intrinsic ultracontractivity and small perturbation for one dimensional generalized diffusion operators abstract we consider a one dimensional generalized diffusion operator g on an open interval which is not necessarily bounded we characterize sp i e 1 is a small perturbation of the operator g in terms of the classification of the boundary points from the view point of diffusion processes and give a sufficient condition for iu i e the associated heat kernel is intrinsically ultracontractive

the ring structure on the cohomology of coordinate subspace arrangements every simplicial complex delta subset2 n on the vertex set n 1 ldots n defines a real resp complex arrangement of coordinate subspaces in mathbb r n resp mathbb c n via the correspondence delta ni sigma mapsto rm span e i i in sigma the linear structure of the cohomology of the complement of such an arrangement is explicitly given in terms of the combinatorics of delta and its links by the goresky macpherson formula here we derive by combinatorial means the ring structure on the integral cohomology in terms of data of delta we provide a non trivial example of different cohomology rings in the real and complex case furthermore we give an example of a coordinate arrangement that yields non trivial multiplication of torsion elements

combinatorial stratification of complex arrangements we present a method for discretizing complex hyperplane arrange ments by encoding their topology into a finite partially ordered set of sign vectors this is used in the following ways 1 a general method is given for constructing regular cell complexes having the homotopy type of the complement of the arrangement 2 for the case of complexified arrangements this specializes to the con struction of salvetti s we study the combinatorial structure of complexified arrangements and the salvetti complex in some detail 3 this general method simultaneously produces cell decompositions of the singularity link this link is shown to have the homotopy type of a wedge of d spheres for arrangements in cd d 4 4 the homology of the link and the cohomology of the complement are computed in terms of explicit bases which are matched by alexander duality this gives a new more elementary and more generally valid proof for the brieskorn orlik solomon theorem and some related results 5 our setup leads to a more general notion of 2 pseudoarrangements which can be thought of as topologically deformed complex arrangements re taining only the essential topological and combinatorial structure we show that all of the above remains true in this generality except for the sign patterns of the orlik solomon relations department of mathematics royal institute of technology s10044 stockholm sweden e mail address andersb nada kth se institut fur mathematik universitat augsburg d 8900 augsburg germany current address institut mittag leffler auravagen 17 s 18262 djursholm sweden address after 1 april 1992 konrad zuse zentrum fur informationstechnik berlin heilbronner str 10 w 1000 berlin 31 germany e mail addresses ziegler ml kva se ziegler zib berlin de this content downloaded from 157 55 39 211 on fri 24 jun 2016 06 44 45 utc all use subject to http about jstor org terms

on the toda and kac van moerbeke systems given a solution of the toda lattice we explicitly construct a solution of the kac van moerbeke system related to each other by a miura type transformation as an illustration of our method we derive the n soliton solutions of the kac van moerbeke lattice

critical phenomena in linear elliptic problems abstract in this article we consider the existence and nonexistence of principal eigenvalues for the linear elliptic equation u v x u in r n and u d 1 2 r n where n 3 and v l 1 loc r n with v 0 in r n we establish that when potential v has strong singularities phenomena which appear under critical exponent nonlinearities brezis and nirenberg are present in linear problems as well

dispersive estimates for rational symbols and local well posedness of the nonzero energy nv equation abstract we consider the cauchy problem for the two dimensional novikov veselov equation integrable via the inverse scattering problem for the schrodinger operator with fixed negative energy the associated linear equation is characterized by a rational symbol which is not a polynomial except when the energy parameter is zero with the help of a complex analysis point of view of the problem we establish uniform decay estimates for the linear solution with gain of almost one derivative and we use this result together with fourier decomposition methods and x s b spaces to prove local well posedness in h s s 1 2

the structure of f pure rings for a reduced f finite ring r of characteristic p 0 and q p e one can write open image in new window where m q has no free direct summands over r we investigate the structure of f finite f pure rings r by studying how the numbers a q grow with respect to q this growth is quantified by the splitting dimension and the splitting ratios of r which we study in detail we also prove the existence of a special prime ideal open image in new window r of r called the splitting prime that has the property that r open image in new window r is strongly f regular we show that this ideal captures significant information with regard to the f purity of r

trudinger moser inequality with remainder terms abstract the paper gives the following improvement of the trudinger moser inequality 0 1 sup u 2 d x u 1 u c 0 e 4 u 2 d x r 2 related to the hardy sobolev mazya inequality in higher dimensions we show 0 1 with u v x u 2 d x for a class of v 0 that includes v r 1 4 r 2 log 1 r 2 max log 1 r 1 which refines two previously known cases of 0 1 proved by adimurthi and druet 2 and by wang and ye 23 in addition we verify 0 1 for u u p 2 as well as give an analogous improvement for the onofri beckner inequality for the unit disk beckner 6

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

the polyhedral product functor a method of decomposition for moment angle complexes arrangements and related spaces this article gives a natural decomposition of the suspension of generalized moment angle complexes or partial product spaces which arise as polyhedral product functors described below the geometrical decomposition presented here provides structure for the stable homotopy type of these spaces including spaces appearing in work of goresky macpherson concerning complements of certain subspace arrangements as well as davis januszkiewicz and buchstaber panov concerning moment angle complexes since the stable decompositions here are geometric they provide corresponding homological decompositions for generalized moment angle complexes for any homology theory

subcriticality and gaugeability of the schrodinger operator we investigate a schrodinger operator 2 v in r d d 3 with a potential v in the class k d satisfying a similar kato condition at infinity and prove an equivalence theorem connecting various conditions on subcriticality strong positivity and gaugeability of the operator

lp norms of non critical schrodinger semigroups we consider schrodinger semigroups e th h v on rn with v c x 2 as x 0 c 12 n 2 2 with h 0 we determine the exact power law divergence of e th p p and of some e th q p as maps from lp to lq the results are expressed most naturally in terms of the power for which there exists a positive resonance such that h 0 x x

critical dissipative estimate for a heat semigroup with a quadratic singular potential and critical exponent for nonlinear heat equations abstract we consider a heat semigroup with an inverse square potential and prove critical dissipative estimates including the endpoint case some application to a nonlinear heat equation with an inverse square potential is also discussed

on the difference between real and complex arrangements if b is an arrangement of linear complex hyperplanes in c d then the following can be constructed from knowledge of its intersection lattice a the cohomology groups of the complement br b the cohomology algebra of the complement os c the fundamental group of the complement if d 2 d the singularity link up to homeomorphism if d 3 e the singularity link up to homotopy type zz if bis more generally a 2 arrangement in ir 2d an arrangement of real subspaces of codimension 2 with even dimensional intersections then the intersection lattice still determines a the cohomology groups of the complement gm and e the homotopy type of the singularity link zz we show however that for 2 arrangements the data b c and d are not deter mined by the intersection lattice they require the knowledge of extra information on sign patterns which can be computed as determinants of linear relations or equiva lently as linking coefficients in the sense of knot theory

coincidences between homological densities predicted by arithmetic abstract motivated by analogies with basic density theorems in analytic number theory we introduce a notion and variations of the homological density of one space in another we use weil s number field function field analogy to predict coincidences for limiting homological densities of various sequences z n d 1 d m x of spaces of 0 cycles on manifolds x the main theorem in this paper is that these topological predictions which seem strange from a purely topological viewpoint are indeed true one obstacle to proving such a theorem is the combinatorial complexity of all possible collisions of points this problem does not arise in the simplest and classical case m n 1 2 of configuration spaces to overcome this obstacle we apply the bjorner wachs theory of lexicographic shellability from algebraic combinatorics

martin compactification of a complete surface with negative curvature in this paper we consider the martin compactification associated with the operator mathcal l delta 1 of a complete non compact surface sigma 2 with negative curvature in particular we investigate positive eigenfunctions with eigenvalue one of the laplace operator delta of sigma 2 and prove a uniqueness result such eigenfunctions are unique up to a positive multiplicative constant if they vanish on the part of the geometric boundary s infty sigma 2 of sigma 2 where the curvature is bounded above by a negative constant and satisfy some growth estimate on the other part of s infty sigma 2 where the curvature approaches zero this uniqueness result plays an essential role in our recent paper cao and he infinitesimal rigidity of steady gradient ricci soliton in dimension three arxiv 1412 2714 2014 preprint in which we prove an infinitesimal rigidity theorem for deformations of certain three dimensional collapsed gradient steady ricci soliton with a non trivial killing vector field

lp norms of nonnegative schrodinger heat semigroup and the large time behavior of hot spots this paper is concerned with the cauchy problem for the heat equation with a potential p tu u v x uin rn 0 u x 0 x in rn where t t n 3 l2 rn and v v x is a smooth nonpositive and radially symmetric function having quadratic decay at the space infinity in this paper we assume that the schrodinger operator h v is nonnegative on l2 rn and give the exact power decay rates of lq norm q 2 of the solution e th of p as t furthermore we study the large time behavior of the solution of p and its hot spots

positive solutions and large time behaviors of schrodinger semigroups simon s problem let h v where v is a multiplication operator by a real valued function v x on rn which is uniformly holder continuous and 1 x 2 2 v x l rn for some 4 the relationship between existence of positive solutions with growth conditions of hg 0 and asymptotic behaviors as t of e th is established using it b simon s problem for h on r2 is solved

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

constructing solutions of the mkdv equation abstract using commutation methods i e n 1 supersymmetry underlying miura s transformation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided

on schrodinger operators with multipolar inverse square potentials abstract positivity essential self adjointness and spectral properties of a class of schrodinger operators with multipolar inverse square potentials are discussed in particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established

small rational model of subspace complement this paper concerns the rational cohomology ring of the complement m of a complex subspace arrangement we start with the de concini procesi differential graded algebra that is a rational model for m inside it we find a much smaller subalgebra d quasi isomorphic to the whole algebra d is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice l whence connecting the de concini procesi model with the goresky macpherson formula for the additive structure of h m the algebra d has a natural integral version that is a good candidate for an integral model of m if the rational local homology of l can be computed explicitly we obtain an explicit presentation of the ring h m q for example this is done for the cases where l is a geometric lattice and where m is a k equal manifold

on cohomology algebras of complex subspace arrangements the integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement we give a combinatorial presentation of the cohomology algebra in the spirit of the orlik solomon result on the cohomology algebras of complex hyperplane arrangements our methods are elementary we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space we describe simplicial cochains that generate the cohomology among them we distinguish a linear basis study cup product multiplication and derive an algebra presentation in terms of generators and relations

the heat kernel of a schrodinger operator with inverse square potential we consider the schrodinger operator h v x with radial potential v which may have singularity at 0 and a quadratic decay at infinity first we study the structure of positive harmonic functions of h and give their precise behavior second under quite general conditions we prove an upper bound for the correspond heat kernel p x y t of the type 0 p x y t ct n 2u min x t u min y t u t 2exp x y 2ct for all x y rn and t 0 where u is a positive harmonic function of h third if u2 is an a2 weight on rn then we prove a lower bound of a similar type

equivariant cohomology and the varchenko gelfand filtration abstract the cohomology of the configuration space of n points in r 3 is isomorphic to the regular representation of the symmetric group which acts by permuting the points we give a new proof of this fact by showing that the cohomology ring is canonically isomorphic to the associated graded of the varchenko gelfand filtration on the cohomology of the configuration space of n points in r 1 along the way we give a presentation of the equivariant cohomology ring of the r 3 configuration space with respect to a circle acting on r 3 via rotation around a fixed line we extend our results to the settings of arbitrary real hyperplane arrangements the aforementioned theorems correspond to the braid arrangement as well as oriented matroids

criticality theory for schrodinger operators with singular potential abstract this paper extends a weighted spectral gap classification for schrodinger operators v to a large class of locally integrable potentials v called balanced potentials here harnack s inequality will not hold in general for such potentials in addition to the standard potentials in the space l l o c p for some p n 2 this class contains potentials for which v is locally bounded with no restriction on v or more generally those for which the positive and negative singularities are separated as well as those for which the interaction of the positive and negative singularities is not too strong we also establish for such potentials under the additional assumption that v l l o c p for some p n 2 versions of the agmon allegretto piepenbrink principle characterising the subcritical critical operators in terms of the cone of positive distributional super solutions

on positivity criticality and the spectral radius of the shuttle operator for elliptic operators in this paper we study the shuttle operator for a second order linear elliptic operator p on a noncompact manifold x zhao has introduced and studied the shuttle operator and its relation to the theory of positive solutions in the case of small perturbations of the laplacian in ir n 3 zhao was motivated by works of chung and varadhan which consider one dimensional schrodinger operators the main purpose of the paper is to extend the above studies we prove that in the general case the spectral radius of the shuttle operator is strictly less equal or strictly greater than 1 if and only if the operator p is respectively subcritical critical or supercritical in x we demonstrate the usefulness of the characterization for proving theorems our approach is purely analytic this research was partially supported by the israel science foundation administered by the israel academy of sciences and humanities and by the fund for the promotion of research at the technion

radial solutions of a semilinear elliptic problem we analyse the set of nonnegative global and radial solutions radial solutions for short of the equation u u p f in r n n 1 where 0 where graphics when f r c r2p 1 p h r with h r o r2p 1 p as r radial solutions continue to exist if h r is sufficiently small at infinity existence however breaks down if h r 0 graphics whenever they exist radial solutions are characterised in terms of their asymptotic behaviour as r

on families of optimal hardy weights for linear second order elliptic operators abstract we construct families of optimal hardy weights for a subcritical linear second order elliptic operator using a one dimensional reduction more precisely we first characterize all optimal hardy weights with respect to one dimensional subcritical sturm liouville operators on a b a b and then apply this result to obtain families of optimal hardy inequalities for general linear second order elliptic operators in higher dimensions as an application we prove a new rellich inequality

quantum scattering at low energies for a class of negative slowly decaying potentials including v x x with 0 2 we study the quantum mechanical scattering theory in the low energy regime using appropriate modifiers of the isozaki kitada type we show that scattering theory is well behaved on the whole continuous spectrum of the hamiltonian including the energy 0 we show that the modified scattering matrices s are welldefined and strongly continuous down to the zero energy threshold similarly we prove that the modified wave matrices and generalized eigenfunctions are norm continuous down to the zero energy if we use appropriate weighted spaces these results are used to derive oscillatory asymptotics of the standard short range and dollard type s matrices for the subclasses of potentials where both kinds of s matrices are defined for potentials whose leading part is x we show that the location of singularities of the kernel of s experiences an abrupt change from passing from positive energies to the limiting energy 0 this change corresponds to the behaviour of the classical orbits under stronger conditions one can extract the leading term of the asymptotics of the kernel of s at its singularities

a probabilistic approach to bounded positive solutions for schrodinger operators with certain classes of potentials consider the equation 1 2 u vu 0 in r d for d 3 for certain classes of potentials v we use probabilistic tools to study the bounded solutions and the positive solutions for a primary motivation is to offer probabilistic intuition for the results

homotopy types of subspace arrangements via diagrams of spaces we prove combinatorial formulas for the homotopy type of the union of the subspaces in an affine compactified affine spherical or projective subspace arrangement from these formulas we derive results of goresky macpherson on the homology of the arrangement and the cohomology of its complement the union of an arrangement can be interpreted as the direct limit of a diagram of spaces over the intersection poset a closely related space is obtained by taking the homotopy direct limit of this diagram our method consists in constructing a combinatorial model diagram over the same poset whose homotopy limit can be compared to the original one by usual homotopy comparison results for diagrams of spaces

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

dirichlet parabolicity and l1 liouville property under localized geometric conditions abstract we shed a new light on the l 1 liouville property for positive superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space we also present examples in any dimension showing that the l 1 liouville property is strictly weaker than the stochastic completeness of the manifold the main tool in our investigations is represented by the potential theory of a manifold with boundary subject to dirichlet boundary conditions the paper incorporates under a unifying viewpoint some old and new aspects of the theory with a special emphasis on global maximum principles and on the role of the dirichlet green s kernel

maximum and anti maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations in this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations we continue the study of some different notions of small perturbations and discuss their relations to comparisons of green s functions refined maximum and anti maximum principles ground state and the decay of eigenfunctions in particular we show that if v is a positive function which is a semismall perturbation of a subcritical schrodinger operator h defined on a domain omega subset mathbb r d and phi k k geq 0 are the dirichlet eigenfunctions of the equation hu lambda vu then for any k geq 0 the function phi k phi 0 is bounded and has a continuous extension up to the martin boundary of the pair omega h where phi 0 is the ground state of h with a principal eigenvalue lambda 0

decompositions of the polyhedral product functor with applications to moment angle complexes and related spaces this article gives a natural decomposition of the suspension of a generalized moment angle complex or partial product space which arises as the polyhedral product functor described below the introduction and application of the smash product moment angle complex provides a precise identification of the stable homotopy type of the values of the polyhedral product functor one direct consequence is an analysis of the associated cohomology for the special case of the complements of certain subspace arrangements the geometrical decomposition implies the homological decomposition in earlier work of others as described below because the splitting is geometric an analogous homological decomposition for a generalized moment angle complex applies for any homology theory implied therefore is a decomposition for the stanley reisner ring of a finite simplicial complex and natural generalizations

sharp condition on the decay of the potential for the absence of a zero energy ground state of the schrodinger equation the authors prove a sharp criterion on the decay of the potential of a schrodinger operator on r3 that ensures the absence of a zero energy ground state this condition complements results due to simon and lieb 1981 they also give an improved version of their results which however is still not optimal

structure of positive solutions to δ v u 0 in rn on etudie la structure de toutes les solutions positives d une equation de schrodinger stationnaire ou le potentiel v est suppose etre une fonction a valeur reelle appartenant a l p loc r n avec p n 2 pour n 2 et p 1 pour n 1 on considere une formule de representation explicite pour des solutions positives leurs asymptotes a l infini et le probleme de dirichlet avec une condition aux limites a l infini

the cohomology rings of complements of subspace arrangements the ring structure of the integral cohomology of complements of real linear subspace arrangements is considered while the additive structure of the cohomology is given in terms of the intersection poset and dimension function by a theorem of goresky and macpherson we describe the multiplicative structure in terms of the intersection poset the dimension function and orientations of the participating subspaces for the class of arrangements without pairs of intersections of codimension one in particular this yields a description of the integral cohomology ring of complex arrangements conjectured by yuzvinsky for general real arrangements a weaker result is obtained the approach is geometric and the methods are elementary

integral representations of nonnegative solutions for parabolic equations and elliptic martin boundaries abstract we consider nonnegative solutions of a parabolic equation in a cylinder d 0 t where d is a noncompact domain of a riemannian manifold under the assumption iu i e the associated heat kernel is intrinsically ultracontractive we establish an integral representation theorem any nonnegative solution is represented uniquely by an integral on d 0 m d 0 t where m d is the martin boundary of d for the associated elliptic operator we apply it in a unified way to several concrete examples to explicitly represent nonnegative solutions we also show that iu implies the condition sp i e the constant function 1 is a small perturbation of the elliptic operator on d

the hypertoric intersection cohomology ring we present a functorial computation of the equivariant intersection cohomology of a hypertoric variety and endow it with a natural ring structure when the hyperplane arrangement associated with the hypertoric variety is unimodular we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category the computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset

optimal hardy weight for second order elliptic operator an answer to a problem of agmon abstract for a general subcritical second order elliptic operator p in a domain r n or noncompact manifold we construct hardy weight w which is optimal in the following sense the operator p w is subcritical in for all 1 null critical in for 1 and supercritical near any neighborhood of infinity in for any 1 moreover if p is symmetric and w 0 then the spectrum and the essential spectrum of w 1 p are equal to 1 and the corresponding agmon metric is complete our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators the constructed hardy weight is given by an explicit simple formula involving two distinct positive solutions of the equation p u 0 the existence of which depends on the subcriticality of p in

asymptotics of green functions and martin boundaries for elliptic operators with periodic coefficients abstract we give the asymptotics at infinity of a green function for an elliptic equation with periodic coefficients on r d basic ingredients in establishing the asymptotics are an integral representation of the green function and the saddle point method we also completely determine the martin compactification of r d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the green function