Department of Mathematics

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A multigrid Krylov method for eigenvalue problems.
(2015-07-31) Yang, Zhao, 1983-; Morgan, Ronald Benjamin, 1958-; Lenells, Jonatan, 1981-
We are interested in computing eigenvalues and eigenvectors of matrices derived from differential equations. They are often large sparse matrices, including both symmetric and non symmetric cases. Restarted Arnoldi methods are iterative methods for eigenvalue problems based on Krylov subspaces. Multigrid methods solve differential equations by taking advantage of the hierarchy of discretizations. A multigrid Krylov method is proposed by combining Arnoldi and multigrid methods. We compare the new approach with other methods, and explore the theory to explain its efficiency.
Adaptive methods for the Helmholtz equation with discontinuous coefficients at an interface.
(2008-03-03T17:35:23Z) Rogers, James W., Jr.; Sheng, Qin (Tim); Mathematics.; Baylor University. Dept. of Mathematics.
In this dissertation, we explore highly efficient and accurate finite difference methods for the numerical solution of variable coefficient partial differential equations arising in electromagnetic wave applications. We are particularly interested in the Helmholtz equation due to its importance in laser beam propagation simulations. The single lens environments we consider involve physical domains with subregions of differing indices of refraction. Coefficient values possess jump discontinuities at the interface between subregions. We construct novel numerical solution methods that avoid computational instability and maintain high accuracy near the interface. The first class of difference methods developed transforms the differential equation problem to a new boundary value problem for which a numerical solution can be readily computed on rectangular subregions with constant wavenumbers. The second class of numerical methods implemented combines adaptive domain transformation with coefficient smoothing to yield a boundary value problem well-suited for numerical solution on a uniform grid in the computational space. The resulting finite difference schemes do not have treat the grid points near the interface as a special case. A novel matrix analysis technique is implemented to examine the stability of these new methods. Computational verifications are carried out.
An explicit description of Pieri inclusions.
(2020-06-09) Miller, John Anderson, II, 1991-; Hunziker, Markus.
By the Pieri rule, the tensor product of an exterior power and a finite-dimensional irreducible representation of a general linear group has a multiplicity-free decomposition. The embeddings of the constituents are called Pieri inclusions and were first studied by Weyman in his thesis and described explicitly by Olver. More recently, these maps have appeared in the work of Eisenbud, Fløstad, and Weyman and of Sam and Weyman to compute pure free resolutions for classical groups. We give a new closed form, non-recursive description of Pieri inclusions. For partitions with a bounded number of distinct parts, the resulting algorithm has polynomial time complexity whereas the previously known algorithm has exponential time complexity.
Applications of full rank factorization to solving matrix equations.
(1992-12) Sykes, Jeffery D.; Mathematics.
In the study of matrices, we are always searching for tools which allow us to simplify our investigations. Because full rank factorizations exist for all matrices and their properties often help to simplify arguments, their uses are abundant. There exist many matrix equations for which solutions are otherwise quite difficult to find. Full rank factorizations and generalized inverses allow us to easily find solutions to many such equations. Their properties can also be used to study the diagonalization of non-square matrices and to develop conditions under which matrices are simultaneously diagonalizable. Finally, the full rank factorization can be used to derive canonical forms and other factorizations such as the singular value decomposition.
Approximation and interpolation with Bernstein polynomials.
(2021-11-03) Allen, Larry J., 1993-; Kirby, Robert C.
Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. In this dissertation, we investigate fundamental problems in approximation theory and numerical analysis involving Bernstein polynomials. We begin by developing a structured decomposition of the inverse of the matrices related to approximation and interpolation. These matrices are highly ill-conditioned, and so we introduce a nonstandard matrix norm to study the conditioning of the matrices, showing that the conditioning in this case is better than in the standard 2-norm. We conclude by giving an algorithm for enforcing bounds constraints on the approximating polynomial. Extensions of the interpolation problem and constrained approximation problem to higher dimensions are also considered.
Asymptotic arc-components in inverse limits of dendrites.
(2011-09-14) Hamilton, Brent (Brent A.); Raines, Brian Edward, 1975-; Mathematics.; Baylor University. Dept. of Mathematics.
We study asymptotic behavior arising in inverse limit spaces of dendrites. In particular, the inverse limit is constructed with a single unimodal bonding map, for which points have unique itineraries and the critical point is periodic. Using symbolic dynamics, sufficient conditions for two rays in the inverse limit space to have asymptotic parameterizations are given. Being a topological invariant, the classification of asymptotic parameterizations would be a useful tool when determining if two spaces are homeomorphic.
Boundary condition dependence of spectral zeta functions.
(2015-07-14) Graham, Curtis W., 1983-; Kirsten, Klaus, 1962-
In this work, we provide the analytic continuation of the spectral zeta function associated with the one-dimensional regular Sturm-Liouville problem and the two-dimensional Laplacian on the annulus. In the one-dimensional setting, we consider general separated and coupled boundary conditions, and on the annulus we restrict our work to Dirichlet-Robin boundary conditions. In both cases, we use our results to calculate the coefficients of the asymptotic expansion of the associated heat kernel. In the one-dimensional case, we additionally use the analytically continued spectral zeta function to compute the determinant of the Sturm-Liouville operator.
Boundary conditions associated with left-definite theory and the spectral analysis of iterated rank-one perturbations.
(2018-05-24) Frymark, Dale, 1992-; Liaw, Constanze.
This dissertation details the development of several analytic tools that are used to apply the techniques and concepts of perturbation theory to other areas of analysis. The main application is an efficient characterization of the boundary conditions associated with the general left-definite theory for differential operators. This theory originated with the groundbreaking work of Littlejohn and Wellman in 2002 which fully determined the `left-definite domains' and spectral properties of powers of self-adjoint Sturm--Liouville operators associated with classical orthogonal polynomials. We will study how the left-definite domains associated with these operators can be explicitly described by classical boundary conditions. Additional applications are made to infinite rank perturbations by successively introducing rank-one perturbations to a self-adjoint operator with absolutely continuous spectrum. The absolutely continuous part of the spectral measure of the constructed operator is controlled and estimated.
Boundary data smoothness for solutions of nonlocal boundary value problems for nth order differential equations.
(2018-04-18) Pennington, Brian C., 1992-; Henderson, Johnny.
The method involves application of Peano's Theorem for initial value problems.
Boundary data smoothness for solutions of nonlocal boundary value problems.
(Mathematical Sciences Publishers., 2011) Lyons, Jeffrey W.; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.
In this dissertation, we investigate boundary data smoothness for solutions of nonlocal boundary value problems over discrete and continuous domains. Essentially, we show that under certain conditions partial derivatives and differences exist, with respect to boundary conditions, for solutions of nonlocal boundary value problems and solve the variational equation in the derivative case and a specific linear difference equation in the difference case. Lastly, we provide a corollary and a few examples as well as some ideas for future work.
Chaos in dendritic and circular Julia sets.
(2016-07-01) Averbeck, Nathan, 1985-; Raines, Brian Edward, 1975-
We demonstrate the existence of various forms of chaos (including transitive distributional chaos, w-chaos, topological chaos, and exact Devaney chaos) on two families of abstract Julia sets: the dendritic Julia sets DT and the "circular" Julia sets ԐT , whose symbolic encoding was introduced by Stewart Baldwin. In particular, suppose one of the two following conditions hold: either fc has a Julia set which is a dendrite, or (provided that the kneading sequence of c is Г-acceptable) that fc has an attracting or parabolic periodic point. Then, by way of a conjugacy which allows us to represent these Julia sets symbolically, we prove that fc exhibits various forms of chaos.
Chaotic properties of set-valued dynamical systems.
(2016-04-04) Tennant, Timothy, 1987-; Raines, Brian Edward, 1975-
In this thesis, many classical results of topological dynamics are adapted to the set-valued case. In particular, focus is given to the notions of topological entropy and the specification property. These properties are defined in the context of set-valued functions, and examples are given both of classical theorems that extend naturally to this setting and of theorems which have no clear analogue. Particular attention is paid to a result which states that if a dynamical system has the specification property, there exists invariant non-atomic measures with full support.
Characterizing smoothness of solutions with respect to boundary data of Dirichlet-nonlocal-Dirichlet boundary value problems for third order ordinary differential equations.
(August 2022) Ickes, Henry, 1992-; Henderson, Johnny.
The goal of our paper is to establish analogues of Peano’s Theorem for Dirichlet-nonlocal-Dirichlet conditions in third order ordinary differential equations. We begin by establishing the difference quotient, followed by a telescoping sum, allowing us to set up a linear system of equations in two unknowns to establish the existence of the limit as h → 0. We finish by employing Cramer’s Rule, and list some quick corollaries for each variation of Peano’s Theorem.
A combinatorial property of Bernstein-Gelfand-Gelfand resolutions of unitary highest weight modules.
(2013-09-24) Hartsock, Gail.; Hunziker, Markus.; Mathematics.; Baylor University. Dept. of Mathematics.
It follows from a formula by Kostant that the difference between the highest weights of consecutive parabolic Verma modules in the Bernstein-Gelfand-Gelfand-Lepowsky resolution of the trivial representation is a single root. We show that an analogous property holds for all unitary representations of simply laced type. Specifically, the difference between consecutive highest weights is a sum of positive noncompact roots all with multiplicity one.
Comparison of smallest eigenvalues and extremal points for third and fourth order three point boundary value problems.
(2011-09-14) Neugebauer, Jeffrey T.; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.
The theory of u₀-positive operators with respect to a cone in a Banach space is applied to the linear differential equations u⁽⁴⁾ + λ₁p(x)u = 0 and u⁽⁴⁾ + λ₂q(x)u = 0, 0 ≤ x ≤ 1, with each satisfying the boundary conditions u(0) = u′(r) = u″(r) = u‴(1) = 0, 0 < r < 1. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained. These results are then extended to the nth order problem using two different methods. One method involves finding the Green's function for –u⁽ⁿ⁾ = 0 satisfying the higher order boundary conditions, and the other involves making a substitution that allows us to work with a variation of the fourth order problem. Extremal points via Krein-Rutman theory are then found. Analogous results are then obtained for the eigenvalue problems u‴ + λ₁p(x)u = 0 and u‴ + λ₂q(x)u = 0, with each satisfying u(0) = u′(r) = u″(1) = 0, 0 < 1/2 < r < 1.
Conventional and asymptotic stabilities of decomposed compact methods for solving highly oscillatory wave problems.
(2018-07-12) Jones, Tiffany Nicole, 1987-; Sheng, Qin (Tim)
This dissertation explores the numerical stabilities of decomposed compact finite difference methods for solving Helmholtz partial differential equation problems with large wave numbers. It is known that many practically effective approximations to highly oscillatory wave solutions are not stable in the traditional von Neumann sense. We will consider three algorithms that numerically approximate Helmholtz problems. In each case, we investigate conventional stability, as well as the physically relevant asymptotic stability where appropriate. For each scheme constructed, radially symmetric transverse fields and standard polar coordinates are considered. A decomposition is implemented to remove the anticipated singularity in the transverse direction for both the paraxial and nonparaxial Helmholtz equations. Compact scheme structures are introduced in the transverse direction to raise the accuracy and efficiency of the schemes developed for laser and subwavelength optical computations. The first scheme approximates a paraxial Helmholtz problem. It is shown that the numerical method introduced is not only highly accurate and efficient due to its straightforward algorithmic structure, but also conventionally stable under reasonable constraints for practical applications. The second and third schemes approximate nonparaxial Helmholtz problems, with the latter being a dual-scale algorithm. While the highly accurate compact algorithms shy away from the conventional stability, they are shown to be asymptotically stable with index one, facilitating the algorithmic effectiveness, reliability, and applicability. Numerical experiments further demonstrate the high reliability of each scheme when implemented in highly oscillatory optical self-focusing optical beam propagation simulations. Computational examples are presented to illustrate our conclusions. Unless otherwise stated, as in Chapter Five, any appropriate matrix norm may be considered.
Diagrams and reduced decompositions for cominuscule flag varieties and affine Grassmannians.
(2010-06-23T12:23:43Z) Pruett, W. Andrew.; Hunziker, Markus.; Mathematics.; Baylor University. Dept. of Mathematics.
We develop a system of canonical reduced decompositions of minimal coset representatives of quotients corresponding to cominuscule flag varieties and affine Grassmannians. This canonical decomposition allows, in the first case, an abbreviated computation of relative R-polynomials. From this, we show that these polynomials can be obtained from unlabelled intervals, and more generally, that Kazhdan-Lusztig polynomials associated to cominuscule flag varieties are combinatorially invariant. In the second case, we are able to provide a list of the rationally smooth Schubert varieties in simply laced affine Grassmannians corresponding to types A, D, and E. The results in this case were obtained independently by Billey and Mitchell in 2008.
Eigenvalue comparison theorems for certain boundary value problems and positive solutions for a fifth order singular boundary value problem.
(2016-03-23) Nelms, Charles F.; Henderson, Johnny.
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.
Existence and uniqueness of solutions of boundary value problems by matching solutions.
(2013-09-24) Liu, Xueyan, 1978.; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.
In this dissertation, we investigate the existence and uniqueness of boundary value problems for the third and nth order differential equations by matching solutions. Essentially, we consider the interval [a, c] of a BVP as the union of the two intervals [a, b] and [b, c], analyze the solutions of the BVP on each, and then match the proper ones to be the unique solution on the whole domain. In the process of matching solutions, boundary value problems with different boundaries, especially at the matching point b, would be quite different for the requirements of conditions on the nonlinear term. We denote the missing derivatives in the boundary conditions at the matching point b by k₁ and k₂. We show how y(ᵏ²)(b) varies with respect to y(ᵏ¹)(b), where y is a solution of the BVP on [a, b] or [b, c]. Under certain conditions on the nonlinear term, we can get a monotone relation between y(ᵏ²)(b) and y(ᵏ¹)(b), on [a, b] and [b, c], respectively. If the monotone relations are different on [a, b] and [b, c], then we can finally get a unique value for y(ᵏ¹)(b) where the k₂nd derivative of two solutions on [a, b] and [b, c] are equal and we can join the two solutions together to obtain the unique solution of our original BVP. If the relations are the same, then we will arrive at the situation that the k₂nd order derivatives of two solutions at b on [a, b] and [b, c] are decreasing with respect to the k₁st derivatives at b at different rates, and by analyzing the relations more in detail, we can finally get a unique value for the k₁st derivative of solutions of BVP's on [a, b] and [b, c], which are matched to be a unique solution of the BVP on [a, c]. In our arguments, we use the Mean Value Theorem and the Rolle's Theorem many times. As the simplest models, third order BVP's are considered first. Then, in the following chapters, nth order problems are studied. Lastly, we provide an example and some ideas for our future work.
Existence of positive solutions to right focal three point singular boundary value problems.
(2013-09-16) Sutherland, Shawn M., 1984-; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.
In this dissertation, we prove the existence of positive solutions to two classes of three point right focal singular boundary value problems of at least fourth order.

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