delay independent stability analysis of linear time delay systems based on frequency discretization this paper studies strong delay independent stability of linear time invariant systems it is known that delay independent stability of time delay systems is equivalent to some frequency dependent linear matrix inequalities to reduce or eliminate conservatism of stability criteria the frequency domain is discretized into several sub intervals and piecewise constant lyapunov matrices are employed to analyze the frequency dependent stability condition applying the generalized kalman yakubovich popov lemma new necessary and sufficient criteria are then obtained for strong delay independent stability of systems with a single delay the effectiveness of the proposed method is illustrated by a numerical example

cayley bacharach schemes and their canonical modules a set of s points in p d is called a cayley bacharach scheme cb scheme if every subset of s 1 points has the same hilbert function we investigate the consequences of this weak uniformity the main result characterizes cb schemes in terms of the structure of the canonical module of their projective coordinate ring from this we get that the hilbert function of a cb scheme x has to satisfy growth conditions which are only slightly weaker than the ones given by harris and eisenbud for points with the uniform position property we also characterize cb schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points

positive forms and stability of linear time delay systems we consider the problem of constructing lyapunov functions for linear differential equations with delays for such systems it is known that exponential stability implies the existence of a positive lyapunov function which is quadratic on the space of continuous functions we give an explicit parameterization of a sequence of finite dimensional subsets of the cone of positive lyapunov functions using positive semidefinite matrices this allows stability analysis of linear time delay systems to be formulated as a semidefinite program

the h vectors of arithmetically gorenstein sets of points on a general sextic surface in p3 abstract we give the possible h vectors of arithmetically gorenstein sets of points on a general sextic surface in p 3 and show the existence of such a set for each possible h vector our methods are centered on the vector bundle techniques developed by several people notably chiantini and faenzi together with extensive use of liaison theory

the cr geometry of weighted extremal kahler and sasaki metrics we establish an equivalence between conformally einstein maxwell kahler 4 manifolds recently studied in apostolov et al j reine angew math 721 109 147 2016 apostolov and maschler j eur math soc 21 1319 1360 2019 futaki and ono j math soc jpn 70 1493 1521 2018 koca et al ann glob anal geom 50 29 46 2016 lebrun einstein maxwell equations extremal kahler metrics and seiberg witten theory in the many facets of geometry a tribute to nigel hitchin oxford university press oxford pp 17 33 2009 lebrun j geom phys 91 171 2015 and lebrun commun math phys 344 621 653 2016 and extremal kahler 4 manifolds in the sense of calabi extremal kahler metrics seminar on differential geometry annals of mathematics studies vol 102 pp 259 290 princeton university press princeton 1982 with nowhere vanishing scalar curvature the corresponding pairs of kahler metrics arise as transversal kahler structures of sasaki metrics compatible with the same cr structure and having commuting sasaki reeb vector fields this correspondence extends to higher dimensions using the notion of a weighted extremal kahler metric apostolov et al in levi kahler reduction of cr structures product of spheres and toric geometry arxiv 1708 05253 apostolov et al in weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 lahdili in j geom anal 29 542 568 2019 lahdili in int math res not arxiv 1710 00235 lahdili in proc lond math soc 119 1065 1114 2019 illuminating and uniting several explicit constructions in kahler and sasaki geometry it also leads to new existence and non existence results for extremal sasaki metrics suggesting a link between the notions of relative weighted k stability of a polarized variety introduced in apostolov et al weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 and lahdili proc lond math soc 119 1065 1114 2019 and relative k stability of the kahler cone corresponding to a sasaki polarization studied in boyer and van coevering math res lett 25 1 19 2018 and collins and szekelyhidi j differ geom 109 81 109 2018

completely positive reformulations for polynomial optimization polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables there is a well established body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of completely positive matrices or its conic dual the cone of copositive matrices as a result of this reformulation approach novel solution schemes for quadratic polynomial optimization problems have been designed by drawing on conic programming tools and the extensively studied cones of completely positive and of copositive matrices in particular this approach has been applied to solve key combinatorial optimization problems along this line of research we consider polynomial optimization problems that are not necessarily quadratic for this purpose we use a natural extension of the cone of completely positive matrices namely the cone of completely positive tensors we provide a general characterization of the class of polynomial optimization problems that can be formulated as a conic program over the cone of completely positive tensors as a consequence of this characterization it follows that recent related results for quadratic problems can be further strengthened and generalized to higher order polynomial optimization problems also we show that the conditions underlying the characterization are conceptually the same regardless of the degree of the polynomials defining the problem to illustrate our results we discuss in further detail special and relevant instances of polynomial optimization problems

analysis of polynomial systems with time delays via the sum of squares decomposition we present a methodology for analyzing robust independent of delay and delay dependent stability of equilibria of systems described by nonlinear delay differential equations by algorithmically constructing appropriate lyapunov krasovskii functionals using the sum of squares decomposition of multivariate polynomials and semidefinite programming we illustrate the methodology using an example from population dynamics

constrained nonlinear polynomial time delay systems a sum of squares approach to estimate the domain of attraction a sum of squares sos approach is proposed to estimate the domain of attraction doa of regulated nonlinear polynomial time delay systems subject to input constraints it will be shown that the joint use of semialgebraic geometric arguments and semidefinite programming methods provides an efficient tool for lyapunov based control system analysis and design for this class of nonlinear delayed systems specifically a two step procedure is presented which consists of deriving first a lyapunov krasovskii functional controller pair complying with the prescribed constraints and computing then an estimate of the doa

a momentum construction for circle invariant kahler metrics examples of kahler metrics of constant scalar curvature are relatively scarce over the past two decades several workers in geometry and physics have used symmetry reduction to construct complete kahler metrics of constant scalar curvature by ode methods one fruitful idea the calabi ansatz is to begin with an hermitian line bundle p l h m g m over a kahler manifold and to search for kahler forms p m dd c f t in some disk subbundle where t is the logarithm of the norm function and f is a function of one variable our main technical result theorem a is the calculation of the scalar curvature for an arbitrary kahler metric g arising from the calabi ansatz this suggests geometric hypotheses which we call a constancy to impose upon the base metric g m and hermitian structure h in order that the scalar curvature of g be specified by solving an ode we show that constancy is necessary and sufficient for the calabi ansatz to work in the following sense under the assumption of constancy the disk bundle admits a one parameter family of complete kahler metrics of constant scalar curvature that restrict to g m on the zero section theorems b and d an analogous result holds for the punctured disk bundle theorem c a simple criterion determines when such a metric is einstein conversely in the absence of constancy the calabi ansatz yields at most one metric of constant scalar curvature in either the disk bundle or the punctured disk bundle theorem e many of the metrics constructed here seem to be new including a complete negative einstein kahler metric on the disk subbundle of a stable vector bundle over a riemann surface of genus at least two and a complete scalar flat kahler metric on c 2

kahler metrics on toric orbifolds a theorem of e lerman and s tolman generalizing a result of t delzant states that compact symplectic toric orbifolds are classified by their moment polytopes together with a positive integer label attached to each of their facets in this paper we use this result and the existence of global actionangle coordinates to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold by means of smooth functions on the corresponding moment polytope this is equivalent to parametrizing all toric kahler metrics and generalizes an analogous result for toric manifolds a simple explicit description of interesting families of extremal kahler metrics arising from recent work of r bryant is given as an application of the approach in this paper the fact that in dimension four these metrics are self dual and conformally einstein is also discussed this gives rise in particular to a one parameter family of self dual einstein metrics connecting the well known eguchi hanson and taub nut metrics

hamiltonian 2 forms in kahler geometry iii extremal metrics and stability this paper concerns the existence and explicit construction of extremal kahler metrics on total spaces of projective bundles which have been studied in many places we present a unified approach motivated by the theory of hamiltonian 2 forms as introduced and studied in previous papers in the series but this paper is largely independent of that theory

extremal kahler metrics on projective bundles over a curve let m p e be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle e over a compact complex curve of genus 2 building on ideas of fujiki 1992 27 we prove that m admits a kahler metric of constant scalar curvature if and only if e is polystable we also address the more general existence problem of extremal kahler metrics on such bundles and prove that the splitting of e as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal kahler metrics in kahler classes sufficiently far from the boundary of the kahler cone the methods used to prove the above results apply to a wider class of manifolds called rigid toric bundles over a semisimple base which are fibrations associated to a principal torus bundle over a product of constant scalar curvature kahler manifolds with fibres isomorphic to a given toric kahler variety we discuss various ramifications of our approach to this class of manifolds

hilbert functions of irreducible arithmetically gorenstein schemes abstract in this paper we compute the hilbert functions of irreducible or smooth and reduced arithmetically gorenstein schemes that are twisted anti canonical divisors on arithmetically cohen macaulay schemes we also prove some folklore results characterizing the hilbert functions of irreducible standard determinantal schemes and we use them to produce a new class of functions that occur as hilbert functions of irreducible or smooth and reduced arithmetically gorenstein schemes in any codimension

reducing the complexity of the sum of squares test for stability of delayed linear systems this technical note considers the problem of reducing the computational complexity associated with the sum of squares approach to stability analysis of time delay systems specifically this technical note considers systems with a large state space but where delays affect only certain parts of the system this yields a coefficient matrix of the delayed state with low rank a common scenario in practice the technical note uses the general framework of coupled differential difference equations with delays in feedback channels this framework includes systems of both the neutral and retarded type the approach is based on recent results which introduced a new lyapunov krasovskii structure which was shown to be necessary and sufficient for stability of this class of systems this technical note shows how exploiting the structure of the new functional can yield dramatic improvements in computational complexity numerical examples are given to illustrate this improvement

central kahler metrics with non constant central curvature the central curvature of a riemannian metric is the determinant of its ricci endomorphism while the scalar curvature is its trace a kahler metric is called central if the gradient of its central curvature is a holornorphic vector field such metrics may be viewed as analogs of the extremal kahler metrics defined by calabi in this work central metrics of non constant central curvature are constructed on various ruled surfaces most notably the first hirzebruch surface this is achieved via the momentum construction of hwang and singer a variant of an ansatz employed by calabi 1979 and by koiso and sakane 1986 non existence real analyticity and positivity properties of central metrics arising in this ansatz are also established

inverse systems of zero dimensional schemes in p n abstract the authors construct the global macaulay inverse system l z for a zero dimensional subscheme z of projective n space p n over an algebraically closed field k from the local inverse systems of the irreducible components of z they show that when z is locally gorenstein a generic element f of degree d apolar to z determines z if d is larger than an invariant z as a consequence of this globalization they show that a natural upper bound for the hilbert function of gorenstein artin quotients of the coordinate ring of z is achieved for large socle degree they also show the uniqueness of generalized additive decompositions of a homogeneous form into powers of linear forms under suitable hypotheses the main tools are elementary but delicate they involve a careful study of how to homogenize a local inverse system and of the behavior of the homogenization under a change of coordinates

regional stabilization of input delayed uncertain nonlinear polynomial systems this paper addresses the problem of local stabilization of nonlinear polynomial control systems subject to time varying input delay and polytopic parameter uncertainty a linear matrix inequality approach based on the lyapunov krasovskii theory is proposed for designing a nonlinear polynomial state feedback controller ensuring the robust local uniform asymptotic stability of the system origin along with an estimate of its region of attraction two convex optimization procedures are presented to compute a stabilizing controller ensuring either a maximized set of admissible initial states for given upper bounds on the delay and its variation rate or a maximized lower bound on the maximum admissible input delay considering a given set of admissible initial states numerical examples demonstrate the potentials of the proposed stabilization approach

the projective geometry of the gale transform abstract the gale transform an involution on sets of points in projective space appears in a multitude of guises and in subjects as diverse as optimization coding theory theta functions and recently in our proof that certain general sets of points fail to satisfy the minimal free resolution conjecture see eisenbud and popescu 1999 invent math 136 419 449 in this paper we reexamine the gale transform in the light of modern algebraic geometry we give a more general definition in the context of finite locally gorenstein subschemes we put in modern form a number of the more remarkable examples discovered in the past and we add new constructions and connections to other areas of algebraic geometry we generalize goppa s theorem in coding theory and we give new applications to castelnuovo theory we also give references to classical and modern sources

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

a delay dependent stability criterion for systems with uncertain time invariant delays this paper provides a new stability criterion for systems with time invariant uncertain delays based on an improved upper bound for the inner product of two vectors a new delay dependent robust stability criterion is derived which is shown by an example less conservative than existing stability criteria

the energy of a kahler class on admissible manifolds on a compact complex manifold of kahler type the energy e of a kahler class is given by the squared l 2 norm of the projection onto the space of holomorphic potentials of the scalar curvature of any kahler metric representing the said class and any one such metric whose scalar curvature has squared l 2 norm equal to e must be an extremal representative of a strongly extremal metric is an extremal metric representing a critical point of e when restricted to the set of kahler classes of fixed positive top cup product we study the existence of strongly extremal metrics and critical points of e on certain admissible manifolds producing a number of nontrivial examples of manifolds that carry this type of metrics and where in many of the cases the class that they represent is one other than the first chern class and some examples of manifolds where these special metrics and classes do not exist we also provide a detailed analysis of the gradient flow of e on admissible ruled surfaces show that this dynamical system can be extended to one beyond the kahler cone and analyze the convergence of solution paths at infinity in terms of conditions on the initial data in particular proving that for any initial data in the kahler cone the corresponding path is defined for all t and converges to a unique critical class of e as time approaches infinity

lyapunov equation for the stability of linear delay systems of retarded and neutral type in this note the delay independent stability of delay systems is studied it is shown that the strong delay independent stability is equivalent to the feasibility of certain linear matrix inequality lmi that is to the existence of a quadratic lyapunov krasovskii functional independent of the nonnegative value of the delay this constitutes the analogue of some well known properties of finite dimensional systems this result is then applied to study delay independent stability of systems with polytopic uncertainties

lmi techniques for optimization over polynomials in control a survey numerous tasks in control systems involve optimization problems over polynomials and unfortunately these problems are in general nonconvex in order to cope with this difficulty linear matrix inequality lmi techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems this survey aims to provide the reader with a significant overview of the lmi techniques that are used in control systems for tackling optimization problems over polynomials describing approaches such as decomposition in sum of squares positivstellensatz theory of moments polya s theorem and matrix dilation moreover it aims to provide a collection of the essential problems in control systems where these lmi techniques are used such as stability and performance investigations in nonlinear systems uncertain systems time delay systems and genetic regulatory networks it is expected that this survey may be a concise useful reference for all readers

stability of vector bundles and extremal metrics on donne des exemples de surfaces reglees m qui n ont pas de champs holomorphes vectoriels non triviaux et qui n admettent pas de metrique de kahler extremale dans une classe de kahler specifique donnee

integral quadratic constraints for delayed nonlinear and parameter varying systems the stability and performance of nonlinear and linear parameter varying lpv time delayed systems are analyzed first the input output behavior of the time delay operator is bounded in the frequency domain by integral quadratic constraints iqcs a simple geometric interpretation is used to derive new iqcs for both constant and varying delays second the performance of nonlinear and lpv delayed systems is bounded using dissipation inequalities that incorporate iqcs the nonlinear or lpv part of the system is treated directly in the analysis and not bounded by iqcs this step makes use of recent results that show under mild technical conditions that an iqc has an equivalent representation as a finite horizon time domain constraint a numerical example with a nonlinear delayed system is provided to demonstrate the effectiveness of the method

computation of an optimal communication schedule in a nonlinear networked control system using sum of squares abstract in this paper we consider the design of a control optimal communication schedule in nonlinear networked control with nonlinear affine plant systems we focus here on periodic communication schedules in control systems with time triggered control communication and assume the existence of predesigned controllers which need to be optimally integrated via the control communication system for this we formulate a quadratic cost function for the nonlinear affine plant the system and the cost are first discretized using an exact mixed discretization approach which allows to lift the problem from the continuous discrete domain into a purely discrete time representation periodicity due to the fixed communication schedule is dealt with the discrete lifting technique to allow for the computation of the cost nonlinearities are approximated in terms of a taylor series which allows the use of the sum of squares framework once a cost function of the networked control system for a given fixed communication schedule is defined it is then possible to compare and choose those sequences that minimize this cost for optimal controller integration

lyapunov stability analysis for nonlinear delay systems abstract sufficient conditions ensuring that a nonlinear system with disturbances having a delay is globally asymptotically stable independent of delay are given the proof carried out relies extensively on a characterization of the stability property in terms of lyapunov function the result is applied to some biological systems and neural networks a stabilizing memoryless controller for a second order system with state delay is also proposed

the dynamics of the energy of a kahler class the energy of a kahler class on a compact complex manifold m j of kahler type is the infimum of the squared l2 norm of the scalar curvature over all kahler metrics representing the class we study general properties of this functional and define its gradient flow over all kahler classes represented by metrics of fixed volume when besides the trivial holomorphic vector field of m j all others have no zeroes we extend it to a flow over all cohomology classes of fixed top cup product we prove that the dynamical system in this space defined by the said flow does not have periodic orbits that its only fixed points are critical classes of a suitably defined extension of the energy function and that along solution curves in the kahler cone the energy is a monotone function if the kahler cone is forward invariant under the flow solutions to the flow equation converge to a critical point of the class energy function we show that this is always the case when the manifold has a signed first chern class we characterize the forward stability of the kahler cone in terms of the value of a suitable time dependent form over irreducible subvarieties of m j we use this result to draw several geometric conclusions including the determination of optimal dimension dependent bounds for the squared l2 norm of the scalar curvature functional