finitely generated groups p adic analytic groups and poincare series this is equivalent to the coefficients apn g satisfing a linear recurrence relation with constant coefficients for sufficiently large n these poincare series were first studied in the case when g is a finitely generated torsion free nilpotent group in a paper by grunewald segal and smith gssm there the authors established that for such a group g p s is a rational function in p8s in that setting the functions g p s are the local factors associated with the dirichlet series

on the structure of normal subgroups of potent p groups abstract let g be a finite p group satisfying g g g4 for p 2 and p 1 g gp for p 2 the main goal of this paper is to show that any normal subgroup n of g lying in g2 is power abelian that is the following holds 1 npk gpk g n 2 k n g n o g p k and 3 n p k n k n

commutator maps measure preservation and t systems let g be a finite simple group we show that the commutator map g g g is almost equidistributed as g 1 this somewhat surprising result has many applications it shows that a for a subset x g we have 1 x g 2 x g o 1 namely is almost measure preserving from this we deduce that almost all elements g 2 g can be expressed as commutators g x y where x y generate g this enables us to solve some open problems regarding t systems and the product replacement algorithm pra graph we show that the number of t systems in g with two generators tends to infinity as g 1 this settles a conjecture of guralnick and pak a similar result follows for the number of connected components of the pra graph of g with two generators some of our results apply for more general finite groups and more general word maps our methods are based on representation theory combining classical character theory with recent results on character degrees and values in finite simple groups in particular the so called witten zeta function g s p 2irr g 1 s plays a key role in the proofs

comparison of ocean acoustic horizontal coherence predicted by path integral approximations and parabolic equation simulation results a line integral approximation to the acoustic path integral has been used to generate predictions for the characteristic length scale of horizontal cross range coherence in long range ocean acoustic propagation these estimates utilize a single range independent sound speed profile and the mean variance as a function of depth of fractional sound speed perturbations due to internal waves the length scales predicted by the integral approximation have been compared to the values generated by parabolic equation simulations through multiple realizations of garrett munk internal waves one of the simulation environments approximates the slice89 experiment transmissions from a 250 hz source were simulated in a deep water transect to a maximum range of 1000km the second environment corresponds to one of the propagation paths in the north pacific acoustic laboratory npal experiment the source in this experiment was bottom mounted near kauai hawaii and the relevant receiver consisted of five vertical line

the levinson recurrence and fast algorithms for solving toeplitz systems of linear equations this work brings together classical polynomial theory as it relates to the levinson recurrence for a hermitian toeplitz operator and matrix theory as it relates to the class of hermitian centro hermitian matrices a new computationally efficient alternative is presented to the levinson recurrence on either the hermitian or skew hermitian polynomial spaces this approach also leads to an entirely new algorithm for solving systems of linear equations when the coefficient matrix is hermitian toeplitz or real symmetric toeplitz analysis of the computational complexity of the algorithms presented is also performed and it is shown that these algorithms lead to significant improvements in the computational complexity as compared to the previously best known recursive algorithms they also provide further insight into the mathematical properties of the structurally rich toeplitz matrices

the split levinson algorithm the classical levinson algorithm for computing the predictor polynomial relative to a real positive definite toeplitz matrix is shown to be redundant in complexity it can be broken down into two simpler algorithms either of which needs only to be processed this result can be interpreted in the framework of the theory of orthogonal polynomials on the real line as follows the symmetric and antisymmetric parts of the predictors relative to the sequence of toeplitz matrices constitute two families of polynomials orthogonal on the interval 1 1 with respect to some even weight functions it turns out that the recurrence relations for these orthogonal polynomials can be used efficiently to compute the desired predictor the resulting split levinson algorithm requires roughly one half the number of multiplications and the same number of additions as the classical levinson algorithm a simple derivation of cybenko s method for computing the pisarenko frequencies is obtained from the recurrence relations underlying the split levinson algorithm

split algorithms for hermitian toeplitz matrices with arbitrary rank profile abstract split algorithms for toeplitz matrices exploit besides the toeplitz structure additional symmetry properties to reduce the number of operations in this paper split levinson and schur algorithms for hermitian toeplitz matrices are presented that work in contrast to previous algorithms without additional conditions like strong nonsingularity the main contribution is the generalization of the split levinson type algorithms of b krishna h krishna and h krishna s morgera to general nonsingular hermitian toeplitz matrices furthermore a schur type counterpart of this algorithm is presented that is also new in the strongly nonsingular case some auxiliary considerations concerning the kernel structure of hermitian toeplitz matrices might be of independent interest

extended deligne lusztig varieties for general and special linear groups abstract we give a generalisation of deligne lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non archimedean local field previously a generalisation was given by lusztig by attaching certain varieties to unramified maximal tori inside borel subgroups in this paper we associate a family of so called extended deligne lusztig varieties to all tamely ramified maximal tori of the group moreover we analyse the structure of various generalised deligne lusztig varieties and show that the unramified varieties including a certain natural generalisation do not produce all the irreducible representations in general on the other hand we prove results which together with some computations of lusztig show that for sl 2 f q 2 with odd q the extended deligne lusztig varieties do indeed afford all the irreducible representations

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

character degrees of p groups and pro p groups in the 1970s isaacs conjectured that there should be a logarithmic bound for the length of solvability of a p group g with respect to the number of different irreducible character degrees of g so far there are just a few partial results for this conjecture in this note we say that a pro p group g has property i if there is a real number d d g that just depends on g such that for any open normal subgroup n d l g n log2 cd g n d we prove that any p adic analytic pro p group has property i we also study the first congruence subgroup g of a classical chevalley group g with respect to the local ring fpt we show that if lie g fp has a non degenerated killing form then g has property i

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

computationally efficient reduced polynomial based algorithms for hermitian toeplitz matrices in this work the mathematical structure is analyzed that is associated with the fast order recursive algorithms for computing the reflection coefficients and the levinson polynomial associated with a hermitian positive definite toeplitz matrix a new form of three term recurrence relation is derived anda computationally efficient alternative to the classical levinson durbin algorithm is obtained the computational complexity of the new algorithm is the same as those of the split algorithms described in the recent literature the new algorithm also provides further insight into the mathematical properties of the structurally rich toeplitz matrices

on the splitting of classical algorithms in linear prediction theory the levinson algorithm has been shown recently to be redundant in complexity it can be broken down into two simpler algorithms either of these split levinson algorithms needs only to be processed in the present paper it is shown that the schur algorithm the lattice algorithm and the normalized lattice algorithm for computing reflection coefficients can be broken down in a similar manner the number of multiplications in each of the resulting split algorithms is reduced with respect to the classical version by a factor lying between 1 5 and 2

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