In this paper, we deal with differential-difference equations of the form $$ f(z)^2+p(z)f(z+c)+h(z)f'(z)+g(z)=d_1e^{\lambda z}+d_2e^{-\lambda z} $$ where $p(z),~ h(z),~ g(z)$ are polynomials, and $c,~ d_1,~d_2, ~\lambda\in \mathbb{C}$ are constants with $d_1 d_2\lambda\not= 0$. By utilizing Nevanlinna's value distribution theory, some sufficient conditions on the nonexistence of entire solutions regarding the equations are provided.

In this paper, a nonsmooth multiobjective programming problem is introduced and studied. By using the generalized Guignard constraint qualification, some stronger Kuhn-Tucker type necessary optimality conditions for efficiency in terms of convexificators are established, in which we are not assuming that the objective functions are directionally differentiable. Moreover, some conditions which ensure that a feasible solution is an efficient solution to nonsmooth multiobjective programming problems are also given. The results presented in this paper improve the corresponding results in the literature.

In this paper, we investigate uniqueness of entire functions of order less than 2 sharing the value 0 with their difference operators and obtain a result as follows: Let $f$ be a transcendental entire function such that $\sigma{(f)}\lt 2$ and $\lambda(f)\lt \sigma{(f)}$. If $f$ and $\Delta^nf$ share the value $0$ CM, then $f$ must be form of $f(z)=Ae^{\alpha z},$ where $A$ and $\alpha$ are two nonzero constants. This result confirms a conjecture posed earlier on the topic.

In this paper, we prove that for a transcendental entire function $f(z)$ of finite order such that $\lambda(f-a(z))\lt \sigma(f)$, where $a(z)$ is an entire function and satisfies $\sigma(a(z))\lt 1$, $n$ is a positive integer, if $\Delta_{\eta}^nf(z)$ and $f(z)$ share entire function $b(z)\,(\,b(z)\not\equiv a(z))$ satisfying $\sigma(b(z))\lt 1$ CM, where $\eta\,(\in\mathbb{C})$ satisfies $\Delta_{\eta}^nf(z)\not\equiv 0$, then $$f(z)=a(z)+ce^{c_1z},$$ where $c,\,c_1$ are two nonzero constants.

We continue the study of the McCoy ring property through examining constant annihilators in the ideals of coefficients of zero-dividing polynomials. In the process we introduce the ideal-McCoy property which is between strongly McCoy and McCoy properties, showing that none of implications can be replaced by an equivalence. We give an example of a right ideal-McCoy ring that is not left ideal-McCoy. We also investigate relations between the ideal-McCoy property and other standard ring theoretic properties. For example, we find possible basic forms of finite right ideal-McCoy rings of minimal order.

In this paper, we consider a problem for the first order canonical Dirac differential equations system with piecewise continuous coefficient and spectral parameter dependent in boundary condition. The asymptotic behavior of eigenvalues, eigenfunctions and normalizing numbers of this system is investigated. The completeness theorem is proved. The spectral expansion formula with respect to eigenvector functions or equivalently Parseval equality is obtained. Weyl solution and Weyl function for the problem are constructed. Uniqueness theorem for inverse problem by the Weyl function and by the spectral data are proved.

A graph $G$ is equitably $k$-colorable if its vertex set can be partitioned into $k$ independent sets, any two of which differ in size by at most 1. We prove a conjecture of Lin and Chang which asserts that for any bipartite graphs $G$ and $H$, their Cartesian product $G\Box H$ is equitably $k$-colorable whenever $k\ge 4$.

In this study, it is introduced a $q-$Stancu type of hybrid summation integral type operators. It is investigated their approximation properties. It is given a weighted approximation theorem and obtained rates of convergence of these operators for continuous functions.

An $L(j,k)$-$labeling$ of a graph is a vertex labeling such that the difference between the labels of adjacent vertices is at least $j$ and that between vertices separated by a distance 2 is at least $k$. The minimum of the spans of all $L(j,k)$-labelings of $G$ is denoted by $\lambda_k^j(G)$. Recently, Haque and Jha [16] proved that if $G$ is a multiple direct product of complete graphs, then $\lambda_k^j(G)$ coincides with the trivial lower bound $(N-1)k$, where $N$ is the order of $G$ and $\frac{j}{k}$ is within a certain bound. In this paper, we suggest a new labeling method for such a graph $G$. With this method, we extend the range of $\frac{j}{k}$ such that $\lambda_k^j(G)= (N-1)k$ holds. Moreover, we obtain the upper bound of $\lambda_k^j(G)$ for the remaining cases in the range $\frac{j}{k}$.

A given discrete constrained extremum problem can be associated with an equivalent continuous one, the equivalence being assured by the equality of their sets of solutions. When the equivalent problem is a penalized one, a crucial question is the size of the penalty parameter. The present paper concerns the case where the problem is a 0-1 extremum (linear) one.

We consider the optimality conditions for the DC (difference of two convex functions) optimization problem with the objective and constraint functions given as DC functions. Adopting convexification technique, the local and global KKT type conditions for this optimization problem are defined. By using properties of the subdifferentials of the involved functions, some sufficient and/or necessary conditions for these two types of optimality conditions are provided.

In this paper we study a Jacobi equation driven by fractional Brownian motion with Hurst index $H\in (\frac{1}{2},1).$ We first prove the existence and uniqueness of the solution. Then we investigate Malliavin differentiability and smoothness of the density of the solution. Finally, we point out that the solution can be approximated by semimartingales.

In this paper, we investigate the Insertion-of-Factors-Property (simply IFP), (quasi-)Baer property, and Armendariz property on skew power series (polynomial) rings and introduce the concept of (strongly) $\sigma$-skew IFP and extend many of related basic results to the wider classes. When a ring $R$ has $\sigma$-skew IFP and $\sigma$ is a monomorphism of $R$ we prove that $R$ is Baer if and only if $R$ is quasi-Baer if and only if $R[[x;\sigma]]$ ($R[x;\sigma]$) is Baer if and only if $R[[x;\sigma]]$ ($R[x;\sigma]$) is quasi-Baer. We also prove that if $R$ is a skew power-serieswise $\sigma$-Armendariz ring then $R$ has strongly $\sigma$-skew IFP and $R[[x;\sigma]]$ has IFP. Several known results follow as consequences of our results. In particular, we provide a $\sigma$-skew power-serieswise Armendariz ring but does not have IFP.

In this paper we prove some fixed point theorems for operators acting in Banach algebras and satisfying conditions expressed mainly with help of weak topology and measures of weak noncompactness. The existence of solutions to some class of operator equations in Banach algebras is also discussed. Some examples are presented to illustrate our results.

Let $A_{i}$ be strictly contractive matrices and let $\lambda_{i}$ be nonnegative real numbers with $\displaystyle{\sum_{i=1}^{m}} \ \lambda_{i} =1$, $ \ i=1,\ldots,m.$ We prove that \begin{equation*} s \left(\frac{I+\displaystyle{\sum_{i=1}^{m}}\lambda_{i}A_{i}}{I- \displaystyle{\sum_{i=1}^{m}} \lambda_{i}A_{i}}\right) \prec_{\rm wlog} \displaystyle{\prod_{i=1}^{m}} \ s \left(\left(\frac{I+ |A_{i}|}{I- |A_{i}|}\right)^{\lambda_{i}}\right), \end{equation*} which generalizes a Lewent type determinantal inequality due to Lin [M. Lin, A Lewent type determinantal inequality, Taiwanese J. Math. 17(2013), 1303-1309]. On the other hand, we also prove \begin{equation*} s \left(\frac{I+\displaystyle{\sum_{i=1}^{m}}\lambda_{i}A_{i}}{I- \displaystyle{\sum_{i=1}^{m}} \lambda_{i}A_{i}} \right) \prec_{\rm wlog} \displaystyle{\sum_{i=1}^{m}} \ \lambda_{i} s \left( \frac{I+|A_{i}|}{I-|A_{i}|} \right). \end{equation*} Here ``$\prec_{\rm wlog}$" stands for weakly log-majorization. In addition, some other related inequalities are also obtained.

In this paper, sufficient conditions for the analytic coefficients of the differential equation \[ (f^{(k)})^{n_k}+A_{k-1}(f^{(k-1)})^{n_{k-1}}+\cdots+A_1(f')^{n_1}+A_0f=0 \] are found such that all analytic solutions belong to a given $H_p^\infty-$ space, or to the Hardy space $H^p.$ The results we obtain are a generalization of some earlier results by Heittokangas, Korhonen and Rättyä.

Let $a,b,c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{2}.$ In 1956, Jeśmanowicz conjectured that for any positive integer $n$, the only solution of $(an)^{x}+(bn)^{y}=(cn)^{z}$ in positive integers is $(x,y,z)=(2,2,2)$. Let $k\geq 1$ be an integer and $F_k=2^{2^k}+1$ be $k$-th Fermat number. In this paper, we show that Jeśmanowicz' conjecture is true for Pythagorean triples $(a,b,c)=(F_k-2,2^{2^{k-1}+1},F_k)$.

In this note we study the ascending chain conditions on principal left (resp. right) ideals of the skew polynomial ring $R[x;\alpha,\delta]$. We give a characterization of skew polynomial rings $R[x;\alpha,\delta]$ that are domains and satisfy the ascending chain condition on principal left (resp. right) ideals. We also prove that if $R$ is an $\alpha$-rigid ring that satisfies the ascending chain condition on right annihilators and ascending chain condition on principal right (resp. left) ideals, then the skew polynomial ring $R[x;\alpha,\delta]$ and skew power series ring $R[[x;\alpha]]$ also satisfy the ascending chain condition on principal right (resp. left) ideals.

In this article we present a combinatorial treatment of normal flatness in analytic spaces, using the idea of equimultiple standard bases. We will prove, using purely combinatorial methods, a characterization theorem for normal flatness. This will lead us to a new proof of a classical theorem on normal flatness, which can be stated by saying that normal flatness at a point along a smooth subspace is equivalent to the Hilbert function being locally constant. Though these topics belong to classical analytic geometry, we believe that this approach is valuable, since it replaces extremely general algebraic theorems by combinatorial objects, obtaining new results and striking the combinatorial nature of the classical (and basic) ideas in the resolution of singularities.

It is known that certain combinatorial convolution sums involving two divisor functions product formulae of arbitrary level can be explicitly expressed as a linear combination of divisor functions. In this article we deal with cases for certain combinatorial convolution sums involving three, four, six and twelve divisor functions product formula and obtain explicit expressions.