on computing the distance to stability for matrices using linear dissipative hamiltonian systems abstract in this paper we consider the problem of computing the nearest stable matrix to an unstable one we propose new algorithms to solve this problem based on a reformulation using linear dissipative hamiltonian systems we show that a matrix a is stable if and only if it can be written as a j r q where j j t r 0 and q 0 that is r is positive semidefinite and q is positive definite this reformulation results in an equivalent optimization problem with a simple convex feasible set we propose three strategies to solve the problem in variables j r q i a block coordinate descent method ii a projected gradient descent method and iii a fast gradient method inspired from smooth convex optimization these methods require o n 3 operations per iteration where n is the size of a we show the effectiveness of the fast gradient method compared to the other approaches and to several state of the art algorithms

zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

representation type of hecke algebras of type a in this paper we provide a complete classification of the representation type for the blocks for the hecke algebra of type a stated in terms of combinatorical data the computation of the complexity of young modules is a key component in the proof of this classification result

fundamental solution of the heat and schrodinger equations with point interaction abstract we find explicit formulae for the fundamental solution for the heat and time dependent schrodinger equations with point interaction in dimension n 3 we discuss in detail the small time asymptotics of these fundamental solutions the case n 2 has the peculiarity of involving logarithmic terms in the expansion

another runner removal theorem for v decomposition numbers of iwahori hecke algebras and q schur algebras abstract let f denote the fock space representation of the quantum group u v sl e the v decomposition numbers are the coefficients when the canonical basis for this representation is expanded in terms of the basis of partitions and the evaluations at v 1 of these polynomials give the decomposition numbers for iwahori hecke algebras and q schur algebras over c james and mathas have proved a theorem which relates v decomposition numbers for different values of e by adding empty runners to the abacus displays for the labelling partitions here we prove a similar theorem which involves adding full runners to these abacus displays

determinantal processes with number variance saturation consider dyson s hermitian brownian motion model after a finite time s where the process is started at n equidistant points on the real line these n points after time s form a determinantal process and has a limit as n this limting determinantal process has the interesting feature that it shows number variance saturation the variance of the number of particles in an interval converges to a limiting value as the length of the interval goes to infinity number variance saturation is also seen for example in the zeros of the riemann function 21 2 the process can also be constructed using non intersecting paths and we consider several variants of this construction one construction leads to a model which shows a transition from a non universal behaviour with number variance saturation to a universal sine kernel behaviour as we go up the line

the ordinary quiver of a weight three block of the symmetric group is bipartite abstract suppose f is a field of characteristic p 5 and that b is a p block of the symmetric group s n of defect 3 we prove that the ext 1 quiver of b is bipartite with the bipartition being described in a simple way using the leg lengths of p hooks of partitions

on vertices of simple modules for symmetric groups of small degrees abstract we present some algorithmic methods for the computation of vertices of indecomposable and simple modules over group algebras in prime characteristic furthermore we apply these to the simple modules of the symmetric groups and determine the vertices of all simple modules of the symmetric groups of degree at most 14 and 15 in characteristic 2 and 3 respectively with one exception

analytical structures and analysis of the simplest fuzzy pi controllers this paper deals with simplest fuzzy pi controllers which employ two fuzzy numbers on the universe of discourse uod of each input variable and three fuzzy numbers on the uod of output variable analytical structures of such controllers are derived using triangular membership functions for fuzzification different combinations of t norms and t conorms different inference methods and center of area coa method for defuzzification properties of these controllers are investigated a comparative study is made on i the fuzzy pi controllers derived and ii on the fuzzy pi controllers and their counterpart conventional pi controller moreover sufficient conditions for bounded input bounded output bibo stability of fuzzy pi control systems are established using the well known small gain theorem

quantum weyl reciprocity and tilting modules quantum weyl reciprocity relates the representation theory of hecke algebras of type a with that of q schur algebras this paper establishes that weyl reciprocity holds integrally i e over the ring q q 1 of laurent polynomials and that it behaves well under base change a key ingredient in our approach involves the theory of tilting modules for q schur algebras new results obtained in that direction include an explicit determination of the ringel dual algebra of a q schur algebra in all cases in particular in the most interesting situation the ringel dual identifies with a natural quotient algebra of the hecke algebra

the probabilistic solution of the third boundary value problem for second order elliptic equations using standard reflected brownian motion srbm and martingales we define in the spirit of stroock and varadhan see s v the probabilistic solution of the boundary value problem begin gathered frac 1 2 delta u qu 0 in d hfill frac partial u partial n cu f on partial d hfill end gathered whered is a bounded domain withc3 boundary andn is the inward unit oormal vector on d the assumptions forq c andf are quite general

hecke algebras of classical type and their representation type the purpose of this article is to determine the representation type for all of the hecke algebras of classical type to do this we combine methods from our previous work which is used to obtain information on their gabriel quivers and recent advances in the theory of finite dimensional algebras the principal computation is for hecke algebras of type b with two parameters then we show that the representation type of hecke algebras is governed by their poincare polynomials

searching for more general weight conjectures using the symmetric group as an example abstract let k be a field of characteristic p and let s n be the symmetric group of degree n assume that n p let a be the centralizer in ks n of ks n 1 we find all simple a modules and relate the number of non projective simple a modules to p local information

free fermions and the classical compact groups there is a close connection between the ground state of non interacting fermions in a box with classical absorbing reflecting and periodic boundary conditions and the eigenvalue statistics of the classical compact groups the associated determinantal point processes can be extended in two natural directions i we consider the full family of admissible quantum boundary conditions i e self adjoint extensions for the laplacian on a bounded interval and the corresponding projection correlation kernels ii we construct the grand canonical extensions at finite temperature of the projection kernels interpolating from poisson to random matrix eigenvalue statistics the scaling limits in the bulk and at the edges are studied in a unified framework and the question of universality is addressed whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is a priori not obvious we complete the picture by constructing a finite temperature extension of the haar measure on the classical compact groups the eigenvalue statistics of the resulting grand canonical matrix models of random size corresponds exactly to the grand canonical measure of free fermions with classical boundary conditions

the heights of irreducible brauer characters in 2 blocks of the symmetric groups abstract we prove that there exists a unique irreducible brauer character of height zero in any 2 blocks of the symmetric group this generalizes the theorem of p fong and g d james that the dimension of every non trivial 2 modular simple module of the symmetric group is even

the breuil mezard conjecture when l neq p let l and p be primes let f qp be a finite extension with absolute galois group gf let f be a finite field of characteristic l and let w gf gln f be a continuous representation let r be the universal framed deformation ring for if l d p then the breuil mezard conjecture as recently formulated by emerton and gee relates the mod l reduction of certain cycles in r to the mod l reduction of certain representations of gln of we state an analogue of the breuil mezard conjecture when l p and we prove it whenever l 2 using automorphy lifting theorems we give a local proof when l is quasibanal for f and is tamely ramified we also analyze the reduction modulo l of the types defined by schneider and zink

augmented lagrangian approach to design of structured optimal state feedback gains we consider the design of optimal state feedback gains subject to structural constraints on the distributed controllers these constraints are in the form of sparsity requirements for the feedback matrix implying that each controller has access to information from only a limited number of subsystems the minimizer of this constrained optimal control problem is sought using the augmented lagrangian method notably this approach does not require a stabilizing structured gain to initialize the optimization algorithm motivated by the structure of the necessary conditions for optimality of the augmented lagrangian we develop an alternating descent method to determine the structured optimal gain we also utilize the sensitivity interpretation of the lagrange multiplier to identify favorable communication architectures for structured optimal design examples are provided to illustrate the effectiveness of the developed method

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

brief some applications of randomized algorithms for control system design in this paper a few difficult problems related to simultaneous stabilization of three plants equivalent to a certain problem related to unit interpolation in h have been addressed through the framework of randomized algorithms these problems which were proposed by blondel simultaneous stabilization of linear systems springer berlin 1994 and blondel and gevers math control signals systems 6 1994 135 concern the existence of a controller

decomposition matrices for low rank unitary groups we study the decomposition matrices of the unipotent blocks of finite special unitary groups sun q for unitary primes larger than n up to few unknown entries we give a complete solution for n 2 10 we also prove a general result for two column partitions when divides q 1 this is achieved using projective modules coming from the adic cohomology of deligne lusztig varieties

two numerical methods for optimizing matrix stability consider the affine matrix family a x a0 m 1 xkak mapping a design vector x r m into the space of n n real matrices we are interested in the question of how to choose x to optimize the stability of the matrix a x a typical motivation is that one wishes to control the stability of the dynamical system a x z a classic example is stabilization by output feedback we take two approaches let x denote the spectral abscissa the largest real part of the eigenvalues of a matrix x as is well known this quantity bounds the asymptotic decay rate of the trajectories of the associated dynamical system z xz our first approach to optimizing stability is to directly minimize the function a x the spectral abscissa x is a continuous but non smooth in fact non lipschitz function of the matrix argument x and finding a global minimizer of a x is hard we introduce a novel random gradient bundle method for approximating local minimizers motivated by recent work on nonsmooth analysis of the function x our second approach is to minimize a related function a x where is a robustness parameter in 0 1 one motivation for the definition of the robust spectral abscissa x is that it bounds transient peaks as well as asymptotic decay of trajectories of xz the function x is lipschitz but non convex for 0 1

stability optimization of hybrid periodic systems via a smooth criterion we consider periodic orbits of controlled hybrid dynamic systems and want to find open loop controls that yield maximally stable limit cycles instead of optimizing the spectral or pseudo spectral radius of the monodromy matrix a which are non smooth criteria we propose a new approach based on the smoothed spectral radius rhoalpha a a differentiable criterion favorable for numerical optimization like the pseudo spectral radius the smoothed spectral radius rhoalpha a converges from above to the exact spectral radius rho a for alphararr 0 its derivatives can be computed efficiently via relaxed lyapunov equations we show that our new smooth stability optimization program based on rhoalpha a has a favorable structure it leads to a differentiable nonlinear optimal control problem with periodicity and matrix constraints for which tailored boundary value problem methods are available we demonstrate the numerical viability of our method using the example of a walking robot model with nonlinear dynamics and ground impacts as a complex open loop stability optimization example

symmetric group blocks of small defect has a specht module s and a p modular irreducible module d the decomposition number s d is de ned to be the composition multiplicity of d in s the following facts areknown about blocks of weight w a if w 0 or 1 then all the decomposition numbers of the block are 0 or 1 b if w 2 and p 2 then all the decomposition numbers of the block are 0 or 1 s2 c if w 4 then some decomposition numbers of a block can be greater than 1 even if p w moreover if w 0 1 or 2 then there is a known method for determining the decompositionnumbers r the situation for the case w 3 is still not properly understood in particular thedecomposition number s 2p 2 2p 2 p 1 1 d 3p 3 2p 1 is yet to be determined for p 5 this is just one of a collection of decomposition numbers forweight 3 which we are unable to evaluate our investigation of blocks of weight 3 grew out of an attempt to improve upon the earlier resultsof martin and russell mr by explicitly calculating the decomposition numbers this led us todiscover various errors and omissions in mr which place in doubt the claim made there that whenp w all the decomposition numbers are 0 or 1 note incidentally that if the decompositionnumbers for a given block are known to be 0 or 1 then the decomposition numbers can in principle be determined by applying schaper s theorem sch jm1 2 basic resultsby using a result of brundan and kleshchev we are able to improve upon the presentation ofseveral of the basic techniques used in s1 s2 mr for estimating decomposition numbers in orderto state these results recall that the diagram of a partition is the set of nodes i j 1 j

source algebras of blocks sources of simple modules and a conjecture of feit abstract we verify a finiteness conjecture of feit on sources of simple modules over group algebras for various classes of finite groups related to the symmetric groups

signed young modules and simple specht modules abstract by a result of hemmer every simple specht module of a finite symmetric group over a field of odd characteristic is a signed young module while specht modules are parametrized by partitions indecomposable signed young modules are parametrized by certain pairs of partitions the main result of this article establishes the signed young module labels of simple specht modules along the way we prove a number of results concerning indecomposable signed young modules that are of independent interest in particular we determine the label of the indecomposable signed young module obtained by tensoring a given indecomposable signed young module with the sign representation as consequences we obtain the green vertices green correspondents cohomological varieties and complexities of all simple specht modules and a class of simple modules of symmetric groups and extend the results of gill on periodic young modules to periodic indecomposable signed young modules

parametric robust structured control design we present a new approach to parametric robust controller design where we compute controllers of arbitrary order and structure which minimize the worst case h infty norm over a pre specified set of uncertain parameters at the core of our method is a nonsmooth minimization method tailored to functions which are semi infinite minima of smooth functions a rich test bench and a more detailed example illustrate the potential of the technique which can deal with complex problems involving multiple possibly repeated uncertain parameters

on donovan s conjecture abstract this article contains reduction theorems for some weaker variants of donovan s conjecture which supposes that for every finite group d of prime order p there are only finitely many morita equivalence classes of p blocks of finite groups having defect groups isomorphic to d i when restricting the conjecture to the case of abelian defect groups it can be reduced to only considering p blocks of a class of groups that are close to being direct products of simple groups ii when restricting the conjecture to the case of principal blocks of finite groups having an abelian sylow p subgroup it can be reduced to only considering p blocks of simple groups a weaker version of donovan s conjecture would suppose that there are only finitely many cartan matrices of p blocks of finite groups with defect group isomorphic to a given one iii this conjecture can be reduced to only considering p blocks of quasisimple groups with centers having orders prime to p

specht modules in the auslander reiten quiver abstract let a be a symmetric algebra over an algebraically closed field we study the position of indecomposable a modules with small socles or heads in auslander reiten components of tree class a we apply the results to specht modules when a is a block of a group algebra of a symmetric group in particular we show that if the block has weight 2 then all specht modules are quasi simple that is they lie at the ends of their components

convergence of the gradient sampling algorithm for nonsmooth nonconvex optimization we study the gradient sampling algorithm of burke lewis and overton for minimizing a locally lipschitz function f on mathbb r n that is continuously differentiable on an open dense subset we strengthen the existing convergence results for this algorithm and introduce a slightly revised version for which stronger results are established without requiring compactness of the level sets of f in particular we show that with probability 1 the revised algorithm either drives the f values to infty or each of its cluster points is clarke stationary for f we also consider a simplified variant in which the differentiability check is skipped and the user can control the number of f evaluations per iteration

projective modules for the symmetric group and young s seminormal form abstract we study the representation theory of the symmetric group s n in positive characteristic p using features of the llt algorithm we give a conjectural description of the projective cover p of the simple module d where is a p restricted partition such that all ladders of the corresponding ladder partition are of order less than p inspired by the recent theory of klr algebras we explain an algorithm that allows us to verify this conjectural description for n 15 at least

martin s conjecture holds for weight 3 blocks of symmetric groups abstract we prove that martin s conjecture holds for weight 3 blocks of symmetric group algebras that if these blocks have abelian defect group then their projective indecomposable modules have a common radical length 7

stabilization via nonsmooth nonconvex optimization nonsmooth variational analysis and related computational methods are powerful tools that can be effectively applied to identify local minimizers of nonconvex optimization problems arising in fixed order controller design we support this claim by applying nonsmooth analysis and methods to a challenging belgian chocolate stabilization problem posed in 1994 find a stable minimum phase rational controller that stabilizes a specified second order plant although easily stated this particular problem remained unsolved until 2002 when a solution was found using an eleventh order controller our computational methods find a stabilizing third order controller without difficulty suggesting explicit formulas for the controller and for the closed loop system which has only one pole with multiplicity 5 furthermore our analytical techniques prove that this controller is locally optimal in the sense that there is no nearby controller with the same order for which the closed loop system has all its poles further left in the complex plane although the focus of the paper is stabilization once a stabilizing controller is obtained the same computational techniques can be used to optimize various measures of the closed loop system including its complex stability radius or hinfin performance

explicit solutions for root optimization of a polynomial family with one affine constraint given a family of real or complex monic polynomials of fixed degree with one affine constraint on their coefficients consider the problem of minimizing the root radius largest modulus of the roots or root abscissa largest real part of the roots we give constructive methods for efficiently computing the globally optimal value as well as an optimal polynomial when the optimal value is attained and an approximation when it is not an optimal polynomial can always be chosen to have at most two distinct roots in the real case and just one distinct root in the complex case examples are presented illustrating the results including several fixed order controller optimal design problems

the principal 3 blocks of four and five dimensional projective special linear groups in non defining characteristic abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he has conjectured that for any prime p if a finite group g has an abelian sylow p subgroup p then the principal p blocks of g and the normalizer ng p of p in g are derived equivalent let q be a power of a prime such that q 2 or 5 mod 9 in this paper we show that broue s conjecture is true for p 3 and for g psl4 q and g psl5 q in these cases g has elementary abelian sylow 3 subgroups of order 9 what we prove here is the following in the case g psl4 q all the principal 3 blocks of g are morita even puig equivalent independently of infinitely many q in the case g psl5 q all the principal 3 blocks of g are morita even puig equivalent to the principal 3 block of ng p independently of infinitely many q

variational analysis of convexly generated spectral max functions the spectral abscissa is the largest real part of an eigenvalue of a matrix and the spectral radius is the largest modulus both are examples of spectral max functions the maximum of a real valued function over the spectrum of a matrix these mappings arise in the control and stabilization of dynamical systems in 2001 burke and overton characterized the regular subdifferential of the spectral abscissa and showed that the spectral abscissa is subdifferentially regular in the sense of clarke when all active eigenvalues are nonderogatory in this paper we develop new techniques to obtain these results for the more general class of convexly generated spectral max functions in particular we extend the burke overton subdifferential regularity result to this class these techniques allow us to obtain new variational results for the spectral radius

painleve transcendent evaluations of finite system density matrices for 1d impenetrable bosons the recent experimental realisation of a one dimensional bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable bose gas of primary concern is the ground state occupation of effective single particle states in the finite system and thus the tendency for bose einstein condensation this requires the computation of the density matrix for the impenetrable bose gas on a circle we evaluate the density matrix in terms of a particular painleve vi transcendent in form and furthermore show that the density matrix satisfies a recurrence relation in the number of particles for the impenetrable bose gas in a harmonic trap and with dirichlet or neumann boundary conditions we give a determinant form for the density matrix a form as an average over the eigenvalues of an ensemble of random matrices and in special cases an evaluation in terms of a transcendent related to painleve v and vi we discuss how our results can be used to compute the ground state occupations

modular lie powers and the solomon descent algebra let v be an r dimensional vector space over an infinite field f of prime characteristic p and let l n v denote the nth homogeneous component of the free lie algebra on v we study the structure of l n v as a module for the general linear group gl r f when n pk and k is not divisible by p and where r n our main result is an explicit 1 1 correspondence multiplicity preserving between the indecomposable direct summands of l k v and the indecomposable direct summands of l n v which are not isomorphic to direct summands of v n our approach uses idempotents of the solomon descent algebras and in addition a correspondence theorem for permutation modules of symmetric groups

adjustment matrices for weight three blocks of iwahori hecke algebras abstract we compute the adjustment matrices for weight 3 blocks of iwahori hecke algebras of type a in characteristic 2 or 3 this enables all the decomposition numbers for weight 3 blocks to be calculated

the quasihereditary structure of the auslander dlab ringel algebra abstract given an arbitrary algebra a we may associate to it a special endomorphism algebra r a introduced by auslander dlab and ringel constructed a heredity chain for r a proving that every algebra a has an associated highest weight theory in this paper we investigate the quasihereditary structure of r a using an axiomatic approach

variational analysis of the abscissa mapping for polynomials the abscissa mapping on the affine variety cmn of monic polynomials of degree n is the mapping that takes a monic polynomial to the maximum of the real parts of its roots this mapping plays a central role in the stability theory of matrices and dynamical systems it is well known that the abscissa mapping is continuous on cmn but not lipschitz continuous furthermore its natural extension to the linear space cpn of polynomials of degree n or less is not continuous in our analysis of the abscissa mapping we use techniques of modern nonsmooth analysis described extensively in variational analysis r t rockafellar and r j b wets springer verlag berlin 1998 using these tools we completely characterize the subderivative and the subgradients of the abscissa mapping and establish that the abscissa mapping is everywhere subdifferentially regular this regularity permits the application of our results in a broad context through the use of standard chain rules for nonsmooth functions our approach is epigraphical and our key result is that the epigraph of the abscissa map is everywhere clarke regular

a robust gradient sampling algorithm for nonsmooth nonconvex optimization let f be a continuous function on rl n and suppose f is continuously differentiable on an open dense subset such functions arise in many applications and very often minimizers are points at which f is not differentiable of particular interest is the case where f is not convex and perhaps not even locally lipschitz but is a function whose gradient is easily computed where it is defined we present a practical robust algorithm to locally minimize such functions based on gradient sampling no subgradient information is required by the algorithm when f is locally lipschitz and has bounded level sets and the sampling radius eps is fixed we show that with probability 1 the algorithm generates a sequence with a cluster point that is clarke eps stationary furthermore we show that if f has a unique clarke stationary point bar x then the set of all cluster points generated by the algorithm converges to bar x as eps is reduced to zero numerical results are presented demonstrating the robustness of the algorithm and its applicability in a wide variety of contexts including cases where f is not locally lipschitz at minimizers we report approximate local minimizers for functions in the applications literature which have not to our knowledge been obtained previously when the termination criteria of the algorithm are satisfied a precise statement about nearness to clarke eps stationarity is available a matlab implementation of the algorithm is posted at http www cs nyu edu overton papers gradsamp alg

from gumbel to tracy widom the tracy widom distribution that has been much studied in recent years can be thought of as an extreme value distribution we discuss interpolation between the classical extreme value distribution exp exp x the gumbel distribution and the tracy widom distribution there is a family of determinantal processes whose edge behaviour interpolates between a poisson process with density exp x and the airy kernel point process this process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by moshe neuberger and shapiro we also consider the deformed gue ensemble m m 0 sqrt 2s v with m 0 diagonal with independent elements and v from gue here we do not see a transition from tracy widom to gumbel but rather a transition from tracy widom to gaussian

sp6 2a is good for the mckay alperin weight and related local global conjectures abstract the so called local global conjectures in the representation theory of finite groups relate the representation theory of g to that of certain proper subgroups such as the normalizers of particular p groups recent results by several authors reduce some of these conjectures to showing that a certain collection of stronger conditions holds for all finite simple groups here we show that g sp 6 2 a is good for these reductions for the mckay conjecture the alperin weight conjecture and their blockwise versions

combining convex concave decompositions and linearization approaches for solving bmis with application to static output feedback a novel optimization method is proposed to minimize a convex function subject to bilinear matrix inequality bmi constraints the key idea is to decompose the bilinear mapping as a difference between two positive semidefinite convex mappings at each iteration of the algorithm the concave part is linearized leading to a convex subproblem applications to various output feedback controller synthesis problems are presented in these applications the subproblem in each iteration step can be turned into a convex optimization problem with linear matrix inequality lmi constraints the performance of the algorithm has been benchmarked on the data from the compleib library

simultaneous stabilization of three or more plants conditions on the positive real axis do not suffice the problem of the simultaneous stabilizability of a finite family of single input single output time invariant systems by a time invariant controller is studied the link between stabilization and avoidance is shown and is used to derive necessary conditions for the simultaneous stabilization of k plants these necessary conditions are proved to be in general not sufficient this result also disproves a long standing conjecture on the stabilizability condition of a single plant with a stable minimum phase controller the main result is to show that unlike the case of two plants the existence of a simultaneous stabilizing controller for more than two plants is not guaranteed by the existence of a controller such that the closed loops have no real unstable poles

james s conjecture holds for weight four blocks of iwahori hecke algebras abstract james s conjecture suggests that in certain cases the decomposition numbers for the iwahori hecke algebra of the symmetric group over a field of prime characteristic and in particular the decomposition numbers for the symmetric group itself coincide with the decomposition numbers for a corresponding iwahori hecke algebra defined over c and hence can be computed using the llt algorithm we prove this conjecture for blocks of weight 4

on the number of constituents of induced modules of ariki koike algebras abstract we examine the crystal graph of the sl e module arising from an sl e categorification to study the defining endo functors of the categorification this yields lower bounds on the number of irreducible constituents of certain objects we use ariki s categorification result on ariki koike algebras to obtain a new lower bound on the number of constituents of their parabolically induced modules in particular this will imply reducibility of every induced module

adjustment matrices for the principal block of the iwahori hecke algebra h5e abstract james s conjecture predicts that the adjustment matrix for blocks of the iwahori hecke algebra of the symmetric group is the identity matrix when the weight of the block is strictly less than the characteristic of the field in this paper we consider the case when the characteristic of the field is greater than or equal to 5 and prove that the adjustment matrix for the principal block of h 5 e is the identity matrix whenever e 4 when e 4 we are able to calculate all but two entries of the adjustment matrix

equivalent blocks of finite general linear groups in non describing characteristic abstract j chuang r kessar and j rickard have proved broue s abelian defect group conjecture for many symmetric groups we adapt the ideas of kessar and chuang towards finite general linear groups represented over non describing characteristic we then describe morita equivalences between certain p blocks of gl n q with defect group c p c p as q varies see theorem 2 here p and q are coprime this generalizes work of s koshitani and m hyoue who proved the same result for principal blocks of gl n q when p 3 1 in a different way

the derived categories of some blocks of symmetric groups and a conjecture of broue abstract a well known conjecture of broue in the representation theory of finite groups involves equivalences of derived categories of blocks the aim of this paper is to verify this conjecture for defect 2 blocks of symmetric groups actually we prove for these blocks a refinement of broue s conjecture due to rickard

second quantization approach to characteristic polynomials in rmt the distribution of the characteristic polynomial z u of n n matrices u in the circular unitary ensemble is studied by the method of second quantization for one dimensional fermions for infinite n the gaussian distribution of z u is established straightforwardly by bosonization a general expression for the n point correlation function of the characteristic polynomial at different points is given by this method the case of finite n is discussed pacs numbers 0210n 0530c 0210l 0210s

defect 3 blocks of symmetric group algebras abstract in this paper we study if the following properties are present in defect 3 blocks of symmetric group algebras k g n where k has characteristic p 5 the ext quiver is bipartite the p i m s have a common loewy length 7 we obtain a criteria for a defect 3 block b of k g n to inherit these properties from a defect 3 block b of k g n 1 with which it forms a 3 1 pair we also verify that the principal blocks of k g n 3 p n 4 p 1 have these properties

cartan matrices and morita equivalence for blocks of the symmetric groups theorem 1 the blocks of the symmetric groups with a given weight w 0 over a field of characteristic p can be collected into families according to the shapes of their cores each family consists of morita equivalent blocks the number of such families is at most flf 1 i l w 1 11 and each family contains a block of some g where n is less than or equal to p2 p 1 2 w 1 2 4 wp

nonintersecting brownian motions on the unit circle we consider an ensemble of n nonintersecting brownian particles on the unit circle with diffusion parameter n 1 2 which are conditioned to begin at the same point and to return to that point after time t but otherwise not to intersect there is a critical value of t which separates the subcritical case in which it is vanishingly unlikely that the particles wrap around the circle and the supercritical case in which particles may wrap around the circle in this paper we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as n and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution we also give a streamlined approach to identifying the pearcey and tacnode processes in scaling limits the formula of the tacnode correlation kernel is new and involves a solution to a lax system for the painleve ii equation of size 2 2 the proofs are based on the determinantal structure of the ensemble asymptotic results for the related system of discrete gaussian orthogonal polynomials and a formulation of the correlation kernel in terms of a double contour integral

a conjecture for q decomposition matrices of cyclotomic v schur algebras abstract the jantzen sum formula for cyclotomic v schur algebras yields an identity for some q analogues of the decomposition matrices of these algebras we prove a similar identity for matrices of canonical bases of higher level fock spaces we conjecture then that those matrices are actually identical for a suitable choice of parameters in particular we conjecture that decomposition matrices of cyclotomic v schur algebras are obtained by specializing at q 1 some transition matrices between the standard basis and the canonical basis of a fock space

weights of multipartitions and representations of ariki koike algebras abstract we consider representations of the ariki koike algebra a q deformation of the group algebra of the complex reflection group c r s n the representations of this algebra are naturally indexed by multipartitions of n and for each multipartition we define a non negative integer called the weight of we prove some basic properties of this weight function and examine blocks of small weight

statistical properties of determinantal point processes in high dimensional euclidean spaces the goal of this paper is to quantitatively describe some statistical properties of higher dimensional determinantal point processes with a primary focus on the nearest neighbor distribution functions toward this end we express these functions as determinants of n n matrices and then extrapolate to n this formulation allows for a quick and accurate numerical evaluation of these quantities for point processes in euclidean spaces of dimension d we also implement an algorithm due to hough et al for generating configurations of determinantal point processes in arbitrary euclidean spaces and we utilize this algorithm in conjunction with the aforementioned numerical results to characterize the statistical properties of what we call the fermi sphere point process for d 1 4 this homogeneous isotropic determinantal point process discussed also in a companion paper s torquato a scardicchio and c e zachary j stat mech theory exp 2008 p11019 i s the high dimensional generalization of the distribution of eigenvalues on the unit circle of a random matrix from the circular unitary ensemble in addition to the nearest neighbor probability distribution we are able to calculate voronoi cells and nearest neighbor extrema statistics for the fermi sphere point process and we discuss these properties as the dimension d is varied the results in this paper accompany and complement analytical properties of higher dimensional determinantal point processes developed in a prior paper

the irreducible representations of the alternating group which remain irreducible in characteristic p let p be an odd prime and an the alternating group of degree n we determine which ordinary irreducible representations of u n remain irreducible in characteristic p verifying the author s conjecture from 2010 given the preparatory work done in 2010 our task is to determine which self conjugate partitions label specht modules for the symmetric group in characteristic p having exactly two composition factors this is accomplished through the use of the robinson brundan kleshchev i restriction functors together with known results on decomposition numbers for the symmetric group and additional results on the mullineux map and homomorphisms between specht modules

sign sequences and decomposition numbers we obtain a closed formula for the v decomposition numbers d v arising from the canonical basis of the fock space representation of uv le where the partition is obtained from by moving some nodes in its young diagram all of which having the same e residue we also show that when these v decomposition numbers are evaluated at v 1 we obtain the corresponding decomposition numbers for the schur algebras and symmetric groups

weight two blocks of iwahori hecke algebras of type b abstract in the representation theory of iwahori hecke algebras of type a and in particular for representations of symmetric groups the notion of the weight of a block introduced by james plays a central role richards determined the decomposition numbers for blocks of weight 2 and here the same task is undertaken for weight two blocks of iwahori hecke algebras of type b using the author s own definition of the weight of a bipartition

schubert calculus and torsion explosion the author observes that certain numbers occurring in schubert calculus for sl also occur as entries in intersection forms controlling decompositions of soergel bimodules in higher rank these numbers grow exponentially this observation gives many counter examples to the expected bounds in lusztig s conjecture on the characters of simple rational modules for sl over fields of positive characteristic the examples also give counter examples to the james conjecture on decomposition numbers for symmetric groups

smoothing methods for nonsmooth nonconvex minimization we consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex not differentiable and perhaps not even locally lipschitz at the solutions such optimization problems arise from wide applications including image restoration signal reconstruction variable selection optimal control stochastic equilibrium and spherical approximations in this paper we focus on smoothing methods for solving such optimization problems which use the structure of the minimization problems and composition of smoothing functions for the plus function x many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods we present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function moreover we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem

specht modules labelled by hook bipartitions i abstract brundan kleshchev and wang equip the specht modules s over the cyclotomic khovanov lauda rouquier algebra h n with a homogeneous z graded basis in this paper we begin the study of graded specht modules labelled by hook bipartitions n m 1 m in level 2 of h n which are precisely the hecke algebras of type b with quantum characteristic at least three we give an explicit description of the action of the khovanov lauda rouquier algebra generators 1 n 1 on the basis elements of s n m 1 m introducing certain specht module homomorphisms we construct irreducible submodules of these specht modules and thereby completely determine the composition series of specht modules labelled by hook bipartitions

decomposition numbers for weight three blocks of symmetric groups and iwahori hecke algebras let f be a field q a non zero element of f and h n h f q g n the iwahori hecke algebra of the symmetric group g n if b is a block of h n of e weight 3 and the characteristic of f is at least 5 we prove that the decomposition numbers for b are all at most 1 in particular the decomposition numbers for a p block of g n of defect 3 are all at most 1

bounds on the dimension of ext for finite groups of lie type abstract let g be a finite group of lie type defined in characteristic p and let k be an algebraically closed field of characteristic r 0 we will assume that r p so we are in the non defining characteristic case let v be a finite dimensional irreducible left kg module in 2011 guralnick and tiep found bounds on the dimension of h 1 g v in non defining characteristic which are independent of v the aim of this paper is to generalize the work of gurlanick and tiep we assume that g is split and use methods of modular harish chandra theory to find bounds on the dimension of ext k g 1 y v where y and v are irreducible kg modules we then use dipper and du s algorithms to illustrate our bounds in a series of examples

general runner removal and the mullineux map abstract we prove a new runner removal theorem for q decomposition numbers of the level 1 fock space of type a e 1 1 generalising earlier theorems of james mathas and the author by combining this with another theorem relating to the mullineux map we show that the problem of finding all q decomposition numbers indexed by partitions of a given weight is a finite computation

nearest stable system using successive convex approximations stability is a crucial property in the study of dynamical systems we focus on the problem of enforcing the stability of a system a posteriori the system can be a matrix or a polynomial either in continuous time or in discrete time we present an algorithm that constructs a sequence of successive stable iterates that tend to a nearby stable approximation x of a given system a the stable iterates are obtained by projecting a onto the convex approximations of the set of stable systems some possible applications for this method are correcting the error arising from some noise in system identification and a possible solver for bilinear matrix inequalities based on convex approximations in the case of polynomials a fair complexity is achieved by finding a closed form solution to first order optimality conditions

on young modules of defect 2 blocks of symmetric group algebras abstract in this paper we obtain the ordinary characters and module structures of the young modules of defect 2 blocks of symmetric group algebras we also study how these modules induce and restrict in a 2 1 pair

on torsion in the intersection cohomology of schubert varieties abstract we prove that the prime torsion in the local integral intersection cohomology of schubert varieties in the flag variety of the general linear group grows exponentially in the rank the idea of the proof is to find a highly singular point in a schubert variety and calculate the euler class of the normal bundle to the miraculously smooth fibre in a particular bott samelson resolution the result is a geometric version of an earlier result established using soergel bimodule techniques

equating decomposition numbers for different primes abstract this paper shows that certain decomposition numbers for the iwahori hecke algebras of the symmetric groups and the q schur algebras at different roots of unity in characteristic zero are equal to prove our results we first establish the corresponding theorem for the canonical basis of the level one fock space and then apply deep results of ariki and varagnolo vasserot

scopes reduction and morita equivalence classes of blocks in finite classical groups ii abstract we extend our work of j algebra 230 2000 378 423 on unipotent blocks of classical groups to blocks of orthogonal groups containing characters labelled by degenerate symbols we also show that for classical groups in the linear prime case there are only finitely many morita types of unipotent blocks with a given defect group