Some computational methods for systems of nonlinear equations and systems of polynomial equations

This paper gives a brief survey and assessment of computational methods for finding solutions to systems of nonlinear equations and systems of polynomial equations. Starting from methods which converge locally and which find one solution, we progress to methods which are globally convergent and find an a priori determinable number of solutions. We will concentrate on simplicial algorithms and homotopy methods. Enhancements of published methods are included and further developments are discussed.     

Operator algebras associated with polymorphisms

A short survey of the problems and developments in the theory of operator algebras associated with semigroup dynamical systems is presented. The main part is an exposition of results concerning the algebras generated by multivalued transformations (polymorphisms). Bibliography: 14 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 23–27.     

Uniqueness of Positive Solutions of Semilinear Elliptic Equations and Related Eigenvalue Problems

In this paper we survey some recent results about the uniqueness of the solution of some semilinear elliptic Dirichlet problems in bounded domains. The presentation aims to emphasize the role of the geometrical properties of the second eigenfunction of the linearized problem in the study of the above question. This motivates the analysis of the asymptotic behaviour of these eigenfunctions and of the relative eigenvalues when the nonlinear term is a power with exponent close to the critical Sobolev exponent. Research supported by MIUR, project “Variational Methods and Nonlinear Differential Equations”. Lecture held in the Seminario Matematico e Fisico on January 31, 2005 Received: June 2005     

Continuous dependence inequalities for a class of quasilinear parabolic problems

In this paper, we establish continuous dependence inequalities for the solutions u(x, t) of a class of nonlinear parabolic initial-boundary value problems and their gradients when the data are subject to variations. Dedicated to Joseph Hersch on the occasion of his 80th birthday (Received: February 24, 2005; revised: March 14, 2005)     

Fractional programming

The paper reviews some of the major developments in fractional programming. The emphasis is on applications and algorithms. Theoretical results are only outlined to the extent as they are needed to describe the algorithms.The survey is subjective in nature. It tries to critically evaluate developments in fractional programming after more than 20 years of research with over 500 articles. The authors hope that this somewhat critical report will serve as a guide for meaningful future work in this growing area of nonlinear programming.     

Controllability of Nonlinear Systems in Banach Spaces: A Survey

This paper presents a survey on research using fixed-point theorems and semigroup theory to study the controllability of nonlinear systems and functional integrodifferential systems in Banach spaces. Also discussed is the use of this technique in K-controllability and boundary controllability problems for nonlinear systems and integrodifferential systems in abstract spaces.     

Stepwise Gauge Equivalence of Differential Operators

In this paper, we study the relation between the notion of gauge equivalence and solutions of certain systems of nonlinear partial differential equations. This relation is based on stepwise gauge equivalence.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 917–929.Original Russian Text Copyright ©2005 by S. P. Khekalo.     

Variational approach to various nonlinear problems in geometry and physics

In this survey, we will summarize the existence results of nonlinear partial differential equations which arises from geometry or physics by using variational method. We use the method to study Kazdan-Warner problem, Chern-Simons-Higgs model, Toda systems, and the prescribed Q-curvature problem in 4-dimension.      

Recent results on blow-up and quenching for nonlinear Volterra equations

In this survey paper, the author examines nonlinear Volterra integral equations of the second kind with solutions that blow-up or quench. The focus is on analytical results, although a few words about numerical solutions for such equations are provided. The integral equations arise in the mathematical modeling of thermal processes within a reactive–diffusive medium. The scope of this review is on the published literature between 1997 and 2005, serving as an update to a previous review by the same author.     

The achievable region method in the optimal control of queueing systems; formulations,bounds and policies

We survey a new approach that the author and his co-workers have developed to formulate stochastic control problems (predominantly queueing systems) asmathematical programming problems. The central idea is to characterize the region of achievable performance in a stochastic control problem, i.e., find linear or nonlinear constraints on the performance vectors that all policies satisfy. We present linear and nonlinear relaxations of the performance space for the following problems: Indexable systems (multiclass single station queues and multiarmed bandit problems), restless bandit problems, polling systems, multiclass queueing and loss networks. These relaxations lead to bounds on the performance of an optimal policy. Using information from the relaxations we construct heuristic nearly optimal policies. The theme in the paper is the thesis that better formulations lead to deeper understanding and better solution methods. Overall the proposed approach for stochastic control problems parallels efforts of the mathematical programming community in the last twenty years to develop sharper formulations (polyhedral combinatorics and more recently nonlinear relaxations) and leads to new insights ranging from a complete characterization and new algorithms for indexable systems to tight lower bounds and nearly optimal algorithms for restless bandit problems, polling systems, multiclass queueing and loss networks.     

Finite approximations of Markov operators

In this paper we survey some recent developments in the numerical analysis of Markov operators, and in particular Frobenius–Perron operators associated with chaotic discrete dynamical systems.     

Rigidity problems in toric topology: A survey

Several rigidity problems in toric topology are addressed in the survey paper by the second and third authors, “Classification Problems of Toric Manifolds via Topology” (in Toric Topology, Am. Math. Soc., Providence, RI, 2008, pp. 273–286). In the present paper, we survey the results on those problems including recent developments.     

Covering of nonlinear maps on a cone in neighborhoods of irregular points

Inverse function theorems for smooth nonlinear maps defined on convex cones in Banach spaces in a neighborhood of an irregular point are considered. The corresponding covering theorem is proved. The proofs are based on a Banach open mapping theorem for convex cones in Banach spaces, which is also proved in the paper. Sufficient conditions for tangency to the zero set of a nonlinear map without a priori regularity assumptions are obtained.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 483–497.Original Russian Text Copyright © 2005 by A. V. Arutyunov.This revised version was published online in April 2005 with a corrected issue number.     

Optimization of a Class of Nonlinear Dynamic Systems: New Efficient Method without Lagrange Multipliers

This paper deals with optimization of a class of nonlinear dynamic systems with n states and m control inputs commanded to move between two fixed states in a prescribed time. Using conventional procedures with Lagrange multipliers, it is well known that the optimal trajectory is the solution of a two-point boundary-value problem. In this paper, a new procedure for dynamic optimization is presented which relies on tools of feedback linearization to transform nonlinear dynamic systems into linear systems. In this new form, the states and controls can be written as higher derivatives of a subset of the states. Using this new form, it is possible to change constrained dynamic optimization problems into unconstrained problems. The necessary conditions for optimality are then solved efficiently using weighted residual methods.     

Multiple periodic solutions for a class of nonlinear nonautonomous wave equations

In this paper, by means of ourZ _p index theory developed recently we study the existence of multiple periodic solutions for asymptotically linear nonautonomous wave equations. All previous known results rely either on the oddness of nonlinear terms, or on autonomous systems, and the best result for the general case is the existence of two nontrivial periodic solutions (Amann, Zehnder, K. C. Chang, S. P. Wu, S. J. Li, Z. Q. Wang). In this paper, under the assumption that the nonlinear term isT/p periodic we discuss multiple periodic solutions of nonautonomous systems and generalize a series of previous results.This research was supported in part by the National Postdoctoral Science Fund.     

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.     

Parallel Newton Two-Stage Multisplitting Iterative Methods for Nonlinear Systems

Parallel Newton two-stage iterative methods to solve nonlinear systems are studied. These algorithms are based on both the multisplitting technique and the two-stage iterative methods. Convergence properties of these methods are studied when the Jacobian matrix is either monotone or an H-matrix. Furthermore, in order to illustrate the performance of the algorithms studied, computational results about these methods on a distributed memory multiprocessor are discussed.This revised version was published online in October 2005 with corrections to the Cover Date.     

Distribution of Nonlinear Congruential Pseudorandom Numbers Modulo Almost Squarefree Integers

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new bound on the s-dimensional discrepancy of nonlinear congruential pseudorandom numbers over the residue ring modulo M for an “almost squarefree” integer M. It is useful to recall that almost all integers are of this type. Moreover, if the generator is associated with a permutation polynomial over we obtain a stronger bound “on average” over all initial values. This bound is new even in the case when M = p is prime.     

Existence of periodic solutions of nonlinear systems with nonlinear boundary conditions

In this paper we consider the periodic solutions of nonlinear parabolic systems with nonlinear boundary conditions. By constructing the Poincare operator, we obtain the existence of -periodic weak solutions under some reasonable assumptions.     

The Management Information System Survey in Rolls-Royce — A Case History

The paper describes the history of a survey conducted within Rolls-Royce Ltd. during 1965 and 1966 of the company's internal communications. The objective was to provide a sound basis for future developments in systems and data processing.