representation zeta functions of compact p adic analytic groups and arithmetic groups we introduce new methods from p adic integration into the study of representation zeta functions associated to compact p adic analytic groups and arithmetic groups they allow us to establish that the representation zeta functions of generic members of families of p adic analytic pro p groups obtained from a global perfect lie lattice satisfy functional equations in the case of semisimple compact p adic analytic groups we exhibit a link between the relevant p adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple lie algebra defined by centraliser dimension based on this algebro geometric description we compute explicit formulae for the representation zeta functions of principal congruence subgroups of the groups sl 3 o where o is a compact discrete valuation ring of characteristic 0 and of the corresponding unitary groups these formulae combined with approximative clifford theory allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type a 2 assuming a conjecture of serre on the congruence subgroup problem we thereby prove a conjecture of larsen and lubotzky on lattices in higher rank semisimple groups for algebraic groups of type a 2 defined over number fields

analytic properties of zeta functions and subgroup growth it has become somewhat of a cottage industry over the last fifteen years to understand the rate of growth of the number of subgroups of finite index in a group g although the story began much before the recent activity grew out of a paper by dan segal in 36 the story so far has been well documented in lubotzky s subsequent survey paper in 30 in 24 the second author of this article segal and smith introduced the zeta function of a group as a tool for understanding this growth of subgroups let an g be the number of subgroups of index n in the finitely generated group g and sn g a1 g an g be the number of subgroups of index n or less the zeta function is defined as the dirichlet series with coefficients an g and has a natural interpretation as a noncommutative generalization of the dedekind zeta function of a number field

layer adapted meshes for convection diffusion problems abstract in the present paper a classification and a survey are given of layer adapted meshes for stationary convection dominated convection diffusion problems for a number of standard numerical schemes theoretical results are presented that demonstrate that the use of properly layer adapted meshes yields robust methods i e methods that perform equally well no matter how dominant the convection we review a number of techniques used in the convergence analysis of these methods

on quantum gauge theories in two dimensions two dimensional quantum yang mills theory is studied from three points of view i by standard physical methods ii by relating it to the largek limit of three dimensional chern simons theory and two dimensional conformal field theory iii by relating its weak coupling limit to the theory of reidemeister ray singer torsion the results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces

on the characteristic map of finite unitary groups abstract in his classic book on symmetric functions macdonald describes a remarkable result by green relating the character theory of the finite general linear group to transition matrices between bases of symmetric functions this connection allows one to analyze the character theory of the general linear group via symmetric group combinatorics using works of ennola kawanaka lusztig and srinivasan this paper describes the analogous setting for the finite unitary group in particular we explain the connection between deligne lusztig theory and ennola s efforts to generalize green s work and from this we deduce various character theoretic results applications include calculating certain sums of character degrees and giving a model of deligne lusztig type for the finite unitary group which parallels results of klyachko and inglis and saxl for the finite general linear group

qualocation for a singularly perturbed boundary value problem a singularly perturbed one dimensional two point boundary value problem of reaction convection diffusion type is considered we generate a c 0 collocation like method by combining galerkin with an adapted quadrature rule using lobatto quadrature and splines of degree r we prove on a shishkin mesh for the qualocation method the same error estimate as for the galerkin technique the result is also important for the practical realization of finite element methods on shishkin meshes using quadrature formulas we report the results of numerical experiments that support the theoretical findings

regular characters of groups of type an over discrete valuation rings let o be a complete discrete valuation ring with finite residue field k of odd characteristic let g be a general or special linear group or a unitary group defined over o and let g denote its lie algebra for every positive integer p let k l be the l th principal congruence subgroup of g o a continuous irreducible representation of g o is called regular of level p if it is trivial on kl 1 and its restriction to kl kl 1 similar or equal to g k consists of characters with g k over bar stabiliser of minimal dimension in this paper we construct the regular characters of g o compute their degrees and show that the latter satisfy ennola duality we give explicit uniform formulae for the regular part of the representation zeta functions of these groups c 2017 published by elsevier inc

a robust layer resolving spline collocation method for a convection diffusion problem abstract we consider finite difference approximation of a singularly perturbed one dimensional convection diffusion two point boundary value problem the problem is numerically treated by a quadratic spline collocation method on a piecewise uniform slightly modified shishkin mesh the position of collocation points is chosen so that the obtained scheme satisfies the discrete minimum principle we prove pointwise convergence of order o n 2 ln 2 n inside the boundary layer and second order convergence elsewhere the uniform convergence of the approximate continual solution is also given further we approximate normalized flux and give estimates of the error at the mesh points and between them the numerical experiments presented in the paper confirm our theoretical results

representation growth of compact linear groups we study the representation growth of simple compact lie groups and of sln o where o is a compact discrete valuation ring as well as the twist representation growth of gln o this amounts to a study of the abscissae of convergence of the corresponding twist representation zeta functions we determine the abscissae for a class of mellin zeta functions which include the witten zeta functions as a special case we obtain a new proof of the theorem of larsen and lubotzky that the abscissa of witten zeta functions is r where r is the rank and the number of positive roots we then show that the twist zeta function of gln o exists and has the same abscissa of convergence as the zeta function of sln o provided n does not divide char o we compute the twist zeta function of gl2 o when the residue characteristic p of o is odd and approximate the zeta function when p 2 to deduce that the abscissa is 1 finally we construct a large part of the representations of sl2 fq t q even and deduce that its abscissa lies in the interval 1 5 2

normal forms for matrices over uniserial rings of length two this paper develops methods to describe the conjugacy classes of gl n r on rn n for a serial ring r of length two the main result is a reduction to a computation in the matrix algebra over the residue class field of r which in some cases can be done theoretically without any actual computation

on p saturable groups a pro p group g is a pf group if it has central series of closed subgroups ni i n with trivial intersection satisfying n1 g and ni g p 1 g ni 1p in this paper we prove that a finitely generated pro p group g is a p saturable group in the sense of lazard if and only if it is a torsion free pf group using this characterization we study certain families of subgroups of p saturable groups for example we prove that any normal subgroup of a p saturable group contained in the frattini is again p saturable

generators relations and coverings of algebraic groups ii theorem 1 1 let g be the group of rational points of a simply connected simple afine algebraic group defined and split over a fmite j eld of q elements then the schur multiplier m of g is trivial with the following exceptions a zf g is a 4 a 2 a 3 c 2 f 2 or g 4 then h4 is z cyclic of order 2 if g is a 9 b 3 or g 3 it is iz b zf g is a 4 resp d 2 then m is the direct product of z resp z with itself

solving semilinear partial differential equations with probabilistic potential theory techniques of probabilistic potential theory are applied to solve lu f u u where u is a signed measure f a possibly discontinuous function and l a second order elliptic or parabolic operator on rj or more generally the infinitesi mal generator of a markov process a1so formulated are sufficient conditions guaranteeing existence of a solution to a countably infinite system of such equations

similarity classes of integral p adic matrices and representation zeta functions of groups of type a2 we compute explicitly dirichlet generating functions enumerating finite dimensional irreducible complex representations of various p adic analytic and adelic profinite groups of type a2 this has consequences for the representation zeta functions of arithmetic groups h k where k is a number field and h is a k form of sl3 assuming that possesses the strong congruence subgroup property we obtain precise uniform estimates for the representation growth of our results are based on explicit uniform formulae for the representation zeta functions of the p adic analytic groups sl3 o and su3 o where o is a compact discrete valuation ring of characteristic 0 these formulae build on our classification of similarity classes of integral p adic 3 3 matrices in gl3 o and gu3 o where o is a compact discrete valuation ring of arbitrary characteristic organising the similarity classes by invariants which we call their shadows allows us to combine the kirillov orbit method with clifford theory to obtain explicit formulae for representation zeta functions in a different direction we introduce and compute certain similarity class zeta functions our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form sl3 o su3 o gl3 o and gu3 o arising from the respective congruence filtrations these formulae are valid in case that the characteristic of o is either 0 or sufficiently large analysis of some of these formulae leads us to observe p adic analogues of ennola duality

on nonlinear reaction diffusion systems abstract this paper presents a qualitative analysis for a coupled system of two reaction diffusion equations under various boundary conditions which arises from a number of physical problems the nonlinear reaction functions are classified into three basic types according to their relative quasi monotone property for each type of reaction functions an existence comparison theorem in terms of upper and lower solutions is established for the time dependent system as well as some boundary value problems three concrete physical systems arising from epidemics biochemistry and engineering are taken as representatives of the basic types of reacting problems through suitable construction of upper and lower solutions various qualitative properties of the solution for each system are obtained these include the existence and bounds of time dependent solutions asymptotic behavior of the solution stability and instability of nontrivial steady state solutions estimates of stability regions and finally the blowing up property of the solution special attention is given to the homogeneous neumann boundary condition

groups acting on rooted trees and their representations on the boundary abstract we consider groups that act on spherically symmetric rooted trees and study the associated representation of the group on the space of locally constant functions on the boundary of the tree we introduce and discuss the new notion of locally 2 transitive actions assuming local 2 transitivity our main theorem yields a precise decomposition of the boundary representation into irreducible constituents the method can be used to study gelfand pairs and enables us to answer a question of grigorchuk to provide examples we analyse in detail the local 2 transitivity of ggs groups moreover our results can be used to determine explicit formulae for zeta functions of induced representations defined by klopsch and the author

on the lie theory of p adic analytic groups the aim of this paper is to fill a small but fundamental gap in the theory of p adic analytic groups we illustrate by example that the now standard notion of a uniformly powerful pro p group is more restrictive than lazard s concept of a saturable pro p group for instance the sylow pro p subgroups of many classical groups are saturable but need not be uniformly powerful extending work of ilani we obtain a correspondence between subgroups and lie sublattices of saturable pro p groups this leads to applications for instance in the subject of subgroup growth

arithmetic groups have rational representation growth let be an arithmetic lattice in a semisimple algebraic group over a number field we show that if has the congruence subgroup property then the number of n dimensional irreducible representations of grows like n where is a rational number

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

character degrees and random walks in finite groups of lie type for a finite group h let irr h denote the set of irreducible characters of h and define the zeta function zeta h t sum chi in irr h chi 1 t for real t 0 we study the asymptotic behaviour of zeta h t for finite simple groups h of lie type and also of a corresponding zeta function defined in terms of conjugacy classes applications are given to the study of random walks on simple groups and on base sizes of primitive permutation groups

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

a uniformly accurate spline collocation method for a normalized flux we are concerned with a two point boundary value problem for a semilinear singularly perturbed reaction diffusion equation with a singular perturbation parameter e our goal is to construct global e uniform approximations of the solution y x and the normalized flux p x e d dx y x using the collocation with the classical quadratic splines u x c1 i on a slightly modified piecewise uniform mesh of shishkin type the constructed approximate solution and normalized flux converge e uniformly with the rate o n 2 ln2 n and o n 1 ln n respectively on the shishkin type mesh and with o n 1 ln 2 n and o ln 3 n when the mesh has to be modified we present numerical experiments in support of these results

representations of automorphism groups of finite o modules of rank two abstract let o be a complete discrete valuation domain with finite residue field in this paper we describe the irreducible representations of the groups aut m for any finite o module m of rank two the main emphasis is on the interaction between the different groups and their representations an induction scheme is developed in order to study the whole family of these groups coherently the results obtained depend on the ring o in a very weak manner mainly through the degree of the residue field in particular a uniform description of the irreducible representations of gl 2 o p l is obtained where p is the maximal ideal of o

on representations of general linear groups over principal ideal local rings of length two abstract we study the irreducible complex representations of general linear groups over principal ideal local rings of length two with a fixed finite residue field we construct a canonical correspondence between the irreducible representations of all such groups that preserves dimensions for general linear groups of order three and four over these rings we construct all the irreducible representations we show that the problem of constructing all the irreducible representations of all general linear groups over these rings is not easier than the problem of constructing all the irreducible representations of all general linear groups over principal ideal local rings of arbitrary length in the function field case

representation growth and rational singularities of the moduli space of local systems let g be a semisimple algebraic group defined over mathbb q p and let gamma be a compact open subgroup of g mathbb q p we relate the asymptotic representation theory of gamma and the singularities of the moduli space of g local systems on a smooth projective curve proving new theorems about both 1 we prove that there is a constant c independent of g such that the number of n dimensional representations of gamma grows slower than n c confirming a conjecture of larsen and lubotzky in fact we can take c 3 cdot text dim e 8 1 745 we also prove the same bounds for groups over local fields of large enough characteristic 2 we prove that the coarse moduli space of g local systems on a smooth projective curve of genus at least lceil c 2 rceil 1 374 has rational singularities for the proof we study the analytic properties of push forwards of smooth measures under algebraic maps more precisely we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities

positive solutions of systems of semilinear elliptic equations the pendulum method abstract conditions are formulated which guarantee the existence of positive solutions for simultaneous systems on an open domain with each constant is a positive measure the main tools used are probabilistic potential theory markov processes and an iterative scheme which is not a generalization of the one used for quasi monotone systems quasi monotonicity is not assumed and new results are obtained