12 research outputs found
-Torsion Points In Finite Abelian Groups And Combinatorial Identities
The main aim of this article is to compute all the moments of the number of
-torsion elements in some type of nite abelian groups. The averages
involved in these moments are those de ned for the Cohen-Lenstra heuristics for
class groups and their adaptation for Tate-Shafarevich groups. In particular,
we prove that the heuristic model for Tate-Shafarevich groups is compatible
with the recent conjecture of Poonen and Rains about the moments of the orders
of -Selmer groups of elliptic curves. For our purpose, we are led to de ne
certain polynomials indexed by integer partitions and to study them in a
combinatorial way. Moreover, from our probabilistic model, we derive
combinatorial identities, some of which appearing to be new, the others being
related to the theory of symmetric functions. In some sense, our method
therefore gives for these identities a somehow natural algebraic context.Comment: 24 page
Multidimensional Matrix Inversions and Elliptic Hypergeometric Series on Root Systems
Multidimensional matrix inversions provide a powerful tool for studying
multiple hypergeometric series. In order to extend this technique to elliptic
hypergeometric series, we present three new multidimensional matrix inversions.
As applications, we obtain a new elliptic Jackson summation, as well as
several quadratic, cubic and quartic summation formulas
Inversion of the Pieri formula for Macdonald polynomials
We give the explicit analytic development of Macdonald polynomials in terms
of "modified complete" and elementary symmetric functions. These expansions are
obtained by inverting the Pieri formula. Specialization yields similar
developments for monomial, Jack and Hall-Littlewood symmetric functions.Comment: 34 page
On Warnaar's elliptic matrix inversion and Karlsson-Minton-type elliptic hypergeometric series
Using Krattenthaler's operator method, we give a new proof of Warnaar's
recent elliptic extension of Krattenthaler's matrix inversion. Further, using a
theta function identity closely related to Warnaar's inversion, we derive
summation and transformation formulas for elliptic hypergeometric series of
Karlsson-Minton-type. A special case yields a particular summation that was
used by Warnaar to derive quadratic, cubic and quartic transformations for
elliptic hypergeometric series. Starting from another theta function identity,
we derive yet different summation and transformation formulas for elliptic
hypergeometric series of Karlsson-Minton-type. These latter identities seem
quite unusual and appear to be new already in the trigonometric (i.e., p=0)
case.Comment: 16 page
A new A_n extension of Ramanujan's 1-psi-1 summation with applications to multilateral A_n series
In this article, we derive some identities for multilateral basic
hypergeometric series associated to the root system A_n. First, we apply
Ismail's argument to an A_n q-binomial theorem of Milne and derive a new A_n
generalization of Ramanujan's 1-psi-1 summation theorem. From this new A_n
1-psi-1 summation and from an A_n 1-psi-1 summation of Gustafson we deduce two
lemmas for deriving simple A_n generalizations of bilateral basic
hypergeometric series identities. These lemmas are closely related to the
Macdonald identities for A_n. As samples for possible applications of these
lemmas, we provide several A_n extensions of Bailey's 2-psi-2 transformations,
and several A_n extensions of a particular 2-psi-2 summation.Comment: LaTeX2e, 26 pages, submitted to Rocky Mount. J. Math., spec. vol.,
conference proceedings of SF2000, Tempe, Arizona, May 29 - June 9, 200
Advanced Determinant Calculus: A Complement
This is a complement to my previous article "Advanced Determinant Calculus"
(S\'eminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the
present article, I share with the reader my experience of applying the methods
described in the previous article in order to solve a particular problem from
number theory (G. Almkvist, J. Petersson and the author, Experiment. Math. 12
(2003), 441-456). Moreover, I add a list of determinant evaluations which I
consider as interesting, which have been found since the appearance of the
previous article, or which I failed to mention there, including several
conjectures and open problems.Comment: AmS-LaTeX, 85 pages; Final, largely revised versio
Macdonald Polynomials and Multivariable Basic Hypergeometric Series
We study Macdonald polynomials from a basic hypergeometric series point of view. In particular, we show that the Pieri formula for Macdonald polynomials and its recently discovered inverse, a recursion formula for Macdonald polynomials, both represent multivariable extensions of the terminating very-well-poised 6φ5 summation formula. We derive several new related identities including multivariate extensions of Jackson's very-well-poised 8φ7 summation. Motivated by our basic hypergeometric analysis, we propose an extension of Macdonald polynomials to Macdonald symmetric functions indexed by partitions with complex parts. These appear to possess nice properties
System identification and model reduction using modulating function techniques
Weighted least squares (WLS) and adaptive weighted least squares (AWLS) algorithms are initiated for continuous-time system identification using Fourier type modulating function techniques. Two stochastic signal models are examined using the mean square properties of the stochastic calculus: an equation error signal model with white noise residuals, and a more realistic white measurement noise signal model. The covariance matrices in each model are shown to be banded and sparse, and a joint likelihood cost function is developed which links the real and imaginary parts of the modulated quantities. The superior performance of above algorithms is demonstrated by comparing them with the LS/MFT and popular predicting error method (PEM) through 200 Monte Carlo simulations. A model reduction problem is formulated with the AWLS/MFT algorithm, and comparisons are made via six examples with a variety of model reduction techniques, including the well-known balanced realization method. Here the AWLS/MFT algorithm manifests higher accuracy in almost all cases, and exhibits its unique flexibility and versatility. Armed with this model reduction, the AWLS/MFT algorithm is extended into MIMO transfer function system identification problems. The impact due to the discrepancy in bandwidths and gains among subsystem is explored through five examples. Finally, as a comprehensive application, the stability derivatives of the longitudinal and lateral dynamics of an F-18 aircraft are identified using physical flight data provided by NASA. A pole-constrained SIMO and MIMO AWLS/MFT algorithm is devised and analyzed. Monte Carlo simulations illustrate its high-noise rejecting properties. Utilizing the flight data, comparisons among different MFT algorithms are tabulated and the AWLS is found to be strongly favored in almost all facets