The first eigenvalue of a spherical cap

We find upper and lower bounds for the first eigenvalue of the Laplacian on the two-sphere from which a disk has been removed, with Dirichlet conditions imposed on the resulting boundary. When the radius of the disk tends to zero our lower bound is sharper than that obtained by Del Grosso, Gerardi, and Marchetti in the preceding paper.     

An upper bound for the first eigenvalue of a spherical cap

By working with suitable test functions, we obtain an upper bound for the principal eigenvalue of a geodesic ball on a sphere of arbitrary dimension. This bound is sharp in the limiting case when the radius of the ball approaches the diameter of the sphere.This research was supported by ARO Grant 28905-MA.     

Bounds for the second largest eigenvalue of a transition matrix

We find an upper bound, with general form, for the second largest eigenvalue of a transition matrix; special cases of which have previously been proposed as upper bounds and others which are new improvements.     

Asymmetric stokes flow past a spherical cap

Ranger's solution of the asymmetric streating Stokes' flow past a spherical cap is analyzed in detail. It is found that most caps exhibit wakes within their concavities, but the dividing streamsurface does not always emanate from the rim. Drag and torque formulae valid for all cap angles are derived and these are in agreement with the known results in the limiting cases when the cap becomes a sphere and a circular disc.

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Zusammenfassung Die Lösung von Ranger für die unsymmetrische Stokes-Strömung um eine Kugelhaube wurde im Detail analysiert. Es wurde gefunden, dass die meisten Kugelhauben in ihrem Hohlraum Albösungsgebiete aufweisen, doch beginnt die Ablösungs-Stromfläche nicht immer am Rand der Haube. Widerstand und Moment wurde für alle Haubenformen berechnet; die bekannten Resultate der Kugel und der Kreisscheibe ergeben sich als Grenzfälle.

     

The Stokes drag for asymmetric flow past a spherical cap

The non symmetric stokes flow past a spherical cap is solved by the method of complementary integral representations. In particular, the drag and couple are evaluated for the hemispherical cup.

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Zusammenfassung Die nicht-symmetrische Stokes-Strömung um eine Kugelhaube wird bestimmt mit Hilfe der Methode der komplementären Integraldarstellungen. Für den besonderen Fall einer Halbkugel-Haube wird der Widerstand und das Moment bestimmt.

     

Bounds for the first eigenvalue of the elastically supported membrane on convex domains

Bartas principle and gradient bounds for the torsion function are the main tools for deriving lower bounds for the first eigenvalue. The optimal domains are an infinite strip, a disk or an annulus in different situations.     

On the second eigenvalue of the Laplace operator on a spherical band

In this paper we prove that the second eigenvalue of the Laplacian for a spherical band on the unit sphere has multiplicity 2. We also show that among all spherical bands of given fixed area less than the second eigenvalue is maximized at the band which is symmetrical with respect to the equator.

     

Numerical integration with polynomial exactness over a spherical cap

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere \mathbbS²\mathbb{S}^2, we discuss tensor product rules with n ²/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n ³) nodes for numerical integration over spherical caps on \mathbbS²\mathbb{S}^2. For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that have O(n ^d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that are exact for all spherical polynomials of degree ≤ n have at least O(n ^d ) nodes and possess a certain regularity property.     

Bounds for the first eigenvalue of the horizontal Laplacian in positively curved sub-Riemannian manifolds

We establish lower bounds for the first non-zero eigenvalue for the natural geometric sub-elliptic Laplacian operator defined on sub-Riemannian manifolds of step 2 that satisfy a positive curvature condition. The methods are very general and can be applied even when the sub-Riemannian geometry has considerable torsion.     

Existence criteria for singular boundary value problems modelling the membrane response of a spherical cap

General existence criteria are presented for singular boundary value problems where our nonlinearity is allowed to change sign. An example in the theory of membrane response of a spherical cap is also discussed.     

Bounds for the sum of distances of spherical sets of small size

We derive upper and lower bounds on the sum of distances of a spherical code of size N in n dimensions when $N=\mathrm{\Theta }(n^{\alpha }),0<\alpha ?2$. The bounds are derived by specializing recent general, universal bounds on energy of spherical sets. We discuss asymptotic behavior of our bounds along with several examples of codes whose sum of distances closely follows the upper bound.     

Bending of a long circular cylinder containing a thin spherical cap cavity

This paper deals with the three dimensional analysis of the stress distribution in a long circular cylinder containing a concentric very thin spherical cap cavity. The central plane of the cavity is perpendicular to the axis of the cylinder, and the cylinder is subjected to bending. The equations of the classical theory of elasticity are solved in terms of an unknown function which is then shown to be the solution of a Fredholm integral equation of the second kind. Numerical solution of the integral equation is obtained. The resultant stress intensity factors are presented in a graphical form for various proximity ratios.     

紧致流形的第一特征值下界

对紧致Riemannian流形(无边或带有凸边界)的第一(Neumann)特征值,用流形的直径和Ricci曲率的下界,给出一些新的下界估计.     

Exact controllability of a spherical cap: Numerical implementation of HUM

This paper deals with the numerical implementation of the exact boundary controllability of the Reissner model for shallow spherical shells (Ref. 1). The problem is attacked by the Hilbert uniqueness method (HUM, Refs. 2–4), and we propose a semidiscrete method for the numerical approximation of the minimization problem associated to the exact controllability problem. The numerical results compare well with the results obtained by a finite difference and conjugate gradient method in Ref. 5.This work was done when the first two authors were at CNR-IAC, Rome, Italy as Graduate Students.     

紧致流形的第一特征值下界

对紧致Riemannian流形(无边或带有凸边界)的第一(Neumann)特征值,用流形的直径和Ricci曲率的下界,给出一些新的下界估计.