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Finite quotient group
Cosets of a normal subgroup N form a finite quotient group, with (gN)(hN)=(g*h)N independent of representatives.
OCME · Bilingual mathematics & derivations
Knowledge objects
Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.
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Conjugacy class in a finite group
A finite group acts on itself by g*x=g*x*g^(-1); the orbits are conjugacy classes partitioning the group.
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Standard normal distribution
The standard normal law has density phi(x)=(2*pi)^(-1/2)*exp(−x^2/2), continuous CDF Phi, mean0 and variance1.
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Mean squared error of an estimator
For scalar parameter theta, MSE_theta(T)=E_theta[(T−theta)^2] is defined under each complete sampling distribution separately.
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Standard form of a linear program
This node fixes one primal convention: maximize c^T*x subject to A*x≤b and x≥0.
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Vertex cover
A vertex subset C covers a finite simple undirected graph if every edge has at least one endpoint in C.
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First isomorphism theorem for finite groups
A finite group homomorphism induces an isomorphism from its kernel quotient to its image, not automatically to the entire codomain.
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Center of a finite group
The center consists of all elements commuting with every group element and is a normal subgroup.
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Bias-variance decomposition
For each parameter value and its sampling law, scalar-estimator MSE equals variance plus squared bias.
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Linear-programming duality
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.
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Max-flow min-cut theorem
In a finite capacitated network the largest feasible flow value equals the least capacity among all source–sink cuts.
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Minimum vertex cover
A minimum vertex cover has least cardinality among all vertex covers of the finite simple graph.
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Many-one reduction
A≤mB means a total computable map f satisfies x∈A exactly when f(x)∈B for every x.
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Contraction mapping theorem
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.
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Intermediate value theorem
A continuous real function on a closed interval attains each value between its endpoint values.
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Taylor remainder theorem
Under sufficient differentiability, a Lagrange remainder and derivative bound control the Taylor approximation error.
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Maximum-likelihood estimator in a finite model
A finite-model MLE chooses a parameter maximizing the observed likelihood; ties use one fixed rule independent of the unknown parameter.
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Sufficient statistic in a finite model
A statistic is sufficient when the conditional sample law given its value does not depend on theta wherever that condition has positive mass.
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Topological ordering of a directed acyclic graph
A finite directed graph has a topological ordering exactly when it has no directed cycle.
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Multiplicative order modulo n
For gcd(a,n)=1, the multiplicative order is the smallest positive k with a^k congruent to1 modulo n.
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Comparison tests for series
Eventual termwise bounds transfer convergence from a nonnegative upper series or divergence from a nonnegative lower series.
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Ratio test for series
For eventually nonzero terms, an absolute successive-ratio limit below1 gives absolute convergence, above1 gives divergence, and equal to1 is inconclusive.
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A uniform limit of continuous functions is continuous
Continuous functions on one common domain have a continuous limit when convergence is uniform on that domain.
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Finite product topology
Open rectangles generate the topology of a finite Cartesian product of topological spaces.