2 edition of Accurate bounds for the eigenvalues of the laplacian and applications to rhombical domains. found in the catalog.
Accurate bounds for the eigenvalues of the laplacian and applications to rhombical domains.
Cleve B. Moler
Published
1969
by Stanford University in Stanford
.
Written in
Edition Notes
| Series | Technical report -- No. CS 121. |
| Contributions | Stanford University. School of Humanities and Sciences. Computer Science Department. |
| The Physical Object | |
|---|---|
| Pagination | 17 p. |
| Number of Pages | 17 |
| ID Numbers | |
| Open Library | OL19991068M |
Laplacian Eigenvalues modified MPS, as mentioned in [4], solution of an ill-conditioned system is required to obtain these coefficients. Our method works best for smooth domains but still delivers an improvement over MPS even for “hostile” domains with singular corners. The methodology described above can, however, be applied to more. 3. Higher eigenvalues usually require very high tessellation levels to be computed accurately. The accuracy of our technique scales like O(λ)comparedtoO ¡ λ3 ¢ for finite elements. We mention another application to PDE’s. The original domain is usually replaced with an approximate one (often a polygon) and the remaining part is thrown away.
Upper bounds on eigenvalues for manifolds Metrics invariant under a group action Submanifolds Upper Bounds on Eigenvalues of the Laplacian: Surfaces and Beyond Emily B. Dryden Bucknell University Texas Geometry and Topology Conference Febru Emily B. Dryden Upper Bounds on Eigenvalues of the Laplacian. the Laplacian on General 2-D Domains Patrick Guidotti∗ James V. Lambers † March 5, Abstract In this paper we address the problem of determining and efficiently computing an approximation to the eigenvalues of the negative Lapla-cian − on a generaldomain Ω ⊂ R2 subject to homogeneousDirichlet or Neumann boundary conditions.
above mentioned bounds and our bounds on some examples of graphs to conclude that our bounds are good in some sense. 2 MAIN RESULTS First, we obtain lower bounds for the Laplacian eigenvalues 1 and 2,of a simple graph by employing the following result, known as the Cauchy interlacing property. For a proof, see [9, p. ]. Lemma 1. the set of all bounded domains of volume V of M. The first result in this subject is the famous Faber-Krahn Theorem [14, 20], originally conjectured by Rayleigh, stating that Euclidean balls minimize the first eigenvalue of the Dirichlet Laplacian among all domains of given volume.
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PDF | On Jan 1,CLEVE B. MOLER published Accurate Bounds for the Eigenvalues of the Laplacian and Applications to Rhombical Domains | Find. ACCURATE BOUNDS FOR THE EIGENVALUES OF THE LAPLACIAN AND APPLICATIONS TO RHOMBICAL DOMAINS* bY Cleve B. Mole. Introduction. By an eigenvalue and eigenfunction of Laplace's operator on a bounded two-dimensional domain G we mean a positive number h and a non-zero function u(x,y) which satisfy (1), and n"(x,d + h'-&y) = 0, (x,y) c G.
In this study, the bounds for eigenvalues of the Laplacian operator on an L-shaped domain are determined. By adopting some special functions in Goerisch method for lower bounds and in traditional Rayleigh–Ritz method for upper bounds, very accurate bounds to eigenvalues for the problem are by: 9.
C.B. Moler, Accurate bounds for the eigenvalues of the Laplacian and applications to rhombical domains, Report # CSComputer Science Department, Stanford University, Google Scholar [MP]Cited by: 2.
[41] Moler, C. B., “Accurate bounds for the eigenvalues of the Laplacian and applications to rhombical domains”, Technical Report CS“ A fast numerical method for ideal fluid flow in domains with multiple stirrers ”, Nonlinearity 31 (); Cited by: 9. Bounds to eigenvalues of the Laplacian on L-shaped domain by variational methods Article in Journal of Computational and Applied Mathematics (4) December with 40 Reads.
() Bounds to eigenvalues of the Laplacian on L-shaped domain by variational methods. Journal of Computational and Applied Mathematics() Computing the Casimir energy using the point-matching method. 6 Eigenvalues of the Laplacian In this section, we consider the following general eigenvalue problem for the Laplacian, ‰ ¡∆v = ‚v x 2 Ω v satisfies symmetric BCs x 2 @Ω: To say that the boundary conditions are symmetric for an open, bounded set Ω in Rn means that.
According to the literature and existing upper/lower bounds on the eigenvalues of the Laplacian as a function of node degrees, it seems there is a graph with this property.
However, I was not able to find such a graph by generating all graph with $4, 5, \dots, 10$ vertices. In this paper, we investigate non-zero positive eigenvalues of the Laplacian with Dirichlet boundary condition in an n-dimentional Euclidean space ℝn, then we obtain an new upper bound of the (k + 1)-th eigenvalue λk+1, which improve the previous estimate which.
We shall study inequalities for eigenvalues of the Laplacian when the spectrum, i. those values of λ for which a non-trivial solution exists, is discrete. We shall conclude the paper with a few applications to the Morse-Smale index for a minimal immersion in R n, n≥3 and a few historical remarks related to Lagrange and his work.
The investigation of eigenvalues and eigenfunctions of the Laplace operator in a bounded domain or a manifold is a subject with a history of more than two hundred years.
This is still a central area in mathematics, physics, engineering, and computer science, and activity has increased dramatically in the past twenty years for several reasons. () A method of estimating lower bounds for eigenvalues of differential operators by the method of fictitious domains.
USSR Computational Mathematics and Mathematical Physics() An algorithm to compute the eigenvalues/functions of the laplacian operator within a region containing a sharp corner.
As in the Dirichlet Laplacian case, Pólya's inequality for the fractional Laplacian on any bounded domain is still an open problem. Moreover, we also investigate the equivalence of several related inequalites for bounded domains by using the convexity, the Lieb–Aizenman procedure (the Riesz iteration), and some transforms such as the Laplace.
Accurate eigenvalues of the Laplacian 9 A test case • Consider the unit square domain with a border of width η that surrounds an inner square of width 1 − 2η. • Assign a very high stiffness s to the border and a constant stiffness of 1 to the inner square.
• It can be proved using classical minimax arguments that as s → ∞, the smaller eigenpairs of this domain. Let G be a simple connected graph of order n, where n ≥ 2. Its normalized Laplacian eigenvalues are 0 = λ 1 ≤ λ 2 ≤ ⋯ ≤ λ n ≤ 2.
In this paper, some new upper and lower bounds on λ n are obtained, respectively. Moreover, connected graphs with λ 2 = 1 (or λ n − 1 = 1) are also characterized. MSCC50, 15A 1,2 has eigenvalues 0 and 2, and so is positive semidefinite, where we recall that a symmetric matrix M is positive semidefinite if all of its eigenvalues are non-negative.
Also recall that this is equivalent to x T Mx ≥ 0, for all x ∈ Rn. It follows immediately that the Laplacian of every graph is positive semidefinite. One way to see. From this, we see that the ratios of Laplacian eigenvalues are scale invariant. (c) Laplacian eigenvalues are translation and rotation invariant.
Features used by Khabou, Hermi, and Rhouma Let Ω be a domain represented by a binary image. Using the Dirichlet-Laplacian eigenvalues for Ω, define three sets of features as follows. F1(Ω. Applications, and Computations Lectures 12+ Laplacian Eigenvalue Problems for General Domains: IV.
Asymptotics of the Eigenvalues Lecturer: Naoki Saito Scribe: Ernest Woei/Allen Xue May 8 & 10, The basic references for this lecture are [1, Sec. ], [2, Sec VI.2] and [3, Sec. 1 Asymptotics of the Eigenvalues. regular graphs) which bounds the number of edges between the two subgraphs of G that are the least connected to one another using the second smallest eigenvalue of the Laplacian of G.
Contents 1. Introduction 1 2. Spectral Theorem for Real Matrices and Rayleigh Quotients 2 3. The Laplacian and the Connected Components of a Graph 5 4.
Cheeger. the k disjoint domains with C Upper Bounds for the Eigenvalues of the Laplacian on Forms 95 4. Estimatesoff(r)and f (r) f(r) LetaRiemannianmanifold(M,g)beasinSection3.
Firstwebegin withestimatesoff(r). Proposition (Estimatesoff(r)).Z. Du and B. Zhou, Upper bounds for the sum of Laplacian eigenvalues of graphs, Linear Algebra Appl. () – Crossref, Google Scholar; 6. E. Fritscher, C. Hoppen, I. Rocha and V.
Trevisan, On the sum of the Laplacian eigenvalues of a tree, Linear Algebra Appl. () – Crossref, Google Scholar; 7. E.The Bounds for Eigenvalues of Normalized Laplacian Matrices and Signless Laplacian Matrices Author: Serife Büyükköse, Sehri Gülçiçek Eski Subject: In this paper, we found the bounds of the extreme eigenvalues of normalized Laplacian matrices and signless Laplacian .
