An Identification Problem for First-Order Degenerate Differential Equations

We study a first-order identification problem in a Banach space. We discuss the nondegenerate and mainly the degenerate case. As a first step, suitable hypotheses on the involved closed linear operators are made in order to obtain unique solvability after reduction to a nondegenerate case; the general case is then handled with the help of new results on convolutions. Some applications to partial differential equations motivate this abstract approach.Communicated by I. GalliganiWork partially supported by MIUR (Ministero dell’ Istruzione, dell’ Università e dalla Ricerca), Project PRIN 2004011204 “Analisi Matematica nei Problemi Inversi,” and by the University of Bologna Funds for Selected Research Topics.     

The Carleman formula for the Helmholtz equation on the plane

We consider the Cauchy problem for the Helmholtz equation in an arbitrary bounded planar domain with Cauchy data only on part of the boundary of the domain. We derive a Carleman-type formula for a solution to this problem and give a conditional stability estimate.     

The Essential Component of the Solution Set for Vector Equilibrium Problems

In the paper, by using Ky Fan’s section theorem, we obtain an existence theorem for vector equilibrium problems. Motivated by the ideas of Kinoshita and McLennan, we introduce the concept of the essential component of the solution set for vector equilibrium problems, and we prove that there exists at least one essential component of the solution set for every vector equilibrium problem satisfying some conditions. Research was partially supported by the Natural Science Foundation of Guangdong Province, P. R. China.     

On the Existence of Optimal Solutions to an Optimal Control Problem

In this paper, some results concerning the existence of optimal solutions to an optimal control problem are derived. The problem involves a quasilinear hyperbolic differential equation with boundary condition and a nonlinear integral functional of action. The assumption of convexity, under which the main theorem is proved, is not connected directly with the convexity of the functional of action. In the proof, the implicit function theorem for multimappings is used.Communicated by L. D. Berkovitz     

Solving the p-Median Problem with a Semi-Lagrangian Relaxation

Lagrangian relaxation is commonly used in combinatorial optimization to generate lower bounds for a minimization problem. We study a modified Lagrangian relaxation which generates an optimal integer solution. We call it semi-Lagrangian relaxation and illustrate its practical value by solving large-scale instances of the p-median problem. This work was partially supported by the Fonds National Suisse de la Recherche Scientifique, grant 12-57093.99 and the Spanish government, MCYT subsidy dpi2002-03330.     

Novel evolutionary models and applications to sequence alignment problems

In this paper, we present a novel graph-theoretical approach for representing a wide variety of sequence analysis problems within a single model. The model allows incorporation of the operations “insertion”, “deletion”, and “substitution”, and various parameters such as relative distances and weights. Conceptually, we refer the problem as the minimum weight common mutated sequence (MWCMS) problem. The MWCMS model has many applications including multiple sequence alignment problem, the phylogenetic analysis, the DNA sequencing problem, and sequence comparison problem, which encompass a core set of very difficult problems in computational biology. Thus the model presented in this paper lays out a mathematical modeling framework that allows one to investigate theoretical and computational issues, and to forge new advances for these distinct, but related problems. Through the introduction of supernodes, and the multi-layer supergraph, we proved that MWCMS is -complete. Furthermore, it was shown that a conflict graph derived from the multi-layer supergraph has the property that a solution to the associated node-packing problem of the conflict graph corresponds to a solution of the MWCMS problem. In this case, we proved that when the number of input sequences is a constant, MWCMS is polynomial-time solvable. We also demonstrated that some well-known combinatorial problems can be viewed as special cases of the MWCMS problem. In particular, we presented theoretical results implied by the MWCMS theory for the minimum weight supersequence problem, the minimum weight superstring problem, and the longest common subsequence problem. Two integer programming formulations were presented and a simple yet elegant decomposition heuristic was introduced. The integer programming instances have proven to be computationally intensive. Consequently, research involving simultaneous column and row generation and parallel computing will be explored. The heuristic algorithm, introduced herein for multiple sequence alignment, overcomes the order-dependent drawbacks of many of the existing algorithms, and is capable of returning good sequence alignments within reasonable computational time. It is able to return the optimal alignment for multiple sequences of length less than 1500 base pairs within 30 minutes. Its algorithmic decomposition nature lends itself naturally for parallel distributed computing, and we continue to explore its flexibility and scalability in a massive parallel environment.     

Numerical simulation of critical dependences for symmetric two-layered Josephson junctions

Partial critical dependences of the form current-magnetic field in a two-layered symmetric Josephson junction are modeled. A numerical experiment shows that, for the zero interaction coefficient between the layers of the junction, jumps of the critical currents corresponding to different distributions of the magnetic fluxes in the layers may appear on the critical curves. This fact allows a mathematical interpretation of the results of some recent experimental results for two-layered junctions as a consequence of discontinuities of partial critical curves.     

Groups with small Dehn functions and bipartite chord diagrams

We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function n²log n. Some of these groups have undecidable conjugacy problem. Our groups are multiple HNN extensions of free groups. We show that n²log n is the smallest Dehn function of a multiple HNN extension of a free group with undecidable conjugacy problem. Both authors were supported in part by the NSF grants DMS 0245600 and DMS 0455881. In addition, the research of the first author was supported in part by the Russian Fund for Basic Research 05-01-00895, the research of the second author was supported in part by the NSF grant DMS 9978802 and the US-Israeli BSF grant 1999298. Received: February 2005; Revision: September 2005; Accepted: September 2005     

Overdetermined problems with possibly degenerate ellipticity, a geometric approach

Given an open bounded connected subset Ω of ℝⁿ, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation −div(A(|∇u|)∇u)=1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under fairly general assumptions on Ω and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle for a suitable P-function, combined with some geometric arguments involving the mean curvature of ∂Ω.