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

regulator constants and the parity conjecture the p parity conjecture for twists of elliptic curves relates multiplicities of artin representations in p selmer groups to root numbers in this paper we prove this conjecture for a class of such twists for example if e is semistable at 2 and 3 k is abelian and k is its maximal pro p extension then the p parity conjecture holds for twists of e by all orthogonal artin representations of mathop mathrm gal k infty mathbb q we also give analogous results when k is non abelian the base field is not and e is replaced by an abelian variety the heart of the paper is a study of relations between permutation representations of finite groups their regulator constants and compatibility between local root numbers and local tamagawa numbers of abelian varieties in such relations

the performance of universal encoding universal coding theory is surveyed from the viewpoint of the interplay between delay and redundancy the price for universality turns out to be acceptably small

variable length markov chains we study estimation in the class of stationary variable length markov chains vlmc on a finite space the processes in this class are still markovian of high order but with memory of variable length yielding a much bigger and structurally richer class of models than ordinary high order markov chains from an algorithmic view the vlmc model class has attracted interest in information theory and machine learning but statistical properties have not yet been explored provided that good estimation is available the additional structural richness of the model class enhances predictive power by finding a better tradeoff between model bias and variance and allowing better structural description which can be of specific interest the latter is exemplified with some dna data a version of the tree structured context algorithm proposed by rissanen in an information theoretical setup is shown to have new good asymptotic properties for estimation in the class of vlmcs this remains true even when the underlying model increases in dimensionality fur thermore consistent estimation of minimal state spaces and mixing properties of fitted models are given we also propose a new bootstrap scheme based on fitted vlmcs we show its validity for quite general stationary categorical time series and for a broad range of statistical procedures

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

bic context tree estimation for stationary ergodic processes context trees of arbitrary stationary ergodic processes with finite alphabets are considered such a process is not necessarily a markov chain so the context tree may be of infinite depth calculated from a sample of size n the bayesian information criterion bic is shown to provide a strongly consistent estimator of the context tree of the process via minimization over hypothetical context trees without any restriction on the hypothetical context trees strong consistency means that the estimated context tree recovers the true one up to a level k eventually almost surely as n tends to infinity under some conditions on the process it is shown that the recovery level k can grow with n at a specific rate determined by the distribution of the process thus the bic estimator can recover the true context tree to larger and larger depths the results include for the special case of k being an arbitrary constant that the strong consistency is satisfied without any assumption on the stationary ergodic process which itself improves the existing results where either the true context tree was assumed to be of finite depth or the depth of the hypothetical context trees was bounded by o log n

consistency of the unlimited bic context tree estimator the bayesian information criterion bic and krichevsky trofimov kt version of minimum description length mdl principle are popular in the study of model selection for order estimation of markov chains both are known to be strongly consistent when there is an upper bound on the order in the unbounded case the bic is also known to be consistent but the kt estimator is consistent only with a bound o log n on the order for context trees a flexible generalization of markov models widely used in data processing the problem is more complicated both in theory and practice given the substantially higher number of possible candidate models imre csiszar and zsolt talata proved the consistency of bic and kt when the hypothetical tree depths are allowed to grow as o log n this correspondence proves that such a restriction is not necessary for finite context sources the bic context tree estimator is strongly consistent even if there is no constraint at all on the size of the chosen tree moreover an algorithm computing the tree minimizing the bic criterion among all context trees in linear time is provided

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

a sequential algorithm for the universal coding of finite memory sources the estimation and universal compression of discrete sources are considered and a sequential algorithm for the universal coding of finite memory sources attaining asymptotically minimum redundancy is presented the algorithm performs an online estimation of the source states and uses an arithmetic code

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

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

the consistency of the bic markov order estimator the bayesian information criterion bic estimates the order of a markov chain with finite alphabet a from observation of a sample path x 1 x 2 x n as that value k k that minimizes the sum of the negative logarithm of the kth order maximum likelihood and the penalty term a k a 1 2 log n we show that k equals the correct order of the chain eventually almost surely as n oo thereby strengthening earlier consistency results that assumed an apriori bound on the order a key tool is a strong ratio typicality result for markov sample paths we also show that the bayesian estimator or minimum description length estimator of which the bic estimator is regarded as an approximation fails to be consistent for the uniformly distributed i i d process

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

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

a universal finite memory source an irreducible parameterization for a finite memory source is constructed in the form of a tree machine a universal information source for the set of finite memory sources is constructed by a predictive modification of an earlier studied algorithm context it is shown that this universal source incorporates any minimal data generating tree machine in an asymptotically optimal manner in the following sense the negative logarithm of the probability it assigns to any long typical sequence generated by any tree machine approaches that assigned by the tree machine at the best possible rate

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

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

a universal data compression system a universal data compression algorithm is described which is capable of compressing long strings generated by a finitely generated source with a near optimum per symbol length without prior knowledge of the source this class of sources may be viewed as a generalization of markov sources to random fields moreover the algorithm does not require a working storage much larger than that needed to describe the source generating parameters

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

context tree estimation for not necessarily finite memory processes via bic and mdl the concept of context tree usually defined for finite memory processes is extended to arbitrary stationary ergodic processes with finite alphabet these context trees are not necessarily complete and may be of infinite depth the familiar bayesian information criterion bic and minimum description length mdl principles are shown to provide strongly consistent estimators of the context tree via optimization of a criterion for hypothetical context trees of finite depth allowed to grow with the sample size n as o logn algorithms are provided to compute these estimators in o n time and to compute them on line for all i les n in o nlogn time