On the motion of rigid bodies in a viscous incompressible fluid

This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.AMS Subject Classification: 49Q10m, 35P15, 49J20.     

Bounds for eigenvalues and condition numbers in the -version of the finite element method

In this paper, we present a theory for bounding the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices arising from the -version of finite element analysis. Bounds are derived for the eigenvalues and the condition numbers, which are valid for stiffness matrices based on a set of general basis functions that can be used in the -version. For a set of hierarchical basis functions satisfying the usual local support condition that has been popularly used in the -version, explicit bounds are derived for the minimum eigenvalues, maximum eigenvalues, and condition numbers of stiffness matrices. We prove that the condition numbers of the stiffness matrices grow like , where is the number of dimensions. Our results disprove a conjecture of Olsen and Douglas in which the authors assert that ``regardless of the choice of basis, the condition numbers grow like or faster". Numerical results are also presented which verify that our theoretical bounds are correct.

     

Differentiable Eigenvalues of Perturbed Quadratic Matrix Functions

Let T(λ, ε ) = λ² + λC + λεD + K be a perturbed quadratic matrix polynomial, where C, D, and K are n × n hermitian matrices. Let λ₀ be an eigenvalue of the unperturbed matrix polynomial T(λ, 0). With the falling part of the Newton diagram of det T(λ, ε), we find the number of differentiable eigenvalues. Some results are extended to the general case L(λ, ε) = λ² + λD(ε) + K, where D(ε) is an analytic hermitian matrix function. We show that if K is negative definite on Ker L(λ₀, 0), then every eigenvalue λ(ε) of L(λ, ε) near λ₀ is analytic.     

Eigenvalues and lambda constants on Riemannian submersions

Given a Riemannian submersion, we study the relation between lambda constants introduced by G. Perelman on the base manifold and the total space of a Riemannian submersion. We also discuss the relationship between the first eigenvalues of Laplacians on the base manifold and that of the total space. The quantities on warped products are discussed in detail.      

The multi-fan graphs are determined by their Laplacian spectra

A multi-fan graph is a graph of the form (P_n1+P_n2+?+P_nk)×b, where b is a universal vertex, and P_n1+P_n2+?+P_nk is the disjoint union of paths P_ni(n_i?1) for i=1,2,…,k. In particular, if k=1, the multi-fan graph P_n1×b is the classical fan graph F_n1+1. It is proved that all the multi-fan graphs are determined by their Laplacian spectra.     

Eigenvalues and discrete boundary value problems for the one-dimensional p-laplacian

In this paper, we characterize the eigenvalues and show existence of positive solutions to discrete boundary value problem (here ?(s)=|s|^p−2s, p>1 and λ>0 is a parameter)

     

Limiting behavior of the eigenvalues of a multivariate F matrix

The spectral distribution of a central multivariate F matrix is shown to tend to a limit distribution in probability under certain conditions as the number of variables and the degrees of freedom tend to infinity.     

Comprehensive study of the correlations that exist among the members of the [n]cyclacene series and the Möbius[n]cyclacene series

Cyclacenes are the smallest substructures of carbon nanotubes used in modelling studies. The systematics that exists between Hückel molecular orbital eigenvalues and eigenvectors of cyclacenes are delineated. This study of cyclacenes combines the interconnection of concepts of complementarity theorem, characteristic and matching polynomial recursion equations, embedding, greater than twofold symmetry and doubly degenerate eigenvalues, open-shell singlet character, and pairing theorem. Proof that cyclacenes have more open-shell (diradical) character than do polyacenes is also provided by the sum total of this work. Mirror-plane scission of even-ring cyclacenes gives linear polyacene fragments. This shows that the properties of the successor linear polyacene must be contained in the precursor cyclacene. Corresponding Möbius[n]cyclacene isomers display contrasting and unusual comparative properties. A partial list of contrasting properties include alternant (cyclacenes) versus nonalternant (Möbius[n]cyclacene) polyenes, presence of Hamiltonian circuits in Möbius[n]cyclacene and presence of oscillatory electronic properties in cyclacenes.     

Eigensolutions to a vibroacoustic interior coupled problem with a perturbation method

In this paper, an efficient and robust numerical method is proposed to solve non-symmetric eigenvalue problems resulting from the spatial discretization with the finite element method of a vibroacoustic interior problem. The proposed method relies on a perturbation method. Finding the eigenvalues consists in determining zero values of a scalar that depends on angular frequency. Numerical tests show that the proposed method is not sensitive to poorly conditioned matrices resulting from the displacement–pressure formulation. Moreover, the computational times required with this method are lower than those needed with a classical technique such as, for example, the Arnoldi method.