Tiling Bricks with Bricks

The problem addressed by the paper is the filling of a large brick with replicas of smaller bricks of different sizes. We solve this problem by reducing it to an algebraic problem about polynomials. As a by-product, we obtain new combinatorial interpretations of the connection constants linking some classical polynomial sequences of combinatorics.     

带裂缝的不同材料拼接平面混合问题

本文讨论了带若干条任意形状裂缝的不同材料拼接平面的混合问题，即已知裂缝一侧的位移和另一侧的外应力求弹性平衡，给出了混合问题的正确提法，问题是用复就方法求解的，并归结为求解某种正则型奇异积分方程组，证明了适当且唯一地选择某些待定常数的值，该方程组有唯一解。     

带裂缝的半平面弹性基本问题

本文用复变方法讨论了半平面内含若干条任意形状裂缝时的弹性基本问题,包括各向同性和各向异性两种情况,把寻求复应力函数的问题归结为求解某种带若干待定常数的正则型奇异积分方程,证明了若适当且唯一地选择这些常数的值,该方程的解存在且唯一。     

Parametric integer linear programming: A synthesis of branch and bound with cutting planes

A branch and bound algorithm is designed to solve the general integer linear programming problem with parametric right-hand sides. The right-hand sides have the form b + θd where b and d are comformable vectors, d consists of nonnegative constants, and θ varies from zero to one.The method consists of first determining all possible right-hand side integer constants and appending this set of integer constants to the initial tableau to form an expanded problem with a finite number of family members. The implicit enumeration method gives a lower bound on the integer solutions. The branch and bound method is used with fathoming tests which allow one family member possibly to fathom other family members. A cutting plane option applies a finite number of cuts to each node before branching. In addition, the cutting plane method is invoked whenever some members are feasible at a node and others are infeasible. The branching and cutting process is repeated until the entire family of problem has been solved.     

Fourier Method for Riemann-Hilbert-Poincaré Problem of Analytic Function

Spectrum problem with Riemann-Hilbert-Poincaré boundary condition is studied. This problem will lead to inhomogeneous Fuchsian differential equations with its right hand side depending on some constants to be determined simultaneously. We find out that the multiplicities of eigenfunctions for different eigenvalues are not necessary the same, which are in sharp contrast to the known results of Riemann-Hilbert problem for analytic functions.     

The Umd Constants of the Summation Operators

The UMD property of a Banach space is one of the most useful properties when one thinks about possible applications. This is in particular due to the boundedness of the vector-valued Hilbert transform for functions with values in such a space. Looking at operators instead of at spaces, it is easy to check that the summation operator does not have the UMD property. The actual asymptotic behavior however of the UMD constants computed with martingales of length n is unknown. We explain, why it would be important to know this behavior, rephrase the problem of finding these UMD constants and give some evidence of how they behave asymptotically.     

A Priori Estimate for the Discontinuous Oblique Derivative Problem for Elliptic Systems

Recently [6] an existence as well as a uniqueness theorem for the discontinuous oblique derivative problem for nonlinear elliptic system of first order in the plane, see [12, 19, 23] was proved, based on some a priori estimate from [20]. This estimate, however, is deduced by reductio ad absurdum. Therefore the constants in this estimate are unknown so that the estimate cannot be used for numerical procedures, e.g. for approximating the solution of a nonlinear problem by solutions of related linear problems, see [24, 3, 4]. In this paper a direct proof of an a priori estimate is given using some variations of results from [14], see also [11], where the constants can explicitely be estimated. For related a priori estimates see [1 – 5, 8, 16, 17, 20, 21, 24 – 26]. A basic reference for the oblique derivative problem is [9].     

ON THE FUNDAMENTAL PROBLEM FOR AN INFINITE ELASTIC PLANE BONDED BY DIFFERENT ANISOTROPIC MATERIALS WITH CRACKS

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Ground States for a Stationary Mean-Field Model for a Nucleon

In this paper we consider a variational problem related to a model for a nucleon interacting with the ω and σ mesons in the atomic nucleus. The model is relativistic, and we study it in a nuclear physics nonrelativistic limit, which is of a very different nature than the nonrelativistic limit in the atomic physics. Ground states are shown to exist for a large class of values for the parameters of the problem, which are determined by the values of some physical constants.     

Scheduling problems in master-slave model

We consider scheduling problems in the master slave model, which was introduced by Sahni in 1996. The goal is to minimize the makespan and the total completion time. It has been shown that the problem of minimizing makespan is NP-hard. Sahni and Vairaktarakis developed some approximation algorithms to generate schedules whose makespan is at most constant times the optimal. In this paper, we show that the problem of minimizing total completion time is NP-hard in the strong sense. Then we develop algorithms to generate schedules whose total completion time and makespan are both bounded by some constants times their optimal values. Research supported in part by the National Science Foundation through grant DMI-0300156.     

On the problem of propagation of nonlinear coupled TE-TM waves in a layer

The problem of simultaneous propagation of two types of electromagnetic waves (TE and TM) in a plane dielectric waveguide filled with a nonlinear medium is considered. These polarized waves have different frequencies and different propagation constants. The physical problem is reduced to a nonlinear two-parameter transmission eigenvalue problem for Maxwell’s equations in a layer. The coupled eigenvalues are coupled propagation constants. A theorem on the existence and localization of coupled eigenvalues corresponding to coupled polarized electromagnetic waves is proved.     

Lyapunov inequalities for the periodic boundary value problem at higher eigenvalues

This paper is devoted to provide some new results on Lyapunov type inequalities for the periodic boundary value problem at higher eigenvalues. Our main result is derived from a detailed analysis on the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the study of some special minimization problems. This allows to obtain the optimal constants. Our applications include the Hill's equation where we give some new conditions on its stability properties and also the study of periodic and nonlinear problems at resonance where we show some new conditions which allow to prove the existence and uniqueness of solutions.     

Analysis of a Stokes interface problem

We consider a stationary Stokes problem with a piecewise constant viscosity coefficient. For the variational formulation of this problem we prove a well-posedness result in which the constants are uniform with respect to the jump in the viscosity coefficient. We apply a standard discretization with a pair of LBB stable finite element spaces. The main result of the paper is an infsup result for the discrete problem that is uniform with respect to the jump in the viscosity coefficient. From this we derive a robust estimate for the discretization error. We prove that the mass matrix with respect to some suitable scalar product yields a robust preconditioner for the Schur complement. Results of numerical experiments are presented that illustrate this robustness property. This author was supported by the German Research Foundation through the guest program of SFB 540     

An experimental survey of a posteriori Courant finite element error control for the Poisson equation

This comparison of some a posteriori error estimators aims at empirical evidence for a ranking of their performance for a Poisson model problem with conforming lowest order finite element discretizations. Modified residual-based error estimates compete with averaging techniques and two estimators based on local problem solving. Multiplicative constants are involved to achieve guaranteed upper and lower energy error bounds up to higher order terms. The optimal strategy combines various estimators.     

测度链上非线性m-点边值问题的正解

设T是一个测度链(时间标架)并且0,T∈T．讨论测度链上m-点边值问题其中a∈Cld((0,T),[0,∞)),f∈C((0,T)X[0,∞),[0,∞)),β,γ∈[0,∞),ξ∈(0,ρ(T)), ai∈[0,∞)(i=1,．．．,m-2)是一些满足适当条件的定常数．借助于锥上的不动点定理,得到了此问题存在单个及多个正解的一些新的更一般的结果．特别地,我们的结果推广并改进了一些已有的结论．     

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.     

Pseudopolynomial time solvability of a quadratic Euclidean problem of finding a family of disjoint subsets

In this paper, a strongly NP-hard problem of finding a family of disjoint subsets with given cardinalities in a finite set of points from a Euclidean space is considered. Minimization of the sum over all required subsets of the sum of the squared distances from the elements of these subsets to their geometric centers is used as the search criterion. It is proved that if the coordinates of the input points are integer and the space dimension and the number of required subsets are fixed (i.e., bounded by some constants), the problem is a pseudopolynomial time solvable one.     

Applications of a space decomposition method to linear and nonlinear elliptic problems

This work presents some space decomposition algorithms for a convex minimization problem. The algorithms has linear rate of convergence and the rate of convergence depends only on four constants. The space decomposition could be a multigrid or domain decomposition method. We explain the detailed procedure to implement our algorithms for a two-level overlapping domain decomposition method and estimate the needed constants. Numerical tests are reported for linear as well as nonlinear elliptic problems. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 717–737, 1998     

Subsonic irrotational flows in multiply connected domains

The solvability of a boundary value problem for a nonlinar elliptic partial differential equation in a plane, bounded, multiply connected domain is investigated. The solution of the problem is equal to unknown constants on inner components of the boundary. We want to determine these constants so that the solution satisfies the so-called trailing conditions. The results have immediate applications in hydrodynamics.     

Attractor dimension estimate for plane shear flow of micropolar fluid with free boundary

This research is motivated by a problem from lubrication theory. We consider a free boundary problem of a two‐dimensional boundary‐driven micropolar fluid flow. The existence of a unique global‐in‐time solution of the problem and the global attractor for the associated semigroup are known. In this paper we estimate the dimension of the global attractor in terms of the given data and the geometry of the domain of the flow by establishing a new version of the Lieb–Thirring inequality with constants depending explicitly on the geometry of the domain. We also obtain some new estimates for the Navier–Stokes shear flows. Copyright © 2005 John Wiley & Sons, Ltd.