l2 analytic torsion abstract in this paper we introduce a new differential invariant called l2 analytic torsion for closed manifolds with positive decay and whose universal covers have trivial l2 cohomology l2 analytic torsion can be thought of as a suitable generalization of the ray singer analytic torsion we establish various functorial properties of l2 analytic torsion and also compute it for odd dimensional closed hyperbolic manifolds with the help of results from fried

a hierarchical nxfem for fictitious domain simulations we suggest a fictitious domain method based on the nitsche xfem method of comput meth appl mech engrg 2002 191 5537 5552 that employs a band of elements adjacent to the boundary in contrast the classical fictitious domain method uses lagrange multipliers on a line surface where the boundary condition is to be enforced the idea can be seen as an extension of the chimera method of esaim math model numer anal 2003 37 495 514 but with a hierarchical representation of the discontinuous solution field the hierarchical formulation is better suited for moving fictitious boundaries since the stiffness matrix of the underlying structured mesh can be retained during the computations our technique allows for optimal convergence properties irrespective of the order of the underlying finite element method

analytic torsion for twisted de rham complexes we dene analytic torsion x e h 2 det h x e h for the twisted de rham complex consisting of the spaces of dieren tial forms on a compact oriented riemannian man ifold x valued in a at vector bundle e with a dieren tial given by r e h where r e is a at connection on e h is an odd degree closed dieren tial form on x and h x e h denotes the cohomology of this z2 graded complex the denition uses pseudo dieren tial operators and residue traces we show that when dim x is odd x e h is independent of the choice of metrics on x and e and of the representative h in the cohomology class h we explain the relation to generalized geometry when h is a 3 form we demonstrate some basic func torial properties when h is a top degree form we compute the torsion dene its simplicial counterpart and prove an analogue of the cheeger mtheorem we also establish the rela tionship of an invariant form of the analytic torsion for t dual circle bundles with 3 form uxes

eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps we study eta invariants of dirac operators and regularized determinants of dirac lapla cians over hyperbolic manifolds with cusps and their relations with selberg zeta functions using the selberg trace formula and a detailed analysis of the unipotent orbital integral we show that the eta and zeta functions defined by the relative traces are regular at the origin so that we can define the eta invariant and the regularized determinant we also show that the selberg zeta function of odd type has a meromorphic extension over c prove a relation of the eta invariant and a certain value of the selberg zeta function of odd type and derive a corresponding functional equation these results generalize the earlier work of john millson to hyperbolic manifolds with cusps we also prove that the selberg zeta function of even type has a meromorphic extension over c relate it to the regularized determinant and obtain a corresponding functional equation

an algorithm for the computation of eigenvalues spectral zeta functions and zeta determinants on hyperbolic surfaces we present a rigorous scheme that makes it possible to compute eigenvalues of the laplace operator on hyperbolic surfaces within a given precision the method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions it can be applied to check whether or not there is an eigenvalue in an e neighborhood of a given number 0 this makes it possible to find all the eigenvalues in a specified interval up to a given precision with rigorous error estimates the method converges exponentially fast with the number of basis functions used combining the knowledge of the eigenvalues with the selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds as an example we calculate the spectral determinant and the casimir energy of the bolza surface and other surfaces

geometric zeta functions of locally symmetric spaces the theory of geometric zeta functions for locally symmetric spaces is generalized to the case of higher rank spaces we show that the zeta functions can be continued to meromorphic functions on the plane describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalizes a conjecture of patterson we also extend the range of zeta functions in considering higher dimensional flats

an hp adaptive mixed discontinuous galerkin fem for nearly incompressible linear elasticity we develop the a posteriori error estimation of mixed hp version discontinuous galerkin finite element methods for nearly incompressible elasticity problems in two space dimensions computable upper and lower bounds on the error measured in terms of a natural mesh dependent energy norm are derived the bounds are explicit in the local mesh sizes and approximation orders and are independent of the locking parameter a series of numerical experiments are presented which demonstrate the performance of the proposed error estimator within an automatic hp adaptive refinement procedure

quasi optimal convergence rate of an adaptive discontinuous galerkin method we analyze an adaptive discontinuous finite element method adfem for symmetric second order linear elliptic operators the method is formulated on nonconforming meshes made of simplices or quadrilaterals with any polynomial degree and in any dimension geq2 we prove that the adfem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops we design a refinement procedure that maintains the level of nonconformity uniformly bounded and prove that the approximation classes using continuous and discontinuous finite elements are equivalent the geometric decay and the equivalence of classes are instrumental in deriving the optimal cardinality of the adfem we show that the adfem and the afem on nonconforming meshes yields a decay rate of energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

twisted perron frobenius theorem and l functions abstract a theorem of perron frobenius type and its twisted version are established in a setting of topological graphs the applications include a partial extension of selberg s results on his zeta functions and a result by parry and pollicott on meromorphic continuations of dynamical zeta functions to certain l functions associated to a dynamical system of anosov type

ruelle zeta function at zero for surfaces we show that the ruelle zeta function for a negatively curved oriented surface vanishes at zero to the order given by the absolute value of the euler characteristic this result was previously known only in constant curvature

guaranteed and robust discontinuous galerkin a posteriori error estimates for convection diffusion reaction problems we propose and study a posteriori error estimates for convection diffusion reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior penalty discontinuous galerkin methods our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction we first derive locally computable estimates for the error measured in the energy semi norm these estimates are evaluated using h div w conforming diffusive and convective flux reconstructions thereby extending the previous work on pure diffusion problems the resulting estimates are semi robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local peclet and damkohler numbers fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy semi norm a dual norm of the skew symmetric part of the differential operator and a suitable contribution of the interelement jumps of the discrete solution numerical experiments are presented to illustrate the theoretical results

hierarchical a posteriori error estimators for mortar finite element methods with lagrange multipliers hierarchical a posteriori error estimators are introduced and analyzed for mortar finite element methods a weak continuity condition at the interfaces is enforced by means of lagrange multipliers the two proposed error estimators are based on a defect correction in higher order finite element spaces and an adequate hierarchical two level splitting the first provides upper and lower bounds for the discrete energy norm of the mortar finite element solution whereas the second also estimates the error for the lagrange multiplier it is shown that an appropriate measure for the nonconformity of the mortar finite element solution is the weighted l2 norm of the jumps across the interfaces

meromorphic zeta functions for analytic flows we extend to hyperbolic flows in all dimensions rugh s results on the meromorphic continuation of dynamical zeta functions in particular we show that the ruelle zeta function of a negatively curved real analytic manifold extends to a meromorphic function on the complex plane

goal oriented error estimation based on equilibrated flux reconstruction for finite element approximations of elliptic problems abstract we propose an approach for goal oriented error estimation in finite element approximations of second order elliptic problems that combines the dual weighted residual method and equilibrated flux reconstruction methods for the primal and dual problems the objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution it is only assumed here that the discretization methods come with a priori error estimates and an equilibrated flux reconstruction algorithm a high order discontinuous galerkin dg method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes one contribution of the paper is to show how the order of the dg method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method a dg method or a mixed raviart thomas method is used for the solution of the primal problem numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

discontinuous galerkin finite element methods for interface problems a priori and a posteriori error estimations discontinuous galerkin dg finite element methods were studied by many researchers for second order elliptic partial differential equations and a priori error estimates were established when the solution of the underlying problem is piecewise h 3 2 epsilon smooth with epsilon 0 however elliptic interface problems with intersecting interfaces do not possess such a smoothness in this paper we establish a quasi optimal a priori error estimate for interface problems whose solutions are only h 1 alpha smooth with alpha in 0 1 and hence fill a theoretical gap of the dg method for elliptic problems with low regularity the second part of the paper deals with the design and analysis of robust residual and recovery based a posteriori error estimators theoretically we show that the residual and recovery estimators studied in this paper are robust with respect to the dg norm i e their reliability and efficiency bounds do not depend on the jump provided that the distribution of coefficients is locally quasi monotone

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

infinitely divisible distributions in turbulence the imbedding of the scale similarity of random fields into the theory of infinitely divisible probability distributions is considered the general probability distribution for the breakdown coefficients of turbulent energy dissipation is obtained along with corresponding similarity exponents related issues of self similarity and asymptotic behavior of statistical characteristics are also considered

on certain definite integrals which arise in automorphic lie theory calculates in closed form a family of definite integrals in b a c n 0 1 2 3 which arise in the calculation of regularized functional determinants associated with a compact space form x of a rank 1 riemannian symmetric space in the special case when n 0 b 1 2 a pi c 1 and x is a riemann surface the integral i0 1 2 pi 1 is known for example in the context of polyakov string theory

analytic torsion of complete hyperbolic manifolds of finite volume abstract in this paper we define the analytic torsion for a complete oriented hyperbolic manifold of finite volume it depends on a representation of the fundamental group for manifolds of odd dimension we study the asymptotic behavior of the analytic torsion with respect to certain sequences of representations obtained by restriction of irreducible representations of the group of isometries of the hyperbolic space to the fundamental group

an improved a priori error analysis of nitsche s method for robin boundary conditions in a previous paper juntunen and stenberg in math comput 78 1353 1374 2009 we have extended nitsche s method in abhandlungen aus dem mathematischen seminar der universitat hamburg 1970 71 for the poisson equation with general robin boundary conditions the analysis required that the solution is in h s with s 3 2 here we give an improved error analysis using a technique proposed by gudi math comput 79 2169 2189 2010

heat kernels on covering spaces and topological invariants it is well known that there are relationships between the heat flow acting on differential forms on a closed oriented manifold m and the topology of m from hodge theory one can recover the betti numbers of m from the heat flow furthermore ray and singer 42 defined an analytic torsion a smooth invariant of acyclic flat bundles on m and conjectured that it equals the classical reidemeister torsion this conjecture was proved to be true independently by cheeger 7 and muller 40 the analytic torsion is nonzero only on odd dimensional manifolds and behaves in some ways as an odd dimensional counterpart of the euler characteristic 20 if m is not simply connected then there is a covering space analog of the betti numbers using the heat flow on the universal cover m one can define the l2 betti numbers of m 1 by taking the trace of the heat kernel not in the ordinary sense but as an element of a certain type ii von neumann algebra more concretely this amounts to integrating the local trace of the heat kernel over a fundamental domain in m in 11 we summarize this theory we consider the covering space analog of the analytic torsion this i analytic torsion has the same relation to the i cohomology as the ordinary analytic torsion bears to de rham cohomology in 111 we define the i analytic torsion 9 m under a technical assumption which we discuss later and prove its basic properties we show that m is a smooth invariant of manifolds whose l2 betti numbers vanish the proof is similar to that of the analogous statement for the ordinary analytic torsion but requires some care because of the possible slow decay of the heat kernels for large time in order to know if there are interesting examples of m we compute it in the case where m admits a hyperbolic metric it is clear that is proportionate to the volume of the hyperbolic metric which is

homological identities for closed orbits the classical lefschetz formula expresses a relationship between the fixed points of certain transformations and homological invariants when applied to continuous systems one obtains the hopf formula relating the singularities of a flow to the euler characteristic of the underlying manifold the main goal of this paper is to prove an analogous formula that relates the closed orbits of a stable flow to a homological invariant of the manifold there are serious obstruction to finding such a formula for flows the first is that there are flows on any closed manifold of dimension 4 of zero euler characteristic with no periodic orbits whatsoever 27 these flows however are unstable by pugh s c a closing lemma 19 so it is natural to restrict attention to flows that satisfy axiom a and the no cycles property since smale has shown that these flows are o stable 23 secondly there are gradient flows on any closed manifold that are axiom a no cycles and have no closed orbits 24 so it is natural to study only nonsingular flows the possibility that a lefschetz type formula for such flows existed was suggested to us by r bowen even nonsingular morse smale flows however are difficult to deal with since asimov s geometric index is not an invariant of the nonsingular homotopy class 2 we must further restrict the class of flows to assume that the closed orbits relate in some manner to the homology of the underlying manifold we define in sect 2 the class of homology proper flows such a flow has the property that closed orbits of large period lie in large homology classes in sect 7 we show that a homology proper axiom a no cycles flow does satisfy a lefschetz type formula that relates invariants of the closed orbits to a manifold invariant this formula is closely related to franks study of smale flows on s 3 in which the linking numbers of closed orbits are used to compute the alexander polynomial of a certain link indeed the manifold invariant used in sect 7 is a close analogue of an alexander polynomial that we call the alexander

the zeta functions of ruelle and selberg for hyperbolic manifolds with cusps in this paper we study the ruelle zeta function and the selberg zeta functions attached to the fundamental representations for real hyperbolic manifolds with cusps in particular we show that they have meromorphic extensions to mathbb c and satisfy functional equations we also derive the order of the singularity of the ruelle zeta function at the origin to prove these results we completely analyze the weighted unipotent orbital integrals on the geometric side of the selberg trace formula when test functions are defined for the fundamental representations

mixed discontinuous galerkin approximation of the maxwell operator we introduce and analyze a discontinuous galerkin discretization of the maxwell operator in mixed form here all the unknowns of the underlying system of partial differential equations are approximated by discontinuous finite element spaces of the same order for piecewise constant coefficients the method is shown to be stable and optimally convergent with respect to the mesh size numerical experiments highlighting the performance of the proposed method for problems with both smooth and singular analytical solutions are presented

self similarity properties of natural images resemble those of turbulent flows we show that the statistics of an edge type variable in natural images exhibits self similarity properties which resemble those of local energy dissipation in turbulent flows our results show that selfsimilarity and extended self similarity hold remarkably for the statistics of the local edge variance and that the very same models can be used to predict all of the associated exponents these results suggest using natural images as a laboratory for testing more elaborate scaling models of interest for the statistical description of turbulent flows the properties we have exhibited are relevant for the modeling of the early visual system they should be included in models designed for the prediction of receptive fields s0031 9007 97 05190 9

analytic torsion versus reidemeister torsion on hyperbolic 3 manifolds with cusps for a non compact hyperbolic 3 manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of mathrm sl 2 mathbb c to the corresponding reidemeister torsion while the analytic torsion is a spectral invariant of the manifold the reidemeister torsion is of combinatorial nature our proof rests on an expression of the analytic torsion in terms of special values of ruelle zeta functions as well as on recent work of pere menal ferrer and joan porti

forms on vector bundles over hyperbolic manifolds and the trace anomaly we study gauge theories based on abelian p forms on real compact hyperbolic manifolds an explicit formula for the trace anomaly corresponding to skew symmetric tensor fields is obtained by using zeta function regularization and the trace tensor kernel formula explicit exact and numerical values of the anomaly for p forms of order up to p 4 in spaces of dimension up to n 10 are then calculated

convergence of adaptive discontinuous galerkin approximations of second order elliptic problems a residual type a posteriori error estimator is introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem with dirichlet neumann type boundary conditions an adaptive algorithm using this estimator together with specific marking and refinement strategies is constructed and shown to achieve any specified error level in the energy norm in a finite number of cycles the convergence rate is in effect linear with a guaranteed error reduction at every cycle results of numerical experiments are presented

analytic torsion and ruelle zeta functions for hyperbolic manifolds with cusps abstract in this paper we derive a relationship of the leading coefficient of the laurent expansion of the ruelle zeta function at s 0 and the analytic torsion for hyperbolic manifolds with cusps here the analytic torsion is defined by a certain regularized trace following melrose r b melrose the atiyah patodi singer index theorem res notes math vol 4 a k peters ltd wellesley ma 1993 this extends the result of fried which was proved for the compact case in d fried analytic torsion and closed geodesics on hyperbolic manifolds invent math 84 3 1986 523 540 to a noncompact case

two level additive schwarz methods for a discontinuous galerkin approximation of second order elliptic problems we present some two level nonoverlapping and overlapping additive schwarz methods for a discontinuous galerkin method for solving second order elliptic problems it is shown that the condition numbers of the preconditioned systems are of the order o frac h h for the nonoverlapping schwarz methods and of the order o frac h delta for the overlapping schwarz methods where h and h stand for the fine mesh size and the coarse mesh size respectively and delta denotes the size of the overlaps between subdomains numerical experiments are provided to gauge the efficiency of the methods and to validate the theory

asymptotics of the spectrum and the selberg zeta function on the space of riemann surfaces letz s r be the selberg zeta function of a compact riemann surfacer we study the behavior ofz s r asr tends to infinity in the moduli space of stable curves the main result is an estimate forz s r valid fors in a neighborhood depending only on the genus ofs 1 our analysis gives an alternate proof of the belavin knizhnik double pole result 5

fictitious domain finite element methods using cut elements i a stabilized lagrange multiplier method we propose a fictitious domain method where the mesh is cut by the boundary the primal solution is computed only up to the boundary the solution itself is defined also by nodes outside the domain

convergence of the discontinuous finite volume method for elliptic problems with minimal regularity this paper investigates convergence of the discontinuous finite volume method dfvm under minimal regularity assumptions on solutions of second order elliptic boundary value problems conventional analysis requires the solutions to be in sobolev spaces h 1 s s 12 here we assume the solutions are in h 1 s s 0 and employ the techniques developed in gudi 2010 18 20 to derive error estimates in a mesh dependent energy norm and the l 2 norm for dfvm the theoretical estimates are illustrated by numerical results which include problems with corner singularity and intersecting interfaces

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

some analytic continuations of the barnes zeta function in two and higher dimensions abstract formulas for the analytic continuation of the barnes zeta function and some affine extensions thereof in two and more dimensions are constructed the expressions are used to deal with determinants of multidimensional harmonic oscillators an example is therewith obtained of the multiplicative anomaly or defect associated with the most common definition due to ray and singer d b ray reidemeister torsion and the laplacian on lens spaces adv math 4 1970 109 126 d b ray i m singer r torsion and the laplacian on riemannian manifolds adv math 7 1971 145 201 d b ray i m singer analytic torsion for complex manifolds ann math 98 1973 154 177 of determinant of a pseudodifferential operator admitting a zeta function m kontsevich s vishik geometry of determinants of elliptic operators in functional analysis on the eve of the 21st century vol 1 1995 pp 173 197

statistics of natural images scaling in the woods we study the statistics of an ensemble of images taken in the woods distributions of local quantities such as contrast are scale invariant and have nearly exponential tails power spectra exhibit scaling with a nontrivial exponent these data limit the information content of natural images and point to the importance of gain control strategies in visual processing

convergence analysis of an adaptive interior penalty discontinuous galerkin method we study the convergence of an adaptive interior penalty discontinuous galerkin ipdg method for a two dimensional model second order elliptic boundary value problem based on a residual type a posteriori error estimator we prove that after each refinement step of the adaptive scheme we achieve a guaranteed reduction of the global discretization error in the mesh dependent energy norm associated with the ipdg method in contrast to recent work on adaptive ipdg methods o karakashian and f pascal convergence of adaptive discontinuous galerkin approximations of second order elliptic problems preprint university of tennessee knoxville tn 2007 the convergence analysis does not require multiple interior nodes for refined elements of the triangulation in fact it will be shown that bisection of the elements is sufficient the main ingredients of the proof of the error reduction property are the reliability and a perturbed discrete local efficiency of the estimator a bulk criterion that takes care of a proper selection of edges and elements for refinement and a perturbed galerkin orthogonality property with respect to the energy inner product the results of numerical experiments are given to illustrate the performance of the adaptive method

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

torsion and closed geodesics on complex hyperbolic manifolds letx be a compact complex manifold covered by complex hyperbolicn space with the induced metric each stable horocycle has a cocomplex structure preserved by the geodesic flow to a closed geodesic one can thus associate a piece of the poincare map with a holomorphic fixed point the resulting atiyah bott fixed point indices together with the length and multiplicity of as a periodic orbit determine the contribution of to certain zeta functionsrp z 0 p n from the leading coefficient ofr p atz 0 and the hodge numbersh ij x we calculate the ray singer bar partial torsiont p x this indicates that the known connections between torsion and the dynamical features of closed orbits continue to hold in the holomorphic category

microcanonical multifractal formalism a geometrical approach to multifractal systems part i singularity analysis multifractal formalism in the microcanonical framework has proved to be a valuable approach to understand and analyze complex signals typically associated with natural phenomena in scale invariant systems in this paper we discuss the multifractal microcanonical formalism in a comprehensive unified way including new theoretical proofs and validation tests on real signals so completing some known gaps in the foundations of this theory we also review the latest advances and describe the present perspectives in this field some technical details on the implementation of involved algorithms and relevant open issues are also discussed

extremal metrics for spectral functions of dirac operators in even and odd dimensions abstract let m n g be a closed smooth riemannian spin manifold and denote by its atiyah singer dirac operator we study the variation of riemannian metrics for the zeta function and functional determinant of 2 and prove finiteness of the morse index at stationary metrics and local extremality at such metrics under general i e not only conformal change of metrics in even dimensions which is also a new case for the conformal laplacian the relevant stability operator is of log polyhomogeneous pseudodifferential type and we prove new results of independent interest on the spectrum for such operators we use this to prove local extremality under variation of the riemannian metric which in the important example when m n g is the round n sphere gives a partial verification of branson s conjecture on the pattern of extremals thus det 2 has a local max max min min when the dimension is 4 k 4 k 1 4 k 2 4 k 3 respectively

a nonconforming finite element method for the stationary navier stokes equations approximations to solutions of the inhomogeneous boundary value problem for the navier stokes equations are constructed via a nonstandard finite element method the velocity field is approximated using piecewise solenoidal functions that are totally discontinuous across interelement boundaries but which are pointwise divergence free on each element the pressure is approximated by c0 functions optimal rates of convergence results are obtained requiring only local quasi uniformity assumptions on the meshes

eta invariant and selberg zeta function of odd type over convex co compact hyperbolic manifolds abstract we show meromorphic extension and give a complete description of the divisors of a selberg zeta function of odd type z o associated to the spinor bundle on an odd dimensional convex co compact hyperbolic manifold h 2 n 1 as a byproduct we do a full analysis of the spectral and scattering theory of the dirac operator on asymptotically hyperbolic manifolds we show that there is a natural eta invariant d associated to the dirac operator d over a convex co compact hyperbolic manifold h 2 n 1 and that exp i d z o 0 thus extending millson s formula to this setting under some assumption on the exponent of convergence of poincare series for the group we also define an eta invariant for the odd signature operator and we show that for schottky 3 dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the zograf factorization formula relating two natural kahler potentials for weil petersson metric on schottky space

heat kernels and super determinants of laplace operators on super riemann surfaces heat kernels of laplacians on superfields of arbitrary tensor weight on super riemann surfaces are constructed and are used to compute the super determinants of these operators in terms of the selberg super zeta function

discontinuous galerkin finite element approximation of nonlinear second order elliptic and hyperbolic systems we develop the convergence analysis of discontinuous galerkin finite element approximations to symmetric second order quasi linear elliptic and hyperbolic systems of partial differential equations in divergence form in a bounded spatial domain in mathbb r d subject to mixed dirichlet neumann boundary conditions optimal order asymptotic bounds are derived on the discretization error in each case without requiring the global lipschitz continuity or uniform monotonicity of the stress tensor instead only local smoothness and a garding inequality are used in the analysis

the fried conjecture in small dimensions we study the twisted ruelle zeta function x s for smooth anosov vector fields x acting on flat vector bundles over smooth compact manifolds in dimension 3 we prove fried conjecture relating reidemeister torsion and x 0 in higher dimensions we show more generally that x 0 is locally constant with respect to the vector field x under a spectral condition as a consequence we also show fried conjecture for anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic 3 manifolds this gives the first examples of non analytic anosov flows and geodesic flows in variable negative curvature where fried conjecture holds true

the multifractal structure of contrast changes in natural images from sharp edges to textures we present a formalism that leads naturally to a hierarchical description of the different contrast structures in images providing precise definitions of sharp edges and other texture components within this formalism we achieve a decomposition of pixels of the image in sets the fractal components of the image such that each set contains only points characterized by a fixed strength of the singularity of the contrast gradient in its neighborhood a crucial role in this description of images is played by the behavior of contrast differences under changes in scale contrary to naive scaling ideas where the image is thought to have uniform transformation properties field 1987 each of these fractal components has its own transformation law and scaling exponents a conjecture on their biological relevance is also given

efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions in recent years a number of finite element methods have been formulated for the solution of partial differential equations on complex geometries based on nonmatching or overlapping meshes examples of such methods are the fictitious domain method the extended finite element method and nitsche s method in all these methods integrals must be computed over cut cells or subsimplices which is challenging to implement especially in three space dimensions in this note we address the main challenges of such an implementation and demonstrate good performance of a fully general code for automatic detection of mesh intersections and integration over cut cells and subsimplices as a canonical example of an overlapping mesh method we consider nitsche s method which we apply to poisson s equation and a linear elastic problem

convergence of an adaptive discontinuous galerkin method for elliptic interface problems abstract we prove a basic error contraction result of an adaptive discontinuous galerkin method for an elliptic interface problem the interface conditions considered model mass transfer of solutes through semi permeable membranes and other filtering processes the adaptive algorithm is based on a residual type a posteriori error estimator with a bulk refinement criterion the a posteriori error bound is derived under the assumption that the triangulation is aligned with the interfaces although crucially extremely general curved element shapes are also allowed resolving the interface geometry exactly as a corollary convergence of the adaptive discontinuous galerkin method for non essential neumann and or robin type boundary conditions posed on general curved boundaries also follows numerical experiments are also presented

reconstructing images from their most singular fractal manifold real world images are complex objects difficult to describe but at the same time possessing a high degree of redundancy a previous on the statistical properties of natural images reveals that natural images can be viewed through different partitions which are essentially fractal in nature one particular fractal component related to the most singular sharpest transitions in the image seems to be highly informative about the whole scene we show how to decompose the image into their fractal components we see that the most singular component is related to but not coincident with the edges of the objects present in the scenes we propose a new simple method to reconstruct the image with information contained in that most informative component we see that the quality of the reconstruction is strongly dependent on the capability to extract the relevant edges in the determination of the most singular set we discuss the results from the perspective of coding proposing this method as a starting point for future developments

an adaptive stabilized conforming finite element method via residual minimization on dual discontinuous galerkin norms abstract we design and analyze a new adaptive stabilized finite element method we construct a discrete approximation of the solution in a continuous trial space by minimizing the residual measured in a dual norm of a discontinuous test space that has inf sup stability we formulate this residual minimization as a stable saddle point problem which delivers a stabilized discrete solution and a residual representation that drives the adaptive mesh refinement numerical results on an advection reaction model problem show competitive error reduction rates when compared to discontinuous galerkin methods on uniformly refined meshes and smooth solutions moreover the technique leads to optimal decay rates for adaptive mesh refinement and solutions having sharp layers

delocalized l2 invariants abstract we define extensions of the l 2 analytic invariants of closed manifolds called delocalized l 2 invariants these delocalized invariants are constructed in terms of a nontrivial conjugacy class of the fundamental group we show that in many cases they are topological in nature we show that the marked length spectrum of an odd dimensional hyperbolic manifold can be recovered from its delocalized l 2 analytic torsion there are technical convergence questions

energy norm a posteriori error estimation for discontinuous galerkin methods in this paper we present a residual based a posteriori error estimate of a natural mesh dependent energy norm of the error in a family of discontinuous galerkin approximations of elliptic problems the theory is developed for an elliptic model problem in two and three spatial dimensions and general nonconvex polygonal domains are allowed we also present some illustrating numerical examples

analytic and topological torsion for manifolds with boundary and symmetry letg be a finite group acting on a riemannian manifoldm by isometries we introduce analytic torsion an m m1 v r z repr g pl torsion pl m m1 v k1 rg poincare torsion pd m m1 v k1 rg and euler characteristic m m1 v repr g for m the disjoint union of m1 and m2 and v an equivariant coefficient system the analytic torsionis defined in terms of the spectrum of the laplace operator the pl torsionis based on the cellular chain complex and poincare torsionmeasures the failure of equivariant poincare duality in the pl setting which does hold in the analytic context denote by repr g the subgroup of repr g generated by the irreducible representations of real or complex type we define an isomorphism

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

data oscillation and convergence of adaptive fem data oscillation is intrinsic information missed by the averaging process associated with finite element methods fem regardless of quadrature ensuring a reduction rate of data oscillation together with an error reduction based on a posteriori error estimators we construct a simple and efficient adaptive fem for elliptic partial differential equations pdes with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance

robust a posteriori error estimates for conforming and nonconforming finite element methods for convection diffusion problems a posteriori error estimation is carried out within a unified framework for various conforming and nonconforming finite element methods for convection diffusion problems our main contribution is finding an appropriate norm to measure the error which incorporates a discrete energy norm a discrete dual semi norm of the convective derivative and jumps of the approximate solution over element faces edges in two dimensions the error estimator is shown to be robust with respect to the peclet number in the sense of the modified norm based on a general error decomposition we show that the key ingredient of error estimation is the estimation on the consistency error related to the particular numerical scheme and the remaining terms can be bounded in a unified way the numerical results are presented to illustrate the robustness and practical performance of the estimator in an adaptive refinement strategy

analytic torsion and closed geodesics on hyperbolic manifolds pour une variete hyperbolique orientee fermee x et une representation orthogonale de 1 x on definit une fonction zeta de type ruelle r s pour re s grand comme le produit sur les geodesiques closes premieres des facteurs det i exp sl ou l est la longueur de on continue analytiquement r s et on calcule son terme principal en s 0

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

horizontal lift of the brownian motion on the hyperbolic plane and the selberg trace formula by using an explicit representation for the horizontal lift of the brownian motion on the poincare upper half plane h2 we show an expression for the heat kernel for the de rham kodaira laplacian on h2 we apply the result to a study on the selberg trace formula

multiscaling and information content of natural color images naive scale invariance is not a true property of natural images natural monochrome images possess a much richer geometrical structure which is particularly well described in terms of multiscaling relations this means that the pixels of a given image can be decomposed into sets the fractal components of the image with well defined scaling exponents turiel and parga neural comput 12 763 2000 here it is shown that hyperspectral representations of natural scenes also exhibit multiscaling properties observing the same kind of behavior a precise measure of the informational relevance of the fractal components is also given and it is shown that there are important differences between the intrinsically redundant red green blue system and the decorrelated one defined in ruderman cronin and chiao j opt soc am a 15 2036 1998

discontinuous finite element methods for interface problems robust a priori and a posteriori error estimates for elliptic interface problems in two and three dimensions this paper studies a priori and residual based a posteriori error estimations for the crouzeix raviart nonconforming and the discontinuous galerkin finite element approximations it is shown that both the a priori and the a posteriori error estimates are robust with respect to the diffusion coefficient i e constants in the error bounds are independent of the jump of the diffusion coefficient the a priori estimates are also optimal with respect to local regularity of the solution moreover we obtained these estimates with no assumption on the distribution of the diffusion coefficient

on l2 error estimate for nonsymmetric interior penalty galerkin approximation to linear elliptic problems with nonhomogeneous dirichlet data in this paper we present improved a priori error estimates for a nonsymmetric interior penalty galerkin method nipg with super penalty for the problem email protected f in w and u g on email protected using piecewise polynomials of degree less than or equal to r our new l 2 error estimate is of order h r r 1 2 when email protected h r 1 2 email protected and is optimal i e of order h r r 1 when email protected h r 1 email protected where h denotes the mesh size numerical experiments are presented to illustrate the theoretical results

anomalies and analytic torsion on hyperbolic manifolds the global additive and multiplicative properties of the laplacian on j forms and related zeta functions are analyzed the explicit form of zeta functions on a product of closed oriented hyperbolic manifolds hd and of the multiplicative anomaly are derived we also calculate in an explicit form the analytic torsion associated with a connected sum of such manifolds

weak galerkin based a posteriori error estimates for second order elliptic interface problems on polygonal meshes abstract in this paper we present a posteriori error estimate of weak galerkin wg finite element methods based on the second order elliptic interface problems this estimate can be applied to polygonal meshes or meshes with hanging nodes the reliability and efficiency of the designed error estimator has been proved in this work extensive numerical tests are performed to validate our algorithm these results demonstrate the effectiveness of the adaptive mesh refinement guided by the proposed error estimator

adaptive finite element simulation of incompressible flows by hybrid continuous discontinuous galerkin formulations in this work we design hybrid continuous discontinuous finite element spaces that permit discontinuities on nonmatching element interfaces of nonconforming meshes then we develop an equal order stabilized finite element formulation for incompressible flows over these hybrid spaces which combines the element interior stabilization of supg type continuous galerkin formulations and the jump stabilization of discontinuous galerkin formulations optimal stability and convergence results are obtained for the adaptive setting we use a standard error estimator and marking strategy numerical experiments show the optimal accuracy of the hybrid algorithm for both uniformly and adaptively refined nonconforming meshes the outcome of this work is a finite element formulation that can naturally be used on nonconforming meshes as discontinuous galerkin formulations while keeping the much lower cpu cost of continuous galerkin formulations

l2 torsion invariants we discuss a generalisation of reidemeister franz torsion which applies to infinite dimensional representations of the fundamental group by utilising the theory of finite von neumann algebras our results apply to a class of closed oriented manifolds which are l2 acyclic in the sense of having trivial l2 cohomology evidence is presented for equality of the essentially combinatorial torsion introduced here with an analytic l2 ray singer torsion defined by the second named author

on determinants of laplacians on riemann surfaces determinants of laplacians on tensors and spinors of arbitrary weights on compact hyperbolic riemann surfaces are computed in terms of values of selberg zeta functions at half integer points

discontinuous galerkin methods for mass transfer through semipermeable membranes a discontinuous galerkin dg method for the numerical solution of initial boundary value multicompartment partial differential equation models interconnected with interface conditions is presented and analyzed the study of interface problems is motivated by models of mass transfer of solutes through semipermeable membranes more specifically a model problem consisting of a system of semilinear parabolic advection diffusion reaction partial differential equations in each compartment equipped with respective initial and boundary conditions is considered nonlinear interface conditions modeling selective permeability congestion and partial reflection are applied to the compartment interfaces an interior penalty dg method is presented for this problem and it is analyzed in the space discrete setting the a priori analysis shows that the method yields optimal a priori bounds provided the exact solution is sufficiently smooth numerical experiments indicate agreement with the theoretical bounds and highlight the stability of the numerical method in the advection dominated regime