Department of Mathematicshttp://hdl.handle.net/2104/47732018-06-18T00:26:21Z2018-06-18T00:26:21ZLocal automorphisms of finitary incidence algebras.http://hdl.handle.net/2104/101392018-05-22T15:19:08Z2017-07-07T00:00:00ZLocal automorphisms of finitary incidence algebras.
Let $R$ be a commutative, indecomposable ring with identity and let $(P,\le)$ be a locally finite partially ordered set. Let $FI(P)$ denote the finitary incidence algebra of $(P,\le)$ over $R$. In this case, the finitary incidence algebra exactly coincides with the incidence space. We will give an explicit criterion for when local automorphisms of $FI(P)$ are actually $R$-algebra automorphisms. We will also show that cases which do not meet that criterion may have nonsurjective local automorphisms- in particular, those maps are not $R$-algebra automorphisms, showing a strict inclusion of the collection of $R$-algebra automorphisms inside the collection of local automorphisms. In fact, the existence of local automorphisms which fail to be $R$-algebra automorphisms will depend on the chosen model of set theory and will require the existence of measurable cardinals. We will discuss local automorphisms of cartesian products as a special case in preparation of the general result. Finally, we will explore the automorphisms and local automorphisms of the $End_R(V),$ where $V$ is a free $R$-module and $R$ is no longer necessarily indecomposable.
2017-07-07T00:00:00ZSolving degenerate stochastic Kawarada partial differential equations via adaptive splitting methods.http://hdl.handle.net/2104/100992018-05-23T12:36:36Z2017-07-08T00:00:00ZSolving degenerate stochastic Kawarada partial differential equations via adaptive splitting methods.
In this dissertation, we explore and analyze highly effective and efficient computational
procedures for solving a class of nonlinear and stochastic partial differential equations. We are particularly interested in degenerate Kawarada equations arising from numerous multiphysics applications including solid fuel combustion and oil pipeline decay. These differential equations are characterized by their strong degeneracies on the boundary, quenching blow-up singularities in the spatial domain, and vibrant stochastic influences in the evolution processes. In addition, physically relevant numerical solutions to Kawarada-type equations must preserve their positivity and monotonicity in time throughout computations, for all valid initial datum. In light of these concerns and challenges, the numerical methods and analysis developed in this dissertation incorporate nonuniform finite difference approximations of the underlying singular differential equation on arbitrary spatial grids. These approximations are then advanced in time via proper adaptive mechanisms and splitting procedures, when multiple dimensions are involved.
Degenerate stochastic Kawarada equations in one-, two-, and three-dimensions are explored. The numerical procedures developed in each case are rigorously shown to not only be unconditionally stable, but also preserve the anticipated solution positivity and monotonicity throughout computations. While the numerical stability is proven in the local linearized sense, the stability analysis has been successfully extended to a novel setting that significantly considers all nonlinear influences. Nonlinear convergence of the quenching numerical solutions, as well as their temporal derivatives, are discussed and assessed through generalized Milne devices for the two-dimensional problem. These explorations offer new insights into the computations of similar singular and stochastic differential equation problems. Intensive simulation experiments and validations are provided.
2017-07-08T00:00:00ZGlazman-Krein-Naimark theory, left-definite theory, and the square of the Legendre polynomials differential operator.http://hdl.handle.net/2104/96542016-09-07T17:46:23Z2016-02-27T00:00:00ZGlazman-Krein-Naimark theory, left-definite theory, and the square of the Legendre polynomials differential operator.
As an application of a general left-definite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the left-definite theory associated with the classical Legendre self-adjoint second-order differential operator A in L² (−1, 1) which has the Legendre polynomials {Pn} ∞ n=0 as eigenfunctions. As a consequence, they explicitly determined the domain D(A2) of the self-adjoint operator A2 . However, this domain, in their characterization, does not contain boundary conditions in its formulation. In fact, this is a general feature of the left-definite approach developed by Littlejohn and Wellman. Yet, the square of the second-order Legendre expression is in the limit-4 case at each endpoint x = ±1 in L2 (−1, 1), so D(A2) should exhibit four boundary conditions. In this thesis, we show that this domain can, in fact, be expressed using four separated boundary conditions using the classical GKN (Glazman-Krein-Naimark) theory. In addition, we determine a new characterization of D(A2) that involves four non-GKN boundary conditions. These new boundary conditions are surprisingly simple and natural, and are equivalent to the boundary conditions obtained from the GKN theory.
2016-02-27T00:00:00ZEigenvalue comparison theorems for certain boundary value problems and positive solutions for a fifth order singular boundary value problem.http://hdl.handle.net/2104/96312016-09-09T15:28:59Z2016-03-23T00:00:00ZEigenvalue comparison theorems for certain boundary value problems and positive solutions for a fifth order singular boundary value problem.
Comparison of smallest eigenvalues for certain two point boundary value problems for a fifth order linear differential equation are first obtained. The results are extended to (2n+1)-order and (3n+2)-order boundary value problems. Methods used for these results involve the theory of $u_0$-positive operators with respect to a cone in conjunction with sign properties of Green's functions. Finally, initial results are established for the existence of positive solutions for singular two point boundary value problems for a fifth order nonlinear differential equation. The methods involve application of a fixed point theorem for decreasing operators.
2016-03-23T00:00:00Z