reliable numerical methods for polynomial matrix triangularization numerical procedures are proposed for triangularizing polynomial matrices over the field of polynomial fractions and over the ring of polynomials they are based on two standard polynomial techniques sylvester matrices and interpolation in contrast to other triangularization methods the algorithms described in this paper only rely on well worked numerically reliable tools they can also be used for greatest common divisor extraction polynomial rank evaluation or polynomial null space computation

discrete j spectral factorization of possibly singular polynomial matrices we develop two state space algorithms for discrete time j spectral factorization of possibly singular possibly non column reduced and with possible zeros at the origin para hermitian matrices by assignment of matrices in optimal lq return difference equality the column degrees of the spectral factor equal the column degrees of the given matrix and the dimension of a discrete time algebraic riccati system equals the sum of the column degrees necessary and sufficient condition for the least order property of our state space realizations is column reduction of the given matrix

propagating sharp group homology decompositions a collection c of subgroups of a finite group g can give rise to three different standard formulas for the cohomology of g in terms of either the subgroups in c or their centralizers or their normalizers we give a short but systematic study of the relationship among such formulas for nine standard collections c of p subgroups obtaining some new formulas in the process to do this we exhibit some sufficient conditions on the poset c which imply comparison results

on polynomial matrix spectral factorization by symmetric extraction we revise the 1963 davis algorithm 2 for the spectral factorization of a para hermitian nonnegative polynomial matrix phi by symmetric factor extraction this algorithm is careless about zeros at infinity by introducing the notion of diagonal reducedness of phi we obtain an easy sufficient test for the absence of zeros at infinity we show then how to get phi diagonally reduced by diagonal excess reduction steps similar to the oono and yasuura steps removing all zeros at infinity and then how to remove synunetrically finite zeros while keeping el diagonally reduced hence free of zeros at infinity this results in a revised symmetric extraction spectral factorization algorithm with monotone degree control an example shows the didactical conceptual simplicity of the method appropriate symmetric extraction is discovered by revising and discovering important particular one sided factor extraction properties of polynomial matrices

controlled 2 blocks of quasisimple groups abstract controlled p blocks introduced by alperin and broue in 1979 were classified for quasisimple groups g for odd primes in 10 we classify the controlled 2 blocks of quasisimple groups g the results imply that every nilpotent 2 block of g has abelian defect groups this agrees with one of the main results proved in 15 we also give an explicit characterization of non controlled 2 blocks of all quasisimple groups this implies that the p 2 version of the block theoretic analogue of glauberman s zj theorem proved by kessar linckelmann and robinson in 27 holds for quasisimple groups g

screening on correlated variables a bayesian approach let x and y be two correlated variables where the measurement of x is easier or less expensive to make than the one on y suppose that the y measurement is required to meet a certain specification limit given that the joint distribution of x y is bivariate normal and given the sufficient statistics from a random sample from this distribution we propose a bayesian method for determining a cutoff limit on x so that with a guaranteed probability y meets its specification limit monte carlo simulations are used to evaluate the bayesian procedure it is seen to be less conservative than the method of owen li and chou 1981 and it is fairly robust under nonnormality an example using data from one time point to predict results at another time point illustrates the application of the method

2 weights for unitary groups this paper gives a description of the local structures of 2 radical subgroups in a finite unitary group and proves alperin s weight conjecture for finite unitary groups when the characteristic of modular representation is even

an improved procedure for screening based on a correlated normally distributed variable to increase the proportion of units within specifications for a performance variable y a correlated variable x is used to screen units it is assumed that x y follows a bivariate normal distribution with unknown parameters estimated by a preliminary sample the problem is to determine the screening criterion such that at a specified confidence level the proportion of acceptable units in the population after screening exceeds a prescribed proportion one current approach is to compute separate confidence bounds for the unknown parameters and then to determine the screening criterion such that the acceptable proportion is achieved for all sets of parameter values consistent with the confidence bounds that approach can yield extremely conservative results that is too many units are rejected an alternative approach based on approximating the distribution of the estimated conditional probability is proposed and shown to result in less stringent screening

screening in a normal model on presente une approche bayesienne a travers l utilisation de probabilites predictives et on l applique a un modele parametrique dans lequel y x a une distribution normale bivariable

the inductive blockwise alperin weight condition for psln q with n q 1 1 abstract this paper is a continuation of our previous paper in li and zhang 2018 13 we prove that if n q 1 1 psl n q satisfies the inductive blockwise alperin weight condition for any prime l different from the defining characteristic

the h2 control problem a general transfer function solution a general solution of the h2 control problem is presented for linear systems described by rational transfer matrices not necessarily proper or stable the control system is considered in the standard configuration which includes the synthesis model of the plant and the controller the h2 control problem consists of internally stabilizing the control system while minimizing the h2 norm of its transfer function the notion of internal stability is based on bounded input bounded output stability and means that subsystems defined by any pair of input and output signals within the control system all are bounded input bounded output stable in this manner the optimal control system is devoid of impulsive as well as non decaying exponential modes the solution proceeds in three steps first the set of all controllers that internally stabilize the control system is parameterized then the subset of the stabilizing controllers that achieve a finite value of the h2 norm of the control system transfer matrix is

a general theory for the wiener hopf design of multivariable control systems multivariable control system design in a quadratic cost setting is treated for a completely general configuration the class of all controllers is determined for which the general configuration is internally asymptotically stable and a quadratic cost functional is finite this controller class is parameterized in terms of an arbitrary real rational strictly proper matrix z s which is analytic in the closed right hand side of the s plane the choice z identical to 0 gives the controller which minimizes the quadratic cost functional

the blocks and weights of finite special linear and unitary groups abstract this paper has two main parts first we give a classification of the l blocks of finite special linear and unitary groups sl n q in the non defining characteristic l 3 second we describe how the l weights of sl n q can be obtained from the l weights of gl n q when l gcd n q and verify the alperin weight conjecture for sl n q under the condition l gcd n q as a step to establish the alperin weight conjecture for all finite groups we prove the inductive blockwise alperin weight condition for any unipotent l block of sl n q if l gcd n q

weights for symmetric and general linear groups abstract an important feature of the theory of finite groups is the number of connections and analogies with the theory of lie groups the concept of a weight has long been useful in the modular representation theory of finite lie groups in the defining characteristic of the group the idea of a weight in the modular representation theory of an arbitrary finite group was recently introduced in alperin proc sympos pure math 41 1987 369 379 where it was conjectured that the number of weights should equal the number of modular irreducible representations moreover this equality should hold block by block the conjecture has created great interest since its truth would have important consequences a synthesis of known results and solutions of outstanding problems in this paper we prove the conjecture first for the modular representations of symmetric groups and second for modular representations in odd characteristic r for the finite general linear groups in the latter case r may be assumed to be different from the defining characteristic p of the group since the result is known when r is p the well known analogy between the representation theory of the symmetric and general linear groups holds here too

the weighted fusion category algebra and the q schur algebra for gl 2 q abstract we show that the weighted fusion category algebra of the principal 2 block b 0 of gl 2 q is the quotient of the q schur algebra s 2 q by its socle for q an odd prime power as a consequence we get a canonical bijection between the set of isomorphism classes of simple k gl 2 q b 0 modules and the set of conjugacy classes of b 0 weights in this case

uno s invariant conjecture for chevalley groups g2 q in nondefining characteristics abstract uno s invariant conjecture for the chevalley groups g 2 q has been verified when the characteristic of the modular representation is distinct from the defining characteristic of the groups thus dade s reductive conjecture and the isaacs navarro conjecture both hold for g 2 q in non defining characteristics

a coprime factorisation and its application in ℋ2 control given a plant transfer matrix as a fraction of two polynomial matrices we present a doubly coprime factorisation identity over the proper stable rational matrices of the plant transfer matrix and

the inductive blockwise alperin weight condition for psl n q and psu n q with cyclic outer automorphism groups abstract in this paper we prove that if n q is such chosen that the outer automorphism groups of psl n q and psu n q are cyclic psl n q and psu n q satisfy the inductive blockwise alperin weight condition for any prime l different from the defining characteristic to do this it suffices to study actions of automorphisms on the brauer characters and weights of gl n q and gu n q for l and to show that the blockwise bijection between brauer characters and weights given in 1 3 4 and 5 is equivariant

an algorithm for polynomial matrix factor extraction an algorithm is described for extracting a polynomial matrix factor featuring any subset of the zeros of a given non singular polynomial matrix it is assumed that the zeros to be extracted are given as input data complex or repeated zeros are allowed the algorithm is based on interpolation and relies upon numerically reliable subroutines only it makes use of a procedure that computes the generalized characteristic vectors of a polynomial matrix at a given point the extracted factor is provided in column and row reduced popov form applications of the algorithm include polynomial matrix interpolation plus minus factorization column and row reduction or computation of the smith form of a polynomial matrix the numerical routines described in this paper are implemented in the new release 2 0 of the polynomial toolbox for matlab

radical 3 subgroups and essential 3 rank of f4 q abstract we classify maximal 3 local subgroups and radical 3 subgroups of g f 4 q when q odd and then determine the essential 3 rank of the frobenius category f d g where d is a sylow 3 subgroup of g