subharmonic solutions in singular systems abstract we consider the problem of bifurcation from homoclinic towards periodic orbits for a periodic singular system of differential equations we use a functional analytic approach and the lyapunov schmidt method to obtain a bifurcation function for this problem which tends as e 0 to the well known melnikov function for the bifurcation towards transverse homoclinic orbits we also study the behavior of the bifurcation functions as the period of the periodic orbit tends to infinity

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

lehmer s problem for polynomials with odd coefficients we prove that if f x ylkzo akxk is a polynomial with no cyclotomic factors whose coefficients satisfy ak 1 mod 2 for 0 2 we also characterize the polynomials that appear as the noncyclotomic part of a polynomial whose coefficients satisfy ak 1 mod p for each ft for a fixed prime p last we prove that the smallest pisot number whose minimal polynomial has odd coefficients is a limit point from both sides of salem 19 numbers whose minimal polynomials have coef ficients in 1 1

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

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

torsion in the matching complex and chessboard complex abstract topological properties of the matching complex were first studied by bouc in connection with quillen complexes and topological properties of the chessboard complex were first studied by garst in connection with tits coset complexes bjorner lovasz vrecica and ivaljevic established bounds on the connectivity of these complexes and conjectured that these bounds are sharp in this paper we show that the conjecture is true by establishing the nonvanishing of integral homology in the degrees given by these bounds moreover we show that for sufficiently large n the bottom nonvanishing homology of the matching complex m n is an elementary 3 group improving a result of bouc and that the bottom nonvanishing homology of the chessboard complex m n n is a 3 group of exponent at most 9 when n 2 mod 3 the bottom nonvanishing homology of m n n is shown to be z 3 our proofs rely on computer calculations long exact sequences representation theory and tableau combinatorics

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

the existence of transverse homoclinic solutions for higher order equations abstract differential equations are considered which contain a small parameter when the parameter is zero the equation is autonomous with a hyperbolic equilibrium and a homoclinic solution no restriction is placed on the dimension of the phase space or the dimension of intersection of the stable and unstable manifolds sufficient conditions on the perturbation terms for producing a transverse homoclinic solution and hence chaos are obtained the technique follows the work of palmer using exponential dichotomies and the method of lyapunov schmidt

some consequences of masser s counting theorem on elliptic curves we use masser s counting theorem to prove a lower bound for the canonical height in powers of elliptic curves we also prove the galois case of the elliptic lehmer problem combining kummer theory and masser s result with bounds on the rank and torsion of some groups of rational points on an elliptic curve

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

discrete morse theory for totally non negative flag varieties abstract in a seminal 1994 paper lusztig 1994 26 lusztig extended the theory of total positivity by introducing the totally non negative part g p 0 of an arbitrary generalized partial flag variety g p he referred to this space as a remarkable polyhedral subspace and conjectured a decomposition into cells which was subsequently proven by the first author rietsch 1998 33 in williams 2007 40 the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron by conjecturing it to be a regular cw complex that is homeomorphic to a closed ball in this article we use discrete morse theory to prove this conjecture up to homotopy equivalence explicitly we prove that the boundaries of the cells are homotopic to spheres and the closures of cells are contractible the latter part generalizes a result of lusztig s 1998 28 that g p 0 the closure of the top dimensional cell is contractible concerning our result on the boundaries of cells even the special case that the boundary of the top dimensional cell g p 0 is homotopic to a sphere is new for all g p other than projective space

pure product polynomials and the prouhet tarry escott problem an n factor pure product is a polynomial which can be expressed in the form n i 1 1 x i for some natural numbers 1 n we define the norm of a polynomial to be the sum of the absolute values of the coefficients it is known that every n factor pure product has norm at least 2n we describe three algorithms for determining the least norm an n factor pure product can have we report results of our computations using one of these algorithms which include the result that every n factor pure product has norm strictly greater than 2n if n is 7 9 10 or 11

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

the colored symmetric and exterior algebras abstract we study colored generalizations of the symmetric algebra and its koszul dual the exterior algebra the symmetric group s n acts on the multilinear components of these algebras while s n acts trivially on the multilinear components of the colored symmetric algebra we use poset topology techniques to understand the representation on its koszul dual we introduce an s n poset of weighted subsets that we call the weighted boolean algebra and we prove that the multilinear components of the colored exterior algebra are s n isomorphic to the top cohomology modules of its maximal intervals we use a technique of sundaram to compute group representations on cohen macaulay posets to give a generating formula for the frobenius series of the colored exterior algebra we exploit that formula to find an explicit expression for the expansion of the corresponding representations in terms of irreducible s n representations we show that the two colored koszul dual algebras are koszul in the sense of priddy

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

subharmonic solutions bifurcated from homoclinic orbits for weakly coupled singular systems abstract the problem of bifurcation from homoclinic solution towards periodic solution was considered for weekly coupled singular systems by using functional analytic approach based on the lyapunov schmidt reduction we obtained some functions h r d 1 r r d the simple roots of the equations h 0 correspond to the existence of subharmonic solutions and if the vector field is 2 period then for any integer m the weakly coupled singular system has 2 m period solution

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

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

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

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

proof of the k pi 1 conjecture for affine artin groups we prove the k pi 1 conjecture for affine artin groups the complexified complement of an affine reflection arrangement is a classifying space this is a long standing problem due to arnol d pham and thom our proof is based on recent advancements in the theory of dual coxeter and artin groups as well as on several new results and constructions in particular we show that all affine noncrossing partition posets are el shellable we use these posets to construct finite classifying spaces for dual affine artin groups we introduce new cw models for the orbit configuration spaces associated with arbitrary coxeter groups we construct finite classifying spaces for the braided crystallographic groups introduced by mccammond and sulway

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

combinatorics of multigraded poincare series for monomial rings abstract backelin proved that the multigraded poincare series for resolving a residue field over a polynomial ring modulo a monomial ideal is a rational function the numerator is simple but until the recent work of berglund there was no combinatorial formula for the denominator berglund s formula gives the denominator in terms of ranks of reduced homology groups of lower intervals in a certain lattice we now express this lattice as the intersection lattice l a i of a subspace arrangement a i use crapo s closure lemma to drastically simplify the denominator in some cases such as monomial ideals generated in degree two and relate golodness to the cohen macaulay property for associated posets in addition we introduce a new class of finite lattices called complete lattices prove that all geometric lattices are complete and provide a simple criterion for golodness of monomial ideals whose lcm lattices are complete

discrete morse functions from lexicographic orders this paper shows how to construct a discrete morse function with a relatively small number of critical cells for the order complex of any finite poset with o and i from any lexicographic order on its maximal chains specifically if we attach facets according to the lexicographic order on maximal chains then each facet contributes at most one new face which is critical and at most one betti number changes facets which do not change the homotopy type also do not contribute any critical faces dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets as one application the mobius function may be computed as the alternating sum of morse numbers the above construction enables us to prove that the poset n s of partitions of a set 1 1 k k with repetition is homotopy equivalent to a wedge of spheres of top dimension when a is a hook shaped partition it is likely that the proof may be extended to a larger class of a and perhaps to all a despite a result of ziegler 1986 which shows that n s is not always cohen macaulay

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

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

dimension and entropy in compact topological groups abstract we study the topological entropy h f of continuous endomorphisms f of compact like groups more specifically we consider the e spectrum e t o p k for a compact like group k namely the set of all values h f when f runs over the set e n d k of all continuous endomorphisms of k we pay particular attention to the class e of topological groups without continuous endomorphisms of infinite entropy i e e t o p k as well as the subclass e 0 of e consisting of those groups k with e t o p k 0 it turns out that the properties of the e spectrum and these two classes are very closely related to the topological dimension we show among others that a compact connected group k with finite dimensional commutator subgroup belongs to e if and only if dim k and we obtain a simple formula involving the entropy function for the dimension of an abelian topological group which is either locally compact or bounded in particular compact examples are provided to show the necessity of the compactness or commutativity conditions imposed for the validity of these results e g compact connected semi simple groups k with dim k and k e 0 or countably compact connected abelian groups with the same property since the class e is not stable under taking closed subgroups or quotients we study also the largest subclasses s e and q e respectively of e having these stability properties we provide a complete description of these two classes in the case of compact groups that are either abelian or connected the counterpart for s e 0 and q e 0 is done as well

on newman polynomials which divide no littlewood polynomial recall that a polynomial p x g z x with coefficients 0 1 and constant term 1 is called a newman polynomial whereas a polynomial with coefficients 1 1 is called a littlewood polynomial is there an algebraic number ol which is a root of some newman polynomial but is not a root of any littlewood polynomial in other words but not equivalently is there a newman polynomial which divides no littlewood polynomial in this paper for each newman polynomial p of degree at most 8 we find a littlewood polynomial divisible by p moreover it is shown that every trinomial l uxa vxb where a b are positive integers and u v 1 1 so in particular every newman trinomial 1 xa xb divides some littlewood polynomial nevertheless we prove that there exist newman polynomials which divide no littlewood polynomial e g x9 x6 x2 x l this example settles the problem 006 07 posed by the first named author at the 2006 west coast number theory conference it also shows that the sets of roots of newman polynomials vjv littlewood polynomials vc and 1 0 1 polynomials v are distinct in the sense that between them there are only trivial relations v v c v and vc c v moreover v jz vc u vjsf the proofs of several main results after some preparation are computational

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

bifurcation from homoclinic to periodic solutions in singular ordinary differential equations abstract problems of bifurcations from homoclinic to periodic orbits are considered for periodic singular differential equations a functional analytic approach together with the method of lyapunov schmidt is used to obtain bifurcation functions for these problems which yield periodic solutions of all integer multiples of the period of the vector field

koszul incidence algebras affine semigroups and stanley reisner ideals abstract we prove a theorem unifying three results from combinatorial homological and commutative algebra characterizing the koszul property for incidence algebras of posets and affine semigroup rings and characterizing linear resolutions of squarefree monomial ideals the characterization in the graded setting is via the cohen macaulay property of certain posets or simplicial complexes and in the more general nongraded setting via the sequential cohen macaulay property

on the co homology of the poset of weighted partitions we consider the poset of weighted partitions w in troduced by dotsenko and khoroshkin in their study of a certain pair of dual operads the maximal intervals of w provide a gen eralization of the lattice n of partitions which we show possesses many of the well known properties of n in particular we prove these intervals are el shellable we show that the mobius invari ant of each maximal interval is given up to sign by the number of rooted trees on node setf1 2 ng having a xed number of de scents we nd combinatorial bases for homology and cohomology and we give an explicit sign twisted sn module isomorphism from cohomology to the multilinear component of the free lie algebra with two compatible brackets we also show that the characteris tic polynomial of w has a nice factorization analogous to that of n

codimension 3 bifurcations of a class of homoclinic loop with saddle point abstract in this work bifurcations in the class that the homoclinic orbit connects the strong stable and strong unstable manifolds of a saddle are investigated for four dimensional vector fields the existence numbers coexistence and incoexistence of 1 homoclinic orbit 1 periodic orbit 2 n homoclinic orbit and 2 n periodic orbit are obtained the approximate expressions of the corresponding bifurcation surfaces and the bifurcation diagrams are also presented

homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations nonautomonous ordinary differential equations depending on two parameters 1 and 2 are considered in rn it is assumed that when both parameters are zero the differential equation is autonomous with a hyperbolic equilibrium and a homoclinic solution no restriction is placed on the dimension of the phase space rn or on the dimension of intersection of the stable and unstable manifolds by means of the method of lyapunov schmidt a bifurcation function h is constructed between two finite dimensional spaces where the zeros of h correspond to homoclinic solutions at nonzero parameter values the independent variables of h consist of scalars 1 2 and a vector where is a phase angle and corresponds to directions other than along the original homoclinic solution tangent to both the stable and unstable manifolds when is fixed the equation h 0 yields in general several bifurcation curves through the origin in the 1 2 plane along which there exists a homoclinic solution when is varied these become a number of wedge shaped regions the theory is applied to two examples one in r6 where the invariant manifolds meet in dimension three and a second in r4 where these manifolds agree

permutahedra and generalized associahedra given a finite coxeter system w s w s and a coxeter element c or equivalently an orientation of the coxeter graph of w we construct a simple polytope whose outer normal fan is n reading s cambrian fan fc settling a conjecture of reading that this is possible we call this polytope the c generalized associahedron our approach generalizes loday s realization of the associahedron a type a c generalized associahedron whose outer normal fan is not the cluster fan but a coarsening of the coxeter fan arising from the tamari lattice to any finite coxeter group a crucial role in the construction is played by the c singleton cones the cones in the c cambrian fan which consist of a single maximal cone from the coxeter fan moreover if w is a weyl group and the vertices of the permutahedron are chosen in a lattice associated to w then we show that our realizations have integer coordinates in this lattice

algebraic study on cameron walker graphs abstract let g be a finite simple graph on n and i g s the edge ideal of g where s k x 1 x n is the polynomial ring over a field k let m g denote the maximum size of matchings of g and i m g that of induced matchings of g it is known that i m g reg s i g m g where reg s i g is the castelnuovo mumford regularity of s i g cameron and walker succeeded in classifying the finite connected simple graphs g with i m g m g we say that a finite connected simple graph g is a cameron walker graph if i m g m g and if g is neither a star nor a star triangle in the present paper we study cameron walker graphs from a viewpoint of commutative algebra first we prove that a cameron walker graph g is unmixed if and only if g is cohen macaulay and classify all cohen macaulay cameron walker graphs second we prove that there is no gorenstein cameron walker graph finally we prove that every cameron walker graph is sequentially cohen macaulay

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

bifurcation of a unique stable periodic orbit from a homoclinic orbit in infinite dimensional systems sous certaines conditions generiques on montre comment une orbite periodique stable unique peut bifurquer a partir d une orbite homocline pour des equations paraboliques semi lineaires et des equations differentielles fonctionnelles a retard

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

a new subgroup lattice characterization of finite solvable groups abstract we show that if g is a finite group then no chain of modular elements in its subgroup lattice l g is longer than a chief series also we show that if g is a nonsolvable finite group then every maximal chain in l g has length at least two more than the chief length of g thereby providing a converse of a result of j kohler our results enable us to give a new characterization of finite solvable groups involving only the combinatorics of subgroup lattices namely a finite group g is solvable if and only if l g contains a maximal chain x and a chain m consisting entirely of modular elements such that x and m have the same length

spectral theory of extended harper s model and a question by erdős and szekeres the extended harper s model proposed by d j thouless in 1983 generalizes the famous almost mathieu operator allowing for a wider range of lattice geometries parametrized by three coupling parameters by permitting 2d electrons to hop to both nearest and next nearest neighboring nnn lattice sites while still exhibiting its characteristic symmetry aubry andre duality previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space one of which is characterized by the positivity of the lyapunov exponent in this paper we complete the picture with a description of the spectral measures over the entire remaining self dual region for all irrational values of the frequency parameter the magnetic flux in the model most notably we prove that in the entire interior of this regime the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the nnn interaction becomes symmetric in physics literature extensive numerical analysis had indicated such spectral collapse however so far not even a heuristic argument for this phenomenon could be provided on the other hand in the remaining part of the self dual region the spectral measures are singular continuous irrespective of such symmetry the analysis requires some rather delicate number theoretic estimates which ultimately depend on the solution of a problem posed by erd s and szekeres on the product prod k 1 n 1 z a k publ de l institut mathematique paris 1950

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

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

linearly independent homoclinic bifurcations parameterized by a small function abstract in this paper we discuss a small nonautonomous perturbation of an autonomous system on r n which has a homoclinic solution regarding the small perturbation as a parameter in an appropriate space of functions we discuss various situations of co existence of homoclinic orbits those conditions of various co existence actually define bifurcation manifolds in the space of functions for linearly independent homoclinic bifurcations

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

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