The Ambarzumyan Theorem states that for the classical Sturm-Liouville problem on $[0,1]$, if the set of Neumann eigenvalue $sigma_N={(npi)^2: nin { f N}cup { 0}}$, then the potential function $q=0$. In this thesis, we study the analogues of Ambarzumyan Theorem for the Sturm-Liouville operators on star-shaped graphs with 3 edges of different lengths. We first solve the direct problem: to find out the set of eigenvalues when $q=0$. Then we use the theory of transformation operators and Raleigh-Ritz inequality to prove the inverse problem. Following Pivovarchik's work on star-shaped graphs of uniform lengths, we analyze the Kirchoff condition in detail to prove our theorems. In particular, we study the cases when the lengths of the 3 edges satisfy $a_1=a_2=frac{1}{2}a_3$ or $a_1=frac{1}{2}a_2=frac{1}{3}a_3$. Furthermore, we work on Neumann boundary conditions as well as Dirichlet boundary conditions. In the latter case, some assumptions about $q$ have to be made.

In this thesis, we study two new inverse nodal problems introduced by Yang, Shen and Shieh respectively. Consider the classical Sturm-Liouville problem: $$ left{ egin{array}{c} -phi'+q(x)phi=la phi phi(0)cosalpha+phi'(0)sinalpha=0 phi(1)coseta+phi'(1)sineta=0 end{array} ight. , $$ where $qin L^1(0,1)$ and $al,ein [0,pi)$. The inverse nodal problem involves the determination of the parameters $(q,al,e)$ in the problem by the knowledge of the nodal points in $(0,1)$. In 1999, X.F. Yang got a uniqueness result which only requires the knowledge of a certain subset of the nodal set. In short, he proved that the set of all nodal points just in the interval $(0,b) (frac{1}{2}<bleq 1)$ is sufficient to determine $(q,al,e)$ uniquely. In this thesis, we show that a twin and dense subset of all nodal points in the interval $(0,b)$ is enough to determine $(q,al,e)$ uniquely. We improve Yang's theorem by weakening its conditions, and simplifying the proof. In the second part of this thesis, we will discuss an inverse nodal problem for the vectorial Sturm-Liouville problem: $$ left{egin{array}{c} -{f y}'(x)+P(x){f y}(x) = la {f y}(x) A_{1}{f y}(0)+A_{2}{f y}'(0)={f 0} B_{1}{f y}(1)+B_{2}{f y}'(1)={f 0} end{array} ight. . $$ Let ${f y}(x)$ be a continuous $d$-dimensional vector-valued function defined on $[0,1]$. A point $x_{0}in [0,1]$ is called a nodal point of ${f y}(x)$ if ${f y}(x_{0})=0$. ${f y}(x)$ is said to be of type (CZ) if all the zeros of its components are nodal points. $P(x)$ is called simultaneously diagonalizable if there is a constant matrix $S$ and a diagonal matrix-valued function $U(x)$ such that $P(x)=S^{-1}U(x)S.$ If $P(x)$ is simultaneously diagonalizable, then it is easy to show that there are infinitely many eigenfunctions which are of type (CZ). In a recent paper, C.L. Shen and C.T. Shieh (cite{SS}) proved the converse when $d=2$: If there are infinitely many Dirichlet eigenfunctions which are of type (CZ), then $P(x)$ is simultaneously diagonalizable. We simplify their work and then extend it to some general boundary conditions.

Gesamthochsch., Diss.--Paderborn, 1979.

No description available.

University, Diss., 2006--Kassel.

Thesis (M.A.)--Marshall University, 2004. / Title from document title page. Document formatted into pages; contains vi, 39 p. Includes abstract. Includes bibliographical references (p. 39).

Thesis (Ph. D.)--University of Oregon, 2000. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 98-99). Also available for download via the World Wide Web; free to University of Oregon users. Address: http://wwwlib.umi.com/cr/uoregon/fullcit?p9963453.

Ulm, Univ., Diss., 2008.

Ulm, Univ. Diss., 2006.

Two common classes of filter functions in use today, Butterworth functions and Chebyshev functions, are based upon solutions to special cases of the Sturm-Liouville equation. Here, solutions to several other special cases of the Sturm-Liouville equation were used to develop filter functions, and the properties of the resulting filters were examined. The following functions were explored: Chebyshev functions of the second kind, untraspherical functions of the second and third kinds, Hermite functions, and Legendre functions. Filter functions were developed for each of the first five polynomials in each series of functions, and magnitude and phase responses were tabulated and plotted. One of the classes of functions, the Hermite functions, led to filters which have a significant advantage over the commonly used Chebyshev filters in passband magnitude response, and were essentially the same as Chebyshev filters in stopband magnitude response and phase response.