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

averaging principle for slow fast stochastic differential equations with time dependent locally lipschitz coefficients abstract this paper is devoted to studying the averaging principle for stochastic differential equations with slow and fast time scales where the drift coefficients satisfy local lipschitz conditions with respect to the slow and fast variables and the coefficients in the slow equation depend on time t and making use of the techniques of time discretization and truncation we prove that the slow component strongly converges to the solution of the corresponding averaged equation

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

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

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

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

averaging principle for one dimensional stochastic burgers equation abstract in this paper we consider the averaging principle for one dimensional stochastic burgers equation with slow and fast time scales under some suitable conditions we show that the slow component strongly converges to the solution of the corresponding averaged equation meanwhile when there is no noise in the slow component equation we also prove that the slow component weakly converges to the solution of the corresponding averaged equation with the order of convergence 1 r for any 0 r 1

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

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

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

the continuous single source location problem with capacity and zone dependent fixed cost models and solution approaches abstract the continuous capacitated single source multi facility weber problem with the presence of facility fixed cost is investigated a new mathematical model which incorporates multi level type capacity or design and facility fixed cost that is capacity based and zone dependent is introduced as no data set exists for this new location problem a new data set based on convex polygons using triangular shape is constructed a generalised two stage heuristic scheme that combines the concept of aggregation an exact method and an enhanced cooper s alternate location allocation method is put forward a framework that embeds variable neighbourhood search is also proposed computational experiments show that these matheuristics produce encouraging results for this class of location problems the proposed approaches are also easily adapted to cater for a recently studied variant namely the single source capacitated multi facility weber problem where they outperform those recently published solution methods

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

analysis of multiscale methods for stochastic differential equations we analyze a class of numerical schemes proposed in 26 for stochastic differential equations with multiple time scales both advective and diffusive time scales are considered weak as well as strong convergence theorems are proven most of our results are optimal they in turn allow us to provide a thorough discussion on the efficiency as well as optimal strategy for the method c 2005 wiley periodicals inc

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

algorithms for the decomposition of a polygon into convex polygons abstract decomposing a non convex polygon into simpler subsets has been a recurrent theme in the literature due to its many applications in this paper we present different algorithms for decomposing a polygon into convex polygons without adding new vertices as well as a procedure which can be applied to any partition to remove the unnecessary edges of a partition by merging the polygons whose union remains a convex polygon although the partitions produced by the algorithms may not have minimal cardinality their easy implementation and their quick cpu times even for polygons with many vertices make them very suitable to be used when optimal decompositions are not required as for instance in constrained optimization problems having as feasible set a non convex polygon optimization problems are usually easier to solve in convex regions and making use of a branch and bound process or other techniques it is not necessary to find the optimal solution in all the subsets so finding convex decomposition with minimal cardinality may be time wasting computational experiments are presented and analyzed

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

decomposing a polygon into simpler components the problem of decomposing a polygon into simpler components is of interest in fields such as computational geometry syntactic pattern recognition and graphics in this paper we consider decompositions which do not introduce steiner points the simpler components we consider are convex polygons spiral polygons star shaped polygons and monotone polygons we apply a technique for improving the efficiency of dynamic programming algorithms in order to achieve polynomial time algorithms for the problems of decomposing a simple polygon into the minimum number of each of the component types using the same technique we are able to exhibit polynomial time algorithms for the problems of decomposing a simple polygon into each of the component types while minimizing the length of the internal edges used to form the decomposition when the polygons are allowed to contain holes many of the problems become np hard

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

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

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

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

averaging principle and systems of singularly perturbed stochastic differential equations by developing certain auxiliary results a modified version of the stochastic averaging principle is developed to investigate dynamical systems consisting of fast and slow phenomena moreover an attempt is made to establish a relationship between the averaging assumption and certain ergodic type properties of the random process determined by an auxiliary system of stochastic differential equations finally an example is given to illustrate the scope of the results

strong averaging principle for two time scale sdes with non lipschitz coefficients abstract this paper deals with averaging principle for two time scale stochastic differential equations sdes with non lipschitz coefficients which extends the existing results from lipschitz to non lipschitz case under suitable conditions the existence of an averaging equation eliminating the fast variable for coupled system is established and as a result the system can be reduced to a single sdes with a modified coefficient which is also non lipschitz moreover it is shown that the slow variable strongly converges to the solution of the corresponding averaging equation

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