modules of finite projective dimension with negative intersection multiplicities let x be the image of x in r the main object of this paper is to study modules of finite length and finite projective dimension over r one consequence of our study will be counterexamples to several conjectures concerning the behavior of intersection multiplicities for modules of finite projective dimension let denote length if s is a noetherian ring m n are finitely generated s modules l m is finite and d pdsm we may define following serre is d zs m n z m n 1 tors m 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

intersection multiplicity over noncommutative algebras abstract in this paper we will prove serre s multiplicity conjectures for large class of connected algebras not necessarily commutative using the intersection multiplicity defined by ext groups instead of tor groups since this new definition of the intersection multiplicity agrees with classical serre s definition for nice commutative rings the result is a noncommutative version of that of peskine and szpiro the result also applies to noncommutative filtered algebras in certain situations

p variation of strong markov processes let t t 0 t be a strong markov process with values in a complete separable metric space x and with transition probability function ps t x dy 0 s t t x x for any h 0 t and a 0 consider the function h a sup ps t x y x y a x x 0 s t s h t it is shown that a certain growth condition on h a as a 0 and h stays fixed implies the almost sure boundedness of the p variation of t where p depends on the rate of growth

connected graded gorenstein algebras with enough normal elements we generalize 12 1 1 and 1 2 to the following situation theorem 1 let a be a connected graded noetherian algebra of injective dimension d such that every nonsimple graded prime factor ring of a contains a homogeneous normal element of positive degree then 1 a is auslander gorenstein and cohen macaulay 2 a has a quasi frobenius quotient ring 3 every minimal prime ideal p is graded andgkdim a p d 4 if moreover a has finite global dimension then a is a domain and a maximal order in its quotient division ring to prove the above we need the following result which is a generalization of 3 2 46 ii theorem 2 let a be a connected graded noetherian as gorenstein algebra of injective dimension d then 1 the last term of the minimal injective resolution of aais isomorphic to a shift of a 2 for every noetherian graded a module m extd m a is finite dimensional over k

the tame infinitesimal groups of odd characteristic abstract given an infinitesimal group g over an algebraically closed field k of characteristic p 3 we provide criteria for the principal block b 0 g of its algebra of distributions to be of tame representation type these are employed in conjunction with galois coverings to determine the structure of g modulo its multiplicative center as well as the quiver and the relations of the algebra b 0 g

hecke algebras of classical type and their representation type the purpose of this article is to determine the representation type for all of the hecke algebras of classical type to do this we combine methods from our previous work which is used to obtain information on their gabriel quivers and recent advances in the theory of finite dimensional algebras the principal computation is for hecke algebras of type b with two parameters then we show that the representation type of hecke algebras is governed by their poincare polynomials

homological properties of balanced cohen macaulay algebras a balanced cohen macaulay algebra is a connected algebra a having a balanced dualizing complex a d in the sense of yekutieli 1992 for some integer d and some graded a a bimodule a we study some homological properties of a balanced cohen macaulay algebra in particular we will prove the following theorem theorem 0 1 let a be a noetherian balanced cohen macaulay algebra and m a nonzero finitely generated graded left a module then 1 m has a finite resolution of the form 0 a l mj a l 1j h m 0 where h is a finitely generated maximal cohen macaulay graded left a module 2 m has finite injective dimension if and only if m has a finite resolution of the form 0 a l mj a l 1j a l 0j m 0 as a corollary we will have the following characterizations of as gorenstein algebras and as regular algebras corollary 0 2 let a be a noetherian balanced cohen macaulay algebra 1 a is as gorenstein if and only if a has finite projective dimension as a graded left a module 2 a is as regular if and only if every finitely generated maximal cohen macaulay graded left a module is free

maps between tree and band modules the indecomposable modules over special biserial algebras are known to be of a specially simple form so called tree and band modules cf 6 2 crawley boevey has shown how to describe the homomorphisms between tree modules cf 3 and our objective in this paper is to extend this result to band modules tree and band modules occur as indecomposable representations of arbitrary algebras nevertheless we follow crawley boevey and consider zero relation algebras as the appropriate context given two tree or band modules we first study certain quiver homomorphisms between their underlying trees or bands and it turns out that a map between the representations is completely described by k linear maps which are associated to these quiver homomorphisms it is interesting to note that in general the classification of maps between representations of a special biserial algebra a is a wild problem in fact the maps are equivalent to the representations of t a for instance take for a the path algebra of an a n 2 6 then t a is wild although the maps between indecomposable representations of a are easily to describe throughout k will be a fixed algebraically closed field maps are written on the left this paper results from a stay at the university of liverpool supported by the deutscher akademischer austauschdienst i am very grateful to sheila brenner for her suggestions

tame generalized canonical algebras abstract recently a new wide class of finite dimensional algebras whose auslander reiten quivers admit infinite families of faithful standard stable tubes called generalized canonical algebras has been introduced in a skowronski colloq math 90 2001 77 93 we give a complete description of all tame generalized canonical algebras and prove that all these algebras have tame derived categories of bounded complexes

the strong p variation of martingales and orthogonal series let 1 p and letx x n n 0 be a sequence of scalars the strongp variation ofx denoted byw p x is defined as w p x sup left left x 0 p sum limits k 1 infty x n k x n k 1 p right 1 p right where the supremum runs over all increasing sequences of integers 0 n 0 n 1 n 2 let 1 p 2 and letm m n n 0 be a martingale inl p our main results are as follows if sigma mathbb e m n m n 1 p infty thenw p m is finite a s and we have mathbb e w p m p leqq c mathbb e m 0 p sum limits n geqq 1 mathbb e m n m n 1 p for some constantc depending only onp on the other hand let n be an arbitrary orthonormal system of functions inl 2 considerx x n n 0 inl 2 and lets n 0 x i i ands s n n 0 we prove that if x n p 1 p 2 thenw p s t for a e t and w p s 2 c x n p 1 p for some constantc each of these results is an extension of a result proved by bretagnolle for sums of independent mean zero r v s the casep 2 in also discussed our proofs use the real interpolation method of lions peetre they admit extensions in the banach space valued case provided suitable assumptions are imposed on the banach space

on bernstein approximants and the f variation of a fuzzy random variable abstract in this paper we study different properties of bernstein approximants of a fuzzy random variable as those in relation with the approximation of fuzzy random variables by means of its n bernstein approximant moreover we define and study the concept of variation for a fuzzy random variable

tame algebras and tits quadratic forms abstract we solve a long standing problem concerning the connection between the tameness of simply connected algebras and the weak nonnegativity of the associated tits integral quadratic forms and derive some consequences

the order of the remainder in derivatives of composition and inverse operators for p variation norms the theory of differentiable statisticals began with work of von mises e g von mises 1936 1947 and filippova 1961 a nonlinear funtional t is defined for examble on distribution functions von mises differentiated t at a distribution funtion f along lines for t to have a g teaux derivative at f means that in the direction of a function h t f th t f tt prime f h 0 t quad rm as t rightarrow 0 1 1 here t prime f is a bounded linear operator on funtions h for example of the form t prime f h int g quad dh hbox for some function g hbox depending on f 1 2

divergent series and serre s intersection formula for graded rings abstract on a smooth variety serre s intersection formula computes intersection multiplicities via an alternating sum of the lengths of tor groups when the variety is singular the corresponding sum can be a divergent series but there are alternate geometric approaches for assigning often fractional intersection multiplicities in some singular settings our motivating question comes from fulton who asks whether an analytic continuation of the divergent series from serre s formula can be related to these fractional multiplicities we apply work of avramov and buchweitz to answer fulton s question in the context of graded rings

frechet differentiability p variation and uniform donsker classes classically a stastistical functional is defined on a space of distribution f on the real line with the supremum norn the values may be real or themselves functions such as the quantile function f 1 nonlinear functionals are studied via their derivatives this paper and a related one dudley 1991a will show that the differentiability properties originally proved by reeds 1976 also treated in fernholz 1983 can be improved substantially

the p variation of partial sum processes and the empirical process the p variation of a function f is the supremum of the sums of the p th powers of absolute increments of f over nonoverlapping intervals let f be a continuous probability distribution function dudley has shown that the p variation of the empirical process is bounded in probability as n to infty if and only if p 2 and for 1 leq p leq 2 the p variation of the empirical process is at least n 1 p 2 and is at most of the order n 1 p 2 log log n p 2 in probability in this paper we prove that the exact order of the 2 variation of the empirical process is log log n in probability and for 1 leq p 2 the p variation of the empirical process is of exact order n 1 p 2 in expectation and almost surely let s j x 1 x 2 dots x j then the p variation of the partial sum process for x 1 x 2 dots x n is defined as that of f on 0 n where f t s j for j 1 t leq j j 1 2 dots n bretagnolle has shown that the expectation of the p variation for independent centered random variables x i with bounded p th moments is of order n for 1 leq p 2 we prove that for p 2 the 2 variation of the partial sum process of i i d centered nonconstant random variables with finite 2 delta moment for some delta 0 is of exact order n log log n in probability

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

special biserial algebras and right grobner bases abstract it is shown that a finite dimensional basic algebra over an algebraically closed field is representation finite special biserial if and only if every module over it has a right grobner basis theory

a narrow over ring adjustment functor such that xi is finitely generated xl is finitely cogenerated 9 is surjective and the a homomorphism cp xl hom an xg adjoint to rp is injective the category of adjusted right r modules will be denoted by adj r note that if r is an artin algebra then x is in adj r if and only if x is finitely generated sot x is a b module and top x is an a module via the natural epimorphisms a t r b in this case adj i r is one of the classes of modules considered by auslander and smal l and therefore adj has the auslander reiten sequences 1 proposition 6 51 the category adjg r was introduced in 25 26 in order to give a module theoretical interpretation of arbitrary bimodule matrix problems in the sense of drozd and roiter it was shown in 26 that if n kop x k mod k is an additive k functor k is a field k and l are additive krull schmidt k categories and mat jv is the category of matrices then there are a faithfully right b traced ring r and a full dense functor 0 mat wl adj r preserving the representation type and such that 0 vanishes only on a small set of trivial idecomposable matrices see also 14 this allows to show in 14 261 that the category mat n has auslander reiten sequences see also 30 moreover a coxeter map is defined in terms of 0 which provides us a useful computa

universal donsker classes and metric entropy when mathfrak f is a universal donsker class then for independent indetically distributed i i d observation mathbf x 1 ldots mathbf x n with an unknown law p for any mathfrak f i in mathfrak f i 1 ldots m quad n 1 2 left mathfrak f 1 left mathbf x 1 right ldots mathfrak f i left mathbf x n right right 1 leq i leq m is asymptotically normal with mean vector n 1 2 left int mathfrak f i left mathbf x n right d mathbf p left x right right 1 leq i leq m and covariance matrix int mathfrak f i mathfrak f j d mathbf p int mathfrak f id mathbf p int mathfrak f jd mathbf p uniformly for mathfrak f i in mathfrak f then for certain statistics formed frome the mathfrak f i left mathbf x k right even where mathfrak f i may be chosen depending on the mathbf x k there will be asymptotic distribution as n rightarrow infty for example for mathbf x 2 statistics where f i are indicators of disjoint intervals depending suitably on mathbf x 1 ldots mathbf x n whose union is the real line mathbf x 2 quadratic forms have limiting distributions roy 1956 and watson 1958 which may however not be mathbf x 2 distributions and may depend on p chernoff and lehmann 1954 universal donsker classes of sets are up to mild measurability conditions just classes satisfying the vapnik cervonenkis comdinatorial conditions defined later in this section donsker the vapnik cervonenkis combinatorial conditions defined later in this section durst and dudley 1981 and dudley 1984 chapter 11 the use of such classes allows a variety of extensions of the roy watson results to general multidimensional sample spaces pollard 1979 and moore and subblebine 1981 vapnik and cervonenkis 1974 indicated application of their families of sets to classification pattern recognition problems more recently the classes have been applied to tree structured classifiacation breiman friedman olshen and stone 1984 chapter 12

sur une integrale pour les processus a alpha variation bornee dans cet article l auteur obtient une formule d ito et l existence de temps locaux pour certains processus de dirichlet a partir des proprietes d une integrale faisant intervenir un processus a variation bornee et un autre a variation bornee cette integrale est definie comme limite de sommes de riemann

the index theorem of topological regular variation and its applications we develop further the topological theory of regular variation of n h bingham a j ostaszewski topological regular variation i slow variation lse cdam 2008 11 there we established the uniform convergence theorem uct in the setting of topological dynamics i e with a group t acting on a homogenous space x thereby unifying and extending the multivariate regular variation literature here working with real time topological flows on homogeneous spaces we identify an index of regular variation which in a normed vector space context may be specified using the riesz representation theorem and in a locally compact group setting may be connected with haar measure

on auslander reiten components of blocks and self injective biserial algebras we investigate the existence of auslander reiten components of euclidean type for special biserial self injective algebras and for blocks of group algebras in particular we obtain a complete description of stable auslander reiten quivers for the tame self injective algebras considered here

on the castelnuovo mumford regularity of the cohomology ring of a group we prove benson s regularity conjecture that the regularity of the cohomology of a finite group is zero we also prove some of his generalizations to other classes of groups dave benson in 2 conjectured that for any finite groupg and any prime p the castelnuovomumford regularity of the cohomology ring h g fp is zero he showed that reg h g fp 0 and succeeded in proving equality when the difference between the dimension and the depth is at most two the purpose of this paper is to prove benson s regularity conjecture as a corollary of the following result theorem 0 1 if a compact lie group g acts on a smooth manifold m with finite dimensional mod p homology then reg h g m fp dimm dimg by taking g to be finite m to be a point and using benson s inequality we obtain a proof of benson s original conjecture corollary 0 2 for any finite group g we have reg h g fp 0 we record here one easily stated consequence proposition 0 3 for any non trivial finite group the cohomology ring h g fp is generated by elements in degree at most g 1 and the relations between them as a graded commutative algebra are generated in degrees at most 2 g 1 this bound is weak although it can be improved somewhat at the cost of a more complicated formulation but previously no such bound was known we go on to prove some other conjectures of benson on the regularity of the cohomology of other classes of groups 3 specifically for compact lie groups and virtual poincare duality groups the proof uses standard techniques in equivariant cohomology based on work of quillen 30 and a paper of duflot 14 david green and simon king have verified benson s conjecture computationally for all groups of order less than 256 20 there is survey of the properties group cohomology from the point of view of commutative algebra by benson 3 1 castelnuovo mumford regularity we will work in the category of z graded rings and modules so maps will respect the grading elements will be homogeneous etc without specific mention for a module m 1991 mathematics subject classification primary 20j06 secondary 57s15 partially supported by a grant from the leverhulme trust

the algebras with discrete derived category abstract we classify the finite dimensional algebras over an algebraically closed field whose bounded derived category does not admit an infinite continuous family of pairwise non isomorphic indecomposable complexes

gaussian characterization of uniform donsker classes of functions it is proved that for classes of functions y satisfying some measurability the empirical processes indexed by f and based on p e 9 s satisfy the central limit theorem uniformly in p e 9w s if and only if the p brownian bridges gop indexed by y are sample bounded and pp uniformly continuous uniformly in p e 9 s uniform exponential bounds for empirical processes indexed by universal bounded donsker and uniform donsker classes of functions are also obtained

laurent coefficients and ext of finite graded modules n m the present investigation started as an attempt to explain this intriguing formula and to nd out whether analogous equalities exist for subsequent coe cients of laurentexpansions around 1 fairly complete answers are presented in sections 3 and 4 theydepend on embedding 1 into an equality of laurent series which predicts an in nitesequence of similar relations among higher laurent coe cients the results are proved insection 1 and are used in section 2 to obtain non trivial lower bounds for the bass numbersof certain graded modules of nite injective dimension

auslander reiten components of symmetric special biserial algebras abstract we provide a combinatorial algorithm for constructing the stable auslander reiten component containing a given indecomposable module of a symmetric special biserial algebra using only information from its underlying brauer graph we also show that the structure of the auslander reiten quiver is closely related to the distinct green walks of the brauer graph and detail the relationship between the precise shape of the stable auslander reiten components for domestic brauer graph algebras and their underlying graph furthermore we show that the specific component containing a given simple or indecomposable projective module for any brauer graph algebra is determined by the edge in the brauer graph associated to the module

robust estimation of parameters in a mixed unbalanced model this paper describes a method of robust estimation of shift and scale parameters in a mixed unbalanced interlaboratory model estimators presented result from easily computable fr echet differentiable functionals which enjoy some optimal properties in a small neighborhood of the model a rigorous treatment of their asymptotic behaviour under departures from the model assumptions and a simulation study are given 1 introduction a method of robust analysis of random effects in mixed models is introduced in the paper by rocke 1991 see also iglewicz 1983 and a general treatment of robust estimators of variance components is presented by fellner 1986 rocke proposes estimators of variance components applying robust scale estimators to residuals and laboratory effects in the rst step he uses huber s method to robustly assess location parameters a similar point of view is taken by lischer 1994 who improves breakdown properties of the estimators he also gives a comprehensive practical motivation for the use of robust methods in interlaboratory experiments the above mentioned estimation technique may lead to high dimensional nonlinear equations and it is inconvenient for assessing variability of estimators a possible improvement may be in the simultaneous estimation of xed effects and components of variation where problems of a rigorous asymptotic treatment smoothness of estimators and their optimality become more feasible in a preliminary study by bednarski zmy slony and zontek 1992 it is shown that fr echet differentiability of statistical functionals easily leads to reasonable robust estimators of variance components and treatment fixed effects in a simple interlaboratory model the basic idea there is to construct a smooth fisher consistent functional t for the parameter de 1 a ee t at the model

speed of convergence of classical empirical processes in p variation norm let f be any distribution function on r and f n be the nth empirical distribution function based on variables i i d f it is shown that for 2 p and a constant c p not depending on f on some probability space there exist f n and brownian bridges b n such that for the wiener young p variation norm p e n 1 2 f n f b n o f p c p n 2 p 2p where b n o f x b n f x the expectation can be replaced by an orlicz norm of exponential order conversely if f is continuous then for any stochastic process v t w continuous in t for almost all w such as b n o f summation over n distinct jumps shows that n 1 2 f n f v p n 2 p 2p so the upper bound in expectation is best possible up to the constant c p in the proof b n is linked to f n by the komlos major and tusnady construction as for the supremum norm p

on the finitistic dimension conjecture of artin algebras abstract let a be an artin algebra and e be an idempotent element of a we prove that if a has representation dimension at most three then the finitistic dimension of eae is finite and deduce that if quasi hereditary algebras have representation dimensions at most three then the finitistic dimension conjecture holds