660 research outputs found
Applicability of the -Analogue of Zeilberger's Algorithm
The applicability or terminating condition for the ordinary case of
Zeilberger's algorithm was recently obtained by Abramov. For the -analogue,
the question of whether a bivariate -hypergeometric term has a -pair
remains open. Le has found a solution to this problem when the given bivariate
-hypergeometric term is a rational function in certain powers of . We
solve the problem for the general case by giving a characterization of
bivariate -hypergeometric terms for which the -analogue of Zeilberger's
algorithm terminates. Moreover, we give an algorithm to determine whether a
bivariate -hypergeometric term has a -pair.Comment: 15 page
On the Lang-Trotter and Sato-Tate Conjectures on Average for Polynomial Families of Elliptic Curves
We show that the reductions modulo primes of the elliptic curve behave as predicted by the Lang-Trotter and
Sato-Tate conjectures, on average over integers and for and reasonably small compared to , provided that are not powers of another polynomial over \Q. For first results of this kind are due to E. Fouvry and M. R. Murty and
have been further extended by other authors. Our technique is different from
that of E. Fouvry and M. R. Murty which does not seem to work in the case of
general polynomials and
Exponential functionals of Levy processes
This text surveys properties and applications of the exponential functional
of real-valued L\'evy processes .Comment: Published at http://dx.doi.org/10.1214/154957805100000122 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Three lectures on Algebraic Microlocal Analysis
These three lectures present some fundamental and classical aspects of
microlocal analysis. Starting with the Sato's microlocalization functor and the
microsupport of sheaves, we then construct a microlocal analogue of the
Hochschild homology for sheaves and apply it to recover index theorems for
D-modules and elliptic pairs. In the third lecture, we construct the
ind-sheaves of temperate and Whitney holomorphic functions and give some
applications to the study of irregular holonomic D-modules.Comment: small correction
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