the brauer group of irreducible coalgebras abstract in this paper we solve a previously formulated conjecture b torrecillas f van oystaeyen and y h zhang j algebra 177 1995 568 that is for a cocommutative irreducible coalgebra c the homomorphism br c br c is injective the proof uses morita takeuchi theory and the linear topology of all closed cofinite left ideals in c as an inmediate consequence br c is a torsion group some cases where the map is an isomorphism are studied it is also deduded from the main result that the inclusion of the coradical c 0 into c induces a monomorphism i br c br c 0 new examples of brauer groups of cocommutative coalgebras may be given using this fact

second order linear plate theories partial differential equations stress resultants and displacements abstract by combining the uniform approximation technique with the pseudo reduction approach a consistent second order plate theory is developed with recourse to neither kinematical assumptions nor to shear correction factors by truncation of the elastic energy the governing partial differential equations and the expressions for the stress resultants are compared with those of other authors free coefficients of the resulting displacement ansatz are determined a posteriori in order to satisfy three dimensional boundary conditions and local equilibrium equations

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

on exact rod beam shaft theories and the coupling among them due to arbitrary material anisotropies abstract we prove the existence of an exact one dimensional representation of the three dimensional theory of linear elasticity for a structural member with constant two fold symmetric cross section without the use of any a priori assumptions we show that the general problem of three dimensional elasticity decouples into four independent one dimensional subproblems for isotropic material a rod a shaft and two orthogonal beam problems a unique decomposition of any three dimensional load case with respect to the direction and the symmetries of the load is introduced it allows us to identify each part of the decomposition as a driving force for one of the four one dimensional subproblems furthermore we show how the coupling behavior of the four subproblems can be derived directly from the sparsity scheme of the stiffness tensor for general anisotropic materials

on sparse hard sets for counting classes abstract in this paper we study one word decreasing self reducible sets which are introduced by lozano and toran 1991 these are the usual self reducible sets with the peculiarity that the self reducibility machine makes at most one query and this is lexicographically smaller than the input we show first that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one word decreasing self reducible using this fact we can prove that for any class k chosen from pp np c p mod2p mod3 p it holds that 1 if there is a sparse pbtt hard set for k then k p and 2 if there is a sparse snbtt hard set for k then k np co np this generalizes the result of ogiwara and watanabe 1991 to the mentioned complexity classes

a comparison of polynomial time completeness notions ladner lynch and selman 1975 showed the differences among the power of several types of polynomial time reductions for dext c 0dtime 2cn but questions concerning complete degrees w r t these reductions have remained open here we show in dext almost all the possible differences between the completeness notions w r t any pair of the following reductions pt ptt pp n tt pc pd pbtt pk tt pm differences between several types of truth table completenesses we introduce the notion f easy subset for each pr reduction we first prove the property that all pr complete sets for dext have f easy subsets where the class f depends on each reduction type r then the construction of a ps complete set which is not pr complete becomes a simple diagonalization argument these constructions also work for all deterministic superpolynomial complexity classes which have a pm complete set in order to construct a set which is pt complete for dext but not ptt complete for dext we use strings which are not polynomially producible from small sized inputs these strings are said to have large generalized kolmogorov complexity cf hartmanis 1983

wide morita contexts relative injectivity and equivalence results abstract we extend morita theory to abelian categories by using wide morita contexts several equivalence results are given for wide morita contexts between abelian categories widely extending equivalence theorems for categories of modules and comodules due to kato muller and berbec in the case of grothendieck categories we derive equivalence results by using quotient categories we apply the general equivalence results to rings with identity rings with local units graded rings doi hopf modules and coalgebras

finiteness conditions co frobenius hopf algebras and quantum groups abstract some finiteness conditions for infinite dimensional coalgebras particularly right or left semiperfect coalgebras or co frobenius hopf algebras are studied as well examples of co frobenius hopf algebras are constructed via a hopf algebra structure on an ore extension of a group algebra and it is shown that bicrossproducts of co frobenius hopf algebras are themselves co frobenius

duality theorems for graded algebras and coalgebras the concepts of smash product and smash coproduct have been proved to be extremely useful tools in the theory of graded rings and graded coalgebras and more recently in the theory of action and coactions of a hopf algebra on an algebra or on a coalgebra once these concepts were defined a natural question appeared how can the smash product resp the smash coproduct be related to a matrix algebra resp to a matrix w x coalgebra the answer is given by the duality theorems proved in 3 for w x graded rings and in 5 for graded coalgebras various duality theorems w x w x for algebras have been proved during the last ten years cf 2 or 10 the purpose of this paper is to study duality for the smash product resp the smash coproduct associated to a g set and a g graded ring resp a g graded coalgebra the mentioned smash product and smash coproduct w x w x w x have been constructed in 10 and 6 it is shown by an example in 12 that for a g graded ring r and a left g set a we do not have a duality

the brauer group of a cocommutative coalgebra we construct a mayer vietoris type exact sequence for the brauer group of a cocommutative irreducible coalgebra c as an application the brauer group of c and its universal connected coalgebra r c are related

a theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity the energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by g kirchhoff in 1850 we show that it arises as a 0 limit of three dimensional nonlinear elasticity theory as the thickness of a plate goes to zero a key ingredient in the proof is a sharp rigidity estimate for maps v v u r n u r n we show that the l 2 distance of rv from a single rotation matrix is bounded by a multiple of the l 2 distance from

a priori estimation of the systematic error of consistently derived theories for thin structures abstract the consistent approximation approach is a dimension reduction technique for the derivation of hierarchies of lower dimensional analytical theories for thin structural members from the three dimensional theory of elasticity it is based on the uniform truncation of the potential energy after a certain power of a geometric scaling factor that describes the relative thinness of the structural member the truncation power defines the approximation order n of the resulting theory in the contribution we deal with the exemplary case of a slender beam like continuum with rectangular cross section we introduce an extension of the consistent approximation approach towards a simultaneous uniform truncation of the complementary energy we show that compatible displacement boundary conditions can be derived from the euler lagrange equations of the complementary energy whereas field equations and stress boundary conditions follow from the euler lagrange equations of the potential energy thus fully defined boundary value problems are obtained for all orders of approximation n we provide an a priori error estimate for the systematic error of the nth order theory it states that the norm difference of the solution of the so derived boundary value problem of order n to the exact solution of the three dimensional theory of elasticity declines like the geometric scaling factor to the power of n 1 thus the approximation character of the approach is proved furthermore we outline why the approach is to prefer over the in engineering mostly used approach of a fixed displacement field ansatz with unknown coefficients we show that the later approach leads in general to theories of higher complexity without increasing the approximation accuracy compared to a consistent approximation

galois correspondences for hopf bigalois extensions abstract lethbe a hopf algebra with bijective antipode over a commutative ringk a righth galois extension ofkis a righth comodule algebraasuch thatk aco hand a certain canonical mapa a a his a bijection we investigate galois connections for hopf galois extensions that can be formulated with the help of an additional hopf algebralover whichais also a leftl galois extension ofk and anl h bicomodule such an additional hopf algebra always exists and is unique up to isomorphism the galois connection between quotient coalgebras and left modules oflandh subcomodule algebras ofainduces a bijection between those quotients over whichlis faithfully coflat and those subalgebras over whichais faithfully flat as a consequence the lattices of hopf subalgebras oflandhover which these are faithfully flat are isomorphic

exponent reduction for projective schur algebras abstract in this paper it is proved that the exponent reduction property holds for all projective schur algebras this was proved in an earlier paper of the authors for a special class the radical abelian algebras the precise statement is as follows let a be a projective schur algebra over a field k and let k denote the maximal cyclotomic extension of k if m is the exponent of a k k then k contains a primitive m th root of unity one corollary of this result is a negative answer to the question of whether or not the projective schur group ps k is always equal to br l k where l is the composite of the maximal cyclotomic extension of k and the maximal kummer extension of k a second consequence is a proof of the brauer witt analogue in characteristic p if char k p 0 then every projective schur algebra over k is brauer equivalent to a radical abelian algebra

exponent reduction for radical abelian algebras abstract let k be a field a radical abelian algebra over k is a crossed product k k where k k t is a radical abelian extension of k t is a subgroup of k which is finite modulo k and h 2 g k is represented by a cocycle with values in t the main result is that if a is a radical abelian algebra over k and m exp a k k where denotes the group of all roots of unity then k contains the m th roots of unity applications are given to projective schur division algebras and projective schur algebras of nilpotent type

the brauer group of a braided monoidal category representation theory of finite groups prompted the introduction of the brauer group of a field and also of its schur subgroups the theory of quadratic forms and spaces introduced the zr2z graded variant of the brauer group together with the typical clifford algebras and related w x representations cf the work of wall 29 the extension of the zr2zgraded theory to gradings by other cyclic groups and consequently to abelian groups replacing the usual clifford algebras by the so called generalized clifford algebras is then natural from the algebraic point of view however extension to nonabelian groups did not seem to be possible at least not by simple modifications of the abelian theory at the same time the generalization of the brauer group of graded algebras obtained w x by long 12 13 made use of hopf algebras but again cocommutativity conditions were present the recent interest in quantum groups motivated the authors to introduce the brauer group of a quantum group as the brauer group of crossed module algebras also called quantum yang w x baxter module algebras in 4 5 the use of the category of crossed w x modules originated in the case of group algebras cf whitehead 30 module algebras or coalgebras of this crossed type were introduced by w x radford in 24 and used by majid in the description of modules over drinfel d quantum doubles the noncocommutativity aspects of the theory concerning the brauer group in terms of crossed module algebras do not have a counterpart in classical theory note that a first unifying categorical theory has been w x obtained by pareigis in 23 this theory deals with the brauer group of a

on sets truth table reducible to sparse sets we study sets that are truth table reducible to sparse sets in polynomial time the principal results are as follows 1 for every integer k 0 there is a set l and a sparse set s such that l

distinguishing conjunctive and disjunctive reducibilities by sparse sets abstract various polynomial time truth table reducibilities are compared by their ability of using sparse oracles to answer queries the reducibilities studied here include conjunctive reducibility bounded conjunctive reducibility disjunctive reducibility bounded disjunctive reducibility truth table reducibility and bounded truth table reducibility for any two reducibilities r p and s p we compare the class of sets r p reducible to sparse sets with the class of sets s p reducible to sparse sets for most pairs of reducibilities r p and s p it is shown that the two associated reduction classes are incomparable unless a trivial inclusive relation holds

bounds on orders of finite subgroups ofpgln k abstract we characterize the pairs k n k a field n a positive integer for which there is a bound on the orders of finite subgroups of pgl n k explicit bounds are given in important cases an application is made to the analogous problem for central simple k algebras of degree n

projective schur algebras of nilpotent type are brauer equivalent to radical algebras let k be a field and g a finite group for any element h g k k denotes the units of k here as a trivial g module we can form the twisted group algebra kg it is isomorphic to the group algebra as a vector space and its multiplication satisfies the rule xu yu xyf u where f g g k is a 2 cocycle representing it is easily checked that the 2 cocycle condition satisfied by f insures the associativity of the multiplication in kg a projective schur algebra over k is by definition a k central simple algebra which is a homomorphic image of a twisted group algebra kg for some finite group g and some h g k equivalently a k central simple algebra a is a projective schur algebra over k if it contains a subgroup of the units of a which is finite modulo k and spans a as k vector space we often use the notation k clearly a projective schur algebra over k determines an element in the brauer group br k of the field k the subgroup in br k generated by and in fact consisting of such elements is called the projective schur group of k and denoted by ps k the notions of projective schur algebra and projective schur group were introduced in 6 by lorenz and opolka this construction can be viewed as the projective analogue of schur algebras and the schur group denoted by s k of the field k see 10 it is natural to restrict attention to special types of groups g our interest here is in k central simple algebras that are homomorphic images of twisted group algebras kg with g nilpotent resp abelian see 6 we call these algebras projective schur algebras of nilpotent type resp abelian type clearly as above these algebras represent classes in br k and we denote by pnil k resp pab k the subgroup they generate in 6 it is shown that if k is a number field and n is the number of roots of unity in k then pab k nbr k where nb denotes the n torsion subgroup of b more generally since a symbol algebra of degree n over k is a homomorphic

relativizing complexity classes with sparse oracles baker gill and solovay constructed sparse sets a and b such that p a np a and np b co np b in contrast to their results we prove that p np if and only if for every tally language t p t np t and that np co np if and only if for every tally language t np t co np t we show that the polynomial hierarchy collapses if and only if there is a sparse set s such that the polynomial hierarchy relative to s collapses similar results are obtained for several other complexity classes

on the hopf schur group of a field let k be any field we consider the hopf schur group of k defined as the subgroup of the brauer group of k consisting of classes that may be represented by homomorphic images of finite dimensional hopf algebras over k we show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative finite dimensional hopf algebras over k respectively as a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a finite dimensional hopf algebra over k revealing so that the hopf schur group can be much larger than the schur group of k

on the projective schur group of a field abstract if k is a field the projective schur group ps k of k is the subgroup of the brauer group br k consisting of those classes which contain a projective schur algebra i e a homomorphic image of a twisted group algebra k g with g finite h 2 g k it has been conjectured by nelis and van oystaeyen j algebra 137 1991 501 518 that ps k br k for all fields k we disprove this conjecture by showing that ps k br k for rational function fields k 0 x where k 0 is any infinite field which is finitely generated over its prime field

a refined nonlinear theory of plates with transverse shear deformation abstract a higher order shear deformation theory of plates accounting for the von karman strains is presented the theory contains the same dependent unknowns as in the hencky mindlin type first order shear deformation theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate exact solutions of simply supported plates are obtained using the linear theory and the results are compared with the exact solutions of 3 d elasticity theory the first order shear deformation theory and the classical plate theory the present theory predicts the deflections stresses and frequencies more accurately when compared to the first order theory and the classical plate theory