Discrete-time relational dynamical systems are mathematical models of possibly nonlinear and nondeterministic, state-based transition systems. They describe the time evolution of forests, viruses, parallel programs or cooperating agents. This thesis develops the compositional analysis of iterated relations: we study dynamical and computational properties of composed systems by combining the individual analyses of their components, simplified by abstraction techniques. We present a structural view of dynamical complexity, and a strict computational hierarchy of systems. Classical case studies are successfully analyzed: low-dimensional chaotic systems (logistic map, Smale horseshoe map, Cantor relation), high-dimensional complex systems (cellular automata), as well as formal systems (paperfoldings, Turing machines).

There is an ongoing massive effort by many researchers to link category theory and geometry, especially homotopy coherence and categorical coherence. This constitutes just a part of the broad undertaking known as categorification as described by Baez and Dolan. This effort has as a partial goal that of understanding the categories and functors that correspond to loop spaces and their associated topological functors. Progress towards this goal has been advanced greatly by the recent work of Balteanu, Fiedorowicz, Schwänzl, and Vogt who show a direct correspondence between k–fold monoidal categories and k–fold loop spaces through the categorical nerve. This thesis pursues the hints of a categorical delooping that are suggested when enrichment is iterated. At each stage of successive enrichments, the number of monoidal products seems to decrease and the categorical dimension to increase, both by one. This is mirrored by topology. When we consider the loop space of a topological space, we see that paths (or 1–cells) in the original are now points (or objects) in the derived space. There is also automatically a product structure on the points in the derived space, where multiplication is given by concatenation of loops. Delooping is the inverse functor here, and thus involves shifting objects to the status of 1–cells and decreasing the number of ways to multiply. Enriching over the category of categories enriched over a monoidal category is defined, for the case of symmetric categories, in the paper on A∞–categories by Lyubashenko. It seems that it is a good idea to generalize his definition first to the case of an iterated monoidal base category and then to define V–(n + 1)–categories as categories enriched over V–n–Cat, the (k−n)–fold monoidal strict (n+1)–category of V–n–categories where k<n ∈ N. We show that for V k–fold monoidal the structure of a (k−n)–fold monoidal strict (n + 1)–category is possessed by V–n–Cat. / Ph. D.

Human languages are not just tools for transmitting cultural ideas, they are themselves culturally transmitted. This single observation has major implications for our understanding of how and why languages around the world are structured the way they are, and also for how scientists should be studying them. Accounting for the origins of what turns out to be such a uniquely human ability is, and should be, a priority for anyone interested in what makes us different from every other lifeform on Earth. The way the scientific community thinks about language has seen considerable changes over the years. In particular, we have witnessed movements away from a purely descriptive science of language, towards a more explanatory framework that is willing to embrace the difficult questions of not just how individual languages are currently structured and used, but also how and why they got to be that way in the first place. Seeing languages as historical entities is, of course, nothing new in linguistics. Seeing languages as complex adaptive systems, undergoing processes of evolution at multiple levels of interaction however, is. Broadly speaking, this thesis explores some of the implications that this perspective on language has, and argues that in addition to furthering our understanding of the processes of biological evolution and the mechanisms of individual learning required specifically for language, we also need to be mindful of the less well-understood cultural processes that mediate between the two. Human communication systems are not just direct expressions of our genes. Neither are they independently acquired by learners anew at every generation. Instead, languages are transmitted culturally from one generation to another, creating an opportunity for a different kind of evolutionary channel to exist. It is a central aim of this thesis to explore some of the adaptive dynamics that such a cultural channel has, and investigate the extent to which certain structural and statistical properties of language can be directly explained as adaptations to the transmission process and the learning biases of speakers. In order to address this aim, this thesis takes an experimental approach. Building on a rich set of empirical results from various computational simulations and mathematical models, it presents a novel methodological framework for exploring one type of cultural transmission mechanism, iterated learning, in the laboratory using human participants. In these experiments, we observe the evolution of artificial languages as they are acquired and then transmitted to new learners. Although there is no communication involved in these studies, and participants are unaware that their learning efforts are being propagated to future learners, we find that many functional features of language emerge naturally from the different constraints imposed upon them during transmission. These constraints can take a variety of forms, both internal and external to the learner. Taken collectively, the data presented here suggest several points: (i) that iterated language learning experiments can provide us with new insights about the emergence and evolution of language; (ii) that language-like structure can emerge as a result of cultural transmission alone; and (iii) that whilst structure in these systems has the appearance of design, and is in some sense ‘created’ by intentional beings, its emergence is in fact wholly the result of non-intentional processes. Put simply, cultural evolution plays a vital role in language. This work extends our framework for understanding it, and offers a new method for investigating it.

A variety of movements and social pressure have driven the need for an increase in environmental awareness, and subsequently fuels the need for individuals to reduce their ecological footprint. Firms are now trying to implement 'eco-friendly' technologies that both build and run their products. How these 'eco-friendly' products will perform in the market is strongly tied to a variety of consumer related influences and decisions, as well as personality type. This thesis presents a model of varied social influence on consumer markets. First we show how varied playing characteristics amongst opponents in the Iterated Prisoner's Dilemma yields a different distribution of strategies. Utilizing two variations of IPD, we map scores to edges based on the agents involved in each edge as one construct of influence. Other types of influence include a homogeneous influence, and a zero influence for comparison of results. We also introduce the Rate of Social Mobility as a basis for initializing random social movement in a network. We show that the social influence of the network in the consumer market plays a vital role in the dynamics of product adoption. In closing we discuss future model refinements, and advances.

Doctor of Philosophy / Department of Mathematics / Charles N. Moore / The main purpose of this thesis is to derive the law of the iterated logarithm for tail sums in various contexts in analysis. The various contexts are sums of Rademacher functions, general dyadic martingales, independent random variables and lacunary trigonometric series. We name the law of the iterated logarithm for tail sums as tail law of the iterated logarithm. We first establish the tail law of the iterated logarithm for sums of Rademacher functions and obtain both upper and lower bound in it. Sum of Rademacher functions is a nicely behaved dyadic martingale. With the ideas from the Rademacher case, we then establish the tail law of the iterated logarithm for general dyadic martingales. We obtain both upper and lower bound in the case of martingales. A lower bound is obtained for the law of the iterated logarithm for tail sums of bounded symmetric independent random variables. Lacunary trigonometric series exhibit many of the properties of partial sums of independent random variables. So we finally obtain a lower bound for the tail law of the iterated logarithm for lacunary trigonometric series introduced by Salem and Zygmund.

Consider an iterated function system (IFS) that does not necessarily contract uniformly, but instead contracts on average after a finite number of iterations. Under some technical assumptions, previous work by Barnsley, Demko, Elton and Geronimo has shown that such an IFS has a unique invariant probability measure, whilst many (such as Peigné, Hennion and Hervé, Guivarc'h and le Page, Santos and Walkden) have shown that (for different function spaces) the transfer operator associated with the IFS is quasi-compact. A result due to Keller and Liverani allows one to deduce whether the transfer operator remains quasi-compact under suitable, small perturbations. The first part of this thesis proves a large deviations result for IFSs that contract on average using skew product transfer operators, a technique used by Broise to prove a similar result for dynamical systems. The remaining chapters introduce a notion of 'coupled IFSs', an analogue of the traditional coupled map lattices where the base, unperturbed behaviour is determined by an underlying dynamical system. We use transfer operators and Keller and Liverani's theorem to prove that quasi-compactness of the transfer operator is preserved for 'product IFSs' under small perturbations and for coupled IFSs. This allows us to prove a central limit theorem with a rate of convergence for the coupled IFS.

In this thesis, fixed point theory is used to construct a fractal type sets and to solve data classification problem. Fixed point method, which is a beautiful mixture of analysis, topology, and geometry has been revealed as a very powerful and important tool in the study of nonlinear phenomena. The existence of fixed points is therefore of paramount importance in several areas of mathematics and other sciences. In particular, fixed points techniques have been applied in such diverse fields as biology, chemistry, economics, engineering, game theory and physics. In Chapter 2 of this thesis it is demonstrated how to define and construct a fractal type sets with the help of iterations of a finite family of generalized F-contraction mappings, a class of mappings more general than contraction mappings, defined in the context of b-metric space. This leads to a variety of results for iterated function system satisfying a different set of contractive conditions. The results unify, generalize and extend various results in the existing literature. In Chapter 3, the theory of support vector machine for linear and nonlinear classification of data and the notion of generalized support vector machine is considered. In the thesis it is also shown that the problem of generalized support vector machine can be considered in the framework of generalized variation inequalities and results on the existence of solutions are established. / FUSION

The aim of this work is the study of infinite conformal iterated function systems. More specifically, we investigate some properties of a limit set J associated to such system, its Hausdorff and packing measure and Hausdorff dimension. We provide necessary and sufficient conditions for such systems to be bi-Lipschitz equivalent. We use the concept of scaling functions to obtain some result about 1-dimensional systems. We discuss particular examples of infinite iterated function systems derived from complex continued fraction expansions with restricted entries. Each system is obtained from an infinite number of contractions. We show that under certain conditions the limit sets of such systems possess zero Hausdorff measure and positive finite packing measure. We include an algorithm for an approximation of the Hausdorff dimension of limit sets. One numerical result is presented. In this thesis we also explore the concept of positively recurrent function. We use iterated function systems to construct a natural, wide class of such functions that have strong ergodic properties.

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The thesis offers an investigation into the analysis of so-called iterated generating functions and the schemes that produce them. Beginning with the study of some ad hoc scheme formulations, the notion of an iterated generating function is introduced and a mechanism to produce arbitrary finite sequences established. The development of schemes to accommodate infinite sequences leads – in the case of the Catalan sequence – to the discovery of what are termed Catalan polynomials whose properties are examined. Results are formulated for these polynomials through the algebraic adaptation of classical root-finding algorithms, serving as a basis for the synthesis of new generalised results for other infinite sequences and their associated polynomials.