an inverse convergence approach for arguments of jacobian elliptic functions abstract computing the value of the jacobian elliptic functions given the argument u and the parameter m is a problem whose solution can be found either tabulated in tables of elliptic functions 1 or by use of existing software such as mathematica etc the inverse problem finding the argument given the jacobian elliptic function and the parameter m is a problem whose solution is found only in tables of elliptic functions standard polynomial inverse interpolation procedures fail due to ill conditioning of the system of the unknowns in this paper we describe a numerical procedure based on the convergence of the unknowns of the solution by the use of arithmetical method as an alternative way of solving the problem the method gives very good results with no significant error in the computation of the argument of the jacobian elliptic function given the jacobian elliptic function and the parameter this new procedure is important in problems involving cavities or inclusions of ellipsoidal shape encountered in the mechanical design of bearings filters and composite materials they are also important in the modeling of porosity of bones this porosity may lead to osteoporosis a disease which affects bone mineral density in humans with bad consequences also these procedures are of importance in problems encountered in the physics discipline such as in the analysis of the dependence of the maximum tunneling current on external magnetic field for large area josephson junctions with overlap boundary conditions

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

on subalgebras of the car algebra abstract every nuclear unital separable c algebra a is unitally and completely isometrically ismorphic to the range of a unital completely positive projection on the car algbra b m 2 m 2 let c be a nuclearly imbeddable separable unital c algebra in the sense of voiculescu vo13 then there exists a c subalgebra e of the car algebra b such that c is isomorphic to a quotient c algebra of e we include one of the possible proofs that exact c algebras are nuclearly imbeddable proofs of g covariant analogs are outlined for compact g some related results are obtained on non selfadjoint operator algebras the unital separable nuclear c algebras not of type i are mutually cb isomorphic and have cb distance 256 separable lindenstrauss spaces are isometric to quotients of the car algebra by a sum of closed left and a closed right ideal

spectral zeta functions of fractals and the complex dynamics of polynomials we obtain formulas for the spectral zeta function of the laplacian on symmetric finitely ramified fractals such as the sierpinski gasket and a fractal laplacian on the interval these formulas contain a new type of zeta function associated with a polynomial rational functions also can appear in this context it is proved that this zeta function has a meromorphic continuation to a half plane with poles contained in an arithmetic progression it is shown as an example that the riemann zeta function is the zeta function of a quadratic polynomial which is associated with the laplacian on an interval the spectral zeta function of the sierpinski gasket is a product of the zeta function of a polynomial and a geometric part the poles of the former are canceled by the zeros of the latter a similar product structure was discovered by m l lapidus for self similar fractal strings

precise and fast computation of jacobian elliptic functions by conditional duplication we developed a new method to compute the cosine amplitude function c equiv mathrm cn u m by using its double argument formula the accumulation of round off errors is effectively suppressed by the introduction of a complementary variable b equiv 1 c and a conditional switch between the duplication of b and c the sine and delta amplitude functions s equiv mathrm sn u m and d equiv mathrm dn u m are evaluated from thus computed b or c by means of their identity relations the new method is sufficiently precise as its errors are less than a few machine epsilons also it is significantly faster than the existing procedures in case of single precision computation it runs more than 50 times faster than bulirsch s sncndn based on the gauss transformation and 2 7 times faster than our previous method based on the simultaneous duplication of s c and d the ratios change to 7 6 and 3 5 respectively in case of the double precision environment

smooth bumps a borel theorem and partitions of smooth functions on p c f fractals we provide two methods for constructing smooth bump functions and for smoothly cutting off smooth functions on fractals one using a probabilistic approach and sub gaussian estimates for the heat operator and the other using the analytic theory for p c f fractals and a fixed point argument the heat semigroup probabilistic method is applicable to a more general class of metric measure spaces with laplacian including certain infinitely ramified fractals however the cutoff technique involves some loss in smoothness from the analytic approach we establish a borel theorem for p c f fractals showing that to any prescribed jet at a junction point there is a smooth function with that jet as a consequence we prove that on p c f fractals smooth functions may be cut off with no loss of smoothness and thus can be smoothly decomposed subordinate to an open cover the latter result provides a replacement for classical partition of unity arguments in the p c f fractal setting

harmonic oscillators on infinite sierpinski gaskets we study the analog of the quantum mechanical harmonic oscillator on infinite blowups of the sierpinski gasket using the standard kigami laplacian our main task is to find a class of potentials analogous to frac 1 2 x x 0 2 on the line we describe a class of potentials u with the properties u 1 u attains a minimum value zero and u at infinity we show how to construct such potentials attaining the minimum value at any prescribed point and we show how to parameterize the class of potentials by a certain surface in mathbb r 3 we obtain estimates for the growth rate of the eigenvalue counting function for u we obtain numerical approximations to the eigenfunctions and in particular observe that the ground state eigenfunction resembles a gaussian function

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

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

derivatives of the restrictions of harmonic functions on the sierpinski gasket to segments we give an explicit derivative computation for the restriction of a harmonic function on sg to segments at specific points of the segments the derivative is zero at points dividing the segment in ratio 1 3 this shows that the restriction of a harmonic function to a segment of sg has the following curious property the restriction has infinite derivatives on a dense subset of the segment at junction points and vanishing derivatives on another dense subset

the cuntz semigroup the elliott conjecture and dimension functions on c algebras we prove that the cuntz semigroup is recovered functorially from the elliott invariant for a large class of c algebras in particular our re sults apply to the largest class of simple c algebras for which k theoretic classification can be hoped for this work has three significant consequences first it provides new conceptual insight into elliott s classification program proving that the usual form of the elliott conjecture is equivalent among z stable algebras to a conjecture which is in general substantially weaker and for which there are no known counterexamples second and third it resolves for the class of algebras above two conjectures of blackadar and handelman concerning the basic structure of dimension functions on c algebras we also prove in passing that the cuntz pedersen semigroup is recovered functorially from the elliott invariant for a large class of simple unital c algebras

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

simple nuclear c algebras not isomorphic to their opposites abstract we show that it is consistent with zermelo fraenkel set theory with the axiom of choice zfc that there is a simple nuclear nonseparable c c algebra which is not isomorphic to its opposite algebra we can furthermore guarantee that this example is an inductive limit of unital copies of the cuntz algebra o 2 o2 or of the canonical anticommutation relations car algebra

on a simple unital projectionless c algebra we construct a unital separable c algebra z as an analog of the hyperfinite type ii 1 factor besides being nuclear simple projectionless and infinite dimensional z has a unique tracial state and is kk equivalent to inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the algebra of complex numbers it is shown that unital endomorphisms on z are approximately inner and that z is isomorphic to the infinite tensor product of its replicas it is also shown that a z a for certain interesting classes of unital simple nuclear c algebras a of real rank zero

power dissipation in fractal feynman sierpinski ac circuits this paper studies the concept of power dissipation in infinite graphs and fractals associated with passive linear networks consisting of non dissipative elements in particular we analyze the so called feynman sierpinski ladder a fractal ac circuit motivated by feynman s infinite ladder that exhibits power dissipation and wave propagation for some frequencies power dissipation in this circuit is obtained as a limit of quadratic forms and the corresponding power dissipation measure associated with harmonic potentials is constructed the latter measure is proved to be continuous and singular with respect to an appropriate hausdorff measure defined on the fractal dust of nodes of the network

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

evaluating elliptic functions and their inverses abstract in 1 the authors described an inverse interpolation scheme for estimating inverse jacobian elliptic functions in an environment like matlab where direct but not inverse evaluation is available here an easily programmed alternative based on the standard arithmetic geometric mean tabloid is developed a novel recursion eliminates need for evaluating tangents and inverse tangents at intermediate steps in the algorithm a similar algorithm is developed for evaluation of elliptic functions when given argument and modulus the latter seems less beneficial especially when standard software evaluation is available numerical results indicate robustness and high accuracy

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

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 simple separable exact c algebra not anti isomorphic to itself we give an example of an exact stably finite simple separable c algebra d which is not isomorphic to its opposite algebra moreover d has the following additional properties it is stably finite approximately divisible has real rank zero and stable rank one has a unique tracial state and the order on projections over d is determined by traces it also absorbs the jiang su algebra z and in fact absorbs the 3 uhf algebra we can also explicitly compute the k theory of d namely k 0 d cong mathbb z tfrac 1 3 with the standard order and k1 d 0 as well as the cuntz semigroup of d namely w d cong mathbb z tfrac 1 3 sqcup 0 infty

counting spanning trees on fractal graphs and their asymptotic complexity using the method of spectral decimation and a modified version of kirchhoff s matrix tree theorem a closed form solution to the number of spanning trees on approximating graphs to a fully symmetri

on the structure of simple c algebras tensored with a uhf algebra ii let d be a simple unital c algebra let b be a uhf algebra and put a b d it is proved that if p q a are projections and p q for all quasitraces on a then p q in the sense of murray and von neumann a more general result involving positive operators in a is also proved if d has finitely many extremal quasi traces and the projections in d k separate these then it is proved that a has real rank zero finally it is proved that provided d is stably finite then each positive state on k0 d lifts to a quasi trace

graded brauer groups of a groupoid with involution abstract we define a group brr g containing in a sense the graded complex and orthogonal brauer groups of a locally compact groupoid g equipped with an involution when the involution is trivial we show that the new group naturally provides a generalisation of donovan karoubi s graded orthogonal brauer group gbro more generally it is shown to be a direct summand of the well known graded complex brauer group in addition we prove that brr g identifies with a direct sum of a real cohomology group and the abelian group extr g s 1 of real graded s 1 central extensions a cohomological picture is then given

direct and inverse interpolation for jacobian elliptic functions zeta function of jacobi and complete elliptic integrals of the second kind abstract computing the value of the jacobian elliptic functions given the argument u and the parameter m is a problem whose solution can be found either tabulated in tables of elliptic functions 1 or by use of existing software such as mathematica etc the inverse problem finding the argument given the jacobian elliptic function and the parameter m is a problem whose solution is found only in tables of elliptic functions standard polynomial inverse interpolation procedures fail due to ill conditioning of the system of linear equations of the unknowns in this paper we describe a numerical procedure for inverse interpolation which gives good results in the computation of the argument of the jacobian elliptic function given the jacobian elliptic function and the parameter also a direct interpolation is described which gives the zeta function of jacobi and the complete elliptic integral of the second kind given the argument and the parameter these new interpolation procedures are important in problems involving cavities or inclusions of ellipsoidal shape encountered in the mechanical design of bearings filters and composite materials they are also important in the modelling of porosity of bones this porosity may lead to osteoporosis a disease which affects bone mineral density in humans with bad consequences

finite group actions on c algebras with the rohlin property i basic properties of finite group actions with the rohlin property on unital c algebras are investigated a characterization of finite group actions with the rohlin property on the cuntz algebra 02 is given in terms of central sequences which may be considered as an equivariant version of e kirchberg and n c phillips s characterization of 02 a large class of symmetries on 02 are classified in terms of the fixed point algebras for conjugacy and the crossed products for cocycle conjugacy model actions of symmetries of 02 are constructed for given k theoretical invariants

extending states on preordered semigroups and the existence of quasitraces on c algebras abstract it is shown that every state on a subsemigroup of a scaled preordered semigroup extends to the whole semigroup extending a theorem of goodearl and handelman as an application every state on k0 a for a unital c algebra a comes from a quasitrace

on non semisplit extensions tensor products and exactness of group c algebras we show the existence of a block diagonal extensionb of the suspensions a of the reduced groupc algebraa c sl 2 such that there is only onec norm on the algebraic tensor productb op b butb is not nuclear even not exact thus the class of exactc algebras is not closed under extensions the existence comes from a new established tensorial duality between the weak expectation property wep of lance and the local variant llp of the lifting property we characterize the local lifting property of separable unitalc algebrasa as follows a has the local lifting property if and only if ext s a is a group wheres a is the suspension ofa if moreovera is the quotient algebra of ac algebra withwep for brevity a isqwep but does not satisfyllp then there exists a quasidiagonal extensionb of the suspensions a by the compact operators such that on the algebraic tensor productb op b there is only onec norm the question if everyc algebra isqwep remains open but we obtain the following results onqwep ac algebrac isqwep if and only ifc isqwep a von neumannii 1 factorn with separable predualn isqwep if and only ifn is a von neumann subfactor of the ultrapower of the hyperfiniteii 1 factor ifg is a maximally almost periodic discrete non amenable group with haagerup s herz schur multiplier constant g 1 then the universal groupc algebrac g is not exact but the reduced groupc albegrac g is exact and isqwep but does not satisfywep andllp we study functiorial properties of the classes ofc algebras satisfyingwep llp resp beingqwep as applications we obtain some unexpected relations between some open questions onc algebras

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

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

besov class via heat semigroup on dirichlet spaces i sobolev type inequalities abstract we introduce heat semigroup based besov classes in the general framework of dirichlet spaces general properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained as a highlight of the paper we obtain a far reaching l p analogue p 1 of the sobolev inequality that was proved for p 2 by n varopoulos under the assumption of ultracontractivity for the heat semigroup the case p 1 is of special interest since it yields isoperimetric type inequalities

free products of c algebras small spatial free products of c algebras are constructed under certain conditions they have properties similar to those proved by paschke and salinas for the algebras c gi g2 where g1 g2 are discrete groups the freeproduct analogs of noncommutative bernoulli shifts are discussed 0 introduction let k be a field consider the category of unital algebras over k it is well known that this category admits coproducts free products of algebras 2 heuristically the free product of algebras is the algebra generated by them with no relations except for the identification of unit elements if k c the complex numbers and we consider unital algebras we can easily define a operation on the free products let a b be unital c algebras and a b their free product which is a unital algebra the question arises in what ways may one define a pre c norm on a b that extends the norms on a and b guided by analogy with tensor products we expect to have a choice among many pre c norms giving rise to many c free products of a and b one natural norm is 1c i i sup 1it c ih st representation of a b the representations of a b are in 1 1 correspondence with pairs of representations of a and b which act on the same hilbert space let a b be the completion of a b in this norm it is easy to see that this construction defines a coproduct in the category of c algebras and that a b is the biggest free product of a and b analogous to the biggest tensor product a 0 b if g 1 g2 are discrete groups we obtain c g1 c g2 c gi g2 where g1 g2 is the free product group and this is analogous to the relation c g1 0 c g2 c g1 x g2 this paper is motivated by the question is there a smallest c product a b in analogy to the smallest tensor product a b satisfying a relation cr gi g2 cr gi cr g2 received by the editors january 28 1981 and in revised form may 19 1981 1980 mathematics subject classification primary 46l05 secondary 46l55 1 982 american mathematical society 0002 9947 82 00001022 04 25

outer automorphisms and reduced crossed products of simple c algebras every outer automorphism of a separable simplec algebra is shown to have a pure state which is mapped into an inequivalent state under this automorphism the reduced crossed product of a simplec algebra by a discrete group of outer automorphisms is shown to be simple

sobolev algebras on lie groups and riemannian manifolds we prove that on any connected unimodular lie group g the space l p g l g where l p g is the sobolev space of order 0 associated with a sublaplacian is an algebra under pointwise product this generalizes results due to strichartz in the euclidean case to bohnke in the case of stratified groups and others a global version of this fact holds for groups with polynomial growth we give similar results for riemannian manifolds with ricci curvature bounded from below respectively nonnegative

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

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

spectral dimension and bohr s formula for schrodinger operators on unbounded fractal spaces we establish an asymptotic formula for the eigenvalue counting function of the schrodinger operator for some unbounded potentials v on several types of unbounded fractal spaces we give sufficient conditions for bohr s formula to hold on metric measure spaces which admit a cellular decomposition and then verify these conditions for fractafolds and fractal fields based on nested fractals in particular we partially answer a question of fan khandker and strichartz regarding the spectral asymptotics of the harmonic oscillator potential on the infinite blow up of a sierpinski gasket

on approximating the arithmetic geometric mean and complete elliptic integral of the first kind abstract in the article we prove that the double inequalities 1 6 p 7 r p 5 p 6 r tanh 1 r 2 r k r 1 6 q 7 r q 5 q 6 r tanh 1 r 2 r q a 1 r 5 q 6 g 1 r a 1 r 6 q 7 g 1 r l 1 r a g m 1 r p a 1 r 5 p 6 g 1 r a 1 r 6 p 7 g 1 r l 1 r hold for all r 0 1 if and only if p 2 1 570796 and q 89 69 1 289855 where k r 0 2 1 r 2 sin 2 t 1 2 d t is the complete elliptic integral of the first kind tanh 1 r log 1 r 1 r 2 is the inverse hyperbolic tangent function r 1 r 2 and a 1 r 1 r 2 g 1 r r l 1 r r 1 log r and a g m 1 r are the arithmetic geometric logarithmic and gaussian arithmetic geometric means of 1 and r respectively

crossed products by finite group actions with the rokhlin property we prove that a number of classes of separable unital c algebras are closed under crossed products by finite group actions with the rokhlin property including a ai algebras at algebras and related classes characterized by direct limit decompositions using semiprojective building blocks b simple unital ah algebras with slow dimension growth and real rank zero c c algebras with real rank zero or stable rank one d simple c algebras for which the order on projections is determined by traces e c algebras whose quotients all satisfy the universal coefficient theorem f c algebras with a unique tracial state along the way we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective

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

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

calculus on the sierpinski gasket i polynomials exponentials and power series abstract we study the analog of power series expansions on the sierpinski gasket for analysis based on the kigami laplacian the analog of polynomials are multiharmonic functions which have previously been studied in connection with taylor approximations and splines here the main technical result is an estimate of the size of the monomials analogous to x n n we propose a definition of entire analytic functions as functions represented by power series whose coefficients satisfy exponential growth conditions that are stronger than what is required to guarantee uniform convergence we present a characterization of these functions in terms of exponential growth conditions on powers of the laplacian of the function these entire analytic functions enjoy properties such as rearrangement and unique determination by infinite jets that one would expect however not all exponential functions eigenfunctions of the laplacian are entire analytic and also many other natural candidates such as the heat kernel do not belong to this class nevertheless we are able to use spectral decimation to study exponentials and in particular to create exponentially decaying functions for negative eigenvalues

calculus on the sierpinski gasket ii point singularities eigenfunctions and normal derivatives of the heat kernel this paper continues the study of fundamental properties of elementary functions on the sierpinski gasket sg related to the laplacian defined by kigami harmonic functions multiharmonic functions and eigenfunctions of the laplacian we describe the possible point singularities of such functions and we use the description at certain periodic points to motivate the definition of local derivatives at these points we study the global behavior of eigenfunctions on all generic infinite blow ups of sg and construct eigenfunctions that decay at infinity in certain directions we study the asymptotic behavior of normal derivatives of dirichlet eigenfunctions at boundary points and give experimental evidence for the behavior of the normal derivatives of the heat kernel at boundary points

approximation of complex symmetric operators an operator t on a complex hilbert space mathcal h is called a complex symmetric operator if there exists a conjugate linear isometric involution c mathcal h rightarrow mathcal h so that ctc t in this paper we study the approximation of complex symmetric operators by virtue of an intensive analysis of compact operators in singly generated c algebras we obtain a complete characterization of norm limits of complex symmetric operators and provide a classification of complex symmetric operators up to approximate unitary equivalence this gives a general solution to the norm closure problem for complex symmetric operators as an application we provide a concrete description of partial isometries which are norm limits of complex symmetric operators

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

expansion in generalized eigenfunctions for laplacians on graphs and metric measure spaces we consider an arbitrary selfadjoint operator on a separable hilbert space to this operator we construct an expansion in generalized eigenfunctions in which the original hilbert space is decomposed as a direct integral of hilbert spaces consisting of general eigenfunctions this automatically gives a plancherel type formula for suitable operators on metric measure spaces we discuss some growth restrictions on the generalized eigenfunctions for laplacians on locally finite graphs the generalized eigenfunctions are exactly the solutions of the cor responding difference equation

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