (cot(x)
3. The attempt at a solution
d/dx (cot(x))
= cos(x)/sin(x)
= (sin(x))(-sin(x)) - (cos(x))(cos(x)) / (sin(x))^2
= -(sin(x))^2 - (cos(x))^2 / (sin(x))^2
= -1 - (cos(x))^2
Does the last step equal 1 + (cos(x))^2 and how come that turns into negative -(csc(x))^2
2. Feb 2, 2013
### jbunniii
I think you dropped some parentheses and this led you to a wrong answer. That first step should be:
$$\frac{d}{dx} \cot(x) = \frac{\sin(x)(-\sin(x)) - \cos(x)\cos(x)}{\sin^2(x)}$$
Now simplify the numerator and see what you end up with.
3. Feb 2, 2013
### Torshi
= (-1)-(cos(x))^2
4. Feb 2, 2013
### jbunniii
Shouldn't the second term be divided by $\sin^2(x)$?
5. Feb 2, 2013
### Torshi
Hold on
-1 - (cos(x))^2 / (sin(x))^2 = -1/(sin(x))^2 = -(csc(x))^2
but...
-(sin(x))^2 - (cos(x))^2 / (sin(x))^2 = 1/(sin(x))^2 = -(csc(x))^2
6. Feb 2, 2013
### jbunniii
How did you get that?
I think there are some parentheses missing again. The expression on the left should be
$$\frac{-\sin^2(x) - \cos^2(x)}{\sin^2(x)}$$
Now you can use the identity $\sin^2(x) + \cos^2(x) = 1$ in the numerator.
7. Feb 2, 2013
### Torshi
My main question is do both of the following equations equal -(csc(x))^2
1/(sin(x))^2 and -1/(sin(x))^2
or which one of those two define -(csc(x))^2 ?
8. Feb 2, 2013
### HallsofIvy
Staff Emeritus
Since, by definition, csc(x)= 1/sin(x), it follows that sin(x)= 1/csc(x) and then that $sin^2(x)= 1/csc^2(x)$.
If you want "-" on the right, you will have have a "-" on the right.
9. Feb 2, 2013
### Staff: Mentor
You haven't actually taken the derivative yet, so the above should be:
=d/dx(cos(x)/sin(x))
Others have already commented about the missing parentheses and the incorrect parts. | 2017-08-17 18:44:28 | {"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": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7397536039352417, "perplexity": 6553.8049965796545}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886103891.56/warc/CC-MAIN-20170817170613-20170817190613-00560.warc.gz"} |
http://bovbjerg.net/cannot-be/type-java-lang-string-cannot-be-converted-to-jsonobject.php | Home > Cannot Be > Type Java.lang.string Cannot Be Converted To Jsonobject
# Type Java.lang.string Cannot Be Converted To Jsonobject
## Contents
What is the significance of the robot in the sand? share|improve this answer answered Apr 23 '12 at 22:45 Nic Raboy 2,2221419 add a comment| up vote 2 down vote I made this change and now it works for me. //BufferedReader When does TNG take place in relation to DS9? For that, I would suggest you to handle your array as below: JSONArray jsonArr = null; String jsonObjRecv = JSONGet.getJSONfromURL(URL_LIST); if(!TextUtils.isEmpty(jsonObjRecv)){ jsonArr = new JSONArray(jsonObjRecv); } else{ Log.w("json", "jsonObjRecv is null"); Source
It looks fine to me and jsonlint.com says it's valid. –MrLore Aug 11 '13 at 16:42 If you print out the String before you try and put it into It is showing" Error parsing data org.json.JSONException: Value success of type java.lang.String cannot be converted to JSONObject" I am completely new to PHP and Json. Thanks Posting to the forum is only allowed for members with active accounts. This is a basic app that retrieves values from a MySQL database using a PHP script and displays them on your android device. i thought about this
## Type Java.lang.string Cannot Be Converted To Jsonobject Android
Weird Problem. Already have an account? Since your JSON data contains Cyrillic characters, they will lead to invalid data that can no longer be parsed as JSON. (In UTF-8, Cyrillic characters required two bytes.
• GO OUT AND VOTE Colleague is starting to become awkward to work with Can you dispel a magic effect you can't perceive?
• I have made a small sample code for you on how to use it : import org.json.simple.JSONArray; import org.json.simple.parser.JSONParser; JSONParser parser_obj = new JSONParser(); JSONArray array_obj = (JSONArray) parser_obj.parse("String from web
• Missing } inserted. \int dx = x + C & Colleague is starting to become awkward to work with Professor Lewin: "Which string will break?" / Me: "That one." / Professor
• share|improve this answer edited Nov 13 '13 at 0:31 answered Nov 13 '13 at 0:24 bgse 1,751917 add a comment| up vote 0 down vote Check if the JSON string itself
• To emailaddress: To name: From name: Extra information in the email body (optional): Email: I am sending you the codedump of JSONException: Value of type java.lang.String cannot be converted to JSONObject
• However, you read it using the ISO-8859-1.
• Shouldn't you use a GET request?
Please update these videos Treehouse. Came across this and tried switching out .toString(); with just .string(); and its working now... Thanks for your solution:) –code4j Nov 13 '12 at 22:41 3 I have the same problem but cannot see how making a substring would make this go away? –EHarpham Nov Value
Unfortunately I am getting the error: JSONException: Value of type java.lang.String cannot be converted to JSONObject Here are my variables and the code that parses the JSON-File: private InputStream is = Org.json.jsonexception: Value Of Type Java.lang.string Cannot Be Converted To Jsonobject Join them; it only takes a minute: Sign up Value results of type java.lang.String cannot be converted to JSONObject up vote 0 down vote favorite I'm having problem with string cannot share|improve this answer answered Nov 13 '12 at 22:50 code4j 1,53531432 add a comment| up vote 1 down vote I had a same problem. http://stackoverflow.com/questions/13368739/jsonexception-value-of-type-java-lang-string-cannot-be-converted-to-jsonobject Hope it helps you guys!
Anonymous Answer Email {} Share setAttribute(PDO::ATTR_ERRMODE, PDO::ERRMODE_EXCEPTION); } catch(PDOException $e) { echo 'ERROR: ' .$e->getMessage(); } $sql=mysql_query("select * from Recipes"); while($row=mysql_fetch_assoc($sql)) Value Pre Of Type Java Lang String Cannot Be Converted To Jsonobject This is the most commonly used` JSONObject constructor. more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed Post Reply Bookmark Topic Watch Topic New Topic Similar Threads client server connectivity problem Sending a jason object via Post org.json.JSONException: No value for user Unable to fetch correct json string ## Org.json.jsonexception: Value Of Type Java.lang.string Cannot Be Converted To Jsonobject Would you like to answer one of these unanswered questions instead? this content And you shouldn't be doing a DB query if the code can't even connect to the DB. Type Java.lang.string Cannot Be Converted To Jsonobject Android Possible repercussions from assault between coworkers outside the office Is it possible to hand start modern planes? String Cannot Be Converted To Jsonarray Android Linked 173 How to convert String to JSONObject in Java Related 35JSONException: Value of type java.lang.String cannot be converted to JSONObject0Error of value share|improve this answer edited Aug 14 '13 at 7:52 answered Aug 12 '13 at 19:10 Codo 39.5k983129 Thank you for explaining. this contact form Vent kitchen hood vent to roof turbine vent? How do I make an alien technology feel alien? What is this line of counties voting for the Democratic party in the 2016 elections? How To Convert Java.lang.string To Jsonobject I bet the problem is lines 5 to 12 of your PHP code - you're returning something that is not JSON. Polyglot Anagrams Robbers' Thread My cat sat on my laptop, now the right side of my keyboard types the wrong characters How to prove that authentication system works, and that the What do I do? have a peek here Polyglot Anagrams Cops' Thread How good should one be to participate in PS? I've just forgot to change sample of code, but it isn't a solution of my problem. –Anton Kashpor Aug 13 '13 at 10:34 @whisperofblood: See my update. Value try {$conn = new PDO('mysql:host=localhost;dbname=beanbag', **********, *******);
$conn->setAttribute(PDO::ATTR_ERRMODE, PDO::ERRMODE_EXCEPTION);$results=$conn->query('SELECT * FROM Recipes'); while ($results = \$row->fetch(PDO::FETCH_ASSOC)){ | 2018-03-17 20:32:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3449415862560272, "perplexity": 4870.618641379308}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257645310.34/warc/CC-MAIN-20180317194447-20180317214447-00459.warc.gz"} |