this paper we want to study the local cohomology modules of R

Abstract. Basic properties of symplectic reflection algebras over an algebraically closed field k of positive characteristic are laid out. These algebras are always finite modules over their centres, in contrast to the situation in characteristic 0. For the subclass of rational Cherednik algebras, we determine the PI-degree and the Goldie rank, and show that the Azumaya and smooth loci of the centre coincide. 1.

this article, we study the following operation.

Partially supported by CRDF under Grant UM1-2567-OD-03 We study modules over the Weyl-Carlitz ring, a counterpart of the Weyl algebra in analysis over local fields of positive characteristic. It is shown that some basic objects of function field arithmetic, like the Carlitz module, Thakur’s hypergeometric polynomials, and analogs of binomial coefficients arising in the function field version of umbral calculus, generate holonomic modules. Key words: Fq-linear function; holonomic module; holonomic function; Carlitz derivative 2 1

We prove, by simple manipulation of commutators, two noncommutative generalizations of the Cauchy–Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant theory and of Turnbull’s Capelli-type identities for symmetric and antisymmetric matrices.

Doctor of Philosophy in Mathematics One of the major goals in the field of symbolic computation of differential equations is to develop algorithms for exact or closed-form solutions. This thesis studies symbolic computation of maximally overdetermined systems of linear partial differential equations by using constructions in the corresponding ring of differential operators vith polynomial coefficients, vhich is called the Weyl algebra D. We develop algorithms to find polynomial solutions, rational function solutions, and more generally holonomic solutions. By holonomic solutions, we mean the following: sometimes the best way to specify a function F is as the solution of a system of differential equations - this is for instance how many special functions are classically described. Our algorithm takes as input the differential equations describing F as vell as the system S that we wish to solve, and returns as output any solutions to S existing within the D-module generated by F. We also study aspects of the opposite problem, namely given a function F, hov can differential equations describing F be produced? We introduce the Weyl closure of an ideal I of the Weyl algebra, vhich is the set of all differential operators annihilating the common holomorphic solutions of I at a generic point. We give an algorithm to compute Weyl closure, vhich has applications to symbolic integration, and vhich ve also use to make a detailed study of ideals in the first Weyl algebra.

Partially supported by CRDF under Grant UM1-2567-OD-03, and by the Ukrainian Foundation for Fundamental

The paper is a survey of recent results in analysis of additive functions over function fields motivated by applications to various classes of special functions including Thakur’s hypergeometric function. We consider basic notions and results of calculus, analytic theory of differential equations with Carlitz derivatives (including a counterpart of regular singularity), umbral calculus, holonomic modules over the Weyl-Carlitz ring.

Abstract. Let α1, α2...,αm be linear forms defined on Cn and X = Cn \ ∪m i=1V (αi), where V (αi) = {p ∈ Cn: αi(p) = 0}. The coordinate ring OX of X is a holonomic An-module, where An is the n-th Weyl algebra and since holonomic An-modules have finite length, OX has finite length. We consider a ”twisted ” variant of this An-module which is also holonomic. Define M β α to be the free rank 1 C[x]α-module on the generator αβ (thought as a multivalued function), where αβ = α β1 1...αβm m and the multi-index β = (β1,...,βm) ∈ Cm. Our main result is the computation of the number of decomposition factors of M β α and their description when n = 2. 1.

Generalized Laplace transformations and integration of hyperbolic systems of linear partial differential equations ∗