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On a three-dimensional lattice, a random walk has less than unity probability of reaching any point (including the starting point) as the number of steps approaches infinity. ...

Let X be a normed space and X^(**)=(X^*)^* denote the second dual vector space of X. The canonical map x|->x^^ defined by x^^(f)=f(x),f in X^* gives an isometric linear ...

Consider a two-dimensional tessellation with q regular p-gons at each polygon vertex. In the plane, (1-2/p)pi=(2pi)/q (1) 1/p+1/q=1/2, (2) so (p-2)(q-2)=4 (3) (Ball and ...

The Reynolds transport theorem, also called simply the Reynolds theorem, is an important result in fluid mechanics that's often considered a three-dimensional analog of the ...

The Riemann sphere, also called the extended complex plane, is a one-dimensional complex manifold C^* (C-star) which is the one-point compactification of the complex numbers ...

The rule which determines the orientation of the cross product u×v. The right-hand rule states that the orientation of the vectors' cross product is determined by placing u ...

The d-dimensional rigidity matrix M(G) of a graph G with vertex count n, edge count m in the variables v_i=(x_1,...,x_d) is the m×(dn) matrix with rows indexed by the edges ...

A rotation group is a group in which the elements are orthogonal matrices with determinant 1. In the case of three-dimensional space, the rotation group is known as the ...

Suppose f(x) is a function of x that is twice differentiable at a stationary point x_0. 1. If f^('')(x_0)>0, then f has a local minimum at x_0. 2. If f^('')(x_0)<0, then f ...

One of the beautiful arrangements of circles found at the Temple of Osiris at Abydos, Egypt (Rawles 1997). The pattern also appears in Italian art from the 13th century ...