In this paper we show how the Zipf-Mandelbrot law is connected to the theory of majorization. Firstly we consider the Csiszár $f$-divergence for the Zipf-Mandelbrot law and then develop important majorization inequalities for these divergences. We also discuss some special cases for our generalized results by using the Zipf-Mandelbrot law. As applications, we present the majorization inequalities for various distances obtaining by some special convex functions in the Csiszár $f$-divergence for Z-M law like the Rényi $\alpha$-order entropy for Z-M law, variational distance for Z-M law, the Hellinger distance for Z-M law, $\chi^{2}$-distance for Z-M law and triangular discrimination for Z-M law. At the end, we give important applications of the Zipf's law in linguistics and obtain the bounds for the Kullback-Leibler divergence of the distributions associated to the English and the Russian languages.

We report on existence of pair of new recurrence relations (or difference equations) for the Meixner-Pollaczek polynomials. Proof of the correctness of these difference equations is also presented. Next, we found that subtraction of the forward shift operator for the Meixner-Pollaczek polynomials from one of these recurrence relations leads to the difference equation for the Meixner-Pollaczek polynomials generated via $\cosh$ difference differentiation operator. Then, we show that, under the limit $\varphi \to 0$, new recurrence relations for the Meixner-Pollaczek polynomials recover pair of the known recurrence relations for the generalized Laguerre polynomials. At the end, we introduced differentiation formula, which expresses Meixner-Pollaczek polynomials with parameters $\lambda>0$ and $0 \lt \varphi \lt \pi$ via generalized Laguerre polynomials.

In the stability theory, the nonlinear equations were not so much investigated. In this note we consider the stability of a nonlinear integral equation with Lipschitzian kernel. The approach is based on monotonicity properties of a nonlinear operator.

We introduce and study a new class of continuous distributions called the transmuted Gompertz-G family which extends the Gompertz class proposed by Alizadeh et al. (2016a). Explicit expressions for the ordinary and incomplete moments, generating function, probability weighted moment, Lorenz and Bonferroni curves, order statistics, Rényi and Shanon entropies, stress strength model moment of residual and reversed residual life and characterizations for the new family are investigated. We discuss the maximum likelihood estimates for the model parameters. The performance of the new family is assesed by means of two applications.

The aim of this short paper is to prove a $\mathsf{TQ}$-Whitehead theorem for nilpotent structured ring spectra. We work in the framework of symmetric spectra and algebras over operads in modules over a commutative ring spectrum. Our main result can be thought of as a $\mathsf{TQ}$-homology analog for structured ring spectra of Dror's generalized Whitehead theorem for topological spaces; here $\mathsf{TQ}$-homology is short for topological Quillen homology. We also prove retract theorems for the $\mathsf{TQ}$-completion and homotopy completion of nilpotent structured ring spectra.

In this paper, we deal with the connection and duplication problems associated with the hybrid Sheffer family. The hybrid Sheffer polynomials are also studied via matrix approach. The properties of these polynomials are established using simple matrix operations. Examples providing the corresponding results for certain members of the hybrid Sheffer family are considered. This article is first attempt in the direction of obtaining connection and duplication coefficients and matrix representations for the hybrid polynomials.

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them have been studied. In this article new sets of logarithmic-Sheffer polynomials are introduced. Connection with Bell numbers are shown.

The aim of this article is to obtain common fixed points of set mappings defined on a family of sets endowed with a graph satisfying $\Theta-$contraction. Our results generalize and extend various results in the existing literature. We also provide some non trivial examples to support the validity of our main results. Some applications to the construction of common fixed points of set mappings in $\varepsilon$-chainable metric space are also discussed.

The purpose of the article is to study the certain results on para-Kenmotsu manifolds equipped with $M$-projective curvature tensor. Here we investigate para-Kenmotsu manifolds satisfying some curvature conditions $\widetilde{M}\cdot R=0,$ $\widetilde{M}\cdot Q=0$ and $Q\cdot \widetilde{M}=0,$ where $R$, $Q$ and $\widetilde{M}$ respectively denote the Riemannian curvature tensor, Ricci operator and $M$-projective curvature tensor.

Let $L=-\Delta+V\left( x\right) $ be a Schrödinger operator, where $\Delta$ is the Laplacian on ${\mathbb{R}^{n}}$, while nonnegative potential $V\left(x\right)$ belonging to the reverse Hölder class. In this paper, using the some conditions on $\varphi\left(x.r\right) $, we dwell on the boundedness of Marcinkiewicz integrals with rough kernel associated with schrödinger operators and commutators generated by these operators and local Campanato functions both on generalized local Morrey spaces and on generalized vanishing local Morrey spaces, respectively. As an application of the above results, the boundedness of parametric Marcinkiewicz integral and its commutator both on generalized local Morrey spaces and on generalized vanishing local Morrey spaces is also obtained.

This paper is devoted to study a class of critical singular degenerate elliptic system with concave-convex nonlinearities and sign-changing weight functions. The existence and multiplicity of nontrivial nonnegative solutions are obtained by the variational.

In this paper, the Szász-Mirakjan-Kantorovich type operators are studied with the help of direct result and weighted approximation properties. Some basic lemmas, theorems are given and proved, at last, we discuss the rate of convergence and a comparison takes place with the Szász-Mirakjan-Kantorovich operators by graphical representations.