modular elements of lattices and topological fibration an l arrangement is a finite collection of i 1 dimensional vector subspaces of an i dimensional vector space v over a field k we restrict our attention to the case that k c for a while the complement m of the hyperplanes is an open manifold in 3 falk and randell studied a libration of m m is called strictly linearly jibered if after a suitable linear change of coordinates the restriction of the projection to the first i 1 coordinates is a fiber bundle projection with base n the complement of an arrangement in c and fiber a complex plane c with finitely many points removed the arrangement is called fiber type when it allows a successive strictly linearly fibrations finally reaching c as base in this case they investigated several topological properties of m here we are interested in a combinatorial characterization of a strictly linear fibration in lattice theoretic terms for this the concept of modular elements defined by stanley s is crucial for example concerning a geometric lattice arising from an essential the intersection of all the elements is the origin arrangement defined over c the existence of l dimensional modular element is equivalent to a strictly linear libration of the complement m 2 15 this is a corollary of theorem 2 9 as another corollary we have a characterization 2 17 that the arrangement is fiber type if and only if it gives rise to an ss supersolvable lattice which is defined also by stanley 9 to be a geometric lattice which is defined also by stanley 9 to be a geometric lattice with an ascending maximal chain consisting of modular elements these are done in section 2 in section 3 we state theorem 3 8 asserting that the algebra a associated with a finite geometric lattice studied in s 63 factors as a tensor product of graded subspaces by a modular element which is theorem 3 8 this has a topological interpretation 3 9 when the lattice originates from an arrangement defined over c the factorization of the cohomological ring of m into that of the base and that of the fiber since 135 oool 8708 86 7 50

exact boundary controllability of vibrating non classical euler bernoulli micro scale beams abstract this study investigates the exact controllability problem for a vibrating non classical euler bernoulli micro beam whose governing partial differential equation pde of motion is derived based on the non classical continuum mechanics in this paper it is proved that via boundary controls it is possible to obtain exact controllability which consists of driving the vibrating system to rest in finite time this control objective is achieved based on the pde model of the system which causes that spillover instabilities do not occur

divisionally free arrangements of hyperplanes let mathcal a mathcal a mathcal a h be the triple of hyperplane arrangements we show that the freeness of mathcal a h and the division of chi mathcal a t by chi mathcal a h t imply the freeness of mathcal a this division theorem improves the famous addition deletion theorem and it has several applications which include a definition of divisionally free arrangements it is a strictly larger class of free arrangements than the classical important class of inductively free arrangements also whether an arrangement is divisionally free or not is determined by the combinatorics moreover we show that a lot of recursively free arrangements to which almost all known free arrangements are belonging are divisionally free

a rank two vector bundle associated to a three arrangement and its chern polynomial abstract we prove that the poincare polynomial a t of an essential central three arrangement a over a field k of characteristic zero is 1 t ct d 0 a where d 0 a is the sheaf associated to the kernel of the jacobian ideal of a and ct is the chern polynomial this shows that a version of terao s theorem invent math 63 1981 159 179 on free arrangements also holds for all three arrangements we also prove that for such an arrangement d 0 a is a vector bundle on p2 and derive an algorithm which computes ct d 0 a from a free resolution of the jacobian ideal of the defining polynomial of a

matroid representations and free arrangements we show that terao s conjecture freeness of the module of logarithmic forms at a hyperplane arrangement is determined by its abstract matroid holds over fields with at most four elements however an example demonstrates that the field characteristic has to be fixed for this

a geometric deletion restriction formula abstract in this paper we recover the characteristic polynomial of an arrangement of hyperplanes by computing the rational equivalence class of the variety defined by the logarithmic ideal of the arrangement the logarithmic ideal was introduced in cohen et al 2011 5 in a study of the critical points of the master function the above result is used to understand the asymptotic behaviour of the hilbert series of the logarithmic ideal as an application we note that a well known formula due to solomon and terao may be expressed as an euler characteristic and at least in the case of tame arrangements deduced from our main theorem

the module of logarithmic p forms of a locally free arrangement for an essential central hyperplane arrangement a v k n 1 we show that 1 a the module of logarithmic one forms with poles along a gives rise to a locally free sheaf on p n if and only if for all x la with rank x dim v the module 1 ax is free our main result says that in this case a t is essentially the chern polynomial the proof is based on a result of solomon terao 16 and a formula we give for the chern polynomial of a bundle e on p n in terms of the hilbert series of m2zh 0 p n i e m if 1 a has projective dimension one and is locally free we give a minimal

homological properties of determinantal arrangements abstract we explore a natural extension of braid arrangements in the context of determinantal arrangements we show that these determinantal arrangements are free divisors additionally we prove that free determinantal arrangements defined by the minors of 2 n matrices satisfy nice combinatorial properties we also study the topology of the complements of these determinantal arrangements and prove that their higher homotopy groups are isomorphic to those of s 3 furthermore we find that the complements of arrangements satisfying those same combinatorial properties above have poincare polynomials that factor nicely

distribution of shortest cycle lengths in random networks we present analytical results for the distribution of shortest cycle lengths dscl in random networks the approach is based on the relation between the dscl and the distribution of shortest path lengths dspl we apply this approach to configuration model networks for which analytical results for the dspl were obtained before we first calculate the fraction of nodes in the network which reside on at least one cycle conditioning on being on a cycle we provide the dscl over ensembles of configuration model networks with degree distributions which follow a poisson distribution erd s renyi network degenerate distribution random regular graph and a power law distribution scale free network the mean and variance of the dscl are calculated the analytical results are found to be in very good agreement with the results of computer simulations

a formula for the characteristic polynomial of an arrangement on etablit une formule permettant de calculer le polynome caracteristique d un arrangement constitue d un ensemble fini d hyperplans en termes de serie de poincarre de modules gradues

rationally smooth schubert varieties and inversion hyperplane arrangements abstract we show that an element w of a finite weyl group w is rationally smooth if and only if the hyperplane arrangement i associated to the inversion set of w is inductively free and the product d 1 1 d l 1 of the coexponents d 1 d l is equal to the size of the bruhat interval e w where e is the identity in w as part of the proof we describe exactly when a rationally smooth element in a finite weyl group has a chain billey postnikov decomposition for finite coxeter groups we show that chain billey postnikov decompositions are connected with certain modular coatoms of i

deletion theorem and combinatorics of hyperplane arrangements we show that the deletion theorem of a free arrangement is combinatorial i e whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice in fact we give a sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials as a corollary we prove that whether a free arrangement has a free filtration is also combinatorial the proof is based on the result about a minimal set of generators of a logarithmic derivation module of a multiarrangement which satisfies the b 2 equality

universal cover of salvetti s complex and topology of simplicial arrangements of hyperplanes let v be a real vector space an arrangement of hyperplanes in v is a finite set a of hyperplanes through the origin a chamber of a is a connected component of v h a h the arrangement a is called simplicial if h a h 0 and every chamber of a is a simplicial cone for an arrangement a of hyperplanes in v we set formule where v c c v is the complexification of v and for h a h c is the complex hyperplane of v c spanned by h let a be an arrangement of hyperplanes of v salvetti constructed a simplicial complex sal a and proved that sal a has the same homotopy type as m a in this paper we give a new short proof of this fact

algebras generated by reciprocals of linear forms let be a finite set of nonzero linear forms in several variables with coefficients in a field k of characteristic zero consider the k algebra c of rational functions generated by 1 then the ring v of differential operators with constant coefficients naturally acts on c we study the graded v module structure of c we especially find standard systems of minimal generators and a combinatorial formula for the poincare series of c our proofs are based on a theorem by brion vergne 4 and results by orlik terao 9

the first two obstructions to the freeness of arrangements abstract in his previous paper the author characterized free arrangements bythe vanishing of cohomology modules of a certain sheaf of graded modules overa polynomial ring thus the homogeneous components of these cohomologymodules can be viewed as obstructions to the freeness of an arrangement inthis paper the first two obstructions are studied in detail in particular thecomponent of degree zero of the first nontrivial cohomology module has a closerelation to formal arrangements and to the operation of truncation this enablesus to prove that in dimension greater than two every free arrangement is formal and not a proper truncation of an essential arrangement introductionin this paper we develop further the ideas from y where certain sheaf co homology was applied to the theory of arrangements let v be an dimensional 2 vector space over an arbitrary field kand sf an arrangement of n 2 hyperplanes in v for each h sfwe fix a linear functional ah v such that kera 77 let s d 0sdbe the graded symmetric algebra of v and der s der s s the gradeds module of derivations of s the following graded submodule of deri wasintroduced by h terao ti t3 and studied in many papers on arrangements d d s 6 der s 0 ah sah h sf if the module d is freethen the arrangement sf is called free let lo be the lattice of all intersections of hyperplanes from sf except

the poincare series of the algebra of rational functions which are regular outside hyperplanes let be a finite set of nonzero linear forms in several variables with coefficients in a field k of characteristic zero consider the k algebra r of rational functions on v which are regular outside ker then the ring r is naturally doubly filtered by the degrees of denominators and of numerators in this paper we give an explicit combinatorial formula for the poincare series in two variables of the associated bigraded vector space r this generalizes the main theorem of h terao algebras generated by reciprocals of linear forms j algebra 250 2002 549 558

entropy measures for networks toward an information theory of complex topologies the quantification of the complexity of networks is today a fundamental problem in the physics of complex systems a possible roadmap to solve the problem is via extending key concepts of information theory to networks in this rapid communication we propose how to define the shannon entropy of a network ensemble and how it relates to the gibbs and von neumann entropies of network ensembles the quantities we introduce here will play a crucial role for the formulation of null models of networks through maximum entropy arguments and will contribute to inference problems emerging in the field of complex networks

on controllability of a rotating string we consider the problem of controllability of a rotating string we say that a set of initial data of the string is controllable if for any initial data of this set by suitable manipulation of the exterior force the string goes to rest the main result of the paper is a description of controllable sets of initial data the equation is not strongly hyperbolic to get exact controllability of such equation in the sharp time interval we use control functions from sobolev spaces with noninteger indices to prove our results we apply the method of moments that has been widely used in control theory of distributed parameter systems since the classical papers of h o fattorini and d l russell we use recent results about exponential bases in sobolev spaces with noninteger indices

on inductively free restrictions of reflection arrangements abstract let w be a finite complex reflection group acting on the complex vector space v and let a w a w v be the associated reflection arrangement in 6 we classified all inductively free reflection arrangements a w the aim of this note is to extend this work by determining all inductively free restrictions of reflection arrangements

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 controllability of a non homogeneous elastic string with memory abstract we are motivated by the problem of control for a non homogeneous elastic string with memory we reduce the problem of controllability to a non standard moment problem the solution of the latter problem is based on an auxiliary riesz basis property result for a family of functions quadratically close to the nonharmonic exponentials this result requires the detailed analysis of an integro differential equation and is of interest in itself for function theory controllability of the string implies observability of a dual system

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

exponential bases in sobolev spaces in control and observation problems for the wave equation the fourier method in control systems reduces the study of controllability observability to the study of related exponential families in this paper we present examples of such systems specifically those for which we can prove that the related exponential families form a riesz basis in corresponding appropriately defined sobolev spaces this makes it possible to choose natural pairs of spaces the state space observability space and the control space state space depending on whether an observation or a control problem is studied respectively so that the observation and control operators are isomorphisms

coxeter multiarrangements with quasi constant multiplicities we study structures of derivation modules of coxeter multiarrangements with quasi constant multiplicities by using the primitive derivation as an application we show that the characteristic polynomial of a coxeter multiarrangement with quasi constant multiplicity is combinatorially computable

simultaneous control problems for systems of elastic strings and beams abstract controllability problems for n strings or beams controlled from a common endpoint are studied we give necessary and sufficient conditions for simultaneous spectral and approximate controllability and describe the space of simultaneously reachable states as a function of the position of the endpoint for each type of controllability result the sharp controllability time is obtained the proofs are based on new results in nonharmonic fourier series which describe riesz bases of exponential divided differences

the number of critical points of a product of powers of linear functions be the complement of a given a complex n vector 2 21 2n e cn consider the multivalued holomorphic function defined on m by studying bethe vectors in statistical mechanics a varchenko 18 conjectured that for generic 2 all critical points of qs are nondegenerate and the number of critical points is equal to the absolute value of the euler characteristic of m m he proved the conjecture for complexified real arrangements in 18 a similar result was known to k aomoto 1 example 1 who stated without proof that for positive 2 the number of critical points of b equals the number of bounded components of the complement of the real arrangement see 2 theorem 4 4 1 1 for details in this paper we prove varchenko s conjecture for all arrangements more generally we give a formula for the number of critical points of the multivalued holomorphic function

statistics of cycles how loopy is your network we study the distribution of cycles of length h in large networks of size n 1 and find it to be an excellent ergodic estimator even in the extreme inhomogeneous case of scale free networks the distribution is sharply peaked around a characteristic cycle length h n our results suggest that h and the exponent might usefully characterize broad families of networks in addition to an exact counting of cycles in hierarchical nets we present a monte carlo sampling algorithm for approximately locating h and reliably determining our empirical results indicate that for small random scale free nets of degree exponent 1 1 and grows as the nets become larger

free arrangements of hyperplanes and supersolvable lattices a finite family of hyperplanes through the origin in c is called an iarrangement of hyperplanes or simply an l arrangement an l arrangement is said to be free if the module of logarithmic vector fields holomorphic vector fields tangent to all hyperplanes in the arrangement is a free module the accurate definition will be given in 2 1 let a be a free i arrangement then a set do d of i 1 nonnegative integers can be defined 2 3 these d d are called the exponents of a on the other hand one can associate a geometric lattice l a with any arrangement a 3 1 the characteristic polynomial xlca t e z t is defined by using the mobius function 3 4 the second author 7 proved that

on a generalization of solomon terao formula for subspace arrangements abstract we investigate in this paper a generalization of solomon terao formula for central equidimensional subspace arrangements we introduce generalized solomon terao functions based on the hilbert poincare series of the modules of multi logarithmic forms and logarithmic multi residues we show that as in the case of hyperplane arrangements these solomon terao functions are polynomial we then prove that if the solomon terao polynomial of the modules of multi residues satisfies a certain property then this polynomial is related to the characteristic polynomial of the subspace arrangement in particular we prove that this generalized solomon terao formula holds for any line arrangement of any codimension

coxeter and crystallographic arrangements are inductively free abstract using the classification of finite weyl groupoids we prove that crystallographic arrangements a large subclass of the class of simplicial arrangements which was recently defined are hereditarily inductively free in particular all crystallographic reflection arrangements are hereditarily inductively free among them the arrangement of type e 8 with little extra work we prove that also all coxeter arrangements are inductively free

the minimal model of the complement of an arrangement of hyperplanes in this paper the methods of rational homotopy theory are applied to a family of examples from singularity theory let a be a finite collection of hyperplanes in cl and let m cl uhea h we say a is a rational k w 1 arrangement if the rational completion of m is aspherical for these arrangements an identity the lcs formula is established relating the lower central series of 7r1 m to the cohomology of m this identity was established by group theoretic means for the class of fiber type arrangements in previous work we reproduce this result by showing that the class of rational k w 1 arrangements contains all fiber type arrangements this class includes the reflection arrangements of types al and bl there is much interest in arrangements for which m is a k w 1 space the methods developed here do not apply directly because m is rarely a nilpotent space we give examples of k w 1 arrangements which are not rational k w 1 for which the lcs formula fails and k w 1 arrangements which are not rational k w 1 where the lcs formula holds it remains an open question whether rational k w 1 arrangements are necessarily k w 1 l introduction an arrangement of hyperplanes is a finite collection of clinear subspaces of dimension 1 l in c1 to such an arrangement a is associated an open 21 manifold the complement m cl u hih e a the connections between the topology of m and the combinatorial geometry of a are the source of much current research in this area the most successful investigations concern the cohomology of m 13 15 whereas the most difficult unsolved problems involve the homotopy groups of m 5 in this paper we study the link between cohomology and homotopy provided by sullivan s theory of minimal models 14 in 6 a numerical relationship was established between 1 m and h m for the class of fiber type arrangements this lcs formula reads as follows ri 1 tn y7n m pm t n l where the n m are the ranks of successive quotients in the lower central series of 1 m and pm t is the poincare polynomial of h m because the sequence n m is related to the l minimal model y of m we conjecture in 5 that the lds formula holds precisely when r determines h m with the methods developed in this paper we can resolve this conjecture specifically we show corollary 3 8 that the lcs formula holds when h 5 is isomorphic to h m arrangements satisfying the latter condition are called rational k 7r l corollary 3 8 may be received by the editors june 1 1987 1980 mathematicts subject clatstsification 1985 revitsion primary 55p62 57m05 secondary 06c10 r 1988 american mathematical society 0002 9947 88 1 00 25 per page

characterization of a free arrangement and conjecture of edelman and reiner we consider a hyperplane arrangement in a vector space of dimension four or higher in this case the freeness of the arrangement is characterized by properties around a fixed hyperplane as an application we prove the freeness of cones over certain truncated affine weyl arrangements which was conjectured by edelman and reiner

on the algebra associated with a geometric lattice abstract let l be a geometric lattice following p orlik and l solomon combinatorics and topology of complements of hyperplanes invent math 56 1980 167 189 we associate with l a graded commutative algebra a l in this paper we introduce a new invariant of the algebra a l which suffices to distinguish algebras for which all other known invariants coincide this result is applied to the study of arrangements of complex hyperplanes with l being the intersection lattice in this case a l is isomorphic to the cohomology algebra of the associated hyperplane complement the goal is to find examples of arrangements with non isomorphic lattices but homotopy equivalent complements the invariant introduced here effectively narrows the list of candidates nevertheless we exhibit combinatorially inequivalent arrangements for which all known invariants including coincide

arrangements of ideal type abstract in 2006 sommers and tymoczko defined so called arrangements of ideal type a i stemming from ideals i in the set of positive roots of a reduced root system they showed in a case by case argument that a i is free if the root system is of classical type or g 2 and conjectured that this is also the case for all types this was established only very recently in a uniform manner by abe barakat cuntz hoge and terao the set of non zero exponents of the free arrangement a i is given by the dual of the height partition of the roots in the complement of i in the set of positive roots generalizing the shapiro steinberg kostant theorem which asserts that the dual of the height partition of the set of positive roots gives the exponents of the associated weyl group our first aim in this paper is to investigate a stronger freeness property of the a i we show that all a i are inductively free with the possible exception of some cases in type e 8 in the same paper from 2006 sommers and tymoczko define a poincare polynomial i t associated with each ideal i which generalizes the poincare polynomial w t for the underlying weyl group w solomon showed that w t satisfies a product decomposition depending on the exponents of w for any coxeter group w sommers and tymoczko showed in a case by case analysis in types a n b n and c n and some small rank exceptional types that a similar factorization property holds for the poincare polynomials i t generalizing the formula of solomon for w t they conjectured that their multiplicative formula for i t holds in all types in our second aim to investigate this conjecture further the same inductive tools we develop to obtain inductive freeness of the a i are also employed here we also show that this conjecture holds inductively in almost all instances with only a small number of possible exceptions

the lower central series of a fiber type arrangement for a certain class fiber type of arrangements including the supersolvable ones of jambu and terao 3 we prove a formula relating the poincare polynomial of the complement with the ranks of successive quotients in the lower central series of the fundamental group such a formula was proved by kohno 5 for the single family of examplesa l

exact controllability and stabilization of a vibrating string with an interior point mass in this article we examine the problems of boundary control and stabilization for a one dimensional wave equation with interior point masses we show that singularities in waves are smoothed one order as they cross a point mass thus in the case of one interior point mass with e g l 2 dirichlet control at the left end the most general reachable space from 0 that one can expect is l 2 times h 1 to the left of the mass and h 1 times l 2 to the right of the mass we show that this is in fact the optimal result modulo certain compatibility conditions several related results for both control and stabilization of such systems are also given

combinatorial construction of logarithmic differential forms abstract h terao has shown that the structure of the module of rational differential forms with at most logarithmic poles at an arrangement of hyperplanes as defined by k saito is very strongly controlled by the combinatorial structure of the arrangement in this paper we demonstrate how the existence of rational logarithmic forms with poles of high order depends on the existence of highly degenerate special position subarrangements the associated combinatorial structures are studied first a strong version of k saito s preparation lemma for logarithmic forms leads to a new simple proof of terao s celebrated addition deletion theorem for hyperplane arrangements with free module of logarithmic 1 forms free arrangements then structural characterizations are developed in two extreme cases for the generation of the module of logarithmic differential 1 forms this module has a triangular basis iff the arrangement is supersolvable strictly linearly fibered and it is generated by forms of degree 1 iff the arrangement is generic in codimension 3 both conditions on the geometry of the arrangement are combinatorial in a very strong sense determined by restricted data on the lattice of intersections of the hyperplanes however examples show that the cardinality and degree sequence of a minimal set of generators for the module of logarithmic 1 forms are not in general determined by this intersection lattice

growth and arrest of topological cycles in small physical networks the chordless cycle sizes of spatially embedded networks are demonstrated to follow an exponential growth law similar to random graphs if the number of nodes formula see text is below a critical value formula see text for covalent polymer networks increasing the network size as measured by the number of cross link nodes beyond formula see text results in a crossover to a new regime in which the characteristic size of the chordless cycles formula see text no longer increases from this result the onset and intensity of finite size effects can be predicted from measurement of formula see text in large networks although such information is largely inaccessible with experiments the agreement of simulation results from molecular dynamics metropolis monte carlo and kinetic monte carlo suggests the crossover is a fundamental physical feature which is insensitive to the details of the network generation these results show random graphs as a promising model to capture structural differences in confined physical networks

the characteristic polynomial of a multiarrangement abstract given a multiarrangement of hyperplanes we define a series by sums of the hilbert series of the derivation modules of the multiarrangement this series turns out to be a polynomial using this polynomial we define the characteristic polynomial of a multiarrangement which generalizes the characteristic polynomial of an arrangement the characteristic polynomial of an arrangement is a combinatorial invariant but this generalized characteristic polynomial is not however when the multiarrangement is free we are able to prove the factorization theorem for the characteristic polynomial the main result is a formula that relates global data to local data of a multiarrangement given by the coefficients of the respective characteristic polynomials this result gives a new necessary condition for a multiarrangement to be free consequently it provides a simple method to show that a given multiarrangement is not free

free arrangements and coefficients of characteristic polynomials ziegler showed that the multirestriction of a free arrangement is also free after ziegler s work several results concerning the reverse direction i e characterizing freeness of an arrangement via that of its multirestriction have appeared in this paper we prove a new characterization of freeness in which the second betti number of the arrangement plays a crucial role

a free resolution of the module of logarithmic forms of a generic arrangement 1 1 the setup of this paper let v be an i dimensional vector space over a field k we assume that the characteristic of k is zero for the positive characteristic case see 4 6 let d be a generic arrangement of ii hyperplanes d is a finite family of one codimensional subspaces of v satisfying 1 n d 1 3 2 every 2 hyperplanes of d intersect only at the origin let s denote the symmetric algebra s v of the dual space v of v then 5 can be considered as the k algebra of all polynomial functions on v let 0 6 q 1 let 524 2 denote the s module of all regular q forms on v then each qq is a free s module of rank i for each he d choose cake v such that ker a h let

some almost exceptional arrangements abstract we introduce and study a family of real hyperplane arrangements that includes the reflection arrangements of types a and d as well as an infinite family of arrangements e n constructed by orlik solomon and terao related to the exceptional arrangements of type e n n 8 we investigate freeness in the sense of terao of these arrangements disproving the conjecture that the arrangements e n are free in general

generalized exponents of a free arrangement of hyperplanes and shepherd todd brieskorn formula we define an n arrangement as a finite family of hyperplanes through the origin in c 1 in 11 and 12 we studied the free arrangement and defined its structure sequence their definitions will be given again in sect 2 in this article we say the generalized exponents instead of the structure sequence let d o d 1 d be the generalized exponents of a free n arrangement x let xj u h our main result is hex

on the connection between exponential bases and certain related sequences in l2 π π a sequence of vectors fn in a separable hilbert space h is a frame if there are positive constants a b such that a f 2 n f fn 2 b f 2 for all f h and fn is a riesz sequence if it is a riesz basis in the closure of the space spanned by the vectors fn the latter of the following two questions has been raised by khrushchev nikolskii and pavlov can every frame of complex exponentials ei nx in l2 be made into a riesz basis by removing from ei nx a suitable collection of the functions ei nx can every riesz sequence ei nx in l2 be made into a riesz basis by adjoining to ei nx a suitable collection of exponentials ei nx ei nx we show that this is indeed so for all frames and riesz sequences ei nx studied so far and we prove that we can always solve both problems in a certain weak sense however our main conclusion is that the answer to both questions is no

hyperplane arrangements and linear strands in resolutions the cohomology ring of the complement of a central complex hyperplane arrangement is the well studied orlik solomon algebra the homotopy group of the complement is interesting complicated and few results are known about it we study the ranks for the lower central series of such a homotopy group via the linear strand of the minimal free resolution of the field c over the orlik solomon algebra

a partial order on the regions of ⁿ dissected by hyperplanes we study a partial order on the regions of r dissected by hyperplanes this includes a computation of the mobius function and in some cases of the homotopy type applications are presented to zonotopes the weak bruhat order on weyl groups and acyclic orientations of graphs 0 introduction let cc h h2 hk be a set of hyperplanes in rw then the components of rw uheh form a set i of open n cells we will call regions traditionally i has been studied in terms of enumeration for instance counting the number of regions and the number of intersections of various dimensions among the hyperplanes in c for a thorough discussion of this problem see zaslavsky s monograph zi and the references therein see also os and te in this paper we will study i from a different perspective our concern will be the way in which one can move from one region to another by walking through one hyperplane at a time to analyze this situation we define a partial order on j relative to a base region where the cover relations correspond to walking through a hyperplane and moving further away from the base the paper is organized as follows in 1 we define the partial order on rfl and begin to study its structure this culminates in theorem 1 11 where an explicit computation of its mbbius function is performed in 11 we examine the special case where all the hyperplanes go through the origin in certain cases we are able to compute the homotopy type of these posets the next three sections are devoted to applications 111 applies our work to a partial order on a zonotope iv deals with the weak bruhat order of a weyl group as a special case of our posets as a corollary we show that the homotopy type of the weak bruhat order is that of a sphere finally v applies our results to a partial order on the acyclic orientations of a graph some of the relationships used in iii iv and v appear in an enumerative context in the excellent article of greene and zaslavsky gz most of our poset notation is standard and can be found in a when more than one partial order is being used we will sometimes use subscripts to indicate where the relation is i e x py means that x is less than or equal to y in the poset p by l p we mean the simplicial complex of chains totally ordered sets in p we will think of p as its geometric realization by p we mean the order dual of p i e received by the editors july 13 1982 and in revised form may 15 1983 1980 mathematics subject classification primary 06a10 51m20 1984 american mathematical society 0002 9947 84 1 00 25 per page

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

higher homotopy groups of complements of complex hyperplane arrangements abstract we generalize results of hattori on the topology of complements of hyperplane arrangements from the class of generic arrangements to the much broader class of hypersolvable arrangements we show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range and we give an explicit z 1 module presentation of p the first non vanishing higher homotopy group we also give a combinatorial formula for the 1 coinvariants of p for affine line arrangements whose cones are hypersolvable we provide a minimal resolution of 2 and study some of the properties of this module for graphic arrangements associated to graphs with no 3 cycles the algorithm for computing 2 is purely combinatorial the fitting varieties associated to 2 may distinguish the homotopy 2 types of arrangement complements with the same 1 and the same betti numbers in low degrees