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A cubic triangular number is a positive integer that is simultaneously cubic and triangular. Such a number must therefore satisfy T_n=m^3 for some positive integers n and m, ...

Given a hereditary representation of a number n in base b, let B[b](n) be the nonnegative integer which results if we syntactically replace each b by b+1 (i.e., B[b] is a ...

If p_1, ..., p_n are positive numbers which sum to 1 and f is a real continuous function that is convex, then f(sum_(i=1)^np_ix_i)<=sum_(i=1)^np_if(x_i). (1) If f is concave, ...

The group Gamma of all Möbius transformations of the form tau^'=(atau+b)/(ctau+d), (1) where a, b, c, and d are integers with ad-bc=1. The group can be represented by the 2×2 ...

If p is a prime >3, then the numerator of the harmonic number H_(p-1)=1+1/2+1/3+...+1/(p-1) (1) is divisible by p^2 and the numerator of the generalized harmonic number ...

The d-analog of a complex number s is defined as [s]_d=1-(2^d)/(s^d) (1) (Flajolet et al. 1995). For integer n, [2]!=1 and [n]_d! = [3][4]...[n] (2) = ...

The q-analog of integration is given by int_0^1f(x)d(q,x)=(1-q)sum_(i=0)^inftyf(q^i)q^i, (1) which reduces to int_0^1f(x)dx (2) in the case q->1^- (Andrews 1986 p. 10). ...

A generalization of the p-adic norm first proposed by Kürschák in 1913. A valuation |·| on a field K is a function from K to the real numbers R such that the following ...

Consider a square wave f(x) of length 2L. Over the range [0,2L], this can be written as f(x)=2[H(x/L)-H(x/L-1)]-1, (1) where H(x) is the Heaviside step function. Since ...

A concordance between knots K_0 and K_1 in S^3 is a locally flat cylinder C=S^1×[0,1] embedded in S^3×[0,1] in such a way that the ends S^1×{1} are embedded in S^3×{i} as ...