Thesis (M.S.)--West Virginia University, 2005. / Title from document title page. Document formatted into pages; contains viii, 80 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 79-80).

In [DS1,2,3], Denker and Sato initiated a new point of view to study the problem. They identified the Sierpinski gasket as a Martin boundary of some canonical Markov chain and used the associated theory to consider the problem. In this thesis, we will extend their result so as to be applicable to all single-point connected monocyclic post critically finite (m.p.c.f.) self-similar sets. / In the first chapter, we review some basic facts of the self-similar sets and the Martin boundaries, and we prove that every m.p.c.f. self-similar set K is homeomorphic to the quotient space of the symbolic space associated with K, moreover, the homeomorphism is a Lipschitz equivalence for some special m.p.c.f. self-similar sets. / In the second chapter, we first prove that the quotient space of the symbolic space associated with K is homeomorphic to the Martin boundary with respect to the state space associated with K if K is a single-point connected m.p.c.f. self-similar set. Combining this result and the result in the first chapter, we conclude that every single-point connected m.p.c.f. self-similar set can be identified with the Martin boundary of some canonical Markov chain. Then for the 3-level Sierpinski gasket, we prove that there exists a one to one relation between the strongly P-harmonic functions on the 3 state space and K-harmonic functions constructed by Kigami. / In the third chapter, we define a new Markov chain on the pentagasket K which is a single-point connected m.p.c.f. self-similar also. Under the new Markov chain, we prove that K can be identified with the Martin boundary of the new Markov chain and that there exists a one to one relation between the strongly P-harmonic functions and the K-harmonic functions. / One of the fundamental problems in fractal analysis is to construct a Laplacian on fractals. Since fractals, like the Sierpinski gasket and the pentagasket, do not have any smooth structures, it is not possible to construct it from the classical point of view. Hence, until now there is no systematic way to define such a notion on the general class of fractals. / There are two approaches for the problem which have achieved some success in certain special situations. The first one is a probabilistic approach via constructing Brownian motions on self-similar sets. The second approach is an analytical one proposed by Kigami. He approximated the underlying self-similar set K by an increasing sequence of finite sets equipped with the discrete Laplacians Hm in a consistent way. He showed that if K is strongly symmetric, then Hm converge to a Laplacian on K. / by Ju, Hongbing. / "March 2008." / Adviser: Lau Ka Sing. / Source: Dissertation Abstracts International, Volume: 70-03, Section: B, page: 1702. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 91-94). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

The class of groups generated by automata have been a source of many counterexamples in group theory. At the same time it is connected to other branches of mathematics, such as analysis, holomorphic dynamics, combinatorics, etc. A question that naturally arises is finding the ways to classify these groups. The task of a complete classification and understanding at the moment seems to be too ambitious, but it is reasonable to concentrate on some smaller subclasses of this class. One approach is to consider groups generated by small automata: the automata with k states over d-letter alphabet (so called, (k,d)-automata) with small values of k and d. Certain steps in this directions have been made already: All groups generated by (2,2)-automata have been classified, and groups generated by (3,2)-automata were studied. In this work we study the class of groups generated by (4,2)-automata. More specifically, we partition all such automata into equivalence classes up to symmetry and minimal symmetry (symmetric and minimally symmetric automata naturally generate isomorphic groups) and classify completely all finite groups generated by automata in this class. We also list all classes generating abelian groups. Another important result of the project is developing a database of (4,2)-automata and computational routines that represent a new effective tool for the search for (4,2)-automata generating groups with specific properties, which hopefully will lead to finding counterexamples of certain conjectures.

Burgers’ Equation ut + cuux = νuxx is a nonlinear partial differential equation which arises in models of traffic and fluid flow. It is perhaps the simplest equation describing waves under the influence of diffusion. We consider the large time behavior of solutions with exponentially localized initial conditions, analyzing the rate of convergence to a known self similar single-hump solution. We use the Cole-Hopf Transformation to convert the problem into a heat equation problem with exponentially localized initial conditions. The solution to this problem converges to a Gaussian. We then find an optimal Gaussian approximation which is accurate to order t−2. Transforming back to Burgers’ Equation yields a solution accurate to order t−2.

This dissertation is devoted to the measure theoretical aspects of the theory of automata and groups generated by them. It consists of two main parts. In the first part we study the action of automata on Bernoulli measures. We describe how a finite-state automorphism of a regular rooted tree changes the Bernoulli measure on the boundary of the tree. It turns out, that a finite-state automorphism of polynomial growth, as defined by Sidki, preserves a measure class of a Bernoulli measure, and we write down the explicit formula for its Radon-Nikodim derivative. On the other hand the image of the Bernoulli measure under the action of a strongly connected finite-state automorphism is singular to the measure itself. The second part is devoted to introduction of measure into the theory of limit spaces of Nekrashevysh. Let G be a group and φ : H → G be a contracting homomorphism from a subgroup H < G of finite index. Nekrashevych associated with the pair (G, φ) the limit dynamical system (JG, s) and the limit G-space XG together with the covering ∪g∈GT · g by the tile T. We develop the theory of selfsimilar measures m on these limit spaces. It is shown that (JG, s,m) is conjugate to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile T has integer measure and we give an algorithmic way to compute it. In addition we give an algorithm to find the measure of the intersection of tiles T ∩ (T · g) for g ∈ G. We present applications to the evaluation of the Lebesgue measure of integral self-affine tiles. Previously the main tools in the theory of self-similar fractals were tools from measure theory and analysis. The methods developed in this disseration provide a new way to investigate self-similar and self-affine fractals, using combinatorics and group theory.

Let μ be a Radon probability measure on M, the d-dimensional Real Euclidean space (where d is a positive integer), and f a measurable function. Let P be the space of sequences whose coordinates are elements in M. Then, for any point x in M, define a function ƒn on M and P that looks at the first n terms of an element of P and evaluates f at the first of those n terms that minimizes the distance to x in M. The measures for which such sequences converge in measure to f for almost every sequence are called Mycielski-regular. We show that the self-similar measure generated by a finite family of contracting similitudes and which up to a constant is the Hausdorff measure in its dimension on an invariant set C is Mycielski-regular.

In this dissertation, we address the statistical analysis under the multiscale framework for the self-similar process. Motivated by the problems arising from geophysics and health informatics, we develop a set of statistical measures as discriminative summaries of the self-similar process. These measures include Multiscale Schur Monotone (MSM) measures, Geometric Attributes of Multifractal Spectrum (GAMFS), Quasi-Hurst exponents, Mallat Model and Tsallis Maxent Model. These measures are used as methods to quantify the difference (or similarities) or as input (feature) vectors in the classification model. As the cornstone of GAMFS, we study the estimation of multifractal spectrum and adopt a Weighted Least Squares (WLS) schemes in the wavelet domain to minimize the heteroskedastic effects , which is inherent because the sample variances of the wavelet coefficients depend on the scale. We also propose a Combined K-Nearest-Neighbor classifier (Comb-K-NN) to address the inhomogeneity of the class attributes, which is indicated by the large variations between subsets of input vectors. The Comb-K-NN classifier stabilizes the variations in the sense of reducing the misclassification rates. Bayesian justifications of Comb-K-NN classifier are provided. GAMFS, Quasi-Hurst exponents, Mallat Model and Tsallis Maxent Model are used in the study of assessing the effects of atmospheric stability on the turbulence measurements in the inertial subrange. We also formulate the criteria for success in evaluating how atmospheric stability alters the MFS of a single flow variable time series as a statistical classification model. We use the multifractal discriminate model as the solution of this problem. Also, high frequency pupil-diameter dynamic measurements, which are well documented as measures of mental workload, are summarized using both GAMFS and MSM. These summaries are further used as the feature vector in the Comb-K-NN classifier. The serious inhomogeneity among subjects in the same user group makes classification difficult. These difficulties are overcome by using Comb-K-NN classifier.

This dissertation is devoted to groups generated by bounded automata and geometric objects related to these groups (limit spaces, Schreier graphs, etc.). It is shown that groups generated by bounded automata are contracting. We introduce the notion of a post-critical set of a finite automaton and prove that the limit space of a contracting self-similar group generated by a finite automaton is post-critically finite (finitely-ramified) if and only if the automaton is bounded. We show that the Schreier graphs on levels of automaton groups can be constructed by an iterative procedure of inflation of graphs. This was used to associate a piecewise linear map of the form fK(v) = minA∈KAv, where K is a finite set of nonnegative matrices, with every bounded automaton. We give an effective criterium for the existence of a strictly positive eigenvector of fK. The existence of nonnegative generalized eigenvectors of fK is proved and used to give an algorithmic way for finding the exponents λmax and λmin of the maximal and minimal growth of the components of f(n) K (v). We prove that the growth exponent of diameters of the Schreier graphs is equal to λmax and the orbital contracting coefficient of the group is equal to 1/λmin . We prove that the simple random walks on orbital Schreier graphs are recurrent. A number of examples are presented to illustrate the developed methods with special attention to iterated monodromy groups of quadratic polynomials. We present the first example of a group whose coefficients λmin and λmax have different values.

This dissertation is devoted to groups generated by bounded automata and geometric objects related to these groups (limit spaces, Schreier graphs, etc.). It is shown that groups generated by bounded automata are contracting. We introduce the notion of a post-critical set of a finite automaton and prove that the limit space of a contracting self-similar group generated by a finite automaton is post-critically finite (finitely-ramified) if and only if the automaton is bounded. We show that the Schreier graphs on levels of automaton groups can be constructed by an iterative procedure of inflation of graphs. This was used to associate a piecewise linear map of the form fK(v) = minA[set]KAv, where K is a finite set of nonnegative matrices, with every bounded automaton. We give an effective criterium for the existence of a strictly positive eigenvector of fK. The existence of nonnegative generalized eigenvectors of fK is proved and used to give an algorithmic way for finding the exponents λmax and λmin of the maximal and minimal growth of the components of fK(n)(v). We prove that the growth exponent of diameters of the Schreier graphs is equal to λmax and the orbital contracting coefficient of the group is equal to 1/λmin . We prove that the simple random walks on orbital Schreier graphs are recurrent. A number of examples are presented to illustrate the developed methods with special attention to iterated monodromy groups of quadratic polynomials. We present the first example of a group whose coefficients λmin and λmax have different values.

The purpose of this paper is to construct sets, measures and energy forms of certain mixed nested fractals which are spatially homogeneous but not strictly self-similar. We start with the constructions of regular nested fractals, such as Sierpinski gaskets and Koch curves, by employing the iterated map system. Then we show that under the open set condition, the unique invariant (self-similar) measure consists with the normalized Hausdorff measure ristricted on the invariant set. The energy forms construced on regular Sierpinski gaskets and Koch curves is also proved to be a closed form. Next, we use the similar idea, by extending the iterated maps system into a general case, to construct the mixture sets, as well as measures and energy forms. It can be seen that the elements so constructed will not have any strict self-similarity, but them indeed satisfy some weak self-similar properties.