# Knowledge objects Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence. [← Previous](/en/knowledge/?page=10#results)[1](/en/knowledge/?page=1#results) … [10](/en/knowledge/?page=10#results) 11 [12](/en/knowledge/?page=12#results) [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=12#results) 302 results · Page 11 of 13 Knowledge object ## [Mobius inversion on finite divisor sets](/en/mko/mko-mobius-inversion-finite-divisors/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) If F(n) sums f(d) over every positive divisor of n, Mobius weights exactly recover f(n). mko-mobius-inversion-finite-divisors · Number theory Knowledge object ## [Pushdown automaton](/en/mko/mko-pushdown-automaton/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) A nondeterministic PDA uses finite control, input and stack transitions; this slice accepts only after reading the whole input in a final state. mko-pushdown-automaton · Mathematics of computer science Knowledge object ## [Lagrange theorem for finite groups](/en/mko/mko-lagrange-theorem-finite-groups/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) Equal-size left cosets of a subgroup H partition the finite group G, so |G|=[G:H]*|H| and |H| divides |G|. mko-lagrange-theorem-finite-groups · Algebra Knowledge object ## [Higher-order derivative](/en/mko/mko-higher-order-derivative/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) Define each existing finite derivative recursively by f^(0)=f and f^(n+1)=(f^(n))'. mko-higher-order-derivative · Analysis Knowledge object ## [Hausdorff space](/en/mko/mko-hausdorff-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) A topological space is Hausdorff if every two distinct points have disjoint open neighborhoods. mko-hausdorff-space · Topology Knowledge object ## [Continuous real distribution function](/en/mko/mko-continuous-real-distribution-function/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) Here a continuous distribution means that its CDF is continuous; the CDF must still be nondecreasing, right-continuous and have endpoint limits0 and1. mko-continuous-real-distribution-function · Probability and statistics Knowledge object ## [Catalan numbers](/en/mko/mko-catalan-numbers/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) With C_0=1, the convolution recurrence and two exact closed forms define the same zero-based Catalan sequence. mko-catalan-numbers · Combinatorics and discrete mathematics Knowledge object ## [Quadratic residue](/en/mko/mko-quadratic-residue/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) Under this odd-prime nonzero convention, a class is a quadratic residue when its normalized nonzero value is a square of a nonzero class modulo p. mko-quadratic-residue · Number theory Knowledge object ## [Nondeterministic finite automaton](/en/mko/mko-nondeterministic-finite-automaton/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) An epsilon-NFA relates a state and input symbol or epsilon to possible successor states; one full-input final path suffices for acceptance. mko-nondeterministic-finite-automaton · Mathematics of computer science Knowledge object ## [Cauchy theorem for finite groups](/en/mko/mko-cauchy-theorem-finite-groups/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) If a prime p divides the order of a finite group G, G contains an element of exact order p. mko-cauchy-theorem-finite-groups · Algebra Knowledge object ## [Taylor polynomial](/en/mko/mko-taylor-polynomial/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) At center a, derivative values of orders0 through n, scaled by their factorials, form the order-n Taylor polynomial. mko-taylor-polynomial · Analysis Knowledge object ## [A compact subset of a Hausdorff space is closed](/en/mko/mko-compact-subset-closed-in-hausdorff/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) If K is compact in the Hausdorff space X, then K is closed in X. mko-compact-subset-closed-in-hausdorff · Topology Knowledge object ## [Improper integral](/en/mko/mko-improper-integral/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) Integrals over unbounded intervals or with singularities are defined by appropriate one-sided limits, with every required piece converging independently. mko-improper-integral · Analysis Knowledge object ## [Radius of convergence of a power series](/en/mko/mko-power-series-radius-of-convergence/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) Around its center c, a power series has absolute convergence inside radius R and divergence outside; endpoints require separate decisions. mko-power-series-radius-of-convergence · Analysis Knowledge object ## [Legendre symbol](/en/mko/mko-legendre-symbol/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) For odd prime p, the Legendre symbol uses0,1,−1 for the zero class, nonzero quadratic residues and nonresidues respectively. mko-legendre-symbol · Number theory Knowledge object ## [Subset construction from an NFA to a DFA](/en/mko/mko-subset-construction-nfa-to-dfa/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) Use epsilon-NFA state subsets as DFA states, defining start, transitions and acceptance with epsilon closure to preserve the language. mko-subset-construction-nfa-to-dfa · Mathematics of computer science Knowledge object ## [Finite normal subgroup](/en/mko/mko-finite-normal-subgroup/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) A subgroup H of a finite group G is normal when gHg^(-1)=H for every g, equivalently when each left coset equals the corresponding right coset. mko-finite-normal-subgroup · Algebra Knowledge object ## [Real exponential function](/en/mko/mko-real-exponential-function/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) Define E(x)=sum over n≥0 of x^n/n!, absolutely convergent for every real x, with E(x+y)=E(x)E(y) and E(x)>0. mko-real-exponential-function · Analysis Knowledge object ## [Analytic ordinary generating function](/en/mko/mko-analytic-ordinary-generating-function/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) An ordinary power series A(x)=sum over n≥0 of a_n*x^n analytically encodes a sequence on an explicitly declared convergence domain. mko-analytic-ordinary-generating-function · Combinatorics and discrete mathematics Knowledge object ## [Euler criterion for quadratic residues](/en/mko/mko-euler-criterion-quadratic-residue/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) For odd prime p and a not divisible by p, a^((p−1)/2) modulo p is the modulo-p representative of the Legendre symbol (a/p). mko-euler-criterion-quadratic-residue · Number theory Knowledge object ## [Turing-machine decision model](/en/mko/mko-turing-machine-decision-model/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) Under fixed deterministic single-tape semantics, a decider must halt on every legal input and return the correct accept/reject answer. mko-turing-machine-decision-model · Mathematics of computer science Knowledge object ## [Gradient vector](/en/mko/mko-gradient-vector/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) In fixed standard Euclidean coordinates, the first partial derivatives of a differentiable real-valued function form an ordered gradient vector. mko-gradient-vector · Analysis Knowledge object ## [Finite flow network](/en/mko/mko-finite-flow-network/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) A nonnegative flow in a finite directed network must respect every edge capacity and conserve flow at each vertex other than source and sink. mko-finite-flow-network · Optimization and operations research Knowledge object ## [Kernel of a finite group homomorphism](/en/mko/mko-finite-group-homomorphism-kernel/?from=%2Fen%2Fknowledge%2F%3Fpage%3D11) For a finite group homomorphism f:G→Q, the full preimage of the target identity is a normal subgroup of G, and nonempty fibers are its cosets. mko-finite-group-homomorphism-kernel · Algebra [← Previous](/en/knowledge/?page=10#results)[1](/en/knowledge/?page=1#results) … [10](/en/knowledge/?page=10#results) 11 [12](/en/knowledge/?page=12#results) [13](/en/knowledge/?page=13#results)[Next →](/en/knowledge/?page=12#results) 302 results · Page 11 of 13