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A Banach space is a complete vector space B with a norm ||·||. Two norms ||·||_((1)) and ||·||_((2)) are called equivalent if they give the same topology, which is equivalent ...

A set of positive integers is double-free if, for any integer x, the set {x,2x} !subset= S (or equivalently, x in S implies 2x not in S). For example, of the subsets of ...

A map u:M->N, between two compact Riemannian manifolds, is a harmonic map if it is a critical point for the energy functional int_M|du|^2dmu_M. The norm of the differential ...

For n a positive integer, expressions of the form sin(nx), cos(nx), and tan(nx) can be expressed in terms of sinx and cosx only using the Euler formula and binomial theorem. ...

The geodesic on an oblate spheroid can be computed analytically, although the resulting expression is much more unwieldy than for a simple sphere. A spheroid with equatorial ...

Let p_n be the nth prime, then the primorial (which is the analog of the usual factorial for prime numbers) is defined by p_n#=product_(k=1)^np_k. (1) The values of p_n# for ...

The "witch of Agnesi" is a curve studied by Maria Agnesi in 1748 in her book Instituzioni analitiche ad uso della gioventù italiana (the first surviving mathematical work ...

Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations. As shown by Cramer's rule, a nonhomogeneous system of ...

The game of life is the best-known two-dimensional cellular automaton, invented by John H. Conway and popularized in Martin Gardner's Scientific American column starting in ...

Given a "good" graph G (i.e., one for which all intersecting graph edges intersect in a single point and arise from four distinct graph vertices), the crossing number is the ...