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Qualitatively, a deep theorem is a theorem whose proof is long, complicated, difficult, or appears to involve branches of mathematics which are not obviously related to the ...

Let |A| be an n×n determinant with complex (or real) elements a_(ij), then |A|!=0 if |a_(ii)|>sum_(j=1; j!=i)^n|a_(ij)|.

If X is any space, then there is a CW-complex Y and a map f:Y->X inducing isomorphisms on all homotopy, homology, and cohomology groups.

The only linear associative algebra in which the coordinates are real numbers and products vanish only if one factor is zero are the field of real numbers, the field of ...

The reciprocity theorem for x^8=q (mod p).

A sequence of approximations a/b to sqrt(n) can be derived by factoring a^2-nb^2=+/-1 (1) (where -1 is possible only if -1 is a quadratic residue of n). Then ...

The conjugate gradient method can be applied on the normal equations. The CGNE and CGNR methods are variants of this approach that are the simplest methods for nonsymmetric ...

Let K be a finite complex, let h:|K|->|K| be a continuous map. If Lambda(h)!=0, then h has a fixed point.

The three circles theorem, also called Hadamard's three circles theorem (Edwards 2001, p. 187), states that if f is an analytic function in the annulus 0<r_1<|z|<r_2<infty, ...

Simplemindedly, a number theoretic transform is a generalization of a fast Fourier transform obtained by replacing e^(-2piik/N) with an nth primitive root of unity. This ...