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A definition assigns properties to some sort of mathematical object. For example, Euclid's Elements starts with a number of definitions, such as "a line is a breadthless ...

A piecewise regular function that 1. Has a finite number of finite discontinuities and 2. Has a finite number of extrema can be expanded in a Fourier series which converges ...

The Gaussian joint variable theorem, also called the multivariate theorem, states that given an even number of variates from a normal distribution with means all 0, (1) etc. ...

The rank of a matrix or a linear transformation is the dimension of the image of the matrix or the linear transformation, corresponding to the number of linearly independent ...

A near-perfect matching is a matching in which a single vertex is left unmatched. Just as perfect matchings can occur only for graphs with an even number of vertices, ...

An ideal I of a partial order P is a subset of the elements of P which satisfy the property that if y in I and x<y, then x in I. For k disjoint chains in which the ith chain ...

The maximal number of regions into which n lines divide a plane are N(n)=1/2(n^2+n+2) which, for n=1, 2, ... gives 2, 4, 7, 11, 16, 22, ... (OEIS A000124), the same maximal ...

Let R+B be the number of monochromatic forced triangles (where R and B are the number of red and blue triangles) in an extremal graph. Then R+B=(n; 3)-|_1/2n|_1/4(n-1)^2_|_|, ...

Let p be a prime number, G a finite group, and |G| the order of G. 1. If p divides |G|, then G has a Sylow p-subgroup. 2. In a finite group, all the Sylow p-subgroups are ...

The companion matrix to a monic polynomial a(x)=a_0+a_1x+...+a_(n-1)x^(n-1)+x^n (1) is the n×n square matrix A=[0 0 ... 0 -a_0; 1 0 ... 0 -a_1; 0 1 ... 0 -a_2; | | ... ... |; ...