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A pair of zips, each zip being half a zipper, which can be zippered up to close a surface along a curve. The concept of a zip-pair can be extremely useful in topological ...

If f is continuous on a closed interval [a,b], then there is at least one number x^* in [a,b] such that int_a^bf(x)dx=f(x^*)(b-a). The average value of the function (f^_) on ...

The set R union {infty}, obtained by adjoining one improper element to the set R of real numbers, is the set of projectively extended real numbers. Although notation is not ...

Given integers a and b with close to 2n bits each, the half-GCD of a and b is a 2×2 matrix [u v; u^' v^'] with determinant equal to -1 or 1 such that ua+vb=r and ...

Using a Chebyshev polynomial of the first kind T(x), define c_j = 2/Nsum_(k=1)^(N)f(x_k)T_j(x_k) (1) = 2/Nsum_(k=1)^(N)f[cos{(pi(k-1/2))/N}]cos{(pij(k-1/2))/N}. (2) Then f(x) ...

With a large enough sample, any outrageous thing is likely to happen (Diaconis and Mosteller 1989). Littlewood (1986) considered an event which occurs one in a million times ...

A Liouville number is a transcendental number which has very close rational number approximations. An irrational number beta is called a Liouville number if, for each n, ...

The function z=f(x)=ln(x/(1-x)). (1) This function has an inflection point at x=1/2, where f^('')(x)=(2x-1)/(x^2(x-1)^2)=0. (2) Applying the logit transformation to values ...

A matrix is a concise and useful way of uniquely representing and working with linear transformations. In particular, every linear transformation can be represented by a ...

Given a set S of n nonnegative integers, the number partitioning problem requires the division of S into two subsets such that the sums of number in each subset are as close ...