Knowledge object
Modular inverse
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.
OCME · Bilingual mathematics & derivations
Knowledge objects
Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.
Knowledge object
Rank-nullity theorem
A linear map T:V→W with finite-dimensional source satisfies dim(V)=dim(ker T)+rank(T).
Knowledge object
Uniform convergence of functions
Uniform convergence requires one witness N(epsilon) controlling every point of the domain simultaneously.
Knowledge object
Absolute convergence of series
A series is absolutely convergent when the sum of absolute terms converges; absolute convergence implies convergence, but the converse does not hold in general.
Knowledge object
Statistical estimator
An estimator is a fixed measurable total statistic independent of the unknown parameter; its observed numerical value is an estimate.
Knowledge object
Chinese remainder theorem
For positive pairwise-coprime moduli, an integer congruence system has a unique solution class modulo their product.
Knowledge object
Spanning tree
A spanning tree is a connected acyclic subgraph containing every vertex of its parent graph.
Knowledge object
Lp convergence
On a common probability space with fixed integer p≥1, Lp convergence means E|X_n−X|^p tends to zero.
Knowledge object
Direct sum of subspaces
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.
Knowledge object
Dual vector space
The dual of a real vector space consists of all real-valued linear functionals; a covector coordinate row depends on a chosen basis.
Knowledge object
Euclidean algorithm
The Euclidean algorithm computes the gcd using exact quotient-remainder divisions, gcd invariance and strictly decreasing nonnegative remainders.
Knowledge object
Euler trails and circuits
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.
Knowledge object
Homeomorphism
A homeomorphism is a bijection that is continuous in both directions; a continuous bijection alone need not be a homeomorphism.
Knowledge object
Sampling distribution
At a fixed parameter, an estimator pushes the sample law forward to its sampling distribution; all parameters give a family of such laws.
Knowledge object
Weak law of large numbers
For IID variables with finite variance, a Chebyshev bound shows the sample mean converges in probability to the common mean.
Knowledge object
Chromatic number
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.
Knowledge object
Cliques and independent sets
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.
Knowledge object
Estimator bias
Bias at theta is the estimator expectation under its theta sampling law minus the target parameter; unbiasedness requires zero bias at every theta.
Knowledge object
Estimator variance
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.
Knowledge object
Fermat's little theorem
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.
Knowledge object
Linear Diophantine equation
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.
Knowledge object
Quotient vector space
Cosets of a subspace W form the quotient vector space V/W, whose operations must be independent of representative choices.
Knowledge object
Finite group action
An action of a finite group on a finite set is a total action table satisfying the identity and multiplication-compatibility laws.
Knowledge object
Supremum and infimum
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.