Lau Ching Yan Ada = 耗散系統中的廣義Sturm-Liouville理論 / 劉正欣. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 156-157). / Text in English; abstracts in English and Chinese. / Lau Ching Yan Ada = Hao san xi tong zhong de guang yi Sturm-Liouville li lun / Liu Zhengxin. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Vibrational motion in physics --- p.1 / Chapter 1.2 --- Normal modes of vibration --- p.2 / Chapter 1.3 --- Boundary conditions --- p.4 / Chapter 1.4 --- The wave equation --- p.6 / Chapter 1.4.1 --- Mechanical waves --- p.7 / Chapter 1.4.2 --- Electromagnetic waves --- p.9 / Chapter 1.5 --- General form of the wave equation --- p.10 / Chapter 1.5.1 --- V(x) as a restoring force --- p.11 / Chapter 1.5.2 --- V(x) in gravitational waves --- p.13 / Chapter 1.5.3 --- V(x) by transformation --- p.16 / Chapter 2 --- Sturm-Liouville systems --- p.18 / Chapter 2.1 --- Introduction --- p.18 / Chapter 2.2 --- Differential operators --- p.19 / Chapter 2.2.1 --- Introduction --- p.19 / Chapter 2.2.2 --- Adjoint operators --- p.20 / Chapter 2.2.3 --- Self-adjoint operators --- p.21 / Chapter 2.2.4 --- More examples --- p.24 / Chapter 2.3 --- Sturm-Liouville boundary-value problems --- p.27 / Chapter 2.4 --- Sturm-Liouville theory --- p.28 / Chapter 2.4.1 --- Real eigenvalues --- p.29 / Chapter 2.4.2 --- Orthogonal eigenfunctions --- p.30 / Chapter 2.4.3 --- Completeness of eigenfunctions --- p.31 / Chapter 2.4.4 --- Interlacing zeros of the eigenfunctions --- p.33 / Chapter 2.5 --- Applications of Sturm-Liouville theory --- p.35 / Chapter 2.5.1 --- Vibrations of a string --- p.36 / Chapter 2.5.2 --- The hydrogen atom --- p.40 / Chapter 3 --- Wave equation with damping --- p.46 / Chapter 3.1 --- Statement of problem --- p.46 / Chapter 3.1.1 --- The equation --- p.46 / Chapter 3.1.2 --- The operator --- p.48 / Chapter 3.1.3 --- Non-self-adjointness --- p.49 / Chapter 3.2 --- Eigenfunctions and Eigenvalues --- p.51 / Chapter 3.3 --- The completeness problem --- p.53 / Chapter 4 --- Green's function solution --- p.55 / Chapter 4.1 --- Introduction --- p.55 / Chapter 4.2 --- Green's function solution --- p.56 / Chapter 4.3 --- Fourier transform --- p.58 / Chapter 4.4 --- Inverse Fourier transform --- p.61 / Chapter 5 --- Proof of completeness --- p.66 / Chapter 5.1 --- WKB approximation --- p.66 / Chapter 5.2 --- "An upper bound for \G(x,y,w)e~iwt\ " --- p.68 / Chapter 5.3 --- Proof of completeness --- p.72 / Chapter 5.3.1 --- The limit when R→∞ --- p.72 / Chapter 5.3.2 --- Eigenfunction expansion --- p.76 / Chapter 6 --- The bilinear map --- p.80 / Chapter 6.1 --- Introduction --- p.80 / Chapter 6.2 --- Evaluation of J1(wj) --- p.82 / Chapter 6.3 --- Self-adjointness of H --- p.84 / Chapter 6.4 --- Properties of the map --- p.87 / Chapter 7 --- Applications --- p.89 / Chapter 7.1 --- Eigenfunction expansion --- p.89 / Chapter 7.2 --- Perturbation theory --- p.94 / Chapter 7.2.1 --- First and second-order corrections --- p.95 / Chapter 7.2.2 --- Example --- p.97 / Chapter 7.2.3 --- Example (Constant r) --- p.102 / Chapter 8 --- Critical points --- p.104 / Chapter 8.1 --- Introduction --- p.104 / Chapter 8.2 --- Conservative cases (Γ = 0) --- p.105 / Chapter 8.3 --- Non-conservative cases (Constant r) --- p.107 / Chapter 8.4 --- Critical points away from imaginary axis --- p.108 / Chapter 9 --- Jordan block and applications --- p.114 / Chapter 9.1 --- Jordan basis --- p.114 / Chapter 9.2 --- An analytical example --- p.117 / Chapter 9.2.1 --- Solving for the extra basis function --- p.117 / Chapter 9.2.2 --- Freedom of choice --- p.118 / Chapter 9.2.3 --- Interpolating function --- p.120 / Chapter 9.3 --- A numerical example --- p.122 / Chapter 9.3.1 --- "Solving for f2,1 " --- p.124 / Chapter 9.3.2 --- Interpolating function --- p.126 / Chapter 9.4 --- Jordan basis expansion --- p.127 / Chapter 9.5 --- Perturbation theory near critical points --- p.131 / Appendices --- p.142 / Chapter A --- WKB approximation --- p.142 / Chapter B --- Green's function (Discontinuous V(x)) --- p.145 / Chapter B.l --- Finite discontinuouity in V(x) --- p.145 / Chapter B.1.1 --- Green's function --- p.145 / Chapter B.1.2 --- "Behaviour of the extra phases Φ, Φ " --- p.147 / Chapter B.2 --- Delta function in --- p.148 / Chapter B.2.1 --- Green's function --- p.148 / Chapter B.2.2 --- "Behaviour of the extra phases Φ, Φ " --- p.150 / Chapter C --- Dual basis --- p.151 / Chapter C.1 --- Matrix representation --- p.152 / Chapter C.2 --- Relation with bilinear map --- p.153 / Chapter C.3 --- Construction of dual basis --- p.154 / Bibliography --- p.156

Thesis (Ph. D.)--University of Wisconsin--Madison, 1934. / Typescript and manuscript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

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In this thesis, we study the optimal estimate of eigenvalue ratios £f_n/£f_m of the Sturm-Liouville equation with Dirichlet boundary conditions on (0, £k). In 2005, Horvath and Kiss [10] showed that £f_n/£f_m≤(n/m)^2 when the potential function q ≥ 0 and is a single-well function. Also this is an optimal upper estimate, for equality holds if and only if q = 0. Their result gives a positive answer to a problem posed by Ashbaugh and Benguria [2], who earlier showed that £f_n/£f_1≤n^2 when q ≥ 0. Here we first simplify the proof of Horvath and Kiss [10]. We use a modified Prufer substitutiony(x)=r(x)sin(£s£c(x)), y'(x)=r(x)£scos(£s£c(x)), where £s = ¡Ô£f. This modified phase seems to be more effective than the phases £p and £r that Horvath and Kiss [10] used. Furthermore our approach can be generalized to study the one-dimensional p-Laplacian eigenvalue problem. We show that for the Dirichlet problem of the equation -[(y')^(p-1)]'=(p-1)(£f-q)y^(p-1), where p > 1 and f^(p-1)=|f|^(p-1)sgn f =|f|^(p-2)f. The eigenvalue ratios satisfies £f_n/£f_m≤(n/m)^p, assuming that q(x) ≥ 0 and q is a single-well function on the domain (0, £k_p). Again this is an optimal upper estimate.

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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.

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.

Tez (Yüksek Lisans) - Süleyman Demirel Üniversitesi, Fen Bilimleri Enstitüsü, Matematik Anabilim Dalı, 2009. / Kaynakça var.