VIBRATIONS OF A MEMBRANE WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY

1995 ◽  
Vol 05 (05) ◽  
pp. 565-585 ◽  
Author(s):  
MIGUEL LOBO ◽  
EUGENIA PÉREZ
Keyword(s):  
Perturbation Theory ◽  
Asymptotic Behavior ◽  
Small Parameter ◽  
Local Problem ◽  
Micro Structure ◽  
Spectral Perturbation Theory ◽  
Concentrated Masses ◽  
The Masses ◽  
Spectral Perturbation ◽  
Small Eigenvalues

We consider the asymptotic behavior of the vibrations of a membrane occupying a domain Ω ⊂ ℝ2. The density, which depends on a small parameter ε, is of order O(1) out of certain regions where it is O(ε−m) with m>0. These regions, the concentrated masses with diameter O(ε), are located near the boundary, at mutual distances O(η), with η=η(ε)→0. We impose Dirichlet (respectively Neumann) conditions at the points of ∂Ω in contact with (respectively, out of) the masses. Depending on the value of the parameter m(m>2, m=2 or m<2) we describe the asymptotic behavior of the eigenvalues. Small eigenvalues, of order O(εm−2) for m>2, are approached via those of a local problem obtained from the micro-structure of the problem, while the eigenvalues of order O(1) are approached through those of a homogenized problem, which depend on the relation between ε and η. Techniques of boundary homogenization and spectral perturbation theory are used to study this problem.

ON VIBRATIONS OF A BODY WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY

1993 ◽  
Vol 03 (02) ◽  
pp. 249-273 ◽  
Author(s):  
MIGUEL LOBO ◽  
EUGENIA PEREZ
Keyword(s):  
Asymptotic Behavior ◽  
Small Parameter ◽  
Critical Size ◽  
Local Problem ◽  
Boundary Values ◽  
Micro Structure ◽  
Spectral Convergence ◽  
Fourier And Laplace Transforms ◽  
Concentrated Masses ◽  
The Masses

We consider the asymptotic behavior of the vibration of a body occupying a region Ω⊂ℝ3. The density, which depends on a small parameter ε, is of order O(1) out of certain regions where it is O(ε–m) with m>2. These regions, the concentrated masses with diameter O(ε), are located near the boundary, at mutual distances O(η), with η=η(ε)→0. We impose Dirichlet (respectively Neumann) conditions at the points of ∂Ω in contact with (respectively, out of) the masses. For the critical size ε=O(η2), the asymptotic behavior of the eigenvalues of order O(εm−2) is described via a Steklov problem, where the ‘mass’ is localized on the boundary, or through the eigenvalues of a local problem obtained from the micro-structure of the problem. We use the techniques of the formal asymptotic analysis in homogenization to determine both problems. We also use techniques of convergence in homogenization, Semigroups theory, Fourier and Laplace transforms and boundary values of analytic functions to prove spectral convergence. In the same framework we study the case m=2 as well as the case when other boundary conditions are imposed on ∂Ω.

scholarly journals Spectral Perturbation Meets Incomplete Multi-view Data

2019 ◽  
Author(s):  
Hao Wang ◽  
Linlin Zong ◽  
Bing Liu ◽  
Yan Yang ◽  
Wei Zhou
Keyword(s):  
Perturbation Theory ◽  
Spectral Clustering ◽  
Matrix Completion ◽  
Similarity Matrix ◽  
Spectral Perturbation Theory ◽  
Solid Fusion ◽  
Incomplete Similarity ◽  
Pic Method ◽  
Risk Bounds ◽  
Spectral Perturbation

Beyond existing multi-view clustering, this paper studies a more realistic clustering scenario, referred to as incomplete multi-view clustering, where a number of data instances are missing in certain views. To tackle this problem, we explore spectral perturbation theory. In this work, we show a strong link between perturbation risk bounds and incomplete multi-view clustering. That is, as the similarity matrix fed into spectral clustering is a quantity bounded in magnitude O(1), we transfer the missing problem from data to similarity and tailor a matrix completion method for incomplete similarity matrix. Moreover, we show that the minimization of perturbation risk bounds among different views maximizes the final fusion result across all views. This provides a solid fusion criteria for multi-view data. We motivate and propose a Perturbation-oriented Incomplete multi-view Clustering (PIC) method. Experimental results demonstrate the effectiveness of the proposed method.

A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

1988 ◽  
Vol 102 ◽  
pp. 343-347
Author(s):  
M. Klapisch
Keyword(s):  
Perturbation Theory ◽  
Small Parameter ◽  
Classification Scheme ◽  
Sum Rules ◽  
Energy Levels ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Formal Expansion ◽  
Atomic Energy Levels ◽  
As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

scholarly journals Spectral deformation in a problem of singular perturbation theory

2019 ◽  
Vol 484 (4) ◽  
pp. 397-400
Author(s):  
S. A. Stepin ◽  
V. V. Fufaev
Keyword(s):  
Perturbation Theory ◽  
Asymptotic Behavior ◽  
Integral Method ◽  
Deformation Parameter ◽  
Liouville Problem ◽  
Singular Perturbation Theory ◽  
Different Types ◽  
Spectral Deformation ◽  
Sturm Liouville ◽  
Spectral Complex

Quasi-classical asymptotic behavior of the spectrum of a non-self-adjoint Sturm–Liouville problem is studied in the case of a one-parameter family of potentials being third-degree polynomials. For this problem, the phase-integral method is used to derive quantization conditions characterizing the asymptotic distribution of the eigenvalues and their concentration near edges of the limit spectral complex. Topologically different types of limit configurations are described, and critical values of the deformation parameter corresponding to type changes are specified.

Sign in / Sign up

Export Citation Format

Share Document