Universal Inequalities for Eigenvalues of the Buckling Problem on Spherical Domains

In this paper we study the eigenvalues of the buckling problem on domains in a unit sphere. We obtain universal bounds on the (k + 1)^th eigenvalue in terms of the first k eigenvalues independent of the domains. Partially supported by FEMAT. Partially supported by CNPq, Pronex and Proex.     

Estimates of the first two buckling eigenvalues on spherical domains

In this paper, we study the first two eigenvalues of the buckling problem on spherical domains. We obtain an estimate of the second eigenvalue in terms of the first eigenvalue, which improves on a recent result obtained by Wang and Xia (2007) [1].     

On Ulam-von Neumann transformations

We define and study Ulam-von Neumann transformations which are certain interval mappings and conjugate toq(x)=1–2x ² on [–1,1]. We use a singular metric on [–1,1] to study a Ulam-von Neumann transformation. This singular metric is universal in the sense that it does not depend on any particular mapping but only on the exponent of this mapping at its unique critical point. We give the smooth classification of Ulam-von Neumann transformations by their eigenvalues at periodic points and exponents and asymmetries.The author is partially supported by a PSC-CUNY grant and a NSF grant.     

The Euler-Poincaré equations and double bracket dissipation

This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincaré) system by a special dissipation term that has Brockett's double bracket form. We show that a formally unstable equilibrium of the unperturbed system becomes a spectrally and hence nonlinearly unstable equilibrium after the perturbation is added. We also investigate the geometry of this dissipation mechanism and its relation to Rayleigh dissipation functions. This work complements our earlier work (Bloch, Krishnaprasad, Marsden and Ratiu [1991, 1994]) in which we studied the corresponding problem for systems with symmetry with the dissipation added to the internal variables; here it is added directly to the group or Lie algebra variables. The mechanisms discussed here include a number of interesting examples of physical interest such as the Landau-Lifschitz equations for ferromagnetism, certain models for dissipative rigid body dynamics and geophysical fluids, and certain relative equilibria in plasma physics and stellar dynamics.Research partially supported by the National Science Foundation PYI grant DMS-91-57556, and AFOSR grant F49620-93-1-0037.Research partially supported by the AFOSR University Research Initiative Program under grants AFOSR-87-0073 and AFOSR-90-0105 and by the National Science Foundation's Engineering Research Centers Program NSFD CDR 8803012.Research partially supported by, DOE contract DE-FG03-92ER-25129, a Fairchild Fellowship at Caltech, and the Fields Institute for Research in the Mathematical Sciences.Research partially supported by NSF Grant DMS 91-42613, DOE contract DE-FG03-92ER-25129, the Fields Institute, the Erwin Schrödinger Institute, and the Miller Institute of the University of California.     

Coupling constant thresholds in nonrelativistic quantum mechanics

We extend the analysis of absorbtion of eigenvalues for the two body case to situations where absorbtion occurs at a two cluster threshold in anN-body system. The result depends on a Birman-Schwinger kernel for such anN-body system, an object which we apply in other ways. In particular, we control the number of discrete eigenvalues in the 0 limit.Research partially supported by U.S.N.S.F. under Grant MCS-78-01885.     

Dilation analyticity in constant electric field

We extend the analysis of Paper I from two body dilation analytic systems in constant electric field toN-body systems in constant electric field. Particular attention is paid to what happens to isolated eigenvalues of an atomic or molecular system in zero field when the field is turned on. We prove that the corresponding eigenvalue of the complex scaled Hamiltonian is stable and becomes a resonance. We study analyticity properties of the levels as a function of the field and also Borel summability.Research partially supported by USNSF grant MCS 78-00101Research partially supported by USNSF grant MCS 78-01885     

Large <Emphasis Type="Italic">n</Emphasis> Limit of Gaussian Random Matrices with External Source,Part II

We continue the study of the Hermitian random matrix ensemble with external source where A has two distinct eigenvalues ±a of equal multiplicity. This model exhibits a phase transition for the value a=1, since the eigenvalues of M accumulate on two intervals for a>1, and on one interval for 0<a<1. The case a>1 was treated in Part I, where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices, that is, limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum, and in terms of the Airy kernel at the edge. In this paper we establish the same results for the case 0<a<1. As in Part I we apply the Deift/Zhou steepest descent analysis to a 3×3-matrix Riemann-Hilbert problem. Due to the different structure of an underlying Riemann surface, the analysis includes an additional step involving a global opening of lenses, which is a new phenomenon in the steepest descent analysis of Riemann-Hilbert problems.The first and third author are supported in part by INTAS Research Network NeCCA 03-51-6637 and by NATO Collaborative Linkage Grant PST.CLG.979738. The first author is supported in part by RFBR 05-01-00522 and the program “Modern problems of theoretical mathematics” RAS(DMS). The second author is supported in part by the National Science Foundation (NSF) Grant DMS-0354962. The third author is supported in part by FWO-Flanders projects G.0176.02 and G.0455.04 and by K.U.Leuven research grant OT/04/24 and by the European Science Foundation Program Methods of Integrable Systems, Geometry, Applied Mathematics (MISGAM) and the European Network in Geometry, Mathematical Physics and Applications (ENIGMA)     

On the Strongly Damped Wave Equation

We prove the existence of the universal attractor for the strongly damped semilinear wave equation, in the presence of a quite general nonlinearity of critical growth. When the nonlinearity is subcritical, we prove the existence of an exponential attractor of optimal regularity, having a basin of attraction coinciding with the whole phase-space. As a byproduct, the universal attractor is regular and of finite fractal dimension. Moreover, we carry out a detailed analysis of the asymptotic behavior of the solutions in dependence of the damping coefficient.Research partially supported the Italian MIUR Research Projects Problemi di Frontiera Libera nelle Scienze Applicate, Aspetti Teorici e Applicativi di Equazioni a Derivate Parziali and Metodi Variazionali e Topologici nello Studio dei Fenomeni Nonlineari. The second author was also supported by the Istituto Nazionale di Alta Matematica F. Severi (INdAM).     

On a theorem of Deift and Hempel

We provide an alternative proof of the main result of Deift and Hempel [1] on the existence of eigenvalues ofv-dimensional Schrödinger operatorsH =H ₀+W in spectral gaps ofH ₀.Research partially supported by USNSF under Grant DMS-8416049On leave of absence from the Institute for Theoretical Physics, University of Graz, A-8010 Graz, Austria; Max Kade Foundation Fellow     

Regularity of harmonic maps with prescribed singularities

In this paper, we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds. This is motivated from one of Hawking's conjectures on the uniqueness of Kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum Einstein equation in general relativity.Research partially supported by a NSF grant DMS-8907849.Research partially supported by a NSF grant     

On the integral construction of the universal R-matrix for quantum current superalgebraUq (oŝp(1,2))

We discuss the problems in the construction of the universal R-matrix for the the Drinfeld current realization of quantum affine superalgebrasUq (oŝp(1,2)), where we try to present the universal R-matrix for the corresponding “Drinfeld” comultiplication in the form of certain integrals over current operators with specially chosen contours. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. Author’s research is partially supported by the Summer Research Fellowship and the Taft Foundation at the University of Cincinnati.     

A Class of Integrable Flows on the Space of Symmetric Matrices

For a given skew symmetric real n × n matrix N, the bracket [X, Y] _N = XNY − YNX defines a Lie algebra structure on the space Sym(n, N) of symmetric n × n real matrices and hence a corresponding Lie-Poisson structure. The purpose of this paper is to investigate the geometry, integrability, and linearizability of the Hamiltonian system , or equivalently in Lax form, the equation on this space along with a detailed study of the Poisson geometry itself. If N has distinct eigenvalues, it is proved that this system is integrable on a generic symplectic leaf of the Lie-Poisson structure of Sym(n, N). This is established by finding another compatible Poisson structure. If N is invertible, several remarkable identifications can be implemented. First, (Sym(n, N), [·, ·]) is Lie algebra isomorphic with the symplectic Lie algebra associated to the symplectic form on given by N ⁻¹. In this case, the system is the reduction of the geodesic flow of the left invariant Frobenius metric on the underlying symplectic group Sp(n, N ⁻¹). Second, the trace of the product of matrices defines a non-invariant non-degenerate inner product on Sym(n, N) which identifies it with its dual. Therefore Sym(n, N) carries a natural Lie-Poisson structure as well as a compatible “frozen bracket” structure. The Poisson diffeomorphism from Sym(n, N) to maps our system to a Mischenko-Fomenko system, thereby providing another proof of its integrability if N is invertible with distinct eigenvalues. Third, there is a second ad-invariant inner product on Sym(n, N); using it to identify Sym(n, N) with itself and composing it with the dual of the Lie algebra isomorphism with , our system becomes a Mischenko- Fomenko system directly on Sym(n, N). If N is invertible and has distinct eigenvalues, it is shown that this geodesic flow on Sym(n, N) is linearized on the Prym subvariety of the Jacobian of the spectral curve associated to a Lax pair formulation with parameter of the system. If, on the other hand, N has nullity one and distinct eigenvalues, in spite of the fact that the system is completely integrable, it is shown that the flow does not linearize on the Jacobian of the spectral curve. Research partially supported by NSF grants CMS-0408542 and DMS-0604307. Research partially supported by the Swiss SCOPES grant IB7320-110721/1, 2005-2008, and MEdC Contract 2-CEx 06-11-22/25.07.2006. Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932. Research partially supported by the Swiss NSF and the Swiss SCOPES grant IB7320-110721/1.     

The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

The purpose of this Letter is to investigate the geometry of new classes of soliton-like solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [10] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and draw some consequences from this setting. Amongst these consequences, one obtains new solutions such as quasiperiodic solutions,n-solitons, solitons with quasiperiodic background, billiard, andn-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow onN-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.Research supported in part by DOE CHAMMP and HPCC programs.Research partially supported by the Department of Energy, the Office of Naval Research and the Fields Institute for Research in the Mathematical Sciences.     

Generalized Diagonal Crossed Products and Smash Products for Quasi-Hopf Algebras. Applications

In this paper we introduce generalizations of diagonal crossed products, two-sided crossed products and two-sided smash products, for a quasi-Hopf algebra H. The results we obtain may then be applied to H ^*-Hopf bimodules and generalized Yetter-Drinfeld modules. The generality of our situation entails that the “generating matrix” formalism cannot be used, forcing us to use a different approach. This pays off because as an application we obtain an easy conceptual proof of an important but very technical result of Hausser and Nill concerning iterated two-sided crossed products.Research partially supported by the EC programme LIEGRITS, RTN 2003, 505078, and by the bilateral projects “Hopf Algebras in Algebra, Topology, Geometry and Physics” and “New techniques in Hopf algebras and graded ring theory” of the Flemish and Romanian Ministries of Research. The first two authors have been also partially supported by the programme CERES of the Romanian Ministry of Education and Research, contract no. 4-147/2004.     

Central limit theorems for the one-dimensional Rayleigh gas with semipermeable barriers

A version of the one-dimensional Rayleigh gas is considered: a point particle of massM (molecule), confined to the unit interval [0,1], is surrounded by an infinite ideal gas of point particles of mass 1 (atoms). The molecule interacts with the atoms and with the walls via elastic collision. Central limit theorems are proved for a wide class of additive functionals of this system (e.g. the number of collisions with the walls and the total length of the molecular path).Research partially supported by the Hungarian National Foundation for Scientific Research, grant No. 819/1     

定跨度变截面梁的弯曲问题

一.引言 在处理梁的弯曲问题时,人们经常利用函数级数来表示有关的各量,并后而得到各该量的近似值。胡海昌曾经指出:在横向载荷和轴向力同时作用下,适宜于用梁的屈曲的本徵函数展开式来表示梁的挠度;其中φ_n是满足所给的梁的支座情况的屈曲本徵函数,a_n是常数系数。他求得一个相当简单的公式以已知的本徵函数和本徵值表示诸系数     

Classification of singular Sobolev connections by their holonomy

For a connection on a principalSU(2) bundle over a base space with a codimension two singular set, a limit holonomy condition is stated. In dimension four, finite action implies that the condition is satisfied and an a priori estimate holds which classifies the singularity in terms of holonomy. If there is no holonomy, then a codimension two removable singularity theorem is obtained.Research partially supported by NSF Grant DMS-8701813Research partially supported by NSF Grant INT-8511481     

A New Coherent States Approach to Semiclassics Which Gives Scott’s Correction

We introduce a new semiclassical calculus by generalizing the standard coherent states. This is applied to the semiclassical expansion for the sum of negative eigenvalues of Schrödinger operators which leads to a new proof of the Scott correction for non-relativistic molecules.Work partially supported by an EU TMR grant, by a grant from the Danish research council, and by MaPhySto – Centre for Mathematical Physics and Stochastics, funded by a grant from The Danish National Research Foundation. ©2003 by the authors. This article may be reproduced in its entirety for non-commercial purposes.     

Large n Limit of Gaussian Random Matrices with External Source, Part I

We consider the random matrix ensemble with an external sourcedefined on n×n Hermitian matrices, where A is a diagonal matrix with only two eigenvalues ±a of equal multiplicity. For the case a>1, we establish the universal behavior of local eigenvalue correlations in the limit n, which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. We use a characterization of the associated multiple Hermite polynomials by a 3×3-matrix Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze the Riemann-Hilbert problem in the large n limit.Dedicated to Freeman Dyson on his eightieth birthdayThe first author was supported in part by NSF Grants DMS-9970625 and DMS-0354962.The second author was supported in part by projects G.0176.02 and G.0455.04 of FWO-Flanders, by K.U.Leuven research grant OT/04/24, and by INTAS Research Network NeCCA 03-51-6637.     

Schrödinger Operators with Random Sparse Potentials. Existence of Wave Operators

We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than 1 and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.The research of the author is supported in part by a Grant-in-Aid for Scientific Research (B) (No. 15340005) from the Ministry of Education, Culture, Sports, Science and Technology.Mathematics Subject classifications (2000). primary, 34M35, secondary, 33E20.This revised version was published online in March 2005 with corrections to the cover date.