model categories of diagram spectra working in the category mathcal t of based spaces we give the basic theory of diagram spaces and diagram spectra these are functors mathcal d longrightarrow mathcal t for a suitable small topological category mathcal d when mathcal d is symmetric monoidal there is a smash product that gives the category of mathcal d spaces a symmetric monoidal structure examples include begin enumerate item prespectra as defined classically item symmetric spectra as defined by jeff smith item orthogonal spectra a coordinate free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category item gamma spaces as defined by graeme segal item mathcal w spaces an analogue of gamma spaces with finite sets replaced by finite cw complexes in the domain category end enumerate we construct and compare model structures on these categories with the caveat that gamma spaces are always connective these categories and their simplicial analogues are quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category monoids in these categories are strict ring spectra often the subcategories of ring spectra module spectra over a ring spectrum and commutative ring spectra are also model categories when this holds the respective categories of ring and module spectra are quillen equivalent and thus have equivalent homotopy categories this allows interchangeable use of these categories in applications 2000 mathematics subject classification primary 55p42 secondary 18a25 18e30 55u35

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

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 models for flow induced deformation of soft biological tissue abstract the behaviour of a deformable porous medium during the flow of fluid under a pressure difference is examined for both infinitesimal and finite deformations models for both cases are solved for the problem of steady one dimensional compression and compared with experimental data from parker et al j appl mech 54 794 800 1987 for a polyurethane sponge the purpose of this study is to identify a simple model which agrees qualitatively with these published results to relate the stress relations for biological tissues to the data for polymer sponges parker et al 1987 a translation of 1 1 kpa was introduced this allows for some structural differences between the two media it was found that the infinitesimal models were adequate up to 20 strain but significant divergence occurred for higher strains a finite deformation model with the permeability depending exponentially on the strain gave the most consistent results and required the fitting of only two parameters

comparison theorems and similarity solution approximations for a nonlinear diffusion equation arising in the study of soft tissue a nonlinear diffusion equation arising from a biphasic continuum model of soft tissue such as for articular cartilage is studied through the use of maximum principles and comparison theorems the quantitative characteristics of the solution are established independent of the particular nonlinear constitutive equation used for the solid fluid interaction in the tissue in conjunction with this similarity and asymptotic methods are employed to obtain approximate solutions to the problem

finite groups of symplectic automorphisms of k3 surfaces in positive characteristic we show that mukai s classication of nite groups which may act symplectically on a complex k3 surface extends to positive characteristic p under assumptions that i the order of the group is coprime to p and ii either the surface or its quotient is not birationally isomorphic to a supersingular k3 surface with artin invariant 1 in the case without assumption ii we classify all possible new groups which may appear we prove that assumption i on the order of the group is always satised if p 11 for p 2 3 5 11 we give examples of k3 surfaces with nite symplectic automorphism groups of order divisible by p which are not contained in mukai s list

nonlinear equation governing flow in a saturated porous medium it is argued that the appropriate generalization of darcy s law when inertia effects are included takes the form p k v c k v v div v 0 where k is the permeability of the medium and the form drag constant c is a coefficient which is independent of the pressure p the seepage velocity v and the density and viscosity of the fluid but which is dependent on the geometry of the medium we formulate a nonlinear extension of brinkman s self consistent theory for the flow of a viscous fluid through a swarm of spherical particles we equate the drag per unit volume given by the right hand side of the first of the above equations to the total drag nd on the n particles contained within that unit volume in an infinite region where d is the drag on a single particle placed in a velocity field v subject to v v grad p 2v k v c k v v div v 0 v is a prescribed constant where is the viscosity without solving these equations we obtain an estimate for c from the known experimental drag law for a solid sphere placed in a uniform stream

tate s conjecture for k3 surfaces of finite height this paper extends the proof 16 of the tate conjecture for ordinary k3 surfaces over a finite field to the more general case of all k3 s of finite height as in 16 our method is to find a lifting of the k3 to characteristic zero with sufficiently many hodge cycles in the ordinary case the so called canonical lifting of deligne and illusie 7 did the job and a study of the galois action on p adic etale cohomology revealed the hodge cycles here we use more general quasi canonical liftings and the action of the crystalline weil group on

a generalization of the burnside basis theorem abstract a b group is a group such that all its minimal generating sets with respect to inclusion have the same size we prove that the class of finite b groups is closed under taking quotients and that every finite b group is solvable via a complete classification of frattini free finite b groups we obtain a general structure theorem for finite b groups applications include new proofs for the characterization of finite matroid groups

morava modules and brown comenetz duality fix a prime number p and let x be a finite spectrum whose n 1 st morava k theory is trivial but whose n th morava k theory is nontrivial n 0 we prove following a method outlined to us by m j hopkins that if 2 p n 2 n 2 the morava module of the brown comenetz dual of the e n localization of x is isomorphic to a suspension of the pontryagin dual of the morava module of x to complete this proof we found it necessary to develop a more canonical construction of certain modified adams spectral sequences this construction should be of independent interest

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

stability of runge kutta methods for quasilinear parabolic problems we consider a quasilinear parabolic problem u t q u t u t u t 0 u 0 d where q w d c x x w w c x is a family of sectorial operators in a banach space x with fixed domain d this problem is discretized in time by means of a strongly a stable 0 2 runge kutta method we prove that the resulting discretization is stable under some natural assumptions on the dependence of q w with respect to w our results are useful for studying in l p norms 1 p many problems arising in applications some auxiliary results for time dependent parabolic problems are also provided

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

a class of 2 local finite spectra which admit a v21 self map abstract at the prime 2 behrens hill hopkins and mahowald showed that m 2 1 4 admits a 32 periodic v 2 self map more recently in joint work with mahowald we showed that a 1 also admits a 32 periodic v 2 self map this leads to the question of whether there exists a finite 2 local complex with periodicity less than 32 we answer this question in the affirmative by producing a class of finite 2 local spectra z all of which admit a 1 periodic v 2 self map

chromatic homotopy theory is asymptotically algebraic inspired by the ax kochen isomorphism theorem we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects we employ this theory to give an asymptotic solution to the approximation problem in chromatic homotopy theory more precisely we show that the ultraproduct of the e n p local categories over any non principal ultrafilter on the set of prime numbers is equivalent to the ultraproduct of certain algebraic categories introduced by franke this shows that chromatic homotopy theory at a fixed height is asymptotically algebraic

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

on some sets of generators of finite groups abstract in several papers finite groups with fixed cardinalities of sets of generators are investigated and sometimes are named b groups groups with all subgroups being b groups are called groups with the basis property in the first part of this paper we correct and generalize some characterizations of groups with the basis property in the second part we consider groups satisfying analogous conditions but only for sets of generators of prime power orders the class of groups introduced here is much larger then the class of groups with the basis property for example it contains all nilpotent groups

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

independent generating sets and geometries for symmetric groups abstract julius whiston showed that the size of an independent generating set in the symmetric group s n is at most n 1 we determine all sets meeting this bound we also give some general remarks on the maximum size of an independent generating set of a group and its relationship to coset geometries for the group in particular we determine all coset geometries of maximum rank for the symmetric group s n for n 6

a mixture approach to the mechanics of skin skin can be considered to be a mixture of a solid and a fluid a general theory for the description of the behaviour of mixtures is presented and applied to a mixture of a solid and a fluid a numerical procedure is presented to solve the non linear field equations describing such a mixture the abilities of the procedure are demonstrated by means of a confined compression test

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

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

crystals and derived local moduli for ordinary k3 surfaces abstract it is shown that the k3 spectra which refine the local rings of the moduli stack of ordinary p primitively polarized k3 surfaces in characteristic p allow for an e structure which is unique up to equivalence this uses the e obstruction theory of goerss and hopkins and the description of the deformation theory of such k3 surfaces in terms of their hodge f crystals due to deligne and illusie furthermore all automorphisms of such k3 surfaces can be realized by e maps which are unique up to homotopy and this can be rigidified to an action if the automorphism group is tame

on solutions to a quasi linear diffusion problem from the study of soft tissue a continuum mechanics model for the unidirectional deformation of soft tissue can be approximated by a quasi linear diffusion equation u t exp u x u xx defined on the scaled spatial interval 0 1 with boundary conditions given by u x 0 0 u 1 t 0 u 0 t g t conditions on g t under which the problem has no global solutions as well as conditions when such a solution exists are given the behavior of the nonglobal solution near the time t t when it ceases to exist is also studied

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

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

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