reachability indices of periodic positive systems via positive shift similarity the problem of the paper is the formulation of the concept of reachability indices of periodic linear positive systems a positive system is one in which the positive orthant is an invariant subset with respect to positive input signals the concept of shift similarity for periodic positive systems is defined and the associated cyclically augmented system is considered the results are the relation of the positive similarity of both representations the formulation of the reachability indices and a detailed example

behaviour of the frobenius map in a noncommutative world abstract we study various properties of the frobenius map x x p on a noncommutative algebra r over a field f of positive characteristic p we call r perfect whenever is surjective more generally we say that r is additively respectively multiplicatively p power closed if for every x y r there exists a positive integer n such that x p n y p n respectively x p n y p n is a p power in r if for example n were a ring homomorphism as when r is commutative then r would be additively and multiplicatively p power closed we show that a finite dimensional unital algebra r is lie nilpotent if and only if r is additively respectively multiplicatively p power closed and each of its simple homomorphic images has schur index 1 since not all finite dimensional perfect division algebras are lie nilpotent the schur index condition cannot be omitted we deduce that similar results hold for the class of all finitely generated pi algebras moreover for this same class we give a positive solution to the following problem reminiscent of the problems of kurosh and levitzki is every finitely generated perfect algebra finite dimensional

on artin s theorem and azumaya algebras a short proof of the existence of central polynomial of matrix rings is given and some of its applications the following characterization of azumaya algebras is proved r is azumaya iff there exist ai bi r i 1 k with ik 1 airbi z r r r and aibi 1 this is proved as a consequence of the following generalization of a theorem due to m artin and generalized by c procesi let r x1 xk be a p i ring a central noetherian subring then r is azumaya iff for every two sided ideal i in r z ri z r i z r where z r denotes the center of r

structure of zariski closed algebras the objective of this paper is to describe the structure of zariski closed algebras which provide a useful generalization to finite dimensional algebras in the study of representable algebras over finite fields our results include a version of wedderburn s principal theorem as well as a more explicit description using representations in terms of gluing in wedderburn components finally we construct generic zariski closed algebras whose description is considerably more complicated than the description of generic algebra of finite dimensional algebras special attention is given to infinite dimensional algebras over finite fields

geometric quantization on the super disk in this article we discuss the geometric quantization on a certain type of infinite dimensional super disk such systems are quite natural when we analyze coupled bosons and fermions the large n limit of a system like that corresponds to a certain super homogeneous space first we define an example of a super homogeneous manifold a super disk we show that it has a natural symplectic form and it can be used to introduce classical dynamics once a hamiltonian is chosen existence of moment maps provide a poisson realization of the underlying symmetry super group these are the natural operators to quantize via methods of geometric quantization and we show that this can be done

quantization of the poisson su 2 and its poisson homogeneous space the 2 sphere we show that deformation quantizations of the poisson structures on the poisson lie groupsu 2 and its homogeneous space the 2 sphere are compatible with woronowicz s deformation quantization ofsu 2 s group structure and podles deformation quantization of 2 sphere s homogeneous structure respectively so in a certain sense the multiplicativity of the lie poisson structure onsu 2 at the classical level is preserved under quantization

feedback invariants of a pair of matrices with prescribed columns abstract this paper studies the possible feedback equivalence classes of a pair of matrices with some prescribed columns and related problems

classification of pairs of linear mappings between two vector spaces and between their quotient space and subspace we classify pairs of linear mappings u v u u v in which u v are finite dimensional vector spaces over a field if and u are their subspaces c 2016 elsevier inc all rights re

controllability and hermite indices of matrix pairs given a pair of matrices a b we can associate to each nice basis of the controllability subspace of a b a set of integers to be called the indices of the basis each set provides partial information about the similarity class of a the relationship between several sets of indices as well as the constraints that they impose on the invariant factors of a are investigated special attention is paid to the controllability and hermite indices of a given pair a b

non perturbative deformation quantization of cartan domains abstract we construct families of non commuting c algebras of quantized functions for bounded irreducible hermitian symmetric spaces for this procedure we use algebras of toeplitz operators defined with respect to a perturbation of the ordinary bergman metric we prove the deformation quantization conditions for these algebras

super toeplitz operators and non perturbative deformation quantization of supermanifolds the purpose of this paper is to construct non perturbative deformation quantizations of the algebras of smooth functions on poisson supermanifolds for the examplesu1 1 andcm n algebras of super toeplitz operators are defined with respect to certain hilbert spaces of superholomorphic functions generators and relations for these algebras are given the algebras can be thought of as algebras of quantized functions and deformation conditions are proven which demonstrate the recovery of the super poisson structures in a semi classical limit

propagation observation and control of waves approximated by finite difference methods this paper surveys several topics related to the observation and control of wave propagation phenomena modeled by finite difference methods the main focus is on the property of observability corresponding to the question of whether the total energy of solutions can be estimated from partial measurements on a subregion of the domain or boundary the mathematically equivalent property of controllability corresponds to the question of whether wave propagation behavior can be controlled using forcing terms on that subregion as is often desired in engineering applications observability controllability of the continuous wave equation is well understood for the scalar linear constant coefficient case that is the focus of this paper however when the wave equation is discretized by finite difference methods the control for the discretized model does not necessarily yield a good approximation to the control for the original continuous problem in other words the classical convergence consistency stability property of a numerical scheme does not suffice to guarantee its suitability for providing good approximations to the controls that might be needed in applications observability controllability may be lost under numerical discretization as the mesh size tends to zero due to the existence of high frequency spurious solutions for which the group velocity vanishes this phenomenon is analyzed and several remedies are suggested including filtering tychonoff regularization multigrid methods and mixed finite element methods we also briefly discuss these issues for the heat beam and schrodinger equations to illustrate that diffusive and dispersive effects may help to retain the observability controllability properties at the discrete level we conclude with a list of open problems and future subjects for research

matrix cartan superdomains super toeplitz operators and quantization we present a general theory of non perturbative quantization of a class of hermitian symmetric supermanifolds the quantization scheme is based on the notion of a super toeplitz operator on a suitable z2 graded hilbert spaces of super holomorphic functions the quantized supermanifold arises as the c algebra generated by all such operators we prove that our quantization framework reproduces the invariant super poisson structure on the classical supermanifold as planck s constant tends to zero

fundamental theorem of state feedback for singular systems abstract the limits of state feedback in altering the dynamics of singular systems are studied a necessary and sufficient condition is given for a list of polynomials to be invariant polynomials of a proper system obtained by state feedback from the singular system the condition consists of inequalities which involve the controllability indices of the singular system and the degrees of the invariant polynomials the development is based on the properties of polynomial matrices a procedure is given for the calculation of a feedback gain which achieves the desired dynamics

nonmatrix varieties and nil generated algebras whose units satisfy a group identity abstract letr denote the group of units of an associative algebrarover an infinite fieldf we prove that ifris unitarily generated by its nilpotent elements thenr satisfies a group identity precisely whenrsatisfies a nonmatrix polynomial identity as an application we examine the group algebrafgof a torsion groupgand the restricted enveloping algebrau l of ap nil restricted lie algebral giambruno sehgal and valenti recently proved that if the group of units fg satisfies a group identity thenfgsatisfies a polynomial identity thus confirming a conjecture of brian hartley we show that in fact fg satisfies a group identity if and only iffgsatisfies a nonmatrix polynomial identity in the case of restricted enveloping algebras we prove thatu l satisfies a group identity if and only ifu l satisfies the engel condition

linear systems with prescribed similarity structural invariants the problem of the existence of linear systems dot x t ax t bu t with prescribed structural invariants for system similarity is studied namely we solve the problem of the existence of such a system with prescribed controllability indices hermite indices and invariant factors when the invariant factors of a which are also invariants under system similarity are given

generalized hermite matrices and complete invariants of strict system equivalence a complete list of invariants for reachable system matrices sigma s left begin array 20 c p s q s v s w s end array right with respect to strict system equivalence s s e is determined by polynomial methods the polynomial input output pairs u y for which there exists a polynomial vector z such that pz qu and y vz wu form a k s module mu sigma it is shown that the unique basis matrix of mu sigma in hermite form yields a complete set of discrete hermite indices resp continuous invariants of s s e the hermite invariants are characterized in state space terms and a realization of sigma s in hermite canonical form is presented nice orders and generalized hermite forms are introduced in order to develop a framework that encompasses hermite invariants and kronecker invariants hermite s theorem is generalized to these matrices finally nice orders are used to single out unique representatives among all minimal bas

kemer s theory for h module algebras with application to the pi exponent abstract let h be a semisimple finite dimensional hopf algebra over a field f of zero characteristic we prove three major theorems 1 the representability theorem which states that every h module associative f algebra w satisfying an ordinary pi has the same h identities as the grassmann envelope of an h f z 2 z module algebra which is finite dimensional over a field extension of f 2 the specht problem for h module ordinary pi algebras that is every h t ideal which contains an ordinary pi contains h polynomials f 1 f s which generate as an h t ideal 3 amitsur s conjecture for h module algebras saying that the exponent of the h codimension sequence of an ordinary pi h module algebra is an integer

quantum riemann surfaces i the unit disc we construct a non commutative algebra c mu bar u which is a quantum deformation of the algebra of continuous functions on the closed unit disc bar u c mu bar u is generated by the toeplitz operators on a suitable hilbert space of holomorphic functions onu

interlacing inequalities and control theory abstract given a p by p matrix a we solve the problem of the existence of a p by q matrix b such that a b has prescribed controllability indices and i p a b has prescribed invariant polynomials the solution of this problem together with an earlier theorem of the author s is used to provide a new proof of the sa thompson interlacing inequalities for invariant polynomials

toeplitz c algebras on bounded symmetric domains in this paper we develop a structure theory for the toeplitz c algebra y for an arbitrary bounded symmetric domain d in c classical or exceptional as the main result it is shown that 5y is a c algebra of type i which is solvable i e admits a finite composition sequence of closed ideals sk in 7 such that the subquotients fk i k are continuous fields of elementary c algebras of compact operators the length of this composition sequence coincides with the rank r of d as a hermitian symmetric space for 0 k r the spectral components sk of ay parametrizing the irreducible representations of y are realized in terms of the boundary components complex analytic faces of d the structure theory for y is based on the fact that the tangent space z c c of d carries a jordan algebraic structure z is a jordan triple system with triple product uv w describing the geometry of d in algebraic terms for example d is the open unit ball with respect to the so called spectral norm on z and the boundary components of d are in 1 1 correspondence with the tripotents e e z partial isometries satisfying ee e 4 e in particular the shilov boundary s of d consists of all tripotents of maximal rank r the jordan theoretic description of d was used in 17 to determine an explicit peter weyl decomposition of the hardy space h2 s cf 15 in terms of the norm functions generalized determinants associated with the jordan triple

feedback invariants via output injection abstract the feedback equivalence classes of matrix pairs a kc b kw obtained by output injection of matrix k on a system a b c w are completely characterized working on arbitrary fields some related problems are also solved

a polynomial approach to the invariant factors assignment problem via state feedback and output injection the characterization of the similarity classes of matrices a bf kc kwf obtained by state feedback and output injection on a system a b c w are revisited new more straighforward proofs are obtained the results hold working on arbitrary fields

interlacing and majorization in invariant factor assignment problems this paper shows some connections between pole assignment in control theory and assignment of invariant factors on matrices with some prescribed submatrices the majorization in the hardy littlewood polya sense and the interlacing inequalities for invariant factors play a fundamental role in the solution of some problems common to both theories

the nilpotency of the radical in a finitely generated p i ring abstract let r x 1 x k be a p i ring satisfying a monic polynomial identity one of its coefficients is 1 where is a central noetherian subring it is proven that n r the nil radical of r is nilpotent as a corollary by taking f a field we settle affirmatively the open problem posed in c procesi rings with polynomial identities p 186 marcel dekker new york 1973 we prove the jacobson radical of a finitely generated p i algebra is nilpotent

matrix pencils completion problems ii in this paper we give new simple and explicit necessary and sufficient conditions for the existence of a matrix pencil with prescribed kronecker invariants and a regular subpencil moreover we explicitly solve the problem of describing the possible kronecker invariants of a pencil with a prescribed subpencil in the case of the existence of an intermediate regular pencil all the results are obtained over algebraically closed fields

minimal varieties of algebras of exponential growth abstract the exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer through this scale we can now classify all minimal varieties of given exponent and of finite basic rank as a consequence we describe the corresponding t ideals of the free algebra and we compute the asymptotics of the related codimension sequences verifying in this setting some known conjectures we also show that the number of these minimal varieties is finite for any given exponent we finally point out some relations between the exponent of a variety and the gelfand kirillov dimension of the corresponding relatively free algebras of finite rank

equivalence of quadruples and feedback equivalence of related pairs in this paper we look for a generalization of a property of matrix pairs to quadruples of matrices it is known that it is possible to analyze the feedback equivalence of matrix pairs in terms of the similarity of the square matrices belonging to sets related to the pairs we will determine when it is possible to analyze the equivalence of matrix quadruples in terms of the feedback equivalence of matrix pairs

a two parameter quantum deformation of the unit disc we define and study a two parameter deformation of the unit disc this deformation is described in terms of a family of type i c algebras c q we study the representation theory for c q and construct an action of the quantized universal enveloping algebra uq sl 2 on c q

feedback invariants of restrictions a polynomial approach the possible feedback invariants of a linear time invariant dynamical system represented by a pair of matrices a b are studied when the only information about the system is that it has an a b invariant subspace s whose restriction is an autonomous system and the quotient is controllable this problem is shown to be related to the problem of designing a control matrix b 1 for a series connected system to have prescribed feedback invariants

supersymmetry and fredholm modules over quantized spaces the purpose of this paper is to apply the framework of non commutative differential geometry to quantum deformations of a class of kahler manifolds for the examples of the cartan domains of type i and flat space we construct fredholm modules over the quantized manifolds using the supercharges which arise in the quantization of supersymmetric generalizations of the manifolds we compute an explicit formula for the chern character on generators of the toeplitz algebra

feedback invariants of matrix quadruple completions abstract given a matrix quadruple a 1 a 2 a 3 a 4 f n 1 n 1 f n 2 n 1 f n 1 n 3 f n 2 n 3 we completely characterize the feedback invariants of the matrix pair a 1 x a 2 y a 3 a 4 for all possible selections of matrices x f n 1 n 2 and y f n 2 n 2

on feedback equivalence and completion problems abstract in 1987 i zaballa characterized the possible similarity classes of a square matrix with some prescribed rows in 1988 the same author characterized the possible feedback equivalence classes of a b where a is a fixed square matrix and b varies firstly in this paper we observe that these results are equivalent that is each one of them can be obtained as a corollary of the other then we apply similar arguments to other inverse problems in particular we study the possible invariant polynomials and the possible characteristic polynomials of a bx yc when x and y vary and we study the linear systems that can become completely observable with a suitable choice of linear state feedback control

on the polynomial representations of current algebras abstract we obtain a sufficient condition for a representation of a lie algebra of the form l g where g is a finite dimensional simple lie algebra over an algebraically closed field f of zero characteristic and is an associative commutative algebra over f to be polynomial as a corollary we classify integrable graded irreducible modules over an extended loop algebra with finite dimensional homogeneous spaces the criteria is applied also to irreducible conformal modules over current algebras

topological classification of linear control systems an elementary analytic approach abstract in this paper we present a unified treatment on the linear differentiable and topological equivalences for time invariant linear control systems governed by ordinary differential equations odes especially we give a new solution to the topological classification problem for linear control systems by means of an elementary analytic approach which differs considerably from j c willems algebraic method developed in willems 1980 17 more precisely based on p brunovsky s results in brunovsky 1970 2 we determine all topological invariants for the linear control system as an interesting by product we show that there exist only finitely many topological equivalence classes for any stabilizable system with given numbers of state and control variables which might be useful for some engineering applications

invariant factor assignment under state feedback and output injection for a given system a b c necessary and sufficient conditions are given to assign invariant factors to the matrix of the system performing a state feedback and an output injection over an arbitrary field

some geometric considerations about the kronecker normal form abstract in the first part of this paper we develop general algorithms from the kronecker normal form of any matrix pencil and give some of their properties in the second part these results are applied to the so called restricted matrix pencil associated with a given c a b system and introduced by jaffe and karcanias 1981 this allows us to make a link with well known algorithms of the geometric approach particularly the a b invariant subspace algorithm and the almost a b controllability subspace algorithm finally relationships between morse s invariants morse 1973 and the invariants of the restricted matrix pencil are exhibited

deformation estimates for the berezin toeplitz quantization deformation estimates for the berezin toeplitz quantization ofcn are obtained these estimates justify the description of ccr k as a first order quantum deformation of ap c0 where ccr is the usualc algebra of boson canonical commutation relations k is the full algebra of compact operators ap is the algebra of almost periodic functions andc0 is the algebra of continuous functions which vanish at infinity

feedback classification of nonlinear single input control systems with controllable linearization normal forms canonical forms and invariants we study the feedback group action on single input nonlinear control systems we follow an approach of kang and krener based on analyzing step by step the action of homogeneous transformations on the homogeneous part of the same degree of the system we construct a dual normal form and dual invariants with respect to those obtained by kang we also propose a canonical form and a dual canonical form and show that two systems are equivalent via a formal feedback if and only if their canonical forms resp their dual canonical forms coincide we give an explicit construction of transformations bringing the system to its normal dual normal canonical and dual canonical forms we illustrate our results by simple examples on mathbb r 3 and mathbb r 4

deformation quantization of heisenberg manifolds form a smooth manifold equipped with a poisson bracket we formulate ac algebra framework for deformation quantization including the possibility of invariance under a lie group of diffeomorphisms preserving the poisson bracket we then show that the much studied non commutative tori give examples of such deformation quantizations invariant under the usual action of ordinary tori going beyond this the main results of the paper provide a construction of invariant deformation quantizations for those poisson brackets on heisenberg manifolds which are invariant under the action of the heisenberg lie group and for various generalizations suggested by this class of examples interesting examples are obtained of simplec algebras on which the heisenberg group acts ergodically

when is a power of the frobenius map on a noncommutative ring a homomorphism abstract let x x p be the frobenius map on an associative unital ring r with prime characteristic p it is well known that whenever r is commutative n is a ring homomorphism for all positive integers n the converse however is not true in general indeed we prove that if r is m engel for some positive integer m then there exists a positive integer n 0 depending only on m such that for all n n 0 n is a ring homomorphism with central image conversely if any one of the following conditions holds n respects addition n respects multiplication n respects lie multiplication or the image of n is commutative then r is m engel for some m depending only on n consequently if is surjective and any one of the aforementioned conditions holds then r must be commutative

supercoherent states super kahler geometry and geometric quantization generalized coherent states provide a means of connecting square integrable representations of a semi simple lie group with the symplectic geometry of some of its homogeneous spaces in the first part of the present work this point of view is extended to the supersymmetric context through the study of the osp 2 2 coherent states these are explicitly constructed starting from the known abstract typical and atypical representations of osp 2 2 their underlying geometries turn out to be those of supersymplectic osp 2 2 homogeneous spaces moment maps identifying the latter with coadjoint orbits of osp 2 2 are exhibited via berezin s symbols when considered within rothstein s general paradigm these results lead to a natural general definition of a super kahler supermanifold the supergeometry of which is determined in terms of the usual geometry of holomorphic hermitian vector bundles over kahler manifolds in particular the supergeometry of the above orbits is interpreted in terms of the geometry of einstein hermitian vector bundles in the second part an extension of the full geometric quantization procedure is applied to the same coadjoint orbits thanks to the super kahler character of the latter this procedure leads to explicit super unitary irreducible representations of osp 2 2 in super hilbert spaces of superholomorphic square integrable sections of prequantum bundles of the kostant type this work lays the foundations of a program aimed at classifying lie supergroups coadjoint orbits and their associated irreducible representations ultimately leading to harmonic superanalysis for this purpose a set of consistent conventions is exhibited