Calculus on the Sierpinski Gasket I: Polynomials, Exponentials and Power Series

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

Conductive Homogeneity of Compact Metric Spaces and Construction of -Energy

In the ordinary theory of Sobolev spaces on domains of $R^n$, the $p$-energy is defined as the integral of $|\nabla{f}|^p$. In this paper, we try to construct $p$-energy on compact metric spaces as a scaling limit of discrete $p$-energies on a series of graphs approximating the original space. In conclusion, we propose a notion called conductive homogeneity under which one can construct a reasonable $p$-energy if $p$ is greater than the Ahlfors regular conformal dimension of the space. In particular, if $p = 2$, then we construct a local regular Dirichlet form and show that the heat kernel associated with the Dirichlet form satisfies upper and lower sub-Gaussian type heat kernel estimates. As examples of conductively homogeneous spaces, we present a new class of square-based self-similar sets and rationally ramified Sierpinski cross, where no diffusion was constructed before.Comment: (Almost) final version reflecting comments from referees and reader

New Bounds for Edge-Cover by Random Walk

We show that the expected time for a random walk on a (multi-)graph $G$ to traverse all $m$ edges of $G$, and return to its starting point, is at most $2m^2$; if each edge must be traversed in both directions, the bound is $3m^2$. Both bounds are tight and may be applied to graphs with arbitrary edge lengths, with implications for Brownian motion on a finite or infinite network of total edge-length $m$

Nontangential limits and Fatou-type theorems on post-critically finite self-similar sets

In this paper we study the boundary limit properties of harmonic functions on $\mathbb R_+\times K$, the solutions $u(t,x)$ to the Poisson equation $$\frac{\partial^2 u}{\partial t^2} + \Delta u = 0,$$ where $K$ is a p.c.f. set and $\Delta$ its Laplacian given by a regular harmonic structure. In particular, we prove the existence of nontangential limits of the corresponding Poisson integrals, and the analogous results of the classical Fatou theorems for bounded and nontangentially bounded harmonic functions.Comment: 22 page

Hydrodynamic limit for a zero-range process in the Sierpinski gasket

We prove that the hydrodynamic limit of a zero-range process evolving in graphs approximating the Sierpinski gasket is given by a nonlinear heat equation. We also prove existence and uniqueness of the hydrodynamic equation by considering a finite-difference scheme.Comment: 24 pages, 1 figur

Characterization of Self-organied Osteocytic Spheroids Using Mouse Osteoblast-like Cells

Osteocyte plays a central role as a commander in the bone to modulate bone remodeling processes. While the osteocyte is known to be differentiated from osteoblasts, understanding in mechanism of the osteocyte differentiation remained still poor. The aim of this study is to elucidate the osteocyte differentiation capability using three-dimensional (3D) cell culture technique. We first fabricated a self-organized spheroid reconstructed by mouse osteoblast-like cells by adjusting the number of subcultured cells in the round-bottom well. Compared to a conventional two-dimensional (2D) monolayer model, the 3D spheroid exerted greater osteocyte gene expressions in vitro within 2 days. As a result of the size-dependent experiment, there might be an appropriate cell-cell and cell-ECM interaction for osteoblast-like cells to induce the osteocytogenesis in the form of 3D spheroid culture. Moreover, the present model showed that the spheroid further exerted the prolonged osteocyte differentiation capability after a long period of incubation, 7 days. In conclusion, we characterized the self-organized osteocytic spheroids reconstructed by osteoblast-like cells and further suggested the potential application of the spheroid as a new in vitro tissue-engineered osteocytic model

A C*-algebra of geometric operators on self-similar CW-complexes. Novikov-Shubin and L^2-Betti numbers

A class of CW-complexes, called self-similar complexes, is introduced, together with C*-algebras A_j of operators, endowed with a finite trace, acting on square-summable cellular j-chains. Since the Laplacian Delta_j belongs to A_j, L^2-Betti numbers and Novikov-Shubin numbers are defined for such complexes in terms of the trace. In particular a relation involving the Euler-Poincare' characteristic is proved. L^2-Betti and Novikov-Shubin numbers are computed for some self-similar complexes arising from self-similar fractals.Comment: 30 pages, 7 figure