This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular notation. The ideas of partially ordered sets, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a Boolean algebra. From this fundamental understanding, we were able to study atoms, Boolean algebra isomorphisms, and Stone’s Representation Theorem for finite Boolean algebras. We also verified and proved many properties involving Boolean algebras and related structures. We then expanded our study to more thoroughly developed theory. This comprehensive theory was more abstract and required the use of a different, more universal, notation. We continued examining least upper and greatest lower bounds but extended our knowledge to subalgebras and families of subsets. The notions of cardinality, cellularity, and pairwise disjoint families were investigated, defined, and then used to understand the Erdös-Tarski Theorem. Lastly, this study concluded with the investigation of denseness and incomparability as well as normal forms and the completion of Boolean algebras.

Lead halide perovskites (LHPs) have forcefully emerged as a promising materials class for next-generation solar cells. The high efficiencies of LHP-based photovoltaics are underpinned by their outstanding optoelectronic properties, including long carrier lifetimes, long carrier diffusion lengths, high radiative efficiencies, and long-lived hot carriers. In conventional semiconductors, high efficiencies are achieved by stringent control over defect densities; higher purity diminishes the number of carrier scattering events and yields better optoelectronic properties. Given the high defect densities of LHPs, these observed behaviors indicate that LHPs are defect-tolerant and disobey this paradigm via dynamic screening of charge carriers. In order to expand the library of defect-tolerant semiconductors, we must elucidate the carrier-lattice interactions that lead to dynamic screening. LHP lattices are highly anharmonic and dynamically disordered, which must play a role in this screening mechanism. This anharmonicity demands a departure from the conventional Fröhlich interaction, which considers the harmonic coupling of a carrier to one phonon, to a picture that incorporates anharmonic phonon-phonon couplings. The objective of this thesis is to investigate the ultrafast anharmonic lattice response associated with dynamic screening of charge carriers. We probe the formation of large polarons in CH3NH3PbBr3 and CsPbBr3 using time-resolved optical Kerr effect spectroscopy. We further investigate the coupling of phonon modes in a model system, CsPbBr3, in the presence of charge carriers using ultrafast coherent phonon spectroscopy.

<p>Предмет&nbsp; истраживања&nbsp; тезе&nbsp; је&nbsp; из области&nbsp; паралелног&nbsp; програмирања,<br />имплементација&nbsp; CFD&nbsp; (Computational Fluid&nbsp; Dynamics)&nbsp; методе&nbsp; на&nbsp; више<br />хетерогених&nbsp; вишејезгарних&nbsp; уређаја истовремено.&nbsp; У&nbsp; раду&nbsp; је&nbsp; приказано<br />неколико&nbsp; алгоритама&nbsp; чији&nbsp; је&nbsp; циљ убрзање&nbsp; CFD&nbsp; симулације&nbsp; на персоналним&nbsp; рачунарима.&nbsp; Показано је&nbsp; да&nbsp; описано&nbsp; решење&nbsp; постиже задовољавајуће&nbsp; перформансе&nbsp; и&nbsp; на HPC&nbsp; уређајима&nbsp; (Тесла&nbsp; графичким картицама).&nbsp; Направљена&nbsp; је симулација&nbsp; у&nbsp; микросервис архитектури&nbsp; која&nbsp; је&nbsp; портабилна&nbsp; и флексибилна и додатно олакшава рад на персоналним рачунарима.</p> / <p>Predmet&nbsp; istraživanja&nbsp; teze&nbsp; je&nbsp; iz oblasti&nbsp; paralelnog&nbsp; programiranja,<br />implementacija&nbsp; CFD&nbsp; (Computational Fluid&nbsp; Dynamics)&nbsp; metode&nbsp; na&nbsp; više<br />heterogenih&nbsp; višejezgarnih&nbsp; uređaja istovremeno.&nbsp; U&nbsp; radu&nbsp; je&nbsp; prikazano<br />nekoliko&nbsp; algoritama&nbsp; čiji&nbsp; je&nbsp; cilj ubrzanje&nbsp; CFD&nbsp; simulacije&nbsp; na personalnim&nbsp; računarima.&nbsp; Pokazano je&nbsp; da&nbsp; opisano&nbsp; rešenje&nbsp; postiže zadovoljavajuće&nbsp; performanse&nbsp; i&nbsp; na HPC&nbsp; uređajima&nbsp; (Tesla&nbsp; grafičkim karticama).&nbsp; Napravljena&nbsp; je simulacija&nbsp; u&nbsp; mikroservis arhitekturi&nbsp; koja&nbsp; je&nbsp; portabilna&nbsp; i fleksibilna i dodatno olakšava rad na personalnim računarima.</p> / <p>The&nbsp; case&nbsp; study&nbsp; of&nbsp; this&nbsp; dissertation belongs&nbsp; to&nbsp; the&nbsp; field&nbsp; of&nbsp; parallel programming,&nbsp; the&nbsp; implementation&nbsp; of CFD&nbsp; (Computational&nbsp; Fluid&nbsp; Dynamics) method&nbsp; on&nbsp; several&nbsp; heterogeneous multiple&nbsp; core&nbsp; devices&nbsp; simultaneously. The&nbsp; paper&nbsp; presents&nbsp; several&nbsp; algorithms aimed&nbsp; at&nbsp; accelerating&nbsp; CFD&nbsp; simulation on common computers. Also it has been shown&nbsp; that&nbsp; the&nbsp; described&nbsp; solution achieves&nbsp; satisfactory&nbsp; performance&nbsp; on<br />HPC&nbsp; devices&nbsp; (Tesla&nbsp; graphic&nbsp; cards). Simulation is&nbsp; created&nbsp;&nbsp;&nbsp; in&nbsp; micro-service architecture that is portable and flexible and&nbsp; makes&nbsp; it&nbsp; easy&nbsp; to&nbsp; test&nbsp; CFD<br />simulations on common computers.</p>

In this thesis, we study the Fisher's zeros in lattice gauge theory. The analysis of singularities in the complex coupling plane is an important tool to understand the critical phenomena of statistical models. The Fisher's zero structure characterizes the scaling properties of the underlying models and has a strong influence on the complex renormalization group transformation flows in the region away from both the strong and weak coupling regimes. By reconstructing the density of states, we try to develop a systematical method to investigate these singularities and we apply the method to SU(2) and U(1) lattice gauge models with a Wilson action in the fundamental representation. We first take the perturbative approach. By using the saddle point approximation, we construct the series expansions of the density of states in both of the strong and weak regimes from the strong and weak coupling expansions of the free energy density. We analyze the SU(2) and U(1) models. The expansions in the strong and weak regimes for the two models indicate both possess finite radii of convergence, suggesting the existence of complex singularities. We then perform the numerical calculations. We use Monte Carlo simulations to construct the numerical density of states of the SU(2) and U(1) models. We also discuss the convergence of the Ferrenberg-Swendsen's method which we use for the SU(2) model and propose a practical method to find the initial values that improve the convergence of the iterations. The strong and weak series expansions are in good agreement with the numerical results in their respective limits. The numerical calculations also enable the discussion of the finite volume effects which are important to the weak expansion. We calculate the Fisher's zeros of the SU(2) and U(1) models at various volumes using the numerical entropy density functions. We compare different methods of locating the zeros. By the assumption of validity of the saddle point approximation, we find that the roots of the second derivative of the entropy density function have an interesting relation with the actual zeros and may possibly reveal the scaling property of the zeros. Using the analytic approximation of the numerical density of states, we are able to locate the Fisher's zeros of the SU(2) and U(1) models. The zeros of the SU(2) stabilize at a distance from the real axis, which is compatible with the scenario that a crossover instead of a phase transition is expected in the infinite volume limit. In contrast, with the precise determination of the locations of Fisher's zeros for the U(1) model at smaller lattice sizes L=4, 6 and 8, we show that the imaginary parts of the zeros decrease with a power law of L-3.07 and pinch the real axis at β= 1.01134, which agrees with results using other methods. Preliminary results at larger volumes indicate a first-order transition in the infinite volume limit.

The analysis of the error of perturbative series by comparing it to the exact solution is an important tool to understand the non-perturbative physics of statistical models. For some toy models, a new method can be used to calculate higher order weak coupling expansion and modified perturbation theory can be constructed. However, it is nontrivial to generalize the new method to understand the critical behavior of high dimensional spin and gauge models. Actually, it is a big challenge in both high energy physics and condensed matter physics to develop accurate and efficient numerical algorithms to solve these problems. In this thesis, one systematic way named tensor renormalization group method is discussed. The applications of the method to several spin and gauge models on a lattice are investigated. theoretically, the new method allows one to write an exact representation of the partition function of models with local interactions. E.g. O(N) models, Z2 gauge models and U(1) gauge models. Practically, by using controllable approximations, results in both finite volume and the thermodynamic limit can be obtained. Another advantage of the new method is that it is insensitive to sign problems for models with complex coupling and chemical potential. Through the new approach, the Fisher's zeros of the 2D O(2) model in the complex coupling plane can be calculated and the finite size scaling of the results agrees well with the Kosterlitz-Thouless assumption. Applying the method to the O(2) model with a chemical potential, new phase diagram of the models can be obtained. The structure of the tensor language may provide a new tool to understand phase transition properties in general.

No description available.

Thesis (Ph. D.)--University of Hong Kong, 2007. / Title proper from title frame. Also available in printed format.

<p>I detta exjobb har en trådlös enhet för vibrationsmätning på maskiner konstruerats. Arbetet har gällt både hårdvara ochmjukvara. I arbetet ingår bland annat implementering av olika digitala filter i den mikrokontroller som valts ochframtagning och implementering av ett kommunikationsprotokoll för trådlös kommunikation. Resultatet har blivit att enkomplett fungerande prototyp på en mätenhet med lovande egenskaper kunnat tas fram.</p>

Graduation date: 1998

In this thesis we consider four topics all related to using problem reformulations in order to solve integer programs, i.e. optimization problems in which the decision variables must be integer. We first consider the polyhedral approach. We start by addressing the question of lifting valid inequalities, i.e. finding a valid inequality for a set Y from the knowledge of a valid inequality for a lower-dimensional restriction X of Y. We simplify and clarify the presentation of the procedure. This allows us to derive conditions under which the computation of the lifting is tractable. The second topic is the study of valid inequalities for the single node flow set. The single node flow set is the problem obtained by considering one node of a fixed charge network flow problem. We derive valid inequalities for this set and various generalizations. Our approach is a systematic procedure using only basic tools of integer programming: fixing and complementing variables, mixed-integer rounding and lifting. The method allows us to explain and generate a large range of inequalities describing the convex hull of such sets. The last two topics are based on non-standard approaches for integer programming. We first show how the group relaxation approach can be used to provide reformulations for the integral basis method. This is based on a study of extended formulations for the group problem. We present four extended formulations and show that the projections of three of these formulations provide the convex hull of the original group problem. Initial computational tests of the approach are also reported. Finally we consider a problem that is difficult for the standard branch-and-bound approach even for small instances. A reformulation based on lattice basis reduction is known to be more effective. However the step to compute the reduced basis is O(n^4) and becomes a bottleneck for small to medium instances. By using the structure of the problem, we show that we can decompose the problem and obtain the basis by taking the kronecker product of two smaller bases easier to compute. Furthermore, if the two small bases are reduced, the kronecker product is also reduced up to a reordering of the vectors. Computational results show the gain from such an approach.