Elliptic Equation with Singular Terms on Riemannian Manifold

In this paper we deal with the following equation: on a three-dimensional Riemannian manifold . We assume that the volume of Σ, the norm , and are small enough. Using a rather simple argument we show the existence of solution to the problem. Dedicated to Gosia and Basia.     

An Infinite Genus Mapping Class Group and Stable Cohomology

We exhibit a finitely generated group whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g with n boundary components, for any g ≥ 0 and n > 0. We construct a representation of into the restricted symplectic group of the real Hilbert space generated by the homology classes of non-separating circles on , which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first universal Chern class in is the pull-back of the Pressley-Segal class on the restricted linear group via the inclusion . L. F. was partially supported by the ANR Repsurf:ANR-06-BLAN-0311.     

Regularity Criteria for the Dissipative Quasi-Geostrophic Equations in Hölder Spaces

We study regularity criteria for weak solutions of the dissipative quasi-geostrophic equation (with dissipation (−Δ) ^γ/2, 0 < γ ≤ 1). We show in this paper that if , or with is a weak solution of the 2D quasi-geostrophic equation, then θ is a classical solution in . This result improves our previous result in [18]. Partially supported by a start-up funding from the Division of Applied Mathematics of Brown University and NSF grant number DMS 0800129. Partially supported by a start-up funding from the College of Natural Sciences of the University of Texas at Austin, NSF grant number DMS 0758247 and an Alfred P. Sloan Research Fellowship.     

Moment Bounds for the Smoluchowski Equation and their Consequences

We prove bounds on moments of the Smoluchowski coagulation equations with diffusion, in any dimension d ≥ 1. If the collision propensities α(n, m) of mass n and mass m particles grow more slowly than , and the diffusion rate is non-increasing and satisfies for some b ₁ and b ₂ satisfying 0 ≤ b ₂ < b ₁ < ∞, then any weak solution satisfies for every and T ∈(0, ∞), (provided that certain moments of the initial data are finite). As a consequence, we infer that these conditions are sufficient to ensure uniqueness of a weak solution and its conservation of mass. This work was performed while A.H. held a postdoctoral fellowship in the Department of Mathematics at U.B.C. This work is supported in part by NSF grant DMS0307021.     

TYZ Expansion for the Kepler Manifold

The main goal of the paper is to address the issue of the existence of Kempf’s distortion function and the Tian-Yau-Zelditch (TYZ) asymptotic expansion for the Kepler manifold - an important example of non-compact manifold. Motivated by the recent results for compact manifolds we construct Kempf’s distortion function and derive a precise TYZ asymptotic expansion for the Kepler manifold. We get an exact formula: finite asymptotic expansion of n − 1 terms and exponentially small error terms uniformly with respect to the discrete quantization parameter ( standing for Planck’s constant and , ). Moreover, the coefficients are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the Kepler manifold. We show that our estimates are sharp by analyzing the nonharmonic behaviour of T _m for . The arguments of the proofs combine geometrical methods, quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic-Gevrey spaces. The first author was supported in part by the project PRIN (Cofin) n. 2006019457 with M.I.U.R., Italy. The second author was supported in part by the M.I.U.R. Project “Geometric Properties of Real and Complex Manifolds”.     

(Co)cyclic (Co)homology of Bialgebroids: An Approach via (Co)monads

For a (co)monad T _l on a category , an object X in , and a functor , there is a (co)simplex in . The aim of this paper is to find criteria for para-(co)cyclicity of Z ^*. Our construction is built on a distributive law of T _l with a second (co)monad T _r on , a natural transformation , and a morphism in . The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun’s axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads and on the category of R-bimodules. The functor Π can be chosen such that is the cyclic R-module tensor product. A natural transformation is given by the flip map and a morphism is constructed whenever T is a (co)module algebra or coring of an R-bialgebroid. The notion of a stable anti-Yetter-Drinfel’d module over certain bialgebroids, the so-called  ×  _R -Hopf algebras, is introduced. In the particular example when T is a module coring of a  ×  _R -Hopf algebra and X is a stable anti-Yetter-Drinfel’d -module, the para-cyclic object Z _* is shown to project to a cyclic structure on . For a -Galois extension , a stable anti-Yetter-Drinfel’d -module T _S is constructed, such that the cyclic objects and are isomorphic. This extends a theorem by Jara and Ştefan for Hopf Galois extensions. As an application, we compute Hochschild and cyclic homologies of a groupoid with coefficients in a stable anti-Yetter-Drinfel’d module, by tracing it back to the group case. In particular, we obtain explicit expressions for (coinciding relative and ordinary) Hochschild and cyclic homologies of a groupoid. The latter extends results of Burghelea on cyclic homology of groups.     

Stability,Convergence to Self-Similarity and Elastic Limit for the Boltzmann Equation for Inelastic Hard Spheres

We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres, in the framework of so-called constant normal restitution coefficients . In the physical regime of a small inelasticity (that is for some constructive ) we prove uniqueness of the self-similar profile for given values of the restitution coefficient , the mass and the momentum; therefore we deduce the uniqueness of the self-similar solution (up to a time translation). Moreover, if the initial datum lies in , and under some smallness condition on depending on the mass, energy and norm of this initial datum, we prove time asymptotic convergence (with polynomial rate) of the solution towards the self-similar solution (the so-called homogeneous cooling state). These uniqueness, stability and convergence results are expressed in the self-similar variables and then translate into corresponding results for the original Boltzmann equation. The proofs are based on the identification of a suitable elastic limit rescaling, and the construction of a smooth path of self-similar profiles connecting to a particular Maxwellian equilibrium in the elastic limit, together with tools from perturbative theory of linear operators. Some universal quantities, such as the “quasi-elastic self-similar temperature” and the rate of convergence towards self-similarity at first order in terms of (1−α), are obtained from our study. These results provide a positive answer and a mathematical proof of the Ernst-Brito conjecture [16] in the case of inelastic hard spheres with small inelasticity.     

Stability of Two Soliton Collision for Nonintegrable gKdV Equations

We continue our study of the collision of two solitons for the subcritical generalized KdV equations

A Strong Szegő Theorem for Jacobi Matrices

We use a classical result of Golinskii and Ibragimov to prove an analog of the strong Szegő theorem for Jacobi matrices on . In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences and find necessary and sufficient conditions on the spectral measure such that and lie in , the linearly-weighted l ² space. An erratum to this article can be found at     

Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzschild Space

We provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzschild background. Our estimate implies that such solutions have asymptotic behavior as long as the source term is bounded in the norm . In particular this gives scattering at small amplitudes for non-linear scalar fields of the form for all 2 < p. This paper is dedicated to the memory of Hope Machedon The second author would like thank MSRI and Princeton University, where a portion of this research was conducted during the Fall of 2005. The second author was also supported by a NSF postdoctoral fellowship.     

Integrable Discrete Nets in Grassmannians

We consider discrete nets in Grassmannians , which generalize Q-nets (maps with planar elementary quadrilaterals) and Darboux nets (-valued maps defined on the edges of such that quadruples of points corresponding to elementary squares are all collinear). We give a geometric proof of integrability (multidimensional consistency) of these novel nets, and show that they are analytically described by the noncommutative discrete Darboux system.      

Wave Decay on Convex Co-Compact Hyperbolic Manifolds

For convex co-compact hyperbolic quotients , we analyze the long-time asymptotic of the solution of the wave equation u(t) with smooth compactly supported initial data f = (f ₀, f ₁). We show that, if the Hausdorff dimension δ of the limit set is less than n/2, then where and . We explain, in terms of conformal theory of the conformal infinity of X, the special cases , where the leading asymptotic term vanishes. In a second part, we show for all the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip . As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f.     

Openness of the Set of Non-characteristic Points and Regularity of the Blow-up Curve for the 1 D Semilinear Wave Equation

We consider here the 1 D semilinear wave equation with a power nonlinearity and with no restriction on initial data. We first prove a Liouville Theorem for that equation. Then, we consider a blow-up solution, its blow-up curve and the set of non-characteristic points. We show that I ₀ is open and that T(x) is C ¹ on I ₀. All these results fundamentally use our previous result in [19] showing the convergence in selfsimilar variables for . This work was supported by a grant from the french Agence Nationale de la Recherche, project ONDENONLIN, reference ANR-06-BLAN-0185.     

The <Emphasis Type="Italic">N</Emphasis> = 1 Triplet Vertex Operator Superalgebras

We introduce a newfamily of C ₂-cofinite N = 1 vertex operator superalgebras , m ≥ 1, which are natural super analogs of the triplet vertex algebra family , p ≥ 2, important in logarithmic conformal field theory. We classify irreducible -modules and discuss logarithmic modules. We also compute bosonic and fermionic formulas of irreducible characters. Finally, we contemplate possible connections between the category of -modules and the category of modules for the quantum group , , by focusing primarily on properties of characters and the Zhu’s algebra . This paper is a continuation of our paper Adv. Math. 217, no.6, 2664–2699 (2008). The second author was partially supported by NSF grant DMS-0802962.     

Local Resolvent Estimates for <Emphasis Type="Italic">N</Emphasis>-body Stark Hamiltonians

For an N-body Stark Hamiltonian , the resolvent estimate holds uniformly in with Re and Im , where , and is a compact interval. This estimate is well known as the limiting absorption principle. In this paper, we report that by introducing the localization in the configuration space, a refined resolvent estimate holds uniformly in with Re and Im . Dedicated to Professor Hideo Tamura on the occasion of his 60th birthday     

Carleman Estimates and Absence of Embedded Eigenvalues

Let L = ?Δ? W be a Schrödinger operator with a potential $W\in L^{\frac{n+1}{2}}(\mathbb{R}^n)Let L = −Δ− W be a Schr?dinger operator with a potential , . We prove that there is no positive eigenvalue. The main tool is an Carleman type estimate, which implies that eigenfunctions to positive eigenvalues must be compactly supported. The Carleman estimate builds on delicate dispersive estimates established in [7]. We also consider extensions of the result to variable coefficient operators with long range and short range potentials and gradient potentials.The first author was partially supported by DFG grant KO1307/1 and also by MSRI for Fall 2005The second author was partially supported by NSF grants DMS0354539 and DMS 0301122 and also by MSRI for Fall 2005     

On the Uniqueness of Weak Weyl Representations of the Canonical Commutation Relation

Let (T, H) be a weak Weyl representation of the canonical commutation relation (CCR) with one degree of freedom. Namely T is a symmetric operator and H is a self-adjoint operator on a complex Hilbert space satisfying the weak Weyl relation: for all (the set of real numbers), e^−itH D(T) ⊂ D(T) (i is the imaginary unit and D(T) denotes the domain of T) and . In the context of quantum theory where H is a Hamiltonian, T is called a strong time operator of H. In this paper we prove the following theorem on uniqueness of weak Weyl representations: Let be separable. Assume that H is bounded below with and , where is the set of complex numbers and, for a linear operator A on a Hilbert space, σ(A) denotes the spectrum of A. Then ( is the closure of T) is unitarily equivalent to a direct sum of the weak Weyl representation on the Hilbert space , where is the multiplication operator by the variable and with . Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation . This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from Japan Society for the Promotion of Science (JSPS).     

A Uniform Quantum Version of the Cherry Theorem

Consider in the operator family . P ₀ is the quantum harmonic oscillator with diophantine frequency vector ω, F ₀ a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space, and . Then there exist independent of and an open set such that if and , the quantum normal form near P ₀ converges uniformly with respect to . This yields an exact quantization formula for the eigenvalues, and for the classical Cherry theorem on convergence of Birkhoff’s normal form for complex frequencies is recovered. Partially supported by PAPIIT-UNAM IN106106-2.     

Group Orbits and Regular Partitions of Poisson Manifolds

We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits. Examples include all varieties of Lagrangian subalgebras of reductive quadratic Lie algebras with Poisson structures defined by Lagrangian splittings of . In the special case of , where is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on defined by arbitrary Lagrangian splittings of . Such Lagrangian splittings have been classified by P. Delorme, and they contain the Belavin–Drinfeld splittings as special cases.