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The convolution of two complex-valued functions on a group G is defined as (a*b)(g)=sum_(k in G)a(k)b(k^(-1)g) where the support (set which is not zero) of each function is ...

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)|.

A theorem proved by É. Cartan in 1913 which classifies the irreducible representations of complex semisimple Lie algebras.

Let K be a finite complex, and let phi:C_p(K)->C_p(K) be a chain map, then sum_(p)(-1)^pTr(phi,C_p(K))=sum_(p)(-1)^pTr(phi_*,H_p(K)/T_p(K)).

Let V!=(0) be a finite dimensional vector space over the complex numbers, and let A be a linear operator on V. Then V can be expressed as a direct sum of cyclic subspaces.

If K is a finite complex and h:|K|->|K| is a continuous map, then Lambda(h)=sum(-1)^pTr(h_*,H_p(K)/T_p(K)) is the Lefschetz number of the map h.

A line bundle is a special case of a vector bundle in which the fiber is either R, in the case of a real line bundle, or C, in the case of a complex line bundle.

A regular surface M subset R^n is called orientable if each tangent space M_p has a complex structure J_p:M_p->M_p such that p->J_p is a continuous function.

A matrix A for which A^(H)=A^(T)^_=A, where the conjugate transpose is denoted A^(H), A^(T) is the transpose, and z^_ is the complex conjugate. If a matrix is self-adjoint, ...

The single bar | is a notation variously used to denote the absolute value |x|, complex modulus |z|, vector norm |x|, determinant |A|, or "divides" (a|b).