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61.
R. Komendarczyk 《Transactions of the American Mathematical Society》2006,358(6):2399-2413
In the 3-dimensional Riemannian geometry, contact structures equipped with an adapted Riemannian metric are divergence-free, nondegenerate eigenforms of the Laplace-Beltrami operator. We trace out a two-dimensional consequence of this fact: there is a close relationship between the topology of the contact structure on a convex surface in the 3-manifold (the dividing curves) and the nodal curves of Laplacian eigenfunctions on that surface. Motivated by this relationship, we consider a topological version of Payne's conjecture for the free membrane problem. We construct counterexamples to Payne's conjecture for closed Riemannian surfaces. In light of the correspondence between the nodal lines and dividing curves, we interpret the conjecture in terms of the tight versus overtwisted dichotomy for contact structures.
62.
A. A. Abramov V. I. Ul’yanova L. F. Yukhno 《Computational Mathematics and Mathematical Physics》2008,48(6):942-945
The nonlinear self-adjoint eigenvalue problem for a Hamiltonian system of two ordinary differential equations is examined under the assumption that the matrix of the system is a monotone function of the spectral parameter. Certain properties of eigenvalues that were previously established by the authors for Hamitonian systems of arbitrary order are now worked out in detail and made more precise for the above system. In particular, a single second-order ordinary differential equation is analyzed. 相似文献
63.
The basic boundary-contact oscillation problems are considered for a three-dimensional piecewise-homogeneous isotropic elastic medium bounded by several closed surfaces. Using Carleman's method, the asymptotic formulas for the distribution of eigenfunctions and eigenvalues are obtained. 相似文献
64.
In this paper we investigate the vector bundle-valued Cauchy and Poisson kernel functions. We compute explicitly matrix-valued eigenfunctions of an invariant differential operator on the classical domain of Type I. Furthermore, a special choice of a vector bundle gives us a matrix-valued Cauchy and Poisson kernel function which satisfies the matrix-valued Laplacian operator on the classical domain of Type I. 相似文献
65.
66.
《Mathematical Methods in the Applied Sciences》2018,41(12):4691-4713
The Euler‐Bernoulli beam model with fully nonconservative boundary conditions of feedback control type is investigated. The output vector (the shear and the moment at the right end) is connected to the observation vector (the velocity and its spatial derivative on the right end) by a 2 × 2 matrix (the boundary control matrix), all entries of which are nonzero real numbers. For any combination of the boundary parameters, the dynamics generator, , of the model is a non–self‐adjoint matrix differential operator in the state Hilbert space. A set of 4 self‐adjoint operators, defined by the same differential expression as on different domains, is introduced. It is proven that each of these operators, as well as , is a finite‐rank perturbation of the same self‐adjoint dynamics generator of a cantilever beam model. It is also shown that the non–self‐adjoint operator, , shares a number of spectral properties specific to its self‐adjoint counterparts, such as (1) boundary inequalities for the eigenfunctions, (2) the geometric multiplicities of the eigenvalues, and (3) the existence of real eigenvalues. These results are important for our next paper on the spectral asymptotics and stability for the multiparameter beam model. 相似文献
67.
Jiayin Qin Xizhang Luo Jiang Xiao Xiao Huang Ruman Qiu 《International Journal of Infrared and Millimeter Waves》1999,20(5):783-789
In this paper, the density matrix equation of three-level system for far-infrared (FIR) Raman transition in vibrational ground state was solved. By means of iteration method and numerical calculation, the spectral characteristics of NH3 molecules FIR laser with TEA CO2-10R(14) laser pumping was studied theoretically. The theoretical result would help us to understand the physical mechanism of optically pumped molecular gases FIR laser with ground-state reversal transition. 相似文献
68.
H. Y. Tian S. Reutskiy C. S. Chen 《Numerical Methods for Partial Differential Equations》2008,24(3):1018-1036
In this article, we introduce a type of basis functions to approximate a set of scattered data. Each of the basis functions is in the form of a truncated series over some orthogonal system of eigenfunctions. In particular, the trigonometric eigenfunctions are used. We test our basis functions on recovering the well‐known Franke's and Peaks functions given by scattered data, and on the extension of a singular function from an irregular domain onto a square. These basis functions are further used in Kansa's method for solving Helmholtz‐type equations on arbitrary domains. Proper one level and two level approximation techniques are discussed. A comparison of numerical with analytic solutions is given. The numerical results show that our approach is accurate and efficient. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 相似文献
69.
It is outlined how the utilization of a basis of projected spin eigenfunctions can lead to increased computational efficiency
in the evaluation of matrix elements and density matrices in spin-coupled valence bond calculations.
Received: 17 September 1997 / Accepted: 23 October 1997 相似文献
70.
We study the spectral probleml(u)=−u″+q(x)u(x)=λu(x),u′(0)=0, u′(π)=mλu(π), where λ andm are a spectral and a physical parameter. Form<0, we associate with the problem a self-adjoint operator in Pontryagin space II1. Using this fact and developing analytic methods of the theory of Sturm-Liouville operators, we study the dynamics of eigenvalues
and eigenfunctions of the problems asm→−0.
Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 163–172, August, 1999. 相似文献