graded morita theory for infinite groups abstract having shown that not all graded rings are graded equivalent via classical morita theory to a skew group ring we extend classical morita theory which is based on rings with identity to a generalized graded morita theory for rings with local units this enables us to give necessary and sufficient conditions for graded morita equivalence between two rings graded by a group we show that the strongly graded property is a graded morita invariant and we show that a graded ring is graded morita equivalent to a skew group ring namely r g g if and only if it is strongly graded these results extend the classical cohen montgomery duality theory to rings graded by infinite groups the fundamental tools include rings with local units and the results of boisen

the brauer splitting theorem and projective representations of finite groups over rings in the theory of representations of finite groups the projective representations arise naturally when one studies the relations between the representations of the group and representations of certain subgroups the role played by the group rings in representation theory is taken by the twisted groups rings when one considers projective representations for representations over algebraically closed fields the theory of schur multipliers provides a very satisfactory tool that may be used to lift projective representations of a finite group to usual representations of a finite central extension of that group furthermore there exists a satisfactory theory of projective characters block theory for projective representations etc for which we may refer to karpilovsky s recent book 3 since for finite groups the twisted group rings defined over commutative rings are azumaya algebras when a mild condition holds it is natural to ask for an equivalent of the brauer splitting theorem for group rings in this paper we derive this splitting theorem for twisted group rings over commutative rings without nontrivial idempotents this is obtained by introducing a suitable extension of the schur multiplier theorem in such a way that this result may be applied to a single representation and over arbitrary fields or rings of the forementioned type along the way we get involved in some cocycle manipulations that will lead to the determination of the center of a twisted group ring in terms of ray classes the results derived in this paper should be of some use in the study of properties and

efficient empirical bayes variable selection and estimation in linear models we propose an empirical bayes method for variable selection and coefficient estimation in linear regression models the method is based on a particular hierarchical bayes formulation and the empirical bayes estimator is shown to be closely related to the lasso estimator such a connection allows us to take advantage of the recently developed quick lasso algorithm to compute the empirical bayes estimate and provides a new way to select the tuning parameter in the lasso method unlike previous empirical bayes variable selection methods which in most practical situations can be implemented only through a greedy stepwise algorithm our method gives a global solution efficiently simulations and real examples show that the proposed method is very competitive in terms of variable selection estimation accuracy and computation speed compared with other variable selection and estimation methods

tweedie s formula and selection bias we suppose that the statistician observes some large number of estimates zi each with its own unobserved expectation parameter i the largest few of the zi s are likely to substantially overestimate their corresponding i s this being an example of selection bias or regression to the mean tweedie s formula first reported by robbins in 1956 offers a simple empirical bayes approach for correcting selection bias this article investigates its merits and limitations in addition to the methodology tweedie s formula raises more general questions concerning empirical bayes theory discussed here as relevance and empirical bayes information there is a close connection between applications of the formula and james stein estimation

time periodic solutions of the compressible magnetohydrodynamic equations abstract the compressible magnetohydrodynamic equations with a time periodic external force in r n are studied in this paper we establish the existence uniqueness and time asymptotic stability of time periodic solutions when n 5 our proof is essentially based on a combination of the energy method and the time decay estimates of solutions to the linearized system

global well posedness of strong solutions to the magnetohydrodynamic equations of compressible flows abstract this paper is concerned with the global well posedness of strong solutions to the compressible mhd equations in a bounded domain r 3 we first establish the local existence and uniqueness of strong solutions with arbitrary initial data and external force field based on the local existence result and some a priori estimates the global existence and large time behavior of strong solutions are proved under the assumptions that the initial data and external force field are small enough furthermore the existence of the time periodic solution is also obtained when the external force field is time periodic

global solutions of nonlinear magnetohydrodynamics with large initial data a free boundary problem for nonlinear magnetohydrodynamics with general large initial data is investigated the existence uniqueness and regularity of global solutions are established with large initial data in h1 it is shown that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time although there is a complex interaction between the hydrodynamic and magnetodynamic effects an existence theorem of global solutions with large discontinuous initial data is also established

heavily separable functors abstract prompted by an example related to the tensor algebra we introduce and investigate a stronger version of the notion of separable functor that we call heavily separable we test this notion on several functors traditionally connected to the study of separability

empirical bayes deconvolution estimates an unknown prior density g theta has yielded realizations theta 1 ldots theta n they are unobservable but each theta i produces an observable value x i according to a known probability mechanism such as x i sim rm po theta i we wish to estimate g theta from the observed sample x 1 ldots x n traditional asymptotic calculations are discouraging indicating very slow nonparametric rates of convergence in this article we show that parametric exponential family modelling of g theta can give useful estimates in moderate sized samples we illustrate the approach with a variety of real and artificial examples covariate information can be incorporated into the deconvolution process leading to a more detailed theory of generalized linear mixed models

reduction techniques for strongly graded rings and finite representation type abstract we present reduction techniques for studying the category of lattices over strongly graded orders in particular we apply these techniques in order to reduce the problem of classifying those strongly graded orders with finite representation type to the case where the coefficient ring is a maximal order in a division ring

group graded rings and duality we give an alternative construction of the duality between finite group actions and group gradings on rings which was shown by cohen and montgomery in 1 this duality is then used to extend known results on skew group rings to corresponding results for large classes of group graded rings finally we modify the construction slightly to handle infinite groups introduction in the first section we give an alternate construction of the duality between finite group actions and group gradings on rings which was shown by cohen and montgomery in 1 this duality is then used to extend results on skew group rings and their modules to corresponding results for large classes of groupgraded rings in the second section we modify the construction slightly to handle infinite groups and apply it to a theorem on prime and semiprime infinite crossed products i would like to thank my thesis advisor d s passman for his guidance and encouragement throughout the writing of this paper 1 finite group graded rings and the smash product let g be a multiplicative group an associative ring with identity is said to be g graded if 1 r e r x xec is a direct sum of additive subgroups r x with r x r y c r xy it follows that necessarily 1r e r 1 so that each r x is a unitary r l bimodule if r e r we write r x for the component of r in r x so that r exgcr x r is said to be strongly g graded if r x r y r xy for all x y e g throughout this section r is assumed to be g graded where g is a finite group with igi n let mc r denote the set of n x n matrices over r with the rows and columns indexed by the elements of g if a c mg r we write ax y for the entry in the x y position of a then if a 1 c mc r the matrix product a 8 is given by 2 x x y e xx zaz y z e c if u c g is any subset of g let r u ex eur x in particular r r g now suppose h c g is a subgroup of g we define r h c mc r by r h a e ms r ax y e r xhy l that is r h ex ygcr xhy l e x y where e x y is the matrix unit with 1r in the x y position and zeroes elsewhere note that r g mc r received by the editors september 27 1984 1980 mathematics subject classification primary 16a03 c 1985 american mathematical society 0002 9947 85 1 00 25 per page

generalized 2 cocycles of finite groups and maximal orders abstract letrbe a dedekind domain with quotient fieldkand letgbe a finite group let g g r 0 be a generalized 2 cocycle i e not necessarily taking its values in the units ofr and consider the generalized twisted group ringr g in this paper we derive criteria forr gto be a graded maximalr order resp a hereditaryr order or a maximalr order ink g

crossed product algebras defined by separable extensions abstract we generalize the classical construction of crossed product algebras defined by finite galois field extensions to finite separable field extensions by studying properties of rings graded by groupoids we are able to calculate the jacobson radical of these algebras we use this to determine when the analogous construction of crossed product orders yield azumaya maximal or hereditary orders in a local situation thereby we generalize results by haile larson and sweedler

navier stokes equations for compressible fluids global existence and qualitative properties of the solutions in the general case we consider the equations which describe the motion of a viscous compressible fluid taking into consideration the case of inflow and or outflow through the boundary by means of some a priori estimates we prove the existence of a global in time solution moreover as a consequence of a stability result we show that there exist a periodic solution and a stationary solution

homological dimension of skew group rings and crossed products abstract in this paper we study the homological dimension of skew group rings and crossed products a sufficient condition for r g a crossed product to have finite right global dimension is given in terms of crossed products over simple artinian factors of r if r is right fbn and left coherent and g is finite some necessary conditions and sufficient conditions for r g a skew group ring of a finite group over a local or semilocal right noetherian ring to have finite right global dimension are also given then in particular if r is commutative noetherian and g is finite we obtain some equivalent conditions for r g a skew group ring to have finite global dimension using work of aljadeff e aljadeff serre s extension theorem for crossed products j london math soc 44 1991 47 54 these results are extended to polycyclic by finite groups

time periodic solutions of compressible navier stokes equations abstract in this paper we study the navier stokes equations with a time periodic external force in r n we show that a time periodic solution exists when the space dimension n 5 under some smallness assumption the main idea is to combine the energy method and the spectral analysis for the optimal decay estimates on the linearized solution operator with the optimal decay estimates we prove the existence and uniqueness of time periodic solution in some suitable function space by the contraction mapping theorem in addition we also study the time asymptotic stability of the time periodic solution

semisimple strongly graded rings is semisimple meaning in this paper semisimple artinian has been studied byseveral authors the most classical result is maschke s theorem for group rings for crossed products over elds there is a satisfactory answer given by aljadeffand robinson 3 another partial answer for skew group rings was given byalfaro et al 1 a reduction of the problem to crossed products over divisionrings was rst given by jespers and okninski 10 and a more constructiveversion was given by haefner and del rio 8 so in order to give a complete answer tothe problem there is still a gap between crossed products over division rings andcrossed products over elds the rst aim of the paper is to ll this gap showingthat the semisimplicity question for crossed products over division rings reducesto the same question for crossed products over elds in theorem 3 2 we makethis reductionand then in theorem3 3 we put togetherall the piecesof the puzzle a strongly

a schneider type theorem for hopf algebroids abstract comodule algebras of a hopf algebroid h with a bijective antipode i e algebra extensions b a by h are studied assuming that a lifted canonical map is a split epimorphism of modules of the non commutative base algebra of h relative injectivity of the h comodule algebra a is related to the galois property of the extension b a and also to the equivalence of the category of relative hopf modules to the category of b modules this extends a classical theorem by h j schneider on galois extensions by a hopf algebra our main tool is an observation that relative injectivity of a comodule algebra is equivalent to relative separability of a forgetful functor a notion introduced and analysed hereby

derivations of graded rings and clifford systems as a counterpart to the commutative galois theory of groups acting on rings one may consider lie algebras acting on rings as lie algebras of derivations a peculiar situation where techniques from both areas meet and blend arises when studying derivations of graded rings and clifford systems some results for strongly graded rings might benefit from a rephrasing in terms of hopf algebra actions the hopf algebras being smash products of dual group algebras and enveloping algebras but we did not go into these matters here for simplicity s sake in the first part of the paper we provide general results concerning r derivations of a graded ring of type g r ejntc r where e is the neutral clement of g into weakly cancellative r bimodules m which need not be graded in case a4 is graded as a left r module a useful criterion for an r derivation d r m to be a graded left r morphism may be given we then investigate when an r derivation is a sum of such graded derivation cg we establish that for a finitely generated abelian group g every r derivation of r into a weakly cancellative graded r bimodule has a decomposition as a sum of graded r derivations in the second section we prove a result relating the r derivations to some galois cohomology group for g and as a consequcncc of this it follows that for a strongly graded ring hochschild cohomology whenever defined coincides with some galois cohomology of the grading group along the way we provide an interesting extension of results about the miyashita auto morphisms of strongly graded rings our approach leads to some applications in the crossed product theory for azumaya algebras or even central simple algebras presenting a new point of view on crossed product structures which seems to deserve further development in the light of these applications and also to the benefit of general applicability the consideration of clifford systems which are not graded rings is important for example every algebra generated by invertible elements may be viewed 485 0021 8693 86 3 00

false discovery rate adjusted multiple confidence intervals for selected parameters often in applied research confidence intervals cis are constructed or reported only for parameters selected after viewing the data we show that such selected intervals fail to provide the assumed coverage probability by generalizing the false discovery rate fdr approach from multiple testing to selected multiple cis we suggest the false coverage statement rate fcr as a measure of interval coverage following selection a general procedure is then introduced offering fcr control at level q under any selection rule the procedure constructs a marginal ci for each selected parameter but instead of the confidence level 1 q being used marginally q is divided by the number of parameters considered and multiplied by the number selected if we further use the fdr controlling testing procedure of benjamini and hochberg for selecting the parameters the newly suggested procedure offers cis that are dual to the testing procedure and are shown to be optimal in the independent case under the positive re

periodic solutions to the compressible magnetohydrodynamic equations in a periodic domain abstract in this paper we study the compressible magnetohydrodynamic system with a time periodic external force in a periodic domain in r n based on the idea in 12 we obtain the existence of a time periodic solution under some smallness and symmetry assumptions on the external force and improve the result from 12 in the following sense we extend the space dimension from 3 to n n 1 the uniqueness of the small periodic solution is also obtained

two clifford s theorems for strongly group graded rings abstract let a be a strongly group graded ring in this paper we obtain the main results as following 1 in a a similar result to clifford s theorem of the decomposition of induced modules from normal subgroups it corrects the result of 4 2 frobenius reciprocity for a 3 we extend the theory of stable modules in two ways one of which is the localization theory of rings and modules in some special cases we can obtain the corresponding results of 1 2 3

frobenius and separable functors for the category of entwined modules over cowreaths ii applications abstract let h be a quasi hopf algebra we apply results obtained in 8 to give necessary and sufficient conditions for the forgetful functor from doi hopf modules two sided hopf modules or yetter drinfeld modules over h to representations of the underlying algebra to be frobenius resp separable we show that in some situations these conditions reduce to the unimodularity and or co semisimplicity of the quasi hopf algebra h

epsilon strongly graded rings separability and semisimplicity abstract we introduce the class of epsilon strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products we determine precisely when an epsilon strongly graded ring is separable over its principal component thereby we simultaneously generalize a result for strongly group graded rings by n st sescu van den bergh and van oystaeyen and a result for unital partial crossed products by bagio lazzarin and paques we also show that the class of unital partial crossed products appears in the class of epsilon strongly graded rings in a fashion similar to how the classical crossed products present themselves in the class of strongly graded rings thereby we obtain in the special case of unital partial crossed products a short proof of a general result by dokuchaev exel and simon concerning when graded rings can be presented as partial crossed products we also provide some interesting classes of examples of separable epsilon strongly graded rings with finite as well as infinite grading groups in particular we obtain an answer to a question raised by le bruyn van den bergh and van oystaeyen in 1988

covers of modules over a fixed ring abstract in this note we find conditions under which it is possible to prove the existence of relative injective covers of any module over the fixed ring r g by means of relative injective covers of modules over the base ring r the same problem is treated for flat covers

finite groups over arithmetical rings and globally irreducible representations abstract given the ring of integersrof an algebraic number fieldk for which natural numbernis there a finite groupg gl n r such thatrg ther span ofg coincides withm n r the ring of n n matrices overr giveng gl n r we show thatrg m n r if and only if the brauer reduction ofgmodulo every prime is absolutely irreducible in addition the question above is fully answered ifnis an odd prime

general maximum likelihood empirical bayes estimation of normal means we propose a general maximum likelihood empirical bayes gmleb method for the estimation of a mean vector based on observations with i i d normal errors we prove that under mild moment conditions on the unknown means the average mean squared error mse of the gmleb is within an infinitesimal fraction of the minimum average mse among all separable estimators which use a single deterministic estimating function on individual observations provided that the risk is of greater order than log n 5 n we also prove that the gmleb is uniformly approximately minimax in regular and weak l p balls when the order of the length normalized norm of the unknown means is between log n k1 n 1 p 2 and n log n k2 simulation experiments demonstrate that the gmleb outperforms the james stein and several state of the art threshold estimators in a wide range of settings without much down side

even unimodular lattices associated with the weil representation of the finite symplectic group soit q une puissance d un nombre premier impair p et soit g le groupe symplectique sp 2n q quand n est pair et q p est un nombre premier avec p 3 mod 4 il existe un treillis integral rationnel g invariant l dans m qui supporte une forme unimodulaire definie positive symetrique paire m est un qg module et q le corps des rationnels

separability and triangulated categories abstract we prove that the category of modules over a separable ring object in a tensor triangulated category admits a unique structure of triangulated category which is compatible with the original one this applies in particular to etale algebras more generally we do this for exact separable monads

monads and comonads on module categories let a be a ring and ma the category of right a modules it is well known in module theory that any a bimodule b is an a ring if and only if the functor ab ma ma is a monad or triple similarly an a bimodule c is an a coring provided the functor ac ma ma is a comonad or cotriple the related categories of modules or algebras of ab and comodules or coalgebras of ac are well studied in the literature on the other hand the right adjoint endofunctors homa b and homa c are a comonad and a monad respectively but the corresponding co module categories did not find much attention so far the category of homa b comodules is isomorphic to the category of b modules while the category of homa c modules called c contramodules by eilenberg and moore need not be equivalent to the category of c comodules the purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background this leads to a deeper understanding and characterisations of algebraic structures such as corings bialgebras and hopf algebras for example it turns out that the categories of c comodules and homa c modules are equivalent provided c is a coseparable coring furthermore we describe equivalences between categories of homa c modules and comodules over a coring d in terms of new galois properties of bicomodules finally we characterise hopf algebras h over a commutative ring r by properties of the functor homr h and the category of mixed homr h bimodules this generalises in particular the fact that a finite dimensional vector space h is a hopf algebra if and only if the dual space h is a hopf algebra

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

ergodipotent maps and commutativity of elements in noncommutative rings and algebras with twisted intertwining a property of algebraic dependence between two commuting elements is shown to hold in a more general setting than that in which it has previously been established key conditions are identified and some methods for establishing them are given moreover the class of algebras with a generalised weyl structure generalising the so called generalised weyl algebras gwas or hyperbolic rings is introduced and studied we also present an interesting class of algebras which are not gwas but share many of their properties by virtue of their generalised weyl structure for these classes of algebras centralisers and algebraic dependence are investigated c 2007 elsevier inc all rights reserved

an empirical bayes method for chi squared data in a thought provoking paper efron investigated the merit and limitation of an empirical bayes method to correct selection bias based on tweedie s formula first reported in the study by robbins t

separable functors for the category of doi hopf modules applications abstract we prove a maschke type theorem for the category of doi hopf modules in fact we give necessary and sufficient conditions for the functor that forgets the c coaction to be separable this leads to a generalized notion of integrals our results can be applied to obtain maschke type theorems for yetter drinfel d modules long dimodules and modules graded by g sets existing maschke type theorems due to doi and the authors are recovered as special cases

the poisson compound decision problem revisited the compound decision problem for a vector of independent poisson random variables with possibly different means has a half century old solution however it appears that the classical solution needs smoothing adjustment we discuss three such adjustments we also present another approach that first transforms the problem into the normal compound decision problem a simulation study shows the effectiveness of the procedures in improving the performance over that of the classical procedure a real data example is also provided the procedures depend on a smoothness parameter that can be selected using a nonstandard cross validation step which is of independent interest finally we mention some asymptotic results

on the differential graded eilenberg moore construction abstract the eilenberg moore construction for modules over a differential graded monad is used to study a question of balmer regarding existence of an exact adjoint pair representing an exact monad a bousfield like localization for differential graded categories is realized as a special case of this construction using drinfeld quotients as applications we study some example coming from g equivariant triangulated categories and twisted derived categories

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

modules graded byg sets duality and finiteness conditions an efficient technique to uncover the relationship between the representations of a group g and the representations of a subgroup h is to consider gradings by grh the class of the left h cosets of g this l w x method of study was initiated by dade in 3 where he describes the structure of the simple objects in the category r gr of g graded r modules for a g graded ring r as a generalization of the classical clifford theory a systematic investigation of modules graded by g sets was started w x w x by nastasescu et al in 9 and then continued in other papers 1 8 5 categorical properties of the category g a r gr of a graded left r modules where r is a g graded ring and a a left g set were studied first and later a series of finiteness conditions for the objects of this category were considered one natural question to ask in this framework is the following if m is an object of g a r gr where a is a finite g set and if n is an r submodule of m then which finiteness conditions i e ascending chain condition descending chain condition existence of krull dimension on n

graded equivalence theory with applications we develop the foundations for graded equivalence theory and apply them to investigate properties such as primeness finite representation type and vertex theory of graded rings the key fact that we prove is that for any two g graded rings r and s such that there is a category equivalence from gr r to gr s that commutes with suspensions then for any subgroup h of g the categories gr h g r and gr h g s of modules graded by the g set of right cosets are also equivalent

group linear empirical bayes estimates for a heteroscedastic normal mean abstractthe problem of estimating the mean of a normal vector with known but unequal variances introduces substantial difficulties that impair the adequacy of traditional empirical bayes estimators by taking a different approach that treats the known variances as part of the random observations we restore symmetry and thus the effectiveness of such methods we suggest a group linear empirical bayes estimator which collects observations with similar variances and applies a spherically symmetric estimator to each group separately the proposed estimator is motivated by a new oracle rule which is stronger than the best linear rule and thus provides a more ambitious benchmark than that considered in the previous literature our estimator asymptotically achieves the new oracle risk under appropriate conditions and at the same time is minimax the group linear estimator is particularly advantageous in situations where the true means and observed variances are empirically dependent to demonstrate the meri

covers and envelopes over gr gorenstein rings abstract in this paper we study the existence of gorenstein injective envelopes and gorenstein projective and flat covers in the category of graded modules and we relate them with the corresponding envelopes and covers in the category of modules

the jespers van oystaeyen conjecture e g 3 8 121 the mother examples of all these generalizations were studied already in 1968 by fossum 6 i e maximal orders over krull domains one of the main problems in the theory is to find a suitable generalization of unique factorization domains and related to this question to find a proper definition of a class group several possible definitions were suggested e g the k theoretic classgroup w a by fossum 6 the normalizing classgroup c l by chamarie 3 and the central classgroup which has been studied extensively by jespers and wouters 7 81 van oystaeyen 22 and later jespers 7 asked whether for a maximal order n over a dedekind domain