affine spaces of strong linearizations for rational matrices and the recovery of eigenvectors and minimal bases abstract the main aim of this paper is two fold first to define and analyze two affine spaces of matrix pencils associated with a realization of an n n rational matrix g and show that almost all of these pencils are strong linearizations of g second to describe the recovery of eigenvectors minimal bases and minimal indices of g from those of the strong linearizations of g the affine spaces of pencils are constructed from the vector spaces of linearizations of matrix polynomials defined and analyzed by mackey et al in siam j matrix anal appl 28 2006 pp 971 1004 we show that eigenvectors minimal bases and minimal indices of g can be easily recovered from those of the strong linearizations of g in particular we construct a symmetric strong linearization of g when g is regular and symmetric we also describe the recovery of pole zero structure of g at infinity from the pole zero structure at infinity of a strong linearization of g

global strong solutions for nonlinear schro dinger equations on considere l existence globale unique des solutions fortes pour les equations de schrodinger non lineaire a interaction en loi de puissance i u t u u p 1 u t r x r n u t 0 x u 0 x x r n ou t 0 r

strichartz type estimates in mixed besov spaces with application to critical nonlinear schrodinger equations abstract we study the cauchy problem for the nonlinear schrodinger equation with power nonlinearity in the fractional order sobolev space h s r n we consider the case where the nonlinear term is h s critical and the differentiability thereof is less than s we prove the existence of time global solutions of the cauchy problem for small initial data to this end we prove strichartz type estimates for linear inhomogeneous schrodinger equations in mixed besov spaces

efficient arnoldi type algorithms for rational eigenvalue problems arising in fluid solid systems we develop and analyze efficient methods for computing damped vibration modes of an acoustic fluid confined in a cavity with absorbing walls capable of dissipating acoustic energy the discretization in terms of pressure nodal finite elements gives rise to a rational eigenvalue problem numerical evidence shows that there are no spurious eigenmodes for such discretization and also confirms that the discretization based on nodal pressures is much more efficient than that based on raviart thomas finite elements for the displacement field the trimmed linearization method is used to linearize the associated rational eigenvalue problem into a generalized eigenvalue problem gep of the form ax lbx for solving the gep we apply arnoldi algorithm to two different types of single matrices b 1a and ab 1 numerical accuracy shows that the application of arnoldi on ab 1 is better than that on b 1a

transformation of j spectral factorization of improper matrices to proper matrices we reduce the general problem of j spectral factorization with improper matrices given by a descriptor system realization to the problem of j spectral factorization with proper matrices

parabolic inductions for pro p iwahori hecke algebras abstract we give some properties of parabolic inductions and their adjoint functors for pro p iwahori hecke algebras

a comparison of solutions to the minimal design problem two of the recent algorithms for solving the minimal design problem in an efficient and apparently stable manner are presented in a common setting it is shown that the resulting algorithm maintains attractive features of both earlier algorithms including efficiency ease of use flexibility in selecting criteria of closeness as well as apparently numerical stability

comments on minimum gain minimum time deadbeat controllers abstract we point out that the problem formulated in the above paper by elabdalla and amin 1 was solved in an earlier paper 3 using a direct approach

computation of transfer function matrices of generalized state space models a reliable computational algorithm is proposed for the evaluation of the transfer function matrices of linear multivariable systems described by generalized state space models the algorithm is based on a new efficient and numerically stable procedure for computing irreducible state space realizations of generalized state space models for each input output channel the proposed algorithm computes an irreducible realization using this procedure then the parameters of the corresponding transfer function gain poles zeros are determined from the resulting irreducible model

robust stabilization of discrete generalized systems the robust stabilization problem for generalized discrete time systems described by polynomial or improper transfer function matrices subject to perturbations acting on the normalized coprime factors is solved the maximum achievable stability margin and the robust stabilizing controller are given in terms of realizations and solutions to appropriate riccati equations

an algorithm for pole assignment in high order multivariable systems an efficient computational method is presented for solving the problem of pole assignment by state feedback in linear multivariable systems with large dimensions a given multiinput system is first transformed to an upper block hessenberg form by means of orthogonal state coordinate transformations the state feedback problem is then reformulated in terms of the sylvester equation the transformed system matrices along with certain assumed block forms for unknown matrices enable the sylvester equation to be decomposed and solved effectively a distinct point of the proposed algorithm is that the solution procedure can be tailored to parallel implementation and is therefore fast computational aspects of the algorithm are discussed and numerical examples are provided

scattering theory in the energy space for a class of non linear wave equations we study the asymptotic behaviour in time of the solutions and the theory of scattering in the energy space for the non linear wave equation square varphi f varphi 0 in n n 3 we prove the existence of the wave operators asymptotic completeness for small initial data and forn 4 asymptotic completeness for arbitrarily large data the assumptions onf cover the case wheref behaves slightly better than a single powerp 1 4 n 2 both near zero and at infinity see 1 5 1 6 and 1 8

unfolding the zero structure of a linear control system abstract this paper is motivated by the problem of controller design for a parameter dependent linear system many compensation techniques depend critically on aspects of the zero structure of the system such as relative degree or the nonminimum phase property however the zero structure is structurally unstable meaning it may change discontinuously with small changes in the parameters thus it is crucial that the designer know what structures and structural transitions are possible this paper uses a miniversal deformation or unfolding of the kronecker form previously reported by the authors and applies it directly to pencils in a canonical form of the system matrix the unfolding is used to explore all zero structures in the neighborhood of a nominal system several examples are presented when possible the results are presented in the form of a bifurcation diagram

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

a geometric approach to the synthesis of failure detection filters a geometric formulation of beard s failure detection filter problem is stated using the concepts of c a invariant and unobservability subspaces the notions of output separable and mutually detectable families of subspaces introduced by beard are also clarified it is shown that mutual detectability is a necessary and sufficient condition for the existence of a detection filter with arbitrarily assignable spectrum moreover it is shown that the failure detection falter problem has a computationally simple solution when the failure events satisfy some mild restrictions finally the complete duality between a generalization of beard s detection filter problem and the restricted control decoupling problem is illustrated

feedback design for regularizing descriptor systems this paper surveys numerical techniques for the regularization of descriptor generalized state space systems by proportional and derivative feedback we review generalizations of controllability and observability to descriptor systems along with definitions of regularity and index in terms of the weierstras canonical form three condensed forms display the controllability and observability properties of a descriptor system the condensed forms are obtained through orthogonal equivalence transformations and rank decisions so they may be computed by numerically stable algorithms in addition the condensed forms display whether a descriptor system is regularizable i e when the system pencil can be made to be regular by derivative and or proportional output feedback and if so what index can be achieved also included is a a new characterization of descriptor systems that can be made to be regular with index 1 by proportional and derivative output feedback

disturbance decoupling for descriptor systems by state feedback we study the disturbance decoupling problem for linear time invariant descriptor systems we give necessary and sufficient conditions for the existence of a solution to the disturbance decoupling problem with or without stability via a proportional and or derivative feedback that also makes the resulting closed loop system regular and or of index at most one all results are proved constructively based on condensed forms that can be computed using orthogonal matrix transformations i e transformations that can be implemented in a numerically stable way

transmission zero assignment using semistate descriptions the problem of finite infinite transmission zero assignment is examined by squaring a system from the outputs to the inputs in particular this problem is studied in two cases state accessible systems and partially state accessible systems it is shown that the problem of transmission zero assignment for state accessible state variable systems is equivalent to a pole placement problem with state feedback for a generalized system which always has a solution in the case of partially state accessible systems it is shown that the transmission zero assignment problem is equivalent to a pole placement problem with output feedback for a generalized system in both cases the block hessenberg form of the system and the extended lower triangular hessenberg form are exploited to formulate the problem

parametric state feedback control with model reduction a parametrization of the class of linear state feedback controllers that assign a set of desired self conjugate eigenvalues to the closed loop system is applied to a method of model reduction by making the maximum number of the closed loop modes unobservable while retaining arbitrary assignment of the remaining modes a lower order transfer function matrix is obtained the technique may be used to reduce a broad class of multivariable systems to first order multivariable systems the main result is a class of mode reducing fixed gain state feedback controllers explicitly specified by a set of free parameters which may be chosen to satisfy additional design requirements

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

the classification of irreducible admissible mod p representations of a p adic gln let f be a finite extension of p using the mod p satake transform we define what it means for an irreducible admissible smooth representation of an f split p adic reductive group over overline mathbb f p to be supersingular we then give the classification of irreducible admissible smooth gln f representations over overline mathbb f p in terms of supersingular representations as a consequence we deduce that supersingular is the same as supercuspidal these results generalise the work of barthel livne for n 2 for general split reductive groups we obtain similar results under stronger hypotheses

numerical aspects of disturbance decoupling by measurement feedback a numerically stable algorithm is given to determine orthogonal similarity transformations such that the system attains a certain condensed form using this condensed form solvability of disturbance decoupling by measurement can be decided by inspection and controllers can be computed by solving linear equations

inclusion principle for descriptor systems the purpose of this paper is to propose an expansion contraction framework for linear constant descriptor systems within the inclusion principle for dynamic systems our primary objective is to provide an explicit characterization of the expansion process whereby a given descriptor system is expanded into the larger space where all its solutions are reproducible by the expanded descriptor system if appropriate initial conditions are selected when a control law is formulated in the expanded space the proposed characterizations provide contractibility conditions for implementation of the control law in the original system a full freedom is provided for selecting appropriate matrices in the proposed expansion contraction control scheme in particular the derived theoretical framework serves as a flexible environment for expansion contraction control design of descriptor systems under overlapping information structure constraints

a parallel additive schwarz preconditioned jacobi davidson algorithm for polynomial eigenvalue problems in quantum dot simulation we develop a parallel jacobi davidson approach for finding a partial set of eigenpairs of large sparse polynomial eigenvalue problems with application in quantum dot simulation a jacobi davidson eigenvalue solver is implemented based on the portable extensible toolkit for scientific computation petsc the eigensolver thus inherits petsc s efficient and various parallel operations linear solvers preconditioning schemes and easy usages the parallel eigenvalue solver is then used to solve higher degree polynomial eigenvalue problems arising in numerical simulations of three dimensional quantum dots governed by schrodinger s equations we find that the parallel restricted additive schwarz preconditioner in conjunction with a parallel krylov subspace method e g gmres can solve the correction equations the most costly step in the jacobi davidson algorithm very efficiently in parallel besides the overall performance is quite satisfactory we have observed near perfect superlinear speedup by using up to 320 processors the parallel eigensolver can find all target interior eigenpairs of a quintic polynomial eigenvalue problem with more than 32 million variables within 12 minutes by using 272 intel 3 0ghz processors

computing the radius of controllability for state space systems abstract in this paper we discuss the problem of computing the nearest uncontrollable system to a given control system represented by a matrix pair a b in order to do so we construct a sequence of structured matrices from given system matrices a and b controllability of the pair a b is equivalent to a condition on the null space dimension of an appropriate matrix in this sequence we show that the dimension of the reachability space is also related to the above condition further it is shown that the nearest structured low rank approximation slra of this structured matrix corresponds to a nearest uncontrollable system to the pair a b

a canonical torsion theory for pro p iwahori hecke modules abstract let f be a locally compact nonarchimedean field with residue characteristic p and g the group of f rational points of a connected split reductive group over f we define a torsion pair in the category mod h of modules over the pro p iwahori hecke k algebra h of g where k is an arbitrary field we prove that under a certain hypothesis the torsionfree class embeds fully faithfully into the category mod i g of smooth k representations of g generated by their pro p iwahori fixed vectors if the characteristic of k is different from p then this hypothesis is always satisfied and the torsionfree class is the whole category mod h if k contains the residue field of f then we study the case g s l 2 f we show that our hypothesis is satisfied and we describe explicitly the torsionfree and the torsion classes if f q p and p 2 then an h module is in the torsion class if and only if it is a union of supersingular finite length submodules it lies in the torsionfree class if and only if it does not contain any nonzero supersingular finite length module if f q p the torsionfree class is the whole category mod h and we give a new proof of the fact that mod h is equivalent to mod i g these results are based on the computation of the h module structure of certain natural cohomology spaces for the pro p iwahori subgroup i of g

the properties of reduced order minimum variance filters for systems with partially perfect measurements the problem of the finite time reduced order minimum variance full state estimation of linear continuous time invariant systems is considered in cases where the output measurement is partially free of corrupting white noise components the structure of the optimal filter is obtained and a link between this structure and the structure of the system invariant zeros is established using expressions that are derived in closed form for the invariant zeros of the system simple sufficient conditions are obtained for the existence of the optimal filter in the stationary case the structure and the transmission properties of the stationary filter for general left invertible systems are investigated a direct relation between the optimal filter and a particular minimum order left inverse of the system is obtained a simple explicit expression for the filter transfer function matrix is also derived the expression provides an insight into the mechanism of the optimal estimation

iterative projection methods for computing relevant energy states of a quantum dot a computational technique for computing relevant energy levels and corresponding wave functions of an electron confined by a 3d quantum dot embedded in a semiconductor matrix are studied assuming an energy and position dependent electron effective mass approximation this problem is governed by a rational eigenvalue problem we discuss the application of iterative projection method of arnoldi and jacobi davidson type projected problems of small dimension are solved efficiently by safeguarded iteration

formulae relating the bernstein and iwahori matsumoto presentations of an affine hecke algebra abstract we consider the anti dominant variants of the elements occurring in the bernstein presentation of an affine hecke algebra h we find explicit formulae for in terms of the iwahori matsumoto generators tw w ranging over the extended affine weyl group of the root system r in the case i r is arbitrary and is a minuscule coweight or ii r is attached to gln and mek where ek is a standard basis vector and m 1 in the above cases certain minimal expressions for play a crucial role such minimal expressions exist in fact for any coweight for gln we give a sheaf theoretic interpretation of the existence of a minimal expression for the corresponding perverse sheaf on the affine schubert variety x t is the push forward of an explicit perverse sheaf on the demazure resolution m x t x t this approach yields for a minuscule coweight of any r or for an arbitrary coweight of gln a conceptual albeit less explicit expression for the coefficient w of the basis element tw in terms of the cohomology of a fiber of the demazure resolution

paper minimal representations of implicit linear systems definitions of external behaviour equivalence and minimality are given for a class of implicit non oriented linear dynamical systems of particular interest in computer aided analysis and design conditions for minimality are stated in terms of the rank properties of submatrices in the singular pencil realization and a geometric analysis is used in the proof a stable library quality numerical algorithm for minimization is included

a geometric approach to proportional plus derivative feedback using quotient and partitioned subspaces abstract in this paper a new characterization of invariant subspaces is presented using the notions of partitioned and quotient subspaces this classification is based on a feedback approach that exhibits the importance of these subspaces for the problem of proportional plus derivative feedback it also provides the ability to decompose the closed loop system into two subsystems that completely characterize the closed loop behavior a feedback free formulation is also considered which opens new horizons for the problem of general semistate feedback by utilizing nonsquare pencils for spectrum assignability the importance of this technique arises from the fact that we deal with reduced order pencils therefore the computational methods are more stable in order to accomplish this we use two generalized lyapunov equations and exploit the generalized hessenberg form

a numerical design of interactors for general descriptor systems in this technical note we develop a numerically reliable method to design an interactor for a general descriptor system relative to a stability region cg and an offending zero set our main result is based on a condensed form this condensed form is independent of the offending zeros and is computed using only orthogonal transformations and hence is numerically backward stable

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

pro p iwahori hecke algebra and supersingular representations the motivation of this paper is the search for a langlands correspondence modulo p we show that the pro p iwahori hecke ring h 1 of a split reductive p adic group g over a local field f of finite residue field fq with q elements admits an iwahori matsumoto presentation and a bernstein z basis and we determine its centre we prove that the ring h 1 is finitely generated as a module over its centre these results are proved in 12 only for the iwahori hecke ring let p be the prime number dividing q and let k be an algebraically closed field of characteristic p a character from the centre of h 1 to k which is as null as possible will be called null the simple h k modules with a null central character are called supersingular when g gl n we show that each simple h k module of dimension n containing a character of the affine subring h aff is supersingular the proof uses the minimal expressions of haines generalized to h 1 and that the number of such modules is equal to the number of irreducible k representations of the weil group wf of dimension n when the action of an uniformizer pf in the hecke algebra side and of the determinant of a frobenius frf in the galois side are fixed i e the number nn q of unitary irreducible polynomials in fq x of degree n one knows that the converse is true by explicit computations when n 2 11 and when n 3 rachel ollivier

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

an algorithm for the computation of 2 d eigenvalues in this paper a definition for the eigenvalues of two dimensional 2 d general discrete state space models is proposed this definition is consistent with that known for 2 d roesser models also a programmable algorithm is developed to compute the 2 d eigenvalues

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

modules simples en caracteristique p de l algebre de hecke du pro p iwahori de gl3 f resume pour f un corps p adique et n n n 2 on ne connait pas de facon generale les representations lisses irreductibles modulo p de gl n f le foncteur des invariants par le pro p iwahori i 1 invite a les apprehender par l etude des modules simples a droite sur l algebre de hecke h f p gl n f i 1 dans le cas n 2 on connait les h f p gl n f i 1 modules simples ayant un caractere central parmi eux on distingue les modules que l on appelle supersinguliers ils ne correspondent pas via le foncteur des invariants par le pro p iwahori a des sous quotients d induites paraboliques de gl 2 f m f vigneras representations modulo p of the p adic group gl 2 f compos math 140 2004 333 358 nous considerons le cas n 3 il s agit de donner la classification des h f p gl 3 f i 1 modules simples ayant un caractere central et de decrire conjecturalement les modules supersinguliers l ensemble de ces modules supersinguliers est en bijection non unique avec l ensemble des f p representations irreductibles de dimension 3 du groupe de weil de f

deflation in eigenvalue assignment of descriptor systems using state feedback it is shown how deflation by unitary equivalence transformations can be used to solve the eigenvalue assignment problem for descriptor systems using state feedback the system is first transformed to a form that reveals its controllability the uncontrollable part of the system if any is then discarded and the assignments are continued with the completely controllable part of the system to give insight into the approach a mathematical method for the solution is derived without considering numerical issues using the idea of deflation it is shown how this method can be modified to derive a numerically efficient algorithm the numerical stability of the algorithm is proved and a numerical example is presented

modified strichartz estimates with an application to the critical nonlinear schrodinger equation abstract the cauchy problem for the critical nonlinear schrodinger equation is considered in the sobolev space of fractional order some modified strichartz estimates are constructed and applied to the problem to obtain small global solutions with less regularity assumption for the nonlinear term

critere d irreductibilite pour les series principales de gln f en caracteristique p resume soient p un nombre premier et f un corps p adique parmi les f p representations lisses irreductibles de gl 2 f ayant un caractere central l barthel et r livne ont classifie les sous quotients d induites paraboliques et montre qu il en existe d autres qu ils n ont pas classifiees et qu ils ont appelees supersingulieres l barthel r livne irreducible modular representations of gl 2 of a local field duke math j 75 2 1994 261 292 l barthel r livne modular representations of gl 2 of a local field the ordinary unramified case j number theory 55 1995 en 2001 c breuil a obtenu la classification de ces dernieres dans le cas particulier ou f q p c breuil sur quelques representations modulaires et p adiques de gl 2 q p i compos math 138 2 2003 165 188 pour n 3 on ne connait pas meme la classification des f p representations lisses irreductibles de gl n f qui sont des sous quotients d induites paraboliques nous nous interessons ici au cas des series principales le cadre de la caracteristique p fait jouer un role particulier a l unique pro p sylow du sous groupe d iwahori de gl n f toute f p representation lisse non nulle de gl n f possede un vecteur non trivial fixe par ce pro p iwahori il est ainsi naturel d approcher les representations lisses de gl n f via l etude des modules sur la f p algebre de hecke du pro p iwahori de gl n f les travaux de m f vigneras sur la structure de cette algebre ont permis de definir ses modules standards m f vigneras pro p iwahori hecke ring and supersingular f p representations math ann 331 3 2005 523 556 erratum in math ann 333 3 2005 699 701 a l aide des modules standards dits reguliers nous etudions l espace des invariants sous l action du pro p iwahori des series principales de gl n f en caracteristique p nous en deduisons un critere d irreductibilite pour ces series principales

singular systems a new approach in the time domain a new approach in the time domain is developed for the study of singular linear systems of the form e dot x ax bu y cx with e singular central to the approach is the fundamental triple alpha e a 1 e alphae a 1 b c with a real number satisfying det alpha e a neq 0 controllability and observability matrices are expressed in terms of the fundamental triple new tests for impulse controllability and impulse observability are also established feedback control problems including pole placement decoupling and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form of u f alpha x dot x v

computation of matrix fraction descriptions of linear time invariant systems in this paper algorithms are presented for obtaining relatively prime matrix fraction descriptions mfd s of linear time invariant systems orthogonal coordinate transformations are used to reduce a given state space model to one whose state matrix is in block hessenberg form the observability controllability indexes of the system are also obtained in the process a recursive algorithm is then described for obtaining a relatively prime mfd from the block hessenberg representation it is then shown that some simplification can be made in the recursive algorithm by first reducing the block hessenberg form by means of a nonsingular triangular coordinate transformation to a more compact form such as the block frobenius form a permutation of the state variables of the block frobenius form yields a canonical representation similar to the luenberger canonical form some numerical and other properties of the algorithms are discussed and the use of the algorithms is illustrated by numerical examples the numerical performance of the algorithms is also compared with that of the structure algorithm of wolovich and falb

hessenberg and hessenberg triangular forms in linear system thcory abstract in this paper we distinguish between free hessenberg forms where only the system matrix is transformed into a special structure and system hessenberg forms where either an input or output matrix is simultaneously simplified we show how the freedom in the choice of the transformation matrix for the free hessenberg form can be used to obtain a unique system hessenberg form we discuss numerical aspects and outline some basic applications for both types of hessenberg forms in control and system theory we show that the system hessenberg form can be extremely sensitive to perturbations a property not necessarily shared by the free hessenberg form several examples are explored and a first order perturbation analysis is given analogous aspects are also discussed for the hessenberg triangular forms resulting from reduction of a matrix pencil the main emphasis of this paper is on systems with one input

computation of structural invariants of generalized state space systems in this paper we develop an algorithm for computing the zeros of a generalized state space model described by the matrix 5 tuple e a b c d where e may be a singular matrix but det a he 0 the characterization of these zeros is based on the system matrix of the corresponding 5 tuple both the characterization and the computational algorithm are extensions of equivalent results for state space models described by the 4 tuples a b c d we also extend these results to the computation of infinite zeros and left and right minimal indices of the system matrix several non trivial numerical examples are included to illustrate the proposed results

linear transformations which leave controllable multiinput descriptor systems controllable abstract a general linear descriptor system e itx ax bu where e a c n n b c n m x c n u c m e singular is called controllable if rank e a b n for all 0 0 c let f c n n m c n n m be a linear transformation form f x cxd we characterize all such linear transformations that leave the set c e a b rank e a b n 0 0 c invariant and show that only the well known transformations that leave controllability invariant can occur

on well posedness for nonlinear schrodinger equations with power nonlinearity in fractional order sobolev spaces abstract we study the well posedness for the nonlinear schrodinger equation nls i t u 1 2 u u p 1 u in r 1 n where p 1 c and prove that nls is locally well posed in h s if 2 s 4 and s 2 p 1 4 n 2 s to obtain a good lower bound for p we systematically use strichartz type estimates in fractional order besov spaces for the time variable

on special representations of p adic reductive groups let f be a non archimedean locally compact field let g be a split connected reductive group over f for a parabolic subgroup q g and a ring l we consider the g representation on the l module c g q l

recovery of eigenvectors of rational matrix functions from fiedler like linearizations abstract linearization is a standard method often used when dealing with matrix polynomials recently the concept of linearization has been extended to rational matrix functions and fiedler like matrix pencils for rational matrix functions have been constructed a linearization l of a rational matrix function g does not necessarily guarantee a simple way of recovering eigenvectors of g from those of l we show that fiedler like pencils of g allow an easy operation free recovery of eigenvectors of g that is eigenvectors of g are recovered from eigenvectors of fiedler like pencils of g without performing any arithmetic operations we also consider fiedler like pencils of the rosenbrock system polynomial s associated with an lti system in state space form ssf and show that the fiedler like pencils allow operation free recovery of eigenvectors of s the eigenvectors of s are the invariant zero directions of the lti system

local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems abstract this paper presents a definition for local linearizations of rational matrices and studies their properties this definition allows to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and poles in subsets of any algebraically closed field and also at infinity this new theory of local linearizations captures and explains rigorously the properties of all the different pencils that have been used from the 1970 s until 2020 for computing zeros poles and eigenvalues of rational matrices particular attention is paid to those pencils that have appeared recently in the numerical solution of nonlinear eigenvalue problems through rational approximation

the eigenstructure of an arbitrary polynomial matrix computational aspects we give a new numerical method to compute the eigenstructure i e the zero structure the polar structure and the left and right null space structure of a polynomial matrix p these structural elements are of fundamental importance in several systems and control problems involving polynomial matrices the approach is more general than previous numerical methods because it can be applied to an arbitrary m n polynomial matrix p with normal rank r smaller than m and or n the algorithm is then shown to compute the structure of the left and right null spaces of p as well the speed and accuracy of this new approach are also discussed

some examples of square integrable representations of semisimple p adic groups we construct irreducible representations of the hecke algebra of an affine weyl group analogous to kilmoyer s reflection representation corresponding to finite weyl groups and we show that in many cases they correspond to a square integrable representation of a simple p adic group

numerical methods for semiconductor heterostructures with band nonparabolicity this article presents numerical methods for computing bound state energies and associated wave functions of three dimensional semiconductor heterostructures with special interest in the numerical treatment of the effect of band nonparabolicity a nonuniform finite difference method is presented to approximate a model of a cylindrical shaped semiconductor quantum dot embedded in another semiconductor matrix a matrix reduction method is then proposed to dramatically reduce huge eigenvalue systems to relatively very small subsystems moreover the nonparabolic band structure results in a cubic type of nonlinear eigenvalue problems for which a cubic jacobi davidson method with an explicit nonequivalence deflation method are proposed to compute all the desired eigenpairs numerical results are given to illustrate the spectrum of energy levels and the corresponding wave functions in rather detail

numerical simulation of three dimensional pyramid quantum dot we present a simple and efficient numerical method for the simulation of the three dimensional pyramid quantum dot heterostructure the pyramid shaped quantum dot is placed in a computational box with uniform mesh in cartesian coordinates the corresponding schrodinger equation is discretized using the finite volume method and the interface conditions are incorporated into the discretization scheme without explicitly enforcing them the resulting matrix eigenvalue problem is then solved using a jacobi davidson based method both linear and non linear eigenvalue problems are simulated the scheme is 2nd order accurate and converges extremely fast the superior performance is a combined effect of the uniform spacing of the grids and the nice structure of the resulting matrices we have successfully simulated a variety of test problems including a quintic polynomial eigenvalue problem with more than 32 million variables

pro p iwahori hecke ring and supersingular representations the motivation of this paper is the search for a langlands correspondence modulo p we show that the pro p iwahori hecke ring open image in new window of a split reductive p adic group g over a local field f of finite residue field fq with q elements admits an iwahori matsumoto presentation and a bernstein z basis and we determine its centre we prove that the ring open image in new window is finitely generated as a module over its centre these results are proved in 11 only for the iwahori hecke ring let p be the prime number dividing q and let k be an algebraically closed field of characteristic p a character from the centre of open image in new window to k which is as null as possible will be called null the simple open image in new window modules with a null central character are called supersingular when g gl n we show that each simple open image in new window module of dimension n containing a character of the affine subring open image in new window is supersingular using the minimal expressions of haines generalized to open image in new window and that the number of such modules is equal to the number of irreducible k representations of the weil group wf of dimension n when the action of an uniformizer pf in the hecke algebra side and of the determinant of a frobenius frf in the galois side are fixed i e the number nn q of unitary irreducible polynomials in fq x of degree n one knows that the converse is true by explicit computations when n 2 10 and when n 3 rachel ollivier