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An n×n magic square for which every pair of numbers symmetrically opposite the center sum to n^2+1. The Lo Shu is associative but not panmagic. The numbers of associative ...

Let P(x) be defined as the power series whose nth term has a coefficient equal to the nth prime p_n, P(x) = 1+sum_(k=1)^(infty)p_kx^k (1) = 1+2x+3x^2+5x^3+7x^4+11x^5+.... (2) ...

An integer n is p-balanced for p a prime if, among all nonzero binomial coefficients (n; k) for k=0, ..., n (mod p), there are equal numbers of quadratic residues and ...

A binary bracketing is a bracketing built up entirely of binary operations. The number of binary bracketings of n letters (Catalan's problem) are given by the Catalan numbers ...

An apodization function given by A(x)=(21)/(50)+1/2cos((pix)/a)+2/(25)cos((2pix)/a), (1) which has full width at half maximum of 0.810957a. This function is defined so that ...

A Tschirnhausen transformation can be used to algebraically transform a general quintic equation to the form z^5+c_1z+c_0=0. (1) In practice, the general quintic is first ...

Using a Tschirnhausen transformation, the principal quintic form can be transformed to the one-parameter form w^5-10cw^3+45c^2w-c^2=0 (1) named after Francesco Brioschi ...

A catastrophe which can occur for four control factors and one behavior axis. The butterfly catastrophe is the universal unfolding of the singularity f(x)=x^6 of codimension ...

The maximum number of regions that can be created by n cuts using space division by planes, cube division by planes, cylinder cutting, etc., is given by N_(max)=1/6(n^3+5n+6) ...

Consider a unit circle and a radiant point located at (mu,0). There are four different regimes of caustics, illustrated above. For radiant point at mu=infty, the catacaustic ...