Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle

We investigate heteroclinic orbits between equilibria and rotating waves for scalar semilinear parabolic reaction-advection-diffusion equations with periodic boundary conditions. Using zero number properties of the solutions and the phase shift equivariance of the equation, we establish a necessary and sufficient condition for the existence of a heteroclinic connection between any pair of hyperbolic equilibria or rotating waves.     

Book Review: The Emerging Consensus in Social Systems Theory. By Kenneth C. Bausch. Kluwer Academic/Plenum Publishers, 2001. ISBN 0-306-46539-6, $69.50

Nonlinear Dynamics, Psychology, and Life Sciences -     

On Constraint Qualifications in Nonsmooth Optimization

We introduce extensions of the Mangasarian-Fromovitz and Abadie constraint qualifications to nonsmooth optimization problems with feasibility given by means of lower-level sets. We do not assume directional differentiability, but only upper semicontinuity of the defining functions. By deriving and reviewing primal first-order optimality conditions for nonsmooth problems, we motivate the formulations of the constraint qualifications. Then, we study their interrelation, and we show how they are related to the Slater condition for nonsmooth convex problems, to nonsmooth reverse-convex problems, to the stability of parametric feasible set mappings, and to alternative theorems for the derivation of dual first-order optimality conditions.In the literature on general semi-infinite programming problems, a number of formally different extensions of the Mangasarian-Fromovitz constraint qualification have been introduced recently under different structural assumptions. We show that all these extensions are unified by the constraint qualification presented here.     

Current correlators to all orders in the quark masses

The contributions to the coefficient functions of the quark and the mixed quark-gluon condensate to mesonic correlators are calculated for the first time to all orders in the quark masses, and to lowest order in the strong coupling constant. Existing results on the coefficient functions of the unit operator and the gluon condensate are reviewed. The proper factorization of short- and long-distance contributions in the operator product expansion is discussed in detail. It is found that to accomplish this task rigorously the operator product expansion has to be performed in terms ofnon-normal-ordered condensates. The resulting coefficient functions are improved with the help of the renormalization group. The scale invariant combination of dimension 5 operators, including mixing with the mass operator, which is needed for the renormalization group improvement, is calculated in the leading order.Supported by the German Bundesministerium für Forschung und Technologie, under the contract 06 TM 761     

Moderate Deviations for Longest Increasing Subsequences: The Lower Tail

We derive a moderate deviation principle for the lower tail probabilities of the length of a longest increasing subsequence in a random permutation. It refers to the regime between the lower tail large deviation regime and the central limit regime. The present article together with the upper tail moderate deviation principle in Ref. 12 yields a complete picture for the whole moderate deviation regime. Other than in Ref. 12, we can directly apply estimates by Baik, Deift, and Johansson, who obtained a (non-standard) Central Limit Theorem for the same quantity.     

Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: the Hull–White model

In the present Note, we study the asymptotic behavior of the distribution density of the stock price process in the Hull–White model. The leading terms in the asymptotic expansions at zero and infinity are found for such a density and the corresponding error estimates are given. Similar problems are solved for time averages of the volatility process, which are also of interest in the study of Asian options. To cite this article: A. Gulisashvili, E.M. Stein, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Upper Bounds to the Number of Vertices in a k-Critically n-Connected Graph

 Let k≤n be positive integers. A finite, simple, undirected graph is called k-critically n-connected, or, briefly, an (n,k)-graph, if it is noncomplete and n-connected and the removal of any set X of at most k vertices results in a graph which is not (n−|X|+1)-connected. We present some new results on the number of vertices of an (n,k)-graph, depending on new estimations of the transversal number of a uniform hypergraph with a large independent edge set. Received: April 14, 2000 Final version received: May 8, 2001     

An Enumerative Geometry for Magic and Magilatin Labellings

A magic labelling of a set system is a labelling of its points by distinct positive integers so that every set of the system has the same sum, the magic sum. Examples are magic squares (the sets are the rows, columns, and diagonals) and semimagic squares (the same, but without the diagonals). A magilatin labelling is like a magic labelling but the values need be distinct only within each set. We show that the number of n × n magic or magilatin labellings is a quasipolynomial function of the magic sum, and also of an upper bound on the entries in the square. Our results differ from previous ones because we require that the entries in the square all be different from each other, and because we derive our results not by ad hoc reasoning but from a general theory of counting lattice points in rational inside-out polytopes. We also generalize from set systems to rational linear forms. Dedicated to the memory of Claudia Zaslavsky, 1917–2006 Received August 10, 2005