an improved stability criterion for discrete time lur e systems with sector and slope restrictions this paper introduces an improved stability criterion for discrete time lur e systems with sector and slope restrictions for the stability criterion a lur e lyapunov functional candidate is constructed to include quadratic terms and integration terms with nonlinearities to handle the resulting integration terms in the lyapunov difference there is proposed a new bound lemma where the integration terms are relaxed into the second order terms by the geometric point of view in addition an appropriate weighting method uses slack variables from the sector and slope restrictions in the overall derivation the stabilization condition is formulated in terms of a parameterized linear matrix inequality plmi which is then converted into the linear matrix inequality an example illustrates that the proposed criterion presents a less conservative result than the previous criteria in the literature

geometry of free loci and factorization of noncommutative polynomials abstract the free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces z n f x m n k g det f x 0 the main theorem of this article shows that f is irreducible if and only if z n f is eventually irreducible a key step in the proof is an irreducibility result for linear pencils arising from this is a free singularity locus nullstellensatz for noncommutative polynomials apart from consequences to factorization in a free algebra the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry

brief paper a revisited tsypkin criterion for discrete time nonlinear lur e systems with monotonic sector restrictions this paper revisits a well known tsypkin criterion for stability analysis of discrete time nonlinear luroe systems when nonlinearities are monotonic and sector restricted by 0 1 where 1 is positive de nite it is shown by kapila and haddad that the system is absolutely stable if a function g 0 z 1 1 mi 1 z 1 k ng z is strictly positive real wherek is nonnegative diagonal and g z represents a transfer function of the linear part of the luroe system with invertible or identically zero g 0 this paper extends this criterion when 1 is positive diagonal by choosing a new lyapunov function to obtain an lmi criterion from a frequency domain interpretation of this lmi criterion another su cient criterion is generated which establishes that the system is absolutely stable if a function g 0 z 1 1 mi 1 z 1 k 1 z k n g z is strictly positive real wherek andk are nonnegative diagonal and orthogonal to each other 1998 elsevier science ltd all rights reserved

inverse numerical range and determinantal representation abstract the numerical range w a of an n n matrix a is the collection of quadratic forms z a over unit sphere 1 the inverse numerical range problem aims to find a unit vector which corresponds to a given point z of the numerical range w a given a matrix a the helton vinnikov theorem produces a symmetric matrix s so that a and s have the same numerical range w a w s in this paper we investigate the inverse numerical range problem for the boundary points and points on the boundary generating curve of the numerical range in place of the construction of unit vectors satisfying z a we express the kernel vector function of the linear pencil x s y s z i n as a function on the abel jacobi variety of the associated elliptic curve of a the kernel function plays a key role for the inverse numerical problem we perform this process when s is a generic 3 3 symmetric matrix

cauchy integral formula for generalized analytic functions in hydrodynamics it is shown that for several classes of generalized analytic functions arising in linearized equations of hydrodynamics and magnetohydrodynamics the cauchy integral formulae follow from the one for generalized holomorphic vectors in a uniform fashion if hydrodynamic fields velocity pressure and vorticity admit representations in terms of corresponding generalized analytic functions those representations and the cauchy integral formulae form two essential parts of the generalized analytic function approach which readily yields either closed form solutions or boundary integral equations this approach is demonstrated for problems of axisymmetric and asymmetric stokes flows two phase axisymmetric stokes flows two dimensional and axisymmetric oseen flows

generalized analytic functions in magnetohydrodynamics an approach of generalized analytic functions to the magnetohydrodynamic mhd problem of an electrically conducting viscous incompressible flow past a solid non magnetic body of revolution is pres

convex matrix inequalities versus linear matrix inequalities most linear control problems lead directly to matrix inequalities mis many of these are badly behaved but a classical core of problems are expressible as linear matrix inequalities lmis in many engineering systems problems convexity has all of the advantages of a lmi since lmis have a structure which is seemingly much more rigid than convex mis there is the hope that a convexity based theory will be less restrictive than lmis how much more restrictive are lmis than convex mis there are two fundamentally different classes of linear systems problems dimension free and dimension dependent a dimension free mi is a mi where the unknowns are g tuples of matrices and appear in the formulas in a manner which respects matrix multiplication most of the classic mis of control theory are dimension free dimension dependent mis have unknowns which are tuples of numbers the two classes behave very differently and this survey describes what is known in each case about the relation between convex mis and lmis the proof techniques involve and necessitate new developments in the field of semialgebraic geometry

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

lmi based stability criteria for discrete time lur e systems with monotonic sector and slope restricted nonlinearities this note presents new lmi based stability criteria for the discrete time multivariable lur e system with nonlinearities that are monotonic sector and slope restricted corresponding lur e lyapunov functions are constructed for such a system the new criteria are expressed in a reasonably general form that can be applied to both non diagonal and diagonal nonlinearities we explicitly compare the new approach to the existing lmi based popov like criteria in the literature and express the results in terms of an integral quadratic constraint iqc the applications of the new criteria to some control problems and strategies are briefly discussed numerical examples are included to show their performance and they are shown to be less conservative than the previous results

real zero polynomials and a horn s problem abstract a horn s problem concerns finding two self adjoint matrices so that a b and a b have prescribed spectrum in this paper we show how it connects to an interpolation problem for two variable real zero polynomials and a tracial moment problem in addition we outline an algorithm to construct a pair a b

the framework of k harmonically analytic functions for three dimensional stokes flow problems part ii the framework of generalized analytic functions arising from the related potentials so called k harmonically analytic functions has been developed in application to three dimensional 3d axially symmetric stokes flow problems cauchy s integral formula for the class of k harmonically analytic functions has been obtained and series representations for k harmonically analytic functions for the regions exterior to sphere and prolate and oblate spheroids have been derived as the central result in the developed framework a solution form representing the velocity field and pressure for 3d axially symmetric stokes flows has been constructed in terms of two 0 harmonically analytic functions it has also been shown that it uniquely determines an external velocity field vanishing at infinity with the obtained solution form the problem of 3d stokes flows due to the axially symmetric translation of a solid body of revolution has been reduced to a boundary value problem for two 0 harmonically analytic function

indefinite hardy spaces on finite bordered riemann surfaces abstract it is well known that one may study hardy spaces with the domain a finite bordered riemann surface rather than the unit disk however for domains with more than one boundary component it is natural to consider besides the usual positive definite inner product on h 2 indefinite inner products obtained by picking up different signs or in the vector valued case different signature matrices when integrating over different components in this paper we obtain a necessary and sufficient condition for such an indefinite inner product to be nondegenerate and show that when this condition is satisfied we actually get a krein space furthermore each holomorphic mapping of the finite bordered riemann surface onto the unit disk which maps boundary to boundary determines an explicit isometric isomorphism between this space and a usual vector valued hardy space on the unit disk with an indefinite inner product defined by an appropriate hermitian matrix as is usual when studying hardy spaces on a multiply connected domain the elements of the space are sections of a vector bundle rather than functions the main point is to construct an appropriate extension of this bundle to the double of the finite bordered riemann surface and to use cauchy kernels for certain vector bundles on a compact riemann surface

a less conservative stability criterion for discrete time lur e systems with sector and slope restrictions this paper proposes a less conservative stability criterion for discrete time lur e systems with sector and slope restrictions by constructing a novel lyapunov functional compared with the lyapunov functional in the literature this paper fully utilizes the sector and slope restrictions in our lyapunov functional which includes integral terms involved with the sector restriction of the nonlinearities phi y i and phi y i 1 integral terms involved with the slope restriction between phi y i and phi y i 1 and a quadratic term with an augmented vector related to available vectors for representing upper and lower bounds of all integral terms the positive definiteness of a matrix appearing in the quadratic term can be relaxed by utilizing the lower bounds of all integral terms based on the lyapunov functional an improved stability criterion is derived in terms of linear matrix inequalities numerical examples show the effectiveness of the proposed criterion

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

representations of lie algebras by non skewselfadjoint operators in hilbert space we study non skewselfadjoint representations of a finite dimensional real lie algebra gg to this end we embed a non skewselfadjoint representation of gg into a more complicated structure that we call a gg operator vessel and that is associated to an overdetermined linear conservative input state output system on the corresponding simply connected lie group gg we develop the frequency domain theory of the system in terms of representations of gg and introduce the joint characteristic function of a gg operator vessel which is the analogue of the classical notion of the characteristic function of a single non selfadjoint operator as the first non commutative example we apply the theory to the lie algebra of the ax bax b group the group of affine transformations of the line

how to count zeroes of polynomials on quadrature domains using the bezout matrix abstract classically the bezout matrix or simply bezoutian of two polynomials is used to locate the roots of the polynomial and in particular test for stability in this paper we develop the theory of bezoutians on real riemann surfaces of dividing type the main result connects the signature of the bezoutian of two real meromorphic functions to the topological data of their quotient which can be seen as the generalization of the classical cauchy index as an application we propose a method to count the number of zeroes of a polynomial in a quadrature domain using the inertia of the bezoutian we provide examples of our method in the case of simply connected quadrature domains

absolute stability criteria for multiple slope restricted monotonic nonlinearities absolute stability criteria such as the classical popov criterion guarantee stability for a class of sector bounded nonlinearities although the sector restriction bounds the admissible class of the nonlinearities the local slope of the nonlinearity may be arbitrarily large in this paper the authors derive absolute stability criteria for multiple slope restricted time invariant monotonic nonlinearities like the popov criterion in the single input single output case the authors results provide a simple graphical interpretation involving a straight line in a modified popov plane

quantum convex support abstract convex support the mean values of a set of random variables is central in information theory and statistics equally central in quantum information theory are mean values of a set of observables in a finite dimensional c algebra a which we call quantum convex support the convex support can be viewed as a projection of the state space of a and it is a projection of a spectrahedron spectrahedra are increasingly investigated at least since the 1990s boom in semi definite programming we recall the geometry of the positive semi definite cone and of the state space we write a convex duality for general self dual convex cones this restricts to projections of state spaces and connects them to results on spectrahedra our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections using natural isomorphisms the result encodes the face lattice of the convex support into a set of projections in a and enables the integration of convex geometry with matrix calculus or algebraic techniques

singular points of the algebraic curves of symmetric hyperbolic forms abstract we investigate singular points of the algebraic curve defined by a hyperbolic form satisfying some symmetric properties we show that all singular points of such a curve are real nodes

the algebraic degree of semidefinite programming given a generic semidefinite program specified by matrices with rational entries each coordinate of its optimal solution is an algebraic number we study the degree of the minimal polynomials of these algebraic numbers geometrically this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices we determine this degree using methods from complex algebraic geometry such as projective duality determinantal varieties and their chern classes

livsic type determinantal representations and hyperbolicity abstract hyperbolic homogeneous polynomials with real coefficients i e hyperbolic real projective hypersurfaces and their determinantal representations play a key role in the emerging field of convex algebraic geometry in this paper we consider a natural notion of hyperbolicity for a real subvariety x p d of an arbitrary codimension l with respect to a real l 1 dimensional linear subspace v p d and study its basic properties we also consider a class of determinantal representations that we call livsic type and a nice subclass of these that we call very reasonable much like in the case of hypersurfaces l 1 the existence of a definite hermitian very reasonable livsic type determinantal representation implies hyperbolicity we show that every curve admits a very reasonable livsic type determinantal representation our basic tools are cauchy kernels for line bundles and the notion of the bezoutian for two meromorphic functions on a compact riemann surface that we introduce we then proceed to show that every real curve in p d hyperbolic with respect to some real d 2 dimensional linear subspace admits a definite hermitian or even definite real symmetric very reasonable livsic type determinantal representation

containment problems for polytopes and spectrahedra we study the computational question of whether a given polytope or spectrahedron s a as given by the positive semidefiniteness region of a linear matrix pencil a x is contained in another s b first we classify the computational complexity extending results on the polytope polytope case by gritzmann and klee to the polytope spectrahedron case for various restricted containment problems np hardness is shown we then study in detail semidefinite conditions to certify containment building upon work by ben tal and nemirovski and helton klep and mccullough in particular we discuss variations of a sufficient semidefinite condition to certify containment of a spectrahedron in a spectrahedron it is shown that these sufficient conditions even provide exact semidefinite characterizations for containment in several important cases including containment of a spectrahedron in a polyhedron moreover in the case of bounded s a the criteria will always succeed in certifying containment of some scale

the lee yang and polya schur programs ii theory of stable polynomials and applications in the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains for such sets this completes the multivariate generalization of the classification program initiated by polya and schur for univariate real polynomials we build on these classification theorems to develop here a theory of multivariate stable polynomials applications and examples show that this theory provides a natural framework for dealing in a uniform way with lee yang type problems in statistical mechanics combinatorics and geometric function theory in one or several variables in particular we answer a question of hinkkanen on multivariate apolarity

deciding polyhedrality of spectrahedra spectrahedra are linear sections of the cone of positive semidefinite matrices which as convex bodies generalize the class of polyhedra in this paper we investigate the problem of recognizing when a spectrahedron is polyhedral we generalize and strengthen results of m v ramana polyhedra spectrahedra and semidefinite programming in topics in semidefinite and interior point methods fields inst commun 18 ams providence ri 1998 pp 27 38 regarding the structure of spectrahedra and we devise a normal form of representations of spectrahedra this normal form is effectively computable and leads to an algorithm for deciding polyhedrality

remarks on the lax conjecture for hyperbolic polynomials abstract a conjecture of lax p lax differential equations difference equations and matrix theory commun pure appl math 11 1958 175 194 says that every hyperbolic polynomial in two space variables is the determinant of a symmetric hyperbolic matrix the conjecture has recently been proved by lewis parillo ramana based on previous work of dubrovin and helton vinnikov in this note we prove related results for polynomials in several space variables which have rotational symmetries

construction of determinantal representation of trigonometric polynomials abstract for a pair of n n hermitian matrices h and k a real ternary homogeneous polynomial defined by f t x y det ti n xh yk is hyperbolic with respect to 1 0 0 the fiedler conjecture or lax conjecture is recently affirmed namely for any real ternary hyperbolic polynomial f t x y there exist real symmetric matrices s 1 and s 2 such that f t x y det ti n xs 1 ys 2 in this paper we give a constructive proof of the existence of symmetric matrices for the ternary forms associated with trigonometric polynomials

exposed faces of semidefinitely representable sets a linear matrix inequality lmi is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite motivated by the fact that diagonal lmis define polyhedra the solution set of an lmi is called a spectrahedron linear images of spectrahedra are called semidefinitely representable sets part of the interest in spectrahedra and semidefinitely representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming such as one can do on polyhedra by linear programming it is known that every face of a spectrahedron is exposed this is also true in the general context of rigidly convex sets we study the same question for semidefinitely representable sets lasserre proposed a moment matrix method to construct semidefinite representations for certain sets our main result is that this method can work only if all faces of the considered set are exposed this necessary condition complements sufficient conditions recently proved by lasserre helton and nie

applications of stable polynomials to mixed determinants johnson s conjectures unimodality and symmetrized fischer products for n x n matrices a and b define eta a b sigma s det a s det b s where the summation is over all subsets of 1 n s is the complement of s and a s is the principal submatrix

positive polynomials and projections of spectrahedra this article is concerned with different aspects of spectrahedra and their projections sets that are important in semidefinite optimization we prove results on the limitations of so called lasserre and theta body relaxation methods for semialgebraic sets and varieties as a special case we obtain the main result of netzer plaumann and schweighofer siam j optim 20 2010 pp 1944 1955 on nonexposed faces we also solve the open problems from that work we further give a unified account of several results on convex hulls of curves and images of polynomial maps we finally prove a positivstellensatz for projections of spectrahedra which exceeds the known results that only work for basic closed semialgebraic sets

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

the analytic structure of oseen flow past a sphere as a function of reynolds number the feasibility of van dyke s 28 proposal that one go from zero to infinite reynolds number r by computer extension of a stokes series in powers of r depends on the analytic structure of the flow as a function of r this structure is investigated in detail for oseen flow past a sphere we find that there is a doubly infinite array of simple pole singularities in the left half complex r plane each of which lies close to one of the n zeros of the modified bessel functions k n 1 2 r for some positive integer n consequently r infty is an accumulation point of poles and hence a nonisolated singularity we have extended the stokes series for the drag coefficient c d r to 66 terms in double precision and have sought the asymptotic behavior of c d r as r to infty neither pade approximants nor an euler transformation formed from our long series give good convergence we suggest that the asymptotic behavior of c d r is primarily as an expansion in powers of r 2 3 though

minimum drag shapes in magnetohydrodynamics a necessary optimality condition for the minimum drag shape for a non magnetic solid body immersed in the uniform flow of an electrically conducting viscous incompressible fluid under the presence of a magnetic field is obtained it is assumed that the flow and magnetic field are uniform and parallel at infinity and that the body and fluid have the same magnetic permeability the condition is derived based on the linearized magnetohydrodynamic mhd equations subject to a constraint on the body s volume and generalizes the existing optimality conditions for the minimum drag shapes for the body in the stokes and oseen flows of a non conducting fluid it is shown that for any hartmann number m reynolds number re and magnetic reynolds number re m the minimum drag shapes are fore and aft symmetric and have conic vertices with an angle of 2 3 the minimum drag shapes are represented in a function series form and the series coefficients are found iteratively with the derived optimality condition at each iteration the mhd problem is solved via the boundary integral equations obtained based on the cauchy integral formula for generalized analytic functions with respect to the equal volume sphere drag reduction as a function of the cowling number s m 2 re m re is smallest at s 1 also in the considered examples the drag values for the minimum drag shapes and equal volume minimum drag spheroids are sufficiently close

on the correlation functions of the vector bundle generalization of the bc system it is shown that the determinants of the correlation functions of the generalized bc system introduced recently are given as pullbacks of the non abelian theta divisor

a multivariable extension of the tsypkin criterion using a lyapunov function approach for analyzing the stability of discrete time systems containing a feedback nonlinearity the tsypkin criterion is the closest analog to the popov criterion which is used for analyzing such systems in continuous time traditionally the proof of this criterion is based upon input output properties and function analytic methods in this paper we extend the tsypkin criterion to multivariable systems containing an arbitrary number of monotonic sector bounded memoryless time invariant nonlinearities along with providing a lyapunov function proof for this classical result

noncommutative polynomials nonnegative on a variety intersect a convex set abstract this paper gives a precise algebraic certificate for a noncommutative polynomial to be nonnegative on a free convex semialgebraic set intersect the variety of a free left ideal

strict convexity of the joint c numerical range abstract let h h 1 h 2 h m be an m tuple of n n hermitian matrices and c c 1 c 2 c n r n the joint c numerical range w c h of h is defined for x x 1 x 2 x m r m the form associated with h is defined as f h t x det ti n x 1 h 1 x 2 h 2 x m h m we prove that the convex hull of w c h is strictly convex if the roots of f h t x 0 have constant multiplicities for all x on the unit sphere in r m the result is illustrated with several examples

semidefinite descriptions of low dimensional separable matrix cones abstract let k e k e be convex cones residing in finite dimensional real vector spaces an element y in the tensor product e e is k k separable if it can be represented as finite sum y l x l x l where x l k and x l k for all l let s n h n q n be the spaces of n n real symmetric complex hermitian and quaternionic hermitian matrices respectively let further s n h n q n be the cones of positive semidefinite matrices in these spaces if a matrix a h mn h m h n is h m h n separable then it fulfills also the so called ppt condition i e it is positive semidefinite and has a positive semidefinite partial transpose the same implication holds for matrices in the spaces s m s n h m s n and for m 2 in the space q m s n we provide a complete enumeration of all pairs n m when the inverse implication is also true for each of the above spaces i e the ppt condition is sufficient for separability we also show that a matrix in q n s 2 is q n s 2 separable if and only if it is positive semidefinite

on the oseen drag on a sphere the method of induced forces is employed to evaluate the oseen drag on a sphere the accuracy of the values obtained for the drag coefficient for large reynolds numbers is increased by application of shanks s transformation on the basis of the improved values we obtain as numerical approximations for the exponent of the first term in the asymptotic expansion values approaching 0 660 for reynolds numbers between 10 2 and 10 6

an algebraic geometry study of the b c system with arbitrary twist fields and arbitrary statistics we present an analysis of the generalb c system including the system on a compact riemann surface of arbitrary genusg 0 by postulating that its correlation functions should only have the singularities imposed by the operator product expansion ope of the system studying a very in fact optimally general form of theb c system we prove rigorously that the standard practice of eliminating zero modes and even the standard lagrangian follow from the analyticity structure dictated by the ope alone we extend the analysis to consider the most general case of the presence of twist e g spin fields we then determine all the possible correlation functions of theb c system with statistics unspecified compatible with the ope on imposing fermi and bose statistics we obtain the correlation functions of the fermionicb c and systems respectively

interlacing families ii mixed characteristic polynomials and the kadison singer problem we use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the kadison singer problem the rst is weaver s conjecture ks 2 which is known to imply kadison singer via a projection paving conjecture of akemann and anderson the second is a formulation due to casazza et al of anderson s original paving conjecture s for which we are able to compute explicit paving bounds the proof involves an analysis of the largest roots of a family of polynomials that we call the mixed characteristic polynomials of a collection of matrices

elementary transformations of pfaffian representations of plane curves abstract let c be a smooth curve in p 2 given by an equation f 0 of degree d in this paper we consider elementary transformations of linear pfaffian representations of c elementary transformations can be interpreted as actions on a rank 2 vector bundle on c with canonical determinant and no sections which corresponds to the cokernel of a pfaffian representation every two pfaffian representations of c can be bridged by a finite sequence of elementary transformations pfaffian representations and elementary transformations are constructed explicitly for a smooth quartic applications to aronhold bundles and theta characteristics are given

the bc system of higher rank revisited the bc system of higher rank introduced recently is examined further it is shown that the correlation functions are connected with certain non abelian functions and it is discussed how quasi determinants arise

finite dimensional de branges spaces on riemann surfaces abstract we study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real riemann surface these spaces are analogues of the spaces introduced by l de branges when the riemann sphere is replaced by a compact real riemann surface of a higher genus in de branges theory an important role is played by resolvent like difference quotient operators r here we introduce generalized difference quotient operators r y for any non constant meromorphic function y on the riemann surface the spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators r y 1 i and r y 2 2 where y 1 and y 2 generate the field of meromorphic functions on the riemann surface which satisfy a supplementary identity analogous to the de branges identity for difference quotients just as the classical de branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint or nonunitary operator the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks

roots of bivariate polynomial systems via determinantal representations we give two determinantal representations for a bivariate polynomial they may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two parameter eigenvalue problem the first determinantal representation is suitable for polynomials with scalar or matrix coefficients and consists of matrices with asymptotic order n 2 4 where n is the degree of the polynomial the second representation is useful for scalar polynomials and has asymptotic order n 2 6 the resulting method to compute the roots of a system of two bivariate polynomials is very competitive with some existing methods for polynomials up to degree 10 as well as for polynomials with a small number of terms

generalized analytic functions in axially symmetric oseen flows a class of generalized analytic functions arising from the oseen equations in the axially symmetric case has been identified for this class of functions the generalized cauchy integral formula has been obtained and a series representation for the region exterior to a sphere has been constructed the velocity field of the axially symmetric oseen flow has been represented in terms of two generalized analytic functions and it has been shown that for an exterior oseen flow problem those functions are uniquely determined provided that they both vanish at infinity also the pressure and vorticity have been determined as the real and imaginary parts of the two functions representing the velocity field and the drag exerted on a solid body of revolution in the axially symmetric oseen flow has been expressed in terms of one of the involved generalized analytic functions the problem of the axially symmetric oseen flow past a solid body of revolution has been reduced to an integral equation based on the gener

zero pole interpolation for matrix meromorphic functions on a compact riemann surface and a matrix fay trisecant identity this paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact riemann surface having a prescribed weil divisor i e having prescribed zeros and poles with directional as well as multiplicity information included in the vector case this new formalism unifies the earlier approach of ball clancey in the setting of trivial bundles over an abstract riemann surface with an earlier approach of the authors where the riemann surface was assumed to be the normalizing riemann surface for an algebraic curve embedded in c 2 with determinantal representation and the vector bundles were assumed to be presented as the kernels of linear matrix pencils the main tool is a version of the cauchy kernel appropriate for flat vector bundles over the riemann surface our formula for the interpolating bundle map in the special case of a single zero and a single pole can be viewed as a generalization of the fay trisecant identity from the usual line bundle case to the vector bundle case in terms of cauchy kernels in particular we obtain a new proof of the fay trisecant identity

stability criteria of sector and slope restricted lur e systems presents stability criteria of sector and slope restricted lur e systems in terms of linear matrix inequalities by fully exploiting inherent properties of sector and slope restrictions in the time domain interpreting the time domain criteria in the frequency domain furthermore supplies simpler expressions several examples show excellent performances of these criteria

the asymptotic stability of nonlinear lur e systems with multiple slope restrictions the authors present the analysis of the asymptotic stability of multiple slope restricted nonlinear lur e systems by providing a lyapunov function they obtain a matrix language criterion in terms of algebraic riccati equations and linear matrix inequalities which are discussed at the point of computational issues additionally they consider the frequency domain interpretation of the result

interpolation problems for holomorphic functions abstract in this note we discuss the nevanlinna pick problem corona problem and caratheodory fejer problem for bounded holomorphic functions

determinantal representations of hyperbolic forms via weighted shift matrices we characterize hyperbolic ternary forms of degrees 3 4 and 5 that admit determinantal representations via cyclic weighted shift matrices

a randomized approximation algorithm for the minimal norm static output feedback problem a new randomized algorithm is suggested for extracting static output stabilizing feedbacks with approximately minimal norm for lti systems the algorithm has two similar stages where in the first one the feasibility problem is solved and in the second one the optimization problem is solved the formulation is unified for the feasibility and for the optimization problems as well as for continuous time or discrete time systems the method is demonstrated by applying it to the hard conjectured to be np hard problem of the minimal gain static output stabilizing feedback and to the hard conjectured to be np hard problem of regional pole placement via static output feedback in non convex or unconnected regions a proof of convergence in probability that captures the two rounds of the algorithm is given and complexity analysis is provided under some mild assumptions

polynomial matrix inequality and semidefinite representation consider a convex set s x in cal d g x succeq 0 where g x is a symmetric matrix whose every entry is a polynomial or rational function d rn is a domain on which g x is defined and g x succeq 0 means g x is positive semidefinite the set s is called semidefinite representable if it equals the projection of a higher dimensional set that is defined by a linear matrix inequality lmi this paper studies sufficient conditions guaranteeing semidefinite representability of s we prove that s is semidefinite representable in the following cases i d rn g x is a matrix polynomial and matrix sos concave ii d is compact convex g x is a matrix polynomial and strictly matrix concave on d iii g x is a matrix rational function and q module matrix concave on d explicit constructions of semidefinite representations are given some examples are illustrated

de branges spaces on compact riemann surfaces and a beurling lax type theorem abstract using the notion of commutative operator vessels this work investigates de branges rovnyak spaces whose elements are sections of a line bundle of multiplicative half order differentials on a compact real riemann surface as a special case we obtain a beurling lax type theorem in the setting of the corresponding hardy space on a finite bordered riemann surface

unitary similarity of the determinantal representation of unitary bordering matrices abstract let f x y z be a hyperbolic ternary form of degree n the helton vinnikov theorem asserts that there exists an n n complex symmetric matrix s such that f s x y z f x y z where the determinantal hyperbolic ternary form of an n n matrix b is defined by f b x y z det x b y b z i n let a be an n n unitary bordering matrix it is known that a is unitarily similar to a symmetric matrix in this paper we investigate the unitary similarity between the unitary bordering matrix a and the helton vinnikov symmetric matrix s admitted the determinantal representation of the ternary form f a x y z satisfying f s x y z f a x y z for n 3 4

the far field oseen velocity expansion consider uniform steady incompressible fluid flow in an unbounded domain past a fixed closed body in the far field the oseen equations approximately hold we give the oseen velocity and pressure expansions in the far field for two and three dimensional flow by using two approaches the first decomposes the velocity into a potential velocity and a wake velocity and was introduced by lamb and goldstein the second approach uses the oseen representation of the velocity and pressure by a green9s integral distribution of singularities called oseenlets these are the force drag and lift singular solutions we show that in general the lamb goldstein approach will not model the oseen velocity in the far field wake in contrast by using oseen9s representation to expand each oseenlet in a taylor series we obtain velocity and pressure expansions everywhere in the far field nevertheless there are exceptions for which the lamb goldstein approach can be used notably i the singular drag and lift solutions ii axisymmetric flow such that the axis of rotational symmetry is parallel to the uniform stream velocity direction iii low reynolds number flow and iv two dimensional flow for these special cases the two approaches are shown to be equivalent

a less conservative lmi condition for stability of discrete time systems with slope restricted nonlinearities many conditions have been found for the absolute stability of discrete time lur e systems in the literature it is advantageous to find convex searches via lmis where possible in this technical note we construct two less conservative lmi conditions for discrete time systems with slope restricted nonlinearities the first condition is derived via lyapunov theory while the second is derived via the theory of integral quadratic constraints iqcs and noncausal zames falb multipliers both conditions are related to the jury lee criterion most appropriate for systems with such nonlinearities and the second generalizes it numerical examples demonstrate a significant reduction in conservatism over competing approaches