singular continuous spectrum for palindromic schrodinger operators we give new examples of discrete schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum if the hullx of the potential is strictly ergodic then the existence of just one potentialx inx for which the operator has no eigenvalues implies that there is a generic set inx for which the operator has purely singular continuous spectrum a sufficient condition for the existence of such anx is that there is az x that contains arbitrarily long palindromes thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inx the class includes well known substitutions like fibonacci thue morse period doubling binary non pisot and ternary non pisot we also show that the operator has no absolutely continuous spectrum for allx x ifx derives from a primitive substitution for potentials defined by circle maps x n 1 j 0 n we show that the operator has purely singular continuous spectrum for a generic subset inx for all irrational and every half open intervalj

constructing higher order methods for obtaining the multiple roots of nonlinear equations this paper concentrates on iterative methods for obtaining the multiple roots of nonlinear equations using the computer algebra system mathematica we construct an iterative scheme and discuss the conditions to obtain fourth order methods from it all the presented fourth order methods require one function and two derivative evaluation per iteration and are optimal higher order iterative methods for obtaining multiple roots we present some special methods from the iterative scheme including some known already numerical examples are also given to show their performance

eikonal equations on the sierpinski gasket we study the eikonal equation on the sierpinski gasket in the spirit of the construction of the laplacian in kigami analysis on fractals cambridge tracts in mathematics 2001 we consider graph eikonal equations on the prefractals and we show that the solutions of these problems converge to a function defined on the fractal set we characterize this limit function as the unique metric viscosity solution to the eikonal equation on the sierpinski gasket according to the definition introduced in giga et al trans am math soc 367 49 66 2015

a third order modification of newton s method for multiple roots in this paper we present a new third order modification of newton s method for multiple roots which is based on existing third order multiple root finding methods numerical examples show that the new method is competitive to other methods for multiple roots

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

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

construction of optimal order nonlinear solvers using inverse interpolation abstract there is a vast literature on finding simple roots of nonlinear equations by iterative methods these methods can be classified by order by the information used or by efficiency there are very few optimal methods that is methods of order 2 m requiring m 1 function evaluations per iteration here we give a general way to construct such methods by using inverse interpolation and any optimal two point method the presented optimal multipoint methods are tested on numerical examples and compared to existing methods of the same order of convergence

spectra and transport in almost periodic dimers we study spectral properties of discrete schrodinger operators with potentials obtained via dimerization of a class of aperiodic sequences it is shown that both the nature of the autocorrelation measure of a regular sequence and the presence of generic full probability singular continuous spectrum in the hull of primitive and palindromic four block substitution potentials are robust under dimerization generic results also hold for circle potentials we illustrate these results with numerical studies of the quantum mean square displacement as a function of time the numerical techniques provide a very fast algorithm for the time evolution of wave packets

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

natural boundaries and spectral theory we present and exploit an analogy between lack of absolutely continuous spectrum for schrodinger operators and nat ural boundaries for power series among our new results are gen eralizations of hecke s example and natural boundary examples for random power series where independence is not assumed

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

the normalized laplacian spectrum of subdivisions of a graph determining and analyzing the spectra of graphs is an important and exciting research topic in mathematics science and theoretical computer science the eigenvalues of the normalized laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects in particular those related to random walks in this paper we give the spectra of the normalized laplacian of iterated subdivisions of simple connected graphs as an example of application of these results we find the exact values of their multiplicative degree kirchhoff index kemeny s constant and number of spanning trees

cauchy transforms of self similar measures the laurent coefficients abstract the cauchy transform of a measure has been used to study the analytic capacity and uniform rectifiability of subsets in c recently lund et al experiment math 7 1998 177 have initiated the study of such transform f of self similar measure in this and the forecoming papers starlikeness and the cauchy transform of some self similar measures in preparation the cauchy transform on the sierpinski gasket in preparation we study the analytic and geometric behavior as well as the fractal behavior of the transform f the main concentration here is on the laurent coefficients an n 0 of f we give asymptotic formulas for an n 0 and for f k z near the support of hence the precise growth rates on an and f k are determined these formulas are connected with some multiplicative periodic functions which reflect the self similarity of and k as a by product we also discover new identities of certain infinite products and series

letter to the editor new higher order methods for solving nonlinear equations with multiple roots for a nonlinear equation f x 0 having a multiple root we consider steffensen s transformation t using the transformation say f q x t qf x for integer q 2 repeatedly we develop higher order iterative methods which require neither derivatives of f x nor the multiplicity of the root it is proved that the convergence order of the proposed iterative method is 1 2 q 2 for any equation having a multiple root of multiplicity m 2 the efficiency of the new method is shown by the results for some numerical examples

singularly continuous spectrum of a self similar laplacian on the half line we investigate the spectrum of the self similar laplacian which generates the so called pq random walk on the integer half line using the method of spectral decimation we prove that the spectral type of the laplacian is singularly continuous whenever p 12 this serves as a toy model for generating singularly continuous spectrum which can be generalized to more complicated settings we hope it will provide more insight into fibonacci type and other weakly self similar models

semilinear equations on fractal blowups abstract the paper concerns existence of a ground state for a nonlinear scalar field equation on a blowup fractal where imbedding of the energy space into l p is not compact in absence of invariant transformations involved in conventional concentration compactness argument the paper develops convergence reasoning based on the fractal s self similarity

new families of nonlinear third order solvers for finding multiple roots in this paper we present two new families of iterative methods for multiple roots of nonlinear equations one of the families require one function and two derivative evaluation per step and the other family requires two function and one derivative evaluation it is shown that both are third order convergent for multiple roots numerical examples suggest that each family member can be competitive to other third order methods and newton s method for multiple roots in fact the second family is even better than the first

exact hausdorff measure and intervals of maximum density for cantor sets consider a linear cantor set k which is the attractor of a linear iterated function system i f s sjx pjx bj j 1 m on the line satisfying the open set condition where the open set is an interval it is known that k has hausdorff dimension ca given by the equation ftl1 pa 1 and that 7 t k is finite and positive where 7h x denotes hausdorff measure of dimension a we give an algorithm for computing 7 t k exactly as the maximum of a finite set of elementary functions of the parameters of the i f s when pi pm or more generally if log p1 and log pm are commensurable the algorithm also gives an interval i that maximizes the density d i 7 ha k n i ii i the hausdorff measure 7 o k is not a continuous function of the i f s parameters we also show that given the contraction parrameters pj it is possible to choose the translation parameters bj in such a way that 7 h k ikia so the maximum density is one most of the results presented here were discovered through computer experiments but we give traditional mathematical proofs

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

localization in general one dimensional random systems i jacobi matrices we consider random discrete schrodinger operators in a strip with a potentialv n n a label in and a finite label across the strip andv an ergodic process we prove thath0 v has only point spectrum with probability one under two assumptions 1 theconditional distribution of v n n 0 1 all conditioned on v n 0 1 all has an absolutely continuous component with positive probability 2 for a e e no lyaponov exponent is zero

families of third and fourth order methods for multiple roots of nonlinear equations this paper presents two families of higher order iterative methods for solving multiple roots of nonlinear equations one is of order three and the other is of order four the presented iterative families all require two evaluations of the function and one evaluation of its first derivative thus the latter is of optimal order the third order family contains several iterative methods known already and different from the optimal fourth order methods for multiple roots known already the presented fourth order family use the modified newton s method as its first step local convergence analyses and some special cases of the presented families are given we also carry out some numerical examples to show their performance

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

exact evaluation of the causal spectrum and localization properties of electronic states on a scale free network abstract a deterministic network with tree structure is considered for which the spectrum of its adjacency matrix can be exactly evaluated by a recursive renormalization approach it amounts to successively increasing number of contributions at any finite step of construction of the tree resulting in a causal chain the resulting eigenvalues can be related the full energy spectrum of a nearest neighbor tight binding model defined on this structure given this association it turns out that further properties of the eigenvectors can be evaluated like the degree of quantum localization of the tight binding eigenstates expressed by the inverse participation ratio ipr it happens that for the current model the ipr s are also suitable to be analytically expressed in terms in corresponding eigenvalue chain the resulting ipr scaling behavior is expressed by the tails of eigenvalue chains as well

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

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

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

euclidean reconstruction and reprojection up to subgroups the necessary and sufficient conditions for being able to estimate scene structure motion and camera calibration from a sequence of images are very rarely satisfied in practice what exactly can be estimated in sequences of practical importance when such conditions are not satisfied in this paper we give a complete answer to this question for every camera motion that fails to meet the conditions we give explicit formulas for the ambiguities in the reconstructed scene motion and calibration such a characterization is crucial both for designing robust estimation algorithms that do not try to recover parameters that cannot be recovered and for generating novel views of the scene by controlling the vantage point to this end we characterize explicitly all the vantage points that give rise to a valid euclidean reprojection regardless of the ambiguity in the reconstruction we also characterize vantage points that generate views that are altogether invariant to the ambiguity all the results are presented using simple notation that involves no tensors nor complex projective geometry and should be accessible with basic background in linear algebra

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

a note on convergence radius of osada s method under holder continuous condition abstract in this paper we revise the proofs of the results obtained in convergence radius of osada s method under holder continuous condition 4 because the remainder of the taylor s expansion used for the obtainment of the local convergence radius is not correct so we perform the complete study in order to modify the equation for getting the local convergence radius the uniqueness radius and the error bounds moreover a dynamical study for the third order osada s method is also developed

convergence radius of the modified newton method for multiple zeros under holder continuous derivative abstract in recent years a lot of iterative methods for finding multiple zeros of nonlinear equations have been presented and analyzed however almost all these studies give no information for the convergence radius of the corresponding method in this paper we give an estimate of the convergence radius of the well known modified newton s method for multiple zeros when the involved function satisfies a holder and center holder continuity condition

operators with singular continuous spectrum i general operators the baire category theorem implies that the family f of dense sets g in fixed metric space x is a candidate for generic sets since it is closed under countable intersections and if x is perfect has no isolated point then a f has uncountable intersections with any open ball in x

on the spectrum of lamplighter groups and percolation clusters let mathfrak g be a finitely generated group and x its cayley graph with respect to a finite symmetric generating set s furthermore let mathfrak h be a finite group and mathfrak h wr mathfrak g the lamplighter group wreath product over mathfrak g with group of lamps mathfrak h we show that the spectral measure plancherel measure of any symmetric switch walk switch random walk on mathfrak h wr mathfrak g coincides with the expected spectral measure integrated density of states of the random walk with absorbing boundary on the cluster of the group identity for bernoulli site percolation on x with parameter mathsf p 1 mathfrak h the return probabilities of the lamplighter random walk coincide with the expected annealed return probabilities on the percolation cluster in particular if the clusters of percolation with parameter mathsf p are almost surely finite then the spectrum of the lamplighter group is pure point this generalizes results of grigorchuk and uk resp dicks and schick regarding the case when mathfrak g is infinite cyclic analogous results relate bond percolation with another lamplighter random walk in general the integrated density of states of site or bond percolation with arbitrary parameter mathsf p is always related with the plancherel measure of a convolution operator by a signed measure on mathfrak h wr mathfrak g where mathfrak h mathbb z or another suitable group

stochastic schrodinger operators and jacobi matrices on the strip we discuss stochastic schrodinger operators and jacobi matrices with wave functions taking values in l so there are 2l lyaponov exponents 1 l 0 l 1 2l 1 our results include the fact that if 1 0 on a set positive measure thenv is deterministic and one that says that e exactly 2j s are zero is the essential support of the a c spectrum of multiplicity 2j

generic singular spectrum for ergodic schrodinger operators we consider schrodinger operators with ergodic potential v n f t n n z where t is a nonperiodic homeomorphism we show that for generic f c the spectrum has no absolutely continuous component the proof is based on approximation by discontinuous potentials which can be treated via kotani theory

operators with singular continuous spectrum ii rank one operators for an operator a with cyclic vector we studya p wherep is the rank one projection onto multiples of if spec a anda has no a c spectrum we prove thata p has purely singular continuous spectrum on for a denseg of s

clustering to minimize the maximum intercluster distance the problem of clustering a set of points so as to minimize the maximum intercluster distance is studied an o kn approximation algorithm where n is the number of points and k is the number of clusters that guarantees solutions with an objective function value within two times the optimal solution value is presented this approximation algorithm succeeds as long as the set of points satisfies the triangular inequality we also show that our approximation algorithm is best possible with respect to the approximation bound if pz np

extended hamiltonian learning on riemannian manifolds theoretical aspects this paper introduces a general theory of extended hamiltonian second order learning on riemannian manifolds as an instance of learning by constrained criterion optimization the dynamical learning equations are derived within the general framework of extended hamiltonian stationary action principle and are expressed in a coordinate free fashion a theoretical analysis is carried out in order to compare the features of the dynamical learning theory with the features exhibited by the gradient based ones in particular gradient based learning is shown to be an instance of dynamical learning and the classical gradient based learning modified by a momentum term is shown to resemble discrete time dynamical learning moreover the convergence features of gradient based and dynamical learning are compared on a theoretical basis this paper discusses cases of learning by dynamical systems on manifolds of interest in the scientific literature namely the stiefel manifold the special orthogonal group the grassmann manifold the group of symmetric positive definite matrices the generalized flag manifold and the real symplectic group of matrices

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

singular continuous spectrum for a class of almost periodic jacobi matrices we consider operators parametrized by a o on 2 with matrix ojy i o i fl o y with an cos 2ttcw 6 if ce is a liouville number and 2 we prove that for a e 0 the operator s spectral measures are all singular continuous we consider the operator on l2 z depending upon three parameters x a 0 1 x a 0 u n u n 1 u n 1 x cos 2iran 6 u n in this note we will sketch the proof of the following result whose detailed proof will appear elsewhere 3 theorem 1 fix a an irrational number obeying

one dimensional schrodinger operators with random decaying potentials we investigate the spectrum of the following random schrodinger operators h omega frac d 2 dt 2 a t f x t omega wheref xt is a markovian potential studied by the russian school 8 we completely describe the transition of the spectrum from pure point type to absolutely continuous type as the decreasing order ofa t grows this is an extension to a continuous case of the result due to delyon simon souillard 6 who deal with the lattice case

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

on the spectrum of the normalized laplacian of iterated triangulations of graphs the eigenvalues of the normalized laplacian of a graph provide information on its topological and structural characteristics and also on some relevant dynamical aspects specifically in relation to random walks in this paper we determine the spectra of the normalized laplacian of iterated triangulations of a generic simple connected graph as an application we also find closed forms for their multiplicative degree kirchhoff index kemeny s constant and number of spanning trees

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

high order nonlinear solver for multiple roots a method of order four for finding multiple zeros of nonlinear functions is developed the method is based on jarratt s fifth order method for simple roots and it requires one evaluation of the function and three evaluations of the derivative the informational efficiency of the method is the same as previously developed schemes of lower order for the special case of double root we found a family of fourth order methods requiring one less derivative thus this family is more efficient than all others all these methods require the knowledge of the multiplicity

the spectral function for sturm liouville problems where the potential is of wigner von neumann type or slowly decaying we consider the linear second order differential equation y q x y 0on 0 with the boundary condition y 0 cos y 0 sin 0forsome 0 we suppose that q x is real valued continuously differentiable and that q x 0 as x with q l1 0 our main object of study is the spectral function associated with and we derive a series expansion for this function valid for 0 where 0 is computable and establish a 1 also computable such that and with 0 have no points of spectral concentration for 1 we illustrate our results with examples in particular we consider the case of the wigner von neumann potential

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

parabolic type equations associated with the dirichlet form on the sierpinski gasket by using analytic tools from stochastic analysis we initiate a study of some non linear parabolic equations on sierpinski gasket motivated by modellings of fluid flows along fractals which can be considered as models of simplified rough porous media unlike the regular space case such parabolic type equations involving non linear convection terms must take a different form due to the fact that convection terms must be singular to the linear part which defines the heat semigroup in order to study these parabolic type equations a new kind of sobolev inequalities for the dirichlet form on the gasket will be established these sobolev inequalities which are interesting on their own and in contrast to the case of euclidean spaces involve two l p norms with respect to two mutually singular measures by examining properties of singular convolutions of the associated heat semigroup we derive the space time regularity of solutions to these parabolic equations under a few technical conditions the burgers equations on the sierpinski gasket are also studied for which a maximum principle for solutions is derived using techniques from backward stochastic differential equations and the existence uniqueness and regularity of its solutions are obtained

the resolvent kernel for pcf self similar fractals for the laplaciandefined on a p c f self similar fractal we give an explicit formula for the resolvent kernel of the laplacian with dirichlet boundary conditions and also with neumann boundary conditions that is we construct a symmetric function g which solves i 1 f x r g x y f y d y the method is similar to kigami s construction of the green kernel in kig01 3 5 and is expressed as a sum of scaled and translated copies of a certain function which may be considered as a fundamental solution of the resolvent equation examples of the explicit resolvent kernel formula are given for the unit interval standard sierpinski gasket an d the level 3 sierpinski gasket s g3

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

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

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

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

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

convergence radius of halley s method for multiple roots under center holder continuous condition recently a new treatment based on taylor s expansion to give the estimate of the convergence radius of iterative method for multiple roots has been presented it has been successfully applied to enlarge the convergence radius of the modified newton s method and osada s method for multiple roots this paper re investigates the convergence radius of halley s method under the condition that the derivative f m 1 of function f satisfies the center holder continuous condition we show that our result can be obtained under much weaker condition and has a wider range of application than that given by bi et al 2011 21

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

optimization on lie manifolds and pattern recognition several pattern recognition problems can be reduced in a natural way to the problem of optimizing a nonlinear function over a lie manifold however optimization on lie manifolds involves in general a large number of nonlinear equality constraints and is hence one of the hardest optimization problems we show that exploiting the special structure of lie manifolds allows one to devise a method for optimizing on lie manifolds in a computationally efficient manner the new method relies on the differential geometry of lie manifolds and the underlying connections between lie groups and their associated lie algebras we describe an application of the new lie group method to the problem of diagnosing malignancy in the cytological extracts of breast tumors the diagnosis method that we present has a mean sensitivity of 98 086 and a predictive index of 0 0602 making it the most accurate and reliable diagnostic method reported thus far

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

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

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

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

absolute continuity for unbounded jacobi matrices with constant row sums we investigate the spectral properties of a class of jacobi matrices in which the subdiagonal entries are quadratics and the row sums are constants

instability of dense point spectrum under finite rank perturbations we prove that if a is a self adjoint operator and r a non negative finite rank operator whose range is cyclic for a then for a generic in the baire sense t in each eigenvalue of the operator has a neighborhood containing no other eigenvalues

some jacobi matrices with decaying potential and dense point spectrum we discuss doubly infinite matrices of the formm ij i j 1 i j 1 v i ij as operators on l2 we present for each 0 examples of potentialsv n with v n o n 1 2 and wherem has only point spectrum our discussion should be viewed as a remark on the recent work of delyon kunz and souillard

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

operators with singular continuous spectrum vii examples with borderline time decay we construct one dimensional potentials v x so that if h d 2 dx 2 v x on l 2 then h has purely singular spectrum but for a dense set d d implies that ith c t 1 2 in t for t 2 this implies the spectral measures have hausdorff dimension one and also following an idea of malozemov molchanov provides counterexamples to the direct extension of the theorem of simon spencer on one dimensional infinity high barriers

fractafolds based on the sierpinski gasket and their spectra we introduce the notion of fractafold which is to a fractal what a manifold is to a euclidean half space we specialize to the case when the fractal is the sierpinski gasket sg we show that each such compact fractafold can be given by a cellular construction based on a finite cell graph g which is 3 regular in the case that the fractafold has no boundary we show explicitly how to obtain the spectrum of the fractafold from the spectrum of the graph using the spectral decimation method of fukushima and shima this enables us to obtain isospectral pairs of nonhomeomorphic fractafolds we also show that although sg is topologically rigid there are fractafolds based on sg that are not topologically rigid

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

convergence radius of osada s method under center holder continuous condition recently a new treatment based on taylor s expansion to give the estimate of the convergence radius of iterative method for multiple roots has been presented it has been successfully applied to enlarge the estimate of the convergence radius of the modified newton s method for multiple roots using the similarly treatment this paper investigates the convergence radius of the osada s method under the condition that the derivative f m 1 of function f satisfies the center holder continuous condition by some examples we show the treatment is simpler and efficient once again the uniqueness ball of solution is also discussed