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By asking a small number of innocent-sounding questions about an unknown number, it is possible to reconstruct the number with absolute certainty (assuming that the questions ...

Three-dimensional generalization of the polyominoes to n dimensions. The number of polycubes N(n) composed of n cubes are 1, 1, 2, 8, 29, 166, 1023, ... (OEIS A000162; Ball ...

Let f be a contraction mapping from a closed subset F of a Banach space E into F. Then there exists a unique z in F such that f(z)=z.

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 ...

Let A be a closed convex subset of a Banach space and assume there exists a continuous map T sending A to a countably compact subset T(A) of A. Then T has fixed points.

Inside a ball B in R^3, {rectifiable currents S in BL area S<=c, length partialS<=c} is compact under the flat norm.

The Dirichlet kernel D_n^M is obtained by integrating the number theoretic character e^(i<xi,x>) over the ball |xi|<=M, D_n^M=-1/(2pir)d/(dr)D_(n-2)^M.

Place a point somewhere on a line segment. Now place a second point and number it 2 so that each of the points is in a different half of the line segment. Continue, placing ...

The French metro metric is an example for disproving apparently intuitive but false properties of metric spaces. The metric consists of a distance function on the plane such ...

A prismatoid having planar sides and the same number of vertices in both of its parallel planes. The faces of a prismoid are therefore either trapezoids or parallelograms. ...