hascasl integrated higher order specification and program development we lay out the design of hascasl a higher order extension of the algebraic specification language casl that serves both as a wide spectrum language for the rigorous specification and development of software in particular but not exclusively in modern functional programming languages and as an expressive standard language for higher order logic distinctive features of hascasl include partial higher order functions higher order subtyping shallow polymorphism and an extensive type class mechanism moreover hascasl provides dedicated specification support for monad based functional imperative programming with generic side effects including a monad based generic hoare logic

a functorial semantics for multi algebras and partial algebras with applications to syntax multi algebras allow for the modelling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results we propose a functorial presentation of various categories of multi algebras and partial algebras analogous to the classical presentation of algebras over a signature as cartesian functors from the algebraic theory over to set we introduce two different notions of theory over a signature both having a structure weaker than cartesian and we consider functors from them to rel or pfn the categories of sets and relations or partial functions respectively next we discuss how the functorial presentation provides guidelines when choosing syntactical notions for a class of algebras and as an application we argue that the natural generalization of usual terms are conditioned terms for partial algebras and term graphs for multi algebras

impact of element level static condensation on iterative solver performance this paper provides theoretical estimates that quantify and clarify the savings associated to the use of element level static condensation as a first step of an iterative solver these estimates are verified numerically the numerical evidence shows that static condensation at the element level is beneficial for higher order methods for lower order methods or when the number of iterations required for convergence is low the setup cost of the elimination as well as its implementation may offset the benefits obtained during the iteration process however as the iteration count e g above 50 or the polynomial order e g above cubics grows the benefits of element level static condensation are significant

recovery based error estimation and adaptivity using high order splines over hierarchical t meshes abstract an adaptive higher order method based on a generalization of polynomial rational splines over hierarchical t meshes pht rht splines is introduced while most problems considered in isogeometric analysis can be solved efficiently when the solution is smooth many non trivial simulations have rough solutions this can be caused for example by the presence of re entrant corners in the domain for such problems a tensor product basis is less suitable for resolving the singularities that appear as refinement propagates throughout the computational domain hierarchical bases and adaptivity allow for a more efficient way of dealing with singularities by adding more degrees of freedom only where they are necessary to improve the approximation in order to drive the adaptive refinement an efficient recovery based error estimator is proposed in this work the estimator produces a recovered solution which is a more accurate approximation than the computed numerical solution several 2d and 3d numerical investigations with pht splines of higher order and greater continuity show good performance compared to uniform refinement in terms of degrees of freedom and computational cost

spectral approximation properties of isogeometric analysis with variable continuity abstract we study the spectral approximation properties of isogeometric analysis with local continuity reduction of the basis such continuity reduction results in a decrement in the interconnection between the degrees of freedom of the mesh which allows for large computational savings during the solution of the resulting linear system the continuity reduction results in extra degrees of freedom that modify the approximation properties of the method the convergence rate of such refined isogeometric analysis is equivalent to that of the maximum continuity basis we show how the breaks in continuity and inhomogeneity of the basis lead to artifacts in the frequency spectra such as stopping bands and outliers and present a unified description of these effects in finite element method isogeometric analysis and refined isogeometric analysis accuracy of the refined isogeometric analysis approximations can be improved by using non standard quadrature rules in particular optimal quadrature rules lead to large reductions in the eigenvalue errors and yield two extra orders of convergence as it occurs in standard isogeometric analysis

boolean restriction categories and taut monads a boolean category is a restriction category if and only if it has one exception and all morphisms are deterministic in the category of sets taut monads are precisely the boolean ones it follows that collection monad types in haskell inherit an assertion calculus based on dynamic logic

conformal blocks and generalized theta functions let yp be the moduli space of rank r vector bundles with trivial determinant on a riemann surface x this space carries a natural line bundle the determinant line bundle s we describe a canonical isomorphism of the space of global sections of sk with the space of conformal blocks defined in terms of representations of the lie algebra 5 c z it follows in particular that the dimension of h yk is given by the verlinde formula

algebraic theories of quasivarieties abstract analogously to the fact that lawvere s algebraic theories of finitary varieties are precisely the small categories with finite products we prove that i algebraic theories of many sorted quasivarieties are precisely the small left exact categories with enough regular injectives and ii algebraic theories of many sorted horn classes are precisely the small left exact categories with enough m injectives where m is a class of monomorphisms closed under finite products and containing all regular monomorphisms we also present a gabriel ulmer type duality theory for quasivarieties and horn classes

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

infinite grassmannians and moduli spaces of g bundles let c be a smooth projective irreducible algebraic curve over c of any genus and g a connected simply connected simple affine algebraic group over c in this paper we elucidate the relationship between 1 the space of vacua conformal blocks defined in conformal field theory using an integrable highest weight representation of the affine kac moody algebra associated to g and 2 the space of regular sections generalised theta functions of a line bundle on the moduli space ta of semistable principal g bundles on c fix a point p in c and let p resp kp be the completion of the local ring p of c at p resp the quotient field of c p let g kp the kp rational points of the algebraic group g be the loop group of g and let g p be the standard maximal parahoric subgroup of then the generalised flag variety x n is an inductive limit of projective varieties in fact of generalised schubert varieties one has a natural r line bundle s on x cf sect 2 2 and the picard group pic x is isomorphic to z which is generated by s proposition 2 3 where f is the universal central extension of by the multiplicative group c cf sect 2 2 by an analogue of the borel weil theorem proved in the kac moody setting by kumar and also by mathieu the space h 162 s of the regular sections of the line bundle y dxo y xo for any d 0 is canonically isomorphic with the full vector space dual v dxo of the integrable highest weight irreducible module v dxo of the affine kacmoody group or of the associated affine kac moody algebra which is a certain one dimensional central extension of the loop algebra lie g with highest weight dxo cf sect 6 1

the cohomology rings of moduli spaces of bundles over riemann surfaces the cohomology of the moduli space ji n d of semistable bundles of coprime rank n and degree d over a fixed riemann surface m of genus g 2 has been much studied over the last two decades and a great deal is known about it it is known that the cohomology has no torsion 2 the betti numbers have been computed using various different methods 2 5 11 and a set of generators for the cohomology ring is known 2 the purpose of this article is to provide a method of deciding when an expression in these generators is a relation and in particular when the rank n is two to prove two conjectures about the relations between the generators due to mumford and to newstead and ramanan newstead and ramanan conjectured that certain relations hold among the generators whereas mumford conjectured that a certain set of relations is a complete set in fact the conjecture of newstead and ramanan has two parts the first of which was proved by gieseker in 1982 4 the second part of their conjecture which is the part examined in this article was given an independent proof recently by thaddeus 19 who has a very different method of obtaining information about the relations in the cohomology ring the computation of the betti numbers of the moduli spaces i n d in 2 stimulated a general method of computing betti numbers of quotients in the sense of mumford s geometric invariant theory 14 of nonsingular complex projective varieties by complex reductive group actions see 8 9 10 similarly the ideas applied in this article to finding generators and relations for the cohomology rings of the moduli spaces i n d also lead to an analogous method of obtaining generators and relations for the rational cohomology rings of geometric invariant theoretic quotients the first section provides background material on the cohomology of the moduli spaces i n d in the second section mumford s conjecture is proved when the rank n is two the method could in principle be used to decide whether the conjecture is true for rank n 2 but appears to become significantly more complicated in the general case the third section gives a method for deciding when an expression in the generators is a relation and the fourth

on quantum gauge theories in two dimensions two dimensional quantum yang mills theory is studied from three points of view i by standard physical methods ii by relating it to the largek limit of three dimensional chern simons theory and two dimensional conformal field theory iii by relating its weak coupling limit to the theory of reidemeister ray singer torsion the results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces

semi free hamiltonian circle actions on 6 dimensional symplectic manifolds assume m is a connected compact 6 dimensional symplectic manifold equipped with a semi free hamiltonian circle action such that the fixed point set consists of isolated points or compact orientable surfaces we restrict attention to the case dim h 2 m 3 we give a complete list of the possible manifolds and determine their equivariant cohomology rings and equivariant chern classes some of these manifolds are classified up to diffeomorphism we also show the existence for a few cases

on the intersection theory of quot schemes and moduli of bundles with sections we consider a class of tautological top intersection products on the moduli space of stable pairs consisting of vector bundles together with n sections on a smooth complex curve c we show that when n is large these intersection numbers can be viewed as arising on the grothendieck quot scheme of coherent sheaf quotients of the rank n trivial sheaf on c the result has applications to the computation of the intersection theory of the moduli space of semistable bundles on c

dispersion optimized quadratures for isogeometric analysis abstract we develop and analyze quadrature blending schemes that minimize the dispersion error of isogeometric analysis up to polynomial order seven with maximum continuity in the span the schemes yield two extra orders of convergence superconvergence on the eigenvalue errors while the eigenfunction errors are of optimal convergence order both dispersion and spectrum analysis are unified in the form of a taylor expansion for eigenvalue errors the resulting schemes increase the accuracy and robustness of isogeometric analysis for wave propagation as well as the differential eigenvalue problems we also derive an a posteriori error estimator for the eigenvalue error based on the superconvergence result we verify with numerical examples the analysis of the performance of the proposed schemes

an energy stable generalized α method for the swift hohenberg equation abstract we propose a second order accurate energy stable time integration method that controls the evolution of numerical instabilities introducing numerical dissipation in the highest resolved frequencies our algorithm further extends the generalized method and provides control over dissipation via the spectral radius we derive the first and second laws of thermodynamics for the swift hohenberg equation and provide a detailed proof of the unconditional energy stability of our algorithm finally we present numerical results to verify the energy stability and its second order accuracy in time

restriction categories ii partial map classification an algebraic characterization of monads which are abstract partial map classifiers is provided without the assumption that the categories of total maps possess products by an abstract partial map classifier we mean a monad whose kleisli category is a full subcategory of a partial map category wherein the induced comonad classifies partial maps in the usual sense a construction of the corresponding actual partial map classifier from an abstract one is described and conditions for an abstract partial map classifier to be a real one are provided the paper uses the notion of a restriction category developed in earlier work and the characterization of these as full subcategories of partial map categories

the cost of continuity performance of iterative solvers on isogeometric finite elements in this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized galerkin weak form specifically we compare performance of linear solvers when discretizing using c 0 b splines which span traditional finite element spaces and c p 1 b splines which represent maximum continuity we provide theoretical estimates for the increase in cost of the matrix vector product as well as for the construction and application of black box preconditioners we accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size h and polynomial order of approximation p in addition to the aforementioned continuity of the basis finally we present timing results for a range of preconditioning options for the laplace problem we conclude that the matrix vector product operation is at most left 33p 2 middle 8 right times more expensive for the more cont

intersection cohomology of quotients of nonsingular varieties let m p be a nonsingular projective variety acted on by a connected complex reductive group g kc via a homomorphism g gl n 1 which is the complexification of a homomorphism k u n 1 geometric invariant theory git gives us a recipe to form a quotient m m g of the set of semistable points and many interesting spaces in algebraic geometry are constructed in this manner mfk94 but often this quotient is singular and hence intersection cohomology with middle perversity is an important topological invariant the purpose of this paper is to present a way to compute the middle perversity intersection cohomology of the singular quotients the choice of an embedding m p provides us with a moment map for the k action and the morse stratification of m with respect to its norm square is kequivariantly perfect with the unique open dense stratum m kir84 hence one can compute the betti numbers for the equivariant cohomology h k m as well as the cup product at least in principle see also kir92 when g acts locally freely on m we get an orbifold m g m g and h k m h m g ih m g

fast isogeometric solvers for explicit dynamics in finite element analysis solving time dependent partial differential equations with explicit time marching schemes requires repeatedly applying the inverse of the mass matrix for mass matrices that can be expressed as tensor products of lower dimensional matrices we present a direct method that has linear computational complexity i e o n o n where nn is the total number of degrees of freedom in the system we refer to these matrices as separable matrices for non separable mass matrices we present a preconditioned conjugate gradient method with carefully designed preconditioners as an alternative we demonstrate that these preconditioners which are easy to construct and cheap to apply o n o n can deliver significant convergence acceleration the performances of these preconditioners are independent of the polynomial order p independence and mesh resolution h independence for maximum continuity b splines as verified by various numerical tests

a domain decomposition method for isogeometric multi patch problems with inexact local solvers abstract in isogeometric analysis the computational domain is often described as multi patch where each patch is given by a tensor product spline nurbs parametrization in this work we propose a feti like solver where local inexact solvers exploit the tensor product structure at the patch level to this purpose we extend to the isogeometric framework the so called all floating variant of feti that allows us to use the fast diagonalization method at the patch level we construct then a preconditioner for the whole system and prove its quasi robustness with respect to the local mesh size h and patch size h precisely the condition number of the preconditioned system is bounded by the square of the logarithm of h h our numerical tests confirm the theory and also show a favourable dependence of the computational cost of the method from the spline degree p

equivariant volumes of non compact quotients and instanton counting motivated by nekrasov s instanton counting we discuss a method for calculating equivariant volumes of non compact quotients in symplectic and hyper kahler geometry by means of the jeffrey kirwan residue formula of non abelian localization in order to overcome the non compactness we use varying symplectic cuts to reduce the problem to a compact setting and study what happens in the limit that recovers the original problem we implement this method for the adhm construction of the moduli spaces of framed yang mills instantons on mathbb r 4 and rederive the formulas for the equivariant volumes obtained earlier by nekrasov shadchin expressing these volumes as iterated residues of a single rational function

isogeometric bddc preconditioners with deluxe scaling a balancing domain decomposition by constraints bddc preconditioner with a novel scaling introduced by dohrmann for problems with more than one variable coefficient and here denoted as deluxe scaling is extended to isogeometric analysis of scalar elliptic problems this new scaling turns out to be more powerful than the standard rho and stiffness scalings considered in a previous isogeometric bddc study our h analysis shows that the condition number of the resulting deluxe bddc preconditioner is scalable with a quasi optimal polylogarithmic bound which is also independent of coefficient discontinuities across subdomain interfaces extensive numerical experiments support the theory and show that the deluxe scaling yields a remarkable improvement over the older scalings in particular for large isogeometric polynomial degree and high regularity

brst model for equivariant cohomology and representatives for the equivariant thom class in this paper the brst formalism for topological field theories is studied in a mathematical setting the brst operator is obtained as a member of a one parameter family of operators connecting the weil model and the cartan model for equivariant cohomology furthermore the brst operator is identified as the sum of an equivariant derivation and its fourier transform using this the mathai quillen representative for the thom class of associated vector bundles is obtained as the fourier transform of a simple brst closed element

isogeometric block feti dp preconditioners for the stokes and mixed linear elasticity systems abstract the aim of this work is to construct and analyze a feti dp type domain decomposition preconditioner for isogeometric discretizations of the stokes and mixed linear elasticity systems this method extends to the isogeometric analysis context the preconditioner previously proposed by tu and li 2015 for finite element discretizations of the stokes system the resulting isogeometric feti dp algorithm is proven to be scalable in the number of subdomains and has a quasi optimal convergence rate bound which is polylogarithmic in the ratio of subdomain and element sizes extensive two dimensional numerical experiments validate the theory investigate the behavior of the preconditioner with respect to both the spline polynomial degree and regularity and show its robustness with respect to domain deformation material incompressibility and presence of elastic coefficient discontinuities across subdomain interfaces

tail recursion through universal invariants abstract tail recursive constructions suggest a new semantics for datatypes which allows a direct match between specifications and tail recursive programs the semantics focuses on loops their fixpoints invariants and convergence convergent models of the natural numbers and lists are examined in detail and under very mild conditions are shown to be equivalent to the corresponding initial algebra models

on the computational efficiency of isogeometric methods for smooth elliptic problems using direct solvers summary we compare the computational efficiency of isogeometric galerkin and collocation methods for partial differential equations in the asymptotic regime we define a metric to identify when numerical experiments have reached this regime we then apply these ideas to analyze the performance of different isogeometric discretizations which encompass c0 finite element spaces and higher continuous spaces we derive convergence and cost estimates in terms of the total number of degrees of freedom and then perform an asymptotic numerical comparison of the efficiency of these methods applied to an elliptic problem these estimates are derived assuming that the underlying solution is smooth the full gauss quadrature is used in each non zero knot span and the numerical solution of the discrete system is found using a direct multi frontal solver we conclude that under the assumptions detailed in this paper higher continuous basis functions provide marginal benefits copyright 2014 john wiley sons ltd

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

unfolding semantics of graph transformation several attempts have been made of extending to graph grammars the unfolding semantics originally developed by winskel for safe petri nets but only partial results were obtained in this paper we fully extend winskel s approach to single pushout grammars providing them with a categorical concurrent semantics expressed as a coreflection between the category of semi weighted graph grammars and the category of prime algebraic domains which factorises through the category of occurrence grammars and the category of asymmetric event structures for general possibly nonsemi weighted single pushout grammars we define an analogous functorial concurrent semantics which however is not characterised as an adjunction similar results can be obtained for double pushout graph grammars under the assumptions that nodes are never deleted

partial morphisms in categories of effective objects abstract this paper is divided in two parts in the first we analyse in great generality data types in relation to partial morphisms we introduce partial function spaces partial cartesian closed categories and complete objects motivate their introduction and show some of their properties in the second part we define the partial cartesian closed category gen of generalized numbered sets prove that it is a good extension of the category of numbered sets and show how it is related to the recursive topos

the index formula for the moduli of g bundles on a curve we prove the formulae conjectured by the first author for the index of k theory classes over the moduli stack of algebraic g bundles on a smooth projective curve the formulae generalize e verlinde s for line bundles and have witten s integrals over the moduli space of stable bundles as their large level limits as an application we prove the newstead ramanan conjecture on the vanishing of high chern classes of certain moduli spaces of semi stable g bundles

a scalable block preconditioning strategy for divergence conforming b spline discretizations of the stokes problem abstract the recently introduced divergence conforming b spline discretizations allow the construction of smooth discrete velocity pressure pairs for viscous incompressible flows that are at the same time inf sup stable and pointwise divergence free when applied to the discretized stokes problem these spaces generate a symmetric and indefinite saddle point linear system the iterative method of choice to solve such system is the generalized minimum residual method this method lacks robustness and one remedy is to use preconditioners for linear systems of saddle point type a large family of preconditioners can be obtained by using a block factorization of the system in this paper we show how the nesting of black box solvers and preconditioners can be put together in a block triangular strategy to build a scalable block preconditioner for the stokes system discretized by divergence conforming b splines besides the known cavity flow problem we used for benchmark flows defined on complex geometries an eccentric annulus and hollow torus of an eccentric annular cross section

relational graph rewritings this note presents a new formalization of graph rewritings which generalizes traditional graph rewritings relational notions of graphs and their rewritings are introduced and several properties about graph rewritings are discussed using relational calculus theory of binary relations single pushout approaches to graph rewritings proposed by raoult and kennaway are compared with our rewritings of relational labeled graph moreover a more general sufficient condition for two rewritings to commute and a theorem concerning critical pairs useful to demonstrate the confluency of graph rewriting systems are also given

algebraic approach to single pushout graph transformation abstract the single pushout approach to graph transformation interprets a double pushout transformation rule of the classical algebraic approach which consists of two total graph morphisms as a single partial morphism from the left to the right hand side the notion of a double pushout diagram for the transformation process can then be substituted by a single pushout diagram in an appropriate category of partial morphisms it can be shown that this kind of transformation generalizes the double pushout framework hence the classical approach can be seen as a special and very important case of the new concept it can be reobtained from the single pushout approach by imposing an application condition on the redices which formulates the gluing conditions in the new setting on the other hand single pushout transformations are always possible even if the gluing conditions for the redex are violated the simpler structure of a direct transformation one pushout diagram instead of two simplifies many proofs hence the whole theory for double pushout transformations including sequential composition parallel composition and amalgamation can be reformulated and generalized in the new framework some constructions provide new effects and properties which are discussed in detail

an equational notion of lifting monad we introduce the notion of an equational lifting monad a commutative strong monad satisfying one additional equation valid for monads arising from partial map classifiers we prove that any equational lifting monad has a representation by a partial map classifier such that the kleisli category of the former fully embeds in the partial category of the latter thus equational lifting monads precisely capture the equational properties of partial maps as induced by partial map classifiers the representation theorem also provides a tool for transferring nonequational properties of partial map classifiers to equational lifting monads it is proved using a direct axiomatization of kleisli categories of equational lifting monads this axiomatization is of interest in its own right

virtual intersections on the quot scheme and vafa intriligator formulas we construct a virtual fundamental class on the quot scheme parametrizing quotients of a trivial bundle on a smooth projective curve we use the virtual localization formula to calculate virtual intersection numbers on quot as a consequence we re prove the vafa intriligator formula our answer is valid even when the quot scheme is badly behaved more intersections of vafa intriligator type are computed by the same method finally we present an application to the nonvanishing of the pontryagin ring of the moduli space of bundles

some results on locally finitely presentable categories we prove that any full subcategory of a locally finitely presentable l f p category having small limits and filtered colimits preserved by the inclusion functor is itself l f p here full may be weakened to full with respect to isomorphisms further we characterize those left exact functors i c d between small categories with finite limits for which the functor i lex d set lex c set induced by composition is full and faithful as an application we prove a theorem on sheaf representations a consequence of which is that for any site c c j on a category c with finite limits defined by a subcanonical grothendieck topology j the closure in lex c set under small limits and filtered colimits of the models of c is the whole of lex c set introduction in this paper we make contributions to the logic of essentially algebraic theories although the terminology will be categorical the motivation is a model theoretical one our interest lies in arriving at results connecting the syntax and the semantics of that logic the definition of essentially algebraic theory can be given conveniently only in categorical terms as is usually done we identify such a theory with a small category having all finite limits the 2 category of all such categories with functors preserving finite limits as morphisms 1 cells and all natural transformations as 2 cells is denoted by lex lex denotes the 2 category with the smallness restriction removed set the category of small sets is the standard theory an object of lex a model of a theory c e lex is a morphism c set in lex lex c set is the category of models of c talking about relations of syntax and semantics amounts roughly to relating c and lex c set following kr we could call our subject the doctrine of cartesian logic the doctrine of cartesian logic corresponds to a logic with existential quantification restricted to cases when unique existence is assured for a detailed account of this correspondence cf c the main fact of the correspondence can be stated simply call a first order theory t over a language l axiomatized by sentences of the form vx p x 3 y0 x y with p and finite conjunctions of atomic formulas 3 1y means there is a unique y such that a lim theory then the categories of models of lim theories with ordinary homomorphisms as morphisms are the same as the categories lex c set for c c lex the connection of the lim theories themselves and objects of lex is a bit less easy to state it is done in c received by the editors may 8 1984 and in revised form april 20 1985 1980 mathematics subject classification 1985 revision primary 03g30 18b99 secon lary 03c20 03c52 d1987 american mathematical society 0002 9947 87 1 00 25 per page

dispersion optimized quadrature rules for isogeometric analysis modified inner products their dispersion properties and optimally blended schemes abstract this paper introduces optimally blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations to quantify the approximation errors when we modify the inner products we generalize the pythagorean eigenvalue theorem of strang and fix the proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non uniform meshes as well as for different polynomial orders and continuity of the basis the optimally blended schemes improve the convergence rate of the method by two orders with respect to the fully integrated galerkin method the proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems

intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a riemann surface let n and d be coprime positive integers and define m n d to be the moduli space of semi stable holomorphic vector bundles of rank n degree d and fixed determinant on a compact riemann surface e this moduli space is a compact kifhler manifold which has been studied from many different points of view for more than three decades see for instance narasimhan and seshadri 41 the subject of this article is the characterization of the intersection pairings in the cohomology ring1 h m n d a set of generators of this ring was described by atiyah and bott in their seminal 1982 paper 2 on the yangmills equations on riemann surfaces where in addition inductive formulas for the betti numbers of m n d obtained earlier using number theoretic methods 13 25 were rederived by poincare duality knowledge of the intersection pairings between products of these generators or equivalently knowledge of the evaluation on the fundamental class of products of the generators completely determines the structure of the cohomology ring in 1991 donaldson 15 and thaddeus 47 gave formulas for the intersection pairings between products of these generators in h m 2 1 in terms of bernoulli numbers then using physical methods witten 50 found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in h m n d for general rank n these generalized his rigorously proved formulas 49 for the symplectic volume of m n d for instance the symplectic volume of m 2 1 is given by

restriction categories i categories of partial maps given a category with a stable system of monics one can form the corresponding category of partial maps to each map in this category there is on the domain of the map an associated idempotent which measures the degree of partiality this structure is captured abstractly by the notion of a restriction category in which every arrow is required to have such an associated idempotent categories with a stable system of monics functors preserving this structure and natural transformations which are cartesian with respect to the chosen monics form a 2 category which we call mcat the construction of categories of partial maps provides a 2 functor par mcat cat we show that par can be made into an equivalence of 2 categories between mcat and a 2 category of restriction categories the underlying ordinary functor par r0 mcat 0 ca t0 of the above 2 functor par turns out to be monadic and from this we deduce the completeness and cocompleteness of the 2 categories of m categories and of restriction categories we also consider the problem of how to turn a formal system of subobjects into an actual system of subobjects a formal system of subobjects is given by a functor into the category slat of semilattices this structure gives rise to a restriction category which via the above equivalence of 2 categories gives an m category this m category contains the universal realization of the given formal subobjects as actual subobjects

overlapping schwarz preconditioners for isogeometric collocation methods isogeometric collocation methods are very recent and promising numerical schemes that preserve the advantages of isogeometric analysis but often exhibit better performances than their galerkin counterparts in the present paper an additive overlapping schwarz method for isogeometric collocation discretizations is introduced and studied the resulting preconditioner accelerated by gmres is shown to be scalable with respect to the number of subdomains and very robust with respect to the isogeometric discretization parameters such as the mesh size and polynomial degree as well as with respect to the presence of discontinuous elliptic coefficients and domain deformations

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

trace functionals on noncommutative deformations of moduli spaces of flat connections abstract we describe an efficient construction of a canonical noncommutative deformation of the algebraic functions on the moduli spaces of flat connections on a riemann surface the resulting algebra is a variant of the quantum moduli algebra introduced by alekseev grosse and schomerus and buffenoir and roche we construct a natural trace functional on this algebra and show that it is related to the canonical trace in the formal index theory of fedosov and nest and tsygan via verlinde s formula

modelling environments in call by value programming languages in categorical semantics there have traditionally been two approaches to modelling environments one by use of finite products in cartesian closed categories the other by use of the base categories of indexed categories with structure each requires modifications in order to account for environments in call by value programming languages there have been two more general definitions along both of these lines the first generalising from cartesian to symmetric premonoidal categories the second generalising from indexed categories with specified structure to categories in this paper we investigate environments in call by value languages by analysing a fine grain variant of moggi s computational calculus giving two equivalent sound and complete classes of models one given by closed freyd categories which are based on symmetric premonoidal categories the other given by closed categories

paracategories i internal paracategories and saturated partial algebras based on the monoid classifier we give an alternative axiomatization of freyd s paracategories which can be interpreted in any bicategory of partial maps assuming furthermore a free monoid monad t in our ambient category and coequalisers satisfying some exactness conditions we give an abstract envelope construction putting paramonoids and paracategories in the more general context of partial algebras we introduce for the latter the crucial notion of saturation which characterises those partial algebras which are isomorphic to the ones obtained from their enveloping algebras we also set up a factorisation system for partial algebras via epimorphisms and monic kleene morphisms and relate the latter to saturation

counts of maps to grassmannians and intersections on the moduli space of bundles we show that intersection numbers on the moduli space of stable bundles of coprime rank and degree over a smooth complex curve can be recovered as highestdegree asymptotics in formulas of vafa intriligator type in particular we explicitly evaluate all intersection numbers appearing in the verlinde formula our results are in agreement with previous computations of witten jeffrey kirwan and liu moreover we prove the vanishing of certain intersections on a suitable quot scheme which can be interpreted as giving equations between counts of maps to the grassmannian

an energy stable time integrator for phase field models we introduce a provably energy stable time integration method for general classes of phase field models with polynomial potentials we demonstrate how taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy stability implicitly which are second order accurate in time the spatial discretization relies on a mixed finite element formulation and isogeometric analysis we also propose an adaptive time stepping discretization that relies on a first order backward approximation to give an error estimator this error estimator is accurate robust and does not require the computation of extra solutions to estimate the error this methodology can be applied to any second order accurate time integration scheme we present numerical examples in two and three spatial dimensions which confirm the stability and robustness of the method the implementation of the numerical schemes is done in petiga a high performance isogeometric analysis framework

isogeometric methods for computational electromagnetics b spline and t spline discretizations in this paper we introduce methods for electromagnetic wave propagation based on splines and on t splines we define spline spaces which form a de rham complex and following the isogeometric paradigm we map them on domains which are piecewise spline or nurbs geometries we analyze their geometric and topological structure as related to the connectivity of the underlying mesh and we present degrees of freedom together with their physical interpretation the theory is then extended to the case of meshes with t junctions leveraging on the recent theory of t splines the use of t splines enhance our spline methods with local refinement capability and numerical tests show the efficiency and the accuracy of the techniques we propose

a preconditioning strategy for linear systems arising from nonsymmetric schemes in isogeometric analysis abstract in the context of isogeometric analysis we consider two discretization approaches that make the resulting stiffness matrix nonsymmetric even if the differential operator is self adjoint these are the collocation method and the recently developed weighted quadrature for the galerkin discretization in this paper we are interested in the solution of the linear systems arising from the discretization of the poisson problem using one of these approaches in sangalli and tani 2016 a well established direct solver for linear systems with tensor structure was used as a preconditioner in the context of galerkin isogeometric analysis yielding promising results in particular this preconditioner is robust with respect to the mesh size h and the spline degree p in the present work we discuss how a similar approach can be applied to the considered nonsymmetric linear systems the efficiency of the proposed preconditioning strategy is assessed with numerical experiments on two dimensional and three dimensional problems

group valued equivariant localization we prove a localization formula for group valued equivariant de rham cohomology of a compact g manifold this formula is a non trivial generalization of the localization formula of berline vergne and atiyah bott for the usual equivariant de rham cohomology we derive from this result a duistermaat heckman formula for group valued moment maps as an application we prove part of witten s conjectures about intersection pairings on moduli spaces of flat connections on 2 manifolds

a category theoretic characterization of functional completeness abstract functional languages are based on the notion of application programs may be applied to data or programs by application one may define algebraic functions and a programming language is functionally complete when any algebraic function f x1 xn is representable i e there is a constant a such that f x1 xn a x1 xn combinatory logic is the simplest type free language which is functionally complete in a sound category theoretic framework the constant a may be considered as an abstract godel number for f when godel numberings are generalized to principal morphisms in suitable categories by this models of combinatory logic are categorically characterized and their relation is given to lambda calculus models within cartesian closed categories finally the partial recursive functionals in any finite higher type are shown to yield models of combinatory logic

the hascasl prologue categorical syntax and semantics of the partial λ calculus we develop the semantic foundations of the specification language hascasl which combines algebraic specification and functional programming on the basis of moggi s partial calculus generalizing lambek s classical equivalence between the simply typed calculus and cartesian closed categories we establish an equivalence between partial cartesian closed categories pccc s and partial theories building on these results we define set theoretic notions of intensional henkin model and syntactic algebra for moggi s partial calculus these models are shown to be equivalent to the originally described categorical models in pccc s via the global element construction the semantics of hascasl is defined in terms of syntactic algebras correlations between logics and classes of categories facilitate reasoning both on the logical and on the categorical side as an application we pinpoint unique choice as the distinctive feature of topos logic in comparison to intuitionistic higher order logic of partial functions which by our results is the logic of pccc s with equality finally we give some applications of the model theoretic equivalence result to the semantics of hascasl and its relation to first order casl

group cohomology construction of the cohomology of moduli spaces of flat connections on 2 manifolds we use group cohomology and the de rham complex on simplicial manifolds to give explicit differential forms representing generators of the cohomology rings of moduli spaces of representations of fundamental groups of 2 manifolds these generators are constructed using the de rham representatives for the cohomology of classifying spaces bk where k is a compact lie group such representatives universal characteristic classes were found by bott and shulman thus our representatives for the generators of the cohomology of moduli spaces are given explicitly in terms of the maurer cartan form this work solves a problem posed by weinstein who gave a corresponding construction following karshon and goldman of the symplectic forms on these moduli spaces we also give a corresponding construction of equivariant differential forms on the extended moduli space x which is a finite dimensional symplectic space equipped with a hamiltonian action of k for which the symplectic reduced space is the moduli space of representations of the 2 manifold fundamental group in k

p multigrid methods and their comparison to h multigrid methods within isogeometric analysis abstract over the years isogeometric analysis has shown to be a successful alternative to the finite element method fem however solving the resulting linear systems of equations efficiently remains a challenging task in this paper we consider a p multigrid method in which coarsening is applied in the spline degree p instead of the mesh width h and compare it to h multigrid methods since the use of classical smoothers e g gauss seidel results in a p multigrid h multigrid method with deteriorating performance for higher values of p the use of an ilut smoother is investigated as well numerical results and a spectral analysis indicate that the use of this smoother exhibits convergence rates essentially independent of h and p for both p multigrid and h multigrid methods in particular we compare both coarsening strategies e g coarsening in h or p adopting both smoothers for a variety of two and three dimensional benchmarks furthermore the ilut smoother is compared to a state of the art smoother hofreither and takacs 2017 using both coarsening strategies finally the proposed p multigrid method is used to solve linear systems resulting from thb spline discretizations

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

categories of partial maps abstract this paper attempts to reconcile the various abstract notions of category of partial maps which appear in the literature first a particular algebraic theory p categories is introduced and a representation theorem proved this gives the authors a coherent framework in which to place the various other definitions both algebraic theories and theories which make essential use of the poset enriched structure of partial maps are discussed proofs of equivalence are given where possible and counterexamples where known the paper concludes with brief sections on the representation of partial maps and on partial algebras

restriction categories as enriched categories restriction categories were introduced as a simple equational axiomatisation for categories of partial maps such as those which arise in the foundations of computability theory a restriction structure on a category is given by an operation obeying four simple axioms that assigns to each morphism of the category an endomorphism of its domain to be thought of as the partial identity representing the degree of definition in this paper we show that restriction categories can be seen as a kind of enriched category this allows their theory to be studied by way of the enrichment unlike most enrichments ours is based not on a monoidal category but rather a weak double category in the sense of grandis pare and provides from a purely mathematical perspective an example of such an enrichment arising in nature beyond exhibiting restriction categories as enriched categories we show that varying the base of this enrichment also allows the important notions of join and range restriction category to be understood in the same manner

addendum to on the variation in the cohomology of the symplectic form of the reduced phase space here we have used the notation rlx for the inner product of the vector field x with the form a and ix m x j m for the x component of a in an earlier paper 4 it was shown that the push forward j dm of the liouville measure dm on m under the momentum mapping j is a piecewise polynomial measure on moreover in case x has isolated isolated zeros on m an explicit formula for the integral

witten s formulas for intersection pairings on moduli spaces of flat g bundles abstract in a 1992 paper j geom phys 9 1992 303 witten gave a formula for the intersection pairings of the moduli space of flat g bundles over an oriented surface possibly with markings in this paper we give a general proof of witten s formula for arbitrary compact simple groups and any markings for which the moduli space has at most orbifold singularities

conformal field theory and the cohomology of the moduli space of stable bundles let g be a compact riemann surface of genus g 2 and let be a line bundle over g of degree 1 then the moduli space of rank 2 stable bundles v over g such that v was shown by seshadri 19 to be a nonsingular projective variety ng its rational cohomology ring h ng is exceedingly rich and despite years of study has never quite been computed in full the object of this paper is to give an essentially complete characterization of this ring or at least to reduce the problem to a matter of linear algebra in particular we find a proof of newstead s conjecture that pg1 ng 0 we also obtain a formula for the volume of ng which can be regarded as a twisted version of the formula for the degree 0 moduli space recently announced by witten the approach we shall take is not from algebraic geometry but from mathematical physics it relies on the su 2 wess zumino witten model which is a functor zk associating a finite dimensional vector space to each riemann surface with marked points the relationship with the moduli space is that when the level k of the functor is even the vector space associated to g with no marked points can be identified with h ng l where l is a fixed line bundle over ng now the work of verlinde provides us with a means of calculating the dimension of any vector space arising from our functor and in particular dimh ng l which we shall denote d g k on the other hand newstead 14 found explicit generators for h ng and we can also express d g k in terms of them using a riemann roch theorem equating the two formulas enables us to evaluate any monomial in the generators on the fundamental class of ng and by poincare duality this is sufficient at least in principle to determine the ring structure of h ng in the discussion above one important point has been skated over the su 2 wzw model is of course associated to bundles of degree 0 not degree 1 so in order to make use of verlinde s work it is necessary to formulate a twisted version of the field theory this is carried out in 2 but the crucial properties of the twisted theory analogous to those which make the ordinary theory a modular functor are not proved in this paper rather we will confine ourselves to exploring the consequences of these claims and hope to return to justify them in a later paper an outline of the remaining sections goes as follows in 3 we review those parts of verlinde s work we shall need show how they must be modified in the twisted case and work out some explicit formulas for d g k in 4 we study the cohomology of ng which we regard throughout as a space of representations via the theorem of narasimhan and seshadri we define newstead s generators and i of h ng

localization for yang mills theory on the fuzzy sphere we present a new model for yang mills theory on the fuzzy sphere in which the configuration space of gauge fields is given by a coadjoint orbit in the classical limit it reduces to ordinary yang mills theory on the sphere we find all classical solutions of the gauge theory and use nonabelian localization techniques to write the partition function entirely as a sum over local contributions from critical points of the action which are evaluated explicitly the partition function of ordinary yang mills theory on the sphere is recovered in the classical limit as a sum over instantons we also apply abelian localization techniques and the geometry of symmetric spaces to derive an explicit combinatorial expression for the partition function and compare the two approaches these extend the standard techniques for solving gauge theory on the sphere to the fuzzy case in a rigorous framework

normal forms for algebras of connections recent years have seen a growing interest towards algebraic structures that are able to express formalisms different from the standard tree like presentation of terms many of these approaches reveal a specific interest towards the application to the distributed and concurrent systems field but an exhaustive comparison between them is sometimes difficult because their presentations can be quite dissimilar this work is a first step towards a unified view focusing on the primitive ingredients of distributed spaces namely interfaces links and basic modules we introduce a general schema for describing a normal form presentation of many algebraic formalisms and show that those normal forms can be thought of as arrows of suitable monoidal categories

multiplicities formula for geometric quantization part ii 1 introduction let p be a compact manifold let h be a compact lie group acting on the right on p we assume that the stabilizer of each element y e p is a finite subgroup of h the space m p h is an orbifold and every orbifold can be presented this way if h acts freely then m is a manifold if ae graded space of h invariant solutions of the h horizontal dirac operator on p twisted by the line bundle let m o 9 be the moment map for the g action assume 0 is a regular value of let mred be the reduced orbifold of m that is mred o g consider the reduced orbifold line bundle rd lu lto g on m oa in the ease where both g andh are torus we prove here the formula

partiality cartesian closedness and toposes presentation equationnelle des categories de morphismes partiels avec etablissement systematique des rapports entre ces structures et les classiques structures totales

refined isogeometric analysis for a preconditioned conjugate gradient solver david pardo has received funding from the project of the spanish ministry of economy and competitiveness with reference mtm2016 76329 r aei feder eu and mtm2016 81697 erc aei the bcam severo ocho accreditation of excellencesev 2013 0323 and the basque government through the berc 2014 2017 program and the consolidated research group grant it649 13 on mathematical modeling simulation and industrial applications m2si this publication was also made possible in part by the european union s horizon 2020 research and innovation programme under the marie sklodowska curie grant agreement no 644602 the csiro professorial chair in computational geoscience at curtin university the deep earth imaging enterprise future science platforms of the commonwealth scientific industrial research organisation csiro of australia the mega grant of the russian federation government n14 y26 31 0013 and the curtin institute for computation the j tinsley oden faculty fellowship research program at the institute for computational engineering and sciences ices of the university of texas at austin has partially supported the visits of vmc to ices the authors acknowledge the texas advanced computing center tacc at the university of texas at austin for providing hpc resources that have contributed to the research results reported within this paper

the moduli space of stable vector bundles over a real algebraic curve we study the spaces of stable real and quaternionic vector bundles on a real algebraic curve the basic relationship is established with unitary representations of an extension of mathbb z 2 by the fundamental group by comparison with the space of real or quaternionic connections some of the basic topological invariants of these spaces are calculated