an isoparametric approach to high order curvilinear boundary layer meshing abstract the generation of high order curvilinear meshes for complex three dimensional geometries is presently a challenging topic particularly for meshes used in simulations at high reynolds numbers where a thin boundary layer exists near walls and elements are highly stretched in the direction normal to flow in this paper we present a conceptually simple but very effective and modular method to address this issue we propose an isoparametric approach whereby a mesh containing a valid coarse discretization comprising of high order triangular prisms near walls is refined to obtain a finer prismatic or tetrahedral boundary layer mesh the validity of the prismatic mesh provides a suitable mapping that allows one to obtain very fine mesh resolutions across the thickness of the boundary layer we describe the method in detail for a high order approximation using modal basis functions discuss the requirements for the splitting method to produce valid prismatic and tetrahedral meshes and provide a sufficient criterion of validity in both cases by considering two complex aeronautical configurations we demonstrate how highly stretched meshes with sufficient resolution within the laminar sublayer can be generated to enable the simulation of flows with reynolds numbers of 10 6 and above

local optimization based simplicial mesh untangling and improvement we present an optimization based approach for mesh untangling that maximizes the minimum area or volume of simplicial elements in a local submesh these functions are linear with respect to the free vertex position thus the problem can be formulated as a linear program that is solved by using the computationally inexpensive simplex method we prove that the function level sets are convex regardless of the position of the free vertex and hence the local subproblem is guaranteed to converge maximizing the minimum area or volume of mesh elements although well suited for mesh untangling is not ideal for mesh improvement and its use often results in poor quality meshes we therefore combine the mesh untangling technique with optimization based mesh improvement techniques and expand previous results to show that a commonly used two dimensional mesh quality criterion can be guaranteed to converge when starting with a valid mesh typical results showing the effectiveness of the combined untangling and smoothing techniques are given for both two and three dimensional simplicial meshes

the topological trace formula the topological trace formula is a computation of the lefschetz number of a hecke correspondence c acting on the weighted cohomology groups defined in ghm of a locally symmetric space x it expresses this lefschetz number as a sum of contributions from fixed point components of c on the reductive borel serre compactifi cation of x the proof uses the lefschetz fixed point formula of gm2

hodge cohomology of gravitational instantons we study the space of l2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type these metrics generalize the geometric structures at infinity of several different well known classes of metrics including asymptotically locally euclidean manifolds the known types of gravitational instantons and also poincare metrics on q rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in kahler manifolds the answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold the l2 signature formula implied by our result is closely related to the one proved by dai 25 and more generally by vaillant 67 and identifies dai s invariant directly in terms of intersection cohomology of differing perversities this work is also closely related to a recent paper of carron 12 and the forthcoming paper of cheeger and dai 17 we apply our results to a number of examples gravitational instantons among them arising in predictions about l2 harmonic forms in duality theories in string theory

discrete series characters and two structures let g be a connected semisimple real lie group with compact cartan subgroup harish chandra gave formulas for discrete series characters which are completely explicit except for certain interger constants appearing in the numerators the main result of this paper is a new formula for these constants using two structures the new formula avoids endoscopy and stable discrete series entirely expressing unaveraged discrete series constants directly in terms of unaveraged discrete series constants corresponding to two structures of noncompact type

simplicial differential forms with poles shadow forms consider a simplicial complex k whitney associated to each simplicial cochain 4 a differential form v 4 on k whitney s map v from simplicial cochains to differential forms is a map of cochain complexes which induces the derham isomorphism we call the forms v 4 whitney forms they have been useful in several contexts most notably integral geometry and rational homotopy theory in this paper we introduce a generalization of whitney forms called shadow forms we associate to certain simplicial chains g with respect to the first derived subdivision k of k a shadow form w on k let us illustrate the construction of shadow forms when k is a single n simplex which we now denote by a in this case w g is defined for k simplices g of ax which do not lie in the boundary of a or for linear combinations of these for example if a is the 2 simplex then either of the 1 simplices or 2 pictured below has an associated shadow form

fourier mukai transform on abelian surfaces we study moduli spaces of stable sheaves on abelian surfaces whose mukai vectors are related by a cohomological fourier mukai transform we show that there is a fourier mukai transform inducing a birational map between them

categorical entropy for fourier mukai transforms on generic abelian surfaces abstract in this note we shall compute the categorical entropy of an autoequivalence on a generic abelian surface

on the m function and borg marchenko theorems for vector valued sturm liouville equations we will consider a vector valued sturm liouville equation of the form r u pu qu lambdawu xis an element of 0 b with p 1 w qis an element ofl loc 1 0 b mxm being hermitian and under some additional conditions on p 1 and w we give an elementary deduction of the leading order term asymptotics for the titchmarsh weyl m function corresponding to this equation in the special case of p w i qis an element ofl 1 0 infinity mxm and the neumann boundary conditions at 0 we will also prove that m 1 root lambda i r i r 1 where r lim n infinity r n sigma n 1 infinity q n for recursively defined sequences r n and q n if qis an element ofl loc 1 0 b mxm 0 bless than or equal toinfinity the same formula is valid with an exponentially small error for large lambda it is clear that expansions of this type are helpful in finding representatives of the kdv invariants for p w i we prove that the spectral measure corresponding to the equation r u lambdau uniquely determines q as well as b and the boundary conditions at 0 and b we finally give a new proof of a local form of the borg marchenko theorem cf gesztesy and simon on local borg marchenko uniqueness results commun math phys 211 273 287 2000 chap 3 a theorem which is due to simon see simon a new approach to inverse spectral theory i fundamental formalism ann math 150 1 29 1999 in the scalar case for applications to physics it is worth mentioning that vector valued sturm liouville equations appear in some problems in magneto hydro dynamics

from h to p efficiently implementing finite and spectral hp element methods to achieve optimal performance for low and high order discretisations the spectral hp element method can be considered as bridging the gap between the traditionally low order finite element method on one side and spectral methods on the other side consequently a major challenge which arises in implementing the spectral hp element methods is to design algorithms that perform efficiently for both low and high order spectral hp discretisations as well as discretisations in the intermediate regime in this paper we explain how the judicious use of different implementation strategies can be employed to achieve high efficiency across a wide range of polynomial orders furthermore based upon this efficient implementation we analyse which spectral hp discretisation which specific combination of mesh size h and polynomial order p minimises the computational cost to solve an elliptic problem up to a predefined level of accuracy we investigate this question for a set of both smooth and non smooth problems

local contribution to the lefschetz fixed point formula for a class of self correspondencesc calledweakly hyperbolic we give a computable formula for the contribution of a fixed point component to the lefschetz number ofc the formula applies to lefschetz numbers of cohomology with coefficients in a constructible complex of sheaves such as intersection homology

the generation of triangular meshes for nurbs enhanced fem summary this paper presents the first method that enables the fully automatic generation of triangular meshes suitable for the so called non uniform rational b spline nurbs enhanced finite element method nefem the meshes generated with the proposed approach account for the computer aided design boundary representation of the domain given by nurbs curves the characteristic element size is completely independent of the geometric complexity and of the presence of very small geometric features the proposed strategy allows to circumvent the time consuming process of de featuring complex geometric models before a finite element mesh suitable for the analysis can be produced a generalisation of the original definition of a nefem element is also proposed enabling to treat more complicated elements with an edge defined by several nurbs curves or more than one edge defined by different nurbs three examples of increasing difficulty demonstrate the applicability of the proposed approach and illustrate the advantages compared with those of traditional finite element mesh generators finally a simulation of an electromagnetic scattering problem is considered to show the applicability of the generated meshes for finite element analysis 2016 the authors international journal for numerical methods in engineering published by john wiley sons ltd

high order unstructured curved mesh generation using the winslow equations we propose a method to generate high order unstructured curved meshes using the classical winslow equations we start with an initial straight sided mesh in a reference domain and fix the position of the nodes on the boundary on the true curved geometry in the interior of the domain we solve the winslow equations using a new continuous galerkin finite element discretization this formulation appears to produce high quality curved elements which are highly resistant to inversion in addition the corresponding nonlinear equations can be solved efficiently using picard iterations even for highly stretched boundary layer meshes compared to several previously proposed techniques such as optimization and approaches based on elasticity analogies this can significantly reduce the computational cost while producing curved elements of similar quality we show a number of examples in both two and three space dimensions including complex geometries and stretched boundary layers and demonstrate the high quality of the generated meshes and the performance of the nonlinear solver

robust untangling of curvilinear meshes this paper presents a technique that allows to untangle high order curvilinear meshes the technique makes use of unconstrained optimization where element jacobians are constrained to lie in a prescribed range through moving log barriers the untangling procedure starts from a possibly invalid curvilinear mesh and moves mesh vertices with the objective of producing elements that all have bounded jacobians bounds on jacobians are computed using the results of johnen et al 2012 2013 1 2 the technique is applicable to any kind of polynomial element for surface volume hybrid or boundary layer meshes a series of examples demonstrate both the robustness and the efficiency of the technique the final example involving a time explicit computation shows that it is possible to control the stable time step of the computation for curvilinear meshes through an alternative element deformation measure

isogeometric analysis on v reps first results abstract inspired by the introduction of volumetric modeling via volumetric representations v reps by massarwi and elber in 2016 in this paper we present a novel approach for the construction of isogeometric numerical methods for elliptic pdes on trimmed geometries seen as a special class of more general v reps we develop tools for approximation and local re parameterization of trimmed elements for three dimensional problems and we provide a theoretical framework that fully justifies our algorithmic choices we validate our approach both on two and three dimensional problems for diffusion and linear elasticity

isogeometric triangular bernstein bezier discretizations automatic mesh generation and geometrically exact finite element analysis isogeometric analysis iga was introduced as a way to bypass the design to analysis bottleneck inherent in the traditional computer aided design cad through finite element analysis fea paradigm however an outstanding problem in the field of iga is that of surface to volume parameterization in cad packages solid objects are represented by a collection of nurbs or t spline bounding surfaces but to perform engineering analysis on real world problems we must find a way to parameterize the volumes of these objects as well this has proven to be difficult using traditional iga as the tensor product nature of trivariate nurbs and t splines limit their ability to create analysis suitable parameterizations of arbitrarily complex volumes to overcome the limitations of trivariate nurbs and t splines we propose the use of rational bernstein bezier tetrahedra to create analysis suitable volumetric parameterizations for isogeometric analysis in this paper which is part one of a two part series we present the methodology for discretizing two dimensional geometries using rational bernstein bezier triangles in addition to presenting finite element analysis methodologies based on rational bernstein bezier triangles we also introduce two new mesh generation strategies for automatically creating high quality geometrically exact curvilinear meshes we assess the quality of our mesh generation schemes using a suite of challenging two dimensional geometries and we verify the accuracy of our proposed numerical discretization approach using the method of manufactured solutions

on the reductive borel serre compactification l p cohomology of arithmetic groups for large p the l 2 cohomology of an arithmetic quotient of an hermitian symmetric space i e of a locally symmetric variety is known to have the topological interpretation as the intersection homology of its baily borel satake compactification in this article we observe that even without the hermitian hypothesis the l p cohomology of an arithmetic quotient for p finite and sufficiently large is isomorphic to the ordinary cohomology of its reductive borel serre compactification we use this to generalize a theorem of mumford concerning homogeneous vector bundles their invariant chern forms and the canonical extensions of the bundles here though we are referring to canonical extensions to the reductive borel serre compactification of any arithmetic quotient to achieve that we give a systematic discussion of vector bundles and chern classes on stratified spaces

automorphisms of positive entropy on some hyperkahler manifolds via derived automorphisms of k3 surfaces abstract we construct examples of hyperkahler manifolds of picard number two with automorphisms of positive entropy via derived automorphisms of k3 surfaces of picard number one our hyperkahler manifolds are constructed as moduli spaces of bridgeland stable objects in derived categories of k3 surfaces then automorphisms of positive entropy are induced by derived automorphisms of positive entropy on k3 surfaces

lefschetz properties for noncompact arithmetic ball quotients we prove a lefschetz property for the restriction of the cohomology of noncompact arithmetic ball quotients to ball quotients of smaller dimension

optimizing the geometrical accuracy of curvilinear meshes this paper presents a method to generate valid high order meshes with optimized geometrical accuracy the high order meshing procedure starts with a linear mesh that is subsequently curved without taking care of the validity of the high order elements an optimization procedure is then used to both untangle invalid elements and optimize the geometrical accuracy of the mesh standard measures of the distance between curves are considered to evaluate the geometrical accuracy in planar two dimensional meshes but they prove computationally too costly for optimization purposes a fast estimate of the geometrical accuracy based on taylor expansions of the curves is introduced an unconstrained optimization procedure based on this estimate is shown to yield significant improvements in the geometrical accuracy of high order meshes as measured by the standard hausdorff distance between the geometrical model and the mesh several examples illustrate the beneficial impact of this method on cfd solutions with a particular role of the enhanced mesh boundary smoothness

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

the dimensions of spaces of siegel cusp forms of general degree abstract in this paper we give a dimension formula for spaces of siegel cusp forms of general degree with respect to neat arithmetic subgroups the formula was conjectured before by several researchers cf 40 43 45 75 the dimensions are expressed by special values of shintani zeta functions for spaces of symmetric matrices at non positive integers this formula was given by shintani for only a small part of the geometric side of the trace formula see 64 to be precise it is the contribution of unipotent elements corresponding to the partitions 2 j 1 2 n 2 j where n denotes the degree and 0 j n hence our work is to show that all the other contributions vanish therefore one finds that shintani s formula means the dimension itself combining our formula and an explicit formula of the shintani zeta functions which was discovered by ibukiyama and saito cf 44 42 61 we can derive an explicit dimension formula for the principal congruence subgroups of level greater than two cf 43 45 in this explicit dimension formula the dimensions are described by degree n weight k level n and the bernoulli numbers b m

stability conditions on k3 surfaces this article contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic k3 surface

discrete series characters and the lefschetz formula for hecke operators this paper consists of three independent but related parts in the rst part xx1 6 we give a combinatorial formula for the constants appearing in the numerators of characters of stable discrete series representations of real groups see x3 as well as an analogous formula for individual discrete series representations see x6 moreover we give an explicit formula theorem 5 1 for certain stable virtual characters on real groups by theorem 5 2 these include the stable discrete series characters and thus we recover the results of x3 in a more natural way in the second part x7 we use the character formula given in theorem 5 1 to rewrite the lefschetz formula of gm for the local contribution at a single xed point component to the trace of a hecke operator on weighted cohomology in the same spirit as that of arthur s lefschetz formula a in terms of stable virtual characters on real groups see theorem 7 14 b we then sum the contributions of the various xed point components and show that in the case of middle weighted cohomology the resulting global lefschetz xed point formula agrees with arthur s lefschetz formula this gives a topological proof of arthur s formula

a distortion measure to validate and generate curved high order meshes on cad surfaces with independence of parameterization the first author was sponsored by cur from diue of the generalitat de catalunya and the european social fund grants fi dgr 11 and be dgr 12 and by the ferran sunyer i balaguer foundation grant fsb 13 the work of the corresponding author was partially supported by the boeing co us air force office of scientific research european commission through the boeing mit alliance the computational math program marie sklodowska curie actions hipermegaflows individual fellowship grant

sato tate theorem for families and low lying zeros of automorphic l functions we consider certain families of automorphic representations over number fields arising from the principle of functoriality of langlands let g be a reductive group over a number field f which admits discrete series representations at infinity let l g widehat g rtimes mathrm gal bar f f be the associated l group and r l g rightarrow mathrm gl d mathbb c a continuous homomorphism which is irreducible and does not factor through mathrm gal bar f f the families under consideration consist of discrete automorphic representations of g mathbb a f of given weight and level and we let either the weight or the level grow to infinity we establish a quantitative plancherel and a quantitative sato tate equidistribution theorem for the satake parameters of these families this generalizes earlier results in the subject notably of sarnak prog math 70 321 331 1987 and serre j am math soc 10 1 75 102 1997 as an application we study the distribution of the low lying zeros of the associated family of l functions l s pi r assuming from the principle of functoriality that these l functions are automorphic we find that the distribution of the 1 level densities coincides with the distribution of the 1 level densities of eigenvalues of one of the unitary symplectic and orthogonal ensembles in accordance with the katz sarnak heuristics we provide a criterion based on the frobenius schur indicator to determine this symmetry type if r is not isomorphic to its dual r vee then the symmetry type is unitary otherwise there is a bilinear form on mathbb c d which realizes the isomorphism between r and r vee if the bilinear form is symmetric resp alternating then r is real resp quaternionic and the symmetry type is symplectic resp orthogonal

curved boundary treatments for the discontinuous galerkin method applied to aeroacoustic propagation a discontinuous galerkin method is applied to unstructured grids to simulate aeroacoustic propagation modeled by the linearized euler equations on triangular and tetrahedral elements with straight edges the quadrature free form of the discontinuous galerkin method is used in addition to the classical linear treatment of wall boundaries two treatments involving a second order representation of the geometry are presented the simulation of acoustic scattering problems shows that the linear treatment can limit the accuracy at high order and demonstrates how the boundary treatment involving curved elements overcomes this restriction the benefits of higher order treatments are also assessed for more realistic geometries namely a high lift airfoil and an elliptical muffler

the intersection cohomology of the satake compactification of mathcal a _g for g le 4 we completely determine the intersection cohomology of the satake compactifications mathcal a 2 mathrm sat mathcal a 3 mathrm sat and mathcal a 4 mathrm sat except for ih 10 mathcal a 4 mathrm sat we also determine all the ingredients appearing in the decomposition theorem applied to the map from a toroidal compactification to the satake compactification in these genera as a byproduct we obtain in addition several results about the intersection cohomology of the link bundles involved

l2 cohomology of arithmetic varieties abstract the l2 cohomology of arithmetic quotients of bounded symmetric domains is studied we establish the conjecture of zucker equating the l2 cohomology of these spaces to the intersection cohomology of their baily borel compactifications

hodge decompositions and dolbeault complexes on normal surfaces give the smooth subset of a normal singular complex projective surface the metric induced from the ambient projective space the l2 cohomology of this incomplete manifold is isomorphic to the surface s intersection cohomology which has a natural hodge decomposition this paper identifies dolbeault complexes whose 0 closed and 0 coclosed forms represent the classes of pure type in the corresponding hodge decomposition of l2 cohomology

high order discontinuous galerkin schemes on general 2d manifolds applied to the shallow water equations an innovating approach is proposed to solve vectorial conservation laws on curved manifolds using the discontinuous galerkin method this new approach combines the advantages of the usual approaches described in the literature the vectorial fields are expressed in a unit non orthogonal local tangent basis derived from the polynomial mapping of curvilinear triangle elements while the convective flux functions are written is the usual 3d cartesian coordinate system the number of vectorial components is therefore minimum and the tangency constraint is naturally ensured while the method remains robust and general since not relying on a particular parametrization of the manifold the discontinuous galerkin method is particularly well suited for this approach since there is no continuity requirement between elements for the tangent basis definition the possible discontinuities of this basis are then taken into account in the riemann solver on inter element interfaces the approach is validated on the sphere using the shallow water equations for computing standard atmospheric benchmarks in particular the williamson test cases are used to analyze the impact of the geometry on the convergence rates for discretization error the propagation of gravity waves is eventually computed on non conventional irregular curved manifolds to illustrate the robustness and generality of the method

accounting for curved domains in mesh adaptation this paper presents a procedure to project vertices created by refining straight sided tetrahedra to appropriate locations on curved model boundaries although many of the vertices introduced by refinement of mesh edges and faces on the boundary of the model can easily be projected to the model geometry there are situations where this projection creates invalid elements to ensure the ability to place those vertices on the model boundary an efficient two step procedure has been developed the first step applies local mesh modification operations such as collapse and swap to incrementally project vertices it is shown that the effective application of mesh modifications that will yield a mesh in which the vertex in question can be projected to the boundary must focus on ensuring the vertex in question moves at least as far as the plane on which the mesh would have element shapes degrade to zero volume the second step applies a generalized local cavity triangulation algorithm to ensure the success of the process when the set of local mesh modifications considered is unable to project the vertex to the boundary copyright 2003 john wiley sons ltd

simultaneous untangling and smoothing of moving grids in this work a technique for simultaneous untangling and smoothing of meshes is presented it is based on an extension of an earlier mesh smoothing strategy developed to solve the computational mesh dynamics stage in fluid structure interaction problems in moving grid problems mesh untangling is necessary when element inversion happens as a result of a moving domain boundary the smoothing strategy formerly published by the authors is defined in terms of the minimization of a functional associated with the mesh distortion by using a geometric indicator of the element quality this functional becomes discontinuous when an element has null volume making it impossible to obtain a valid mesh from an invalid one to circumvent this drawback the functional proposed is transformed in order to guarantee its continuity for the whole space of nodal coordinates thus achieving the untangling technique this regularization depends on one parameter making the recovery of the original functional possible as this parameter tends to 0 this feature is very important consequently it is necessary to regularize the functional in order to make the mesh valid then it is advisable to use the original functional to make the smoothing optimal finally the simultaneous untangling and smoothing technique is applied to several test cases including 2d and 3d meshes with simplicial elements as an additional example the application of this technique to a mesh generation case is presented copyright 2008 john wiley sons ltd

partial identification of the potential from phase shifts we consider the three dimensional inverse scattering with fixed energy in the spherically symmetrical case we give a characterization of the sequences of phase shifts for two potentials which can be different only in a ball of radius a in other words we study how the large distance interaction influences the asymptotical behavior of the phase shifts we also characterize the tail of the potential by the growth order of the scattering amplitude f t for large t

moduli spaces of stable sheaves on abelian surfaces in this paper we consider basic problems on moduli spaces of stable sheaves on abelian surfaces our main assumption is the primitivity of the associated mukai vector we determine the deformation types albanese maps bogomolov factors and their weight 2 hodge structures we also discuss the deformation types of moduli spaces of stable sheaves on k3 surfaces

on entropy for autoequivalences of the derived category of curves abstract to an exact endofunctor of a triangulated category with a split generator the notion of entropy is given by dimitrov haiden katzarkov kontsevich which is a possibly negative infinite real valued function of a real variable it is important to evaluate the value of the entropy at zero in relation to the topological entropy in this paper we study the entropy at zero of an autoequivalence of the derived category of a complex smooth projective curve and prove that it coincides with the natural logarithm of the spectral radius of the induced automorphism on its numerical grothendieck group

the noether lefschetz conjecture and generalizations we prove the noether lefschetz conjecture on the moduli space of quasi polarized k3 surfaces this is deduced as a particular case of a general theorem that states that low degree cohomology classes of arithmetic manifolds of orthogonal type are dual to the classes of special cycles i e sub arithmetic manifolds of the same type for compact manifolds this was proved in 3 here we extend the results of 3 to non compact manifolds this allows us to apply our results to the moduli spaces of quasi polarized k3 surfaces

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

a minimal element distortion strategy for computational mesh dynamics mesh motion strategy is one of the key points in many fluid structure interaction fsi problems due to the increasing application of fsi to solve the current challenging engineering problems this topic has become of great interest there are several different strategies to solve this problem some of them use a discrete and lumped spring mass system to propagate the boundary motion into the volume mesh and many others use an elastostatic problem to deform the mesh in all these strategies there is always risk of producing an invalid mesh i e a mesh with some elements inverted normally this condition is irreversible and once an invalid mesh is obtained it is difficult to continue in this paper the mesh motion strategy is defined as an optimization problem by its definition this strategy can be classified as a particular case of an elastostatic problem where the material constitutive law is defined in terms of the minimization of certain energy functional that takes into account the degree of element distortion some advantages of this strategy are its natural tendency to high quality meshes its robustness and its straightforward extension to 3d problems several examples included in this paper show these capabilities even though this strategy seems to be very robust it is not able to recover a valid mesh starting from an invalid one this improvement is left for future work copyright 2006 john wiley sons ltd

additivity and non additivity for perverse signatures a well known property of the signature of closed oriented 4n dimensional manifolds is novikov additivity which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface then the signature of the original manifold equals the sum of the signatures of the resulting manifolds with boundary wall showed that this property is not true of signatures on manifolds with boundary and that the dierence from additivity could be described as a certain maslov triple index perverse signatures are signatures dened for any stratied pseudomanifold using the intersection homology groups of goresky and macpherson in the case of witt spaces the middle perverse signature is the same as the witt signature this paper proves a generalization to perverse signatures of wall s non additivity theorem for signatures of manifolds with boundary under certain topological conditions on the dividing hypersurface novikov additivity for perverse signatures may be deduced as a corollary in particular siegel s version of novikov additivity for witt signatures is a special case of this corollary

intersection numbers of hecke cycles on hilbert modular varieties let o be the ring of integers of a totally real number field e and set g res e q gl 2 fix an ideal c 0 for each ideal m 0 let t m denote the mth hecke operator associated to the standard compact open subgroup u 0 c of g a f setting x 0 c g q g a k u 0 c where k is a certain subgroup of g r we use t m to define a hecke cycle z m ih 2 e q x 0 c x 0 c here ih denotes intersection homology we use zucker s conjecture proven by looijenga and independently by saper and stern to obtain a formula relating the intersection number z m z n to the trace of t m o t n considered as an endomorphism of the space of hilbert cusp forms on u 0 c

h multigrid agglomeration based solution strategies for discontinuous galerkin discretizations of incompressible flow problems abstract in this work we exploit agglomeration based h multigrid preconditioners to speed up the iterative solution of discontinuous galerkin discretizations of the stokes and navier stokes equations as a distinctive feature h coarsened mesh sequences are generated by recursive agglomeration of a fine grid admitting arbitrarily unstructured grids of complex domains and agglomeration based discontinuous galerkin discretizations are employed to deal with agglomerated elements of coarse levels both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated for the sake of efficiency coarse grid operators are inherited through element by element l 2 projections avoiding the cost of numerical integration over agglomerated elements specific care is devoted to the projection of viscous terms discretized by means of the br2 dg method we demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels the numerical solution of steady and unsteady linear and non linear problems is considered tackling challenging 2d test cases and 3d real life computations on parallel architectures significant execution time gains are documented

optimization of a regularized distortion measure to generate curved high order unstructured tetrahedral meshes we present a robust method for generating high order nodal tetrahedral curved meshes the approach consists of modifying an initial linear mesh by first introducing high order nodes second displacing the boundary nodes to ensure that they are on the cad surface and third smoothing and untangling the mesh obtained after the displacement of the boundary nodes to produce a valid curved high order mesh the smoothing algorithm is based on the optimization of a regularized measure of the mesh distortion relative to the original linear mesh this means that whenever possible the resulting mesh preserves the geometrical features of the initial linear mesh such as shape stretching and size we present several examples to illustrate the performance of the proposed algorithm furthermore the examples show that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements we consider cases with polynomial approximations up to degree ten large deformations of the curved boundaries concave boundaries and highly stretched boundary layer elements the meshes obtained are suitable for high order finite element analyses

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

hamiltonian point of view of non euclidean geometry and elliptic functions abstract this paper offers a new way of looking at the classical geometries and the theory of elliptic functions through hamiltonian systems on lie groups in particular the paper shows that i the classical models of non euclidean geometries are canonically induced by bi invariant sub riemannian metrices on lie groups which act by left actions on the underlying space ii there is a class of canonical variational problems on lie groups g whose projections on homogeneous spaces g k generalize euler s elasticae and include all curves of constant curvature and all functions of weierstrass iii complex lie groups unify non euclidean geometries and complex elasticae offer a distinctive look at the elliptic functions

weighted cohomology of arithmetic groups let g be a semisimple algebraic group defined over the rational numbers k a maximal compact subgroup ofg g r and g q a neat arithmetic subgroup letx g k be the locally symmetric space associated to and e the local system on x constructed out of a finite dimensional irreducible algebraic representation e of g fix a maximally q split torus s in g s is assumed to be nontrivial so that x is noncompact let a s r 0 and by a slight abuse of notation let x a denote the group of rational characters of s choose a minimal rational parabolic subgroup p0 s the choice determines a notion of positivity in x a and a set of positive roots among the roots of s in g let 0 be the half sum of the positive roots weighted cohomology is an invariant of introduced by m goresky g harder and r macpherson 12 in the study 15 16 12 14 of the trace of hecke operators in the cohomology of for each p x a q there are weight profiles p and p upper and lower p see 1 5 and groups w ph i e and w h i e the notion of profile here differs from but is equivalent to that of 12 see remark 1 2 below if p q 2 0 then the profiles p and q are dual and the corresponding groups are in poincare duality for very positive p one gets the compactly supported cohomology h i c x e and dually for very negative p the full cohomology h i x e the definition of the weighted cohomology groups is somewhat involved so i will attempt to give some idea of the motivation behind it detailed definitions are recalled in section 1 associated to p and p are complexes of sheaves wpc e and wpc e on the reductive borel serre compactification x of x let p be a rational parabolic subgroup np its unipotent radical sp the split centre of p np and xp the p boundary component in x the idea behind the construction in 12 is as follows the stalk cohomology of the direct image ri e by i x x at x xp is

kazhdan lusztig algorithms for nonlinear groups and applications to kazhdan patterson lifting we establish an algorithm to compute characters of irreducible harish chandra modules for a large class of nonalgebraic lie groups roughly speaking the class of groups consists of those obtained as nonlinear double covers of linear groups in harish chandra s class we then apply this theory to study a particular group the universal cover of the real general linear group and discover a symmetry of the character computations encoded in a character multiplicity duality using this duality theory we reinterpret a kind of representation theoretic shimura correspondence for the general linear group geometrically and find that it is dual to an analogous lifting for indefinite unitary groups it seems likely that this example is illustrative of a general framework for studying similar correspondences

geometrical validity of curvilinear finite elements in this paper we describe a way to compute accurate bounds on jacobian determinants of curvilinear polynomial finite elements our condition enables to guarantee that an element is geometrically valid i e that its jacobian determinant is strictly positive everywhere in its reference domain it also provides an efficient way to measure the distortion of curvilinear elements the key feature of the method is to expand the jacobian determinant using a polynomial basis built using bezier functions that has both properties of boundedness and positivity numerical results show the sharpness of our estimates

the l2 lefschetz numbers of hecke operators introduction 5 1 hecke operators and l2 cohomology 52 the spectral decomposition of cohomology 6 3 application of the trace formula 54 the functions pa t 5 5 proof of theorem 5 1 6 the main formula appendix bibliography