Foundations

If F=∇f with f of class C¹, then ∫C F·dr=f(endpoint)−f(start), independent of the piecewise smooth path.

For planar F=(P,Q) of class C¹ on a neighborhood of a bounded region D, ∮∂D(P dx+Q dy)=∫D(Qₓ−Pᵧ)dA and outward flux ∮∂D(P dy−Q dx)=∫D(Pₓ+Qᵧ)dA.

For −(py′)′+qy=λwy on [a,b] with homogeneous Dirichlet endpoints, real p,p′,q,w continuous, p,w>0 and q≥0, eigenvalues increase to infinity and eigenfunctions are unique up to scale.

For the stated regular Sturm–Liouville problem and continuous piecewise smooth f, coefficients cₙ=∫f yₙw dx/∫yₙ²w dx give f=Σcₙyₙ at interior points.

If a language is regular, some p≥1 ensures every accepted word x with |x|≥p decomposes as uvw with |uv|≤p, |v|≥1 and uvᵐw accepted for all m≥0.

The language {0ⁿ1ⁿ:n≥0} is not regular.

A DFA's language is empty iff no accepting state is reachable from its start.

A finite context-free grammar has decidable membership: convert to Chomsky normal form and enumerate bounded derivations for a nonempty input word.

For a finite CFG, chart Iⱼ stores dotted productions [A→β·γ,k], where k is the starting position.

For X with density f and measurable g, E[g(X)]=∫R g(x)f(x)dx when ∫|g(x)|f(x)dx<∞.

For nonnegative X with finite mean and a>0, P(X≥a)≤E[X]/a.

If √n(θ̂−θ) converges in distribution to N(0,v) and g is differentiable at θ with g′(θ)≠0, then √n(g(θ̂)−g(θ)) converges to N(0,[g′(θ)]²v).

When testing m≥1 hypotheses, reject each only at a valid individual level α/m.

For simple null and alternative laws with common mass/density support, a test rejecting for sufficiently large alternative/null likelihood ratio is most powerful against that alternative among tests of no greater null rejection probability.

For nested parametric models, Λ is the null-model supremum likelihood divided by the full-model supremum likelihood.

For one multinomial sample in an r×c table, independence fits expected counts Eᵢⱼ=Nᵢ·N·ⱼ/n.

For k groups let B=Σnᵢ(x̄ᵢ−x̄)² and W=Σᵢⱼ(xᵢⱼ−x̄ᵢ)².

The bijections of an n-element set form Sₙ under composition, with n! elements.

The parity of the number of transpositions in a permutation factorization is invariant.

Elements a,b of G are conjugate when b=gag⁻¹ for some g∈G; this is an equivalence relation.

A left action G×X→X satisfies e·x=x and (gh)·x=g·(h·x).

For a left action, Orb(x)={g·x:g∈G} and Stab(x)={g:g·x=x}.

For finite G, C_G(x)={g:gx=xg} is the conjugation stabilizer and Z(G) consists of elements commuting with every g.

If |G|=pᵏ with prime p and k≥1, the class equation implies p divides |Z(G)|, so Z(G) is nontrivial.

Aut(G) consists of group isomorphisms G→G under composition.

If a prime p divides the order of finite G, G contains an element of order p and hence a cyclic subgroup of order p.

For finite |G|=pᵏm, prime p, k≥1 and p∤m, Sylow p-subgroups have order pᵏ.

For distinct primes p<q, a group of order pq has one Sylow q-subgroup, so it is normal.

For a finite group acting on finite X, the number of orbits is |G|⁻¹Σg∈G|Fix(g)|, where Fix(g)={x:g·x=x}.

G×H uses coordinatewise multiplication.