Convergence towards attractors for a degenerate Ginzburg-Landau equation

We study a real Ginzburg-Landau equation, in a bounded domain of with a variable, generally non-smooth diffusion coefficient having a finite number of zeroes. By using the compactness of the embeddings of the weighted Sobolev spaces involved in the functional formulation of the problem, and the associated energy equation, we show the existence of a global attractor. The extension of the main result in the case of an unbounded domain is also discussed, where in addition, the diffusion coefficient has to be unbounded. Some remarks for the case of a complex Ginzburg-Landau equation are given.Received: May 6, 2002; revised: October 3, 2002     

Convex Embeddings and Bisections of 3-Connected Graphs1

Given two disjoint subsets T ₁ and T ₂ of nodes in an undirected 3-connected graph G = (V, E) with node set V and arc set E, where and are even numbers, we show that V can be partitioned into two sets V ₁ and V ₂ such that the graphs induced by V ₁ and V ₂ are both connected and holds for each j = 1,2. Such a partition can be found in time. Our proof relies on geometric arguments. We define a new type of convex embedding of k-connected graphs into real space R ^k-1 and prove that for k = 3 such an embedding always exists. ¹ A preliminary version of this paper with title Bisecting Two Subsets in 3-Connected Graphs appeared in the Proceedings of the 10th Annual International Symposium on Algorithms and Computation, ISAAC 99, (A. Aggarwal, C. P. Rangan, eds.), Springer LNCS 1741, 425–434, 1999.     

Explicit relation between the solutions of the heat and the Hermite heat equation

There are lots of results on the solutions of the heat equation but much less on those of the Hermite heat equation due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005). Supported partially by 973 project (2004CB318000)     

On Shadows Of Intersecting Families

The shadow minimization problem for t-intersecting systems of finite sets is considered. Let be a family of k-subsets of . The -shadow of is the set of all (k-)-subsets contained in the members of . Let be a t-intersecting family (any two members have at least t elements in common) with . Given k,t,m the problem is to minimize (over all choices of ). In this paper we solve this problem when m is big enough.     

On the Cauchy problem of a weakly dissipative μ-Hunter–Saxton equation

In this paper, we study the Cauchy problem of a weakly dissipative μ-Hunter–Saxton equation. We first establish the local well-posedness for the weakly dissipative μ-Hunter–Saxton equation by Kato's semigroup theory. Then, we derive the precise blow-up scenario for strong solutions to the equation. Moreover, we present some blow-up results for strong solutions to the equation. Finally, we give two global existence results to the equation.     

Properties of the solutions of the Ginzburg-Landau equation on the bifurcation branch

Generalizing previous results of M. Comte and P. Mironescu, it is shown that for degree d large enough (such that ), there is a bifurcation branch in the set of the solutions of the Ginzburg-Landau equation, emanating from the branch of radial solutions at the critical value _d of the parameter. Moreover, the solutions on the bifurcation branch admit exactly d zeroes, and the energy on the bifurcation branch is strictly smaller than the energy on the radial branch.     

Time periodic solutions to a nonlinear wave equation with <Emphasis Type="Italic">x</Emphasis>-dependent coefficients

In this paper, we study the problem of time periodic solutions to the nonlinear wave equation with x-dependent coefficients on under the boundary conditions a ₁ y(0, t)+b ₁ y _x (0, t) = 0, ( for i = 1, 2) and the periodic conditions y(x, t + T) = y(x, t), y _t (x, t + T) = y _t (x, t). Such a model arises from the forced vibrations of a bounded nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For , we establish the existence of time periodic solutions in the weak sense by utilizing some important properties of the wave operator with x-dependent coefficients. This work was supported by the 985 Project of Jilin University, the Specialized Research Fund for the Doctoral Program of Higher Education, and the Science Research Foundation for Excellent Young Teachers of College of Mathematics at Jilin University.     

Explicit Bounded-Degree Unique-Neighbor Concentrators

Here we solve an open problem considered by various researchers by presenting the first explicit constructions of an infinite family of bounded-degree ‘unique-neighbor’ concentrators Γ; i.e., there are strictly positive constants α and ε, such that all Γ = (X,Y,E(Γ)) ∈ satisfy the following properties. The output-set Y has cardinality times that of the input-set X, and for each subset S of X with no more than α|X| vertices, there are at least ε|S| vertices in Y that are adjacent in Γ to exactly one vertex in S. Also, the construction of is simple to specify, and each has fewer than edges. We then modify to obtain explicit unique-neighbor concentrators of maximum degree 3. * Supported by NSF grant CCR98210-58 and ARO grant DAAH04-96-1-0013.     

Well-posedness and sharp uniform decay rates at the L 2(Ω)-Level of the Schrödinger equation with nonlinear boundary dissipation

We prove that the Schr?dinger equation defined on a bounded open domain of and subject to a certain attractive, nonlinear, dissipative boundary feedback is (semigroup) well-posed on L₂(Ω) for any n = 1, 2, 3, ..., and, moreover, stable on L₂(Ω) for n = 2, 3, with sharp (optimal) uniform rates of decay. Uniformity is with respect to all initial conditions contained in a given L₂(Ω)-ball. This result generalizes the corresponding linear case which was proved recently in [L-T-Z.2]. Both results critically rely—at the outset—on a far general result of interest in its own right: an energy estimate at the L₂(Ω)-level for a fully general Schr?dinger equation with gradient and potential terms. The latter requires a heavy use of pseudo-differential/micro-local machinery [L-T-Z.2, Section 10], to shift down the more natural H¹(Ω)-level energy estimate to the L₂(Ω)-level. In the present nonlinear boundary dissipation case, the resulting energy estimate is then shown to fit into the general uniform stabilization strategy, first proposed in [La-Ta.1] in the case of wave equations with nonlinear (interior and) boundary dissipation.     

Gauge functions,Eikonal equations and Bôcher’s theorem on stratified Lie groups

Let \({\mathcal{L} = \sum_{i=1}^m X_i^2}\) be a real sub-Laplacian on a Carnot group \({\mathbb{G}}\) and denote by \({\nabla_\mathcal{L} = (X_1,\ldots,X_m)}\) the intrinsic gradient related to \({\mathcal{L}}\). Our aim in this present paper is to analyze some features of the \({\mathcal{L}}\)-gauge functions on \({\mathbb{G}}\), i.e., the homogeneous functions d such that \({\mathcal{L}(d^\gamma) = 0}\) in \({\mathbb{G} \setminus \{0\}}\) , for some \({\gamma \in \mathbb{R} \setminus \{0\}}\). We consider the relation of \({\mathcal{L}}\)-gauge functions with: the \({\mathcal{L}}\)-Eikonal equation \({|\nabla_\mathcal{L} u| = 1}\) in \({\mathbb{G}}\); the Mean Value Formulas for the \({\mathcal{L}}\)-harmonic functions; the fundamental solution for \({\mathcal{L}}\); the Bôcher-type theorems for nonnegative \({\mathcal{L}}\)-harmonic functions in “punctured” open sets \({\dot \Omega:= \Omega \setminus \{x_0\}}\).     

Supersaturation For Ramsey-Turán Problems

For an l-graph , the Turán number is the maximum number of edges in an n-vertex l-graph containing no copy of . The limit is known to exist [8]. The Ramsey–Turán density is defined similarly to except that we restrict to only those with independence number o(n). A result of Erdős and Sós [3] states that as long as for every edge E of there is another edge E′of for which |E∩E′|≥2. Therefore a natural question is whether there exists for which . Another variant proposed in [3] requires the stronger condition that every set of vertices of of size at least εn (0<ε<1) has density bounded below by some threshold. By definition, for every . However, even is not known for very many l-graphs when l>2. We prove the existence of a phenomenon similar to supersaturation for Turán problems for hypergraphs. As a consequence, we construct, for each l≥3, infinitely many l-graphs for which . We also prove that the 3-graph with triples 12a, 12b, 12c, 13a, 13b, 13c, 23a, 23b, 23c, abc, satisfies . The existence of a hypergraph satisfying was conjectured by Erdős and Sós [3], proved by Frankl and R?dl [6], and later by Sidorenko [14]. Our short proof is based on different ideas and is simpler than these earlier proofs. * Research supported in part by the National Science Foundation under grants DMS-9970325 and DMS-0400812, and an Alfred P. Sloan Research Fellowship. † Research supported in part by the National Science Foundation under grants DMS-0071261 and DMS-0300529.     

Anti-selfdual Lagrangians II: unbounded non self-adjoint operators and evolution equations

New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary boundary value problems of the form as well ass dissipative initial value evolutions of the form where is a convex potential on an infinite dimensional space, A is a linear operator and is any scalar. The framework developed in the above mentioned paper reformulates these problems as and respectively, where is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by transport operators with or without a diffusion term. Nassif Ghoussoub research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The author gratefully acknowledges the hospitality and support of the Centre de Recherches Mathématiques in Montréal where this work was initiated. Leo Tzou’s research was partially supported by a doctoral postgraduate scholarship from the Natural Science and Engineering Research Council of Canada.     

Matching Polynomials And Duality

Let G be a simple graph on n vertices. An r-matching in G is a set of r independent edges. The number of r-matchings in G will be denoted by p(G, r). We set p(G, 0) = 1 and define the matching polynomial of G by and the signless matching polynomial of G by .It is classical that the matching polynomials of a graph G determine the matching polynomials of its complement . We make this statement more explicit by proving new duality theorems by the generating function method for set functions. In particular, we show that the matching functions and are, up to a sign, real Fourier transforms of each other.Moreover, we generalize Foatas combinatorial proof of the Mehler formula for Hermite polynomials to matching polynomials. This provides a new short proof of the classical fact that all zeros of µ(G, x) are real. The same statement is also proved for a common generalization of the matching polynomial and the rook polynomial.     

Uniqueness in the characteristic Cauchy problem of the Klein–Gordon equation and tame restrictions of generalized functions

We show that every tempered distribution, which is a solution of the (homogenous) Klein–Gordon equation, admits a “tame” restriction to the characteristic (hyper)surface {x ⁰ + x ⁿ = 0} in (1 + n)-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space which we have introduced in (Ullrich in J. Math. Phys. 45, 2004). Moreover, we show that every element of appears as the “tame” restriction of a solution of the (homogeneous) Klein–Gordon equation.     

On A Hypergraph Turán Problem Of Frankl

Let be the 2k-uniform hypergraph obtained by letting P₁, . . .,P_r be pairwise disjoint sets of size k and taking as edges all sets P_i∪P_j with i ≠ j. This can be thought of as the ‘k-expansion’ of the complete graph K_r: each vertex has been replaced with a set of size k. An example of a hypergraph with vertex set V that does not contain can be obtained by partitioning V = V1 ∪V₂ and taking as edges all sets of size 2k that intersect each of V₁ and V₂ in an odd number of elements. Let denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. For n sufficiently large we prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no has at most as many edges as . Sidorenko has given an upper bound of for the Tur′an density of for any r, and a construction establishing a matching lower bound when r is of the form 2^p+1. In this paper we also show that when r=2^p+1, any -free hypergraph of density looks approximately like Sidorenko’s construction. On the other hand, when r is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turán density of to , where c(r) is a constant depending only on r. The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal–Katona theorem and properties of Krawtchouck polynomials. * Research supported in part by NSF grants DMS-0355497, DMS-0106589, and by an Alfred P. Sloan fellowship.     

On the non-existence of global solutions to the Cauchy problem for semilinear parabolic equations and inequalities

We generalize and improve recent non-existence results for global solutions to the Cauchy problem for the inequality as well as for the equation u_t = Δu + |u|^q in the half-space . Received: 16 September 2005     

Localization Operators and Exponential Weights for Modulation Spaces

We study localization operators within the framework of ultradistributions. More precisely, given a symbol a and two windows φ₁, φ₂, we investigate the multilinear mapping from to the localization operator Results are formulated in terms of modulation spaces with weights which may have exponential growth. We give sufficient and necessary conditions for a to be bounded or to belong to a Schatten class. As an application, we study symbols defined by ultra-distributions with compact support, that give trace class localization operators.     

Complete Minors In <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">s,s</Emphasis></Subscript>-Free Graphs

We prove that for a fixed integer s2 every K _s,s -free graph of average degree at least r contains a K _p minor where . A well-known conjecture on the existence of dense K _s,s -free graphs would imply that the value of the exponent is best possible. Our result implies Hadwigers conjecture for K _s,s -free graphs whose chromatic number is sufficiently large compared with s.