on the asymptotic exactness of error estimators for linear triangular finite elements this paper deals with a posteriori error estimates for piecewise linear finite element approximations of elliptic problems we analyze two estimators based on recovery operators for the gradient of the approximate solution by using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution one of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation this estimator is equivalent to the error in the energy norm under rather general conditions however we show that for the asymptotic exactness the regularity assumption on the mesh is not merely technical while doing this we analyze the relation between superconvergence and asymptotic exactness for some particular examples

weak c cleft extensions weak entwining structures and weak hopf algebras abstract we formulate the concept of weak cleft extension for a weak entwining structure in a braided monoidal category c with equalizers and coequalizers we prove that if a is a weak c cleft extension then there is an isomorphism of algebras between a and a subobject of the tensor product of a c and c where a c is a subalgebra of a also we prove the corresponding dual results and linking the information of this two parts we obtain a general property for a pair morphisms f c a and g a c of algebras and coalgebras satisfying certain conditions finally as particular instances we get the results of fernandez and rodriguez the theorems of radford majid and bespalov in the case of hopf algebras with projection and the ones obtained by alonso and gonzalez for weak hopf algebras living in a symmetric category with split idempotents for example the weak theorem of blattner cohen and montgomery for weak hopf algebras with coalgebra splitting is one of them

piecewise polynomial interpolation in muckenhoupt weighted sobolev spaces and applications we develop a constructive piecewise polynomial approximation theory in weighted sobolev spaces with muckenhoupt weights for any polynomial degree the main ingredients to derive optimal error estimates for an averaged taylor polynomial are a suitable weighted poincare inequality a cancellation property and a simple induction argument we also construct a quasi interpolation operator built on local averages over stars which is well defined for functions in l 1 l1 we derive optimal error estimates for any polynomial degree on simplicial shape regular meshes on rectangular meshes these estimates are valid under the condition that neighboring elements have comparable size which yields optimal anisotropic error estimates over n n rectangular domains the interpolation theory extends to cases when the error and function regularity require different weights we conclude with three applications nonuniform elliptic boundary value problems elliptic problems with singular sources and fractional powers of elliptic operators

a space time trefftz discontinuous galerkin method for the acoustic wave equation in first order formulation we introduce a space time trefftz discontinuous galerkin method for the first order transient acoustic wave equations in arbitrary space dimensions extending the one dimensional scheme of kretzschmar et al ima j numer anal 36 1599 1635 2016 test and trial discrete functions are space time piecewise polynomial solutions of the wave equations we prove well posedness and a priori error bounds in both skeleton based and mesh independent norms the space time formulation corresponds to an implicit time stepping scheme if posed on meshes partitioned in time slabs or to an explicit scheme if posed on tent pitched meshes we describe two trefftz polynomial discrete spaces introduce bases for them and prove optimal high order h convergence bounds

an exact sequence interpretation of the lie bracket in hochschild cohomology we give an interpretation of the lie bracket and the divided squaring operations in the hochschild cohomology of an associative algebra in terms of exact sequences of bimodules we construct natural loops in the category of extensions whose homotopy classes represent the lie bracket and squaring operations 1991 ams math subj class 16e40

drinfeld center of enriched monoidal categories abstract we define the drinfeld center of a monoidal category enriched over a braided monoidal category and show that every modular tensor category can be realized in a canonical way as the drinfeld center of a self enriched monoidal category we also give a generalization of this result for important applications in physics

global hochschild co homology of singular spaces abstract we introduce hochschild co homology of morphisms of schemes or analytic spaces and study its fundamental properties in analogy with the cotangent complex we introduce the so called derived hochschild complex of a morphism the hochschild cohomology and homology groups are then the ext and tor groups of that complex we prove that these objects are well defined extend the known cases and have the expected functorial and homological properties such as graded commutativity of hochschild cohomology and existence of the characteristic homomorphism from hochschild cohomology to the graded centre of the derived category

jackson type inequalities for h p clouds and error estimates abstract the interest in meshfree methods for solving boundary value problems has grown rapidly in recent years a meshless method that has attracted much interest in the community of computational mechanics is the h p clouds method for this kind of applications it is fundamental to analyze the orders of approximation in this paper we prove jackson type inequalities for h p cloud functions these inequalities set up a general framework for the theoretical analysis of high order error estimates of the h p clouds method with the same remarkable features of finite element theory

exact sequences hochschild cohomology and the lie module structure over the m relative center abstract in this article we present actions by central elements on hochschild cohomology groups with arbitrary bimodule coefficients as well as an interpretation of these actions in terms of exact sequences since our construction utilizes the monoidal structure that the category of bimodules possesses we will further recognize that these actions are compatible with monoidal functors and thus as a consequence are invariant under morita equivalences by specializing the bimodule coefficients to the underlying algebra itself our efforts in particular yield a description of the degree n 0 part of the lie bracket in hochschild cohomology and thereby close a gap in earlier work by s schwede

support varieties and hochschild cohomology rings we define a support variety for a finitely generated module over an artin algebra lambda over a commutative artinian ring k with lambda flat as a module over k in terms of the maximal ideal spectrum of the algebra rm hh lambda of lambda this is modelled on what is done in modular representation theory and the varieties defined in this way are shown to have many of the same properties as those for group rings in fact the notions of a variety in our sense and those for principal and non principal blocks are related by a finite surjective map of varieties for a finite dimensional self injective algebra over a field the variety is shown to be an invariant of the stable component of the auslander reiten quiver moreover we give information on nilpotent elements in rm hh lambda give a thorough discussion of the ring rm hh lambda on a class of nakayama algebras give a brief discussion of a possible notion of complexity and make a comparison with support varieties for complete intersections

modified taylor reproducing formulas and h p clouds we study two different approximations of a multivariate function f by operators of the form n i 1 r f x i wi x wi x where w i is an m reproducing partition of unity and t f x i x are modified taylor polynomials of degree r expanded at x i the first approximation was introduced by xuli 2003 in the univariate case and generalized for convex domains by guessab et al 2005 the second one was introduced by duarte 1995 and proved in the univariate case in this paper we first relax the guessab s convexity assumption and we prove duarte s reproduction formula in the multivariate case then we introduce two related reproducing quasi interpolation operators in sobolev spaces a weighted error estimate and jackson s type inequalities for h p cloud function spaces are obtained last numerical examples are analyzed to show the approximative power of the method

piecewise solenoidal vector fields and the stokes problem nonconforming finite element approximations to solutions of the stokes equations are constructed optimal rates of convergence are proved for the velocity and pressure approximations for the pressure approximation c 0 piecewise polynomial functions are used the class of vector fields used to approximate the velocity field have piecewise polynomial components discontinuous across interelement boundaries on each triangle these vector fields satisfy the incompressibility condition pointwise it is shown that these piecewise solenoidal vector fields possess optimal approximation properties to smooth solenoidal vector fields on domains with curved boundaries

milnor moore categories and monadic decomposition abstract in this paper monoidal hom lie algebras lie color algebras lie superalgebras and other type of generalized lie algebras are recovered by means of an iterated construction known as monadic decomposition of functors which is based on eilenberg moore categories to this aim we introduce the notion of milnor moore category as a monoidal category for which a milnor moore type theorem holds we also show how to lift the property of being a milnor moore category whenever a suitable monoidal functor is given and we apply this technique to provide examples

idempotents and one sided units lattice invariants and a semigroup of functors on the category of monoids abstract for a monoid m we denote by g m the group of units e m the submonoid generated by the idempotents and g l m and g r m the submonoids consisting of all left or right units writing m for the monoidal category of monoids g e g l and g r are all monoidal functors m m there are other natural functors associated to submonoids generated by combinations of idempotents and one or two sided units the above functors generate a monoid with composition as its operation we show that this monoid has size 15 and describe its algebraic structure we also show how to associate certain lattice invariants to a monoid and classify the lattices that arise in this fashion a number of examples are discussed throughout some of which are essential for the proofs of the main theoretical results

an algorithmic proof theory for hypergeometric ordinary and q multisum integral identities it is shown that every proper hypergeometric multisum integral identity orq identity with a fixed number of summations and or integration signs possesses a short computer constructible proof we give a fast algorithm for finding such proofs most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here we give many examples of the method including computer generated proofs of identities of mehta dyson selberg hille hardy q saalschutz and others the prospect of using the method for proving multivariate identities that involve an arbitrary number of summations integrations is discussed

reflection fusion categories abstract we introduce the notion of a reflection fusion category which is a type of a g crossed category generated by objects of frobenius perron dimension 1 and p where p is an odd prime we show that such categories correspond to orthogonal reflection groups over f p this allows us to use the known classification of irreducible reflection groups over finite fields to classify irreducible reflection fusion categories

finite element differential forms on curvilinear cubic meshes and their approximation properties we study the approximation properties of a wide class of finite element differential forms on curvilinear cubic meshes in n n dimensions specifically we consider meshes in which each element is the image of a cubical reference element under a diffeomorphism and finite element spaces in which the shape functions and degrees of freedom are obtained from the reference element by pullback of differential forms in the case where the diffeomorphisms from the reference element are all affine i e mesh consists of parallelotopes it is standard that the rate of convergence in l 2 l 2 exceeds by one the degree of the largest full polynomial space contained in the reference space of shape functions when the diffeomorphism is multilinear the rate of convergence for the same space of reference shape function may degrade severely the more so when the form degree is larger the main result of the paper gives a sufficient condition on the reference shape functions to obtain a given rate of convergence

algebra extensions and nonsingularity this paper is concerned with a notion of nonsingularity for noncommutative algebras which arises naturally in connection with cyclic homology let us consider associative unital algebras over the complex numbers we call an algebra a quasi free when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism a r i where i is a nilpotent ideal in r can be lifted to a homomorphism a r if we restrict to the category of finitely generated commutative algebras then this lifting property characterizes smooth algebras the ones corresponding to nonsingular affine varieties in this way quasi free algebras appear as noncommutative analogues of smooth algebras stretching the analogy we can even regard quasi free algebras as analogues of manifolds one of the aims of this paper is to develop the analogy further by showing that quasi free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds to give an example let us consider the analogue of an embedding an extension a r i where a and r are quasi free algebras playing the role of the submanifold and ambient manifold respectively in the manifold situation i i2 is the module of linear functions on the nor2 mal bundle and the symmetric algebra sa iii is the algebra of polynomial functions now in passing from commutative to noncommutative algebras the symmetric algebra of a module is replaced by the tensor algebra of a bimodule

injective hopf bimodules cohomologies of infinite dimensional hopf algebras and graded commutativity of the yoneda product abstract we prove that the category of hopf bimodules over any hopf algebra has enough injectives which enables us to extend some results on the unification of hopf bimodule cohomologies of r taillefer phd thesis 2001 arxiv preprint math qa 0005019 to the infinite dimensional case we also prove that the cup product defined on these cohomologies is graded commutative unlike the algebra case see s schwede j reine angew math 498 1998 153 172 these methods do not give a non trivial gerstenhaber algebra structure on the cohomology we consider we also comment that the other approach to finding such a structure that we know of see m farinati a solotar arxiv preprint math kt 0207243 also gives a trivial gerstenhaber algebra structure

differential graded categories and deligne conjecture abstract we prove a version of the deligne conjecture for n fold monoidal abelian categories a over a field k of characteristic 0 assuming some compatibility and non degeneracy conditions for a the output of our construction is a weak leinster n 1 algebra over k a relaxed version of the concept of leinster n algebra in a lg k the difference between the leinster original definition and our relaxed one is apparent when n 1 for n 1 both concepts coincide we believe that there exists a functor from weak leinster n 1 algebras over k to c e n 1 k algebras well defined when k q and preserving weak equivalences for the case n 1 such a functor is constructed in 31 by elementary simplicial methods providing together with this paper a complete solution for 1 monoidal abelian categories our approach to deligne conjecture is divided into two parts the first part completed in the present paper provides a construction of a weak leinster n 1 algebra over k out of an n fold monoidal k linear abelian category provided the compatibility and non degeneracy condition are fulfilled the second part still open for n 1 is a passage from weak leinster n 1 algebras to c e n 1 k algebras as an application we prove in theorem 8 1 that the gerstenhaber schack complex of a hopf algebra over a field k of characteristic 0 admits a structure of a weak leinster 2 1 algebra over k extending the yoneda structure it relies on our earlier construction 30 of a 2 fold monoidal structure on the abelian category of tetramodules over a bialgebra

error estimates for 3 d narrow finite elements we obtain error estimates for finite element approximations of the lowest degree valid uniformly for a class of three dimensional narrow elements first for the lagrange interpolation we prove optimal error estimates both in order and regularity in l p for p 2 for p 2 it is known that this result is not true applying extrapolation results we obtain an optimal order error estimate for functions sligthly more regular than h 2 these results are valid both for tetrahedral and rectangular elements second for the case of rectangular elements we obtain optimal in order and regularity error estimates for an average interpolation valid for functions in w 1 s p with 1 p and 0 s 1

on polynomial approximation in sobolev spaces we obtain new bounds for the error in the polynomial approximation in sobolev spaces in an open bounded set which is star shaped with respect to every point in a set of positive measure b subset omega this estimate follows by applying the hardy littlewood maximal function

group cohomology and gauge equivalence of some twisted quantum doubles the purpose of the present paper is to investigate relationships that exist between certain quasi hopf algebras namely twisted quantum doubles of a finite abelian group and their module categories this is closely related to aspects of group cohomology concerning the relationship of the bar complex to some other complexes defined and studied by eilenberg and maclane mac52 em50a and em50b although our results are purely homological and algebraic in nature the motivation for studying these questions derives mainly from connections with conformal field theory cft and vertex operator algebras dw90 dpr92 ms89 dl93 we will thus review background motivation although it is hardly necessary for an understanding of our main results and then give a more detailed overview of the paper it is not necessary for the reader to appreciate what a vertex operator algebra voa is cf bor86 flm88 fhl93 but only to understand this voas have a representation theory hence a linear module category and it is important to understand the nature of this category it is expected that the module category v mod for a voa v has the structure of a braided monoidal category js93 roughly this means that there is a notion of tensor product of v modules and that the tensor product of three modules is associative by an isomorphism that depends on the particular modules moreover tensor product of modules is commutative again via an isomorphism that depends on the particular modules particularly nice voas called rational have the additional property that v mod is a semisimple category so that v has only a finite number of isomorphism classes of simple modules and every object in v mod is a direct sum of simple objects the simplest voas of all from this perspective are called holomorphic they are rational and have but one simple module namely the adjoint module v itself thus if v is a holomorphic voa then v mod is equivalent as a semi simple braided monoidal category to the category of vector spaces

a two line algorithm for provingq hypergeometric identities abstract we show that q hypergeometric identities k f n k 1 can be proved by checking that they are correct for only finitely many n say values of n we give a specific a priori formula for n as a polynomial of degree 24 in the parameters of f n k we see this because of the presence of q the estimates of n can be made smaller than the general estimates that were found in the author s thesis 8 as an example of the method we show that the q vandermonde identity can be proved by only checking that its first 2358 cases i e values of n are correct by direct computation

a local regularization operator for triangular and quadrilateral finite elements this paper develops a local regularization operator on triangular or quadrilateral finite elements built on structured or unstructured meshes this operator is a variant of the regularization operator of clement however ours is constructed via a local projection in a reference domain we prove in this paper that it has the same optimal approximation properties as the standard interpolation operator and we present some applications

on zeilberger s algorithm and its q analogue abstract gosper s and zeilberger s algorithms for summation of terminating hypergeometric series as well as the q versions of these algorithms are described in a very rigorous way the paper is a companion to maple v procedures implementing these algorithms it concludes with the help information for these procedures

a modular invariant bulk theory for the c 0 triplet model a proposal for the bulk space of the logarithmic cal w 2 3 triplet model at central charge zero is made the construction is based on the idea that one may reconstruct the bulk theory of a rational conformal field theory from its boundary theory the resulting bulk space is a quotient of the direct sum of projective representations which is isomorphic as a vector space to the direct sum of tensor products of the irreducible representations with their projective covers as a consistency check of our analysis we show that the partition function of the bulk theory is modular invariant and that the boundary state analysis is compatible with the proposed annulus partition functions of this model

dg analogues of the zuckerman functors and the dual zuckerman functors i abstract we study the category of dg harish chandra modules over arbitrary commutative rings and generalize the induction functor the production functor the zuckerman functor and the dual zuckerman functor

on the lie algebra structure of the first hochschild cohomology of gentle algebras and brauer graph algebras abstract in this paper we determine the first hochschild homology and cohomology with different coefficients for gentle algebras and we give a geometrical interpretation of these co homologies using the ribbon graph of a gentle algebra as defined in 32 we give an explicit description of the lie algebra structure of the first hochschild cohomology of gentle and brauer graph algebras with multiplicity one based on trivial extensions of gentle algebras and we show how the hochschild cohomology is encoded in the brauer graph in particular we show that except in one low dimensional case the resulting lie algebras are all solvable

homological epimorphisms recollements and hochschild cohomology with a conjecture by snashall solberg in view abstract we show that recollements of module categories give rise to homomorphisms between the associated hochschild cohomology algebras which preserve the strict gerstenhaber structure i e the cup product the graded lie bracket and the squaring map we review various long exact sequences in hochschild cohomology and apply our results in order to realise that the occurring maps preserve the strict gerstenhaber structure as well as a byproduct we generalise a known long exact cohomology sequence of koenig nagase to arbitrary surjective homological epimorphisms we use our observations to motivate and formulate a variation of the finite generation conjecture by snashall solberg

infinite loop spaces and coherence for symmetric monoidal bicategories abstract this paper proves three different coherence theorems for symmetric monoidal bicategories first we show that in a free symmetric monoidal bicategory every diagram of 2 cells commutes second we show that this implies that the free symmetric monoidal bicategory on one object is equivalent as a symmetric monoidal bicategory to the discrete symmetric monoidal bicategory given by the disjoint union of the symmetric groups third we show that every symmetric monoidal bicategory is equivalent to a strict one we give two topological applications of these coherence results first we show that the classifying space of a symmetric monoidal bicategory can be equipped with an e structure second we show that the fundamental 2 groupoid of an e n space n 4 has a symmetric monoidal structure these calculations also show that the fundamental 2 groupoid of an e 3 space has a sylleptic monoidal structure