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Summary  

New results for the eigenvalue ratios of vibrating strings are presented partially in connection with previous results concerning Schrdinger operators.

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Abstract  

Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q 1(t)q 2(t)2 ... q m (t)m, where each q i (t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of q j (t). We give an algorithm to construct the polynomials q i (t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) =
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop \sum \limits_{i = 1}^n$$ \end{document}
i| of G, where λ1, λ2, ..., λn are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients.
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47 – 50 . [19] Wu , J. 2009 Power sums of Hecke eigenvalues and application Acta Arith. 137 333

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. [8] Fulton , C. T. 1977 Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions Proc. R. Soc. Edinburgh A77

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In this paper we formulate and prove that there exists the first positive eigenvalue of the eigenvalue problem with oblique derivative for the Laplace-Beltrami operator on the unit sphere. The firrst eigenvalue plays a major role in studying the asymptotic behaviour of solutions of oblique derivative problems in cone-like domains. Our work is motivated by the fact that the precise solutions decreasing rate near the boundary conical point is dependent on the first eigenvalue.

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Abstract  

We establish several comparison results on the eigenvalue gap for vibrating strings with symmetric single-well densities or symmetric double-well densities.

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Studia Scientiarum Mathematicarum Hungarica
Authors:
Carlos M. da Fonseca
,
Victor Kowalenko
, and
László Losonczi

pentadiagonal matrices, their determinants, eigenvalues and linear systems of equations for finding the eigenvectors. This work had originated from the pioneering study on the last of these topics by Egerváry’s Ph.D. advisor, Leopold Fejér [ 10 ]. In particular

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Abstract  

For vibrating strings with symmetric single-well densities, it is known that the ratio λ211 is maximized when the density is constant. In this note, we extend this result to a class of symmetric densities.

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