semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

generalised absolute stability and sum of squares this paper introduces a general framework for analysing nonlinear systems using absolute stability theory and sum of squares programming the technique decomposes a vector field into a system with a polynomial vector field in feedback with a nonlinear memoryless term which is contained in a generalised sector inequality with polynomial bounds this decomposition can be used to model uncertainty in the nonlinearity or to bound difficult to analyse terms by simpler polynomial functions such as time varying non polynomial or higher order nonlinearities conditions for stability and regions of attraction are found using polynomial and lur e type lyapunov functions which generalise those used for the derivation of the multivariable circle and popov criteria in classical absolute stability the technique extends both absolute stability theory and the applicability of sum of squares programming the usefulness of the technique is demonstrated with illustrative examples

primitive cohomology of hopf algebras abstract primitive cohomology of a hopf algebra is defined by using a modification of the cobar construction of the underlying coalgebra among many of its applications two classifications are presented firstly we classify all non locally pi pointed hopf algebra domains of gelfand kirillov dimension two and secondly we classify all pointed hopf algebras of rank one the first classification extends some results of brown goodearl and others in an ongoing project to understand all hopf algebras of low gelfand kirillov dimension the second generalizes results of krop radford and wang you chen which classified hopf algebras of rank one under extra hypothesis properties and algebraic structures of the primitive cohomology are discussed

robust region of attraction estimation we propose a method to compute invariant subsets of the region of attraction for asymptotically stable equilibrium points of polynomial dynamical systems with bounded parametric uncertainty parameter independent lyapunov functions are used to characterize invariant subsets of the robust region of attraction a branch and bound type refinement procedure reduces the conservatism we demonstrate the method on an example from the literature and uncertain controlled short period aircraft dynamics

on the estimation of asymptotic stability regions state of the art and new proposals this paper deals with the problem of the estimation of regions of asymptotic stability for continuous autonomous nonlinear systems the first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields in the second part certain topological considerations are first developed and the trajectory reversing method is then presented together with a theorem on which it is based in the final part several examples of application are reported showing the efficiency of the proposed technique for low order second and third systems

brief paper stability and robustness analysis of nonlinear systems via contraction metrics and sos programming a wide variety of stability and performance questions about linear dynamical systems can be reformulated as convex optimization problems involving linear matrix inequalities lmis these techniques have been recently extended to nonlinear systems with polynomial or rational dynamics through the use of sum of squares sos programming in this paper we further extend the class of systems that can be analyzed with convexity based methods we show how to analyze the robust stability properties of uncertain nonlinear systems with polynomial or rational dynamics via contraction analysis and sos programming since the existence of a global contraction metric is a sufficient condition for global stability of an autonomous system we develop an algorithm for finding such contraction metrics using sos programming the search process is made computationally tractable by relaxing matrix definiteness constraints the feasibility of which indicates the existence of a contraction metric to sos constraints on polynomial matrices we illustrate our results through examples from the literature and show how our contraction based approach offers advantages when compared with traditional lyapunov analysis

a generalization of zubov s method to perturbed systems a generalization of zubov s theorem on representing the domain of attraction via the solution of a suitable partial differential equation is presented for the case of perturbed systems with a singular fixed point for the construction it is necessary to consider solutions in the viscosity sense as a consequence maximal robust lyapunov functions can be characterized as viscosity solutions

on the performance of nonlinear dynamical systems under parameter perturbation we present a method for analysing the deviation in transient behaviour between two parameterised families of nonlinear odes as initial conditions and parameters are varied within compact sets over which stability is guaranteed this deviation is taken to be the integral over time of a user specified positive definite function of the difference between the trajectories for instance the l 2 norm we use sum of squares programming to obtain two polynomials which take as inputs the possibly differing initial conditions and parameters of the two families of odes and output upper and lower bounds to this transient deviation equality can be achieved using symbolic methods in a special case involving linear time invariant parameter dependent systems we demonstrate the utility of the proposed methods in the problems of model discrimination and location of worst case parameter perturbation for a single parameterised family of ode models

estimating the domain of attraction for uncertain polynomial systems estimating the domain of attraction da of equilibrium points is a problem of fundamental importance in systems engineering several approaches have been proposed for the case of known polynomial systems allowing one to find the largest estimate of the da leda for a given lyapunov function lf however the problem of estimating the robust da rda that is the da guaranteed for all possible uncertainties in an uncertain system it is still an unsolved problem in this paper lmi methods are proposed for estimating the rda in the case of systems depending polynomially in the state and in the uncertainty which is supposed to belong to a polytope specifically the issue of computing the robust leda rleda that is the intersection of all ledas is considered for common and parameter dependent lfs providing constant and parameter dependent lower bounds the computation of approximations with simple shape of the rleda in the case of parameter dependent lfs is also discussed

classification of connected hopf algebras of dimension p 3 i abstract let p be a prime and k be an algebraically closed field of characteristic p in this paper we provide the classification of connected hopf algebras of dimension p 3 except for the case when the primitive space of the hopf algebra is a two dimensional abelian restricted lie algebra each isomorphism class is presented by generators x y z with relations and hopf algebra structures let be the multiplicative group of p 2 p 1 th roots of unity when the primitive space is one dimensional and p is odd there is an infinite family of isomorphism classes which is naturally parameterized by a k 1

stability regions of nonlinear dynamical systems a constructive methodology a constructive methodology for estimating stability regions of general nonlinear dynamical systems is developed the constructive methodology starts with a given lyapunov function either a global or a local lyapunov function and yields a sequence of lyapunov functions which are then used to estimate the stability region the resulting sequence of estimated stability regions is shown to be a strictly monotonic increasing sequence and yet each of them remains inside the entire stability region the significance of this methodology includes its ability to reduce significantly the conservativeness in estimating the stability regions its computational efficiency its adaptability and its sound theoretical basis

paper maximal lyapunov functions and domains of attraction for autonomous nonlinear systems in this paper we present various theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system these methods are based on the concept of a maximal lyapunov function which is introduced in this paper a partial differential equation characterizing a maximal lyapunov function is derived and the relationships of this equation as compared to zubov s partial differential equation are discussed an iterative procedure is given for solving the new partial differential equation this procedure yields lyapunov function candidates that are rational functions rather than polynomials the method is applied to four two dimensional examples and one three dimensional example and it is shown that the estimates obtained using this method are in many cases substantially better than those obtained using known methods

hochschild cohomology and the coradical filtration of pointed coalgebras applications abstract in this paper we show that there is a close connection between the coradical filtration of a pointed coalgebra and the hochschild cohomology of that coalgebra with coefficients in some one dimensional bicomodules as an application for a given prime numberpand an algebraically closed fieldkof characteristic 0 we classify all pointed hopf algebras of dimensionp3overk

pure subgroups of torsion free groups in this paper we show that certain new notions of purity stronger than the classical concept are relevant to the study of torsion free abelian groups in particular implications of purity a concept introduced in one of our recent papers are investigated we settle an open question posed by nongxa by proving that the union of an ascending countable sequence of pure subgroups is completely decomposable provided the subgroups are this result is false for ordinary purity the principal result of the paper however deals with e purity a concept stronger than purity but weaker than the usual notion of strong purity our main theorem which has a number of corollaries including the recent result of nongxa that strongly pure subgroups of separable groups are again separable states that a e pure subgroup of a k group is itself a k group among other results is the negative resolution of the conjecture valid in the countable case that a strongly pure subgroup of a completely decomposable group is again completely decomposable

finite quantum groups and cartan matrices abstract we consider a class of hopf algebras having as an invariant a generalized cartan matrix for a hopf algebra of this kind we prove that it is finite dimensional if and only if its generalized cartan matrix is actually a finite cartan matrix under some mild hypothesis these results allow us to classify all the finite dimensional coradically graded pointed hopf algebras whose coradical has odd prime dimension p we also characterize coradically graded pointed hopf algebras of order p 4

a decomposition technique for nonlinear dynamical system analysis a method for analyzing large scale nonlinear dynamical systems by decomposing them into coupled lower order subsystems that are sufficiently simple for computational analysis is presented it is shown that the decomposition approach can be used to scale the sum of squares programming framework for nonlinear systems analysis the method constructs subsystem lyapunov functions which are used to form a composite lyapunov function for the whole system further computational savings are achieved if a method based on sparsity maximization is used to obtain the subsystem lyapunov functions

connected hopf algebras of dimension p2 abstract let h be a finite dimensional connected hopf algebra over an algebraically closed field k of characteristic p 0 we provide the algebra structure of the associated graded hopf algebra gr h then we study the case when h is generated by a hopf subalgebra k and another element and the case when h is cocommutative when h is a restricted universal enveloping algebra we give a specific basis for the second term of the hochschild cohomology of the coalgebra h with coefficients in the trivial h bicomodule k finally we classify all connected hopf algebras of dimension p 2 over k

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

brief paper local stability analysis using simulations and sum of squares programming the problem of computing bounds on the region of attraction for systems with polynomial vector fields is considered invariant subsets of the region of attraction are characterized as sublevel sets of lyapunov functions finite dimensional polynomial parametrizations for lyapunov functions are used a methodology utilizing information from simulations to generate lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed the suitability of lyapunov function candidates is assessed solving linear sum of squares optimization problems qualified candidates are used to compute invariant subsets of the region of attraction and to initialize various bilinear search strategies for further optimization we illustrate the method on small examples from the literature and several control oriented systems

region of attraction analysis with integral quadratic constraints abstract a general framework is presented to estimate the region of attraction of attracting equilibrium points the system is described by a feedback connection of a nonlinear polynomial system and a bounded operator the input output behavior of the operator is characterized using an integral quadratic constraint this allows to analyze generic problems including for example hard nonlinearities and different classes of uncertainties adding to the state of practice in the field which is typically limited to polynomial vector fields the iqc description is also nonrestrictive with the main result given for both hard and soft factorizations optimization algorithms based on sum of squares techniques are then proposed with the aim to enlarge the inner estimates of the roa numerical examples are provided to show the applicability of the approaches these include a saturated plant where bounds on the states are exploited to refine the sector description and a case study with parametric uncertainties for which the conservativeness of the results is reduced by using soft iqcs

local stability analysis for uncertain nonlinear systems we propose a method to compute provably invariant subsets of the region of attraction for the asymptotically stable equilibrium points of uncertain nonlinear dynamical systems we consider polynomial dynamics with perturbations that either obey local polynomial bounds or are described by uncertain parameters multiplying polynomial terms in the vector field this uncertainty description is motivated by both incapabilities in modeling as well as bilinearity and dimension of the sum of squares programming problems whose solutions provide invariant subsets of the region of attraction we demonstrate the method on three examples from the literature and a controlled short period aircraft dynamics example

stability region analysis using polynomial and composite polynomial lyapunov functions and sum of squares programming we propose using bilinear sum of squares programming for obtaining inner bounds of roas for dynamical systems with polynomial vector fields we search for polynomial as well as composite lyapunov functions comprised of pointwise maximums of polynomial functions results for several examples from the literature are presented using the proposed methods and the penbmi solver

isomorphism classes of finite dimensional connected hopf algebras in positive characteristic abstract we classify all finite dimensional connected hopf algebras with large abelian primitive spaces we show that they are hopf algebra extensions of restricted enveloping algebras of certain restricted lie algebras for any abelian matched pair associated with these extensions we construct a cohomology group which classifies all the extensions up to equivalence moreover we present a 1 1 correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points where the group respects the automorphisms of the abelian matched pair and the exceptional points represent those restricted lie algebra extensions

region of attraction estimation using invariant sets and rational lyapunov functions this work addresses the problem of estimating the region of attraction ra of equilibrium points of nonlinear dynamical systems the estimates we provide are given by positively invariant sets which are not necessarily defined by level sets of a lyapunov function moreover we present conditions for the existence of lyapunov functions linked to the positively invariant set formulation we propose connections to fundamental results on estimates of the ra are presented and support the search of lyapunov functions of a rational nature we then restrict our attention to systems governed by polynomial vector fields and provide an algorithm that is guaranteed to enlarge the estimate of the ra at each iteration

analysis of autocatalytic networks in biology autocatalytic networks in particular the glycolytic pathway constitute an important part of the cell metabolism changes in the concentration of metabolites and catalyzing enzymes during the lifetime of the cell can lead to perturbations from its nominal operating condition we investigate the effects of such perturbations on stability properties e g the extent of regions of attraction of a particular family of autocatalytic network models numerical experiments demonstrate that systems that are robust with respect to perturbations in the parameter space have an easily verifiable in terms of proof complexity region of attraction properties motivated by the computational complexity of optimization based formulations we take a compositional approach and exploit a natural decomposition of the system induced by the underlying biological structure into a feedback interconnection of two input output subsystems a small subsystem with complicating nonlinearities and a large subsystem with simple dynamics this decomposition simplifies the analysis of large pathways by assembling region of attraction certificates based on the input output properties of the subsystems it enables numerical as well as analytical construction of block diagonal lyapunov functions for a large family of autocatalytic pathways