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Bipartite graph
A finite undirected simple graph is bipartite when all vertices split into two disjoint parts and every edge crosses between the parts.
OCME · Bilingual mathematics & derivations
Knowledge objects
Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.
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Convergence in distribution
Convergence in distribution means that F_n tends pointwise to F at every continuity point of the limiting CDF.
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Quotient ring of a finite ring
Additive cosets of a proper two-sided ideal in a finite ring form a quotient ring when its operations are independent of representative choice.
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Graph coloring
A proper k-coloring uses at most k colors and gives different colors to the endpoints of every edge.
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Real vector subspace
A nonempty subset of a real vector space is a subspace when it contains zero and is closed under vector addition and multiplication by every real scalar.
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Topological closure and interior
The closure of a set is the intersection of all closed supersets, and its interior is the union of all open subsets, relative to the specified topology.
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Open cover and subcover
An open cover consists of open sets in the specified topology whose union contains the target; a subcover selects from that original cover and still covers the same target.
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Almost-sure convergence
Random variables on the same probability space converge almost surely when they converge pointwise outside an exceptional set of probability zero.
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Graph matching
A matching in a finite graph is an edge subset whose edges share no endpoints; the empty matching is allowed.
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Euler totient function
Euler phi(n) counts the representatives coprime to n in a complete residue-representative system, retaining the convention phi(1)=1.
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Deterministic finite automaton
A DFA consists of finite states, a finite alphabet, a total transition function, one initial state and a subset of accepting states.
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Dimension of a finite-dimensional vector space
The dimension of a finite-dimensional vector space is the cardinality of any finite basis; different bases have the same cardinality.
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Kernel of a linear map
The kernel of T consists of all source vectors x satisfying T(x)=0 and forms a subspace of the source.
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Image of a linear map
The image of a linear map is the set of all outputs T(x); for a matrix map it is the full column span.
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Topological boundary
The boundary of A is closure(A) minus interior(A), equivalently closure(A) intersected with the closure of its complement.
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Compact topological space
A space is compact when every open cover has a finite subcover; every finite topological space is compact.
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Bayes theorem
With positive conditioning denominators, Bayes theorem reverses conditional probabilities using the same intersection together with priors and likelihoods.
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Regular language
A language is regular when a complete DFA recognizes all and exactly its words.
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Matrix rank
Matrix rank is the dimension of the row or column space and can be computed as the number of pivots after exact rank-preserving elimination.
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Pointwise convergence of functions
A function sequence on one domain converges pointwise when its value sequence converges at each fixed x; the witness N may depend on x and epsilon.
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Infinite series
An infinite series is defined through the limit of its partial-sum sequence; terms tending to zero are necessary but not sufficient for convergence.
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Parametric statistical model
A parametric statistical model assigns each parameter theta a complete probability law P_theta on the same finite measurable sample space.
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Connected topological space
A space is connected when it has no separation into two disjoint nonempty open sets, equivalently no nontrivial clopen subset.
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Maximum matching
A maximum matching has the greatest edge count among all valid matchings of the complete finite graph; it may be nonunique and need not be perfect.