on copositive matrices abstract a criterion for copositive matrices is given and for n 3 the set of all copositive matrices is determined in terms of matrix elements copositive matrices are applied to the problem of excluding periodic solutions of certain algebraic differential equations

exclusion and persistence in deterministic and stochastic chemostat models abstract we first introduce and analyze a variant of the deterministic single substrate chemostat model in this model microbe removal and growth rates depend on biomass concentration with removal terms increasing faster than growth terms using a comparison principle we show that persistence of all species is possible in this scenario then we turn to modelling the influence of random fluctuations by setting up and analyzing a stochastic differential equation in particular we show that random effects may lead to extinction in scenarios where the deterministic model predicts persistence on the other hand we also establish some stochastic persistence results

harmonic analysis on nilpotent groups and singular integrals ii singular kernels supported on submanifolds on etudie des operateurs de convolution singuliers sur des groupes de lie nilpotents generaux et plus specifiquement des operateurs dont les noyaux ont pour supports des varietes de faible dimension et qui peuvent aussi contenir des facteurs polynomiaux exponentiels oscillatoires

criteria for copositive matrices abstract some finite criteria for copositive copositive plus and strictly copositive matrices are proposed and compared with existing determinantal tests the basic mathematical tool is principal pivoting

copositivity detection by difference of convex decomposition and ω subdivision we present three new copositivity tests based upon difference of convex d c decompositions and combine them to a branch and bound algorithm of subdivision type the tests employ lp or convex qp techniques but also can be used heuristically using appropriate test points we also discuss the selection of efficient d c decompositions and propose some preprocessing ideas based on the spectral d c decomposition we report on first numerical experience with this procedure which are very promising

sub hermitian geometry and the quantitative newlander nirenberg theorem abstract given a finite collection of c 1 complex vector fields on a c 2 manifold m such that they and their complex conjugates span the complexified tangent space at every point the classical newlander nirenberg theorem gives conditions on the vector fields so that there is a complex structure on m with respect to which the vector fields are t 0 1 in this paper we give intrinsic diffeomorphic invariant necessary and sufficient conditions on the vector fields so that they have a desired level of regularity with respect to this complex structure i e smooth real analytic or have zygmund regularity of some finite order by addressing this in a quantitative way we obtain a holomorphic analog of the quantitative theory of sub riemannian geometry initiated by nagel stein and wainger we call this sub hermitian geometry moreover we proceed more generally and obtain similar results for manifolds which have an associated formally integrable elliptic structure this allows us to introduce a setting which generalizes both the real and complex theories

restriction of fourier transforms to curves an endpoint estimate with affine arclength measure consider the fourier restriction operators associated to curves in r d 3 we prove for various classes of curves the endpoint restricted strong type estimate with respect to affine arclength measure on the curve an essential ingredient is an interpolation result for multilinear operators with symmetries acting on sequences of vector valued functions

models of degenerate fourier integral operators and radon transforms fourier integral operators whose lagrangians project with singularities arise frequently in many areas of analysis and geometry 3 5 9 10 18 20 however there are as yet few analytic tools available for their study and even their simplest regularity properties with respect to the singularities of the lagrangian are still obscure 4 5 13 16 in this paper we study a class of oscillatory integrals ta and fourier integral operators r which can be expected to model the higher order singularities of the lagrangian they have homogeneous polynomial phases in two variables of order n and indeed the case n 3 is the model for lagrangians which project with whitney folds 3 10 11 the main difficulty which sets the higher order degeneracy cases n 4 apart from the the lower ones n 2 3 is that the critical varieties which arise there are usually not smooth manifolds a systematic study of these oscillatory integrals was begun in 14 the main idea in that work was to treat the critical varieties as smooth manifolds away from a lower dimensional subvariety and to keep track of the distance to this lower dimensional subvariety to achieve this we introduced a method of stationary phase which exhibited clearly the dependence on the distance between critical points the method gave sharp bounds on the size of the kernel k x y of tat but no information on its phase and the resulting cancellations it does not seem possible to refine this approach to the phase level and this suggests looking instead for decompositions of the operators ta which can incorporate indirectly the required cancellations the main goal of this paper is to introduce such decompositions these decompositions which we will try to describe momentarily reflect the singular nature of the critical points and are powerful enough to yield sharp bounds for the above oscillatory integrals and radon transforms although the models under consideration are very

separation and relaxation for cones of quadratic forms let p subseteq mathfrak r n be a pointed polyhedral cone in this paper we study the cone mathcal c rm cone xx t x in p of quadratic forms understanding the structure of mathcal c is important for globally solving np hard quadratic programs over p we establish key characteristics of mathcal c and construct a separation algorithm for mathcal c provided one can optimize with respect to a related cone over the boundary of p this algorithm leads to a nonlinear representation of mathcal c and a class of tractable relaxations for mathcal c each of which improves a standard polyhedral semidefinite relaxation of mathcal c the relaxation technique can further be applied recursively to obtain a hierarchy of relaxations and for constant recursive depth the hierarchy is tractable we apply this theory to two important cases p is the nonnegative orthant in which case mathcal c is the cone of completely positive matrices and p is the homogenized cone of the box 0 1 n through various results and examples we demonstrate the strength of the theory for these cases for example we achieve for the first time a separation algorithm for 5 5 completely positive matrices

frobenius morphisms and representations of algebras by introducing frobenius morphisms f on algebras a and their modules over the algebraic closure f q of the finite field f q of q elements we establish a relation between the representation theory of a over f q and that of the f fixed point algebra a f over f q more precisely we prove that the category mod a f of finite dimensional a f modules is equivalent to the subcategory of finite dimensional f stable a modules and when a is finite dimensional we establish a bijection between the isoclasses of indecomposable a f modules and the f orbits of the isoclasses of indecomposable a modules applying the theory to representations of quivers with automorphisms we show that representations of a modulated quiver or a species over f q can be interpreted as f stable representations of the corresponding quiver over f q we further prove that every finite dimensional hereditary algebra over f q is morita equivalent to some a f where a is the path algebra of a quiver q over f q and f is induced from a certain automorphism of q a close relation between the auslander reiten theories for a and a f is established in particular we prove that the auslander reiten modulated quiver of a f is obtained by folding the auslander reiten quiver of a finally by taking frobenius fixed points we are able to count the number of indecomposable representations of a modulated quiver over f q with a given dimension vector and to generalize kac s theorem for all modulated quivers and their associated kac moody algebras defined by symmetrizable generalized cartan matrices

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

almost copositive matrices abstract a real n n matrix is termed almost copositive if it is not copositive but all its principal submatrices of order n 1 are a real copositive n n matrix is called almost copositive plus almost strictly copositive if it is not copositive plus strictly copositive but all its principal submatrices of order n 1 are such matrices are of crucial importance in deriving criteria for copositivity the properties of these matrix classes are investigated the emphasis being on the less known class of almost copositive plus matrices the basic mathematical tools used are principal pivoting and quadratic programming

criteria for copositive matrices of order four abstract it is proved that the copositivity of a symmetric matrix of order n is equivalent to the copositivity of two symmetric matrices of order n 1 if the matrix has a row whose off diagonal elements are all nonpositive this result is used for deriving criteria for copositive matrices of order 3 and 4

multi parameter singular radon transforms ii the lp theory the purpose of this paper is to study the l2 boundedness of operators of the form f x f t x k t dt where t x is a c function defined on a neighborhood of the origin in t x n n satisfying 0 x x is a c cut off function supported on a small neighborhood of 0 n and k is a multi parameter singular kernel supported on a small neighborhood of 0 n the goal is given an appropriate class of kernels k to give conditions on such that every operator of the above form is bounded on l2 the case when k is a calderon zygmund kernel was studied by christ nagel stein and wainger we generalize their conditions to the case when k has a multi parameter structure for example when k is given by a product kernel even when k is a calderon zygmund kernel our methods yield some new results this is the first paper in a three part series the later two of which are joint with e m stein the second paper deals with the related question of lp boundedness while the third paper deals with the special case when is real analytic

algorithms for determining the copositivity of a given symmetric matrix abstract andersson et al l e andersson g chang t elfving criteria for copositive matrices using simplices and barycentric coordinates linear algebra appl 220 1995 9 30 gives necessary and sufficient conditions for symmetric matrices of order n 5 to be copositive and yang and li s yang x li some simple criteria for copositive matrices in proceedings of the seventh international conference on matrix theory and applications advances in matrix theory and applications world academic union 2006 gives necessary and sufficient conditions for symmetric z matrices of any order to be strictly copositive and some simpler sufficient conditions for symmetric matrices of any order to be copositive or strictly copositive or to be not copositive based on these known results we will present six algorithms of determining whether a given symmetric matrix is strictly copositive or copositive or not copositive the algorithms for matrices of order 3 4 5 6 or 7 are efficient because they quickly give the complete answer for the test the algorithm for matrices of order n 8 is not guaranteed to produce an answer but usually does produce one

copositive matrices and simpson s paradox abstract given a finite population characterized by two attributes a and b a factor c with n levels one case of simpson s paradox sp occurs when a and b are positively associated within each level of c but they are negatively associated or independent in the population given an attribute k let k be its complement assume the conditional proportions of the combinations of attributes ab a b a b a b respectively within each level of c are known to the analyst but the proportions of the n subpopulations corresponding to the n levels of c in the population are not known to the analyst the problem is to find conditions under which sp occurs and find the probability of sp the first part of the problem is solved completely for all n 2 using properties of copositive matrices and the theorems of cottle habetler and lemke 1970 and of pereira 1972 the second part of the problem is solved partially

testing the definiteness of matrices on polyhedral cones abstract the principal pivoting scheme for quadratic programming is used to derive finite criteria for testing whether a real symmetric matirx is nonnegative definite nonnegative definite plus or positive definite on a convex polyhedral cone the paper is an extension of a recent one where the author gave criteria for different classes of copositive matrices

on the nonlinear brascamp lieb inequality we prove a nonlinear variant of the general brascamp lieb inequality our proof consists of running an efficient or tight induction on scales argument which uses the existence of gaussian near extremizers to the underlying linear brascamp lieb inequality lieb s theorem in a fundamental way a key ingredient is an effective version of lieb s theorem which we establish via a careful analysis of near minimizers of weighted sums of exponential functions instances of this inequality are quite prevalent in mathematics and we illustrate this with some applications in harmonic analysis

a note on weakly active constraints in connection with nonconvex quadratic programming abstract in general it is very difficult to determine a true local minimizer in nonconvex quadratic programming the main problems arise if we have so called weakly active constraints we discuss an efficient method for checking the local optimality or determining a direction of descent

criteria for copositive matrices using simplices and barycentric coordinates abstract we present criteria for verifying the copositivity of an n n matrix given that all its principal submatrices of order n 1 are copositive for n 4 5 necessary and sufficient conditions for copositivity are given based on the sign distribution of the off diagonal elements of a single row here it is only assumed that one principal submatrix is copositive

a test for copositive matrices abstract the paper presents necessary and sufficient conditions that a symmetric matrix be copositive or strictly copositive the conditions are given in terms of the eigenvalues and eigenvectors of the principal submatrices of the given matrix

optimal frames for erasures we study frames from the viewpoint of coding theory we introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting

a kakeya type problem for circles we prove full hausdorff dimension in a variant of the kakeya problem involving circles in the plane and also sharp estimates for the relevant maximal function these results can also be formulated in terms of the wave equation in two space variables a novelty in our approach is the use of ideas from computational geometry

the copositive completion problem abstract an n n real symmetric matrix a is called strictly copositive if x t ax 0 0 whenever x r n satisfies x 0 x 0 and x 0 the strictly copositive matrix completion problem asks which partial strictly copositive matrices have a completion to a strictly copositive matrix we prove that every partial strictly copositive matrix has a strictly copositive matrix completion and give a lower bound on the values used in the completion we answer affirmatively an open question whether an n n copositive matrix a a ij with all diagonal entries a ii 1 stays copositive if each off diagonal entry of a is replaced by min a ij 1

algorithmic copositivity detection by simplicial partition we present new criteria for copositivity of a matrix i e conditions which ensure that the quadratic form induced by the matrix is nonnegative over the nonnegative orthant these criteria arise from the representation of the quadratic form in barycentric coordinates with respect to the standard simplex and simplicial partitions thereof we show that as the partition gets finer and finer the conditions eventually capture all strictly copositive matrices we propose an algorithmic implementation which considers several numerical aspects as an application we present results on the maximum clique problem we also briefly discuss extensions of our approach to copositivity with respect to arbitrary polyhedral cones

singular and maximal radon transforms analysis and geometry part 2 geometric theory 8 curvature introduction 8 1 three notions of curvature 8 2 theorems 8 3 examples 9 curvature some details 9 1 the exponential representation 9 2 diffeomorphism invariance 9 3 curvature condition cy 9 4 two lemmas 9 5 double fibration formulation 10 equivalence of curvature conditions 10 1 invariant submanifolds and deficient lie algebras 10 2 vanishing jacobians 10 3 construction of invariant submanifolds

on degenerations and extensions of finite dimensional modules abstract we derive a cancellation theorem for degenerations of modules that says in particular that projective or injective common direct summands can always be neglected combining the cancellation result with the existence of almost split sequences we characterize the orbit closure of a module living on preprojective components by the fact that the dimension of the homomorphism space to any other module does not decrease for representation directed algebras whence in particular for path algebras of dynkin quivers we provide an alternative proof which shows in addition that any minimal degenerationnofmcomes from an exact sequence with middle termmwhose end terms add up ton by a careful examination the same is true for degenerations of matrix pencils having used so far the existence of certain extensions to obtain degenerations we then turn the tables and use degenerations to produce a lot of interesting short exact sequences in particular we show that any non simple indecomposable over a tame quiver is an extension of an indecomposable and a simple

the size of lp improving measures abstract a measure on the locally compact abelian group g is lp improving if for some 1 p p g l r g l space properties of such are discussed if is lp improving and non negative and f is a bounded borel function then f is lp improving the non negativity of and boundedness of f are essential as is shown in several ways in particular there exists on each infinite compact abelian group an lp improving such that is not lp improving the distribution function of every lp improving measure on the circle group is shown to satisfy a lipschitz condition but that is not sufficient for a measure to be lp improving except in restricted cases the gelfand transform on of an lp improving measure is shown to be small in a quantitative sense from that the strong continuity of lp improving measures follows the spectrum of an lp improving measure as an operator on lp is shown to be precisely the closure of related results and open problems are included

uniform estimates for fourier restriction to polynomial curves in ℝ d we prove uniform l p to l q bounds for fourier restriction to polynomial curves in bbb r d with affine arclength measure in the conjectured range

hilbert integrals singular integrals and radon transforms i on introduit deux nouvelles classes d operateurs on etudie leur relation et on montre comment on peut les appliquer a l etude des problemes aux valeurs limites

uniform bounds for convolution and restricted x ray transforms along degenerate curves abstract we establish endpoint lebesgue space bounds for convolution and restricted x ray transforms along curves satisfying fairly minimal differentiability hypotheses with affine and euclidean arclengths we also explore the behavior of certain natural interpolants and extrapolants of the affine and euclidean versions of these operators

a variational approach to copositive matrices this work surveys essential properties of the so called copositive matrices the study of which has been spread over more than fifty five years special emphasis is given to variational aspects related to the concept of copositivity in addition some new results on the geometry of the cone of copositive matrices are presented here for the first time

on the numerical realization of the exact penalty method for quadratic programming algorithms abstract optimization problems with quadratic nonconvex objective functions and linear inequalities are considered an active constraint strategy is used to define a sequence of equality constrained problems under discussion is how the subproblems can be solved efficiently

a generalization of bourgain s circular maximal theorem where d r is the normalized surface measure on r s1 it is easy to see that m is not bounded on l2 see example 1 1 below a well known result of bourgain 1 asserts that m is bounded on lp for 2 p we will consider the question of boundedness of m and m from lp to lq unless stated to the contrary we will be dealing only with functions defined on r2 absolute constants will be denoted by c and the notation will mean up to a constant

canonical basis for type a4 i monomial elements the global crystal basis or canonical basis plays an important role in the theory of the quantum groups and their representations the tight monomials are the simplest elements in the canonical basis based on lusztig and xi s work the regions of tight monomials in quantized enveloping algebra of type a4 are determined in this paper 2003 elsevier science usa all rights reserved

radon transforms and finite type conditions the purpose of this paper is to study averaging operators of radon transform type we shall formulate suitable finite type conditions and prove lp sobolev and lplq estimates the results will be essentially sharp for operators associated with families of curves in the plane let xc and t be smooth manifolds dimx3 nl dim2 nr and let m be a submanifold in xc x t with conormal bundle n m we denote by t the codimension of m we shall always assume that the projections lry m x and 7rt m 42 are surjective with rank dirx nl rank dr9 nr this in particular implies that n m c t x o x t o where 0 refers to the zero sections in t x and t 2 respectively this is the usual assumption for the canonical relation associated with lagrangian distributions arising as kernels of fourier integral operators the assumptions on dirx d7rt imply that for fixed x e x y e 2 the sets amx y e t x y e mi m x ye2x x y em i

universal l p improving for averages along polynomial curves in low dimensions we prove sharp lp lq estimates for averaging operators along general polynomial curves in two and three dimensions these operators are translation invariant given by convolution with the so called affine arclength measure of the curve and we obtain universal bounds over the class of curves given by polynomials of bounded degree our method relies on a geometric inequality for general vector polynomials together with a combinatorial argument due to m christ almost sharp lorentz space estimates are obtained as well

the pythagorean theorem i the finite case the pythagorean theorem and variants of it are studied the variations evolve to a formulation in terms of noncommutative conditional expectations on von neumann algebras that displays the theorem as the basic result of noncommutative metric euclidean geometry the emphasis in the present article is finite dimensionality both discrete and continuous

quadratic programming criteria for copositive matrices abstract finding out whether a real symmetric n n matrix a is not copositive is an np complete problem thus it is not surprising that there is no efficient algorithm for determining the copositivity class of a in the worst case the computational effort required by the available matrix theoretic criteria grows exponentially with n in the present paper some criteria based on quadratic programming are proposed for copositive copositive plus and strictly copositive matrices these criteria supplement the existing ones being more efficient in some cases where the matrix contains large positive definite submatrices on the other hand matrix theoretic criteria are superior if the matrix contains only small positive definite submatrices

nonlocal inversion formulas for the x ray transform etude de la reconstruction d une fonction f x sur r n a partir de ses integrales sur des plans k dimensionnels cas de k 1

tight monomials and the monomial basis property abstract we generalize a criterion for tight monomials of quantum enveloping algebras associated with symmetric generalized cartan matrices and a monomial basis property of those associated with symmetric classical cartan matrices to their respective symmetrizable case we then link the two by establishing that a tight monomial is necessarily a monomial defined by a weakly distinguished word as an application we develop an algorithm to compute all tight monomials in the rank 2 dynkin case

an algorithm for determining copositive matrices abstract in this paper we present an algorithm of simple exponential growth called copomatrix for determining the copositivity of a real symmetric matrix the core of this algorithm is a decomposition theorem which is used to deal with simplicial subdivision of t y m t y 0 on the standard simplex m where each component of the vector is 1 0 or 1

restriction of the fourier transform to some complex curves abstract the purpose of this paper is to prove a fourier restriction estimate for certain 2 dimensional surfaces in r 2 d d 3 these surfaces are defined by a complex curve z of simple type which is given by a mapping of the form z z z z 2 z d 1 z where z is an analytic function on a domain c this is regarded as a real mapping z x y x y from r 2 to r 2 d our results cover the case z z n for any nonnegative integer n in all dimensions d 3 the main result is a uniform estimate valid when d 3 where z may be taken to be an arbitrary polynomial of degree at most n it is uniform in the sense that the operator norm is independent of the coefficients of the polynomial these results are analogues of the uniform restricted strong type estimates in 5 valid for polynomial curves of simple type and some other classes of curves in r d d 3

uniform estimates for the x ray transform restricted to polynomial curves abstract we establish near optimal mixed norm estimates for the x ray transform restricted to polynomial curves with a weight that is a power of the affine arclength the bounds that we establish depend only on the spatial dimension and the degree of the polynomial some of our results are new even in the well curved case

dynamics of stochastic seis epidemic model with varying population size abstract we introduce the stochasticity into a deterministic model which has state variables susceptible exposed infected with varying population size in this paper the infected individuals could return into susceptible compartment after recovering we show that the stochastic model possesses a unique global solution under building up a suitable lyapunov function and using generalized ito s formula the densities of the exposed and infected tend to extinction when some conditions are being valid moreover the conditions of persistence to a global solution are derived when the parameters are subject to some simple criteria the stochastic model admits a stationary distribution around the endemic equilibrium which means that the disease will prevail to check the validity of the main results numerical simulations are demonstrated as end of this contribution

fourier restriction to polynomial curves i a geometric inequality we prove a fourier restriction result for general polynomial curves in bbb r d measuring the fourier restriction with respect to the affine arclength measure of the curve we obtain a universal estimate for the class of all polynomial curves of bounded degree our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right applications of this geometric inequality to other problems in euclidean harmonic analysis have recently been established

stochastic permanence of an siqs epidemic model with saturated incidence and independent random perturbations abstract this article discusses a stochastic siqs epidemic model with saturated incidence we assume that random perturbations always fluctuate at the endemic equilibrium the existence of a global positive solution is obtained by constructing a suitable lyapunov function under some suitable conditions we derive the stochastic boundedness and stochastic permanence of the solutions of a stochastic siqs model some numerical simulations are carried out to check our results

fourier integral operators with cusp singularities we study the boundedness properties on lebesgue and sobolev spaces of fourier integral operators associated with canonical relations such that at least one of the projections is a simple whitney cusp in the process we obtain decay estimates for oscillatory integral operators whose symplectic relations have the same singular structure such singularities occur generically for averages over lines and curves in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i on l 2 we show that the operators lose 1 3 derivative to obtain sharp results off of l 2 we need to impose an additional transversality condition satisfied by many geometric averaging operators which leads to the notion of a strong cusp these estimates can be further improved if we impose curvature conditions on the cusp surface one application is the l 2 comp l 2 1 6 loc and l 12 7 comp l 2 loc boundedness of restrictions of the x ray transform on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i to four dimensional families of lines satisfying a natural curvature and torsion condition

almost definiteness of matrices on polyhedral cones let a be a real symmetric n n matrix and let k a proper subset of rn be a polyhedral cone then a is called almost nonnegative definite on k if it is not nonnegative definite on k but is nonnegative definite of order n 1 on k if a is nonnegative definite on k it is termed almost nonnegative definite plus almost positive definite on k if it is not nonnegative definite plus positive definite on k but is nonnegative definite plus positive definite of order n 1 on k such almost definite matrices are of crucial importance in deriving criteria for conditional definiteness we give necessary and sufficient conditions for almost definite matrices we also determine breaking rays of these kinds of matrices i e rays in k on which the conditional definiteness class in question is violated the basic mathematical tools used are principal pivoting and quadratic programming the paper is an extension of a recent one where the author studied almost copositive matrices

persistence and distribution of a stochastic susceptible infected removed epidemic model with varying population size abstract in this paper the dynamics of a stochastic susceptible infected removed model in a population with varying size is investigated we firstly show that the stochastic epidemic model has a unique global positive solution with any positive initial value then we verify that random perturbations lead to extinction when some conditions are being valid moreover we prove that the solution of the stochastic epidemic model is persistent in the mean by building up a suitable lyapunov function and using generalized ito s formula further the stochastic epidemic model admits a stationary distribution around the endemic equilibrium when parameters satisfy some sufficient conditions to end this contribution and to check the validity of the main results numerical simulations are separately carried out to illustrate these results

extended affine surface area during the twentieth century many of the notions of classical differential geometry have been extended so as to be defined on the boundaries of arbitrary convex bodies without smoothness requirements for example the projection measures quermassintegrals are the natural extensions of the integrals of mean curvature see santa16 27 pp 215 2321 for a discussion the surface area measures of aleksandrov fenchel jessen are extensions of the indefinite integrals of the elementary symmetric functions of the principal radii of curvature while federer s curvature measures when restricted to convex hypersurfaces are extensions of the indefinite integrals of the elementary symmetric functions of the principal curvatures see schneider 30 pp 27 331 one of the important concepts of classical differential geometry which has been difficult to extend is afline surface area this has often led to difficulties which when circumvented have been done so only by considerable effort and ingenuity for example suppose we are given a measure on the unit sphere which satisfies the hypothesis of the minkowski problem the solution of the minkowski problem guarantees the existence of a convex body whose surface area measure is the given measure from the given measure we can find the surface area of the body the minkowski mixed volume inequality can be used to obtain various upper bounds for the volume of the body however we cannot even inquire about the affine surface area of the body unless we can somehow convince ourselves that the convex body is sufficiently smooth results regarding aftine surface area have a surprising number of applications for example a number of geometric inequalities which appear to have no connection with affine surface area are in fact consequences of inequalities involving afline surface area see for example petty 25 and also 151 since polytopes have zero afftne surface area it seems quite surprising that the notion of afline surface area arises frequently in the area of polytope approximation see the survey of gruber 9 recently leichtweiss 11 121 presented a definition of extended afftne

a copositivity probe abstract the paper explores ways of determining whether a given symmetric matrix is copositive in particular a computational procedure is proposed for determining if it exists a representation of the matrix as a sum of a positive semidefinite matrix and a nonnegative matrix the procedure is found to be successful in a significant number of cases

on the classes of fully copositive and fully semimonotone matrices abstract in this paper we consider the class c 0 f of fully copositive and the class e 0 f of fully semimonotone matrices we show that c 0 f matrices with positive diagonal entries are column sufficient we settle a conjecture made by murthy and parthasarathy to the effect that a c 0 f q 0 matrix is positive semidefinite by providing a counterexample we finally consider e 0 f matrices introduced by cottle and stone math program 27 1983 191 213 and partially address stone s conjecture to the effect that e 0 f q 0 p 0 by showing that e 0 f d c matrices are p 0 where d c is the doverspike class of matrices

canonical basis for typeb2 abstract a quantized enveloping algebra has a remarkable basis called the canonical basis or global crystal basis for type a 2 the canonical basis is described explicitly the purpose of this paper is to find the canonical basis for type b 2 the main result is theorem 2 2

dynamics of a stochastic sir epidemic model with saturated incidence in this paper the dynamics of a stochastic sir epidemic model with saturated incidence is investigated firstly we prove that the system has a unique global positive solution with any positive initial value then we verify that random effect may lead the disease to extinction under a simple condition thirdly we establish a sufficient condition for persistence in the mean of the disease moreover we show that there is a stationary distribution to the stochastic system under certain parametric restrictions finally some numerical simulations are carried out to confirm the analytical results

norm estimates for k plane transforms and geometric inequalities abstract the article is devoted to remarkable interrelation between the norm estimates for k plane transforms in weighted and unweighted l p spaces and geometric integral inequalities for cross sections of measurable sets in r n a similar interrelation is studied for more general j plane to k plane transforms on affine grassmannians and their compact modifications the article contains a series of new integral geometric inequalities with sharp constants explicit equalities conjectures and open problems

remarks on second order conditions in connection with the algorithm of beale for quadratic programming abstract the algorithm of beale is considered it is discussed in which case the solution point of beale s algorithm is a local minimizer in this algorithm a general form of second order conditions arises

block pivoting and shortcut strategies for detecting copositivity abstract in recently proposed quadratic optimization algorithms copositivity detection procedures are frequently employed which deliver a feasible direction yielding a negative value of the considered quadratic form if the answer is negative to improve the computational performance of this routine here 1 recursive characterizations of copositivity are presented which enable efficient reduction of the dimension of the problem using block pivoting techniques and 2 shortcut strategies are described which are connected with diagonalization

asymptotic properties of hybrid diffusion systems in response to the increasing needs for control and optimization of hybrid systems this work is concerned with such asymptotic properties as recurrence also known as weak stochastic stability in the literature and ergodicity of regime switching diffusions using liapunov functions necessary and sufficient conditions for positive recurrence are developed then ergodicity of positive recurrent regime switching diffusions is obtained by constructing cycles using the associated discrete time markov chains

endpoint lp lq bounds for integration along certain polynomial curves we establish strong type endpoint lp rd lq rd bounds for the operator given by convolution with affine arclength measure on polynomial curves for d 4 the bounds established depend only on the dimension d and the degree of the polynomial

spectral theory of copositive matrices abstract let a r n n we provide a block characterization of copositive matrices with the assumption that one of the principal blocks is positive definite haynsworth and hoffman showed that if r is the largest eigenvalue of a copositive matrix then r for all other eigenvalues of a we continue their study of the spectral theory of copositive matrices and show that a copositive matrix must have a positive vector in the subspace spanned by the eigenvectors corresponding to the nonnegative eigenvalues moreover if a symmetric matrix has a positive vector in the subspace spanned by the eigenvectors corresponding to its nonnegative eigenvalues then it is possible to increase the nonnegative eigenvalues to form a copositive matrix a without changing the eigenvectors we also show that if a copositive matrix has just one positive eigenvalue and n 1 nonpositive eigenvalues then a has a nonnegative eigenvector corresponding to a nonnegative eigenvalue

principal pivot transforms of some classes of matrices in this paper we study and characterize various classes of matrices that are defined based on principal pivot transforms we show that matrices in these classes have nonnegative principal minors

weak type 1 1 bounds for rough operators ii in this paper we consider four operators arising in harmonic analysis in r n 1 they share one feature each is known to be bounded on lp for all 1 p o but for each it has been an open question whether a weak type 1 1 bound also holds in two cases we show that the weak 1 1 bound does hold and for the others we establish weaker estimates which are still sharper than the lp boundedness although the four operators are quite distinct they all involve convolution kernels which are related to the singular integrals of calderon and zygmund but which lack the smoothness required for the usual analysis in each case this defect is mitigated by the presence of some form of curvature consider the bochner riesz means

l p improving bounds for averages along curves we establish local l p l q mapping properties for averages on curves the exponents are sharp except for endpoints let n 2 and let m1 and m2 be two smooth n 1 dimensional manifolds 1 each containing a preferred origin 0m1 and 0m2 we shall abuse notation and write 0 for both 0m1 and 0m2 for the purposes of integration we shall place a smooth riemannian metric on m1 and m2 although the exact choice of this metric will not be relevant all our considerations shall be local to the origin 0 we are interested in the local l p improving properties of averaging operators on curves before we give the rigorous description of these operators let us first give an informal discussion informally we assume that we have a smooth assignment x2 7 x2 taking points in m2 to curves in m1 with a corresponding dual assignment x1 x1 taking points in m1 to curves in m2 such that x1 x2 x2 1 we then form the operator r taking functions 2 on m1 to functions on m2 defined

symmetric nonnegative tensors and copositive tensors abstract we first prove two new spectral properties for symmetric nonnegative tensors we prove a maximum property for the largest h eigenvalue of a symmetric nonnegative tensor and establish some bounds for this eigenvalue via row sums of that tensor we show that if an eigenvalue of a symmetric nonnegative tensor has a positive h eigenvector then this eigenvalue is the largest h eigenvalue of that tensor we also give a necessary and sufficient condition for this we then introduce copositive tensors this concept extends the concept of copositive matrices symmetric nonnegative tensors and positive semi definite tensors are examples of copositive tensors the diagonal elements of a copositive tensor must be nonnegative we show that if each sum of a diagonal element and all the negative off diagonal elements in the same row of a real symmetric tensor is nonnegative then that tensor is a copositive tensor some further properties of copositive tensors are discussed

on adapted coordinate systems the notion of an adapted coordinate system for a given real analytic function introduced by v i arnol d plays an important role for instance in the study of asymptotic expansions of oscillatory integrals in two dimensions a n varchenko gave sufficient conditions for the adaptness of a given coordinate system and proved the existence of an adapted coordinate system for analytic functions without multiple components varchenko s proof is based on a two dimensional resolution of singularities result in this article we present a more elementary approach to these results which is based on the puiseux series expansion of roots of the given function this approach is inspired by the work of d h phong and e m stein on the newton polyhedron and oscillatory integral operators it applies to arbitrary real analytic functions and even to arbitrary smooth functions of finite type in particular we show that varchenko s conditions are in fact necessary and sufficient for the adaptedness of a given coordinate system and that adapted coordinates always exist in two dimensions even in the smooth finite type setting for analytic functions a construction of adapted coordinates by means of puiseux series expansions of roots has already been carried out in work by d h phong e m stein and j a sturm on the growth and stability of real analytic function as we learned after the completion of this paper in contrast to their work however our proof more closely follows varchenko s algorithm for the construction of an adapted coordinate system which turns out to be useful for the extension to the smooth setting