Higher degree layer potentials for non-smooth domains with arbitrary topology

We consider layer potentials associated with the Hodge-Laplacian acting on differential forms of arbitrary degree defined on Lipschitz subdomains of a Riemannian manifold. The main emphasis is on the interplay between the mapping properties of such layer potentials and the topology of the underlying domain.Partially supported by a UMC Research Board GrantPartially supported by NSF grant DMS-9870018     

Approximation in the mean by bounded analytic functions

Let 1<p< and . LetC _q denote the Bessel capacity in the plane. Let be the set of homomorphisms ofH (G) such that (z)= and letNP denote the set of points in G for which is not a peak set forH (G). In this note, we show that ifC _q (NP)=0, thenH (G) is dense inL _a ^p (G), the Bergman space overG.Partially supported by NSF DMS-9401234     

Constraint-Based Job Shop Scheduling with IILOG SCHEDULER

We introduce constraint-based scheduling and discuss its main principles. An approximation algorithm based on tree search is developed for the job shop scheduling problem using ILOG SCHEDULER. A new way of calculating lower bounds on the makespan of the job shop scheduling problem is presented and we show how such results can be used within a constraint-based approach. An empirical performance analysis shows that the algorithm we developed performs well. Finally, taking the job shop scheduling problem as a start point, we discuss how constraint-based scheduling can be used to solve more general scheduling problems.     

On the unique finite (SI) decomposition of the Jordan operator ⊕ i=1 n S(θ) for certain inner function θ

In this paper, we have proven that for the Jordan blockS() withS() (SI), _i=1 ⁿ S() =S() ⁽ⁿ⁾ (n 1) has unique finite (SI) decomposition up to a similarity. As result, we obtain that ifV is a Volterra operator onH=L ²([0, 1]), thenV ⁽ⁿ⁾ has unique finite (SI) decomposition.This project was supported by National Natural Science Foundation of China.     

Accurate scalar products and optimization

This paper describes a procedure for computing accurate scalar products of real-valued vectors. Anada implementation of this procedure is used to demonstrate the extent to which the technique can improve the convergence of the conjugate gradient algorithm. We also give a brief discussion of the costs and limitations of accurate arithmetic operations.     

The chebop system for automatic solution of differential equations

In Matlab, it would be good to be able to solve a linear differential equation by typing u = L\f, where f, u, and L are representations of the right-hand side, the solution, and the differential operator with boundary conditions. Similarly it would be good to be able to exponentiate an operator with expm(L) or determine eigenvalues and eigenfunctions with eigs(L). A system is described in which such calculations are indeed possible, at least in one space dimension, based on the previously developed chebfun system in object-oriented Matlab. The algorithms involved amount to spectral collocation methods on Chebyshev grids of automatically determined resolution. AMS subject classification (2000)  65L10, 65M70, 65N35     

Modelling the dynamics of stochastic local search on <Emphasis Type="Italic">k</Emphasis>-<Emphasis Type="SmallCaps">sat</Emphasis>

A new analytical tool is presented to provide a better understanding of the search space of k-sat. This tool, termed the local value distribution , describes the probability of finding assignments of any value q′ in the neighbourhood of assignments of value q. The local value distribution is then used to define a Markov model to model the dynamics of a corresponding stochastic local search algorithm for k-sat. The model is evaluated by comparing the predicted algorithm dynamics to experimental results. In most cases the fit of the model to the experimental results is very good, but limitations are also recognised.     

Collapsing monoids consisting of permutations and constants

In this paper we determine all collapsing transformation monoids that contain at least one unary constant operation and whose nonconstant operations are permutations. Furthermore, we find an infinite family of transformation monoids that consist of at least three unary constant operations and some permutations for which the corresponding monoidal intervals are 2-element chains. This research is supported by Hungarian National Foundation for Scientific Research grant nos. T 37877 and K 60148.     

Boundary structure of hyperbolic 3-manifolds admitting toroidal fillings at large distance

We show that if a hyperbolic 3-manifold M has two toroidal Dehn fillings with distance at least 3, then ∂M consists of at most three tori. As a result, we can obtain an optimal estimate for the number of exceptional slopes on hyperbolic 3-manifolds with boundary a union of at least 4 tori. S. Lee was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-314-C00024). M. Teragaito was supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 19540089.     

Invariant subspaces of submarkovian semigroups

We characterize the invariance under a submarkovian semigroup of a measurable subset by capacity conditions on its boundary.      

A Cayley theorem for distributive lattices

A Cayley-like representation theorem for distributive lattices is proved. Support of the research of the first author by the Czech Government Research Project MSM 6198959214 is gratefully acknowledged.     

Correlated equilibrium and concave games

This paper shows that if a game satisfies the sufficient condition for the existence and uniqueness of a pure-strategy Nash equilibrium provided by Rosen (Econometrica 33:520, 1965), then the game has a unique correlated equilibrium, which places probability one on the unique pure-strategy Nash equilibrium. In addition, it shows that a weaker condition suffices for the uniqueness of a correlated equilibrium. The condition generalizes the sufficient condition for the uniqueness of a correlated equilibrium provided by Neyman (Int J Game Theory 26:223, 1997) for a potential game with a strictly concave potential function. I thank the editor, an associate editor, and an anonymous referee for detailed comments and suggestions, which have substantially improved this paper. Special thanks are due to the referee for pointing out Lemmas 4 and 5. I acknowledge financial support by The Japan Economic Research Foundation and by MEXT, Grant-in-Aid for Scientific Research. All remaining errors are mine.     

On the structure of pseudo-holomorphic discs with totally real boundary conditions

In this paper, we study the structure ofJ-holomorphic discs in relation to the Fredholm theory of pseudo-holomorphic discs with totally real boundary conditions in almost complex manifolds (M, J). We prove that anyJ-holomorphic disc with totally real boundary condition that is injective in the interior except at a discrete set of points, which we call a “normalized disc,” must either have some boundary point that is regular and has multiplicity one, or satisfy that its image forms a smooth immersed compact surface (without boundary) with a finite number of self-intersections and a finite number of branch points. In the course of proving this theorem, we also prove several theorems on the local structure of boundary points ofJ-holomorphic discs, and as an application we give a complete treatment of the transverslity result for Floer’s pseudo-holomorphic trajectories for Lagrangian intersections in symplectic geometry. This paper is supported in part by NSF Grant DMS 9215011.     

Simple random order methods to share costs

A simple random order method (SROM) is an extension of Webers random order values (ROVs), which allows the convex weights on orderings of agents to depend on the set of agents with strictly positive demands. Thus, a SROM permits different coalitions of agents adopting exogenously different ROVs to take into account the differences in, for example, bargaining abilities, rights or status of the agents in a cost sharing problem. Within the family of additive methods satisfying the dummy axiom, we characterize SROMs by Measurement Invariance in the discrete cost sharing model where demands are indivisible, and Ordinality in the continuous model where demands are divisible, respectively.I am indebted to Yves Sprumont for initiating and providing the starting point of this paper, and for his great help in this work. I thank Hervé Moulin, Ahmet Alkan, Anirban Kar, an associate editor, and a referee whose comments greatly improved the paper. I gratefully acknowledge the support from Sabanci University Research Fund.     

The heat flow and harmonic maps between complete manifolds

We consider the existence, uniqueness and convergence for the long time solution to the harmonic map heat equation between two complete noncompact Riemannian manifolds, where the target manifold is assumed to have nonpositive curvature. As an application, we solve the Dirichlet problem at infinity for proper harmonic maps between two hyperbolic manifolds for a class of boundary maps. The boundary map under consideration has finite many points at which either it is not differentiable or has vanishing energy density.     

Jackson’s integral of the Hurwitz zeta function

We give a Jacksonq-integral analogue of Euler’s logarithmic sine integral established in 1769 from several points of view, specifically from the one relating to the Hurwitz zeta function. Partially supported by Grant-in-Aid for Exploratory Research No. 15654003. Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340003. Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340012.     

Relaxation in Reactive Flows

We consider convective systems in a bounded domain, in which viscous fluids described by the Stokes system are coupled using the Boussinesq approximation to a reaction-advection-diffusion equation for the temperature. We show that the resulting flows possess relaxation-enhancing properties in the sense of [CoKRZ]. In particular, we show that solutions of the nonlinear problems become small when gravity is sufficiently strong due to the improved interaction with the cold boundary. As an application, we deduce that the explosion threshold for power-like nonlinearities tends to infinity in the large Rayleigh number limit. We also discuss the behavior of the principal eigenvalues of the corresponding advection-diffusion problem and the quenching phenomenon for reaction-diffusion equations. Received: March 2007, Revision: May 2007, Accepted: May 2007     

Soap films and covering spaces

A new mathematical model of soap films is proposed, called the “covering space model.” The two sides of a film are modeled as currents on different sheets of a covering space branching along the film boundary. Hence a film may be seen as the minimal cut separating one sheet of the covering space from the others. The film is thus the oriented boundary of one sheet, which represents the exterior of the film. As oriented boundaries, films may be calibrated with differential forms on the covering space, a version of the min-cut, max-flow duality of network theory. This model applies to unoriented films, films with singularities, films touching only part of a knotted curve, films that deformation retract to their boundaries, and other examples that have proved troublesome for previous soap film models. Communicated by Frederick Almgren     

On the geometry of Nash equilibria and correlated equilibria

It is well known that the set of correlated equilibrium distributions of an n-player noncooperative game is a convex polytope that includes all the Nash equilibrium distributions. We demonstrate an elementary yet surprising result: the Nash equilibria all lie on the boundary of the polytope.We are grateful to Francoise Forges, Dan Arce, the editors, and several anonymous referees for helpful comments. This research was supported by the National Science Foundation under grant 98–09225 and by the Fuqua School of Business.The use of correlated mixed strategies in 2-player games was discussed by Raiffa (1951), who noted: it is a useful concept since it serves to convexify certain regions [of expected payoffs] in the Euclidean plane. (p. 8)Received: April 2002 / Revised: November 2003     

Amalgamation,interpolation and epimorphisms in algebraic logic

In his landmark paper on amalgamation published in Algebra Universalis in 1971, Don Pigozzi posed some open questions in connection with amalgamation of subclasses of cylindric algebras. Some of these questions were originally raised by Comer, Daigneault, Johnson, McKenzie and others. In this paper we give answers to all these as well as a number of other related questions. Most of the solutions were found by the authors of this paper. However, a few were contributed by others who will of course be given due credit at the appropriate points. This paper is dedicated to Don Pigozzi on his retirement. Presented by R. W. Quackenbush. The first author’s research was supported by the Hungarian National Foundation for Scientific Research grants no T30314 and T23234. The second author’s research was supported by the Hungarian National Foundation for Scientific Reserach OKTA grant no T30314. All the new results in this article were announced by Judit Madarász in the Workshop on Abstract Algebraic Logic, held in Centre de Recerca Matematica Bellaterra, Spain July 1–5, 1997. Received December 15, 2002; accepted in final form January 3, 2006.