Let Fqt be the finite field with qt elements and let F*qt be its multiplicative group. We study the diagonal equation axq−1 + byq−1 = c, where a,b and c ∈ F*qt. This equation can be written as xq−1+αyq−1 = β, where α, β ∈ F ∗ q t . Let Nt(α, β) denote the number of solutions (x,y) ∈ F*qt × F*qt of xq−1 + αyq−1 = β and I(r; a, b) be the number of monic irreducible polynomials f ∈ Fq[x] of degree r with f(0) = a and f(1) = b. We show that Nt(α, β) can be expressed in terms of I(r; a, b), where r | t and a, b ∈ F*q are related to α and β. A recursive formula for I(r; a, b) will be given and we illustrate this by computing I(r; a, b) for 2 ≤ r ≤ 4. We also show that N3(α, β) can be expressed in terms of the number of monic irreducible cubic polynomials over Fq with prescribed trace and norm. Consequently, N3(α, β) can be expressed in terms of the number of rational points on a certain elliptic curve. We give a proof that given any a, b ∈ F*q and integer r ≥ 3, there always exists a monic irreducible polynomial f ∈ Fq[x] of degree r such that f(0) = a and f(1) = b. We also use the result on N2(α, β) to construct a new family of planar functions.

When an intense femtosecond laser pulse propagates in a gas, it undergoes filamentation, a spectacular process where the pulse spatial, spectral and temporal characteristics change considerably. A thin short-lived plasma column is formed in the wake of the propagating pulse. My PhD work has been dedicated to the further understanding of the filamentation process. In a first part, I compare the properties of a usual filament with those of a filament formed by a femtosecond laser pulse with a Bessel beam profile. Using a laser pulse of same intensity and duration, I show that a Bessel beam can form a longer and more uniform plasma column in air, but that the plasma density is significantly lower. In a second part, I show that it is possible to increase considerably the lifetime of the plasma column, using a dual femtosecond/nanosecond laser pulse technique. To obtain an increased lifetime over a significant segment of a plasma column, I rely on the properties of Bessel beams in the nonlinear regime developed in the first chapter. In a third part, I study the dynamics of free electrons that are produced in the filamentation process. To do this, I have developed a specially designed current probe. Experiments reveal a very rich behaviour. The longitudinal displacements of electrons in the plasma column depend sensitively on the nature of the gas and its pressure as well as on the laser polarization of the laser. I propose a model to explain this behaviour. The direction of electron flow results from the competition between pure laser forces and a Coulomb wake field force. In the last chapter, I study filamentation in a Helium gas. This required improving the laser characteristics in order to reach the necessary power for filamentation. Improved characteristics have been achieved by implementing a planar compression stage which shortened the laser pulse from 50 fs to 10 fs without appreciable energy loss. The first experimental evidence for filamentation in He is presented at the end of the thesis. Agreement is found with a numerical simulation.

Worst-case analysis is often meaningless in practice. Some problems never reach the anticipated worst-case complexity. Other solutions get bogged down with impractical constants during implementation, despite having favorable asymptotic running times. In this thesis, we investigate these contrasts in the context of finding maximum flows in planar digraphs. We suggest analytic techniques that adapt to the problem instance, and present a structural property that concludes equivalence between shortest paths and maximum st-flow in planar graphs. The best known algorithm for maximum st-flow in directed planar graphs is an augmenting- paths algorithm with O(n) iterations. Using dynamic trees, each iteration can be implemented in O(log n) time. Long before, Itai and Shiloach showed that when s and t are on the boundary of a common face, the O(n)-iteration augmenting-paths algorithm is equivalent to Dijkstra's algorithm in the graph���s dual: the max st-planar st-flow problem can be solved with one single-source shortest-path computation. In this thesis we show that (a) when s and t are separated by p faces, the max st-flow can be found with at most 2p single-source shortest-path computations, which, using the linear-time shortest-paths algorithm for planar graphs, results in an O(np)-time algorithm, and (b) that the equivalence between augmenting-paths and Dijkstra's extends to the most general non-st-planar digraphs, using their half-infinite universal cover graph. / Graduation date: 2013

In the context of transistor size miniaturization the motivation of this work was focused on the fabrication process of planar double gate devices. We proposed in this work three process flows based on the use of buried mask which could allow the fabrication of self-aligned planar double gate transistors. The novel concept of buried mask consists into modifying the etch rate of a buried polysilicon or oxide layer. This etch rate modification being defined by ion implantation, etch stop or scacrificial zones aligned with the implantation mask can thus be fabricated. This technique solve the alignment of the front and back gate. Ion implantation causes damages to the implanted target, and is used to dope semiconductor material. If the implanted atoms have a small radii they can induce stress to the implanted lattice. These three consequences of ion implantation, damage, doping and stress are used to modify the etch rate of oxide and polysilicon. High etching selectivity are reached, which allow the fabrication of a localized buried sacrificial or etch stop zone, called buried mask. The definition of the buried mask being done by ion implantation, it opens the possibility to fabricate a buried mask aligned with the implantation mask. Although some more work has to be invested to fabricate planar double gate MOS using buried mask in polysilicon, this concept of buried mask, which could also be called anisotropic wet and vapor etching, is foreseen as a very promising technique in MEMS micromachining and for bio sensor applications.

Thesis (Ph.D. (Zoology))--North-West University, Potchefstroom Campus, 2008.

In the study of the Degasperis-Procesi dierential equation, an eigenvalue problem called the cubic string occurs. This is a third order generalization of the second order problem describing the eigenmodes of a vibrating string. In this thesis we study the eigenfunctions of the cubic string for discrete and continuous mass distributions, using the theory of total positivity, via a combinatorial approach with planar networks.

Little is known about the molecular components and interactions of the planar cell polarity pathway that regulate neuronal polarity. This study uses a prkl-1 induced backwards locomotion defect as an array to perform a prkl-1 suppressor screen in C. elegans looking for new components of the planar cell polarity pathway involved in the neuronal polarization of VC4 and VC5. The screen discovered twelve new alleles of vang-1, one new allele of fntb-1 and five new mutations in unknown polarity genes. fntb-1 encodes for the worm ortholog of Farnesyltransferase β-subunit and is important for neuronal polarization. Acting cell and non-cell autonomously, fntb-1 regulates the function and localization of prkl-1 through the recognition of a CAAX motif. Therefore, fntb-1 modifies prkl-1 to regulate the neuronal polarity of VC4 and VC5.

This dissertation addresses three research challenges in planar lightwave circuit (PLC) optical signal processing. 1. Dynamic localization, a relatively new class of quantum phenomena, has not been demonstrated in any system to date. To address this challenge, the quantum system was mapped to the optical domain using a set of curved, coupled PLC waveguides in aluminum gallium arsenide (AlGaAs). The devices demonstrated, for the first time, exact dynamic localization in any system. These experiments motivate further mappings of quantum phenomena in the optical domain, leading toward the design of novel optical signal processing devices using these quantum-analog effects. 2. The PLC microresonator promises to reduce PLC device size and increase optical signal processing functionality. Microresonators in a parallel cascaded configuration, called "side coupled integrated spaced sequence of resonators" (SCISSORs), could offer very interesting dispersion compensation abilities, if a sufficient number of rings is present to produce fully formed "Bragg" gaps. To date, a SCISSOR with only three rings has been reported in a high-index material system. In this work, one, two, four and eight-ring SCISSORs were fabricated in AlGaAs. The eight-ring SCISSOR succeeded in producing fully formed Bragg peaks, and offers a platform to study interesting linear and nonlinear phenomena such as dispersion compensators and gap solitons. 3. PLCs are ideal candidates to satisfy the projected performance requirements of future microchip interconnects. In addition to data routing, these PLCs must provide over 100-bit switchable delays operating at ~ 1 Tbit/s. To date, no low loss optical device has met these requirements. To address this challenge, an ultrafast, low loss, switchable optically controllable delay line was fabricated in AlGaAs, capable of delaying 126 bits, with a bit-period of 1.5 ps. This successful demonstrator offers a practical solution for the incorporation of optics with microelectronics systems. The three aforementioned projects all employ, in their unique way, the coupling of light between PLC waveguides in AlGaAs. This central theme is explored in this dissertation in both its two- and multi-waveguide embodiments.

Non-planar lifting surfaces offer potentially significant gains in aerodynamic efficiency by lowering induced drag. Non-aerodynamic considerations, such as structures can impact the overall efficiency. Here, a panel method and equivalent beam finite element model are used to explore non-planar configurations taking into account the coupling between aerodynamics and structures. A single discipline aerodynamic optimization and a multidisciplinary aerostructural optimization are investigated. Due to the complexity of the design space and the presence of multiple local minima, an augmented Lagrangian particle swarm optimizer is used. The aerodynamic optimum solution found for rectangular lifting surfaces is a box wing, while allowing for sweep and taper yields a joined wing. Adding parasitic drag in the aerodynamic model reduces the size of the non--planar elements. The aerostructural optimal solution found is a winglet configuration when the span is constrained and a wing rake when there is no such constraint.

Cerebral small vessel disease (SVD) is highly prevalent in older adults and is a predictor of stroke, dementia, and death. SVD is also associated with cognitive dysfunction, gait problems, and urinary incontinence. SVD is diagnosed based on white matter hyperintensities on T2 weighted scans. This thesis investigates the cardiac frequency component of resting state functional magnetic resonance imaging data in young healthy adults, older healthy adults, and older adults with pronounced SVD. A cardiac pulsatility metric is defined, and a tissue type contrast is observed between white matter, grey matter, and cerebrospinal fluid. Aging and disease effects are observed on cardiac pulsatility in white matter. The increased pulsatility may reflect the pathology of venous collagenosis and draining vein stenosis. Developing a better understanding of the etiology of SVD is an important step towards treating the disease.