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In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

main (disambiguation). [imagelink] Illustration showing f, a function from the pink domain X to the blue codomain Y. The yellow oval inside Y is the image of f. Both the image and the codomain are sometimes called the range of f. <span>In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or value for each member of the domain. [1] Conversely, the set of values the function takes on as output is termed the image of th

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The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite.

s 7 Generalization 8 Implementations in programming languages 9 See also 10 Notes 11 References 12 External links 12.1 History of science 12.2 Information 12.3 Computer code 12.4 Use of the matrix in simulation 12.5 Online calculators <span>Statement[edit source] The Cholesky decomposition of a Hermitian positive-definite matrix A is a decomposition of the form A = L L ∗ , {\displaystyle \mathbf {A} =\mathbf {LL} ^{*},} where L is a lower triangular matrix with real and positive diagonal entries, and L* denotes the conjugate transpose of L. Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition. [2] If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL* if the diagonal entries of L are allowed to be zero. [3] When A has real entries, L has real entries as well, and the factorization may be written A = LL T . [4] The Cholesky decomposition is unique when A is positive definite; there is only one lower triangular matrix L with strictly positive diagonal entries such that A = LL*. However, the decomposition need not be unique when A is positive semidefinite. The converse holds trivially: if A can be written as LL* for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite. LDL decomposition[edit source] A closely related variant of the classical Cholesky decomposition is the LDL decomposition, A =

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The forward-backward algorithm In the first pass, the forward–backward algorithm computes a set of forward probabilities which provide, for all , the probability of ending up in any particular state given the first observations in the sequence, i.e. . In the second pass, the algorithm computes a set of backward probabilities which provide the probability of observing the remaining observations given any starting point , i.e. . Thes

cific instance of this class. Contents [hide] 1 Overview 2 Forward probabilities 3 Backward probabilities 4 Example 5 Performance 6 Pseudocode 7 Python example 8 See also 9 References 10 External links Overview[edit source] <span>In the first pass, the forward–backward algorithm computes a set of forward probabilities which provide, for all k ∈ { 1 , … , t } {\displaystyle k\in \{1,\dots ,t\}} , the probability of ending up in any particular state given the first k {\displaystyle k} observations in the sequence, i.e. P ( X k | o 1 : k ) {\displaystyle P(X_{k}\ |\ o_{1:k})} . In the second pass, the algorithm computes a set of backward probabilities which provide the probability of observing the remaining observations given any starting point k {\displaystyle k} , i.e. P ( o k + 1 : t | X k ) {\displaystyle P(o_{k+1:t}\ |\ X_{k})} . These two sets of probability distributions can then be combined to obtain the distribution over states at any specific point in time given the entire observation sequence: P ( X k | o 1 : t ) = P ( X k | o 1 : k , o k + 1 : t ) ∝ P ( o k + 1 : t | X k ) P ( X k | o 1 : k ) {\displaystyle P(X_{k}\ |\ o_{1:t})=P(X_{k}\ |\ o_{1:k},o_{k+1:t})\propto P(o_{k+1:t}\ |\ X_{k})P(X_{k}|o_{1:k})} The last step follows from an application of the Bayes' rule and the conditional independence of o k + 1 : t {\displaystyle o_{k+1:t}} and o 1 : k {\displaystyle o_{1:k}} given X k {\displaystyle X_{k}} . As outlined above, the algorithm involves three steps: computing forward probabilities computing backward probabilities computing smoothed values. The forward and backward steps m

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edit source] For all the different settings of the Poisson point process, the two key properties [b] of the Poisson distribution and complete independence play an important role. [25] [45] Homogeneous Poisson point process[edit source] <span>If a Poisson point process has a parameter of the form Λ = ν λ {\displaystyle \textstyle \Lambda =\nu \lambda } , where ν {\displaystyle \textstyle \nu } is Lebegues measure, which assigns length, area, or volume to sets, and λ {\displaystyle \textstyle \lambda } is a constant, then the point process is called a homogeneous or stationary Poisson point process. The parameter, called rate or intensity, is related to the expected (or average) number of Poisson points existing in some bounded region, [49] [50] where rate is usually used when the

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If a Poisson point process has a parameter of the form , where is Lebegues measure, and is a constant, then the point process is called a homogeneous or stationary Poisson point process.

edit source] For all the different settings of the Poisson point process, the two key properties [b] of the Poisson distribution and complete independence play an important role. [25] [45] Homogeneous Poisson point process[edit source] <span>If a Poisson point process has a parameter of the form Λ = ν λ {\displaystyle \textstyle \Lambda =\nu \lambda } , where ν {\displaystyle \textstyle \nu } is Lebegues measure, which assigns length, area, or volume to sets, and λ {\displaystyle \textstyle \lambda } is a constant, then the point process is called a homogeneous or stationary Poisson point process. The parameter, called rate or intensity, is related to the expected (or average) number of Poisson points existing in some bounded region, [49] [50] where rate is usually used when the

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If a Poisson point process has a parameter of the form , where is Lebegues measure, and is a constant, then the point process is called a homogeneous or stationary Poisson point process.

edit source] For all the different settings of the Poisson point process, the two key properties [b] of the Poisson distribution and complete independence play an important role. [25] [45] Homogeneous Poisson point process[edit source] <span>If a Poisson point process has a parameter of the form Λ = ν λ {\displaystyle \textstyle \Lambda =\nu \lambda } , where ν {\displaystyle \textstyle \nu } is Lebegues measure, which assigns length, area, or volume to sets, and λ {\displaystyle \textstyle \lambda } is a constant, then the point process is called a homogeneous or stationary Poisson point process. The parameter, called rate or intensity, is related to the expected (or average) number of Poisson points existing in some bounded region, [49] [50] where rate is usually used when the

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e i k 0 t {\displaystyle e^{ik_{0}t}\,} <span>In mathematics, a degenerate distribution is a probability distribution in a space (discrete or continuous) with support only on a space of lower dimension. If the degenerate distribution is univariate (involving only a single random variable) it is a deterministic distribution and takes only a single value. Examples include a two-headed co

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In mathematics, a degenerate distribution is a probability distribution in a space (discrete or continuous) with support only on a space of lower dimension.

e i k 0 t {\displaystyle e^{ik_{0}t}\,} <span>In mathematics, a degenerate distribution is a probability distribution in a space (discrete or continuous) with support only on a space of lower dimension. If the degenerate distribution is univariate (involving only a single random variable) it is a deterministic distribution and takes only a single value. Examples include a two-headed co

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In mathematics, a degenerate distribution is a probability distribution in a space (discrete or continuous) with support only on a space of lower dimension.

e i k 0 t {\displaystyle e^{ik_{0}t}\,} <span>In mathematics, a degenerate distribution is a probability distribution in a space (discrete or continuous) with support only on a space of lower dimension. If the degenerate distribution is univariate (involving only a single random variable) it is a deterministic distribution and takes only a single value. Examples include a two-headed co

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