一类n维中立型泛函微分方程的周期解

通过对NFDE周期系统；周期解的讨论，给出了其周期解界的估计式，结合不动点原理研究了下列系统：周期解的存在性，唯一性等问题，得到一些新的结果．     

某二阶微分方程周期解的存在性

本文研究非线性微分方程组周期解的存在性。此系统在许多振动问题中出现。系统(1)当(y)=y时周期解或极     

Lienard型系统解的定性行为

该文讨论非线性系统狓＝１犪（狓）（犺（狔）－犉（狓）），　狔＝－犪（狓）犵（狓） （Ｅ）解的一些定性行为，获得了系统（Ｅ）为振动，全局渐近稳定，全局中心的充要条件和周期解的存在的充分条件．     

具有周期脉冲的分数阶发展方程周期mild解的存在性

该文讨论了具有分段Caputo导数和周期脉冲的分数阶发展方程,建立了具有周期脉冲的相关线性发展方程周期mild解的存在性和唯一性.借助线性脉冲周期问题解算子的表达式,利用算子半群理论和不动点定理,证明了半线性脉冲周期问题周期mild解的一些新的存在性结果.     

Poincaré非线性振动理论在连续介质力学中的推广(Ⅰ)——基本理论与方法*

本文将离散介质的Poincaré非线性振动理论[1]向连续介质力学推广,做了初步尝试。首先讨论在非共振与共振情况下,连续介质线性强迫振动周期解,及其周期解存在条件。进而运用线性理论结果,将Poincaré理论中的主要结论推广到连续介质非线性振动问题中去。此外,本文提出并建议用偏微分方程直接摄动与加权积分方法,计算共振区内的周期解。     

非线性振动系统周期解的数值分析

用直接数值积分法求非线性振动系统的周期解,求解时对初始条件进行迭代,使它与终点条件相一致.积分时间区间(即周期)或运动方程中的某些参数,也可在迭代过程中随同变化,积分方法是变步长的. 用这种“打靶”法求周期解,所需计算工作量相对较少.其中误差主要来源于数值积分,故不难估计并控制它足够小.这种方法可处理各种类型的振动问题,如单自由度和多自由度系统的自由无阻尼振动、强迫振动、自激振动和参数振动等等;也能求得不稳定解和那些对参数变动十分敏感的解.解的稳定性根据相关的周期系数微分方程来研究.求共振曲线或其他振动特性曲线时,利用插值方法并自动调节步长来定出迭代始值. 为了阐明这种方法的通用性,计算了若干例子.非线性的描述可用解析函数或任何其他形式,例如分段线性函数.文中还就所得周期解指出了非线性振动的一些值得注意的性质.部分计算结果与已有的近似解或实验结果作了比较.     

一类时滞反应扩散方程的周期解和概周期解

通过构造上、下控制函数，结合上、下解方法及相应的单调迭代方法研究了一类时滞反应扩散方程，证明了在反应项非单调时，如果一雏边值问题存在一对周期（或概周期）上、下解，则方程一定存在唯一的周期（或概周期）解．并给出了二维边值问题周期（或概周期）解存在唯一性的充分条件．推广了已有的一些结果。     

中立型Lotka-Volterra系统正周期解的存在性

利用迭合度理论的连续定理，讨论了一类中立型系统的正周期解的存在性．得到了正周期解存在的一些充分条件．     

某些高维非线性周期系统的周期解

本文用李雅普诺夫函数法和 Brouwer 不动点定理究研某些高维非线性周期系统的周期解问题,得到了一些存在唯一稳定周期解的充分条件。     

一类具有偏差变元的泛函微分方程周期解的存在性

利用重合度理论研究了一类具有偏差变元的泛函微分方程周期解的存在性问题,得到了该方程具有周期解的一些充分条件.结论丰富了现有文献的结果.     

Complexity of a scheduling problem with controllable processing times

We consider the problem of scheduling a set of independent jobs on a single machine so as to minimize the total weighted completion time, subject to the constraint that the total compression cost is less than or equal to a fixed amount. The complexity of this problem is mentioned as an open problem. In this note we show that the problem is NP-hard.     

The Complexity Analysis of the Inverse Center Location Problem

Given a feasible solution, the inverse optimization problem is to modify some parameters of the original problem as little as possible, and sometimes also with bound restrictions on these adjustments, to make the feasible solution become an optimal solution under the new parameter values. So far it is unknown that for a problem which is solvable in polynomial time, whether its inverse problem is also solvable in polynomial time. In this note we answer this question by considering the inverse center location problem and show that even though the original problem is polynomially solvable, its inverse problem is NP–hard.     

About the Power Law of the PageRank Vector Component Distribution. Part 1. Numerical Methods for Finding the PageRank Vector

In Part 1 of this paper we consider the web-page ranking problem, also known as the problem of finding the PageRank vector, or the Google problem. We discuss the link between this problem and the ergodic theorem and describe different numerical methods to solve this problem together with their theoretical background, such asMarkov chain Monte Carlo and equilibrium in a macrosystem.     

Problem difficulty of real instances of convoy planning

Moving men and materials in large numbers and quantities is a long-standing military problem faced by all arms. An important part of this is the routing of convoys so that they reach their correct destinations in the shortest time. The optimization problem at the heart of this problem is referred to as the convoy movement problem. Previous work on the convoy movement problem has made the assumption that the problem is difficult in practice because of the NP-hardness of the problem in combination with the limited success of early approaches based on genetic algorithms. As a result subsequent work has focused on mathematical programming-based methods, principally Lagrangian relaxation. In this paper, we demonstrate that a straightforward reformulation of the problem renders the real-world like instances, used to benchmark previous approaches, amenable to solution by simple heuristics. The main lessons learnt from this work is that analysis of the problem in conjunction with simple algorithms can, in practice, yield surprisingly effective solutions.     

Numerical solutions of the m-membranes problem

In this paper numerical approximation for the m-membrane problem is considered. We make a change of variables that leads to a different expression of the quadratic functional that allows after discretizing the problem to reformulate it as finite dimensional bound constrained quadratic problem. To our knowledge this is the first paper on numerical approximation of the m-membrane problem. We reformulate the m-membrane problem as a bound constraint quadratic minimization problem. The bound constraint quadratic form is solved with the gradient projection method.     

Characterization of duality for a generalized quasi-equilibrium problem

In this paper, the duality theory of a generalized quasi-equilibrium problem (also called generalized Ky Fan quasi-inequality) is investigated by using the image space approach. Generalized quasi-equilibrium problem is transformed into a minimization problem. The minimization problem is further reformulated as an image problem by virtue of linear/nonlinear separation function. The dual problem of the image problem is constructed in the image space, then zero duality gap between the image problem and its dual problem is derived under saddle point condition as well as the equivalent regular linear/nonlinear separation condition. Finally, some more sufficient conditions guaranteeing zero duality gap are also proposed.     

A linear programming reformulation of the standard quadratic optimization problem

The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note, we show that the SQO problem may be reformulated as an (exponentially sized) linear program (LP). This reformulation also suggests a hierarchy of polynomial-time solvable LP’s whose optimal values converge finitely to the optimal value of the SQO problem. The hierarchies of LP relaxations from the literature do not share this finite convergence property for SQO, and we review the relevant counterexamples.     

Nonclassical pseudospectral method for the solution of brachistochrone problem

In this paper, nonclassical pseudospectral method is proposed for solving the classic brachistochrone problem. The brachistochrone problem is first formulated as a nonlinear optimal control problem. Properties of nonclassical pseudospectral method are presented, these properties are then utilized to reduce the computation of brachistochrone problem to the solution of algebraic equations. Using this method, the solution to the brachistochrone problem is compared with those in the literature.     

Some remarks on the dynamics of cell membranes

Motivated by the desire to model the entry of 1,25D into a cell by receptor mediated endocytosis, we have formulated the problem as the dynamics of a bilayer membrane. We have discussed setting the problem as a variational problem using the Helfrich modeling of the bilayer in terms of the free energy. Using a Lagrangian formulation we arrive at the Euler–Lagrange equations for the system. The model we have used depends on the amount of reagent in the neighborhood of the upper membrane. The problem thereby reduces to a moving boundary problem, which is dependent on a diffusion equation for a region changing with time. In order to solve this problem we seek the correct Neumann function for this altered. This is accomplished by deriving a Hadamard variational formula for the diffusion equation. We also offer an iterative procedure for solving this non-linear problem.     

葡萄酒质量的评价模型

针对2012年“高教社杯”全国大学生数学建模竞赛A题“葡萄酒的评价”进行建模,首先介绍了问题的背景;然后分析了对几个问题的理解,并给出部分问题的具体模型和解答结果;接着分析了竞赛论文中普遍存在的问题。最后提出值得进一步研究的几个问题。