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Let f:M|->N be a map between two compact, connected, oriented n-dimensional manifolds without boundary. Then f induces a homomorphism f_* from the homology groups H_n(M) to ...

A bubble is a minimal-energy surface of the type that is formed by soap film. The simplest bubble is a single sphere, illustrated above (courtesy of J. M. Sullivan). More ...

Bürmann's theorem deals with the expansion of functions in powers of another function. Let phi(z) be a function of z which is analytic in a closed region S, of which a is an ...

There are two curves known as the butterfly curve. The first is the sextic plane curve given by the implicit equation y^6=x^2-x^6 (1) (Cundy and Rollett 1989, p. 72; left ...

Given two normal subgroups G_1 and G_2 of a group, and two normal subgroups H_1 and H_2 of G_1 and G_2 respectively, H_1(G_1 intersection H_2) is normal in H_1(G_1 ...

Given a chord PQ of a circle, draw any other two chords AB and CD passing through its midpoint. Call the points where AD and BC meet PQ X and Y. Then M is also the midpoint ...

A binary unit of information equal to 8 bits. Unfortunately, the storage of binary numbers in computers is not entirely standardized. Because computers store information in ...

A C-matrix is a symmetric (C^(T)=C) or antisymmetric (C^(T)=-C) C_n (-1,0,1)-matrix with diagonal elements 0 and others +/-1 that satisfies CC^(T)=(n-1)I, (1) where I is the ...

The nth cabtaxi number is the smallest positive number that can be written in n ways as a sum of two (not necessarily positive) cubes. The name derived from the taxicab ...

A Calkin-Wilf tree is a special type of binary tree obtained by starting with the fraction 1/1 and iteratively adding a/(a+b) and (a+b)/b below each fraction a/b. The ...