rational singularities and almost split sequences the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the finite dimen sional representations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules of the quotient singularity associated with g over the past decade almost split sequences have been playing an increasingly important role in the representation theories of finite dimensional algebras and classical orders see 5 and 3 8 for basic existence theorems in these contexts while they have been known for some time to exist in higher dimensional situations 3 it has not been at all clear how they related to singularity theory if at all the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the fimte dimensional complex represen tations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules over the quotient singularity r associated with g as in the case of finite dimensional algebras the ar quiver of reflexive r modules is defined in terms of the almost split sequences of reflexive r modules in the case g c sl 2 c mckay observed that the underlying graph of the mckay quiver with the trivial module removed is isomorphic to the desingulariza tion graph of the associated singularity various explanations of this phenomenon have been given by knorrer gonzalez sprinberg verdier 6 and artin verdier 1 which along the way have established most explicitly in 1 a natural one to one correspondence between the indecomposable reflexive r modules and the nodes of the desingularization graph but why the almost split sequences describe the edge of the desingularization graph still remains to be explained an effort has been made to make this paper as self contained as possible in particular no prior knowledge of almost split sequences is required before describing the contents of the six sections of this paper we fix some notation throughout this paper g is a finite group k an algebraically closed field of characteristic not dividing the order of g and v a two dimensional k representation of g setting s k x y the k algebra of formal power series the two dimen sional representation v gives a linear action of g on s as a group of k algebra automorphisms we denote by s g the skew group ring given by this action

on involutions in symmetric groups and a conjecture of lusztig abstract let w s be a coxeter system equipped with a fixed automorphism of order 2 which preserves s lusztig and with vogan in some special cases have shown that the space spanned by set of twisted involutions i e elements w w with w w 1 was naturally endowed with a module structure of the hecke algebra of w s with two distinguished bases which can be viewed as twisted analogues of the well known standard basis and kazhdan lusztig basis the transition matrix between these bases defines a family of polynomials p y w which can be viewed as twisted analogues of the well known kazhdan lusztig polynomials of w s lusztig has conjectured that this module is isomorphic to the right ideal of the hecke algebra with hecke parameter u 2 associated to w s generated by the element x w w u l w t w in this paper we prove this conjecture in the case when id and w s n the symmetric group on n letters our methods are expected to be generalised to all the other finite crystallographic coxeter groups

the combinatorics of twisted involutions in coxeter groups the open intervals in the bruhat order on twisted involutions in a coxeter group are shown to be pl spheres this implies results conjectured by f incitti and sharpens the known fact that these posets are gorenstein over z 2 we also introduce a boolean cell complex which is an analogue for twisted involutions of the coxeter complex several classical coxeter complex properties are shared by our complex when the group is finite it is a shellable sphere shelling orders being given by the linear extensions of the weak order on twisted involutions furthermore the h polynomial of the complex coincides with the polynomial counting twisted involutions by descents in particular this gives a type independent proof that the latter is symmetric

noncommutative rational double points in this paper we study noncommutative analogues of rational double points the approach is to consider the action of a finite group g on certain noncommutative analogues of k x y which were studied by artin and stafford regular local rings of dimension 2 manuscript an explicit description in terms of generators and relations is given for a large class of such algebras when g is cyclic finally we show that these algebras are as gorenstein of dimension two have finite representation type and in many cases are regular in codimension one

group actions on stanley reisner rings and invariants of permutation groups abstract let h be a group of permutations of x 1 x n and let q h x 1 x 2 x n denote the ring of h invariant polynomials in x 1 x 2 x n with rational coefficients combinatorial methods for the explicit construction of free bases for q h x 1 x 2 x n as a module over the symmetric polynomials are developed the methods are developed by studying the action of the symmetric group on the stanley reisner ring of the subset lattice some general results are also obtained by studying the action of a coxeter group on the stanley reisner ring of the corresponding coxeter complex in the case of a weyl group a purely combinatorial construction of certain invariants first considered by r steinberg topology 14 1975 173 177 is obtained some applications to representation theory are also included

canonical singularities of orders over surfaces we define and study canonical singularities of orders over surfaces these are non commutative analogues of kleinian singularities that arise naturally in the minimal model program for orders over surfaces d chan and c ingalls invent math 161 2005 427 452 we classify canonical singularities of orders using their minimal resolutions which we define we describe them explicitly as invariant rings for the action of a finite group on a full matrix algebra over a regular local ring we also prove that they are gorenstein describe their auslander reiten quivers and note a simple version of the mckay correspondence

positivity conjectures for kazhdan lusztig theory on twisted involutions the finite case abstract let w s be any coxeter system and let w w be an involution of w which preserves the set of simple generators s lusztig and vogan have shown that the corresponding set of twisted involutions i e elements w w with w 1 w naturally generates a module of the hecke algebra of w s with two distinguished bases the transition matrix between these bases defines a family of polynomials p y w which one can view as a twisted analogue of the much studied family of kazhdan lusztig polynomials of w s the polynomials p y w can have negative coefficients but display several conjectural positivity properties of interest which parallel positivity properties of the kazhdan lusztig polynomials this paper reports on some calculations which verify four such positivity conjectures in several finite cases of interest in particular for the non crystallographic coxeter systems of types h 3 and h 4

optimal estimates of the diffusion coefficient of a single brownian trajectory modern developments in microscopy and image processing are revolutionizing areas of physics chemistry and biology as nanoscale objects can be tracked with unprecedented accuracy the goal of single particle tracking is to determine the interaction between the particle and its environment the price paid for having a direct visualization of a single particle is a consequent lack of statistics here we address the optimal way of extracting diffusion constants from single trajectories for pure brownian motion it is shown that the maximum likelihood estimator is much more efficient than the commonly used least squares estimate furthermore we investigate the effect of disorder on the distribution of estimated diffusion constants and show that it increases the probability of observing estimates much smaller than the true average value

the residue complex of a noncommutative graded algebra where x rx x is the set of points of dimension q the q skeleton q qy1 and k x is an injective hull of the residue field k x the coboundary a operator d k x a k y is nonzero precisely when y is an immediate a a w x specialization of x for a discussion of the commutative theory see rd w x and ye2 in this paper we propose a definition of the residue complex r of a noncommutative noetherian graded k algebra a s k a a 1 2 we begin in section 1 with the generalized auslander gorenstein a g condition this condition can be checked whenever a has a dualizing complex if a is gorenstein i e has finite injective dimension it reduces

fractal probability laws we explore six classes of fractal probability laws defined on the positive half line weibull frechet levy hyper pareto hyper beta and hyper shot noise each of these classes admits a unique statistical power law structure and is uniquely associated with a certain operation of renormalization all six classes turn out to be one dimensional projections of underlying poisson processes which in turn are the unique fixed points of poissonian renormalizations the first three classes correspond to linear poissonian renormalizations and are intimately related to extreme value theory weibull frechet and to the central limit theorem levy the other three classes correspond to nonlinear poissonian renormalizations pareto s law commonly perceived as the universal fractal probability distribution is merely a special case of the hyper pareto class

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

from shape to randomness a classification of langevin stochasticity abstract the langevin equation perhaps the most elemental stochastic differential equation in the physical sciences describes the dynamics of a random motion driven simultaneously by a deterministic potential field and by a stochastic white noise the langevin equation is in effect a mechanism that maps the stochastic white noise input to a stochastic output a stationary steady state distribution in the case of potential wells and a transient extremum distribution in the case of potential gradients in this paper we explore the degree of randomness of the langevin equation s stochastic output and classify it a la mandelbrot into five states of randomness ranging from infra mild to ultra wild we establish closed form and highly implementable analytic results that determine the randomness of the langevin equation s stochastic output based on the shape of the langevin equation s potential field

measuring statistical evenness a panoramic overview motivated by the question how equal is the distribution of wealth within a given human population economics devised an impressive toolbox of quantitative measures of societal egalitarianism including the lorenz curve and the following indices gini pietra hoover amato hirschman theil and atkinson these quantitative measures considered in the broader context of general data sets with positive values are in effect general gauges of statistical evenness while the application of gini s index grew beyond economics and reached diverse fields of science the aforementioned evenness toolbox has largely remained within the confines of the social sciences the aim of this paper is to expose this evenness toolbox to the physics community by presenting a comprehensive evenness based approach to a fundamental problem in science the measurement of statistical heterogeneity

existence theorems for dualizing complexes over non commutative graded and filtered rings abstract in this note we prove existence theorems for dualizing complexes over graded and filtered rings thereby generalizing some results by zhang yekutieli and jorgensen

on the physical interpretation of statistical data from black box systems abstract in this paper we explore the physical interpretation of statistical data collected from complex black box systems given the output statistics of a black box system and considering a class of relevant markov dynamics which are physically meaningful we reverse engineer the markov dynamics to obtain an equilibrium distribution that coincides with the output statistics observed this reverse engineering scheme provides us with a conceptual physical interpretation of the black box system investigated five specific reverse engineering methodologies are developed based on the following dynamics langevin geometric langevin diffusion growth collapse and decay surge in turn these methodologies yield physical interpretations of the black box system in terms of conceptual intrinsic forces temperatures and instabilities the application of these methodologies is exemplified in the context of the distribution of wealth and income in human societies which are outputs of the complex black box system called the economy

dualizing complexes over noncommutative graded algebras the motivation for the work presented in this paper was two questions posed by m artin let a be a d dimensional regular graded algebra over the field k with augmentation ideal m see def 4 13 denote by hi a resp hi a the dth left resp right local cohomology of a at the ideal m i e hi a lim extd a m a and similarly for hi a as left modules and as right modules hi a and h a are both isomorphic to a hom a k up to a twist in degrees all operations are taking place in the graded category

on selfsimilar levy random probabilities abstract we explore a class of random probabilities induced by the normalization of selfsimilar levy random measures random measures whose probability laws are governed by stable one sided levy distributions various statistical properties of these random probabilities are analyzed i moment structure ii auto covariance structure iii one dimensional and multidimensional tail probabilities and iv behavior in limiting cases for the levy smirnoff case corresponding to the selfsimilarity index of order 1 2 an explicit analytic formula for the multidimensional probability density functions is derived last a comparison between the class of selfsimilar levy random probabilities and the dirichlet random probability induced by the normalization of the gamma random measure is conducted showing the former to be far more robust than the latter

on the fractal characterization of paretian poisson processes abstract paretian poisson processes are poisson processes which are defined on the positive half line have maximal points and are quantified by power law intensities paretian poisson processes are elemental in statistical physics and are the bedrock of a host of power law statistics ranging from pareto s law to anomalous diffusion in this paper we establish evenness based fractal characterizations of paretian poisson processes considering an array of socioeconomic evenness based measures of statistical heterogeneity we show that amongst the realm of poisson processes which are defined on the positive half line and have maximal points paretian poisson processes are the unique class of fractal processes exhibiting scale invariance the results established in this paper are diametric to previous results asserting that the scale invariance of poisson processes with respect to physical randomness based measures of statistical heterogeneity is characterized by exponential poissonian intensities

bar operators for quasiparabolic conjugacy classes in a coxeter group abstract the action of a coxeter group w on the set of left cosets of a standard parabolic subgroup deforms to define a module m j of the group s iwahori hecke algebra h with a particularly simple form rains and vazirani have introduced the notion of a quasiparabolic set to characterize w sets for which analogous deformations exist a motivating example is the conjugacy class of fixed point free involutions in the symmetric group deodhar has shown that the module m j possesses a certain antilinear involution called the bar operator and a certain basis invariant under this involution which generalizes the kazhdan lusztig basis of h the well known significance of this basis in representation theory makes it natural to seek to extend deodhar s results to the quasiparabolic setting in general the obstruction to finding such an extension is the existence of an appropriate quasiparabolic analogue of the bar operator in this paper we consider the most natural definition of a quasiparabolic bar operator and develop a theory of quasiparabolic kazhdan lusztig bases under the hypothesis that such a bar operator exists giving content to this theory we prove that a bar operator in the desired sense does exist for quasiparabolic w sets given by twisted conjugacy classes of twisted involutions finally we prove several results classifying the quasiparabolic conjugacy classes in a coxeter group

dualizing complexes over noncommutative local rings abstract we prove an existence theorem for dualizing complexes over noncommutative noetherian complete semilocal algebras which generalizes van den bergh s existence theorem in the graded case using the dualizing complex noncommutative versions of bass theorem and the no holes theorem are proved we also prove that noetherian complete semilocal algebras satisfying polynomial identities are catenary

fixed points of involutive automorphisms of the bruhat order abstract applying a classical theorem of smith we show that the poset property of being gorenstein over z 2 is inherited by the subposet of fixed points under an involutive poset automorphism as an application we prove that every interval in the bruhat order on twisted involutions in an arbitrary coxeter group has this property and we find the rank function this implies results conjectured by f incitti we also show that the bruhat order on the fixed points of an involutive automorphism induced by a coxeter graph automorphism is isomorphic to the bruhat order on the fixed subgroup viewed as a coxeter group in its own right

the mo bius function of factor order abstract intervals in the factor ordering of a free monoid are investigated it was shown by farmer 1982 that such intervals are contractible or homotopy spheres in case is the empty word we observe here that the same is true in general this implies that the mobius function of factor order takes values in 0 1 1 a recursive rule for this mobius function is given which allows efficient computation via the knuth morris pratt algorithm the mobius function of subword order was studied in bjorner 1990 we give here a simpler proof a parity changing involution for its combinatorial interpretation

integral u deformed involution modules abstract let w s be a coxeter system and an automorphism of w with order 2 and s s lusztig and vogan 20 23 have introduced a u deformed version m u of kottwitz s involution module over the iwahori hecke algebra h u w with hecke parameter u 2 where u is an indeterminate lusztig has proved that m u is isomorphic to the left h u w submodule of h u generated by x w w w u l w t w where h u is the vector space consisting of all formal possibly infinite sums x w c x t x c x q u for each x he speculated that one can extend this by replacing u with any c 0 1 1 in this paper we give a positive answer to his speculation for any k 0 1 1 and any w where k is an arbitrary ground field

proportionate vs disproportionate distribution of wealth of two individuals in a tempered paretian ensemble abstract we study the distribution p of the random variable x 1 x 1 x 2 where x 1 and x 2 are the wealths of two individuals selected at random from the same tempered paretian ensemble characterized by the distribution x x x 1 where 0 is the pareto index and x is the cut off function we consider two forms of x a bounded function x 1 for l x h and zero otherwise and a smooth exponential function x exp l x x h in both cases x has moments of arbitrary order we show that for 1 p always has a unimodal form and is peaked at 1 2 so that most probably x 1 x 2 for 0 1 we observe a more complicated behavior which depends on the value of l h in particular for c a certain threshold value p has a three modal for a bounded x and a bimodal m shape for an exponential x form which signifies that in such ensembles the wealths x 1 and x 2 are disproportionately different

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

symmetry theorems for ext vanishing abstract it was proved by avramov and buchweitz that if a is a commutative local complete intersection ring with finitely generated modules m and n then the ext groups between m and n vanish from some step if and only if the ext groups between n and m vanish from some step this paper shows that the same is true under the weaker conditions that a is gorenstein and that m and n have finite complete intersection dimension the result is also proved if a is gorenstein and has finite cohen macaulay type similar results are given for non commutative complete semi local algebras

sliding blocks with random friction and absorbing random walks with the purpose of explaining recent experimental findings we study the distribution a lambda of distances lambda traversed by a block that slides on an inclined plane and stops due to friction a simple model in which the friction coefficient mu is a random function of position is considered the problem of finding a lambda is equivalent to a first passage time problem for a one dimensional random walk with nonzero drift whose exact solution is well known from the exact solution of this problem we conclude that a for inclination angles theta less than theta c tan av mu the average traversed distance av lambda is finite and diverges when theta to theta c as av lambda sim theta c theta 1 b at the critical angle a power law distribution of slidings is obtained a lambda sim lambda 3 2 our analytical results are confirmed by numerical simulation and are in partial agreement with the reported experimental results we discuss the possible reasons for the remaining discrepancies

topology of posets with special partial matchings abstract special partial matchings spms are a generalisation of brenti s special matchings let a pircon be a poset in which every non trivial principal order ideal is finite and admits an spm thus pircons generalise marietti s zircons we prove that every open interval in a pircon is a pl ball or a pl sphere it is then demonstrated that bruhat orders on certain twisted identities and quasiparabolic w sets constitute pircons together these results extend a result of can cherniavsky and twelbeck prove a conjecture of hultman and confirm a claim of rains and vazirani

mean field theory for a simple model of evolution a simple dynamical model for darwinian evolution on its slowest time scale is analyzed its mean field theory is formulated and solved a random neighbor version of the model is simulated as is a one dimensional version in one dimension the dynamics can be described in terms of a repetitious random walker and anomalous diffusion with exponent 0 4 in all cases the model self organizes to a robust critical attractor