periodicity in a delayed ratio dependent predator prey system abstract with the help of a continuation theorem based on gaines and mawhin s coincidence degree easily verifiable criteria are established for the global existence of positive periodic solutions of a delayed ratio dependent predator prey system in a periodic environment

matrix sum of squares relaxations for robust semi definite programs we consider robust semi definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description on the basis of matrix sum of squares decompositions we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart with a novel matrix version of putinar s sum of squares representation for positive polynomials on compact semi algebraic sets we prove asymptotic exactness of the relaxation family under a suitable constraint qualification if the uncertainty region is a compact polytope we provide a new duality proof for the validity of putinar s constraint qualification with an a priori degree bound on the polynomial certificates finally we point out the consequences of our results for constructing relaxations based on the so called full block s procedure which allows to apply recently developed tests in order to computationally verify the exactness of possibly small sized relaxations

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

a type d structure in khovanov homology abstract we describe the first part of a gluing theory for the bigraded khovanov homology with z coefficients this part associates a type d structure to a tangle properly embedded in a half space and proves that the homotopy class of the type d structure is an invariant of the isotopy class of the tangle the construction is modeled off bordered heegaard floer homology but uses only combinatorial diagrammatic methods

global optimization of rational functions a semidefinite programming approach we consider the problem of global minimization of rational functions on unconstrained case and on an open connected semi algebraic subset of or the partial closure of such a set constrained case we show that in the univariate case n 1 these problems have exact reformulations as semidefinite programming sdp problems by using reformulations introduced in the phd thesis of jibetean 16 this extends the analogous results by nesterov 13 for global minimization of univariate polynomials for the bivariate case n 2 we obtain a fully polynomial time approximation scheme fptas for the unconstrained problem if an a priori lower bound on the infimum is known by using results by de klerk and pasechnik 1 for the np hard multivariate case we discuss semidefinite programming based relaxations for obtaining lower bounds on the infimum by using results by parrilo 15 and lasserre 12

iterated rings of bounded elements and generalizations of schmudgen s positivstellensatz let a be a commutative r algebra of finite transcendence degree d 2 n we investigate the relationship between the subring of geometrically bounded elements

global optimization of polynomials using gradient tentacles and sums of squares we consider the problem of computing the global infimum of a real polynomial f on mathbb r n every global minimizer of f lies on its gradient variety i e the algebraic subset of mathbb r n where the gradient of f vanishes if f attains a minimum on mathbb r n it is therefore equivalent to look for the greatest lower bound of f on its gradient variety nie demmel and sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds based on these certificates they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite programs and thus be solved efficiently we deal here with the more general case when f is bounded from below but does not necessarily attain a minimum in this case the method of nie demmel and sturmfels might yield completely wrong results in order to overcome this problem we replace the gradient variety by larger semialgebraic subsets of mathbb r n which we call gradient tentacles it now gets substantially harder to prove the existence of the necessary sums of squares certificates

non negative hereditary polynomials in a free algebra we prove a non negative stellensatz and a null stellensatz for a class of polynomials called hereditary polynomials in a free algebra

the l problem of moments in two dimensions abstract the classical solution to the l problem of moments due to n i akhiezer and m g krein is partially generalized to the two dimensional case a parallel between the apparently quite different one dimensional and two dimensional l problems of moments is established by means of the theories of the phase shift and of the principal function this point of view brings into the field the geometry of hilbert spaces and the analysis of the pairs of not necessarily commuting self adjoint operators acting on them

probleme des moments sur un compact de rn et decomposition de polynomes a plusieurs variables on donne la solution du probleme des moments pour un compact d interieur non vide quelconque de r n on utilise cette resolution pour obtenir un resultat de decomposition des polynomes strictement positifs sur un compact d interieur non vide

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

a general representation theorem for partially ordered commutative rings an extension of the kadison dubois representation theorem is proved this extends both the classical version 3 and the preordering version given by jacobi in 5 it is then shown how this can be used to sharpen the results on representations of strictly positive polynomials given by jacobi and prestel in 6

multiplicity of positive periodic solutions to a generalized delayed predator prey system with stocking abstract in this paper by using the continuation theorem of coincidence degree theory the existence of multiple positive periodic solutions for a generalized delayed predator prey system with stocking is established when our result is applied to a delayed predator prey system with nonmonotonic functional response and stocking we establish the sufficient condition for the existence of multiple positive periodic solutions for the system

optimization of polynomials on compact semialgebraic sets a basic closed semialgebraic subset s of r n is defined by simultaneous polynomial inequalities g 1 ge 0 dotsc g m ge 0 we give a short introduction to lasserre s method for minimizing a polynomial f on a compact set s of this kind it consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs we give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ast of f on s which is constructive and elementary in the case where f possesses a unique minimizer x ast we prove that every sequence of nearly optimal solutions of the successive relaxations gives rise to a sequence of points in r n converging to x ast

error bounds for some semidefinite programming approaches to polynomial minimization on the hypercube we consider the problem of minimizing a polynomial on the hypercube 0 1 n and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the positivstellensatz of schmudgen math ann 289 1991 pp 203 206 the main tool we employ is bernstein approximations of polynomials which also gives constructive proofs and degree bounds for positivity certificates on the hypercube

moment problems for multi sequences of operators abstract we give an explicit solution to the scalar moment problem on semi algebraic compact subsets of r n and apply this result to the study of some operator multi sequences

a real nullstellensatz for free modules abstract let a be the algebra of all n n matrices with entries from r x 1 x d and let g 1 g m f a we will show that f a v 0 for every a r d and v r n such that g i a v 0 for all i if and only if f belongs to the smallest real left ideal of a which contains g 1 g m here a left ideal j of a is real if for every h 1 h k a such that h 1 t h 1 h k t h k j j t we have that h 1 h k j we call this result the one sided real nullstellensatz for matrix polynomials we first prove by induction on n that it holds when g 1 g m f have zeros everywhere except in the first row this auxiliary result can be formulated as a real nullstellensatz for the free module r x 1 x d n

convexity in semialgebraic geometry and polynomial optimization we review several and provide new results on the theory of moments sums of squares and basic semialgebraic sets when convexity is present in particular we show that under convexity the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence a highly desirable feature as convex problems are in principle easier to solve in addition if a basic semialgebraic set mathbf k is convex but its defining polynomials are not we provide two algebraic certificates of convexity which can be checked numerically the second is simpler and holds if a sufficient and almost necessary condition is satisfied it also provides a new condition for mathbf k to have semidefinite representation for this we use and extend some of the recent results from the author and helton and nie math program to appear finally we show that when restricting to a certain class of convex polynomials the celebrated jensen s inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures

moment problems for operator polynomials abstract haviland s theorem states that given a closed subset k in r n each functional l r x r positive on pos k p r x p k 0 admits an integral representation by a positive borel measure schmudgen proved that in the case of compact semialgebraic set k it suffices to check positivity of l on a preordering t having k as the non negativity set further he showed that the compactness of k is equivalent to the archimedianity of t the aim of this paper is to extend these results from functionals on the usual real polynomials to operators mapping from the real matrix or operator polynomials into r m n r or b k

positive periodic solutions for the neutral ratio dependent predator prey model by using a continuation theorem based on coincidence degree theory we obtain some new sufficient conditions for the existence of positive periodic solutions for the neutral ratio dependent predator prey model with holling type ii functional response

conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone we investigate the completely positive semidefinite cone mathcal cs n a new matrix cone consisting of all n times n matrices that admit a gram representation by positive semidefinite matrices of any size in particular we study relationships between this cone and the completely positive and the doubly nonnegative cone and between its dual cone and trace positive noncommutative polynomials we use this new cone to model quantum analogues of the classical independence and chromatic graph parameters alpha g and chi g which are roughly obtained by allowing variables to be positive semidefinite matrices instead of 0 1 scalars in the programs defining the classical parameters we can formulate these quantum parameters as conic linear programs over the cone mathcal cs n using this conic approach we can recover the bounds in terms of the theta number and define further approximations by exploiting the link to trace positive polynomials

the strict positivstellensatz for global analytic functions and the moment problem for semianalytic sets analytic functions strictly positive on a global semianalytic set x f1 0 fk 0 in rn are characterized as functions expressible as g a0 a1f1 akfk for strictly positive global analytic functions a0 ak the proof is elementary using the fact that the analytic functions are dense in c rn r in the whitney topology the same proof works for nash functions this is an improvement of the standard analytic version of stengle s positivstellensatz in two directions the hypothesis is weaker there is no requirement that x be compact and the conclusion is stronger several applications are given including i a new proof of the weak positivstellensatz for semianalytic sets and ii the solution of thek moment problem for basic closed semianalytic

polyhedra related to integer convex polynomial systems this paper deals with the reformulation of a polynomial integer program for deducing a linear integer relaxation of such a program a class of polyhedra that are associated with nonlinear functions is introduced a characterization of the family of polynomials for which our approach leads to an equivalent linear integer program is given finally the family of so called integer convex polynomials is defined and polyhedra related to such a polynomial are investigated

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

finsler s lemma for matrix polynomials abstract finsler s lemma characterizes all pairs of symmetric n n real matrices a and b which satisfy the property that v t a v 0 for every nonzero v r n such that v t b v 0 we extend this characterization to all symmetric matrices of real multivariate polynomials but we need an additional assumption that b is negative semidefinite outside some ball we also give two applications of this result to noncommutative real algebraic geometry which for n 1 reduce to the usual characterizations of positive polynomials on varieties and on compact sets

positivstellensatze for real function algebras we look for algebraic certificates of positivity for functions which are not necessarily polynomial functions similar questions were examined earlier by lasserre and putinar positivity and optimization for semi algebraic functions to appear proposition 1 and by putinar a striktpositivestellensatz for measurable functions corrected version to appear theorem 2 1 we explain how these results can be understood as results on hidden positivity the required positivity of the functions implies their positivity when considered as polynomials on the real variety of the respective algebra of functions this variety is however not directly visible in general we show how algebras and quadratic modules with this hidden positivity property can be constructed we can then use known results for example jacobi s representation theorem jacobi in math z 237 259 273 2001 theorem 4 or the krivine stengle positivstellensatz marshall in positive polynomials and sums of squares mathematical surveys and monographs 146 2008 page 25 to obtain certificates of positivity relative to a quadratic module of an algebra of real valued functions our results go beyond the results of lasserre and putinar for example when dealing with non continuous functions the conditions are also easier to check we explain the application of our result to various sorts of real finitely generated algebras of semialgebraic functions the emphasis is on the case where the quadratic module is also finitely generated our results also have application to optimization of real valued functions using the semidefinite programming relaxation methods pioneered by lasserre siam j optim 11 3 796 817 2001 lasserre in moments positive polynomials and their applications imperial college press london 2009 lasserre and putinar in positivity and optimization for semi algebraic functions to appear marshall in positive polynomials and sums of squares mathematical surveys and monographs 146 2008 page 25

closure of the cone of sums of 2d powers in commutative real topological algebras abstract let r be a unitary commutative r algebra and k x r hom r r closed with respect to the product topology we consider r endowed with the topology t k induced by the family of seminorms a a for k and a r in case k is compact we also consider the topology induced by a k sup k a for a r if k is zariski dense then those topologies are hausdorff in this paper we prove that the closure of the cone of sums of 2d powers r 2 d with respect to those two topologies is equal to psd k a r a 0 for all k in particular any continuous linear functional l on the polynomial ring r r x r x 1 x n with l h 2 d 0 for each h r x is integration with respect to a positive borel measure supported on k finally we give necessary and sufficient conditions to ensure the continuity of a linear functional with respect to those two topologies

the k moment problem with densities given a compact basic semi algebraic set mathbf k subset mathbb r n a rational fraction f mathbb r n to mathbb r and a sequence of scalars y y we investigate when y alpha int mathbf k x alpha f d mu holds for all alpha in mathbb n n i e when y is the moment sequence of some measure fd supported on k this yields a set of convex linear matrix inequalities lmi we also use semidefinite programming to develop a converging approximation scheme to evaluate the integral fd when the moments of are known and f is a given rational fraction numerical expreriments are also provided we finally provide again lmi conditions on the moments of two measures nu mu with support contained in k to have d nu f d mu for some rational fraction f

archimedean operator theoretic positivstellensatze abstract we prove a general archimedean positivstellensatz for hermitian operator valued polynomials and show that it implies the multivariate fejer riesz theorem of dritschel rovnyak and positivstellensatze of ambrozie vasilescu and scherer hol we also obtain several generalizations of these and related results the proof of the main result depends on an extension of the abstract archimedean positivstellensatz for algebras that is interesting in its own right

sums of squares on reducible real curves we ask whether every polynomial function that is non negative on a real algebraic curve can be expressed as a sum of squares in the coordinate ring scheiderer has classified all irreducible curves for which this is the case for reducible curves we show how the answer depends on the configuration of the irreducible components and give complete necessary and sufficient conditions we also prove partial results in the more general case of finitely generated preorderings and discuss applications to the moment problem for semialgebraic sets

extending positive definiteness the main result of this paper gives criteria for extendibility of mappings defined on symmetric subsets of semigroups to positive definite ones by specifying the mappings in question we obtain new solutions of relevant issues in harmonic analysis concerning truncations of some important multivariate moment problems like complex two sided complex and multidimensional trigonometric moment problems in addition unbounded subnormality and existence of unitary power dilation of several contractions is treated as an application of our general scheme

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

positive periodic solutions in delayed gause type predator prey systems abstract by using a continuation theorem based on coincidence degree theory we establish easily verifiable criteria for the existence of positive periodic solutions in delayed gause type predator prey systems some known results are shown to be special cases of the presented paper

a categorification of the jones polynomial author s khovanov mikhail abstract we construct a bigraded cohomology theory of links whose euler characteristic is the jones polynomial

periodic solutions of a neutral impulsive predator prey model with beddington deangelis functional response with delays in this paper we consider a neutral predator prey model with the beddington deangelis functional response and impulsive effect sufficient conditions are obtained for the existence of positive periodic solutions by a systematic qualitative analysis some known results in the literature are generalized

distinguished representations of non negative polynomials abstract let g 1 g r r x 1 x n such that the set k g 1 0 g r 0 in r n is compact we study the problem of representing polynomials f with f k 0 in the form f s 0 s 1 g 1 s r g r with sums of squares s i with particular emphasis on the case where f has zeros in k assuming that the quadratic module of all such sums is archimedean we establish a local global condition for f to have such a representation vis a vis the zero set of f in k this criterion is most useful when f has only finitely many zeros in k we present a number of concrete situations where this result can be applied as another application we solve an open problem from s kuhlmann et al positivity sums of squares and the multi dimensional moment problem ii adv geometry in press on one dimensional quadratic modules

how fast do polynomials grow on semialgebraic sets abstract we study the growth of polynomials on semialgebraic sets for this purpose we associate a graded algebra with the set and address all kinds of questions about finite generation we show that for a certain class of sets the algebra is finitely generated this implies that the total degree of a polynomial determines its growth on the set at least modulo bounded polynomials we however also provide several counterexamples where there is no connection between total degree and growth in the plane we give a complete answer to our questions for certain simple sets and we provide a systematic construction for examples and counterexamples some of our counterexamples are of particular interest for the study of moment problems since none of the existing methods seems to be able to decide the problem there we finally also provide new three dimensional sets for which the algebra of bounded polynomials is not finitely generated

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

pade approximation for a multivariate markov transform methods of pade approximation are used to analyse a multivariate markov transform which has been recently introduced by the authors the first main result is a characterization of the rationality of the markov transform via hankel determinants the second main result is a cubature formula for a special class of measures

existence of multiple positive periodic solutions of delayed predator prey models with functional responses in this paper by applying the continuation theorem of coincidence degree theory we establish some new criteria for the existence of multiple positive periodic solutions for the delayed predator prey model x t x t r t a t x t b t f x t y t y t y t c t f x email protected d t when functional response function f is monotonic or nonmonotonic

a conic programming approach to generalized tchebycheff inequalities consider the problem of finding optimal bounds on the expected value of piecewise polynomials over all measures with a given set of moments this is a special class ofgeneralized tchebycheff inequalities in probability theory we study this problem within the framework of conic programming relying on a general approximation scheme for conic programming we show that these bounds can be numerically computed or approximated via semidefinite programming we also describe some applications of this class of generalized tchebycheff inequalities in probability finance and inventory theory

permanence and stability in non autonomous predator prey lotka volterra systems with feedback controls the main purpose of this article is considering whether or not the feedback controls have an influence on a non autonomous predator prey lotka volterra type system general criteria on permanence are established which is described by an integral form and independent of some feedback controls by constructing suitable lyapunov functionals a set of easily verifiable sufficient conditions are derived for the global stability of any positive solution to the model

a representation theorem for certain partially ordered commutative rings let a be a commutative ring with 1 let p subset a be a preordering of higher level i e 0 1 in p 1 not in p p p subset p p cdot p subset p and a 2n subset p for some n in n and let m subset a be an archimedean p module i e 1 in m 1 not in m m m subset m p cdot m subset m and forall a in a exists n in n n a in m we endow x m varphi in rm hom a r mid varphi m subset r with the weak topology with respect to all mappings widehat a x m rightarrow r widehat a varphi varphi a and consider the representation phi m a rightarrow cal c x m r a mapsto widehat a we find that x m is a non empty compact hausdorff space further we prove that

real radical initial ideals abstract we explore the consequences of an ideal i r x 1 x n having a real radical initial ideal both for the geometry of the real variety of i and as an application to sums of squares representations of polynomials we show that if in w i is real radical for a vector w in the tropical variety then w is in the logarithmic set of the real variety we also give algebraic sufficient conditions for w to be in the logarithmic limit set of a more general semialgebraic set if in addition w r 0 n then the corresponding quadratic module is stable in particular if in w i is real radical for some w r 0 n then r x 1 x n 2 i is stable this provides a method for checking the conditions for stability given by powers and scheiderer 2001 ps

positivity and optimization for semi algebraic functions we describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of borel measurable functions with particular emphasis on algebras generated by semi algebraic functions in this case the standard global optimization problem with constraints given by elements of the same algebra is reduced via a natural change of variables to the better understood case of polynomial optimization a collection of simple examples and numerical experiments complement the theoretical parts of the article

khovanov homology and the symmetry group of a knot we introduce an invariant of tangles in khovanov homology by considering a natural inverse system of khovanov homology groups as application we derive an invariant of strongly invertible knots this invariant takes the form of a graded vector space that vanishes if and only if the strongly invertible knot is trivial while closely tied to khovanov homology and hence the jones polynomial we observe that this new invariant detects non amphicheirality in subtle cases where khovanov homology fails to do so in fact we exhibit examples of knots that are not distinguished by khovanov homology but owing to the presence of a strong inversion may be distinguished using our invariant this work suggests a strengthened relationship between khovanov homology and heegaard floer homology by way of two fold branched covers that we formulate in a series of conjectures

polynomial programming lp relaxations also converge we consider the global minimization of a multivariate polynomial on a compact semialgebraic set using a result of krivine j analyse math 12 1964 pp 307 326 c r acad sci paris 258 1964 pp 3417 3418 it is shown that lp relaxations based on products of the original constraints in the spirit of the reformulation linearization technique procedure of sherali and adams converge to the global optimum this extends to compact semialgebraic sets previous results known for polytopes and 0 1 programs the asymptotic behavior of the lp relaxations is also analyzed and it is shown that in important cases finite convergence cannot occur

positive polynomials and tame preorderings from the geometric point of view it is one of the essential merits of theartinschreier theory of ordered fields to have shed new light on certain problems concerning the behaviour of regular functions on an irreducible variety v over r this eventually led to a purely algebraic characterization of those f r v which are positive or nonnegative on v r reg a first example is provided by e artin s positive solution of hilbert s 17 th problem in the seventies classical artin schreier theory was extended to commutative rings and generalizations of artin s result were proven for the purpose of this introduction we consider only a special situation let us fix polynomials f1 fk r x r x1 xn