The Golden Rectangle

Fibonacci(Leonardo Pisano) was an outstanding mathematician and scholar whose interests lay particularly in solving practical problems.In Chapter 2,you came across the Fibonacci series.The series has a number of interesting properties.If we label each number(term) in the series using t1,     

Geometry and Inductive Reasoning

As a child you learned by experimenting with the natural world about you．You learn ed how to walk,talk,and hold your spoon, all by trial and error．After many tries,you e ventually learned how to ride your bicycle, discovering it was easier to keep your balance     

Problem solving

Problem solving is an essential skill in both school and work.A model for it was established by the charismatic teacher and mathematician George Polya in How to solve it.This book,first published in 1945,has sold more than 1 million copies and is available in 17 languages.Using a four-step procedure for problem solving,Polya's book demonstrates how to think clearly in any field.     

Being a Good Problem Solver（1）

When solving problems, You need not feel as if you are lost in a maze. Strategies can help you find solutions. What Is an Algorithm?you may have learned to solve any equation of the form x a = b for x. The method was to add -a to each side and then simplify. This method is an example of an algorithm. An algorithm is a sequence of steps that leads to a desired result. Not all algorithms are short. The algorithm called long division can involve many steps.     

EXPERIMENTAL STUDY OF THE ASYNCHRONOUS MULTISPLITTING RELAXATION METHODS FOR THE LINEAR COMPLEMENTARITY PROBLEMS

We study the numerical behaviours of the relaxed asynchronous multisplitting methods for the linear complementarity problems by solving some typical problems from practical applications on a real multiprocessor system. Numerical results show that the parallel multisplitting relaxation methods always perform much better than the corresponding sequential alternatives, and that the asynchronous multisplitting relaxation methods often outperform their corresponding synchronous counterparts. Moreover, the two-sweep relaxed multisplitting methods have better convergence properties than their corresponding one-sweep relaxed ones in the sense that they have larger convergence domains and faster convergence speeds. Hence, the asynchronous multisplitting unsymmetric relaxation iterations should be the methods of choice for solving the large sparse linear complementarity problems in the parallel computing environments.     

A PROJECTION-TYPE METHOD FOR SOLVING VARIOUS WEBER PROBLEMS

This paper investigates various Weber problems including unconstrained Weber problems and constrained Weber problems under l1, l2 and l∞-norms. First with a transformation technique various Weber problems are turned into a class of monotone linear variational inequalities. By exploiting the favorable structure of these variational inequalities, we present a new projection-type method for them. Compared with some other projection-type methods which can solve monotone linear variational inequality, this new projection-type method is simple in numerical implementations and more efficient for solving this class of problems; Compared with some popular methods for solving unconstrained Weber problem and constrained Weber problem, a singularity would not happen in this new method and it is more reliable by using this new method to solve various Weber problems.     

Learning to sample initial solution for solving 0-1 discrete optimization problem by local search

Local search methods are convenient alternatives for solving discrete optimization problems(DOPs).These easy-to-implement methods are able to find approximate optimal solutions within a tolerable time limit.It is known that the quality of the initial solution greatly affects the quality of the approximated solution found by a local search method. In this paper, we propose to take the initial solution as a random variable and learn its preferable probability distribution. The aim is to sample a g...     

On an extension of the H k mean curvature flow

In this note,we generalize an extension theorem in [Le-Sesum] and [Xu-Ye-Zhao] of the mean curvature flow to the Hk mean curvature flow under some extra conditions.The main difficulty in proving the extension theorem is to find a suitable version of Michael-Simon inequality for the Hk mean curvature flow,and to do a suitable Moser iteration process.These two problems are overcome by imposing some extra conditions which may be weakened or removed in our forthcoming paper.On the other hand,we derive some estimates for the generalized mean curvature flow,which have their own interesting.     

THE NUMERICAL METHODS FOR SOLVING EULER SYSTEM OF EQUATIONS IN REPRODUCING KERNEL SPACE H~2(R)

1. IntroductionIn recede years, more and more people are interested in solving Euler system of equations.They presented various methods to simulate the flow of the complicated fluid field. It is wellknown that Euler system of equations has described many practical engineering problems, suchas spherically symmetric flow, the flow inside a pipe, whose sectional area changed slowly, theradius of curvature is large, sectional area is small and so on. And it not only describes theincompressible id…     

TRANSPORTATION NETWORKS： OLD AND NEW

This paper shows a number of problems in pure and applied mathematics that are solved by constructing transportation networks Moreover, it also shows thatall the solutions are characterized by forbidden configuratlozz which are not minors.However, all the characterisations are much related to the graphic method which was found by Chinese for solving a kind of the transportation problem in the fifties.     

A LQP BASED INTERIOR PREDICTION-CORRECTION METHOD FOR NONLINEAR COMPLEMENTARITY PROBLEMS

To solve nonlinear complementarity problems （NCP）, at each iteration, the classical proximal point algorithm solves a well-conditioned sub-NCP while the Logarithmic-Quadratic Proximal （LQP） method solves a system of nonlinear equations （LQP system）. This paper presents a practical LQP method-based prediction-correction method for NCP. The predictor is obtained via solving the LQP system approximately under significantly relaxed restriction, and the new iterate （the corrector） is computed directly by an explicit formula derived from the original LQP method. The implementations are very easy to be carried out. Global convergence of the method is proved under the same mild assumptions as the original LQP method. Finally, numerical results for traffic equilibrium problems are provided to verify that the method is effective for some practical problems.     

A Penalty Approach for Generalized Nash Equilibrium Problem

The generalized Nash equilibrium problem (GNEP) is a generalization of the standard Nash equilibrium problem (NEP),in which both the utility function and the strategy space of each player depend on the strategies chosen by all other players.This problem has been used to model various problems in applications.However,the convergent solution algorithms are extremely scare in the literature.In this paper,we present an incremental penalty method for the GNEP,and show that a solution of the GNEP can be found by solving a sequence of smooth NEPs.We then apply the semismooth Newton method with Armijo line search to solve latter problems and provide some results of numerical experiments to illustrate the proposed approach.     

ON DE FINETTI'S OPTIMAL IMPULSE DIVIDEND CONTROL PROBLEM UNDER CHAPTER 11 BANKRUPTCY

Motivated by recent advances made in the study of dividend control and risk management problems involving the U.S.bankruptcy code,in this paper we follow [44] to revisit the De Finetti dividend control problem under the reorganization process and the regulator’s intervention documented in U.S.Chapter 11 bankruptcy.We do this by further accommodating the fixed transaction costs on dividends to imitate the real-world procedure of dividend payments.Incorporating the fixed transaction costs transfor...     

Measuring the Size(1)

How do you measure the size of a one-dimensional figure?The size of a line segment is given by its length.How do you measure the size of two-dimensional figures?Rectangles and other polygons are two-dimensional.There are two common ways of measuring these figures.Perimeter measures the boundary.Area measures the space inside the figure.     

THREE PROBLEMS IN SEARCHING FOR A MOVING TARGET BETWEEN TWO SITES

Suppose that a moving target moves randomly between two sites and its movement is modeled by a homogeneous Markov chain.We consider three classical problems:(1)what kind of strategies are valid(2) what strategy is the optimal(3) what is the infimum of expected numbers of looks needed to detect the target Problem(3) is thoroughly solved,and some partial solutions to problems(1) and(2) are achieved.     

SOME NEW RESULTS IN MULTIVARIABLE LINEAR SYSTEM ——Some Achievements in China, 1980—1984

From 1980—1984 some new results on multivariable linear system had been obtained by Chinese authors. The author of this paper gave out in 5 section the concept of (A, B)-characteristic subspace and its applications, the minimal polynomial matrix and its properties and the equivalence between state-space model and polynomial matrix and other corresponding results, some applications of Yokoyama canonical form especially in solving the disturbance decoupling problem, the properties of external skew prime polynomial matrices and their applications to output regulation, the internal model principle in frequence domain and robust control, and finally some other results were introduced.     

An algorithm for solving new trust region subproblem with conic model

The new trust region subproblem with the conic model was proposed in 2005, and was divided into three different cases. The first two cases can be converted into a quadratic model or a convex problem with quadratic constraints, while the third one is a nonconvex problem. In this paper, first we analyze the nonconvex problem, and reduce it to two convex problems. Then we discuss some dual properties of these problems and give an algorithm for solving them. At last, we present an algorithm for solving the new trust region subproblem with the conic model and report some numerical examples to illustrate the efficiency of the algorithm.     

基于DNA计算的Hamilton图

DNA computing is a novel method for solving a class of intractable computationalproblems in which the computing can grow exponentially with problem size. Up to now, manyaccomplishments have been achieved to improve its performance and increase its reliability.Hamilton Graph Problem has been solved by means of molecular biology techniques. A smallgraph was encoded in molecules of DNA, and the “operations” of the computation wereperformed with standard protocols and enzymes. This work represents further evidence forthe ability of DNA computing to solve NP-complete search problems.     

RELIABILITY ANALYSIS FOR A REPAIRABLE PARALLEL SYSTEM WITH TIME-VARYING FAILURE RATES

To solve a real problem :how to calculate the reliability of a system with time-varying failure rates in industry systems,this paper studies a model for the load-sharing parallel system with time-varying failure rates,and obtains calculating formulas of reliability and availability of the system by solving differential equations. In this paper, the failure rates are expressed in polynomial configuration. The constant,linear and Weibull failure rate are in their special form. The polynomial failure rates provide flexibility in modeling the practical time-varying failure rates.     

ERROR ESTIMATES FOR THE RECURSIVE LINEARIZATION OF INVERSE MEDIUM PROBLEMS

This paper is devoted to the mathematical analysis of a general recursive linearization algorithm for solving inverse medium problems with multi-frequency measurements. Under some reasonable assumptions, it is shown that the algorithm is convergent with error estimates. The work is motivated by our effort to analyze recent significant numerical results for solving inverse medium problems. Based on the uncertainty principle, the recursive linearization allows the nonlinear inverse problems to be reduced to a set of linear problems and be solved recursively in a proper order according to the measurements. As an application, the convergence of the recursive linearization algorithm [Chen, Inverse Problems 13（1997）, pp.253-282] is established for solving the acoustic inverse scattering problem.