The displacement operator is related to the displaced parity operator through a two dimensional Fourier transform. Both operators are important operators in phase space and the trace of both with respect to the density operator gives the Wigner functions (displaced parity operator) and Weyl functions (displacement operator). The generalisation of the parity-displacement operator relationship considered here is called the bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional displacement operators lead to the novel concept of bi-fractional coherent states. The generalisation from Fourier transform to fractional Fourier transform can be applied to other phase space functions. The case of the Wigner-Weyl function is considered and a generalisation is given, which is called the bi-fractional Wigner functions, H(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to give the bi-fractional Q−functions and bi-fractional P−functions respectively. The generalisation is likewise applied to the Moyal star product and Berezin formalism for products of non-commutating operators. These are called the bi-fractional Moyal star product and bi-fractional Berezin formalism. Finally, analysis, applications and implications of these bi-fractional transforms to the Heisenberg uncertainty principle, photon statistics and future applications are discussed.

The displacement operator is related to the displaced parity operator through a two dimensional Fourier transform. Both operators are important operators in phase space and the trace of both with respect to the density operator gives the Wigner functions (displaced parity operator) and Weyl functions (displacement operator). The generalisation of the parity-displacement operator relationship considered here is called the bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional displacement operators lead to the novel concept of bi-fractional coherent states. The generalisation from Fourier transform to fractional Fourier transform can be applied to other phase space functions. The case of the Wigner-Weyl function is considered and a generalisation is given, which is called the bi-fractional Wigner functions, H(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to give the bi-fractional Q−functions and bi-fractional P−functions respectively. The generalisation is likewise applied to the Moyal star product and Berezin formalism for products of non-commutating operators. These are called the bi-fractional Moyal star product and bi-fractional Berezin formalism. Finally, analysis, applications and implications of these bi-fractional transforms to the Heisenberg uncertainty principle, photon statistics and future applications are discussed.

This work revolves around the use of fractional order calculus in control science. Techniques such as fractional order universal adaptive stabilization (FO-UAS), and the fascinating results of their application to real-world systems, are presented initially. A major portion of this work deals with fractional order modeling and control of real-life systems like heat flow, fan and plate, and coupled tank systems. The fractional order models and controllers are not only simulated, they are also emulated using analog hardware. The main aim of all the above experimentation is to develop a fractional order controller for plasma position control of the Saskatchewan torus-1, modified (STOR-1M) tokamak at the Utah State University (USU) campus. A new method for plasma position estimation has been formulated. The results of hardware emulation of plasma position and its control are also presented. This work performs a small scale test measuring controller performance, so that it serves as a platform for future research efforts leading to real-life implementation of a plasma position controller for the tokamak.

AN ABSTRACT OF THE DISSERTATION OFSaleh Aljurbua, for the Doctor of Philosophy degree in APPLIED MATHEMATICS, presented on January 27th, 2021, at Southern Illinois University Carbondale. TITLE: THE EXISTENCE OF SOLUTIONS FOR FRACTIONAL DIFFERENTIAL EQUATIONS FOR ORDER q ∈ (n − 1, n], n ∈ N, WITH ANTIPERIODIC BOUNDARY CONDITIONS MAJOR PROFESSOR: Dr. Mingqing Xiao Differential equations play a major role in natural science, physics and technology. Fractional differential equations (FDE) gained a lot of popularity in the past three decades and they became very important in economics, physics and chemistry. In fact, fractional integrals and derivatives became essential and made a significant contribution in dynamical systems which simulate it. They fill the gaps between the integer-types of integrations and derivatives in the classical settings. This work consists of four Chapters. The first Chapter will be covering background, preliminary and fundamental tools used in our dissertation topic. The second Chapter consists of the existence of solutions for nonlinear fractional differential equations of some specific orders with antiperiodic boundary conditions followed by the main topic which is the existence of solutions for nonlinear fractional differential equations of order q ∈ (n−1, n], n ∈ N with antiperiodic boundary conditions of a continuous function f(t, x(t)). Moreover, definitions, theorems and some lemmas will be provided. v In the third Chapter, we offer some examples to illustrate our approach in the main topic. Finally, the fourth Chapter includes the summary and perspective researches.

Master of Science / Department of Statistics / James J. Higgins / Completely random and randomized block designs involving n factors at each of two levels are used to screen for the effects of a large number of factors. With such designs it may not be possible either because of costs or because of time to run each treatment combination more than once. In some cases, only a fraction of all the treatments may be run. With a large number of factors and limited observations, even one outlier can adversely affect the results. Robust regression methods are designed to down-weight the adverse affects of outliers. However, to our knowledge practitioners do not routinely apply robust regression methods in the context of fractional replication of 2^n factorial treatment structures. The purpose of this report is examine how robust regression methods perform in this context.

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本論文會討論有關於二維的耗散準地轉方程，特別是有關於存在性及規律性的問題。有關的討論主要取決於該方程的分數冪。這份論文中將會介紹一些最近有關耗散準地轉方程的結果。 / This paper is discussing about problems in the 2-D Dissipative Quasi-Geostrophic equation, mainly the existence and regularity results, depending on the fractional power. We will introduce the recent results in this topics. / Detailed summary in vernacular field only. / Kwan, Danny. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 65-68). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Main Results in QG --- p.9 / Chapter 2.1 --- Definitions --- p.9 / Chapter 2.2 --- Subcritical Case (γ>[1/2]) --- p.10 / Chapter 2.3 --- Critical Case (γ=[1/2]) --- p.10 / Chapter 2.4 --- Supercritical Case (γ<[1/2]) --- p.11 / Chapter 3 --- The Proofs of Main Results --- p.12 / Chapter 3.1 --- Some Previous Results --- p.12 / Chapter 3.2 --- Subcritical Case (γ>[1/2]) --- p.14 / Chapter 3.2.1 --- Proof of Theorem 1 --- p.14 / Chapter 3.2.2 --- Proof of Theorem 2 --- p.19 / Chapter 3.2.3 --- Proof of Corollary 1 --- p.25 / Chapter 3.2.4 --- Summary for Subcritical Case --- p.27 / Chapter 3.3 --- Critical Case (γ=[1/2]) --- p.28 / Chapter 3.3.1 --- Proof of Theorem 3 --- p.28 / Chapter 3.3.2 --- Proof of Theorem 4 --- p.36 / Chapter 3.3.3 --- Summary for Critical Case --- p.41 / Chapter 3.4 --- Supercritical Case (γ<[1/2]) --- p.41 / Chapter 3.4.1 --- Proof of Theorem 5 --- p.41 / Chapter 3.4.2 --- Proof of Theorem 6 --- p.50 / Chapter 3.4.3 --- Proof of Theorem 7 --- p.54 / Chapter 3.4.4 --- Summary for Supercritical Case --- p.64 / Chapter 4 --- Further Development --- p.65

This thesis presents results in two apparently disparate mathematical fields which can both be examined -- and even united -- by means of pure analysis. Fractional calculus is the study of differentiation and integration to non-integer orders. Dating back to Leibniz, this idea was considered by many great mathematical figures, and in recent decades it has been used to model many real-world systems and processes, but a full development of the mathematical theory remains incomplete. Many techniques for partial differential equations (PDEs) can be extended to fractional PDEs too. Three chapters below cover my results in this area: establishing the elliptic regularity theorem, Malgrange-Ehrenpreis theorem, and unified transform method for fractional PDEs. Each one is analogous to a known result for classical PDEs, but the proof in the general fractional scenario requires new ideas and modifications. Fractional derivatives and integrals are not uniquely defined: there are many different formulae, each of which has its own advantages and disadvantages. The most commonly used is the classical Riemann-Liouville model, but others may be preferred in different situations, and now new fractional models are being proposed and developed each year. This creates many opportunities for new research, since each time a model is proposed, its mathematical fundamentals need to be examined and developed. Two chapters below investigate some of these new models. My results on the Atangana-Baleanu model proposed in 2016 have already had a noticeable impact on research in this area. Furthermore, this model and the results concerning it can be extended to more general fractional models which also have certain desirable properties of their own. Fractional calculus and zeta functions have rarely been united in research, but one chapter below covers a new formula expressing the Lerch zeta function as a fractional derivative of an elementary function. This result could have many ramifications in both fields, which are yet to be explored fully. Zeta functions are very important in analytic number theory: the Riemann zeta function relates to the distribution of the primes, and this field contains some of the most persistent open problems in mathematics. Since 2012, novel asymptotic techniques have been applied to derive new results on the growth of the Riemann zeta function. One chapter below modifies some of these techniques to prove asymptotics to all orders for the Hurwitz zeta function. Many new ideas are required, but the end result is more elegant than the original one for Riemann zeta, because some of the new methodologies enable different parts of the argument to be presented in a more unified way. Several related problems involve asymptotics arbitrarily near a stationary point. Ideally it should be possible to find uniform asymptotics which provide a smooth transition between the integration by parts and stationary phase methods. One chapter below solves this problem for a particular integral which arises in the analysis of zeta functions.

In this thesis, as an extension to [1], we propose a prioritized multi-layer network coding scheme for collaborative packet recovery in hybrid (interweave and underlay) cellular cognitive radio networks. This scheme allows the uncoordinated collaboration between the collocated primary and cognitive radio base-stations in order to minimize their own as well as each other’s packet recovery overheads, thus by improving their throughput. The proposed scheme ensures that each network’s performance is not degraded by its help to the other network. Moreover, it guarantees that the primary network’s interference threshold is not violated in the same and adjacent cells. Yet, the scheme allows the reduction of the recovery overhead in the collocated primary and cognitive radio networks. The reduction in the cognitive radio network is further amplified due to the perfect detection of spectrum holes which allows the cognitive radio base station to transmit at higher power without fear of violating the interference threshold of the primary network. For the secondary network, simulation results show reductions of 20% and 34% in the packet recovery overhead, compared to the non-collaborative scheme, for low and high probabilities of primary packet arrivals, respectively. For the primary network, this reduction was found to be 12%. Furthermore, with the use of fractional cooperation, the average recovery overhead is further reduced by around 5% for the primary network and around 10% for the secondary network when a high fractional cooperation probability is used.

We consider how the optimal decision can be made if the optimality criterion of maximizing profit changes to that of maximizing return on investment for the general uncapacitated facility location problem. We show that the inherent structure of the proposed model can be exploited to make a significant computational reduction.