Lower bounds for the first eigenvalue of the magnetic Laplacian

We consider a Riemannian cylinder Ω endowed with a closed potential 1-form A and study the magnetic Laplacian ${\mathrm{\Delta }}_A$ with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate.     

On two conjectures in the fixed membrane eigenvalue problem

In this paper we give conditions sufficient to insure that the following two results hold for eigenfunctions of the fixed membrane problem defined over a bounded region of the plane:

1. The first eigenfunction has only one interior critical point.
2. The second eigenfunction has no closed interior nodal curve.

     

Lower bounds for the first eigenvalue of the Dirac operator on compact Riemannian manifolds

Some new, improved, curvature depending lower bounds for the first eigenvalue of the Dirac operator on compact Riemannian manifolds are proved. If certain curvature conditions are satisfied, then these lower bounds are also useful in cases where the scalar curvature has zeros or attains negative values. This implies stronger vanishing theorems for harmonic spinors.     

Lower eigenvalue bounds for pencils of matrices

A procedure is set up for obtaining lower eigenvalue bounds for pencils of matrices A—vB where A is a Stieltjes matrix and B is positive definite, under assumptions suitable for the estimation of asymptotic convergence rates of locally perturbed factorization iterative schemes. Using these results and a formerly developed approach for estimating upper bounds, we widely confirm Gustafsson's conjecture concerning the nonnecessity of Axelsson's perturbations. In so doing, we however keep local perturbations, thereby enlarging the number of applications where their sufficiency is proven; their necessity remains, on the other hand, an open question.     

Explicit bounds of the first eigenvalue

It is proved that the general formulas, obtained recently for the lower bound of the first eigenvalue, can be further bounded by one or two constants depending on the coefficients of the corresponding operators only. Moreover, the ratio of the upper and lower bounds is no more than four.     

Pertubation bounds for the definite generalized eigenvalue problem

Let A and B be Hermitian matrices, and let c(A, B) = inf{|x^H(A + iB)x|:6 = 1}. The eigenvalue problem Ax = λBx is called definite if c(A, B)>0. It is shown that a definite problem has a complete system of eigenvectors and that its eigenvalues are real. Under pertubations of A and B, the eigenvalues behave like the eigenvalues of a Hermitian matrix in the sense that there is a 1-1 pairing of the eigenvalues with the perturbed eigenvalues and a uniform bound for their differences (in this case in the chordal metric). Pertubation bounds are also developed for eigenvectors and eigenspaces.     

Isoperimetric bounds for the first eigenvalue of the Laplacian

In this paper, generalizing an earlier result by Payne–Rayner, we prove an isoperimetric lower bound for the first eigenvalue of the Laplacian in the fixed membrane problem on a compact minimal surface in a Euclidean space R ⁿ with weakly connected boundary. We also prove an isoperimetric upper bound for the first eigenvalue of the Laplacian of an embedded closed hypersurface in R ⁿ .     

Lower bounds for the quadratic assignment problem

We investigate the classical Gilmore-Lawler lower bound for the quadratic assignment problem. We provide evidence of the difficulty of improving the Gilmore-Lawler bound and develop new bounds by means of optimal reduction schemes. Computational results are reported indicating that the new lower bounds have advantages over previous bounds and can be used in a branch-and-bound type algorithm for the quadratic assignment problem.     

An isoperimetric inequality for the first eigenfunction in the fixed membrane problem

Summary An isoperimetric inequality is obtained which relates theL ₁-andL ₂-integrals of the first eigenfunction in the problem of the vibrating clamped membrane.

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Zusammenfassung Für das Problem der eingespannten schwingenden Membran wird zwischen demL ₁-und demL ₂-Integral der ersten Eigenfunktion eine isoperimetrische Ungleichung hergeleitet.

     

Lower bounds of principal eigenvalue in dimension one

For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case (i.e., the ergodic case) is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case (one-dimensional diffusions) and ends at the discrete one (birth-death processes). Possible generalization of the results studied here is discussed at the end of the paper.     

Lower bounds for the axial three-index assignment problem

This paper describes new bounding methods for the axial three-index assignment problem (3AP). For calculating 3AP lower bounds, we use a projection method followed by a Hungarian algorithm, based on a new Lagrangian relaxation. We also use a cost transformation scheme, which iteratively transforms 3AP costs in a series of equivalent 3APs, which provides the possibility of improving the 3AP lower bound. These methods produce efficiently computed relatively tight lower bound.     

Lower bounds for Turán's problem

Turán's problem is to determine the maximum numberT(n,k,t) oft-element subsets of ann-set without a complete sub-hypergraph onk vertices, i.e., allt-subsets of ak-set. It is proved that fora≥1 fixed andt sufficiently largeT(n, t+a,t)>(1-a(a+4+o(1))logt/( _a ^t )( _t ⁿ holds     

An alternative proof of lower bounds for the first eigenvalue on manifolds

Recently, Andrews and Clutterbuck [1] gave a new proof of the optimal lower eigenvalue bound on manifolds via modulus of continuity for solutions of the heat equation. In this short note, we give an alternative proof of Theorem 2 in [1]. More precisely, following Ni's method (Section 6 of [5]), we give an elliptic proof of this theorem.     

Upper and lower bounds for the first Dirichlet eigenvalue of a triangle

We prove some new upper and lower bounds for the first Dirichlet eigenvalue of a triangle in terms of the lengths of its sides.