Harnack inequality for some classes of Markov processes

In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps. Mathematics Subject Classification (2000): Primary 60J45, 60J75, Secondary 60J25.This work was completed while the authors were in the Research in Pairs program at the Mathematisches Forschungsinstitut Oberwolfach. We thank the Institute for the hospitality.The research of this author is supported in part by NSF Grant DMS-9803240.The research of this author is supported in part by MZT grant 0037107 of the Republic of Croatia.     

Decay rates at low and high frequencies for a plate equation with feedback concentrated in interior curves

In this paper we study the stabilization of plate vibrations by means of piezoelectric actuators. In this situation the geometric control condition of Bardos, Lebeau and Rauch [6] is not satisfied. We prove that we have exponential stability for the low frequencies but not for the high frequencies. We give an explicit decay rate for regular initial data at high frequencies while clarifying the behavior of the constant which intervenes in this estimation there function of the frequency of cut n. The method used is based on some trace regularity which reduces stability to some observability inequalities for the corresponding undamped problem. Moreover, we show numerically at low frequencies, that the optimal location of the actuator is the center of the domain Ω.     

On Hyers-Ulam stability for a class of functional equations

Summary In this paper we prove some stability theorems for functional equations of the formg[F(x, y)]=H[g(x), g(y), x, y]. As special cases we obtain well known results for Cauchy and Jensen equations and for functional equations in a single variable. Work supported by M.U.R.S.T. Research funds (60%).     

A pattern theorem for lattice clusters

We consider general classes of lattice clusters, including various kinds of animals and trees on different lattices. We prove that if a given local configuration (pattern) of sites and bonds can occur in large clusters, then for some constantc>0, it occurs at leastcn times in most clusters of sizen. An analogous theorem for self-avoiding walks was proven in 1963 by Kesten [9]. We use the pattern theorem to prove the convergence of lim _n a _n+1 /a _n , wherea _n is the number of clusters of sizen, up to translation. The results also apply to weighted sums, and in particular, we can takea _n to be the probability that the percolation cluster containing the origin consists of exactlyn sites. Another consequence is strict inequality of connective constants for sublattices and for certain subclasses of clusters.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.The author was visiting the Fields Institute for Research in Mathematical Sciences, Toronto, Canada, while writing this paper.     

Toeplitz Operators on Analytic Besov Spaces

We characterize complex measures μ on the unit disk for which the Toeplitz operator T _μ is bounded or compact on the analytic Besov spaces B _p with 1 ≤ p < ∞. Research supported in part by NSF grant, DMS 0200587 (first author); and by a NSERC grant (third author).     

Some remarks on the Schr?dinger equation with a potential in LrtLsx

We study the dispersive properties of the linear Schr?dinger equation with a time-dependent potential V(t,x). We show that an appropriate integrability condition in space and time on V, i.e. the boundedness of a suitable L^r_tL^s_x norm, is sufficient to prove the full set of Strichartz estimates. We also construct several counterexamples which show that our assumptions are optimal, both for local and for global Strichartz estimates, in the class of large unsigned potentials V ∈L^r_tL^s_x. Support. The authors are partially supported by the Research Training Network (RTN) HYKE and by grant HPRN-CT-2002-00282 from the European Union. The third author is supported also by INDAM     

Some excluded-minor theorems for a class of polymatroids

Problems involving representability are among the most frequently studied of all the problems in matroid theory. This paper considers the corresponding class of problems for polymatroids. A polymatroidh on the setS is representable over a free matroid or is Boolean if there is a map fromS into the set of subsets of a setV which preserves rank, that is for all subsetsA ofS, . The class of Boolean polymatroids is minor-closed and in this paper we investigate the excluded minors of this class. In particular, we determine all such Boolean excluded minors that are 2-polymatroids.This research was partially supported by a grant from the Louisiana Education Quality Support Fund Through the Board of RegentsThis research was supported by a grant from the Commonwealth of Australia through the Australian Research Council     

On fixing elements in matroid minors

LetF be a collection of 3-connected matroids which is (3, 1)-rounded, that is, whenever a 3-connected matroidM has a minor in F ande is an element ofM, thenM has a minor in F whose ground set contains.e. The aim of this note is to prove that, for all sufficiently largen, the collection ofn-element 3-connected matroids having some minor inF is also (3, 1)-rounded.This research was partially supported by the National Science Foundation under Grant No. DMS-8500494.     

Blow-up Results for Some Nonlinear Delay Differential Equations

In this work we study the blow up phenomena for some scalar delay differential equations. In particular, we make connection with the blow up of ordinary differential equations that are related to the delay differential equations. The first author is supported by a Grant from TWAS under contract No: 03-030 RG/MATHS/AF/AC. The second author is supported by a grant from the Lebanese National Council for Scientific Research.     

Hyponormality of Toeplitz operators with rational symbols

In this article we introduce a notion of `division' for rational functions and then give a criterion for hyponormality of (f, g are rational functions) in the cases where g divides f. Furthermore we show that we may assume, without loss of generality, that g divides f when we consider the hyponormality of . Supported in part by a grant from Faculty Research Fund, Sungkyunkwan University, 2004. Supported in part by a grant (R14-2003-006-01000-0) from the Korea Research Foundation.     

Algebraic and combinatorial properties of zircons

In this paper we introduce and study a new class of posets, that we call zircons, which includes all Coxeter groups partially ordered by Bruhat order. We prove that many of the properties of Coxeter groups extend to zircons often with simpler proofs: in particular, zircons are Eulerian posets and the Kazhdan-Lusztig construction of the Kazhdan-Lusztig representations can be carried out in the context of zircons. Moreover, for any zircon Z, we construct and count all balanced and exact labelings (used in the construction of the Bernstein-Gelfand-Gelfand resolutions in the case that Z is the Weyl group of a Kac-Moody algebra). Partially supported by the program “Gruppi di trasformazioni e applicazioni”, University of Rome “La Sapienza”. Part of this research was carried out while the author was a member of the Institut Mittag-Leffler of the Royal Swedish Academy of Sciences.     

Equivalent factor matroids of graphs

The factor matroid of a graphG is the matric matroid of the vertex-edge incidence matrix ofG interpreted over the real numbers. This paper presents a constructive characterization of the graphs hat have the same factor matroid as a given 4-connected bipartite graph.Research partially supported by NSF Grant ESS-8307796 and Office of Naval Research Grant N00014-86-K-0689.     

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.     

A note on a generalized form of the Laplacian and of sub-Laplacians

In this note we prove a recent conjecture of Hasson [11]: we show that, for a locally integrable function u, a sufficient condition to be harmonic is that $ \lim\limits_{r\to 0^+} r^{-2}(M_{r}u-u) = 0 $ in the weak sense of distributions (M _r being the averaging operator on balls of radius r). We also extend this and other results to the setting of sub-Laplacians on Carnot groups.Investigation supported by University of Bologna. Funds for selected research topics.     

<Emphasis Type="Bold">C</Emphasis><Superscript><Emphasis Type="Bold">1,</Emphasis><Emphasis Type="Italic">α</Emphasis></Superscript> regularity for infinity harmonic functions in two dimensions

We propose a new method for showing C ^{1, α} regularity for solutions of the infinity Laplacian equation and provide full details of the proof in two dimensions. The proof for dimensions n ≥ 3 depends upon some conjectured local gradient estimates for solutions of certain transformed PDE. LCE is supported in part by NSF Grant DMS-0500452. OS was supported in part by the Miller Institute for Basic Research in Science, Berkeley.     

The dimension of suborders of the Boolean lattice

We consider the order dimension of suborders of the Boolean latticeB _n . In particular we show that the suborder consisting of the middle two levels ofB _n dimension at most of 6 log₃ n. More generally, we show that the suborder consisting of levelss ands+k ofB _n has dimensionO(k ² logn).The research of the second author was supported by Office of Naval Research Grant N00014-90-J-1206.The research of the third author was supported by Grant 93-011-1486 of the Russian Fundamental Research Foundation.     

Proof of a conjecture of Kahn for non-binary matroids

This note proves a conjecture of Kahn by showing that ifX is a 3-element independent set in a 3-connected non-binary matroid M, thenM has a connected non-binary minor havingX as a basis. This research was partially supported by an LSU Summer Research Grant.     

A note on the existence, uniqueness and symmetry of par-balanced realizations

We give a proof of the realization theorem of N.J. Young which states that analytic functions which are symbols of bounded Hankel operators admit par-balanced realizations. The main tool used in this proof is the induced Hilbert spaces and a lifting lemma of Kreîn-Reid-Lax-Dieudonné. Alternatively one can use the Loewner inequality. A short proof of the uniqueness of par-balanced realizations is included. As an application, it is proved that par-balanced realizations of real symmetric transfer functions areJ-self-adjoint.Research supported in part by the Romanian Academy grant GAR-6645/1996.This research was supported in part by NSF grant DMS-9501223.     

Irreducibility and Mixed Boundary Conditions

In this paper we consider positive semigroups on L^p(Ω) generated by elliptic operators A subject to mixed Dirichlet-Neumann boundary conditions on non-smooth domains Ω. We show in particular that these semigroups as well as those generated by multiplicative perturbations bA of A are irreducible, provided b ∈ L^∞(Ω) is real and satisfies b ≥ δ for some δ > 0. In memoriam Helmut H. Schaefer     

An asymptotic formula for the voltage potential in the case of a near-surface conductivity inclusion

We rigorously derive an asymptotic expansion of the steady-state voltage potentials in the presence of a conductivity inclusion of small volume that is close to a planar surface. This new formula is motivated by the practically important inverse problem of imaging a conductivity inclusion near a planar interface. Partly supported by ACI Jeunes Chercheurs (0693) from the Ministry of Education and Scientific Research, France. Partially supported by grant R02-2003-000-10012-0 from the Korea Science and Engineering Foundation. Received: December 2, 2003; revised: January 18, 2004