computing finite galois groups arising from automorphic forms abstract we study modulo p reduction of compatible systems of p adic representations of the absolute galois group of q arising from an algebraic automorphic representation in particular we prove that there is a field extension of q with the galois group g 2 p ramified at 5 and p only for a set of primes p of density one

prime power graphs for groups of lie type abstract we associate a weighted graph g to each finite simple group g of lie type we show that with an explicit list of exceptions g determines g up to isomorphism and for these exceptions g nevertheless determines the characteristic of g this result was motivated by algorithmic considerations we prove that for any finite simple group g of lie type input as a black box group with an oracle to compute the orders of group elements g and the characteristic of g can be computed by a monte carlo algorithm in time polynomial in the input length the characteristic is needed as part of the input in a previous constructive recognition algorithm for g

groups with the basis property abstract we study finite groups for which every minimal generating set has the same cardinality a group has the basis property if it and every subgroup satisfies this condition on minimal generating sets we classify all finite groups with the basis property

lago a computationally efficient approach for statistical detection we study a general class of statistical detection problems where the underlying objective is to detect items belonging to a rare class from a very large database we propose a computationally efficient method to achieve this goal our method consists of two steps in the first step we estimate the density function of the rare class alone with an adaptive bandwidth kernel density estimator the adaptive choice of the bandwidth is inspired by the ancient chinese board game known today as go in the second step we adjust this density locally depending on the density of the background class nearby we show that the amount of adjustment needed in the second step is approximately equal to the adaptive bandwidth from the first step which gives us additional computational savings we name the resulting method lago for locally adjusted go kernel density estimator we then apply lago to a real drug discovery dataset and compare its performance with a number of existing and popular methods

a polynomial time reduction algorithm for groups of semilinear or subfield class abstract we present a las vegas algorithm for finding a nontrivial reduction of groups that are irreducible with m generators and either lie in the subfield class of matrix or projective groups or are semilinear or have non absolutely irreducible derived group let r a denote the cost of producing a random element from a matrix algebra a and r h g denote the cost of producing a random element in the normal closure of a group h by a group g then the algorithm runs in o d 3 m d log log d log q r a log log d r h g d log q finite field operations we also characterise the absolutely irreducible groups g over arbitrary fields whose derived group consists only of scalars and prove probabilistic generation results about matrix groups

tensor products are projective geometries abstract given a finite dimensional vector space v we construct a family of projective geometries whose flats are certain subspaces of v and show that there is a one to one correspondence between this family of projective geometries and the set of equivalence classes of tensor decompositions of v this provides a practical method for finding a tensor decomposition of a finite dimensional kg module or proving that no nontrivial tensor decomposition exists

finite affine groups cycle indices hall littlewood polynomials and probabilistic algorithms abstract the study of asymptotic properties of the conjugacy class of a random element of the finite affine group leads one to define a probability measure on the set of all partitions of all positive integers four different probabilistic understandings of this measure are given three using symmetric function theory and one using markov chains this leads to non trivial enumerative results cycle index generating functions are derived and are used to compute the large dimension limiting probabilities that an element of the affine group is separable cyclic or semisimple and to study the convergence to these limits the semisimple limit involves both rogers ramanujan identities this yields the first examples of such computations for a maximal parabolic subgroup of a finite classical group

comparison of semi simplifications of galois representations abstract let be the absolute galois group of a global field let 1 and 2 be two p adic finite dimensional representations of then there exists a finite number of primes q such that if the characteristic polynomials of 1 frob q and 2 frob q are equal then 1 and 2 have isomorphic semi simplifications and so the same l functions we give a method to compute a sufficient list of primes based on the ramification and the dimension of the representations we use then the result to prove a conjecture by b van geemen and j top

sequentially refined latin hypercube designs reusing every point the use of iteratively enlarged latin hypercube designs for running computer experiments has recently gained popularity in practice this approach conducts an initial experiment with a computer code using a latin hypercube design and then runs a follow up experiment with additional runs elaborately chosen so that the combined design set for the two experiments forms a larger latin hypercube design this augmenting process can be repeated multiple stages where in each stage the augmented design set is guaranteed to be a latin hypercube design we provide a theoretical framework to put this approach on a firm footing numerical examples are given to corroborate the derived theoretical results supplementary materials for this article are available online

large sample properties of simulations using latin hypercube sampling latin hypercube sampling mckay conover and beckman 1979 is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables the asymptotic variance of such an estimate is obtained the estimate is also shown to be asymptotically normal asymptotically the variance is less than that obtained using simple random sampling with the degree of variance reduction depending on the degree of additivity in the function being integrated a method for producing latin hypercube samples when the components of the input variables are statistically dependent is also described these techniques are applied to a simulation of the performance of a printer actuator

modification of the maximin and ϕp phi criteria to achieve statistically uniform distribution of sampling points this article proposes a sampling technique that delivers robust designs that is point sets selected from a design domain in the shape of a unit hypercube the designs are guaranteed to provide a

power series coefficients for probabilities in finite classical groups abstract it is shown that a wide range of probabilities and limiting probabilities in finite classical groups have integral coefficients when expanded as a power series in q 1 moreover it is proved that the coefficients of the limiting probabilities in the general linear and unitary cases are equal modulo 2 the rate of stabilization of the finite dimensional coefficients as the dimension increases is discussed

sudoku based space filling designs sudoku is played by millions of people across the globe it has simple rules and is very addictive the game board is a nine by nine grid of numbers from one to nine several entries within the grid are provided and the remaining entries must be filled in subject to no row column or three by three subsquare containing duplicate numbers by exploiting these three types of uniformity we propose an approach to constructing a new type of design called a sudoku based space filling design such a design can be divided into groups of subdesigns so that the complete design and each subdesign achieve maximum uniformity in univariate and bivariate margins examples are given illustrating the proposed construction method applications of such designs include computer experiments with qualitative and quantitative factors linking parameters in engineering and crossvalidation copyright 2011 oxford university press

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

fast constructive recognition of black box orthogonal groups abstract we present an algorithm that constructively recognises when a given black box group is a nontrivial homomorphic image of the orthogonal group e d q for known e d and q the algorithm runs in polynomial time assuming oracles for handling sl 2 q subgroups and discrete logarithms in f q

construction of nested space filling designs new types of designs called nested space filling designs have been proposed for conducting multiple computer experiments with different levels of accuracy in this article we develop several approaches to constructing such designs the development of these methods also leads to the introduction of several new discrete mathematics concepts including nested orthogonal arrays and nested difference matrices

an approach to the construction of asymmetrical orthogonal arrays abstract use of asymmetrical orthogonal arrays oa s for planning industrial experiments with mixed levels has become increasingly popular in addition to applications to experimental investigations in many disciplines they can also be used in the balanced repeated replications method of inference from general stratified survey samples in this article we propose a general approach to the construction of asymmetrical oa s with economic run size and flexibility in the choice of factor levels as applications of this approach we construct several general classes of arrays that include numerous existing classes of arrays as special cases a large number of known asymmetrical oa s can be reproduced by our approach in a unified and simple manner many new arrays are obtained a list of asymmetrical oa s with moderate run size is given our approach consists of three steps we first construct an oa as the generalized kronecker sum of an oa and some difference matrices we then add to the constructed oa co

an analogue of serre s conjecture for galois representations and hecke eigenclasses in the mod p cohomology of gl n ℤ

abundant p singular elements in finite classical groups abstract in 1995 isaacs kantor and spaltenstein proved that for a finite simple classical group g defined over a field with q elements and for a prime divisor p of g distinct from the characteristic the proportion of p singular elements in g elements with order divisible by p is at least a constant multiple of 1 1 p e where e is the order of q modulo p motivated by algorithmic applications we define a subfamily of p singular elements called p abundant elements which leave invariant certain large subspaces of the natural g module we find explicit upper and lower bounds for the proportion of p abundant elements in g and prove that it approaches a positive limiting value as the dimension of g tends to infinity it turns out that the limiting proportion of p abundant elements is at least a constant multiple of the isaacs kantor spaltenstein lower bound for the proportion of all p singular elements

computations of siegel modular forms of genus two we explain the basic notions and theorems for doing computations in the theory of siegel modular forms of degree two on the full modular group and of even weight this synopsis concludes with a handy and computationally realistic algorithm for tabulating the fourier coefficients of such forms and the euler factors of their spinor zeta functions in the second part of this paper we present and discuss some of the results of actual computations which we performed following this algorithm we point out two theoretically striking phenomena that are implied by the results of these computations

galois representations with conjectural connections to arithmetic cohomology in this paper we extend a conjecture of a ash and w sinnott relating niveau 1 galois representations to the mod p cohomology of congruence subgroups ofsln z to include galois representations of higher niveau we then present computational evidence for our conjecture in the case n 3 in the form of three dimensional galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture our examples include galois representations with nontrivial weight and level as well as irreducible three dimensional representations that are in no obvious way related to lower dimensional representations in addition we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture

strong orthogonal arrays of strength two plus strong orthogonal arrays were recently introduced and studied in he and tang biometrika100 2013 254 260 as a class of space filling designs for computer experiments to enjoy the benefits of better space filling properties when compared to ordinary orthogonal arrays strong orthogonal arrays need to have strength three or higher which may require run sizes that are too large for experimenters to afford to address this problem we introduce a new class of arrays called strong orthogonal arrays of strength two plus these arrays while being more economical than strong orthogonal arrays of strength three still enjoy the better two dimensional space filling property of the latter among the many results we have obtained on the characterizations and constructions of strong orthogonal arrays of strength two plus worth special mention is their intimate connection with second order saturated designs

on the stability of periodic solutions for differential equations with a fredholm operator at the highest derivative abstract on the basis of a classic floquet theory variant the stability of periodic solutions at andronov hopf bifurcation for autonomous nonlinear differential equations of s th order s 1 with a degenerate fredholm operator at the highest derivative in banach spaces is investigated stability criteria based on the perturbation theory and newton diagram method are suggested

walks on generating sets of groups we study a markov chain on generating n tuples of a fixed group which arises in algorithms for manipulating finite groups the main tools are comparison of two markov chains on different but related state spaces and combinatorics of random paths the results involve group theoretical parameters such as the size of minimal generating sets the number of distinct generating k tuples for different k s and the maximal diameter of the group

on the principle of reduced stability abstract the principle of reduced stability says that the stability of bifurcating stationary or periodic solutions is given by the finite dimensional bifurcation equation obtained by the method of lyapunov schmidt to be more precise the linearized stability is governed by the linearization of the bifurcation equation about the bifurcating branch of solutions and in particular by the signs of the real parts of the perturbation of the eigenvalues along this branch this principle is true for simple eigenvalue bifurcation whereas it may be false for higher dimensional bifurcation equations a condition for the validity of that principle is given a counterexample shows that it cannot be dropped in general

geometric families of 4 dimensional galois representations with generically large images we study compatible families of four dimensional galois representations constructed in the etale cohomology of a smooth projective variety we prove a theorem asserting that the images will be generically large if certain conditions are satisfied we only consider representations with coefficients in an imaginary quadratic field we apply our result to an example constructed by jasper scholten a non selfdual 4 dimensional galois representation www math uiuc edu algebraic number theory 0183 1999 obtaining a family of linear groups and one of unitary groups as galois groups over mathbb q

on the images of modular and geometric three dimensional galois representations we study compatible families of three dimensional galois representations constructed in the etale cohomology of a smooth projective variety we prove a theorem asserting that the residual images will be generically large if certain easy to check conditions are satisfied we only consider representations with coefficients in an imaginary quadratic field for primes inert in this field the residual representations when irreducible are unitary we apply our result to an example constructed by van geemen and top obtaining a family of special linear groups and one of special unitary groups as galois groups over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i we also consider the case of cohomological modular forms for a congruence subgroup of sl inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i assuming clozel s conjecture stating that a geometric family of three dimensional galois representations can be attached to them we verify for three examples the conditions guaranteeing generically large images and explicitly bound the finite set of primes with non maximal image we also discuss what intrinsic conditions a modular form should verify to guarantee that the images of the attached galois representations will be generically large

uniform coverage designs for molecule selection in screening for drug discovery chemists often select a large subset of molecules from a very large database e g select 1 000 molecules from 100 000 to generate diverse leads for drug optimization highly active compounds in several structurally different chemical classes are sought molecules can be characterized by numerical descriptors and the chosen subset should cover the descriptor space or subspaces formed by several descriptors we propose a method that concentrates on low dimensional subspaces a criterion for uniformity of coverage and a fast exchange algorithm to optimize the criterion these methods are illustrated by using a national cancer institute database

orthogonal array based latin hypercubes abstract in this article we use orthogonal arrays oa s to construct latin hypercubes besides preserving the univariate stratification properties of latin hypercubes these strength r oa based latin hypercubes also stratify each r dimensional margin therefore such oa based latin hypercubes provide more suitable designs for computer experiments and numerical integration than do general latin hypercubes we prove that when used for integration the sampling scheme with oa based latin hypercubes offers a substantial improvement over latin hypercube sampling

on the images of the galois representations attached to genus 2 siegel modular forms we address the problem of the determination of the images of the galois representations attached to genus 2 siegel cusp forms of level 1 having multiplicity one these representations are symplectic we prove that the images are as large as possible for almost every prime if the siegel cusp form is not a maass spezialform and verifies two easy to check conditions mathematics subject classification 11f80 11f46

computing matrix group decompositions with respect to a normal subgroup abstract we describe an algorithm which can be used to investigate whether a matrix group defined over a finite field decomposes with respect to a normal subgroup defined as the normal closure of a given set of matrices the possible decompositions correspond to classes in aschbacher s classification of subgroups of the general linear group

potential automorphy and change of weight we prove an automorphy lifting theorem for l adic representations where we impose a new condition at l which we call potentential diagonalizability this result allows for change of weight and seems to be substantially more exible than previous theorems along the same lines we derive several applications for instance we show that any irreducible totally odd essentially self dual regular weakly compatible system of l adic representations of the absolute galois group of a totally real eld is potentially automorphic and hence is pure and its l function has meromorphic continuation to the whole complex plane and satises the expected functional equation

estimating proportions of elements in finite groups of lie type abstract we show that certain closure properties on subsets q of a finite group g of lie type enable the ratio q g to be determined by finding the proportions of elements of q in the maximal tori of g and the proportions of certain related subsets of the weyl group we prove fundamental results about these subsets q including those necessary for moving between these groups and their automorphism groups normal subgroups and central quotients as a sample application of these new techniques we derive upper and lower bounds for the proportion of elements with order divisible by certain natural sets of primes for the finite classical groups thereby improving existing results

a recognition algorithm for special linear groups contain th speciase l linea sl rfr grou q pamplifying the problem he said he would be happy to have a program that couldbe used fo r genera l purpose practical computatio in th range e rf n 30 charle ssims suggested tha onte should be more ambitiou ass to degree sa y d 70 bu trestrict onesel tfo the case q 2 in this pape wr e shall propos ae method whichwe believ teo be practica fol r value osf d up t somewhero e aroun 60d or 70 andwhich is no particularlt y sensitiv to thee size of q w muse t emphasize howeverthat our ideas hav noe t yet at th time e of writing been implemente andd tested until adequate evidenc ha accumulates e d such claim nost mus b takee t n veryseriously ou algorithr ims partly probabilistic it whes respons n ies positiv iettells us that the group generate bdy x certainly does contai sl rfn q but if theoutput is negative that assertion will usually b onle mady e with probabilit 1y e where e is a pre determined small positive number the analogous proble fomr permutatio n group is tso decide fo r a given se t xof permutation osf a set o sizf e n whethe thr e subgroup generate bdy x containsthe alternating group alt n an elegan t probabilistic algorith fo recognizinr m galternating and symmetric group hass been in use fo manr y years le t u cals l anelement of th symmetrie c group sym n if on primae of cycle its has prims elength p non oef the othe r cycles have length divisibl by p ane d ps n 3 themethod exploit thse following facts i if g i alt zs or sym n the an good proportio onf th elemente s o f g areprima ii if g does no contait n alt n and if g i primitivs a e permutation groupthen g contains no such elements the forme irs no hart d to prove see fo exampler 20 lectur e 11 th lattee rassertion comes from a well known old theorem of jordan see fo exampler 30 theorem 13 9 accordingl the ide ya i s to tes it f g i primitives whic cahn bedone pretty quickl byy fo exampler a method of atkinson i f no thet n certainlyg is neither alt n no sym n r i f g i primitives th e crucial ste isp to make asuitable numbe orf independent random selection of elements osf g no mant yare needed see fo r exampl 20 e som if e prima permutation show up thens gis identified correctl as alt ny o r sym n i f not the n with high probabilit g ydoes not contain alt n thus ther is a smale l probability tha the algoritht m willfail to identify alt n or sym n this procedur wa firsts e programme ford

orthogonal column latin hypercubes and their application in computer experiments abstract latin hypercubes have been frequently used in conducting computer experiments in this paper a class of orthogonal latin hypercubes that preserves orthogonality among columns is proposed applying an orthogonal latin hypercube design to a computer experiment benefits the data analysis in two ways first it retains the orthogonality of traditional experimental designs the estimates of linear effects of all factors are uncorrelated not only with each other but also with the estimates of all quadratic effects and bilinear interactions second it can facilitate nonparametric fitting procedures because one can select good space filling designs within the class of orthogonal latin hypercubes according to selection criteria

testing matrix groups for primitivity abstract we describe an algorithm which seeks to decide whether or not a matrix group defined over a finite field acts to preserve blocks of imprimitivity and if so to find a block system implementations of the algorithm are publicly available

black box exceptional groups of lie type if a black box group is known to be isomorphic to an exceptional simple group of lie type of twisted rank 1 other than any f4 q over a field of known size a las vegas algorithm is given to produce a constructive isomorphism in view of its timing this algorithm yields an upgrade of all known nearly linear time monte carlo permutation group algorithms to las vegas algorithms when the input group has no composition factor isomorphic to any group f4 q or g2 q

construction of 2 m4 n designs via a grouping scheme on developpe une methode pour grouper les 2 k 1 effets factoriels dans un plan factoriel a deux niveaux dans des ensembles disjoints mutuellement s t st ou st est l interaction generalisee des effets s et t comme application on construit des tableaux orthogonaux oa 2 k 2 m 4 n 2 de taille 2 k m contraintes avec 2 niveaux et n contraintes avec 4 niveaux satisfaisant m 3n 2 k 1 et de puissance 2 le nombre maximum de contraintes avec 4 niveaux dans la construction ne peut plus etre ameliore dans ce sens le schema de groupement qu on donne est optimal on discute les avantages de la presente approche par rapport a d autres methodes de construction

a computer assisted analysis of some matrix groups abstract we use algorithms developed recently for the study of linear groups to investigate a sequence of matrix groups defined over gf 2 these are images of representations of certain finitely presented groups considered by soicher in a study of simplicial complexes related to the suzuki sequence graphs