An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg

We present a direct analytic approach to the Guillemin-Sternberg conjecture [GS] that `geometric quantization commutes with symplectic reduction', which was proved recently by Meinrenken [M1], [M2] and Vergne [V1], [V2] et al. Besides providing a new proof of this conjecture, our methods also lead immediately to further extensions in various contexts. Oblatum 3-IX-1996 & 4-VIII-1997     

Réduction symplectique et quantification

We present a direct analytic proof of the Guillemin-Sternberg geometric quantization conjecture [2]. Further extensions are also obtained.     

Geometric bounds on the linearization domain and analytic dependence on parameters for families of analytic vector fields in a neighborhood of a singular point

We study families of holomorphic vector fields, holomorphically depending on parameters, in a neighborhood of an isolated singular point. When the singular point is in the Poincaré domain for every vector field of the family we prove, through a modification of classical Sternberg's linearization argument, cf. Nelson (1969) [7] too, analytic dependence on parameters of the linearizing maps and geometric bounds on the linearization domain: each vector field of the family is linearizable inside the smallest Euclidean sphere which is not transverse to the vector field, cf. Brushlinskaya (1971) [2], Ilyashenko and Yakovenko (2008) [5] for related results. We also prove, developing ideas in Martinet (1980) [6], a version of Brjuno's Theorem in the case of linearization of families of vector fields near a singular point of Siegel type, and apply it to study some 1-parameter families of vector fields in two dimensions.     

Eisenstein series on quaternion half-space of degree N

In Kim [7], we studied an Eisenstein series on quaternion half-space of degree 2. By calculating the Siegel series using the method of Karel [5], we obtained the analytic continuation and functional equation of the Eisenstein series. In this note we study an Eisenstein series on quaternion half-space of degreen. By calculating the Siegel series in an analogous way as in Shimura [15] and Kitaoka [8], we obtain singular modular forms of weightk, k<2n and 4/k. Furthermore, we obtain the analytic continuation and functional equation of the Eisenstein series.     

关于多裂纹圆柱体的扭转*

本文在文[1]基础上,导出了含有任意分布裂纹系的圆柱扭曲函数的解析表达式,从而把问题化为以未知位错密度函数表示的奇异积分方程组.文中利用奇异积分方程的数值方法[2,7],对带有多根裂纹的圆柱的抗扭刚度和应力强度因子作了若干数值计算.此外,本文还首次将裂纹切割法[5]推广用于求解矩形柱的扭转,数值结果表明方法是成功的.     

Exact solution of a boundary value problem describing the uniform cylindrical or spherical piston motion

In this paper, we construct the exact solution for fluid motion caused by the uniform expansion of a cylindrical or spherical piston into still air. Following Lighthill [1], we introduce velocity potential into the analysis and seek a similarity form of the solution. We find both numerical and analytic solutions of the second order nonlinear differential equation, with the boundary conditions at the shock and at the piston. The results obtained from the analytic solutions justify numerical solution and the approximate solution of Lighthill [1]. We find that although the approximate solution of Lighthill [1] gives remarkably good numerical results, the analytic form of that solution is not mathematically satisfactory. We also find that in case of spherical piston motion Lighthill’s [1] solution differs significantly from that of our analytic and numerical solutions. We use Pade′ approximation to extend the radius of convergence of the series solution. We also carry out some local analysis at the boundary to obtain some singular solutions.     

A maximal inequality for stochastic convolution integrals on hilbert spaces and space-time regularity of linear stochastic partial differential equations

We consider optimal control problems for one-dimensional diffusion processes [ILM0001] where the control processes υ_t are increasing, positive, and adapted. Several types of expected cost structures associated with each policy υ(.) are adopted, e.g. discounted cost, long term average cost and time average cost. Our work is related to [2,6,12,14,16 and 21], where diffusions are allowed to evolve in the whole space, and to [13] and [20], where diffusions evolve only in bounded regions. We shall present some analytic results about value functions, mainly their characterizations, by simple dynamic programming arguments. Several simple examples are explicitly solved to illustrate the singular behaviour of our problems.     

Asymptotic expansion of stochastic flows

Summary We study the asymptotic expansion in small time of the solution of a stochastic differential equation. We obtain a universal and explicit formula in terms of Lie brackets and iterated stochastic Stratonovich integrals. This formula contains the results of Doss [6], Sussmann [15], Fliess and Normand-Cyrot [7], Krener and Lobry [10], Yamato [17] and Kunita [11] in the nilpotent case, and extends to general diffusions the representation given by Ben Arous [3] for invariant diffusions on a Lie group. The main tool is an asymptotic expansion for deterministic ordinary differential equations, given by Strichartz [14].     

SOLUTION OF A RESEARCH PROBLEM

ＳＯＬＵＴＩＯＮＯＦＡＲＥＳＥＡＲＣＨＰＲＯＢＬＥＭＷＵＳＨＩＱＵＡＮ（巫世权）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＮａｔｉｏｎａｌＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ，Ｃｈａｎｇｓｈａ４１００７３，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ...     

Flots et series de Taylor stochastiques

Summary We study the expansion of the solution of a stochastic differential equation as an (infinite) sum of iterated stochastic (Stratonovitch) integrals. This enables us to give a universal and explicit formula for any invariant diffusion on a Lie group in terms of Lie brackets, as well as a universal and explicit formula for the brownian motion on a Riemannian manifold in terms of derivatives of the curvature tensor. The first of these formulae contains, and extends to the non nilpotent case, the results of Doss [6], Sussmann [17], Yamato [18], Fliess and Normand-Cyrot [7], Krener and Lobry [19] and Kunita [11] on the representation of solutions of stochastic differential equations.     

Singular stationary measures are not always fractal

We know of few explicit results to insure that stationary measures are simultaneously (i) singular, (ii) nonatomic, (iii) with interval support, and (iv) unique. Such results would appear useful, to further separate the analytic notion of singular from the geometric notion of fractal. We prove two general theorems, one for maps of [0,1] into [0, 1], the other for 2×2 random matrices. In each setting, we study measures supported on two points of the transformation space, and we provide sufficient conditions to insure that the stationary measures satisfy (i)–(iv).     

Fourier-Feynman transforms and the first variation

In this paper we complete the following four objectives: 1. We obtain an integration by parts formula for analytic Feynman integrals. 2. We obtain an integration by parts formula for Fourier-Feynman transforms. 3. We find the Fourier-Feynman transform of a functionalF from a Banach algebra after it has been multiplied byn linear factors. 4. We evaluate the analytic Feynman integral of functionals like those described in 3 above. A very fundamental result by Cameron and Storvick [5, Theorem 1], in which they express the analytic Feynman integral of the first variation of a functionalF in terms of the analytic Feynman integral ofF multiplied by a linear factor, plays a key role throughout this paper.     

Singuläre S-hermitesche Rand-Eigenwertprobleme

The theory of singular self-adjoint eigenvalue problems developed by Weyl [13], Stone [12], Kodaira [2] and others has been generalized by A. Schneider [8], [9], [10], [11] to real S-hermitian systems of differential equations with real boundary conditions. Here the theory of singular S-hermitian boundary-value problems for arbitrary complex systems of differential equations with complex boundary conditions is developed. Moreover the boundary conditions are allowed to depend linearly on the eigenvalue-parameter.     

Rankin-Cohen brackets and formal quantization

In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results [A. Connes, H. Moscovici, Rankin-Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J. 4 (1) (2004) 111-130, 311]. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation [D. Zagier, Modular forms and differential operators, in: K.G. Ramanathan Memorial Issue, Proc. Indian Acad. Sci. Math. Sci. 104 (1) (1994) 57-75] of modular forms, and relate this deformation to the Weyl-Moyal product. We also show that the projective structure introduced by Connes and Moscovici is equivalent to the existence of certain geometric data in the case of foliation groupoids. Using the methods developed by the second author [X. Tang, Deformation quantization of pseudo (symplectic) Poisson groupoids, Geom. Funct. Anal. 16 (3) (2006) 731-766], we reconstruct a universal deformation formula of the Hopf algebra H₁ associated to codimension one foliations. In the end, we prove that the first Rankin-Cohen bracket RC₁ defines a noncommutative Poisson structure for an arbitrary H₁ action.     

Mehler's formula and functional calculus Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

We show that Mehler's formula can be used to handle several formulas involving the quantization of singular Hamiltonians. In particular, we diagonalize in the Hermite basis the Weyl quantization of the characteristic function of several domains of the phase space.     

An additive formula for Samuel multiplicities on Hilbert spaces of analytic functions

We establish a short exact sequence to relate the germ model of invariant subspaces of a Hilbert space of vector-valued analytic functions and the sheaf model of the corresponding coinvariant subspaces. As a consequence we obtain an additive formula for Samuel multiplicities. As an application, we give a different proof for a formula relating the fibre dimension and the Samuel multiplicity which is first proved in Fang (2005) [11]. The feature of the new proof is that the analytic arguments in Fang (2005) [11] are now subsumed by algebraic machinery.     

Regularization for Higher Order Singular Integral Equations

By means of the definition of Hadamard principal value permutation formula and composition formula for higher order singular integrals developing in [5], the regularization problems for higher order singular integral equations with variable coefficients are discussed, and the higher order singular integral equations with constant coefficient are solved. It is the first time to treat the high order singular integral equations of arbitrary degree in the theory of singular integral equations.      

On the colored Tutte polynomial of a graph of bounded treewidth

We observe that a formula given by Negami [Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987) 601-622] for the Tutte polynomial of a k-sum of two graphs generalizes to a colored Tutte polynomial. Consequently, an algorithm of Andrzejak [An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998) 39-54] may be directly adapted to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has also been proven by Makowsky [Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width, Discrete Appl. Math. 145 (2005) 276-290], using a different algorithm based on logical techniques.