subdivisions and local h vectors in part i a general theory of f vectors of simplicial subdivisions or triangulations of simplicial complexes is developed based on the concept of lo cal h vector as an application we prove that the h vector of a cohen macaulay complex increases under quasi geometric subdivision thus establishing a spe cial case of a conjecture of kalai and this author techniques include com mutative algebra homological algebra and the intersection homology of toric varieties in part ii we extend the work of part i to more general situations first a formal generalization of subdivision is given based on incidence alge bras special cases are then developed in particular one based on subdivisions of eulerian posets and involving generalized h vectors other cases deal with kazhdan lusztig polynomials ehrhart polynomials and a q analogue of eule rian posets many applications and examples are given throughout department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address rstan math mit edu this content downloaded from 157 55 39 210 on sat 30 jul 2016 05 59 16 utc all use subject to http about jstor org terms

zero temperature limit for interacting brownian particles i motion of a single body we consider a system of interacting brownian particles in r d with a pairwise potential which is radially symmetric of finite range and attains a unique minimum when the distance of two particles becomes a 0 the asymptotic behavior of the system is studied under the zero temperature limit from both microscopic and macroscopic aspects if the system is rigidly crystallized namely if the particles are rigidly arranged in an equal distance a the crystallization is kept under the evolution in macroscopic time scale then assuming that the crystal has a definite limit shape under a macroscopic spatial scaling the translational and rotational motions of such shape are characterized

face rings of simplicial complexes with singularities the face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is nonsingular i e has the homology of a wedge of spheres of the expected dimension this is derived from an enumerative result for local cohomology of face rings modulo generic linear forms as compared with local cohomology of the face ring itself the enumerative result is generalized to squarefree modules a concept of cohen macaulay in codimension c is defined and characterized for arbitrary finitely generated modules and coherent sheaves for the face ring of an r dimensional complex it is equivalent to nonsingularity of in dimension r c for a coherent sheaf on projective space this condition is shown to be equivalent to the same condition on any single generic hyperplane section the characterization of nonsingularity in dimension m via finite local cohomology thus generalizes from face rings to arbitrary graded modules

a classification of the face numbers of buchsbaum simplicial posets the family of buchsbaum simplicial posets generalizes the family of simplicial cell manifolds the h vector of a simplicial complex or simplicial poset encodes the combinatorial and topological data of its face numbers and the reduced betti numbers of its geometric realization novik and swartz showed that the h vector of a buchsbaum simplicial poset satisfies certain simple inequalities in this paper we show that these necessary conditions are in fact sufficient to characterize the h vectors of buchsbaum simplicial posets with prescribed betti numbers

weight monodromy conjecture for p adically uniformized varieties the aim of this paper is to prove the weight monodromy conjecture deligne s conjecture on the purity of monodromy filtration for varieties p adically uniformized by the drinfeld upper half spaces of any dimension the ingredients of the proof are to prove a special case of the hodge standard conjecture and apply a positivity argument of steenbrink m saito to the weight spectral sequence of rapoport zink as an application by combining our results with the results of schneider stuhler we compute the local zeta functions of p adically uniformized varieties in terms of representation theoretic invariants we also consider a p adic analogue by using the weight spectral sequence of mokrane

buchsbaumness of ordinary powers of two dimensional square free monomial ideals abstract let s k x 1 x 2 x n be a polynomial ring let i be a stanley reisner ideal in s of a pure simplicial complex of dimension one in this paper we study the buchsbaum property of s i r for any integer r 0 our first purpose is giving a characterization of ext modules ext s p s m t s j for any monomial ideal j where m t x 1 t x 2 t x n t in terms of certain simplicial complexes then we consider the buchsbaum property of s i r the main tool to check the buchsbaumness is the surjectivity criterion we see the behavior of the canonical map from ext s p s m t s i r to h m p s i r from the view point of reduced cohomology groups of simplicial complexes

applications of dualizing complexes to buchsbaum rings let a m k denote a local noetherian ring with d dim a admitting a normalized dualizing complex d by well known theorems on local duality a is cohen macaulay ring if and only if the d th place truncated dualizing complex r d is isomorphic to the zero complex in the derived category d a see section 1 for more precise statements in the following we consider a class of local rings for which the truncated dualizing complex t d is isomorphic to a complex of vector spaces over a m k there have been several efforts to get results about local rings which generalize the cohen macaulay property see for instance 7 13 14 a connection to our consideration is given in the following sense in 2 3 we prove one of our main theorems the complex t d of a is isomorphic to a complex of vector spaces over k if and only if a is buchsbaum ring in the sense of 13 141 they are those rings for which the difference l a d e x a between length and multiplicity of a system of parameters x does ndt depend on x see also 2 1 for a brief summary of essential facts about buchsbaum rings and relation to a conjecture of d a buchsbaum in 7 m hochster and j l roberts have studied purity of the frobenius map and related things to show that the local cohomology modules of certain rings of such type behave nicely applying our criterion 3 1 to this situation we can show in 4 1 that such rings have a truncated dualizing complex t d which is isomorphic to a complex of vector spaces over k following the examples given in 7 we get a number of buchsbaum rings for instance let vc ip be a projective variety defined by an ideal which is generated by square free monomials then v is arithmetically buchsbaum if and only if v is a cohen macaulay variety our theorem 2 3 is related to a result of r kiehl in 8 using the surjective map criterion of 141 he showed that the canonical module of a is buchsbaum if t d is isomorphic to a complex of vector spaces over k this yields in 8 that the local rings at the cusps of hilbert modular groups are factorial buchsbaum rings which are not cohen macaulay 61 oool 8708 82 040061 17 05 00 o

exponential stability of nonautonomous linear differential equations with linear perturbations by liao methods in this paper the author considers by liao methods the stability of lyapunov exponents of a nonautonomous linear differential equations dx dt a t x t x r rn with linear small perturbations it is proved that if a t is a upper triangular real n by n matrix valued function on r continuous and uniformly bounded and if there is a relatively dense sequence ti 0 in r say 0 t0 t1 ti such that a lim supi 1ti i 0i 1max1 k n ti ti 1akk t dt 0 then there corresponds to any e 0 a constant 0 such that for every linear equations b dz dt b t z t z r rn satisfying supt r b t a t where the real n by n matrix valued function b t is continuous with respect to t r one has lim supt 1tlog z t z 0 a e z 0 rn where z t z t z 0 is the solution of eq b with z 0 z 0 for the nonuniformly expanding case there is a similar statement

lifting inequalities for polytopes we present a method of lifting linear inequalities for the flag f vector of polytopes to higher dimensions known inequalities that can be lifted using this technique are the non negativity of the toric g vector and that the simplex minimizes the cd index we obtain new inequalities for six dimensional polytopes in the last section we present the currently best known inequalities for dimensions 5 through 8

on simple polytopes letp be a simpled polytope ind dimensional euclidean space mathbb e d and let p be the subalgebra of the polytope algebra generated by the classes of summands ofp it is shown that the dimensions of the weight spaces r p of p are theh numbers ofp which describe the dehn sommerville equations between the numbers of faces ofp and reflect the duality between r p and d r p moreover p admits a lefschetz decomposition under multiplication by the element of 1 p corresponding top itself which yields a proof of the necessity of mcmullen s conditions in theg theorem on thef vectors of simple polytopes the lefschetz decomposition is closely connected with the new hodge riemann minkowski quadratic inequalities between mixed volumes which generalize minkowski s second inequality also proved are analogous generalizations of the aleksandrov fenchel inequalities a striking feature is that these are obtained without using brunn minkowski theory indeed the brunn minkowski theorem without characterization of the cases of equality can be deduced from them the connexion found between p and the face ring of the dual simplicial polytopep enables this ring to be looked at in two ways and a conjectured formulation of theg theorem in terms of a gale diagram ofp is also established

ext and local cohomology modules of face rings of simplicial posets abstract there are a large number of theorems detailing the homological properties of the stanley reisner ring of a simplicial complex here we attempt to generalize some of these results to the case of a simplicial poset by investigating the combinatorics of certain modules associated with the face ring of a simplicial poset from a topological viewpoint we extend some results of miyazaki and grabe to a wider setting

simplicial moves on balanced complexes abstract we introduce a notion of cross flips local moves that transform a balanced i e properly d 1 colored triangulation of a combinatorial d manifold into another balanced triangulation these moves form a natural analog of bistellar flips also known as pachner moves specifically we establish the following theorem any two balanced triangulations of a closed combinatorial d manifold can be connected by a sequence of cross flips along the way we prove that for every m d 2 and any closed combinatorial d manifold m two m colored triangulations of m can be connected by a sequence of bistellar flips that preserve the vertex colorings

cohen macaulayness of large powers of stanley reisner ideals abstract we prove that for m 3 the symbolic power i m of a stanley reisner ideal is cohen macaulay if and only if the simplicial complex is a matroid similarly the ordinary power i m is cohen macaulay for some m 3 if and only if i is a complete intersection these results solve several open questions on the cohen macaulayness of ordinary and symbolic powers of stanley reisner ideals moreover they have interesting consequences on the cohen macaulayness of symbolic powers of facet ideals and cover ideals

almost periodic and almost automorphic solutions of linear differential difference equations without favard s separation condition ii abstract the well known favard s theorem states that the linear differential equation 1 x a t x f t with bohr almost periodic coefficients admits at least one bohr almost periodic solution if it has a bounded solution the main assumption in this theorem is the separation among bounded solutions of homogeneous equations 2 x b t x where b h a b b t lim n a t t n if there are bounded solutions which are non separated sometimes almost periodic solutions do not exist r johnson r ortega and m tarallo v zhikov and b levitan in this paper we prove that linear differential equation 1 with levitan almost periodic coefficients has a unique levitan almost periodic solution if it has at least one bounded solution and the bounded solutions of the homogeneous equation 3 x a t x are homoclinic to zero i e lim t t 0 for all bounded solutions of 3 if the coefficients of 1 are bohr almost periodic and all bounded solutions of all limiting equations 2 are homoclinic to zero then eq 1 admits a unique almost automorphic solution the analogue of this result for difference equations is also given we study the problem of existence of bohr levitan almost periodic solutions of eq 1 in the framework of general non autonomous dynamical systems cocycles

the signature of a toric variety we identify a combinatorial quantity the alternating sum of the h vector defined for any simple polytope as the signature of a toric variety this quantity was introduced by r charney and m davis in their work which in particular showed that its nonnegativity is closely related to a conjecture of h hopf on the euler characteristic of a nonpositively curved manifold we prove positive or nonnegative lower bounds for this quantity under geometric hypotheses on the polytope and in particular resolve a special case of their conjecture these hypotheses lead to ampleness or weaker conditions for certain line bundles on toric divisors and then the lower bounds follow from calculations using the hirzebruch signature formula moreover we show that under these hypotheses on the polytope the ith l class of the corresponding toric variety is 1 i times an effective class for any i

the polytope algebra let f be an ordered field and let p denote the family of all convex polytopes in the d dimensional vector space v over f the universal abelian group corresponding to the translation invariant valuations on p has generators p for p p with 0 satisfying the relations v p q p p q whenever p q p q p and t p t p for p p and t v with multiplication induced by m p q p q is almost a graded commutative algebra over f in that dr 0 r with 0 z r a vector space over f r 1 and r s r s r s 0 r 0 forr d the dilatation d p p for p p and f is such that x rx for x r and 0 negative dilatations arise from the euler map e p p f 1 dimf f the sum extending over all faces f of p since x rx for x r and 0 separating group homomorphisms for are the frame functionals which give the volumes of the faces of polytopes determined by successive support hyperplanes in sequences of directions two isomorphisms on are described one related to cones of outer normal vectors and the other to the polytope groups obtained from by discarding polytopes of dimension less than d various applications of the polytope algebra are given including a theory of mixed polytopes which has implications for mixed valuations

almost periodic and almost automorphic solutions of linear differential equations the paper is dedicated to the study problem of existence of almost periodic respectively almost automorphic recurrent solutions of linear differential equations u t a t y u t f t y y y in banach space with almost periodic respectively almost automorphic recurrent coefficients where y r is a dynamical system on the metric space y in particular we prove that if the operator a y is positive and auto adjoint then for the equation 1 u t a t y u t 0 one of the following alternative is fulfilled i there exists a complete trajectory of 1 with constant positive norm ii the trivial solution of equation 1 is uniformly asymptotically stable this problem we investigate in the framework of general linear non autonomous dynamical system we apply our general results also for the functional differential equations and difference equations

rigidity percolation in a field rigidity percolation with g degrees of freedom per site is analyzed on randomly diluted erdos renyi graphs with average connectivity gamma in the presence of a field h in the gamma h plane the rigid and flexible phases are separated by a line of first order transitions whose location is determined exactly this line ends at a critical point with classical critical exponents analytic expressions are given for the densities n f of uncanceled degrees of freedom and gamma r of redundant bonds upon crossing the coexistence line gamma r and n f are continuous although their first derivatives are discontinuous we extend for the case of nonzero field a recently proposed hypothesis namely that the density of uncanceled degrees of freedom is a free energy for rigidity percolation analytic expressions are obtained for the energy entropy and specific heat some analogies with a liquid vapor transition are discussed particularizing to zero field we find that the existence of a g 1 core is a necessary condition for rigidity percolation with g degrees of freedom at the transition point gamma c maxwell counting of degrees of freedom is exact on the rigid cluster and on the g 1 rigid core i e the average coordination of these subgraphs is exactly 2g although gamma c the average coordination of the whole system is smaller than 2g gamma c is found to converge to 2g for large g i e in this limit maxwell counting is exact globally as well

regression to the mean for the bivariate binomial distribution regression to the mean rtm occurs when subjects having relatively high or low measurements are remeasured and found closer to the population mean this phenomenon can potentially lead to an inaccurate conclusion in a pre post study design expressions are available for quantifying rtm when the distribution of pre and post observations are bivariate normal and bivariate poisson however situations exist where the response variables are the number of successes in a fixed number of trials and follow the bivariate binomial distribution in this article expressions for quantifying rtm effects are derived when the underlying distribution is the bivariate binomial unlike the normal and poisson distributions the correlation between pre and post observations can be either negative or positive under the bivariate binomial distribution and the severity of rtm is greater in the former case the percentage relative difference is used to highlight the differences in quantifying rtm under the bivariate binomial distribution and normal and poisson approximations to the bivariate binomial distribution expressions for estimating rtm using the method of maximum likelihood along with its asymptotic distribution are derived a simulation study is conducted to empirically assess the statistical properties of the rtm estimator and its asymptotic distribution data examples using the number of obese individuals and the number of nonconforming cardboard cans are discussed

on the generalized lower bound conjecture for polytopes and spheres in 1971 mcmullen and walkup posed the following conjecture which is called the generalized lower bound conjecture if p is a simplicial d polytope then its h vector h0 h1 hd satisfies h 0 leq h 1 leq ldots leq h left lfloor d left 2 right right rfloor moreover if hr 1 hr for some r leq frac 1 2 d then p can be triangulated without introducing simplices of dimension d r

a generalized lower bound theorem for balanced manifolds a simplicial complex of dimension d 1 is said to be balanced if its graph is d colorable juhnke kubitzke and murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes we establish a generalization of their result to balanced triangulations of closed homology manifolds and balanced triangulations of orientable homology manifolds with boundary under an additional assumption that all proper links of these triangulations have the weak lefschetz property as a corollary we show that if delta is an arbitrary balanced triangulation of any closed homology manifold of dimension d 1 ge 3 then 2h 2 delta d 1 h 1 delta ge 4 d atopwithdelims 2 tilde beta 1 delta tilde beta 0 delta thus verifying a conjecture by klee and novik to prove these results we develop the theory of flag h vectors

complete guiding sets for a class of almost periodic differential equations abstract a functional differential equation admitting a complete guiding set always has an a priori bounded solution the introduction of this concept and the idea of reducibility allow the authors to apply a well known method for the autonomous case to a non autonomous problem a previous result of ortega and tineo is extended to a non hyperbolic semilinear functional differential equation with almost periodic coefficients in its lineal part the non hyperbolicity condition is established in terms of the sacker sell spectrum of the associated linear system

quantifying the regression to the mean effect in poisson processes regression to the mean rtm can occur whenever an extreme observation is selected from a population and a later observation is closer to the population mean a consequence of this phenomenon is that natural variability can be mistaken as real change simple expressions are available to quantify rtm when the underlying distribution is bivariate normal however there are many real world situations which are better approximated as a poisson process examples include the number of hard disk failures during a year the number of cargo ships damaged by waves daily homicide counts in california and the number of deaths per quarter attributable to acquired immune deficiency syndrome in australia in this paper we derive expressions for quantifying rtm effects for the bivariate poisson distribution for both the homogeneous and inhomogeneous cases statistical properties of our derivations have been evaluated through a simulation study the asymptotic distributions of rtm estimators have been derived the rtm effect for the number of people killed in road accidents in different regions of new south wales australia is estimated using maximum likelihood

euclidean distance geometry and applications euclidean distance geometry is the study of euclidean geometry based on the concept of distance this is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in euclidean space realizing those given distances we survey the theory of euclidean distance geometry and its most important applications with special emphasis on molecular conformation problems

on bounded solutions of almost periodic linear differential systems has an almost periodic solution with frequency module contained in the joint frequency module of a and f whenever it has a bounded solution the method of proof was his so called minimax method favard s result has been extended to equations in banach space and partial differential equations amerio and prouse 11 sibuya 171 sell 151 zhikov and levitan 191 however apart from some results of bochner 3 and cameron 4 cf also fink 10 p 1081 no conditions have been given which ensure that bounded solutions of the homogeneous system 1 itself are almost periodic as shown by an example of lillo 13 p 4061 cf also bochner 3 p 285 it can happen that all the solutions of 1 are bounded that all systems in the hull have the favard property and still no nontrivial solution is almost periodic it is the main purpose of this paper to show nevertheless that favard s minimax method can be used to show that under certain conditions some bounded solutions of 1 are almost periodic in particular if a t a t is real with u f 0 for i j and all systems in the hull have the favard property then 1 has a nontrivial almost periodic solution if it has a nontrivial bounded solution whose components are nonnegative this

effect of regression to the mean in the presence of within subject variability regression to the mean arises often in statistical applications where the units chosen for study relate to some observed characteristic in the extreme of its distribution gardner and heady attribute the effect of regression to the mean to measurement errors they assume the model yi u ei where u is a fixed within subject component and ei is the random measurement error they suggest several replicate measurements to reduce the regression effect under the assumption that the measurement errors ei are independent within subjects while measurement errors play an important role in regression to the mean one should not overlook within subject variation in this paper we consider a model to estimate the regression effect in the presence of correlated within subject effects as well as independent measurement errors

graph rigidity for unitarily invariant matrix norms abstract a rigidity theory is developed for bar joint frameworks in linear matrix spaces endowed with a unitarily invariant matrix norm analogues of maxwell s counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the matroidal class of k l sparse graphs for suitable k and l an edge colouring technique is developed to characterise infinitesimal rigidity for product norms and then applied to show that the graph of a minimally rigid bar joint framework in the space of 2 2 symmetric respectively hermitian matrices with the trace norm admits an edge disjoint packing consisting of a euclidean rigid graph and a spanning tree

mechanics of materials with periodic truss or frame micro structures this paper describes the mechanics of materials with periodic skeletal micro structures in infinite domains the principal technical results consist of certain korn type inequalities that provide upper and lower bounds for the linear elastic strain energy in the material using these inequalities existence and uniqueness results for the equations of linear elastic equilibrium are derived and some asymptotic properties of the solutions are described particular attention is paid to the question of when a lattice structure can accurately be modeled as a pin jointed truss and when a rigid node frame model must be employed a practical technique for how to distinguish between the two types of material is given and the distinct differences in their mechanical behavior are described

infinitesimally rigid polyhedra ii modified spherical frameworks in the first paper alexandrov s theorem was studied and extended to show that convex polyhedra form statically rigid frameworks in space when built with plane rigid faces this second paper studies two modifications of these polyhedral frameworks i block polyhedral frameworks with some discs as open holes other discs as spacbrigid blocks and the remaining faces plane rigid and ii extended polyhedral frameworks with individually added bars shafts and selected edges removed inductive methods are developed to show the static rigidity of particular patterns of holes and blocks and of extensions in general realizations of the polyhedron the methods are based on proof techniques for steinitz s theorem and a related coordinatization of the proper realizations of a 3 connected spherical polyhedron sample results show that a a single k gonal block and a k gonal hole yield static rigidity if and only if the block and hole are k connected in a vertex sense and b a 4 connected triangulated sphere with one added bar is a statically rigid circuit removing any one bar leaves a minimal statically rigid framework the results are also interpreted as a description of which dihedral angles in a triangulated sphere will flex when one bar is removed

ergodic properties and rotation number for linear hamiltonian systems abstract this paper is concerned with the dynamical behavior of the solutions of a class of linear hamiltonian systems including those to which kotani s theory applies we first present a symplectic l 2 perron transformation which takes these systems into skew symmetric form this allows us to study the average of the trajectories and the fourier coefficients of the solutions in addition from the construction of two invariant complex lagrange planes the differentiability of the rotation number is analyzed

the morse spectrum of linear flows on vector bundles for a linear flow on a vector bundle e s a spectrum can be defined in the following way for a chain recurrent component m on the projective bundle pe consider the exponential growth rates associated with finite time e t chains in m and define the morse spectrum mo m over m as the limits of these growth rates as e 0 and t the morse spectrum mo of is then the union over all componentsm pe this spectrum is a synthesis of the topological approach of selgrade and salamon zehnder with the spectral concepts based on exponential growth rates such as the oseledec spectrum or the dichotomy spectrum of sacker sell it turns out that mo contains all lyapunov exponents of for arbitrary initial values and the mo m are closed intervals whose boundary points are actually lyapunov exponents using the fact that is cohomologous to a subflow of a smooth linear flow on a trivial bundle one can prove integral representations of all morse and all lyapunov exponents via smooth ergodic theory a comparison with other spectral concepts shows that in general the morse spectrum is contained in the topological spectrum and the dichotomy spectrum but the spectral sets agree if the induced flow on the base space is chain recurrent however even in this case the associated subbundle decompositions of e may be finer for the morse spectrum than for the dynamical spectrum if one can show that the closure of the floquet spectrum i e the lyapunov spectrum based on periodic trajectories in pe agrees with the morse spectrum then one obtains equality for the floquet the entire oseledec the lyapunov and the morse spectrum we present an example flows induced by c vector fields with hyperbolic chain recurrent components on the projective bundle where this fact can be shown using a version of bowen s shadowing lemma

exponential dichotomy rotation number and linear differential operators with bounded coefficients on considere la relation entre la dichotomie exponentielle de certains systemes differentiels lineaires ou aux differences et le spectre des operateurs differentiels lineaires auto adjoints associes l on etudie une facon systematique d enumerer les intervalles ouverts dans la resolvante de l

favard theory for the adjoint equation and fredholm alternative abstract fredholm alternative is a classical tool of periodic linear equations allowing to describe the existence of periodic solutions of an inhomogeneous equation in terms of the adjoint equation a few partial extensions have been proposed in the literature for recurrent equations our aim is to point out that they have a common root and discuss whether such a root gives rise to a general fredholm type alternative sacker sell spectral theory and favard theory are main ingredients in this discussion a considerable effort is devoted to understand how favard theory is affected by adjunction at least for planar equations

global attractors in fiber bundles and right invariant systems abstract this article deals with the notions of lyapunov stability and attraction of semiflows on fiber bundles the focus of the studies is the question on the existence of global attractors in bundles associated to principal bundles applications to right invariant systems are presented

infinitesimally rigid polyhedra i statics of frameworks from the time of cauchy mathematicians have studied the motions of convex polyhedra with the faces held rigid while changes are allowed in the dihedral angles in the 1940s alexandrov proved that even with additional vertices along the natural edges and with an arbitrary triangulation of the natural faces on these vertices such polyhedra are infinitesimally rigid in this paper the dual and equivalent concept of static rigidity for frameworks is used to describe the behavior of bar and joint frameworks built around convex and other polyhedra the static techniques introduced provide a new simplified proof of alexandrov s theorem as well as an essential extension which characterizes the static properties of frameworks built with more general patterns on the faces including frameworks with vertices interior to the faces the static techniques are presented and employed in a pattern appropriate to the extension of an arbitrary statically rigid framework built around any polyhedron nonconvex toroidal etc the techniques are also applied to derive the static rigidity of tensegrity frameworks with cables and struts in place of bars and the static rigidity of frameworks projectively equivalent to known polyhedral frameworks finally as an exercise to give an additional perspective to the results in 3 space detailed analogues of alexandrov s theorem are presented for convex 4 polytopes built as bar and joint frameworks in 4 space

socles of buchsbaum modules complexes and posets abstract the socle of a graded buchsbaum module is studied and is related to its local cohomology modules this algebraic result is then applied to face enumeration of buchsbaum simplicial complexes and posets in particular new necessary conditions on face numbers and betti numbers of such complexes and posets are established these conditions are used to settle in the affirmative kuhnel s conjecture for the maximum value of the euler characteristic of a 2k dimensional simplicial manifold on n vertices as well as kalai s conjecture providing a lower bound on the number of edges of a simplicial manifold in terms of its dimension number of vertices and the first betti number

cantor spectrum for schrodinger operators with potentials arising from generalized skew shifts we consider continuous sl 2 r cocycles over a strictly ergodic home omorphism which fibers over an almost periodic dynamical system generalized skew shifts we prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an so 2 r cocycle using this we show that if a cocycle s homotopy class does not display a certain obstruc tion to uniform hyperbolicity then it can be c 0 perturbed to become uniformly hyperbolic for cocycles arising from schrodinger operators the obstruction van ishes and we conclude that uniform hyperbolicity is dense which implies that for a generic continuous potential the spectrum of the corresponding schrodinger operator is a cantor set 1 statement of the results

topological finiteness for edge vertex enumeration the number of pl homeomorphism types of combinatorial manifolds in a fixed dimension with an upper bound on g2 is finite

linear programming the simplex algorithm and simple polytopes in the first part of the paper we survey some far reaching applications of the basic facts of linear programming to the combinatorial theory of simple polytopes in the second part we discuss some recent developments concerning the simplex algorithm we describe subexponential randomized pivot rules and upper bounds on the diameter of graphs of polytopes 1997 the mathematical programming society inc published by elsevier science b v

on face numbers of manifolds with symmetry abstract necessary conditions on the face numbers of cohen macaulay simplicial complexes admitting a proper action of the cyclic group z p z of a prime order are given this result is extended further to necessary conditions on the face numbers and the betti numbers of buchsbaum simplicial complexes with a proper z p z action adin s upper bounds on the face numbers of cohen macaulay complexes with symmetry are shown to hold for all d 1 dimensional buchsbaum complexes with symmetry on n 3 d 2 vertices a generalization of kuhnel s conjecture on the euler characteristic of 2 k dimensional manifolds and sparla s analog of this conjecture for centrally symmetric 2 k manifolds are verified for all 2 k manifolds on n 6 k 3 vertices connections to the upper bound theorem are discussed and its new version for centrally symmetric manifolds is established

real quadrics in cn complex manifolds and convex polytopes in this paper we investigate the topology of a class of non kahler compact complex manifolds generalizing that of hopf and calabi eckmann manifolds these manifolds are diffeomorphic to special systems of real quadrics cn which are invariant with respect to the natural action of the real torus s1 n onto cn the quotient space is a simple convex polytope the problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes we prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich we also resolve an associated wall crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope finally as a nice consequence we obtain that affine non kahler compact complex manifolds can have arbitrary amount of torsion in their homology groups contrasting with the kahler situation

three dimensional normal pseudomanifolds with relatively few edges abstract let be a d dimensional normal pseudomanifold d 3 a relative lower bound for the number of edges in is that g 2 of is at least g 2 of the link of any vertex when this inequality is sharp has relatively minimal g 2 for example whenever the one skeleton of equals the one skeleton of the star of a vertex then has relatively minimal g 2 subdividing a facet in such an example also gives a complex with relatively minimal g 2 we prove that in dimension three these are the only examples as an application we determine the combinatorial and topological type of 3 dimensional with relatively minimal g 2 whenever has two or fewer singularities the topological type of any such complex is a pseudocompression body a pseudomanifold version of a compression body complete combinatorial descriptions of with g 2 2 are due to kalai 12 g 2 0 nevo and novinsky 13 g 2 1 and zheng 20 g 2 2 in all three cases is the boundary of a simplicial polytope zheng observed that for all d 0 there are triangulations of s d r p 2 with g 2 3 she asked if this is the only nonspherical topology possible for g 2 3 as another application of relatively minimal g 2 we give an affirmative answer when is 3 dimensional

alexander duality for stanley reisner rings and squarefree nn graded modules abstract let s k x1 xn be a polynomial ring and let s be its canonical module first we will define squarefreeness for n n graded s modules a stanley reisner ring k s i its syzygy module syzi k and extis k s are always squarefree this notion will simplify some standard arguments in the stanley reisner ring theory next we will prove that the i linear strand of the minimal free resolution of a stanley reisner ideal i s has the same information as the module structure of extis k s where is the alexander dual of in particular if k has a linear resolution we can describe its minimal free resolution using the module structure of the canonical module of k which is cohen macaulay in this case we can also give a new interpretation of a result of herzog and co workers which states that k is sequentially cohen macaulay if and only if i is componentwise linear

hard lefschetz theorem for nonrational polytopes the hard lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope one can construct the intersection cohomology combinatorially from the polytope hence it is well defined even for nonrational polytopes when there is no variety associated to it we prove the hard lefschetz theorem for the intersection cohomology of a general polytope

rigidity and the lower bound theorem 1 for an arbitrary triangulated d 1 manifold without boundaryc withf 0 vertices andf 1 edges define gamma c f 1 df 0 left begin array 20 c d 1 2 end array right barnette proved that c 0 we use the rigidity theory of frameworks and in particular results related to cauchy s rigidity theorem for polytopes to give another proof for this result we prove that ford 4 if c 0 thenc is a triangulated sphere and is isomorphic to the boundary complex of a stacked polytope other results a we prove a lower bound conjectured by bjorner for the number ofk faces of a triangulated d 1 manifold with specified numbers of interior vertices and boundary vertices b ifc is a simply connected triangulatedd manifold d 4 and lk v c 0 for every vertexv ofc then c 0 lk v c is the link ofv inc c letc be a triangulatedd manifold d 3 then ske11 d 2 can be embedded in skel1 c iff c 0 d is thed dimensional simplex d ifp is a 2 simpliciald polytope then f 1 p geqq df 0 p left begin array 20 c d 1 2 end array right related problems concerning pseudomanifolds manifolds with boundary and polyhedral manifolds are discussed