Microelectromechanical Systems are ubiquitous in modern technology, with applications ranging from accelerometers in smartphones to ultra-high precision motion stages used for atomically-precise positioning. With the appropriate selection of materials and device design, MEMS resonators with ultra-high quality factors can be fabricated at minimal cost. As the sizes of such resonators decrease, however, their mechanical, electrical, and material properties can no longer be treated as linear, as can be done for larger-scale devices. Unfortunately, adding nonlinear effects to a system changes its dynamics from exactly-solvable to only solvable in specific cases, if at all. Despite (and because of) these added complications, nonlinear effects open up an entirely new world of behaviors that can be measured or taken advantage of to create even more advanced technologies. In our resonators, oscillations are induced and measured using aluminum nitride transducers. I used this mechanism for several separate highly-sensitive experiments. In the first, I demonstrate the incredible sensitivity of these resonators by actuating a mechanical resonant mode using only the force generated by the radiation pressure of a laser at room temperature. In the following three experiments, which use similar mechanisms, I demonstrate information transfer and force measurements by taking advantage of the nonlinear behavior of the resonators. When nonlinear resonators are strongly driven, they exhibit sum and difference frequency generation, in which a large carrier signal can be mixed with a much smaller modulation to produce signals at sum and difference frequencies of the two signals. These sum and difference signals are used to detect information encoded in the modulation signal using optical radiation pressure and acoustic pressure waves. Finally, in my experiments, I probe the nonlinear nature of the piezoelectric material rather than take advantage of the nonlinear resonator behavior. The relative sizes of the linear and nonlinear portions of the piezoelectric constant can be determined because the force applied to the resonator by a transducer is independent of the dielectric constant. This method allowed me to quantify the nonlinear constants.

Usamos as regras de soma da QCD para obter a massa dos pentaquarks \" \'teta\' POT.+\"(1540) e \"\'ksi\' POT.--\" (1862), a largura de decaimento da \"\'teta\' POT.+\" e a constante de acoplamento e fator de forma presentes no vertice J/ \'psi\' \"DD POT.*\". No estudo dos pentaquarks, usamos dois tipos de campos interpolantes contendo dois diquarks altamente correlacionados. Obtemos as massas consistentes com os dados experimentais, porém esta regra de soma têm uma grande contribuição do contínuo e a OPE não é muito boa. Obtivemos uma largura de decaimento compatível com os valores experimentais desde que sejam subtraídos diagramas que representam a \" \'teta\' POT.+\" como um estado ligado K - n. No estudo do vértice vertice J/ \'psi\' \"DD POT.*\" calculamos o fator de forma e a constante de acoplamento, considerando três casos: D off-shell, \"D POT.*\" off-shell e J/ \'psi\' off-shell. A constante de acoplamento é a mesma nesses três casos, porém o fator de forma depende da escolha da partícula off-shell, onde para a J/ \'psi\' off-shell o fator de forma é bem mais duro que o fator de forma obtido para os outros casos. Também comparamos os nossos resultados com outros métodos: o modelo de quark méson constituintes e o modelo de quarks relativísticos constituintes / We use the QCD sum rules to obtain the masses of the pentaquarks _+(1540) and _−−(1862), _+ decay width and the coupling constant and the form factor for the J/ DD* vertex.In the study of the pentaquarks, we use two kinds of interpolating fields, containing two highly correlated diquarks. We get the masses in a good agreement with the experimental value, but this sum rule has a large continuuum contribution and the OPE convergence is not so good. We get the decay width compatible with the experimental value, since we subtract the diagrams that represent _+ as a K − n bound state. In the study of J/ DD* vertex, we calculate the form factor and the coupling constant considering three cases: D off-shell, D* off-shell and J/ off-shell. The coupling constant is the same in those three cases, however the form factor depends on the choice of the particle off-shell, where for the J/ off-shell the form factor is much harder than the form factor obtained for the other cases. We also compare our results with other methods: the constituent quark meson model and the relativistic constituent quark model.

Syftet med denna kvalitativa studien är att analysera vilka strategier SUM-elever väljer att använda sig av vid uträkning av additionsuppgifter i relation till räkneförmågan. I studien identifierades även likheter och skillnader mellan SUM-eleverna.  Med hjälp av två diagnoser har eleverna kategoriserats in i fyra elevgrupper utifrån vilka strategier de använt och hur väl de förstod dem. Eleverna använde sig av både konkreta och abstrakta strategier i både huvud- och skriftliga beräkningar. Utifrån analys kan det konstateras i studien att elevgrupperna befinner sig i Piagets olika utvecklingsstadier. De elevgrupper som använde sig av färre strategier befinner sig i det konkreta operationella stadiet eftersom de kan tänka mer formellt så länge det blir konkret för dem. De elevgrupper som använde sig av fler strategier befinner sig i ett högre utvecklingsstadie eftersom de kan tänka formellt utan att använda sig av konkreta hjälpmedel. Till sist drogs paralleller mellan elevgrupperna för att kunna se likheter och skillnader.

Usamos as regras de soma da QCD para obter a massa dos pentaquarks \" \'teta\' POT.+\"(1540) e \"\'ksi\' POT.--\" (1862), a largura de decaimento da \"\'teta\' POT.+\" e a constante de acoplamento e fator de forma presentes no vertice J/ \'psi\' \"DD POT.*\". No estudo dos pentaquarks, usamos dois tipos de campos interpolantes contendo dois diquarks altamente correlacionados. Obtemos as massas consistentes com os dados experimentais, porém esta regra de soma têm uma grande contribuição do contínuo e a OPE não é muito boa. Obtivemos uma largura de decaimento compatível com os valores experimentais desde que sejam subtraídos diagramas que representam a \" \'teta\' POT.+\" como um estado ligado K - n. No estudo do vértice vertice J/ \'psi\' \"DD POT.*\" calculamos o fator de forma e a constante de acoplamento, considerando três casos: D off-shell, \"D POT.*\" off-shell e J/ \'psi\' off-shell. A constante de acoplamento é a mesma nesses três casos, porém o fator de forma depende da escolha da partícula off-shell, onde para a J/ \'psi\' off-shell o fator de forma é bem mais duro que o fator de forma obtido para os outros casos. Também comparamos os nossos resultados com outros métodos: o modelo de quark méson constituintes e o modelo de quarks relativísticos constituintes / We use the QCD sum rules to obtain the masses of the pentaquarks _+(1540) and _−−(1862), _+ decay width and the coupling constant and the form factor for the J/ DD* vertex.In the study of the pentaquarks, we use two kinds of interpolating fields, containing two highly correlated diquarks. We get the masses in a good agreement with the experimental value, but this sum rule has a large continuuum contribution and the OPE convergence is not so good. We get the decay width compatible with the experimental value, since we subtract the diagrams that represent _+ as a K − n bound state. In the study of J/ DD* vertex, we calculate the form factor and the coupling constant considering three cases: D off-shell, D* off-shell and J/ off-shell. The coupling constant is the same in those three cases, however the form factor depends on the choice of the particle off-shell, where for the J/ off-shell the form factor is much harder than the form factor obtained for the other cases. We also compare our results with other methods: the constituent quark meson model and the relativistic constituent quark model.

The results section of my thesis includes three chapters. The first two chapters are on theoretical game theory. In both chapters, by mathematical modelling and game theoretical tools, I am predicting the behaviour of the players in some real world issues. Hoteling-Downs model plays an important role in the modern political interpretations. The first chapter of this study investigates an extension of Hoteling-Downs model to have multi-dimensional strategy space and asymmetric candidates. Chapter 3 looks into the inspection game where the inspections are not the same in the series of sequential inspections. By modelling the game as a series of recursive zero-sum games I find the optimal strategy of the players in the equilibrium. The forth chapter investigates direct optimization methods for large scale problems. Using Matlab implementations of Genetic and Nelder-Mead algorithms, I compare the efficiency and accuracy of the most famous direct optimization methods for unconstraint optimization problems based on differing number of variables.

For many combinatorial objects we can associate a natural probability distribution on the members of the class, and we can then call the resulting class a class of random structures. Random structures form good models of many real world problems, in particular real networks and disordered media. For many such problems, the systems under consideration can be very large, and we often care about whether a property holds most of the time. In particular, for a given class of random structures, we say that a property holds with high probability if the probability that that property holds tends to one as the size of the structures increase. We examine several classes of random structures with real world applications, and look at some properties of each that hold with high probability. First we look at percolation in 3 dimensional lattices, giving a method for producing rigorous confidence intervals on the percolation threshold. Next we look at random geometric graphs, first examining the connectivity thresholds of nearest neighbour models, giving good bounds on the threshold for a new variation on these models useful for modelling wireless networks, and then look at the cop number of the Gilbert model. Finally we look at the structure of random sum-free sets, in particular examining what the possible densities of such sets are, what substructures they can contain, and what superstructures they belong to.

Visible Light Communications (VLC) has recently emerged as a viable solution for solving the spectrum shortage problem. The idea is to use artificial light sources as medium to communicate with portable devices. In particular, the light sources can be switched on and off with a very high-frequency corresponding to 1s and 0s of digital communication. The high-frequency on-off switching can be detected by electronic devices but not the human eyes, and hence will not affect the light sources' illumination functions. In VLC, if a receiver is equipped with photodiodes that count the number of arriving photons, the channels can be modeled as Poisson channels. Unlike Gaussian channels that are suitable for radio spectrum and have been intensively investigated, Poisson channels are more challenging and are not that well understood. The goal of this thesis is to characterize the fundamental limits of various Poisson channels that models different scenarios in VLC. We first focus on single user Poisson fading channels with time-varying background lights. Our model is motivated by indoor optical wireless communication systems, in which the noise level is affected by the strength of the background light. We study both the single-input single-output (SISO) and the multiple-input and multiple-output (MIMO) channels. For each channel, we consider scenarios with and without delay constraints. For the case without a delay constraint, we characterize the optimal power allocation scheme that maximizes the ergodic capacity. For the case with a strict delay constraint, we characterize the optimal power allocation scheme that minimizes the outage probability. We then extend the study to the multi-user Poisson channels and analyze the sum-rate capacity of two-user Poisson multiple access channels (MAC). We first characterize the sum-rate capacity of the non-symmetric Poisson MAC when each transmitter has a single antenna. We show that, for certain channel parameters, it is optimal for a single-user to transmit to achieve the sum-rate capacity. This is in sharp contrast to the Gaussian MAC, in which both users must transmit, either simultaneously or at different times, in order to achieve the sum-rate capacity. We then characterize the sum-rate capacity of the Poisson MAC with multiple antennas at each transmitter. By converting a non-convex optimization problem with a large number of variables into a non-convex optimization problem with two variables, we show that the sum-rate capacity of the Poisson MAC with multiple transmit antennas is equivalent to a properly constructed Poisson MAC with a single antenna at each transmitter. We further analyze the sum-rate capacity of two-user Poisson MIMO multiple-access channels (MAC), when both the transmitters and the receiver are equipped with multiple antennas. We first characterize the sum-rate capacity of the Poisson MAC when each transmitter has a single antenna and the receiver has multiple antennas. We show that similar to Poisson MISO-MAC channels, for certain channel parameters, it is optimal for a single user to transmit to achieve the sum-rate capacity, and for certain channel parameters, it is optimal for both users to transmit. We then characterize the sum-rate capacity of the channel where both the transmitters and the receiver are equipped with multiple antennas. We show that the sum-rate capacity of the Poisson MAC with multiple transmit antennas is equivalent to a properly constructed Poisson MAC with a single antenna at each transmitter.

Tong Kwong-Bun. / Thesis submitted in: December 2002. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 77-[80]). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- An Overview --- p.1 / Chapter 1.2 --- Tree Search --- p.2 / Chapter 1.2.1 --- Minimax Algorithm --- p.2 / Chapter 1.2.2 --- The Alpha-Beta Algorithm --- p.4 / Chapter 1.2.3 --- Alpha-Beta Enhancements --- p.5 / Chapter 1.2.4 --- Selective Search --- p.9 / Chapter 1.3 --- Construction of Evaluation Function --- p.16 / Chapter 1.4 --- Contribution of the Thesis --- p.17 / Chapter 1.5 --- Structure of the Thesis --- p.19 / Chapter 2 --- The Probabilistic Forward Pruning Framework --- p.20 / Chapter 2.1 --- Introduction --- p.20 / Chapter 2.2 --- The Generalized Probabilistic Forward Cuts Heuristic --- p.21 / Chapter 2.3 --- The GPC Framework --- p.24 / Chapter 2.3.1 --- The Alpha-Beta Algorithm --- p.24 / Chapter 2.3.2 --- The NegaScout Algorithm --- p.25 / Chapter 2.3.3 --- The Memory-enhanced Test Algorithm --- p.27 / Chapter 2.4 --- Summary --- p.27 / Chapter 3 --- The Fast Probabilistic Forward Pruning Framework --- p.30 / Chapter 3.1 --- Introduction --- p.30 / Chapter 3.2 --- The Fast GPC Heuristic --- p.30 / Chapter 3.2.1 --- The Alpha-Beta algorithm --- p.32 / Chapter 3.2.2 --- The NegaScout algorithm --- p.32 / Chapter 3.2.3 --- The Memory-enhanced Test algorithm --- p.35 / Chapter 3.3 --- Performance Evaluation --- p.35 / Chapter 3.3.1 --- Determination of the Parameters --- p.35 / Chapter 3.3.2 --- Result of Experiments --- p.38 / Chapter 3.4 --- Summary --- p.42 / Chapter 4 --- The Node-Cutting Heuristic --- p.43 / Chapter 4.1 --- Introduction --- p.43 / Chapter 4.2 --- Move Ordering --- p.43 / Chapter 4.2.1 --- Quality of Move Ordering --- p.44 / Chapter 4.3 --- Node-Cutting Heuristic --- p.46 / Chapter 4.4 --- Performance Evaluation --- p.48 / Chapter 4.4.1 --- Determination of the Parameters --- p.48 / Chapter 4.4.2 --- Result of Experiments --- p.50 / Chapter 4.5 --- Summary --- p.55 / Chapter 5 --- The Integrated Strategy --- p.56 / Chapter 5.1 --- Introduction --- p.56 / Chapter 5.2 --- "Combination of GPC, FGPC and Node-Cutting Heuristic" --- p.56 / Chapter 5.3 --- Performance Evaluation --- p.58 / Chapter 5.4 --- Summary --- p.63 / Chapter 6 --- Conclusions and Future Works --- p.64 / Chapter 6.1 --- Conclusions --- p.64 / Chapter 6.2 --- Future Works --- p.65 / Chapter A --- Examples --- p.67 / Chapter B --- The Rules of Chinese Checkers --- p.73 / Chapter C --- Application to Chinese Checkers --- p.75 / Bibliography --- p.77

An algebra $A$ over a field $\K$ is said to be \textit{invertible} if it has a basis $\B$ consisting only of of units; if $\B^{-1}$ is again a basis, $A$ is \textit{invertible-2}, or \textit{I2}. The question of when an invertible algebra is necessarily I2 arises naturally. The study of these algebras was recently initiated by Lòpez-Permouth, Moore, Szabo, Pilewski \cite{lopezIJM}, \cite{lopez1}. In this work, we prove several positive results on this problem, answering also some questions and generalizing a few results from these papers. We show that every field is an I2 algebra over any subfield, and that any subalgebra of the rational functions field $K(x)$ that strictly contains $K[x]$, with $K$ an algebraically closed field, has a symmetric basis $\B=\B^{-1}$. Using this, we expand the class of examples of algebras known to be invertible or I2 with several classes, such as semiprimary rings over fields $K\neq \F_2$ satisfying some additional mild conditions. We also show that every commutative, finitely generated, invertible algebra without zero divisors is almost I2 in the sense that it becomes I2 after localization at a single element.

Med den kvalitativa metoden design-research har vi i vår studie, likt en bro kombinerat teori och praktik genom att genomföra tre lektioner, i en cyklisk process med en årskurs 3. Vårt syfte var att skapa förståelse för hur en lektionssekvens kan främja SUM-elevers inlärning av rimlighetsbedömning samt lämpligt val av  subtraktionsstrategi, genom inkludering i den ordinarie undervisningen. Lektionssekvensen har bestått av utvalda matematiska centrala moment, vilka skapat design-principer som varit lektionernas utgångspunkt. Dessa principer har prövats, efter analys förändrats och anpassats under studien gång. Vårt resultat visar att begreppsförståelse måste läras in på en konkret nivå, där uppgifter är anpassade efter elevens verklighet i ett känt sammanhang. Korta och tydliga genomgångar, samt modellering hjälper till med begreppsutveckling av uppskattning och rimlighetsbedömning. Studiens slutsats är att SUM-elever utvecklas bäst genom reflektion och resonemang, i par, där kamraten har något högre kunskapsnivå. Reflektion och resonemang ger djupare kunskap och begrepp kan då förstås i ett sammanhang. Läraren har en viktig roll genom att vara SUM-elevers tankestöd samt ställa kontrollfrågor av typen; ”Är det rimligt?” för att uppmana elever att tänka efter om exempelvis strategin eller svaret kan vara möjligt. Representationsformen tallinje har vi sett som ett viktigt redskap för att SUM-elever ska synliggöra sambandet mellan talen som ska subtraheras på ett konkret sätt, för att se om talen är nära och då förstå att strategin ”räkna uppåt” är mest lämplig att använda. De steg som skiljer dessa tal blir då enklare att räkna och eleverna har förutsättning att göra en rimlighetsbedömning.