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There are two definitions of Bernoulli polynomials in use. The nth Bernoulli polynomial is denoted here by B_n(x) (Abramowitz and Stegun 1972), and the archaic form of the ...

Given two groups G and H, there are several ways to form a new group. The simplest is the direct product, denoted G×H. As a set, the group direct product is the Cartesian ...

Informally, an elliptic curve is a type of cubic curve whose solutions are confined to a region of space that is topologically equivalent to a torus. The Weierstrass elliptic ...

Any two rectilinear figures with equal area can be dissected into a finite number of pieces to form each other. This is the Wallace-Bolyai-Gerwien theorem. For minimal ...

The two recursive sequences U_n = mU_(n-1)+U_(n-2) (1) V_n = mV_(n-1)+V_(n-2) (2) with U_0=0, U_1=1 and V_0=2, V_1=m, can be solved for the individual U_n and V_n. They are ...

A factorization algorithm which works by expressing N as a quadratic form in two different ways. Then N=a^2+b^2=c^2+d^2, (1) so a^2-c^2=d^2-b^2 (2) (a-c)(a+c)=(d-b)(d+b). (3) ...

Find the tunnel between two points A and B on a gravitating sphere which gives the shortest transit time under the force of gravity. Assume the sphere to be nonrotating, of ...

A Gaussian integer is a complex number a+bi where a and b are integers. The Gaussian integers are members of the imaginary quadratic field Q(sqrt(-1)) and form a ring often ...

Scherk's two minimal surfaces were discovered by Scherk in 1834. They were the first new surfaces discovered since Meusnier in 1776. Beautiful images of wood sculptures of ...

A collection of identities which hold on a Kähler manifold, also called the Hodge identities. Let omega be a Kähler form, d=partial+partial^_ be the exterior derivative, ...