Modified Moment Problem in Complex Domains

We study the moment problem in a complex domain. The results obtained generalize and sharpen the well-known Kaz'min result. We also consider applications of these results to the Abel--Goncharov problem (the uniqueness and construction problem) and to the completeness problem for systems of analytic functions.     

The combinatorial complexity of masterkeying

We consider the combinatorial complexity of the algorithmic design of mechanical master key locking systems. Such locking systems use pin tumblers and profile elements (wards) to realize the functional dependencies given by a key/cylinder matrix. We prove that even very restricted versions of the masterkeying problem are NP-complete. We show that the general masterkeying problem can be defined by an integer linear program whose number of variables and restrictions is polynomial in the size of the key/cylinder matrix and the size of the locking system. Heuristic solutions are also discussed.     

Merit Functions for Complementarity and Related Problems: A Survey

Merit functions have become important tools for solving various mathematical problems arising from engineering sciences and economic systems. In this paper, we are surveying basic principles and properties of merit functions and some of their applications. As a particular case we will consider the nonlinear complementarity problem (NCP) and present a collection of different merit functions. We will also introduce and study a class of smooth merit functions for the NCP.     

Bifurcations of limit circles and center conditions for a class of non-analytic cubic Z 2 polynomial differential systems

In this paper, bifurcations of limit cycles at three fine focuses for a class of Z ₂-equivariant non-analytic cubic planar differential systems are studied. By a transformation, we first transform nonanalytic systems into analytic systems. Then sufficient and necessary conditions for critical points of the systems being centers are obtained. The fact that there exist 12 small amplitude limit cycles created from the critical points is also proved. Henceforth we give a lower bound of cyclicity of Z ₂-equivariant non-analytic cubic differential systems.     

Asymptotic stability of viscous contact wave for the inflow problem of the heat-conductive ideal gas without viscosity

This paper is devoted to studying the inflow problem governed by the non-viscous and heat-conductive gas dynamic system in the one-dimensional half space. We establish the unique global-in-time existence and the asymptotic stability of the viscous contact wave. The contact discontinuity in the linearly degenerate field is less stable, and the dissipative mechanism for non-viscous systems is also weaker than that of viscous systems, these all make the problem more challenging. We used the weighted energy estimates to overcome those difficulties. Some technical discussions were created carefully by taking good advantage of properties of the supersonic region and the viscous contact wave.     

Some results on systems of quasi-variational inclusion problems and systems of generalized quasi-variational inclusion problems

In this paper, we study systems of quasi-variational inclusion problem and systems of quasi-variational disclusion problem. From the existence theorems of solution for these two types of problems, we study various types of systems of quasi-variational inclusion problems, systems of quasi-equilibrium problems, systems of quasi-KKM theorem, abstract economics and system of KKM theorem. We also show their equivalent relations. We study further existence theorems of solution for generalized quasi-variational inclusion problem. Our results are different from any existence result in the literature.     

Symmetry preserving partial eigenvalue assignment for undamped structural systems

In this paper, the partial eigenvalue assignment problem for undamped structural systems by output feedback control where the output matrix is also a designing parameter is considered. We propose a method to solve this problem in which the unwanted eigenvalues are move to desired values and all other eigenpairs remain unchanged. In addition, our method can preserve symmetry of the systems. Numerical example shows that the proposed method is effective.     

The inverse problem of the calculus of variations: The use of geometrical calculus in Douglas's analysis

The main objective of this paper is to work out a full-scale application of the integrability analysis of the inverse problem of the calculus of variations, as developed in recent papers by Sarlet and Crampin. For this purpose, the celebrated work of Douglas on systems with two degrees of freedom is taken as the reference model. It is shown that the coordinate-free, geometrical calculus used in Sarlet and Crampin's general theoretical developments provides effective tools also to do the practical calculations. The result is not only that all subcases distinguished by Douglas can be given a more intrinsic characterization, but also that in most of the cases, the calculations can be carried out in a more efficient way and often lead to sharper conclusions.

     

Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems

Whether or not the general asymmetric variational inequality problem can be formulated as a differentiable optimization problem has been an open question. This paper gives an affirmative answer to this question. We provide a new optimization problem formulation of the variational inequality problem, and show that its objective function is continuously differentiable whenever the mapping involved in the latter problem is continuously differentiable. We also show that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem. We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems.     

Extensions of the generalized moment theorem and applications to distributed-parameter control systems

In this paper, we present extensions to the generalized moment theorem and apply it to optimal control problems for a certain class of distributed-parameter systems. We also apply it to the time-optimal control problem and extend the results of Ref. 1 pertaining to the largest controllable set, so that we can discuss the problem of recoverability for some distributed-parameter systems.The author wishes to express his gratitude to Professor P. K. C. Wang for his guidance and suggestions.     

The existence, uniqueness and stability of positive periodic solution for periodic reaction-diffusion system

1. IntroductionLet fi e RN be a bounded open domain with smooth boundary afl, and for eachi E {1, 2,', m}, let Li be a second order differelltial operator defined byand Bi be a boundary operator given bywhere % denotes the outward normal derivative of m on an.We consider the following boundary value problem of the reaction-diffusion system withtime delaywhere i = 1, 2,'. I m) mr = "i(x, t -- r), r 2 0 is a constant. If r ~ 0, it means that system(I) does not include the terms of time lag…     

Interface Problems for Elliptic Systems

Interface problems for elliptic systems of second order partial differential equations are studied. The main result is that the solution in the neighborhood of the singular point can be divided into two parts one of which is a solution to the homogeneous system with constant coefficients, and the other one possesses higher regularity.     

Two general centre producing systems for the Poincar\'{e} problem

We consider the polynomial system $\td{x}{t}=-y-ax^{s+3}y^{n-s-3}-bx^{s+1}y^{n-s-1},$\, $\td{y}{t}=x+cx^{s+2}y^{n-s-2} + dx^sy^{n-s}$ where $n \ge 3$ is an odd integer and $s=0, \dots, n-3$ is an even integer. We calculate the first three nonzero Lyapunov coefficients for the system and obtain a Gr\"obner basis for the ideal generated by them. Potential centre conditions for the system are obtained by setting the basis elements equal to zero and solving the resulting system. This gives five basic solutions and within this set we find two well known classes of centres and three new centre producing systems. One of the three is a variant of one of the other new systems, so we obtain two general independent systems which produce multiple centre conditions for each $n \ge 5.$     

A RIEMANN-HILBERT PROBLEM IN A RIEMANN SURFACE

One of the inspirations behind Peter Lax’s interest in dispersive integrable systems, as the small dispersion parameter goes to zero, comes from systems of ODEs discretizing 1-dimensional compressible gas dynamics [17]. For example, an understanding of the asymptotic behavior of the Toda lattice in different regimes has been able to shed light on some of von Neumann’s conjectures concerning the validity of the approximation of PDEs by dispersive systems of ODEs. Back in the 1990s several authors have worked on the long time asymptotics of the Toda lattice [2, 7, 8, 19]. Initially the method used was the method of Lax and Levermore [16], reducing the asymptotic problem to the solution of a minimization problem with constraints (an "equilibrium measure" problem). Later, it was found that the asymptotic method of Deift and Zhou (analysis of the associated Riemann-Hilbert factorization problem in the complex plane) could apply to previously intractable problems and also produce more detailed information. Recently, together with Gerald Teschl, we have revisited the Toda lattice; instead of solutions in a constant or steplike constant background that were considered in the 1990s we have been able to study solutions in a periodic background. Two features are worth noting here. First, the associated Riemann-Hilbert factorization problem naturally lies in a hyperelliptic Riemann surface. We thus generalize the Deift-Zhou "nonlinear stationary phase method" to surfaces of nonzero genus. Second, we illustrate the important fact that very often even when applying the powerful Riemann-Hilbert method, a Lax-Levermore problem is still underlying and understanding it is crucial in the analysis and the proofs of the Deift-Zhou method!     

The studies of systems of variational inclusions problems and variational disclusions problems with applications

In this paper, we study existence theorems of solutions for systems of variational inclusions problems and systems of variational disclusions problems. From these existence results, we establish existence theorems of solutions for systems of generalized vector quasiequilibrium problems and systems of quasioptimization problems.     

Solution of large eigenvalue problems in electronic structure calculations

Predicting the structural and electronic properties of complex systems is one of the outstanding problems in condensed matter physics. Central to most methods used in molecular dynamics is the repeated solution of large eigenvalue problems. This paper reviews the source of these eigenvalue problems, describes some techniques for solving them, and addresses the difficulties and challenges which are faced. Parallel implementations are also discussed.Work supported by NSF grants DMR-9217287 and ASC 95-04038, and by the Minnesota Supercomputer Institute     

Hill regions of charged three-body systems

For charged three-body systems, we discuss the configurations and orientations that are admissible for given values of the conserved total energy and angular momentum. The admissible configurations and orientations are discussed on a configuration space that is reduced by the translational, rotational and dilation symmetries of charged three-body systems. We consider the examples of the charged three-body systems given by the helium atom (two electrons and a nucleus) and the compound of two electrons and one positron. For comparison, the well known example of the Newtonian gravitational three-body system is discussed for the scheme presented in this paper first.     

Existences theorems of systems of vector quasi-equillibrium problems and mathematical programs with equilibrium constraint

In this paper, we introduce systems of vector quasi-equilibrium problems and prove the existence of their solutions. As applications of our results, we derive the existence theorems for solution of system of vector quasi-saddle point problem, the existences theorems of a solution of system of generalized quasi-minimax inequalities, the mathematical program with equilibrium constraint, semi-infinite and bilevel problems.     

Point‐wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19 :1263–1317; Nonlinear Anal. 1997; 28 :1299–1322; Chin. Ann. Math. 2004; 25B :37–56). We give a new, very simple proof of this result and also give a sharp point‐wise decay estimate of the solution. Then, we consider the mixed initial‐boundary‐value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12 (1):59–78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point‐wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

On the Cauchy problem for a capillary drop. Part I: irrotational motion

The Cauchy problem for the motion of a liquid drop under surface tension is solved locally in time on the basis of a general abstract existence theorem for Hamiltonian systems which seems to be of interest also in other areas. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.