endotrivial modules for finite groups of lie type a in nondefining characteristic let g be a finite group such that mathop text sl nolimits n q subseteq g subseteq mathop text gl nolimits n q and z be a central subgroup of g in this paper we determine the group t g z consisting of the equivalence classes of endotrivial k g z modules where k is an algebraically closed field of characteristic p such that p does not divide q the results in this paper complete the classification of endotrivial modules for all finite groups of untwisted lie type a initiated earlier by the authors

parametric h sub spl infin loopshaping and weighted mixed sensitivity minimization generalized h sub spl infin loopshaping is developed for a particular weighted mixed sensitivity minimization task a free parameter is introduced to tradeoff minimizations of sensitivity and complementary sensitivity in the h sub spl infin performance index a quadratically convergent algorithm is derived for computing the optimal performance index and a state space solution is obtained for suboptimal controllers these results are applicable to stabilization of uncertain systems where normalized coprime factors involve parametrically weighted h sub spl infin norm bounded perturbations controller reduction is also investigated for the parametric h sub spl infin loopshaping robust stability condition for using the reduced order controller is established and an estimate for the associated h sub spl infin performance is obtained similar results are derived for normalized linear quadratic gaussian lqg control

an h minimax approach to the design of robust control systems abstract a realistic feedback design problem is posed based on the minimization of a weighted combination of the sensitivity and complementary sensitivity matrices a solution is obtained which makes use of the recently proposed methods for minimizing the sensitivity function alone

time varying versus time invariant compensation for rejection of persistent bounded disturbances and robust stabilization it is shown that time varying compensation does not improve the optimal rejection of persistent bounded disturbances this result is obtained by exploiting a key observation that any time varying compensator which yields a given degree of disturbance rejection must do so uniformly over time thereby removing any advantage of time variation this key observation is exploited to show that time varying compensation does not improve the optimal rejection of disturbances regardless of the norm used to measure the disturbances thus absolutely summable finite energy or persistent bounded disturbances may be treated in the same manner it is shown that time varying compensation does not help in the bounded input bounded output robust stabilization of time invariant plants with unstructured uncertainty in doing so it is also shown that the small gain theorem is both necessary and sufficient for the bounded input bounded output stability of certain linear time varying plants subject to unstructured linear time varying perturbations

a connection between normalized coprime factorizations and linear quadratic regulator theory given a transfer matrix described by a minimal state space triple a method is given for computing state space realizations for the numerator and denominator of a normalized stable right coprime factorization for the transfer matrix the method involves the solution of an algebraic riccati equation it allows the use of existing computational state space algorithms in finding normalized stable right coprime factorizations and avoids explicit calculations of spectral factors

robust stabilization of linear multivariable systems relations to approximation the stabilization of a linear system modelled as g a where g is a known rational transfer function and a is a perturbation satisfying w a2 g1 or equivalently there does not exist g with fewer poles in c than g such that w1 g g w2 a simple characterization of all robustly stabilizing controllers is then derived and state space formulae for maximally robust controllers are given finally reduced order controllers are considered

on the robust stability of linear time invariant plants with unstructured uncertainty we consider general families of linear time invariant plants described by a nominal plant model with unstructured uncertainty we show under a rather weak assumption that when a family of systems is not robustly stabilizable by any linear time invariant feedback controller then no nonlinear time varying controller can robustly stabilize the given family

reduced order controller design using coprime factor model reduction two procedures for reduced order controller design that incorporate coprime factor model reduction techniques and a related robust stabilization problem are proposed it is shown that robust stability can be achieved with these controllers the results obtained show how to reduce the controller degree systematically they clearly indicate how the maximum stability margin influences allowable model reduction error

extremal results in sparse pseudorandom graphs abstract szemeredi s regularity lemma is a fundamental tool in extremal combinatorics however the original version is only helpful in studying dense graphs in the 1990s kohayakawa and rodl proved an analogue of szemeredi s regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs many of the key applications of szemeredi s regularity lemma use an associated counting lemma in order to prove extensions of these results which also apply to sparse graphs it remained a well known open problem to prove a counting lemma in sparse graphs the main advance of this paper lies in a new counting lemma proved following the functional approach of gowers which complements the sparse regularity lemma of kohayakawa and rodl allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph we use this to prove sparse extensions of several well known combinatorial theorems including the removal lemmas for graphs and groups the erd s stone simonovits theorem and ramsey s theorem these results extend and improve upon a substantial body of previous work

on endo trivial modules for p solvable groups in this note we will prove a conjecture of carlson et al 1 thus resolving the question left open in 1 and completing the classification of torsion endo trivial modules for p solvable groups namely we will prove that if g is a finite p nilpotent group which contains a non cyclic elementary abelian p subgroup and k is an algebraically closed field of characteristic p then all simple endo trivial kg modules are 1 dimensional in fact we do rather more we prove the analogous result directly in the case that g is p solvable and contains an elementary abelian p subgroup of order p2 carlson mazza and thevenaz had reduced the proof of this result for p solvable g to the p nilpotent case and had proved the result in the solvable case but our method is somewhat different our proof does require the classification of finite simple groups specifically we require the well known fact that the outer automorphism group of a finite simple group of order prime to p has cyclic sylow p subgroups see for example theorem 7 1 2 of gorenstein et al 4 let us recall that a kg module m is endo trivial if m m k n where n is a projective kg module if g is divisible by p then any endo trivial kg module has dimension prime to p the vertex of any indecomposable endo trivial kg module is a sylow p subgroup of g we remark that if m is an endo trivial kg module which is not 1 dimensional then a sylow p subgroup of g acts faithfully on m m and hence acts faithfully on m

reduced order compensators via balancing and central control design for a structural control problem real time control of a physical system necessitates controllers that are low order in this article we compare two balanced truncation methods as a means of designing low order controllers for a nonlinear cable mass system the first is the standard technique of balanced truncation the second linear quadratic gaussian lqg balanced truncation can be thought of as balancing based on the controller and states that are important from the perspective of control and filter design are retained the control design applied to each reduced order model is the central controller we provide an overview of the central controller and devote attention to the design of this controller in the presence of balancing also described in this article is a method for reducing computational time in solving algebraic riccati equations for the design of low order lqg balanced controllers

irreducible specht modules are signed young modules recently donkin defined signed young modules as a simultaneous generalization of young and twisted young modules for the symmetric group we show that in odd characteristic if a specht module s is irreducible then s is a signed young module thus the set of irreducible specht modules coincides with the set of irreducible signed young modules this provides evidence for our conjecture that the signed young modules are precisely the class of indecomposable self dual modules with specht filtrations the theorem is false in characteristic two

an algorithmic regularity lemma for hypergraphs in this paper we will consider the problem of designing an efficient algorithm that finds an epsilon regular partition of an l uniform hypergraph

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

simple endotrivial modules for quasi simple groups we investigate simple endotrivial modules of finite quasi simple groups and classify them in several important cases this is motivated by a recent result of robinson 41 showing that simple endotrivial modules of most groups come from quasi simple groups

on the decomposition numbers of the ree groups f42 q2 in non defining characteristic abstract we compute the l modular decomposition matrices of the simple ree groups f 4 2 q 2 where q 2 2 2 n 1 and n 0 for all primes l 3 up to some entries in the unipotent and one entry in the non unipotent characters we determine the smallest degree of a nontrivial irreducible l modular representation of f 4 2 q 2 for all primes l 3 and also obtain results on the 3 modular decomposition matrices of f 4 2 q 2

closed formulae for a parametric mixed sensitivity problem abstract mixed sensitivity optimization is a two block problem which normally requires an iterative solution the purpose of this paper is to give an explicit closed formula for the optimal cost in the case of a parametric mixed sensitivity problem we also show that all the controllers which satisfy an arbitrary h norm bound may be parameterized in terms of the solutions to the standards h 2 riccati equations we hope that these results will help progress towards closed formulae for the optimal cost in a wider class of problems

a fast approximation algorithm for computing thefrequencies of subgraphs in a given graph in this paper we give an algorithm which given a labeled graph on n vertices and a list of all labeled graphs on k vertices provides for each graph h of this list an approximation to the number of induced copies of h in g with total error small this algorithm has running time o n 1 over log log n cdot m n where m n is the time needed to square a n by n matrix with 0 1 entries over the integers the main tool in designing this algorithm is a variant of the regularity lemma of szemeredi

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

controller reduction via stable factorization and balancing abstract a controller reduction procedure based on a representation of a controller as a matrix function defined using stable proper transfer functions and employing a balancing technique is studied in this paper for a certain right coprime factorization of an lqg designed controller k s n s d 1 s we approximate using a balancing technique the pair d s n s t by a low order pair d1 s n1 s t defining a factorization of the reduced order controller k 1 s n1 s d1 1 s we show that reducing the controller order in this way is motivated in a natural way which leads to the expectation of both good stability properties and good accuracy of approximation of closed loop behaviour this is also demonstrated in some examples

brauer trees for the schur cover of the symmetric group abstract we determine the brauer trees of the faithful blocks of weight 1 of the schur covers of the symmetric group and the alternating group in odd characteristic

robust controllers for uncertain linear multivariable systems abstract this paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems in the first part it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty in the second part in contrast with the first part it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known the type of plant uncertainty is the so called stable factor uncertainty and necessary and sufficient conditions are given for robust stabilization in the third part the model of uncertainty is a ball in the space of rational matrices metrized by the so called graph metric and sufficient conditions for robust stabilization are derived

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

optimal h multivariable robust controllers and the relationship to lqg design problems a new solution for the multivariable minimax optimal robust design problem is presented the relationship between lqg and h design methods is considered and the h solution is obtained from the solution of a related lqg problem the approach provides insights into the relationship between the h and lqg design philosophies the h solution involves the same type of steps which are involved in the polynomial based solutions of the lqg problem namely spectral factorization and the solution of linear diophantine equations a simplified design procedure is described which requires only the solution of a standard eigenvalue eigenvector problem

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

approximate hypergraph partitioning and applications szemeredi s regularity lemma is a cornerstone result in extremal combinatorics it roughly asserts that any dense graph is composed of a finite number of pseudorandom graphs the regularity lemma has found many applications in theoretical computer science and thus a lot of attention was given to designing algorithmic versions of this lemma our main results in this paper are the following i we introduce a new approach to the problem of constructing regular partitions of graphs which results in a surprisingly simple o n time algorithmic version of the regularity lemma thus improving over the previous o n 2 time algorithms furthermore unlike all the previous approaches for this problem see n alon and a naor siam j comput 35 2006 pp 787 803 r a duke h lefmann and v rodl siam j comput 24 1995 pp 598 620 a frieze and r kannan electron j combin 6 1999 article 17 a frieze and r kannan the regularity lemma and approximation schemes for dense problems in proceedings of the 37th annual symposium on foundations of computer science burlington vt 1996 ieee computer society press los alamitos ca 1996 pp 12 20 and y kohayakawa v rodl and l thoma siam j comput 32 2003 pp 1210 1235 which only guaranteed to find tower size partitions our algorithm will find a small regular partition if one exists in the graph ii for any constant r geq3 we give an o n time randomized algorithm for constructing regular partitions of r uniform hypergraphs thus improving the previous o n 2r 1 time deterministic algorithms a czygrinow and v rodl siam j comput 30 2000 pp 1041 1066 a frieze and r kannan the regularity lemma and approximation schemes for dense problems in proceedings of the 37th annual symposium on foundations of computer science burlington vt 1996 ieee computer society press los alamitos ca 1996 pp 12 20 these two results are obtained as an application of an efficient algorithm for approximating partition problems of hypergraphs which we obtain here given a directed hypergraph with bounded edge arities a set of constraints on the set sizes and densities of a possible partition of its vertex set and an approximation parameter we provide in o n time a partition approximating the constraints if a partition satisfying them exists we can also test in o 1 time for the existence of such a partition given the approximation parameter this algorithm extends the result of goldreich goldwasser and ron for graph partition problems o goldreich s goldwasser and d ron j acm 45 1998 pp 653 750 and encompasses more recent hypergraph related results such as the maximal constraint satisfaction approximation of g andersson and l engebretsen random structures algorithms 21 2002 pp 14 32

the graph metric for unstable plants and robustness estimates for feedback stability in this paper a graph metric is defined that provides a measure of the distance between unstable multivariable plants the graph metric induces a graph topology on unstable plants which is the weakest possible topology in which feedback stability is robust using the graph metric it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of rhp poles if the perturbed and unperturbed systems have the same rhp poles then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations as an application of these results the design of stabilizing controllers for unstable singularly perturbed systems is studied finally the relationship of the graph metric to the gap metric introduced by zames and el sakkary is studied in detail in particular it is shown that the robustness results of zames and el sakkary do not enable one to conclude the causality of the perturbed system whereas the present results do

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

bibo stability robustness in the presence of coprime factor perturbations a necessary and sufficient condition for robustly stabilizing a family of plants described by perturbations of fixed coprime factors of a plant is given the computation of the largest stability margin is discussed via solving a nonsquare l sup 1 optimal control problem an algorithm for obtaining lower approximations of the minimum value of the optimization problem mu sub 0 is proposed this together with the standard algorithm which provides upper approximations allows mu sub 0 to be computed within any degree of accuracy

the general distance problem in h optimal control theory abstract the general distance problem that arises in h optimal control theory is considered transformations that reduce the general h problem to the general distance problem are reviewed and an iterative scheme called iteration is considered in which the general distance problem is reduced to a standard best approximation problem the iteration is viewed as a problem of finding the zero crossing of a function this function is shown to be continuous monotonically decreasing convex and bounded by some very simple functions these properties make it possible to obtain very rapid convergence of the iterative process the issues of model reduction and approximation in h synthesis will also be addressed

robust stabilization of delay systems by approximation of coprime factors given a delay system with transfer function g s h2 s h1 s where h1 s 01npi s e is and h2 s 02nqi s e is with 0 0 1 n1 0 0 n2 the pi being polynomials of degree i and i 0for i 0 and the qi polynomials of degree di 0 for each i the robust stabilization of a class of perturbed coprime factors of this system is considered asymptotic estimates are obtained based on recent results on the approximation and stabilization of normalized coprime factors an explicit formula is given for the normalized coprime factor stability margin for the case of a multivariable transfer function of the form g s e str s with r rational

normalised coprime factorizations for nonstrictly proper systems some earlier results of d g meyer and g f franklin see ibid vol ac 32 pp 227 8 march 1987 are extended and a procedure is derived for obtaining directly a normalized right coprime factorization of a system that may not be strictly proper

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

model reduction with relative multiplicative error bounds and relations to controller reduction balanced model reduction with a priori relative multiplicative error bounds in l sub spl infin norm is studied inverse weighted balanced truncation iwbt is proposed that is a variant of the enns algorithm it is shown that iwbt is equivalent to the balanced stochastic truncation bst this equivalence leads to the extension of an improved relative multiplicative error bound for bst obtained by wang and safonov to general nonsquare transfer matrices natural relations between iwbt and observer based controller reduction are investigated which motivate the simultaneous reduction of the plant and controller robust stability conditions for simultaneous reduction of the plant and the controller are established for the closed loop system consisting of full order plant and reduced order controller as a by product robust stability and stabilization problems are solved for general coprime factor plant descriptions with relative multiplicative h sub spl infin norm bounded uncertainties similar results are obtained for general coprime factors of the feedback controller involving relative multiplicative h sub spl infin norm bounded uncertainties

systematic identification of statistically significant network measures we present a graph embedding space i e a set of measures on graphs for performing statistical analyses of networks key improvements over existing approaches include discovery of motif hubs multiple overlapping significant subgraphs computational efficiency relative to subgraph census and flexibility the method is easily generalizable to weighted and signed graphs the embedding space is based on scalars functionals of the adjacency matrix representing the network scalars are global involving all nodes although they can be related to subgraph enumeration there is not a one to one mapping between scalars and subgraphs improvements in network randomization and significance testing we learn the distribution rather than assuming gaussianity are also presented the resulting algorithm establishes a systematic approach to the identification of the most significant scalars and suggests machine learning techniques for network classification

the stable category and invertible modules for infinite groups abstract we construct a well behaved stable category of modules for a large class of infinite groups we then consider its picard group which is the group of invertible or endotrivial modules we show how this group can be calculated when the group acts on a tree with finite stabilisers

an optimal algorithm for checking regularity we present a deterministic algorithm cal a that in o m2 time verifies whether a given m by m bipartite graph g is regular in the sense of szemeredi regular partitions of graphs in problemes combinatoires et theorie des graphes orsay 1976 colloques internationaux cnrs 260 cnrs paris 1978 pp 399 401 in the case in which g is not regular enough our algorithm outputs a witness to this irregularity algorithm cal a may be used as a subroutine in an algorithm that finds an varepsilon regular partition of a given n vertex graph gamma in time o n2 this time complexity is optimal up to a constant factor and improves upon the bound o m n proved by alon et al the algorithmic aspects of the regularity lemma j algorithms 16 1994 pp 80 109 where m n o n2 376 is the time required to square a 0 1 matrix over the integers our approach is elementary except that it makes use of linear sized expanders to accomplish a suitable form of deterministic sampling

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

a controller degree bound for h 0e control problems of the second kind this paper is a continuation of our work on mathcal h infty optimal control problems which may be embedded in the linear fractional configuration of fig 1 in two previous articles 19 20 a controller degree bound was established for problems in which both p 12 s and p 21 s are square problems of the first kind if the mcmillan degree of p s is n it was shown that there exist mathcal h infty optimal controllers with mcmillan degree no greater than n 1 here we switch our attention to problems of the second kind that is we allow p 12 s to have more rows than columns with p 21 s square or alternatively we allow p 21 s to have more columns than rows with p 12 s square our main result shows that the degree bound derived previously for problems of the first kind carries over to problems of the second kind without change in addition to the controller degree bound our analysis suggests a number of modifications which are easily made to cur

simple endotrivial modules for linear unitary and exceptional groups motivated by a recent result of robinson showing that simple endotrivial modules essentially come from quasi simple groups we classify such modules for finite special linear and unitary groups as well as for exceptional groups of lie type our main tool is a lifting result for endotrivial modules obtained in a previous paper which allows us to apply character theoretic methods as one application we prove that the ell rank of quasi simple groups possessing a faithful simple endotrivial module is at most 2 as a second application we complete the proof that principal blocks of finite simple groups cannot have loewy length 4 thus answering a question of koshitani kulshammer and sambale our results also imply a vanishing result for irreducible characters of special linear and unitary groups