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For a braid with M strands, R components, P positive crossings, and N negative crossings, {P-N<=U_++M-R if P>=N; P-N<=U_-+M-R if P<=N, (1) where U_+/- are the smallest number ...

A set S in a metric space (S,d) is bounded if it has a finite generalized diameter, i.e., there is an R<infty such that d(x,y)<=R for all x,y in S. A set in R^n is bounded ...

A particular solution to a differential equation corresponding to a specific value of the equation's free parameters. For example, the integral curves of the differential ...

The projective special linear group PSL_n(q) is the group obtained from the special linear group SL_n(q) on factoring by the scalar matrices contained in that group. It is ...

Let one grain of wheat be placed on the first square of a chessboard, two on the second, four on the third, eight on the fourth, etc. How many grains total are placed on an ...

The simple continued fraction for pi is given by [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...] (OEIS A001203). A plot of the first 256 terms of the ...

The Heilbronn triangle problem is to place n>=3 points in a disk (square, equilateral triangle, etc.) of unit area so as to maximize the area Delta(n) of the smallest of the ...

The usual number of scalar operations (i.e., the total number of additions and multiplications) required to perform n×n matrix multiplication is M(n)=2n^3-n^2 (1) (i.e., n^3 ...

A plane path on a set of equally spaced lattice points, starting at the origin, where the first step is one unit to the north or south, the second step is two units to the ...

Just as the ratio of the arc length of a semicircle to its radius is always pi, the ratio P of the arc length of the parabolic segment formed by the latus rectum of any ...