Search Results for ""

The permanent of an n×n integer matrix with all entries either 0 or 1 is 0 iff the matrix contains an r×s submatrix of 0s with r+s=n+1. This result follows from the ...

The amazing identity for all theta, where Gamma(z) is the gamma function. Equating coefficients of theta^0, theta^4, and theta^8 gives some amazing identities for the ...

A set of identities involving n-dimensional visible lattice points was discovered by Campbell (1994). Examples include product_((a,b)=1; ...

The Robertson-Seymour theorem, also called the graph minor theorem, is a generalization of the Kuratowski reduction theorem by Robertson and Seymour, which states that the ...

If a function f(x) is continuous on a closed interval [a,b], then f(x) has both a maximum and a minimum on [a,b]. If f(x) has an extremum on an open interval (a,b), then the ...

Let the squares square ABCD and square AB^'C^'D^' share a common polygon vertex A. The midpoints Q and S of the segments B^'D and BD^' together with the centers of the ...

Tarski's theorem says that the first-order theory of reals with +, *, =, and > allows quantifier elimination. Algorithmic quantifier elimination implies decidability assuming ...

Given any real number theta and any positive integer N, there exist integers h and k with 0<k<=N such that |ktheta-h|<1/N. A slightly weaker form of the theorem states that ...

For all n, there exists a k such that the kth term of the Goodstein sequence G_k(n)=0. In other words, every Goodstein sequence converges to 0. The secret underlying ...

Mergelyan's theorem can be stated as follows (Krantz 1999). Let K subset= C be compact and suppose C^*\K has only finitely many connected components. If f in C(K) is ...