hard lefschetz theorem for valuations complex integral geometry and unitarily invariant valuations we obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets a version of the hard lefschetz theorem using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations it implies new integral geometric formulas for real submanifolds in hermitian spaces generalizing the classical kinematic formulas in euclidean

a semilinear parabolic system arising in the theory of superconductivity arises in the theory of superconductivity of liquids proposed by abrahams and tsuneto in 11 system 1 has also been briefly treated as an example by chueh conley and smaller in 61 throughout this paper we shall assume that d is a bounded domain in r n 1 2 or 3 with smooth boundary x a denotes the laplacian all the results of the paper which hold for n 1 also hold when a is replaced by any second order symmetric elliptic differential expression with positive spectrum because of the very symmetric nature of l it is reasonable to expect solutions of 1 to have nice properties we show in this paper that this is in fact the case in section 2 we give a complete description of steady state solutions of 1 satisfying dirichlet or neumann boundary conditions we show that often all steady states are of the form u u t au for some constant a where u is the solution of a single differential equation it is possible using standard techniques for scalar nonlinear boundary value problems to obtain lots of information about the solutions of this single equation and so about solutions of 1 in 6 1 it is shown that all solutions of 1 are attracted towards the set u v u v 1 in section 3 we show that 1 has a lyapunov function and use this fact and the standard theory of w limit sets to prove that any solution of 1 tends as t 03 towards the family of all steady state solutions of 1 often in the case of scalar equations it can be shown that

a predator prey system with stage structure for predator and nonlocal delay abstract this paper deals with the behavior of solutions to the reaction diffusion system under homogeneous neumann boundary condition which describes a prey predator model with nonlocal delay sufficient conditions for the global stability of each equilibrium are derived by the lyapunov functional and the results show that the introduction of stage structure into predator positively affects the coexistence of prey and predator numerical simulations are performed to illustrate the results

nash wings and real prime divisors let v be an affine variety defined over an arbitrary real closed field r with affine coordinate ring a r v and function field k r v then open semialgebraic subsets of the regular points of v r define what we call oeometric preorders on k the study of affine semialgebraic geometry corresponds to the study of commutative algebra in k in the presence of such a preorder the theorem which motivated this paper is

counterexamples to some conjectures on the number of solutions of nonlinear equations on construit trois contre exemples resolvant par la negative certains problemes d analyse non lineaire

strengthened existence and uniqueness conditions for search directions in semidefinite programming abstract primal dual interior point p d i p methods for semidefinite programming sdp are generally based on solving a system of matrix equations for a newton type search direction for a symmetrization of the optimality conditions these search directions followed the derivation of similar p d i p methods for linear programming lp among these a computationally interesting search direction is the aho direction however in contrast to the lp case existence and uniqueness of the aho search direction is not guaranteed under the standard nondegeneracy assumptions two different sufficient conditions are known that guarantee the existence and uniqueness independent of the specific linear constraints the first is given by shida shindoh kojima and is based on the semidefiniteness of the symmetrization of the product sx at the current iterate the second is a centrality condition given first by monteiro zanjacomo and then improved by monteiro todd in this paper we revisit and strengthen both of the above mentioned sufficient conditions we include characterizations for existence and uniqueness in the matrix equations arising from the linearization of the optimality conditions as well we present new results on the relationship between the kronecker product and the symmetric kronecker product that arise from these matrix equations we conclude with a proof that the existence and uniqueness of the aho direction is a generic property for every sdp problem and extend the results to the general monteiro zhang family of search directions

long time dynamics of an sirs reaction diffusion epidemic model abstract in this paper we first deal with a diffusive sirs epidemic system in spatially heterogeneous and time periodic environment with no flux boundary condition we establish the uniform persistence if the basic reproduction number r 0 1 when the diffusion rates are equal and the system parameters are constant we obtain the global stability of the unique endemic equilibrium if r 0 1 and that of the diseases free equilibrium if r 0 1 we also study the sirs epidemic system with spontaneous infection in addition to disease transmission our results show that spontaneous infection enhances the persistence of disease the uniform bounds of solutions which are derived by a feedback argument play a fundamental role in our analysis

spatiotemporal patterns in a reaction diffusion model with the degn harrison reaction scheme abstract spatial and temporal patterns generated in ecological and chemical systems have become a central object of research in recent decades in this work we are concerned with a reaction diffusion model with the degn harrison reaction scheme which accounts for the qualitative feature of the respiratory process in a klebsiella aerogenes bacterial culture we study the global stability of the constant steady state existence and nonexistence of nonconstant steady states as well as the hopf and steady state bifurcations in particular our results show the existence of turing patterns and inhomogeneous periodic oscillatory patterns while the system parameters are all spatially homogeneous these results also exhibit the critical role of the system parameters in leading to the formation of spatiotemporal patterns

elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation abstract we derive a result on the limit of certain sequences of principal eigenvalues associated with some elliptic eigenvalue problems this result is then used to give a complete description of the global structure of the curves of positive steady states of a parameter dependent diffusive version of the classical logistic equation in particular we characterize the bifurcation values from infinity to positive steady states the stability of the positive steady states as well as the asymptotic behaviour of positive solutions is also discussed

varying total population enhances disease persistence qualitative analysis on a diffusive sis epidemic model abstract this paper performs qualitative analysis on an sis epidemic reaction diffusion system with a linear source in spatially heterogeneous environment the main feature of our model lies in that its total population number varies compared to its counterpart proposed by allen et al 2 the uniform bounds of solutions are derived based on which the threshold dynamics in terms of the basic reproduction number is established and the global stability of the unique endemic equilibrium is discussed when spatial environment is homogeneous in particular the asymptotic profile of endemic equilibria is determined if the diffusion rate of the susceptible or infected population is small or large the theoretical results show that a varying total population can enhance persistence of infectious disease and therefore the disease becomes more threatening and harder to control

qualitative analysis for a diffusive predator prey model with a transmissible disease in the prey population in this paper we study a diffusive predator prey model with a transmissible disease in the prey population we offer a complete discussion of the dynamical properties under the homogeneous neumann boundary condition we analyze local and global stabilities of nonnegative constant steady states and long time behaviors of the positive solutions moreover we study the existence nonexistence and bifurcation of nonconstant positive stationary solutions finally some numerical simulations are presented to support and strengthen our analytical analysis

semialgebraic topology over a real closed field ii basic theory of semialgebraic spaces our first goal in this paper is to develop a basic language for the theory of 4 4 semialgebraic spaces over an arbitrary real closed field r 6 9 this language seems to be very convenient for topological considerations in the space x r of rational points of an algebraic variety x over r we then study paths and path components in semialgebraic spaces 10 13 expanding and completing the results of part i of the paper 6 we use the notations and terminology from part i without further explanation we refer the reader to the introduction 1 in part i to get an idea about the program pursued here we hope to continue our work in the near future by research on the homology cohomology and homotopy of semialgebraic spaces a n essential purpose of the present paper is to lay firm ground for all that thus we feel it necessary to proceed carefully and even pedantically in all foundational matters especially in 6 and 7

orlicz centroid bodies the sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body from below by the volume of the star body itself is the busemann petty centroid inequality a decade ago the lp analogue of the classical busemannpetty centroid inequality was proved here the definition of the centroid body is extended to an orlicz centroid body of a star body and the corresponding analogue of the busemann petty centroid inequality is established for convex bodies the centroids of the intersections of an origin symmetric body with half spaces form the surface of a convex body this centroid body is a concept that dates back at least to dupin the classical affine isoperimetric inequality that relates the volume of a convex body with that of its centroid body was conjectured by blaschke and established in a landmark work by petty 53 petty s inequality became known as the busemann petty centroid inequality because in establishing his inequality petty not only made critical use of busemann s random simplex inequality but as petty stated he reinterpreted it see e g the books by gardner 12 leichtweiss 25 schneider 55 and thompson 58 for reference the concept of a centroid body had a natural extension in what became known as the lp brunn minkowski theory and its dual this theory had its origins in the early 1960s when firey see e g schneider 55 introduced his concept of lp compositions of convex bodies three decades later in 34 and 35 these firey minkowski lp combinations were shown to lead to an embryonic lp brunn minkowski theory this theory and its dual has witnessed a rapid growth see e g 1 9 17 23 26 32 34 44 46 48 54 56 57 59 62 the lp analogues of centroid bodies became a central focus within the lp brunn minkowski theory and its dual and establishing the lpanalogue of the busemann petty centroid inequality became a major goal this was accomplished by the authors of the present paper in 37 with an independent approach presented by campi and gronchi 3 the lp centroid bodies quickly became objects of interest in asymptotic mathematics subject classification mr 52a40 research supported in part by nsf grant dms 0706859 proof copy 1 not for distribution 2 erwin lutwak deane yang and gaoyong zhang geometric analysis see e g 10 11 24 49 50 51 52 and even the theory of stable distributions see e g 48 using concepts introduced by ludwig 29 haberl and schuster 21 were led to establish asymmetric versions of the lp busemann petty centroid inequality that for bodies that are not origin symmetric are stronger than the lp busemann petty centroid inequality presented in 37 and 3 the asymmetric inequalities obtained by haberl and schuster are ideally suited for non symmetric bodies this can be seen by looking at the haberl schuster version of the lp analogue of the classical blaschke santalo inequality that was presented in 46 while for origin symmetric bodies the lp extension of 46 does recover the original blaschke santalo inequality as p for arbitrary bodies only the haberl schuster version does so the works of haberl and schuster 21 see also 22 ludwig and reitzner 32 and ludwig 31 have demonstrated the clear need to move beyond the lp brunn minkowski theory to what we are calling an orlicz brunn minkowski theory this need is not only motivated by compelling geometric considerations such as those presented in ludwig and reitzner 32 but also by the desire to obtain sobolev bounds see 22 of a far more general nature this paper is the second in a series intended to develop a few of the elements of an orlicz brunn minkowski theory and its dual here we define the orlicz centroid body establish some of its basic properties and most importantly establish what we call the orlicz busemannpetty centroid inequality for orlicz centroid bodies we consider convex r 0 such that 0 0 this means that must be decreasing on 0 and increasing on 0 we require that one of these is happening strictly so i e is either strictly decreasing on 0 or strictly increasing on 0 the class of such will be denoted by c if k is a star body see section 1 for precise definitions with respect to the origin in rn with volume k and c then we define the orlicz centroid body k of k as the convex body whose support function at x rn is given by h k x inf 0 1 k k x y dy 1 where x y denotes the standard inner product of x and y in rn and the integration is with respect to lebesgue measure in rn when p t t p with p 1 then

witt equivalence of algebraic function fields over real closed fields we examine the conditions for two algebraic function fields over real closed fields to be witt equivalent we show that there are only two witt classes of algebraic function fields with a fixed real closed field of constants real and non real ones the first of them splits further into subclasses corresponding to the tame equivalence this condition has a natural interpretation in terms of both orderings the associated harrison isomorphism maps 1 pt fans onto 1 pt fans and geometry and topology of associated real curves the bijection of points is a homeomorphism and these two curves have the same number of semi algebraically connected components finally we derive some immediate consequences of those theorems in particular we describe all the witt classes of algebraic function fields of genus 0 and 1 over the fixed real closed field

the six grothendieck operations on o minimal sheaves in this paper we develop the formalism of the grothendieck six operations on o minimal sheaves the grothendieck formalism allows us to obtain o minimal versions of i derived projection formula ii universal coefficient formula iii derived base change formula iv kunneth formula v local and global verdier duality

strategy and stationary pattern in a three species predator prey model abstract in this paper we study a strongly coupled system of partial differential equations which models the dynamics of a two predator one prey ecosystem in which the prey exercises a defense switching mechanism and the predators collaboratively take advantage of the prey s strategy we demonstrate the emergence of stationary patterns for this system and show that it is due to the cross diffusion that arises naturally in the model as far as the authors are aware this is the first example of stationary patterns in a predator prey system arising solely from the effect of cross diffusion

stability in a diffusive food chain model with michaelis menten functional response abstract this paper deals with the behavior of positive solutions to a reaction diffusion system with homogeneous neumann boundary conditions describing a three species food chain a sufficient condition for the local asymptotical stability is given by linearization and also a sufficient condition for the global asymptotical stability is given by a lyapunov function our result shows that the equilibrium solution is globally asymptotically stable if the net birth rate of the first species is big enough and the net death rate of the third species is neither too big nor too small

dynamics of a general prey predator model with prey stage structure and diffusive effects in this paper we propose and study the dynamics of a diffusive prey predator model with general functional response and stage structure for the prey firstly we consider the asymptotical stability of equilibrium points and hopf bifurcation for the reduced ode system secondly the existence and uniform boundedness of global solutions and stability of equilibrium points for the corresponding reaction diffusion system are discussed finally we establish the existence and the nonexistence of nonconstant positive steady states of this reaction diffusion system which indicates the effect of large diffusivity our results shows the importance of the diffusion rate of the predator species i e d 3 the large diffusion rate of the predator alone will help the generation of patterns however a large diffusion rate of the immature prey species or a large diffusion rate of the mature prey species can lead to the nonexistence of spatial patterns

stability analysis of diffusive predator prey model with modified leslie gower and holling type iii schemes abstract the stability of a diffusive predator prey model with modified leslie gower and holling type iii schemes is investigated a threshold property of the local stability is obtained for a boundary steady state and sufficient conditions of local stability and un stability for the positive steady state are also obtained furthermore the global asymptotic stability of these two steady states are discussed our results reveal the dynamics of this model system

uniform bounds on complexity and transfer of global properties of nash functions we show that the complexity of semialgebraic sets and mappings can be used to parametrize nash sets and mappings by nash families from this we deduce uniform bounds on the complexity of nash functions that lead to first order descriptions of many properties of nash functions and a good behaviour under real closed field extension e g primary decomposition as a distinguished application we derive the solution of the extension and global equations problems over arbitrary real closed fields in particular over the field of real algebraic numbers this last fact and a technique of change of base are used to prove that the artin mazur description holds for abstract nash functions on the real spectrum of any commutative ring and solve extension and global equations in that abstract setting to complete the view we prove the idempotency of the real spectrum and an abstract version of the separation problem we also discuss the conditions for the rings of abstract nash functions to be noetherian

semialgebraic topology over a real closed field i paths and components in the set of rational points of an algebraic variety we fix once and for all a real closed base field r by a variety x over r we mean a separated algebraic scheme x over r for all problems attacked here we could equally well assume that x is also reduced since we are only interested in the set x r of rational points of x notice that for every closed point x of x the residue class field k x ojmx either coincides with r i e x is rational or is isomorphic to r t we call the points x of x with k x r the real points of x and the other closed points the complex points of x r is a topological field a basis of open sets being given by the open intervals

travelling waves in radially symmetric parabolic systems on montre que toutes les solutions ondes planes de propagation ou stationnaire de certains systemes paraboliques a symetrie radiale peuvent etre trouvees en resolvant une equation differentielle scalaire associee

on the integral hodge conjecture for real varieties i we formulate the real integral hodge conjecture a version of the integral hodge conjecture for real varieties and raise the question of its validity for cycles of dimension 1 on uniruled and calabi yau threefolds and on rationally connected varieties we relate it to the problem of determining the image of the borel haefliger cycle class map for 1 cycles with the problem of deciding whether a real variety with no real point contains a curve of even geometric genus and with the problem of computing the torsion of the chow group of 1 cycles of real threefolds new results about these problems are obtained along the way

global bifurcation structure of a one dimensional ginzburg landau model we consider an equation of a simplified ginzburg landau model of superconductivity in a one dimensional ring the equation for a complex order parameter has two real parameters and related to the magnitude of an applied magnetic field and the ginzburg landau parameter respectively the purpose of this paper is to reveal a global bifurcation structure for the equation in the parameter space in particular we show that there exist modulating amplitude solutions which bifurcate from constant amplitude solutions and how the bifurcation branches of such solutions continue or disappear as varies we also determine the minimizer of the energy functional

moments and valuations all measurable and rm sl n covariant vector valued valuations on convex polytopes containing the origin in their interiors are completely classified the moment vector is shown to be essentially the only such valuation

bifurcation and stability of periodic traveling waves for a reaction diffusion system v plane having a stable critical point at 1 o and unstable critical points at 0 o and cos sin moreover if we take 0 rc then flows starting with 3 i rc make large excursions before decaying to 1 0 whereas those with 0 decay monotonically to l o this excitability is a feature in common with the nonlinear vector field in the hodgkin huxley equations and also with the simpler fitzhugh nagumo equations we begin by describing some results of numerical simulations we made of

estimates for the eigenvalues of the jordan product of hermitian matrices n alikakos and p w bates department of mathematics heriot watt university edinburgh scotland submitted by richard a brualdi abstract given hermitian matrices a and b professor taussky todd posed the problem of estimating the eigenvalues of their jordan product ab ba here we establish bounds for all the eigenvalues of the jordan product when both a and b are positive definite at the same time we give a more straightforward proof and an improvement of estimates given by d w nicholson for the smallest eigenvalue 0 introduction given hermitian matrices a and b their

a holling s type ii prey predator model with stage structure and nonlocal delay abstract in this paper a prey predator model with reaction diffusion is investigated under homogenous neumann boundary condition by taking food ingestion and species moving into account model is further coupled with holling s type ii function response and nonlocal delay sufficient conditions for the global stability of three equilibria i e positive semi trivial and trivial steady states are mainly derived by lyapunov functional respectively results show that intra specific competition benefits the coexistence of prey and predator numerical simulations are performed to illustrate the analytical results

projective modules over smooth affine varieties over real closed fields abstract let x spec a be a smooth affine variety of dimension n 2 over the field r of real numbers let p be a projective a module of rankn such that its nth chern class c n p ch 0 x is zero in this set up bhatwadekar das mandal showed amongst many other results that p a q in the case that either n is odd or the topological space x r of real points of x does not have a compact connected component in this paper we prove that similar results hold for smooth affine varieties over an arbitrary real closed field r the proof is algebraic and does not make use of tarski s principle nor of the earlier result for r