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COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol.2(2002), No.2, pp.171–185 c© Institute of Mathematics of the National Academy of Sciences of Belarus SUPERCONVERGENCE POSTPROCESSING FOR EIGENVALUES1 MILENA R. RACHEVA Department of Mathematics, Technical University of Gabrovo 5300 Gabrovo, Bulgaria E-mail: mracheva@mail.com ANDREY B. ANDREEV Department of Informatics, Technical University of Gabrovo, Central Laboratory for Parallel Processing, Bulgarian Academy of Sciences Sofia, Bulgaria E-mail: andreev@tugab.bg Abstract — The main goal of this paper is to

References [1] W.N. Anderson Jr. and T.D. Morley, Eigenvalues of the Laplacian of a graph, Linear Multilinear Algebra 18 (1985) 141-145. doi:10.1080/03081088508817681 [2] D.P. Bertsekas, Nonlinear Programming: Second Edition (Athena Scientific, Belmont, Massachusetts 1999) page 278, Proposition 3.1.1. [3] B. Bollobas and V. Nikiforov, Graphs and Hermitian matrices: eigenvalue interlac- ing, Discrete Math. 289 (2004) 119-127. doi:10.1016/j.disc.2004.07.011 [4] A.E. Brouwer and W.H. Haemers, Spectra of Graphs (Springer Verlag, 2011). [5] S. Butler, Interlacing for

Energy Eigenvalue Level Motion with Two Parameters W illi-Hans Steeb, Yorick Hardy, and Ruedi Stoop International School for Scientific Computing, Rand Afrikaans University, RO. Box 524, Auckland Park 2006, South Africa Reprint requests to Prof. W.-H. Steeb; E-mail: whs@na.rau.ac.za Z. Naturforsch. 56 a, 565-567 (2001); received April 17, 2001 From the eigenvalue equation H\\ipn(\)) = En(X)\xpn(\)) where H\ = Ho + XV one can derive an autonomous system of first order ordinary differential equations for the eigenvalues En(A) and the matrix elements K „ n

On Laplacian Eigenvalues of a Graph Bo Zhou Department of Mathematics, South China Normal University, Guangzhou 510631, P.R. China Reprint requests to B. Z.; e-mail: zhoubo@scnu.edu.cn Z. Naturforsch. 59a, 181 – 184 (2004); received November 11, 2003 Let G be a connected graph with n vertices and m edges. The Laplacian eigenvalues are denoted by µ1(G) ≥ µ2(G) ≥ ·· · ≥ µn−1(G) > µn(G) = 0. The Laplacian eigenvalues have important appli- cations in theoretical chemistry. We present upper bounds for µ1(G)+ · · ·+ µk(G) and lower bounds for µn−1(G)+ · · ·+µn−k(G) in

References [1] K. Fan, On a theorem of Weyl concerning eigenvalues of linear transformations, I. Proc. Nat. Acad. Sci. U. S. A. 35 (1949), 652–655. [2] S. Friedland, Extremal eigenvalue problems, Bull. Brazilian Math. Soc. 9 (1978), 13-40. [3] S. Friedland, A generalization of the Motzkin-Taussky theorem, Linear Algebra Appl. 36 (1981), 103-109. [4] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, Second edition, 1952. [5] T. Kato, A Short Introduction to Perturbation Theory for Linear Operators, Springer-Verlag, 2nd ed

Adv. Nonlinear Anal. 2 (2013), 91–125 DOI 10.1515/anona-2012-0202 © de Gruyter 2013 A class of degenerate elliptic eigenvalue problems Marcello Lucia and Friedemann Schuricht Abstract. We consider a general class of eigenvalue problems where the leading ellip- tic term corresponds to a convex homogeneous energy function that is not necessarily differentiable. We derive a strong maximum principle and show uniqueness of the first eigenfunction. Moreover we prove the existence of a sequence of eigensolutions by using a critical point theory in metric spaces. Our

References [1] F. Belardo and S.K. Simić, On the Laplacian coefficients of signed graphs , Linear Algebra Appl. 475 (2015) 94–113. doi:10.1016/j.laa.2015.02.007 [2] C.J. Colbourn and J.H. Dinitz (Ed(s)), Handbook of Combinatorial Designs (Chapman and Hall/CRC, Boca Raton, 2007). doi:10.1201/9781420010541 [3] E.R. van Dam, Regular graphs with four eigenvalues , Linear Algebra Appl. 226–228 (1995) 139–162. doi:10.1016/0024-3795(94)00346-F [4] E. Ghasemian and G.H. Fath-Tabar, On signed graphs with two distinct eigenvalues , Filomat 31 (2017) 6393–6400. doi

1 Introduction Let Ω be a bounded domain with piecewise smooth boundary ∂ ⁡ Ω $\partial\Omega$ in an n -dimensional Euclidean space ℝ n $\mathbb{R}^{n}$ . Let λ i $\lambda_{i}$ be the i -th eigenvalue of the Dirichlet eigenvalue problem of the poly-Laplacian with arbitrary order (1.1) { ( - Δ ) l ⁢ u = λ ⁢ u in ⁢ Ω , u = ∂ ⁡ u ∂ ⁡ ν = ⋯ = ∂ l - 1 ⁡ u ∂ ⁡ ν l - 1 = 0 on ⁢ ∂ ⁡ Ω , $\left\{\begin{aligned} &\displaystyle(-\Delta)^{l}u=\lambda u&&\displaystyle% \text{in }\Omega,\\ &\displaystyle u=\frac{\partial u}{\partial\nu}=\cdots=\frac{\partial^{l-1}u

special dependence on the spectral parameter are called transmission eigenvalue problems. Recently such problems have attracted much attention in connection with the inverse acoustic scattering theory (see [ 19 , 20 , 10 , 8 , 1 , 2 , 9 , 11 , 12 , 7 ] and the references therein). In this work we study inverse spectral problems for the eigenvalue problem R ⁢ ( a , q ) {R(a,q)} . The obtained results can be easily reformulated in terms of Q ⁢ ( b , n ) {Q(b,n)} in case when R ⁢ ( a , q ) {R(a,q)} is obtained from a problem Q ⁢ ( b , n ) {Q(b,n)} . We note

Advanced Nonlinear Studies 5 (2005), 573–585 Variational Eigenvalues of Degenerate Eigenvalue Problems for the Weighted p-Laplacian An Lê Mathematics Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720 e-mail: anle@msri.org Klaus Schmitt Department of Mathematics, University of Utah, Salt Lake city, Utah 84112 e-mail: schmitt@math.utah.edu Received 7 September 2005 Communicated by Shair Ahmad Abstract We prove the existence of nondecreasing sequences of positive eigenvalues of the homogeneous degenerate quasilinear eigenvalue problem − div