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Given a sequence of real numbers a_n, the supremum limit (also called the limit superior or upper limit), written lim sup and pronounced 'lim-soup,' is the limit of ...

A parallelohedron is a space-filling polyhedron that fills space using an infinite number of similarly situated copies (Tutton 1964, pp. 567 and 723; Coxeter 1973, pp. ...

The absolute value of a real number x is denoted |x| and defined as the "unsigned" portion of x, |x| = xsgn(x) (1) = {-x for x<=0; x for x>=0, (2) where sgn(x) is the sign ...

Although Descartes originally used the term "imaginary number" to refer to what is today known as a complex number, in standard usage today, "imaginary number" means a ...

The perspective image of an infinite checkerboard. It can be constructed starting from any triangle DeltaOXY, where OX and OY form the near corner of the floor, and XY is the ...

The nullity of a linear transformation f:V->W of vector spaces is the dimension of its null space. The nullity and the map rank add up to the dimension of V, a result ...

For a normed space (X,||·||), define X^~ to be the set of all equivalent classes of Cauchy sequences obtained by the relation {x_n}∼{y_n} if and only if lim_(n)||x_n-y_n||=0. ...

Given a complex Hilbert space H with associated space L(H) of continuous linear operators from H to itself, the bicommutant M^('') of an arbitrary subset M subset= L(H) is ...

Let V(r) be the volume of a ball of radius r in a complete n-dimensional Riemannian manifold with Ricci curvature tensor >=(n-1)kappa. Then V(r)<=V_kappa(r), where V_kappa is ...

Given a complex Hilbert space H with associated space L(H) of continuous linear operators from H to itself, the commutant M^' of an arbitrary subset M subset= L(H) is the ...