可修复人机系统的指数稳定性

研究了两相同部件温储备可修的人机系统,运用C_0半群的相关理论,对系统主算子的谱界进行估值.估算系统的算子产生的半群的增长界,然后运用了共尾的概念及相关的理论,得到了系统算子A+B的谱界与系统算子产生的半群的增长界相同.进而运用相关代数知识证得,0为系统算子的简单本征值,并分析了系统算子的谱分布,得到系统的指数稳定性.并研究了系统算子预解式的特性.对任意给定的δ0,γ=a+bi,-μ+δa_1≤a≤a_2,得到lim|b|→∞‖R(γ;A+B)‖=0.进而得到在~sRγ≥a_1的右半平面内相应于系统算子A+B的谱点由有限个本征值组成.     

Analytic proof of dual variational formula for the first eigenvalue in dimension one

The first non-zero eigenvalue is the leading term in the spectrum of a self-adjoint operator. It plays a critical role in various applications and is treated in a large number of textbooks. There is a well-known variational formula for it (called the Min-Max Principle) which is especially effective for an upper bound of the eigenvalue. However, for the lower bound of the spectral gap, some dual variational formulas have been obtained only very recently. The original proofs are probabilistic. Some analytic proofs in one-dimensional case are proposed and certain extension is made. Project supported in part by the National Natural Science Foundation of China (Grant No. 19631060), Qiu Shi Science & Technology Foundation, DPFIHE, MCSEC and MCMCAS.     

Particular features of implementation of an unsaturated numerical method for the exterior axisymmetric Neumann problem

Using Babenko’s profound ideas, we construct a fundamentally new unsaturated numerical method for solving the spectral problem for the operator of the exterior axisymmetric Neumann problem for Laplace’s equation. We estimate the deviation of the first eigenvalue of the discretized problem from the eigenvalue of the Neumann operator. More exactly, the unsaturated discretization of the spectral Neumann problem yields an algebraic problem with a good matrix, i.e., a matrix inheriting the spectral properties of the Neumann operator. Thus, its spectral portrait lacks “parasitic” eigenvalues provided that the discretization error is sufficiently small. The error estimate for the first eigenvalue involves efficiently computable parameters, which in the case of C ^∞-smooth data provides a foundation for a guaranteed success.     

Guaranteed lower eigenvalue bounds for the biharmonic equation

The computation of lower eigenvalue bounds for the biharmonic operator in the buckling of plates is vital for the safety assessment in structural mechanics and highly on demand for the separation of eigenvalues for the plate’s vibrations. This paper shows that the eigenvalue provided by the nonconforming Morley finite element analysis, which is perhaps a lower eigenvalue bound for the biharmonic eigenvalue in the asymptotic sense, is not always a lower bound. A fully-explicit error analysis of the Morley interpolation operator with all the multiplicative constants enables a computable guaranteed lower eigenvalue bound. This paper provides numerical computations of those lower eigenvalue bounds and studies applications for the vibration and the stability of a biharmonic plate with different lower-order terms.     

Spectral Analysis of the Differential Operator in Wavelet Bases

We analyze the spectral properties of the differential operator in wavelet bases. The problem is studied on a periodic domain, with periodized wavelets. An algorithm for finding the eigenvalue function of the differential operator is presented, and general conditions that ensure a "nicely behaving" eigenvalue function are derived.     

Deforming the point spectra of one-dimensional Dirac operators

We provide a method of inserting and removing any finite number of prescribed eigenvalues into spectral gaps of a given one-dimensional Dirac operator. This is done in such a way that the original and deformed operators are unitarily equivalent when restricted to the complement of the subspace spanned by the newly inserted eigenvalue. Moreover, the unitary transformation operator which links the original operator to its deformed version is explicitly determined.

     

Error bound on the eigenvalue of the difference analog of the spectral problem in a cylindrical coordinate system

We consider the eigenvalue problem for the operator defined in a rectangle whose vertical left-hand side coincides with the z-axis. A difference scheme is constructed by an integrointerpolation method. An error bound is obtained for the simple eigenvalue of the difference analog in weighted generalized spaces W ₂ ² and W ₂ ³ .Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 29–35, 1986.     

Eigenvalue estimates for the Dirac operator on Kähler–Einstein manifolds of even complex dimension

In the case of a Kähler–Einstein manifold of positive scalar curvature and even complex dimension, an improved lower bound for the first eigenvalue of the Dirac operator is given. It is shown by a general construction that there are manifolds for which this new lower bound itself is the first eigenvalue.     

A Minimax Principle for the Eigenvalues in Spectral Gaps

A minimax principle is derived for the eigenvalues in the spectralgap of a possibly non-semibounded self-adjoint operator. Itallows the nth eigenvalue of the Dirac operator with Coulombpotential from below to be bound by the nth eigenvalue of asemibounded Hamiltonian which is of interest in the contextof stability of matter. As a second application it is shownthat the Dirac operator with suitable non-positive potentialhas at least as many discrete eigenvalues as the Schrödingeroperator with the same potential.     

板几何中具反射边界条件的迁移算子的谱分析

在Lp(1 p<∞)空间上研究了板几何中具反射边界条件下各向异性、连续能量、非均匀介质的迁移方程,证明了该迁移算子产生C0半群的Dyson-Phillips展开式的二阶余项在Lp(1相似文献   

一类人与出租车构成的排队模型主算子的谱特征

运用C0-半群理论研究一类人与出租车构成的排队模型主算子的谱特征.首先证明0是对应于该排队模型的主算子的几何重数为1的特征值,其次证明在虚轴上除了0以外其他所有点都属于该算子的豫解集,然后证明0是该主算子共轭算子的特征值.     

On the CR analogue of Obata’s theorem in a pseudohermitian 3-manifold

In this paper, we first prove the CR analogue of Obata’s theorem on a closed pseudohermitian 3-manifold with zero pseudohermitian torsion. Secondly, instead of zero torsion, we have the CR analogue of Li-Yau’s eigenvalue estimate on the lower bound estimate of positive first eigenvalue of the sub-Laplacian in a closed pseudohermitian 3-manifold with nonnegative CR Paneitz operator P ₀. Finally, we have a criterion for the positivity of first eigenvalue of the sub-Laplacian on a complete noncompact pseudohermitian 3-manifold with nonnegative CR Paneitz operator. The key step is a discovery of integral CR analogue of Bochner formula which involving the CR Paneitz operator. This research was supported in part by the NSC of Taiwan.     

Eigenvalues of the Wentzell–Laplace operator and of the fourth order Steklov problems

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell–Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a bounded domain in a Euclidean space. We study some fourth order Steklov problems and obtain isoperimetric upper bound for the first eigenvalue of them. We also find all the eigenvalues and eigenfunctions for two kind of fourth order Steklov problems on a Euclidean ball.     

Borg-type theorem for the missing eigenvalue problem

In this paper, we discuss the inverse spectral problem for Sturm–Liouville operators for the missing eigenvalue problem. We show that a Borg-type theorem for the missing eigenvalue problem of the Sturm–Liouville operator holds by the Weyl $m$-function.     

On the spectral properties of odd-order differential operators with integrable potential

We consider a boundary value problem with irregular boundary conditions for a differential operator of arbitrary odd order. The potential in this operator is assumed to be an integrable function. We suggest a method for studying the spectral properties of differential operators with integrable coefficients. We analyze the asymptotic behavior of solutions of the differential equation in question for large values of the spectral parameter. The eigenvalue asymptotics for the considered differential operator is obtained.     

Quaternionic Killing Spinors

In [17] we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kähler manifolds. In the present article we study the limiting case, i.e. manifolds where the lower bound is attained as an eigenvalue. We give an equivalent formulation in terms of a quaternionic Killing equation and show that the only symmetric quaternionic Kähler manifolds with smallest possible eigenvalue are the quaternionic projective spaces.     

Lower bounds for the logarithmic Sobolev constant avoiding uniform lower bounds on the Ricci curvature

In this paper we obtain a lower bound for the logarithmic Sobolev constant of the operator on C^∞(M) given by L^U f = Δ f - (?U|?f), where U ? C^∞(M), M being a finite dimensional compact Riemannian manifold without boundary, in terms of the spectral gap of L^U and the lowest eigenvalue of the operator -L^U + V, where V is a function related to U and the Ricci curvature of M. Under suitable conditions and being U ≡ 0, this result improves a previous one by J.-D. DEUSCHEL and D.W. STROOCK (J. Funct. Anal. 92 (1990), 30–48).     

Hyperbolicity of periodic solutions of functional differential equations with several delays

We study conditions for the hyperbolicity of periodic solutions to nonlinear functional differential equations in terms of the eigenvalues of the monodromy operator. The eigenvalue problem for the monodromy operator is reduced to a boundary value problem for a system of ordinary differential equations with a spectral parameter. This makes it possible to construct a characteristic function. We prove that the zeros of this function coincide with the eigenvalues of the monodromy operator and, under certain additional conditions, the multiplicity of a zero of the characteristic function coincides with the algebraic multiplicity of the corresponding eigenvalue.     

On the Existence of Trapped Modes in Channels of Arbitrary Cross-section

This article establishes the existence of trapped-mode solutions of a linearized water-wave problem. The fluid occupies a symmetric horizontal channel that is uniform everywhere apart from a confined region which either contains a thin vertical plate spanning the depth of the channel or has indentations in the channel walls. A trapped mode corresponds to an eigenvalue of a non-local Neumann–Dirichlet operator for an elliptic boundary-value problem in the fluid domain. The existence of such an eigenvalue is established by generalizing previous results concerning spectral theory for differential operators to this non-local operator. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

The Laplacian spectral radius of graphs

The Laplacian spectral radius of a graph is the largest eigenvalue of the associated Laplacian matrix. In this paper, we improve Shi’s upper bound for the Laplacian spectral radius of irregular graphs and present some new bounds for the Laplacian spectral radius of some classes of graphs.