Partial differential equations (PDEs) are used to describe multi-dimensional physical phenomena. However, some of these phenomena are described by a more general class of systems called fractional systems. Indeed, fractional calculus has emerged as a new tool for modeling complex phenomena thanks to the memory and hereditary properties of fraction derivatives. In this thesis, we explore a class of controllers and estimators that respond to some control and estimation challenges for both PDE and FPDE. We first propose a backstepping controller for the flow control of a first-order hyperbolic PDE modeling the heat transfer in parabolic solar collectors. While backstepping is a well-established method for boundary controlled PDEs, the process is less straightforward for in-domain controllers. One of the main contributions of this thesis is the development of a new integral transformation-based control algorithm for the study of reference tracking problems and observer designs for fractional PDEs using the extended backstepping approach. The main challenge consists of the proof of stability of the fractional target system, which utilizes either an alternative Lyapunov method for time FPDE or a fundamental solution for the error system for reference tracking, and observer design of space FPDE. Examples of applications involving reference tracking of FPDEs are gas production in fractured media and solute transport in porous media. The designed controllers, require knowledge of some system’s parameters or the state. However, these quantities may be not measurable, especially, for space-evolving PDEs. Therefore, we propose a non-asymptotic and robust estimation algorithm based on the so-called modulating functions. Unlike the observers-based methods, the proposed algorithm has the advantage that it converges in a finite time. This algorithm is extended for the state estimation of linear and non-linear PDEs with general non-linearity. This algorithm is also used for the estimation of parameters and disturbances for FPDEs. This thesis aims to design an integral transformation-based algorithm for the control and estimation of PDEs and FDEs. This transformation is defined through a suitably designed function that transforms the identification problem into an algebraic system for non-asymptotic estimation purposes. It also maps unstable systems to stable systems to achieve control goals.

This dissertation is motivated by the increasing applications of high-dimensional large-scale data sets in various fields and lack of theoretical understanding of the existing algorithms as well as lack of efficient algorithms in many cases. Hence, identifying the geometrical properties of data sets is essential for many data processing tasks, such as data retrieval and denoising. In Part I, we derive the fundamental limits on the sampling rate required to study three important problems (i) low-rank data completion, (ii) rank estimation, and (iii) data clustering. In Chapter 2 we characterize the geometrical conditions on the sampling pattern, i.e., locations of the sampled entries, for finite and unique completability of a low-rank tensor, assuming that its rank vector is given or estimated. To this end, we propose a manifold analysis and study the independence of a set of polynomials defined based on the sampling pattern. Then, using the polynomial analysis, we derive a lower bound on the sampling rate such that it guarantees that the proposed conditions on the sampling patterns for finite and unique completability hold true with high probability. Then, in Chapter 3, we study the problem of rank estimation, where a data structure is partially sampled and we propose a geometrical analysis on the sampling pattern to estimate the true value of rank for various data structures by providing extremely tight lower and upper bounds on the rank value. And in Chapters 4 and 5, we make use of the developed tools to obtain a lower bound on the sampling rate to be able to correctly cluster a union of sampled matrices or tensors by identifying their corresponding unknown subspaces. In Part II, first in Chapter 6, motivated by the algebraic tools developed in Part I, we develop a data completion algorithm based on solving a set of polynomial equations using Newton's method, that is effective especially when the sampling rate is low. Then, in Chapter 7, we consider a data structure consisting of a union of nested low-rank matrix or tensor subspaces, and develop a structured alternating minimization-based approach for completing such data, that is capable of taking advantage of multiple rank constraints simultaneously to achieve faster convergence and higher recovery accuracy.

Personal mobile devices (PMDs) initiated a multi-dimensional paradigmatic shift in personal computing and personal information collection fueled by the indispensability of the Internet and the increasing functionality of the devices. From 2005 to 2016, the perceived necessity of conducting transactions on the Internet moved from optional to indispensable. The context of these transactions changes from traditional desktop and laptop computers, to the inclusion of smartphones and tablets (PMDs). However, the traditional privacy calculus published by (Dinev and Hart 2006) was conceived before this technological and contextual change, and several core assumptions of that model must be re-examined and possibly adapted or changed to account for this shift. This paradigm shift impacts the decision process individuals use to disclose personal information using PMDs. By nature of their size, portability, and constant proximity to the user, PMDs collect, contain, and distribute unprecedented amounts of personal information. Even though the context within which people are sharing information has changed significantly, privacy calculus research applied to PMDs has not moved far from the seminal work by Dinev and Hart (2006). The traditional privacy calculus risk-benefit model is limited in the PMD context because users are unaware of how much personal information is being shared, how often it is shared, or to whom it is shared. Furthermore, the traditional model explains and predicts intent to disclose rather than actual disclosure. However, disclosure intentions are a poor predictor of actual information disclosure. Because of perceived indispensability of the information and the inability to assess potential risk, the deliberate comparison of risks to benefits prior to disclosure—a core assumption of the traditional privacy calculus—may not be the most effective basis of a model to predict and explain disclosure. The present research develops a Personal Mobile Device Privacy Calculus model designed to predict and explain disclosure behavior within the specific context of actual disclosure of personal information using PMDs.

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In a software development cycle, programs go through many iterations. Identifying and understanding program changes is a tedious but necessary task for programmers, especially when software is developed in a collaborative environment. Existing tools used by the programmers either lack in finding the structural differences, or report the differences as atomic changes, such as updates of individual syntax tree nodes. Programmers frequently use program restructuring techniques, such as refactorings that are composed of several individual atomic changes. Current version differencing tools omit these high-level changes, reporting just the set of individual atomic changes. When a large number of refactorings are performed, the number of reported atomic changes is very large. As a result, it will be very difficult to understand the program differences. This problem can be addressed by reporting the program differences as composite changes, thereby saving programmers the effort of navigating through the individual atomic changes. This thesis proposes a methodology to explore the atomic changes reported by existing version differencing tools to infer composite changes. First, we will illustrate the different approaches that can be used for representing object language program differences using a variation representation. Next we will present the process of composite change inference from the structured representation of atomic changes. This process describes patterns that specify the expected structure of an expression corresponding to each composite change that has to be inferred. The information in patterns is then used to design the change inference algorithm. The composite changes inferred from a given expression are annotated in the expression, allowing the changes to be reported as desired. / Graduation date: 2012

>Magister Scientiae - MSc / The aim of the thesis is to research the difficulties that first year mainstream mathematics students at UWC experience when solving Related Rates problems in calculus. In chapter 2, an in-depth study was made of the nature of Related Rates problems by studying a number of examples. The findings of this study are summarized in section 2.12. The study adopted the same model of the solution of all types of Related Rates that was used by Martin (2000) for the solution of geometric Related Rates problems. In chapter 3 of this thesis, many examples were used to illustrate how the seven step solution procedure of the Standard Solution model is applied. In the literature review in chapter 4, the underlying concepts which underpin Related Rates problems are identified and specific examples of research on each of these concepts are given. For example, the review of the literature on word problems is done comprehensively and covers extensively the range of issues involved in this topic. Drawing on the work in chapter 2 on the nature of Related Rates problems, it is explained in chapter 5 why this study is underpinned by Constructivism as a theoretical basis. Chapter 6 of the thesis is devoted to answering the 3 research questions posed in chapter 1. The thesis contains many worked examples of Related Rates problems which can be used by the lecturers assigned to the MAT105 course.

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This thesis considers the problem of program correctness within a rich theory of dependent types, the Extended Calculus of Constructions (ECC). This system contains a powerful programming language of higher-order primitive recursion and higher-order intuitionistic logic. It is supported by Pollack's versatile LEGO implementation, which I use extensively to develop the mathematical constructions studied here. I systematically investigate Burstall's notion of deliverable, that is, a program paired with a proof of correctness. This approach separates the concerns of programming and logic, since I want a simple program extraction mechanism. The Sigma-types of the calculus enable us to achieve this. There are many similarities with the subset interpretation of Martin-Löf type theory. I show that deliverables have a rich categorical structure, so that correctness proofs may be decomposed in a principled way. The categorical combinators which I define in the system package up much logical book-keeping, allowing one to concentrate on the essential structure of algorithms. I demonstrate our methodology with a number of small examples, culminating in a machine-checked proof of the Chinese remainder theorem, showing the utility of the deliverables idea. Some drawbacks are also encountered. I consider also semantic aspects of deliverables, examining the definitions in an abstract setting, again firmly based on category theory. The aim is to overcome the clumsiness of the language of categorical combinators, using dependent type theories and their interpretation in fibrations. I elaborate a concrete instance based on the category of sets, which generalises to an arbitrary topos. In the process, I uncover a subsystem of ECC within which one may speak of deliverables defined over the topos. In the presence of enough extra structure, the interpretation extends to the whole of ECC. The wheel turns full circle.