# Knowledge objects

Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.

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Knowledge object



## [Modular inverse](/en/mko/mko-modular-inverse/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

For n≥2, a has a multiplicative inverse modulo n exactly when gcd(a,n)=1; the inverse is unique as a residue class modulo n.

mko-modular-inverse · Number theory



Knowledge object



## [Rank-nullity theorem](/en/mko/mko-rank-nullity-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

A linear map T:V→W with finite-dimensional source satisfies dim(V)=dim(ker T)+rank(T).

mko-rank-nullity-theorem · Algebra



Knowledge object



## [Uniform convergence of functions](/en/mko/mko-uniform-convergence-of-functions/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

Uniform convergence requires one witness N(epsilon) controlling every point of the domain simultaneously.

mko-uniform-convergence-of-functions · Analysis



Knowledge object



## [Absolute convergence of series](/en/mko/mko-absolute-convergence-of-series/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

A series is absolutely convergent when the sum of absolute terms converges; absolute convergence implies convergence, but the converse does not hold in general.

mko-absolute-convergence-of-series · Analysis



Knowledge object



## [Statistical estimator](/en/mko/mko-statistical-estimator/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

An estimator is a fixed measurable total statistic independent of the unknown parameter; its observed numerical value is an estimate.

mko-statistical-estimator · Probability and statistics



Knowledge object



## [Chinese remainder theorem](/en/mko/mko-chinese-remainder-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

For positive pairwise-coprime moduli, an integer congruence system has a unique solution class modulo their product.

mko-chinese-remainder-theorem · Number theory



Knowledge object



## [Spanning tree](/en/mko/mko-spanning-tree/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

A spanning tree is a connected acyclic subgraph containing every vertex of its parent graph.

mko-spanning-tree · Combinatorics and discrete mathematics



Knowledge object



## [Lp convergence](/en/mko/mko-lp-convergence/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

On a common probability space with fixed integer p≥1, Lp convergence means E|X_n−X|^p tends to zero.

mko-lp-convergence · Probability and statistics



Knowledge object



## [Direct sum of subspaces](/en/mko/mko-direct-sum/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

The sum of two subspaces is an internal direct sum when every vector in that sum has a unique decomposition; covering an ambient space is a separate spanning obligation.

mko-direct-sum · Algebra



Knowledge object



## [Dual vector space](/en/mko/mko-dual-vector-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

The dual of a real vector space consists of all real-valued linear functionals; a covector coordinate row depends on a chosen basis.

mko-dual-vector-space · Algebra



Knowledge object



## [Euclidean algorithm](/en/mko/mko-euclidean-algorithm/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

The Euclidean algorithm computes the gcd using exact quotient-remainder divisions, gcd invariance and strictly decreasing nonnegative remainders.

mko-euclidean-algorithm · Number theory



Knowledge object



## [Euler trails and circuits](/en/mko/mko-eulerian-trail/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

An Euler trail uses each edge once; on connected nonzero-edge support, zero odd-degree vertices give a circuit and two give an open Euler trail.

mko-eulerian-trail · Combinatorics and discrete mathematics



Knowledge object



## [Homeomorphism](/en/mko/mko-homeomorphism/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

A homeomorphism is a bijection that is continuous in both directions; a continuous bijection alone need not be a homeomorphism.

mko-homeomorphism · Topology



Knowledge object



## [Sampling distribution](/en/mko/mko-sampling-distribution/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

At a fixed parameter, an estimator pushes the sample law forward to its sampling distribution; all parameters give a family of such laws.

mko-sampling-distribution · Probability and statistics



Knowledge object



## [Weak law of large numbers](/en/mko/mko-weak-law-large-numbers/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

For IID variables with finite variance, a Chebyshev bound shows the sample mean converges in probability to the common mean.

mko-weak-law-large-numbers · Probability and statistics



Knowledge object



## [Chromatic number](/en/mko/mko-chromatic-number/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

The chromatic number of a finite simple graph is the minimum number of colors in a proper vertex coloring; equality needs both upper and lower certificates.

mko-chromatic-number · Combinatorics and discrete mathematics



Knowledge object



## [Cliques and independent sets](/en/mko/mko-clique-and-independent-set/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

Every pair of distinct vertices of a clique is adjacent; every pair of an independent set is nonadjacent. Inclusion-maximal and maximum-cardinality are different.

mko-clique-and-independent-set · Combinatorics and discrete mathematics



Knowledge object



## [Estimator bias](/en/mko/mko-estimator-bias/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

Bias at theta is the estimator expectation under its theta sampling law minus the target parameter; unbiasedness requires zero bias at every theta.

mko-estimator-bias · Probability and statistics



Knowledge object



## [Estimator variance](/en/mko/mko-estimator-variance/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

At each theta, estimator variance is the second central moment of its sampling law around its own mean; it is not bias or MSE by itself.

mko-estimator-variance · Probability and statistics



Knowledge object



## [Fermat's little theorem](/en/mko/mko-fermat-little-theorem/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

For a prime p, a^p is congruent to a modulo p; if p does not divide a, a^(p−1) is congruent to1 modulo p. The converse fails.

mko-fermat-little-theorem · Number theory



Knowledge object



## [Linear Diophantine equation](/en/mko/mko-linear-diophantine-equation/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

A nondegenerate integer equation a*x+b*y=c has a solution exactly when gcd(a,b) divides c; all solutions have one integer parameter.

mko-linear-diophantine-equation · Number theory



Knowledge object



## [Quotient vector space](/en/mko/mko-quotient-vector-space/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

Cosets of a subspace W form the quotient vector space V/W, whose operations must be independent of representative choices.

mko-quotient-vector-space · Algebra



Knowledge object



## [Finite group action](/en/mko/mko-finite-group-action/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

An action of a finite group on a finite set is a total action table satisfying the identity and multiplication-compatibility laws.

mko-finite-group-action · Algebra



Knowledge object



## [Supremum and infimum](/en/mko/mko-supremum-and-infimum/?from=%2Fen%2Fknowledge%2F%3Fpage%3D9)

For a nonempty appropriately bounded real set, the supremum is its least upper bound and the infimum its greatest lower bound; either need not belong to the set.

mko-supremum-and-infimum · Analysis





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