Communications in Mathematical Research CMR Volume 29, Number 1

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Volume 29, Number 1

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Existence of Positive Solutions for Higher Order Boundary Value Problem on Time Scales

Dapeng Xie, Yang Liu & Mingzhe Sun

Commun. Math. Res., 29 (2013), pp. 1-13.

Preview Full PDF 3531 33378 Abstract

Abstract

In this paper, we investigate the existence of positive solutions of a class higher order boundary value problems on time scales. The class of boundary value problems educes a four-point (or three-point or two-point) boundary value problems, for which some similar results are established. Our approach relies on the Krasnosel'skii fixed point theorem. The result of this paper is new and extends previously known results.

An Extended Multiple Hardy-Hilbert’s Integral Inequality

Yong Hong

Commun. Math. Res., 29 (2013), pp. 14-22.

Preview Full PDF 2936 33401 Abstract

Abstract

In this paper, by introducing the norm $∥x∥_α (x ∈ \boldsymbol{R}^n)$, a multiple Hardy-Hilbert's integral inequality with the best constant factor and it's equivalent form are given.

Superlinear Fourth-Order Elliptic Problem Without Ambrosetti and Rabinowitz Growth Condition

Yuanhong Wei, Xiaojun Chang & Yue Lü

Commun. Math. Res., 29 (2013), pp. 23-31.

Preview Full PDF 3266 34182 Abstract

Abstract

This paper deals with superlinear fourth-order elliptic problem under Navier boundary condition. By using the mountain pass theorem and suitable truncation, a multiplicity result is established for all $λ > 0$ and some previous result is extended.

Strong Consistency of $M$ Estimator in Linear Model for $\widetilde{φ}$-Mixing Samples

Xuejun Wang, Shuhe Hu, Jimin Ling, Yunfei Wei & Zhuqiang Chen

Commun. Math. Res., 29 (2013), pp. 32-40.

Preview Full PDF 3013 32754 Abstract

Abstract

The strong consistency of $M$ estimator of regression parameter in linear model for $\widetilde{φ}$-mixing samples is discussed by using the classic Rosenthal type inequality. We get the strong consistency of $M$ estimator under lower moment condition, which generalizes and improves the corresponding ones for independent sequences.

$\mathcal{F}$-Perfect Rings and Modules

Bo Lu

Commun. Math. Res., 29 (2013), pp. 41-50.

Preview Full PDF 2881 33065 Abstract

Abstract

Let $R$ be a ring, and let $(\mathcal{F}, C)$ be a cotorsion theory. In this article, the notion of $\mathcal{F}$-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring $R$ is said to be right $\mathcal{F}$-perfect if $F$ is projective relative to $R$ for any $F ∈ \mathcal{F}$. We give some characterizations of $\mathcal{F}$-perfect rings. For example, we show that a ring $R$ is right $\mathcal{F}$-perfect if and only if $\mathcal{F}$-covers of finitely generated modules are projective. Moreover, we define $\mathcal{F}$-perfect modules and investigate some properties of them.

Dynamics and Long Time Convergence of the Extended Fisher-Kolmogorov Equation under Numerical Discretization

Jue Wang

Commun. Math. Res., 29 (2013), pp. 51-60.

Preview Full PDF 3005 33240 Abstract

Abstract

We present a numerical study of the long time behavior of approximation solution to the Extended Fisher–Kolmogorov equation with periodic boundary conditions. The unique solvability of numerical solution is shown. It is proved that there exists a global attractor of the discrete dynamical system. Furthermore, we obtain the long-time stability and convergence of the difference scheme and the upper semicontinuity $d(\mathcal{A}_{h,τ} ,\mathcal{A}) → 0$. Our results show that the difference scheme can effectively simulate the infinite dimensional dynamical systems.

Existence and Blow-up of Solutions for a Nonlinear Parabolic System with Variable Exponents

Yunzhu Gao & Wenjie Gao

Commun. Math. Res., 29 (2013), pp. 61-67.

Preview Full PDF 2856 32376 Abstract

Abstract

In this paper, we study a nonlinear parabolic system with variable exponents. The existence of classical solutions to an initial and boundary value problem is obtained by a fixed point theorem of the contraction mapping, and the blow-up property of solutions in finite time is obtained with the help of the eigenfunction of the Laplace equation and a delicate estimate.

The Étale Homology and the Cycle Maps in Adic Coefficients

Ting Li

Commun. Math. Res., 29 (2013), pp. 68-87.

Preview Full PDF 2834 33068 Abstract

Abstract

In this article, we define the $ℓ$-adic homology for a morphism of schemes satisfying certain finiteness conditions. This homology has these functors similar to the Chow groups: proper push-forward, flat pull-back, base change, cap-product, etc. In particular, on singular varieties, this kind of $ℓ$-adic homology behaves much better than the classical $ℓ$-adic cohomology. As an application, we give a much easier approach to construct the cycle maps for arbitrary algebraic schemes over fields. And we prove that these cycle maps kill the algebraic equivalences and commute with the Chern action of locally free sheaves.

A Joint Density Function in the Renewal Risk Model

Huai Xu & Ling Tang

Commun. Math. Res., 29 (2013), pp. 88-96.

Preview Full PDF 2848 32621 Abstract

Abstract

In this paper, we consider a general expression for $ϕ(u, x, y)$, the joint density function of the surplus prior to ruin and the deficit at ruin when the initial surplus is $u$. In the renewal risk model, this density function is expressed in terms of the corresponding density function when the initial surplus is 0. In the compound Poisson risk process with phase-type claim size, we derive an explicit expression for $ϕ(u, x, y)$. Finally, we give a numerical example to illustrate the application of these results.

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