Reconstruction of boundary regimes in the inverse problem of thermal convection of a high-viscosity fluid

A problem of reconstruction of boundary regimes in a model for free convection of a high-viscosity fluid is considered. A variational method and a quasi-inversion method are suggested for solving the problem in question. The variational method is based on the reduction of the original inverse problem to some equivalent variational minimum problem for an appropriate objective functional and solving this problem by a gradient method. When realizing the gradient method for finding a minimizing element of the objective functional, an iterative process actually reducing the original problem to a series of direct well-posed problems is organized. For the quasi-inversion method, the original differential model is modified by means of introducing special additional differential terms of higher order with small parameters as coefficients. The new perturbed problem is well-posed; this allows one to solve this problem by standard methods. An appropriate choice of small parameters gives an opportunity to obtain acceptable qualitative and quantitative results in solving the inverse problem. A comparison of the methods suggested for solving the inverse problem is made with the use of model examples.     

Penalty method for solving a class of stochastic differential variational inequalities with an application

The aim of this paper is to study the penalty method for solving a class of stochastic differential variational inequalities (SDVIs). The penalty problem for solving SDVIs is first constructed and the convergence of the sequences generated by the penalty problem is proved under some mild conditions. As an application, the convergence of the sequences generated by the penalty problem is obtained for solving a stochastic migration equilibrium problem with movement cost.     

Global minimization of a generalized linear multiplicative programming

This article presents a simplicial branch and bound algorithm for globally solving generalized linear multiplicative programming problem (GLMP). Since this problem does not seem to have been studied previously, the algorithm is apparently the first algorithm to be proposed for solving such problem. In this algorithm, a well known simplicial subdivision is used in the branching procedure and the bound estimation is performed by solving certain linear programs. Convergence of this algorithm is established, and some experiments are reported to show the feasibility of the proposed algorithm.     

The impact of a conceptual model-based mathematics computer tutor on multiplicative reasoning and problem-solving of students with learning disabilities

The study explored the impact of Please Go Bring Me-COnceptual Model-based Problem Solving (PGBM-COMPS) computer tutoring system on multiplicative reasoning and problem solving of students with learning disabilities. The PGBM-COMPS program focused on enhancing the multiplicative reasoning and problem solving through nurturing fundamental mathematical ideas and moving students above and beyond the concrete level of operation. This is achieved by taking advantages of the constructivist approach from mathematics education and explicit conceptual model-based problem solving approach from special education. Participants were three elementary students with learning disabilities (LD). A mixed method design was employed to investigate the effect of the PGBM-COMPS program on enhancing students’ multiplicative reasoning and problem solving. It was found that the PGBM-COMPS program significantly improved participating students’ problem solving performance not only on researcher developed criterion tests but also on a norm-referenced standardized test. Qualitative and quantities data from this study indicate that, in addition to nurturing fundamental concept of composite units, it is necessary to help students to understand underlying problem structures and move toward mathematical model-based problem representation and solving for generalized problem solving skills.     

“电池剩余放电时间预测”问题解析

本文就2016年"高教社杯"全国大学生数学建模竞赛C题"电池剩余放电时间预测"给出了一种求解方法,并针对学生在参赛论文中出现的问题作了简要的说明与点评。为保证求解的连贯性,论文的前一部分是问题的求解,后一部分是参赛论文的点评。     

Global Maximization of a Generalized Concave Multiplicative Function

This article presents a branch-and-bound algorithm for globally solving the problem (P) of maximizing a generalized concave multiplicative function over a compact convex set. Since problem (P) does not seem to have been studied previously, the algorithm is apparently the first algorithm to be proposed for solving this problem. It works by globally solving a problem (P1) equivalent to problem (P). The branch-and-bound search undertaken by the algorithm uses rectangular partitioning and takes place in a space which typically has a much smaller dimension than the space to which the decision variables of problem (P) belong. Convergence of the algorithm is shown; computational considerations and benefits for users of the algorithm are given. A sample problem is also solved.     

“接触式轮廓仪的自动标注”的问题解析

就2020年"高教社杯"全国大学生数学建模竞赛D题"接触式轮廓仪的自动标注"给出了一种求解方法,并针对学生在参赛论文中出现的问题作了简要的说明与点评.为保证求解的连贯性,论文的前一部分是问题的求解,后一部分是参赛论文的点评.     

一类多商品设施选址问题的基于线性松弛解的启发式方法

多商品设施选址问题是众多设施选址问题中一类重要而困难的问题.在这一问题中,顾客的需求可能包含不止一种商品.对于大规模问题,成熟的商业求解器往往不能在满意的时间内找到高质量的可行解.研究了无容量限制的单货源多商品设施选址问题的一般形式,并给出了应用于此类问题的两个启发式方法.这两个方法基于原选址问题的线性规划松弛问题的最优解,分别通过求解紧问题和邻域搜索的方式给出了原问题的一个可行上界.理论分析指出所提方法可以实施于任意可行问题的实例.数值结果表明所提方法可以显著地提高求解器求解此类设施选址问题的求解效率.     

General fractional variational problem depending on indefinite integrals

In this report, we consider two kind of general fractional variational problem depending on indefinite integrals include unconstrained problem and isoperimetric problem. These problems can have multiple dependent variables, multiorder fractional derivatives, multiorder integral derivatives and boundary conditions. For both problems, we obtain the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Also, we apply the Rayleigh-Ritz method for solving the unconstrained general fractional variational problem depending on indefinite integrals. By this method, the given problem is reduced to the problem for solving a system of algebraic equations using shifted Legendre polynomials basis functions. An approximate solution for this problem is obtained by solving the system. We discuss the analytic convergence of this method and finally by some examples will be showing the accurately and applicability for this technique.     

Problem solving as a challenge for mathematics education in The Netherlands

This paper deals with the challenge to establish problem solving as a living domain in mathematics education in The Netherlands. While serious attempts are made to implement a problem-oriented curriculum based on principles of realistic mathematics education with room for modelling and with integrated use of technology, the PISA 2003 results suggest that this has been successful in educational practice only to a limited extent. The main difficulties encountered include institutional factors such as national examinations and textbooks, and issues concerning design and training. One of the main challenges is the design of good problem solving tasks that are original, non-routine and new to the students. It is recommended to pay attention to problem solving in primary education and in textbook series, to exploit the benefits of technology for problem solving activities and to use the schools’ freedom to organize school-based examinations for types of assessment that are more appropriate for problem solving.     

Solving a kind of boundary-value problem for ordinary differential equations using Fermi—The next generation CUDA computing architecture

The aim of this paper is to show that a special kind of boundary value problem for solving second-order ordinary differential equations can be efficiently solved on modern heterogeneous computer architectures based on CPU and GPU Fermi processors. Such a problem reduces to the problem of solving a large tridiagonal system of linear equations with an almost Toeplitz structure. The considered algorithm is based on the recently developed divide and conquer method for solving linear recurrence systems with constant coefficients.     

A numerical solution of problems in calculus of variation using direct method and nonclassical parameterization

In this paper a direct method for solving variational problems using nonclassical parameterization is presented. A nonclassical parameterization based on nonclassical orthogonal polynomials is first introduced to reduce a variational problem to a nonlinear mathematical programming problem. Then, using the Lagrange multiplier technique the problem is converted to that of solving a system of algebraic equations. Illustrative examples are given to demonstrate the validity and applicability of the technique.     

An improvement in the Gavish-Shlifer algorithm for a class of transportation scheduling problems

In [2] Gavish and Shlifer present an algorithm for solving a class of transportation scheduling problems which includes the delivery problem, the school bus problem, and others. Their algorithm is based on the ‘savings method’ of Clarke and Wright. By solving a sequence of assignment problems, upper bounds may be generated for the original problem and the optimal solution determined through a branch and bound procedure. However, for certain problems the CPU time becomes excessive. In this paper we show that the bounds may be improved by solving a related maximum matching problem instead of the assignment problem. The result is that fewer branches need to be investigated. Computational results are presented indicating that considerably less CPU time is needed to solve problems using this approach than with the approach of Gavish and Shlifer.     

解线性互补问题的预处理加速模Gauss-Seidel迭代方法

本文提出了解线性互补问题的预处理加速模系Gauss-Seidel迭代方法,当线性互补问题的系统矩阵是M-矩阵时证明了方法的收敛性,并给出了该预处理方法关于原方法的一个比较定理.数值实验显示该预处理迭代方法明显加速了原方法的收敛.     

A branch and bound network approach to the canonical constrained entropy problem

In this paper, we present a branch and bound algorithm for solving the constrained entropy mathematical programming problem. Unlike other methods for solving this problem, our method solves more general problems with inequality constraints. The advantage of the proposed technique is that the relaxed problem solved at each node is a singly constrained network problem. The disadvantage is that the relaxed problem has twice as many variables as the original problem. An application to regional planning is given, and an example problem is solved.     

Evaluation model of problem solving

Problem solving is a style of thinking, which transforms a given problem to the goal state through a so-called PS (problem solving) path. Different from the traditional GPS (General Problem Solver) approach, the focus in this paper is placed on how to judge the performance of PS paths, that is, the evaluation problem of problem solving.A series of PS paths point from the given source problem to the destination goal, then form a PSN (PS Network). This paper proposes an elaborated CPSN (Coordinate Problem Solving Network) as the evaluation model of problem solving. In CPSN, each problem is assigned a unique coordinate and then each PS path can have an evaluation vector. Several examples show such arrangement can give more insight to PS paths.Furthermore, an incremental learning algorithm is developed for the update of CPSN. When a new PS path is obtained, it is not necessary to recalculate the whole CPSN. Examples show such algorithm provides a more efficient way in finding new PS paths.     

Calculation of the derivative of an inaccurately defined function by means of feedback laws

The problem of calculating the derivative of an inaccurately defined function is considered. This problem is one of the “classical” problems in mathematical analysis. Various algorithms for solving this problem have been proposed by many authors. In the present study, constructions of positional control theory are invoked to solve this problem. Due to the presence of error in the measurement of the values of a function, the exact calculation of the derivative seems to be impossible. In view of this feature, a problem solving algorithm is presented that is based on an appropriate modification of the method of models controlled by feedback laws and is robust with respect to informational noise and measurement errors.     

Comparison of metaheuristic algorithms for Examination Timetabling Problem

SA, TS, GA and ACS are four of the main algorithms for solving challenging problems of intelligent systems. In this paper we consider Examination Timetabling Problem that is a common problem for all universities and institutions of higher education. There are many methods to solve this problem, In this paper we use Simulated Annealing, Tabu Search, Genetic Algorithm and Ant Colony System in their basic frameworks for solving this problem and compare results of them with each other.