# Knowledge objects Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence. [← Previous](/en/knowledge/?page=11#results)[1](/en/knowledge/?page=1#results) … [11](/en/knowledge/?page=11#results) 12 [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=13#results) 302 results · Page 12 of 13 Knowledge object ## [Finite quotient group](/en/mko/mko-finite-quotient-group/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) Cosets of a normal subgroup N form a finite quotient group, with (gN)(hN)=(g*h)N independent of representatives. mko-finite-quotient-group · Algebra Knowledge object ## [Conjugacy class in a finite group](/en/mko/mko-finite-group-conjugacy-class/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) A finite group acts on itself by g*x=g*x*g^(-1); the orbits are conjugacy classes partitioning the group. mko-finite-group-conjugacy-class · Algebra Knowledge object ## [Standard normal distribution](/en/mko/mko-standard-normal-distribution/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) The standard normal law has density phi(x)=(2*pi)^(-1/2)*exp(−x^2/2), continuous CDF Phi, mean0 and variance1. mko-standard-normal-distribution · Probability and statistics Knowledge object ## [Mean squared error of an estimator](/en/mko/mko-estimator-mean-squared-error/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) For scalar parameter theta, MSE_theta(T)=E_theta[(T−theta)^2] is defined under each complete sampling distribution separately. mko-estimator-mean-squared-error · Probability and statistics Knowledge object ## [Standard form of a linear program](/en/mko/mko-linear-program-standard-form/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) This node fixes one primal convention: maximize c^T*x subject to A*x≤b and x≥0. mko-linear-program-standard-form · Optimization and operations research Knowledge object ## [Vertex cover](/en/mko/mko-vertex-cover/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) A vertex subset C covers a finite simple undirected graph if every edge has at least one endpoint in C. mko-vertex-cover · Combinatorics and discrete mathematics Knowledge object ## [First isomorphism theorem for finite groups](/en/mko/mko-first-isomorphism-theorem-finite-groups/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) A finite group homomorphism induces an isomorphism from its kernel quotient to its image, not automatically to the entire codomain. mko-first-isomorphism-theorem-finite-groups · Algebra Knowledge object ## [Center of a finite group](/en/mko/mko-finite-group-center/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) The center consists of all elements commuting with every group element and is a normal subgroup. mko-finite-group-center · Algebra Knowledge object ## [Bias-variance decomposition](/en/mko/mko-bias-variance-decomposition/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) For each parameter value and its sampling law, scalar-estimator MSE equals variance plus squared bias. mko-bias-variance-decomposition · Probability and statistics Knowledge object ## [Linear-programming duality](/en/mko/mko-linear-program-duality/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) For primal maximize c^T*x with A*x≤b,x≥0, the dual minimizes b^T*y with A^T*y≥c,y≥0; feasible pairs satisfy weak duality. mko-linear-program-duality · Optimization and operations research Knowledge object ## [Max-flow min-cut theorem](/en/mko/mko-max-flow-min-cut-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) In a finite capacitated network the largest feasible flow value equals the least capacity among all source–sink cuts. mko-max-flow-min-cut-theorem · Optimization and operations research Knowledge object ## [Minimum vertex cover](/en/mko/mko-minimum-vertex-cover/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) A minimum vertex cover has least cardinality among all vertex covers of the finite simple graph. mko-minimum-vertex-cover · Combinatorics and discrete mathematics Knowledge object ## [Many-one reduction](/en/mko/mko-many-one-reduction/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) A≤mB means a total computable map f satisfies x∈A exactly when f(x)∈B for every x. mko-many-one-reduction · Mathematics of computer science Knowledge object ## [Contraction mapping theorem](/en/mko/mko-contraction-mapping-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) A self-map on a nonempty complete metric space with one global contraction constant q<1 has a unique fixed point and convergent Picard iterates. mko-contraction-mapping-theorem · Analysis Knowledge object ## [Intermediate value theorem](/en/mko/mko-intermediate-value-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) A continuous real function on a closed interval attains each value between its endpoint values. mko-intermediate-value-theorem · Analysis Knowledge object ## [Taylor remainder theorem](/en/mko/mko-taylor-remainder-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) Under sufficient differentiability, a Lagrange remainder and derivative bound control the Taylor approximation error. mko-taylor-remainder-theorem · Analysis Knowledge object ## [Maximum-likelihood estimator in a finite model](/en/mko/mko-finite-maximum-likelihood-estimator/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) A finite-model MLE chooses a parameter maximizing the observed likelihood; ties use one fixed rule independent of the unknown parameter. mko-finite-maximum-likelihood-estimator · Probability and statistics Knowledge object ## [Sufficient statistic in a finite model](/en/mko/mko-finite-sufficient-statistic/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) A statistic is sufficient when the conditional sample law given its value does not depend on theta wherever that condition has positive mass. mko-finite-sufficient-statistic · Probability and statistics Knowledge object ## [Topological ordering of a directed acyclic graph](/en/mko/mko-topological-ordering-dag/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) A finite directed graph has a topological ordering exactly when it has no directed cycle. mko-topological-ordering-dag · Mathematics of computer science Knowledge object ## [Multiplicative order modulo n](/en/mko/mko-multiplicative-order-mod-n/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) For gcd(a,n)=1, the multiplicative order is the smallest positive k with a^k congruent to1 modulo n. mko-multiplicative-order-mod-n · Number theory Knowledge object ## [Comparison tests for series](/en/mko/mko-series-comparison-test/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) Eventual termwise bounds transfer convergence from a nonnegative upper series or divergence from a nonnegative lower series. mko-series-comparison-test · Analysis Knowledge object ## [Ratio test for series](/en/mko/mko-ratio-test-series/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) For eventually nonzero terms, an absolute successive-ratio limit below1 gives absolute convergence, above1 gives divergence, and equal to1 is inconclusive. mko-ratio-test-series · Analysis Knowledge object ## [A uniform limit of continuous functions is continuous](/en/mko/mko-uniform-limit-continuity-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) Continuous functions on one common domain have a continuous limit when convergence is uniform on that domain. mko-uniform-limit-continuity-theorem · Analysis Knowledge object ## [Finite product topology](/en/mko/mko-finite-product-topology/?from=%2Fen%2Fknowledge%2F%3Fpage%3D12) Open rectangles generate the topology of a finite Cartesian product of topological spaces. mko-finite-product-topology · Topology [← Previous](/en/knowledge/?page=11#results)[1](/en/knowledge/?page=1#results) … [11](/en/knowledge/?page=11#results) 12 [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=13#results) 302 results · Page 12 of 13