analytical bounds for block approximate factorization methods abstract we derive simple analytical upper bounds on the spectral condition number associated with the preconditioning of large sparse linear systems by the so called modified block incomplete factorizations the results prove to handle a large class of discrete second order elliptic partial differential equations and support the conclusion that sparse approximate block matrix factorizations provide powerful preconditioners

fractal reinforcement of elastic membranes we describe a homogenization model of an elastic membrane reinforced by the inclusion of a fractal string we follow a variational approach consisting in proving the convergence of certain energy functionals this leads to the spectral convergence of a sequence of weighted second order elliptic partial differential operators to a singular elliptic operator with a fractal term

spectral triples for the sierpinski gasket abstract we construct a family of spectral triples for the sierpinski gasket k for suitable values of the parameters we determine the dimensional spectrum and recover the hausdorff measure of k in terms of the residue of the volume functional a tr a d s at its abscissa of convergence d d which coincides with the hausdorff dimension d h of the fractal we determine the associated connes distance showing that it is bi lipschitz equivalent to the distance on k induced by the euclidean metric of the plane and show that the pairing of the associated fredholm module with odd k theory is non trivial when the parameters belong to a suitable range the abscissa of convergence d of the energy functional a tr d s 2 d a 2 d s 2 takes the value d e log 12 5 log 2 which we call energy dimension and the corresponding residue gives the standard dirichlet form on k

sub gaussian estimates of heat kernels on infinite graphs we prove that a two sided sub gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the green kernel admits a uniform polynomial decay

composite media and asymptotic dirichlet forms we introduce a metric topology on the space of dirichlet forms and study compactness and closure properties of families of local and non local forms

distributions of localized eigenvalues of laplacians on post critically finite self similar sets abstract in this paper we study distributions of eigenvalues corresponding to localized eigenfunctions of laplacians on p c f self similar sets precisely we divide the eigenvalue counting function x of a laplacian into two parts w x and f x where w x is the counting function of localized eigenvalues and f x is the counting function of non localized global eigenvalues we study asymptotic behaviors of w x and f x as x it is shown that w x x d s 2 where d s is the spectral exponent on the other hand for a class of laplacians including the standard laplacian on the sierpinski gasket f x x f for some f d s 2 so localized eigenfunctions dominate global eigenfunctions in such cases

asymptotically one dimensional diffusions on the sierpinski gasket and the abc gaskets diffusion processes on the sierpinski gasket and theabc gaskets are constructed as limits of random walks in terms of the associated renormalization group the present method uses the inverse trajectories which converge to unstable fixed points corresponding to the random walks on one dimensional chains in particular non degenerate fixed points are unnecessary for the construction a limit theorem related to the discrete time multi type non stationary branching processes is applied

martingale dimensions for fractals we prove that the martingale dimensions for canonical diffusion processes on a class of self similar sets including nested fractals are always one this provides an affirmative answer to the conjecture of s kusuoka publ res inst math sci 25 1989 659 680

diffusion on multifractals abstract we introduce a class of random fields with variable mean square regularity order defined on multifractal domains the local singularity order of these random fields then depends on the initial variable mean square regularity order and on the variable local singularity exponent of the multifractal measure defining the local dimension of the domain considered the theory is developed in a generalized framework through the covariance factorization using the tools of reproducing kernel hilbert spaces and fractional sobolev spaces of variable order

some properties of laplacians on fractals abstract kigami has defined an analog of the laplacian on a class of self similar fractals including the familiar sierpinski gasket we study properties of this operator we show that there is a maximal principle for solutions of certain nonlinear equations of the form u x f x u x we discuss the extension of the laplacian to non compact fractal blow ups and show that it is essentially self adjoint and we prove an analog of liouville s theorem in some cases we also give an explicit algorithm for solving the dirichlet problem for certain domains in the sierpinski gasket and give a characterization of all harmonic functions on those domains

what is not in the domain of the laplacian on sierpinski gasket type fractals abstract we consider the analog of the laplacian on the sierpinski gasket and related fractals constructed by kigami a function f is said to belong to the domain of if f is continuous and f is defined as a continuous function we show that if f is a nonconstant function in the domain of then f2 is not in the domain of we give two proofs of this fact the first is based on the analog of the pointwise identity f2 2f f f 2 where we show that f 2 does not exist as a continuous function in fact the correct interpretation of f2 is as a singular measure a result due to kusuoka we give a new proof of this fact the second is based on a dichotomy for the local behavior of a function in the domain of at a junction point x0 of the fractal in the typical case nonvanishing of the normal derivative we have upper and lower bounds for f x f x0 in terms of d x x0 for a certain value and in the nontypical case vanishing normal derivative we have an upper bound with an exponent greater than 2 this method allows us to show that general nonlinear functions do not operate on the domain of

a general incomplete block matrix factorization method abstract a general method for incomplete factorization of m matrices in block matrix form is presented the method generalizes previous methods for pointwise incomplete factorization methods of varga 1960 of wo nicki 1973 and of meijerink and van der vorst 1977 as well as a method for tridiagonal block matrices also we present a new and more general result on the existence of so called modified incomplete factorizations which satisfy a generalized rowsum criterion

on the resistance of the sierpinski carpet let f text n be the nth stage in the construction of the sierpinski carpet let r text n be the electrical resistance of f text n when the left and right sides are each short circuited and a voltage is applied between them we prove that there exists a constant rho such that textstyle frac 1 4 rho n leq r n leq 4 rho n the motivation for this result came from the problem of establishing a the existence and b the value of the spectral dimension of the sierpinski carpet in this and a subsequent paper we settle a and give bounds for b

energy inequalities for cutoff functions and some applications let x d m be a metric measure space with a local regular dirichlet form we establish necessary and sufficient conditions for upper heat kernel bounds with subdiffusive space time exponent to hold this characterization is stable under rough isometries that is it is preserved under bounded perturbations of the dirichlet form further we give a criterion for stochastic completeness in terms of a sobolev inequality for cutoff functions as an example we show that this criterion applies to an anomalous diffusion on a geodesically incomplete fractal space where the well established criterion in terms of volume growth fails

small time gaussian behavior of symmetric diffusion semi groups this work is involved with the short time asymptotics of diffusion semigroups in a general setting a generalization of fang s version of varadhan s formula is proven for general dirichlet spaces that are local and conservative the intrinsic metric appearing in the formula is characterized by pointwise distance for canonical dirichlet spaces on loop groups

fredholm modules on p c f self similar fractals and their conformal geometry the aim of the present work is to show how using the differential calculus associated to dirichlet forms it is possible to construct non trivial fredholm modules on post critically finite fractals by regular harmonic structures d r the modules are d s summable the summability exponent d s coinciding with the spectral dimension of the generalized laplacian operator associated with d r the characteristic tools of the noncommutative infinitesimal calculus allow to define a d s energy functional which is shown to be a self similar conformal invariant

local extinction versus local exponential growth for spatial branching processes let x be either the branching diffusion corresponding to the operator lu u2 u on d rd where x 0 and 0 is bounded from above or the superprocess corresponding to the operator lu u u2 on d rd with 0 and is bounded from above but no restriction on its sign let c denote the generalized principal eigenvalue for the operator l on d we prove the following dichotomy either c 0 and x exhibits local extinction or c 0 and there is exponential growth of mass on compacts of d with rate c for superdiffusions this completes the local extinction criterion obtained by pinsky ann probab 24 1996 237 267 and a recent result on the local growth of mass under a spectral assumption given by englander and turaev ann probab 30 2002 683 722 the proofs in the above two papers are based on pde techniques however the proofs we offer are probabilistically conceptual for the most part they are based on spine decompositions or immortal particle representations along with martingale convergence and the law of large numbers further they are generic in the sense that they work for both types of processes

on some versions of incomplete block matrix factorization iterative methods abstract various forms of preconditioning matrices for iterative acceleration methods are discussed the preconditioning is based on two versions of incomplete block matrix factorization

ordering strategies for modified block incomplete factorizations in this study the aim is to first investigate how the rate of convergence of unperturbed modified block incomplete factorization methods depends on the discrete partial differential equation pde to solve and next on this basis to provide simple practical rules for easily selecting orderings that result in rapid convergence it emerges from the analysis here that for discrete pdes ordering schemes that optimize the rate of convergence strongly depend on the variation of both the pde coefficients and the mesh sizes the arguments made which bring to light the appreciable potentialities of modified methods also display insight into why block versions are more robust than pointwise ones

a trace theorem for dirichlet forms on fractals abstract we consider a trace theorem for self similar dirichlet forms on self similar sets to self similar subsets in particular we characterize the trace of the domains of dirichlet forms on sierpinski gaskets and sierpinski carpets to their boundaries where the boundaries are represented by triangles and squares that confine the gaskets and the carpets as an application we construct diffusion processes on a collection of fractals called fractal fields these processes behave as an appropriate fractal diffusion within each fractal component of the field

conformal energy conformal laplacian and energy measures on the sierpinski gasket on the sierpinski gasket sg and related fractals we define a notion of conformal energy e and conformal laplacian for a given conformal factor based on the corresponding notions in riemannian geometry in dimension n 2 we derive a differential equation that describes the dependence of the effective resistances of on we show that the spectrum of dirichlet or neumann has similar asymptotics compared to the spectrum of the standard laplacian and also has similar spectral gaps provided the function does not vary too much we illustrate these results with numerical approximations we give a linear extension algorithm to compute the energy measures of harmonic functions with respect to the standard energy and as an application we show how to compute the l p dimensions of these measures for integer values of p 2 we derive analogous linear extension algorithms for energy measures on related fractals

asymptotic formula on average path length of fractal networks modeled on sierpinski gasket abstract in this paper we introduce a new method to construct evolving networks based on the construction of the sierpinski gasket using self similarity and renewal theorem we obtain the asymptotic formula for average path length of our evolving networks

upper estimate of martingale dimension for self similar fractals we study upper estimates of the martingale dimension dm of diffusion processes associated with strong local dirichlet forms by applying a general strategy to self similar dirichlet forms on self similar fractals we prove that dm 1 for natural diffusions on post critically finite self similar sets and that dm is dominated by the spectral dimension for the brownian motion on sierpinski carpets

brownian motion on a homogeneous random fractal we introduce a simple random fractal based on the sierpinski gasket and construct a brownian motion upon the fractal the properties of the process on the sierpinski gasket are modified by the random environment a sample path construction of the process via time truncation is used which is a direct construction of the process on the fractal from the associated dirichlet forms we obtain estimates on the resolvent and transition density for the process and hence a value for the spectral dimension which satisfiesd s 2d f dw a branching process in a random environment can be used to deduce some of the sample path properties of the process

taylor approximations on sierpinski gasket type fractals for a class of fractals that includes the familiar sierpinski gasket there is now a theory involving laplacians dirichlet forms normal derivatives green s functions and the gauss green integration formula analogous to the theory of analysis on manifolds this theory was originally developed as a by product of the construction of stochastic processes analogous to brownian motion but has been given by a direct analytic construction in the work of kigami until now this theory has not provided anything analogous to the gradient of a function or a local taylor approximation in this paper we construct a family of derivatives which includes the known normal derivative at vertex points in the graphs that approximate the fractal and obtain taylor approximations at these points we show that a function in the domain of n can be locally well approximated by an n harmonic function solution of nu 0 one novel feature of this result is that it requires several different estimates to describe the optimal rate of approximation

on the conditioning analysis of block approximate factorization methods abstract the paper is devoted to the conditioning analysis of modified block incomplete factorizations of a given stieltjes matrix we obtain new results improve other theories and compare all existing upper bounds through numerical experiments applied to discrete elliptic pdes our results show that an o h 1 spectral bound can be achieved for a large class of problems

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

energy and laplacian on hanoi type fractal quantum graphs this article studies potential theory and spectral analysis on compact metric spaces which we refer to as fractal quantum graphs these spaces can be represented as a possibly infinite union of one dimensional intervals and a totally disconnected possibly uncountable compact set which roughly speaking represents the set of junction points classical quantum graphs and fractal spaces such as the hanoi attractor are included among them we begin with proving the existence of a resistance form on the hanoi attractor and go on to establish heat kernel estimates and upper and lower bounds on the eigenvalue counting function of laplacians corresponding to weakly self similar measures on the hanoi attractor these estimates and bounds rely heavily on the relation between the length and volume scaling factors of the fractal we then state and prove a necessary and sufficient condition for the existence of a resistance form on a general fractal quantum graph finally we extend our spectral results to a large class of weakly self similar fractal quantum graphs

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

brownian motion penetrating fractals an application of the trace theorem of besov spaces abstract for a closed connected set f in rn assume that there is a local regular dirichlet form a symmetric diffusion process on f whose domain is included in a lipschitz space or a besov space on f under some condition for the order of the space and the newtonian 1 capacity of f we prove that there exists a symmetric diffusion process on rn which moves like the process on f and like brownian motion on rn outside f as an application we will show that when f is a nested fractal or a sierpinski carpet whose hausdorff dimension is greater than n 2 we can construct brownian motion penetrating the fractal for the proof we apply the technique developed in the theory of besov spaces

pure point spectrum for the laplacian on unbounded nested fractals abstract we prove for the class of nested fractals introduced by t lindstrom 1990 memoirs amer math soc 420 that the integrated density of states is completely created by the so called neuman dirichlet eigenvalues the corresponding eigenfunctions lead to eigenfunctions with compact support on the unbounded set and we prove that for a large class of blow ups the set of neuman dirichlet eigenfunctions is complete leading to a pure point spectrum with compactly supported eigenfunctions this generalizes previous results of h teplyaev 1998 j funct anal 159 537 567 on the sierpinski gasket our methods are elementary and use only symmetry arguments via the representations of the symmetry group of the set

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

on axelsson s perturbations abstract a nice perturbation technique was introduced by axelsson and further developed by gustafsson to prove that factorization iterative methods are able under appropriate conditions to reach a convergence rate larger by an order of magnitude than that of classical schemes gustafsson observed however that the perturbations introduced to prove this result seemed actually unnecessary to reach it in practice in the present work on the basis of eigenvalue bounds recently obtained by the author we offer an alternative approach which brings a partial confirmation of gustafsson s conjecture

branching exit markov systems and superprocesses superprocesses under the name continuous state branching processes appeared first in a pioneering work of s watanabe j math kyoto univ 8 1968 141 167 deep results on paths of the super brownian motion were obtained by dawson perkins le gall and others in earlier papers a superprocess was interpreted as a markov process x t in the space of measures this is not sufficient for a probabilistic approach to boundary value problems a reacher model based on the concept of exit measures was introduced by e b dynkin probab theory related fields 89 1991 89 115 a model of a superprocess as a system of exit measures from time space open sets was systematically developed in 1993 e b dynkin ann probab 21 1993 1185 1262 in particular branching and markov properties of such a system were established and used to investigate partial differential equations in the present paper we show that the entire theory of superprocesses can be deduced from these properties

harmonic analysis for resistance forms abstract in this paper we define the green functions for a resistance form by using effective resistance and harmonic functions then the green functions and harmonic functions are shown to be uniformly lipschitz continuous with respect to the resistance metric making use of this fact we construct the green operator and the measure valued laplacian the domain of the laplacian is shown to be a subset of uniformly lipschitz continuous functions while the domain of the resistance form in general consists of uniformly 1 2 holder continuous functions

semilinear pdes on self similar fractals a laplacian may be defined on self similar fractal domains in terms of a suitable self similar dirichlet form enabling discussion of elliptic pdes on such domains in this context it is shown that that semilinear equations such as u u p 0 with zero dirichlet boundary conditions have non trivial non negative solutions if 0 1 or if 2 and 1 p 2 2 where is the intrinsic dimension or spectral dimension of the system thus the intrinsic dimension takes the r o le of the euclidean dimension in the classical case in determining critical exponents of semilinear problems

harmonic calculus on fractals a measure geometric approach ii differentiation of functions w r t finite atomless measures with compact support on the real line is introduced the related harmonic calculus is similar to that of the classical lebesgue case as an application we obtain the weyl exponent for the spectral asymptotics of the laplacians w r t linear cantor type measures with arbitrary weights

on singularity of energy measures on self similar sets we provide general criteria for energy measures of regular dirichlet forms on self similar sets to be singular to bernoulli type measures in particular every energy measure is proved to be singular to the hausdorff measure for canonical dirichlet forms on 2 dimensional sierpinski carpets

analysis on products of fractals for a class of post critically finite p c f fractals which includes the sierpinski gasket sg there is a satisfactory theory of analysis due to kigami including energy harmonic functions and laplacians in particular the laplacian coincides with the generator of a stochastic process constructed independently by probabilistic methods the probabilistic method is also available for non p c f fractals such as the sierpinski carpet in this paper we show how to extend kigami s construction to products of p c f fractals since the products are not themselves p c f this gives the first glimpse of what the analytic theory could accomplish in the non p c f setting there are some important differences that arise in this setting it is no longer true that points have positive capacity so functions of finite energy are not necessarily continuous also the boundary of the fractal is no longer finite so boundary conditions need to be dealt with in a more involved manner all in all the theory resembles pde theory while in the p c f case it is much closer to ode theory

conditioning analysis of modified block incomplete factorizations abstract we improve the conditioning analysis of modified block incomplete factorizations of stieltjes matrices letting n denote the number of diagonal blocks our results show that the spectral condition number is bounded by n for a large class of two dimensional pdes

products of random matrices and derivatives on p c f fractals abstract we define and study intrinsic first order derivatives on post critically finite fractals and prove differentiability almost everywhere with respect to self similar measures for certain classes of fractals and functions we apply our results to extend the geography is destiny principle to these cases and also obtain results on the pointwise behavior of local eccentricities on the sierpinski gasket previously studied by oberg strichartz and yingst and the authors we also establish the relation of the derivatives to the tangents and gradients previously studied by strichartz and the authors our main tool is the furstenberg kesten theory of products of random matrices

transition density estimates for diffusion processes on post critically finite self similar fractals the recent development of analysis on fractal spaces is physically motivated by the study of diffusion in disordered media the natural questions that arise concern the existence and uniqueness of a suitable laplace operator and the behaviour of the associated heat semigroup on a space which is fractal the classes of fractals for which these questions were rst answered were classes of exactly self similar fractals with strong spatial symmetry such as nested or af ne nested fractals see for example 2 8 the existence of a laplacian and estimates on the heat kernel were obtained by considering the associated diffusion process and using the symmetry of the space the uniqueness of the laplacian for nested and af ne nested fractals has recently been solved through consideration of their dirichlet forms 23 in 15 the framework of post critically nite which we abbreviate to p c f self similar sets was introduced in order to capture the notion of exactly selfsimilar nitely rami ed fractals as used in the physics literature finitely rami ed fractals have the property that the intersection of any connected subset of the fractal with the rest of the set should occur only at a nite number of points this makes these structures much easier to analyse than in nitely rami ed sets such as the sierpinski carpet 5 the p c f self similar sets do not have spatial symmetry in general and have provided a mathematical test bed for analysis on fractals in this paper we will obtain uniform short time estimates on the heat kernel associated with a natural laplacian on the fractal a laplacian can be constructed on a p c f self similar fractal as a limit of discrete laplacians on graph approximations to the fractal based on the rami cation points we construct these operators via their dirichlet forms which can be set on any l space with a full measure n in 23 criteria are given which give a partial answer to the existence and uniqueness of laplace type operators on p c f self similar sets in a series of papers 16 17 18 20 25 7 some of the interesting spectral properties of p c f sets have been elucidated it has been shown that there can exist localized eigenfunctions if there is a high degree of symmetry in the set and corresponding laplace operator this corresponds to oscillation in the leading order term for the asymptotics of the eigenvalue counting function such behaviour occurs for nested fractals 7 and p c f sets with strong harmonic

energy measures on p c f self similar sets abstract on connected post critically finite p c f self similar sets we give a linear extension method to compute the energy measures of harmonic functions with respect to the standard energy and as an application we also compute the l 2 dimensions of these measures on some p c f self similar sets

transition density estimates for brownian motion on affine nested fractals a class of affine nested fractals is introduced which have different scale factors for different similitudes but still have the symmetry assumptions of nested fractals for these fractals estimates on the transition density for the brownian motion are obtained using the associated dirichlet form an upper bound for the diagonal can be found using a nash type inequality then probabilistic techniques are used to obtain the off diagonal bound the approach differs from previous treatments as it uses only the dirichlet form and no estimates on the resolvent the bounds obtained are expressed in terms of an intrinsic metric on the fractal

self similarity of volume measures for laplacians on p c f self similar fractals our main goal in this paper is to obtain a precise analogue of weyl s asymptotic formula for the eigenvalue distribution of laplacians on a certain class of finitely ramified or p c f self similar fractals building in particular on the work of 7 9 22 24 our main result consists in precisely identifying for the class of decimable fractals the volume measures constructed by the second author in 24 for general p c f fractals and showing that they are self similar

the hot spots conjecture for the sierpinski gasket abstract there are many works on the hot spots conjecture for domains in euclidean space since the conjecture was posed by j rauch in 1974 in this paper using spectral decimation we prove that the conjecture holds on the sierpinski gasket i e every eigenfunction of the second smallest eigenvalue of the neumann laplacian introduced by j kigami attains its maximum and minimum on the boundary

nonlinear wave equations on a class of bounded fractal sets the nonlinear wave equation u tt u f u with given initial data and zero boundary conditions on a class of bounded self similar fractal sets is investigated the sobolev type inequality is the starting point of this work which holds for a class of fractals including the well known sierpinski gasket we obtain the global existence of strong solutions for suitable f if the spectral dimension ds of the fractal satisfies ds 2 the key is to construct the wave propagator and hilbert spaces of functions on the fractal the main difficulty in obtaining the global existence of a weak solution is establishing a priori estimates depending on a regularity property for f t he regularity property of a weak solution is obtained through a fine analysis in which the sobolev type inequality plays a crucial role 2002 elsevier science usa all rights reserved

conditioning analysis of block incomplete factorizations and its application to elliptic equations the paper deals with eigenvalue estimates for block incomplete fac torization methods for symmetric matrices first some previous results on upper bounds for the maximum eigenvalue of preconditioned matrices are generalized to each eigenvalue second upper bounds for the maximum eigenvalue of the preconditioned matrix are further estimated which presents a substantial im provement of earlier results finally the results are used to estimate bounds for every eigenvalue of the preconditioned matrices in particular for the maximum eigenvalue when a modified block incomplete factorization is used to solve an elliptic equation with variable coefficients in two dimensions the analysis yields a new upper bound of type h 1 for the condition number of the preconditioned matrix and shows clearly how the coefficients of the differential equation influ ence the positive constant

eigenvalues and eigenfunctions of one dimensional fractal laplacians defined by iterated function systems with overlaps under the assumption that a self similar measure defined by a one dimensional iterated function system with overlaps satisfies a family of second order self similar identities introduced by strichartz et al we obtain a method to discretize the equation defining the eigenvalues and eigenfunctions of the corresponding fractal laplacian this allows us to obtain numerical solutions by using the finite element method we also prove that the numerical eigenvalues and eigenfunctions converge to the true ones and obtain estimates for the rates of convergence we apply this scheme to the fractal laplacians defined by the well known infinite bernoulli convolution associated with the golden ratio and the 3 fold convolution of the cantor measure the iterated function systems defining these measures do not satisfy the open set condition or the post critically finite condition we use second order self similar identities to analyze the measures

a probabilistic approach to one class of nonlinear differential equations we establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes as an application we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations

existence and conditioning properties of sparse approximate block factorizations a particular class of sparse approximate block lu factorizations is investigated sufficient conditions are determined for the existence of b lp 1 u for a given coefficient matrix a bounds on the spectral condition number of b 1 a are obtained for symmetric b and a when applied to discrete multidimensional elliptic partial differential equations these results lead under reasonable assumptions to o h 1 spectral condition numbers

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

harmonic calculus on p c f self similar sets the main object of this paper is the laplace operator on a class of fractals first we establish the concept of the renormalization of difference operators on post critically finite p c f for short self similar sets which are large enough to include finitely ramified self similar sets and extend the results for sierpinski gasket given in 10 to this class under each invariant operator for renormalization the laplace operator green function dirichlet form and neumann derivatives are explicitly constructed as the natural limits of those on finite pre self similar sets which approximate the p c f self similar sets also harmonic functions are shown to be finite dimensional and they are characterized by the solution of an infinite system of finite difference equations

on the eigenvalue distribution of a class of preconditioning methods a class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationax b wherea is a symmetric positive definite matrix the methods are based on an incomplete factorization of the matrixa and include both pointwise and blockwise factorization we study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofc 1a wherec is the preconditioning matrix we also show graphic representations of the eigenvalues and present numerical tests of the methods

denker sato type markov chains on self similar sets in denker and sato potential anal 14 211 232 2001 publ rims 35 769 794 1999 math nachr 241 32 55 2002 studied certain markov chain on the symbolic spaces of the sierpinski gasket sg they showed that the martin boundary is homeomorphic to sg and used the potential theory on the martin boundary to induce a harmonic structure on sg in this paper we consider a more general denker sato type markov chain associated with self similar sets k with the open set condition the chain is defined on the augmented tree of the symbolic space such tree was introduced by kaimanovich it is hyperbolic in the sense of gromov kaimanovich in random walks on sierpinski graphs hyperbolicity and stochastic homogenization birhauser basel 2003 lau and wang in indiana univ math j 58 1777 1795 2009 we show that the martin boundary the hyperbolic boundary and the self similar set k are homeomorphic the hitting distribution of the chain is also obtained

lipschitz algebras and derivations ii exterior differentiation abstract basic aspects of differential geometry can be extended to various non classical settings lipschitz manifolds rectifiable sets sub riemannian manifolds banach manifolds wiener space etc although the constructions differ in each of these cases one can define a module of measurable 1 forms and a first order exterior derivative we give a general construction which applies to any metric space equipped with a finite measure and produces the desired result in all of the above cases it agrees with cheeger s construction when the latter is defined it also applies to an important class of dirichlet spaces where however the known first order differential calculus in general differs from ours although the two are related

harmonic analysis on the pascal graph abstract in this paper we completely determine the spectral invariants of an auto similar planar 3 regular graph using the same methods we study the spectral invariants of a natural compactification of this graph

lower eigenvalue bounds for pencils of matrices abstract eigenvalue bounds are obtained for pencils of matrices a vb where a is a stieltjes matrix and b is positive definite under assumptions suitable for the estimation of asymptotic convergence rates of factorization iterative methods where b represents the approximate factorization of a the upper bounds obtained depend on the connectivity structure of the matrices involved which enters through matrix graph considerations in addition a more classical argument is used to obtain a lower bound potential applications of these results include a partial confirmation of gustafsson s conjecture concerning the nonnecessity of axelsson s perturbations

modified block approximate factorization strategies two variants of modified incomplete block matrix factorization with additive correction are proposed for the iterative solution of large linear systems of equations both rigorous theoretical support and numerical evidence are given displaying their efficiency when applied to discrete second order partial differential equations pdes even in the case of quasi singular problems

estimates for the resolvent kernel of the laplacian on p c f self similar fractals and blowups one of the main features of analysis on post critically finite self similar pcfss sets is that it is possible to understand the behavior of the laplacian and its inverse the green operator in terms of the self similar structure of the set indeed a major step in the approach to analysis on self similar fractals via dirichlet forms was kigami s proof 8 10 that for a self similar dirichlet form the green kernel can be written explicitly as a series in which each term is a rescaling of a single expression via the self similar structure in 6 this result was extended to show that the resolvent kernel of the laplacian meaning the kernel of z 1 can also be written as a self similar series for suitable values of z c part of the motivation for that work was that it gives a new understanding of functions of the laplacian such as the heat operator et by writing them as integrals of the resolvent the purpose of the present work is to establish estimates that permit the above approach to be carried out we study the functions occurring in the series decomposition from 6 see theorem 3 3 below for this decomposition and give estimates on their decay from this we determine estimates on the resolvent kernel and on kernels of operators defined as integrals of the resolvent kernel in particular we recover the sharp upper estimates for the heat kernel see theorem 10 2 that were proved for pcfss sets by hambly and kumagai 4 by probabilistic methods see also 1 13 3 for earlier results of this type on less general classes of sets it is important to note that the preceding authors were able to prove not just upper estimates but also lower bounds for the heat kernel and therefore were able to prove sharpness of their bounds our methods permit sharp bounds for the resolvent kernel on the positive real axis but we do not know how to obtain these globally in the complex plane or how to obtain lower estimates for the heat kernel from them therefore in this direction our results are not as strong as those obtained in 4 however in other directions we obtain more information than that known from heat kernel estimates and we hope that our approach will complement the existing probabilistic methods in particular we are able to obtain resolvent bounds on any ray in c other than the negative real axis where the spectrum lies while standard calculations from heat kernel bounds only give these estimates in a half plane a further consequence of our approach is that we extend in theorem 9 7 the decomposition from 6 to the case of blowups which are non compact sets with local structure equivalent to that of the underlying self similar sets the blowup of a pcfss set bears the same relation to the original set as the real line bears to the unit interval see 17 for details the structure of the paper is as follows in section 3 we recall some basic features of analysis on pcfss sets as well as the main result of 6 which is the decomposition of the resolvent as a weighted sum of piecewise eigenfunctions section 4 then discusses

energy measures and indices of dirichlet forms with applications to derivatives on some fractals we introduce the concept of index for regular dirichlet forms by means of energy measures and discuss its properties in particular it is proved that the index of strong local regular dirichlet forms is identical with the martingale dimension of the associated diffusion processes as an application a class of self similar fractals is taken up as an underlying space we prove that first order derivatives can be defined for functions in the domain of the dirichlet forms and their total energies are represented as the square integrals of the derivatives

exact spectrum of the laplacian on a domain in the sierpinski gasket abstract for a certain domain in the sierpinski gasket sg whose boundary is a line segment a complete description of the eigenvalues of the laplacian with an exact count of dimensions of eigenspaces under the dirichlet and neumann boundary conditions is presented the method developed in this paper is a weak version of the spectral decimation method due to fukushima and shima since for a lot of bad eigenvalues the spectral decimation method can not be used directly let 0 x x be the eigenvalue counting functions of the laplacian associated to sg and respectively we prove a comparison between 0 x and x says that 0 0 x x c x log 2 log 5 log x for sufficiently large x for some positive constant c as a consequence x g log x x log 3 log 5 o x log 2 log 5 log x as x for some right continuous discontinuous log 5 periodic function g r r with 0 inf r g sup r g moreover we explain that the asymptotic expansion of x should admit a second term of the order log 2 log 5 that becomes apparent from the experimental data this is very analogous to the conjectures of weyl and berry

function spaces on fractals abstract we construct function spaces analogs of holder zygmund besov and sobolev spaces on a class of post critically finite self similar fractals in general and the sierpinski gasket in particular based on the laplacian and effective resistance metric of kigami this theory is unrelated to the usual embeddings of these fractals in euclidean space and so our spaces are distinct from the function spaces of jonsson and wallin although there are some coincidences for small orders of smoothness we show that the laplacian acts as one would expect an elliptic pseudodifferential operator of order d 1 on a space of dimension d to act where d is determined by the growth rate of the measure of metric balls we establish some sobolev embedding theorems and some results on complex interpolation on these spaces

dirichlet forms on fractals poincare constant and resistance we study dirichlet forms associated with random walks on fractal like finite grahs we consider related poincare constants and resistance and study their asymptotic behaviour we construct a markov semi group on fractals as a subsequence of random walks and study its properties finally we construct self similar diffusion processes on fractals which have a certain recurrence property and plenty of symmetries

hyperfunctions and spectral zeta functions of laplacians on self similar fractals we investigate the spectral zeta function of fractal differential operators such as the laplacian on the unbounded i e infinite sierpinski gasket and a self similar sturm liouville operator associated with a fractal self similar measure on the half line in the latter case sabot discovered the relation between the spectrum of this operator and the iteration of a rational map of several complex variables called the renormalization map we obtain a factorization of the spectral zeta function of such an operator expressed in terms of the dirac delta hyperfunction a geometric zeta function and the zeta function associated with the dynamics of the corresponding renormalization map viewed either as a polynomial function on in the first case or in the second case as a polynomial on the complex projective plane our first main result extends to the case of the fractal laplacian on the unbounded sierpinski gasket a factorization formula obtained by the second author for the spectral zeta function of a fractal string and later extended by teplyaev to the bounded i e finite sierpinski gasket and some other decimable fractals furthermore our second main result generalizes these factorization formulas to the renormalization maps of several complex variables associated with fractal sturm liouville operators moreover as a corollary in the very special case when the underlying self similar measure is the lebesgue measure on 0 1 we obtain a representation of the riemann zeta function in terms of the dynamics of a certain polynomial in thereby extending to several variables an analogous result by teplyaev

transition densities for brownian motion on the sierpinski carpet upper and lower bounds are obtained for the transition densitiesp t x y of brownian motion on the sierpinski carpet these are of the same form as those which hold for the sierpinski gasket in addition the joint continuity ofp t x y is proved the existence of the spectral dimension is established and the einstein relation connecting the spectral dimension the hausdorff dimension and the resistance exponent is shown to hold

weyl s problem for the spectral distribution of laplacians on p c f self similar fractals we establish an analogue of weyl s classical theorem for the asymptotics of eigenvalues of laplacians on a finitely ramified i e p c f self similar fractalk such as for example the sierpinski gasket we consider both dirichlet and neumann boundary conditions as well as laplacians associated with bernoulli type multifractal measures onk from a physical point of view we study the density of states for diffusions or for wave propagation in fractal media more precisely let x be the number of eigenvalues less thanx then we show that x is of the order ofxds 2 asx where the spectral exponent ds is computed in terms of the geometric as well as analytic structures ofk further we give an effective condition that guarantees the existence of the limit ofx ds 2 x asx this condition is in some sense generic in addition we define in terms of the above spectral exponents and calculate explicitly the spectral dimension ofk

a survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices eigenvalue and condition number estimates for preconditioned iteration matrices provide the information required to estimate the rate of convergence of iterative methods such as preconditioned conjugate gradient methods in recent years various estimates have been derived for perturbed modified block incomplete factorizations we survey and extend some of these and derive new estimates in particular we derive upper and lower estimates of individual eigenvalues and of condition number this includes a discussion that the condition number of preconditioned second order elliptic difference matrices is o h 1 some of the methods are applied to compute certain parameters involved in the computation of the preconditioner

derivations and dirichlet forms on fractals abstract we study derivations and fredholm modules on metric spaces with a local regular conservative dirichlet form in particular on finitely ramified fractals we show that there is a non trivial fredholm module if and only if the fractal is not a tree i e not simply connected this result relates fredholm modules and topology refines and improves known results on p c f fractals we also discuss weakly summable fredholm modules and the dixmier trace in the cases of some finitely and infinitely ramified fractals including non self similar fractals if the so called spectral dimension is less than 2 in the finitely ramified self similar case we relate the p summability question with estimates of the lyapunov exponents for harmonic functions and the behavior of the pressure function

on the asymptotics of the eigenvalue counting function for random recursive sierpinski gaskets we consider natural laplace operators on random recursive affine nested fractals based on the sierpinski gasket and prove an analogue of weyl s classical result on their eigenvalue asymptotics the eigenvalue counting function n is shown to be of order ds 2 as where we can explicitly compute the spectral dimension d s moreover the limit n ds 2 will typically exist and can be expressed as a deterministic constant multiplied by a random variable this random variable is a power of the limiting random variable in a suitable general branching process and has an interpretation as the volume of the fractal