blocks of small defect abstract let g be a finite solvable group let p be a prime such that p 5 and o p g 1 and we denote g p p n then g contains a block of defect less than or equal to 3 n 5 let g be a finite solvable group and let p a be the largest power of p dividing 1 for an irreducible character of g we show that g f g p p 3 a for p 5

groups with just one character degree divisible by a given prime the ito michler theorem asserts that if no irreducible character of a finite group g has degree divisible by some given prime p then a sylow p subgroup of g is both normal and abelian in this paper we relax the hypothesis and we assume that there is at exactly one multiple of p that occurs as the degree of an irreducible character of g we show that in this situation a sylow p subgroup of g is almost normal in g and it is almost abelian

brauer characters and normal sylow p subgroups abstract in this paper we study some variations of the well known ito michler theorem for p brauer characters using various inequalities involving the p brauer character degrees of finite groups several new criteria for the existence of a normal sylow p subgroup of finite groups are obtained using the p parts and p parts of the p brauer character degrees

on heights of characters of finite groups abstract in this paper we first state a conjecture on the lower bound of the maximal height of characters in a p block of a finite group then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group for all blocks of a quasi simple group g with g z g isomorphic to a 6 a 7 or a simple group of lie type with an exceptional covering group for all blocks of a symmetric group as well as for all blocks of finite general linear or unitary groups for the proof of the case with symmetric groups it relates to an open question of olsson on the existence of t core partitions where t 3 is an integer as a byproduct our investigation on heights of characters of the symmetric groups and of the general linear or unitary groups also gives evidence for the isaacs moreto navarro tiep conjecture claiming that the number of distinct irreducible character degrees of a sylow p subgroup p of an arbitrary finite group g is at most one more than the number of irreducible character degrees of g that are multiples of p

brauer s height zero conjecture for quasi simple groups abstract we show that brauer s height zero conjecture holds for blocks of finite quasi simple groups this result is used in navarro spath s reduction of this conjecture for general groups to the inductive alperin mckay condition for simple groups

non vanishing elements of finite groups let g be a finite group and let irr g denote the set of irreducible complex characters of g an element x of g is non vanishing if for every in irr g we have x 0 we prove that if x is a non vanishing element of g and the order of x is coprime to 6 then x lies in the fitting subgroup of g

defect zero p blocks for finite simple groups we classify those finite simple groups whose brauer graph or decomposition matrix has a p block with defect 0 completing an investigation of many authors the only finite simple groups whose defect zero p blocks remained unclassified were the alternating groups an here we show that these all have a p block with defect 0 for every prime p 5 this follows from proving the same result for every symmetric group sn which in turn follows as a consequence of the t core partition conjecture that every non negative integer possesses at least one t core partition for any t 4 for t 17 we reduce this problem to lagrange s theorem that every non negative integer can be written as the sum of four squares the only case with t 17 that was not covered in previous work was the case t 13 this we prove with a very different argument by interpreting the generating function for t core partitions in terms of modular forms and then controlling the size of the coefficients using deligne s theorem nee the weil conjectures we also consider congruences for the number of p blocks of sn proving a conjecture of garvan that establishes certain multiplicative congruences when 5 p 23 by using a result of serre concerning the divisibility of coefficients of modular forms we show that for any given prime p and positive integer m the number of p blocks with defect 0 in sn is a multiple of m for almost all n we also establish that any given prime p divides the number of p modularly irreducible representations of sn for almost all n 1996 american mathematical society

formulas for partition k tuples with t cores abstract let a t k n denote the number of partition k tuples of n where each partition is t core in this paper we establish formulas of a t k n for some values of t and k by employing the method of modular forms which extends wang s result for t 3 and k 2 3

the heights of irreducible brauer characters in 2 blocks of the symmetric groups abstract we prove that there exists a unique irreducible brauer character of height zero in any 2 blocks of the symmetric group this generalizes the theorem of p fong and g d james that the dimension of every non trivial 2 modular simple module of the symmetric group is even

bilinear generalized approximate message passing part ii applications in this paper we extend the generalized approximate message passing g amp approach originally proposed for high dimensional generalized linear regression in the context of compressive sensing to the generalized bilinear case in part i of this two part paper we derived our bilinear g amp big amp algorithm as an approximation of the sum product belief propagation algorithm in the high dimensional limit and proposed an adaptive damping mechanism that aids convergence under finite problem sizes an expectation maximization em based method to automatically tune the parameters of the assumed priors and two rank selection strategies here in part ii we discuss the specializations of big amp to the problems of matrix completion robust pca and dictionary learning and present the results of an extensive empirical study comparing big amp to state of the art algorithms on each problem our numerical results using both synthetic and real world datasets demonstrate that em big amp yields excellent reconstruction accuracy often best in class while maintaining competitive runtimes

an iterative construction of solutions of the tap equations for the sherrington kirkpatrick model we propose an iterative scheme for the solutions of the tap equations in the sherrington kirkpatrick model which is shown to converge up to and including the de almeida thouless line the main tool is a representation of the iterations which reveals an interesting structure of them this representation does not depend on the temperature parameter but for temperatures below the de almeida thouless line it contains a part which does not converge to zero in the limit

some congruences for partitions that are p cores a number of linear congruences modulo r are proved for the number of partitions that are p cores where p is prime 5 p 23 and r is any prime divisor of 1 2 p 1 analogous results are derived for the number of irreducible p modular representations of the symmetric group s n the congruences are proved using the theory of modular forms

self conjugate t core partitions sums of squares and p blocks of an we prove that if t is an integer with t 8 or t 10 then every integer n 2 has a self conjugate t core partition this result has consequences in the representation theory of alternating groups and has a version as a theorem about the representation of integers by sums of squares we also give an infinite sequence of integers that have no self conjugate 9 core partition

vertices for characters of p solvable groups suppose that g is a finite p solvable group we associate to every irreducible complex character x irr g of g a canonical pair q where q is a p subgroup of g and irr q uniquely determined by x up to g conjugacy this pair behaves as a green vertex and partitions irr g into families of characters using the pair q we give a canonical choice of a certain p radical subgroup r of g and a character irr r associated to x which was predicted by some conjecture of g r robinson

a meshfree method for elasticity problems with interfaces in this work we develop a meshfree method based on fundamental solutions basis functions for a transmission problem in linear elasticity the addressed problem consists in given the displacement field on the boundary compute the corresponding displacement field of an elastic object which has piecewise constant lame coefficients the lame coefficients are assumed to be constant in non overlapping subdomains and on the corresponding interface interior boundaries non homogeneous jump conditions on the displacement and on the traction vectors are considered the main properties of the method are analyzed and illustrated with several numerical simulations in 2d domains

quasi isolated blocks and brauer s height zero conjecture this paper has two main results firstly we complete the parametrisation of all p blocks of nite quasi simple groups by nding the so called quasi isolated blocks of exceptional groups of lie type for bad primes this relies on the explicit decomposition of lusztig induction from suitable levi subgroups our second major result is the proof of one direction of brauer s long standing height zero conjecture on blocks of nite groups using the

a variation on theorems of jordan and gluck abstract gluck proved that any finite group g has an abelian subgroup a such that g a is bounded by a polynomial function of the largest degree of the complex irreducible characters of g this improved on a previous bound of isaacs and passman in this paper we present a variation of this result that looks at the number of prime factors all these results in turn may be seen as variations on the classical theorem of jordan on linear groups

characterizing normal sylow p subgroups by character degrees abstract suppose that g is a finite group let p be a prime and let p syl p g we prove that p is normal in g if and only if all the irreducible constituents of the permutation character 1 p g have degree not divisible by p

matrix decomposition mfs algorithms for elasticity and thermo elasticity problems in axisymmetric domains in this work we propose an efficient matrix decomposition algorithm for the method of fundamental solutions when applied to three dimensional boundary value problems governed by elliptic systems of partial differential equations in particular we consider problems arising in linear elasticity in axisymmetric domains the proposed algorithm exploits the block circulant structure of the coefficient matrices and makes use of fast fourier transforms the algorithm is also applied to problems in thermo elasticity several numerical experiments are carried out

a meshfree numerical method for acoustic wave propagation problems in planar domains with corners and cracks the numerical solution of acoustic wave propagation problems in planar domains with corners and cracks is considered since the exact solution of such problems is singular in the neighborhood of the geometric singularities the standard meshfree methods based on global interpolation by analytic functions show low accuracy in order to circumvent this issue a meshfree modification of the method of fundamental solutions is developed where the approximation basis is enriched by an extra span of corner adapted non smooth shape functions the high accuracy of the new method is illustrated by solving several boundary value problems for the helmholtz equation modelling physical phenomena from the fields of room acoustics and acoustic resonance

the error probability of sparse superposition codes with approximate message passing decoding sparse superposition codes or sparse regression codes are a recent class of codes for reliable communication over the awgn channel at rates approaching the channel capacity approximate message passing amp decoding a computationally efficient technique for decoding sparcs has been proven to be asymptotically capacity achieving for the awgn channel in this paper we refine the asymptotic result by deriving a large deviation bound on the probability of an amp decoding error this bound gives insight into the error performance of the amp decoder for large but finite problem sizes giving an error exponent as well as guidance on how the code parameters should be chosen at finite block lengths for an appropriate choice of code parameters we show that for any fixed rate less than the channel capacity the decoding error probability decays exponentially in n log n 2t where t the number of amp iterations required for successful decoding is bounded in terms of the gap from capacity

meshless simulation of anti plane crack problems by the method of fundamental solutions using the crack green s function abstract in this paper we present a meshless method of fundamental solutions using the analytical crack green s function to solve anti plane crack problems the proposed scheme is a simple powerful and effective collocation method for crack problems since it only requires the boundary discretization without special treatments of the crack three typical numerical examples namely a central crack an off central crack and a central slant crack are presented and discussed to illustrate the accuracy convergence and stability of the proposed method

explicit formulas for partition pairs and triples with 3 cores abstract let a 3 n resp b 3 n denote the number of partition pairs resp triples of n where each partition is 3 core by applying ramanujan s 1 1 formula and bailey s 6 6 formula we find the explicit formulas for a 3 n and b 3 n using these formulas we confirm a conjecture of xia and establish many arithmetic identities satisfied by a 3 n and b 3 n

on the orders of zeros of irreducible characters abstract let g be a finite group and p a prime number we say that an element g in g is a vanishing element of g if there exists an irreducible character of g such that g 0 the main result of this paper shows that if g does not have any vanishing element of p power order then g has a normal sylow p subgroup also we prove that this result is a generalization of some classical theorems in character theory of finite groups

high dimensional robust m estimation asymptotic variance via approximate message passing in a recent article el karoui et al proc natl acad sci 110 36 14557 14562 2013 study the distribution of robust regression estimators in the regime in which the number of parameters p is of the same order as the number of samples n using numerical simulations and highly plausible heuristic arguments they unveil a striking new phenomenon namely the regression coefficients contain an extra gaussian noise component that is not explained by classical concepts such as the fisher information matrix we show here that that this phenomenon can be characterized rigorously using techniques that were developed by the authors for analyzing the lasso estimator under high dimensional asymptotics we introduce an approximate message passing amp algorithm to compute m estimators and deploy state evolution to evaluate the operating characteristics of amp and so also m estimates our analysis clarifies that the extra gaussian noise encountered in this problem is fundamentally similar to phenomena already studied for regularized least squares in the setting n p

character correspondences in finite general linear unitary and symmetric groups let r 0 be a prime integer and let g be a finite general linear group gl n q a unitary group u n q with q qg or a symmetric group s n of degree n the partition of the irreducible ordinary characters of s n into r blocks of s n was given by brauer s and robinson s solution of nakayama s conjecture see 7 p 245 in their fundamental paper 5 fong and srinivasan have recently classified the r blocks of gl n q and u n q for all primes r 2 with r q 1 using these classifications of the r blocks of g we show in this article that there is a natural one to one correspondence b between the irreducible characters of height zero of an r block b of g with defect group r and the irreducible characters of height zero of the brauer correspondent b of b in n ng r theorem 4 10 in particular ko b ko b where ko b denotes the number of all irreducible characters of b with height h t z 0 therefore alperin s conjecture on the numbers of irreducible characters of height zero is verified for all general linear unitary and symmetric groups if g s n then theorem 4 10 also holds for r 2 in order to establish the character correspondence three other correspondences are studied the product of which is b in sect 1 we construct for every r block b of g with defect group r a subgroup g of g with a sylow r subgroup r such that there is a natural height preserving one to one correspondence t between the set irr b of all irreducible characters of b and the set irr 0 of all irreducible characters of the principal r block o of d reduction theorems 1 9 and 1 10 the map respects the geometric conjugacy classes of characters if 0 denotes the principal r block of n nd and if b is the brauer correspondent of b in n n6 r then by theorems 3 8 and 3 10 the block ideals o and b are morita equivalent have the same decomposition numbers and there is a natural height preserving one to one correspondence a between irr bo and irr b

non negative principal component analysis message passing algorithms and sharp asymptotics principal component analysis pca aims at estimating the direction of maximal variability of a high dimensional data set a natural question is does this task become easier and estimation more accurate when we exploit additional knowledge on the principal vector we study the case in which the principal vector is known to lie in the positive orthant similar constraints arise in a number of applications ranging from the analysis of gene expression data to spike sorting in neural signal processing in the unconstrained case the estimation performances of pca have been precisely characterized using the random matrix theory under a statistical model known as the spiked model it is known that the estimation error undergoes a phase transition as the signal to noise ratio crosses a certain threshold unfortunately tools from the random matrix theory have no bearing on the constrained problem despite this challenge we develop an analogous characterization in the constrained case within a one spike model in particular 1 we prove that the estimation error undergoes a similar phase transition albeit at a different thresholds in signal to noise ratio that we determine exactly 2 we prove that unlike in the unconstrained case the estimation error depends on the spike vector and characterize the least favorable vectors and 3 we show that a non negative principal component can be approximately computed under the spiked model in nearly linear time this despite the fact that the problem is non convex and in general np hard to solve exactly

optimal number of measurements in a linear system with quadratically decreasing snr we consider the design of a linear sensing system with a fixed energy budget assuming that the sampling noise is the dominant noise source the energy constraint implies that the signal energy per measurement decreases linearly with the number of measurements when the maximum sampling rate of the sampling circuit is chosen to match the designed sampling rate the assumption on the noise implies that its variance increases approximately linearly with the sampling rate number of measurements therefore the overall snr per measurement decreases quadratically in the number of measurements our analysis shows that in this setting there is an optimal number of measurements this is in contrast to the standard case where noise variance remains unchanged with sampling rate in which case more measurements imply better performance our results are based on a state evolution technique of the well known approximate message passing algorithm we consider both the sparse e g bernoulli gaussian and least favorable distributions and the non sparse e g gaussian distribution settings in both the real and complex domains numerical results corroborate our analysis

stress intensity factor computation using the method of fundamental solutions mixed mode problems the method of fundamental solutions is applied to the computation of stress intensity factors in linear elastic fracture mechanics the displacements are approximated by linear combinations of the fundamental solutions of the cauchy navier equations of elasticity and the leading terms for the displacement near the crack tip two algorithms are developed one using a single domain and one using domain decomposition numerical results are given copyright 2006 john wiley sons ltd

cranks and t cores new statistics on partitions calledcranks are defined which combinatorially prove ramanujan s congruences for the partition function modulo 5 7 11 and 25 explicit bijections are given for the equinumerous crank classes the cranks are closely related to thet core of a partition usingq series some explicit formulas are given for the number of partitions which aret cores some related questions for self conjugate and distinct partitions are discussed

on p parts of character degrees and conjugacy class sizes of finite groups abstract let g be a finite group and irr g the set of irreducible complex characters of g let e p g be the largest integer such that p e p g divides 1 for some irr g we show that g f g p p k e p g for a constant k this settles a conjecture of a moreto 13 conjecture 4 we also study the related problems of the p parts of conjugacy class sizes of finite groups

on vanishing class sizes in finite groups abstract let g be a finite group an element g of g is called a vanishing element if there exists an irreducible character of g such that g 0 in this case we say that the conjugacy class of g is a vanishing conjugacy class in this paper we discuss some arithmetical properties concerning the sizes of the vanishing conjugacy classes in a finite group

on boundary integral equations for crack problems a ubiquitous linear boundary value problem in mathematical physics involves solving a partial differential equation exterior to a thin obstacle one typical example is the scattering of scalar waves by a curved crack or rigid strip neumann boundary condition in two dimensions this problem can be reduced to an integrodifferential equation which is often regularized we adopt a more direct approach and prove that the problem can be reduced to a hypersingular boundary integral equation similar reductions will obtain in more complicated situations computational schemes for solving this equation are described with special emphasis on smoothness requirements extensions to three dimensional problems involving an arbitrary smooth bounded crack in an elastic solid are discussed

on the computation of two dimensional stress intensity factors using the boundary element method general two dimensional linear elastic fracture problems are investigated using the boundary element method the r displacement and 1 r traction behaviour near a crack tip are incorporated in special crack elements stress intensity factors of both modes i and ii are obtained directly from crack tip nodal values for a variety of crack problems including straight and curved cracks in finite and infinite bodies a multidomain approach is adopted to treat cracks in an infinite body the body is subdivided into two regions an infinite part with a finite hole and a finite inclusion numerical results compared with exact solution whenever possible are accurate even with a coarse discretization

lie type like groups abstract a finite simple group of lie type in defining characteristic p has exactly two p blocks the principal block and a block of defect zero consisting of the steinberg character whose degree is the p part of the order of the group in this paper we characterize finite groups g which have exactly the principal p block and a p block of defect zero consisting of an irreducible character of degree g p

finite sample analysis of approximate message passing algorithms approximate message passing amp refers to a class of efficient algorithms for statistical estimation in high dimensional problems such as compressed sensing and low rank matrix estimation this paper analyzes the performance of amp in the regime where the problem dimension is large but finite for concreteness we consider the setting of high dimensional regression where the goal is to estimate a high dimensional vector beta 0 from a noisy measurement y a beta 0 w amp is a low complexity scalable algorithm for this problem under suitable assumptions on the measurement matrix a amp has the attractive feature that its performance can be accurately characterized in the large system limit by a simple scalar iteration called state evolution previous proofs of the validity of state evolution have all been asymptotic convergence results in this paper we derive a concentration inequality for amp with gaussian matrices with independent and identically distributed i i d entries and finite dimension n times n the result shows that the probability of deviation from the state evolution prediction falls exponentially in n this provides theoretical support for empirical findings that have demonstrated excellent agreement of amp performance with state evolution predictions for moderately large dimensions the concentration inequality also indicates that the number of amp iterations t can grow no faster than order log n log log n for the performance to be close to the state evolution predictions with high probability the analysis can be extended to obtain similar non asymptotic results for amp in other settings such as low rank matrix estimation

explicit bounds for the number of p core partitions in this article we derive explicit bounds on ct n the number of t core partitions of n in the case when t p is prime we express the generating function f z as the sum

radical 3 subgroups of f4 q with q even abstract in a recent paper an and huang classified the radical 3 subgroups of f 4 q for q odd here we use a similar approach to classify the radical 3 subgroups of f 4 q with q even as an application we determine the essential rank of the frobenius category f d f 4 q for q even where d is a sylow 3 subgroup of f 4 q

bilinear generalized approximate message passing part i derivation in this paper we extend the generalized approximate message passing g amp approach originally proposed for high dimensional generalized linear regression in the context of compressive sensing to the generalized bilinear case which enables its application to matrix completion robust pca dictionary learning and related matrix factorization problems here in part i of a two part paper we derive our bilinear g amp big amp algorithm as an approximation of the sum product belief propagation algorithm in the high dimensional limit where central limit theorem arguments and taylor series approximations apply and under the assumption of statistically independent matrix entries with known priors in addition we propose an adaptive damping mechanism that aids convergence under finite problem sizes an expectation maximization em based method to automatically tune the parameters of the assumed priors and two rank selection strategies in part ii of the paper we will discuss the specializations of em big amp to the problems of matrix completion robust pca and dictionary learning and we will present the results of an extensive empirical study comparing em big amp to state of the art algorithms on each problem

on vanishing criteria that control finite group structure abstract many results have been established that show how arithmetic conditions on conjugacy class sizes affect group structure a conjugacy class in g is called vanishing if there exists some irreducible character of g which evaluates to zero on the conjugacy class the aim of this paper is to show that for some classical results it is enough to consider the same arithmetic conditions on the vanishing conjugacy classes of the group

mathematical foundation of the mfs for certain elliptic systems in linear elasticity the method of fundamental solutions mfs is a trefftz type technique in which the solution of an elliptic boundary value problem is approximated by a linear combination of translates of fundamental solutions with singularities placed on a pseudo boundary i e a surface embracing the domain of the problem under consideration in this work we develop a mathematical framework for the numerical implementation of the mfs in elliptic systems we obtain density results with respect to the c l norms which establish the applicability of the method in certain systems arising from the theory of elastostatics and thermo elastostatics the domains in our density results may possess holes and they satisfy the segment condition

formulas for cubic partition with 3 cores abstract let c 3 n denote the number of cubic partitions of n with 3 cores in this paper we establish the arithmetic properties and formulas for c 3 n by employing bailey s 6 6 formula and theta function identities