This work introduces a novel inversion-based neurocontroller for solving control problems involving uncertain nonlinear systems which could also compensate for multi-valued systems. The approach uses recent developments in neural networks, especially in the context of modelling statistical distributions, which are applied to forward and inverse plant models. Provided that certain conditions are met, an estimate of the intrinsic uncertainty for the outputs of neural networks can be obtained using the statistical properties of networks. More generally, multicomponent distributions can be modelled by the mixture density network. Based on importance sampling from these distributions a novel robust inverse control approach is obtained. This importance sampling provides a structured and principled approach to constrain the complexity of the search space for the ideal control law. The developed methodology circumvents the dynamic programming problem by using the predicted neural network uncertainty to localise the possible control solutions to consider. Convergence of the output error for the proposed control method is verified by using a Lyapunov function. Several simulation examples are provided to demonstrate the efficiency of the developed control method. The manner in which such a method is extended to nonlinear multi-variable systems with different delays between the input-output pairs is considered and demonstrated through simulation examples.

This dissertation is a summary of the graduate study in the past few years. In first part, we develop efficient spectral methods for the spectral fractional Laplacian equation and parabolic PDEs with spectral fractional Laplacian on rectangular domains. The key idea is to construct eigenfunctions of discrete Laplacian (also referred to Fourier-like basis) by using the Fourierization method. Under this basis, the nonlocal fractional Laplacian operator can be trivially evaluated, leading to very efficient algorithms for PDEs involving spectral fractional Laplacian. We provide a rigorous error analysis for the proposed methods, as well as ample numerical results to show their effectiveness.<div><br>In second part, we propose a method suitable for the computation of quasiperiodic interface, and apply it to simulate the interface between ordered phases in Lifschitz-Petrich model, which can be quasiperiodic. The function space, initial and boundary conditions are carefully chosen such that it fix the relative orientation and displacement, and we follow a gradient flow to let the interface and its optimal structure. The gradient flow is discretized by the scalar auxiliary variable (SAV) approach in time, and spectral method in space using quasiperiodic Fourier series and generalized Jacobi<br>polynomials. We use the method to study interface between striped, hexagonal and dodecagonal phases, especially when the interface is quasiperiodic. The numerical examples show that our method is efficient and accurate to successfully capture the interfacial structure.</div>

<div>In the first part of the present work, we will study the harmonic maps onto Teichm\"uller space. We will formulate a general Bochner type formula for harmonic maps into Teichm\"uller space. We will also prove the existence theorem of equivariant harmonic maps from a symmetric space with finite volume into its Weil-Petersson completion $\overline{\mathcal{T}}$, by deforming an almost finite energy map in the sense of Saper into a finite energy map.</div><div><br></div><div>In the second part of the work, we discuss the superrigidity of mapping class group. We will provide a geometric proof of both the high rank and the rank one superrigidity of mapping class groups due to Farb-Masur and Yeung. </div>

<div>Let <i>K₀</i>(Var<i>_k</i>) be the Grothendieck group of varieties over a field <i>k</i>. We construct an exact category, denoted Add(Var_<i>k</i>)_<i>S</i>, such that there is a surjection <i>K₀</i>(Var<i>k</i>)→<i>K₀</i>(Add(Var<i>_k</i>)_<i>S</i>).If we consider only zero dimensional varieties, then this surjection is an isomorphism. Like <i>K₀</i>(Var<i>_k</i>), the group K_<i>0</i>(Add(Var_<i>k</i>)<i>_S</i>) is also generated by isomorphism classes of varieties,and we construct motivic measures on <i>K₀</i>(Add(Var<i>_k</i>)<i>_S</i>) including the Euler characteristic if <i>k</i>=<i>C</i>, and point counting measures and the zeta function if <i>k</i> is finite.<br></div>

<p>This thesis is concerned with the problem of characterizing the orbits of certain probability density functions under the action of the Euclidean group. Our motivating application is the recognition of a point configuration where the coordinates of the points are measured under noisy conditions. Consider a random variable X in R^d with probability density function ρ(x). Let x₁ and x₂ be independent random samples following ρ(x). Define ∆ as the squared Euclidean distance between x₁ and x₂. It has previously been shown that two distributions ρ(x) and ρ(x) consisting of Dirac delta distributions in generic positions that have the same respective distributions of ∆ are necessarily related by a rigid motion. That is, there exists some rigid motion g in the Euclidean group E(d) such that ρ(x) = ρ(g · x) for all x ∈ R^d . To account for noise in the measurements, we assume X is a random variable in R^d whose density is a k-component mixture of Gaussian distributions with means in generic position. We further assume that the covariance matrices of the Gaussian components are equal and of the form Σ = σ²1_d with  0 ≤ σ² ∈ R. In Theorem 3.1.1 and Theorem 3.2.1, we prove that, when σ² is known, generic k-component Gaussian mixtures are uniquely reconstructible up to a rigid motion from the density of ∆. A more general formulation is proven in Theorem 3.2.3. Similarly, when σ² is unknown, we prove in Theorem 4.1.1 and Theorem 4.1.2 that generic equally-weighted k-component Gaussian mixtures with k = 1 and k = 2 are uniquely reconstructible up to a rigid motion from the distribution of ∆. There are at most three non-equivalent equally weighted 3-component Gaussian mixtures up to a rigid motion having the same distribution of ∆, as proven in Theorem 4.1.3. In Theorem 4.1.4, we present a test to check if, for a given k and d, the number of non-equivalent equally-weighted k-component Gaussian mixtures in R^d having the same distribution of ∆ is at most (k choose 2) + 1. Numerical computations showed that distributions with k = 4, 5, 6, 7 such that d ≤ k −2 and (k, d) = (8, 1) pass the test, and thus have a finite number of reconstructions up to a rigid motion. When σ² is unknown and the mixture weights are also unknown, we prove in Theorem 4.2.1 that there are at most four non-equivalent 2-component Gaussian mixtures up to a rigid motion having the same distribution of ∆. </p>

<p>We consider a novel setting for local harmonic analysis on reductive groups motivated by Langlands functoriality conjecture. To this end, we characterize certain non-linear Schwartz spaces on tori and reductive groups in spectral terms, and develop some of their structure in the unramified case, and we derive estimates of their moderate growth at infinity. We also consider non-linear Fourier transforms, and calculate their action on tame supercuspidal representations of $GL_2(F)$ in terms of inducing cuspidal data.</p>

This dissertation consists of three research projects of kinetic models: a structure preserving scheme for Poisson-Nernst-Planck equations and two efficient spectral methods for multi-dimensional Boltzmann equation.<br><br>The Poisson-Nernst-Planck (PNP) equations is widely used to describe the dynamics of ion transport in ion channels. We introduce a structure-preserving semi-implicit finite difference scheme for the PNP equations in a bounded domain. A general boundary condition for the Poisson equation is considered. The fully discrete scheme is shown to satisfy the following properties: mass conservation, unconditional positivity, and energy dissipation (hence preserving the steady-state). <br><br>Numerical approximation of the Boltzmann equation presents a challenging problem due to its high-dimensional, nonlinear, and nonlocal collision operator. Among the deterministic methods, the Fourier-Galerkin spectral method stands out for its relative high accuracy and possibility of being accelerated by the fast Fourier transform. In this dissertation, we studied the state of the art in the fast Fourier method and discussed its limitation. Next, we proposed a new approach to implement the Fourier method, which can resolve those issues. <br><br>However, the Fourier method requires a domain truncation which is unphysical since the collision operator is defined in whole space R^d . In the last part of this dissertation, we introduce a Petrov-Galerkin spectral method for the Boltzmann equation in the unbounded domain. The basis functions (both test and trial functions) are carefully chosen mapped Chebyshev functions to obtain desired convergence and conservation properties. Furthermore, thanks to the close relationship of the Chebyshev functions and the Fourier cosine series, we can construct a fast algorithm with the help of the non-uniform fast Fourier transform (NUFFT).<br>

<p> In this thesis, both direct and inverse elastic scattering problems are considered. For a given incident wave, the direct problem is to determine the displacement of wave field from the known structure, which could be an obstacle or a surface in this thesis; The inverse problem is to determine the structure from the measurement of displacement on an artificial boundary.</p><p>In the second chapter, we consider the scattering of an elastic plane wave by a rigid obstacle, which is immersed in a homogeneous and isotropic elastic medium in two dimensions. Based on a Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the scattering problem is formulated as a boundary value problem of the elastic wave equation in a bounded domain. By developing a new duality argument, an a posteriori error estimate is derived for the discrete problem by using the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator which decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is proposed to solve the elastic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are chosen through the finite element discretization error.<br></p><p>In chapter 3, we extend the argument developed in chapter 2 to elastic surface grating problem, where the surface is assumed to be periodic and elastic rigid; Then, we treat the obstacle scattering in three dimensional space; The direct problem is shown to have a unique weak solution by examining its variational formulation. The domain derivative is studied and a frequency continuation method is developed for the inverse problem. Finally, in chapter 4, a rigorous mathematical model and an efficient computational method are proposed to solve the inverse elastic surface scattering problem which arises from the near-field imaging of periodic structures. The surface is assumed to be a small and smooth perturbation of an elastically rigid plane. By placing a rectangle slab of a homogeneous and isotropic elastic medium with larger mass density above the surface, more propagating wave modes can be utilized from the far-field data which contributes to the reconstruction resolution. Requiring only a single illumination, the method begins with the far-to-near field data conversion and utilized the transformed field expansion to derive an analytic solution for the direct problem, which leads to an explicit inversion formula for the inverse problem; Moreover, a nonlinear correction scheme is developed to improve the accuracy of the reconstruction; Numerical examples are presented to demonstrate the effectiveness of the proposed methods for solving the questions mentioned above.<br></p>

n previous work, the author and his collaborators developed a relationship in the SL(2,C) representation theories of two finitely generated groups with isomorphicprofinite completions assuming a certain strong representation rigidity for one of thegroups. This was then exploited as one part of producing examples of lattices in SL(2,C) which are profinitely rigid. In this article, the relationship is extended to representations in any connected reductive algebraic groups under a weaker representation rigidity hypothesis. The results are applied to lattices in higher rank Liegroups where we show that for some such groups, including SL(n,Z) forn≥3, they are either profinitely rigid, or they contain a proper Grothendieck subgroup.

Picture the life of a modern person in the western world: They wake up in the morning and check their social networking sites; they drive to work on roads that connect cities to each other; they make phone calls, send emails and messages to colleagues, friends, and family around the world; they use electricity flowing through power-lines; they browse the Internet, searching for information. All of these typical daily activities rely on the structure of networks. A network, in this case, is a set of nodes (people, web pages, etc) connected by edges (physical connection, collaboration, etc). The term graph is sometimes used to represent a more abstract structure - but here we use the terms graph and network interchangeably. The field of network analysis concerns studying and understanding networks in order to solve problems in the world around us. Graph models are used in conjunction with the study of real-world networks. They are used to study how well an algorithm may do on a real-world network, and for testing properties that may further produce faster algorithms. The first piece of this dissertation is an experimental study which explores features of real data, specifically power-law distributions in degrees and spectra. In addition to a comparison between features of real data to existing results in the literature, this study resulted in a hypothesis on power-law structure in spectra of real-world networks being more reliable than that in the degrees. The theoretical contributions of this dissertation are focused primarily on generating realistic networks through existing and novel graph models. The two graph models presented are called HyperKron and the Triangle Generalized Preferential Attachment model. Both of the models incorporate higher-order structure - leading to more sophisticated properties not examined in traditional models. We use the second of our models to further validate the hypothesis on power-laws in the spectra. Due to the structure of our model, we show that the power-law in the spectra is more resilient to sub-sampling. This gives some explanation for why we see power-laws more frequently in the spectra in real world data.