unbounded perturbation of the exponential dichotomy for evolution equations abstract in this paper we prove that the exponential dichotomy for evolution equations in banach spaces is not destroyed if we perturb the equation by small unbounded linear operator this is done by employing a skew product semiflow technique and a perturbation principle from linear operator theory finally we apply these results to partial parabolic equations and functional differential equations

the essential spectrum of advective equations the geometric optics stability method is extended to a general class of linear advective pde s with pseudodifferential bounded perturbation we give a new short proof of vishik s formula for the essential spectral radius we show that every point in the dynamical spectrum of the corresponding bicharacteristic amplitude system contributes a point into the essential spectrum of the pde generic spectral pictures are obtained in sobolev spaces of sufficiently large smoothness applications to instability are presented

perturbation and an abstract characterization of evolution semigroups abstract evans and howland used an abstract characterization of evolution semigroups and a perturbation result for c 0 semigroups to treat non autonomous schrodinger equations and problems in scattering theory here we proceed in a similar way at first we characterize evolution semigroups on spaces of vector valued functions e x which are induced by a strongly continuous evolution family u t s t s defined on the banach space x from this and a perturbation result of voigt we obtain a sufficient condition for an operator on l p r x to be the generator of an evolution semigroup this will be applied to non autonomous heat equations with absorption

existence and roughness of the exponential dichotomy for skew product semiflow in banach spaces abstract in this paper we introduce a concept of exponential dichotomy for skew product semiflow in infinite dimensional banach spaces which is an extension of the classic concept for evolution operators this concept is used to study the roughness property of the skew product semiflow also we introduce the concept of discrete skew product and give a necessary and sufficient condition for this discrete skew product to have a discrete dichotomy after that we give necessary and sufficient conditions for the existence of exponential dichotomy for skew product semiflow moreover we prove that the exponential dichotomy for skew product semiflow is not destroyed by small perturbation finally we apply these results to parabolic partial differential equations and functional differential equations

two linear transformations each tridiagonal with respect to an eigenbasis of the other let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v satisfying both conditions below 1 i there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal 2 ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal we call such a pair a leonard pair on v refining this notion a bit we introduce the concept of a leonard system we give a complete classification of leonard systems integral to our proof is the following result we show that for any leonard pair a a on v there exists a sequence of scalars taken from k such that both 0 a a2a aa a a a2 aa a a a 0 a a 2a a aa aa 2 a a aa a where r s means rs sr the sequence is uniquely determined by the leonard pair if the dimension of v is at least 4 we conclude by showing how leonard systems correspond to q racah and related polynomials from the askey scheme

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

spectral theory for linearizations of dynamical systems the description of the asymptotic growth rates for the solution trajectories of a system of differential equations is a central problem in the qualitative theory of dynamical systems these growth rates are described by the eigenvalues of the jacobian matrix for trajectories near a critical point by the floquet exponents for trajectories close to a periodic orbit and generally by the lyapunov characteristic exponents in a neighborhood of an arbitrary trajectory in the examples the asymptotic behavior of the solution trajectories is given by the association of a spectrum and this is a recurrent theme in the method of linearization sacker and sell io recently introduced a spectral theory for differential systems of quite general type the linear skew product flows see section 3 their work is in the spirit of the classical linearization theory and is closely tied to the idea of the lyapunov exponents we prefer to use the word spectrum literally identifying it with the spectrum of an operator on a banach space in general we consider a one parameter group 4 of homeomorphisms of a compact metric space m which for example may be the solution flow of a differential equation on a smooth manifold second we assume that a continuous vector bundle e m is defined i e a locally trivial continuous map n e m with linear fibers which are isomorphic to either real n dimensional space r or complex n dimensional space c in case is the flow of a vector field x on m this vector bundle could be the tangent bundle tm finally we assume that there is group of homeomorphisms on e with n 4 and such that i e t e c is a linear isomorphism for each x e m for instance in case 4 is smooth cp could be the tangent flow on tm given by the derivative of 4 using the compactness of m we define a finsler on e which continuously assigns a norm denoted i to each fiber e with this assignment the space r e of continuous sections of e functions q m e such that 7 o q x x forms a banach space with norm iiqll supxem u x 155 1 i the topology of t e is

quadratic maximization and semidefinite relaxation in this paper we study a class of quadratic maximization problems and their semidefinite programming sdp relaxation for a special subclass of the problems we show that the sdp relaxation provides an exact optimal solution another subclass which is hard guarantees that the sdp relaxation yields an approximate solution with a worst case performance ratio of 0 87856 this is a generalization of the well known result of goemans and williamson for the maximum cut problem finally we discuss extensions of these results in the presence of a certain type of sign restrictions

new conditions for non uniform behaviour of linear cocycles over flows abstract we give a characterization of tempered exponential dichotomies for linear cocycles over flows in terms of the spectral properties of certain linear operators we consider noninvertible linear cocycles acting on infinite dimensional spaces and our approach avoids the use of lyapunov norms finally we apply obtained results to give new conditions for uniform exponential stability of linear cocycles

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

lp r x lq r x admissibility and exponential dichotomies of cocycles we analyze the existence of eventually no past exponential and ordinary dichotomies of an exponentially bounded strongly continuous cocycle t t r over a continuous semiflow our main tool is the lp r x lq r x admissibility condition i e there exist p q 1 p q 1 such that for each input f lp r x and there exists x x such that the output u x f r x u t x f t x 0t s t s f s ds belongs to lq r x we prove that the above admissibility condition implies that the output is bounded above by the input but nonuniformly with respect to requiring that the boundedness to be uniform with respect to we prove that the above admissibility condition assures the existence of a no past exponential dichotomy for t t r variants for ordinary dichotomy and also complete characterizations for the exponential dichotomy of cocycles are obtained it is worth to note that we involve a concept of a no past exponential dichotomy for cocycles weaker than the well known concept defined by sacker and sell 1994 33 our definition of exponential dichotomy follows partially the definition given in chow and leiva 1996 2 in the sense that we allow the unstable subspace to have infinite dimension the main difference is that we do not assume a priori that the cocycle is invertible on the unstable space actually we do not even assume that the unstable space is invariant under the cocycle thus we generalize some known results due to perron 1930 23 daleckij and krein 1974 6 massera and schaffer 1966 14 nguyen van minh rabiger and r schnaubelt 1998 18

dual polar graphs the quantum algebra uq sl2 and leonard systems of dual q krawtchouk type abstract in this paper we consider how the following three objects are related i the dual polar graphs ii the quantum algebra u q sl 2 iii the leonard systems of dual q krawtchouk type for convenience we first describe how ii and iii are related for a given leonard system of dual q krawtchouk type we obtain two u q sl 2 module structures on its underlying vector space we now describe how i and iii are related let denote a dual polar graph fix a vertex x of and let t t x denote the corresponding subconstituent algebra by definition t is generated by the adjacency matrix a of and a certain diagonal matrix a a x called the dual adjacency matrix that corresponds to x by construction the algebra t is semisimple we show that for each irreducible t module w the restrictions of a and a to w induce a leonard system of dual q krawtchouk type we now describe how i and ii are related we obtain two u q sl 2 module structures on the standard module of we describe how these two u q sl 2 module structures are related each of these u q sl 2 module structures induces a c algebra homomorphism u q sl 2 t we show that in each case t is generated by the image together with the center of t using the combinatorics of we obtain a generating set l f r k of t along with some attractive relations satisfied by these generators

penalized maximum likelihood estimation the baum welch algorithm diagonal balancing of symmetric matrices and applications to training acoustic data we study penalized maximum likelihood estimation methods for nonparametric density estimation and propose their use in training acoustic models for speech recognition several algorithms for the numerical solution of the optimization problems that we encounter are proposed and analyzed

structure of thin irreducible modules of a q polynomial distance regular graph let be a q polynomial distance regular graph with vertex set x diameter d 3 and adjacency matrix a fix x x and let a a x be the corresponding dual adjacency matrix recall that the terwilliger algebra t t x is the subalgebra of matx c generated by a and a let w denote a thin irreducible t module it is known that the action of a and a on w induces a linear algebraic object known as a leonard pair over the past decade many results have been obtained concerning leonard pairs in this paper we apply these results to obtain a detailed description of w in our description we do not assume that the reader is familiar with leonard pairs everything will be proved from the point of view of our results are summarized as follows let ei i 0d be a q polynomial ordering of the primitive idempotents of and let ei i 0d be the dual primitive idempotents of with respect to x let r t and d be the endpoint dual endpoint and diameter of w respectively let uandv be nonzero vectors in etw and er w respectively we show that er i aiv i 0d and et ia iu i 0d are bases for w that are orthogonal with respect to the standard hermitian dot product we display the matrix representations of a and a with respect to these bases we associate with w two sequences of polynomials pi i 0dand pi i 0d we show that for 0 i d pi a v er i aiv and pi a u et ia iu next we show that er i u i 0d and et iv i 0d are orthogonal bases for w we call these the standard basis and dual standard basis for w respectively we display the matrix representations of a and a with respect to these bases the entries in these matrices will play an important role in our theory we call these the intersection numbers and dual intersection numbers of w using these numbers we compute all inner products involving the standard and dual standard bases we also use these numbers to define two normalizations ui vi resp ui vi for pi resp pi using the orthogonality of the standard and dual standard bases we show that for each of the sequences pi i 0d pi i 0d ui i 0d ui i 0d vi i 0d vi i 0d the polynomials involved are orthogonal and we display the orthogonality relations we also show that each of the sequences satisfy a three term recurrence and a relation known as the askey wilson duality we then turn our attention to two more bases for w we find the matrix representations of a and a with respect to these bases from the entries of these matrices we obtain two sequences of scalars known as the first split sequence and second split sequence of w we associate with w a sequence of scalars called the parameter array this sequence consists of the eigenvalues of the restriction of a to w the eigenvalues of the restriction of a to w the first split sequence of w and the second split sequence of w we express all the scalars and polynomials associated with w in terms of its parameter array we show that the parameter array of w is determined by r t d and one more free parameter from this we conclude that the isomorphism class of w is determined by these four parameters finally we apply our results to the case in which has q racah type or classical parameters

polynomials nonnegative on a grid and discrete optimization we characterize the real valued polynomials on r n that are nonnegative not necessarily strictly positive on a grid k of points of r n in terms of a weighted sum of squares whose degree is bounded and known in advance we also show that the mimimization of an arbitrary polynomial on k a discrete optimization problem reduces to a convex continuous optimization problem of fixed size the case of concave polynomials is also investigated the proof is based on a recent result of curto and fialkow on the k moment problem

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

evolution semigroups translation algebras and exponential dichotomy of cocycles abstract we study the exponential dichotomy of an exponentially bounded strongly continuous cocycle over a continuous flow on a locally compact metric space acting on a banach space x our main tool is the associated evolution semigroup on c 0 x we prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup to show the latter equivalence we establish the spectral mapping annular hull theorem for the evolution semigroup in addition dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c 0 z x here we develop banach algebra techniques and study weighted translation algebras that contain the evolution operators these results imply that dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space also exponential dichotomy follows from pointwise discrete dichotomies with uniform constants finally we extend to our situation the classical perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded continuous mild solutions to the inhomogeneous equation

the terwilliger algebras of grassmann graphs abstract let j q n m denote the grassmann graph with vertex set x and diameter min m n m fix a vertex x x let t t x denote the terwilliger algebra of j q n m corresponding to x in this paper we study the structure of t under the assumption that m 3 and n 2 m let u q sl 2 be the quantum enveloping algebra of sl 2 and let q be the q tetrahedron algebra we first obtain an action of u q sl 2 on the standard module of j q n m then we display a c algebra homomorphism u q sl 2 t and show that t is generated by the image of and some central elements of t as an application we also display an action of q on the standard module of j q n m these results are obtained by using the theory of leonard pairs

the terwilliger algebras of johnson graphs abstract let j n m denote the johnson graph with vertex set x fix a vertex x x let t t x denote the terwilliger algebra of j n m corresponding to x in this paper we study the structure of t under the assumption that m 3 and n 4 m let u sl 2 be the universal enveloping algebra of sl 2 we display a c algebra homomorphism u sl 2 t we show that t is generated by the image of and some central elements of t we obtain these results using the theory of leonard pairs

a semidefinite relaxation scheme for multivariate quartic polynomial optimization with quadratic constraints we present a general semidefinite relaxation scheme for general n variate quartic polynomial optimization under homogeneous quadratic constraints unlike the existing sum of squares approach which relaxes the quartic optimization problems to a sequence of typically large linear semidefinite programs sdp our relaxation scheme leads to a possibly nonconvex quadratic optimization problem with linear constraints over the semidefinite matrix cone in mathbb r n times n it is shown that each alpha factor approximate solution of the relaxed quadratic sdp can be used to generate in randomized polynomial time an o alpha factor approximate solution for the original quartic optimization problem where the constant in o cdot depends only on problem dimension in the case where only one positive definite quadratic constraint is present in the quartic optimization problem we present a randomized polynomial time approximation algorithm which can provide a guaranteed relative approximation ratio of 1 o n 2

complex quadratic optimization and semidefinite programming in this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the hermitian complex form a special case of the problem that we study corresponds to the max 3 cut model used in a recent paper of goemans and williamson j comput system sci 68 2004 pp 442 470 we first develop a closed form formula to compute the probability of a complex valued normally distributed bivariate random vector to be in a given angular region this formula allows us to compute the expected value of a randomized with a specific rounding rule solution based on the optimal solution of the complex semidefinite programming relaxation problem in particular we present an m 2 1 cos frac 2 pi m 8 pi approximation algorithm and then study the limit of that model in which the problem remains np hard we show that if the objective is to maximize a positive semidefinite hermitian form then the randomization rounding procedure guarantees a worst case performance ratio of pi 4 approx 0 7854 which is better than the ratio of 2 pi approx 0 6366 for its counterpart in the real case due to nesterov furthermore if the objective matrix is real valued positive semidefinite with nonpositive off diagonal elements then the performance ratio improves to 0 9349

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

eigenvalues and invariants of tensors abstract a tensor is represented by a supermatrix under a co ordinate system in this paper we define e eigenvalues and e eigenvectors for tensors and supermatrices by the resultant theory we define the e characteristic polynomial of a tensor an e eigenvalue of a tensor is a root of the e characteristic polynomial in the regular case a complex number is an e eigenvalue if and only if it is a root of the e characteristic polynomial we convert the e characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor i e they are invariant under co ordinate system changes we call them principal invariants of that tensor the maximum number of principal invariants of mth order n dimensional tensors is a function of m and n we denote it by d m n and show that d 1 n 1 d 2 n n d m 2 m for m 3 and d m n m n 1 m for m n 3 we also define the rank of a tensor all real eigenvectors associated with nonzero e eigenvalues are in a subspace with dimension equal to its rank

existence and robustness of exponential dichotomy of linear skew product semiflows over semiflows in this paper we investigate the exponential dichotomy of linear skew product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on banach bundles we characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy

evolutionary semigroups and dichotomy of linear skew product flows on locally compact spaces with banach fibers abstract we study evolutionary semigroups generated by a strongly continuous semi cocycle over a locally compact metric space acting on banach fibers this setting simultaneously covers evolutionary semigroups arising from non autonomous abstract cauchy problems and c 0 semigroups and linear skew product flows the spectral mapping theorem for these semigroups is proved the hyperbolicity of the semigroup is related to the exponential dichotomy of the corresponding linear skew product flow to this end a banach algebra of weighted composition operators is studied the results are applied in the study of roughness of the dichotomy dichotomy and solutions of nonhomogeneous equations green s function for a linear skew product flow pointwise dichotomy versus global dichotomy and evolutionary semigroups along trajectories of the flow

discretized characterizations of exponential dichotomy of linear skew product semiflows over semiflows using the method of discretization we investigate the necessary and sufficient conditions for the existence of exponential dichotomy of linear skew product semiflows over semiflows through the existence of discrete exponential dichotomy of the discretized linear skew product semiflows we then apply the obtained results to consider the roughness of exponential dichotomy of linear skew product semiflows

leonard pairs and the q racah polynomials abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following two conditions i there exists a basis for v with respect to which the matrix representing a is irreducible tridiagonal and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is irreducible tridiagonal wecall such a pair a leonard pair on v in the appendix to linear algebra appl 330 2001 p 149 we outlined a correspondence between leonard pairs and a class of orthogonal polynomials consisting of the q racah polynomials and some related polynomials of the askey scheme we also outlined how for the polynomials in this class the 3 term recurrence difference equation askey wilson duality and orthogonality can be obtained in a uniform manner from the corresponding leonard pair the purpose of this paper is to provide proofs for the assertions which we made in that appendix

criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations abstract we prove that the admissibility of any pair of vector valued schaffer function spaces satisfying a very general technical condition implies the existence of a no past exponential dichotomy for an exponentially bounded strongly continuous cocycle over a semiflow roughly speaking the class of schaffer function spaces consists in all function spaces which are invariant under the right shift and therefore our approach addresses most of the possible pairs of admissible spaces complete characterizations for the exponential dichotomy of cocycles are also obtained moreover we involve a concept of a no past exponential dichotomy for cocycles weaker than the classical concept defined by sacker and sell 1994 in 23 our definition of exponential dichotomy follows partially the definition given by chow and leiva 1996 in 4 in the sense that we allow the unstable subspace to have infinite dimension the main difference is that we do not assume a priori that the cocycle is invertible on the unstable space actually we do not even assume that the unstable space is invariant under the cocycle thus we generalize some known results due to o perron 1930 14 j daleckij and m krein 1974 7 j l massera and j j schaffer 1966 11 n van minh f rabiger and r schnaubelt 1998 26

on maximization of quadratic form over intersection of ellipsoids with common center we demonstrate that if a1 a m are symmetric positive semidefinite n n matrices with positive definite sum and a is an arbitrary symmetric n n matrix then the relative accuracy in terms of the optimal value of the semidefinite relaxation max x tr ax mid tr a ix le1 i 1 m x succeq0 eqno hbox sdp of the optimization program x tax to max mid x ta ix le 1 i 1 m eqno hbox p is not worse than 1 frac 1 2 ln 2m 2 it is shown that this bound is sharp in order as far as the dependence on m is concerned and that a feasible solution x to p with x tax ge frac opt hbox rm sdp 2 ln 2m 2 eqno can be found efficiently this somehow improves one of the results of nesterov 4 where bound similar to is established for the case when all ai are of rank 1

semidefinite relaxation bounds for indefinite homogeneous quadratic optimization this paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and its semidefinite programming sdp relaxation we consider two quadratic optimization models 1 min x c x mid x a k x ge 1 k 0 1 ldots m x in mathbb f n and 2 max x c x mid x a k x le 1 k 0 1 ldots m x in mathbb f n where mathbb f is either the real field mathbb r or the complex field mathbb c and a k c are symmetric matrices for the minimization model 1 we prove that if the matrix c and all but one of the a k s are positive semidefinite then the ratio between the optimal value of 1 and its sdp relaxation is upper bounded by o m 2 when mathbb f mathbb r and by o m when mathbb f mathbb c moreover when two or more of the a k s are indefinite this ratio can be arbitrarily large for the maximization model 2 we show that if c and at most one of the a k s are indefinite while other a k s are positive semidefinite then the ratio between the optimal value of 2 and its sdp relaxation is bounded from below by o 1 log m for both the real and the complex case this result improves the bound based on the so called approximate s lemma of ben tal nemirovski and roos siam j optim 13 2002 pp 535 560 when two or more of the a k s in 2 are indefinite we derive a general bound in terms of the problem data and the sdp solution for both optimization models we present examples to show that the derived approximation bounds are essentially tight

schaffer spaces and uniform exponential stability of linear skew product semiflows we study the exponential stability of linear skew product semiflows on locally compact metric space with banach fibers our main tool is the admissibility of a pair of the so called schaffer spaces this characterization is a very general one it includes as particular cases many interesting situations among them we can mention some results due to clark datko latushkin van minh montgomery smith randolph rabiger schnaubelt

evolutionary semigroups and lyapunov theorems in banach spaces we present a spectral mapping theorem for continuous semigroups of operators on any banach space e the condition for the hyperbolicity of a semigroup on e is given in terms of the generator of an evolutionary semigroup acting in the space of e valued functions the evolutionary semigroup generated by the propagator of a nonautonomous differential equation in e is also studied a discrete technique for investigating the evolutionary semigroup is developed and applied in describing the hyperbolicity exponential dichotomy of the nonautonomous equation

approximating quadratic programming with bound and quadratic constraints we consider the problem of approximating the global maximum of a quadratic program qp subject to bound and simple quadratic constraints based on several early results we show that a 4 7 approximate solution can be obtained in polynomial time

approximation bounds for quadratic optimization with homogeneous quadratic constraints we consider the np hard problem of finding a minimum norm vector in n dimensional real or complex euclidean space subject to m concave homogeneous quadratic constraints we show that a semidefinite programming sdp relaxation for this nonconvex quadratically constrained quadratic program qp provides an o m 2 approximation in the real case and an o m approximation in the complex case moreover we show that these bounds are tight up to a constant factor when the hessian of each constraint function is of rank 1 namely outer products of some given so called steering vectors and the phase spread of the entries of these steering vectors are bounded away from pi 2 we establish a certain constant factor approximation depending on the phase spread but independent of m and n for both the sdp relaxation and a convex qp restriction of the original np hard problem finally we consider a related problem of finding a maximum norm vector subject to m convex homogeneous quadratic constraints we show that an sdp relaxation for this nonconvex qp provides an o 1 ln m approximation which is analogous to a result of nemirovski e math program 86 1999 pp 463 473 for the real case

cocycles and mane sequences with an application to ideal fluids abstract exponential dichotomy of a strongly continuous cocycle is proved to be equivalent to existence of a mane sequence either for or for its adjoint as a consequence we extend some of the classical results to general banach bundles the dynamical spectrum of a product of two cocycles one of which is scalar is investigated and applied to describe the essential spectrum of the euler equation in an arbitrary spacial dimension

approximation algorithms for homogeneous polynomial optimization with quadratic constraints in this paper we consider approximation algorithms for optimizing a generic multi variate homogeneous polynomial function subject to homogeneous quadratic constraints such optimization models have wide applications e g in signal processing magnetic resonance imaging mri data training approximation theory and portfolio selection since polynomial functions are non convex the problems under consideration are all np hard in general in this paper we shall focus on polynomial time approximation algorithms in particular we first study optimization of a multi linear tensor function over the cartesian product of spheres we shall propose approximation algorithms for such problem and derive worst case performance ratios which are shown to be dependent only on the dimensions of the model the methods are then extended to optimize a generic multi variate homogeneous polynomial function with spherical constraint likewise approximation algorithms are proposed with provable approximation performance ratios furthermore the constraint set is relaxed to be an intersection of co centered ellipsoids namely we consider maximization of a homogeneous polynomial over the intersection of ellipsoids centered at the origin and propose polynomial time approximation algorithms with provable worst case performance ratios numerical results are reported illustrating the effectiveness of the approximation algorithms studied