In this paper, we study zeros of difference product $f(z)^n \Delta f(z)$ ($n \geq 2$), and the value distribution of difference product $f(z) \Delta f(z)$, where $f(z)$ is a transcendental entire function of finite order, $\Delta f(z) = f(z+c)-f(z)$, where $c$ ($\not= 0$) is a constant such that $f(z+c) \not\equiv f(z)$.

In 1999, Arnal et al. [Numerical linear algebra and its applications, 6(1999): 79-92] introduced the non-stationary matrix multisplitting algorithms for almost linear systems and studied the convergence of them. In this paper, we generalize Arnal’s algorithms and study the non-stationary matrix multisplitting multi-parameters methods for almost linear systems. The parameters can be adjusted suitably so that the convergence property of methods can be substantially improved. Furthermore, the convergence results of our new method in this paper are weaker than those of Arnal’s. Finally, numerical examples show that our new convergence results are better and more efficient than Arnal’s.

It is proved that if a centralizer near-ring $N$ has a nonzero derivation, then $N$ is a near-field.

We study the existence of ergodic retractions for semigroups of mappings in strictly convex Banach spaces. We prove, for instance, the following theorem. Let $(X,\|\cdot\|)$ be a strictly convex Banach space and let $\Gamma$ be a norming set for $X$. Let $C$ be a bounded and convex subset of $X$, and suppose $C$ is compact in the $\Gamma$-topology. If $\mathcal S$ is a right amenable semigroup, $\varphi=\{T_s:s\in\mathcal S\}$ is a semigroup on $C$ with a nonempty set $F=F(\varphi)$ of common fixed points, and each $T_s$ is ($F$-quasi-) nonexpansive, then there exists an ($F$-quasi-) nonexpansive retraction $R$ from $C$ onto $F$ such that $RT_s=T_sR=R$ for each $s\in \mathcal S$, and every $\Gamma$-closed, convex and $\varphi$-invariant subset of $C$ is also $R$-invariant.

We improve several results recently established by Dejun Zhao for the Landau’s constants.

We introduce para-Eisenstein series as a second analogue of classical elliptic Eisenstein series in the framework of Drinfeld modular forms and show that they share many properties with ordinary Eisenstein series.

In this paper, we prove that Latif’s fixed point statements for $w$-distance or other weak distances are indeed equivalent with some weak distance variants of Caristi’s fixed point statement, Ekeland’s variational principle statement and Takahashi’s nonconvex minimization statement.

In this paper we study the behaviour of linear diameter preserving mappings when defined between subalgebras of continuous functions. Namely, we obtain a representation of such mappings as the sum of a weighted composition operator and a linear functional on, at least, the Choquet boundaries of the algebras under consideration. In particular, we give a complete description when we consider several classical function algebras.

Global projection methods have been used for solving numerous large matrix equations, but nothing has been known on if and how this method can be proposed for solving large eigenproblems. In this paper, a global Arnold method is proposed for large eigenproblems. It computes certain F-Ritz pairs that are used to approximate some eigenpairs. The global Arnoldi method inherits convergence properties of the standard Arnoldi method applied to a larger matrix whose distinct eigenvalues are the eigenvalues of the original given matrix. As an application, assuming that A is diagonalizable, we show that the global Arnoldi method is able to solve multiple eigenvalue problems. To be practical, we develop an implicitly restarted global Arnoldi algorithm with certain F-shifts suggested. In particular, this algorithm can be adaptively used to solve multiple eigenvalue problems. Numerical experiments show that the algorithm is efficient for the eigenproblem and is reliable for quite ill-conditioned multiple eigenproblems.

Boundedness of multilinear operators on weighted Herz-Morrey spaces is established in this paper. The weak estimates on endpoints are also derived. As a special case, the conclusions can apply to multilinear Calderón-Zygmund operators.

In this paper we investigate a notion of extended well-posedness in vector optimization. Appropriate asymptotically minimizing sequences, when both the objective function and the feasible region are subject to perturbation are introduced. We show that convex problems, i.e. problems in which both the objective function and the perturbations are $C$−convex, are extended wellposed. Further, we characterize the proposed well-posedness notion both in terms of linear and nonlinear scalarization.

Fan-Sato[8] proved a tranceference theorem with respect to the multilinear operators on ${\bf R}^n$. Also Blasco-Villarroya[3] proved the similar result with repect to the biilinear operators on ${\bf Z}^2$. In this paper, we prove a tranceference theorem of the bilinear operators between Jacobi series and Hankel transforms.

Suppose $G$ is a planar graph. Let $H_G$ be the graph with vertex set $V(H_G) = \{ C:C$ is a cycle of G with $|C|\in \{4,6,7\} \}$ and $E(H_G) = \{ C_i C_j: C_i$ and $C_j$ are adjacent in $G\}$. We prove that if any $3$-cycles and $5$-cycles are not adjacent to $i$-cycles for $3 \leq i \leq 7$, and $H_G$ is a forest, then $G$ is $3$-colourable.

In the present article the geometry of semi-Riemannian manifolds with nonholonomic constraints is studied. These manifolds can be considered as analogues to the sub-Riemannian manifolds, where the positive definite metric is substituted by a nondegenerate metric. We study properties of the exponential map, the Christoffel symbols and other differential operators are introduced. We study solutions of the Hamiltonian system and their projections into the underlying manifold. The explicit formulae were found for a specific example of a semi-Riemannian manifold with nonholonomic constraints.

In this paper, we study a reaction-diffusion system of a competitorcompetitor-mutualist model with Neumann boundary condition. Using iteration method, we investigate the global asymptotic stability of the unique positive constant steady-state solution under some assumptions. We also give some sufficient conditions under which there are no nonconstant positive steady-state solution exist.

The first part of the paper deals with finding the heat kernel by probabilistic methods for the 1-dimensional elliptic differential operator $\frac{1}{2} (1+x^2) \frac{d^2}{dx^2} + (\sqrt{1+x^2}+\frac{x}{2}) \frac{d}{dx}$. In the second part we apply the same method to the 2-dimensional operator $\frac{1}{2} (\frac{\partial^2}{\partial x_1^2} + 2\sqrt{1+x_2^2} \frac{\partial ^2}{\partial x_1 \partial x_2} + (1+x_2^2) \frac{\partial^2}{\partial x_2^2}) + \frac{1}{2} x_2 \partial x_2$ and provide explicit formulas for its heat kernel.

In the paper, the conditions of normal solvability of some boundary value problems are obtained for a class of operator-differential equations of elliptic type on a semi-axis in weight spaces. The principal part of this equation contains a multiple characteristics operator. All conditions are expressed only by the properties of the operators of the given equation.

This paper extends our recent result about the distribution of normalized zero-sets of random entire functions[Y–200811] to the context of random entire functions with small random perturbation.

In this note, we consider a multi-valued version of a generalized system (GS) called the multi-valued generalized system (MGS). Using the Fan-Browder fixed point theorem and Brouwer’s fixed point theorem as basic tools, we provide existence theorems on (MGS) with and without monotonicity, respectively.

We consider the continuous $q$-Jacobi polynomials $\{P_n^{(\alpha,\beta)}(\cdot|q)\}_{n=0}^{\infty}$, extending the variable and the parameters beyond classical considerations. For those new allowed values of the parameters for which Favard's theorem fails to work, we construct inner products in which the presence of the Askey-Wilson divided difference operator provides the $q$-Sobolev character of the non-standard orthogonality for the corresponding family.

Based on the total H$_{1}$-integrability concept, which is established in this paper, we shall try to show that at any point of a compact interval $(a,b]$ in $\mathbb{R}$, at which a point function $F$ has no a discontinuity, $F$ is the total H$_{1}$-indefinite integral of a function $dF_{ex}$ being the limit of $\Delta F_{ex}(I)$, where $I \subseteq [a,b]$, on $[a,b]$, without additional hypotheses on $F$. A residue function of $F$ is introduced. The paper ends with a few of examples that illustrate the theory.

We find relations between the Levi-Civita connection and a semi-symmetric metric connection of the warped product $M = M_{1} \times_{f} M_{2}$. We obtain some results of Einstein warped product manifolds with a semi-symmetric metric connection.

This work considers the resolution of the Hull and White interest rate model. A deterministic process is adopted to model the random behavior of interest rate variation as a deterministic perturbation. It shows that the interest rate function and the yield function of the Hull and White interest rate model can be obtained by solving a nonlinear semi-infinite programming problem. A relaxed cutting plane algorithm is then proposed for the resulting optimization problem. The features of the proposed method are tested using a set of real data and compared with some commonly used spline fitting methods.

In this paper, we establish some inequalities for differentiable convex mappings whose derivatives in absolute value are convex. This result is connected with Fejér’s inequality holding for convex mappings. Some applications for the weighted trapezoidal formula, $r−moment$, and the expectation of a symmetric and continuous random variable are given.

In this paper we characterize mean Lipschitz spaces in terms of some $L^p$-mean oscillation on the unit disc.

This work develops almost sure and complete convergence of randomly weighted sums of independent random elements in a real separable Banach space. Sufficient conditions for almost sure convergence and complete convergence in the sense defined by Hsu and Robbins are provided. Examples showing that the conditions cannot be removed or weakened are given. It is also demonstrated that some of the known theorems in the literature are special cases of our results.

This work concerns mean-field models, which are formulated using stochastic differential equations. Different from the existing formulations, a random switching process is added. The switching process can be used to describe the random environment and other stochastic factors that cannot be explained in the usual diffusion models. The added switching component makes the formulation more realistic, but it adds difficulty in analyzing the underlying processes. Several properties of the mean-field models are provided including regularity, nonnegativity, finite moments, and continuity. In addition, the paper addresses the issue when the switching takes place an order of magnitude faster than that of the continuous state. It derives a limit that is an average with respect to the invariant measure of the switching process.

A $k$-edge-weighting of a graph $G$ is a mapping $w: E(G) \to \{1,2,\ldots, k\}$. An edge-weighting $w$ induces a vertex coloring $f_w: V(G) \to \mathbb{N}$ defined by $f_w(v) = \sum_{v \in e} w(e)$. An edge-weighting $w$ is vertex-coloring if $f_w(u) \ne f_w(v)$ for any edge $uv$. The current paper studies the parameter $\mu(G)$, which is the minimum $k$ for which $G$ has a vertex-coloring $k$-edge-weighting. Exact values of $\mu(G)$ are determined for several classes of graphs, including trees and $r$-regular bipartite graph with $r \ge 3$.

Some new results related to Jensen’s celebrated inequality for convex functions defined on convex sets in linear spaces are given. Applications for the arithmetic mean-geometric mean inequality are provided as well.

In this article we prove that a subset $E$ of the open unit disc $U$ is a dominating set for the subclass $N^+$ of the Nevanlinna class $N$ if and only if $E$ is nontangentially dense. When $E$ is a compact subset of $U$, we also give a complete characterization of $E$ to be a dominating set for the subclass $N^-$. Here $N^-$ denotes the proper subclass of $N$ that consists of all of the reciprocal of the singular inner functions.

In this paper we used the method of parabola approximation to study some nonlinear differential equations. We derive exact, explicit solutions to the parabolic equations and use this analytical results in the numerical computations for the general equations. We then draw the comparison of between the solutions of original and approximated equations. Moreover, we apply such method to the population growth problem. The error of the difference between the solutions of the differential equations and the numerical results caused by the discrete approximations is reasonable.

Recently, Pan et al. [11] developed the merit function method for SOCCP based on the generalized Fischer-Burmeister (FB) function. This note is along the same line. Indeed, we study a penalized version of the generalized FB function and provide a basic theoretical property that the level set of the merit function induced by the penalized version of the generalized FB function is bounded under suitable assumptions. The proof relies on trace inequalities.

In this paper, we introduce two implicit and explicit hybrid viscositylike approximation methods for solving a general monotone variational inequality, which covers their monotone variational inequality with $C = H$ as a special case. We use the contractions to regularize the general monotone variational inequality, where the monotone operators are the generalized complements of nonexpansive mappings and the solutions are sought in the set of fixed points of another nonexpansive mapping. Such general monotone variational inequality includes some monotone inclusions and some convex optimization problems to be solved over the fixed point sets of nonexpansive mappings. Both implicit and explicit hybrid viscosity-like approximation methods are shown to be strongly convergent. In the meantime, these results are applied to deriving the strong convergence theorems for a general monotone variational inequality with minimization constraint. An application in hierarchical minimization is also included.