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#multivariate-normal-distribution

One definition is that a random vector is said to be *k*-variate normally distributed if every linear combination of its *k* components has a univariate normal distribution.

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a }}\mathbf {t} {\Big )}} In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. <span>One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly)

#multivariate-normal-distribution

The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value

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e definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. <span>The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. Contents [hide] 1 Notation and parametrization 2 Definition 3 Properties 3.1 Density function 3.1.1 Non-degenerate case 3.1.2 Degenerate case 3.2 Higher moments 3.3 Lik

#multivariate-normal-distribution

The mutual information of a distribution is a special case of the Kullback–Leibler divergence in which is the full multivariate distribution and is the product of the 1-dimensional marginal distributions

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al {CN}}_{0}\|{\mathcal {CN}}_{1})=\operatorname {tr} \left({\boldsymbol {\Sigma }}_{1}^{-1}{\boldsymbol {\Sigma }}_{0}\right)-k+\ln {|{\boldsymbol {\Sigma }}_{1}| \over |{\boldsymbol {\Sigma }}_{0}|}.} Mutual information[edit source] <span>The mutual information of a distribution is a special case of the Kullback–Leibler divergence in which P {\displaystyle P} is the full multivariate distribution and Q {\displaystyle Q} is the product of the 1-dimensional marginal distributions. In the notation of the Kullback–Leibler divergence section of this article, Σ 1

#multivariate-normal-distribution

In the bivariate case the expression for the mutual information is:

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ldsymbol {\rho }}_{0}} is the correlation matrix constructed from Σ 0 {\displaystyle {\boldsymbol {\Sigma }}_{0}} . <span>In the bivariate case the expression for the mutual information is: I ( x ; y ) = − 1 2 ln ( 1 − ρ 2 ) . {\displaystyle I(x;y)=-{1 \over 2}\ln(1-\rho ^{2}).} Cumulative distribution function[edit source] The notion of cumulative distribution function (cdf) in dimension 1 can be extended in two ways to the multidimensional case, based

#multivariate-normal-distribution

Conditional distributions

If *N*-dimensional **x** is partitioned as follows

and accordingly **μ** and **Σ** are partitioned as follows

then the distribution of **x**_{1} conditional on **x**_{2} = **a** is multivariate normal (**x**_{1} | **x**_{2} = **a**) ~ *N*( **μ** , **Σ** ) where

and covariance matrix

This matrix is the Schur complement of **Σ**_{22} in **Σ**. This means that to calculate the conditional covariance matrix, one inverts the overall covariance matrix, drops the rows and columns corresponding to the variables being conditioned upon, and then inverts back to get the conditional covariance matrix. Here is the generalized inverse of .

Note that knowing that **x**_{2} = **a** alters the variance, though the new variance does not depend on the specific value of **a**; perhaps more surprisingly, the mean is shifted by ; compare this with the situation of not knowing the value of **a**, in which case **x**_{1} would have distribution .

An interesting fact derived in order to prove this result, is that the random vectors and are independent.

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y two or more of its components that are pairwise independent are independent. But, as pointed out just above, it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent. <span>Conditional distributions[edit source] If N-dimensional x is partitioned as follows x = [ x 1 x 2 ] with sizes [ q × 1 ( N − q ) × 1 ] {\displaystyle \mathbf {x} ={\begin{bmatrix}\mathbf {x} _{1}\\\mathbf {x} _{2}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}} and accordingly μ and Σ are partitioned as follows μ = [ μ 1 μ 2 ] with sizes [ q × 1 ( N − q ) × 1 ] {\displaystyle {\boldsymbol {\mu }}={\begin{bmatrix}{\boldsymbol {\mu }}_{1}\\{\boldsymbol {\mu }}_{2}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}} Σ = [ Σ 11 Σ 12 Σ 21 Σ 22 ] with sizes [ q × q q × ( N − q ) ( N − q ) × q ( N − q ) × ( N − q ) ] {\displaystyle {\boldsymbol {\Sigma }}={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{12}\\{\boldsymbol {\Sigma }}_{21}&{\boldsymbol {\Sigma }}_{22}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times q&q\times (N-q)\\(N-q)\times q&(N-q)\times (N-q)\end{bmatrix}}} then the distribution of x 1 conditional on x 2 = a is multivariate normal (x 1 | x 2 = a) ~ N(μ, Σ) where μ ¯ = μ 1 + Σ 12 Σ 22 − 1 ( a − μ 2 ) {\displaystyle {\bar {\boldsymbol {\mu }}}={\boldsymbol {\mu }}_{1}+{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\left(\mathbf {a} -{\boldsymbol {\mu }}_{2}\right)} and covariance matrix Σ ¯ = Σ 11 − Σ 12 Σ 22 − 1 Σ 21 . {\displaystyle {\overline {\boldsymbol {\Sigma }}}={\boldsymbol {\Sigma }}_{11}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}{\boldsymbol {\Sigma }}_{21}.} [13] This matrix is the Schur complement of Σ 22 in Σ. This means that to calculate the conditional covariance matrix, one inverts the overall covariance matrix, drops the rows and columns corresponding to the variables being conditioned upon, and then inverts back to get the conditional covariance matrix. Here Σ 22 − 1 {\displaystyle {\boldsymbol {\Sigma }}_{22}^{-1}} is the generalized inverse of Σ 22 {\displaystyle {\boldsymbol {\Sigma }}_{22}} . Note that knowing that x 2 = a alters the variance, though the new variance does not depend on the specific value of a; perhaps more surprisingly, the mean is shifted by Σ 12 Σ 22 − 1 ( a − μ 2 ) {\displaystyle {\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\left(\mathbf {a} -{\boldsymbol {\mu }}_{2}\right)} ; compare this with the situation of not knowing the value of a, in which case x 1 would have distribution N q ( μ 1 , Σ 11 ) {\displaystyle {\mathcal {N}}_{q}\left({\boldsymbol {\mu }}_{1},{\boldsymbol {\Sigma }}_{11}\right)} . An interesting fact derived in order to prove this result, is that the random vectors x 2 {\displaystyle \mathbf {x} _{2}} and y 1 = x 1 − Σ 12 Σ 22 − 1 x 2 {\displaystyle \mathbf {y} _{1}=\mathbf {x} _{1}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\mathbf {x} _{2}} are independent. The matrix Σ 12 Σ 22 −1 is known as the matrix of regression coefficients. Bivariate case[edit source] In the bivariate case where x is partitioned into X 1 and X 2 , the conditional distribution of X 1 given X 2 is [14]

#multivariate-normal-distribution

To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions and linear algebra.^{[16]}

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) {\displaystyle \operatorname {E} (X_{1}\mid X_{2}##BAD TAG##\rho E(X_{2}\mid X_{2}##BAD TAG##} and then using the properties of the expectation of a truncated normal distribution. Marginal distributions[edit source] <span>To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions and linear algebra. [16] Example Let X = [X 1 , X 2 , X 3 ] be multivariate normal random variables with mean vector μ = [μ 1 , μ 2 , μ 3 ] and covariance matrix Σ (standard parametrization for multivariate

#multivariate-normal-distribution

If **Y** = **c** + **BX** is an affine transformation of where **c** is an vector of constants and **B** is a constant matrix, then **Y** has a multivariate normal distribution with expected value **c** + **Bμ** and variance **BΣB**^{T} .

*Corollaries: sums of Gaussian are Gaussian, marginals of Gaussian are Gaussian.*

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{\displaystyle {\boldsymbol {\Sigma }}'={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{13}\\{\boldsymbol {\Sigma }}_{31}&{\boldsymbol {\Sigma }}_{33}\end{bmatrix}}} . Affine transformation[edit source] <span>If Y = c + BX is an affine transformation of X ∼ N ( μ , Σ ) , {\displaystyle \mathbf {X} \ \sim {\mathcal {N}}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}),} where c is an M × 1 {\displaystyle M\times 1} vector of constants and B is a constant M × N {\displaystyle M\times N} matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB T i.e., Y ∼ N ( c + B μ , B Σ B T ) {\displaystyle \mathbf {Y} \sim {\mathcal {N}}\left(\mathbf {c} +\mathbf {B} {\boldsymbol {\mu }},\mathbf {B} {\boldsymbol {\Sigma }}\mathbf {B} ^{\rm {T}}\right)} . In particular, any subset of the X i has a marginal distribution that is also multivariate normal. To see this, consider the following example: to extract the subset (X 1 , X 2 , X 4 )

#multivariate-normal-distribution

The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean.

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implies that the variance of the dot product must be positive. An affine transformation of X such as 2X is not the same as the sum of two independent realisations of X. Geometric interpretation[edit source] See also: Confidence region <span>The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean. [17] Hence the multivariate normal distribution is an example of the class of elliptical distributions. The directions of the principal axes of the ellipsoids are given by the eigenvec

#multivariate-normal-distribution

The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix **Σ**. The squared relative lengths of the principal axes are given by the corresponding eigenvalues.

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urs of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean. [17] Hence the multivariate normal distribution is an example of the class of elliptical distributions. <span>The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix Σ. The squared relative lengths of the principal axes are given by the corresponding eigenvalues. If Σ = UΛU T = UΛ 1/2 (UΛ 1/2 ) T is an eigendecomposition where the columns of U are unit eigenvectors and Λ is a diagonal matrix of the eigenvalues, then we have

#multivariate-normal-distribution

The distribution *N*(**μ**, **Σ**) is in effect *N*(0, **I**) scaled by **Λ**^{1/2}, rotated by **U** and translated by **μ**.

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{\mu }}+\mathbf {U} {\mathcal {N}}(0,{\boldsymbol {\Lambda }}).} Moreover, U can be chosen to be a rotation matrix, as inverting an axis does not have any effect on N(0, Λ), but inverting a column changes the sign of U's determinant. <span>The distribution N(μ, Σ) is in effect N(0, I) scaled by Λ 1/2 , rotated by U and translated by μ. Conversely, any choice of μ, full rank matrix U, and positive diagonal entries Λ i yields a non-singular multivariate normal distribution. If any Λ i is zero and U is square, the re

#fourier-analysis

n probability theory and statistics, the **characteristic function** of any real-valued random variable completely defines its probability distribution.

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aracteristic function of a uniform U(–1,1) random variable. This function is real-valued because it corresponds to a random variable that is symmetric around the origin; however characteristic functions may generally be complex-valued. I<span>n probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides the basis of

#fourier-analysis

If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function.

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c around the origin; however characteristic functions may generally be complex-valued. In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. <span>If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There ar

#multivariate-normal-distribution

In the bivariate case where **x** is partitioned into *X*_{1} and *X*_{2}, the conditional distribution of *X*_{1} given *X*_{2} is

where is the correlation coefficient between *X*_{1} and *X*_{2}.

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mathbf {y} _{1}=\mathbf {x} _{1}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\mathbf {x} _{2}} are independent. The matrix Σ 12 Σ 22 −1 is known as the matrix of regression coefficients. Bivariate case[edit source] <span>In the bivariate case where x is partitioned into X 1 and X 2 , the conditional distribution of X 1 given X 2 is [14] X 1 ∣ X 2 = x 2 ∼ N ( μ 1 + σ 1 σ 2 ρ ( x 2 − μ 2 ) , ( 1 − ρ 2 ) σ 1 2 ) . {\displaystyle X_{1}\mid X_{2}=x_{2}\ \sim \ {\mathcal {N}}\left(\mu _{1}+{\frac {\sigma _{1}}{\sigma _{2}}}\rho (x_{2}-\mu _{2}),\,(1-\rho ^{2})\sigma _{1}^{2}\right).} where ρ {\displaystyle \rho } is the correlation coefficient between X 1 and X 2 . Bivariate conditional expectation[edit source] In the general case[edit source] (

Tags

#multivariate-normal-distribution

Question

In the bivariate case, the conditional mean of *X*_{1} given *X*_{2} is **[...]**

Answer

\( \mu_1 + \frac{\sigma_1}{\sigma_2} \rho (x_2 - \mu_2) \)

*where is the correlation coefficient between X*_{1} and X_{2}.

Apparently both the correlation and variance should play a part!

Apparently both the correlation and variance should play a part!

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In the bivariate case where x is partitioned into X 1 and X 2 , the conditional distribution of X 1 given X 2 is where is the correlation coefficient between X 1 and X 2 .

mathbf {y} _{1}=\mathbf {x} _{1}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\mathbf {x} _{2}} are independent. The matrix Σ 12 Σ 22 −1 is known as the matrix of regression coefficients. Bivariate case[edit source] <span>In the bivariate case where x is partitioned into X 1 and X 2 , the conditional distribution of X 1 given X 2 is [14] X 1 ∣ X 2 = x 2 ∼ N ( μ 1 + σ 1 σ 2 ρ ( x 2 − μ 2 ) , ( 1 − ρ 2 ) σ 1 2 ) . {\displaystyle X_{1}\mid X_{2}=x_{2}\ \sim \ {\mathcal {N}}\left(\mu _{1}+{\frac {\sigma _{1}}{\sigma _{2}}}\rho (x_{2}-\mu _{2}),\,(1-\rho ^{2})\sigma _{1}^{2}\right).} where ρ {\displaystyle \rho } is the correlation coefficient between X 1 and X 2 . Bivariate conditional expectation[edit source] In the general case[edit source] (

Tags

#fourier-analysis

Question

If a random variable admits a probability density function, then the [...] is the Fourier transform of the probability density function.

Answer

characteristic function

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If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function.

c around the origin; however characteristic functions may generally be complex-valued. In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. <span>If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There ar

Tags

#fourier-analysis

Question

Answer

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n probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution.

aracteristic function of a uniform U(–1,1) random variable. This function is real-valued because it corresponds to a random variable that is symmetric around the origin; however characteristic functions may generally be complex-valued. I<span>n probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides the basis of

Tags

#multivariate-normal-distribution

Question

The distribution *N*(**μ**, **Σ**) is in effect *N*(0, **I**) scaled by **[...]** , rotated by **[...]** and translated by **[...]** .

Answer

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The distribution N(μ, Σ) is in effect N(0, I) scaled by Λ 1/2 , rotated by U and translated by μ.

{\mu }}+\mathbf {U} {\mathcal {N}}(0,{\boldsymbol {\Lambda }}).} Moreover, U can be chosen to be a rotation matrix, as inverting an axis does not have any effect on N(0, Λ), but inverting a column changes the sign of U's determinant. <span>The distribution N(μ, Σ) is in effect N(0, I) scaled by Λ 1/2 , rotated by U and translated by μ. Conversely, any choice of μ, full rank matrix U, and positive diagonal entries Λ i yields a non-singular multivariate normal distribution. If any Λ i is zero and U is square, the re

Tags

#multivariate-normal-distribution

Question

The directions of the principal axes of the ellipsoids are given by **[...]** of the covariance matrix **Σ**

Answer

the eigenvectors

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The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix Σ. The squared relative lengths of the principal axes are given by the corresponding eigenvalues.

urs of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean. [17] Hence the multivariate normal distribution is an example of the class of elliptical distributions. <span>The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix Σ. The squared relative lengths of the principal axes are given by the corresponding eigenvalues. If Σ = UΛU T = UΛ 1/2 (UΛ 1/2 ) T is an eigendecomposition where the columns of U are unit eigenvectors and Λ is a diagonal matrix of the eigenvalues, then we have

Tags

#multivariate-normal-distribution

Question

[...] of the principal axes are given by the corresponding eigenvalues.

Answer

The squared relative lengths

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The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix Σ. The squared relative lengths of the principal axes are given by the corresponding eigenvalues.

urs of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean. [17] Hence the multivariate normal distribution is an example of the class of elliptical distributions. <span>The directions of the principal axes of the ellipsoids are given by the eigenvectors of the covariance matrix Σ. The squared relative lengths of the principal axes are given by the corresponding eigenvalues. If Σ = UΛU T = UΛ 1/2 (UΛ 1/2 ) T is an eigendecomposition where the columns of U are unit eigenvectors and Λ is a diagonal matrix of the eigenvalues, then we have

Tags

#multivariate-normal-distribution

Question

The equidensity contours of a non-singular multivariate normal distribution are [...] centered at the mean.

Answer

ellipsoids

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The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean.

implies that the variance of the dot product must be positive. An affine transformation of X such as 2X is not the same as the sum of two independent realisations of X. Geometric interpretation[edit source] See also: Confidence region <span>The equidensity contours of a non-singular multivariate normal distribution are ellipsoids (i.e. linear transformations of hyperspheres) centered at the mean. [17] Hence the multivariate normal distribution is an example of the class of elliptical distributions. The directions of the principal axes of the ellipsoids are given by the eigenvec

Tags

#multivariate-normal-distribution

Question

If **Y** = **c** + **BX** is an affine transformation,

then**Y** has a multivariate normal distribution with expected value **[...]**

then

Answer

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If Y = c + BX is an affine transformation of where c is an vector of constants and B is a constant matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB T . Corollaries: sums of Gaussian are Gaussian, marginals of Gaussian are Gaussian.

{\displaystyle {\boldsymbol {\Sigma }}'={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{13}\\{\boldsymbol {\Sigma }}_{31}&{\boldsymbol {\Sigma }}_{33}\end{bmatrix}}} . Affine transformation[edit source] <span>If Y = c + BX is an affine transformation of X ∼ N ( μ , Σ ) , {\displaystyle \mathbf {X} \ \sim {\mathcal {N}}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}),} where c is an M × 1 {\displaystyle M\times 1} vector of constants and B is a constant M × N {\displaystyle M\times N} matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB T i.e., Y ∼ N ( c + B μ , B Σ B T ) {\displaystyle \mathbf {Y} \sim {\mathcal {N}}\left(\mathbf {c} +\mathbf {B} {\boldsymbol {\mu }},\mathbf {B} {\boldsymbol {\Sigma }}\mathbf {B} ^{\rm {T}}\right)} . In particular, any subset of the X i has a marginal distribution that is also multivariate normal. To see this, consider the following example: to extract the subset (X 1 , X 2 , X 4 )

Tags

#multivariate-normal-distribution

Question

If **Y** = **c** + **BX**, then **Y** has variance **[...]**

Answer

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><head> If Y = c + BX is an affine transformation of where c is an vector of constants and B is a constant matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB T . Corollaries: sums of Gaussian are Gaussian, marginals of Gaussian are Gaussian. <html>

{\displaystyle {\boldsymbol {\Sigma }}'={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{13}\\{\boldsymbol {\Sigma }}_{31}&{\boldsymbol {\Sigma }}_{33}\end{bmatrix}}} . Affine transformation[edit source] <span>If Y = c + BX is an affine transformation of X ∼ N ( μ , Σ ) , {\displaystyle \mathbf {X} \ \sim {\mathcal {N}}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}),} where c is an M × 1 {\displaystyle M\times 1} vector of constants and B is a constant M × N {\displaystyle M\times N} matrix, then Y has a multivariate normal distribution with expected value c + Bμ and variance BΣB T i.e., Y ∼ N ( c + B μ , B Σ B T ) {\displaystyle \mathbf {Y} \sim {\mathcal {N}}\left(\mathbf {c} +\mathbf {B} {\boldsymbol {\mu }},\mathbf {B} {\boldsymbol {\Sigma }}\mathbf {B} ^{\rm {T}}\right)} . In particular, any subset of the X i has a marginal distribution that is also multivariate normal. To see this, consider the following example: to extract the subset (X 1 , X 2 , X 4 )

Tags

#multivariate-normal-distribution

Question

To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to [...]

Answer

drop the irrelevant variables

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To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions an

) {\displaystyle \operatorname {E} (X_{1}\mid X_{2}##BAD TAG##\rho E(X_{2}\mid X_{2}##BAD TAG##} and then using the properties of the expectation of a truncated normal distribution. Marginal distributions[edit source] <span>To obtain the marginal distribution over a subset of multivariate normal random variables, one only needs to drop the irrelevant variables (the variables that one wants to marginalize out) from the mean vector and the covariance matrix. The proof for this follows from the definitions of multivariate normal distributions and linear algebra. [16] Example Let X = [X 1 , X 2 , X 3 ] be multivariate normal random variables with mean vector μ = [μ 1 , μ 2 , μ 3 ] and covariance matrix Σ (standard parametrization for multivariate

#multivariate-normal-distribution

the distribution of **x**_{1} conditional on **x**_{2} = **a** is multivariate normal (**x**_{1} | **x**_{2} = **a**) ~ *N*( **μ** , **Σ** ) where and covariance matrix

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Conditional distributions If N-dimensional x is partitioned as follows and accordingly μ and Σ are partitioned as follows then the distribution of x 1 conditional on x 2 = a is multivariate normal (x 1 | x 2 = a) ~ N( μ , Σ ) where and covariance matrix This matrix is the Schur complement of Σ 22 in Σ. This means that to calculate the conditional covariance matrix, one inverts the overall covariance matrix, drops t

y two or more of its components that are pairwise independent are independent. But, as pointed out just above, it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent. <span>Conditional distributions[edit source] If N-dimensional x is partitioned as follows x = [ x 1 x 2 ] with sizes [ q × 1 ( N − q ) × 1 ] {\displaystyle \mathbf {x} ={\begin{bmatrix}\mathbf {x} _{1}\\\mathbf {x} _{2}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}} and accordingly μ and Σ are partitioned as follows μ = [ μ 1 μ 2 ] with sizes [ q × 1 ( N − q ) × 1 ] {\displaystyle {\boldsymbol {\mu }}={\begin{bmatrix}{\boldsymbol {\mu }}_{1}\\{\boldsymbol {\mu }}_{2}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}} Σ = [ Σ 11 Σ 12 Σ 21 Σ 22 ] with sizes [ q × q q × ( N − q ) ( N − q ) × q ( N − q ) × ( N − q ) ] {\displaystyle {\boldsymbol {\Sigma }}={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{12}\\{\boldsymbol {\Sigma }}_{21}&{\boldsymbol {\Sigma }}_{22}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times q&q\times (N-q)\\(N-q)\times q&(N-q)\times (N-q)\end{bmatrix}}} then the distribution of x 1 conditional on x 2 = a is multivariate normal (x 1 | x 2 = a) ~ N(μ, Σ) where μ ¯ = μ 1 + Σ 12 Σ 22 − 1 ( a − μ 2 ) {\displaystyle {\bar {\boldsymbol {\mu }}}={\boldsymbol {\mu }}_{1}+{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\left(\mathbf {a} -{\boldsymbol {\mu }}_{2}\right)} and covariance matrix Σ ¯ = Σ 11 − Σ 12 Σ 22 − 1 Σ 21 . {\displaystyle {\overline {\boldsymbol {\Sigma }}}={\boldsymbol {\Sigma }}_{11}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}{\boldsymbol {\Sigma }}_{21}.} [13] This matrix is the Schur complement of Σ 22 in Σ. This means that to calculate the conditional covariance matrix, one inverts the overall covariance matrix, drops the rows and columns corresponding to the variables being conditioned upon, and then inverts back to get the conditional covariance matrix. Here Σ 22 − 1 {\displaystyle {\boldsymbol {\Sigma }}_{22}^{-1}} is the generalized inverse of Σ 22 {\displaystyle {\boldsymbol {\Sigma }}_{22}} . Note that knowing that x 2 = a alters the variance, though the new variance does not depend on the specific value of a; perhaps more surprisingly, the mean is shifted by Σ 12 Σ 22 − 1 ( a − μ 2 ) {\displaystyle {\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\left(\mathbf {a} -{\boldsymbol {\mu }}_{2}\right)} ; compare this with the situation of not knowing the value of a, in which case x 1 would have distribution N q ( μ 1 , Σ 11 ) {\displaystyle {\mathcal {N}}_{q}\left({\boldsymbol {\mu }}_{1},{\boldsymbol {\Sigma }}_{11}\right)} . An interesting fact derived in order to prove this result, is that the random vectors x 2 {\displaystyle \mathbf {x} _{2}} and y 1 = x 1 − Σ 12 Σ 22 − 1 x 2 {\displaystyle \mathbf {y} _{1}=\mathbf {x} _{1}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\mathbf {x} _{2}} are independent. The matrix Σ 12 Σ 22 −1 is known as the matrix of regression coefficients. Bivariate case[edit source] In the bivariate case where x is partitioned into X 1 and X 2 , the conditional distribution of X 1 given X 2 is [14]

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#multivariate-normal-distribution

Question

the distribution of **x**_{1} conditional on **x**_{2} = **a** is multivariate normal (**x**_{1} | **x**_{2} = **a**) ~ *N*( **μ** , **Σ** ) where **μ** **[...]** and covariance matrix **Σ** **[...]**

Answer

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the distribution of x 1 conditional on x 2 = a is multivariate normal (x 1 | x 2 = a) ~ N( μ , Σ ) where and covariance matrix

y two or more of its components that are pairwise independent are independent. But, as pointed out just above, it is not true that two random variables that are (separately, marginally) normally distributed and uncorrelated are independent. <span>Conditional distributions[edit source] If N-dimensional x is partitioned as follows x = [ x 1 x 2 ] with sizes [ q × 1 ( N − q ) × 1 ] {\displaystyle \mathbf {x} ={\begin{bmatrix}\mathbf {x} _{1}\\\mathbf {x} _{2}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}} and accordingly μ and Σ are partitioned as follows μ = [ μ 1 μ 2 ] with sizes [ q × 1 ( N − q ) × 1 ] {\displaystyle {\boldsymbol {\mu }}={\begin{bmatrix}{\boldsymbol {\mu }}_{1}\\{\boldsymbol {\mu }}_{2}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times 1\\(N-q)\times 1\end{bmatrix}}} Σ = [ Σ 11 Σ 12 Σ 21 Σ 22 ] with sizes [ q × q q × ( N − q ) ( N − q ) × q ( N − q ) × ( N − q ) ] {\displaystyle {\boldsymbol {\Sigma }}={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{12}\\{\boldsymbol {\Sigma }}_{21}&{\boldsymbol {\Sigma }}_{22}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}q\times q&q\times (N-q)\\(N-q)\times q&(N-q)\times (N-q)\end{bmatrix}}} then the distribution of x 1 conditional on x 2 = a is multivariate normal (x 1 | x 2 = a) ~ N(μ, Σ) where μ ¯ = μ 1 + Σ 12 Σ 22 − 1 ( a − μ 2 ) {\displaystyle {\bar {\boldsymbol {\mu }}}={\boldsymbol {\mu }}_{1}+{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\left(\mathbf {a} -{\boldsymbol {\mu }}_{2}\right)} and covariance matrix Σ ¯ = Σ 11 − Σ 12 Σ 22 − 1 Σ 21 . {\displaystyle {\overline {\boldsymbol {\Sigma }}}={\boldsymbol {\Sigma }}_{11}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}{\boldsymbol {\Sigma }}_{21}.} [13] This matrix is the Schur complement of Σ 22 in Σ. This means that to calculate the conditional covariance matrix, one inverts the overall covariance matrix, drops the rows and columns corresponding to the variables being conditioned upon, and then inverts back to get the conditional covariance matrix. Here Σ 22 − 1 {\displaystyle {\boldsymbol {\Sigma }}_{22}^{-1}} is the generalized inverse of Σ 22 {\displaystyle {\boldsymbol {\Sigma }}_{22}} . Note that knowing that x 2 = a alters the variance, though the new variance does not depend on the specific value of a; perhaps more surprisingly, the mean is shifted by Σ 12 Σ 22 − 1 ( a − μ 2 ) {\displaystyle {\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\left(\mathbf {a} -{\boldsymbol {\mu }}_{2}\right)} ; compare this with the situation of not knowing the value of a, in which case x 1 would have distribution N q ( μ 1 , Σ 11 ) {\displaystyle {\mathcal {N}}_{q}\left({\boldsymbol {\mu }}_{1},{\boldsymbol {\Sigma }}_{11}\right)} . An interesting fact derived in order to prove this result, is that the random vectors x 2 {\displaystyle \mathbf {x} _{2}} and y 1 = x 1 − Σ 12 Σ 22 − 1 x 2 {\displaystyle \mathbf {y} _{1}=\mathbf {x} _{1}-{\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\mathbf {x} _{2}} are independent. The matrix Σ 12 Σ 22 −1 is known as the matrix of regression coefficients. Bivariate case[edit source] In the bivariate case where x is partitioned into X 1 and X 2 , the conditional distribution of X 1 given X 2 is [14]

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#multivariate-normal-distribution

Question

In the bivariate normal case the expression for the mutual information is [...]

Answer

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In the bivariate case the expression for the mutual information is:

ldsymbol {\rho }}_{0}} is the correlation matrix constructed from Σ 0 {\displaystyle {\boldsymbol {\Sigma }}_{0}} . <span>In the bivariate case the expression for the mutual information is: I ( x ; y ) = − 1 2 ln ( 1 − ρ 2 ) . {\displaystyle I(x;y)=-{1 \over 2}\ln(1-\rho ^{2}).} Cumulative distribution function[edit source] The notion of cumulative distribution function (cdf) in dimension 1 can be extended in two ways to the multidimensional case, based

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#multivariate-normal-distribution

Question

The mutual information of a distribution is a special case of the Kullback–Leibler divergence in which is **[...]** and is **[...]**

Answer

the full multivariate distribution, the product of the 1-dimensional marginal distributions

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The mutual information of a distribution is a special case of the Kullback–Leibler divergence in which is the full multivariate distribution and is the product of the 1-dimensional marginal distributions

al {CN}}_{0}\|{\mathcal {CN}}_{1})=\operatorname {tr} \left({\boldsymbol {\Sigma }}_{1}^{-1}{\boldsymbol {\Sigma }}_{0}\right)-k+\ln {|{\boldsymbol {\Sigma }}_{1}| \over |{\boldsymbol {\Sigma }}_{0}|}.} Mutual information[edit source] <span>The mutual information of a distribution is a special case of the Kullback–Leibler divergence in which P {\displaystyle P} is the full multivariate distribution and Q {\displaystyle Q} is the product of the 1-dimensional marginal distributions. In the notation of the Kullback–Leibler divergence section of this article, Σ 1

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#multivariate-normal-distribution

Question

The multivariate normal distribution is often used to describe correlated real-valued random variables each of which [...]

Answer

clusters around a mean value

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The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value

e definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. <span>The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. Contents [hide] 1 Notation and parametrization 2 Definition 3 Properties 3.1 Density function 3.1.1 Non-degenerate case 3.1.2 Degenerate case 3.2 Higher moments 3.3 Lik

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#multivariate-normal-distribution

Question

a random vector is said to be *k*-variate normally distributed if [...] has a univariate normal distribution.

Answer

every linear combination of its *k* components

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One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution.

a }}\mathbf {t} {\Big )}} In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. <span>One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly)

#singular-value-decomposition

In linear algebra, the **singular-value decomposition** (**SVD**) generalises the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any matrix via an extension of the polar decomposition.

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nto three simple transformations: an initial rotation V ∗ , a scaling Σ along the coordinate axes, and a final rotation U. The lengths σ 1 and σ 2 of the semi-axes of the ellipse are the singular values of M, namely Σ 1,1 and Σ 2,2 . <span>In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. Formally, the singular-value decomposition of an m × n {\d

#singular-value-decomposition

Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is a rectangular diagonal matrix with non-negative real numbers on the diagonal, and is an real or complex unitary matrix.

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ositive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. <span>Formally, the singular-value decomposition of an m × n {\displaystyle m\times n} real or complex matrix M {\displaystyle \mathbf {M} } is a factorization of the form U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U {\displaystyle \mathbf {U} } is an m × m {\displaystyle m\times m} real or complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is a m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V {\displaystyle \mathbf {V} } is an n × n {\displaystyle n\times n} real or complex unitary matrix. The diagonal entries σ i {\displaystyle \sigma _{i}} of

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#singular-value-decomposition

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Answer

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In linear algebra, the singular-value decomposition (SVD) generalises the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any matrix via an extension of the polar deco

nto three simple transformations: an initial rotation V ∗ , a scaling Σ along the coordinate axes, and a final rotation U. The lengths σ 1 and σ 2 of the semi-axes of the ellipse are the singular values of M, namely Σ 1,1 and Σ 2,2 . <span>In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. Formally, the singular-value decomposition of an m × n {\d

#singular-value-decomposition

Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form

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Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is a rectangular diagonal matrix with non-negative real numbers on the diagonal, and is an real or complex unitary matrix.

ositive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. <span>Formally, the singular-value decomposition of an m × n {\displaystyle m\times n} real or complex matrix M {\displaystyle \mathbf {M} } is a factorization of the form U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U {\displaystyle \mathbf {U} } is an m × m {\displaystyle m\times m} real or complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is a m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V {\displaystyle \mathbf {V} } is an n × n {\displaystyle n\times n} real or complex unitary matrix. The diagonal entries σ i {\displaystyle \sigma _{i}} of

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#singular-value-decomposition

Question

singular-value decomposition factorises an matrix M to the form **[...]**

Answer

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Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form

ositive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. <span>Formally, the singular-value decomposition of an m × n {\displaystyle m\times n} real or complex matrix M {\displaystyle \mathbf {M} } is a factorization of the form U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U {\displaystyle \mathbf {U} } is an m × m {\displaystyle m\times m} real or complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is a m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V {\displaystyle \mathbf {V} } is an n × n {\displaystyle n\times n} real or complex unitary matrix. The diagonal entries σ i {\displaystyle \sigma _{i}} of

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#singular-value-decomposition

Question

With a factorization of the form , , , represent **[...]**

Answer

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Formally, the singular-value decomposition of an real or complex matrix is a factorization of the form , where is an real or complex unitary matrix, is a rectangular diagonal matrix with non-negative real numbers on the diagonal, and is an real or complex unitary matrix.

ositive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. <span>Formally, the singular-value decomposition of an m × n {\displaystyle m\times n} real or complex matrix M {\displaystyle \mathbf {M} } is a factorization of the form U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U {\displaystyle \mathbf {U} } is an m × m {\displaystyle m\times m} real or complex unitary matrix, Σ {\displaystyle \mathbf {\Sigma } } is a m × n {\displaystyle m\times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V {\displaystyle \mathbf {V} } is an n × n {\displaystyle n\times n} real or complex unitary matrix. The diagonal entries σ i {\displaystyle \sigma _{i}} of

#kalman-filter

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into account; P k ∣ k − 1 {\displaystyle P_{k\mid k-1}} is the corresponding uncertainty. <span>Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, one of the primary developers of its theory. The Kalman filter has numerous applications in technology. A common application is for guidanc

#matrix-decomposition

In numerical analysis and linear algebra, **LU decomposition** (where 'LU' stands for 'lower upper', and also called **LU factorization**) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix.

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on - Wikipedia ocultar siempre | ocultar ahora LU decomposition From Wikipedia, the free encyclopedia Jump to: navigation, search <span>In numerical analysis and linear algebra, LU decomposition (where 'LU' stands for 'lower upper', and also called LU factorization) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. The LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of lin

#matrix-decomposition

Computers usually solve square systems of linear equations using the LU decomposition, and it is also a key step when inverting a matrix, or computing the determinant of a matrix.

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rization) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. The LU decomposition can be viewed as the matrix form of Gaussian elimination. <span>Computers usually solve square systems of linear equations using the LU decomposition, and it is also a key step when inverting a matrix, or computing the determinant of a matrix. The LU decomposition was introduced by mathematician Tadeusz Banachiewicz in 1938. [1] Contents [hide] 1 Definitions 1.1 LU factorization with Partial Pivoting 1.2 LU facto

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#matrix-decomposition

Question

Computers usually solve square systems of linear equations using [...]

Answer

the LU decomposition

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Computers usually solve square systems of linear equations using the LU decomposition, and it is also a key step when inverting a matrix, or computing the determinant of a matrix.

rization) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. The LU decomposition can be viewed as the matrix form of Gaussian elimination. <span>Computers usually solve square systems of linear equations using the LU decomposition, and it is also a key step when inverting a matrix, or computing the determinant of a matrix. The LU decomposition was introduced by mathematician Tadeusz Banachiewicz in 1938. [1] Contents [hide] 1 Definitions 1.1 LU factorization with Partial Pivoting 1.2 LU facto

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In numerical analysis and linear algebra, LU decomposition (where 'LU' stands for 'lower upper', and also called LU factorization) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix.

on - Wikipedia ocultar siempre | ocultar ahora LU decomposition From Wikipedia, the free encyclopedia Jump to: navigation, search <span>In numerical analysis and linear algebra, LU decomposition (where 'LU' stands for 'lower upper', and also called LU factorization) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. The LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of lin

#calculus

**Euler's formula**, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x

where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians.

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s formula half-lives exponential growth and decay Defining e proof that e is irrational representations of e Lindemann–Weierstrass theorem People John Napier Leonhard Euler Related topics Schanuel's conjecture v t e <span>Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more

#singular-value-decomposition

The geometric content of the SVD theorem can thus be summarized as follows: for every linear map

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m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map

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#singular-value-decomposition

Question

geometrically SVD finds **[...]** for every linear map *T* : *K*^{n} → *K*^{m}

Answer

orthonormal bases of K^{n} and K^{m}

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SVD as change of coordinates The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases,

m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map

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#singular-value-decomposition

Question

Geometrically SVD finds orthonormal bases of K^{n} and K^{m }for every linear map *T* : *K*^{n} → *K*^{m} such that T maps the i-th basis vector of K^{n} to a non-negative multiple of the i-th basis vector of K^{m} , and sends the left-over basis vectors to [...].

Answer

zero

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ollows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to <span>zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. <span><body><html>

m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map

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#singular-value-decomposition

Question

With SVD geometrically every linear map *T* : *K*^{n} → *K*^{m} is represented by a diagonal matrix with [...] entries.

Answer

non-negative real diagonal

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h that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with <span>non-negative real diagonal entries. <span><body><html>

m , n ) , {\displaystyle T(\mathbf {V} _{i})=\sigma _{i}\mathbf {U} _{i},\qquad i=1,\ldots ,\min(m,n),} where σ i is the i-th diagonal entry of Σ, and T(V i ) = 0 for i > min(m,n). <span>The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : K n → K m one can find orthonormal bases of K n and K m such that T maps the i-th basis vector of K n to a non-negative multiple of the i-th basis vector of K m , and sends the left-over basis vectors to zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. To get a more visual flavour of singular values and SVD factorization — at least when working on real vector spaces — consider the sphere S of radius one in R n . The linear map T map

#matrix

In linear algebra, a square matrix *A* is called **diagonalizable** if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix *P* such that *P*^{−1}*AP* is a diagonal matrix.

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Diagonalizable matrix From Wikipedia, the free encyclopedia Jump to: navigation, search This article is about matrix diagonalisation in linear algebra. For other uses, see Diagonalisation. <span>In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1 AP is a diagonal matrix. If V is a finite-dimensional vector space, then a linear map T : V → V is called diagonalizable if there exists an ordered basis of V with respect to which T is represented by a diagona

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#matrix

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In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1 AP is a diagonal matrix.

Diagonalizable matrix From Wikipedia, the free encyclopedia Jump to: navigation, search This article is about matrix diagonalisation in linear algebra. For other uses, see Diagonalisation. <span>In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1 AP is a diagonal matrix. If V is a finite-dimensional vector space, then a linear map T : V → V is called diagonalizable if there exists an ordered basis of V with respect to which T is represented by a diagona

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#spanish

Question

use conditional simple to *[...]*:

*María me dijo que estaría en casa para las 11, pero no ha aparecido aún.*

Answer

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To refer to the future from a moment in the past: María me dijo que estaría en casa para las 11, pero no ha aparecido aún (María told me she’d be at home by 11, but she hasn’t turned up yet).

e can postpone our trip for some hours, but we would arrive quite late). To express uncertainty in the past: No sabía si estarías en la oficina, por eso no te llamé (I didn’t know whether you’d be at the office. That’s why I didn’t call you). <span>To refer to the future from a moment in the past: María me dijo que estaría en casa para las 11, pero no ha aparecido aún (María told me she’d be at home by 11, but she hasn’t turned up yet). <span><body><html>

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#variational-inference

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Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning.

f references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (September 2010) (Learn how and when to remove this template message) <span>Variational Bayesian methods are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables (usually termed "data") as well as unknown parameters and latent variables, with various

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#spanish

Question

- El Ministro de Economía [...] a su cargo. (Él no quiere seguir trabajando como Ministro)

Answer

renunció

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- El Ministro de Economía renunció a su cargo. (Él no quiere seguir trabajando como Ministro)

Cancel 0 comment(s) Show previous comments Please enter between 2 and 2000 characters. Characters remaining: 2000 Submit Cancel Answers Time: oldest to newest Time: newest to oldest Votes: highest to lowest [imagelink] Hola Svetlana,<span>- Ella rechazó su ayuda pues él no tenía buenas intenciones. (Ella no quiso aceptar la ayuda que él le ofrecía)- El Ministro de Economía renunció a su cargo. (Él no quiere seguir trabajando como Ministro)- Aunque él le explicó sus razones, ella le negó su ayuda. (Ella no quiso ayudarlo).Espero sea de ayuda! Please enter between 2 and 2000 characters. If you copy an answer from another italki page, please include the URL of the original page. Characters remaining: 1673 U

#distributions

The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the *latent* random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution").

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istribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables. <span>The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution"). Contents [hide] 1 Definition 2 Properties 3 Applications 3.1 Testing 3.2 Overdispersion modeling 3.3 Bayesian inference 3.4 Convolution 4 Computation 5 Examples 6 See als

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#distributions

Question

The compound distribution is the result of **[...]** the *latent* random variables representing the parameters of the parametrized distribution

Answer

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The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution").

istribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables. <span>The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution"). Contents [hide] 1 Definition 2 Properties 3 Applications 3.1 Testing 3.2 Overdispersion modeling 3.3 Bayesian inference 3.4 Convolution 4 Computation 5 Examples 6 See als

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#distributions

Question

The compound distribution is integrated over [...] that represents the parameters of the parametrized distribution

Answer

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The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution").

istribution) is the probability distribution that results from assuming that a random variable is distributed according to some parametrized distribution, with (some of) the parameters of that distribution themselves being random variables. <span>The compound distribution ("unconditional distribution") is the result of marginalizing (integrating) over the latent random variable(s) representing the parameter(s) of the parametrized distribution ("conditional distribution"). Contents [hide] 1 Definition 2 Properties 3 Applications 3.1 Testing 3.2 Overdispersion modeling 3.3 Bayesian inference 3.4 Convolution 4 Computation 5 Examples 6 See als

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#kalman-filter

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hat uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating <span>a joint probability distribution over the variables for each timeframe. <span><body><html>

into account; P k ∣ k − 1 {\displaystyle P_{k\mid k-1}} is the corresponding uncertainty. <span>Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, one of the primary developers of its theory. The Kalman filter has numerous applications in technology. A common application is for guidanc

#topology

a set is open if it doesn't contain any of its boundary points

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set is an abstract concept generalizing the idea of an open interval in the real line. The simplest example is in metric spaces, where open sets can be defined as those sets which contain a ball around each of their points (or, equivalently, <span>a set is open if it doesn't contain any of its boundary points); however, an open set, in general, can be very abstract: any collection of sets can be called open, as long as the union of an arbitrary number of open sets is open, the intersection o

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a set is open if it doesn't contain any of its boundary points

set is an abstract concept generalizing the idea of an open interval in the real line. The simplest example is in metric spaces, where open sets can be defined as those sets which contain a ball around each of their points (or, equivalently, <span>a set is open if it doesn't contain any of its boundary points); however, an open set, in general, can be very abstract: any collection of sets can be called open, as long as the union of an arbitrary number of open sets is open, the intersection o

Tags

#logic

Question

Instead of justification of ideas, early modern authors emphasise the role of **[...]** and [...]

Answer

novelty and individual discovery

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Instead, early modern authors emphasise the role of novelty and individual discovery

tually unthinkable before the wide availability of printed books) was well-established. Moreover, as indicated by the passage from Descartes above, the very term ‘logic’ came to be used for something other than what the scholastics had meant. <span>Instead, early modern authors emphasise the role of novelty and individual discovery, as exemplified by the influential textbook Port-Royal Logic (1662), essentially, the logical version of Cartesianism, based on Descartes’s conception of mental operations and the prima

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#stochastics

Question

With an nonhomogeneous Poisson process, the [...] of points of the process is no longer constant.

Answer

average density

*The density is determined by the parameter, obviously.*

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body> If the parameter constant of the Poisson process is replaced with some non-negative integrable function of , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. <body><html>

sses. [49] The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. [102] [103] <span>If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t {\displaystyle t} , the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. [104] Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randoml

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#matrix

Question

a square matrix *A* is called **diagonalizable** if there exists an invertible matrix *P* such that *[...]* is a diagonal matrix.

Answer

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In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1 AP is a diagonal matrix.

Diagonalizable matrix From Wikipedia, the free encyclopedia Jump to: navigation, search This article is about matrix diagonalisation in linear algebra. For other uses, see Diagonalisation. <span>In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1 AP is a diagonal matrix. If V is a finite-dimensional vector space, then a linear map T : V → V is called diagonalizable if there exists an ordered basis of V with respect to which T is represented by a diagona

#multivariate-normal-distribution

If the covariance matrix is not full rank, then the multivariate normal distribution is degenerate and does not have a density.

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operatorname {sgn}(\rho ){\frac {\sigma _{Y}}{\sigma _{X}}}(x-\mu _{X})+\mu _{Y}.} This is because this expression, with sgn(ρ) replaced by ρ, is the best linear unbiased prediction of Y given a value of X. [4] Degenerate case[edit] <span>If the covariance matrix Σ {\displaystyle {\boldsymbol {\Sigma }}} is not full rank, then the multivariate normal distribution is degenerate and does not have a density. More precisely, it does not have a density with respect to k-dimensional Lebesgue measure (which is the usual measure assumed in calculus-level probability courses). Only random vectors

Tags

#multivariate-normal-distribution

Question

If [...], then the multivariate normal distribution is degenerate and does not have a density.

Answer

the covariance matrix is not full rank

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If the covariance matrix is not full rank, then the multivariate normal distribution is degenerate and does not have a density.

operatorname {sgn}(\rho ){\frac {\sigma _{Y}}{\sigma _{X}}}(x-\mu _{X})+\mu _{Y}.} This is because this expression, with sgn(ρ) replaced by ρ, is the best linear unbiased prediction of Y given a value of X. [4] Degenerate case[edit] <span>If the covariance matrix Σ {\displaystyle {\boldsymbol {\Sigma }}} is not full rank, then the multivariate normal distribution is degenerate and does not have a density. More precisely, it does not have a density with respect to k-dimensional Lebesgue measure (which is the usual measure assumed in calculus-level probability courses). Only random vectors

#metric-space

In mathematics, the **Banach fixed-point theorem** (also known as the **contraction mapping theorem**) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.

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Banach fixed-point theorem - Wikipedia Banach fixed-point theorem From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. The theorem is named after Stefan Banach (1892–1945), and was first stated by him in 1922. [1] Contents [hide] 1 Statement 2 Proofs 2.1 Banach's original proof 2.2 Shorter

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#metric-space

Question

the **Banach fixed-point theorem **guarantees [...] of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.

Answer

the existence and uniqueness of fixed points

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d><head> In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. <html>

Banach fixed-point theorem - Wikipedia Banach fixed-point theorem From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. The theorem is named after Stefan Banach (1892–1945), and was first stated by him in 1922. [1] Contents [hide] 1 Statement 2 Proofs 2.1 Banach's original proof 2.2 Shorter

Tags

#metric-space

Question

Answer

the **contraction mapping theorem**

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In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive metho

Banach fixed-point theorem - Wikipedia Banach fixed-point theorem From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. The theorem is named after Stefan Banach (1892–1945), and was first stated by him in 1922. [1] Contents [hide] 1 Statement 2 Proofs 2.1 Banach's original proof 2.2 Shorter

#metric-space

* Definition.* Let (

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was first stated by him in 1922. [1] Contents [hide] 1 Statement 2 Proofs 2.1 Banach's original proof 2.2 Shorter proof 3 Applications 4 Converses 5 Generalizations 6 See also 7 Notes 8 References Statement[edit source] <span>Definition. Let (X, d) be a metric space. Then a map T : X → X is called a contraction mapping on X if there exists q ∈ [0, 1) such that d ( T ( x ) , T ( y ) ) ≤ q d ( x , y ) {\displaystyle d(T(x),T(y))\leq qd(x,y)} for all x, y in X. Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furth

#metric-space

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x ) , T ( y ) ) ≤ q d ( x , y ) {\displaystyle d(T(x),T(y))\leq qd(x,y)} for all x, y in X. <span>Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element x 0 in X and define a sequence {x n } by x n = T(x n−1 ), then x n → x*. Remark 1. The following inequalities are equivalent and describe the speed of convergence: d

Tags

#metric-space

Question

* Definition.* Let (

Answer

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Definition . Let (X, d) be a metric space. Then a map T : X → X is called a contraction mapping on X if there exists q ∈ [0, 1) such that for all x, y in X.

was first stated by him in 1922. [1] Contents [hide] 1 Statement 2 Proofs 2.1 Banach's original proof 2.2 Shorter proof 3 Applications 4 Converses 5 Generalizations 6 See also 7 Notes 8 References Statement[edit source] <span>Definition. Let (X, d) be a metric space. Then a map T : X → X is called a contraction mapping on X if there exists q ∈ [0, 1) such that d ( T ( x ) , T ( y ) ) ≤ q d ( x , y ) {\displaystyle d(T(x),T(y))\leq qd(x,y)} for all x, y in X. Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furth

Tags

#metric-space

Question

* Definition.* Let (

Answer

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Definition . Let (X, d) be a metric space. Then a map T : X → X is called a contraction mapping on X if there exists q ∈ [0, 1) such that for all x, y in X.

was first stated by him in 1922. [1] Contents [hide] 1 Statement 2 Proofs 2.1 Banach's original proof 2.2 Shorter proof 3 Applications 4 Converses 5 Generalizations 6 See also 7 Notes 8 References Statement[edit source] <span>Definition. Let (X, d) be a metric space. Then a map T : X → X is called a contraction mapping on X if there exists q ∈ [0, 1) such that d ( T ( x ) , T ( y ) ) ≤ q d ( x , y ) {\displaystyle d(T(x),T(y))\leq qd(x,y)} for all x, y in X. Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furth

Tags

#metric-space

Question

Let (*X*, *d*) be a non-empty complete metric space with a contraction mapping *T* : *X* → *X*. Then *T* admits *[...]*

Answer

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Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element x 0 in X and define a sequence {x n } by x n = T(x n−1 ), then x n → x* . <

x ) , T ( y ) ) ≤ q d ( x , y ) {\displaystyle d(T(x),T(y))\leq qd(x,y)} for all x, y in X. <span>Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element x 0 in X and define a sequence {x n } by x n = T(x n−1 ), then x n → x*. Remark 1. The following inequalities are equivalent and describe the speed of convergence: d

Tags

#metric-space

Question

When a complete metric space admits a contraction mapping *T* : *X* → *X*. The fixed point for the map can be found as follows: start with *[...]* and define a sequence {*x*_{n}} by *x*_{n} = *T*(*x*_{n−1}), then *x*_{n} → *x** .

Answer

an arbitrary element *x*_{0} in *X*

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Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with <span>an arbitrary element x 0 in X and define a sequence {x n } by x n = T(x n−1 ), then x n → x* . <span><body><html>

x ) , T ( y ) ) ≤ q d ( x , y ) {\displaystyle d(T(x),T(y))\leq qd(x,y)} for all x, y in X. <span>Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element x 0 in X and define a sequence {x n } by x n = T(x n−1 ), then x n → x*. Remark 1. The following inequalities are equivalent and describe the speed of convergence: d

Tags

#metric-space

Question

Using the **Banach Fixed Point Theorem** the fixed point *x** can be found by starting with an arbitrary element *x*_{0} in *X* and define a sequence {*x*_{n}} by [...], then *x*_{n} → *x** .

Answer

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empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element x 0 in X and define a sequence {x n } by <span>x n = T(x n−1 ), then x n → x* . <span><body><html>

x ) , T ( y ) ) ≤ q d ( x , y ) {\displaystyle d(T(x),T(y))\leq qd(x,y)} for all x, y in X. <span>Banach Fixed Point Theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T : X → X. Then T admits a unique fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element x 0 in X and define a sequence {x n } by x n = T(x n−1 ), then x n → x*. Remark 1. The following inequalities are equivalent and describe the speed of convergence: d

#kalman-filter

The underlying model of Kalman filter is similar to a hidden Markov model except that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions.

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stimate in the special case that all errors are Gaussian-distributed. Extensions and generalizations to the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. <span>The underlying model is similar to a hidden Markov model except that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Contents [hide] 1 History 2 Overview of the calculation 3 Example application 4 Technical description and context 5 Underlying dynamical system model 6 Details 6.1 Predict

Tags

#kalman-filter

Question

In Kalman filter [...] variables have Gaussian distributions.

Answer

all latent and observed

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The underlying model of Kalman filter is similar to a hidden Markov model except that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions.

stimate in the special case that all errors are Gaussian-distributed. Extensions and generalizations to the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. <span>The underlying model is similar to a hidden Markov model except that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Contents [hide] 1 History 2 Overview of the calculation 3 Example application 4 Technical description and context 5 Underlying dynamical system model 6 Details 6.1 Predict

#dot-product

In mathematics, the **dot product** or **scalar product**^{[note 1]} is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number.

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ia Jump to: navigation, search "Scalar product" redirects here. For the abstract scalar product, see Inner product space. For the product of a vector and a scalar, see Scalar multiplication. <span>In mathematics, the dot product or scalar product [note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used and often called inner product (or rarely projection product); see also inner product s

#dot-product

if **a** and **b** are orthogonal, then the angle between them is 90° and

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‖ cos ( θ ) , {\displaystyle \mathbf {a} \cdot \mathbf {b} =\|\mathbf {a} \|\ \|\mathbf {b} \|\cos(\theta ),} where θ is the angle between a and b. In particular, <span>if a and b are orthogonal, then the angle between them is 90° and a ⋅ b = 0. {\displaystyle \mathbf {a} \cdot \mathbf {b} =0.} At the other extreme, if they are codirectional, then the angle between them is 0° and a ⋅ b

#logic

The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

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f mathematics). Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. [1] <span>The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order

Tags

#lebesgue-integration

Question

Riemann integration does not interact well with [...]

Answer

taking limits of sequences of functions

*Think the Cantor devil's staircase. Integration is essentials the limit of sums.*

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However, Riemann integration does not interact well with taking limits of sequences of functions

e of easily calculated areas that converge to the integral of a given function. This definition is successful in the sense that it gives the expected answer for many already-solved problems, and gives useful results for many other problems. <span>However, Riemann integration does not interact well with taking limits of sequences of functions, making such limiting processes difficult to analyze. This is important, for instance, in the study of Fourier series, Fourier transforms, and other topics. The Lebesgue integral is bet

Tags

#lebesgue-integration

Question

Answer

a suitable class of *measurable* subsets

*The choice of unmeasurable sets can lead to the Banach-Tarski paradox*

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er set theory developments showed (see non-measurable set), it is actually impossible to assign a length to all subsets of ℝ in a way that preserves some natural additivity and translation invariance properties. This suggests that picking out <span>a suitable class of measurable subsets is an essential prerequisite. <span><body><html>

a useful abstraction of the notion of length of subsets of the real line—and, more generally, area and volume of subsets of Euclidean spaces. In particular, it provided a systematic answer to the question of which subsets of ℝ have a length. <span>As later set theory developments showed (see non-measurable set), it is actually impossible to assign a length to all subsets of ℝ in a way that preserves some natural additivity and translation invariance properties. This suggests that picking out a suitable class of measurable subsets is an essential prerequisite. The Riemann integral uses the notion of length explicitly. Indeed, the element of calculation for the Riemann integral is the rectangle [a, b] × [c, d], whose area is calculated to be

Tags

#dot-product

Question

Answer

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In mathematics, the dot product or scalar product [note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number.

ia Jump to: navigation, search "Scalar product" redirects here. For the abstract scalar product, see Inner product space. For the product of a vector and a scalar, see Scalar multiplication. <span>In mathematics, the dot product or scalar product [note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used and often called inner product (or rarely projection product); see also inner product s

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if a and b are orthogonal, then the angle between them is 90° and

‖ cos ( θ ) , {\displaystyle \mathbf {a} \cdot \mathbf {b} =\|\mathbf {a} \|\ \|\mathbf {b} \|\cos(\theta ),} where θ is the angle between a and b. In particular, <span>if a and b are orthogonal, then the angle between them is 90° and a ⋅ b = 0. {\displaystyle \mathbf {a} \cdot \mathbf {b} =0.} At the other extreme, if they are codirectional, then the angle between them is 0° and a ⋅ b

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ead><head> Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals, which are mappings from a set of functions to the real numbers. elementary calculus is about infinitesimally small changes in the values of functions without changes in the function itself, calculus of variations is about infinitesimally small

l Line integral Surface integral Volume integral Jacobian Hessian matrix Specialized[hide] Fractional Malliavin Stochastic Variations Glossary of calculus[show] Glossary of calculus v t e <span>Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals, which are mappings from a set of functions to the real numbers. [Note 1] Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–L

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The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

f mathematics). Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. [1] <span>The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order

Tags

#logic

Question

The unifying themes in mathematical logic include the study of the expressive power of formal systems and [...].

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The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

f mathematics). Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. [1] <span>The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order

Tags

#logic

Question

The unifying themes in [...] include the study of the expressive power of formal systems and the deductive power of formal proof systems.

Answer

mathematical logic

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The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

f mathematics). Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. [1] <span>The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order

Tags

#lebesgue-integration

Question

a measurable simple function is [...description...]

Answer

a finite linear combination of indicator functions

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A finite linear combination of indicator functions where the coefficients a k are real numbers and the sets S k are measurable, is called a measurable simple function.

μ = μ ( S ) . {\displaystyle \int 1_{S}\,\mathrm {d} \mu =\mu (S).} Notice that the result may be equal to +∞, unless μ is a finite measure. Simple functions: <span>A finite linear combination of indicator functions ∑ k a k 1 S k {\displaystyle \sum _{k}a_{k}1_{S_{k}}} where the coefficients a k are real numbers and the sets S k are measurable, is called a measurable simple function. We extend the integral by linearity to non-negative measurable simple functions. When the coefficients a k are non-negative, we set ∫ (

Tags

#lebesgue-integration

Question

To assign a value to [...], the only reasonable choice is to set:

Answer

the integral of the indicator function 1_{S} of a measurable set *S* consistent with the given measure μ

*Notice that the result may be equal to +∞ , unless μ is a finite measure.*

Trick: just read the expression from left to right

Trick: just read the expression from left to right

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To assign a value to the integral of the indicator function 1 S of a measurable set S consistent with the given measure μ, the only reasonable choice is to set: Notice that the result may be equal to +∞ , unless μ is a finite measure.

x ) {\displaystyle \int _{E}f\,\mathrm {d} \mu =\int _{E}f\left(x\right)\,\mathrm {d} \mu \left(x\right)} for measurable real-valued functions f defined on E in stages: Indicator functions: <span>To assign a value to the integral of the indicator function 1 S of a measurable set S consistent with the given measure μ, the only reasonable choice is to set: ∫ 1 S d μ = μ ( S ) . {\displaystyle \int 1_{S}\,\mathrm {d} \mu =\mu (S).} Notice that the result may be equal to +∞, unless μ is a finite measure. Simple functions: A finite linear combination of indicator functions ∑ k a

Tags

#lebesgue-integration

Question

To assign a value to the integral of the indicator function 1_{S} of a measurable set *S* consistent with the given measure μ, the only reasonable choice is to set [...]

Answer

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scheduled repetition interval | last repetition or drill |

To assign a value to the integral of the indicator function 1 S of a measurable set S consistent with the given measure μ, the only reasonable choice is to set: Notice that the result may be equal to +∞ , unless μ is a finite measure.

x ) {\displaystyle \int _{E}f\,\mathrm {d} \mu =\int _{E}f\left(x\right)\,\mathrm {d} \mu \left(x\right)} for measurable real-valued functions f defined on E in stages: Indicator functions: <span>To assign a value to the integral of the indicator function 1 S of a measurable set S consistent with the given measure μ, the only reasonable choice is to set: ∫ 1 S d μ = μ ( S ) . {\displaystyle \int 1_{S}\,\mathrm {d} \mu =\mu (S).} Notice that the result may be equal to +∞, unless μ is a finite measure. Simple functions: A finite linear combination of indicator functions ∑ k a

#topology

a **topological space** may be defined as

- a set of points, along with
- a set of neighbourhoods for each point, satisfying
- a set of axioms relating points and neighbourhoods.

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n>Topological space - Wikipedia Topological space From Wikipedia, the free encyclopedia Jump to: navigation, search In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, c

#topology

The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence.

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, search In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. <span>The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. [1] Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a centra

Tags

#topology

Question

Answer

topological space

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a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

n>Topological space - Wikipedia Topological space From Wikipedia, the free encyclopedia Jump to: navigation, search In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, c

#probability-theory

A random variable is defined as a function that maps outcomes to numerical quantities (labels)

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In the case of the coin, there are only two possible outcomes, namely heads or tails. Since one of these outcomes must occur, either the event that the coin lands heads or the event that the coin lands tails must have non-zero probability. <span>A random variable is defined as a function that maps outcomes to numerical quantities (labels), typically real numbers. In this sense, it is a procedure for assigning a numerical quantity to each physical outcome, and, contrary to its name, this procedure itself is neither random

Tags

#probability-theory

Question

A [...] is defined as a function that maps outcomes to numerical quantities (labels)

Answer

random variable

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A random variable is defined as a function that maps outcomes to numerical quantities (labels)

In the case of the coin, there are only two possible outcomes, namely heads or tails. Since one of these outcomes must occur, either the event that the coin lands heads or the event that the coin lands tails must have non-zero probability. <span>A random variable is defined as a function that maps outcomes to numerical quantities (labels), typically real numbers. In this sense, it is a procedure for assigning a numerical quantity to each physical outcome, and, contrary to its name, this procedure itself is neither random

Tags

#probability-theory

Question

A random variable is defined as a function that maps outcomes to [...]

Answer

numerical quantities (labels)

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A random variable is defined as a function that maps outcomes to numerical quantities (labels)

In the case of the coin, there are only two possible outcomes, namely heads or tails. Since one of these outcomes must occur, either the event that the coin lands heads or the event that the coin lands tails must have non-zero probability. <span>A random variable is defined as a function that maps outcomes to numerical quantities (labels), typically real numbers. In this sense, it is a procedure for assigning a numerical quantity to each physical outcome, and, contrary to its name, this procedure itself is neither random

Tags

#measure-theory

Question

[...] means that the support of the measure forms a compact set

Answer

the measure has compact support

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You often see written “the measure has compact support” to note that the support of the measure forms a compact (=closed and bounded) set

Tags

#topology

Question

Answer

length and angle

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A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured.

e state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space. <span>The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point o

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Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.

e state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space. <span>The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point o

#incremental-reading

The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading.

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ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

#incremental-reading

The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention.

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last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditio

ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

#incremental-reading

With incremental reading, you ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable to traditional book reading.

status | not read | reprioritisations | ||
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last reprioritisation on | reading queue position [%] | |||

started reading on | finished reading on |

The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading.

ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

Tags

#incremental-reading

Question

incremental reading helps balance [...] and **[...]**

Answer

speed and retention

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention.

ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

Tags

#incremental-reading

Question

With incremental reading, you ensure high-retention of [...]

Answer

the most important pieces of text

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

With incremental reading, you ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable to typical of traditional book reading.

ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

Tags

#incremental-reading

Question

With incremental reading, the majority of time should still be spent on reading at [...].

Answer

normal speed

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scheduled repetition interval | last repetition or drill |

With incremental reading, you ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable to traditional book reading.

ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

Tags

#incremental-reading

Question

With incremental reading, [...how much...] time will be spent reading at speeds comparable to traditional book reading.

Answer

a large proportion

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

With incremental reading, you ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable to traditional book reading.

ng and 10% of your time on adding most important findings to SuperMemo, your reading speed will actually decline only by some 10%, while the retention of the most important pieces will be as high as programmed in SuperMemo (up to 99%). <span>The concept of incremental reading introduced in SuperMemo 2000 provides you with a precise tool for finding the optimum balance between speed and retention. You will ensure high-retention of the most important pieces of text, while a large proportion of time will be spent reading at speeds comparable or higher than those typical of traditional book reading. It is worth noting that the learning speed limit in high-retention learning is imposed by your memory. If one-book-per-year sounds like a major disappointment, the roots of this lay

#calculus

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Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imagina

s formula half-lives exponential growth and decay Defining e proof that e is irrational representations of e Lindemann–Weierstrass theorem People John Napier Leonhard Euler Related topics Schanuel's conjecture v t e <span>Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more

#calculus

Euler's formula states that with the argument x given in radians.

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span> Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. <span><body><html>

s formula half-lives exponential growth and decay Defining e proof that e is irrational representations of e Lindemann–Weierstrass theorem People John Napier Leonhard Euler Related topics Schanuel's conjecture v t e <span>Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more

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#calculus

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Answer

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Euler's formula establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

s formula half-lives exponential growth and decay Defining e proof that e is irrational representations of e Lindemann–Weierstrass theorem People John Napier Leonhard Euler Related topics Schanuel's conjecture v t e <span>Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more

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#calculus

Question

Euler's formula states that [...], with the argument x given in radians

Answer

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Euler's formula states that with the argument x given in radians.

s formula half-lives exponential growth and decay Defining e proof that e is irrational representations of e Lindemann–Weierstrass theorem People John Napier Leonhard Euler Related topics Schanuel's conjecture v t e <span>Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more

#topology

The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined.

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set can be open (called the discrete topology), or no set can be open but the space itself and the empty set (the indiscrete topology). In practice, however, open sets are usually chosen to be similar to the open intervals of the real line. <span>The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets. Each choice of o

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#topology

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The notion of an [...] provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined.

Answer

open set

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The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined.

set can be open (called the discrete topology), or no set can be open but the space itself and the empty set (the indiscrete topology). In practice, however, open sets are usually chosen to be similar to the open intervals of the real line. <span>The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets. Each choice of o

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#topology

Question

The notion of an open set provides a fundamental way to speak of [...] in a topological space, without explicitly having a concept of distance defined.

Answer

nearness of points

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The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined.

set can be open (called the discrete topology), or no set can be open but the space itself and the empty set (the indiscrete topology). In practice, however, open sets are usually chosen to be similar to the open intervals of the real line. <span>The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets. Each choice of o

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#topology

Question

The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of [...] defined.

Answer

distance

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The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined.

set can be open (called the discrete topology), or no set can be open but the space itself and the empty set (the indiscrete topology). In practice, however, open sets are usually chosen to be similar to the open intervals of the real line. <span>The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets. Each choice of o

#topology

The definition of a topological space relies only upon set theory

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The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence.

, search In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. <span>The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. [1] Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a centra

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#topology

Question

The definition of a topological space relies only upon [...]

Answer

set theory

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The definition of a topological space relies only upon set theory

, search In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. <span>The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. [1] Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a centra

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#topology

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Answer

topological space

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The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence.

, search In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. <span>The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. [1] Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a centra

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#topology

Question

topological space allows for the definition of concepts such as **[...CCC...]** .

Answer

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The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence.

, search In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. <span>The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. [1] Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a centra

#topology

Each choice of open sets for a space is called a topology.

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pological space, without explicitly having a concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets. <span>Each choice of open sets for a space is called a topology. Although open sets and the topologies that they comprise are of central importance in point-set topology, they are also used as an organizational tool in other important branches of mat

#topology

In particular, sets of the form (-ε, ε) give us a lot of information about points close to *x* = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to *x*.

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ese points approximate x to a greater degree of accuracy compared to when ε = 1. The previous discussion shows, for the case x = 0, that one may approximate x to higher and higher degrees of accuracy by defining ε to be smaller and smaller. <span>In particular, sets of the form (-ε, ε) give us a lot of information about points close to x = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x. This innovative idea has far-reaching consequences; in particular, by defining different collections of sets containing 0 (distinct from the sets (-ε, ε)), one may find different result

#topology

In general, one refers to the family of sets containing 0, used to approximate 0, as a **neighborhood basis**; a member of this neighborhood basis is referred to as an **open set**.

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e find that in some sense, every real number is distance 0 away from 0. It may help in this case to think of the measure as being a binary condition, all things in R are equally close to 0, while any item that is not in R is not close to 0. <span>In general, one refers to the family of sets containing 0, used to approximate 0, as a neighborhood basis; a member of this neighborhood basis is referred to as an open set. In fact, one may generalize these notions to an arbitrary set (X); rather than just the real numbers. In this case, given a point (x) of that set, one may define a collection of sets &q

#topology

When difining nearness between points with open balls, the measure of distance becomes a binary condition

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ot;measuring distance", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of R. Thus, we find that in some sense, every real number is distance 0 away from 0. <span>It may help in this case to think of the measure as being a binary condition, all things in R are equally close to 0, while any item that is not in R is not close to 0. In general, one refers to the family of sets containing 0, used to approximate 0, as a neig

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#topology

Question

When difining nearness between points with open balls, the measure of distance becomes a [...]

Answer

binary condition

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When difining nearness between points with open balls, the measure of distance becomes a binary condition

ot;measuring distance", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of R. Thus, we find that in some sense, every real number is distance 0 away from 0. <span>It may help in this case to think of the measure as being a binary condition, all things in R are equally close to 0, while any item that is not in R is not close to 0. In general, one refers to the family of sets containing 0, used to approximate 0, as a neig

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#topology

Question

When difining [...] with open balls, the measure of distance becomes a binary condition

Answer

nearness between points

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When difining nearness between points with open balls, the measure of distance becomes a binary condition

ot;measuring distance", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of R. Thus, we find that in some sense, every real number is distance 0 away from 0. <span>It may help in this case to think of the measure as being a binary condition, all things in R are equally close to 0, while any item that is not in R is not close to 0. In general, one refers to the family of sets containing 0, used to approximate 0, as a neig

#topology

Sets that can be constructed as the intersection of countably many open sets are denoted **G**_{δ} sets.

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u } the open sets. Note that infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form (−1/n, 1/n), where n is a positive integer, is the set {0} which is not open in the real line. <span>Sets that can be constructed as the intersection of countably many open sets are denoted G δ sets. The topological definition of open sets generalizes the metric space definition: If one begins with a metric space and defines open sets as before, then the family of all open sets is

#topology

infinite intersections of open sets need not be open.

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τ {\displaystyle \tau } is in τ {\displaystyle \tau } ) We call the sets in τ {\displaystyle \tau } the open sets. Note that <span>infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form (−1/n, 1/n), where n is a positive integer, is the set {0} which is not open in the real line. Sets that can be constructed as

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#topology

Question

[...] of open sets need not be open.

Answer

infinite intersections

*The axiom of sigma-algebra uses infinite (countable) intersections.*

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infinite intersections of open sets need not be open.

τ {\displaystyle \tau } is in τ {\displaystyle \tau } ) We call the sets in τ {\displaystyle \tau } the open sets. Note that <span>infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form (−1/n, 1/n), where n is a positive integer, is the set {0} which is not open in the real line. Sets that can be constructed as

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Each choice of open sets for a space is called a topology.

pological space, without explicitly having a concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets. <span>Each choice of open sets for a space is called a topology. Although open sets and the topologies that they comprise are of central importance in point-set topology, they are also used as an organizational tool in other important branches of mat

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Each choice of open sets for a space is called a topology.

pological space, without explicitly having a concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets. <span>Each choice of open sets for a space is called a topology. Although open sets and the topologies that they comprise are of central importance in point-set topology, they are also used as an organizational tool in other important branches of mat

#topology

A complement of an open set (relative to the space that the topology is defined on) is called a closed set.

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l space. There are, however, topological spaces that are not metric spaces. Properties[edit] The union of any number of open sets, or infinitely many open sets, is open. [2] The intersection of a finite number of open sets is open. [2] <span>A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed. [3] Uses[edit] Open sets have a fundamental im

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A complement of an open set (relative to the space that the topology is defined on) is called a closed set.

l space. There are, however, topological spaces that are not metric spaces. Properties[edit] The union of any number of open sets, or infinitely many open sets, is open. [2] The intersection of a finite number of open sets is open. [2] <span>A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed. [3] Uses[edit] Open sets have a fundamental im

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A complement of an open set (relative to the space that the topology is defined on) is called a closed set.

l space. There are, however, topological spaces that are not metric spaces. Properties[edit] The union of any number of open sets, or infinitely many open sets, is open. [2] The intersection of a finite number of open sets is open. [2] <span>A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed. [3] Uses[edit] Open sets have a fundamental im

#topology

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ntained in U. Metric spaces[edit] A subset U of a metric space (M, d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x, y) < ε, y also belongs to U. Equivalently, <span>U is open if every point in U has a neighborhood contained in U. This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space. Topological spaces[edit] In general topological spaces, the open

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Generally, unless both the objective function and the feasible region are convex in a minimization problem, there may be several local minima

energy functional. A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an optimal solution. In mathematics, conventional optimization problems are usually stated in terms of minimization. <span>Generally, unless both the objective function and the feasible region are convex in a minimization problem, there may be several local minima. A local minimum x* is defined as a point for which there exists some δ > 0 such that for all x where ‖ x −

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#optimization

Question

most algorithms for solving nonconvex problems can't distinguish between [...]

Answer

local and global optima

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A large number of algorithms proposed for solving nonconvex problems—including the majority of commercially available solvers—are not capable of making a distinction between locally optimal solutions and globally optimal solutions

onvex problem, if there is a local minimum that is interior (not on the edge of the set of feasible points), it is also the global minimum, but a nonconvex problem may have more than one local minimum not all of which need be global minima. <span>A large number of algorithms proposed for solving nonconvex problems—including the majority of commercially available solvers—are not capable of making a distinction between locally optimal solutions and globally optimal solutions, and will treat the former as actual solutions to the original problem. Global optimization is the branch of applied mathematics and numerical analysis that is concerned with the develo

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#calculus

Question

Answer

Euler's formula

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Euler's formula establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

s formula half-lives exponential growth and decay Defining e proof that e is irrational representations of e Lindemann–Weierstrass theorem People John Napier Leonhard Euler Related topics Schanuel's conjecture v t e <span>Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x e i x = cos x + i sin x , {\displaystyle e^{ix}=\cos x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more

#differential-equations

ODEs that are linear differential equations have exact closed-form solutions that can be added and multiplied by coefficients.

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quation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable. [1] <span>ODEs that are linear differential equations have exact closed-form solutions that can be added and multiplied by coefficients. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, e

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#differential-equations

Question

ODEs that are linear differential equations have [...] that can be added and multiplied by coefficients.

Answer

exact closed-form solutions

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ODEs that are linear differential equations have exact closed-form solutions that can be added and multiplied by coefficients.

quation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable. [1] <span>ODEs that are linear differential equations have exact closed-form solutions that can be added and multiplied by coefficients. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, e

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#differential-equations

Question

ODEs that are [...] have exact closed-form solutions that can be added and multiplied by coefficients.

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ODEs that are linear differential equations have exact closed-form solutions that can be added and multiplied by coefficients.

quation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable. [1] <span>ODEs that are linear differential equations have exact closed-form solutions that can be added and multiplied by coefficients. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, e

#differential-equations

exact and analytic solutions of nonlinear ODEs are usually in series or integral form.

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olutions that can be added and multiplied by coefficients. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, <span>exact and analytic solutions of ODEs are in series or integral form. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analyti

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#differential-equations

Question

exact and analytic solutions of nonlinear ODEs are usually in [...] form.

Answer

series or integral

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exact and analytic solutions of nonlinear ODEs are usually in series or integral form.

olutions that can be added and multiplied by coefficients. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, <span>exact and analytic solutions of ODEs are in series or integral form. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analyti

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#calculus-of-variations

Question

the necessary condition of functional extremum is **[...description...]**

Answer

functional derivative equals zero.

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the necessary condition of extremum is functional derivative equal zero. the weak formulation of the necessary condition of extremum is an integral with an arbitrary function δf .

pedia Jump to: navigation, search In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. <span>Accordingly, the necessary condition of extremum (functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf. The fundamental lemma of the calculus of variations is typically used to transform this weak formulation into the strong formulation (differential equation), free of the integration with arbitrary function. The proof usually exploits the possibility to choose δf concentrated on an interval on which f keeps sign (positive or negative). Several versions of the lemma are in use. Basic version

#topology

Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics.

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tical space that allows for the definition of concepts such as continuity, connectedness, and convergence. [1] Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. <span>Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. Contents [hide] 1 History 2 Definition 2.1 De

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#topology

Question

Being so general, [...] are a central unifying notion and appear in virtually every branch of modern mathematics.

Answer

topological spaces

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Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics.

tical space that allows for the definition of concepts such as continuity, connectedness, and convergence. [1] Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. <span>Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. Contents [hide] 1 History 2 Definition 2.1 De

#topology

In topology, a branch of mathematics, a **topological manifold** is a topological space (which may also be a separated space) which locally resembles real *n*-dimensional space in a sense defined below.

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Topological manifold - Wikipedia Topological manifold From Wikipedia, the free encyclopedia Jump to: navigation, search In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. A manifold can mean a topological manifold, or more frequently, a topolog

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#topology

Question

Answer

topological manifold

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In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below.

Topological manifold - Wikipedia Topological manifold From Wikipedia, the free encyclopedia Jump to: navigation, search In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. A manifold can mean a topological manifold, or more frequently, a topolog

#topology

A *topological space* is an ordered pair (*X*, *τ*), where *X* is a set and *τ* is a collection of subsets of *X*, satisfying the following axioms:^{[7]}

- The empty set and
*X*itself belong to*τ*. - Any (finite or infinite) union of members of
*τ*still belongs to*τ*. - The intersection of any finite number of members of
*τ*still belongs to*τ*.

The elements of *τ* are called **open sets** and the collection *τ* is called a **topology** on *X*.

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three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

#topology

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A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ.

three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

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#topology

Question

Answer

topological space

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A topological space is an ordered pair (X, τ), where X is a set and τ is a topology of X

three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

Tags

#topology

Question

The elements of **[...]** are called **open sets**

Answer

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ion of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. <span>The elements of τ are called open sets and the collection τ is called a topology on X. <span><body><html>

three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

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A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ.

three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

#topological-space

A topological space *X* is said to be **disconnected** if it is the union of two disjoint nonempty open sets.

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ected spaces 2 Examples 3 Path connectedness 4 Arc connectedness 5 Local connectedness 6 Set operations 7 Theorems 8 Graphs 9 Stronger forms of connectedness 10 See also 11 References 12 Further reading Formal definition[edit source] <span>A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with it

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#topological-space

Question

Answer

disconnected

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A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets.

ected spaces 2 Examples 3 Path connectedness 4 Arc connectedness 5 Local connectedness 6 Set operations 7 Theorems 8 Graphs 9 Stronger forms of connectedness 10 See also 11 References 12 Further reading Formal definition[edit source] <span>A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with it

Tags

#topology

Question

A topology must satisfy axioms of **[...12...]**

Answer

inclusion, complete under infinite union and finit intersection

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A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ.

three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

#stochastics

In mathematics, the **Kronecker delta** is a function of two variables that equals 1 if the variables are equal, and 0 otherwise:

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pedia Kronecker delta From Wikipedia, the free encyclopedia Jump to: navigation, search Not to be confused with the Dirac delta function, nor with the Kronecker symbol. <span>In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: δ i j = {

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In mathematics, the Kronecker delta is a function of two variables that equals 1 if the variables are equal, and 0 otherwise:

pedia Kronecker delta From Wikipedia, the free encyclopedia Jump to: navigation, search Not to be confused with the Dirac delta function, nor with the Kronecker symbol. <span>In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: δ i j = {

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In mathematics, the Kronecker delta is a function of two variables that equals 1 if the variables are equal, and 0 otherwise:

pedia Kronecker delta From Wikipedia, the free encyclopedia Jump to: navigation, search Not to be confused with the Dirac delta function, nor with the Kronecker symbol. <span>In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: δ i j = {

Tags

#topology

Question

A complement of an open set is always relative to **[...]**

Answer

a certain topology in a certain space

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A complement of an open set (relative to the space that the topology is defined on) is called a closed set.

l space. There are, however, topological spaces that are not metric spaces. Properties[edit] The union of any number of open sets, or infinitely many open sets, is open. [2] The intersection of a finite number of open sets is open. [2] <span>A complement of an open set (relative to the space that the topology is defined on) is called a closed set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed. [3] Uses[edit] Open sets have a fundamental im

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#kalman-filter

Question

Answer

linear quadratic estimation

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Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend

into account; P k ∣ k − 1 {\displaystyle P_{k\mid k-1}} is the corresponding uncertainty. <span>Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, one of the primary developers of its theory. The Kalman filter has numerous applications in technology. A common application is for guidanc

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#singular-value-decomposition

Question

Answer

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In linear algebra, the singular-value decomposition (SVD) generalises the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any matrix via an extension of the polar deco

nto three simple transformations: an initial rotation V ∗ , a scaling Σ along the coordinate axes, and a final rotation U. The lengths σ 1 and σ 2 of the semi-axes of the ellipse are the singular values of M, namely Σ 1,1 and Σ 2,2 . <span>In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any m × n {\displaystyle m\times n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. Formally, the singular-value decomposition of an m × n {\d

#matrix-decomposition

a complex square matrix A is **normal** if

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Normal matrix - Wikipedia Normal matrix From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, a complex square matrix A is normal if A ∗ A = A A ∗ {\displaystyle A^{*}A=AA^{*}} where A ∗ is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose. A real square matrix A satisfies A ∗ = A T , and is therefo

#matrix-decomposition #similar-matrix

A matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation *A*^{∗}*A* = *AA*^{∗} is diagonalizable.

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displaystyle A^{*}A=AA^{*}} where A ∗ is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose. A real square matrix A satisfies A ∗ = A T , and is therefore normal if A T A = AA T . <span>A matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation A ∗ A = AA ∗ is diagonalizable. The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means

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a complex square matrix A is normal if

Normal matrix - Wikipedia Normal matrix From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, a complex square matrix A is normal if A ∗ A = A A ∗ {\displaystyle A^{*}A=AA^{*}} where A ∗ is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose. A real square matrix A satisfies A ∗ = A T , and is therefo

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a complex square matrix A is normal if

Normal matrix - Wikipedia Normal matrix From Wikipedia, the free encyclopedia Jump to: navigation, search In mathematics, a complex square matrix A is normal if A ∗ A = A A ∗ {\displaystyle A^{*}A=AA^{*}} where A ∗ is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose. A real square matrix A satisfies A ∗ = A T , and is therefo

#matrix

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over the real number field, see orthogonal matrix. For the restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1, see unitarity. In mathematics, <span>a complex square matrix U is unitary if its conjugate transpose U ∗ is also its inverse—that is, if U ∗ U = U U ∗

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#matrix

Answer

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a complex square matrix U is unitary if its conjugate transpose U ∗ is also its inverse

over the real number field, see orthogonal matrix. For the restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1, see unitarity. In mathematics, <span>a complex square matrix U is unitary if its conjugate transpose U ∗ is also its inverse—that is, if U ∗ U = U U ∗

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#matrix

Question

Answer

unitary

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a complex square matrix U is unitary if its conjugate transpose U ∗ is also its inverse

over the real number field, see orthogonal matrix. For the restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1, see unitarity. In mathematics, <span>a complex square matrix U is unitary if its conjugate transpose U ∗ is also its inverse—that is, if U ∗ U = U U ∗

#similar-matrix

two *n*-by-*n* matrices A and B are **similar** if for some invertible *n*-by-*n* matrix P .

Similar matrices represent the same linear operator under two (possibly) different bases, with P being the change of basis matrix.^{[1]}^{[2]}

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From Wikipedia, the free encyclopedia Jump to: navigation, search For other uses, see Similarity (geometry) and Similarity transformation (disambiguation). Not to be confused with similarity matrix. <span>In linear algebra, two n-by-n matrices A and B are called similar if B = P − 1 A P {\displaystyle B=P^{-1}AP} for some invertible n-by-n matrix P. Similar matrices represent the same linear operator under two (possibly) different bases, with P being the change of basis matrix. [1] [2] A transformation A ↦ P −1 AP is called a similarity transformation or conjugation of the matrix A. In the general linear group, similarity is therefore the same as conjugacy, and simi

#similar-matrix #spectral-theorem

In the definition of similarity, if the matrix *P* can be chosen to be a permutation matrix then *A* and *B* are **permutation-similar;** if *P* can be chosen to be a unitary matrix then *A* and *B* are **unitarily equivalent.** The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix. Specht's theorem states that two matrices are unitarily equivalent if and only if they satisfy certain trace equalities.

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ices over L. This is so because the rational canonical form over K is also the rational canonical form over L. This means that one may use Jordan forms that only exist over a larger field to determine whether the given matrices are similar. <span>In the definition of similarity, if the matrix P can be chosen to be a permutation matrix then A and B are permutation-similar; if P can be chosen to be a unitary matrix then A and B are unitarily equivalent. The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix. Specht's theorem states that two matrices are unitarily equivalent if and only if they satisfy certain trace equalities. See also[edit source] Canonical forms Matrix congruence Matrix equivalence Notes[edit source] Jump up ^ Beauregard & Fraleigh (1973, pp. 240–243) Jump up ^ Bronson (1970

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#similar-matrix

Question

two *n*-by-*n* matrices A and B are **similar **if **[...]** for some invertible *n*-by-*n* matrix P .

Answer

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two n-by-n matrices A and B are similar if for some invertible n-by-n matrix P . Similar matrices represent the same linear operator under two (possibly) different bases, with P being the change of basis matrix. [1] [2] &#

From Wikipedia, the free encyclopedia Jump to: navigation, search For other uses, see Similarity (geometry) and Similarity transformation (disambiguation). Not to be confused with similarity matrix. <span>In linear algebra, two n-by-n matrices A and B are called similar if B = P − 1 A P {\displaystyle B=P^{-1}AP} for some invertible n-by-n matrix P. Similar matrices represent the same linear operator under two (possibly) different bases, with P being the change of basis matrix. [1] [2] A transformation A ↦ P −1 AP is called a similarity transformation or conjugation of the matrix A. In the general linear group, similarity is therefore the same as conjugacy, and simi

Tags

#similar-matrix

Question

two *n*-by-*n* matrices A and B are **[...]** ** **if for some invertible *n*-by-*n* matrix P .

Answer

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two n-by-n matrices A and B are similar if for some invertible n-by-n matrix P . Similar matrices represent the same linear operator under two (possibly) different bases, with P being the change of basis matrix. [1] [2] &#

From Wikipedia, the free encyclopedia Jump to: navigation, search For other uses, see Similarity (geometry) and Similarity transformation (disambiguation). Not to be confused with similarity matrix. <span>In linear algebra, two n-by-n matrices A and B are called similar if B = P − 1 A P {\displaystyle B=P^{-1}AP} for some invertible n-by-n matrix P. Similar matrices represent the same linear operator under two (possibly) different bases, with P being the change of basis matrix. [1] [2] A transformation A ↦ P −1 AP is called a similarity transformation or conjugation of the matrix A. In the general linear group, similarity is therefore the same as conjugacy, and simi

Tags

#similar-matrix

Question

Similar matrices represent the same linear operator under [...]

Answer

(possibly) different bases

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two n-by-n matrices A and B are similar if for some invertible n-by-n matrix P . Similar matrices represent the same linear operator under two (possibly) different bases, with P being the change of basis matrix. [1] [2]

From Wikipedia, the free encyclopedia Jump to: navigation, search For other uses, see Similarity (geometry) and Similarity transformation (disambiguation). Not to be confused with similarity matrix. <span>In linear algebra, two n-by-n matrices A and B are called similar if B = P − 1 A P {\displaystyle B=P^{-1}AP} for some invertible n-by-n matrix P. Similar matrices represent the same linear operator under two (possibly) different bases, with P being the change of basis matrix. [1] [2] A transformation A ↦ P −1 AP is called a similarity transformation or conjugation of the matrix A. In the general linear group, similarity is therefore the same as conjugacy, and simi

Tags

#similar-matrix

Question

Similar matrices represent the same **[...]** under (possibly) different bases

Answer

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two n-by-n matrices A and B are similar if for some invertible n-by-n matrix P . Similar matrices represent the same linear operator under two (possibly) different bases, with P being the change of basis matrix. [1] [2]

From Wikipedia, the free encyclopedia Jump to: navigation, search For other uses, see Similarity (geometry) and Similarity transformation (disambiguation). Not to be confused with similarity matrix. <span>In linear algebra, two n-by-n matrices A and B are called similar if B = P − 1 A P {\displaystyle B=P^{-1}AP} for some invertible n-by-n matrix P. Similar matrices represent the same linear operator under two (possibly) different bases, with P being the change of basis matrix. [1] [2] A transformation A ↦ P −1 AP is called a similarity transformation or conjugation of the matrix A. In the general linear group, similarity is therefore the same as conjugacy, and simi

#similar-matrix #spectral-theorem

The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix.

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> In the definition of similarity, if the matrix P can be chosen to be a permutation matrix then A and B are permutation-similar; if P can be chosen to be a unitary matrix then A and B are unitarily equivalent. The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix. Specht's theorem states that two matrices are unitarily equivalent if and only if they satisfy certain trace equalities. <body><html>

ices over L. This is so because the rational canonical form over K is also the rational canonical form over L. This means that one may use Jordan forms that only exist over a larger field to determine whether the given matrices are similar. <span>In the definition of similarity, if the matrix P can be chosen to be a permutation matrix then A and B are permutation-similar; if P can be chosen to be a unitary matrix then A and B are unitarily equivalent. The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix. Specht's theorem states that two matrices are unitarily equivalent if and only if they satisfy certain trace equalities. See also[edit source] Canonical forms Matrix congruence Matrix equivalence Notes[edit source] Jump up ^ Beauregard & Fraleigh (1973, pp. 240–243) Jump up ^ Bronson (1970

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#similar-matrix #spectral-theorem

Question

The [...] says that every normal matrix is unitarily equivalent to some diagonal matrix.

Answer

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix.

ices over L. This is so because the rational canonical form over K is also the rational canonical form over L. This means that one may use Jordan forms that only exist over a larger field to determine whether the given matrices are similar. <span>In the definition of similarity, if the matrix P can be chosen to be a permutation matrix then A and B are permutation-similar; if P can be chosen to be a unitary matrix then A and B are unitarily equivalent. The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix. Specht's theorem states that two matrices are unitarily equivalent if and only if they satisfy certain trace equalities. See also[edit source] Canonical forms Matrix congruence Matrix equivalence Notes[edit source] Jump up ^ Beauregard & Fraleigh (1973, pp. 240–243) Jump up ^ Bronson (1970

Tags

#similar-matrix #spectral-theorem

Question

Answer

unitarily equivalent

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scheduled repetition interval | last repetition or drill |

The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix.

ices over L. This is so because the rational canonical form over K is also the rational canonical form over L. This means that one may use Jordan forms that only exist over a larger field to determine whether the given matrices are similar. <span>In the definition of similarity, if the matrix P can be chosen to be a permutation matrix then A and B are permutation-similar; if P can be chosen to be a unitary matrix then A and B are unitarily equivalent. The spectral theorem says that every normal matrix is unitarily equivalent to some diagonal matrix. Specht's theorem states that two matrices are unitarily equivalent if and only if they satisfy certain trace equalities. See also[edit source] Canonical forms Matrix congruence Matrix equivalence Notes[edit source] Jump up ^ Beauregard & Fraleigh (1973, pp. 240–243) Jump up ^ Bronson (1970

#matrix-decomposition

A matrix is normal if and only if it is unitarily similar to a diagonal matrix

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A matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation A ∗ A = AA ∗ is diagonalizable.

displaystyle A^{*}A=AA^{*}} where A ∗ is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose. A real square matrix A satisfies A ∗ = A T , and is therefore normal if A T A = AA T . <span>A matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation A ∗ A = AA ∗ is diagonalizable. The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means

Tags

#matrix-decomposition

Question

Answer

normal

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A matrix is normal if and only if it is unitarily similar to a diagonal matrix

displaystyle A^{*}A=AA^{*}} where A ∗ is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose. A real square matrix A satisfies A ∗ = A T , and is therefore normal if A T A = AA T . <span>A matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix A satisfying the equation A ∗ A = AA ∗ is diagonalizable. The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As in the matrix case, normality means

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#fourier-analysis

Question

If a random variable admits a probability density function, then the characteristic function is the [...] of the probability density function.

Answer

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If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function.

c around the origin; however characteristic functions may generally be complex-valued. In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. <span>If a random variable admits a probability density function, then the characteristic function is the Fourier transform of the probability density function. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There ar

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#lebesgue-integration

Question

a measurable simple function can be represented as [...formula...]

Answer

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A finite linear combination of indicator functions where the coefficients a k are real numbers and the sets S k are measurable, is called a measurable simple function.

μ = μ ( S ) . {\displaystyle \int 1_{S}\,\mathrm {d} \mu =\mu (S).} Notice that the result may be equal to +∞, unless μ is a finite measure. Simple functions: <span>A finite linear combination of indicator functions ∑ k a k 1 S k {\displaystyle \sum _{k}a_{k}1_{S_{k}}} where the coefficients a k are real numbers and the sets S k are measurable, is called a measurable simple function. We extend the integral by linearity to non-negative measurable simple functions. When the coefficients a k are non-negative, we set ∫ (

Tags

#lebesgue-integration

Question

a **[...]** is a finite linear combination of indicator functions

Answer

measurable simple function

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A finite linear combination of indicator functions where the coefficients a k are real numbers and the sets S k are measurable, is called a measurable simple function.

μ = μ ( S ) . {\displaystyle \int 1_{S}\,\mathrm {d} \mu =\mu (S).} Notice that the result may be equal to +∞, unless μ is a finite measure. Simple functions: <span>A finite linear combination of indicator functions ∑ k a k 1 S k {\displaystyle \sum _{k}a_{k}1_{S_{k}}} where the coefficients a k are real numbers and the sets S k are measurable, is called a measurable simple function. We extend the integral by linearity to non-negative measurable simple functions. When the coefficients a k are non-negative, we set ∫ (

#topology

sets of the form (-ε, ε) give us a lot of information about points close to *x* = 0

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In particular, sets of the form (-ε, ε) give us a lot of information about points close to x = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x.

ese points approximate x to a greater degree of accuracy compared to when ε = 1. The previous discussion shows, for the case x = 0, that one may approximate x to higher and higher degrees of accuracy by defining ε to be smaller and smaller. <span>In particular, sets of the form (-ε, ε) give us a lot of information about points close to x = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x. This innovative idea has far-reaching consequences; in particular, by defining different collections of sets containing 0 (distinct from the sets (-ε, ε)), one may find different result

Tags

#topology

Question

sets of the form [...] give us a lot of information about points close to *x* = 0

Answer

(-ε, ε)

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scheduled repetition interval | last repetition or drill |

sets of the form (-ε, ε) give us a lot of information about points close to x = 0

ese points approximate x to a greater degree of accuracy compared to when ε = 1. The previous discussion shows, for the case x = 0, that one may approximate x to higher and higher degrees of accuracy by defining ε to be smaller and smaller. <span>In particular, sets of the form (-ε, ε) give us a lot of information about points close to x = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x. This innovative idea has far-reaching consequences; in particular, by defining different collections of sets containing 0 (distinct from the sets (-ε, ε)), one may find different result

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Sets that can be constructed as the intersection of countably many open sets are denoted G δ sets.

u } the open sets. Note that infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form (−1/n, 1/n), where n is a positive integer, is the set {0} which is not open in the real line. <span>Sets that can be constructed as the intersection of countably many open sets are denoted G δ sets. The topological definition of open sets generalizes the metric space definition: If one begins with a metric space and defines open sets as before, then the family of all open sets is

Tags

#topology

Question

Sets that can be constructed as the intersection of countably many open sets are denoted **[...]**

Answer

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Sets that can be constructed as the intersection of countably many open sets are denoted G δ sets.

u } the open sets. Note that infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form (−1/n, 1/n), where n is a positive integer, is the set {0} which is not open in the real line. <span>Sets that can be constructed as the intersection of countably many open sets are denoted G δ sets. The topological definition of open sets generalizes the metric space definition: If one begins with a metric space and defines open sets as before, then the family of all open sets is

Tags

#topology

Question

A set* U* is open if [...description...].

Answer

every point in *U* has a neighborhood contained in *U*

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U is open if every point in U has a neighborhood contained in U.

ntained in U. Metric spaces[edit] A subset U of a metric space (M, d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x, y) < ε, y also belongs to U. Equivalently, <span>U is open if every point in U has a neighborhood contained in U. This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space. Topological spaces[edit] In general topological spaces, the open

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#differential-equations

Question

Answer

added and multiplied by coefficients

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ODEs that are linear differential equations have exact closed-form solutions that can be added and multiplied by coefficients.

quation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable. [1] <span>ODEs that are linear differential equations have exact closed-form solutions that can be added and multiplied by coefficients. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, e