Cohomology of the Vector Fields Lie Algebra and Modules of Differential Operators on a Smooth Manifold

Let M be a smooth manifold, the space of polynomial on fibers functions on T*M (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, Vect(M), of vector fields on M with coefficients in the space of linear differential operators on . This cohomology space is closely related to the Vect(M)-modules, (M), of linear differential operators on the space of tensor densities on M of degree .     

Commuting Differential and Difference Operators Associated to Complex Curves I

The purpose of this paper is to define functional realizations of the Khizhnik–Zamolochikov–Bernard (KZB) connection on the bundle of conformal blocks over the moduli space of curves.     

Semisimple Lie Algebras of Differential Operators

Starting from the commutation relations in a complex semisimple Lie algebra , one may obtain a space of vector fields on Euclidean space such that and are isomorphic when is equipped with the usual Lie bracket between vector fields and the isotropy subalgebra of is a Borel subalgebra . Furthermore, one may adjoin to the vector fields in multiplication operators to obtain an -parameter family of distinct presentations of as spaces of differential operators, where is the dual of a Cartan subalgebra. Some of these presentations will preserve a space of polynomials on Euclidean space, and, in fact, all the finite-dimensional representations of can be presented in this way. All of this is carried out explicitly for arbitrary . In doing so, one discovers there is a Lie group of diffeomorphisms of the unipotent subgroup N complementary to B which acts on these presentations and preserves a certain notion of weight.     

Abstract Differential Geometry, Differential Algebras of Generalized Functions, and de Rham Cohomology

Abstract differential geometry is a recent extension of classical differential geometry on smooth manifolds which, however, does no longer use any notion of Calculus. Instead of smooth functions, one starts with a sheaf of algebras, i.e., the structure sheaf, considered on an arbitrary topological space, which is the base space of all the sheaves subsequently involved. Further, one deals with a sequence of sheaves of modules, interrelated with appropriate differentials, i.e., suitable Leibniz sheaf morphisms, which will constitute the differential complex. This abstract approach captures much of the essence of classical differential geometry, since it places a powerful apparatus at our disposal which can reproduce and, therefore, extend fundamental classical results. The aim of this paper is to give an indication of the extent to which this apparatus can go beyond the classical framework by including the largest class of singularities dealt with so far. Thus, it is shown that, instead of the classical structure sheaf of algebras of smooth functions, one can start with a significantly larger, and nonsmooth, sheaf of so-called nowhere dense differential algebras of generalized functions. These latter algebras, which contain the Schwartz distributions, also provide global solutions for arbitrary analytic nonlinear PDEs. Moreover, unlike the distributions, and as a matter of physical interest, these algebras can deal with the vastly larger class of singularities which are concentrated on arbitrary closed, nowhere dense subsets and, hence, can have an arbitrary large positive Lebesgue measure. Within the abstract differential geometric context, it is shown that, starting with these nowhere dense differential algebras as a structure sheaf, one can recapture the exactness of the corresponding de Rham complex, and also obtain the short exponential sequence. These results are the two fundamental ingredients in developing differential geometry along classical, as well as abstract lines. Although the commutative framework is used here, one can easily deal with a class of singularities which is far larger than any other one dealt with so far, including in noncommutative theories.     

Invariant differential operators and the range of the matrix Radon transform

In this paper we characterize the range of the matrix Radon transform by invariant differential operators. This generalizes analogous results for the d-plane transform in Rn.     

一类二阶二次变系数微分算子的不变子空间及其应用

研究了一类二阶二次变系数微分算子的不变子空间,讨论了这类微分算子不变子空间的应用,并给出了具体应用的一些例子.在这些例子中,构造了大量变系数非线性演化方程的精确解.     

辛矩阵和四阶微分算子自共轭边界条件的基本型

给出了辛矩阵的定义,讨论了它的性质,并通过使用辛矩阵的方法研究四阶自共轭的边界条件,得到了四阶自共轭边界条件的基本型,从而使得其它各种自共轭的边界条件都可以通过基本型的辛变换得到.     

一类二阶与四阶微分算子积的自伴性

利用矩阵运算及算子的基本理论,讨论了由微分算式L_1=D~((2))+q_1(t)和L_2=D~((4))+q_2(t)其中(D=d/dx,t∈I=[a,b])生成的两个微分算子L_i(i=1,2)积L_1L_2的自伴性问题,并在常型情形下,获得了积算子自伴的充分必要条件.     

Generators of Rings of Differential Operators on a Class of k—algebras

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On the Essential Spectrum of a Class of Matrix Differential Operators

We study the essential spectrum of a class of nonelliptic matrix partial differential operators related to a linear magnetohydrodynamic model.     

A priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.     

Hilbert空间L2[a,b]上m个微分算子积的自伴性

考虑 [a ,b]( -∞   

Hilbert空间L^2[α，6]上m个微分算子积的自伴性

考虑[a,b](-∞＜a＜b＜∞)上n阶复值系数正则对称微分算式ly＝∑n j＝0 aj(t)y(j).本文首先给出由lmy(m∈N且m≥2)生成的微分算子T(lm)自伴边条件一种新的描述,其次研究了由微分算式ly生成的m个微分算子Tk(l)(k=1,…,m)的积Tm(l)…T2(l)T1(l)的自伴性并获得其自伴的充分必要条件.     

一类再生Hilbert空间上偏微分算子的有界性(英文)

研究了一类再生Hilbert空间上偏微分算子的有界性,得到了偏微分算子有界的一个充分必要条件,推广了文献[1]中的结果.     

微分算子的对称扩张及Friedrichs扩张的辛几何刻画

本文在加权Hilbert空间L2(I,r(x))(I=(a,6),-∞≤a 0)中,利用辛几何,刻画了n阶对称微分算式的最小算子的对称扩张(含自伴扩张)及 Friedrichs扩张,分别获得了其扩张为对称扩张、Friedrichs扩张的充分必要条件．     

Hypercyclic Sequences of Differential and Antidifferential Operators

In this paper, we provide some extensions of earlier results about hypercyclicity of some operators on the Fréchet space of entire functions of several complex variables. Specifically, we generalize in several directions a theorem about hyper- cyclicity of certain infinite order linear differential operators with constant coefficients and study the corresponding property for certain kinds of “antidifferential” operators which are introduced in the paper. In addition, the existence of hypercyclic functions for certain sequences of differential operators with additional properties, for instance, boundedness or with some nonvanishing derivatives, is established.     

On Computing Gröbner Bases in Rings of Differential Operators with Coefficients in a Ring

Following the definition of Gr?bner bases in rings of differential operators given by Insa and Pauer (1998), we discuss some computational properties of Gr?bner bases arising when the coefficient set is a ring. First we give examples to show that the generalization of S-polynomials is necessary for computation of Gr?bner bases. Then we prove that under certain conditions the G-S-polynomials can be reduced to be simpler than the original one. Especially for some simple case it is enough to consider S-polynomials in the computation of Gr?bner bases. The algorithm for computation of Gr?bner bases can thus be simplified. Last we discuss the elimination property of Gr?bner bases in rings of differential operators and give some examples of solving PDE by elimination using Gr?bner bases. This work was supported by the NSFC project 60473019.     

2n阶微分算子乘积自伴的充分必要条件

给定Hilbert空间L2[a,∞)上两个由2n阶对称微分算式生成的微分算子Li(i=1,2),该文给出了乘积算子L2L1是自伴算子的一个充分必要条件．