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Annotation 1729501203724

#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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Multivariate normal distribution - Wikipedia
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. 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)

Annotation 1729503300876

#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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Multivariate normal distribution - Wikipedia
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. 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

Annotation 1729504873740

#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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Multivariate normal distribution - Wikipedia
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] 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

Annotation 1729506970892

#multivariate-normal-distribution

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

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Multivariate normal distribution - Wikipedia
ldsymbol {\rho }}_{0}} is the correlation matrix constructed from Σ 0 {\displaystyle {\boldsymbol {\Sigma }}_{0}} . 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

Annotation 1729523485964

#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 x₁ conditional on x₂ = a is multivariate normal (x₁ | x₂ = a) ~ N( μ , Σ ) where

and covariance matrix

This matrix is the Schur complement of Σ₂₂ 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 x₂ = 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 x₁ would have distribution .

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

The matrix Σ₁₂Σ₂₂⁻¹ is known as the matrix of regression

...

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Multivariate normal distribution - Wikipedia
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. 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]

Annotation 1729525845260

#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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Multivariate normal distribution - Wikipedia
) {\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] 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

Annotation 1729527942412

#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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Multivariate normal distribution - Wikipedia
{\displaystyle {\boldsymbol {\Sigma }}'={\begin{bmatrix}{\boldsymbol {\Sigma }}_{11}&{\boldsymbol {\Sigma }}_{13}\\{\boldsymbol {\Sigma }}_{31}&{\boldsymbol {\Sigma }}_{33}\end{bmatrix}}} . Affine transformation[edit source] 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 )

Annotation 1729530039564

#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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Multivariate normal distribution - Wikipedia
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 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

Annotation 1729532136716

#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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Multivariate normal distribution - Wikipedia
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. 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

Annotation 1729534496012

#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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Multivariate normal distribution - Wikipedia
{\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. 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

Annotation 1729553632524

#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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Characteristic function (probability theory) - Wikipedia
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. In 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

Annotation 1729555729676

#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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Characteristic function (probability theory) - Wikipedia
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. 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

Annotation 1729578011916

#multivariate-normal-distribution

In the bivariate case where x is partitioned into X₁ and X₂, the conditional distribution of X₁ given X₂ is

where is the correlation coefficient between X₁ and X₂.

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Multivariate normal distribution - Wikipedia
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] 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] (

Flashcard 1729607109900

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#multivariate-normal-distribution

Question

In the bivariate case, the conditional mean of X₁ given X₂ is [...]

Answer

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

where is the correlation coefficient between X₁ and X₂.
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 .

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#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.

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#fourier-analysis

Question

the characteristic function of any real-valued random variable completely defines its [...].

Answer

probability distribution

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

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#multivariate-normal-distribution

Question

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

Answer

Λ^1/2, U , μ.

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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.

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#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.

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#multivariate-normal-distribution

Question

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

Answer

The squared relative lengths

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#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.

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Flashcard 1729672908044

Tags

#multivariate-normal-distribution

Question

Y = c + BX is an affine transformation,
then Y has a multivariate normal distribution with expected value [...]

Answer

c + Bμ

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

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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.

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Flashcard 1729674480908

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#multivariate-normal-distribution

Question

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

Answer

BΣB^T

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

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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>

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#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

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Annotation 1729678413068

#multivariate-normal-distribution

the distribution of x₁ conditional on x₂ = a is multivariate normal (x₁ | x₂ = 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

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Flashcard 1729680772364

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#multivariate-normal-distribution

Question

the distribution of x₁ conditional on x₂ = a is multivariate normal (x₁ | x₂ = 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

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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:

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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

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Flashcard 1729699122444

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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

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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.

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Annotation 1729709870348

#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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Singular-value decomposition - Wikipedia
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 . 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

Annotation 1729711967500

#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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Singular-value decomposition - Wikipedia
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. 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

Flashcard 1729714326796

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#singular-value-decomposition

Question

[...] generalises eigendecomposition of a positive semidefinite normal matrix to any

matrix

Answer

singular-value decomposition (SVD)

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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

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Annotation 1729716686092

#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.

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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

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Question

With a factorization of the form

represent [...]

Answer

real or complex unitary matrix

real or complex unitary matrix.

rectangular diagonal matrix

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#kalman-filter

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.

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Kalman filter - Wikipedia
into account; P k ∣ k − 1 {\displaystyle P_{k\mid k-1}} is the corresponding uncertainty. 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

Annotation 1729833078028

#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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LU decomposition - Wikipedia
on - Wikipedia ocultar siempre | ocultar ahora LU decomposition From Wikipedia, the free encyclopedia Jump to: navigation, search 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

Annotation 1729835175180

#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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LU decomposition - Wikipedia
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. 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.

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Flashcard 1730172554508

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#matrix-decomposition

Question

LU decomposition factors a matrix as the product of [...] and [...] .

Answer

a lower and an upper triangular matrixmatrix.

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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.

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Annotation 1731026881804

#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

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.

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Euler's formula - Wikipedia
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 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

Annotation 1731435564300

#singular-value-decomposition

SVD as change of coordinates

The geometric content of the SVD theorem can thus be summarized as follows: for every linear map T : Kⁿ → K^m one can find orthonormal bases of Kⁿ and K^m such that T maps the i-th basis vector of Kⁿ 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.

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Singular-value decomposition - Wikipedia
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). 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

Flashcard 1731451292940

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#singular-value-decomposition

Question

geometrically SVD finds [...] for every linear map T : Kⁿ → K^m

Answer

orthonormal bases of Kⁿ 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,

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Flashcard 1731453127948

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#singular-value-decomposition

Question

Geometrically SVD finds orthonormal bases of Kⁿ and K^mfor every linear map T : Kⁿ → K^m such that T maps the i-th basis vector of Kⁿ 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 zero. With respect to these bases, the map T is therefore represented by a diagonal matrix with non-negative real diagonal entries. <body><html>

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Flashcard 1731454700812

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#singular-value-decomposition

Question

With SVD geometrically every linear map T : Kⁿ → 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 non-negative real diagonal entries. <body><html>

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Annotation 1731457584396

#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⁻¹AP is a diagonal matrix.

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Diagonalizable matrix - Wikipedia
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. 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

Flashcard 1731460205836

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#matrix

Question

a square matrix A is called diagonalizable if it is similar to [...]

Answer

a diagonal 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.

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Flashcard 1731484323084

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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

refer to the future from the past

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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).

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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). 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). <body><html>

Flashcard 1731535703308

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#variational-inference

Question

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

Answer

intractable integrals

In Bayesian inference this manifests as calculating marginal posteriors

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

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Variational Bayesian methods - Wikipedia
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) 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

Flashcard 1732530801932

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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)

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Unknown title
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,- 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

Annotation 1732621765900

#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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Compound probability distribution - Wikipedia
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. 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

Flashcard 1732623338764

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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

marginalizing out

Also called integrating over

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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").

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Flashcard 1732624911628

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#distributions

Question

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

Answer

latent random variables

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#kalman-filter

Question

Kalman filtering estimats a [...] over the variables for each timeframe.

Answer

joint probability distribution

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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 a joint probability distribution over the variables for each timeframe. <body><html>

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Annotation 1735706676492

#topology

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

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Open set - Wikipedia
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, 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

Flashcard 1735708773644

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#topology

Question

a set is open if it doesn't contain any of its [...]

Answer

boundary points

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

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#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

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The rise and fall and rise of logic | Aeon Essays
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. 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

Flashcard 1736031472908

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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>

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Stochastic process - Wikipedia
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] 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

Flashcard 1736175914252

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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

P⁻¹AP

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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.

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Annotation 1738538093836

#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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Multivariate normal distribution - Wikipedia
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] 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

Flashcard 1738588949772

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#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.

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Annotation 1738844278028

#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 - 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

Flashcard 1738846375180

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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>

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Flashcard 1738856074508

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#metric-space

Question

the Banach fixed-point theorem is also known as [...]

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

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Annotation 1738858695948

#metric-space

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.

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Banach fixed-point theorem - Wikipedia
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] 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

Annotation 1738860268812

#metric-space

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 x₀ in X and define a sequence {x_n} by x_n = T(x_n−1), then x_n → x* .

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Banach fixed-point theorem - Wikipedia
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*). 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

Flashcard 1738862628108

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#metric-space

Question

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

Answer

contraction mapping

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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.

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Flashcard 1738864200972

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#metric-space

Question

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.

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.

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Flashcard 1738867084556

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#metric-space

Question

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

Answer

a unique fixed-point x* 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 an arbitrary element x 0 in X and define a sequence {x n } by x n = T(x n−1 ), then x n → x* . <

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Flashcard 1738868919564

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#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 x₀ 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 an arbitrary element x 0 in X and define a sequence {x n } by x n = T(x n−1 ), then x n → x* . <body><html>

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Flashcard 1738870492428

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#metric-space

Question

Using the Banach Fixed Point Theorem the fixed point x* can be found by starting with an arbitrary element x₀ in X and define a sequence {x_n} by [...], then x_n → x* .

Answer

x_n = T(x_n−1)

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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 x n = T(x n−1 ), then x n → x* . <body><html>

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Annotation 1739411819788

#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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Kalman filter - Wikipedia
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. 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

Flashcard 1739414965516

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.

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#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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Dot product - Wikipedia
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. 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

Annotation 1741785533708

#dot-product

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

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Dot product - Wikipedia
‖ 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, 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

Annotation 1743937998092

#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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Mathematical logic - Wikipedia
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] 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

Flashcard 1744152694028

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

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Lebesgue integration - Wikipedia
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. 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

Flashcard 1744165539084

Tags

#lebesgue-integration

Question

the Banach-Tarski paradox suggests that picking out [...] is an essential prerequisite.

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 a suitable class of measurable subsets is an essential prerequisite. <body><html>

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Lebesgue integration - Wikipedia
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. 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

Flashcard 1744250473740

Tags

#dot-product

Question

[...] takes two equal-length sequences of numbers and returns a single number.

Answer

dot product

or scalar product

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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.

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Flashcard 1744253357324

Tags

#dot-product

Question

if a and b are [...], then the angle between them is 90° and

Answer

orthogonal

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

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Flashcard 1744268561676

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#calculus-of-variations

Question

In Calculus of variations functionals are mappings from [...] to the real numbers.

Answer

a set of functions

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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

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Calculus of variations - Wikipedia
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 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

Flashcard 1744276163852

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#logic

Question

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

Answer

the expressive power of formal systems

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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.

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Flashcard 1744277736716

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#logic

Question

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

Answer

the deductive power of formal proof systems

How does it relate to the inductive process of Bayesian reasoning?

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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.

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Flashcard 1744279309580

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#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.

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Flashcard 1744289533196

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#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.

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Lebesgue integration - Wikipedia
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 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 ∫ (

Flashcard 1744294251788

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

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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.

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Lebesgue integration - Wikipedia
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: 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

Flashcard 1744295824652

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

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

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Annotation 1748729466124

#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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Topological space - Wikipedia
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

Annotation 1748731301132

#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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Topological space - Wikipedia
, 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, 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

Flashcard 1748733660428

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#topology

Question

a [...] may be defined by a set of of points, neighbourhoods, and axioms.

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.

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Annotation 1748751224076

#probability-theory

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

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Random variable - Wikipedia
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. 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

Flashcard 1748753059084

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#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)

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#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)

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#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

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Flashcard 1749100399884

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#topology

Question

the structure of an inner product allows [...] to be measured.

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.

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Hilbert space - Wikipedia
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. 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

Flashcard 1749109574924

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#hilbert-space

Question

Hilbert spaces are complete: there are [...] to allow the techniques of calculus to be used.

Answer

enough limits in the space

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

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Annotation 1753267178764

#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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FAQ: Incremental 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%). 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

Annotation 1753270062348

#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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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

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Annotation 1753271635212

#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.

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#incremental-reading

Question

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

Answer

speed and retention

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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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Flashcard 1753276353804

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#incremental-reading

Question

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

Answer

the most important pieces of text

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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.

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Flashcard 1753278713100

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#incremental-reading

Question

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

Answer

normal speed

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#incremental-reading

Question

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

Answer

a large proportion

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#calculus

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

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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

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Annotation 1753283431692

#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. <body><html>

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Flashcard 1753285790988

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#calculus

Question

Euler's formula establishes the fundamental relationship between the trigonometric functions and [...]

Answer

the complex exponential function.

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

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Flashcard 1753288150284

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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.

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#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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Open set - Wikipedia
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. 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

Flashcard 1753325899020

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#topology

Question

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.

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Flashcard 1753327471884

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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.

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Flashcard 1753329044748

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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.

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Annotation 1753334549772

#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.

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#topology

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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

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#topology

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[...] is the most general notion of a mathematical space

Answer

topological space

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#topology

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

Answer

continuity, connectedness, and convergence

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#topology

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

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Open set - Wikipedia
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. 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

Annotation 1754607258892

#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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Open set - Wikipedia
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. 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

Annotation 1754608831756

#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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Open set - Wikipedia
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. 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

Annotation 1754610404620

#topology

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

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Open set - Wikipedia
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. 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

Flashcard 1754613550348

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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

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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

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#topology

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

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Open set - Wikipedia
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. 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

Annotation 1754618268940

#topology

infinite intersections of open sets need not be open.

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Open set - Wikipedia
τ {\displaystyle \tau } is in τ {\displaystyle \tau } ) We call the sets in τ {\displaystyle \tau } 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. Sets that can be constructed as

Flashcard 1754619841804

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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.

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Flashcard 1754629278988

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#topology

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Each choice of open sets for a space is called a [...].

Answer

topology

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

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Flashcard 1754630851852

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#topology

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Each [...] for a space is called a topology.

Answer

choice of open sets

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

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#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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Open set - Wikipedia
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] 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

Flashcard 1755466304780

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#topology

Question

[...] is called a closed set.

Answer

A complement of an open set

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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.

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Flashcard 1755467877644

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#topology

Question

A complement of an open set is called a [...].

Answer

closed set

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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.

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Annotation 1755469450508

#topology

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

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Open set - Wikipedia
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, 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

Flashcard 1755492257036

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#optimization

Question

Generally, unless both [...] are convex there may be several local minima

Answer

the objective function and the feasible region

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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

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Mathematical optimization - Wikipedia
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. 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 −

Flashcard 1755496189196

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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

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Mathematical optimization - Wikipedia
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. 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

Flashcard 1756368080140

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#calculus

Question

[...] establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

Answer

Euler's formula

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

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Annotation 1758189980940

#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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Ordinary differential equation - Wikipedia
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] 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

Flashcard 1758192078092

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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.

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Flashcard 1758196534540

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#differential-equations

Question

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

Answer

linear differential equations

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

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Annotation 1758198107404

#differential-equations

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

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Ordinary differential equation - Wikipedia
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, 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

Flashcard 1758200466700

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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.

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Flashcard 1758278847756

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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 .

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Fundamental lemma of calculus of variations - Wikipedia
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. 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

Annotation 1758466805004

#topology

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

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Topological space - Wikipedia
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. 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

Flashcard 1758469688588

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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.

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#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 - 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

Flashcard 1758474931468

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#topology

Question

a [...] is a topological space which locally resembles real n-dimensional space in some sense

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.

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Annotation 1758491184396

#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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Topological space - Wikipedia
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. 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 (

Annotation 1758493281548

#topology

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

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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.

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Flashcard 1758495640844

Tags

#topology

Question

A [...] is an ordered pair (X, τ), where X is a set and τ is a topology of X

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

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Flashcard 1758497213708

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#topology

Question

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

Answer

A topology

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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. The elements of τ are called open sets and the collection τ is called a topology on X. <body><html>

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Flashcard 1758499835148

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#topology

Question

A [...] is a collection of subsets of X satisfying certain axioms (inclusion, infinite union, finit intersection).

Answer

topology

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Annotation 1758504291596

#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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Connected space - Wikipedia
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] 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

Flashcard 1758506913036

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#topological-space

Question

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

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.

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Flashcard 1759925898508

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#topology

Question

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

Answer

inclusion, complete under infinite union and finit intersection

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Annotation 1760008736012

#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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Kronecker delta - Wikipedia
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. 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 = {

Flashcard 1760011619596

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#stochastics

Question

the Kronecker delta is a function of two variables that [...]

Answer

equals 1 if the variables are equal, and 0 otherwise:

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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:

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Flashcard 1760013192460

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#stochastics

Question

the [...] is a function of two variables that equals 1 if the variables are equal, and 0 otherwise:

Answer

Kronecker delta

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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:

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Flashcard 1767389662476

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#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.

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Flashcard 1782267383052

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#kalman-filter

Question

Kalman filtering is known as [...]

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

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Flashcard 1791689886988

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#singular-value-decomposition

Question

singular-value decomposition (SVD) generalises [...] of a positive semidefinite normal matrix to any

matrix

Answer

eigendecomposition

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Annotation 1791694343436

#matrix-decomposition

a complex square matrix A is normal if

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Normal matrix - Wikipedia
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

Annotation 1791695916300

#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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Normal matrix - Wikipedia
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 . 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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#matrix-decomposition

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a complex square matrix A is normal if [...]

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a complex square matrix A is normal if

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#matrix-decomposition

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a complex square matrix A is [...] if

Answer

normal

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a complex square matrix A is normal if

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#matrix

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

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Unitary matrix - Wikipedia
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, 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

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a complex square matrix U is unitary if [...description...]

Answer

its conjugate transpose U^∗ is also its inverse

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

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#matrix

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a complex square matrix U is [...] if its conjugate transpose U^∗ is also its inverse

Answer

unitary

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

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#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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Matrix similarity - Wikipedia
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. 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

Annotation 1791717936396

#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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Matrix similarity - Wikipedia
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. 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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two n-by-n matrices A and B are similar if [...] for some invertible n-by-n matrix P .

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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] &#

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#similar-matrix

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

Answer

similar

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Flashcard 1791725276428

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#similar-matrix

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

Answer

(possibly) different bases

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Flashcard 1791728422156

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#similar-matrix

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

Answer

linear operator

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#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>

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Flashcard 1791738907916

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#similar-matrix #spectral-theorem

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

Answer

spectral theorem

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

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#similar-matrix #spectral-theorem

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The spectral theorem says that every normal matrix is [...] to some diagonal matrix.

Answer

unitarily equivalent

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

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#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.

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Flashcard 1791744412940

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#matrix-decomposition

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A matrix is [...] if and only if it is unitarily similar to a diagonal matrix

Answer

normal

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

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#fourier-analysis

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

Answer

Fourier transform

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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.

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#lebesgue-integration

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

Answer

where the coefficients a_k are real numbers and the sets S_k are measurable

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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.

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Flashcard 1791819910412

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#lebesgue-integration

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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.

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Annotation 1791847959820

#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.

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Flashcard 1791849532684

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#topology

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

Answer

(-ε, ε)

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

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Flashcard 1791851105548

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#topology

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Sets that can be constructed as [...] are denoted G_δ sets.

Answer

the intersection of countably many open sets

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

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#topology

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

Answer

G_δ sets

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

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#topology

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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.

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#differential-equations

Question

ODEs that are linear differential equations have exact closed-form solutions that can be [...].

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.

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