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A diagram lemma which states that, given the commutative diagram of additive Abelian groups with exact rows, the following holds: 1. If f_0 is surjective, and f_1 and f_3 are ...

An important result in valuation theory which gives information on finding roots of polynomials. Hensel's lemma is formally stated as follows. Let (K,|·|) be a complete ...

If S is any nonempty partially ordered set in which every chain has an upper bound, then S has a maximal element. This statement is equivalent to the axiom of choice. Renteln ...

The Riemann-Lebesgue Lemma, sometimes also called Mercer's theorem, states that lim_(n->infty)int_a^bK(lambda,z)Csin(nz)dz=0 (1) for arbitrarily large C and "nice" ...

A fundamental structural result in extremal graph theory due to Szemerédi (1978). The regularity lemma essentially says that every graph can be well-approximated by the union ...

The standard Gauss measure of a finite dimensional real Hilbert space H with norm ||·||_H has the Borel measure mu_H(dh)=(sqrt(2pi))^(-dim(H))exp(1/2||h||_H^2)lambda_H(dh), ...

If a contour in the complex plane is curved such that it separates the increasing and decreasing sequences of poles, then ...

A diagram lemma which states that the above commutative diagram of Abelian groups and group homomorphisms with exact rows gives rise to an exact sequence This commutative ...

_2F_1(a,b;c;1)=((c-b)_(-a))/((c)_(-a))=(Gamma(c)Gamma(c-a-b))/(Gamma(c-a)Gamma(c-b)) for R[c-a-b]>0, where _2F_1(a,b;c;x) is a (Gauss) hypergeometric function. If a is a ...

Given two normal subgroups G_1 and G_2 of a group, and two normal subgroups H_1 and H_2 of G_1 and G_2 respectively, H_1(G_1 intersection H_2) is normal in H_1(G_1 ...