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

a generalization of coxeter groups root systems and matsumoto s theorem the root systems appearing in the theory of lie superalgebras and nichols algebras admit a large symmetry extending properly the one coming from the weyl group based on this observation we set up a general framework in which the symmetry object is a groupoid we prove that in our context the groupoid is generated by simple reflections and coxeter relations in a broad sense this answers a question of serganova our weak version of the exchange condition allows us to prove matsumoto s theorem therefore the word problem is solved for the groupoid

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

quantum symmetric kac moody pairs the present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable kac moody algebras the resulting quantum symmetric pairs are right coideal subalgebras of quantized enveloping algebras they give rise to triangular decompositions including a quantum analog of the iwasawa decomposition and they can be written explicitly in terms of generators and relations moreover their centers and their specializations are determined the constructions follow g letzter s theory of quantum symmetric pairs for semisimple lie algebras the main additional ingredient is the classification of involutive automorphisms of the second kind of symmetrizable kac moody algebras due to kac and wang the resulting theory comprises various classes of examples which have previously appeared in the literature such as q onsager algebras and the twisted q yangians introduced by molev ragoucy and sorba

some properties of exact tests for unit roots summary this paper is concerned with the null hypothesis that errors in a regression equation for time series data follow a random walk we examine the power properties of most powerful invariant tests for the unit root null hypotheses against exact stationary and nonstationary first order autoregressive models the analysis shows the importance of a constant term and a linear trend variable in certain cases the implications of the results for models estimated using seasonal data are briefly discussed

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

exponents for b stable ideals let g be a simple algebraic group over the complex numbers containing a borel subgroup b given a b stable ideal i in the nilradical of the lie algebra of b we define natural numbers m 1 m 2 m k which we call ideal exponents we then propose two conjectures where these exponents arise proving these conjectures in types a n b n c n and some other types when 0 we recover the usual exponents of g by kostant 1959 and one of our conjectures reduces to a well known factorization of the poincare polynomial of the weyl group the other conjecture reduces to a well known result of arnold brieskorn on the factorization of the characteristic polynomial of the corresponding coxeter hyperplane arrangement

inference in linear time series models with some unit roots this paper considers estimation and hypothesis testing in linear time series models when some or all of the variables have unit roots our motivating example is a vector autoregression with some unit roots in the companion matrix which might include polynomials in time as regressors in the general formulation the variable might be integrated or cointegrated of arbitrary orders and might have drifts as well we show that parameters that can be written as coefficients on mean zero nonintegrated regressors have jointly normal asymptotic distributions converging at the rate t 2 in general the other coefficients including the coefficients on polynomials in time will have nonnormal asymptotic distributions the results provide a formal characterization of which t or f tests such as granger causality tests will be asymptotically valid and which will have nonstandard limiting distributions

new large rank nichols algebras over nonabelian groups with commutator subgroup z2 abstract in this article we explicitly construct new finite dimensional indecomposable nichols algebras with dynkin diagrams of type a n c n d n e 6 7 8 f 4 over any group g of order 2 l with commutator subgroup isomorphic to z 2 the construction is generic in the sense that the type just depends on the rank and center of g and thus positively answers for all groups of this class a question raised by susan montgomery in 1995 22 4 our construction uses the new notion of a covering nichols algebra as a special case of a covering hopf algebra 18 and produces non faithful nichols algebras we give faithful examples of doi twists for type a 3 c 3 d 4 c 4 f 4 over several nonabelian groups of order 16 and 32 these are hence the first known examples of nondiagonal finite dimensional indecomposable nichols algebras of rank 2 over nonabelian groups

the development version of the chevie package of gap3 abstract i describe the current state of the development version of the chevie package which allows to transform the theories of coxeter groups reductive algebraic groups complex reflection groups hecke algebras braid monoids etc into actual computations examples are given showing the code to check some results of lusztig

additional critical values and asymptotic representations for seasonal unit root tests this paper is concerned with tests for seasonal unit roots in a univariate time series process the paper extends the procedures and tables of critical values due to hylleberg et al 1990 and ghysels et al 1994 to obtain tests which are similar exactly and asymptotically with respect to both the initial values of the process and the possibility of differential seasonal drift under the null hypothesis of a seasonal unit root representations are derived for the limiting distributions of the test statistics in this and other cases of interest these representations provide an explanation for the similarity between critical values for the statistics in different scenarios the seasonal unit root properties of real seasonally unadjusted uk non durable consumption expenditure are re examined

a smoothness priors state space modeling of time series with trend and seasonality abstract a smoothness priors modeling of time series with trends and seasonalities is shown an observed time series is decomposed into local polynomial trend seasonal globally stationary autoregressive and observation error components each component is characterized by an unknown variance white noise perturbed difference equation constraint the constraints or bayesian smoothness priors are expressed in state space model form trading day factors are also incorporated in the model a kalman predictor yields the likelihood for the unknown variances hyperparameters likelihoods are computed for different constraint order models in different subsets of constraint equation model classes akaike s minimum aic procedure is used to select the best model fitted to the data within and between the alternative model classes smoothing is achieved by using a fixed interval smoother algorithm examples are shown

distributed delta 1 coloring in linear in delta time the distributed delta 1 coloring problem is one of the most fundamental and well studied problems in distributed algorithms starting with the work of cole and vishkin in 1986 a long line of gradually improving algorithms has been published the state of the art running time prior to our work is o delta log delta log n due to kuhn and wattenhofer proceedings of the 25 th annual acm symposium on principles of distributed computing denver co 2006 pp 7 15 linial proceedings of the 28 th annual ieee symposium on foundation of computer science los angeles ca 1987 pp 331 335 proved a lower bound of frac 1 2 log n for the problem and szegedy and vishwanathan proceedings of the 25th annual acm symposium on theory of computing san diego ca 1993 pp 201 207 provided a heuristic argument that shows that algorithms from a wide family of locally iterative algorithms are unlikely to achieve a running time smaller than theta delta log delta we present a de

deterministic distributed vertex coloring in polylogarithmic time consider an n vertex graph g v e of maximum degree and suppose that each vertex v v hosts a processor the processors are allowed to communicate only with their neighbors in g the communication is synchronous that is it proceeds in discrete rounds in the distributed vertex coloring problem the objective is to color g with 1 or slightly more than 1 colors using as few rounds of communication as possible the number of rounds of communication will be henceforth referred to as running time efficient randomized algorithms for this problem are known for more than twenty years alon et al 1986 luby 1986 specifically these algorithms produce a 1 coloring within o log n time with high probability on the other hand the best known deterministic algorithm that requires polylogarithmic time employs o 2 colors this algorithm was devised in a seminal focs 87 paper by linial 1987 its running time is o log n in the same article linial asked whether one can color with significantly less than 2 colors in deterministic polylogarithmic time by now this question of linial became one of the most central long standing open questions in this area in this article we answer this question in the affirmative and devise a deterministic algorithm that employs 1 o 1 colors and runs in polylogarithmic time specifically the running time of our algorithm is o f log log n for an arbitrarily slow growing function f 1 we can also produce an o 1 coloring in o log log n time for an arbitrarily small constant 0 and an o coloring in o e log n time for an arbitrarily small constant e 0 our results are in fact far more general than this in particular for a graph of arboricity a our algorithm produces an o a1 coloring for an arbitrarily small constant 0 in time o log a log n

seasonal integration and cointegration this paper develops tests for roots in linear time series which have a modulus of one but which correspond to seasonal frequencies critical values for the tests are generated by monte carlo methods or are shown to be available from dickey fuller or dickey hasza fuller critical values representations for multivariate processes with combinations of seasonal and zero frequency unit roots are developed leading to a variety of autoregressive and error correction representations the techniques are used to examine cointegration at different frequencies between consumption and income in the u k

distributed δ 1 coloring in sublogarithmic rounds we give a new randomized distributed algorithm for 1 coloring in the local model running in o log 2o log log n rounds in a graph of maximum degree this implies that the 1 coloring problem is easier than the maximal independent set problem and the maximal matching problem due to their lower bounds of min log n log log n log log log by kuhn moscibroda and wattenhofer podc 04 our algorithm also extends to list coloring where the palette of each node contains 1 colors we extend the set of distributed symmetry breaking techniques by performing a decomposition of graphs into dense and sparse parts

on nichols algebras over basic hopf algebras this is a contribution to the classification of finite dimensional hopf algebras over an algebraically closed field mathbb k of characteristic 0 concretely we show that a finite dimensional hopf algebra whose hopf coradical is basic is a lifting of a nichols algebra of a semisimple yetter drinfeld module and we explain how to classify nichols algebras of this kind we provide along the way new examples of nichols algebras and hopf algebras with finite gelfand kirillov dimension

tests for seasonal moving average unit root in arima models abstract procedures to test for a unit root in the seasonal moving average ma operator for seasonal autoregressive integrated moving average arima models are examined such procedures are useful to distinguish deterministic from stochastic seasonal patterns the simple integrated seasonal ma 1 model z t y t yt s 1 b s t where y t is the observed seasonal series and t is a white noise sequence is considered and tests for h 0 1 versus h 1 1 are studied the locally best invariant unbiased and point optimal invariant tests are investigated the finite sample and asymptotic distributions of the test statistics under both the null hypothesis and local alternatives are derived quantiles of the null distributions are tabulated for quarterly and monthly data cases and a limited study of the powers of the tests is reported the test procedures are extended to more general seasonal arma models of the form z t 1 b s nt where n t follows a general stationary and invertible

pattern avoidance and rational smoothness of schubert varieties abstract letwbe an element of the weyl groupsn and letxwbe the schubert variety associated towin the flag manifoldsln c b lakshmibai and sandhya showed thatxwis smooth if and only ifwavoids the patterns 4231 and 3412 using two tests for rational smoothness from carrell and peterson we show that rational smoothness ofxwis characterized by pattern avoidance for typesbandcas well a key step in the proof of this result is a sequence of rules for factoring the poincare polynomials for the cohomology ring ofxw generalizing the work of gasharov patterns can also be used to characterize actual smoothness of schubert varieties

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

gaussian tests for seasonal unit roots based on cauchy estimation and recursive mean adjustments abstract we propose tests for seasonal unit roots whose limiting null distributions are always standard normal regardless of the period of seasonality and types of mean adjustments the seasonal models of dickey hasza and fuller 1984 journal of american statistical association 79 355 367 dhf and hylleberg engle granger and yoo 1990 journal of econometrics 44 215 238 hegy are considered for estimating parameters related to the seasonal unit roots regressor signs are used as instrumental variables while recursive sample means are used for adjusting the seasonal means in addition to normality of the limiting null distributions in seasonal mean models the recursive mean adjustment provides the new tests with locally higher powers than those of the existing tests of dhf and hegy based on the ordinary least squares estimators if data have a strong linear time trend the recursive mean adjustment is a source of both power gains of some tests for local alternatives and power losses of all tests for other alternatives limiting normality allow evaluation of p values and testing joint significance of subsets of seasonal unit roots

locality in distributed graph algorithms this paper concerns a number of algorithmic problems on graphs and how they may be solved in a distributed fashion the computational model is such that each node of the graph is occupied by a processor which has its own id processors are restricted to collecting data from others which are at a distance at most t away from them in t time units but are otherwise computationally unbounded this model focuses on the issue of locality in distributed processing namely to what extent a global solution to a computational problem can be obtained from locally available data three results are proved within this model a 3 coloring of an n cycle requires time omega log n this bound is tight by previous work of cole and vishkin any algorithm for coloring the d regular tree of radius r which runs for time at most 2r 3 requires at least omega sqrt d colors in an n vertex graph of largest degree delta an o delta 2 coloring may be found in time o log n

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

median unbiased estimation of coefficient variance in a time varying parameter model abstract this article considers inference about the variance of coefficients in time varying parameter models with stationary regressors the gaussian maximum likelihood estimator mle has a large point mass at 0 we thus develop asymptotically median unbiased estimators and asymptotically valid confidence intervals by inverting quantile functions of regression based parameter stability test statistics computed under the constant parameter null these estimators have good asymptotic relative efficiencies for small to moderate amounts of parameter variability we apply these results to an unobserved components model of trend growth in postwar u s per capita gross domestic product the mle implies that there has been no change in the trend growth rate whereas the upper range of the median unbiased point estimates imply that the annual trend growth rate has fallen by 0 9 per annum since the 1950s

comparison of alternative functional forms in production abstract in this paper a bayesian approach to choosing between two non nested multivariate regression systems is developed the bayesian approach involves the calculation of the posterior probabilities of alternative hypotheses and formation of a posterior odds ratio specific results are obtained for the comparison of a multivariate regression and seemingly unrelated regression model these odds ratio results are applied to the choice between translog and fourier flexible form cost share equation systems when calculated for aggregate u s manufacturing cost data the odds ratios favor the fourier flexible form over the translog form even in small samples the fourier series expansion appears to be an effective approximation technique

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

asymmetry and nonstationarity for a seasonal time series model abstract tests for symmetry and seasonal unit roots are developed for an extended model of hylleberg et al 1990 seasonal integration and cointegration journal econometrics 44 215 238 which can represent both partial seasonal unit roots and threshold effects methods based on ordinary least squares ols estimation and instrumental variable iv estimation are proposed and compared for adjusting mean functions ordinary mean adjustment and recursive mean adjustment are both considered several tests are constructed from various combination of estimation schemes and mean adjustment schemes among the tests the tests based on iv estimation are recommended because they have very simple limiting null distributions and have finite sample power properties comparable to those based on the olse the recommended tests are applied to a us unemployment rate data set and find evidences for both nonstationarities associated with zero frequency and threshold effects

improved distributed algorithms for coloring interval graphs with application to multicoloring trees we give a distributed 1 epsilon approximation algorithm for the minimum vertex coloring problem on interval graphs which runs in the mathcal local model and operates in mathrm o frac 1 epsilon log n rounds if nodes are aware of their interval representations then the algorithm can be adapted to the mathcal congest model using the same number of rounds prior to this work only constant factor approximations using mathrm o log n rounds were known 12 linial s ring coloring lower bound implies that the dependency on log n cannot be improved we further prove that the dependency on frac 1 epsilon is also optimal

distributed coloring algorithms for triangle free graphs vertex coloring is a central concept in graph theory and an important symmetry breaking primitive in distributed computing whereas degree graphs may require palettes of 1 colors in the worst case it is well known that the chromatic number of many natural graph classes can be much smaller in this paper we give new distributed algorithms to find k coloring in graphs of girth 4 triangle free graphs girth 5 and trees the parameter k can be at most 1 4 o 1 ln in triangle free graphs and at most 1 o 1 ln in girth 5 graphs and trees where o 1 is a function of specifically for sufficiently large we can find such a coloring in o k log n time moreover for any we can compute such colorings in roughly logarithmic time for triangle free and girth 5 graphs and in o log log log n time on trees as a byproduct our algorithm shows that the chromatic number of triangle free graphs is at most 4 o 1 ln which improves on jamall s recent bound of 67 o 1 ln finally we show that 1 coloring for triangle free graphs can be obtained in sublogarithmic time for any

arrangements defined by unitary reflection groups let v be a complex vector space of dimension i an arrangement in v is a finite set d of hyperplanes all containing the origin let l l d be the set of intersections of elements of r partially order l by reverse inclusion so that l has v as its minimal element and d as its set of atoms the poset l is a finite geometric lattice with rank function r x dim v x xe l without loss of generality we assume that n h h 0 is the maximal element of l and thus l has rank i the characteristic polynomial z l t of l is defined by

multiple unit roots in periodic autoregression in this paper we propose a model selection strategy for a univariate periodic autoregressive time series which involves tests for one or more unit roots and for parameter restrictions corresponding to seasonal unit roots and multiple unit roots at the zero frequency examples of models that are considered are variants of the seasonal unit roots model and the periodic integration model we show that the asymptotic distributions of various test statistics are the same as well known distributions which are already tabulated we apply our strategy to three empirical series to illustrate its ease of use we find that evidence for seasonal unit roots based on nonperiodic models disappears when periodic representations are considered

testing for nonstationary parameter specifications in seasonal time series models let yt be an autoregressive process satisfying yt aiyt a2yt d a3yt d i et where et t o is a sequence of iid o u2 random variables and d 2 such processes have been used as parametric models for seasonal time series typical values of d are 2 4 and 12 corresponding to time series observed semi annually quarterly and monthly respectively if a 1 2 1 a 3 1 then aladyt et where a yt denotes yt yt r if a 12 3 1 1 1 the process is nonstationary and the theory for stationary autoregressive processes does not apply a methodology for testing the hypothesis as 2 13 1 1 1 is presented and percentiles for test statistics are obtained extensions are presented for multiplicative processes for higher order processes and for processes containing deterministic trend and seasonal components

ad nilpotent ideals of a borel subalgebra ii we provide an explicit bijection between the ad nilpotent ideals of a borel subalgebra of a simple lie algebra g and the orbits of q h 1 q under the weyl group q being the coroot lattice and h the coxeter number of g from this result we deduce in a uniform way a counting formula for the ad nilpotent ideals

time varying periodicity in intraday volatility abstractwe develop a nonparametric test for whether return volatility exhibits time varying intraday periodicity using a long time series of high frequency data our null hypothesis commonly adopt

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 fast network decomposition algorithm and its applications to constant time distributed computation abstract a partition c 1 c 2 c q of g v e into clusters of strong respectively weak diameter d such that the supergraph obtained by contracting each c i is l colorable is called a strong resp weak d l network decomposition network decompositions were introduced in a seminal paper by awerbuch goldberg luby and plotkin in 1989 awerbuch et al showed that strong d l network decompositions with d l exp o log n log log n can be computed in distributed deterministic time o d even more importantly they demonstrated that network decompositions can be used for a great variety of applications in the message passing model of distributed computing the result of awerbuch et al was improved by panconesi and srinivasan in 1992 in the latter result d l exp o log n and the running time is o d as well in another remarkable breakthrough linial and saks in 1992 showed that weak o log n o log n network decompositions can be computed in distributed randomized time o log 2 n much more recently barenboim 2012 devised a distributed randomized constant time algorithm for computing strong network decompositions with d o 1 however the parameter l in his result is o n 1 2 in this paper we drastically improve the result of barenboim and devise a distributed randomized constant time algorithm for computing strong o 1 o n network decompositions as a corollary we derive a constant time randomized o n approximation algorithm for the distributed minimum coloring problem improving the previously best known o n 1 2 approximation guarantee we also derive other improved distributed algorithms for a variety of problems most notably for the extremely well studied distributed minimum dominating set problem currently there is no known deterministic polylogarithmic time algorithm we devise a deterministic polylogarithmic time approximation algorithm for this problem addressing an open problem of lenzen and wattenhofer 2010

generic singularities of certain schubert varieties let g be a connected semisimple algebraic group b a borel subgroup t a maximal torus in b with weyl group w and q a subgroup containing b for w in w let x wq denote the schubert variety overline bwq q for y in w such that x yq subseteq x wq one knows that byq q admits a t stable transversal in x wq which we denote by cal n yq wq we prove that under certain hypotheses cal n yq wq is isomorphic to the orbit closure of a highest weight vector in a certain weyl module we also obtain a generalisation of this result under slightly weaker hypotheses further we prove that our hypotheses are satisfied when q is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight and x yq is an irreducible component of the boundary of x wq that is the complement of the open orbit of the stabiliser in g of x wq as a consequence we describe the singularity of x wq along byq q and obtain that the boundary of x wq equals its singular locus

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

classification of arithmetic root systems arithmetic root systems are invariants of nichols algebras of diagonal type with a certain finiteness property they can also be considered as generalizations of ordinary root systems with rich structure and many new examples on the other hand nichols algebras are fundamental objects in the construction of quantized enveloping algebras in the noncommutative differential geometry of quantum groups and in the classification of pointed hopf algebras by the lifting method of andruskiewitsch and schneider in the present paper arithmetic root systems are classified in full generality as a byproduct many new finite dimensional pointed hopf algebras are obtained

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

testing for unit roots in seasonal time series abstract regression estimators of coefficients in seasonal autoregressive models are described the percentiles of the distributions for time series that have unit roots at the seasonal lag are computed by monte carlo integration for finite samples and by analytic techniques and monte carlo integration for the limit case the tabled distributions may be used to test the hypothesis that a time series has a seasonal unit root

an instrumental variable approach for tests of unit roots and seasonal unit roots in asymmetric time series models the unit root tests of caner and hansen econometrica 69 2001 1555 for asymmetric time series models based on the ordinary least squares estimator olse have asymptotic null distributions that depend on parameters of asymmetry we resolve this parameter dependency by adopting the instrumental variable estimation of shin and lee j business economic statist 19 2001 233 and the recursive mean adjustment of shin and so j time series anal 22 2001b 595 if the threshold parameter is known the limiting null distribution of the proposed wald test does not depend on any nuisance parameter and is chi squared if the threshold parameter is unknown and is estimated from data then under threshold effect the limiting null distributions of the proposed wald tests are chi squared as those for model with known threshold parameter whereas under no threshold effect they are chi squared only conditionally on the weak limit of the estimated threshold parameter our wald tests extend to seasonal models retaining the chi square asymptotics regardless of the parameter of seasonality which is not the case of the olse based tests moreover the proposed tests can be modified into one sided wald tests which are significantly more powerful than the tests of caner and hansen 2001 we apply our method to the monthly us unemployment rate and find some evidences of seasonal unit roots suggesting nonstationarity rather than the strong stationarity of caner and hansen 2001

systematic sampling and temporal aggregation in time series models abstract within the class of armax models we consider the effects systematic sampling or temporal aggregation may have on the dynamic relationships between variables these include changes in lag lengths and causal ordering and may occur even in simple models some implications for the modelling of time series are noted for the subclass of arima models we also analyse the consequences of sampling or aggregating seasonal models

recursively free reflection arrangements abstract let a a w be the reflection arrangement of the finite complex reflection group w by terao s famous theorem the arrangement a is free in this paper we classify all reflection arrangements which belong to the smaller class of recursively free arrangements moreover for the case that w admits an irreducible factor isomorphic to g 31 we obtain a new computer free proof for the non inductive freeness of a w since our classification implies the non recursive freeness of the reflection arrangement a g 31 we can prove a conjecture by abe about the new class of divisionally free arrangements which he recently introduced

a distributed 2 e approximation for vertex cover in o log δ e log log δ rounds we present a simple deterministic distributed 2 e approximation algorithm for minimum weight vertex cover which completes in o log elog log rounds where is the maximum degree in the graph for any e 0 that is at most o 1 for a constant e this implies a constant approximation in o log log log rounds which contradicts the lower bound of kmw10

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

the locality of distributed symmetry breaking symmetry breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open in this article we work in the local model where the input graph and underlying distributed network are identical and study the randomized complexity of four fundamental symmetry breaking problems on graphs computing miss maximal independent sets maximal matchings vertex colorings and ruling sets a small sample of our results includes the following an mis algorithm running in o log2 2o log log n time where is the maximum degree this is the first mis algorithm to improve on the 1986 algorithms of luby and alon babai and itai when log n l l 2 log n and comes close to the log log log lower bound of kuhn moscibroda and wattenhofer a maximal matching algorithm running in o log log 4log n time this is the first significant improvement to the 1986 algorithm of israeli and itai moreover its dependence on is nearly optimal a 1 coloring algorithm requiring o log 2o log log n time improving on an o log log n time algorithm of schneider and wattenhofer a method for reducing symmetry breaking problems in low arboricity degeneracy graphs to low degree graphs roughly speaking the arboricity or degeneracy of a graph bounds the density of any subgraph corollaries of this reduction include an o log n time maximal matching algorithm for graphs with arboricity up to 2 log n and an o log 2 3n time mis algorithm for graphs with arboricity up to 2 log n 1 3 each of our algorithms is based on a simple but powerful technique for reducing a randomized symmetry breaking task to a corresponding deterministic one on a poly log n size graph

on regression based tests for seasonal unit roots in the presence of periodic heteroscedasticity we analyse regression based tests for seasonal unit roots when the shocks are periodically heteroscedastic we show using the case of quarterly data to illustrate that the limiting marginal null distributions of tests for unit roots at the zero and nyquist frequencies are unaffected by the periodic heteroscedasticity however tests at the harmonic seasonal frequencies are shown to be either unaffected or to display a shift in their limiting distribution depending on the specific nature of the periodic heteroscedasticity in extreme cases certain of these limiting distributions are degenerate while others are simple functions of the well known dickey fuller distributions monte carlo evidence shows that the asymptotic theory provides a very good prediction for the finite sample behaviour of the unit root test statistics we show that tests with approximately correct size may be obtained by simulating their sampling distributions using periodic variance parameters estimated from the data in hand though laborious this procedure seems to be the best available since more conservative approaches sacrifice significant power

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

deterministic δ 1 coloring in sublinear in δ time in static dynamic and faulty networks we study the distributed 1 vertex coloring and 2 1 edge coloring problems these problems are among the most important and intensively studied problems in distributed computing despite very intensive research in the last 30 years no deterministic algorithms for these problems with sublinear in time have been known so far moreover for more restricted scenarios and some related problems there are lower bounds of goos et al 2014 hirvonen and suomela 2012 kuhn and wattenhofer 2006 szegedy and vishwanathan 1993 the question of the possibility to devise algorithms that overcome this challenging barrier is one of the most fundamental questions in distributed symmetry breaking barenboim and elkin 2009 2011 goos et al 2014 hirvonen and suomela 2012 kuhn 2009 panconesi and rizzi 2001 in this article we settle this question for 1 vertex coloring and 2 1 edge coloring by devising deterministic algorithms that require o 3s4log log n time in the static dynamic and faulty settings the term log n is unavoidable in view of the lower bound of linial 1987 moreover for 1 o 1 vertex coloring and 2 o 1 edge coloring we devise algorithms with o s log n deterministic time this is roughly a quadratic improvement comparing to the state of the art that requires o log n time barenboim and elkin 2009 kuhn 2009 panconesi and rizzi 2001 our results are actually more general than that since they apply also to a variant of the list coloring problem that generalizes ordinary coloring our results are obtained using a novel technique for coloring partially colored graphs also known as fixing we partition the uncolored parts into a small number of subgraphs with certain helpful properties then we color these subgraphs gradually using a technique that employs constructions of polynomials in a novel way our construction is inspired by the algorithm of linial 1987 for ordinary o 2 coloring however it is a more sophisticated construction that differs from that of linial 1987 in several important respects these new insights in using systems of polynomials allow us to significantly speed up the o coloring algorithms moreover they allow us to devise algorithms with the same running time also in the more complicated settings of dynamic and faulty networks

inductive freeness of ziegler s canonical multiderivations for reflection arrangements abstract let a be a free hyperplane arrangement in 1989 ziegler showed that the restriction a of a to any hyperplane endowed with the natural multiplicity is then a free multiarrangement we initiate a study of the stronger freeness property of inductive freeness for these canonical free multiarrangements and investigate them for the underlying class of reflection arrangements more precisely let a a w be the reflection arrangement of a complex reflection group w by work of terao each such reflection arrangement is free thus so is ziegler s canonical multiplicity on the restriction a of a to a hyperplane we show that the latter is inductively free as a multiarrangement if and only if a itself is inductively free

bootstrapping the hegy seasonal unit root tests this paper proposes bootstrap versions of the seasonal unit root tests of inter alia hylleberg engle granger and yoo j econometrics 44 1990 215 238 hegy we report a simulation study of the properties of both the conventional and bootstrapped seasonal unit root tests when applied to series having higher order serial correlation and or periodic heteroscedasticity both of which are known to severely distort the significance level of the conventional tests our results demonstrate that the bootstrap provides good approximations to the statistics null distributions moreover the bootstrap corrects the adverse effects of data dependent lag selection seen in the conventional augmented hegy tests the bootstrapped tests have comparable power to infeasible exactly significance level corrected lag augmented hegy tests and their use is recommended

randomized distributed edge coloring via an extension of the chernoff hoeffding bounds certain types of routing scheduling and resource allocation problems in a distributed setting can be modeled as edge coloring problems we present fast and simple randomized algorithms for edge coloring a graph in the synchronous distributed point to point model of computation our algorithms compute an edge coloring of a graph g with n nodes and maximum degree delta with at most 1 6 delta o log 1 delta n colors with high probability arbitrarily close to 1 for any fixed delta 0 they run in polylogarithmic time the upper bound on the number of colors improves upon the 2 delta 1 coloring achievable by a simple reduction to vertex coloring to analyze the performance of our algorithms we introduce new techniques for proving upper bounds on the tail probabilities of certain random variables the chernoff hoeffding bounds are fundamental tools that are used very frequently in estimating tail probabilities however they assume stochastic independence among certain random variables which may not always hold our results extend the chernoff hoeffding bounds to certain types of random variables which are not stochastically independent we believe that these results are of independent interest and merit further study

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

exact tests in single equation autoregressive distributed lag models abstract for hypotheses on the coefficient values of the lagged dependent variables in the arx class of dynamic regression models test procedures are developed which yield exact inference for given up to an unknown scale factor distribution of the innovation errors they include exact tests on the maximum lag length for structural change and on the presence of seasonal or multiple unit roots i e they cover situations where usually asymptotic and non exact t f aoc adf or hegy tests are employed the various procedures are demonstrated and compared in illustrative empirical models and the approach is critically discussed

a simple parallel algorithm for the maximal independent set problem two basic design strategies are used to develop a very simple and fast parallel algorithms for the maximal independent set mis problem the first strategy consists of assigning identical copies o

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 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

root systems and weyl groupoids for nichols algebras motivated by work of kac and lusztig we define a root system and a weyl groupoid for a large class of semisimple yetter drinfeld modules over an arbitrary hopf algebra the ob tained combinatorial structure fits perfectly into an existing frame work of generalized root systems associated to a family of cartan matrices and provides novel insight into nichols algebras we demonstrate the power of our construction with new results on nichols algebras over finite non abelian simple groups and sym metric groups

local computation lower and upper bounds the question of what can be computed and how efficiently is at the core of computer science not surprisingly in distributed systems and networking research an equally fundamental question is what can be computed in a distributed fashion more precisely if nodes of a network must base their decision on information in their local neighborhood only how well can they compute or approximate a global optimization problem in this paper we give the first polylogarithmic lower bound on such local computation for optimization problems including minimum vertex cover minimum connected dominating set maximum matching maximal independent set and maximal matching in addition we present a new distributed algorithm for solving general covering and packing linear programs for some problems this algorithm is tight with the lower bounds whereas for others it is a distributed approximation scheme together our lower and upper bounds establish the local computability and approximability of a large class of problems characterizing how much local information is required to solve these tasks

symmetry breaking depending on the chromatic number or the neighborhood growth we deterministically compute a d 1 coloring and a maximal independent set mis in time o d 1 2 q 1 h log n for d 1 email protected email protected 1 i h where d j is defined as the maximal number of nodes within distance j for a node and d email protected 1 our greedy coloring and mis algorithms improve the state of the art algorithms running in o d log n for a large class of graphs i e graphs of moderate neighborhood growth with h 36 we also state and analyze a randomized coloring algorithm in terms of the chromatic number the run time and the used colors our algorithm runs in time o email protected log n for email protected email protected log 1 1 l o g nn and email protected o d log 1 1 l o g nn for graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest d 1 coloring algorithm running in time o email protected logn the algorithm works without knowledge of g and uses less than d colors i e 1 1 o g d with high probability to the best of our knowledge this is the first distributed algorithm for such general graphs taking the chromatic number g into account we also improve on the state of the art deterministic computation of 2 c ruling sets

smooth points of t stable varieties in g b and the peterson map let g be a connected semi simple algebraic group defined over an algebraically closed field k and let t b p be respectively a maximal torus a borel subgroup and a parabolic subgroup of g inspired by a beautiful result of dale peterson describing the singular locus of a schubert variety in g b we characterize the t fixed points in the singular locus of an arbitrary irreducible t stable subvariety of g p a t variety for short peterson s result cf the deformation theorem 1 says that if k then a schubert variety x g b is smooth at a t fixed point x if and only if it is smooth at every t fixed point y x in the bruhat chevalley order on the fixed point set x t and all the limits c x x lim z x t z x z c c t of the zariski tangent spaces t z x of x coincide as c varies over the set of all t stable curves in x with c t x y where y x using this peterson showed that if g is simply laced and defined over then every rationally smooth point of a schubert variety in g b is smooth more generally the deformation c x x is defined for any k variety x with a t action provided c is what we call good i e c is a curve of the form c overline tz where z is a smooth point of x x t and x c t our first main result theorem 1 4 says that if x x is an attractive fixed point then x is smooth at x if and only if there exist at least two good c containing x such that c x x te x x where te x x denotes the span of the tangent lines of the t stable curves in x containing x in addition if x is cohen macaulay at x and c x x te x x for even one good c then x is smooth at x our second main result theorem 1 6 says that if x is a t variety in g p where g is simply laced then c x x te x x for each good c this is not true for general g but when g has no g2 factors then c x x is always contained in the linear span of the reduced tangent cone to x at x these results lead to several descriptions of the smooth fixed points of a t variety in g p and in particular they give simple proofs of peterson s results valid for any algebraically closed field we also show cf example 7 1 that there can exist t stable subvarieties in g b where g is simply laced which have rationally smooth t fixed points in their singular loci

seasonal bvar models abstract many authors have stressed the importance of using seasonally unadjusted data in modelling time series however few seasonal multivariate methodologies are currently available in this paper we extend the bayesian vector autoregressive bvar methodology of litterman 1979 1984 1986 doan litterman and sims 1984 and sims 1989 to the context of seasonal time series we examine some possible modifications to the regular litterman s priors along three time domain specifications often encountered in seasonal time series modelling we assess the forecasting performance of the seasonal bvar models in the context of a monthly model of the u s economy 1967 1 1989 9 comprising both seasonal and nonseasonal variables

yetter drinfeld modules over bosonizations of dually paired hopf algebras abstract let r r be a dual pair of hopf algebras in the category of yetter drinfeld modules over a hopf algebra h with bijective antipode we show that there is a braided monoidal isomorphism between rational left yetter drinfeld modules over the bosonizations of r and of r respectively as an application of this very general category isomorphism we obtain a natural proof of the existence of reflections of nichols algebras of semi simple yetter drinfeld modules over h

a simplicial complex of nichols algebras we translate the concept of restriction of an arrangement in terms of hopf algebras in consequence every nichols algebra gives rise to a simplicial complex decorated by nichols algebras with restricted root systems as applications some of these nichols algebras provide weyl groupoids which do not arise for diagonal nichols algebras and in fact we realize all root systems of finite weyl groupoids of rank greater than three further our result explains the root systems of the folded nichols algebras over nonabelian groups and of generalized satake diagrams

on periodic structures and testing for seasonal unit roots abstract the standard testing procedures for seasonal unit roots developed so far have been based mainly on time invariant autoregressive integrated moving average arima processes with ar polynomials involving seasonal differencing one attractive alternative is to use periodic arma models convenient procedures are presented for testing for the presence of unit roots at the zero and seasonal frequencies in periodic time series the limiting distributions are derived and tabulated simulation evidence illustrates the advantages of allowing for periodicity the tests are illustrated via applications to macroeconomic and ozone level data