We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. Then, we use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.

Primarily via constructivist teaching experiment methodology, units coordination (Steffe, 1992) has emerged as a useful construct for modeling students' psychological constructions pertaining to several mathematical domains, including counting sequences, whole number multiplicative conceptions, and fractions schemes. I describe how consideration of units coordination as a Piagetian (1970b) structure is useful for modeling units coordination across contexts. In this study, I extend teaching experiment methodology (Steffe and Thompson, 2000) to model the dynamics of students' units coordinating activity across contexts within a teaching experiment, using the construct of propensity to coordinate units. Two video-recorded teaching experiments involving pairs of sixth-grade students were analyzed to form a model of the dynamics of students' units coordinating activity. The modeling involved separation of transcriptions into chunks that were coded dichotomously for the units coordinating activity of a single student in each dyad. The two teaching experiments were used to form 5 conjectures about the output of the model that were then tested with a third teaching experiment. The results suggest that modeling units coordination activity via the construct of propensity to coordinate units was useful for describing patterns in the students' perturbations during the teaching sessions. The model was moderately useful for identifying sequences of interactions that support growth in units coordination. Extensions, modifications, and implications of the modeling approach are discussed. / Ph. D.

Moderne statistische und ökonometrische Methoden behandeln erfolgreich stilisierte Fakten auf den Finanzmärkten. Die vorgestellten Techniken erstreben die Dynamik von Finanzmarktdaten genauer als traditionelle Ansätze zu verstehen. Wirtschaftliche und finanzielle Vorteile sind erzielbar. Die Ergebnisse werden hier in praktischen Beispielen ausgewertet, die sich vor allem auf die Prognose von Finanzmarktdaten fokussieren. Unsere Anwendungen umfassen: (i) die Modellierung und die Vorhersage des Liquiditätsangebotes, (ii) die Lokalisierung des ’Multiplicative Error Model’ und (iii) die Erbringung von Evidenz für den empirischen Zustandsfaktorparadox über Landern. / Modern methods in statistics and econometrics successfully deal with stylized facts observed on financial markets. The presented techniques aim to understand the dynamics of financial market data more accurate than traditional approaches. Economic and financial benefits are achievable. The results are here evaluated in practical examples that mainly focus on forecasting of financial data. Our applications include: (i) modelling and forecasting of liquidity supply, (ii) localizing multiplicative error models and (iii) providing evidence for the empirical pricing kernel paradox across countries.

In this report, we present some of the results achieved by investigating two-dimensional monomial cellular automata modulo m, where m is a non-zero positive integer. Throughout the experiments, we work with the von Neumann neighbourhood and apply the same local rule based on modular multiplication. The purpose of the study is to examine the behaviour of these cellular automata in three different environments, (i.e. the infinite plane, the finite plane and the torus), by means of elementary number theory. We notice how the distance between each pair of cells with state 0 influences the evolution of the automaton and the convergence of its configurations. Similar impact is perceived when the cells attain the values of Euler's-<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cphi" />function or of integers with common divisors with m, when m &gt; 2. Alongside with the states of the cells, the evolution of the automaton, as well as the convergence of its configurations, are also decided by the values attributed to m, whether it is a prime, a prime power or a multiple of primes and/or prime powers.

Multiplicative reasoning permeates many mathematical topics, for example fractions and functions. Hence there is consensus on the importance of acquiring multiplicative reasoning. Multiplication is typically introduced as repeated addition, but when it is extended to include multi-digits and decimals a more general view of multiplication is required. There are conflicting reports in previous research concerning students’ understandings of multiplication. For example, repeated addition has been suggested both to support students’ understanding of calculations and as a hindrance to students’ conceptualisation of the two-dimensionality of multiplication. The relative difficulty of commutativity and distributivity is also debated, and there is a possible conflict in how multiplicative reasoning is described and assessed. These inconsistencies are addressed in a study with the aim of understanding more about students’ understandings of multiplication when it is expanded to comprise multi-digits and decimals. Understanding is perceived as connections between representations of different types of knowledge, linked together by reasoning. Especially connections between three components of multiplication were investigated; models for multiplication, calculations and arithmetical properties. Explicit reasoning made the connections observable and externalised mental representations. Twenty-two students were recurrently interviewed during five semesters in grades five to seven to find answers to the overarching research question: What do students’ responses to different forms of multiplicative tasks in the domain of multi-digits and decimals reveal about their understandings of multiplication? The students were invited to solve different forms of tasks during clinical interviews, both individually and in pairs. The tasks involved story telling to given multiplications, explicit explanations of multiplication, calculation problems including explanations and justifications for the calculations and evaluation of suggested calculation strategies. Additionally the students were given written word problems to solve. The students’ understandings of multiplication were robustly rooted in repeated addition or equally sized groups. This was beneficial for their understandings of calculations and distributivity, but hindered them from fluent use of commutativity and to conceptualise decimal multiplication. The robustness of their views might be explained by the introduction to multiplication, which typically is by repeated addition and modelled by equally sized groups. The robustness is discussed in relation to previous research and the dilemma that more general models for multiplication, such as rectangular area, are harder to conceptualise than models that are only susceptible to natural numbers. The study indicated that to evaluate and explain others’ calculation strategies elicited more reasoning and deeper mathematical thinking compared to evaluating and explaining calculations conducted by the students themselves. Furthermore, the different forms of tasks revealed various lines of reasoning and to get a richly composed picture of students’ multiplicative reasoning and understandings of multiplication, a wide variety of forms of tasks is suggested. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.</p>

In the usual definition of a Euclidean domain, a ring has a norm function whose codomain is the positive integers. It was noticed by Motzkin in 1949 that the codomain could be replaced by any well-ordered set. This motivated the study of transfinite Euclidean domains in which the codomain of the norm function is replaced by the class of ordinals. We prove that there exists a (transfinitely valued) Euclidean Domain with Euclidean order type for every indecomposable ordinal. Modifying the construction, we prove that there exists a Euclidean Domain with no multiplicative norm. Following a definition of Clark and Murty, we define a set of admissible primes. We develop an algorithm that can be used to find sets of admissible primes in the ring of integers of quadratic extensions of the rationals and provide some examples.

Systolic architectures are capable of achieving high throughput by maximizing pipelining and by eliminating global data interconnects. Recursive algorithms with regular data flows are suitable for systolization. The computation of multiplicative inversion using algorithms based on EEA (Extended Euclidean Algorithm) are particularly suitable for systolization. Implementations based on EEA present a high degree of parallelism and pipelinability at bit level which can be easily optimized to achieve local data flow and to eliminate the global interconnects which represent most important bottleneck in todays sub-micron design process. The net result is to have high clock rate and performance based on efficient systolic architectures. This thesis examines high performance but also scalable implementations of multiplicative inversion or field division over Galois fields <i>GF</i>(2<i>^m</i>) in the specific case of cryptographic applications where field dimension <i>m</i> may be very large (greater than 400) and either <i>m</i> or defining irreducible polynomial may vary. For this purpose, many inversion schemes with different basis representation are studied and most importantly variants of EEA and binary (Stein's) GCD computation implementations are reviewed. A set of common as well as contrasting characteristics of these variants are discussed. As a result a generalized and optimized variant of EEA is proposed which can compute division, and multiplicative inversion as its subset, with divisor in either <i>polynomial</i> or <i>triangular</i> basis representation. Further results regarding Hankel matrix formation for double-basis inversion is provided. The validity of using the same architecture to compute field division with polynomial or triangular basis representation is proved. Next, a scalable unidirectional bit serial systolic array implementation of this proposed variant of EEA is implemented. Its complexity measures are defined and these are compared against the best known architectures. It is shown that assuming the requirements specified above, this proposed architecture may achieve a higher clock rate performance w. r. t. other designs while being more flexible, reliable and with minimum number of inter-cell interconnects. The main contribution at system level architecture is the substitution of all counter or adder/subtractor elements with a simpler distributed and free of carry propagation delays structure. Further a novel restoring mechanism for result sequences of EEA is proposed using a double delay element implementation. Finally, using this systolic architecture a CMD (Combined Multiplier Divider) datapath is designed which is used as the core of a novel systolic elliptic curve processor. This EC processor uses affine coordinates to compute scalar point multiplication which results in having a very small control unit and negligible with respect to the datapath for all practical values of <i>m</i>. The throughput of this EC based on this bit serial systolic architecture is comparable with designs many times larger than itself reported previously.

Image noise reduction, or denoising, is an active area of research, although many of the techniques cited in the literature mainly target additive white noise. With an emphasis on signal-dependent noise, this thesis presents the General Adaptive Monte Carlo Bayesian Image Denoising (GAMBID) algorithm, a model-free approach based on random sampling. Testing is conducted on synthetic images with two different signal-dependent noise types as well as on real synthetic aperture radar and ultrasound images. Results show that GAMBID can achieve state-of-the-art performance, but suffers from some limitations in dealing with textures and fine low-contrast features. These aspects can by addressed in future iterations when GAMBID is expanded to become a versatile denoising framework.

Systolic architectures are capable of achieving high throughput by maximizing pipelining and by eliminating global data interconnects. Recursive algorithms with regular data flows are suitable for systolization. The computation of multiplicative inversion using algorithms based on EEA (Extended Euclidean Algorithm) are particularly suitable for systolization. Implementations based on EEA present a high degree of parallelism and pipelinability at bit level which can be easily optimized to achieve local data flow and to eliminate the global interconnects which represent most important bottleneck in todays sub-micron design process. The net result is to have high clock rate and performance based on efficient systolic architectures. This thesis examines high performance but also scalable implementations of multiplicative inversion or field division over Galois fields <i>GF</i>(2<i>^m</i>) in the specific case of cryptographic applications where field dimension <i>m</i> may be very large (greater than 400) and either <i>m</i> or defining irreducible polynomial may vary. For this purpose, many inversion schemes with different basis representation are studied and most importantly variants of EEA and binary (Stein's) GCD computation implementations are reviewed. A set of common as well as contrasting characteristics of these variants are discussed. As a result a generalized and optimized variant of EEA is proposed which can compute division, and multiplicative inversion as its subset, with divisor in either <i>polynomial</i> or <i>triangular</i> basis representation. Further results regarding Hankel matrix formation for double-basis inversion is provided. The validity of using the same architecture to compute field division with polynomial or triangular basis representation is proved. Next, a scalable unidirectional bit serial systolic array implementation of this proposed variant of EEA is implemented. Its complexity measures are defined and these are compared against the best known architectures. It is shown that assuming the requirements specified above, this proposed architecture may achieve a higher clock rate performance w. r. t. other designs while being more flexible, reliable and with minimum number of inter-cell interconnects. The main contribution at system level architecture is the substitution of all counter or adder/subtractor elements with a simpler distributed and free of carry propagation delays structure. Further a novel restoring mechanism for result sequences of EEA is proposed using a double delay element implementation. Finally, using this systolic architecture a CMD (Combined Multiplier Divider) datapath is designed which is used as the core of a novel systolic elliptic curve processor. This EC processor uses affine coordinates to compute scalar point multiplication which results in having a very small control unit and negligible with respect to the datapath for all practical values of <i>m</i>. The throughput of this EC based on this bit serial systolic architecture is comparable with designs many times larger than itself reported previously.

Image noise reduction, or denoising, is an active area of research, although many of the techniques cited in the literature mainly target additive white noise. With an emphasis on signal-dependent noise, this thesis presents the General Adaptive Monte Carlo Bayesian Image Denoising (GAMBID) algorithm, a model-free approach based on random sampling. Testing is conducted on synthetic images with two different signal-dependent noise types as well as on real synthetic aperture radar and ultrasound images. Results show that GAMBID can achieve state-of-the-art performance, but suffers from some limitations in dealing with textures and fine low-contrast features. These aspects can by addressed in future iterations when GAMBID is expanded to become a versatile denoising framework.