Related papers: The gamma function and a certain sequence of diffe…

We prove some new results and unify the proofs of old ones involving complete monotonicity of expressions involving gamma and $q$-gamma functions, $0 < q < 1$. Each of these results implies the infinite divisibility of a related probability…

We introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings; that is, we study the number of ways a permutation can be decomposed into a product of a given number of 2-cycles, 3-cycles, etc.…

We establish a one-to-one correspondence between the set of minimal exponential families of dimension n defined on a finite sample space {\Omega} and the affine Grassmannian associated to an appropriate vector space of functions.

We consider nonregular fractions of factorial experiments for a class of linear models. These models have a common general mean and main effects, however they may have different 2-factor interactions. Here we assume for simplicity that…

The Carmichael lambda function $\lambda(n)$ is defined to be the smallest positive integer $m$ such that $a^m$ is congruent to 1 modulo $n,$ for all $a$ and $n$ relatively prime. The function $\lambda_k(n)$ is defined to be the $k$th…

The break-by-one gamma distribution has a probability density function resembling the Schechter function, but with the small-argument behavior modified so it is normalizable in commonly arising cases where the Schechter function is not. Its…

To factor an integer N, given that it is equal to the product of two primes, it suffices to find an integer d satisfying a certain simple numerical test. In this approach, the factorization problem equates to the problem of designing an…

Using a probabilistic approach, we derive some interesting combinatorial identities involving gamma and beta functions. These results generalize certain well-known combinatorial identities involving binomial coefficients and special…

Let $f(n)$ be the number of distinct exponents in the prime factorization of the natural number $n$. We prove some results about the distribution of $f(n)$. In particular, for any positive integer $k$, we obtain that $$ \#\{n \leq x : f(n)…

Functional data analysis is becoming increasingly popular to study data from real-valued random functions. Nevertheless, there is a lack of multiple testing procedures for such data. These are particularly important in factorial designs to…

For $n\ge 3$ let $f(n)$ be the least positive integer $k$ such that $\binom nk>\frac{2^n}{n+1}$. In this paper we investigate the properties of $f(n)$.

We use probabilistic methods to find lower bounds on the maximum number, in a graph with domination number \gamma, of dominating sets of size \gamma. We find that we can randomly generate a graph that, w.h.p., is dominated by almost all…

Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$…

As a generalization of the results [KW3],we study the functional equation of the higher Selberg zeta function for congruence subgroups. To obtain the gamma factor of this function, we introduce a higher Dirichlet $L$-function. Then we…

Let $f$ be a real-valued function of a single variable such that it is positive over the primes. In this article, we construct a factorial, $n!_f$, associated to $f$, called the associated Legendre formula, or $f$-factorial, and show,…

The article discusses selected problems related to both principal component analysis (PCA) and factor analysis (FA). In particular, both types of analysis were compared. A vector interpretation for both PCA and FA has also been proposed.…

The present research deals with generalizations of the Salem function with arguments defined in terms of certain alternating expansions of real numbers. The special attention is given to modelling such functions by systems of functional…

We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. For $m = 1$, this is the classical…

We adapt the rectangular splitting technique of Paterson and Stockmeyer to the problem of evaluating terms in holonomic sequences that depend on a parameter. This approach allows computing the $n$-th term in a recurrent sequence of suitable…

We use the periodicity properties of generalized Gauss sums to factor numbers. Moreover, we derive rules for finding the factors and illustrate this factorization scheme for various examples. This algorithm relies solely on interference and…