# Knowledge objects Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence. [← Previous](/en/knowledge/?page=7#results)[1](/en/knowledge/?page=1#results) … [7](/en/knowledge/?page=7#results) 8 [9](/en/knowledge/?page=9#results) … [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=9#results) 302 results · Page 8 of 13 Knowledge object ## [Bipartite graph](/en/mko/mko-bipartite-graph/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) A finite undirected simple graph is bipartite when all vertices split into two disjoint parts and every edge crosses between the parts. mko-bipartite-graph · Combinatorics and discrete mathematics Knowledge object ## [Convergence in distribution](/en/mko/mko-convergence-in-distribution/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) Convergence in distribution means that F_n tends pointwise to F at every continuity point of the limiting CDF. mko-convergence-in-distribution · Probability and statistics Knowledge object ## [Quotient ring of a finite ring](/en/mko/mko-finite-quotient-ring/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) Additive cosets of a proper two-sided ideal in a finite ring form a quotient ring when its operations are independent of representative choice. mko-finite-quotient-ring · Algebra Knowledge object ## [Graph coloring](/en/mko/mko-graph-coloring/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) A proper k-coloring uses at most k colors and gives different colors to the endpoints of every edge. mko-graph-coloring · Combinatorics and discrete mathematics Knowledge object ## [Real vector subspace](/en/mko/mko-vector-subspace/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) 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. mko-vector-subspace · Algebra Knowledge object ## [Topological closure and interior](/en/mko/mko-closure-and-interior/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) 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. mko-closure-and-interior · Topology Knowledge object ## [Open cover and subcover](/en/mko/mko-open-cover-and-subcover/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) 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. mko-open-cover-and-subcover · Topology Knowledge object ## [Almost-sure convergence](/en/mko/mko-almost-sure-convergence/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) Random variables on the same probability space converge almost surely when they converge pointwise outside an exceptional set of probability zero. mko-almost-sure-convergence · Probability and statistics Knowledge object ## [Graph matching](/en/mko/mko-graph-matching/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) A matching in a finite graph is an edge subset whose edges share no endpoints; the empty matching is allowed. mko-graph-matching · Combinatorics and discrete mathematics Knowledge object ## [Euler totient function](/en/mko/mko-euler-totient-function/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) Euler phi(n) counts the representatives coprime to n in a complete residue-representative system, retaining the convention phi(1)=1. mko-euler-totient-function · Number theory Knowledge object ## [Deterministic finite automaton](/en/mko/mko-finite-automaton/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) A DFA consists of finite states, a finite alphabet, a total transition function, one initial state and a subset of accepting states. mko-finite-automaton · Mathematics of computer science Knowledge object ## [Dimension of a finite-dimensional vector space](/en/mko/mko-vector-space-dimension/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) The dimension of a finite-dimensional vector space is the cardinality of any finite basis; different bases have the same cardinality. mko-vector-space-dimension · Algebra Knowledge object ## [Kernel of a linear map](/en/mko/mko-linear-map-kernel/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) The kernel of T consists of all source vectors x satisfying T(x)=0 and forms a subspace of the source. mko-linear-map-kernel · Algebra Knowledge object ## [Image of a linear map](/en/mko/mko-linear-map-image/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) The image of a linear map is the set of all outputs T(x); for a matrix map it is the full column span. mko-linear-map-image · Algebra Knowledge object ## [Topological boundary](/en/mko/mko-topological-boundary/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) The boundary of A is closure(A) minus interior(A), equivalently closure(A) intersected with the closure of its complement. mko-topological-boundary · Topology Knowledge object ## [Compact topological space](/en/mko/mko-compact-topological-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) A space is compact when every open cover has a finite subcover; every finite topological space is compact. mko-compact-topological-space · Topology Knowledge object ## [Bayes theorem](/en/mko/mko-bayes-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) With positive conditioning denominators, Bayes theorem reverses conditional probabilities using the same intersection together with priors and likelihoods. mko-bayes-theorem · Probability and statistics Knowledge object ## [Regular language](/en/mko/mko-regular-language/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) A language is regular when a complete DFA recognizes all and exactly its words. mko-regular-language · Mathematics of computer science Knowledge object ## [Matrix rank](/en/mko/mko-matrix-rank/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) 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. mko-matrix-rank · Algebra Knowledge object ## [Pointwise convergence of functions](/en/mko/mko-pointwise-convergence-of-functions/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) 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. mko-pointwise-convergence-of-functions · Analysis Knowledge object ## [Infinite series](/en/mko/mko-infinite-series/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) An infinite series is defined through the limit of its partial-sum sequence; terms tending to zero are necessary but not sufficient for convergence. mko-infinite-series · Analysis Knowledge object ## [Parametric statistical model](/en/mko/mko-parametric-statistical-model/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) A parametric statistical model assigns each parameter theta a complete probability law P_theta on the same finite measurable sample space. mko-parametric-statistical-model · Probability and statistics Knowledge object ## [Connected topological space](/en/mko/mko-connected-topological-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) A space is connected when it has no separation into two disjoint nonempty open sets, equivalently no nontrivial clopen subset. mko-connected-topological-space · Topology Knowledge object ## [Maximum matching](/en/mko/mko-maximum-matching/?from=%2Fen%2Fknowledge%2F%3Fpage%3D8) 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. mko-maximum-matching · Combinatorics and discrete mathematics [← Previous](/en/knowledge/?page=7#results)[1](/en/knowledge/?page=1#results) … [7](/en/knowledge/?page=7#results) 8 [9](/en/knowledge/?page=9#results) … [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=9#results) 302 results · Page 8 of 13