lattice valued spaces fuzzy convergence abstract an alternative set of axioms is given for the study of lattice valued convergence spaces these axioms extend the notion of a probabilistic convergence space in the special case when the lattice is l 0 1 the category described here is shown to be topological cartesian closed and extensional

real interpolation of spaces of differential forms in this paper we study interpolation of hilbert spaces of differential forms using the real method of interpolation we show that the scale of fractional order sobolev spaces of differential l forms in hs with exterior derivative in hs can be obtained by real interpolation our proof heavily relies on the recent discovery of smoothed poincare lifting for differential forms m costabel and a mcintosh on bogovskii and regularized poincare integral operators for de rham complexes on lipschitz domains math z 265 2 297 320 2010 they enable the construction of universal extension operators for sobolev spaces of differential forms which in turns pave the way for a fourier transform based proof of equivalences of k functionals

the zeta function of the laplacian on certain fractals we prove that the zeta function of the laplacian a on self similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane we characterise the poles compute their residues and give expressions for some special values of the zeta function furthermore we discuss the presence of oscillations in the eigenvalue counting function thereby answering a question posed by j kigami and m lapidus for this class of fractals

regularity of the maxwell equations in heterogeneous media and lipschitz domains abstract this note establishes regularity estimates for the solution of the maxwell equations in lipschitz domains with non smooth coefficients and minimal regularity assumptions the argumentation relies on elliptic regularity estimates for the poisson problem with non smooth coefficients

dirac operators and the calculation of the connes metric on arbitrary infinite graphs as an outgrowth of our investigation of non regular spaces within the context of quantum gravity and non commutative geometry we develop a graph hilbert space framework on arbitrary infinite graphs and use it to study spectral properties of graph laplacians and graph dirac operators we define a spectral triplet sharing most of the properties of what connes calls a spectral triple with the help of this scheme we derive an explicit expression for the connes distance function on general directed or undirected graphs we derive a series of a priori estimates and calculate it for a variety of examples of graphs as a possibly interesting side we show that the natural setting for approaching such problems may be the framework of non linear programming or optimization we compare our results arrived at within our particular framework with those of other authors and show that the seeming differences depend on the use of different graph geometries and or dirac operators

on the negative spectrum of the hierarchical schrodinger operator abstract this paper is devoted to the spectral theory of the schrodinger operator on the simplest fractal dyson s hierarchical lattice an explicit description of the spectrum eigenfunctions resolvent and parabolic kernel are provided for the unperturbed operator i e for the dyson hierarchical laplacian positive spectrum is studied for the perturbations of the hierarchical laplacian since the spectral dimension of the operator under consideration can be an arbitrary positive number the model allows a continuous phase transition from recurrent to transient underlying markov process this transition is also studied in the paper

lattice valued convergence spaces and regularity we define a regularity axiom for lattice valued convergence spaces where the lattice is a complete heyting algebra to this end we generalize the characterization of regularity by a dual form of a diagonal condition we show that our axiom ensures that a regular t1 space is separated and that regularity is preserved under initial constructions further we present an extension theorem for a continuous mapping from a subspace to a regular space a characterization in the restricted case that the lattice is a complete boolean algebra in terms of the closure of an l filter is given

diagonal conditions for lattice valued uniform convergence spaces we define diagonal conditions for lattice valued uniform convergence spaces and show that these conditions are preserved under initial constructions further the forgetful functor from the category of stratified lattice valued uniform convergence spaces into the category of stratified lattice valued limit spaces maps these conditions to corresponding conditions we especially define uniform regularity and characterize this condition by certain closures of lattice valued filters as an application we generalize an extension theorem for uniformly continuous mappings

on p robust saturation for hp afem for the poisson problem in two dimensions we consider the standard adaptive finite element loop solve estimate mark refine with estimate being implemented using the p robust equilibrated flux estimator and mark being drfler marking as a refinement strategy we employ p refinement we investigate the question by which amount the local polynomial degree on any marked patch has to be incremented in order to achieve a p independent error reduction we show that the analysis can be transferred from the patches to a reference triangle and therein we provide clear cut computational evidence that any increment proportional to the polynomial degree for any fixed proportionality constant yields the desired reduction the resulting adaptive method can be turned into an instance optimal hp adaptive method by the addition of a coarsening routine

the lower and upper p topological p regular modifications for lattice valued convergence spaces in this paper the notions of lower and upper topological regular modifications with respect to a stratified l generalized convergence structure and those with respect to a stratified l strong generalized convergence structure are defined and studied the basic properties and characterizations are presented in particular it is proved that the lower modifications behave reasonably well relative to final structures whereas the upper modifications exhibit comparable behavior relative to initial structures

guaranteed and robust a posteriori bounds for laplace eigenvalues and eigenvectors a unified framework this paper develops a general framework for a posteriori error estimates in numerical approximations of the laplace eigenvalue problem applicable to all standard numerical methods guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given as well as on the energy error in the approximation of the associated eigenvector the bounds are valid under the sole condition that the approximate i th eigenvalue lies between the exact i 1 th and i 1 th eigenvalue where the relative gaps are sufficiently large we give a practical way how to check this the precision of the resulting estimates depends on these relative gaps our bounds feature no unknown solution regularity or polynomial degree dependent constant are optimally convergent efficient and polynomial degree robust under a further explicit a posteriori minimal resolution condition the multiplicative constant in our estimates can be reduced by a fixed factor moreover when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants this multiplicative constant can be brought to the optimal value of 1 with mesh refinement applications of our framework to nonconforming discontinuous galerkin and mixed finite element approximations of arbitrary polynomial degree are provided along with numerical illustrations our key ingredient are equivalences between the i th eigenvalue error the associated eigenvector energy error and the dual norm of the residual we extend them in an appendix to the generic class of bounded below self adjoint operators with compact resolvent

the divergence equation in rough spaces abstract we aim at extending the existence theory for the equation div v f in a bounded or exterior domain with homogeneous dirichlet boundary conditions to a class of solutions which need not have a trace at the boundary typically the weak solutions that we shall consider will belong to some besov space b p q s with s 1 1 p 1 p after generalizing the notion of a solution for this equation we propose an explicit construction by means of the classical bogovski formula this construction enables us to keep track of a marginal information about the trace of solutions in particular it ensures that the trace is zero if f is smooth enough we expect our approach to be of interest for the study of rough solutions to systems of fluid mechanics

asymptotics of the transition probabilities of the simple random walk on self similar graphs it is shown explicitly how self similar graphs can be obtained as blow up constructions of finite cell graphs c this yields a larger family of graphs than the graphs obtained by discretising continuous self similar fractals for a class of symmetrically self similar graphs we study the simple random walk on a cell graph c starting at a vertex v of the boundary of c it is proved that the expected number of returns to v before hitting another vertex in the boundary coincides with the resistance scaling factor using techniques from complex rational iteration and singularity analysis for green functions we compute the asymptotic behaviour of the n step transition probabilities of the simple random walk on the whole graph the results of grabner and woess for the sierpinski graph are generalised to the class of symmetrically self similar graphs and at the same time the error term of the asymptotic expression is improved finally we present a criterion for the occurrence of oscillating phenomena of the n step transition probabilities

locally conservative discontinuous petrov galerkin finite elements for fluid problems we develop a locally conservative formulation of the discontinuous petrov galerkin finite element method dpg for convection diffusion type problems using lagrange multipliers to exactly enforce conservation over each element we provide a proof of convergence as well as extensive numerical experiments showing that the method is indeed locally conservative we also show that standard dpg while not guaranteed to be conservative is nearly conservative for many of the benchmarks considered the new method preserves many of the attractive features of dpg but turns the normally symmetric positive definite dpg system into a saddle point problem

dirac and magnetic schrodinger operators on fractals abstract in this paper we define local dirac operators and magnetic schrodinger hamiltonians on fractals and prove their essential self adjointness to do so we use the concept of 1 forms and derivations associated with dirichlet forms as introduced by cipriani and sauvageot and further studied by the authors jointly with rockner ionescu and rogers for simplicity our definitions and results are formulated for the sierpinski gasket with its standard self similar energy form we point out how they may be generalized to other spaces such as the classical sierpinski carpet

a limiting case for the divergence equation we consider the equation text div mathbb y f with f a zero average function on the torus mathbb t d in their seminal paper bourgain and brezis j am math soc 16 2 393 426 2003 electronic proved the existence of a solution mathbb y in w 1 d cap l infty for a datum f in l d we extend their result to the critical sobolev spaces w s p with s 1 p d and p ge 2 more generally we prove a similar result in the scale of triebel lizorkin spaces we also consider the equation text div mathbb y f in a bounded domain varomega subject to zero dirichlet boundary condition

p topologicalness and p regularity for lattice valued convergence spaces in this paper p topologicalness and p regularity for jager s stratified l generalized convergence spaces and those for boustique et al s stratified l convergence spaces are studied through two different methods one method generalizes the well known fischer s dual diagonal condition the other method uses the neighborhood closure of stratified l filter in addition the p topologicalness and p regularity w r t initial constructions and final constructions are also discussed

weak uncertainty principle for fractals graphs and metric measure spaces we develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by okoudjou and strichartz we assume either an appropriate measure growth condition with respect to the effective resistance metric or in the absence of such a metric we assume the poincare inequality and reverse volume doubling property we also consider the weak uncertainty inequality in the context of nash type inequalities our results can be applied to a wide variety of metric measure spaces including graphs fractals and manifolds

equilibrated residual error estimates are p robust abstract equilibrated residual error estimators applied to high order finite elements are analyzed the estimators provide always a true upper bound for the energy error we prove that also the efficiency estimate is robust with respect to the polynomial degrees a complete theory is provided for tensor product elements in the case of simplicial elements the theorem was first based on a conjecture but the conjecture was quite recently proved by costabel and mcintosh m costabel a mcintosh on bogovskii and regularized poincare integral operators for de rham complexes on lipschitz domains irmar preprint 08 40 rennes august 2008 numerical evidence for the p robustness is provided

quasioptimality of some spectral mixed methods in this paper we construct a sequence of projectors into certain polynomial spaces satisfying a commuting diagram property with norm bounds independent of the polynomial degree using the projectors we obtain quasioptimality of some spectral mixed methods including the raviart thomas method and mixed formulations of maxwell equations we also prove some discrete friedrichs type inequalities involving curl

the regularity of the stokes operator and the fujita kato approach to the navier stokes initial value problem in lipschitz domains abstract we study the regularity of the navier stokes equations in arbitrary lipschitz domains

stratified lmn convergence tower spaces we develop a general theory of convergence for lattice valued spaces based on the concept of s stratified lm filters for different choices of the frames l and m different kinds of filters arise and suitable choices of the lattice n allow to view many existing fuzzy and probabilistic convergence spaces as special examples of our stratified lmn convergence tower spaces the stratification requires a certain relation between the frames l and m and we use a so called stratification mapping to this end the stratification condition is used to show that the resulting category is fiber small and hence cartesian closedness is equivalent to the existence of natural function spaces we give several examples for our spaces for different choices of the lattices l m and n with a special focus on enriched lm fuzzy topological spaces finally we study diagonal axioms and regularity

lattice valued convergence spaces extending the lattice context we define a category of stratified l generalized convergence spaces for the case where the lattice is an enriched cl premonoid we then investigate some of its categorical properties and those of its subcategories in particular the stratified l principal convergence spaces and the stratified l topological convergence spaces for some results we need to introduce a new condition on the lattice which is always true in the case where the lattice is a frame but not always true in the more general case as examples where we may apply the more general lattice context we examine the stratified l topological spaces and probabilistic limit spaces we show that the category of stratified l topological spaces is a reflective subcategory of our category and that the category of probabilistic limit spaces under a t norm is both a reflective and a coreflective subcategory of our category if we choose the lattice context appropriately

compactification of lattice valued convergence spaces we define compactness for stratified lattice valued convergence spaces and show that a tychonoff theorem is true further a generalization of the classical richardson compactification is given this compactification has a universal property

on diagonal completion of lattice valued diagonal cauchy spaces we define and study diagonal axioms for lattice valued cauchy spaces a completion of a weakly diagonal lattice valued cauchy space is constructed which is weakly diagonal and the coarsest among such completions it is at the same time at least as fine as any weakly regular completion

sharp hodge decompositions maxwell s equations and vector poisson problems on nonsmooth three dimensional riemannian manifolds we solve three basic potential theoretic problems hodge decompositions for vector fields poisson problems for the hodge laplacian and inhomogeneous maxwell equations in arbitrary lipschitz subdomains of a smooth compact three dimensional riemannian manifold in each case we derive sharp estimates on sobolev besov scales and establish integral representation formulas for the solution the proofs rely on tools from harmonic analysis and algebraic topology such as calderon zygmund theory and de rham theory

gradients on fractals in this paper we define and study a gradient on p c f post critically finite or finitely ramified fractals we use dirichlet energy form analysis developed for such fractals by kigami we consider both nondegenerate and degenerate harmonic structures where a nonzero harmonic function can be identically zero on an open set we show that the energy is equal to the integral of a certain seminorm of the gradient if the harmonic structure is weakly nondegenerate this result was proved by kusuoka in a different form we show that for a c 1 function on the sierpin ski gasket the gradient considered here and kusuoka s gradient essentially coincide with a gradient considered by kigami the gradient at a junction point was studied by strichartz in relation to the taylor approximation on fractals he also proved the existence of the gradient almost everywhere with respect to the hausdorff bernoulli measure for a function in the domain of the laplacian in this paper we obtain certain continuity properties of the gradient for a function in the domain of the laplacian as an appendix we prove an estimate of the local energy of harmonic functions which was stated by strichartz as a hypothesis 2000 academic press

modifications lattice valued structures several important topological concepts such as regularity local compactness and local boundedness are investigated in the category of lattice valued convergence spaces these definitions coincide with the usual ones in familiar categories

spectral analysis for convolution operators on locally compact groups we consider operators h of convolution with measures on locally compact groups we characterize the spectrum of h by constructing auxiliary operators whose kernel contain the pure point and singular subspaces of h respectively the proofs rely on commutator methods

largest and smallest t2 compactifications of lattice valued convergence spaces we give necessary and sufficient conditions for a lattice valued convergence space to possess a largest or a smallest t2 compactification

a note on regularity an alternative characterization of gahler s definition of regularity of a lattice valued convergence space is given in particular a space is regular exactly whenever convergence of filters remains invariant with respect to taking closures verification employs a slight modification of the work by jager

pure point spectrum of the laplacians on fractal graphs abstract we establish the pure point spectrum of the laplacians on two point self similar fractal graphs all eigenvalues have infinite multiplicity and a countable system of orthonormal eigenfunctions with compact support is complete in the corresponding hilbert space

on bogovskiĭ and regularized poincare integral operators for de rham complexes on lipschitz domains we study integral operators related to a regularized version of the classical poincare path integral and the adjoint class generalizing bogovski s integral operator acting on differential forms in mathbb r n we prove that these operators are pseudodifferential operators of order 1 the poincare type operators map polynomials to polynomials and can have applications in finite element analysis for a domain starlike with respect to a ball the special support properties of the operators imply regularity for the de rham complex without boundary conditions using poincare type operators and with full dirichlet boundary conditions using bogovski type operators for bounded lipschitz domains the same regularity results hold and in addition we show that the cohomology spaces can always be represented by fancyscript c infty functions

spectral analysis on infinite sierpinski gaskets we study the spectral properties of the laplacian on infinite sierpin ski gaskets we prove that the laplacian with the neumann boundary condition has pure point spectrum moreover the set of eigenfunctions with compact support is complete the same is true if the infinite sierpin ski gasket has no boundary but is false for the laplacian with the dirichlet boundary condition in all these cases we describe the spectrum of the laplacian and all the eigenfunctions with compact support to obtain these results first we prove certain new formulas for eigenprojectors of the laplacian on finite sierpin ski pre gaskets then we prove that the spectrum of the discrete laplacian on a sierpin ski lattice is pure point and the eigenfunctions are localized 1998 academic press

hp adaptation driven by polynomial degree robust a posteriori error estimates for elliptic problems we devise and study experimentally adaptive strategies driven by a posteriori error estimates to select automatically both the space mesh and the polynomial degree in the numerical approximation of diffusion equations in two space dimensions the adaptation is based on equilibrated flux estimates these estimates are presented here for inhomogeneous dirichlet and neumann boundary conditions for spatially varying polynomial degree and for mixed rectangular triangular grids possibly containing hanging nodes they deliver a global error upper bound with constant one and up to data oscillation error lower bounds on element patches with a generic constant only dependent on the mesh regularity and with a computable bound we numerically asses the estimates and several hp adaptive strategies using the interior penalty discontinuous galerkin method asymptotic exactness is observed for all the symmetric nonsymmetric odd degrees and incomplete variants on non nested unstructured triangular grids for a smooth solution and uniform refinement exponential convergence rates are reported on nonmatching triangular grids for the incomplete version on several benchmarks with a singular solution and adaptive refinement

optimal error estimation for bfh rmcurl conforming p interpolation in two dimensions in this paper we prove an optimal error estimate for the bf h rm curl conforming projection based p interpolation operator introduced in l demkowicz and i babuska siam j numer anal 41 2003 pp 1195 1208 this result is proved on the reference element either triangle or square k for regular vector fields in bf h r rm curl k with arbitrary r 0 the formulation of the result in the bf h rm div conforming setting which is relevant for the analysis of high order boundary element approximations for maxwell s equations is provided as well

a uniformly stable fortin operator for the taylor hood element we construct a new fortin operator for the lowest order taylor hood element which is uniformly stable both in l 2 and h 1 the construction which is restricted to two space dimensions is based on a tight connection between a subspace of the taylor hood velocity space and the lowest order nedelec edge element general shape regular triangulations are allowed for the h 1 stability while some mesh restrictions are imposed to obtain the l 2 stability as a consequence of this construction a uniform inf sup condition associated the corresponding discretizations of a parameter dependent stokes problem is obtained and we are able to verify uniform bounds for a family of preconditioners for such problems without relying on any extra regularity ensured by convexity of the domain

regularity lattice valued cauchy spaces a category of lattice valued cauchy spaces is defined and its properties are investigated the relationship between this category and the recent work by jager is presented the notion of regularity in the cauchy space context is the primary emphasis here in particular the category consisting of the regular cauchy spaces is shown to be bireflective in the category of all lattice valued cauchy spaces having cauchy continuous maps as its morphisms moreover completions are also investigated

lattice valued convergence diagonal axioms diagonal axioms are defined in the category whose objects consist of all the lattice valued convergence spaces when the latter is linearly ordered a diagonal condition is given which characterizes those objects in the category that are determined by a probabilistic convergence space which is topological

vibration modes of 3n gaskets and other fractals we rigorously study eigenvalues and eigenfunctions vibration modes on the class of self similar symmetric finitely ramified fractals which include the sierpinski gasket and other 3n gaskets we consider the classical laplacian on fractals which generalizes the usual one dimensional second derivative is the generator of the self similar diffusion process and has possible applications as the quantum hamiltonian we develop a theoretical matrix analysis including analysis of singularities which allows us to compute eigenvalues eigenfunctions and their multiplicities exactly we support our theoretical analysis by symbolic and numerical computations our analysis in particular allows the computation of the spectral zeta function on fractals and the limiting distribution of eigenvalues i e integrated density of states we consider such examples as the level 3 sierpinski gasket a fractal 3 tree and the diamond fractal