a class of blocks behaving like blocks of p solvable groups abstract we define a class of blocks having similar properties to blocks of p solvable groups and show that a version of the fong swan theorem holds for irreducible brauer characters in such blocks we also show that the height of an irreducible character in such block is bounded by the exponent of the central quotient of a defect group which in particular implies that if further the defect groups are abelian then every irreducible character in the block has height zero

controllability and observability for non autonomous evolution equations the averaged hautus test abstract we consider the observability problem for non autonomous evolution systems i e the operators governing the system depend on time we introduce an averaged hautus condition and prove that for skew adjoint operators it characterizes exact observability next we extend this to more general class of operators under a growth condition on the associated evolution family we give an application to the schrodinger equation with time dependent potential and the damped wave equation with a time dependent damping coefficient

equivalent conditions for stabilizability of infinite dimensional systems with admissible control operators in this paper we study the optimizability of infinite dimensional systems with admissible control operators we show that under a weak condition such a system is optimizable if and only if the system can be split into an exponentially stable subsystem and an unstable subsystem that is exactly controllable in finite time the state space of the unstable subsystem equals the span of all unstable generalized eigenvectors of the original system this subsystem can be infinite dimensional furthermore the unstable poles satisfy a summability condition the state space of the exponentially stable subsystem is given by all vectors for which the action of the original c0 semigroup is stable

zj controlled blocks of quasisimple groups for odd primes abstract we investigated the converses of glauberman s zj theorem and its block theoretic analogue proved by kessar linckelmann and robinson for a finite group g we proved that a block b of g with a defect group d is zj controlled that is n g z j d controls b fusion in g if and only if b is controlled provided that g is quasisimple and the characteristic p of the modular representation is odd this implies by a result of 5 that the converses of glauberman s zj theorem and its block theoretic analogue both hold in these cases as an application we obtain all q d p free quasisimple groups

nilpotent blocks of quasisimple groups for odd primes we investigate the nilpotent blocks of positive defect of the quasisimple groups for odd primes in particular it is shown that every nilpotent block of a quasisimple group has abelian defect groups a conjecture of puig concerning the recognition of nilpotent blocks is also shown for these groups

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

riesz bases and exact controllability of c0 groups with one dimensional input operators abstract this paper considers linear infinite dimensional systems with c 0 group generators and one dimensional admissible input operators the exact controllability and riesz basis generation property are discussed the corresponding results of jacob and zwart advances in mathematical systems theory birkhauser boston ma 2000 under the assumption of algebraic simplicity for eigenvalues of the generator are generalized to the case in which the eigenvalues are allowed to be algebraically multiple but with a uniform bound on the multiplicity

the regularity of the wave equation with partial dirichlet control and colocated observation in this paper we analyze a multidimensional controlled wave equation on a bounded domain subject to partial dirichlet control and colocated observation by means of a partial fourier transform it is shown that the system is well posed and regular in the sense of d salamon and g weiss the corresponding feedthrough operator is found to be the identity operator on the input space

finding correlations in subquadratic time with applications to learning parities and the closest pair problem given a set of n d dimensional boolean vectors with the promise that the vectors are chosen uniformly at random with the exception of two vectors that have pearson correlation coefficient hamming distance d 1 2 how quickly can one find the two correlated vectors we present an algorithm which for any constant 0 and constant 0 runs in expected time o n5 4 nd applications and extensions of this basic algorithm yield significantly improved algorithms for several other problems approximate closest pair for any sufficiently small constant 0 given n d dimensional vectors there exists an algorithm that returns a pair of vectors whose euclidean or hamming distance differs from that of the closest pair by a factor of at most 1 and runs in time o n2 the best previous algorithms including locality sensitive hashing have runtime o n2 o learning sparse parities with noise given samples from an instance of the learning parities with noise problem where each example has length n the true parity set has size at most k n and the noise rate is there exists an algorithm that identifies the set of k indices in time n 3 k poly 1 1 2 0 4 improves upon the results of grigorescu et al 2011 that give a runtime of n 1 2 2 o 1 k 2 poly 1 1 2 learning k juntas with noise given uniformly random length n boolean vectors together with a label which is some function of just k n of the bits perturbed by noise rate return the set of relevant indices leveraging the reduction of feldman et al 2009 our result for learning k parities implies an algorithm for this problem with runtime n 3 k poly 1 1 2 learning k juntas without noise given uniformly random length n boolean vectors together with a label which is some function of k n of the bits return the set of relevant indices using a modification of the algorithm of mossel et al 2004 and employing our algorithm for learning sparse parities with noise via the reduction of feldman et al 2009 we obtain an algorithm for this problem with runtime n 4 k poly n

admissibility in input elements for diagonal semigroups on l 2 abstract we characterize the admissible unbounded input elements for diagonal semigroups on l 2 the result verifies the necessity of a condition whose sufficience was previously established by l f ho and d l russell

a general necessary condition for exact observability suppose a generates an exponentially stable strongly continuous semigroup on the hilbert space x y is another hilbert space and c d a rightarrow y is an admissible observation operator for this semigroup this means that to any initial state in x we can associate an output function in l 2 0 infty y this paper gives a necessary condition for the exact observability of the system defined by a and c this condition called bf e is related to the hautus lemma from finite dimensional systems theory it is an estimate in terms of the operators a and c alone in particular it makes no reference to the semigroup this paper shows that bf e implies approximate observability and if a is bounded it implies exact observability this paper conjectures that bf e is in fact equivalent to exact observability the paper also shows that for diagonal semigroups bf e takes on a very simple form and applies the results to sequences of complex exponential functions

vertices for characters of p solvable groups suppose that g is a finite p solvable group we associate to every irreducible complex character x irr g of g a canonical pair q where q is a p subgroup of g and irr q uniquely determined by x up to g conjugacy this pair behaves as a green vertex and partitions irr g into families of characters using the pair q we give a canonical choice of a certain p radical subgroup r of g and a character irr r associated to x which was predicted by some conjecture of g r robinson

the green correspondence and normal subgroups in a recent paper harris and knorr 6 have discovered a connection between the brauer correspondence and the covering relation between blocks of a finite group and blocks of its normal subgroups green s module point of view of blocks 4 suggests that this result has a moduletheoretic proof and that there is a more general result on indecomposable modules we shall show that this is the case by giving a short moduletheoretic proof of the harris kniirr theorem and then use the ideas of the proof to formulate and establish a result relating the green correspondence is with normal subgroups let us establish some notation let g be a finite group and h a normal subgroup let k be a field of prime characteristic p and b a block of h so b is an indecomposable algebra summand of kh we shall regard b as a k h x hi module in the usual way let d be a defect group of b k n d and b equal the block of k with bf b that is the brauer correspondent of b finally set l n d so k is a normal subgroup of l

vertices for irreducible characters of a class of blocks we observe that navarro s definition of a vertex for an irreducible character of a p solvable group may be extended to irreducible characters in p blocks with defect groups contained in a normal p solvable subgroup and show that this definition is independent of the choice of n we show that the fundamental properties of navarro s vertices generalize and as a corollary show that the vertices of the irreducible brauer characters in blocks of the above form are radical and are intersections of pairs of sylow p subgroups

necessary conditions for exact controllability with a finite dimensional input space we derive necessary conditions for exact controllability of infinite dimensional systems described by x ax bu where the generator a has a riesz basis of eigenvectors and the input space is finite dimensional these conditions are in terms of the eigenvalues of a and the degree of unboundedness of b but no further information about b is needed our results easily imply lack of exact controllability for a wide range of distributed parameter systems it is known from partial differential equation pde examples that the addition of damping into an exactly controllable second order system can destroy exact controllability we discuss the effect of damping on such systems using our general conditions we also give simple new proofs for results about the lack of exact controllability of hyperbolic systems in more than one space dimension

the operator carleson measure criterion for admissibility of control operators for diagonal semigroups on 1 2 abstract suppose a control system is described by a diagonal semigroup on the state space l 2 and by an unbounded control operator b defined on the input space l 2 we formulate a condition called the operator carleson measure criterion and show that this condition is necessary for the admissibility of b if the semigroup is analytic or invertible then the condition is sufficient as well our results extend the ones of ho russell and weiss concerning the case where the input space is one dimensional

anchors of irreducible characters given a prime number p every irreducible character of a finite group g determines a unique conjugacy class of p subgroups of g which we will call the anchors of this invariant has been considered by barker in the context of finite p solvable groups besides proving some basic properties of these anchors we investigate the relation to other p groups which can be attached to irreducible characters such as defect groups vertices in the sense of j a green and vertices in the sense of g navarro

necessary conditions for the exact observability of systems on hilbert spaces abstract in this paper we introduce the zero class of admissible observation operators for which the norm of the output map tends to zero as the time tends to zero we show that for such an observation operator exact observability implies that the semigroup of the system is bounded from below in addition if the semigroup generator has no residual spectrum then the semigroup can be extended to a c 0 group

exact observability of diagonal systems with a finite dimensional output operator in this paper equivalent conditions for exact observability of diagonal systems with a finite dimensional output operator are given one of these equivalent conditions is the conjecture of russell and weiss siam j control opt 32 1 1994 1 23 the other conditions are in terms of the eigenvalues and the lyapunov solutions of finite dimensional subsystems

controlled blocks of the finite quasisimple groups for odd primes abstract we classify controlled blocks introduced by alperin and broue in 1979 for all quasisimple groups g for odd primes the results imply that every nilpotent block of g has abelian defect groups which in turn is one of the main results proved in an and eaton 2011 6 we also give an explicit characterization of non controlled blocks of all quasisimple groups g for odd primes this implies the block theoretic analogue of glauberman s zj theorem for g proved by kessar linckelmann and robinson 2002 18

observability of polynomially stable systems abstract for finite dimensional systems the hautus test is a well known and easily checkable condition for observability russell and weiss a general necessary condition for exact observability siam j control optim 32 1994 1 23 suggested an infinite dimensional version of the hautus test which is necessary for exact infinite time observability and sufficient for approximate infinite time observability of exponentially stable systems in this paper the notion of observability is studied for polynomially stable systems several known results for exponentially stable systems are extended to the setting of polynomially stable systems by means of an example the obtained results are illustrated

counterexamples concerning observation operators for c0 semigroups this paper concerns systems of the form dot x t ax t y t cx t where a generates a c 0 semigroup two conjectures which were posed in 1991 and 1994 are shown not to hold the first conjecture by g weiss states that if the range of c is one dimensional then c is admissible if and only if a certain resolvent estimate holds the second conjecture by d russell and g weiss states that a system is exactly observable if and only if a test similar to the hautus test for finite dimensional systems holds the c 0 semigroup in both counterexamples is analytic and possesses a basis of eigenfunctions using the a c pair from the second counterexample we construct a generator a e on a hilbert space such that si a e is uniformly left invertible but its semigroup does not have this property

approximate and approximate null controllability of a class of piecewise linear markov switch systems abstract we propose an explicit easily computable algebraic criterion for approximate null controllability of a class of general piecewise linear switch systems with multiplicative noise this gives an answer to the general problem left open in goreac and martinez 2015 the proof relies on recent results in confortola et al 2015 allowing to reduce the dual stochastic backward system to a family of ordinary differential equations second we prove by examples that the notion of approximate controllability is strictly stronger than approximate null controllability a sufficient criterion for this stronger notion is also provided the results are illustrated on a model derived from repressed bacterium operon given in krishna et al 2005 and reduced in crudu et al 2009

exponential stabilization of well posed systems by colocated feedback we consider well posed linear systems whose state trajectories satisfy dot x ax bu where u is the input and a is an essentially skew adjoint and dissipative operator on the hilbert space x this means that the domains of a and a are equal and a a q where q geq 0 is bounded on x the control operator b is possibly unbounded but admissible and the observation operator of the system is b such a description fits many wave and beam equations with colocated sensors and actuators and it has been shown for many particular cases that the feedback u ka y v with ka 0 stabilizes the system strongly or even exponentially here y is the output of the system and v is the new input we show by means of a counterexample that if b is sufficiently unbounded then such a feedback may be unsuitable the closed loop semigroup may even grow exponentially our counterexample is a simple regular system with feedthrough operator zero however we prove that if the original system is exactly controllable and observable and if ka is sufficiently small then the closed loop system is exponentially stable