essentially normal homogeneous quotient modules on the polydisc abstract in the present paper we give a complete characterization of all essentially normal homogeneous quotient modules of the hardy space on the unit polydisc it turns out that our approach also applies to quotients of weighted bergman modules

additive invariants on the hardy space over the polydisc abstract in recent years various advances have been made with respect to the nevanlinna pick kernels especially on the symmetric fock space while the development on the hardy space over the polydisc is relatively slow in this paper several results known on the symmetric fock space are proved for the hardy space over the polydisc the known proofs on the symmetric fock space make essential use of the nevanlinna pick properties specifically we study several integer valued numerical invariants which are defined on an arbitrary invariant subspace of the vector valued hardy spaces over the polydisc these invariants include the samuel multiplicity curvature fiber dimension and a few others a tool used to overcome the difficulty associated with non nevanlinna pick kernels is tauberian theory

almost sure and pth moment stability and stabilization of regime switching jump diffusion systems this work focuses on regime switching jump diffusions which include three classes of random processes brownian motions poisson processes and markov chains first a scalar linear system is treated as a benchmark model then stabilization of systems with one sided linear growth is considered next nonlinear systems that have a finite explosion time are treated in which regularization explosion suppression and stabilization are achieved by introducing appropriate diffusions together with poisson and markov chain perturbations this work reveals the impact of various random effects on the underlying systems for almost sure and p th moment stability and provides insight on stability and stabilization of switching jump diffusion systems

subalgebras of c algebras iii multivariable operator theory a d contraction is a d tuple t1 td of mutually commuting operators acting on a common hilbert space h such that t1 1 t2 2 td d 1 2 d for all 1 2 d h these are the higher dimensional counterparts of contractions we show that many of the operator theoretic aspects of function theory in the unit disk generalize to the unit ball bd in complex d space including von neumann s inequality and the model theory of contractions these results depend on properties of the d shift a distinguished d contraction which acts on a new h2 space associated with bd and which is the higher dimensional counterpart of the unilateral shift h 2 and the d shift are highly unique indeed by exploiting the noncommutative choquet boundary of the d shift relative to its generated c algebra we find that there is more uniqueness in dimension d 2 than there is in dimension one

on the stability of jump diffusions with markovian switching in this paper we consider the stability for a class of jump diffusions with markovian switching we first construct them successively and show that they can be associated with some appropriate generators and they are non explosive we then prove their feller continuity by the coupling methods furthermore we also prove their strong feller continuity by making use of the relation between the transition probabilities of jump diffusions and the corresponding diffusions finally we also investigate their exponential ergodicity

c algebra extension theory and duality we develop a duality theory introduced by paschke to give a simplified account of the main results of the brown douglas fillmore extension theory and the relative k homology theory of baum and douglas

essential normality of polynomial generated submodules hardy space and beyond abstract recently douglas and wang proved that for each polynomial q the submodule q of the bergman module generated by q is essentially normal 11 using improved techniques we show that the hardy space analogue of this result holds and more

joint extensions in families of contractive commuting operator tuples abstract in this paper we systematically study extension questions in families of commuting operator tuples that are associated with the unit ball in c d

l 2 properties of the overline partial and the overline partial neumann operator on spaces with isolated singularities let x be a hermitian complex space of pure dimension with only isolated singularities and pi m rightarrow x a resolution of singularities let omega subset subset x be a domain with no singularities in the boundary omega omega setminus mathrm sing x and omega pi 1 omega we relate l 2 properties of the overline partial and the overline partial neumann operator on omega to properties of the corresponding operators on omega where the situation is classically well understood outside some middle degrees there are compact solution operators for the overline partial equation on omega exactly if there are such operators on the resolution omega and the overline partial neumann operator is compact on omega exactly if it is compact on omega

recursive algorithms for stock liquidation a stochastic optimization approach this work develops stochastic optimization algorithms for a class of stock liquidation problems the stock liquidation rules are based on hybrid geometric brownian motion models allowing regime changes that are modulated by a continuous time finite state markov chain the optimal selling policy is of threshold type and can be obtained by solving a set of two point boundary value problems the total number of equations to be solved is the same as the number of states of the underlying markov chain to reduce the computational burden using a stochastic optimization approach recursive algorithms are constructed to approximate the optimal threshold values convergence and rates of convergence of the algorithm are studied simulation examples are presented and the computation results are compared with the analytic solutions finally the algorithms are tested using real market data

a harmonic analysis approach to essential normality of principal submodules guo and the second author have shown that the closure i in the drury arveson space of a homogeneous principal ideal i in c z1 zn is essentially normal in this note the authors extend this result to the closure of any principal polynomial ideal in the bergman space in particular the commutators and cross commutators of the restrictions of the multiplication operators are shown to be in the schatten p class for p n the same is true for modules generated by polynomials with vector valued coefficients further the maximal ideal space xi of the resulting c algebra for the quotient module is shown to be contained in z i bn where z i is the zero variety for i and to contain all points in bn that are limit points of z i bn finally the techniques introduced enable one to study a certain class of weight bergman spaces on the ball

beurling type quotient modules over the bidisk and boundary representations this paper mainly concerns beurling type quotient modules of h2 d2 over the bidisk by establishing a theorem of function theory over the bidisk it is shown that a beurling type quotient module is essentially normal if and only if the corresponding inner function is a rational inner function having degree at most 1 1 furthermore we apply this result to the study of boundary representations of toeplitz algebras over quotient modules it is proved that the identity representation of c sz sw is a boundary representation of b sz sw in all nontrivial cases this extends a result of arveson to toeplitz algebras on beurling type quotient modules over the bidisk cf w arveson subalgebras of c algebras acta math 123 1969 141 224 w arveson subalgebras of c algebras ii acta math 128 1972 271 308 the paper also establishes k homology defined by beurling type quotient modules over the bidisk

hilbert polynomials and arveson s curvature invariant abstract we define and study hilbert polynomials for certain holomorphic hilbert spaces we obtain several estimates for these polynomials and their coefficients our estimates inspire us to investigate the connection between the leading coefficients of hilbert polynomials for invariant subspaces of the symmetric fock space and arveson s curvature invariant for coinvariant subspaces we are able to obtain some formulas relating the curvature invariant with other invariants in particular we prove that arveson s version of the gauss bonnet chern formula is true when the invariant subspaces are generated by any polynomials

essential normality essential norms and hyperrigidity abstract let s s 1 s d denote the compression of the d shift to the complement of a homogeneous ideal i of c z 1 z d arveson conjectured that s is essentially normal in this paper we establish new results supporting this conjecture and connect the notion of essential normality to the theory of the c envelope and the noncommutative choquet boundary the unital norm closed algebra b i generated by s 1 s d modulo the compact operators is shown to be completely isometrically isomorphic to the uniform algebra generated by polynomials on v z i b d where z i is the variety corresponding to i consequently the essential norm of an element in b i is equal to the sup norm of its gelfand transform and the c envelope of b i is identified as the algebra of continuous functions on v b d which means it is a complete invariant of the topology of the variety determined by i in the ball motivated by this determination of the c envelope of b i we suggest a new more qualitative approach to the problem of essential normality we prove the tuple s is essentially normal if and only if it is hyperrigid as the generating set of a c algebra which is a property closely connected to arveson s notion of a boundary representation we show that most of our results hold in a much more general setting in particular for most of our results the ideal i can be replaced by an arbitrary not necessarily homogeneous invariant subspace of the d shift

computing beltrami fields for solving the nonlinear equations governing force free fields an iterative methodology based on the splitting of the problem is described on the basis of this splitting three families of subproblems have to be solved numerically the first problem consists to find a potential field a mixed hybrid method is used to solve it the second problem which is a curl div system is solved by means of a mixed method the last problem is a transport equation which is approximated using a streamline diffusion technique numerical three dimensional experiments and results are given to illustrate the efficiency of the method

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

stability of a random diffusion with linear drift we consider a linear system with markovian switching which is perturbed by gaussian type noise if the linear system is mean square stable then we show that under certain conditions the perturbed system is also stable we also show that under certain conditions the linear system with markovian switching can be stabilized by such noisy perturbation

the dirac operator of a commuting d tuple abstract given a commuting d tuple t t1 td of otherwise arbitrary operators on a hilbert space there is an associated dirac operator dt significant attributes of the d tuple are best expressed in terms of dt including the taylor spectrum and the notion of fredholmness in fact all properties of t derive from its dirac operator we introduce a general notion of dirac operator in dimension d 1 2 that is appropriate for multivariable operator theory we show that every abstract dirac operator is associated with a commuting d tuple and that two dirac operators are isomorphic iff their associated operator d tuples are unitarily equivalent by relating the curvature invariant introduced in a previous paper to the index of a dirac operator we establish a stability result for the curvature invariant for pure d contractions of finite rank it is shown that for the subcategory of all such t that are a fredholm and and b graded the curvature invariant k t is stable under compact perturbations we do not know if this stability persists when t is fredholm but ungraded although there is concrete evidence that it does

an analytic grothendieck riemann roch theorem we extend the boutet de monvel toeplitz index theorem to complex manifolds with isolated singularities following the relative k homology theory of baum douglas and taylor for manifolds with boundary we apply this index theorem to study the arveson douglas conjecture let bmbm be the unit ball in cmcm and i an ideal in the polynomial algebra c z1 zm c z1 zm we prove that when the zero variety zizi is a complete intersection space with only isolated singularities and intersects with the unit sphere s2m 1s2m 1 transversely the representations of c z1 zm c z1 zm on the closure of i in la2 bm and also the corresponding quotient space qiqi are essentially normal furthermore we prove an index theorem for toeplitz operators on qiqi by showing that the representation of c z1 zm c z1 zm on the quotient space qiqi gives the fundamental class of the boundary zi s2m 1zi s2m 1 in the appendix we prove with kai wang that if f la2 bm vanishes on zi bmzi bm then f is contained inside the closure of the ideal i in la2 bm

multiplier algebras of complete nevanlinna pick spaces dilations boundary representations and hyperrigidity abstract we study reproducing kernel hilbert spaces on the unit ball with the complete nevanlinna pick property through the representation theory of their algebras of multipliers we give a complete description of the representations in terms of the reproducing kernels while representations always dilate to representations of the ambient c algebra we show that in our setting we automatically obtain coextensions in fact we show that in many cases this phenomenon characterizes the complete nevanlinna pick property we also deduce operator theoretic dilation results which are in the spirit of work of agler and several other authors moreover we identify all boundary representations compute the c envelopes and determine hyperrigidity for certain analogues of the disc algebra finally we extend these results to spaces of functions on homogeneous subvarieties of the ball

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

stability for a random evolution equation with gaussian perturbation in this paper we consider a random evolution equation which is perturbed by gaussian type noise and show how to construct its coupled solutions on the basis of the coupling results we discuss the asymptotic flatness and the stability in total variation norm for the solutions of the equation in addition we also prove the feller continuity and the existence and uniqueness of invariant measures of the solutions

numerical solutions of regime switching jump diffusions abstract this work develops numerical algorithms for approximating the solutions of stochastic differential equations that involve switching jump diffusion processes in which the switching is a random process that depends on the jump diffusion being non standard due to the jump diffusion dependent switching makes the problem far more difficult to deal with using decreasing step sizes we construct the algorithm which is in the spirit of euler maruyama method to prove the convergence we first derive the tightness of the algorithm then we establish the strong convergence the strong convergence is in the sense of usual numerical consideration for solutions of stochastic differential equations that is we consider uniform mean square convergence in a finite interval finally numerical examples are provided for demonstration

the curvature of a hilbert module over ℂ z1 zd a notion of curvature is introduced in multivariable operator theory the curvature invariant of a hilbert module over z1 zd is a nonnegative real number which has significant extremal properties which tends to be an integer and which is hard to compute directly it is shown that for graded hilbert modules the curvature agrees with the euler characteristic of a certain finitely generated algebraic module over the appropriate polynomial ring this result is a higher dimensional operator theoretic counterpart of the gauss bonnet formula which expresses the average gaussian curvature of a compact oriented riemann surface as the alternating sum of the betti numbers of the surface and it solves the problem of calculating the curvature of graded hilbert modules the proof of that result is based on an asymptotic formula which expresses the curvature of a hilbert module in terms that allow its comparison to a corresponding asymptotic expression for the euler characteristic

essential normality for quotient modules and complex dimensions abstract for analytic sets m introduced in 9 we show that the corresponding quotient module q of the bergman module is p essentially normal for all p dim c m which verifies the geometric arveson douglas conjecture in this case this result makes it possible to study the helton howe trace invariants on both q and the corresponding submodule r

quotient modules for some hilbert modules over the bidisk in this note we will prove the essential normality of some type of quotient hilbert modules over the bidisk moreover we will study some properties of the compression of the multiplication operators to the quotient hilbert modules

numerical solutions of sdes with markovian switching and jumps under non lipschitz conditions abstract in this paper the numerical solutions to stochastic differential equations with markovian switching and jumps under non lipschitz conditions are considered firstly we study the existence and uniqueness of solutions to this class of equations then we present the moment bounds of the exact and numerical solutions it is also proved that the strong rate of convergence of the euler scheme is equal to 1 2 with the drift coefficient satisfying one sided lipschitz condition while the diffusion and jump coefficients satisfy global lipschitz conditions some numerical examples are given to confirm the theoretical results

essentially normal hilbert modules and k homology this paper mainly concerns the essential normality of graded submodules essentially all of the basic hilbert modules that have received attention over the years are p essentially normal including the d shift hilbert module the hardy and bergman modules of the unit ball arveson conjectured graded submodules over the unit ball inherit this property and provided motivations to seek an affirmative answer some positive results have been obtained by arveson and douglas however the problem has been resistant in dimensions d 2 3 this paper shows that the arveson s conjecture is true in any dimension the paper also gives an affirmative answer in the case of the graded principal submodule finally the paper is associated with k homology invariants arising from graded quotient modules by which geometry of the quotient modules and geometry of algebraic varieties are connected in dimensions d 2 3 it is shown that k homology invariants determined by graded quotients are nontrivial the paper also establishes results on p smoothness of k homology elements and gives an explicit expression for k homology invariant in dimension d 2

geometric arveson douglas conjecture abstract we prove the arveson douglas essential normality conjecture for graded hilbert submodules that consist of functions vanishing on a given homogeneous subvariety of the ball smooth away from the origin our main tool is the theory of generalized toeplitz operators of boutet de monvel and guillemin

a new index theorem for monomial ideals by resolutions abstract we prove an index theorem for the quotient module of a monomial ideal we obtain this result by resolving the monomial ideal by a sequence of essentially normal hilbert modules each of which is a direct sum of weighted bergman spaces on balls

quotients of standard hilbert modules we initiate a study of hilbert modules over the polynomial algebra that are obtained by completing with respect to an inner product having certain natural properties a standard hilbert module is a finite multiplicity version of one of these standard hilbert modules occupy a position analogous to that of free modules of finite rank in commutative algebra and their quotients by submodules give rise to universal solutions of nonlinear relations essentially all of the basic hilbert modules that have received attention over the years are standard including the hilbert module of the shift the hardy and bergman modules of the unit ball modules associated with more general domains in and those associated with projective algebraic varieties we address the general problem of determining when a quotient of an essentially normal standard hilbert module is essentially normal this problem has been resistant our main result is that it can be linearized in that the nonlinear relations defining the submodule can be reduced appropriately to linear relations through an iteration procedure and we give a concrete description of linearized quotients

fredholm and invertible tuples of operators the deformation problem using j l taylor s definition of joint spectrum we study fredholm and invertible tuples of operators on a hilbert space we give the foundations for a several variables theory including a natural generalization of atkinson s theorem and an index which well behaves we obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length we then consider the deformation problem and solve it for the class of almost doubly commuting fredholm pairs with a semi fredholm coordinate 1 introduction 1 let t be a bounded linear operator on a banach space t is said to be invertible if there exists an operator s on such that ts st 1 the identity operator on 9c by the open mapping theorem this is equivalent to ker t 0 and r t range of t the last formulation does not rely upon the existence of an inverse for t but rather on the action of the operator t when t is replaced by an tuple of commuting operators several definitions of nonsingular ity exist j l taylor 19 has obtained one which reflects the actions of the operators by considering the koszul complex associated with the tuple 2 in this paper we develop a general several variables theory on the basis of taylor s work and study commuting and almost commuting commuting mod ulo the compacts tuples of operators on a hilbert space we obtain a characterization of joint invertibility in terms of the invertibility of a single operator which is essential for our approach from that we get a number of corollaries which generalize nicely the known elementary results in one variable at the same time the referred characterization allows us to define a continuous invariant under compact perturbations integer valued index on the class of fred holm tuples those almost commuting tuples which are invertible in the calkin algebra this index extends the classical one for fredholm operators we prove that an almost commuting tuple of essentially normal operators with all commu tators in trace class has index zero 2 and that a natural generalization of atkinson s theorem holds for tuples

stability of regime switching jump diffusion processes abstract this work studies the stability of regime switching jump diffusion processes in a finite or a countably infinite state space some criteria with sufficient conditions for stability and instability are provided based on characterizing the stability property of the processes in any fixed state through constants under common measurements also some variational formula of these constants are given moreover some examples of nonlinear regime switching jump diffusion processes are provided to show the usefulness and sharpness of these criteria

the structure of inner multipliers on spaces with complete nevanlinna pick kernels let k be the reproducing kernel for a hilbert space h k of analytic functions on bd the open unit ball in cd d 1 k is called a complete np kernel if k0 1 and if 1 1 k z is positive definite on bd bd let d be a separable hilbert space and consider h k d h k d and think of it as a space of d valued h k functions a theorem of mccullough and trent j funct anal 178 2000 226 249 partially extends the beurling lax halmos theorem for the invariant subspaces of the hardy space h2 d they show that if k is a complete np kernel and if d is a separable hilbert space then for any scalar multiplier invariant subspace m of h k d there exists an auxiliary hilbert space e and a multiplication operator h k e h k d such that is a partial isometry and m h k e such multiplication operators are called inner multiplication operators and they satisfy the orthogonal projection onto m in this paper we shall show that for many interesting complete np kernels the analogy with the beurling lax halmos theorem can be strengthened we show that for almost every z bd the nontangential limit z of the multiplier bd b e d associated with an inner multiplication operator is a partial isometry and that rank z is equal to a constant almost everywhere the result applies to certain weighted dirichlet spaces and to the space h2d which is determined by the kernel k z 11 z d z bd in particular our result implies that the curvature invariant of arveson proc natl acad sci usa96 1999 11 096 11 099 of a pure contractive hilbert module of finite rank is an integer this answers a question of w arveson proc natl acad sci usa96 1999 11096 11099

estimates for hilbertian koszul homology abstract the objective of this paper is to give new kind of estimates for hilbertian koszul homology inspired by commutative algebra in multivariable fredholm theory

pth moment asymptotic stability of stochastic delayed hybrid systems with levy noise the problem of pth moment asymptotic stability analysis is considered for stochastic delayed hybrid systems with levy noise by virtue of ito s formula and m matrix theories we propose some sufficient conditions to guarantee the asymptotic stability and exponential stability of the system the criterion of mean square asymptotic stability is derived as well for delayed neural networks with levy noise a numerical example is provided to show the usefulness of the proposed asymptotic stability criterion

on woltjer s variational principle for force free fields the existence of minimizers for woltjer s variational principle is established and that the minimizers are force free fields this method has the nature of a constructive implicit function theorem and handles successfully the nonconvex constraint of constant total helicity domains of arbitrary connectivity are allowed as well as nonhomogeneous boundary conditions and periods

curvature invariant on noncommutative polyballs abstract in this paper we develop a theory of curvature resp multiplicity invariant for tensor products of full fock spaces f 2 h n 1 f 2 h n k and also for tensor products of symmetric fock spaces f s 2 h n 1 f s 2 h n k this is an attempt to find a more general framework for these invariants and extend some of the results obtained by arveson for the symmetric fock space by the author and kribs for the full fock space and by fang for the hardy space h 2 d k over the polydisc to prove the existence of the curvature and its basic properties in these settings requires a new approach based on noncommutative berezin transforms and multivariable operator theory on polyballs and varieties as well as summability results for completely positive maps the results are presented in the more general setting of regular polyballs

toeplitz algebras subnormal tuples and rigidity on reproducing c z1 zd modules this paper mainly considers toeplitz algebras subnormal tuples and rigidity concerning reproducing c z1 zd modules by making use of arveson s boundary representation theory we find there is more rigidity in several variables than there is in single variable we specialize our attention to reproducing c z1 zd modules with u invariant kernels by examining the spectrum and the essential spectrum of the d tuple mz1 mzd and deducing an exact sequence of c algebras associated with toeplitz algebra finally we deal with toeplitz algebras defined on arveson submodules and rigidity of arveson submodules

existence of three dimensional toroidal mhd equilibria with nonconstant pressure we establish an existence result for the three dimensional mhd equations x b x b p b 0 b n t 0 with p const in tori t without symmetry more precisely our theorems insure the existence of sharp boundary solutions for tori whose departure from axisymmetry is sufficiently small they allow for solutions to be constructed with an arbitrary number of pressure jumps

stabilization and destabilization of hybrid systems of stochastic differential equations this paper aims to determine whether or not a stochastic feedback control can stabilize or destabilize a given nonlinear hybrid system new methods are developed and sufficient conditions on the stability and instability for hybrid stochastic differential equations are provided these results are then used to examine stochastic stabilization and destabilization

spreading code optimization and adaptation in cdma via discrete stochastic approximation the aim of this paper is to develop discrete stochastic approximation algorithms that adaptively optimize the spreading codes of users in a code division multiple access cdma system employing linear minimum mean square error mmse receivers the proposed algorithms are able to adapt to slowly time varying channel conditions one of the most important properties of the algorithms is their self learning capability they spend most of the computational effort at the global optimizer of the objective function tracking analysis of the adaptive algorithms is presented together with mean square convergence an adaptive step size algorithm is also presented for optimally adjusting the step size based on the observations numerical examples illustrating the performance of the algorithms in multipath fading channels show substantial improvement over heuristic algorithms

a minimal stabilisation procedure for mixed finite element methods stabilisation methods are often used to circumvent the difficulties associated with the stability of mixed finite element methods stabilisation however also means an excessive amount of dissipation or the loss of nice conservation properties it would thus be desirable to reduce these disadvantages to a minimum we present a general framework not restricted to mixed methods that permits to introduce a minimal stabilising term and hence a minimal perturbation with respect to the original problem to do so we rely on the fact that some part of the problem is stable and should not be modified sections 2 and 3 present the method in an abstract framework section 4 and 5 present two classes of stabilisations for the inf sup condition in mixed problems we present many examples most arising from the discretisation of flow problems section 6 presents examples in which the stabilising terms is introduced to cure coercivity problems

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100