Within the context of rank-based interacting particle systems, we study the fluctuations in a symmetric ensemble around its stable distribution. This system is inspired by the classic Atlas model but represents its opposite pole since both the highest- and lowest-ranked particles will have non-zero drifts. In the first part of the dissertation, we derive a fine asymptotic analysis that includes a Law of Large Numbers. The lack of monotonicity of the ensemble requires that we develop alternative tools to those traditionally used in the analysis of the Atlas model. In the second part of the dissertation, we characterize the system’s fluctuations and show that, as the number of particles goes to infinity, they converge weakly to the mild solution of the Additive Stochastic Heat Equation on the real line with a symmetric initial condition. To establish this result, we use the technique proposed by Dembo and Tsai, 2017, where the Empirical Measure Process is used as a proxy for the ensemble’s fluctuations. We expect that a combination of our work, and the available knowledge about the Atlas model, could help draw a full picture of how a finite rank-based interacting particle system with a general drift structure fluctuates around its stationary distribution as the number of particles goes to infinity, a long-standing open question in the field.

This work concerns two new theorems which count the maximum possible number of independent generators of a certain form which leave an ordinary differential equation of second or third order covariant. Sophus Lic has derived such theorems for a particular class of transformations. The new theorems contain Lic’s theorems as a suboase, and are therefore called ‘generalized” theorems.

We prove the existence and nonexistence of solutions for the semilinear problem ∆u + K(r)f(u) = 0 with various boundary conditions on the exterior of the ball in R^N such that lim r→∞u(r) = 0. Here f : R → R is an odd locally lipschitz non-linear function such that there exists a β > 0 with f < 0 on (0, β), f > 0 on (β, ∞), and K(r) \equiv r^−α for some α > 0.

"The purpose of this paper will be to develop a semi-automatic process for numerical solution of ordinary differential equations, associated commonly with the names of Runge and Kutta, which by its essential features can be characterized as an iterative 'method of successive substitutions'"--Introduction. / Typescript. / "August, 1959." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: H. C. Griffith, Professor Directing Paper. / Includes bibliographical references (leaf 48).

Differential equations, in particular partial differential equations, are used to mathematically describe many physical phenomenon. The importance of being able to solve these types of equations can therefore not be overstated. This thesis is going to elucidate one method, the method of characteristics, which can in some cases be used to solve partial differential equations. To further the reader’s understanding on the method this paper will provide some important insights on differential equations as well as show examples on how the method of characteristics can be used to solve partial differential equations of various complexity. We will also in this paper present some important geometric complications for linear partial differential equations which one might have to take into consideration when using the method.

Stochastic partial differential equations (SPDEs) are a central object of study in the field of stochastic analysis. Their study involves a number of different tools coming from probability theory, functional analysis, harmonic analysis, statistical mechanics, and dynamical systems. Conversely SPDEs are an extremely useful paradigm to study scaling limit phenomena encountered throughout many other areas of mathematics and physics. The present thesis is concerned mainly with one particular SPDE called the Kardar-Parisi-Zhang (KPZ) equation, which appears universally as a fluctuation limit of height profiles of microscopic models such as interacting particle systems, directed polymers, and corner growth models. Such limit results are deemed instances of ``weak KPZ universality," a field born from the seminal paper of Bertini and Giacomin. We extend results on weak KPZ universality in a number of different directions. In one direction, we prove a version of Bertini-Giacomin's result in a half-space by adapting their methods to this setting, thus extending a result of Corwin and Shen and completing the final step towards the proof of a conjecture about fluctuation behavior of half-space KPZ. In another direction, we also prove a result for the free energy for directed polymers in an octant converging to the KPZ equation in a half-space with a nontrivial normalization at the boundary. In a third direction, we return to the whole-space regime and extend the Bertini-Giacomin result to the case of several different initial data coupled together, proving joint convergence of ASEP with its basic coupling to KPZ driven by the same realization of its noise. Finally we prove a ``nonlinear" version of the law of the iterated logarithm for the KPZ equation in a weak-noise but strong-nonlinearity regime. Beyond their intrinsic purpose, one application of all these extensions and generalizations is to take limits of known results and identities for discrete systems and pass them to the limit to obtain nontrivial information about the KPZ equation itself, which is a well-known methodology launched by I. Corwin and coauthors.

In this dissertation, we investigate a very interesting class of quasi-linear stochastic partial differential equations. The main purpose of this article is to prove an existence result for such type of stochastic differential equations with non-standard growth conditions. The main difficulty in the present problem is that the existence cannot be easily retrieved from the well known results under Lipschitz type of growth conditions [42]. / Dissertation (MSc)--University of Pretoria, 2010. / Mathematics and Applied Mathematics / unrestricted

In this thesis, we mainly study preconditioning techniques for all-at-once linear systems arising from discretization of three types of time-dependent advection-diffusion equation: linear diffusion equation, constant-coefficients advection-diffusion equation, time-fractional sub-diffusion equation. The proposed preconditioners are used with Krylov subspace solvers. The preconditioner developed for linear diffusion equation is based on -circulant ap- proximation of temporal discretization. Diagonalizability, clustering of spectrum and identity-plus-low-rank decomposition are derived for the preconditioned matrix. We also show that generalized minimal residual (GMRES) solver for the preconditioned system has a linear convergence rate independent of matrix-size. The preconditioner for constant-coefficients advection-diffusion equation is based on approximating the discretization of advection term with a matrix diagonalizable by sine transform. Eigenvalues of the preconditioned matrix are proven to be lower and upper bounded by positive constants independent of discretization parameters. Moreover, as the preconditioner is based on spatial approximation, it is also applicable to steady-state problem. We show that GMRES for the preconditioned steady-state problem has a linear convergence rate independent of matrix size. The preconditioner for time-fractional sub-diffusion equation is based on approximat- ing the discretization of diffusion term with a matrix diagonalizable by sine transform. We show that the condition number of the preconditioned matrix is bounded by a constant independent of discretization parameters so that the normalized conjugate gradient (NCG) solver for the preconditioned system has a linear convergence rate independent of discretization parameters and matrix size. Fast implementations based on fast Fourier transform (FFT), fast sine transform (FST) or multigrid approximation are proposed for the developed preconditioners. Numerical results are reported to show the performance of the developed preconditioners

In this thesis, we mainly study preconditioning techniques for all-at-once linear systems arising from discretization of three types of time-dependent advection-diffusion equation: linear diffusion equation, constant-coefficients advection-diffusion equation, time-fractional sub-diffusion equation. The proposed preconditioners are used with Krylov subspace solvers. The preconditioner developed for linear diffusion equation is based on -circulant ap- proximation of temporal discretization. Diagonalizability, clustering of spectrum and identity-plus-low-rank decomposition are derived for the preconditioned matrix. We also show that generalized minimal residual (GMRES) solver for the preconditioned system has a linear convergence rate independent of matrix-size. The preconditioner for constant-coefficients advection-diffusion equation is based on approximating the discretization of advection term with a matrix diagonalizable by sine transform. Eigenvalues of the preconditioned matrix are proven to be lower and upper bounded by positive constants independent of discretization parameters. Moreover, as the preconditioner is based on spatial approximation, it is also applicable to steady-state problem. We show that GMRES for the preconditioned steady-state problem has a linear convergence rate independent of matrix size. The preconditioner for time-fractional sub-diffusion equation is based on approximat- ing the discretization of diffusion term with a matrix diagonalizable by sine transform. We show that the condition number of the preconditioned matrix is bounded by a constant independent of discretization parameters so that the normalized conjugate gradient (NCG) solver for the preconditioned system has a linear convergence rate independent of discretization parameters and matrix size. Fast implementations based on fast Fourier transform (FFT), fast sine transform (FST) or multigrid approximation are proposed for the developed preconditioners. Numerical results are reported to show the performance of the developed preconditioners

In my Dissertation, I consider the system of thermoelasticity endowed with poly- convex energy. I will present the equations in their mathematical and physical con- text, and I will explain the relevant research in the area and the contributions of my work. First, I embed the equations of polyconvex thermoviscoelasticity into an aug- mented, symmetrizable, hyperbolic system which possesses a convex entropy. Using the relative entropy method in the extended variables, I show convergence from ther- moviscoelasticity with Newtonian viscosity and Fourier heat conduction to smooth solutions of the system of adiabatic thermoelasticity as both parameters tend to zero and convergence from thermoviscoelasticity to smooth solutions of thermoelasticity in the zero-viscosity limit. In addition, I establish a weak-strong uniqueness result for the equations of adiabatic thermoelasticity in the class of entropy weak solutions. Then, I prove a measure-valued versus strong uniqueness result for adiabatic poly- convex thermoelasticity in a suitable class of measure-valued solutions, de ned by means of generalized Young measures that describe both oscillatory and concentra- tion e ects. Instead of working directly with the extended variables, I will look at the parent system in the original variables utilizing the weak stability properties of certain transport-stretching identities, which allow to carry out the calculations by placing minimal regularity assumptions in the energy framework. Next, I construct a variational scheme for isentropic processes of adiabatic polyconvex thermoelasticity. I establish existence of minimizers which converge to a measure-valued solution that dissipates the total energy. Also, I prove that the scheme converges when the limit- ing solution is smooth. Finally, for completeness and for the reader's convenience, I present the well-established theory for local existence of classical solutions and how it applies to the equations at hand.