Iterative character constructions for algebra groups

We construct a family of orthogonal characters of an algebra group which decompose the supercharacters defined by Diaconis and Isaacs (2008) [6]. Like supercharacters, these characters are given by nonnegative integer linear combinations of Kirillov functions and are induced from linear supercharacters of certain algebra subgroups. We derive a formula for these characters and give a condition for their irreducibility; generalizing a theorem of Otto (2010) [20], we also show that each such character has the same number of Kirillov functions and irreducible characters as constituents. In proving these results, we observe as an application how a recent computation by Evseev (2010) [7] implies that every irreducible character of the unitriangular group UTn(q) of unipotent n×n upper triangular matrices over a finite field with q elements is a Kirillov function if and only if n?12. As a further application, we discuss some more general conditions showing that Kirillov functions are characters, and describe some results related to counting the irreducible constituents of supercharacters.     

Noncommutative Symmetric Functions Iv: Quantum Linear Groups and Hecke Algebras at q = 0

We present representation theoretical interpretations ofquasi-symmetric functions and noncommutative symmetric functions in terms ofquantum linear groups and Hecke algebras at q = 0. We obtain inthis way a noncommutative realization of quasi-symmetric functions analogousto the plactic symmetric functions of Lascoux and Schützenberger. Thegeneric case leads to a notion of quantum Schur function.     

双曲群,自动机群与群的Dehn函数

本文第一部分对组台群论的发展史作了一个简要的回顾．然后重点介绍近20年来组合群论领域中的三个热点课题：即双曲群，自动机群和群的Dehn函数的有关概念，并对相关的重要研究成果给予概述，其中包括作者本人在群的二阶Dehn函数的研究工作中的若干成果．最后提出9个公开问题。     

The growth of H-harmonic functions on the Heisenberg group

We discuss the relationship between the frequency and the growth of H-harmonic functions on the Heisenberg group.Precisely,we prove that an H-harmonic function must be a polynomial if its frequency is globally bounded.Moreover,we show that a class of H-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.     

The Hopf graph algebra and renormalization group equations

We study the renormalization group equations implied by the Hopf graph algebra. The vertex functions are regarded as vectors in the dual space of the Hopf algebra. The renormalization group equations for these vertex functions are equivalent to those for individual Feynman integrals. The solution of the renormalization group equations can be represented in the form of an exponential of the beta function. We clearly show that the exponential of the one-loop beta function allows finding the coefficients of the leading logarithms for individual Feynman integrals. The calculation results agree with those obtained in the parquet approximation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 22–32, April, 2005.     

Reidemeister zeta functions of low-dimensional almost-crystallographic groups are rational

We prove that the Reidemeister zeta functions of automorphisms of crystallographic groups with diagonal holonomy ?₂ are rational. As a result, we obtain that Reidemeister zeta functions of automorphisms of almost-crystallographic groups up to dimension 3 are rational.     

On the group of polynomial functions in a group

Let G be a group and let n be a positive integer. A polynomial function in G is a function from G ⁿ to G of the form , where f(x ₁, . . . , x _n ) is an element of the free product of G and the free group of rank n freely generated by x ₁, . . . , x _n . There is a natural definition for the product of two polynomial functions; equipped with this operation, the set of polynomial functions is a group. We prove that this group is polycyclic if and only if G is finitely generated, soluble, and nilpotent-by-finite. In particular, if the group of polynomial functions is polycyclic, then necessarily it is nilpotent-by-finite. Furthermore, we prove that G itself is polycyclic if and only if the subgroup of polynomial functions which send (1, . . . , 1) to 1 is finitely generated and soluble.      

Landau's necessary density conditions for LCA groups

We derive necessary conditions for sampling and interpolation of bandlimited functions on a locally compact abelian group in line with the classical results of H. Landau for bandlimited functions on Rd. Our conditions are phrased as comparison principles involving a certain canonical lattice.     

On periodic matrix-valued Weyl-Titchmarsh functions

We consider a certain class of Herglotz-Nevanlinna matrix-valued functions which can be realized as the Weyl-Titchmarsh matrix-valued function of some symmetric operator and its self-adjoint extension. New properties of Weyl-Titchmarsh matrix-valued functions as well as a new version of the functional model for such realizations are presented. In the case of periodic Herglotz-Nevanlinna matrix-valued functions, we provide a complete characterization of their realizations in terms of the corresponding functional model. We also obtain properties of a symmetric operator and its self-adjoint extension which generate a periodic Weyl-Titchmarsh matrix-valued function. We study pairs of operators (a symmetric operator and its self-adjoint extension) with constant Weyl-Titchmarsh matrix-valued functions and establish connections between such pairs of operators and representations of the canonical commutation relations for unitary groups of operators in Weyl's form. As a consequence of such an approach, we obtain the Stone-von Neumann theorem for two unitary groups of operators satisfying the commutation relations as well as some extension and refinement of the classical functional model for generators of those groups. Our examples include multiplication operators in weighted spaces, first and second order differential operators, as well as the Schrödinger operator with linear potential and its perturbation by bounded periodic potential.     

Positive definite temperature functions on the Heisenberg group

We characterize positive definite temperature functions, i.e., positive definite solutions of the heat equation, on the Heisenberg group in terms of the initial values. We also obtain an integral representation for positive definite and U(n)-invariant temperature functions with polynomial growth, where U(n) is the group of all n× n unitary matrices.     

Conjugacy Length in Group Extensions

Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study the behavior of conjugacy length functions under group extensions, introducing the twisted and restricted conjugacy length functions. We apply these results to show that certain abelian-by-cyclic groups have linear conjugacy length function and certain semidirect products ?^d ? ?^k have at most exponential (if k > 1) or linear (if k = 1) conjugacy length functions.     

Difference Equations for Functions Analytic Beyond Several Squares

We consider some set of squares constructed for the primitive periods 1 and i and sufficiently distant from each other. In a neighborhood of this set we study a four-element difference equation with constant coefficients whose linear stifts are generators of the corresponding doubly periodic group and their inverses. A solution is sought in the class of functions analytic beyond this set and vanishing at infinity. We show that the solvability of the problem depends essentially not only on the choice of the coefficients but also on the disposition of the squares.     

Smoothness and analyticity of perturbation expansions in qed

We consider the ground state of an atom in the framework of non-relativistic qed. We assume that the ultraviolet cutoff is of the order of the Rydberg energy and that the atomic Hamiltonian has a non-degenerate ground state. We show that the ground state energy and the ground state are k-times continuously differentiable functions of the fine structure constant and respectively the square root of the fine structure constant on some nonempty interval [0,ck).     

On the solutions of Knizhnik-Zamolodchikov system

We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions. We prove that under some conditions the solution of the KZ system is rational too. We give the method of constructing the corresponding rational solution. We deduce the asymptotic formulas for the solution of the KZ system when the parameter ρ is an integer.      

基于李群方法的贝塞尔函数数学实现

基于李群的表示理论,首先讨论了欧拉群的表示及其性质;然后,从该群的表示理论出发,分别导出了第一类贝塞尔函数的积分形式和幂级数形式.该研究表明了群方法可以求解对称边界问题的解析波函数,并为用群方法求解电磁场问题创造了条件.     

Quadratic transform of complete ideals in regular local rings

In an upcoming article we study harmonic analysis on the quantum E(2) group within an algebraic framework: we explicitly construct Fourier transforms between quantum E(2) and its Pontryagin dual, involving q-Bessel functions as kernel, prove Plancherel &; inversion formulas etc. In the present paper we propose an algebraic setting in which to perform harmonic analysis on non-compact, non-discrete quantum groups and in particular on quantum E(2). We are mainly concerned with the construction of positive and faithful invariant functionals on an algebraic level, KMS properties, etc.     

Symmetries and solutions to the thin film equations

We study symmetries and solutions of the generalized fourth-order nonlinear partial differential equations which arise from studies of thin liquid films. It is shown that the equations admit extended scaling and rotation symmetries and a class of generalized conditional symmetries for certain coefficient functions. Exact solutions associated to the symmetries are obtained.     

Generating functions for generalized Hermite polynomials associated with parabolic cylinder functions

ABSTRACT

In this article, we first give some basic properties of generalized Hermite polynomials associated with parabolic cylinder functions. We next use Weisner? group theoretic method and operational rules method to establish new generating functions for these generalized Hermite polynomials. The operational methods we use allow us to obtain unilateral, bilinear and bilateral generating functions by using the same procedure. Applications of generating functions obtained by Weisner? group theoretic method are discussed.     

Real Liouville Extensions

We characterize linear differential equations defined over a real differential field with a real closed field of constants C, which are solvable by real Liouville functions, as those having a differential Galois group whose identity component is solvable and C-split.     

Anomalous scaling in the model of turbulent advection of a vector field

We consider the model of turbulent advection of a passive vector field ϕ by a two-dimensional random velocity field uncorrelated in time and having Gaussian statistics with a powerlike correlator. The renormalization group and operator product expansion methods show that the asymptotic form of the structure functions of the ϕ field in the inertial range is determined by the fluctuations of the energy dissipation rate. The dependence of the asymptotic form on the external turbulence scale is essential and has a powerlike form (anomalous scaling). The corresponding exponents are determined by the spectrum of the anomalous dimension matrices of operator families consisting of gradients of ϕ. We find a basis constructed from powers of the dissipation and enstrophy operators in which these matrices have a triangular form in all orders of the perturbation theory. In the two-loop approximation, we evaluate the anomalous-scaling exponents for the structure functions of an arbitrary order. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 467–487, March, 2006.