Well‐posedness and asymptotic behaviour of a non‐dissipative beam equation with a quotient space method

A beam equation model is studied, for which no suitable dissipative norm can be associated but only a semi‐norm, vanishing on some subspace. Considering a quotient by that subspace creates dissipativity which guarantees well posedness of this problem. Asymptotic behaviour is also obtained by using the quotient. The linear problem, involving high‐order multiplier is studied and then the non‐linear problem, thanks to the artificial problem method. Copyright © 2003 John Wiley & Sons, Ltd.     

Uniform LP duality for semidefinite and semi-infinite programming

Recently, a semidefinite and semi-infinite linear programming problem (SDSIP), its dual (DSDSIP), and uniform LP duality between (SDSIP) and (DSDSIP) were proposed and studied by Li et al. (Optimization 52:507–528, 2003). In this paper, we show that (SDSIP) is an ordinary linear semi-infinite program and, therefore, all the existing results regarding duality and uniform LP duality for linear semi-infinite programs can be applied to (SDSIP). By this approach, the main results of Li et al. (Optimization 52:507–528, 2003) can be obtained easily.     

Steady-State Solutions to the Equations of Motion of Second-Grade Fluids with General Navier Type Slip Boundary Conditions in Holder Spaces

We consider a boundary-value problem for the stationary flow of an incompressible second-grade fluid in a bounded domain. The boundary condition allows for no-slip, Navier type slip, and free slip on different parts of the boundary. We first establish the well-posedness of a linear auxiliary problem by means of a fixed-point argument in which the problem is decomposed into a Stokes-type problem and two transport equations. Then we use the method of successive approximations to prove the unique solvability of the nonlinear problem with a sufficiently small body force in Holder spaces. Bibliography: 17 titles.Dedicated to Professor Vsevolod A. Solonnikov on his seventieth birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 306, 2003, pp. 210–228.     

On the Convergence of the Generalized Linear Least Squares Algorithm

This paper considers the issue of parameter estimation for biomedical applications using nonuniformly sampled data. The generalized linear least squares (GLLS) algorithm, first introduced by Feng and Ho (1993), is used in the medical imaging community for removal of bias when the data defining the model are correlated. GLLS provides an efficient iterative linear algorithm for the solution of the non linear parameter estimation problem. This paper presents a theoretical discussion of GLLS and introduces use of both Gauss Newton and an alternating Gauss Newton for solution of the parameter estimation problem in nonlinear form. Numerical examples are presented to contrast the algorithms and emphasize aspects of the theoretical discussion. AMS subject classification (2000) 65F10.R. A. Renaut: This work was partially supported by the Arizona Center for Alzheimer’s Disease Research, by NIH grant EB 2553301 and for the second author by NSF CMG-02223.Received December 2003. Revised November 2004. Communicated by Lars Eldén.     

A note on linear complementarity problems and multiple objective programming

Kostreva and Wiecek [3] introduced a problem called LCP-related weighted problem in connection with a multiple objective programming problem, and suggested that a given linear complementarity problem (LCP) can be solved by solving the LCP-related weighted problem associated with it. In this note we provide several clarifications of the claims made in [3]. Finally, we feel that solving any LCP by the approach given in [3] may not be as useful as it is claimed.Mathematics Subject Classification (2000): 90C33Received: October 1998 / Revised version: August 2003     

Equivariant cosymmetry and front solutions of the Dubreil–Jacotin–Long equation. Part 1: Boussinesq limit

The problem of two-dimensional internal travelling waves in a perfect fluid with smooth density being close to linear stratification is considered. Approximate front solutions connecting uniform flow with a conjugate shear flow of the first mode are constructed. It is demonstrated that the number of the front branches essentially depends on the fine-scale stratification for linear density background. To cite this article: N. Makarenko, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Minimum convex piecewise linear cost tension problem on quasi-<Emphasis Type="Italic">k</Emphasis> series-parallel graphs

Some hypermedia synchronization issues request the resolution of the minimum convex piecewise linear cost tension problem (CPLCT problem) on directed graphs that are close to two-terminal series-parallel graphs (TTSP-graphs), the so-called quasi-k series-parallel graphs (k-QSP graphs). An aggregation algorithm has already been introduced for the CPLCT problem on TTSP-graphs. We propose here a reconstruction method, based on the aggregation and the well-known out-of-kilter techniques, to solve the problem on k-QSP graphs. One of the main steps being to decompose a graph into TTSP-subgraphs, methods based on the recognition of TTSP-graphs are thoroughly discussed.Received: October 2003, Revised: July 2004, MSC classification: 90C35, 05C85     

On a variational problem with lack of compactness related to the Strichartz inequality

We study a variational problem from nonlinear fiber optics which strongly lacks compactness, due to the absence of a priori bounds in spaces different from . A method is established how to restore this missing compactness by means of the dispersive properties inherent in the problem.Received: 30 October 2002, Accepted: 15 April 2003, Published online: 1 July 2003     

Capacities of quantum channels and how to find them

 We survey what is known about the information transmitting capacities of quantum channels, and give a proposal for how to calculate some of these capacities using linear programming. Received: March 14, 2003 / Accepted: April 14, 2003 Published online: June 5, 2003     

On the Carathéodory–Fejér Interpolation Problem for Generalized Schur Functions

The solutions of the Carathéodory–Fejér interpolation problem for generalized Schur functions can be parametrized via a linear fractional transformation over the class of classical Schur functions. The linear fractional transformation of some of these functions may have a pole (simple or multiple) in one or more of the interpolation points or not satisfy one or more interpolation conditions, hence not all Schur functions can serve as a parameter. The set of excluded parameters is characterized in terms of the related Pick matrix.Research was supported by the Summer Research Grant from the College of William and MarySubmitted: June 26, 2002 Revised: January 31, 2003     

Initial boundary value problems for linear viscoelastic flows generated by integro-differential equations

An estimate of the velocity field is obtained for the equation of motion of incompressible media. With the help of this estimate, the integro-differential equations that describe the motion of linear viscoelastic fluids in the two-dimensional case are studied. The existence is proved for a weak, global in time, solution of the Cauchy problem and of the initial boundary value problem with periodic boundary conditions. Bibliography: 18 titles.Dedicated to Olga Aleksandrovna Ladyzhenskaya on the occasion of her jubilee__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 90–98.     

Septic spline solutions of sixth-order boundary value problems

Septic spline is used for the numerical solution of the sixth-order linear, special case boundary value problem. End conditions for the definition of septic spline are derived, consistent with the sixth-order boundary value problem. The algorithm developed approximates the solution and their higher-order derivatives. The method has also been proved to be second-order convergent. Three examples are considered for the numerical illustrations of the method developed. The method developed in this paper is also compared with that developed in [M. El-Gamel, J.R. Cannon, J. Latour, A.I. Zayed, Sinc-Galerkin method for solving linear sixth order boundary-value problems, Mathematics of Computation 73, 247 (2003) 1325–1343], as well and is observed to be better.     

Error bounds for linear complementarity problems of DB-matrices

Doubly B-matrices (DB-matrices), which properly contain B-matrices, are introduced by Peña (2003) [2]. In this paper we present error bounds for the linear complementarity problem when the matrix involved is a DB-matrix and a new bound for linear complementarity problem of a B-matrix. The numerical examples show that the bounds are sharp.     

Upper and lower bounds for the fixed spectrum frequency assignment problem

A survey of the results described in the authors PhD thesis (Montemanni 2001) is presented. The thesis, which was supervised by Prof. Derek H. Smith and Dr. Stuart M. Allen, has been defended in January 2002 at the University of Glamorgan (U.K.). The thesis proposes new heuristic algorithms, based on well-known meta-heuristic paradigms, and new lower bounding techniques, based on linear programming, for the fixed spectrum frequency assignment problem.Received: May 2003, Revised: May 2003, AMS classification: 90C27, 90C59,05C90, 90C05Roberto Montemanni: Present address: Istituto Dalle Molle di Studi sullIntelligenza Artificiale (IDSIA), Galleria 2, 6928 Manno-Lugano, Switzerland (e-mail: roberto@idsia.ch)     

Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem

We consider the problem of designing a reflector that transforms a spherical wave front with a given intensity into an output front illuminating a prespecified region of the far-sphere with prescribed intensity. In earlier approaches, it was shown that in the geometric optics approximation this problem is reduced to solving a second order nonlinear elliptic partial differential equation of Monge–Ampere type. We show that this problem can be solved as a variational problem within the framework of Monge–Kantorovich mass transfer problem. We develop the techniques used by the authors in their work Optical Design of Two-Reflector Systems, the Monge–Kantorovich Mass Transfer Problem and Fermat's Principle [Preprint, 2003], where the design problem for a system with two reflectors was considered. An important consequence of this approach is that the design problem can be solved numerically by tools of linear programming. A known convergent numerical scheme for this problem was based on the construction of very special approximate solutions to the corresponding Monge–Ampere equation. Bibliography: 14 titles.     

Global Solution Approach for a Nonconvex MINLP Problem in Product Portfolio Optimization

The rigorous and efficient determination of the global solution of a nonconvex MINLP problem arising from product portfolio optimization introduced by Kallrath (2003) is addressed. The objective of the optimization problem is to determine the optimal number and capacity of reactors satisfying the demand and leading to a minimal total cost. Based on the model developed by Kallrath (2003), an improved formulation is proposed, which consists of a concave objective function and linear constraints with binary and continuous variables. A variety of techniques are developed to tighten the model and accelerate the convergence to the optimal solution. A customized branch and bound approach that exploits the special mathematical structure is proposed to solve the model to global optimality. Computational results for two case studies are presented. In both case studies, the global solutions are obtained and proved optimal very efficiently in contrast to available commercial MINLP solvers.     

Initial boundary value problem for a class of non‐linear strongly damped wave equations

The paper studies the existence, asymptotic behaviour and stability of global solutions to the initial boundary value problem for a class of strongly damped non‐linear wave equations. By a H00.5ptk‐Galerkin approximation scheme, it proves that the above‐mentioned problem admits a unique classical solution depending continuously on initial data and decaying to zero as t→+∞as long as the non‐linear terms are sufficiently smooth; they, as well as their derivatives or partial derivatives, are of polynomial growth order and the initial energy is properly small. Copyright © 2003 John Wiley & Sons, Ltd.     

Neural networks,linear functions and neglected non-linearity

The multiplicity of approximation theorems for Neural Networks do not relate to approximation of linear functions per se. The problem for the network is to construct a linear function by superpositions of non-linear activation functions such as the sigmoid function. This issue is important for applications of NNs in statistical tests for neglected nonlinearity, where it is common practice to include a linear function through skip-layer connections. Our theoretical analysis and evidence point in a similar direction, suggesting that the network can in fact provide linear approximations without additional assistance. Our paper suggests that skip-layer connections are unnecessary, and if employed could lead to misleading results.Received: August 2002, Revised: March 2003, AMS Classification: 82c32The authors are grateful to Prof. Mick Silver and to GFK Marketing for help with the provision of data.     

Cyclic group blocking polyhedra

Gomory (Linear Algebra Appl 2:451–558, 1969) gave a subadditive characterization of the facets of the group polyhedron. Although there are exponentially many facets (see Gomory and Johnson in Math Program 3:359–389, 1972, Example 4.6) and exponentially many vertices for the group polyhedron for the master cyclic group problem, Gomory’s characterization of the non-trivial facets has polynomially many subadditive inequalities, in fact of order |G|² for a finite Abelian group G. We reduce this subadditive inequality system to its minimal representation by a triple system of the same order and show the dimensionality of the polytope of non-trivial facets. The system of all triples corresponds to all solution vectors of length three into which every solution vector can be decomposed. Our minimal representation leads to a characterization of the vertices of length three. Gomory et al. (Math Program 96:321–339, 2003) introduced a shooting experiment involving solving the shooting linear program repeatedly to find important facets. We develop a topological network flow model of the dual problem of the shooting linear program in a reverse procedure from the decomposition of solution vectors into triples. Hunsaker (2003) gave a knapsack shooting experiment, which we use to find a simple pattern for the most hit knapsack facets.     

Polynomial time approximation schemes for dense instances of minimum constraint satisfaction

It is known that large fragments of the class of dense Minimum Constraint Satisfaction (MIN‐CSP) problems do not have polynomial time approximation schemes (PTASs) contrary to their Maximum Constraint Satisfaction analogs. In this paper we prove, somewhat surprisingly, that the minimum satisfaction of dense instances of kSAT ‐formulas, and linear equations mod 2, Ek‐LIN2, do have PTASs for any k. The MIN‐Ek‐LIN2 problems are equivalent to the k‐ary versions of the Nearest Codeword problem, the problem which is known to be exceedingly hard to approximate on general instances. The method of solution of the above problems depends on the development of a new density sampling technique for k‐uniform hypergraphs which could be of independent interest. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 73–91, 2003