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Discrete-time stochastic process
A discrete-time stochastic process is a time-indexed family of random variables on one probability space; marginal distributions do not determine its joint law.
OCME · Bilingual mathematics & derivations
Knowledge objects
Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.
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Multivariable real-valued function
A multivariable real-valued function maps inputs in an explicitly declared real-vector domain to real numbers; coordinate roles and domain are function data.
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Linear independence and bases
Linear independence excludes nontrivial zero-sum coefficient relations; a basis is both independent and spanning for the declared space.
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Measurable function
With fixed source and target sigma-algebras, a function is measurable when the preimage of every target measurable set belongs to the source sigma-algebra.
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Finite-state Markov chain
A finite time-homogeneous Markov chain uses a row-stochastic transition kernel and, when needed, an initial law to specify finite path probabilities.
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Partial derivative
A partial derivative fixes every other coordinate and takes the one-variable derivative along one coordinate slice as h tends to zero.
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Finite convex combination
A finite convex combination weights a finite nonempty ordered vector list by exact nonnegative weights whose total is exactly one.
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Partial differential equation
A partial differential equation relates a multivariable unknown function and one or more partial derivatives; residuals and principal-part classification require a fixed operator convention.
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Convex set
A set is convex when the entire segment formed by every pair of its points and every lambda in [0,1] remains in the set.
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Boundary-value problem
A boundary-value problem specifies both an interior differential equation and boundary values or derivative data; successful candidate checks do not establish general well-posedness.
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Convex function
A function on a convex domain is convex when its value at every two-point convex combination is at most the same weighted combination of endpoint values.
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Convex optimization problem
Convex optimization minimizes a convex objective on a convex feasible domain; problem qualification, feasibility, evaluation and global optimality are distinct obligations.
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Finite abelian group
A finite group is abelian when its complete Cayley table satisfies commutativity for every pair of elements.
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Real normed vector space
A real vector space equipped with a positive-definite, absolutely homogeneous real-valued norm satisfying the triangle inequality.
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Cauchy sequence
A sequence in a metric space is Cauchy if, for every positive epsilon, all pairs of sufficiently late terms are less than epsilon apart.
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Finite joint distribution
Value tuples of random variables on one common finite probability space, together with exact masses, specify a finite joint distribution.
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Covariance
Under one finite joint distribution, covariance is E[XY]−E[X]E[Y]; zero covariance does not in general imply independence.
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Graph isomorphism
An isomorphism of finite simple graphs is a vertex bijection preserving both adjacency and nonadjacency for every pair.
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Equivalence relation
A relation on a fixed carrier is an equivalence relation when it fully satisfies reflexivity, symmetry and transitivity.
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Finite unital ring
On one complete finite carrier, this unital-ring protocol requires an additive abelian group, associative multiplication, both distributive laws and a multiplicative identity different from zero.
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Norm-induced metric
On the same real normed vector space, taking the norm of a vector difference defines a metric whose axioms follow universally from the norm axioms.
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Complete metric space
A metric space is complete when every Cauchy sequence converges to a point still belonging to that space.
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Real bilinear form
A bilinear form on a real vector space preserves addition and real scalar multiplication separately in each input.
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Linear map between real vector spaces
A map between real vector spaces is linear when it preserves linear combinations for all vectors and real scalars.