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1

Comparison estimates on the first eigenvalue of a quasilinear elliptic system

Abolarinwa Abimbola, Azami Shahroud

Journal of Applied Analysis

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2020

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Vol. 26, nr 2

273--285

EN

We study a system of quasilinear eigenvalue problems with Dirichlet boundary conditions on complete compact Riemannian manifolds. In particular, Cheng comparison estimates and the inequality of Faber-Krahn for the first eigenvalue of a (p, q)-Laplacian are recovered. Lastly, we reprove a Cheeger-type estimate for the p-Laplacian, 1 < p < ∞, from where a lower bound estimate in terms of Cheeger’s constant for the first eigenvalue of a (p, q)-Laplacian is built. As a corollary, the first eigenvalue converges to Cheeger’s constant as p, q → 1, 1.

2

The Kummer confluent hypergeometric function and some of its applications in the theory of azimuthally magnetized circular ferrite waveguides

Georgiev G. N., Georgieva-Grosse M. N.

Journal of Telecommunications and Information Technology

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2005

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nr 3

112-128

EN

Examples of the application of the confluent hypergeometric functions in miscellaneous areas of the theoretical physics are presented. It is suggested these functions to be utilized as a universal means for solution of a large number of problems, leading to: cylindrical, incomplete gamma, Coulomb wave, Airy, Kelvin, Bateman, Weber's parabolic cylinder, logarithmic-integral and exponential integral functions, generalized Laguerre, Poisson-Charlier and Hermit polynomials, integral sine and cosine, Fresnel and probability integrals, etc. (whose complete list is given), which are their special cases. The employment of such an approach would permit to develop general methods for integration of these tasks, to generalize results of different directions of physics and to find the common features of various phenomena, governed by equations, pertaining to the same family. Emphasis is placed here on the use of the Kummer function in the field of microwaves: the cases of normal and slow rotationally symmetric TE modes propagation in the azimuthally magnetized circular ferrite waveguide are considered. Lemmas on the properties of the argument, real and imaginary parts, and positive purely imaginary (real) zeros of the function mentioned in the complex (real) domain, of importance in the solution of boundary-value problem stated for normal (slow) waves, are substantiated analytically or numerically. A theorem for the identity of positive purely imaginary and real zeros of the complex respectively real Kummer function for certain parameters, is proved numerically. Tables and graphs support the results established. The terms for wave transmission are obtained as four bilaterally open intervals of variation of the quantities, specifying the fields. It turns out that the normal (slow) modes may exist in one (two) region(s). The theoretically predicted phase curves for the first waves of the two TE sets examined show that the structure explored is suitable for ferrite control components design.

3

Foundations of the theory of open waveguides

Nosich A.I.

Journal of Telecommunications and Information Technology

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2000

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nr 1-2

42-45

EN

The theory of electromagnetic wave eigenmodes propagating on open dielectric and metallic waveguides has been reviewed. The main steps of different theoretical approaches to the problem are outlined and discussed. The unsolved problems and also directions of future development are pointed out.

4

Sub-gradient algorithms for computation of extreme eigenvalues of a real symmetric matrix

Yassine A.

Control and Cybernetics

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1998

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Vol. 27, no 3

387-415

EN

The computation of eigenvalues of a matrix is still of importance from both theoretical and practical points of view. This is a significant problem for numerous industrial and scientific situations, notably in dynamics of structures (e.g. Gerardin, 1984), physics (e.g. Rappaz, 1979), chemistry (e.g. Davidson, 1983), economy (e.g. Morishima, 1971; Neumann, 1946), mathematics (e.g. Golub, 1989; Chatelin, 1983, 1984, 1988). The study of eigenvalue problems remains a delicate task, which generally presents numerical difficulties in relation to its sensivity to roundoff errors that may lead to numerical unstabilities, particularly if the eigenvalues are not well separated. In this paper, new subgradient-algorithms for computation of extreme eigenvalues of a symmetric real matrix are presented. Those algorithms are based on stability of Lagrangian duality for non-convex optimization and on duality in the difference of convex functions. Some experimental results which prove the robustness and efficiency of our algorithms are provided.

5

Approximation of the eigenvalue problem for elliptic operator by finite element method with numerical integration

Lewińska Ewa

Matematyka Stosowana : matematyka dla społeczeństwa

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1994

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Nr 37

3--14

EN

The author considers the effect of numerical integration in the case of solving a two-dimensional eigenvalue problem for the second-order elliptic differential operator via the finite element method. It is proved that the optimal estimates for eigenfunctions (namely, the estimates of the same order as the optimal estimates for the classical finite element approximation without numerical integration) are valid under the assumption that the precision of the numerical quadrature is the same as that for the corresponding boundary value problem.