# 臺灣博碩士論文加值系統

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 我們藉由某些特殊形式的矩陣函數的分解法去研究關於Jacobi矩陣與Stieltjes弦的反問題。關於Stieltjes弦，我們也試圖估計Stieltjes弦的第一固有值。我們證明，當弦之密度數列之上下界與總質量固定時，我們可以決定使Stieltjes弦的第一固有值為極大或極小的密度數列為何。除此之外，我們也證明幾個關於Stieltjes弦的第一、第二固有值之比值的比較定理。
 Applying the factorization of some related matrix functions, we investigate some inverse problems with mixed spectral data for Jacobi matrices and Stieltjes strings. Besides, we prove a discrete analogue of Borg's theorem for the Green's matrix. We also study the first eigenvalue, and the ratio of the first two eigenvalues of the Stieltjes string equation. With certain restrictions on the class of density sequences $p$, we determine the shapes of the extremal density sequence for the first eigenvalue, and the minimum for the ratio of the first two eigenvalues.
 1. Introduction 12. Preliminaries 43. Decompositions of the Jacobi continued fraction and the Stieltses continuedfraction 64. Some inverse spectral problems with mixed spectral data 105. The extrema of the rst eigenvalue 176. The bounds for the ratio of the rst and second eigenvalues 23References 30
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Soc., vol 127, 6, 1805--1803.Huang, M.-J. (2007) {\em The eigenvalue gap for vibrating strings with symmetric densities}, Acta Math. Hungar., vol 123, vol 117, 4, 341--348.Huang, M.-J. (2009) {\em A note on the eigenvalue ratio of vibrating strings}, Acta Math. Hungar., vol 123, 3, 265--271.Keller, J. B. (1976) {\em The minimum ratio of two eigenvalues}, SIAM J. Appl. Math., 31, 485--491.Kre\v\i n, M. G. (1951) {\em Solution of the inverse Sturm-Liouville problem}, Dokl. Akad. Nauk SSSR (N. S.), vol 76, 21--24.Kre\v\i n, M. G. (1955) {\em On certain problems on the maximum and minimum of characteristic values and on. Lyapunov zones of stability}, Amer. Math. Soc. Transl., 1, 163--187.Lanczos, C. (1950) {\em An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators}, J. of Research of the National Bureau of Standards, 45, 4, 255--282.Levinson, N. (1949) {\em The inverse Sturn-Liouville problem}, Mar. Tidsskr., 3, 25--30.Levitan, B. M. 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(2007) {\em On some inverse spectral problems related to the Ambarzumyan problem and the dual string of the string equation}, Inverse Problems, 23, 2417--2436.
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