Depth of field is a naturally occurring effect in lenses describing the distance between theclosest and furthest object that appears in focus. The effect is commonly used in film andphotography to direct a viewers focus, give a scene more complexity, or to improve aes-thetics. In computer graphics, the same effect is possible, but since there are no naturaloccurrences of lenses in the virtual world, other ways are needed to achieve it. There aremany different approaches to simulate depth of field, but not all are suited for real-time usein computer games. In this thesis, multiple methods are explored and compared to achievedepth of field in real-time with a focus on computer games. The aspect of bokeh is alsocrucial when considering depth of field, so during the thesis, a method to simulate a bokeheffect similar to reality is explored. Three different methods based on the same approachwas implemented to research this subject, and their time and memory complexity weremeasured. A questionnaire was performed to measure the quality of the different meth-ods. The result is three similar methods, but with noticeable differences in both quality andperformance. The results give the reader an overview of different methods and directionsfor implementing it on their own, based on which requirements suits them.

This thesis brings a reader an overview about historical and modern world of cryptographic methods, as well evaluates actual state of cryptographic algorithm progressions, which are used in applications nowadays. The aim of the work describes common symmetric, asymmetric encryption methods, cryptographic hash functions and as well pseudorandom number generators, authentication protocols and protocols for building VPNs. This document also shows the basics of the successful modern cryptanalysis and reveals algorithms that shouldn't be used and which algorithms are vulnerable. The reader will be also recommended an overview of cryptographic algorithms that are expected to stay safe in the future.

The discovery of the entropy production paradox (Hoffmann et al., 1998) raised basic questions about the nature of irreversibility in the regime between diffusion and waves. First studied in the form of spatial movements of moments of H functions, pseudo propagation is the pre-limit propagation-like movements of skewed probability density function (PDFs) in the domain between the wave and diffusion equations that goes over to classical partial differential equation propagation of characteristics in the wave limit. Many of the strange properties that occur in this extraordinary regime were thought to be connected in some manner to this form of proto-movement. This paper eliminates pseudo propagation by employing a similar evolution equation that imposes spatial unimodal symmetry on evolving PDFs. Contrary to initial expectations, familiar peculiarities emerge despite the imposed symmetry, but they have a distinct character.

Un système couvrant est un ensemble fini de progressions arithmétiques avec la propriété que chaque entier appartient à au moins une des progressions. L’étude des systèmes couvrants a été initié par Erdős dans les années 1950, et il posa dans les années qui suivirent plusieurs questions sur ces objets mathématiques. Une de ses questions les plus célèbres est celle du plus petit module : est-ce que le plus petit module de tous les systèmes couvrants avec modules distinct est borné uniformément? En 2015, Hough a montré que la réponse était affirmative, et qu’une borne admissible est 1016. En se basant sur son travail, mais en simplifiant la méthode, Balister, Bollobás, Morris, Sahasrabudhe et Tiba on réduit cette borne a 616, 000. Leur méthode a menée a plusieurs applications supplémentaires. Entre autres, ils ont compté le nombre de système couvrant avec un nombre fixe de module. La première partie de ce mémoire vise a étudier une question similaire. Nous allons essayer de compter le nombre de système couvrant avec un ensemble de module fixé. La technique que nous utiliserons nous mènera vers l’étude des symmétries de système couvrant. Dans la seconde partie, nous répondrons à des variantes du problème du plus petit module. Nous regarderons des bornes sur le plus petit module d’un système couvrant de multiplicité s, c’est-à-dire un système couvrant dans lequel chaque module apparait au plus s fois. Nous utiliserons ensuite ce résultat afin montrer que le plus petit module d’un système couvrant de multiplicité 1 d’une progression arithmétique est borné, ainsi que pour montrer que le n-eme plus petit module dans un système couvrant de multiplicité 1 est borné. / A covering system is a finite set of arithmetic progressions with the property that every integer belongs to at least one of them. The study of covering systems was started by Erdős in the 1950’s, and he asked many questions about them in the following years. One of the most famous questions he asked was if the minimum modulus of a covering system with distinct moduli is bounded uniformly. In 2015, Hough showed that it is at most 1016. Following on his work, but simplifying the method, Balister, Bollobás, Morris, Sahasrabudhe and Tiba showed that it is at most 616, 000. Their method led them to many further applications. Notably, they counted the number of covering systems with a fixed number of moduli. The first part of this thesis seeks to study a related question, that is to count the number of covering systems with a given set of moduli. The technique developped to do this for some sets will lead us to look at symmetries of covering systems. The second part of this thesis will look at variants of the minimum modulus problem. Notably, we will be looking at bounds on the minimum modulus of a covering system of multiplicity s, that is a covering system in which each moduli appears at most s times, as well as bounds on the minimum modulus of a covering system of multiplicity 1 of an arithmetic progression, and finally look at bounds for the n-th smallest modulus in a covering system.

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With the development of electric vehicle (EV) charging systems, energy storage systems (ESS), data center power supplies, and solid-state transformer (SST) systems, the fixed-ratio isolated DC-DC converter, namely the DC transformer (DCX), has gained significant popularity. Similar to the passive AC transformer, DCX can bidirectionally convey DC power with very high efficiency. Due to zero-voltage switching (ZVS) and a small root mean square (RMS) current, the open-loop CLLC resonant converter operating at the resonant frequency is a promising candidate for DCX with a constant voltage transfer ratio. In Chapter 2, to solve unsmooth bidirectional power flow and current distortion in the traditional CLLC-DCX with synchronization rectification (SR) modulation, a dual-active-synchronization (DAS) modulation is adopted with identical driving signals on both sides. First, the switching transition of this modulation is thoroughly analyzed considering the large switch's output capacitances. After comparing different transitions, a so-called sync-ZVS transition is more desirable with ZVS, has no deadtime conduction loss, and almost has load-independent voltage gain. An axis and center symmetric (ACS) method is proposed to achieve this switching transition. Based on this method, an overall design procedure of CLLC-DCX with DAS modulation is also proposed. However, designing a high-power and high-frequency transformer for CLLC-DCX presents significant challenges due to the trade-off between thermal management, leakage inductance minimization, and insulation requirements. To overcome this trade-off between power rating and operation frequency, a scalable electronic-embedded transformer (EET) with a low-voltage bridge integrated into the transformer windings is proposed in Chapter 3. The EET addresses the challenge through simple open-loop control and natural current sharing, enabling easy parallel connection and scaling to different power ratings. Based on this concept, a bidirectional, EET-based DC transformer (EET-DCX) is proposed to solve the transformer-level paralleling and resonant point shift issues in traditional LLC-DCX designs. By employing the embedded full bridge, the EET-DCX effectively cancels out the impedance of the leakage inductance, ensuring optimal operation at any frequency. Additionally, the EET-DCX retains the inherent advantages of the LLC-DCX, such as load-independent voltage gain, simple open-loop control, full-load range ZVS, and low circulating current. Leveraging these advantages, the proposed EET-DCX solution has the potential to push the boundaries of transformer performance to the MHz operation frequency range with hundreds of kilowatts of power capability. Moreover, to address the significant RMS current problem of the CLLC-DCX, a trapezoidal current modulation is also proposed in Chapter 3. Compared to the CLLC-DCX with a sinusoidal current, an EET-DCX with a trapezoidal current can reduce the total conduction loss by up to 23%. This total conduction loss includes semiconductor loss on both high-voltage and low-voltage bridges and transformer winding loss. In light of this EET concept, another resonant commutation (RC) EET-DCX is proposed to streamline the circuit. First, it replaces the embedded full bridge with a low-voltage bidirectional AC switch. Second, it introduces a resonant current commutation to realize a quasi-trapezoidal transformer current with a smaller RMS value. Compared to the triangular current produced by the original EET-DCX, the RMS current can be decreased by 15%. By incorporating only one embedded bidirectional AC switch, the high-frequency transformer leakage inductance impedance is fully neutralized. As a result, the rated power of the proposed RC EET-DCX can be readily scaled up through transformer-level parallelism. Furthermore, the RC EET-DCX maintains the benefits of a typical LLC/CLLC-DCX, including load-independent voltage gain, full load range ZVS, and low circulating current. However, either in EET-DCX or RC EET-DCX, the trapezoidal current modulation will increase the voltage stress on the low-voltage full bridge or bidirectional AC switch, especially when the leakage inductance is large and variable, such as in the high-power wireless charging application. To address this trade-off between RMS current and voltage stress, this paper proposes the concept of a hybrid resonant-type EET-DCX with a series resonant capacitor. Following this concept, two specific topologies, hybrid EET-DCX and hybrid RC EET-DCX, are proposed. The main difference between these topologies is that the former adopts a full bridge. In a hybrid RC EET-DCX, a resonant current commutation scheme is developed. Among these topologies, since the passive capacitor can mainly cancel the leakage inductance impedance, the full bridge or AC switch only needs to handle the remaining impedance. Thus, the voltage stress on active components can be dramatically decreased. Additionally, these two proposed topologies can retain all the advantages of previous EET-DCX designs, including natural current sharing, load-independent voltage gain, simple open-loop control, and full-load range ZVS. The comparison between these two topologies is thoroughly studied. Finally, a 12-kW DCX testbench is built to verify all the analysis and performance in Chapter 3. If output voltage regulation is required, DCX can cooperate with other voltage regulators to realize high conversion efficiency and power density. In Chapter 4, two DCX applications are implemented: an 18-kW 98.8% peak efficiency EV battery charger with partial power processing and a 50-kW symmetric 3-level buck-boost converter with common-mode (CM) noise reduction. In the first battery charger, a large portion of the power is handled by an 18 kW CLLC-DCX, and the remaining partial power goes through a 3-phase interleaved buck converter. The proposed switching transition optimization in Chapter 2 is adopted in the 18-kW CLLC-DCX to realize 98.8% peak efficiency. To handle the step-up and step-down cases at the same time, a symmetric 3-level buck-boost converter with coupled inductors is also studied as a post regulator. With symmetric topology and quadrangle current control, the converter can achieve a CM noise reduction and full load range ZVS with a small RMS current. To further optimize the performance and simplify the control, a mid-point bridging with a better CM noise reduction and a split capacitor voltage auto-balance is implemented. A 50-kW prototype is built to verify the above analysis. To summarize, Chapter 2 first proposes a switching transition optimization for CLLC-DCX. Later, to address the intrinsic trade-off between transformer rating power and frequency, an EET concept and its corresponding soft-switching DCX family are found in Chapter 3. Finally, to handle voltage regulation, two examples for practical applications are studied in Chapter 4 —one is an 18-kW partial power converter, and the other is a 50-kW 3-L buck-boost converter. Finally, Chapter 5 will draw conclusions and illustrate future work. / Doctor of Philosophy / With the development of electric vehicle (EV) charging systems, energy storage systems (ESS), data center power supply, and solid-state transformer (SST) systems, the fixed-ratio isolated dc-dc converter, namely dc transformer (DCX), has gained significant popularity. However, designing a high-performance DCX still has many challenges, such as large dead time loss, poor current sharing, and sensitivity to parameter tolerance. Firstly, the state-of-the-art resonant CLLC-DCX is optimized in Chapter 2. With an optimal switching frequency and dead time, both the primary and secondary sides of zero voltage switching (ZVS) can begin and finish simultaneously, which means dead time loss caused by current through the body diode can be eliminated. Therefore, the efficiency of CLLC-DCX can be improved. However, designing a high-power and high-frequency CLLC-DCX transformer still presents significant challenges due to the trade-off between thermal management, leakage inductance minimization, and insulation requirements. To overcome this trade-off, in Chapter 3, a scalable electronic-embedded transformer (EET) concept with a low-voltage bridge integrated into the transformer windings is proposed. The EET addresses the challenge through its simple open-loop control and natural current sharing, enabling easy parallel connection and scaling to different power ratings. In light of this EET concept, a new family of soft-switching DCXs is proposed for different applications, such as high-power wireless charging systems. All these EET-based DCXs retain the merits of typical CLLC-DCX, such as small circulating current ringing, small turn-off current, full load range ZVS, and load-independent gain. After realizing a desirable design for DCX, Chapter 4 presents two DCX applications with voltage regulation. Firstly, an 18 kW 98.8% peak efficiency battery charger is designed with partial power processing. Most of the power will go through an optimized DCX, and the remaining small portion of power will go through a 3-phase interleaved buck converter. On the other hand, DCX can also be adopted as a front-end or rear-end converter in a typical two-state DC-DC converter. As for another stage, a non-isolated DC-DC converter with a large output range can be used to handle voltage regulation. Following this structure, a 50-kW symmetric 3-L buck-boost converter with coupled inductors and reduced common emission is proposed. To summarize, the state-of-the-art CLLC-DCX is optimized in Chapter 2. Afterward, a new concept of EET-DCX and its corresponding DCX family is proposed in Chapter 3. After obtaining an optimized DCX, two practical applications with DCX are implemented in Chapter 4. Finally, Chapter 5 will draw conclusions and illustrate future work.

The purpose of the thesis is to study completeness of abstract spaces. In particular, we study completeness in partial metric spaces, partial metric type spaces, dislocated metric spaces, dislocated metric type spaces and symmetric spaces that are generalizations of metric spaces. It is well known that complete metric spaces have a wide range of applications. For instance, the classical Banach contraction principle is phrased in the context of complete metric spaces. Analogously, the Banach's xed point theorem and xed point results for Lipschitzian maps are discussed in this context, namely in, partial metric spaces and metric type spaces. Finally, xed point results are presented for symmetric spaces / Mathematical Sciences / Ph. D. (Mathematics)

The purpose of the thesis is to study completeness of abstract spaces. In particular, we study completeness in partial metric spaces, partial metric type spaces, dislocated metric spaces, dislocated metric type spaces and symmetric spaces that are generalizations of metric spaces. It is well known that complete metric spaces have a wide range of applications. For instance, the classical Banach contraction principle is phrased in the context of complete metric spaces. Analogously, the Banach's xed point theorem and xed point results for Lipschitzian maps are discussed in this context, namely in, partial metric spaces and metric type spaces. Finally, xed point results are presented for symmetric spaces. / Geography / Ph. D. (Mathematics)

In this thesis we present research on mathematical properties of methods for solv- ing symmetric systems of linear equations that arise in various optimization problem formulations and in methods for solving such problems. In the first and third paper (Paper A and Paper C), we consider the connection be- tween the method of conjugate gradients and quasi-Newton methods on strictly convex quadratic optimization problems or equivalently on a symmetric system of linear equa- tions with a positive definite matrix. We state conditions on the quasi-Newton matrix and the update matrix such that the search directions generated by the corresponding quasi-Newton method and the method of conjugate gradients respectively are parallel. In paper A, we derive such conditions on the update matrix based on a sufficient condition to obtain mutually conjugate search directions. These conditions are shown to be equivalent to the one-parameter Broyden family. Further, we derive a one-to-one correspondence between the Broyden parameter and the scaling between the search directions from the method of conjugate gradients and a quasi-Newton method em- ploying some well-defined update scheme in the one-parameter Broyden family. In paper C, we give necessary and sufficient conditions on the quasi-Newton ma- trix and on the update matrix such that equivalence with the method of conjugate gra- dients hold for the corresponding quasi-Newton method. We show that the set of quasi- Newton schemes admitted by these necessary and sufficient conditions is strictly larger than the one-parameter Broyden family. In addition, we show that this set of quasi- Newton schemes includes an infinite number of symmetric rank-one update schemes. In the second paper (Paper B), we utilize an unnormalized Krylov subspace frame- work for solving symmetric systems of linear equations. These systems may be incom- patible and the matrix may be indefinite/singular. Such systems of symmetric linear equations arise in constrained optimization. In the case of an incompatible symmetric system of linear equations we give a certificate of incompatibility based on a projection on the null space of the symmetric matrix and characterize a minimum-residual solu- tion. Further we derive a minimum-residual method, give explicit recursions for the minimum-residual iterates and characterize a minimum-residual solution of minimum Euclidean norm. / I denna avhandling betraktar vi matematiska egenskaper hos metoder för att lösa symmetriska linjära ekvationssystem som uppkommer i formuleringar och metoder för en mängd olika optimeringsproblem. I första och tredje artikeln (Paper A och Paper C), undersöks kopplingen mellan konjugerade gradientmetoden och kvasi-Newtonmetoder när dessa appliceras på strikt konvexa kvadratiska optimeringsproblem utan bivillkor eller ekvivalent på ett symmet- risk linjärt ekvationssystem med en positivt definit symmetrisk matris. Vi ställer upp villkor på kvasi-Newtonmatrisen och uppdateringsmatrisen så att sökriktningen som fås från motsvarande kvasi-Newtonmetod blir parallell med den sökriktning som fås från konjugerade gradientmetoden. I den första artikeln (Paper A), härleds villkor på uppdateringsmatrisen baserade på ett tillräckligt villkor för att få ömsesidigt konjugerade sökriktningar. Dessa villkor på kvasi-Newtonmetoden visas vara ekvivalenta med att uppdateringsstrategin tillhör Broydens enparameterfamilj. Vi tar också fram en ett-till-ett överensstämmelse mellan Broydenparametern och skalningen mellan sökriktningarna från konjugerade gradient- metoden och en kvasi-Newtonmetod som använder någon väldefinierad uppdaterings- strategi från Broydens enparameterfamilj. I den tredje artikeln (Paper C), ger vi tillräckliga och nödvändiga villkor på en kvasi-Newtonmetod så att nämnda ekvivalens med konjugerade gradientmetoden er- hålls. Mängden kvasi-Newtonstrategier som uppfyller dessa villkor är strikt större än Broydens enparameterfamilj. Vi visar också att denna mängd kvasi-Newtonstrategier innehåller ett oändligt antal uppdateringsstrategier där uppdateringsmatrisen är en sym- metrisk matris av rang ett. I den andra artikeln (Paper B), används ett ramverk för icke-normaliserade Krylov- underrumsmetoder för att lösa symmetriska linjära ekvationssystem. Dessa ekvations- system kan sakna lösning och matrisen kan vara indefinit/singulär. Denna typ av sym- metriska linjära ekvationssystem uppkommer i en mängd formuleringar och metoder för optimeringsproblem med bivillkor. I fallet då det symmetriska linjära ekvations- systemet saknar lösning ger vi ett certifikat för detta baserat på en projektion på noll- rummet för den symmetriska matrisen och karaktäriserar en minimum-residuallösning. Vi härleder även en minimum-residualmetod i detta ramverk samt ger explicita rekur- sionsformler för denna metod. I fallet då det symmetriska linjära ekvationssystemet saknar lösning så karaktäriserar vi en minimum-residuallösning av minsta euklidiska norm. / <p>QC 20150519</p>

This study focusses on modelling important aspects of the evolution of pulsar wind nebulae using two different approaches. The first uses a hydrodynamic model to simulate the morphological evolution of a spherically-symmetric composite supernova remnant that is expanding into a homogeneous interstellar medium. In order to extend this model, a magnetic field is included in a kinematic fashion, implying that the reaction of the fluid on the magnetic field is taken into account, while neglecting any counter-reaction of the field on the fluid. This approach is valid provided that the ratio of electromagnetic to particle energy in the nebula is small, or equivalently, for a large plasma β environment. This model therefore allows one to not only calculate the evolution of the convection velocity but also, for example, the evolution of the average magnetic field. The second part of this study focusses on calculating the evolution of the energy spectra of the particles in the nebula using a number of particle evolution models. The first of these is a spatially independent temporal evolution model, similar to the models that can be found in the literature. While spatially independent models are useful, a large part of this study is devoted to developing spatially dependent models based on the Fokker-Planck transport equation. Two such models are developed, the first being a spherically-symmetric model that includes the processes of convection, diffusion, adiabatic losses, as well as the non-thermal energy loss processes of synchrotron radiation and inverse Compton scattering. As the magnetic field geometry can lead to the additional transport process of drift, the previous model is extended to an axisymmetric geometry, thereby allowing one to also include this process. / PhD (Space Physics), North-West University, Potchefstroom Campus, 2014