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 Gao, Jun.
 2017.
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KolmogrovPetrovskiiPiskunov (KPP) equations are a class of nonlinear parabolic equations which are used to model various biological, ecological, and physical phenomena. In particular it is used as a model for population dynamics. Originally it was studied by Kolmogrov, Petrovskii and Piskunov in 1937. This thesis investigates one type of integrodifferential equation: the nonlocal KPP equation, used in population dynamics. For the nonlocal KPP equation, we prove estimates regarding the front location and in particular introduce logarithmic correction. We also do a nonrigorous analysis to show the profile convergence of solution to the travelling wave.
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3781 2017 G  Inlibrary use 
 Shabani, Beniada.
 2017.
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 Book — 1 online resource.
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Advectionreactiondiffusion equations (ARD) are nonlinear elliptic or parabolic equations that are used to model a variety of systems in natural sciences and engineering, ranging from biological (population dynamics) to chemical and astrophysical systems (reactions in a fluid). They describe the evolution of a normalized quantity, such as a local density of a population or a temperature, as a result of three processes: transport inside the domain (advection), creation or depletion due to an energy source (reaction), and spatial movement (diffusion). These equations exhibit interesting behaviors such as growth/decay, spreading and mixing. Over the past decades there has been significant progress towards understanding and quantifying the behavior of ARD equations. A great focus was shown in equations of biological invasions, more specifically in the phenomenon of spreading, which happens as a result of the invasion of an unstable equilibrium state by a positive stable one. There are two main threads of research in this field: (1) frontlike spreading, which leads to the study of traveling fronts, their speeds of propagation and stability properties, and (2) multidirectional spreading, arising from initial data that have compact support or fast decay in every direction. In this thesis we study the problem of spreading rates for the Cauchy problem in multidimensional periodic FisherKPP equations. Localized initial data give rise to an invasion that will happen typically at different speeds and profiles in each unit direction, but independent on the size and distribution of the original mass. The main result will be on precise asymptotics for the location of level sets of solutions for these data. The trajectory of the thesis will be as follows: In the first part, after a brief introduction of the problem and past results, we study the linearized Dirichlet equation in a half space moving at a constant speed derived from the slowest traveling fronts. This provides intuition for the location of the fronts in the compactly supported case, as well as concrete bounds that can be compared to the solution of the original problem. In the second part we prove the main result by controlling the propagation of the FisherKPP solution using sub and supersolutions constructed from the linearized fronts.
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Online 3. Two models on limit order trading [electronic resource] [2016]
 Ren, Weiluo.
 2016.
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In this thesis we study two limit order trading models based on the one of Avellaneda and Stoikov (2008). For the first one, we study AvellanedaStoikov model under the condition when the underlying price is mean reverting. Our main result is that when time is far from the terminal, the optimal price for bid and ask limit orders is constant, which means that it does not track the underlying price. Numerical simulations confirm this behavior. When the underlying price is mean reverting, then for times sufficiently far from terminal, it is more advantageous to focus on the mean price and ignore fluctuations around it. Mean reversion suggests that limit orders will be executed with some regularity, and this is why they are optimal. We also explore intermediate time regimes where limit order prices are influenced by the inventory of outstanding orders. The duration of this intermediate regime depends on the liquidity of the market as measured by specific parameters in the model. The second model we study is a price impact model where the underlying price is affected by the mean field terms representing existing limit orders in the market. The main result on this model is that there are two different regimes for the representative trader in the solution of MFG: In the beginning, he trades aggressively which results in a large price impact; after a short period of time, he trades, by expectation, approximately symmetric about the underlying price. As a function of time, the price impact decays exponentially in the solution of MFG. Moreover the expectations of optimal prices for the representative trader are constants symmetric about a level close to the terminal value of expected underlying price.
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3781 2016 R  Inlibrary use 
 Tsai, LiCheng.
 2016.
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In this dissertation we consider random growth phenomenon related to the KardarParisiZhang (KPZ) equation. We focus particularly on the weak universality of interacting particle systems. That is, under suitable scalings, a certain class of onedimensional, weakly irreversible interacting particle systems converge to the KPZ equation. Our discussion emphasizes on both the inter and intromodel universality. For the former, we derive the exact microscopic HopfCole transformation for the 4parameter family of Higher Spin Exclusion Processes (HSEPs) introduced by Corwin and Petrov (2016). This is done by exploiting the close relationship between HopfCole transformation and duality. As the HSEPs sit above most of the known integrable models in the KPZ class, we thus obtain the exact microscopic HopfCole transformation for all lowerlevel models. To demonstrate the weak universality, we further consider a particular weak scaling of the HSEPs, and show the convergence to the KPZ equation. This expands the relatively small number of systems for which weak universality of the KPZ equation has been demonstrated. As for intromodel universality, we analyze a class of nonnearestneighbor exclusion processes and the corresponding growth models. Our approach is to exploit an approximate HopfCole transformation, to which end we identify the main nonlinearity and eliminate it by imposing a gradient type condition. For hopping range at most 3, using the generalized transformation, we prove the convergence of the exclusion process toward the KPZ equation.
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Online 5. Mean field games with common noise [electronic resource] [2015]
 Ahuja, Saran.
 2015.
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Mean Field Games (MFG) are a limit of stochastic differential games with a large number of identical players. They were proposed and first studied by Lasry and Lions and independently by Caines, Huang, and Malhame in 2006. They have attracted a lot of interest in the past decades due to their application in many fields. By assuming independence among each agent, taking the limit as N goes to infinity reduces a problem to a fullycoupled system of forwardbackward partial differential equations (PDE). The backward one is a HamiltonJacobiBellman (HJB) equation for the value function of each player while the forward one is the FokkerPlanck (FP) equation for the evolution of the players distribution. This limiting system is more tractable and one can use its solution to approximate the Nash equilibrium strategy of Nplayer games. In this thesis, we consider the MFG model in the presence of common noise, relaxing the usual independence assumption of individual random noise. The presence of common noise clearly adds an extra layer of complexity to the problem as the distribution of players now evolves stochastically. Our first task is proving existence and uniqueness of a Nash equilibrium strategy for this game, showing wellposedness of MFG with common noise. We use a probabilistic approach, namely the Stochastic Maximum Principle (SMP), instead of a PDE approach. This approach gives us a forwardbackward stochastic differential equation (FBSDE) of McKeanVlasov type instead of coupled HJBFP equations. This was first done by Carmona and Delarue in the case of no common noise and we extend their results to MFG with common noise. We are able to extend their results under a linearconvexity framework and a weak monotonicity assumption on the cost functions. In addition to wellposedness results, we also prove the Markov property of McKeanVlasov FBSDE by proving the existence of a decoupling function. In the second part of this thesis, we consider MFG models when the common noise is small. For simplicity, we assume a quadratic running cost function while keeping a general terminal cost function satisfying the same assumptions as in the first part. Our goal is to give an approximation of Nash equilibrium of this game using the solution from the original MFG with no common noise, which could be described through a finitedimensional system of PDEs. We characterize the first order approximation terms as the solution to a linear FBSDE of meanfield type. We then show that the solution to this FBSDE is a centered Gaussian process with respect to the common noise. By assuming regularity of the decoupling function of the 0MFG problem, we can find an explicit solution showing that they are in the form of a stochastic integral with respect to the common noise with the integrands adapted to the information from the 0MFG only. We then are able to compute the covariance function explicitly.
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Online 6. Propagation phenomena in reactionadvectiondiffusion equations [electronic resource] [2015]
 Henderson, Christopher.
 2015.
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 Book — 1 online resource.
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Reactionadvectiondiffusion (RAD) equations are a class of nonlinear parabolic equations which are used to model a diverse range of biological, physical, and chemical phenomena. Originally introduced in the early twentieth century as a model for population dynamics, they have been used in recent years in diverse contexts including climate change, criminal behavior, and combustion. These equations are characterized by the combination of three behaviors: spreading, stirring, and growth/decay. The main focus of mathematical research into RAD equations over the past century has been in characterizing the propagation of solutions. Indeed, these equations are characterized by the invasion of an unstable state by a stable state at a constant rate (for instance, the invasion of empty space by a population until the environmental carrying capacity is reached). In general, this can be characterized by the existence, uniqueness, and stability of traveling wave solutions, or solutions with a fixed profile which move at a constant speed in time. In general, the speed and shape of these traveling waves gives us the speed with which the stable state invades the unstable state. This thesis assumes the following trajectory, investigating two specific RAD equations: the FisherKPP equation, used in population dynamics, and a coupled reactiveBoussinesq system, used to model combustion in a fluid. For the former equation, we prove results regarding the precise spreading rate, and for the latter equation, we prove an existence result for a special solution that generalizes the traveling wave. In the first part of this thesis, we prove two results quantifying the precise speed of spreading for solutions to the Cauchy problem of the FisherKPP equation. The first of these results, concerning localized initial data, provides intuition for a lower order term obtained nonrigorously in. Specifically, we prove a quantitative convergencetoequilibrium result in a related model, which has been used as a close approximation of the FisherKPP equation. The second of these results, concerning nonlocalized initial data and building on the work of Hamel and Roques, quantifies the superlinear in time spreading of the population. Here we compute the highest order term in the spreading for a broad class of initial data. In the second part of this thesis, we look at a coupled system that models combustion in a fluid, and we prove a qualitative propagation result. Unlike classical models, this relatively new system accounts for the effect of advection induced by the buoyancy force that results from the evolution of the temperature. Essentially, this means that we take into account the phenomenon that ``hot air rises.'' We exhibit a generalized traveling wave solution of this system, called a pulsating front, in twodimensional periodic domains. To our knowledge, this is the first result regarding the existence of ``pulsating fronts'' in a coupled system.
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3781 2015 H  Inlibrary use 
Online 7. Deviation inequalities for eigenvalues of deformed random matrices [electronic resource] [2013]
 Peng, Minyu.
 2013.
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We consider two models of deformed random matrices. The first model is the deformed GOE. Let G be a n by n matrix in the Gaussian Orthogonal Ensemble (GOE) with off diagonal entries having variance $\frac{\sigma^2}{n}$ and let P be a n by n real symmetric matrix whose non zero eigenvalues are $\theta_1 \geq \cdots \geq \theta_r > 0$. We establish nonasymptotic deviation inequalities for the extreme eigenvalues of A = P + G of the following type: $P(\lambda_i (A)  \lambda_{\theta_i}[verticle line] \geq t) \leq C_1 n^{ri+1} e^{C_2 nt^2 / \sigma^2}, 1\leq i \leq r$, where $\lambda_{\theta_i} = \theta_i + \frac{\sigma^2}{\theta_i}$ or $2\sigma$, depending on $\theta_i > \sigma$ or $0 < \theta_i \leq\sigma$, with $C_1, C_2$ positive constants independent of n, r and t. The second model studied is the spiked population model, which is sometimes called Laguerre Orthogonal Ensemble (LOE). We establish similar deviation inequalities for extreme eigenvalues of matrices in this model. Unlike classical approach in the study of extreme eigenvalues of large dimensional random matrices, which relies on the moment method and the eigenvalue density formulae, our approach is based on the minimax characterization of eigenvalues and comparison inequalities for Gaussian processes.
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Online 8. Sparse and lowrank structures in robust principal component analysis, compressed sensing with corruptions, and phase retrieval [electronic resource] [2013]
 Li, Xiaodong, 1985
 2013.
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In this dissertation, we discuss three related problems: robust principal component analysis, compressed sensing with corruptions, and signal recovery from quadratic measurements. Robust principal component analysis (RPCA) is a computational framework that aims to decompose a matrix as the sum of a lowrank component and a sparse component. We prove that under mild technical assumptions, a tractable convex optimization works for an exact decomposition. In the case in which there are missing entries, we propose another convex optimization and show that a lowrank matrix can be exactly recovered from a small portion of entries with a constant proportion of corruptions. Further, this dissertation improves some existing results in compressed sensing with grossly corrupted measurements in the literature. For both the Gaussian and nonGaussian sensing matrices, we derive theorems showing that by weighted L1 minimization, a sparse signal can be recovered from a few linear observations with constant proportion of corruptions. Finally, we discuss the problem of signal recovery from intensity measurements. For general signal recovery, we improve existing results in the literature, showing that the signal can be recovered exactly up to a global phase factor provided the number of measurements is in the order of the dimension of the signal. For sparse signal recovery, we derive both necessary and sufficient conditions for the number of Gaussian measurements, such that a natural convex relaxation works for the recovery of the signal.
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3781 2013 L  Inlibrary use 
 Shkolnikov, Mykhaylo.
 2011, c2012.
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The following dissertation studies several classes of competing particle systems, which are of importance in statistical physics and mathematical finance. Two main examples are systems which go by the names rankbased market models and volatilitystabilized market models in stochastic portfolio theory. Motivated by the applications, we analyze the behavior of the ordered particles, the shape of the invariant distributions for the processes of spacings between consecutive particles, the global behavior of the particle systems as the number of particles becomes large, as well as the corresponding systems of infinitely many particles.
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Online 10. Filtering and parameter estimation for partially observed generalized Hawkes processes [electronic resource] [2011]
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We consider the nonlinear filtering problem for partially observed Generalized Hawkes Processes, which can be applied in the context of portfolio credit risk. The problem belongs to the larger class of hidden Markov models, where the counting process is observed at discrete points in time and the observations are sparse, while the intensity driving process in unobservable. We construct the conditional distribution of the process given the information filtration and we discuss the analytical and numerical properties of the corresponding filters. In particular, we study the sensitivity of the filters with respect to the parameters of the model, and we obtain a monotonicity result with respect to the jump and the volatility terms driving the intensity. Using the scaled process, we provide necessary and sufficient conditions for the frequency of time observations in terms of the parameters of the model, to ensure a good performance of the filter. We also address the problem of parameter estimation for the Generalized Hawkes Process in the framework of the EM algorithm, and we analyze the effect of the selfexciting feature of our process on the asymptotic and numerical properties of the estimators.
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Online 11. Topology of spaces of microimages, and an application to texture discrimination. [electronic resource] [2011]
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Using the fact that most 3by3 pixel highcontrast patches from natural images accumulate around a space with the topology of a Klein bottle, we present in this thesis a novel method for texture representation and discrimination. Given a texture image, most of its highcontrast patches can be projected onto the aforementioned Klein bottle. We analyze this sample in terms of its underlying probability density function, which we show can be represented via Fourierlike coefficients. These coefficients in turn, can be estimated with high confidence from the sample. Dissimilarity measures are defined on the set of estimated coefficients, and performance of the method is tested on a large collection of texture images.
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