A new approach to Kneser's Theorem, which achieves a simplification of the analysis through the introduction of maximal sets, the basic sequence of maximal e-transformations, and the limit set, B*, is presented. For two sets of non-negative integers, A and B, with C∈A⋂B, the maximal sets, Aᴹ and Bᴹ, are the largest supersets of A and B, respectively, such that Aᴹ + Bᴹ = A + B. By shifting from A and B to Aᴹ and Bᴹ to initiate the analysis, the maximal properties of Aᴹ and Bᴹ are exploited to simplify the analysis. A maximal e-transformation is a Kneser e-transformation in which the image sets are maximized in order to preserve the properties of maximal sets. The basic sequence of maximal e-transformation is a specific sequence of maximal a-transformations which is exclusively used throughout the analysis. B* is the set of all non-negative elements of sM which are not deleted by any transformation in the basic sequence of maximal e-transformations. Whether or not B* = {O} divides the analysis into two cases. One significant result is that B* = {O} implies δ (A + B) = δ (A, B) where δ(A + B) is asymptotic density of A + B and δ (A, B) is the two-fold asymptotic density of A and B. The second major result describes the structure of A + B when δ(A + B) < δ(A, B). With B* ≠ {0} it is shown, using only elementary properties of greatest common divisor and residue classes, that there exists C⊆ A+ B, 0εC, such that δ(C) ≥ δ(A, B) -1/g where g is the greatest common divisor of B* and C is asymptotically equal to C^(g), the union of all residue classes, mod g, which have a representative in C. The existence of C provides the crucial step in obtaining an equivalent form of Kneser’s Theorem: If A and B are two subsets of non-negative integer, 0εA⋂B, and δ(A + B) < δ(A, B), then there exists a positive integer g such that A + B is asymptotically equal to (A + B)^(g) and δ(A + B) = δ ((A + B)^(g)) ≥ δ (A^(g) , B^(g)) - 1/g ≥ δ(A, B) -1/g. / Ph. D.

The behavior of fishes in and around power plant discharges has not been adequately explored despite the recent proliferation of electric generating stations. This study investigated the response of fish to heated and/or chlorinated waters in a field laboratory. Carp (Cyprinus carpio), channel catfish (Ictalurus punctatus) and golden shiners (Notemigonus crysoleucas) were tested at 6, 12, 18, 24 and 30-C acclimation temperatures in a steep concentration gradient. Residence time in the treated water was determined to be the most simple and sensitive index of avoidance behavior. TRC, CRC, and FRC avoidance thresholds varied between species and acclimation temperatures. HOCl avoidance thresholds were consistent between acclimation temperatures, suggesting that HOCl is the primary constituent of the TRC controlling avoidance behavior. Trials were conducted at 12 and 24 C in which the treated water was heated to a preferred temperature, derived in a shallow horizontal temperature gradient. At low chlorine doses fish significantly preferred the warmer water, whereas at high chlorine doses, the fish were repelled by the treated water. The TRC, CRC, and FRC avoidance thresholds varied between fish species, but the HOCl avoidance thresholds were similar for all species at 24 C. The avoidance thresholds for the carp and golden shiners were well below 96hrLC₅₀ values, signifying that they can avoid dangerous solutions before they are adversely affected. The channel catfish avoidance thresholds are slightly greater than reported 96hrLC₅₀ values, indicating a potential for mortalities in power plant discharges. This mortality has not been observed in the field and alternate explanations are discussed. Present interim criteria for chlorinated discharges may not adequately protect all warmwater fish. This study indicates that HOCl may greatly influence fish behavior and suggests that HOCl criteria be adopted for use in conjunction with current total residual chlorine criteria. / Master of Science

Observational surveys which probe our universe deeper and deeper into the nonlinear regime of structure formation are becoming increasing accurate. This makes numerical simulations an essential tool for theory to be able to predict phenomena at comparable scales. In the first part of this thesis we study the behaviour of cosmological models involving a scalar field. We are particularly interested in the existence of fixed points of the dynamical system and the behaviour of the system in their vicinity. Upon addition of spatial curvature to the single-scalar field model with an exponential potential, canonical kinetic term, and a matter fluid, we demonstrate the existence of two extra fixed points that are not present in the case without curvature. We also analyse the evolution of the equation-of-state parameter. In the second part, we numerically simulate collisionless particles in the weak field approximation to General Relativity, with large gradients of the fields and relativistic velocities allowed. To reduce the complexity of the problem and enable high resolution simulations, we consider the spherically symmetric case. Comparing numerical solutions to the exact Schwarzschild and Lemaître-Tolman-Bondi solutions, we show that the scheme we use is more accurate than a Newtonian scheme, correctly reproducing the leading-order post-Newtonian behaviour. Furthermore, by introducing angular momentum, configurations corresponding to bound objects are found. In the final part, we simulate the conditions under which one would expect to form ultracompact minihalos, dark matter halos with a steep power-law profile. We show that an isolated object exhibits the profile predicted analytically. Embedding this halo in a perturbed environment we show that its profile becomes progressively more similar to the Navarro-Frenk-White profile with increasing amplitude of perturbations. Next, we boost the power spectrum at a very early redshift during radiation domination on a chosen scale and simulate clustering of dark matter particles at this scale until low redshift. In this scenario halos form earlier, have higher central densities, and are more compact.

In this work we explore three-point statistics applied to the large-scale structure in our Universe. Three-point statistics, such as the bispectrum, encode information not accessible via the standard analysis method-the power spectrum-and thus provide the potential for greatly improving current constraints on cosmological parameters. They also present us with additional challenges, and we focus on two of these arising from a measurement as well as modelling point of view. The first challenge we address is the covariance matrix of the bispectrum, as its precise estimate is required when performing likelihood analyses. Covariance matrices are usually estimated from a set of independent simulations, whose minimum number scales with the dimension of the covariance matrix. Because there are many more possibilities of finding triplets of galaxies than pairs, compared to the power spectrum this approach becomes rather prohibitive. With this motivation in mind, we explore a novel alternative to the bispectrum: the line correlation function (LCF). It specifically targets information in the phases of density modes that are invisible to the power spectrum, making it a potentially more efficient probe than the bispectrum, which measures a combination of amplitudes and phases. We derive the covariance properties and the impact of shot noise for the LCF and compare these theoretical predictions with measurements from N-body simulations. Based on a Fisher analysis we assess the LCF's sensitivity on cosmological parameters, finding that it is particularly suited for constraining galaxy bias parameters and the amplitude of fluctuations. As a next step we contrast the Fisher information of the LCF with the full bispectrum and two other recently proposed alternatives. We show that the LCF is unlikely to achieve a lossless compression of the bispectrum information, whereas a modal decomposition of the bispectrumcan reduce the size of the covariancematrix by at least an order of magnitude. The second challenge we consider in this work concerns the relation between the dark matter field and luminous tracers, such as galaxies. Accurate knowledge of this galaxy bias relation is required in order to reliably interpret the data gathered by galaxy surveys. On the largest scales the dark matter and galaxy densities are linearly related, but a variety of additional terms need to be taken into account when studying clustering on smaller scales. These have been fully included in recent power spectrumanalyses, whereas the bispectrummodel relied on simple prescriptions that were likely extended beyond their realm of validity. In addition, treating power spectrumand bispectrum on different footings means that the two models become inconsistent on small scales. We introduce a new formalism that allows us to elegantly compute the lacking bispectrum contributions from galaxy bias, without running into the renormalization problem. Furthermore, we fit our new model to simulated data by implementing these contributions into a likelihood code. We show that they are crucial in order to obtain results consistent with those fromthe power spectrum, and that the bispectrum retains its capability of significantly reducing uncertainties in measured parameters when combined with the power spectrum.

Laser techniques, such as gain / Q switching, mode-locking, have successfully overcome the energy restriction of gain clamping in the stead-state operated lasers, and allowed the generation of giant pulses with short pulse durations. However, gain saturation further limits the amount of stored energy in a gain medium, and therefore limits the possible maximum pulse energy obtained by laser techniques. Here we circumvent both gain clamping and the capacity limitation of energy storage by operating the double-quantum-well laser diode chips on ultrafast gain-switching model using femtosecond (fs) laser pulses as the optical pump. The advantage of our pumping approach is that the fs pulse can instantly produce a very large number of carriers, and therefore enable the formation of non-equilibrium coherent e-h BCS-like condensate state in a large energy region from the lowest QW subband edges to the highest subband and then obtain the ultra-broadband superrandiant pulses. / Superradiance (SR) or the coherent spontaneous emission is not a new quantum optics phenomenon, which has been proposed in 1954 by R. Dicke, even earlier than the invention of laser. It is famous as by its ultrashort duration, high peak power, high coherence and high timing jitter. Recently, femtosecond SR pulses have been generated from semiconductors. This investigation has revived both theoretical and experimental studies of SR emission. / In this thesis, we have demonstrated the generation of intense, delayed SR pulses from the InP based double quantum well laser diode at room temperature. The 1040 nm femtosecond laser was applied as the optical pumping source, and when the pump power is high enough, the cooperative recombination of e-h pairs from higher order quantum energy levels can occur to generate SR bursts earlier than the cooperative emission from the lower quantum energy levels. Then, ultra-broadband TM polarized SR pulses have been firstly generated at room temperature. Our experiments also provide a well prospect of ultra-high energy light pulse generation based on SR, besides, the ultra-broadband spectrum is promising for applications in a diverse range of fields, including optical coherent tomography and spectroscopy. / Graphene, a truly 2D material, has stimulated a vast amount of research in recent years. In our work, we have wet transferred the CVD grown monolayer graphene onto the top of our LD chips. With the combination of graphene on top of QW LD, the evanescent field of TM polarized modes can well interact with top graphene layer, and therefore, produce a dramatically modulation of the output power, and optical spectra of output pulses. The graphene-on-DQW LD will be promising to explore the novel devices, such as optical modulator, which will greatly promote the applications of SR emission in near future. / 激光技術，諸如增益 / Q調製，鎖模，已經成功克服了穩態激光器存在的增益箝制現象，從而產生了脈寬窄，峰值功率高的巨脈衝。然而，增益飽和現象又進一步限制了增益介質的儲能上限，因而設定了激光技術產生的脈衝的能量上限。在這裡，我們以飛秒激光為光泵,使量子阱激光器工作在超快增益調製模式, 從而有效地規避了增益箝制和儲能限制。这种泵浦方式的优势在于可以瞬间产生极大量的载流子，從而可以在一个很宽的能带区间内（从量子阱的最低能级到最高能级）形成非平衡相干类BCS状态的电子空穴对的凝聚态，从而产生宽频超辐射。 / 超輻射（SR），或稱為相干的自發輻射並非是一個新的量子光學現象，早在1954年就被由R. Dicke提出了，比發明激光器還早。SR以它脈衝時間短，峰值功率高，相干性好，以及脈衝定時抖動高等特點而聞名。近來，不同的研究組報導了他們用半導體材料產生了SR, 復興了理論和實驗雙方面對SR的研究。 / 在這篇論文中，我們論證了如何用飛秒激光泵浦雙量子阱激光二極管產生高功率帶有延時的超辐射脈衝。當泵功率足夠高時，高量子能級的電子空穴對先於低能級的電子空穴對發生超輻射，從而第一次在室溫下得到了超寬頻譜的超輻射脈衝。此外，我們證明了超輻射光是TM偏振的。根據我們的結果可以預測關於SR的超高能量脈衝的研究以及相关应用具有光明的發展前景，比如，SR可用於包括光學相干斷層成像和光譜學等的各種各樣的先進領域。 / 石墨烯，真正的2D材料，近年來激發了巨大的研究熱潮。我們用濕法轉移的方法將化學氣相沉積生長的單層石墨烯鋪到我們的樣品表面。這使TM偏振模式的漸逝場能很好地與石墨烯層作用，因此，顯著調製SR脈衝的功率和頻譜。石墨烯與多量子阱激光二極管的結合將大大推動了新型器件的研究，如光調製器等，這將大大改善SR的應用前景。 / Liu, Jingjing. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2015. / Includes bibliographical references. / Abstracts also in Chinese. / Title from PDF title page (viewed on 21, December, 2016). / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

This thesis studies two types of problems in financial derivatives pricing. The first type is the free boundary problem, which can be formulated as a partial differential equation (PDE) subject to a set of free boundary condition. Although the functional form of the free boundary condition is given explicitly, the location of the free boundary is unknown and can only be determined implicitly by imposing continuity conditions on the solution. Two specific problems are studied in details, namely the valuation of fixed-rate mortgages and CEV American options. The second type is the multi-dimensional problem, which involves multiple correlated stochastic variables and their governing PDE. One typical problem we focus on is the valuation of basket-spread options, whose underlying asset prices are driven by correlated geometric Brownian motions (GBMs). Analytic approximate solutions are derived for each of these three problems. / For each of the two free boundary problems, we propose a parametric moving boundary to approximate the unknown free boundary, so that the original problem transforms into a moving boundary problem which can be solved analytically. The governing parameter of the moving boundary is determined by imposing the first derivative continuity condition on the solution. The analytic form of the solution allows the price and the hedging parameters to be computed very efficiently. When compared against the benchmark finite-difference method, the computational time is significantly reduced without compromising the accuracy. The multi-stage scheme further allows the approximate results to systematically converge to the benchmark results as one recasts the moving boundary into a piecewise smooth continuous function. / For the multi-dimensional problem, we generalize the Kirk (1995) approximate two-asset spread option formula to the case of multi-asset basket-spread option. Since the final formula is in closed form, all the hedging parameters can also be derived in closed form. Numerical examples demonstrate that the pricing and hedging errors are in general less than 1% relative to the benchmark prices obtained by numerical integration or Monte Carlo simulation. By exploiting an explicit relationship between the option price and the underlying probability distribution, we further derive an approximate distribution function for the general basket-spread variable. It can be used to approximate the transition probability distribution of any linear combination of correlated GBMs. Finally, an implicit perturbation is applied to reduce the pricing errors by factors of up to 100. When compared against the existing methods, the basket-spread option formula coupled with the implicit perturbation turns out to be one of the most robust and accurate approximation methods. / 本論文為金融衍生產品定價的兩類問題作出了研究。第一類是自由邊界問題，它可以制定一個受制於自由邊界條件的偏微分方程式(PDE)，雖然當中自由邊界條件的函數形式是已知的，但自由邊界的位置是未知的，只能通過為實際解施加連續性條件作隱式確定。這裡為兩個具體問題進行了研究，分別是固定利率按揭合約(fixed-rate mortgages)定價和方差恆彈性模型的美式期權(CEV American options)定價。第二類是多維問題，它涉及到多個相關隨機變量及他們引申出的多維PDE。這裡為一個典型例子進行了研究，稱為籃子差異期權(basket-spread options)，其基礎資產價格由相關的幾何布朗運動驅動。我們為這三個問題提出了解析近似解。 / 對於上述的自由邊界問題，我們提出了一項參數移動邊界來近似模仿未知的自由邊界，使原來的自由邊界問題轉化為移動邊界問題，從而提出一種解析近似解。控制移動邊界的參數是通過滿足近似解的一階導數連續性條件來定。得到了解析近似解令當中的衍生產品定價和避險參數能有效快速地計算出，相比於有限差分法(finite-difference method)，精度保持了但計算時間顯著降低。再透過應用一個多階段方案，將移動邊界重鑄成一項分段光滑的連續函數，能有系統地將近似解的結果逼近有限差分法的結果。 / 對於上述的多維問題，我們從Kirk(1995)的二維差異期權(spread option)近似解定價公式推廣到多維的籃子差異期權。由於最終的定價公式是封閉形式，所有避險參數也從而得到封閉式近似解。從一些模擬例子顯示出，近似解的定價和避險參數，與通過數值積分法(numerical integration)或蒙地卡羅模擬法(Monte Carlo simulation)獲得的基準值比較，只有小於百分之一的誤差。此外，透過利用一種期權價格和相關基礎變量的概率分佈關係，我們進一步推論出一項籃子差異變量的近似解分佈函數，這可應用到任何多維幾何布朗運動的線性組合變量分佈。最後，我們提出一種隱式攝動方法，把定價誤差減少高達一百倍，跟現有的近似解定價方法相比，這是其中一種最健全和準確的籃子差異期權定價方法。 / Lau, Chun Sing = 自由邊界和多維的金融衍生產品定價問題 : 解析近似解 / 劉振聲. / Thesis Ph.D. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 174-186). / Abstracts also in Chinese. / Title from PDF title page (viewed on 12, September, 2016). / Lau, Chun Sing = Zi you bian jie he duo wei de jin rong yan sheng chan pin ding jia wen ti : jie xi jin si jie / Liu Zhensheng. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.