Observations on the tanh-coth expansion method for finding solutions to nonlinear evolution equations

The ‘tanh-coth expansion method’ for finding solitary travelling-wave solutions to nonlinear evolution equations has been used extensively in the literature. It is a natural extension to the basic tanh-function expansion method which was developed in the 1990s. It usually delivers three types of solution, namely a tanh-function expansion, a coth-function expansion, and a tanh-coth expansion. It is known that, for every tanh-function expansion solution, there is a corresponding coth-function expansion solution. It is shown that there is a tanh-coth expansion solution that is merely a disguised version of the coth solution. In many papers, such tanh-coth solutions are erroneously claimed to be ‘new’. However, other tanh-coth solutions may be delivered that are genuinely new in the sense that they would not be delivered via the basic tanh-function method. Similar remarks apply to tan, cot and tan-cot expansion solutions.     

Convergence of a large time step generalization of Godunov's method for conservation laws

A natural generalization of Godunov's method for Courant numbers larger than 1 is obtained by handling interactions between neighboring Riemann problems linearly, i.e., by allowing waves to pass through one another with no change in strength or speed. This method is well defined for arbitrarily large Courant numbers and can be written in conservation form. It follows that if a sequence of approximations converges to a limit u(x,t) as the mesh is refined, then u is a weak solution to the system of conservation laws. For scalar problems the method is total variation diminishing and every sequence contains a convergent subsequence. It is conjectured that in fact every sequence converges to the (unique) entropy solution provided the correct entropy solution is used for each Riemann problem. If the true Riemann solutions are replaced by approximate Riemann solutions which are consistent with the conservation law, then the above convergence results for general systems continue to hold.     

Affine-Periodic Solutions and Pseudo Affine-Periodic Solutions for Differential Equations with Exponential Dichotomy and Exponential Trichotomy

It is proved that every $(Q,T)$-affine-periodic differential equation has a $(Q,T)$-affine-periodic solution if the corresponding homogeneous linear equation admits exponential dichotomy or exponential trichotomy. This kind of ``periodic'' solutions might be usual periodic or quasi-periodic ones if $Q$ is an identity matrix or orthogonal matrix. Hence solutions also possess certain symmetry in geometry. The result is also extended to the case of pseudo affine-periodic solutions.     

Functionality defense through diversity: a design framework to multitier systems

Diversification is one of the most effective approaches to defend multitier systems against attacks, failure, and accidents. However, designing such a system with effective diversification is a challenging task because of stochastic user and attacker behaviors, combinatorial-explosive solution space, and multiple conflicting design objectives. In this study, we present a systematic framework for exploring the solution space, and consequently help the designer select a satisfactory system solution. A simulation model is employed to evaluate design solutions, and an artificial neural network is trained to approximate the behavior of the system based on simulation output. Guided by a trained neural network, a multi-objective evolutionary algorithm (MOEA) is proposed to search the solution space and identify potentially good solutions. Our MOEA incorporates the concept of Herbert Simon??s satisficing. It uses the decision maker??s aspiration levels for system performance metrics as its search direction to identity potentially good solutions. Such solutions are then evaluated via simulation. The newly-obtained simulation results are used to refine the neural network. The exploration process stops when the result converges or a satisfactory solution is found. We demonstrate and validate our framework using a design case of a three-tier web system.     

On the regularization of mixed complementarity problems

A variational inequality problem (VIP) satisfying a constraint qualification can be reduced to a mixed complementarity problem (MOP). Monotonicity of the VIP implies that the MOP is also monotone. Introducing regularizing perturbations, a sequence of strictly monotone mixed complementarity problems is generated. It is shown that, if the original problem is solvable, the sequence of computable inexact solutions of the strictly monotone MCP's is bounded and every accumulation point is a solution. Under an additional condition on the precision used for solving each subproblem, the sequence converges to the minimum norm solution of the MCP.     

Jacobi-Davidson方法中的修正方程和对应的精化方法

Jacobi-Davidson方法的核心之一是求解用以合理扩展投影子空间的线性修正方程组,众多文献均认为该方程是自然有解的.本文详细研究了修正方程,证明它可能无解,并给出了解存在的条件.同时,为克服近似特征向量的可能不收敛性,提出了精化的Jacobi-Davidson方法,建立了对应的修正方程.     

A stochastic local search algorithm with adaptive acceptance for high-school timetabling

Automating high school timetabling is a challenging task. This problem is a well known hard computational problem which has been of interest to practitioners as well as researchers. High schools need to timetable their regular activities once per year, or even more frequently. The exact solvers might fail to find a solution for a given instance of the problem. A selection hyper-heuristic can be defined as an easy-to-implement, easy-to-maintain and effective ‘heuristic to choose heuristics’ to solve such computationally hard problems. This paper describes the approach of the team hyper-heuristic search strategies and timetabling (HySST) to high school timetabling which competed in all three rounds of the third international timetabling competition. HySST generated the best new solutions for three given instances in Round 1 and gained the second place in Rounds 2 and 3. It achieved this by using a fairly standard stochastic search method but significantly enhanced by a selection hyper-heuristic with an adaptive acceptance mechanism.     

The optimal exchange algorithm and comparisons with the generalized remes algorithm † ‡

This paper is concerned with the general problem of the determination of the best uniform approximation of a given function. A special case is the calculation of the minimax solution of an overdetermined linear system. Single point exchange algorithms produce successive approximate solutions for such problems. An example is furnished by the generalized Remes algorithm, which includes both the original Remes algorithm and the Stiefel algorithm as special cases. The optimal exchange algorithm is similar, but it has the important feature that every exchange is optimal in a certain desirable sense. It is proved that eventually the optimal exchange and generalized Remes algorithms coincide. However, early exchanges in the latter algorithm can be far from optimal and quite inefficient. A combination of the optimal exchange and generalized Remes algorithms is suggested as a reasonable strategy.     

More on oscillation ofnth-order equations

In this paper we prove that a higher-order differential equation with one middle term has every bounded solution oscillatory. Moreover, the behavior of unbounded solutions is given. Two other results dealing with positive solutions are also given.     

Justification of the adiabatic principle for hyperbolic Ginzburg-Landau equations

We study the adiabatic limit in hyperbolic Ginzburg-Landau equations which are the Euler-Lagrange equations for the Abelian Higgs model. By passing to the adiabatic limit in these equations, we establish a correspondence between the solutions of the Ginzburg-Landau equations and adiabatic trajectories in the moduli space of static solutions, called vortices. Manton proposed a heuristic adiabatic principle stating that every solution of the Ginzburg-Landau equations with sufficiently small kinetic energy can be obtained as a perturbation of some adiabatic trajectory. A rigorous proof of this result has been found recently by the first author.     

ON HERMITIAN POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATION X-A^＊X^-2 A=I

The Hermitian positive definite solutions of the matrix equation X-A^＊X^-2 A=I are studied. A theorem for existence of solutions is given for every complex matrix A. A solution in case A is normal is given. The basic fixed point iterations for the equation are discussed in detail. Some convergence conditions of the basic fixed point iterations to approximate the solutions to the equation are given.     

School redistricting: embedding GIS tools with integer programming

The paper deals with a school redistricting problem in which blocks of a city must be assigned to schools according to diverse criteria. Previous approaches are reviewed and some desired properties of a good school districting plan are established. An optimization model together with a geographic information system environment are then proposed for finding a solution that satisfies these properties. A prototype of the system is described, some implementation issues are discussed, and two real-life examples from the city of Philadelphia are studied, one corresponding to a relatively easy to solve problem, and the other to a much harder one. The trade-offs in the solutions are analysed and feasibility questions are discussed. The results of the study strongly suggest that ill-defined spatial problems, such as school redistricting, can be addressed effectively by an interaction between objective analysis and subjective judgement.     

Solutions of the Hom-Yang–Baxter Equation from Monoidal Hom-(Co)Algebra Structures

A method for constructing solutions of the Hom-Yang–Baxter equations is presented. Thus, methods yields a so-called α-involutory solution of the Hom-Yang–Baxter equation for every monoidal Hom-(co)algebra structure on a space. Characterizations for solutions of Hom-Yang–Baxter equations arising from monoidal Hom-(co)algebra structures are given, and a monoidal Hom-(co)algebra structure which produces such a solution is constructed.     

基于正弦型光滑打磨函数对0-1规划问题的连续化求解方法

传统的求解0-1规划问题方法大多属于直接离散的解法.现提出一个包含严格转换和近似逼近三个步骤的连续化解法:(1)借助阶跃函数把0-1离散变量转化为[0,1]区间上的连续变量;(2)对目标函数采用逼近折中阶跃函数近光滑打磨函数,约束条件采用线性打磨函数逼近折中阶跃函数,把0-1规划问题由离散问题转化为连续优化模型;(3)利用高阶光滑的解法求解优化模型.该方法打破了特定求解方法仅适用于特定类型0-1规划问题惯例,使求解0-1规划问题的方法更加一般化.在具体求解时,采用正弦型光滑打磨函数来逼近折中阶跃函数,计算效果很好.     

On the integer-valued variables in the linear vertex packing problem

Given a graph with weights on vertices, the vertex packing problem consists of finding a vertex packing (i.e. a set of vertices, no two of them being adjacent) of maximum weight. A linear relaxation of one binary programming formulation of this problem has these two well-known properties: (i) every basic solution is (0, 1/2, 1)-valued, (ii) in an optimum linear solution, an integer-valued variable keeps the same value in an optimum binary solution.As an answer to an open problem from Nemhauser and Trotter, it is shown that there is a unique maximal set of variables which are integral in optimal (VLP) solutions.This research was supported by National Research Council of Canada GRANT A8528 and RD 804.     

Smooth and convergent ε-approximations of the first boundary-value problem for equations of the Kelvin-Voight and oldroyd fluids

Smooth convergent ε-approximations (11)–(13) for the equations of Oldroyd (8) and Kelvin-Voight (9), (10) fluids are constructed. It is shown that the first initial boundary-value problem for two-dimensional system (11) and three-dimensional systems (12) and (13) for every ε>0 has a unique classical solution, and as ε → 0 these solutions converge to classical solutions of the first initial boundary-value problem for Eqs. (8), (9), and (10) respectively. Bibliography: 10 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 246–255. Translated by O. A. Ivanov.     

Fast,accurate and robust adaptive finite difference methods for fractional diffusion equations

The computation time required by standard finite difference methods with fixed timesteps for solving fractional diffusion equations is usually very large because the number of operations required to find the solution scales as the square of the number of timesteps. Besides, the solutions of these problems usually involve markedly different time scales, which leads to quite inhomogeneous numerical errors. A natural way to address these difficulties is by resorting to adaptive numerical methods where the size of the timesteps is chosen according to the behaviour of the solution. A key feature of these methods is then the efficiency of the adaptive algorithm employed to dynamically set the size of every timestep. Here we discuss two adaptive methods based on the step-doubling technique. These methods are, in many cases, immensely faster than the corresponding standard method with fixed timesteps and they allow a tolerance level to be set for the numerical errors that turns out to be a good indicator of the actual errors.     

An indirect genetic algorithm for set covering problems

This paper presents a new type of genetic algorithm for the set covering problem. It differs from previous evolutionary approaches first because it is an indirect algorithm, ie the actual solutions are found by an external decoder function. The genetic algorithm itself provides this decoder with permutations of the solution variables and other parameters. Second, it will be shown that results can be further improved by adding another indirect optimisation layer. The decoder will not directly seek out low cost solutions but instead aims for good exploitable solutions. These are then post-optimised by another hill-climbing algorithm. Although seemingly more complicated, we will show that this three-stage approach has advantages in terms of solution quality, speed and adaptability to new types of problems over more direct approaches. Extensive computational results are presented and compared to the latest evolutionary and other heuristic approaches to the same data instances.     

Oscillation theorem for higher-order linear differential equations with periodic coefficients

In this paper, the zeros of solutions for higher-order linear differential equations with periodic coefficients are studied. It is shown that under certain hypotheses, the convergence exponent of zeros of the product of every fundamental solution is infinite.     

On the quality of heuristic solutions to a 19 × 19 quadratic assignment problem

Recently Murtagh et al. communicated the best (?) known solution to a 19 × 19 quadratic assignment problem, presented and solved earlier by Elshafei. Their approaches are based on the application of quadratic programming and on the construction of a good initial solution respectively. Their solutions are confronted with one better solution obtained by Lashkari and Jaisingh by means of linear assignment approximations, and many better solutions obtained by the present author through the application of exchange heuristics. The results stress the nonconvex character of the problem, and show the rather misleading nature of the concept: ‘good initial solution’. The quality of initial and final solution prove to be rather weakly (or even negatively!) correlated. A simple, but broad and randomized search results in significantly, better solutions, while simple statistical aids provide an estimate of the quality of these solutions. Finally we make remarks concerning some practical issues.