On the Global Wellposedness to the 3-D Incompressible Anisotropic Navier-Stokes Equations

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations (NS _ν) with initial data in the scaling invariant Besov space, here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations (ANS _ν), where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, and Then with initial data in the scaling invariant space we prove the global wellposedness for (ANS _ν) provided the norm of initial data is small enough compared to the horizontal viscosity. In particular, this result implies the global wellposedness of (ANS _ν) with high oscillatory initial data (1.2).     

Editorial

The production of charmed mesons ,D ^± , andD ^*± is studied in a sample of 478,000 hadronicZ decays. The production rates are measured to be

Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions

We consider the point process of zeroes of certain Gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r, as . For the planar Gaussian analytic function, , we show that this probability is asymptotic to . For the hyperbolic Gaussian analytic functions, , we show that this probability decays like .In the planar case, we also consider the problem posed by Mikhail Sodin² on moderate and very large deviations in a disk of radius r, as . We partially solve the problem by showing that there is a qualitative change in the asymptotics of the probability as we move from the large deviation regime to the moderate.Research supported by NSF grant #DMS-0104073 and NSF-FRG grant #DMS-0244479.     

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.     

Dyson’s Constants in the Asymptotics of the Determinants of Wiener-Hopf-Hankel Operators with the Sine Kernel

Let stand for the integral operators with the sine kernels acting on L ²[0,α]. Dyson conjectured that the asymptotics of the Fredholm determinants of are given by

Power Series for Solutions of the 3 {\mathcal D}-Navier-Stokes System on R 3

In this paper we study the Fourier transform of the -Navier-Stokes System without external forcing on the whole space R ³. The properties of solutions depend very much on the space in which the system is considered. In this paper we deal with the space of functions where and c (k) is bounded, . We construct the power series which converges for small t and gives solutions of the system for bounded intervals of time. These solutions can be estimated at infinity (in k-space) by .     

On the Rate of Convergence to Equilibrium of the Andersen Thermostat in Molecular Dynamics

It has been shown in E and Li (Comm. Pure. Appl. Math., 2007, in press) that the Andersen dynamics is uniformly ergodic. Exponential convergence to the invariant measure is established with an error bound of the form

Level-rank duality of WZW models in conformal field theory

We consider the decomposition of the conformal blocks under the conformal embeddings. The case (â is an affine extension of the abelian subalgebra of the central elements ofgl(lr)) is studied in detal. The reciprocal decompositions of -modules induce a pairing between the spaces of conformal blocks of and Wess-Zumino-Witten models on the Riemann sphere. The completeness of the pairing is shown. Hence it defines aduality between two spaces.Dedicated to Professor Masahisa Adachi on his 60th birthday     

Global Regularity for the Navier-Stokes-Maxwell System with Fractional Diffusion

In this paper, we study the global regularity for the Navier-Stokes-Maxwell system with fractional diffusion. Existence and uniqueness of global strong solution are proved for \(\alpha \geqslant \frac {3}{2}\). When 0 < α < 1, global existence is obtained provided that the initial data \(\|u_{0}\|_{H^{\frac {5}{2}-2\alpha }}+\|E_{0}\|_{H^{\frac {5}{2}-2\alpha }}+\|B_{0}\|_{H^{\frac {5}{2}-2\alpha }}\) is sufficiently small. Moreover, when \(1<\alpha <\frac {3}{2}\), global existence is obtained if for any ε >?0, the initial data \(\|u_{0}\|_{H^{\frac {3}{2}-\alpha +\varepsilon }}+\|E_{0}\|_{H^{\frac {3}{2}-\alpha +\varepsilon }}+\|B_{0}\|_{H^{\frac {3}{2}-\alpha +\varepsilon }}\) is small enough.     

On the Modified Korteweg–De Vries Equation

We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg–de Vries equation , with initial data . We assume that the coefficient is real, bounded and slowly varying function, such that , where . We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space . In comparison with the previous paper (Internat. Res. Notices 8 (1999), 395–418), here we exclude the condition that the integral of the initial data u ₀ is zero. We prove the time decay estimates and for all , where . We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.     

A Counterexample to Dispersive Estimates for Schrödinger Operators in Higher Dimensions

In dimension n > 3 we show the existence of a compactly supported potential in the differentiability class , for which the solutions to the linear Schrödinger equation in,

General theory of renormalization of gauge theories in nonlinear gauges

We discuss the general theory of renormalization of unbroken gauge theories in the nonlinear gauges in which the gauge-fixing term is of the form We show that higher loop renormalization modifiesfα [A] to contain ghost terms of the form and show how the corresponding ghost terms are deduced fromfα [A, c, c] uniquely. We show that the theory can be renormalized while preserving a modified form of BRS invariance by multiplicative and independent renormalizations onA, c, g, η, ζ, τ. We briefly discuss the independence of the renormalized S-matrix from η,ζ, τ.     

On the Product of Real Spectral Triples

The product of two real spectral triples and , the first of which is necessarily even, was defined by A.Connes as given by and, in the even-even case, by . Generically it is assumed that the real structure obeys the relations , , , where the -sign table depends on the dimension n modulo 8 of the spectral triple. If both spectral triples obey Connes' >-sign table, it is seen that their product, defined in the straightforward way above, does not necessarily obey this -sign table. In this Letter, we propose an alternative definition of the product real structure such that the -sign table is also satisfied by the product.     

Bosons in Disc-Shaped Traps: From 3D to 2D

We present a mathematically rigorous analysis of the ground state of a dilute, interacting Bose gas in a three-dimensional trap that is strongly confining in one direction so that the system becomes effectively two-dimensional. The parameters involved are the particle number, , the two-dimensional extension, , of the gas cloud in the trap, the thickness, of the trap, and the scattering length a of the interaction potential. Our analysis starts from the full many-body Hamiltonian with an interaction potential that is assumed to be repulsive, radially symmetric and of short range, but otherwise arbitrary. In particular, hard cores are allowed. Under the premises that the confining energy, ~ 1/h ², is much larger than the internal energy per particle, and a/h→ 0, we prove that the system can be treated as a gas of two-dimensional bosons with scattering length a _2D = hexp(−(const.)h/a). In the parameter region where , with the mean density, the system is described by a two-dimensional Gross-Pitaevskii density functional with coupling parameter ~ Na/h. If the coupling parameter is and thus independent of a. In both cases Bose-Einstein condensation in the ground state holds, provided the coupling parameter stays bounded.     

On the Yangian Y $$(\mathfrak{g}\mathfrak{l}_{m|n} )$$ and its quantum Berezinian

Jonathan Brundan and Alexander Kleshchev recently introduced a new family of presentations for the Yangian Y of the general linear Lie algebra . In this article, we extend some of their ideas to consider the Yangian Y of the Lie superalgebra . In particular, we give a new proof of the result by Nazarov that the quantum Berezinian is central. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

The energy levels for simplified anharmonic oscillators

The energy levels for quantum mechanical oscillators with interaction imitatingx (for integer >2) are found by perturbative methods in finite number of dimensions. It is argued that in the limit of infinite dimensional space the coefficients in the expansion for the energy of theith level are growing with the perturbation ordern like . For the ground state (i=1) this reproduces estimates established for anharmonic oscillators.     

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).     

Dynamical Determinants via Dynamical Conjugacies for Postcritically Finite Polynomials

We give an analogue of Levin–Sodin–Yuditskii's study of the dynamical Ruelle determinants of hyperbolic rational maps in the case of subhyperbolic quadratic polynomials. Our main tool is to reduce to an expanding situation. We do so by applying a dynamical change of coordinates on the domains of a Markov partition constructed from the landing ray at the postcritical repelling orbit. We express the dynamical determinants as the product of an (entire) determinant with an explicit expression involving the postcritical repelling orbit, thus explaining the poles in d (z).     

A Variational Principle in the Dual Pair of Reproducing Kernel Hilbert Spaces and an Application

Given a positive definite, bounded linear operator A on the Hilbert space ₀≔l ²(E), we consider a reproducing kernel Hilbert space ₊ with a reproducing kernel A(x,y). Here E is any countable set and A(x,y), x,y∊ E, is the representation of A w.r.t. the usual basis of ₀. Imposing further conditions on the operator A, we also consider another reproducing kernel Hilbert space ₋ with a kernel function B(x,y), which is the representation of the inverse of A in a sense, so that ₋⊃ ₀⊃ ₊ becomes a rigged Hilbert space. We investigate the ratios of determinants of some partial matrices of A and B. We also get a variational principle on the limit ratios of these values. We apply this relation to show the Gibbsianness of the determinantal point process (or fermion point process) defined by the operator A(I+A)⁻¹ on the set E. 2000 Mathematics Subject Classification: Primary: 46E22 Secondary: 60K35