Poisson Reduction by Distributions

It is shown that the singular Poisson reduction procedure can be improved for a large class of situations. In addition, Poisson reduction of orbit type manifolds is carried out in detail.     

Singular Reduction of Poisson Manifolds

The conditions under which it is possible to reduce a Poisson manifold via a regular foliation have been completely characterized by Marsden and Ratiu. In this Letter we show that this characterization can be generalized in a natural way to the singular case and, as a corollary, we obtain that when the singular distribution is given by the tangent spaces to the orbits created by a Hamiltonian Lie group action, one reproduces the Universal Reduction Procedure of Arms, Cushman, and Gotay.     

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.     

Nambu-Poisson dynamics of superintegrable systems

After an introduction to Nambu-Poisson dynamics (NPD), some applications of NPD in finite-dimensional (superintegrable) and infinite-dimensional (extended quantum mechanics and hydrodynamics) systems are considered. The text was submitted by the author in English.     

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t₀=t is naturally understood where t₀ is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink–anti-kink and soliton–soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.     

Drop-Size Distributions Produced by Flat-Spray Nozzles

This study deals with measurements of the drop-size distributions of different types of fan spray atomisers using a scattered-light particle size counting analyser. The drop-size distributions were determined at various locations in the spray cone. These local distributions change systematically from the fan's axis to its border. Superimposing these local distributions, adequately weighted, one acquires the entire distribution of all the drops. A comparison of the experimental results is made with those yielded by mathematical equations.     

Estimation of Nanoparticle Size Distributions by Image Analysis

Knowledge of the nanoparticle size distribution is important for the interpretation of experimental results in many studies of nanoparticle properties. An automated method is needed for accurate and robust estimation of particle size distribution from nanoparticle images with thousands of particles. In this paper, we present an automated image analysis technique based on a deformable ellipse model that can perform this task. Results of using this technique are shown for both nearly spherical particles and more irregularly shaped particles. The technique proves to be a very useful tool for nanoparticle research.     

Spin Holonomy of Einstein Manifolds

Berger's Theorem classifies the linear holonomy groups of irreducible, simply connected Riemannian manifolds. For physical applications, however, it is at least as important to have a classification of the possible spin holonomy groups (defined by the parallel transport of spinors) of non-simply-connected manifolds. We establish a complete classification of the spin holonomy groups of all compact, locally irreducible, Einstein Riemannian spin manifolds of non-negative scalar curvature.     

Injectivity Radius of Lorentzian Manifolds

Motivated by the application to general relativity we study the geometry and regularity of Lorentzian manifolds under natural curvature and volume bounds, and we establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geometric, and are formulated via a reference Riemannian metric that we canonically associate with a given observer (p, T) –where p is a point of the manifold and T is a future-oriented time-like unit vector prescribed at p only. The proofs are based on a generalization of arguments from Riemannian geometry. We first establish estimates on the reference Riemannian metric, and then express them in terms of the Lorentzian metric. In the context of general relativity, our estimate on the injectivity radius of an observer should be useful to investigate the regularity of spacetimes satisfying Einstein field equations.     

Kan Replacement of Simplicial Manifolds

We establish a functor Kan from local Kan simplicial manifolds to weak Kan simplicial manifolds. It gives a solution to the problem of extending local Lie groupoids to Lie 2-groupoids.     

Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the Formality conjecture), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.     

Quantization of Multiply Connected Manifolds

The standard (Berezin-Toeplitz) geometric quantization of a compact Kähler manifold is restricted by integrality conditions. These restrictions can be circumvented by passing to the universal covering space, provided that the lift of the symplectic form is exact. I relate this construction to the Baum-Connes assembly map and prove that it gives a strict quantization of the original manifold. I also propose a further generalization, classify the required structure, and provide a means of computing the resulting algebras. These constructions involve twisted group C*-algebras of the fundamental group which are determined by a group cocycle constructed from the cohomology class of the symplectic form. This provides an algebraic counterpart to the Morita equivalence of a symplectic manifold with its fundamental group.     

Rigidity of Asymptotically Hyperbolic Manifolds

In this paper, we prove a rigidity theorem of asymptotically hyperbolic manifolds only under the assumptions on curvature. Its proof is based on analyzing asymptotic structures of such manifolds at infinity and a volume comparison theorem.The first author’s research is partially supported by NSF grant of China.The second author’s research is partially supported by an NSF grant and a Simon fund.     

Regular ZnO nanopillar arrays by nanosphere photolithography

Highly regular vertical ZnO nanopillar arrays were hydrothermally grown through a nucleation window pattern generated by nanosphere photolithography. The in-plane intensity modulation of the exposing ultraviolet light in the photoresist was performed by Stöber silica or polystyrene nanospheres in the masking Langmuir–Blodgett monolayer. By comparing six different nanosphere diameters in the 180–700 nm range only those with diameter above the exposure wavelength of 405 nm generate a pattern in the thin photoresist layer. The pattern quality is improving with increasing diameter, therefore, the masking for nanopillar growth was demonstrated with 700 nm polystyrene nanospheres. The results of the nanosphere photolithography were supported by finite-difference time-domain calculations. This growth approach was shown to have the potential for low-cost, low-temperature, large area fabrication of ZnO pillars or nanowires enabling a precise engineering of geometry.     

Theory of Holomorphic Maps of Two-Dimensional Complex Manifolds to Toric Manifolds and Type-A Multi-String Theory

JETP Letters - We study the field theory localizing to holomorphic maps from a complex manifold of complex dimension 2 to a toric target (a generalization of A model). Fields are realized as maps...     

激光雷达观测北京大气气溶胶的垂直分布

利用激光雷达对北京2005年4月26-30日大气气溶胶进行了连续观测,对气溶胶光学特性进行了测量和分析,讨论了大气污染物气溶胶和沙尘气溶胶的垂直分布,以及沙尘气溶胶的传输和沉降过程.因此,激光雷达是研究阴霾天气和沙尘天气特征的有效手段.     

On Quantization of Complex Symplectic Manifolds

Let X be a complex symplectic manifold. By showing that any Lagrangian subvariety has a unique lift to a contactification, we associate to X a triangulated category of regular holonomic microdifferential modules. If X is compact, this is a Calabi-Yau category of complex dimension dim X + 1. We further show that regular holonomic microdifferential modules can be realized as modules over a quantization algebroid canonically associated to X.     

Strict Quantizations of Almost Poisson Manifolds

We show the existence of (non-Hermitian) strict quantization for every almost Poisson manifold.