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Structured light has shown great promises and impacts in various fields such as microscopy, optical trapping, sensing, classical communication, high dimensional quantum information, quantum key distribution (QKD) and quantum metrology. In this Thesis, we will be discussing two projects, in the context of structured light, one about the stability of 3D optical topologies and the other about the reconstruction of strongly correlated bi-photon states. In the introduction, we will be reviewing some of the basic concepts needed to follow the two projects. Starting from the spatial modes and their characteristics, we discuss about some interferometric approaches to reconstruct the phase and the amplitude of an electromagnetic field, either for a classical laser beam or for photons. Moreover, we briefly explain the spontaneous parametric down conversion (SPDC) process that is commonly used for generating entangled bi-photon states. In the second chapter, after giving an overview about singular optics and optical topologies, we explore the effect of different types of aberrations on three different 3D optical structures, Trefoil knot, Cinquefoil knot and Hopf Link. In this work we show that these structures are robust under aberrations with the highest strength if the defining aperture of the aberration is big enough. In addition, we discuss about how these aberrations will modify the structure if we decrease the size of the aperture and in which aperture the structures start to break and therefore, we investigate the impact of the strength under such apertures. In the third chapter, we return to the SPDC process, discuss the thin-crystal approximation, spatial mode correlations of an entangled bi-photon state, and coincidence imaging performed with time stamping cameras. We propose an approach to characterize an unknown two-photon state using interference between the unknown bi-photon state and a reference bi-photon state in addition to coincidence imaging. We show that this approach is faster, scalable and loss-free compared to projective measurements. Finally, in the conclusion, we put together the results of these two projects and discuss about the future work that can follow up on what is done so far.

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The problem of edge disjoint path routing arises from applications in distributed memory parallel computing. We examine this problem in both the directed hypercube and two-dimensional grid topologies. Complexity results are obtained for these problems where the routing must consist entirely of shortest length paths. Additionally, approximation algorithms are presented for the case when the routing request is of a special form known as a permutation. Permutations simply require that no vertex in the graph may be used more than once as either a source or target for a routing request. Szymanski conjectured that permutations are always routable in the directed hypercube, and this remains an open problem.

<p> This thesis deals with the lattice of all topologies which may be put on an arbitrary set E. The structure of the lattice is investigated together with its lattice properties. A chapter is included on the co-atoms of the lattice, the ultra-topologies, wherein various topological properties which do and do not hold are investigated. Various topological properties are considered as to which topologies are minimal and maximal such and also which topological properties are preserved under lattice operations and relations.</p> / Thesis / Master of Science (MSc)

This dissertation systematically explores multiple-input converters (MICs) configuration and topologies, and then proposes improvements on certain beneficial MICs—time-sharing MICs and soft-switching MICs for distributed generation (DG) applications with high voltage transfer ratio. Compared with other MIC families which are derived from same input and output cells, time-sharing MICs have the fewest circuit components. However, time-sharing MICs lack for bi-directional power flow capability due to their special input switches requirement. In addition, their hard-switching characteristic leads to a low efficiency problem when isolation is necessary. The dissertation investigates into time-sharing MIC input switch selection, which leads to a new driving strategy and new input switch combinations. With the new input switch combinations, bi-directional and high efficiency time-sharing MICs are made possible. Besides isolated time-sharing MIC, Soft-switching MICs might also be a common choice for high voltage transfer ratio DG applications. However, the enormous amount of circuit components makes the soft-switching MICs become less attractive. An input cell reduction method is introduced in this dissertation to greatly reduce the component count of isolated MICs, including soft-switching MICs. In addition to the improvement on existing MIC families, a new push-pull connected MIC family is proposed in this dissertation as another choice of high voltage transfer ratio DG applications. Moreover, a comparison among MIC families is made to provide a sense of topologies selection in certain applications. Prototypes of time-sharing dual-input (DI) SEPICs, a push-pull connected DI-Boost converter, and a DI full-bridge (FB) converter are built to verify aspects discussed in this dissertation. Bi-directional power flow capability of time-sharing MIC is confirmed with a time-sharing DI-SEPIC and a soft-switching time-sharing MIC is realized by an isolated time-sharing DI-SEPIC with an active clamping leg. Maximum power point tracking control feasibility in these converters is evaluated with real photovoltaic modules that are connected to the push-pull connected DI-Boost converter that uses a perturb-and-observe method. Finally, an efficiency comparison is made between time-sharing MIC and push-pull connected MIC. / text

Η εργασία αυτή, αναφέρεται σε τοπολογίες σε χώρους συναρτήσεων και δομείται ως εξής: Στο κεφάλαιο 1 γίνεται μια εισαγωγή και δίνονται βασικές έννοιες των τοπολογικών χώρων. Στο κεφάλαιο 2 μελετώνται κυρίως η compact open και η σημειακή ανοικτή τοπολογία. Στο κεφάλαιο 3 μελετώνται οι συνδετικά συνεχείς και οι διαχωριστικές τοπολογίες. Στο κεφάλαιο 4 μελετώνται οι τοπολογίες scott και isbell. Το κεφάλαιο 5 αναφέρεται σε ανοικτά προβλήματα των χώρων συναρτήσεων. Στο τέλος της εργασίας υπάρχει εκτενής βιβλιογραφία. / This task is about topologies on function spaces.In chapter 1 there is an introduction in topological spaces. In chapter 2 we study the compact open and point open topologies. In chapter 3 we study jointly continuous and splitting topologies. In chapter 4 we study Scott and Isbell topologies. Finally, in chapter 5 there is a reference in some open problems on function spaces.

As is the case for DC-DC converters, multi-phase converters require both high-quality power control and high power-density. Although a higher switching frequency not only improves the quality of the converter output but also decreases the size of the converter, it increases switching losses and electromagnetic interference (EMI) noise. Since the soft-switching topologies reduce the switching losses of the converter main switches, the topologies make converters partially independent from the switching frequency. However, the conventional soft-switching topologies have already proposed most of the possible ways to improve converter performance. In addition, the trends of the newly generated power devices reduce the advantages of soft-switching topologies. This critical situation surrounding soft-switching topologies gives research motivations: What features of soft-switching topologies facilitate their practical applications? Given this motivation, the dissertation discusses two aspects = simplifying auxiliary circuits and accounting for the effects of soft-switching operations on the converter control. Engineers working with medium- and high-power multi-phase converters require simplified soft-switching topologies that have the same level of performance as the conventional soft-switching topologies. This demand is the impetus behind one of the research objectives = simplifying the auxiliary circuits of Zero-Voltage-Transition (ZVT) inverters. Simplifying the auxiliary circuits results in both a smaller number of and lower cost for auxiliary components, without any negative impact on performance. This dissertation proposes two major concepts for the simplification - the Single-Switch Single-Leg (S3L) ZVT cell and the Phase-Lock (PL) concept. Throughout an effort to eliminate circulating currents of inductor-coupled (IC) ZVT converters, the S3L ZVT cell is developed. The proposed cell allows a single auxiliary switch to achieve zero-voltage conditions for both the top and bottom main switches, and it achieves the same level of performance as the conventional ZVT cell, as well. This proposal makes IC ZVT topologies more attractive to multi-phase converter applications. Because all of the top main switches generally have identical sequences for zero-voltage turn-on commutations, one auxiliary switch might handle the commutations of all of the top main switches. This possibility introduces the PL concept, which allows the two auxiliary switches to provide a zero-voltage condition for any main switch commutation. In order to compensate for restrictions of this concept, a modified space-vector modulation (SVM) scheme also is introduced. A soft-switching topology changes the duty ratios of the converter, which affects the controllability of the converter. Therefore, this dissertation selects resolution of this issue as one of the research objectives. This dissertation derives the generalized timing equations of ZVT operations, and the generalized equations formulize the effect of ZVT operation on both duty ratios and DC current. Moreover, the effect of SVM schemes is also investigated. An average model of the ZVT converter is developed using both the timing analysis and the investigation of SVM schemes, and small-signal analysis using the average model predicts the steady-state characteristics of the converter. / Ph. D.

Complex networks like the Internet, peer-to-peer systems, and emerging sensor and ad-hoc networks are large distributed decentralized communication systems arising repeatedly in today's technology. In such networks it is critical to characterize network performance as the size of the network scales. The focus of my work is to relate basic network performance metrics to structural characteristics of underlying network topologies, and to develop protocols that reinforce and exploit desired structural characteristics. For the case of the Internet at the Autonomous System level, we relate the graph theoretic notions of conductance and spectrum to network clustering and network congestion. In particular, we show how spectral analysis can identify clusters, and how the presence of clusters affects congestion. This is important for network prediction and network simulation. For the case of peer-to-peer networks we relate conductance and spectral gap to the fundamental questions of searching and topology maintenance. We propose new protocols for maintaining peer-to-peer networks with good conductance and low network overhead. We compare the performance of the traditional method of search by flooding to searching by random walks. We isolate cases of practical interest, such as clustered and dynamic network topologies, where the latter have superior performance. The improvement in the performance can be directly quantified in terms of the conductance of the underlying topology. We introduce further hybrid search schemes, of which flooding and random walks are special instances, which aim to improve the performance of searching by using locally maintained information about the network topology.