In this paper we establish a very general and useful theorem which interconnects the Laplace transform and the generalized Weyl fractional integral operator involving the multivariable H-function of related functions of several variables. Our main theorem involves a multidimensional series with essentially arbitrary sequence of complex numbers. By suitably assigning different values to these sequences, one can easily evaluate the generalized Weyl fractional integral operator of special functions of several variables. We have illustrated it for Srivastava-Daoust multivariable hypergeometric function. On account of general nature of this function a number of results involving special functions of one or more variables can be obtained merely by specializing the parameters.

The classical Lotka-Volterra system of two first-order nonlinear differential equations was investigated by Lotka [25] on chemical reactions, Lotka [26] on rhythmical reactions in physiology, Lotka [27] on parasitology, Volterra [46] on fishing activity in the upper Adriatic Sea, and Kozyreff [23] and Erneux and Kozyreff [10] on laser dynamics in the vicinity of the Hopf bifurcation. A functional relation between two dependent variables has been known to describe its periodic behavior in the phase plane. We first solve this equation explicitly for one dependent variable in terms of the other, and then obtain two integral representations of the period having a singularity of the square root type at each endpoint of the integration. Our notations are based on Lambert's W functions, which are two inverse functions of $x \to x\exp(x)$ restricted to $(-\infty, -1]$ and $[-1, \infty)$, respectively. A power series of the period is constructed in small energy for arbitrary number of terms by virtue of expansions of Lambert's W functions near the branch point $x = -\exp(-1)$. Our result settles the discrepancy of various approximate results in the literature, and are further compared with numerical results of computing the period by applying the Gauss-Tschebyscheff integration rule of the first kind.

Let $\mu$ be a probability measure on the real line with finite moments of all orders. Suppose the linear span of polynomials is dense in $L^2(\mu)$. Then there exists a sequence $\{P_n\}_{n=0}^\infty$ of orthogonal polynomials with respect to $\mu$ such that $P_n$ is a polynomial of degree $n$ with leading coefficient $1$ and the equality $(x-\alpha_n) P_n(x) = P_{n+1}(x) + \omega_n P_{n-1}(x)$ holds, where $\alpha_n$ and $\omega_n$ are Szeg\"o-Jacobi parameters. In this paper we use the concepts of pre-generating function, multiplicative renormalization, and generating function to derive $\{P_n, \alpha_n, \omega_n\}$ from a given $\mu$. Two types of pre-generating functions are studied. We apply our method to the special distributions such as Gaussian, Poisson, gamma, uniform, arcsine, semi-circle, and beta-type to derive $\{P_n, \alpha_n, \omega_n\}$. Moreover, we show that the corresponding polynomials $P_n$'s are exactly the classical polynomials such as Hermite, Charlier, Laguerre, Legendre, Chebyshev of the first kind, Chebyshev of the second kind, and Gegenbauer. We also apply our method to study the negative binomial distributions.

We consider a local stable reduction of a family of curves with smooth fibers except a central fiber that has a singularity like $(y^a-x^b)(y^p-x^q)$.

A coupled functional differential systems depending on two parameters is considered. It is shown that there are three mutually exclusive and exhaustive subsets Θ1, Γ and Θ2 of the parameter space such that there exist at least two positive periodic solutions associated with pairs in Θ1, at least one positive periodic solution associated with Γ and none associated with Θ2.

In the paper, an important necessary and sufficient condition for a commonly used non-orthogonal estimating function to be fixed-sample optimal is proposed. The class of all fixed-sample optimal non-orthogonal estimating functions is characterized under the proposed condition. A simple counterexample without any fixed-sample optimal non-orthogonal estimating function is constructed to show that the proposed condition does not necessarily hold. The usefulness and applicability of the proposed method are illustrated by two classical examples with many nuisance parameters.

In this paper, we investigate a minimax complex programming problem. Several sufficient Optimality conditions are established under the framework of generalized convexity for analytic functions. Employing the sufficient optimality conditions, we have proved the weak, strong and strict converse duality theorems for the complex minimax programming problem.

Sufficient conditions for controllability of neutral functional integrodifferential infinite delay systems in Banach spaces are established. The results are obtained by using the analytic semigroup theory and the Nussbaum fixed point theorem. An example is provided to illustrate the theory.

In this paper, we found a new result by relaxing the condition of [15, Corollary 2]. As its application, we have obtained some new minimax inequalities of Ky Fan and minimax theorems in the spaces without linear structure.

For a $v$-set $V$, let ${\cal A}$ be a collection of $r \times c$ arrays with elements in $V$. A pair $(V, {\cal A})$ is called an {\it $r\times c$ grid-block packing} if every two distinct points $i$ and $j$ in $V$ occur together at most once in the same row or in the same column of arrays in ${\cal A}$. And an $r\times c$ grid-block packing $(V, {\cal A})$ is said to be {\it resolvable} if the collection of arrays ${\cal A}$ can be partitioned into sub-classes $\boldsymbol{R}_1$, $\boldsymbol{R}_2$, $\ldots$, $\boldsymbol{R}_t$ such that every point of $V$ is contained in precisely one array of each class. These packings have originated from the use of DNA library screening. In this paper, we give some constructions of resolvable $r\times c$ grid-block packings and give a brief survey of their application to DNA library screening.

In this paper, motivated by the study of the wide diameter and the Rabin number of graphs, we define the generalized $k$-diameter of $k$-connected graphs, and show that every $k$-regular $k$-connected graph on $n$ vertices has the generalized $k$-diameter at most $n/2$ and this upper bound cannot be improved when $n=4k-6+i(2k-4)$.

A graph $G$ is equitably $k$-choosable if, for any $k$-uniform list assignment $L$, $G$ admits a proper coloring $\pi$ such that $\pi(v)\in L(v)$ for all $v\in V(G)$ and each color appears on at most $\lceil |G|/k\rceil$ vertices. It was conjectured in [8] that every graph $G$ with maximum degree $\Delta$ is equitably $k$-choosable whenever $k\ge \Delta+1$. We prove the conjecture for the following cases: (i) $\Delta \le 3$; (ii) $k\ge (\Delta-1)^2$. Moreover, equitably 2-choosable graphs are completely characterized.

In this paper, we investigate one-sided unit-regular ideals of regular rings. Let $I$ be a purely infinite, simple and essential ideal of a regular ring $R$. It is shown that $R$ is one-sided unit-regular if and only if so is $R/I$. Also we prove that every square matrix over one-sided unit-regular ideals of regular rings admits a diagonal matrix with idempotent entries.