Products of Quasi-Involutions in Unitary Groups

Given a regular ^–-hermitian form on an n-dimensional vector space V over a commutative field K of characteristic 2 ( ). Call an element of the unitary group a quasi-involution if is a product of commuting quasi-symmetries (a quasi-symmetry is a unitary transformation with a regular (n–1)-dimensional fixed space). In the special case of an orthogonal group every quasi-involution is an involution. Result: every unitary element is a product of five quasi-involutions. If K is algebraically closed then three quasi-involutions suffice.     

Designs in Grassmannian Spaces and Lattices

We introduce the notion of a t-design on the Grassmann manifold of the m-subspaces of the Euclidean space ⁿ . It generalizes the notion of antipodal spherical design which was introduced by P. Delsarte, J.-M. Goethals and J.-J. Seidel. We characterize the finite subgroups of the orthogonal group which have the property that all its orbits are t-designs. Generalizing a result due to B. Venkov, we prove that, if the minimal m-sections of a lattice L form a 4-design, then L is a local maximum for the Rankin function _n,m .     

p-Adic Automorphic Functions for the Unitary Group in Three Variables

Let G be a special unitary group in three variables over a local field and, a co-compact subgroup of G. We construct, -automorphic functions on the rigid analytic space of points in the projective plane stable under the action of tori in G.This research was partially supported by an ARC grant.2000 Mathematics Subject Classification: 11F55, 11F85, 20G25, 22E35     

Log-Polynomial Period Functions for Hecke Groups

In this article we shall determine the automorphic integrals of positive and negative integral weight associated with the full modular group and some Hecke groups. This will be done by using the Hecke Correspondence. We will also give a characterization of multiplier systems of real weight for Hecke groups.     

Globally Irreducible Representations of Finite Groups and Integral Lattices

The notion of globally irreducible representations of finite groups has been introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell--Weil lattices of elliptic curves. In this paper we first give a necessary condition for global irreducibility. Then we classify all globally irreducible representations of L ₂(q) and ²B₂(q), and of the majority of the 26 sporadic finite simple groups. We also exhibit one more globally irreducible representation, which is related to the Weil representation of degree (pⁿ-1)/2 of the symplectic group Sp_2n(p) (p 1 (mod 4) is a prime). As a consequence, we get a new series of even unimodular lattices of rank 2(pⁿ–1). A summary of currently known globally irreducible representations is given.     

On Subgroups and Quotient Groups of L-fuzzy Topological Groups

In paper [1], it gave the concept and basic properties of L-fuzzy topological groups. This paper will enter the concept of quotient groups of L-fuzzy topological groups and will prove some properties of subgroups and quotient groups. The persent work is the continuation of [1] and hence all the conventions in [1] still hold good in the present paper.     

Poisson Measures for Topological Groups and Their Representations

A class of unitary strongly continuous representations of infinite-dimensional groups such as geometric loop and diffeomorphism groups of real, complex and non-Archimedean manifolds is investigated. This class is constructed by producing Poisson measures P_m on configuration spaces of infinite-dimensional topological groups with the help of quasi-invariant measures m. Their irreducibility, equivalence and inequivalence is investigated.     

Deformation of Irreducible Unitary Representations of Discrete Series of Hermitian Symmetric Simple Lie Groups in the Class of Pure Pseudorepresentations

Mathematical Notes -     

Rate of convergence to the semi-circle law for the Deformed Gaussian Unitary Ensemble

It is shown that the Kolmogorov distance between the expected spectral distribution function of an n × n matrix from the Deformed Gaussian Ensemble and the distribution function of the semi-circle law is of order O(n ^−2/3+v ). Research supported by the DFG-Forschergruppe FOR 399/1. Partially supported by INTAS grant N 03-51-5018, by RFBF grant N 02-01-00233, by RFBR-DFG grant N 04-01-04000, by RF grant of the leading scientific schools NSh-4222.2006.1.     

Equivalence Groups for First-Order Balance Equations and Applications to Electromagnetism

We investigate the groups of equivalence transformations for first-order balance equations involving an arbitrary number of dependent and independent variables. We obtain the determining equations and find their explicit solutions. The approach to this problem is based on a geometric method that depends on Cartan's exterior differential forms. The general solutions of the determining equations for equivalence transformations for first-order systems are applied to a class of the Maxwell equations of electrodynamics.     

A Measure of Simultaneous Approximation for Quasi-Modular Functions

We improve on a recent result of Saradha giving a transcendence measure for the quotient of a period of an elliptic curve defined over by its associated quasi-period. In an (almost successful) attempt to include in a single measure both this result and that obtained by Reyssat in 1980, we blend into the modular method ideas related to modular but also hypergeometric functions, as appearing e.g. in André's work, as well as some Galois considerations.     

Root Systems and Length Functions

We introduce root systems for those imprimitive complex reflection groups which are generated by involutory reflections, and study the associated length functions. These have many properties in common with the usual length functions for finite Weyl groups.     

几种辅助函数方法及应用

本通过举例说明的方法说明了寻求辅助函数的几种方法及其在解题中的作用。     

Two-periodic waves and asymptotic property for generalized 2D Toda lattice equation

In this paper, two-periodic wave solutions are constructed for the (2 + 1)-dimensional generalized Toda lattice equation by using Hirota bilinear method and Riemann theta function. At the same time, we analyze in details asymptotic properties of the two-periodic wave solutions and give their asymptotic relations between the periodic wave solutions and the soliton solutions.     

双变量部分theta函数的乘积公式与性质

本文研究一类新的双变量部分theta函数,它是经典部分theta函数的推广,主要围绕这类函数的乘积公式、递推关系、级数展开等性质展开讨论.作为主要结果,我们建立了任意两个双变量部分theta函数的乘积公式,推广了Andrews-Warnaar经典部分theta函数的乘积公式,发现了双变量部分theta函数所满足的二阶递...     

F值函数的微分及其在FNN中的应用

本文引进了集值函数的ｓ－可微和模糊值（Ｆ值）函数的Ｆｓ－可微概念。给出了这两种可微性的几个判别条件。最后研究并得到了一类模糊神经网络（ＦＮＮ）的Ｆｓ－可微性和连续性。     

Extreme Bloch Functions and Summation Methods for Bloch Functions

In this paper we construct Bloch functions F for which the set {z | e sup _{|ζ| < 1} |F'(ζ)| ( 1 - |ζ| ² ) = |F'(z)| ( 1 - |z| ² )} is an analytic Jordan curve tangential to the unit disk in some points. It is proved that, using such functions, we can derive analogs to the Taylor expansion for Bloch functions in cases where the Taylor expansion does not converge. October 15, 1997. Date revised: March 12, 1998. Date accepted: June 18, 1998.     

New Modular Relations for the Göllnitz-Gordon Functions

We attempt to obtain new modular relations for the Göllnitz-Gordon functions by techniques which have been used by L. J. Rogers, G. N. Watson, and D. Bressoud to prove some of Ramanujan's 40 identities. Also, we give new proofs for some modular relations for the Göllnitz-Gordon functions which have been previously established by using results from L. Rogers and D. Bressoud. Finally, we give applications of those new modular relations to the theory of partitions.