complexity and multiple complexes in this chapter we introduce the notion of the complexity of a module and explore some related ideas complexity was first defined by jon alperin in the late 1970 s and it helped to motivate much of the development of the homological properties of modules in fact the complexity of a finitely generated kg module m is a rather crude invariant in that it only measures the polynomial rate of growth of a minimal projective resolution of the module the original theorem of alperin and evens which extended quillen s dimension theorem to modules was proved only in terms of complexity later it was extended to support varieties as well the theorem computes the complexity of a kg module m as the maximum of the complexities of the restrictions of m to the elementary abelian subgroups of the group g

a putnam fuglede commutativity property for hilbert space operators abstract given hilbert space operators a b b h define a b and a b in b b h by a b x a x x b and a b x a x b x for each x b h an operator a b h satisfies the putnam fuglede properties respectively notation a pf respectively a pf if for every isometry v b h for which the equation a v x 0 respectively a v x 0 has a non trivial solution x b h the solution x also satisfies a v x 0 respectively a v x 0 we prove that an operator a b h is in pf if and only if it is in pf

on the shape of a set of points in the plane a generalization of the convex hull of a finite set of points in the plane is introduced and analyzed this generalization leads to a family of straight line graphs alpha shapes which seem to capture the intuitive notions of fine shape and crude shape of point sets it is shown that a shapes are subgraphs of the closest point or furthest point delaunay triangulation relying on this result an optimal o n log n algorithm that constructs alpha shapes is developed

brownian limits local limits and variance asymptotics for convex hulls in the ball schreiber and yukich ann probab 36 2008 363 396 establish an asymptotic representation for random convex polytope geometry in the unit ball bd d 2 in terms of the general theory of stabilizing functionals of poisson point processes as well as in terms of generalized paraboloid growth processes this paper further exploits this connection introducing also a dual object termed the paraboloid hull process via these growth processes we establish local functional limit theorems for the properly scaled radius vector and support functions of convex polytopes generated by high density poisson samples we show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius vector and support functions generalized paraboloid growth processes coupled with general techniques of stabilization theory yield brownian sheet limits for the defect volume and mean width functionals finally we provide explicit variance asymptotics and central limit theorems for the k face and intrinsic volume functionals

random polytopes and the efron stein jackknife inequality let k be a smooth convex body the convex hull of independent random points in k is a random polytope estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained the essential step is the use of the efron stein jackknife inequality for the variance of symmetric statistics consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope a conjecture of barany concerning random and best approximation of convex bodies is confirmed analogous results for random polytopes with vertices on the boundary of the convex body are given

variance asymptotics and scaling limits for gaussian polytopes let k n be the convex hull of i i d random variables distributed according to the standard normal distribution on mathbb r d we establish variance asymptotics as n rightarrow infty for the re scaled intrinsic volumes and k face functionals of k n k in 0 1 ldots d 1 resolving an open problem weil and wieacker handbook of convex geometry vol b pp 1391 1438 north holland elsevier amsterdam 1993 variance asymptotics are given in terms of functionals of germ grain models having parabolic grains with apices at a poisson point process on mathbb r d 1 times mathbb r with intensity e h dh dv the scaling limit of the boundary of k n as n rightarrow infty converges to a festoon of parabolic surfaces coinciding with that featuring in the geometric construction of the zero viscosity solution to burgers equation with random input

cohomology of tails tate vogel cohomology and noncommutative serre duality over koszul quiver algebras the main result of the paper shows that under koszul duality between quiver algebras cohomology of tails is identified with graded vogel cohomology as an application a new proof of the noncommutative serre duality over generalized artin schelter regular koszul quiver algebras is given it is deduced from a similar formula over an arbitrary i e not necessarily koszul frobenius algebra which turns out to be equivalent to the auslander reiten formula as another application it is shown that over a generalized artin schelter regular koszul quiver algebra any algebra automorphism appearing in the noncommutative serre duality formula is closely related under koszul duality to the nakayama automorphism of the koszul dual algebra

a nonparametric approach to the estimation of lengths and surface areas fil cuevas antonio universidad de san andres departamento de matematica y ciencias argentina

central limit theorems for random polytopes let k be a smooth convex set the convex hull of independent random points in k is a random polytope central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity one essential step is to determine the precise asymptotic order of the occurring variances

asymptotical minimax recovery of sets with smooth boundaries in this paper optimal rates of convergence are derived for estimates of sets in n dimensional black and white pictures under smoothness conditions it is assumed that the boundaries of the black regions have a smooth parameterisation that is that the boundaries are given by smooth functions from the sphere s n 1 into r n furthermore classes of convex regions are considered two models are studied edge estimation models motivated by image segmentation problems and density support estimation

the combinatorial structure of random polytopes abstract choose n random points in r d let p n be their convex hull and denote by f i p n the number of i dimensional faces of p n a general method for computing the expectation of f i p n i 0 d 1 is presented this generalizes classical results of efron in the case i 0 and renyi and sulanke in the case i d 1 to arbitrary i for random points chosen in a smooth convex body a limit law for f i p n is proved as n for random points chosen in a polytope the expectation of f i p n is determined as n this implies an extremal property for random points chosen in a simplex

rationality of the poincare series for koszul algebras abstract in this paper we study the rationality of the poincare series of a finitely generated graded left module over a koszul connected algebra in particular we show the following result let a be a koszul connected algebra and a be its koszul dual suppose that a is noetherian and having a balanced dualizing complex if a is either 1 an artinian algebra 2 a graded quotient algebra of a noetherian as regular algebra or 3 an fbn algebra e g a noetherian pi algebra then the poincare series of every finitely generated graded left a module is a rational function over the complex numbers

variance asymptotics for random polytopes in smooth convex bodies let k subset mathbb r d be a smooth convex set and let mathcal p lambda be a poisson point process on mathbb r d of intensity lambda the convex hull of mathcal p lambda cap k is a random convex polytope k lambda as lambda rightarrow infty we show that the variance of the number of k dimensional faces of k lambda when properly scaled converges to a scalar multiple of the affine surface area of k similar asymptotics hold for the variance of the number of k dimensional faces for the convex hull of a binomial process in k

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

minimax estimation of the volume of a set under the rolling ball condition abstractwe consider the problem of estimating the volume of a compact domain in a euclidean space based on a uniform sample from the domain we assume that the domain has a boundary with positive reach we propose a data splitting approach to correct the bias of the plug in estimator based on the sample convex hull we show that this simple estimator achieves a minimax lower bound that we derive some numerical experiments corroborate our theoretical findings

weakly supercyclic operators inspired by a recent result that a weakly hypercyclic operator may fail to be norm hypercyclic we show there exists a weakly supercyclic operator that fails to be norm supercyclic moreover despite a classical result of hilden and wallen that every unilateral weighted backward shift is supercyclic we show such an operator may have a weakly supercyclic vector that is not a norm supercyclic vector in addition to these results we extend a result of kitai by showing a hyponormal operator cannot be weakly hypercyclic

the variety of an indecomposable module is connected let g be a finite group and let k be an algebraically closed field of characteristic p 0 the connections between the modular representation theory of finite groups and cohomology theory is currently a topic of great interest in particular cohomology theory has been used to attach to each finitely generated kgmodule m a homogeneous affine variety v m if we let 17 m be the corresponding projective variety then our main result is the following

records the maximal layer and uniform distributions in monotone sets abstract consider a nondecreasing nonnegative integrable function f on 0 1 draw an independent sample x 1 y 1 x n y n of size n from the uniform distribution under f and let n n be the number of records in the sample where x i y i is a record when for all j i either x j x i or y j y i we study the dependence upon f of the constant c in the asymptotic formula e n n c n and show that whenever 0 1 f 0 n n e n n 1 in probability the results are related to the expected time analysis of algorithms for finding the collection of all records i e the maximal layer

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

a plug in approach to support estimation we suggest a new approach based on the use of density estimators for the problem of estimating the compact support of a multivariate density this subject motivated in terms of pattern analysis by grenander has interesting connections with detection and clustering a natural class of density based estimators is defined universal consistency results and convergence rates are established for these estimators with respect to the usual measure based metric d between sets further convergence rates with respect to both d and the hausdorff metric d h are also obtained under some fairly intuitive shape restrictions

absolute relative and tate cohomology of modules of finite gorenstein dimension we study finitely generated modules m over a ring r noetherian on both sides if m has finite gorenstein dimension mbox g dim rm in the sense of auslander and bridger then it determines two other cohomology theories besides the one given by the absolute cohomology functors rm ext n r m relative cohomology functors rm ext n mathcal g m are defined for all non negative integers n they treat the modules of gorenstein dimension 0 as projectives and vanish for n mbox g dim rm tate cohomology functors widehat rm ext n r m are defined for all integers n all groups widehat rm ext n r m n vanish if m or n has finite projective dimension comparison morphisms varepsilon mathcal g n colon rm ext n mathcal g m to rm ext n r m and varepsilon r n colon rm ext n r m to widehat rm ext n r m link these functors we give a self contained treatment of modules of finite g dimension establish basic properties of relative and tate cohomology and embed the comparison morphisms into a canonical long exact sequence 0 to rm ext 1 mathcal g m to cdots to rm ext n mathcal g m to rm ext n r m to widehat rm ext n r m to rm ext n 1 mathcal g m to cdots we show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings 2000 mathematical subject classification 16e05 13h10 18g25

bishop s property β a commutativity theorem and the dynamics of class a s t operators abstract given a hilbert space operator a b h with polar decomposition a u a the class a s t 0 s t 1 consists of operators a b h such that a 2 t a t a 2 s a t t t s every class a s t operator is paranormal prominent amongst the subclasses of a s t operators are the class a 1 2 1 2 consisting of w hyponormal operators and the class a 1 1 consisting of semi quasihyponormal 16 p 93 or class a operators our aim here is threefold we prove that a s t operators satisfy i bishop s property thereby providing a proof of 6 theorem 3 1 and ii a putnam fuglede commutativity theorem thereby answering a question posed in 18 conjecture 2 4 we prove also an extension of 3 theorem 3 4 to prove that iii if an a s t operator is weakly supercyclic then it is a scalar multiple of a unitary operator

variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points we show that the random point measures induced by vertices in the convex hull of a poisson sample on the unit ball when properly scaled and centered converge to those of a mean zero gaussian field we establish limiting variance and covariance asymptotics in terms of the density of the poisson sample similar results hold for the point measures induced by the maximal points in a poisson sample the approach involves introducing a generalized spatial birth growth process allowing for cell overlap

on nonparametric estimation of density level sets let x 1 x n be independent identically distributed observations from an unknown probability density f consider the problem of estimating the level set g g f x r 2 f x from the sample x 1 x n under the assumption that the boundary of g has a certain smoothness we propose piecewise polynomial estimators of g based on the maximization of local empirical excess masses we show that the estimators have optimal rates of convergence in the asymptotically minimax sense within the studied classes of densities we find also the optimal convergence rates for estimation of convex level sets a generalization to the n dimensional case where n 2 is given

limit theorems for functionals of convex hulls in 4 a central limit theorem for the number of vertices of the convex hull of a uniform sample from the interior of a convex polygon is derived this is done by approximating the process of vertices of the convex hull by the process of extreme points of a poisson point process and by considering the latter process of extreme points as a markov process for a particular parametrization we show that this method can also be applied to derive limit theorems for the boundary length and for the area of the convex hull this extents results of renyi and sulanke 1963 and buchta 1984 and shows that the boundary length and the area have a strikingly different probabilistic behavior

boundary density and voronoi set estimation for irregular sets in this paper we study the inner and outer boundary densities of some sets with self similar boundary having minkowski dimension s d 1 in mathbb r d these quantities turn out to be crucial in some problems of set estimation theory as we show here for the voronoi approximation of the set with a random input constituted by n iid points in some larger bounded domain we prove that some classes of such sets have positive inner and outer boundary density and therefore satisfy berry essen bounds in n s 2d for kolmogorov distance the von koch flake serves as an example and a set with cantor boundary as a counter example we also give the almost sure rate of convergence of hausdorff distance between the set and its approximation

nonparametric estimation of surface integrals the estimation of surface integrals on the boundary of an unknown body is a challenge for nonparametric methods in statistics with powerful applications to physics and image analysis among other fields provided that one can determine whether random shots hit the body cuevas et al ann statist 35 2007 1031 1051 estimate the boundary measure the boundary length for planar sets and the surface area for 3 dimensional objects via the consideration of shots at a box containing the body the statistics considered by these authors as well as those in subsequent papers are based on the estimation of minkowski content and depend on a smoothing parameter which must be carefully chosen for the same sampling scheme we introduce a new approach which bypasses this issue providing strongly consistent estimators of both the boundary measure and the surface integrals of scalar functions provided one can collect the function values at the sample points examples arise in experiments in which the density of the body can be measured by physical properties of the impacts or in situations where such quantities as temperature and humidity are observed by randomly distributed sensors our method is based on random delaunay triangulations and involves a simple procedure for surface reconstruction from a dense cloud of points inside and outside the body we obtain basic asymptotics of the estimator perform simulations and discuss via google earth s data an application to the image analysis of the aral sea coast and its cliffs

nonvanishing cohomology and classes of gorenstein rings abstract we give counterexamples to the following conjecture of auslander given a finitely generated module m over an artin algebra there exists a positive integer n m such that for all finitely generated modules n if ext i m n 0 for all i 0 then ext i m n 0 for all i n m some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local gorenstein rings

distribution estimators and confidence intervals for stereological volumes assessing the precision of volume estimates from systematic samples is a question of great practical importance but statistically a challenging task due to the strong spatial dependence of the data and typically small sample sizes the approach taken in this paper is more ambitious than earlier methodologies the goal of which was estimation of the variance of a volume estimator rather than estimation of the distribution of we shall show that bootstrap methods yield consistent estimators of the distribution of and also suggest a variety of confidence intervals for the true volume our new methodology covers cases where serial sections are exactly periodic as well as instances where the physical slicing procedure introduces errors in the placement of the sampling points measurement errors within sections are also taken into account the performance of the method is illustrated by a simulation study with synthetic data and also applied to real datasets

symmetry in the vanishing of ext over stably symmetric algebras abstract a frobenius algebra over a field k is called symmetric if the nakayama automorphism is an inner automorphism a stably symmetric algebra is defined to be a generalization of a symmetric k algebra in this paper we will study symmetry in the vanishing of ext for such algebras r namely for all finitely generated r modules m and n ext r i m n 0 for all i 0 if and only if ext r i n m 0 for all i 0 we show that a certain class of noetherian stably symmetric gorenstein algebras such as the group algebra of a finite group and the exterior algebra k n when n is odd have this symmetry using serre duality we also show that every exterior algebra k n whether n is even or odd has this symmetry for graded modules using koszul duality

cohomological symmetry in triangulated categories abstract we give a criterion for cohomological symmetry in a triangulated category as an application we show that such cohomological symmetry holds for all pairs of modules over any exterior algebra

variance asymptotics and scaling limits for random polytopes abstract let k be a convex set in r d and let k be the convex hull of a homogeneous poisson point process p of intensity on k when k is a simple polytope we establish scaling limits as for the boundary of k in a vicinity of a vertex of k and we give variance asymptotics for the volume and k face functional of k k 0 1 d 1 resolving an open question posed in 17 the scaling limit of the boundary of k and the variance asymptotics are described in terms of a germ grain model consisting of cone like grains pinned to the extreme points of a poisson point process on r d 1 r having intensity d e d h d h d v

on the area and perimeter of a random convex hull in a bounded convex set suppose k is a compact convex set in 2 and x i 1 i n is a random sample of points in the interior of k under general assumptions on k and the distribution of the x i we study the asymptotic properties of certain statistics of the convex hull of the sample

random points on the boundary of smooth convex bodies the convex hull of n independent random points chosen on the boundary of a convex body k c r d according to a given density function is a random polytope the expectation of its i th intrinsic volume for i 1 d is investigated in the case that the boundary of k is sufficiently smooth asymptotic expansions for these expected intrinsic volumes as n oo are derived

central limit theorems for random polytopes in a smooth convex set let k be a smooth convex set with volume one in choose n random points in k independently according to the uniform distribution the convex hull of these points denoted by kn is called a random polytope we prove that several key functionals of kn satisfy the central limit theorem as n tends to infinity

supporting points processes and some of their applications we introduce a stochastic point process of s supporting points and prove that upon rescaling it converges to a gaussian field the notion of s supporting points specializes for adequately chosen s to pareto or more generally cone extremal points or to vertices of convex hulls or to centers of generalized voronoi tessellations in the models of large scale structure of the universe based on burgers equation the central limit theorems proven here imply i a the asymptotic normality for the number of convex hull vertices in large poisson sample from a simple polyhedra or for the number of pareto vector extremal points in poisson samples with independent coordinates

on the variance of random polytopes abstract a random polytope is the convex hull of uniformly distributed random points in a convex body k a general lower bound on the variance of the volume and f vector of random polytopes is proved also an upper bound in the case when k is a polytope is given for polytopes as for smooth convex bodies the upper and lower bounds are of the same order of magnitude the results imply a law of large numbers for the volume and f vector of random polytopes when k is a polytope

set estimation from reflected brownian motion we study the problem of estimating a compact set s rd from a trajectory of a reflected brownian motion in s with reflections on the boundary of s this problem has relevant applications in ecology in estimating the home range of an animal based on tracking data

measuring mass concentrations and estimating density contour clusters an excess mass approach by using empirical process theory the so called excess mass approach is studied it can be applied to various statistical problems especially in higher dimensions such as testing for multimodality estimating density contour clusters estimating nonlinear functionals of a density density estimation regression problems and spectral analysis we mainly consider the problems of testing for multimodality and estimating density contour clusters but the other problems also are discussed the excess mass over mathbb c is defined as a supremum of a certain functional defined on mathbb c where mathbb c is a class of subsets of the d dimensional euclidean space comparing excess masses over different classes mathbb c yields information about the modality of the underlying probability measure f this can be used to construct tests for multimodality if f has a density f the maximizing sets of the excess mass are level sets or density contour clusters of f provided they lie in mathbb c the excess mass and the density contour clusters can be estimated from the data asymptotic properties of these estimators and of the test statistics are studied for general classes mathbb c including the classes of balls ellipsoids and convex sets

lacunarity of self similar and stochastically self similar sets let k be a self similar set in rdd of hausdorff dimension d and denote by ik c l the d dimensional lebesgue measure of its 6 neighborhood we study the limiting behavior of the quantity e d d ik e i as 6 e 0 it turns out that this quantity does not have a limit in many interesting cases including the usual ternary cantor set and the sierpinski carpet we also study the above asymptotics for stochastically self similar sets the latter results then apply to zero sets of stable bridges which are stochastically self similar in the sense of the present paper and then more generally to level sets of stable processes specifically it follows that if kt is the zero set of a realvalued stable process of index a c 1 2 run up to time t then e siit e i converges to a constant multiple of the local time at 0 simultaneously for all t 0 on a set of probability one the asymptotics for deterministic sets are obtained via the renewal theorem the renewal theorem also yields asymptotics for the mean 1e ik c in the random case while the almost sure asymptotics in this case are obtained via an analogue of the renewal theorem for branching random walks

support varieties and cohomology over complete intersections quillen s geometric approach to the cohomology of finite groups 30 has revolutionized modular representation theory the ideology and techniques involved have been generalized and extended to representations of various hopf algebras culminating in the recent work of friedlander and suslin 17 on finite group schemes in this paper we develop geometric methods for the study of finite modules over a local complete intersection r if r is such a ring and m is its maximal ideal then the m adic completion r has the form q f where f is a regular sequence and q is a regular local ring that can be taken to be a ring of formal power series over a field or a discrete valuation ring the least number of equations needed to cut out r from a regular local ring equals the codimension of r where codim r r m dim r and r m denotes the minimal number of generators of an r module m in 5 a cone that is a homogeneous algebraic set vr m is attached to each finite r module m and used to study its minimal free resolution here we prove

algebras that satisfy auslander s condition on vanishing of cohomology auslander conjectured that every artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules the failure of this conjecture by a 2003 counterexample due to jorgensen and ega motivates the consideration of the class of rings that do satisfy auslander s condition we call them ac rings and show that an ac artin algebra that is left gorenstein is also right gorenstein furthermore the auslander reiten conjecture is proved for ac rings and auslander s g dimension is shown to be functorial for ac rings that are commutative or have a dualizing complex

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

hereditarily polaroid operators svep and weyl s theorem abstract a banach space operator t b x is hereditarily polaroid t hp if every part of t is polaroid hp operators have svep it is proved that if t b x has svep and r b x is a riesz operator which commutes with t then t r satisfies generalized a browder s theorem if in particular r is a quasi nilpotent operator q then both t q and t q satisfy generalized a browder s theorem furthermore if q is injective then also t q satisfies weyl s theorem if a b x is an algebraic operator which commutes with the polynomially hp operator t then t n is polaroid and has svep f t n satisfies generalized weyl s theorem for every function f which is analytic on a neighbourhood of t n and f t n satisfies generalized a weyl s theorem for every function f which is analytic on and constant on no component of a neighbourhood of t n

asymmetry of ext groups in this paper we will use techniques of noncommutative projective geometry to construct examples of algebras r over a field k not satisfying the following two types of symmetric behaviors of ext groups ee for any pair of finitely generated r modules m n dimkextri m n for all i n if and only if dimkextri n m for all i n ee for any pair of finitely generated r modules m n extri m n 0 for all i 0 if and only if extri n m 0 for all i 0 in particular and contrary to the commutative case we give a simple example of a noncommutative noetherian gorenstein frobenius local algebra satisfying uac uniform auslander condition but not satisfying ee

limit theorems for convex hulls it is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally after rescaling to a strongly mixing markov process as the sample size tends to infinity the structure of the limiting markov process is determined explicitly and from this a central limit theorem for the number of vertices of the convex hull is derived similar results are given for uniform samples from the unit disk