# Department of Mathematics

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Item Oscillating satellites about the straight line equilibrium points.(1952) Clayton, Robert LeeThe problem as studied in this thesis is the behavior of a satellite at the three equilibrium points. When a satellite is displaced from rest at one of these points it will either oscillate or rapidly leave the system. The equations of motion were first set up between the satellite and two other rotating bodies. These equations were then expanded in series and certain restrictions placed upon them. With these restrictions, constants of integration were found, and the integrated equations of motion were also found. An application was then made to the Solar System in which the Sun and Jupiter were taken as the major bodies. The masses of these two bodies were then substituted into the equations of motion and equations were found for the three points of equilibrium. The orbits were then plotted at the three points.Item 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.Item Existence of positive solutions to singular right focal boundary value problems.(Orlando, FL : International Publications., 2005-05) Maroun, Mariette.; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.In this dissetation, we seek positive solutions for the n^th order ordinary differential equation, y^(n)=f(x,y), satisfying the right focal boundary conditions, y^(i)(0)=y^(n-2)(p)=y^(n-1)(1)=0, i=0,...,n-3, where p is a fixed value between 1/2 and 1, and where f(x,y) has certain singulrities at x=0, at y=0, and possibly at y=infinity.Item Uniqueness implies uniqueness and existence for nonlocal boundary value problems for fourth order differential equations.(2006-07-07T21:11:00Z) Ma, Ding.; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.In this dissertation, we are concerned with uniqueness and existence of solutions of certain types of boundary value problems for fourth order differential equations. In particular, we deal with uniqueness implies uniqueness and uniqueness implies existence questions for solutions of the fourth order ordinary differential equation, y⁴ = f (x, y, y¹, yⁿ, yᵐ) , satisfying nonlocal 5-point boundary conditions given by y(x₁) = y₁, y(x₂) = y₂, y(x₃) = y₃, y(x) - y(x₅) = y₄ , where a < x₁ < x₂ < x₃ < x₄ < x₅ < b, and y₁, y₂, y₃, y₄ ∈ R. We also consider solutions of this fourth order differential equation satisfying nonlocal 4-point and 3-point boundary conditions given by y(x₁) = y₁, y'(x₁) = y₂, y(x₂) = y₃, y(x₃) - y(x₄) = y₄ , y(x₁) = y₁, y'(x₁) = y₂, y''(x₁) = y₃, y(x₂) - y(x₃) =y₄.Item Uniqueness implies uniqueness and existence for nonlocal boundary value problems for third order ordinary differential equations.(2006-07-29T16:00:19Z) Gray, Michael Jeffery.; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.For the third order ordinary differential equation, $y'''=f(x,y,y',y'')$, it is assumed that, for some $m\geq 4$, solutions of nonlocal boundary value problems satisfying \[y(x_1)=y_1,\ y(x_2)=y_2,\] \[y(x_m)-\sum_{i=3}^{m-1} y(x_{i})=y_3,\] $aItem 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.Item Restarting the Lanczos algorithm for large eigenvalue problems and linear equations.(2008-10-02T18:39:19Z) Nicely, Dywayne A.; Morgan, Ronald Benjamin, 1958-; Mathematics.; Baylor University. Dept. of Mathematics.We are interested in computing eigenvalues and eigenvectors of large matrices and in solving large systems of linear equations. Restarted versions of both the symmetric and nonsymmetric Lanczos algorithms are given. For the symmetric case, we give a method called Lan-DR that simultaneously solves linear equations and computes eigenvalues and eigenvectors. The use of approximate eigenvectors deflates eigenvalues. Maintaining the orthogonality of the Lanczos vectors is a concern. We suggest an approach that is a combination of Parlett and Scott's idea of selective orthogonalization and Simon's partial orthogonalization. For linear systems with multiple right-sides, eigenvectors computed during the solution of the first right-hand side can be used to give much faster convergence of the second and subsequent right-hand sides. A restarted version of the nonsymmetric Lanczos algorithm is developed. Both the right and left eigenvectors are computed while systems of linear equations are solved. We also investigate a restarted two-sided Arnoldi. We compare expense and stability of this approach with restarted nonsymmetric Lanczos.Item The left-definite spectral analysis of the legendre type differential equation.(2010-02-02T20:08:14Z) Tuncer, Davut.; Littlejohn, Lance, 1951-; Mathematics.; Baylor University. Dept. of Mathematics.Littlejohn and Wellman developed a general abstract left-definite theory for a self-adjoint operator A that is bounded below in a Hilbert space (H,(‧,‧)). More specifically, they construct a continuum of Hilbert spaces {(H_r,(‧,‧)_r)}_r>0 and, for each r>0, a self-adjoint restriction A_r of A in H_r. The Hilbert space H_r is called the rth left-definite Hilbert space associated with the pair (H,A) and the operator A_r is called the rth left-definite operator associated with (H,A). We apply this left-definite theory to the self-adjoint Legendre type differential operator generated by the fourth-order formally symmetric Legendre type differential expression ℓ[y](x):=((1-x²)²y″(x))″-((8+4A(1-x²))y′(x))′ +λy(x), where the numbers A and λ are, respectively, fixed positive and non-negative parameters and where x ∈ (-1,1).Item 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.Item Indecomposability in inverse limits.(2010-10-08T16:32:06Z) Williams, Brian R. (Brian Robert), 1982-; Ryden, David James, 1971-; Mathematics.; Baylor University. Dept. of Mathematics.Topological inverse limits play an important in the theory of dynamical systems and in continuum theory. In this dissertation, we investigate classical inverse limits of Julia sets and set-valued inverse limits of arbitrary compacta. Using the theory of Hubbard trees, the trunk of the Julia set of a postcriticallly finite polynomial is introduced. Using this trunk, a characterization of indecomposability is provided for inverse limits of post-critically finite polynomials restricted to their Julia sets. Inverse limits with upper semicontinuous set-valued bonding maps are also examined. We provide necessary and sufficient conditions for inverse limits of upper semicontinuous functions to have the full projection property, answering a question posed by Ingram. The full projection property is an important tool in the study of indecomposable inverse limits. A characterization of the full projection property for arbitrary compacta is given based solely on the dynamics of the bonding functions and a second characterization is given for the class of continuum-valued maps of trees that are residual-preserving.Item 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.Item 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.Item 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.Item Global SL(2,R) representations of the Schrödinger equation with time-dependent potentials.(2012-08-08) Franco, Jose A.; Sepanski, Mark R. (Mark Roger); Mathematics.; Baylor University. Dept. of Mathematics.We study the representation theory of the solution space of the one-dimensional Schrödinger equation with time-dependent potentials that possess sl₂-symmetry. We give explicit local intertwining maps to multiplier representations and show that the study of the solution space for potentials of the form V (t, x) = g₂(t)x²+g₁(t)x+g₀ (t) reduces to the study of the potential free case. We also show that the study of the time-dependent potentials of the form V (t, x) = λx⁻² + g₂(t)x² + g₀(t) reduces to the study of the potential V (t, x) = λx⁻². Therefore, we study the representation theory associated to solutions of the Schrödinger equation with this potential only. The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category.Item Spectral functions for generalized piston configurations.(2012-11-29) Morales-Almazán, Pedro Fernando.; Kirsten, Klaus, 1962-; Mathematics.; Baylor University. Dept. of Mathematics.In this work we explore various piston configurations with different types of potentials. We analyze Laplace-type operators P=-g^ij ∇^E_i ∇ ^E_j+V where V is the potential. First we study delta potentials and rectangular potentials as examples of non-smooth potentials and find the spectral zeta functions for these piston configurations on manifolds I x N, where I is an interval and N is a smooth compact Riemannian d-1 dimensional manifold. Then we consider the case of any smooth potential with a compact support and develop a method to find spectral functions by finding the asymptotic behavior of the characteristic function of the eigenvalues for P. By means of the spectral zeta function on these various configurations, we obtain the Casimir force and the one-loop effective action for these systems as the values at s= -½ and the derivative at s = 0. Information about the heat kernel coefficients can also be found in the spectral zeta function in the form of residues, which provide an indirect way of finding this geometric information about the manifold and the operator.Item Orbit structures of homeomorphisms.(2012-11-29) Sherman, Casey L.; Raines, Brian Edward, 1975-; Mathematics.; Baylor University. Dept. of Mathematics.In this dissertation we answer the following question: If X is a Cantor set and T: X → to X is a homeomorphism, what possible orbit structures can T have? The answer is given in terms of the orbit spectrum of T. If X is a Cantor set, then there is a homeomorphism T : X → to X with σ(T) = (0, ζ, σ₁, σ₂, σ₃, …) if and only if one of the following holds: 1) ζ = 0, there exists k ∈ N and a set {n₁ … ,nk} with σ _{n_i} > 0 for each 1 ≤ i ≤ k such that if σ _j > 0 then there exists i ∈ {1, 2, …, k} with n_i|j and there is an m ∈ N with σ _{mj} = c. 2) 1 ≤ ζ < c, {n: σ_ n= c} is infinite, and ∑ σ_ n : σ_ {mn} < c { for all m∈N} ≤ ζ, or 3) ζ = c.Item 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.Item On a ring associated to F[x].(2013-09-24) Aceves, Kelly Fouts.; Dugas, Manfred.; Mathematics.; Baylor University. Dept. of Mathematics.For a ﬁeld F and the polynomial ring F [x] in a single indeterminate, we deﬁne Ḟ [x] = {α ∈ End_F(F [x]) : α(ƒ) ∈ ƒF [x] for all ƒ ∈ F [x]}. Then Ḟ [x] is naturally isomorphic to F [x] if and only if F is inﬁnite. If F is ﬁnite, then Ḟ [x] has cardinality continuum. We study the ring Ḟ[x] for ﬁnite ﬁelds F. For the case that F is ﬁnite, we discuss many properties and the structure of Ḟ [x].Item 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.Item 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.

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