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Consider a knot as being formed from two tangles. The following three operations are called mutations. 1. Cut the knot open along four points on each of the four strings ...

A number theoretic character, also called a Dirichlet character (because Dirichlet first introduced them in his famous proof that every arithmetic progression with relatively ...

The golden ratio phi can be written in terms of a nested radical in the beautiful form phi=sqrt(1+sqrt(1+sqrt(1+sqrt(1+...)))), (1) which can be written recursively as ...

A partially ordered set (or poset) is a set taken together with a partial order on it. Formally, a partially ordered set is defined as an ordered pair P=(X,<=), where X is ...

An n-he (a term coined by Brendan Owen) is a shape formed from a polyhex by removing half of each hexagon in such a way that the remaining pieces are connected (Clarke). The ...

Consider the expression 3×7+2^2. This expression has value (3×7)+(2^2)=25 due to what is called operator precedence (or "order of operations"). Precedence of common operators ...

A formula of first-order logic is in prenex normal form if it is of the form Q_1x_1...Q_nx_nM, (1) where each Q_i is a quantifier forall ("for all") or exists ("exists") and ...

A set A of integers is productive if there exists a partial recursive function f such that, for any x, the following holds: If the domain of phi_x is a subset of A, then f(x) ...

The nth Ramanujan prime is the smallest number R_n such that pi(x)-pi(x/2)>=n for all x>=R_n, where pi(x) is the prime counting function. In other words, there are at least n ...

Let phi(n) be any function, say analytic or integrable. Then int_0^inftyx^(s-1)sum_(k=0)^infty(-1)^kx^kphi(k)dx=(piphi(-s))/(sin(spi)) (1) and ...