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In the theory of transfinite ordinal numbers, 1. Every well ordered set has a unique ordinal number, 2. Every segment of ordinals (i.e., any set of ordinals arranged in ...

For any system that seeks to minimize a function, only seven different local forms of catastrophe "typically" occur for four or fewer variables: 1. Fold catastrophe, 2. Cusp ...

An axiomatic system is said to be categorical if there is only one essentially distinct representation for it. In particular, the names and types of objects within the system ...

A Gaussian quadrature-like formula for numerical estimation of integrals. It uses weighting function W(x)=1 in the interval [-1,1] and forces all the weights to be equal. The ...

Let r and s be positive integers which are relatively prime and let a and b be any two integers. Then there is an integer N such that N=a (mod r) (1) and N=b (mod s). (2) ...

The pedal curve of a unit circle with parametric equation x = cost (1) y = sint (2) with pedal point (x,y) is x_p = cost-ycostsint+xsin^2t (3) y_p = ...

Let E be a simply connected compact set in the complex plane. By the Riemann mapping theorem, there is a unique analytic function ...

A recursively enumerable set A is creative if its complement is productive. Creative sets are not recursive. The property of creativeness coincides with completeness. Namely, ...

Consider the distribution of distances l between a point picked at random in the interior of a unit cube and on a face of the cube. The probability function, illustrated ...

A Dyck path is a staircase walk from (0,0) to (n,n) that lies strictly below (but may touch) the diagonal y=x. The number of Dyck paths of order n is given by the Catalan ...