two conditions concerning common quadratic lyapunov functions for linear systems concerning a pair of linear systems some triangle conditions for the existence of a common quadratic lyapunov function are presented in this paper these conditions are derived from a criterion for judging the semipositiveness of a linear map defined on symmetric matrices

the sensitivity of the stable lyapunov equation we present an analysis of the sensitivity of the solution of the lyapunov equation a x xa w where a is stable this analysis leads to a spectral norm bound on the relative perturbation of the solution which is optimal for a certain class of estimates and which is essentially equivalent to the frobenius norm bound obtained from the associated kronecker product system the latter bound can be expressed in terms of operatorname sep a a and is known to accurately reflect the sensitivity of the lyapunov problem but it is hard to interpret in terms of the original matrix a in contrast the spectral norm bound which we derive is directly related to the minimal l 2 damping of the dynamical system dot z az moreover this dynamical link with the sensitivity problem leads to a new method of systematically investigating the norm behavior of e at as well as providing a wealth of information about control theoretic aspects of dot z az when a is the closed loop state matrix

on a common quadratic lyapunov function for widely distant systems a sufficient condition concerning the construction of a common quadratic lyapunov function for widely distant linear systems is presented the condition is set on a mutual relation matrix whose size is just equal to the number of the coefficient matrices

on the number of homomorphisms from a finite group to a general linear group abstract we study the number of homomorphisms from a finite group to a general linear group over a finite field in particular we give a generating function of such numbers then the rogers ramanujan identities are applicable

trace norm bounds for stable kyapunov operators abstract certain aspects of stable lyapunov operators can be easily studied by exploiting the linearity of the trace operator and its invariance under reversal of order in matrix products for example sharp upper and lower bounds on the trace of solutions to the stable lyapunov equation can be obtained by applying the trace operator to a well known integral representation of these solutions other applications include using the connection between dual norms and the trace operator to obtain new results on the norms of lyapunov operators associated with the conditioning of solutions to the riccati equation in this regard trace norm results can be obtained from well known spectral norm results since the trace and spectral norms are dual to each other a somewhat deeper analysis involving the power method gives monotonically decreasing upper bounds on the frobenius norms of these lyapunov operators these upper bounds complement the usual monotonically increasing lower bounds associated with the power method and provide a nice means of assessing the accuracy of the resulting frobenius norm estimates

the probability distribution of word maps on finite groups abstract word maps provide a wealth of information about finite groups we examine the connection between the probability distribution induced by a word map and the underlying structure of a finite group we show that a finite group is nilpotent if and only if every surjective word map has fibers of uniform size moreover we show that probability distributions themselves are sufficient to identify nilpotent groups and these same distributions can be used to determine abelian groups up to isomorphism in addition we answer a question of amit and vishne

singularity conditions for the non existence of a common quadratic lyapunov function for pairs of third order linear time invariant dynamic systems abstract the condition that a finite collection of stable matrices a 1 a m has no common quadratic lyapunov function cqlf is formulated as a hierarchy of singularity conditions for block matrices involving a number of unknown parameters these conditions are applied to the case of two stable 3 3 matrices where they are used to derive necessary and sufficient conditions for the non existence of a cqlf

to how many simultaneous hypothesis tests can normal student s t or bootstrap calibration be applied in the analysis of microarray data and in some other contemporary statistical problems it is not uncommon to apply hypothesis tests in a highly simultaneous way the number n say of tests used can be much larger than the sample sizes n to which the tests are applied yet we wish to calibrate the tests so that the overall level of the simultaneous test is accurate often the sampling distribution is quite different for each test so there may not be an opportunity to combine data across samples in this setting how large can n be as a function of n before level accuracy becomes poor here we answer this question in cases where the statistic under test is of student s t type we show that if either the normal or student t distribution is used for calibration then the level of the simultaneous test is accurate provided that log n increases at a strictly slower rate than n 1 3 as n diverges on the other hand if bootstrap methods are used for calibration then we may choose log n almost as large as

stability robustness for linear state space models a lyapunov mapping approach some pythagorean like conditions for the existence of a common lyapunov solution for two given matrices are presented these conditions are set on the lyapunov mappings associated with given matrices the key is the semipositiveness of linear mappings defined on symmetric matrices from the viewpoint of the existence of a common lyapunov solution some known stability robustness results are reviewed and refined

on the cauchy index of linear systems the cauchy index for linear siso systems is given by the signature of the cross gramian matrix w co defined by fernando and nicholson 1 this index is useful in the qualitative understanding of systems and the importance of this index in internally balanced model reduction is illustrated

hall s relations in finite groups let h be a finite group and let be an automorphism of h whose order divides n hall proved that the number of elements x of h that satisfy the relation xx x 2 x n 1 1 is a multiple of gcd n h for a prime factor p of gcd n h if such a number is not a multiple of gcd pn h then a sylow p subgroup of h is exceptional

necessary and sufficient conditions for stabilization of discrete time planar switched systems abstract this paper discusses the stabilization problem for single input discrete time planar switched systems we establish necessary and sufficient conditions for the stabilization of planar switched systems a series of linear inequalities are presented for describing the set of all common quadratic lyapunov functions our results are not only easily testable but also constructive

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

the number of subgroups of a finite group let a be a finite group and let p be a prime suppose that ps is the highest power of p dividing a a where a is the commutator subgroup of a and that the type 1 2 with 1 2 of a

innovated higher criticism for detecting sparse signals in correlated noise higher criticism is a method for detecting signals that are both sparse and weak although first proposed in cases where the noise variables are independent higher criticism also has reasonable performance in settings where those variables are correlated in this paper we show that by exploiting the nature of the correlation performance can be improved by using a modified approach which exploits the potential advantages that correlation has to offer indeed it turns out that the case of independent noise is the most difficult of all from a statistical viewpoint and that more accurate signal detection for a given level of signal sparsity and strength can be obtained when correlation is present we characterize the advantages of correlation by showing how to incorporate them into the definition of an optimal detection boundary the boundary has particularly attractive properties when correlation decays at a polynomial rate or the correlation matrix is toeplitz

approximate balanced truncation model reduction for large scale systems the complexity of a large scale system makes the computations of reduced order models based on balancing impractical however typically there is a weak coupling between the subsystems of a large scale system this is used here to derive approximate balanced truncation reduced order models for large scale systems with reliable and tractable computations

degrees of controllability and observability for close modes of flexible space structures flexible structures usually possess vibration modes with close but not exactly repeated natural frequencies quantifying the controllability and observability of such modes is a nontrivial task this contrasts with the case of modes that are either exactly repeated or widely separated the paper derives expressions for the degrees of controllability and observability of near repeated modes by examining the closed form grammians of flexible structures the results obtained quantify the role of damping and frequency separation in controllability and observability

hom a g abstract in this paper we prove that for any finite group g and any finite abelian group a the number of homomorphism from a to g is divisible by the greatest common divisor of a and g this result is a generalization of an old theorem of frobenius on the number of solutions of the equation x n 1 on a finite group

hom a g iv abstract in this paper we prove that for any finite groups a and g the number of homomorphisms from a to g is divisible by the greatest common divisor of a a a a and g where a is the commutator subgroup of a and a a is the frattini subgroup of a a

the simes method for multiple hypothesis testing with positively dependent test statistics abstract the simes method for testing intersection of more than two hypotheses is known to control the probability of type i error only when the underlying test statistics are independent although this method is more powerful than the classical bonferroni method it is not known whether it is conservative when the test statistics are dependent this article proves that for multivariate distributions exhibiting a type of positive dependence that arise in many multiple hypothesis testing situations the simes method indeed controls the probability of type i error this extends some results established very recently in the special case of two hypotheses

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

direct computation of stability bound for systems with polytopic uncertainties the stability of linear systems subject to possibly fast time varying uncertainties is analyzed a method is developed to compute the upper bound of uncertainty size allowed for retaining quadratic stability the algorithm consists of choosing an extremum among a finite number of values it is proved that although the outer level of the algorithm is a nonconvex optimization problem no local minimum distinct from the global minimum can exist an illustrative example is presented

stabilization of switched linear systems with mode dependent time varying delays abstract in this paper we investigate the problem of stabilization via state feedback and or state based switching for switched linear systems with mode dependent time varying delays by using the multiple lyapunov functional method we establish sufficient conditions that guarantee the switched system is stabilizable via state feedback and or switching under time varying delays with appropriate upper bounds the main results are presented in terms of linear matrix inequalities lmis which generalize some known results and can be easily tested by using the matlab s lmi tool box

on quadratic lyapunov functions a topological structure as a subset of 0 2 spl pi sup l spl times spl ropf sub sup n 1 is proposed for the set of quadratic lyapunov functions qlfs of a given stable linear system a necessary and sufficient condition for the existence of a common qlf of a finite set of stable matrices is obtained as the positivity of a certain integral the structure and the conditions are considerably simplified for planar systems it is also proved that a set of block upper triangular matrices share a common qlf if each set of diagonal blocks share a common qlf

asymptotic comparison of step down and step up multiple test procedures based on exchangeable test statistics in this paper interest is focused on some theoretical investigations concerning the comparison of two popular multiple test procedures so called step down and step up procedures in terms of their defining critical values such procedures can be applied for example to multiple comparisons with a control in the definition of the critical values for these procedures order statistics play a central role for k n 0 fixed we consider the joint cumulative distribution function cdf p y 1 n c 1 y n k n c n k of the first n k order statistics and the cdf p y n k n c n k of the k 1 th largest order statistic y n k n of n random variables y1 y n belonging to a sequence of exchangeable real valued random variables our interest is focused on the asymptotic behavior of these cdf s and their interrelation if n tends to it turns out that they sometimes behave completely differently compared with the iid case treated in finner and roters so that positive results are only possible under additional assumptions concerning the underlying distribution we consider different sets of assumptions which then allow analogous results for the exchangeable case recently dalal and mallows derived a result concerning the monotonicity of a certain set of critical values in connection with the joint cdf of order statistics in the iid case we give a counterexample for the exchangeable case underlining the difficulties occurring in this situation as an application we consider the comparison of certain step down and step up procedures in multiple comparisons with a control the results of this paper yield a more theoretical explanation of the superiority of the step up procedure which has been observed earlier by comparing tables of critical values as a byproduct we are able to quantify the tightness of the bonferroni inequality in connection with maximum statistics

hierarchical recognition of sparse patterns in large scale simultaneous inference summary we study how to accurately separate signals from noisy data and determine the patterns of the selected signals controlling the inflation of false positive errors is an important issue in largescale simultaneous inference but has not been addressed in the pattern recognition literature we develop a decision theoretic framework and formulate the sparse pattern recognition problem as a simultaneous inference problem with multiple decision trees oracle and adaptive classifiers are proposed for maximizing the expected number of true positives subject to a constraint on the overall false positive rate existing results on multiple testing are extended by allowing more than two states of nature hierarchical decision making and new error rate concepts

an implicit deflation method for ill conditioned sylvester and lyapunov equations this paper describes an effective algorithm for solving ill conditioned sylvester or lyapunov equations these equations can be solved by conventional methods for ill conditioned linear systems however such methods are not efficient since they require on the order of n 6 arithmetic operations on order n 2 data for these ill conditioned matrix equations an implicit deflation algorithm is proposed to implement a certain svd based minimum norm least squares approximate solution the method is practical since it relies only on the ability to solve a sylvester or lyapunov equation certain practical details are also discussed

robust stability of state space models with structured uncertainties a method for robust eigenvalue location analysis of linear state space models affected by structured real parametric perturbations is proposed the approach based on algebraic matrix properties deals with state space models in which system matrix entries are perturbed by polynomial functions of a set of uncertain physical parameters a method converting the robust stability problem into nonsingularity analysis of a suitable matrix is proposed the method requires a check of the positivity of a multinomial form over a hyperrectangular domain in parameter space this problem which can be reduced to finding the real solutions of a system of polynomial equations simplifies considerably when cases with one or two uncertain parameters are considered for these cases necessary and sufficient conditions for stability are given in terms of the solution of suitable real eigenvalue problems

on optimality of stepdown and stepup multiple test procedures consider the multiple testing problem of testing k null hypotheses where the unknown family of distributions is assumed to satisfy a certain monotonicity assumption attention is restricted to procedures that control the familywise error rate in the strong sense and which satisfy a monotonicity condition under these assumptions we prove certain maximin optimality results for some well known stepdown and stepup procedures

dependency and false discovery rate asymptotics some effort has been undertaken over the last decade to provide conditions for the control of the false discovery rate by the linear step up procedure lsu for testing n hypotheses when test statistics are dependent in this paper we investigate the expected error rate eer and the false discovery rate fdr in some extreme parameter configurations when n tends to infinity for test statistics being exchangeable under null hypotheses all results are derived in terms of p values in a general setup we present a series of results concerning the interrelation of simes rejection curve and the limiting empirical distribution function of the p values main objects under investigation are largest limiting crossing points between these functions which play a key role in deriving explicit formulas for eer and fdr as specific examples we investigate equi correlated normal and t variables in more detail and compute the limiting eer and fdr theoretically and numerically a surprising limit behavior occurs if these models tend to independence

crossed homomorphisms from rank 2 abelian to exceptional p groups abstract in this paper we shall verify the conjecture z 1 a g 0 mod gcd a a g of t asai t yoshida j algebra 160 1993 273 285 in the case where a is an abelian group of rank 2 and g is an exceptional p group for a prime p namely one of the following groups for p 2 a cyclic 2 group a dihedral 2 group a generalized quaternion 2 group and a semidihedral 2 group while for p 2 a cyclic p group

stabilization of uncertain systems with norm bounded uncertainty a control a robust stabilization problem in a state space setting is treated it is assumed that the states are available for feedback using a fixed lyapunov function approach quadratic stability it is shown that an open loop stabilizability condition is equivalent to the existence of a stabilizing memoryless linear state feedback controller as a consequence it is shown that the existence of a quadratically stabilizing nonlinear time varying dynamic state feedback controller implies the existence of a quadratically stabilizing memoryless linear time invariant state feedback compensator

technical communique improvement of system order reduction via balancing using the method of singular perturbations in this paper we first present several alternative techniques for system order reduction based on system balancing by employing the method of singular perturbations the techniques obtained have the same robustness accuracy evaluated with respect to the h norm of the reduced order system as the two techniques existing in the control literature known as the direct truncation and balancing residualization methods the newly proposed techniques preserve the exact dc gain as the original system and produce very good to excellent accuracy at low and medium frequencies we also present a simple formula that fixes the dc gain for the direct truncation method in the second part of the paper we propose how to perform order reduction of a balanced system using theory of singular perturbations that can produce very good accuracy at high frequencies particularly for systems that have lightly damped highly oscillatory modes the result obtained for such kind of systems requires that the modes with higher hankel singular values should be neglected and those with smaller hankel singular values should be retained this is a counter intuitive result in the spirit of h theory but an intuitive result in the spirit of theory of singular perturbations the second part of the paper raises an important question about validity of the system robustness estimate based on the h norm for linear systems with lightly damped highly oscillatory modes

on the simultaneous diagonal stability of linear discrete time systems abstract this paper studies the set of time invariant linear discrete time systems in which each system has a diagonal quadratic lyapunov function first it is shown that there is generally no common diagonal quadratic lyapunov function for such a set of systems even if the set is assumed to be commutative it is also shown that the commutativity assures the existence of a common diagonal quadratic lyapunov function inside the set of 2 2 systems or the set of nonnegative systems then two simple topological results are presented concerning the simultaneous diagonal stability on the set of nonnegative systems the first is a measure of the difference of matrices that assures the simultaneous diagonal stability the second is a measure of the commutativity of matrices

multiple hypotheses testing and expected number of type i errors the performance of multiple test procedures with respect to error control is an old issue assuming that all hypotheses are true we investigate the behavior of the expected number of type i errors ene as a characteristic of certain multiple tests controlling the familywise error rate fwer or the false discovery rate fdr at a prespecified level we derive explicit formulas for the distribution of the number of false rejections as well as for the ene for single step step down and step up procedures based on independent p values moreover we determine the corresponding asymptotic distributions of the number of false rejections as well as explicit formulae for the ene if the number of hypotheses tends to infinity in case of fwer control we mostly obtain poisson distributions and in one case a geometric distribution as limiting distributions in case of fdr control we obtain limiting distributions which are apparently not named in the literature surprisingly the ene is bounded by a small number regardless of the number of hypotheses under consideration finally it turns out that in case of dependent test statistics the ene behaves completely differently compared to the case of independent test statistics

stabilization of planar switched systems this paper considers the problem of stabilization of single input planar switched systems we assume the switching law is observable a formula is presented which provides a necessary and sufficient condition for the system to be quadratically stabilizable a set of linear inequalities are given to describe the set of all quadratic lyapunov functions the solvability and the control design technique are clearly described in a straightforward computation algorithm

a conjecture of frobenius and the sporadic simple groups ii a conjecture of frobenius which has been reduced to the classification of finite simple groups is verified for the sporadic simple groups let g be a finite group and n be a positive integer dividing igi let ln g x e g xn 1 then by a theorem of frobenius 6 one knows that iln g i cnn for some integer cn frobenius conjectured that ln g forms a subgroup of g provided iln g i n see 2 zemlin 25 has reduced the conjecture to the classification of finite simple groups which is now complete see 8 the author has verified the conjecture for the fischer griess monster f1 and the fischer baby monster f2 in 24 the purpose of this note is to prove the following theorem the conjecture of frobenius is true for all the sporadic simple groups the proof of our theorem has been carried out in the following way with the use of a computer let g be one of the sporadic simple groups by 24 we may assume that g f1 and g f2 let f g t be the number of elements of order t in g and ord g order of x i x e g tables of f g t are given in the appendix see the supplements section at the end of this issue for f g t the reader is referred to the following papers ml1 m22 m23 burgoyne and fong 1 m12 m24 frobenius 5 ii janko 14 hj j2 hall and wales 8 hjm j3 janko 15 j4 janko 16 his frame 4 suz wright 23 mcl 3 finkelstein 3 rud rudvalis 19 hhm held 10 lys lyons 17 received june 21 1982 revised march 19 1984 and july 11 1984 1980 mathematics subject classification primary 20d05

on the kalman yacubovich popov lemma and common lyapunov solutions for matrices with regular inertia in this paper we extend the classical lefschetz version of the kalman yacubovich popov kyp lemma to the case of matrices with general regular inertia we then use this result to derive an easily verifiable spectral condition for a pair of matrices with the same regular inertia to have a common lyapunov solution cls extending a recent result on cls existence for pairs of hurwitz matrices

balanced actuator and sensor placement for flexible structures properties of flexible structures in modal and balanced coordinates are derived from these properties the root mean square law of the hankel singular values is obtained it says that the squares of the hankel singular values of a set of sensors and actuators is approximately a root mean square sum of the squares of the hankel singular values for each individual sensor and actuator this property allows us to evaluate each actuator and sensor in terms of the joint controllability and observability and to serve as a tool of the actuator and or sensor placement methodology

a step up multiple test procedure abstract we consider the problem of simultaneously testing k 2 hypotheses on parameters 1 k in a typical application the s may be a set of contrasts for instance a set of orthogonal contrasts among population means or a set of differences between k treatment means and a standard treatment mean it is assumed that least squares estimators 1 k are available that are jointly normally distributed with a common variance known up to a scalar namely the error variance 2 and a common known correlation an independent 2 distributed unbiased estimator of 2 is also assumed to be available we propose a step up multiple test procedure for this problem which tests the t statistics for the k hypotheses in order starting with the least significant one and continues as long as an acceptance occurs by contrast the step down approach which is usually used starts with the most significant and continues as long as a rejection occurs critical constants required by this step up procedure to co

on a conjecture about the norm of lyapunov mappings abstract the norm properties of lyapunov mappings and their restrictions on symmetric and skew symmetric subspaces are investigated for non negative non positive and tridiagonal matrices this paper gives an affirmative answer to an open conjecture which says that the norm of a lyapunov mapping over the space of real square matrices equals that of its restricted map over the subspace of real symmetric matrices in addition matrix transformation and norm relationships on certain matrices arising from the lyapunov mapping and its restrictions are provided

discrete time norms of flexible structures abstract the paper presents and analyses the h 2 h and hankel norms of flexible structures the analysis is conducted for the discrete time models of structures and compared with the continuous time results the structural state space models are presented in modal co ordinates closed form expressions for norms of structural modes are obtained and norms of a structure are determined from the modal norms the relationships between the hankel h and h 2 modal norms are derived in addition the paper shows that the discrete time hankel and h norms converge to the continuous time counterparts when the sampling time approaches zero however the h 2 norm does not

stability robustness bounds for linear state space models with structured uncertainty in this note we consider the robust stability analysis problem in linear state space models we consider systems with structured uncertainty some lower bounds on allowable perturbations which maintain the stability of a nominally stable system are derived these bounds are shown to be less conservative than the existing ones

on the sensitivity of the coupled continuous time lyapunov equation the sensitivity of coupled continuous time lyapunov equations is analyzed through three types of norms of the inverse lyapunov like operator this analysis leads to bounds on the relative perturbation of the solution the results extend the bounds obtained for the standard lyapunov equation in hewcer et al 1988

robust stability of uncertain discrete impulsive switching systems this paper studies discrete impulsive switching systems and uncertain linear discrete switching systems it establishes some stability and robust stability criteria for such systems by using the method of lyapunov functions and linear matrix inequalities lmi these criteria are verifiable by using the lmi toolbox of matlab

some results on false discovery rate in stepwise multiple testing procedures the concept of false discovery rate fdr has been receiving increasing attention by researchers in multiple hypotheses testing this paper produces some theoretical results on the fdr in the context of stepwise multiple testing procedures with dependent test statistics it was recently shown by benjamini and yekutieli that the benjamini hochberg step up procedure controls the fdr when the test statistics are positively dependent in a certain sense this paper strengthens their work by showing that the critical values of that procedure can be used in a much more general stepwise procedure under similar positive dependency it is also shown that the fdr controlling benjamini liu step down procedure originally developed for independent test statistics works even when the test statistics are positively dependent in some sense an explicit expression for the fdr of a generalized stepwise procedure and an upper bound to the fdr of a step down procedure are obtained in terms of probability distributions of ordered components of dependent random variables before establishing the main results 1 introduction the main purpose of this paper is to present some formulas and inequalities providing theoretical insight into when and what stepwise procedures control the false discovery rate fdr at a prespecified level in multiple hypotheses testing the fdr first introduced in multiple testing by benjamini and hochberg 1995 is the expected proportion of erroneously rejected null hypotheses with v and r representing respectively the number of true null hypotheses rejected and the total number of null hypotheses rejected in a multiple testing procedure let q v r if r 0a nd 0i fr 0 then e q is the fdr of that procedure as they have argued it is an appropriate error rate to control in many practical situations particularly where a large number of null hypotheses are involved they also put forward an fdr controlling stepup procedure that is more powerful than comparable procedures controlling the traditional familywise error fwe rate when the underlying test statistics are independent benjamini and liu 1999 recently introduced a step down procedure with the same property the concept of fdr is now being used in a wide variety of applications abramovich and benjamini 1996 basford and tukey 1997 drigalenko

on the sensitivity of the coupled discrete time lyapunov equation the sensitivity of coupled discrete time lyapunov equations is analyzed through three types of norms of the inverse lyapunov like operator this analysis leads to bounds on the relative perturbation of the solution these results extend the results for standard lyapunov equation presented by gahinet et al 1990

new bound on the sensitivity of the solution of the lyapunov equation abstract a new bound on the size of the variation of the solution of the normal lyapunov equation under variation of the data is derived in contrast to a first order bound for differential perturbation the new bound is exact and still holds under large data variation it is shown that the new bound is less conservative than those currently available especially in the case of a high dimensional lyapunov equation

a survey of extreme point results for robustness of control systems abstract this paper surveys a subset of the body of research which was sparked by kharitonov s theorem the focal point is extreme point results for robust stability and robust performance that is we give conditions under which satisfaction of a performance specification is ascertained for a family of systems by only checking a finite subset of the extreme members of this family the results which are surveyed apply mainly to systems with structured real parametric uncertainty in addition a number of counterexamples are given to illustrate cases for which an extreme point result does not hold for such cases a solution via the so called edge theorem is often possible

on the structure of balanced and other principal representations of siso systems a new matrix w co which can be considered as a cross gramian matrix which contains information about both controllability and observability is defined for single input single output linear systems using this matrix the structural properties of linear systems are studied in the context of principal component analysis the matrix w co can be used in obtaining balanced and other principal representations without computation of the controllability and the observability gramians the importance of this matrix in model order reduction is highlighted

principal component analysis of flexible systems open loop case a generic class of flexible systems characterized by finitely many lightly damped harmonic oscillators is analyzed by means of the open loop principal component analysis that is singular value analysis and gramian balancing as the main result it is shown that as the damping ratio goes to zero the balanced state coordinates are decoupled and coincide with the modal coordinates further simple formulas expressing the asymptotic singular values as functions of the modal parameters are derived

non abelian sylow subgroups of finite groups of even order we have been able to prove that every nonabelian sylow subgroup of a finite group of even order contains a nontrivial element which commutes with an involution the proof depends upon the consequences of the classification of finite simple groups the purpose of this note is to announce 4 main theorem every nonabelian sylow subgroup of a finite group of even order contains a nontrivial element which commutes with an involution let g be a finite group and g the prime graph of g g is the graph such that the vertex set is the set of prime divisors of g and two distinct vertices p and r are joined by an edge if and only if there exists an element of order pr in g let n g be the number of connected components of g and dg p r the distance between two vertices p and r of g it has been proved that n g 6 in 12 10 11 9 theorem 1 let g be a finite group of even order and p a prime divisor of g if dg 2 p 2 then a sylow p subgroup of g is abelian theorem 1 is a restatement of the main theorem in terms of the prime graph g corollary 1 let g be a finite group of even order and p a prime divisor of g if is a connected component of g p not containing 2 then a sylow r subgroup of g is abelian for any r there is a certain relation between a subgraph g p of g and brauer characters of p modular representations of g see 3 theorem 2 let g be a finite nonabelian simple group and p an odd prime divisor of g then dg 2 p 1 or 2 provided dg 2 p the significance of the prime graphs of finite groups can be found in 1 3 5 6 7 8 14 15 we apply the classification of finite simple groups see 1 2 4 7 10 13 it has been proved that a minimal counterexample to theorem 1 received by the editors october 20 1997 1991 mathematics subject classification primary 20d05 20d06 20d20

numerical methods of lyapunov function construction and their application to the absolute stability problem the minimax approach is used to study numerical selection of a lyapunov function from a multiparameter class of functions a numerical method is suggested for plotting an absolute stability region in the space of system gains for a control system with several nonstationary nonlinearities

on a notion of breadth in the sense of frobenius abstract given a finite group g and an integer e 1 dividing the order of g the size of the set l e g x g x e 1 was studied originally by frobenius in order to find restrictions on the structure of g the aim of the present paper is to classify groups by b g max l e g e e div exp g where div exp g is the set of all divisors of the exponent exp g of g we will show general statements regarding the center and the central quotient of g by looking at b g this improves some recent contributions of w meng and j shi in 7

the perfect order subset conjecture for simple groups abstract this paper verifies the perfect order subset conjecture for simple groups for all but one family of finite simple groups specifically for each nonabelian finite simple group g there is some n such that the cardinality of the nonempty subset of all elements of order n in g does not divide the order of g unless g is an orthogonal group of plus type in dimension 4n for some n 2

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

some results on quadratic stability of switched systems with interval uncertainty in this paper we present some results on the quadratic stability of switched systems with uncertain parameters we show that the quadratic stability of a class of switched uncertain systems may be readily verified using simple algebraic conditions

commuting matrices equilibrium points for control systems with single saturated input in this paper commutative matrices of multiple input multiple output mimo linear systems are considered the existence of the feedback matrices of a commutative state matrix set in the mimo closed loops is reduced to the existence of an invariant subspace of a matrix a the existence of feedback matrices in systems in open loop is equivalent to the existence of the solution of matrix equations denoted by kronecker products by defining new equilibrium points the relationship between equilibrium points are discussed for a linear system with a single saturated input four criteria for equilibrium points are outlined for such linear systems finally four interesting examples including their corresponding simulink plots are shown to illustrate the above results