Singularities of minima: a walk on the wild side of the Calculus of Variations

I will report on some recent developments concerning the problem of estimating the Hausdorff dimension of the singular sets of solutions to elliptic and variational problems. Emphasis will be given on some open issues. Connections with measure data problems will be outlined.     

From the 1-2-3 conjecture to the Riemann hypothesis

This survey presents some combinatorial problems with number-theoretic flavor. Our journey starts from a simple graph coloring question, but at some point gets close to dangerous territory of the Riemann Hypothesis. We will mostly focus on open problems, but we will also provide some simple proofs, just for adorning.     

Scalarization in vector optimization

In this paper some scalar optimization problems are presented whose optimal solutions are also solutions of a general vector optimization problem. This will be done for weakly minimal and minimal solutions, respectively. Finally the results will be applied to a certain class of approximation problems.     

Dynamic shortest paths and transitive closure: Algorithmic techniques and data structures

In this paper, we survey fully dynamic algorithms for path problems on general directed graphs. In particular, we consider two fundamental problems: dynamic transitive closure and dynamic shortest paths. Although research on these problems spans over more than three decades, in the last couple of years many novel algorithmic techniques have been proposed. In this survey, we will make a special effort to abstract some combinatorial and algebraic properties, and some common data-structural tools that are at the base of those techniques. This will help us try to present some of the newest results in a unifying framework so that they can be better understood and deployed also by non-specialists.     

The Solvability of Nonlinear Split Ordered Variational Inequality Problems in Partially Ordered Vector Spaces

The concept of nonlinear split ordered variational inequality problems on partially ordered Banach spaces extends the concept of the linear split vector variational inequality problems on Banach spaces, while the latter is a natural extension of vector variational inequality problems on Banach spaces. In this article, we prove the solvability of some nonlinear split vector variational inequality problems by using fixed-point theorems on partially ordered Banach spaces. It is important to notice that in the results obtained in this article, the considered mappings are not required to have any type of continuity and they just satisfy some order-monotonic conditions. Consequently, both the solvability of linear split vector variational inequality problems and vector variational inequality problems will be immediately obtained from the solvability of nonlinear split vector variational inequality problems. We will apply these results to solving nonlinear split vector optimization problems. The underlying spaces of the considered variational inequality problems may just be vector spaces which do not have topological structures, the considered mappings are not required to satisfy any continuity conditions, which just satisfy some order-increasing conditions.     

关于最佳局部逼近

The purpose of this paper is to introduce certain aspects concerning the problem of best local approximations. Especially in multipoint best local approximation, there are many intuitively quite clear yet unsolved interesting problems. We will discuss some recent results and some unsolved problems.     

Inverse modelling problems in linear algebra undergraduate courses

This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different presentations will be discussed. Finally, several results will be presented and some conclusions proposed.     

空间轴对称问题边界元法的程序处理

弹性力学的空间轴对称问题可以化为两个变量(r,z)的二维问题求解,但又比平面问题略复杂些.在轴对称问题边界元的程序处理上也会相应带来些麻烦.文章具体介绍了作者在调试轴对称问题边界元程序中遇到的一些问题及处理方法,并给出数值结果加以验证.     

Using Repertory Grids for Problem Construction

In the last couple of years, O.R. workers have expressed interest in the use of repertory grids as a part of their armoury for tackling messy problems in organizations. This paper is designed to describe some of our experiences of using grids to aid the process of problem construction. It will also express some of our views of the theoretical and technical aspects of their use for working on problems.The principles behind the use of grids invite the design of other methods for managing qualitative problem data. The paper will briefly introduce some we have found helpful.     

MPC for the Burgers Equation Based on an LQG Design

We consider optimal control problems for semilinear parabolic PDEs where process and measurement noise can occur. We discuss the solution of such problems by using a Model Predictive Control (MPC) strategy. For the resulting sub-problems we will use a Linear Quadratic Gaussian (LQG) design. Thus we will discuss the efficient implementation of the LQG approach since it is the major computational part in the MPC scheme for this class of optimal control problems. We will present some numerical results for the Burgers equation. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

国际油价波动对我国经济影响的CGE分析

石油价格变动会引起一个经济体全方位的反应.本文利用一个中国CGE模型—MCHUGE模型,研究油价上涨20%对我国经济发展的影响.研究结果表明,石油生产、加工、冶炼以及部分化工等跟原油关系最直接的产业将从石油涨价中获益;以石油为原料的行业由于成本不容易转嫁,大部分会受到石油涨价的影响;其他替代能源产业因需求大幅上升而涨价;汽车和机械制造业将受到一定的冲击.在此基础上,我们提出了一些政策建议.     

外表设计及其案例研究（Ⅱ）

本篇前部分介绍参数设计的一些案例，这些案例被用来对外表设计诸方法进行实际考核和比较；后部分介绍对外表设计诸方法进行比较所采用的一些准则及其有关问题．     

Solutions of the autonomous Kirchhoff type equations in RN

Recently, the existence, the multiplicity and the concentration of solutions for the Kirchhoff type equations or systems have been extensively established by a number of authors. Such problems contain a nonlocal term which implies that the equation or system is no longer a pointwise identity. Thus, some researchers think that the nonlocal phenomenon will cause some mathematical difficulties. In this note, however, we will give a simple transformation so that the solutions of the autonomous Kirchhoff type equation or system are easily obtained by using the known solutions of the corresponding local equation or system. In particular, some qualitative properties of solutions for the local problems are also inherited.     

From ODE to DDE

In this paper, by considering ordinary differential equation (ODE) as a special case and a starting point of delay differential equation (DDE), we will show that some typical topological methods such as continuation theorems can be applied to detect some dynamics of DDE like periodic solutions. Several problems will be presented.      

Finsler几何的回顾与进展

本文回顾了近年来Finsler几何的进展．特别，我们描述了Finsler流形上几何不变量所满足的基本方程及其应用，并提出了相关的未解决的问题。     

Transfer of boundary conditions for difference equations

It is well-known that the idea of transferring boundary conditions offers a universal and, in addition, elementary means how to investigate almost all methods for solving boundary value problems for ordinary differential equations. The aim of this paper is to show that the same approach works also for discrete problems, i.e., for difference equations. Moreover, it will be found out that some results of this kind may be obtained also for some particular two-dimensional problems.     

A two-player competitive discrete location model with simultaneous decisions

In this paper we will describe and study a competitive discrete location problem in which two decision-makers (players) will have to decide where to locate their own facilities, and customers will be assigned to the closest open facilities. We will consider the situation in which the players must decide simultaneously, unsure about the decisions of one another, and we will present the problem in a franchising environment. Most problems described in the literature consider sequential rather than simultaneous decisions. In a competitive environment, most problems consider that there is a set of known and already located facilities, and new facilities will have to be located, competing with the existing ones. In the presence of more than one decision-maker, most problems found in the literature belong to the class of Stackelberg location problems, where one decision-maker, the leader, locates first and then the other decision-maker, the follower, locates second, knowing the decisions made by the first. These types of problems are sequential and differ significantly from the problem tackled in this paper, where we explicitly consider simultaneous, non-cooperative discrete location decisions. We describe the problem and its context, propose some mathematical formulations and present an algorithmic approach that was developed to find Nash equilibria. Some computational tests were performed that allowed us to better understand some of the features of the problem and the associated Nash equilibria. Among other results, we conclude that worsening the situation of a player tends to benefit the other player, and that the inefficiency of Nash equilibria tends to increase with the level of competition.     

谱问题及发展方程族的等价性

本文讨论了四对谱问题、相应保谱发展方程族以及它们 Lax表示的等价性 ,并利用这种等价关系导出了一些方程的 Lax表示 .     

VALUE DISTRIBUTION THEORY AND DIOPHANTINE APPROXIMATION

In this paper, we will introduce some problems and results between Diophantine approximation and value distribution theory.     

高等数学求极限教学的一些建议

分析极限教学中出现的一些问题,并给出相应建议．