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The first and second Zagreb indices for a graph with vertex count n and vertex degrees v_i for i=1, ..., n are defined by Z_1=sum_(i=1)^nv_i^2 and Z_2=sum_((i,j) in ...

In the English language, the probability of encountering the rth most common word is given roughly by P(r)=0.1/r for r up to 1000 or so. The law breaks down for less frequent ...

If 1<=b<a and (a,b)=1 (i.e., a and b are relatively prime), then a^n-b^n has at least one primitive prime factor with the following two possible exceptions: 1. 2^6-1^6. 2. ...

Let a random n×n (0,1)-matrix have entries which are 1 (with probability p) or 0 (with probability q=1-p) and numbers are assigned to the edges of a grid. A b-cluster is an ...

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 constants C_n defined by C_n=[int_0^infty|d/(dt)((sint)/t)^n|dt]-1. (1) These constants can also be written as the sums C_n=2sum_(k=1)^infty(1+x_k^2)^(-n/2), (2) and ...

The h-statistic h_r is the unique symmetric unbiased estimator for a central moment of a distribution <h_r>=mu_r. (1) In addition, the variance var(h_r)=<(h_r-mu_r)^2> (2) is ...

The q-analog of the binomial theorem (1-z)^n=1-nz+(n(n-1))/(1·2)z^2-(n(n-1)(n-2))/(1·2·3)z^3+... (1) is given by (1-z/(q^n))(1-z/(q^(n-1)))...(1-z/q) ...

There are several q-analogs of the cosine function. The two natural definitions of the q-cosine defined by Koekoek and Swarttouw (1998) are given by cos_q(z) = ...

D_q=1/(1-q)lim_(epsilon->0)(lnI(q,epsilon))/(ln(1/epsilon),) (1) where I(q,epsilon)=sum_(i=1)^Nmu_i^q, (2) epsilon is the box size, and mu_i is the natural measure. The ...