shifted simplicial complexes are laplacian integral we show that the combinatorial laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra we give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences generalizing a theorem of merris for graphs we also conjecture a majorization inequality for the spectra of these laplace operators in an arbitrary simplicial complex with equality achieved if and only if the complex is shifted this generalizes a conjecture of grone and merris for graphs

on a class of special linear systems of mathbb p 3 in this paper we deal with linear systems of p 3 through fat points we consider the behavior of these systems under a cubic cremona transformation that allows us to produce a class of special systems which we conjecture to be the only ones

combinatorial decompositions of a class of rings in this paper we introduce some combinatorial methods for the study of graded rings and use these methods to analyze a class of rings that are of importance in combinatorics our methods are based on a unique representation theorem which is shown section 2 to hold for any graded algebra and which in a sense is an extension of the direct sum decomposition for cohen macaulay algebras this unique representation is reminiscent of a similar result obtained by rees 20 both results yield a decomposition of the algebra however our result differs from that of rees in that it carries more information about the algebra our main results are concerned with a class of rings see section 3 for the definition associated to simplicial complexes there are several natural operations that can be performed on simplicial complexes to obtain new complexes which in turn have an interpretation for the corresponding rings we show sections 4 5 and 6 that our basic decompositions can be transferred during these operations from the original rings to the newly constructed ones the operations studied here are the chain transform or barycentric subdivision section 4 rank selection both for partially ordered sets and for simplicial complexes section s and localization section 6 as we develop these tools we give some applications in section 5 we give a new topological characterization of the cohen macaulay property for partially ordered sets an immediate consequence of which is the rankselection theorem baclawski 2 this characterization is closely related to those obtained by garsia in 14

a categorification of biclosed sets of strings abstract we consider a closure space known as biclosed sets of strings of a gentle algebra of finite representation type palu pilaud and plamondon proved that the collection of all biclosed sets of strings forms a lattice and moreover that this lattice is congruence uniform many interesting examples of finite congruence uniform lattices may be represented as the lattice of torsion classes of an associative algebra we introduce a generalization the lattice of torsion shadows and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows when every indecomposable module over the gentle algebra is a brick finite congruence uniform lattices admit an alternate partial order known as the core label order in many cases the core label order of a congruence uniform lattice is isomorphic to a lattice of wide subcategories of an associative algebra analogous to torsion shadows we introduce wide shadows and prove that the core label order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows

on a notion of speciality of linear systems in p n 2014 american mathematical society given a linear system in pn with assigned multiple general points we compute the cohomology groups of its strict transforms via the blow up of its linear base locus this leads us to give a new definition of expected dimension of a linear system which takes into account the contribution of the linear base locus and thus to introduce the notion of linear speciality we investigate such a notion giving sufficient conditions for a linear system to be linearly non special for an arbitrary number of points and necessary conditions for a small number of points

a broad class of shellable lattices abstract we introduce a new class of lattices the modernistic lattices and their duals the comodernistic lattices we show that every modernistic or comodernistic lattice has shellable order complex we go on to exhibit a large number of examples of co modernistic lattices we show comodernism for two main families of lattices that were not previously known to be shellable the order congruence lattices of finite posets and a weighted generalization of the k equal partition lattices we also exhibit many examples of co modernistic lattices that were already known to be shellable to begin with the definition of modernistic is a common weakening of the definitions of semimodular and supersolvable we thus obtain a unified proof that lattices in these classes are shellable subgroup lattices of solvable groups form another family of comodernistic lattices that were already proven to be shellable we show not only that subgroup lattices of solvable groups are comodernistic but that solvability of a group is equivalent to the comodernistic property on its subgroup lattice indeed the definition of comodernistic exactly requires on every interval a lattice theoretic analogue of the composition series in a solvable group thus the relation between comodernistic lattices and solvable groups resembles in several respects that between supersolvable lattices and supersolvable groups

subspace arrangements over finite fields cohomological and enumerative aspects abstract the enumeration of points on or off the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function we discuss the basic relations between these two points of view counting points is also related to the l adic cohomology of the arrangement as a variety we describe the eigenvalues of the frobenius map acting on this cohomology which corresponds to a finer decomposition of the zeta function the l adic cohomology groups and their decomposition into eigenspaces are shown to be fully determined by combinatorial data finally it is shown that the zeta function is determined by the topology of the corresponding complex variety in some important cases

alexander duality and stanley depth of multigraded modules abstract we apply miller s theory on multigraded modules over a polynomial ring to the study of the stanley depth of these modules several tools for stanley s conjecture are developed and a few partial answers are given for example we show that taking the alexander duality twice but with different centers is useful for this subject generalizing a result of apel we prove that stanley s conjecture holds for the quotient by a cogeneric monomial ideal

hilbert series and hilbert depth of squarefree veronese ideals abstract in this paper we obtain explicit formulas for the hilbert series and hilbert depth of squarefree veronese ideals in a standard graded polynomial ring

grobner bases of syzygies and stanley depth abstract let f be any free resolution of a finitely generated z n graded module over the polynomial ring k x 1 x n we show that for a suitable term order on f then for 0 p n the initial module of the p 1 th syzygy module z p f p is generated by terms m i e i where the m i are monomials in k x p 1 x n also for a large class of free resolutions f encompassing eliahou kervaire resolutions we show that a grobner basis for z p is given by the boundaries of generators of f p 1 we apply the above to give lower bounds for the stanley depth of the p 1 th syzygy module z p in particular showing it is greater or equal to p 1 we also show that if i is any squarefree ideal in k x 1 x n the stanley depth of i is at least of order 2 n

a characterization of solvability for finite groups in terms of their frame abstract the frame of a group is the poset of conjugacy classes of all its proper subgroups in this paper we will prove that a finite group is solvable if and only if every collection of maximal elements of its frame has a well defined meet and the poset consisting of all such meets including the meet of the empty set is a modular lattice

structure of the loday ronco hopf algebra of trees abstract loday and ronco defined an interesting hopf algebra structure on the linear span of the set of planar binary trees they showed that the inclusion of the hopf algebra of non commutative symmetric functions in the malvenuto reutenauer hopf algebra of permutations factors through their hopf algebra of trees and these maps correspond to natural maps from the weak order on the symmetric group to the tamari order on planar binary trees to the boolean algebra we further study the structure of this hopf algebra of trees using a new basis for it we describe the product coproduct and antipode in terms of this basis and use these results to elucidate its hopf algebraic structure in the dual basis for the graded dual hopf algebra our formula for the coproduct gives an explicit isomorphism with a free associative algebra we also obtain a transparent proof of its isomorphism with the non commutative connes kreimer hopf algebra of foissy and show that this algebra is related to non commutative symmetric functions as the commutative connes kreimer hopf algebra is related to symmetric functions

topology of order complexes of intervals in subgroup lattices abstract we conjecture that the order complex of an open interval in the subgroup lattice of a finite group has the homotopy type of a wedge of spheres and prove that if h g is a minimal counterexample to this conjecture then either g is almost simple or g hn where n is the unique minimal normal subgroup of g n is non abelian and h n 1

saturation and castelnuovo mumford regularity abstract let i be a homogeneous ideal of the polynomial ring k x 0 x n where k is an arbitrary field avoiding the construction of a minimal graded free resolution of i we provide effective methods for computing the castelnuovo mumford regularity of i that also compute other cohomological invariants of k x 0 x n i

how to compute the stanley depth of a monomial ideal abstract let j i be monomial ideals we show that the stanley depth of i j can be computed in a finite number of steps we also introduce the fdepth of a monomial ideal which is defined in terms of prime filtrations and show that it can also be computed in a finite number of steps in both cases it is shown that these invariants can be determined by considering partitions of suitable finite posets into intervals

stanley depth of complete intersection monomial ideals and upper discrete partitions abstract let i be an m generated complete intersection monomial ideal in s k x 1 x n we show that the stanley depth of i is n m 2 we also study the upper discrete structure for monomial ideals and prove that if i is a squarefree monomial ideal minimally generated by 3 elements then the stanley depth of i is n 1

linear decision trees subspace arrangements and mobius functions let p be a set in rn given a point x we want to test if x e p our model of computation is a linear decision tree a rooted ternary tree t where with each interior node v is associated a linear function lv x l j ajxj b and the three edges connecting v to its descendants are labelled starting from the root we move down the tree at each internal node v we check whether iv x 0 and follow the appropriately labelled edge leaves are labelled yes and no and arriving at a leaf we read off the answer to the question is x e p linear decision trees are sometimes surprisingly powerful devices we mention here the result that some np complete problems like the knapsack problem have polynomial depth linear decision trees mh lower bound results on the size or depth of linear decision trees usually depend on counting the number of connected components of p or r n p this method was introduced by dobkin and lipton dl and further developed by steele and yao sy ben or bo and others in this paper we describe topological methods that can be applied to obtain lower bounds also if these sets are connected the basic idea of this paper was first explored in bly the euler characteristic of p or r n p was used to estimate the number of leaves in a linear decision tree for p here we improve and extend the topological methods of bly in particular we show that the euler characteristic can be replaced by the sum of the betti numbers of r n p in bl y the method was applied to the case when p is a subspace arrangement i e a union of a finite number of affine subspaces the euler characteristic of the union is essentially the mobius function of the intersection lattice of these subspaces in this paper using a formula of goresky and macpherson gm we work with the betti numbers of the complement of this union and thereby obtain better bounds the motivating problem in bly was the k equal problem given n real numbers xl xn decide if some k of them are equal it was shown that ev ry linear decision tree for the k equal problem has size at least 2njk 0 n and consequently depth at least q nlog 2njk which is best possible we

on borel fixed ideals generated in one degree abstract we construct a shellable polytopal cell complex that supports a minimal free resolution of a borel fixed ideal which is borel generated by just one monomial in s k x 1 x 2 x n this includes the case of powers of the homogeneous maximal ideal x 1 x 2 x n as a special case in our most general result we prove that for any borel fixed ideal i generated in one degree there exists a polytopal cell complex that supports a minimal free resolution of i

generalized quotients in coxeter groups for w s a coxeter group we study sets of the form w v w e w i l wv 1 w i v for all v e v where v c w such sets w v here called generalized quotients are shown to have much of the rich combinatorial structure under bruhat order that has previously been known only for the case when v c s i e for minimal coset representatives modulo a parabolic subgroup we show that bruhat intervals in w v for general v c w are lexicographically shellable the mobius function on w v under bruhat order takes values in 1 0 1 for finite groups w generalized quotients are the same thing as lower intervals in the weak order this is however in general not true connections with the weak order are explored and it is shown that w v is always a complete meet semilattice and a convex order ideal as a subset of w under weak order descent classes di w e w il1 ws 1 w xt s e i for all s e s i c s are also analyzed using generalized quotients it is shown that each descent class as a poset under bruhat order or weak order is isomorphic to a generalized quotient under the corresponding ordering the latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics for instance the set of standard young tableaux of a fixed shape or the set of linear extensions of a rooted forest suitably interpreted form generalized quotients we prove a factorization result for the quotients that come from rooted forests which shows that algebraically these quotients behave as a system of minimal coset representatives of a subset which is in general not a subgroup we also study the rank generating function for certain quotients in the symmetric group

on linear systems of p 3 through multiple points abstract we prove a conjecture about the dimension of linear systems of surfaces of degree d in p 3 through at most eight multiple points in general position

global existence for semilinear evolution equations with nonlocal conditions abstract in this paper we study the global existence of solutions for semilinear evolution equations with nonlocal conditions via a fixed point analysis approach using the leray schauder alternative we derive conditions under which a solution exists globally

why the characteristic polynomial factors we survey three methods for proving that the characteristic polynomial of a finite ranked lattice factors over the nonnegative integers and indicate how they have evolved recently the first technique uses geometric ideas and is is based on zaslavsky s theory of signed graphs the second approach is algebraic and employs results of saito and terao about free hyperplane arrangements finally we consider a purely combinatorial theorem of stanley about supersolvable lattices and its generalizations

on a new class of abstract integral equations and applications abstract in this paper we introduce a new class of abstract integral equations which enables us to study in a unified manner several different types of differential equations

characteristic polynomials of subspace arrangements and finite fields let a be any subspace arrangement in rndefined over the integers and let fqdenote the finite field withqelements letqbe a large prime we prove that the characteristic polynomial a q of a counts the number of points in fnqthat do not lie in any of the subspaces ofa viewed as subsets of fnq this observation which generalizes a theorem of blass and sagan about subarrangements of the bnarrangement reduces the computation of a q to a counting problem and provides an explanation for the wealth of combinatorial results discovered in the theory of hyperplane arrangements in recent years the basic idea has its origins in the work of crapo and rota 1970 we find new classes of hyperplane arrangements whose characteristic polynomials have simple form and very often factor completely over the nonnegative integers

shellable nonpure complexes and posets ii this is a direct continuation of shellable nonpure complexes and posets i which appeared in transactions of the american mathematical society 348 1996 1299 1327 8 interval generated lattices and dominance order in this section and the following one we will continue exemplifying the applicability of lexicographic shellability to nonpure posets let f i1 i2 in be a family of intervals of integers by which is meant sets of the form a b a a 1 b a b we assume that there are no containments among these intervals and that they are ordered so that their left and right endpoints are increasing let l f be the lattice of all sets that are unions of subfamilies of f ordered by inclusion such interval generated lattices l f were introduced and studied by greene g define an edge labeling of l f as follows if a b is a covering and a max b a then a b a if a 1 a and a is the left endpoint of some i f

the alexander duality functors and local duality with monomial support abstract alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated n n graded module the functors associated with alexander duality provide a duality on the level of free and injective resolutions and numerous bass and betti number relations result as corollaries a minimal injective resolution of a module m is equivalent to the injective resolution of its alexander dual and contains all of the maps in the minimal free resolution of m over every z n graded localization results are obtained on the interaction of duality for resolutions with cellular resolutions and lcm lattices using injective resolutions theorems of eagon reiner and terai are generalized to all n n graded modules the projective dimension of m equals the support regularity of its alexander dual and m is cohen macaulay if and only if its alexander dual has a support linear free resolution alexander duality is applied in the context of the z n graded local cohomology functors hii for squarefree monomial ideals i in the polynomial ring s proving a duality directly generalizing local duality which is the case when i m is maximal in the process a new flat complex for calculating local cohomology at monomial ideals is introduced showing as a consequence that terai s formula for the hilbert series of hii s is equivalent to hochster s for hn i m s i

eigenspace arrangements of reflection groups the lattice of intersections of reflecting hyperplanes of a complex reflection group w may be considered as the poset of 1 eigenspaces of the elements of w in this paper we replace 1 with an arbitrary eigenvalue and study the topology and homology representation of the resulting poset after posing the main question of whether this poset is shellable we show that all its upper intervals are geometric lattices and then answer the question in the affirmative for the infinite family g m p n of complex reflection groups and the first 31 of the 34 exceptional groups by constructing cl shellings in addition we completely determine when these eigenspaces of w form a k 1 resp free arrangement for the symmetric group we also extend the combinatorial model available for its intersection lattice to all other eigenvalues by introducing balanced partition posets presented as particular upper order ideals of dowling lattices study the representation afforded by the top co homology group and give a simple map to the posets of pointed d divisible partitions 1 main question and results let v denote an n dimensional c vector space a reflection in v is any nonidentity element r in gl v of finite order that fixes some hyperplane hr and a finite subgroup w of gl v is called a reflection group if it is generated by reflections example the action of the symmetric group sn on n 1 2 n gives rise to a faithful action on c via ei e i where e1 e2 en denote the standard basis vectors since the transpositions act as reflections and generate the group this representation realizes sn as a reflection group in gl c we shall refer to it as the defining representation for an element g w and root of unity let v g denote the eigenspace of g in v define e w to be the w poset partially ordered set of all such eigenspaces v g g w ordered by reverse inclusion with w action given by h v g v hgh the eigenvalue 1 gives the lattice lw of intersections of reflecting hyperplanes for w see 29 lemma 4 4 the minimal elements of e w i e the inclusion maximal eigenspaces forw are the focus of springer s theory of regular elements 35 and each has dimension equal to the number a d of degrees d1 d2 dn of w that are divisible by d the order of see proposition 3 7 below when w is crystallographic the poset e w itself appears in broue malle and michel s sylow theory 8 2010 mathematics subject classification primary 20f55 05e10 05e18 55p20 14n20

computing the stanley depth in the first chapter some definitions and necessary results from commutative algebra are given also some detail are given which leads us towards the stanley decompositions of multigraded s modules where s is a polynomi ring in n varables over a field f in the end of the chapter we give some important results about the stanledecompositions and stanley conjecture in chapter 2 for the given monomial primary ideals and of s we gave an upper bound for the stanley depth of s which is reached if are irreducible also we showed that stanley conjecture holds for i being some irreducible monomial ideals of s these results are published in our paper 23 for integers 1 n consider the ideal i n s in chapter 3 we gave an upper bound for the stanley depth of the ideal we gave similar upper bounds for the stanley depth of the ideal where in 2 is the square free veronese ideal of degree 2 in n variables these results are from our paper 11

stanley depth of multigraded modules abstract the stanley s conjecture on cohen macaulay multigraded modules is studied especially in dimension 2 in codimension 2 similar results were obtained by herzog soleyman jahan and yassemi as a consequence of our results stanley s conjecture holds in 5 variables

free resolutions of lex ideals over a koszul toric ring in this paper we study the minimal free resolution of lex ideals over a koszul toric ring in particular we study in which toric ring r all lex ideals are componentwise linear we give a certain necessity and sufficiency condition for this property and show that lex ideals in a strongly koszul toric ring are componentwise linear in addition it is shown that in the toric ring arising from the segre product p p every hilbert function of a graded ideal is attained by a lex ideal and that lex ideals have the greatest graded betti numbers among all ideals having the same hilbert function

group actions on arrangements of linear subspaces and applications to configuration spaces for an arrangement of linear subspaces in ri that is invariant under a finite subgroup of the general linear group gin ir we develop a formula for the g module structure of the cohomology of the complement m a our formula specializes to the well known goresky macpherson theorem in case g 1 but for g 1 the formula shows that the g module structure of the complement is not a combinatorial invariant as an application we are able to describe the free part of the cohomology of the quotient space ma4ig our motivating examples are arrangements in cn that are invariant under the action of sn by permuting coordinates a particular case is the k equal arrangement first studied by bj6rner lovasz and yao motivated by questions in complexity theory in these cases ma4 and ma sn are spaces of ordered and unordered point configurations in cn many of whose properties are reduced by our formulas to combinatorial questions in partition lattices more generally we treat point configurations in rd and provide explicit results for the k equal and the k divisible cases

postulation of general quintuple fat point schemes in p 3 we study the postulation of a general union y of double triple quartuple and quintuple points of p 3 in characteristic 0 we prove that y has good postulation in degree d 11 the proof is based on the combination of the horace differential lemma with a computer assisted proof we also classify the exceptions in degree 9 and 10

existence of solutions of nonlinear differential equations with nonlocal conditions in this paper we investigate the existence of mild solutions to first order semilinear differential equations in banach spaces with nonlocal conditions we shall rely on a fixed point theorem for compact maps due to schaefer

stanley depth of squarefree monomial ideals in this paper we answer a question posed by y h shen we prove that if i is an m generated squarefree monomial ideal in the polynomial ring s k x1 xn with k a field then sdepthi n m 2 the proof is inductive and uses the correspondence between a stanley decomposition of a monomial ideal and a partition of a particular poset into intervals established by herzog vladoiu and zheng

on the stanley depth of powers of edge ideals abstract let k be a field and s k x 1 x n be the polynomial ring in n variables over k let g be a graph with n vertices assume that i i g is the edge ideal of g and p is the number of its bipartite connected components we prove that for every positive integer k the inequalities sdepth i k i k 1 p and sdepth s i k p hold as a consequence we conclude that s i k satisfies stanley s inequality for every integer k n 1 also it follows that i k i k 1 satisfies stanley s inequality for every integer k 0 furthermore we prove that if i g is a non bipartite graph or ii at least one of the connected components of g is a tree with at least one edge then i k satisfies stanley s inequality for every integer k n 1 moreover we verify a conjecture of the author in special cases

rings with lexicographic straightening law the concept of a straightening law as a means of analyzing the structure of particular commutative algebras has appeared independently in the work of a number of authors the purpose of this paper is to introduce a systematic theory of algebras endowed with a structure of a lexicographic straightening law based on a partially ordered set or lexicographic ring for short much of the work in the literature dealing with such rings fits quite naturally in the context of our theory and we sketch some of this work moreover some of our later results in particular the betti number bound in section 6 may actually be new results even for the special cases of the known examples of lexicographic rings the concept of a lexicographic ring as an interesting area of study was suggested to this author by deconcini who conjectured that a lexicographic ring is cohen macaulay if the underlying partially ordered set is so we prove this result in two different ways it has recently come to our attention that this conjecture has also been proved by deconcini et al 131 using deformation theory methods the results of this paper are arranged in two parts the first part consisting of sections 2 through 4 is more elementary and uses the technique of combinatorial decompositions as developed by garsia and this author the second part sections 6 and 7 requires some knowledge of homological algebra methods in ring theory the intermediate section 5 discusses some of the known examples of lexicographic rings in this section we also sketch some of the ways that our theory may be extended to more general contexts the main result of the first part is the transfer theorem 4 3 which enables one to transfer combinatorial decompositions from the underlying partially ordered set or more precisely from the corresponding stanley reisner ring to the lexicographic ring as a result those concepts that are expressible in terms of combinatorial decompositions are also

stanley depths of certain stanley reisner rings abstract let be a simplicial complex on n s k x 1 x n the polynomial ring in n variables over a field k and k s i the stanley reisner ring of with respect to k it is proved that the stanley conjecture holds for k i e sdepth s k depth s k when i has four associated prime ideals it is also obtained that sdepth s k size s i

prime filtrations of monomial ideals and polarizations abstract we show that an arbitrary monomial ideal i is pretty clean if and only if its polarization i p is clean this yields a new characterization of pretty clean monomial ideals in terms of the arithmetic degree and it also implies that a multicomplex is shellable if and only the simplicial complex corresponding to its polarization is non pure shellable we also discuss stanley decompositions in relation to prime filtrations

linear systems on the blow up of p1 n abstract in this note we study linear systems on the blow up of p 1 n at r points in very general position we prove that the fibers of the projections p 1 n p 1 s 1 s n 1 can give contribution to the speciality of the linear system this allows us to give a new definition of expected dimension of a linear system in p 1 n which we call fiber dimension finally we state a conjecture about linear systems on p 1 3

on abstract differential equations with state dependent non local conditions abstract in this paper we introduce the concept of state dependent nonlocal condition and study the existence and uniqueness of solution for a general class of abstract differential equations with state dependent delay subject to this class of conditions in this type of problem the nonlocal condition can be dependent on times or time intervals determined by the unknown solution some examples on partial differential equations with state dependent delay arising in population dynamics are presented

euler flag enumeration of whitney stratified spaces abstract the flag vector contains all the face incidence data of a polytope and in the poset setting the chain enumerative data it is a classical result due to bayer and klapper that for face lattices of polytopes and more generally eulerian graded posets the flag vector can be written as a c d index a non commutative polynomial which removes all the linear redundancies among the flag vector entries this result holds for regular cw complexes we relax the regularity condition to show the c d index exists for whitney stratified manifolds by extending the notion of a graded poset to that of a quasi graded poset this is a poset endowed with an order preserving rank function and a weighted zeta function this allows us to generalize the classical notion of eulerianness and obtain a c d index in the quasi graded poset arena we also extend the semi suspension operation to that of embedding a complex in the boundary of a higher dimensional ball and study the simplicial shelling components

some combinatorial and algebraic properties of coxeter complexes and tits buildings a des groupes de coxeter et des groupes a bn paire sont associes certaines structures combinatoires les complexes de coxeter et les constructions de tits on etudie ces structures du point de vue des complexes de cohen macaulay et des complexes decorticables

collisions of fat points and applications to interpolation theory abstract we address the problem to determine the flat limit of the collision of fat points in p n we give a description of the limit scheme in many cases in particular in low dimension and multiplicities the problem turns out to be closely related with interpolation theory and as an application we exploit collisions to prove new cases of laface ugaglia conjecture

on the shellability of the order complex of the subgroup lattice of a finite group we show that the order complex of the subgroup lattice of a finite group g is nonpure shellable if and only if g is solvable a by product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups

the homology representations of the k equal partition lattice we determine the character of the action of the symmetric group on the homology of the induced subposet of the lattice of partitions of the set 1 2 nr obtained by restricting block sizes to the set 1 k k 1 a plethystic formula for the generating function of the frobenius characteristic of the representation is given we combine techniques from the theory of nonpure shellability recently developed by bj6rner and wachs with symmetric function techniques developed by sundaram for determining representations on the homology of subposets of the partition lattice

weakly defective varieties a projective variety x is k weakly defective when its intersection with a general k 1 tangent hyperplane has no isolated singularities at the k 1 points of tangency if x is k defective i e if the k secant variety of x has dimension smaller than expected then x is also k weakly defective the converse does not hold in general a classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension we start this classification here describing all weakly defective irreducible surfaces our method also provides a new proof of the classical terracini s classification of k defective surfaces

a generalization of rota s nbc theorem abstract we generalize rota s theorem characterizing the mobius function of a geometric lattice in terms of subsets of atoms containing no broken circuit and give applications to the weak bruhat order of a finite coxeter group and the tamari lattices we also give a direct proof of the fact that in the geometric case any total order of the atoms can be used simple involutions are used in both proofs finally we show how involutions can be used in similar situations specifically in a special case of rota s crosscut theorem as well as in related proofs of walker on hall s theorem and reiner on characteristic and poincare polynomials

poset fiber theorems suppose that f p q is a poset map whose fibers f 1 qless than or equal to q are sufficiently well connected our main result is a formula expressing the homotopy type of p in terms of q and the fibers several fiber theorems from the literature due to babson baclawski and quillen are obtained as consequences or special cases homology cohen macaulay and equivariant versions are given and some applications are discussed

dimension filtration sequential cohen macaulayness and a new polynomial invariant of graded algebras let k be a field and let a be a standard n graded k algebra using numerical information of some invariants in the primary decomposition of 0 in a namely the so called dimension filtration we ass

decomposable clutters and a generalization of simon s conjecture abstract each equigenerated squarefree monomial ideal in the polynomial ring s k x 1 x n represents a family of subsets of n called a uniform clutter in this paper we introduce a class of uniform clutters called decomposable clutters whose associated ideal has linear quotients and hence linear resolution over all fields we show that chordality of these clutters guarantees the correctness of a conjecture raised by r s simon 23 on extendable shellebility of d skeletons of a simplex n for all d we then prove this conjecture for d n 3

clusters coxeter sortable elements and noncrossing partitions we introduce coxeter sortable elements of a coxeter group for finite we give bijective proofs that coxeter sortable elements are equinumerous with clusters and with noncrossing partitions we characterize coxeter sortable elements in terms of their inversion sets and in the classical cases in terms of permutations

nonzero solutions of nonlinear systems of differential equations via fixed point theorems for multivalued maps in the present paper we consider a nonlinear system of differential equations x a t x x f t x and a condition x e b x t 0 where a t x is a n x n real matrix defined and continuous on a x r a denotes a compact interval of r f a x r r is continuous and b is a banach space of continuous functions to solve this problem we use a comparison technique and we reduce the existence of solutions for this problem to that of a fixed point of a suitably defined multivalued map s see also anichini l kartsatos 3 and schmitt 4 in these papers the eilenberg montgomery futed point theorem is applied first we consider the problem x a x x and the condition using the eilenberg montgomery theorem 3 1 of 3 fixed point theorem we improve