Diffusive limit of a tagged particle in asymmetric simple exclusion processes

Invariance principles are proved under diffusive scaling for the centered position of a tagged particle in the simple exclusion process with asymmetric nonzero drift jump probabilities in dimensions d ≥ 3. The method of proof is by martin‐gale techniques which rely on the fact that symmetric random walks are transient in high dimensions. © 2000 John Wiley & Sons, Inc.     

Metastability of Ginzburg-Landau model with a conservation law

The hydrodynamics of Ginzburg-Landau dynamics has previously been proved to be a nonlinear diffusion equation. The diffusion coefficient is given by the second derivative of the free energy and hence nonnegative. We consider in this paper the Ginzburg-Landau dynamics with long-range interactions. In this case the diffusion coefficient is nonnegative only in the metastable region. We prove that if the initial condition is in the metastable region, then the hydrodynamics is governed by a nonlinear diffusion equation with the diffusion coefficient given by the metastable curve. Furthermore, the lifetime of the metastable state is proved to be exponentially large.     

Taming symplectic forms and the Calabi-Yau equation

We study the Calabi–Yau equation on symplectic manifolds.We show that Donaldson's conjecture on estimates for this equationin terms of a taming symplectic form can be reduced to an integralestimate of a scalar potential function. Under a positive curvaturecondition, we show that the conjecture holds.     

Obstructions to the Existence of Sasaki–Einstein Metrics

We describe two simple obstructions to the existence of Ricci-flat Kähler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kähler–Einstein metrics on Fano orbifolds. We present several families of hypersurface singularities that are obstructed, including 3-fold and 4-fold singularities of ADE type that have been studied previously in the physics literature. We show that the AdS/CFT dual of one obstruction is that the R–charge of a gauge invariant chiral primary operator violates the unitarity bound.     

+c–face incident pentagon mutually pumped phase conjugator

The mutually pumped phase conjugator with +c-face incident geometry is first demonstrated in a photorefractive pentagon-shaped 0°-cut BaTiO₃ crystal. Designed by choosing the appropriate geometry such as the orientation between two mutual incoherent beams and the special cut crystals c-axis, a novel configuration can be formed in the crystal when the mutually pumped phase conjugation is established. This configuration provides more efficient interaction, which provides fast response and high stability as well as large angular acceptance and high image resolution. PACS 42.65.HW; 42.70.Nq     

Fabrication of high-aspect-ratio sub-diffraction-limit microstructures by two-photon-absorption photopolymerization

Micro-cell structures with side-walls as thin as 0.70 m and aspect ratios as high as 6.7 are fabricated by single-layer writing through two-photon-absorption (TPA) photopolymerization. The use of a moderate-numerical-aperture (N.A.) objective lens to obtain a much more elongated voxel and an in situ ultraviolet (UV) pre-exposure step to improve the sensitivity of TPA photopolymerization are the main factors responsible for the high aspect ratio and sub-diffraction-limit resolution that are achieved. PACS 42.70.Gi; 81.05.Lg; 82.35.-x     

An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry

We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. In particular, we prove completeness of the solutions of the separated ODEs. This integral representation is a suitable starting point for a detailed analysis of the long-time dynamics of scalar waves in the Kerr geometry. Research supported by NSERC grant # RGPIN 105490-2004. Research supported in part by the NSF, Grant No. DMS-010-3998. Research supported in part by the NSF, Grant No. 33-585-7510-2-30.     

Scalar curvature,non-abelian group actions,and the degree of symmetry of exotic spheres

It is proved that if a compact manifold admits a smooth action by a compact, connected, non-abelian Lie group, then it admits a metric of positive scalar curvature. This result is used to prove that if ∑ ⁿ is an exoticn-sphere which does not bound a spin manifold, then the only possible compact connected transformation groups of ∑ ⁿ are tori of dimension ≤[(n+1)/2]. Research partially supported by the Sloan Foundation and NSF Grant GP-34785X.