jacobi identity for vertex algebras in higher dimensions vertex algebras in higher dimensions introduced previously by nikolov provide an algebraic framework for investigating axiomatic quantum field theory with global conformal invariance we develop further the theory of such vertex algebras by introducing formal calculus techniques and investigating the notion of polylocal fields we derive a jacobi identity which together with the vacuum axiom can be taken as an equivalent definition of vertex algebra

the truncated complex moment problem let 2n denote a sequence of complex numbers 00 01 10 0 2n 2n 0 00 0 ij ji and let k denote a closed subset of the complex plane c the truncated complex k moment problem for entails determining whether there exists a positive borel measure on c such that ij zizj d 0 i j 2n and supp k for k kp a semi algebraic set determined by a collection of complex polynomials p pi z z mi 1 we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix m n and the localizing matrices mpi we prove that there exists a rankm n atomic representing measure for 2n supported in kp if and only if m n 0 and there is some rankpreserving extension m n 1 for which mpi n ki 0 where deg pi 2ki or 2ki 1 1 i m

a spinor like representation of the contact superconformal algebra k 4 in this work we construct an embedding of a nontrivial central extension of the contact superconformal algebra k 4 into the lie superalgebra of pseudodifferential symbols on the supercircle s1 2 associated with this embedding is a one parameter family of spinor like tiny irreducible representations of k 4 realized just on 4 fields instead of the usual 16

new generalized simple lie algebras of cartan type over a field with characteristic 0 abstract we construct four new series of generalized simple lie algebras of cartan type using the mixtures of grading operators and down grading operators our results in this paper are further generalizations of those in osborn s work j algebra 185 1996 820 835

cubic column relations in truncated moment problems for the truncated moment problem associated to a complex sequence 2n ij i j z i j 2n to have a representing measure it is necessary for the moment matrix m n to be positive semidefinite and for the algebraic variety v to satisfy rankm n cardv as well as a consistency condition the riesz functional vanishes on every polynomial of degree at most 2n that vanishes on v in previous work with l fialkow and h m moller the first named author proved that for the extremal case rankm n cardv positivity and consistency are sufficient for the existence of a representing measure in this paper we solve the truncated moment problem for cubic column relations in m 3 of the form z3 itz uz u t r we do this by checking consistency for u t in the open cone determined by 0 u t 2 u we first prove that the algebraic variety has exactly 7 points and rankm 3 7 we then apply the above mentioned result to obtain a concrete computable necessary and sufficient condition for the existence of a representing measure

conformal and lie superalgebras motivated from free fermionic fields in this paper we construct six families of conformal superalgebras of infinite type motivated from free quadratic fermonic fields with derivatives and we prove their simplicity the lie superalgebras generated by these conformal superalgebras are proven to be simple except for a few special cases in the general linear superalgebras and the type q lie superalgebras in which these lie superalgebras have a one dimensional centre and the quotient lie superalgebras modulo the centre are simple certain natural central extensions of these families of conformal superalgebras are also given moreover we prove that these conformal superalgebras are generated by their finite dimensional subspaces of minimal weight in a certain sense it is shown that a conformal superalgebra is simple if and only if its generated lie superalgebra does not contain a proper nontrivial ideal with a one variable structure

singularities of energy minimizing maps from the ball to the sphere we study maps from the unit ball b in r3 to the unit sphere s2 in r3 which minimize dirichlet s energy integral varepsilon varphi int b nabla varphi 2 dv

non extremal sextic moment problems abstract positive semidefiniteness recursiveness and the variety condition of a moment matrix are necessary and sufficient conditions to solve the quadratic and quartic moment problems also positive semidefiniteness combined with another necessary condition consistency is a sufficient condition in the case of extremal moment problems i e when the rank of the moment matrix denoted by r and the cardinality of the associated algebraic variety denoted by v are equal however these conditions are not sufficient for non extremal sextic or higher order truncated moment problems in this paper we settle three key instances of the non extremal i e r v sextic moment problem as follows when r 7 positive semidefiniteness consistency and the variety condition guarantee the existence of a 7 atomic representing measure when r 8 we construct two determining algorithms corresponding to the cases v 9 and v to accomplish this we generalize the rank reduction technique developed in previous work where we solved the nonsingular quartic moment problem and found an explicit way to build a representing measure

relative entropy and the multivariable multidimensional moment problem entropy like functionals on operator algebras have been studied since the pioneering work of von neumann umegaki lindblad and lieb the best known are the von neumann entropy rho trace rhologrho and a generalization of the kullback leibler distance s rhoparsigma trace rhologrho rhologsigma referred to as quantum relative entropy and used to quantify distance between states of a quantum system the purpose of this paper is to explore and s as regularizing functionals in seeking solutions to multivariable and multidimensional moment problems it will be shown that extrema can be effectively constructed via a suitable homotopy the homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems this is accomplished by a renormalization of a riemannian metric induced by entropy functionals as an application we discuss the inverse problem of describing power spectra which are consistent with second order statistics which has been the main motivation behind the present work

random matrices in 2d laplacian growth and operator theory since it was first applied to the study of nuclear interactions by wigner and dyson almost 60 years ago random matrix theory rmt has developed into a field of its own within applied mathematics and is now essential to many parts of theoretical physics from condensed matter to high energy the fundamental results obtained so far rely mostly on the theory of random matrices in one dimension the dimensionality of the spectrum or equilibrium probability density in the last few years this theory has been extended to the case where the spectrum is two dimensional or even fractal with dimensions between 1 and 2 in this paper we review these recent developments and indicate some physical problems where the theory can be applied

restriction operators balayage and doubly orthogonal systems of analytic functions abstract systems of analytic functions which are simultaneously orthogonal over each of two domains were apparently first studied in particular cases by walsh and szego and in full generality by bergman in principle these are very interesting objects allowing application to analytic continuation that is not restricted as weierstrassian continuation via power series either by circular geometry or considerations of locality however few explicit examples are known and in general one does not know even gross qualitative features of such systems the main contribution of the present paper is to prove qualitative results in a quite general situation it is by now very well known that the phenomenon of double orthogonality is not restricted to bergman spaces of analytic functions nor even indeed has it any intrinsic relation to analyticity its essence is an eigenvalue problem arising whenever one considers the operator of restriction on a hilbert space of functions on some set to a subset thereof provided this restriction is injective and compact however in this paper only hilbert spaces of analytic functions are considered especially bergman spaces in the case of the hardy spaces fisher and micchelli discovered remarkable qualitative features of doubly orthogonal systems and we have shown how based on the classical potential theoretic notion of balayage and its modern generalizations one can deduce analogous results in the bergman space set up but with restrictions imposed on the geometry of the considered domains and measures these were not needed in the fisher micchelli analysis but are necessary here as shown by examples from a more constructive point of view we study the bergman restriction operator between the unit disk and a compactly contained quadrature domain and show that the representing kernel of this operator is rational and it is expressible as an inversion followed by a logarithmic derivative in terms of the polynomial equation of the boundary of the inner domain

partially finite programming in l_1 and the existence of maximum entropy estimates best entropy estimation is a technique that has been widely applied in many areas of science it consists of estimating an unknown density from some of its moments by maximizing some measure of the entropy of the estimate this problem can be modelled as a partially finite convex program with an integrable function as the variable a complete duality and existence theory is developed for this problem and for an associated extended problem which allows singular measure theoretic solutions this theory explains the appearance of singular components observed in the literature when the burg entropy is used it also provides a unified treatment of existence conditions when the burg boltzmann shannon or some other entropy is used as the objective some examples are discussed

nonlinear theory of defects in nematic liquid crystals phase transition and flow phenomena this is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals it also includes some new results concerning disclinations line defects which have not been published elsewhere

extremal solutions of the two dimensionall problem of moments by using the theory of the principal function of a hyponormal operator with rank one self commutator one classifies some extremal points of the solutions of the truncatedl problem of moments in two real variables these extremal elements called degenerated solutions of thel problem are proved to be in a natural bijective correspondence with the bounded quadrature domains in the complex plane the main result of the paper characterizes these degenerated solutions from three distinct perspectives complex analysis moment problems and operator theory the interplay between the three points of view each with specific techniques constitutes the main body of the paper

sums of squares on real algebraic curves given an affine algebraic variety v over with real points v compact and a non negative polynomial function f v with finitely many real zeros we establish a local global criterion for f to be a sum of squares in v we then specialize to the case where v is a curve the notion of virtual compactness is introduced and it is shown that in the local global principle compactness of v can be relaxed to virtual compactness the irreducible curves on which every non negative polynomial is a sum of squares are classified all results are extended to the more general framework of preorders moreover applications to the k moment problem from analysis are given in particular schmudgen s solution of the k moment problem for compact k is extended for dim k 1 to the case when k is virtually compact

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

the resultant on compact riemann surfaces we introduce a notion of the resultant of two meromorphic functions on a compact riemann surface and demonstrate its usefulness in several respects for example we exhibit several integral formulas for the resultant relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact riemann surface as a particular application the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the schottky double of the domain

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

on a diagonal pade approximation in two complex variables a special type diagonal pade approximation for a class of hermitian power series in two variables is related to a canonical strong operator topology finite rank approximation of cyclic operators the expected convergence of the process uniform or in measure is derived from operator theory facts

dimensional stability in truncated moment problems abstract an approach to truncated moment problems is developed via the riesz functional and an assumed dimensional stability of its associated hilbert spaces although equivalent to a concept of flatness introduced by r curto and l fialkow the dimensional stability discussed in this paper has a different geometric aspect and leads to statements parallel to those of the quoted authors as well as to some newer ones obtained by simpler arguments a stability equation giving a local characterization of the dimensional stability is also presented

real separated algebraic curves quadrature domains ahlfors type functions and operator theory abstract the aim of this paper is to inter relate several algebraic and analytic objects such as real type algebraic curves quadrature domains functions on them and rational matrix functions with special properties and some objects from operator theory such as vector toeplitz operators and subnormal operators our tools come from operator theory but some of our results have purely algebraic formulation we make use of xia s theory of subnormal operators and of the previous results by the author in this direction we also correct in section 5 some inaccuracies in the works of d v yakubovich subnormal operators of finite type i xia s model and real algebraic curves in c 2 rev mat iberoamericana 14 1998 95 115 d v yakubovich subnormal operators of finite type ii structure theorems rev mat iberoamericana 14 1998 623 681 by the author

topology of quadrature domains we address the problem of topology of quadrature domains namely we give upper bounds on the connectivity of the domain in terms of the number of nodes and their multiplicities in the quadrature identity

finite conformal modules over n 2 3 4 superconformal algebras in this paper we continue the study of representation theory of formal distribution lie superalgebras initiated by cheng and kac asian j math 1 181 193 1997 2 153 156 1998 erratum we study finite verma type conformal modules over the n 2 n 3 and the two n 4 superconformal algebras and also find explicitly all singular vectors in these modules from our analysis of these modules we obtain a complete list of finite irreducible conformal modules over the n 2 n 3 and the two n 4 superconformal algebras

global existence of solutions of the liquid crystal flow for the oseen frank model in r2 abstract in the first part of this paper we establish the global existence of solutions of the liquid crystal gradient flow for the well known oseen frank model the liquid crystal flow is a prototype of equations from the ericksen leslie system in the hydrodynamic theory and generalizes the heat flow for harmonic maps into the 2 sphere the ericksen leslie system is a system of the navier stokes equations coupled with the liquid crystal flow in the second part of this paper we also prove the global existence of solutions of the ericksen leslie system for a general oseen frank model in r 2

a new look at entropy for solving linear inverse problems entropy based methods are widely used for solving inverse problems particularly when the solution is known to be positive here we address linear ill posed and noisy inverse problems of the form z ax n with a general convex constraint x spl isin x where x is a convex set although projective methods are well adapted to this context we study alternative methods which rely highly on some information based criteria our goal is to clarify the role played by entropy in this field and to present a new point of view on entropy using general tools and results coming from convex analysis we present then a new and broad scheme for entropic based inversion of linear noisy inverse problems this scheme was introduced by navaza in 1985 in connection with a physical modeling for crystallographic applications and further studied by dacunha castelle and gamboa 1990 important features of this paper are i a unified presentation of many well known reconstruction criteria ii proposal of new criteria for reconstruction under various prior knowledge and with various noise statistics iii a description of practical inversion of data using the aforementioned criteria and iv a presentation of some reconstruction results

maximum entropy in the generalized moment problem the generalized moment problem in the framework of the maximum entropy approach is considered a proof for the existence conditions of the solution is provided

selected topics on quadrature domains this is a selection of facts old and recent about quadrature domains the text written in the form of a survey is addressed to non experts and covers a variety of phenomena related to quadratur

a new n 6 superconformal algebra in this paper we construct a newn 6 superconformal algebra which extends the virasoro algebra by theso 6 current algebra by 6 odd primary fields of conformal weight 3 2 and by 10 odd primary fields of conformal weight 1 2 the commutation relations of this algebra which we will refer to asck 6 are represented by short distance operator product expansions ope we constructck 6 as a subalgebra of theso 6 superconformal algebra k6 thus giving it a natural representation as first order differential operators on the circle withn 6 extended symmetry we show thatck 6 has no nontrivial central extensions

generalized interpolation in h infty with a complexity constraint in a seminal paper sarason generalized some classical interpolation problems for h infinity functions on the unit disc to problems concerning lifting onto h 2 of an operator t that is defined on k

positivity of riesz functionals and solutions of quadratic and quartic moment problems abstract we employ positivity of riesz functionals to establish representing measures or approximate representing measures for truncated multivariate moment sequences for a truncated moment sequence y we show that y lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set k r n if and only if the associated riesz functional l y is k positive for a determining set k we prove that if l y is strictly k positive then y admits a representing measure supported in k as a consequence we are able to solve the truncated k moment problem of degree k in the cases i n k 2 4 and k r 2 ii n 1 k 2 and k is defined by one quadratic equality or inequality in particular these results solve the truncated moment problem in the remaining open cases of hilbert s theorem on sums of squares

quadratic conformal superalgebras abstract in this paper we shall classify quadratic conformal superalgebras by certain compatible pairs of a lie superalgebra and a novikov superalgebra four general constructions of such pairs are given moreover we shall classify such pairs related to simple novikov algebras

a convex optimization approach to the rational covariance extension problem in this paper we present a convex optimization problem for solving the rational covariance extension problem given a partial covariance sequence and the desired zeros of the modeling filter the poles are uniquely determined from the unique minimum of the corresponding optimization problem in this way we obtain an algorithm for solving the covariance extension problem as well as a constructive proof of georgiou s seminal existence result and his conjecture a stronger version of which we have resolved in byrnes et al ieee trans automat control ac 40 1995 pp 1841 1857

a new approach to spectral estimation a tunable high resolution spectral estimator traditional maximum entropy spectral estimation determines a power spectrum from covariance estimates here we present a new approach to spectral estimation which is based on the use of filter banks as a means of obtaining spectral interpolation data such data replaces standard covariance estimates a computational procedure for obtaining suitable pole zero arma models from such data is presented the choice of the zeros ma part of the model is completely arbitrary by suitable choices of filter bank poles and spectral zeros the estimator can be tuned to exhibit high resolution in targeted regions of the spectrum

singularities of energy minimizing maps from the ball to the sphere examples counterexamples and bounds we study maps qp from domains q in r3 to s2 which have prescribed boundary value functions 4 and which minimize dirichlet s integral energy such p s can have isolated singular points and our principal concern is the number and arrangement of such singularities our main result is that the number of singular points is bounded above by a constant times dirichlet s integral of 4 on d q and we show that this linear law is the best possible in two different senses furthermore singularities of minimizing q s sometimes have peculiar and counterintuitive properties i we illustrate by example that the mapping area jacobian integral of 4 can be zero although qp has many singular points ii we also show how to construct uniquely minimizing qp s having many singular points stacked up vertically near d q like bubbles in a pan of water about to boil iii finally we show by example that symmetries of q and 4 do not insure corresponding symmetries of qp and its singularities in particular singularities in p can be unstable under small perturbations of 4

superconformal algebras and transitive group actions on quadrics a classification of physical superconformal algebras is given the list consists of seven algebras the virasoro algebra the neveu schwarz algebra then 2 3 and 4 algebras the superalgebra of all vector fields on then 2 supercircle and a new algebrack6 constructed in 3 the proof relies heavily on the classification of all connected subgroups ofson c which act transitively on the quadric v v 1

regularity of a free boundary with application to the pompeiu problem in the unit ball b 0 1 let u and a domain in n solve the following overdetermined problem u in b 0 1 02 u jruj 0 inb 0 1 n where denotes the characteristic function and the equation is satisfled in the sense of distributions if the complement of does not develop cusp singularities at the origin then we prove is analytic in some small neighborhood of the origin the result can be modifled to yield for more general divergence form operators as an application of this then we obtain the regularity of the boundary of a domain without the pompeiu property provided its complement has no cusp singularities

singular dimensions of the n 2 superconformal algebras i verma modules of superconfomal algebras can have singular vector spaces with dimensions greater than 1 following a method developed for the virasoro algebra by kent we introduce the concept of adapted orderings on superconformal algebras we prove several general results on the ordering kernels associated to the adapted orderings and show that the size of an ordering kernel implies an upper limit for the dimension of a singular vector space we apply this method to the topological n 2 algebra and obtain the maximal dimensions of the singular vector spaces in the topological verma modules 0 1 2 or 3 depending on the type of verma module and the type of singular vector as a consequence we prove the conjecture of gato rivera and rosado on the possible existing types of topological singular vectors 4 in chiral verma modules and 29 in complete verma modules interestingly we have found two dimensional spaces of singular vectors at level 1 finally by using the topological twists and the spectral flows we also obtain the maximal dimensions of the singular vector spaces for the neveu schwarz n 2 algebra 0 1 or 2 and for the ramond n 2 algebra 0 1 2 or 3

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

positivity sums of squares and the multi dimensional moment problem let k be the basic closed semi algebraic set in r n defined by some finite set of polynomials s and t the preordering generated by s for k compact f a polynomial in n variables nonnegative on k and real e 0 we have that f e t in particular the k moment problem has a positive solution in the present paper we study the problem when k is not compact for n 1 we show that the k moment problem has a positive solution if and only if s is the natural description of k see section 1 for n 2 we show that the k moment problem fails if k contains a cone of dimension 2 on the other hand we show that if k is a cylinder with compact base then the following property holds f r x f 0 on k q t such that real 0 f eq t this property is strictly weaker than the one given in schmudgen 1991 but in turn it implies a positive solution to the k moment problem using results of marshall 2001 we provide many noncompact examples in hypersurfaces for which holds finally we provide a list of 8 open problems

hyponormal operators with rank two self commutators abstract in this paper it is shown that if t l h satisfies i t is a pure hyponormal operator ii t t is of rank two and iii ker t t is invariant for t then t is either a subnormal operator or the putinar s matricial model of rank two more precisely if t ker t t has a rank one self commutator then t is subnormal and if instead t ker t t has a rank two self commutator then t is either a subnormal operator or the kth minimal partially normal extension t k k of a k 1 hyponormal operator t k which has a rank two self commutator for any k z hence in particular every weakly subnormal or 2 hyponormal operator with a rank two self commutator is either a subnormal operator or a finite rank perturbation of a k hyponormal operator for any k z

landau de gennes theory of nematic liquid crystals the oseen frank limit and beyond we study global minimizers of a continuum landau de gennes energy functional for nematic liquid crystals in three dimensional domains subject to uniaxial boundary conditions we analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly in w1 2 to a global minimizer predicted by the oseen frank theory for uniaxial nematic liquid crystals with constant order parameter moreover the convergence is uniform in the interior of the domain away from the singularities of the limiting oseen frank global minimizer we obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the landau de gennes global minimizer we also study the interplay between biaxiality and uniaxiality in landau de gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions

signal estimation via selective harmonic amplification music redux the technique known as multiple signal classification music is a semi empirical way to obtain pseudo spectra that highlight the spectral energy distribution of a time series it is based on a certain canonical decomposition of a toeplitz matrix formed out of an estimated autocorrelation sequence the purpose of this paper is to present an analogous canonical decomposition of the state covariance matrix of a stable linear filter driven by a given time series accordingly the paper concludes with a modification of music the new method starts with filtering the time series and then estimating the covariance of the state of the filter this step in essence improves the signal to noise ratio snr by amplifying the contribution to the actual value of the state covariance of a selected harmonic interval where spectral lines are expected to reside then the method capitalizes on the canonical decomposition of the filter state covariance to retrieve information on the location of possible spectral lines the framework requires uniformly spaced samples of the process

a generalized entropy criterion for nevanlinna pick interpolation with degree constraint we present a generalized entropy criterion for solving the rational nevanlinna pick problem for n 1 interpolating conditions and the degree of interpolants bounded by n the primal problem of maximizing this entropy gain has a very well behaved dual problem this dual is a convex optimization problem in a finite dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n the criterion requires a selection of a monic schur polynomial of degree n it follows that this class of monic polynomials completely parameterizes all such rational interpolants and it therefore provides a set of design parameters for specifying such interpolants the algorithm is implemented in a state space form and applied to several illustrative problems in systems and control namely sensitivity minimization maximal power transfer and spectral estimation

freely generated vertex algebras and non linear lie conformal algebras we introduce the notion of a non linear lie conformal superalgebra and prove a pbw theorem for its universal enveloping vertex algebra we also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a unique up to isomorphism non linear lie conformal superalgebra this correspondence will be applied in the subsequent work to the problem of classification of finitely generated simple graded vertex algebras

spectral estimation via selective harmonic amplification the state covariance of a linear filter is characterized by a certain algebraic commutativity property with the state matrix of the filter and also imposes a generalized interpolation constraint on the power spectrum of the input process this algebraic property and the relationship between state covariance and the power spectrum of the input allow the use of matrix pencils and analytic interpolation theory for spectral analysis several algorithms for spectral estimation are developed with resolution higher than state of the art

ginzburg landau minimizers with prescribed degrees capacity of the domain and emergence of vortices let omega subset mathbb r 2 be a simply connected domain let omega be a simply connected subdomain of omega and set a omega setminus overline omega let j be the class of complex valued maps on the annular domain a with degree 1 both on partial omega and on partial omega we consider the variational problem of minimizing the ginzburg landau energy e lambda among all maps in j because only the degrees of the map are prescribed on the boundary the set j is not closed under weak h 1 convergence we show that the attainability of the minimum of e lambda over j is determined by the value of cap a where cap a is the h 1 capacity of the domain a in contrast it is known that the existence of minimizers of e lambda among the maps with a prescribed dirichlet boundary data does not depend on this geometric characteristic when cap a ge pi such an a is denoted as either subcritical if cap a pi or critical if cap a pi we show that the global minimizers of e lambda exist for each lambda 0 and that they are vortexless when lambda is large assuming that lambda to infty we demonstrate that the minimizers of e lambda converge in h 1 a and even better to an mathbb s 1 valued harmonic map which we explicitly identify when cap a lambda 0 or ii the global minimizers exist for each lambda 0 we conjecture that the second case never occurs further for large lambda we establish that the minimizing sequences minimizers in supercritical domains develop exactly two vortices a vortex of degree 1 near partial omega and a vortex of degree 1 near partial omega

solving moment problems by dimensional extension polynomial inequalities the second part of the paper uses the representation results of positive functionals on certain spaces of rational functions developed in the flrst part for decomposing a polynomial which is positive on such a semi algebraic set into a canonical sum of squares of rational functions times explicit multipliers let n 1 be a flxed integer due to the fact that for n 1 not every nonnegative polynomial in r n can be written as a sum of squares of polynomials see for instance 2 x6 3 the moment problems in n variables are more di cult than the classical one variable problems this very intriguing territory has been investigated by many authors see 2 7 12 and their references although characterizations for measures whose support lies in an arbitrary generally unbounded semi algebraic set do not seem to exist the present paper starts from an idea of the second author see 19 about solving moment problems by a change of basis via an embedding of r n into a submanifold of a higher dimensional euclidean space rougly speaking we prove that certain n 1 dimensional extensions of a moment sequence are naturally characterized by positivity conditions and moreover these extensions parametrize all possible solutions of the moment problem to be more speciflc let fi z r n x fi d x fi 2 z n

the structure of state covariances and its relation to the power spectrum of the input we study the relationship between power spectra of stationary stochastic inputs to a linear filter and the corresponding state covariances and identify the structure of positive semidefinite matrices that qualify as state covariances of the filter this structure is best revealed by a rank condition pertaining to the solvability of a linear equation involving the state covariance and the system matrices we then characterize all input power spectra consistent with any specific state covariance the parametrization of input spectra is achieved through a relation to solutions of an analytic interpolation problem which is analogous but not equivalent to a matricial nehari problem

linear analysis of quadrature domains iii the linear data of a quadrature domain introduced in the first paper in this series is further investigated the matricial structure of all hyponormal operators with finite rank self commutators which is inductively constructed from such linear data is discussed in particular a conservation law of a determinantal function in the iterative process of defining these matrices is exploited the spectrum of the determining left upper corner of some of these operators is analyzed in connection with the existence of generalized quadrature formulas on planar domains

solution of the truncated moment problem with variety y x 3 we show that positivity consistency and the variety condition which are sufficient to solve the truncated moment problem on planar curves of degree 2 are not sufficient for curves of higher degree using new partly algorithmic conditions based on positive moment matrix extensions we present a concrete solution to the truncated moment problem on the curve y x 3 we also use moment matrix extensions to solve in a less concrete sense truncated moment problems on curves of the form y g x and yg x 1 g x leading to degree bounded weighted sum of squares representations for polynomials which are strictly positive on such curves

duality relationships for entropy like minimization problems this paper considers the minimization of a convex integral functional over the positive cone of an l p space subject to a finite number of linear equality constraints such problems arise in spectral estimation where the bjective function is often entropy like and in constrained approximation the lagrangian dual problem is finite dimensional and unconstrained under a quasi interior constraint qualification the primal and dual values are equal with dual attainment examples show the primal value may not be attained conditions are given that ensure that the primal optimal solution can be calculated directly from a dual optimum these conditions are satisfied in many examples

on the moment problem of closed semi algebraic sets the moment problem for compact semi algebraic sets has been solved in s1 see pd section 6 4 for a refinement of the result in the terminology explained below this means that each defining sequence f of a compact semialgebraic set kf has property smp on the other hand for many noncompact semi algebraic sets for instance sets containing a cone of dimension two km ps the moment problem is not solvable only very few noncompact semi algebraic sets classes of real algebraic curves so km ps and cylinder sets mc are known to have a positive solution of the moment problem in this paper we study semi algebraic setskf such that there exist polynomials h1 hn which are bounded on the set kf our main result theorem 1 reduces the moment problem for the set kf to the moment problem for the fiber sets kf c where c is real algebraic variety x r h1 x 1 hn x n from this theorem new classes of non compact closed semi algebraic sets are obtained for which the moment problem has an affirmative solution combined with a result of v powers and c scheiderer ps it follows that tube sets around certain real algebraic curves have property smp see theorem 9 let f f1 fk be a finite set of polynomials fj r x r x1 xd and let k kf be the associated closed semi algebraic subset defined by

realization of power spectra from partial covariance sequences in this paper we address the problem of realization of a spectral density function from incomplete information about the underlying stochastic process the standing assumption is the availability of an incomplete partial sequence of covariance samples of the process we study the set of rational extensions of this finite sequence to an infinite covariance function that agrees with the available samples the classical theory of orthogonal polynomials with respect to the unit circle and the theory of moments have been utilized extensively in a variety of engineering problems including the one we are dealing with these have been known to provide a unifying framework for a variety of current spectral estimation techniques maximum entropy method pisarenko s harmonic decomposition etc in this work we consider and study the set of all covariance realizations of dimension lower than or equal to the length of the partial sequence and equal to the dimension of the maximum entropy realization the me solution is a point in this set other points correspond to pole zero models a general formula is obtained for recursively updated pole zero models of dimension increasing with the data record information about the zeros is obtained from the asymptotic behavior of the partial autocorrelation coefficients our approach combines techniques from the theory of orthogonal polynomials on the unit circle the theory of moments and also techniques from degree theory topology our objective is to develop a theory that will provide a frame for constructing recursively pole zero realizations of increasing dimension

solution of the general moment problem via a one parameter imbedding this paper presents a computational theory for the general scalar moment problem the formalism is sufficiently general to encompass problems in sensor arrays with arbitrary geometry and dynamics and in nonuniform multidimensional sampling given a finite set of moments the theory provides a test for the existence of a positive measure which is consistent with such data at the same time the theory also provides a characterization of all such consistent positive measures it should be noted that classical results e g in the theory of the trigonometric moment problem hamburger stieljes nevanlinna pick interpolation etc are not applicable to the general setting sought herein where there is no natural shift operator in the space spanned by the integration kernels the centerpiece of the theory is a differential equation which depends on the given finite set of moments and on an arbitrary positive function spl psi which plays the role of a free parameter the differential equation has an exponentially attractive point of equilibrium if and only if there exists a consistent positive measure for each spl psi the fixed point determines a corresponding measure suitable choice of spl psi allows recovering any measure which is consistent with the data the fixed point of the differential equation corresponds to an extremum of an entropy like functional and the differential equation is constructed via an appropriate homotopy that follows changes in the lagrange multipliers from a convenient starting value to a value for the multipliers that corresponds to the given moments

a robust solver using a continuation method for nevanlinna pick interpolation with degree constraint this note modifies a previous algorithm for solving a certain convex optimization problem introduced by byrnes georgiou and lindquist to determine any nevanlinna pick interpolant satisfying degree constraint the modified algorithm is based on a continuation method with predictor corrector steps and it turns out to be quite efficient and numerically robust

metastability and dynamics of discrete topological singularities in two dimensions a γ convergence approach this paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions based on convergence we consider discrete systems described by scalar functions defined on a square lattice and governed by periodic interaction potentials our main motivation comes from xy spin systems described by the phase parameter and screw dislocations described by the displacement function for these systems we introduce a discrete notion of vorticity as the lattice spacing tends to zero we derive the first order limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices as a byproduct of this analysis we show that such systems exhibit increasingly many metastable configurations of singularities therefore we propose a variational approach to the depinning and dynamics of discrete vortices based on minimizing movements we show that letting first the lattice spacing and then the time step of the minimizing movements tend to zero the vortices move according with the gradient flow of the renormalized energy as in the continuous ginzburg landau framework

the complex moment problem and subnormality a polar decomposition approach abstract it has been known that positive definiteness does not guarantee a bisequence to be a complex moment however it turns out that positive definite extendibility does theorems 1 and 22 and this is the main theme of this paper the main tool is generally understood polar decomposition to strengthen applicability of our approach we work out a criterion for positive definite extendibility in a fairly wide context theorems 9 and 29 all this enables us to prove characterizations of subnormality of unbounded operators having invariant domain theorems 37 and 39 and their further applications theorems 41 and 43 and a description of the complex moment problem on real algebraic curves theorems 52 and 56 the latter question is completed in the appendix in which we relate the complex moment problem to the two dimensional real one with emphasis on real algebraic sets

multidimensional mem spectral estimation the problem of multidimensional maximum entropy method mem spectral estimation from nonuniformly spaced correlation measurements is investigated a necessary and sufficient condition is derived for the existence and uniqueness of the mem spectral estimate in its usual form it is shown that this condition is not satisfied in many multidimensional problems of interest although it is satisfied in the important practical case of spectral supports composed of a finite number of points when the existence condition is satisfied calculation of the mem estimate reduces to the solution of a finite dimensional convex optimization problem the application of standard optimization techniques to this problem results in iterative computational algorithms which are guaranteed to converge the algorithms so obtained are compared to those previously proposed and a spectral estimation example is presented