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

noncoercive variational inequalities some applications ever since the fundamental paper of lions and stampacchia io the subject of variational inequalities has become a very active area of study and become an important tool in the study of nonlinear partial differential equations which are subject to various types of constraints see e g l 5 8 111 in lo the authors consider first the case of coercive forms and then using elliptic regularization problems involving noncoercive but nonnegative forms we shall use a similar approach to study variational inequalities involving noncoercive bilinear forms and nonlinear functionals which are nice perturbations of lower semi continuous convex functionals furthermore our bilinear forms need not necessarily have a finite dimensional kernel our investigation has been motivated by some work of baiocchi gastaldi and tomarelli see 3 61 where variational inequalities involving so called compact coercive bilinear forms are studied we shall introduce here a somewhat more general concept that of a p coercive bilinear form and study variational inequalities involving such forms and nonlinear functionals which are compact and bounded perturbations of lower semi continuous convex functionals in the next section we shall introduce the necessary terminology prove some preliminary results and give the statement of the abstract problem we then shall state and prove our main abstract result and conclude the paper with applications including one to the study of a flat plate on an elastic foundation with unilateral boundary conditions see e g l for the terminology and one to the study of unilateral problems in semipermeable media see 5

pullback attractors for asymptotically compact non autonomous dynamical systems abstract first we introduce the concept of pullback asymptotically compact non autonomous dynamical system as an extension of the similar concept in the autonomous framework our definition is different from that of asymptotic compactness already used in the theory of random and non autonomous dynamical systems as developed by crauel flandoli kloeden schmalfuss amongst others which means the existence of a random or time dependent family of compact attracting sets next we prove a result ensuring the existence of a pullback attractor for a non autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets this attractor is minimal and in most practical applications it is unique finally we illustrate the theory with a 2d navier stokes model in an unbounded domain

positive solutions of variational inequalities a degree theoretic approach if the mapping a is strictly monotone and coercive is equivalent to a fixed point equation u m u in k cf section 1 suppose that m is continuous and maps bounded subsets of k into compact ones then due to the fact that k is a retract of x it is possible to define a fixed point index of m with properties similar to those associated with the leray schauder degree 1 sect 111 in section 1 we introduce another index which is based on the previous one but has the advantage of being directly related to inequality rather than to the fixed point equation u m u in section 2 we apply this new index in order to show that if c is a closed convex subset of k then under suitable assumptions upon f the set of solutions of the inequality

pullback attractors of a multi valued process generated by parabolic differential equations with unbounded delays abstract the theory of nonautonomous multi valued dynamical systems and a method of asymptotic compactness based on the concept of the kuratowski measure of the noncompactness of a bounded set are used to prove the existence of pullback attractors in the weighted space c v for the multi valued process associated with the nonautonomous parabolic differential equation with infinite delay for which the uniqueness of solutions need not hold

a selected pullback attractor abstract we introduce a new class of multivalued processes generated by generalized processes and a new concept of pullback attractor named a selected pullback attractor and we establish suitable conditions for the existence of such a selected pullback attractor finally we give and application to nonautonomous problems with dynamic boundary conditions

the critical forms of the attractors for semigroups and the existence of critical attractors the critical forms of the minimal closed global attractor m and the minimal closed global b attractor m for operator semigroups are given i e if the set x x x x attracts each point x in the phase space x then it must be the unique minimal closed global attractor m and if the set b b x x attracts each bounded set in x then it must be the unique minimal closed global b attractor m on this basis the sufficient conditions for the existence of m and m interrelation between m and m are discussed we introduce some classes of semigroups and study a variety of relations of them and the different classes of semigroups considered by ladyzhenskaya hale and temam when a semigroup9s continuity is not required it is proved that the semigroups with b acp introduced by us coincide with the asymptotically smooth semigroups the semigroups of class ak or b ak respectively and that if these semigroups are b bounded and x x x b then each of them possesses the unique non empty minimal invariant compact global attractor x x x x and the unique non empty minimal invariant compact global b attractor m b b x x m omega o where 0is a small number furthermore the sufficient conditions for m and m are given it is further proved that these conditions which guarantee the existence of the compact invariant global b attractor m are not only sufficient but also necessary so in order to guarantee a semigroup to have such attractor m the conditions which are put on the semigroup cannot be weakened again these results will be useful in studying the existence and construction of attractors

on n translation algebras abstract motivated by iyama s higher representation theory we introduce n translation quivers and n translation algebras the classical z q construction of the translation quiver is generalized to construct an n 1 translation quiver from an n translation quiver using trivial extension and smash product we find a sufficient and necessary condition for the quadratic dual of n translation algebras to have n almost split sequences in the category of its projective modules

a global attractor for a p x laplacian parabolic problem abstract in this work we study the asymptotic behavior of a p x laplacian parabolic problem of the form u t p x u b u in a bounded smooth domain in r n under dirichlet boundary conditions we prove under suitable assumptions the existence of a compact invariant global b attractor in l 2

nonzero solutions for a class of set valued variational inequalities in reflexive banach spaces in this paper we study the existence of nonzero solutions for a class of set valued variational inequalities involving set contractive mappings by using the fixed point index approach in reflexive banach spaces some new existence theorems of nonzero solutions for this class of set valued variational inequalities are established

returning arrows for self injective algebras and artin schelter regular algebras abstract we show that when constructing twisted trivial extensions for a graded self injective algebra the returning arrows appear in the quiver and in the meantime the complexity increases by one in koszul cases and the representation dimension also increases by one under certain additional conditions we apply our results to koszul artin schelter regular algebras and obtain a family of extensions of a koszul artin schelter regular algebra among them one is central extension and one is calabi yau

weak pullback attractors of setvalued processes weak pullback attractors are defined for nonautonomous setvalued processes and their existence and upper semicontinuous convergence under perturbation is established unlike strong pullback attractors invariance and pullback attraction here are required only for at least one trajectory rather than all trajectories at each starting point the concept is useful in for example continuous time control systems and is related to that of viability

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

pbw deformation theory and regular central extensions a deformation u of a graded k algebra a is said to be of pbw type if gru is a it has been shown for koszul and n koszul algebras that the deformation is pbw if and only if the relations of u satisfy a jacobi type condition in particular for these algebras the determination of the pbw property is a finite and explicitly determined linear algebra problem we extend these results to an arbitrary graded k algebra using the notion of central extensions of algebras and a homological constant attached to a which we call the complexity of a

long time behavior for a nonlinear parabolic problem with variable exponents abstract this paper addresses the question of the asymptotic behavior of solutions to the p x laplacian problem u t div u p x 2 u f x u g with general assumptions on f x u and the exponent p x we prove the existence of global attractors in proper spaces then we consider the fractal dimension of global attractors for the problem under suitable conditions we show that the problem admits an infinite dimensional global attractor

coverings and truncations of graded self injective algebras abstract let be a graded self injective algebra we describe its smash product k z with the group z its beilinson algebra and their relationship starting with we construct algebras with finite global dimension called slice algebras we show that their trivial extensions are all isomorphic and their repetitive algebras are the same as k z there exist mutations similar to the bgp reflections for the slice algebras

nonzero solutions for generalized variational inequalities by index the existence of nonzero solutions for a class of variational inequalities is studied by fixed point index approach for multivalued mappings in finite dimensional spaces and reflexive banach spaces

asymptotic behaviour of nonautonomous difference inclusions abstract nonautonomous difference inclusions are formulated as difference inclusion cocycles which generalize set valued semigroups corresponding to autonomous difference inclusions and the existence of a difference inclusion cocycle attractor is established under the assumption of upper semi continuity of the cocycle mappings if the difference inclusion cocycle has an absorbing set several illustrative examples are provided

graded selfinjective and koszul algebras abstract in this paper we continue our work on koszul algebras initiated in earlier studies the consideration about the existence of almost split sequences for koszul modules appeared in our early work and only a partial answer is known koszul duality relates finite dimensional algebras of infinite global dimension with infinite dimensional algebras such that the injective dimension of the graded simples is finite two questions naturally arise the existence of almost split sequences for infinite dimensional graded algebras and the relations between the global dimension of a graded algebra and the injective dimension of the graded simples in the first part of the paper we deal with the first matter in the second we consider the relations between the graded global dimension and the maximum of the injective dimensions of the graded simples proving that in the noetherian case they coincide in the third part we consider finite dimensional graded selfinjective quiver algebras and prove that for such algebras all indecomposable projective modules have the same lowey length in the forth and last part of the paper we specialize to selfinjective koszul algebras and characterize their yoneda algebras proving they constitute non commutative versions of the commutative regular algebras