representation dimension and finitely generated cohomology abstract we consider selfinjective artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators for such an algebra we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules this provides a unified approach to computing lower bounds for the representation dimension of group algebras exterior algebras and artin complete intersections we also obtain new examples of classes of algebras with arbitrarily large representation dimension

categorical formulation of finite dimensional quantum algebras we describe how frobenius monoids give the correct categorical description of certain kinds of finite dimensional quantum algebras we develop the concept of an involution monoid and use it to construct a correspondence between finite dimensional c algebras and certain types of frobenius monoids in the category of hilbert spaces using this technology we recast the spectral theorems for commutative c algebras and for normal operators into an explicitly categorical language and we examine the case that the results of measurements do not form finite sets but rather objects in a finite boolean topos we describe the relevance of these results for topological quantum field theory

the method of space time conservation element and solution element a new approach for solving the navier stokes and euler equations a new numerical framework for solving conservation laws is being developed this new framework differs substantially in both concept and methodology from the well established methods i e finite difference finite volume finite element and spectral methods it is conceptually simple and designed to overcome several key limitations of the above traditional methods a two level scheme for solving the convection diffusion equation u t a u x 2u x2 0 0 is constructed and used to illuminate major differences between the present method and those mentioned above this explicit scheme referred to as the a scheme has two independent marching variables unj and ux nj which are the numerical analogues of u and u x at j n respectively the a scheme has the unusual property that its stability is limited only by the cfl condition i e it is independent of also it can be shown that the amplification factors of the a scheme are identical to those of the leapfrog scheme if o and to those of the dufort frankel scheme if a o these coincidences are unexpected because the a scheme and the above classical schemes are derived from completely different perspectives and the a scheme does not reduce to the above classical schemes in the limiting cases the a scheme is extended to solve the 1d time dependent navier stokes equations of a perfect gas stability of this explicit solver also is limited only by the cfl condition in spite of the fact that it does not use i any techniques related to the high resolution upwind methods and ii any ad hoc parameter the current navier stokes solver is capable of generating highly accurate shock tube solutions particularly for high reynolds number flows shock discontinuities can be resolved within one mesh interval the inviscid 0 a scheme is reversible in time it also is neutrally stable i e free from numerical dissipation such a scheme generally cannot be extended to solve the euler equations thus the inviscid version is modified stability of this modified scheme referred to as the a scheme is limited by the cfl condition and 0 1 where is a special parameter that controls numerical dissipation moreover if 0 the amplification factors of the a scheme are identical to those of the leapfrog scheme which has no numerical dissipation on the other hand if 1 the two amplification factors of the a scheme become the same function of the courant number and the phase angle unexpectedly this function also is the amplification factor of the highly diffusive lax scheme note that because the lax scheme is very diffusive and it uses a mesh that is staggered in time a two level scheme using such a mesh is often associated with a highly diffusive scheme the a scheme which also uses a mesh staggering in time demonstrates that it can also be a scheme with no numerical dissipation the euler extension of the a scheme has stability conditions similar to those of the a epsiv scheme itself it has the unusual property that numerical dissipation at all mesh points can be controlled by a set of local parameters moreover it is capable of generating accurate shock tube solutions with the cfl number ranging from close to 1 to 0 022

the time line interpolation method for large time step godunov type schemes abstract this paper describes the use of the time line interpolation procedure for the design of large time step godunov type schemes for systems of hyperbolic conservation laws in one dimension these schemes are based on a specific procedure to characterize the left and right states of the riemann problems at the cell interfaces when the courant number associated with the waves exceeds unity to do so the time line interpolation technique is used constant and linear reconstruction techniques are presented sonic or critical points are seen to be a source of difficulty in the algorithm and an appropriate treatment is proposed the algorithms are applied to the linear advection equation to the inviscid burgers equation and to the set of hyperbolic conservation laws that describe shallow water flow in one dimension these simulations show the superiority of the linear time reconstruction over the use of a constant time reconstruction when the linear reconstruction technique is used the modulus of the amplification factor of the scheme is equal to unity for all wave numbers inducing oscillations in the computed profile owing to the phase error the introduction of a slope limiter allows these oscillations to be eliminated but yields numerical diffusion thus restricting the range of applications of the scheme

the geometry of tensor calculus i this paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals mathematics subject classifications 1991 18d10 52b11 53a45 57m25 68q10 82b23

finite volume treatment of dispersion relation preserving and optimized prefactored compact schemes for wave propagation in developing suitable numerical techniques for computational aero acoustics the dispersion relation preserving drp scheme by tam and co workers and the optimized prefactored compact opc scheme by ashcroft and zhang have shown desirable properties of reducing both dissipative and dispersive errors these schemes originally based on the finite difference attempt to optimize the coefficients for better resolution of short waves with respect to the computational grid while maintaining pre determined formal orders of accuracy in the present study finite volume formulations of both schemes are presented to better handle the nonlinearity and complex geometry encountered in many engineering applications linear and nonlinear wave equations with and without viscous dissipation have been adopted as the test problems highlighting the principal characteristics of the schemes and utilizing linear and nonlinear wave equations with different wavelengths as the test cases the performance of these approaches is documented for the linear wave equation there is no major difference between the drp and opc schemes for the nonlinear wave equations the finite volume version of both drp and opc schemes offers substantially better solutions in regions of high gradient or discontinuity

on some classes of nonstationary axially symmetric solutions to the navier stokes equations in the paper the navier stokes equations are studied in axially symmetric cases of nonstationary motion with rotation of incompressible viscous fluids the problem is reduced to a nonlinear system of three partial differential equations for three unknown functions of the cylindrical coordinates r and z and time t the three functions are sought in the form of power series in r with coefficients depending on t and z for the unknown coefficients recurrence relations are obtained which contain three arbitrary functions the relations are examined in three particular cases in which they give analytical solutions to the navier stokes equations for any values of the coordinates t z and r

the phantom cover of a module abstract a morphism f m n of left r modules is a phantom morphism if for any morphism g a m with a finitely presented the composition fg factors through a projective module equivalently tor 1 x f 0 for every right r module x it is proved that every r module possesses a phantom cover whose kernel is pure injective if mod r is the category of finitely presented right r modules modulo projectives then the association m tor 1 m is a functor from the category of left r modules to that of the flat functors on mod r the phantom cover is used to characterize when this functor is faithful or full it is faithful if and only if the flat cover of every module has a pure injective kernel this is equivalent to the flat cover being the phantom cover the question of fullness is only reasonable when the functor is restricted to the subcategory of cotorsion modules this restriction is full if and only if every phantom cover of a cotorsion module is pure injective

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

regular article the space time conservation element and solution element method a new high resolution and genuinely multidimensional paradigm for solving conservation laws a new high resolution and genuinely multidimensional numerical method for solving conservation laws is being developed it was designed to avoid the limitations of the traditional methods and was built from ground zero with extensive physics considerations nevertheless its foundation is mathematically simple enough that one can build from it a coherent robust efficient and accurate numerical framework two basic beliefs that set the new method apart from the established methods are at the core of its development the first belief is that in order to capture physics more efficiently and realistically the modeling focus should be placed on the original integral form of the physical conservation laws rather than the differential form the latter form follows from the integral form under the additional assumption that the physical solution is smooth an assumption that is difficult to realize numerically in a region of rapid change such as a boundary layer or a shock the second belief is that with proper modeling of the integral and differential forms themselves the resulting numerical solution should automatically be consistent with the properties derived from the integral and differential forms e g the jump conditions across a shock and the properties of characteristics therefore a much simpler and more robust method can be developed by avoiding the explicit use of the above derived properties specifically to capture physics as fully as possible the method requires that i space and time be unified and treated as a single entity ii both local and global flux conservation in space and time be enforced and iii a multidimensional scheme be constructed without using the dimensional splitting approach such that multidimensional effects and source terms which are scalars can be modeled more realistically to simplify mathematics and broaden its applicability as much as possible the method attempts to use the simplest logical structures and approximation techniques specifically i it uses a staggered space time mesh such that flux at any interface separating two conservation elements can be evaluated internally in a simpler and more consistent manner without using a separate flux model ii it avoids the use of many well established techniques such as riemann solvers flux splittings and monotonicity constraints such that the complications and possibly even the limitations associated with them may be avoided and iii it avoids the use of special techniques that are not applicable to more general problems furthermore triangles in 2d space and tetrahedrons in 3d space are used as the basic building blocks of the spatial meshes such that the method i can be used to construct 2d and 3d nondissipative schemes in a natural manner and ii is compatible with the simplest unstructured meshes note that while numerical dissipation is required for shock capturing it may also result in annihilation of small disturbances such as sound waves and in the case of flow with a large reynolds number may overwhelm physical dissipation to overcome this difficulty two different and mutually complementary types of adjustable numerical dissipation are introduced in the present development

phantom maps and purity in modular representation theory iii abstract in the third installment of this paper we exhibit two phantom maps between kg modules whose composite is not projective whenever k is an uncountable field of characteristic p and g is a finite group of p rank at least 2

a local space time conservation scheme and its application in shock wave propagation abstract in this paper a local space time conservation scheme based on non staggered grids is introduced which is a variation of space time conservation element and solution element ce se scheme it inherits most features and advantages of ce se method including unified treatment of space and time and high accuracy resolution of hyperbolic conservation equations moreover riemann solvers are not needed to capture shocks and dimensional splitting methods are not needed in the multi dimensional schemes the stability of the present scheme is verified through von neumann analysis moreover several shock wave problems including one two and three dimensional cases are simulated by the present scheme by carefully comparing the present scheme s numerical results with exact solutions experimental results original ce se scheme s numerical results and third order eno scheme s numerical results it can be conclude that the present scheme is efficient and accurate

symmetric algebras in categories of corepresentations and smash products we investigate frobenius algebras and symmetric algebras in the monoidal category of right comodules over a hopf algebra h for the symmetric property h is assumed to be cosovereign if h is finite dimensional and a is an h comodule algebra we uncover the connection between a and the smash product a h with respect to the frobenius and symmetric properties

comatrix corings galois corings descent theory and a structure theorem for cosemisimple corings we study the corings whose category of right comodules has a finitely generated projective generator in order to extricate the structure of these corings we introduce the notion of comatrix coring comatrix corings allow as well to give a complete description of all cosemisimple corings

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

application of space time ce se method to shallow water magnetohydrodynamic equations in this article we apply a space time conservation element and solution element ce se method for the approximate solution of shallow water magnetohydrodynamic smhd equations in one and two space dimensions these equations model the dynamics of nearly incompressible conducting fluids for which the evolution is nearly two dimensional with magnetic equilibrium in the third direction in this article we are using a variant of the ce se method developed by zhang et al a space time conservation element and solution element method for solving the two dimensional unsteady euler equations using quadrilateral and hexahedral meshes j comput phys 175 2002 168 199 this method uses structured and unstructured quadrilateral and hexahedral meshes in two and three space dimensions respectively in this method a single conservation element at each grid point is employed for solving conservation laws no matter in one two and three space dimensions the present scheme use the conservation element to calculate flow variables only while the gradients of flow variables are calculated by central differencing reconstruction procedure we give both one and two dimensional test computations a qualitative comparison reveals an excellent agreement with previous published results of wave propagation method and evolution galerkin schemes the one and two dimensional computations reported in this paper demonstrate the remarkable versatility of the present ce se scheme

an analysis of finite difference and finite volume formulations of conservation laws abstract finite difference and finite volume formulations are analyzed in order to clear up the confusion concerning their application to the numerical solution of conservation laws a new coordinate free formulation of systems of conservation laws is developed which clearly distinguishes the role of physical vectors from that of algebraic vectors which characterize the system the analysis considers general types of equations potential euler and navier stokes three dimensional unsteady flows with time varying grids are described using a single consistent nomenclature for both formulations grid motion due to a non inertial reference frame as well as flow adaptation is covered in comparing the two formulations it is found useful to distinguish between differences in numerical methods and differences in grid definition the former plays a role for non cartesian grids and results in only cosmetic differences in the manner in which geometric terms are handled the differences in grid definition for the two formulations is found to be more important since it affects the manner in which boundary conditions zonal procedures and grid singularities are handled at computational boundaries the proper interpretation of strong and weak conservation law forms for quasi one dimensional and axisymmetric flows is brought out

phantom covering ideals in categories without enough projective morphisms abstract we give sufficient conditions to ensure that the ideal e of e phantom maps in a locally presentable exact category a e is a special pre covering ideal where e is an exact substructure of a e as a byproduct we infer the existence of various covering ideals in categories of sheaves which have a meaningful geometrical motivation in particular we deal with a zariski local notion of phantom maps in categories of sheaves we would like to point out that our approach is necessarily different from 18 as the categories involved in most of the examples we are interested in do not have enough projective morphisms

almost split sequences for relatively projective modules the paper deals with almost split sequences introduced in 2 for the category mod a of finitely generated modules over an artin algebra a almost split sequences were later found in the category of lattices over an order l 43 as well as in certain subcategories of mod a 6 l 3 it is generally recognized that if almost split sequences exist the subcategory has nice properties we are concerned with the subcategory of relatively projective modules let r be a field or a dedekind domain with the field of quotients k and let a and a be finite dimensional r algebras or r orders respectively with a mapped into a via an r algebra map i a a here we understand orders and lattices in the sense of 1 p 85 example b namely a is an r order if it is a noetherian r algebra projective as an r module and x k or a is a self injective ring a mod denotes the category of finitely generated left a modules if r is a field or the category of left a lattices if r is a dedekind domain where a left a module h4 is a lattice if it is a finitely generated projective r module such that k ox m is a projective

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

smashing subcategories and the telescope conjecture an algebraic approach we prove a modified version of ravenel s telescope conjecture it is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra this result is based on a new characterization of smashing subcategories which leads in addition to a classification of these subcategories in terms of the category of finite spectra the approach presented here is purely algebraic it is based on an analysis of pure injective objects in a compactly generated triangulated category and covers therefore also situations arising in algebraic geometry and representation theory

ideal approximation theory abstract let a e be an exact category and f ext a subfunctor a morphism in a is an f phantom if the pullback of an e conflation along is a conflation in f if the exact category a e has enough injective objects and projective morphisms it is proved that an ideal i of a is special precovering if and only if there is a subfunctor f ext with enough injective morphisms such that i is the ideal of f phantom morphisms a crucial step in the proof is a generalization of salce s lemma for ideal cotorsion pairs if i is a special precovering ideal then the ideal cotorsion pair i i generated by i in a e is complete this theorem is used to verify 1 that the ideal cotorsion pair cogenerated by the pure injective modules of r mod is complete 2 that the ideal cotorsion pair cogenerated by the contractible complexes in the category of complexes ch r mod is complete and using auslander and reiten s theory of almost split sequences 3 that the ideal cotorsion pair cogenerated by the jacobson radical jac mod of the category mod of finitely generated representations of an artin algebra is complete

a new category of hermitian upwind schemes for computational acoustics a new high order upwind scheme is developed to solve linear acoustics in order to produce a compact scheme with a high level of precision a hermite interpolation is used an algorithm in three stages is built in order to optimize the procedure of discretization in the reconstruction stage a local spatial interpolator based on an upwind stencil is defined in the decomposition stage the calculation of a wave model adapted to the decomposition of the time derivatives in simple wave contributions is derived then the local spatial interpolator is used in order to calculate the wave strengths for each wave in the evolution stage we update by means of a procedure similar to that of cauchy kovaleskaya the discrete variable and its derivatives by using the information brought by each simple wave specific boundary conditions are finally established in this way a third order upwind hermitian scheme may be constructed d p3 scheme a one dimensional analysis in fourier series shows the principal properties of such a scheme the effectiveness and the exactitude of the new compact scheme are shown by their applications to several two dimensional problems and by comparisons to competing finite difference method for wave propagation

qf functors and co monads abstract one reason for the universal interest in frobenius algebras is that their characterisation can be formulated in arbitrary categories a functor k a b between categories is frobenius if there exists a functor g b a which is at the same time a right and left adjoint of k a monad f on a is a frobenius monad provided the forgetful functor a f a is a frobenius functor where a f denotes the category of f modules with these notions an algebra a over a field k is a frobenius algebra if and only if a k is a frobenius monad on the category of k vector spaces the purpose of this paper is to find characterisations of quasi frobenius algebras by just referring to constructions available in any categories to achieve this we define qf functors between two categories by requiring conditions on pairings of functors which weaken the axioms for adjoint pairs of functors qf monads on a category a are those monads f for which the forgetful functor u f a f a is a qf functor applied to module categories or grothendieck categories our notions coincide with definitions first given k morita and others further applications show the relations of qf functors and qf monads with frobenius exact categories

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

a space time conservation element and solution element method for solving the two and three dimensional unsteady euler equations using quadrilateral and hexahedral meshes in this paper we report a version of the space time conservation element and solution element ce se method in which the 2d and 3d unsteady euler equations are simulated using structured or unstructured quadrilateral and hexahedral meshes respectively in the present method mesh values of flow variables and their spatial derivatives are treated as independent unknowns to be solved for at each mesh point the value of a flow variable is obtained by imposing a flux conservation condition on the other hand the spatial derivatives are evaluated using a finite difference weighted average procedure note that the present extension retains many key advantages of the original ce se method which uses triangular and tetrahedral meshes respectively for its 2d and 3d applications these advantages include efficient parallel computing ease of implementing nonreflecting boundary conditions high fidelity resolution of shocks and waves and a genuinely multidimensional formulation without the need to use a dimensional splitting approach in particular because riemann solvers the cornerstones of the godunov type upwind schemes are not needed to capture shocks the computational logic of the present method is considerably simpler to demonstrate the capability of the present method numerical results are presented for several benchmark problems including oblique shock reflection supersonic flow over a wedge and a 3d detonation flow

polycategories via pseudo distributive laws in this paper we give a novel abstract description of szabo s polycategories we use the theory of double clubs a generalisation of kelly s theory of clubs to pseudo or weak double categories to construct a pseudo distributive law of the free symmetric strict monoidal category pseudocomonad on mod over itself qua pseudomonad and show that monads in the two sided kleisli bicategory of this pseudo distributive law are precisely symmetric polycategories

transport of pollutant in shallow flows a space time ce se scheme abstract this paper focuses on implementation of space time ce se scheme for computing the transport of a passive pollutant by a flow the flow model comprises of the saint venant system of shallow water equations and the pollutant propagation is described by a transport equation the one dimensional and two dimensional flow equations are numerically investigated using the ce se scheme a number of test problems are presented to check the accuracy and efficiency of the proposed scheme the results of ce se scheme are compared with the central scheme both the schemes are found to be in close agreement however our proposed ce se scheme accurately captures shocks and discontinuous profiles

phantom maps in the stable module category abstract we study phantom maps in the stable module category stmod kg wherekis a field andgis a finite group in this article we almost exclusively deal with maps out of countably generated modules we show that the space ph kg m n of stabilized phantom maps m n has an expression as a lim 1 space which allows some control on its vanishing then we present a situation where all maps are phantom and need not be trivial we provide explicit details for such a particular situation finally we construct a universal phantom map we use it to show that the composite of two phantom maps is trivial and to characterize the modules with no nontrivial outbound phantom maps

aeroacoustics of t junction merging flow this paper reports a numerical study of the aeroacoustics of merging flow at t junction the primary focus is to elucidate the acoustic generation by the flow unsteadiness the study is conducted by performing direct aeroacoustic simulation approach which solves the unsteady compressible navier stokes equations and the perfect gas equation of state simultaneously using the conservation element and solution element method for practical flows the reynolds number based on duct width is usually quite high 105 in order to properly account for the effects of flow turbulence a large eddy simulation methodology together with a wall modeling derived from the classical logarithm wall law is adopted the numerical simulations are performed in two dimensions and the acoustic generation physics at different ratios of side branch to main duct flow velocities vr 0 5 0 67 1 0 2 0 are studied both the levels of unsteady interactions of merging flow structures and the efficiency of acoustic generation are obser

monoidal bicategories and hopf algebroids why are groupoids such special categories the obvious answer is because all arrows have inverses yet this is precisely what is needed mathematically to model symmetry in nature the relation between the groupoid and the physical object is expressed by an action the presence of inverses means that actions of a groupoid g behave much better than actions of an arbitrary category the totality of actions of g on vector spaces forms a category mod g of modules the feature of mod g which epitomises the fact that g is a groupoid is that the internal hom in mod g is calculated in a particularly simple way more precisely the functor out of mod g which forgets the actions preserves not only the monoidal structure but also the closed structure with this as a guiding principle we develop a general concept of autonomous pseudomonoid which includes ordinary hopf algebras indeed hopf algebroids and autonomous compact rigid monoidal categories this is intended to elucidate the interaction between hopf algebras and autonomous monoidal categories in tannaka duality as appearing in js2 d2 for example given a topological monoid m it is explained in js2 section 8 why the monoidal category of finite dimensional representations of m is equivalent to the monoidal category of finite dimensional comodules over the bialgebra r m of representative functions on m this provides evidence that when regarding a hopf algebra as a quantum group it is the finitedimensional comodules rather than modules which should be regarded as the representations of the group in dealing with comodules we are using the coalgebra structure of the hopf algebra h a hopf algebroid is an additive category that is algebra with several objects that is algebroid with a comonoidal structure modules over a hopf algebroid make sense but comodules are not appropriate what we need is a notion where the article no ai971649

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

transfinite adams representability abstract we consider the following problems in a well generated triangulated category t let be a regular cardinal and t t the full subcategory of compact objects is every functor h t op ab that preserves products of h t x t for some x in t is every natural transformation t x t t y t of the form t f t for some f x y in t if the answer to both questions is positive we say that t satisfies adams representability a classical result going back to brown and adams shows that the stable homotopy category satisfies 0 adams representability the case 0 is well understood thanks to the work of christensen keller and neeman in this paper we develop an obstruction theory to decide whether t satisfies adams representability we derive necessary and sufficient conditions of homological nature and we compute several examples in particular we show that there are rings satisfying adams representability for all 0 and rings which do not satisfy adams representability for any 0 moreover we exhibit rings for which the answer to both questions is no for all 2

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

frobenius algebras of corepresentations and group graded vector spaces abstract we consider frobenius algebras in the monoidal category of right comodules over a hopf algebra h if h is a group hopf algebra we study a more general frobenius type property uncover the structure of graded frobenius algebras and investigate graded symmetric algebras the graded frobenius concept is related to frobenius functors

combining a monad and a comonad we give a systematic treatment of distributivity for a monad and a comonad as arises in giving category theoretic accounts of operational and denotational semantics and in giving an intensional denotational semantics we do this axiomatically in terms of a monad and a comonad in a 2 category giving accounts of the eilenberg moore and kleisli constructions we analyse the eight possible relationships deducing that two pairs are isomorphic but that the other pairs are all distinct we develop those 2 categorical definitions necessary to support this analysis

graded semisimple algebras are symmetric abstract we study graded symmetric algebras which are the symmetric monoids in the monoidal category of vector spaces graded by a group we show that a finite dimensional graded semisimple algebra is graded symmetric the center of a symmetric algebra is not necessarily symmetric but we prove that the center of a finite dimensional graded division algebra is symmetric provided that the order of the grading group is not divisible by the characteristic of the base field

galois functors and entwining structures abstract galois comodules over a coring can be characterised by properties of the relative injective comodules they motivated the definition of galois functors over some comonad or monad on any category and in the first section of the present paper we investigate the role of the relative injectives projectives in this context then we generalise the notion of corings derived from an entwining of an algebra and a coalgebra to the entwining of a monad and a comonad hereby a key role is played by the notion of a grouplike natural transformation g i g generalising the grouplike elements in corings we apply the evolving theory to hopf monads on arbitrary categories and to opmonoidal monads with antipode on autonomous monoidal categories named hopf monads by bruguieres and virelizier which can be understood as an entwining of two related functors as well known for any set g the product g defines an endofunctor on the category of sets and this is a hopf monad if and only if g allows for a group structure in the final section the elements of this case are generalised to arbitrary categories with finite products leading to galois objects in the sense of chase and sweedler

frobenius monads and pseudomonoids six equivalent definitions of frobenius algebra in a monoidal category are provided in a monoidal bicategory a pseudoalgebra is frobenius if and only if it is star autonomous autonomous pseudoalgebras are also frobenius what it means for a morphism of a bicategory to be a projective equivalence is defined this concept is related to strongly separable frobenius algebras and weak monoidal morita equivalence wreath products of frobenius algebras are discussed

dimensions of triangulated categories via koszul objects lower bounds for the dimension of a triangulated category are provided these bounds are applied to stable derived categories of artin algebras and of commutative complete intersection local rings as a consequence one obtains bounds for the representation dimensions of certain artin algebras

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

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

gpu accelerated cese method for 1d shock tube problems in the present study a gpu accelerated 1d space time cese method is developed and applied to shock tube problems with and without condensation we have demonstrated how to implement the cese algorithm to solve 1d shock tube problems using an older generation gpu the nvidia 9800 gt with relatively limited memory to optimize the code performance we used shared memory and solved the inter block boundary problem in two ways namely the branch scheme and the overlapping scheme the implementations of these schemes are discussed in detail and their performances are compared for the sod shock tube problems for the sod problem without condensation the speedup over an intel cpu e7300 is 23 for the branch scheme and 41 for the overlapping scheme respectively while for problems with condensation both schemes achieve higher acceleration ratios 53 and 71 respectively the higher speedup of the condensation case can be ascribed to the source term calculation which has a local dependence on the mesh point and the source kernel has a higher acceleration ratio

ideals in triangulated categories phantoms ghosts and skeleta we begin by showing that in a triangulated category specifying a projective class is equivalent to specifying an ideal i of morphisms with certain properties and that if i has these properties then so does each of its powers we show how a projective class leads to an adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence we use this to study various ideals in the stable homotopy category we examine phantom maps skeletal phantom maps superphantom maps and ghosts a ghost is a map which induces the zero map of homotopy groups we show that ghosts lead to a stable analogue of the lusternik schnirelmann category of a space and we calculate this stable analogue for low dimensional real projective spaces we also give a relation between ghosts and the hopf and kervaire invariant problems in the case ofa modules over ana ring spectrum the ghost spectral sequence is a universal coefficient spectral sequence from the phantom projective class we derive a generalized milnor sequence for filtered diagrams of finite spectra and from this it follows that the group of phantom maps fromxtoycan always be described as alim 1group the last two sections focus on algebraic examples in the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of kelly on the vanishing of composites of such maps we also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non trivially

powers of the phantom ideal the mono epi me exact structure on the morphisms of an exact category a e is introduced and used to prove ideal versions of salce s lemma christensen s ghost lemma and wakamatsu s lemma for an exact category salce s lemma establishes a bijective correspondence i i between the class of special precovering ideals of a e and that of its special preenveloping ideals me extensions of morphisms are used to define an extension i j of ideals christensen s lemma asserts that the class of special precovering respectively special preenveloping ideals is closed under products and extensions and that the bijective correspondence of salce s lemma satisfies ij j i and i j j i wakamatsu s lemma asserts that if a covering ideal i is closed under me extensions then it is a special precovering ideal as an application it is proved that if g is a finite group and is the ideal of phantom morphisms in the category k g mod then n 1 is the object ideal generated by projective modules where n is the nilpotency index of the jacobson radical j if r is a semiprimary ring with jn 0 then n is generated by projective modules for a right coherent ring r over which every cotorsion left r module has a coresolution of length n by pure injective modules n 1 is generated by flat modules

the fundamental theorem for weak braided bimonads abstract the theories of hopf bialgebras and weak hopf bialgebras have been introduced for vector space categories over fields and make heavily use of the tensor product as first generalisations these notions were formulated for monoidal categories with braidings if needed the present authors developed a theory of bimonads and hopf monads h on arbitrary categories a employing distributive laws allowing for a general form of the fundamental theorem for hopf algebras for bimonads h properties of braided hopf bialgebras were captured by requiring a yang baxter operator h h h h the purpose of this paper is to extend the features of weak hopf bialgebras to this general setting including an appropriate form of the fundamental theorem this subsumes the theory of braided hopf algebras based on weak yang baxter operators as considered by alonso alvarez and others

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

solving the mhd equations by the space time conservation element and solution element method we apply the space time conservation element and solution element cese method to solve the ideal mhd equations with special emphasis on satisfying the divergence free constraint of magnetic field i e b 0 in the setting of the cese method four approaches are employed i the original cese method without any additional treatment ii a simple corrector procedure to update the spatial derivatives of magnetic field b after each time marching step to enforce b 0 at all mesh nodes iii a constraint transport method by using a special staggered mesh to calculate magnetic field b and iv the projection method by solving a poisson solver after each time marching step to demonstrate the capabilities of these methods two benchmark mhd flows are calculated i a rotated one dimensional mhd shock tube problem and ii a mhd vortex problem the results show no differences between different approaches and all results compare favorably with previously reported data

discontinuous galerkin bgk method for viscous flow equations one dimensional systems this paper is about the construction of a bgk navier stokes bgk ns solver in the discontinuous galerkin dg framework since in the dg formulation the conservative variables and their slopes can be updated simultaneously the flow evolution in each element involves only the flow variables in the nearest neighboring cells instead of using the semidiscrete approach in the runge kutta discontinuous galerkin rkdg method the current dg bgk method integrates the governing equations in time as well due to the coupling of advection and dissipative terms in the gas kinetic formulation the dg bgk method solves the viscous governing equations directly numerical examples for the one dimensional compressible navier stokes solutions will be presented

application of the space time conservation element and solution element method to one dimensional convection diffusion problems abstract in the space time conservation element and solution element ce se method the independent marching variables used comprise not only the mesh values of the physical dependent variables but also in contrast to a typical numerical method the mesh values of the spatial derivatives of these physical variables the use of the extra marching variables results from the need to construct the two level explicit and nondissipative schemes which are at the core of the ce se development it also results from the need to minimize the stencil while maintaining accuracy in this paper using the 1d a scheme as an example the effect of this added complication on consistency accuracy and operation count is assessed as part of this effort an equivalent yet more efficient form of the a scheme in which the independent marching variables are the local fluxes tied to each mesh point is introduced also the intriguing relations that exist among the a leapfrog and dufort frankel schemes are further explored in addition the redundancy of the leapfrog dufort frankel and lax schemes and the remedy for this redundancy are discussed this paper is concluded with the construction and evaluation of a ce se solver for the inviscid burgers equation

ghosts in modular representation theory a ghost over a finite p group g is a map between modular representations of g which is invisible in tate cohomology motivated by the failure of the generating hypothesis the statement that ghosts between finite dimensional g representations factor through a projective we define the ghost number of kg to be the smallest integer l such that the composite of any l ghosts between finite dimensional g representations factors through a projective in this paper we study ghosts and the ghost numbers of p groups we begin by showing that a weaker version of the generating hypothesis where the target of the ghost is fixed to be the trivial representation k holds for all p groups we then compute the ghost numbers of all cyclic p groups and all abelian 2 groups with c2 as a summand we obtain bounds on the ghost numbers for abelian p groups and for all 2 groups which have a cyclic subgroup of index 2 using these bounds we determine the finite abelian groups which have ghost number at most 2 our methods involve techniques from group theory representation theory triangulated category theory and constructions motivated from homotopy theory

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

nonsymmetric autonomous categories abstract the theory of autonomus categories was originally developed for symmetric monoidal categories since it has recently been seen as supplying models for linear logic and since there is interest in nonsymmetric linear logic to provide for example models of temporal dependencies there is also interest in a nonsymmetric version of autonomous categories this paper provides the definition and develops some equivalent forms of the definition which are a bit different in the nonsymmetric situation it also gives a nonsymmetric version of the chu construction which has proven so useful in the past

matrices relations and group representations abstract motivated by the search for a good notion of a selfdual object in a symmetric monoidal category we give an abstract notion of commutative separable algebra in a symmetric monoidal category v such that when v is the category of modules over a commutative ring k we obtain the usual commutative separable algebras developing the calculus based on such an abstract notion we are able to prove that the dual category of separable algebras in any compact closed additive category with coequalizers is in fact a boolean pretopos we apply this result to give a simple characterization of the categories of continuous representations of profinite groups in discrete finite dimensional k vector spaces finally we show that the same calculus can be applied to symmetric monoidal categories of relations to give an essentially algebraic characterization of such categories

the space time cese scheme for shallow water equations incorporating variable bottom topography and horizontal temperature gradients abstract the effects of bottom topography and horizontal temperature gradients on the shallow water flows are theoretically investigated the considered systems of partial differential equations pdes are non strictly hyperbolic and non conservative due to the presence of non conservative differential terms on the right hand side the solutions of these model equations are very challenging for a numerical scheme thus our primary goal is to introduce an improved numerical scheme which can handle the non conservative differential terms efficiently and accurately in this paper the space time conservation element and solution element cese method is extended to approximate these model equations the proposed scheme has capability to overcome all difficulties posed by this nonlinear system of pdes the performance of the scheme is analyzed by considering several case studies of practical interest and the results of suggested scheme are compared with those of central nt scheme the accuracy of the scheme is verified qualitatively and quantitatively

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

weak bimonads and weak hopf monads we define a weak bimonad as a monad t on a monoidal category m with the property that the eilenberg moore category mt is monoidal and the forgetful functor mt m is separable frobenius whenever m is also cauchy complete a simple set of axioms is provided that characterizes the monoidal structure of mt as a weak lifting of the monoidal structure of m the relation to bimonads and the relation to weak bimonoids in a braided monoidal category are revealed we also discuss antipodes obtaining the notion of weak hopf monad

correspondences of ribbon categories much of algebra and representation theory can be formulated in the general framework of tensor categories the aim of this paper is to further develop this theory for braided tensor categories several results are established that do not have a substantial counterpart for symmetric tensor categories in particular we exhibit various equivalences involving categories of modules over algebras in ribbon categories finally we establish a correspondence of ribbon categories that can be applied to and is in fact motivated by the coset construction in conformal quantum field theory

representation dimension of exterior algebras we determine the representation dimension of exterior algebras this provides the first known examples of representation dimension 3 we deduce that the loewy length of the group algebra over f2 of a finite group is strictly bounded below by the 2 rank of the group a conjecture of benson a key tool is the use of the concept of dimension of a triangulated category

lie monads and dualities abstract we study dualities between lie algebras and lie coalgebras and their respective co representations to allow a study of dualities in an infinite dimensional setting we introduce the notions of lie monads and lie comonads as special cases of yb lie algebras and yb lie coalgebras in additive monoidal categories we show that strong dualities between lie algebras and lie coalgebras are closely related to iso morphisms between associated lie monads and lie comonads in the case of a duality between two hopf algebras in the sense of takeuchi we recover a duality between a lie algebra and a lie coalgebra in the sense defined in this note by computing the primitive and the indecomposable elements respectively

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