probabilistic generation of finite simple groups ii abstract in earlier work it was shown that each nonabelian finite simple group g has a conjugacy class c such that whenever 1 x g the probability is greater than 1 10 that g x y for a random y c much stronger asymptotic results were also proved here we show that allowing equality the bound 1 10 can be replaced by 13 42 and excluding an explicitly listed set of simple groups the bound 2 3 holds we use these results to show that any nonabelian finite simple group g has a conjugacy class c such that if x 1 x 2 are nontrivial elements of g then there exists y c such that g x 1 y x 2 y similarly aside from one infinite family and a small explicit finite set of simple groups g has a conjugacy class c such that if x 1 x 2 x 3 are nontrivial elements of g then there exists y c such that g x 1 y x 2 y x 3 y we also prove analogous but weaker results for almost simple groups

anatomy of the monster ii abstract we summarise much of what is currently known about the subgroup structure of the fischer griess monster introduction the publication of the atlas of finite groups 1 henceforth referred to as the atlas marked the end of a chapter in the history of group theory in more ways than one the classification of finite simple groups had been completed and using new computational techniques the last of the sporadic groups had been constructed and enough progress had been made in classifying their maximal subgroups to make the rest seem like a tidying up job since the publication of the atlas these computational methods have been pushed further the classification of the maximal subgroups for fi 22 th fi 23 j 4 and fi 24 was completed by kleidman linton parker and wilson in 7 8 9 10 12 13 14 this means that appendix 2 of the atlas of brauer characters 11 when collated with the atlas itself completes the listing for all sporadic groups except the baby monster and monster for the baby monster four new maximal subgroups s 6 l 3 4 2 2 s 6 s 6 4 l 2 31 and l 2 49 2 3 were given in appendix 2 of 11 and wilson has since found four more m 11 l 3 3 l 2 17 2 and l 2 ll 2 for references see 23 24 25 with work on the 2 locals by meierfrankenfeld and shpektorov 15 an end to the baby monster problem is in sight with a lot of work also having been done on maximal subgroups of generic groups this leaves only the monster unfortunately the degree of its smallest representation over any field 196882 is too big for present day computational techniques to make any headway

diffraction of elastic waves by a penny shaped crack consider an infinite elastic solid containing a penny shaped crack we suppose that time harmonic elastic waves are incident on the crack and are required to determine the scattered displacement field u i in this paper we describe a new method for solving the corresponding linear boundary value problem for u i which we denote by s we begin by defining an elastic double layer we prove that any solution of s can be represented by an elastic double layer whose density satisfies certain conditions we then introduce various green functions and define a new crack green function g ij that is discontinuous across the crack next we use g ij to derive a fredholm integral equation of the second kind for the discontinuity in u i across the crack we prove that this equation always has a unique solution hence we are able to prove that the original boundary value problem s always possesses a unique solution and that this solution has an integral representation as an elastic double layer whose density solves an integral equation of the second kind

analysis of repeated measurements using nonparametric smoothers and randomization tests the mixed model analysis of variance anova which is commonly applied to repeated measurements taken over time depends on specialized assumptions about the error distribution and fails to exploit information contained in the ordering of the data points over time this paper describes a procedure that overcomes these disadvantages while preserving familiar features of the mixed model anova group profiles are estimated by nonparametric smoothing of observed mean profiles group and time main effects and the group by time interaction effect are tested using randomization tests results of zerbe 1979 journal of the american statistical association 74 215 221 are used to construct f test approximations for the randomization tests of the group and group by time effects a new approximate f test for time effect is proposed a simulation study demonstrates that the approximations perform well and that smoothing increases the power of the tests for time main effect and group by time interaction the procedure is applied to data on hormone levels in cows

subgroups of maximal rank in finite exceptional groups of lie type the purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of lie type which we call subgroups of maximal rank these subgroups play a prominent role both in the classification of local maximal subgroups in 9 theorem 1 and in the reduction theorem for subgroups in 28 theorem 2 in 9 it is shown that any local maximal subgroup of a finite exceptional group of lie type is either a subgroup of maximal rank or one of a small list of exceptions and in 28 using this result it is proved that any maximal subgroup is either of maximal rank or almost simple or in a known list to describe these subgroups we require the following notation let g be a simple adjoint algebraic group over an algebraically closed field k of characteristic p 0 and let o be an endomorphism of g such that l ga is a finite exceptional simple group of lie type over fq where q p let a be a group such that f x l the group aut l is generated by ga together with field and graph automorphisms see 5 38 all of which extend to morphisms of the abstract group g commuting with o thus there is a subgroup x of caut c a such that x x o and so x acts on the set of a stable subsets of g for a a stable subset y we write nx y for the stabilizer in x of y if d is a a stable closed connected reductive subgroup of g containing a maximal torus t of g and m nx d we call m a subgroup of maximal rank in x in this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in x

linear groups with orders having certain large prime divisors in this paper we obtain a classification of those subgroups of the finite general linear group gld q with orders divisible by a primitive prime divisor of qe 1 for some in the course of the analysis we obtain new results on modular representations of finite almost simple groups in particular in the last section we obtain substantial extensions of the results of landazuri and seitz on small cross characteristic representations of some of the finite classical groups 1991 mathematics subject classification primary 20g40 secondary 20c20 20c33 20c34 20e99

assessment of dpoae test retest difference curves via hierarchical gaussian processes distortion product otoacoustic emissions dpoae testing is a promising alternative to behavioral hearing tests and auditory brainstem response testing of pediatric cancer patients the central goal of this study is to assess whether significant changes in the dpoae frequency emissions curve dp gram occur in pediatric patients in a test retest scenario this is accomplished through the construction of normal reference charts or credible regions that dp gram differences lie in as well as contour probabilities that measure how abnormal or in a certain sense rare a test retest difference is a challenge is that the data were collected over varying frequencies at different time points from baseline and on possibly one or both ears a hierarchical structural equation gaussian process model is proposed to handle the different sources of correlation in the emissions measurements wherein both subject specific random effects and variance components governing the smoothness and variability of each child s gaussian process are coupled together

nonparametric regression analysis of growth curves in recent years nonparametric curve estimates have been extensively explored in theoretical work there has however been a certain lack of convincing applications in particular involving comparisons with parametric techniques the present investigation deals with the analysis of human height growth where longitudinal measurements were collected for a sample of boys and a sample of girls evidence is presented that kernel estimates of acceleration and velocity of height and of height itself might offer advantages over a parametric fitting via functional models recently introduced for the specific problem considered both approaches are biased but the parametric one shows qualitative and quantitative distortion which both are not easily predictable data analytic problems involved with kernel estimation concern the choice of kernels the choice of the smoothing parameter and also whether the smoothing parameter should be chosen uniformly for all subjects or individually

a solvable version of the baer suzuki theorem suppose that g is a finite group and x g has prime order p 5 then x is contained in the solvable radical of g o g if and only if x x g is solvable for all g g if g is an almost simple group and x g has prime order p 5 then this implies that there exists g g such that x x g is not solvable in fact this is also true when p 3 with very few exceptions which are described explicitly

miscellanea random coefficient autoregressive models for longitudinal data summary stable autoregressive models with exogenous covariates can be used to mimic nonstationary stabilising growth profiles in longitudinal data if different individuals have very different trajectories the autoregressive parameters can be interpreted as random variables varying from one individual to the next if all random effects are assumed gaussian the model will be quite easy to estimate although the marginal distributions of the observations will no longer be gaussian however this kind of model can only be used when the observations are equally spaced

recovering gradients from sparsely observed functional data summary the recovery of gradients of sparsely observed functional data is a challenging ill posed inverse problem given observations of smooth curves e g growth curves at isolated time points the aim is to provide estimates of the underlying gradients or growth velocities to address this problem we develop a bayesian inversion approach that models the gradient in the gaps between the observation times by a tied down brownian motion conditionally on its values at the observation times the posterior mean and covariance kernel of the growth velocities are then found to have explicit and computationally tractable representations in terms of quadratic splines the hyperparameters in the prior are specified via nonparametric empirical bayes with the prior precision matrix at the observation times estimated by constrained l1 minimization the infinitessimal variance of the brownian motion prior is selected by cross validation the approach is illustrated using both simulated and real data examples

a comparison of gcv and gml for choosing the smoothing parameter in the generalized spline smoothing problem the partially improper prior behind the smoothing spline model is used to obtain a generalization of the maximum likelihood gml estimate for the smoothing parameter then this estimate is compared with the generalized cross validation gcv estimate both analytically and by monte carlo methods the comparison is based on a predictive mean square error criteria it is shown that if the true unknown function being estimated is smooth in a sense to be defined then the gml estimate undersmooths relative to the gcv estimate and the predictive mean square error using the gml estimate goes to zero at a slower rate than the mean square error using the gcv estimate if the true function is rough then the gcv and gml estimates have asymptotically similar behavior a monte carlo experiment was designed to see if the asymptotic results in the smooth case were evident in small sample sizes mixed results were obtained for n 32 gcv was somewhat better than gml for n 64 and gcv was decidedly superior for n 128 in the n 32 case gcv was better for smaller sigma 2 and the comparison close for larger sigma 2 the theoretical results are shown to extend to the generalized spline smoothing model which includes the estimate of functions given noisy values of various integrals of them

a successive differences method for growth curves with missing data and random observation times abstract incomplete growth curve data can be analyzed by the successive differences sd method which uses the difference in consecutive pairs of observations for all subjects having two or more repeated measures we have generalized it to handle varying observation times as well by partitioning the time interval spanned by all repeated measures into subintervals the model is developed including test statistics for the hypotheses of parallelism and of no change in response level over time assuming parallelism this generalized sd method is then applied to repeated serum cholesterol measurements on a subsample of 1 072 men from a prospective cohort study of 8 006 hawaiian japanese men living on oahu

the maximal subgroups of the chevalley groups g2 q with q odd the ree groups 2g2 q and their automorphism groups

a bayesian model for growth curve analysis an experiment involves k subjects where for subject i ni values yi1 yi2 y ini of a random variable y are observed at times ti1 ti2 t ini assume that yij f tij eij where eij are independently and identically distributed i i d n 0 sigma2 we consider the estimation of the function f and the testing of the homogeneity hypothesis that for formula see text does not depend on t the function f i t is modelled as a gaussian process which seeks to quantify the notions that for each i f i t is a slowly changing function of t and that for i is not equal to j f i t and f j t are in some sense similar we propose to estimate f i t by its posterior mean given all of the data this bayes estimate is shown to be equivalent to a particular form of penalised likelihood estimation we consider data based methods for setting the parameters of the gaussian process prior develop a test of the homogeneity hypothesis report the results of a monte carlo study illustrating the effectiveness of the proposed methodology and apply the methods to a study of variations in temperature and blood pressure over the course of the menstrual cycle

further solvable analogues of the baer suzuki theorem and generation of nonsolvable groups abstract let g be an almost simple group we prove that if x g has prime order p 5 then there exists an involution y such that x y is not solvable also if x is an involution then there exist three conjugates of x that generate a nonsolvable group unless x belongs to a short list of exceptions which are described explicitly we also prove that if x has order 6 or 9 then there exist two conjugates that generate a nonsolvable group

application of nonlinear models with random coefficients to growth data summary the application of nonlinear random coefficient models to the analysis of growth curve data is described the approach is further developed for the estimation of mean growth curves and their variability from mark recapture data when the age of an animal at first capture is unknown but the time between successive captures is known these methods are of wide applicability as shown by the analysis of data on the growth of emus dromaius novaehollandiae noisy scrub birds atrichornis clamosus and whelks dicathais aegaota simulations of a mark recapture experiment on the rock spiny lobster panulirus argus in mexico using a known growth curve showed the effectiveness of the approach in estimating both the population s growth curve and the variability in individual growth

a generalisation of a theorem of wielandt abstract in 1974 helmut wielandt proved that in a finite group g a subgroup a is subnormal if and only if it is subnormal in a g for all g g in this paper we prove that the subnormality of an odd order nilpotent subgroup a of g is already guaranteed by a seemingly weaker condition a is subnormal in g if for every conjugacy class c of g there exists c c for which a is subnormal in a c we also prove the following property of finite non abelian simple groups if a is a subgroup of odd prime order p in a finite almost simple group g then there exists a cyclic p subgroup of f g which does not normalise any non trivial p subgroup of g that is generated by conjugates of a

factorizations of simple algebraic groups we determine all factorizations of simple algebraic groups as the product of two maximal closed connected subgroups additional results are established which drop the maximality assumption and applications are given to the study of subgroups of classical groups transitive on subspaces of a given dimension