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.     

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.     

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.

     

Numerical integration over a disc. A new Gaussian quadrature formula

Summary. We construct a quadrature formula for integration on the unit disc which is based on line integrals over distinct chords in the disc and integrates exactly all polynomials in two variables of total degree . Received August 8, 1996 / Revised version received July 2, 1997     

Numerical integration over a semi-infinite interval,using the lognormal distribution

Summary A method is described for numerical integration over a semiinfinite interval using a Gaussian formula, with the corresponding set of orthogonal polynomials constructed from a lognormal weight function.The lognormal weight function and hence the coefficients of the polynomials are functions of two arbitrary parameters; the mean and the logarithmic variance. The method is found to be of particular use for integration of bell-shaped or sharply spiked functions. Rapidly convergent results can be obtained in these cases, since the lognormal distribution can be used to provide a good approximation to the actual function to be integrated, by suitable choice of the two arbitrary parameters. Two examples are given for integrals with known solutions.     

Bounds for the First eigenvalue of a spherical cap

We study upper and lower bounds for the lowest eigenvalueλ of the Laplace operator on a spherical capC _θ inm-dimensional space (m ≥ 3). We prove that these bounds are sharp by finding asymptotic expressions forλ asθ → π and asθ → 0.     

Verifying exactness of relaxations for robust semi-definite programs by solving polynomial systems

In this paper, robust semi-definite programs are considered with the goal of verifying whether a particular LMI relaxation is exact. A procedure is presented showing that verifying exactness amounts to solving a polynomial system. The main contribution of the paper is a new algorithm to compute all isolated solutions of a system of polynomials. Standard techniques in computational algebra, often referred to as Stetter’s method [H.J. Stetter, Numerical Polynomial Algebra, SIAM, 2004], involve the computation of a Gröbner basis of the ideal generated by the polynomials and further require joint eigenvector computations in order to arrive at the zeros of the polynomial system. Our algorithm does neither require structural knowledge on the polynomial system, nor does it rely on the computation of joint eigenvectors.     

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.

     

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.     

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.     

Automatic integration over a sphere

We present an algorithm for automatic integration over an N-dimensional sphere. The quadrature formula is obtained by using a trapezoidal rule after a non-linear transformation, and allows to deal with integrand singularities on the surface or in the centre of the sphere.At the basis of the theoretical development lie the construction and the selection of suitable transformations.The algorithm is cast into an automatic integration program coded as a Standard Fortran sub-routine.     

Numerical solution of the Poisson equation over hypercubes using reduced Chebyshev polynomial bases

We describe how to use new reduced size polynomial approximations for the numerical solution of the Poisson equation over hypercubes. Our method is based on a non-standard Galerkin method which allows test functions which do not verify the boundary conditions. Numerical examples are given in dimensions up to 8 on solutions with different smoothness using the same approximation basis for both situations. A special attention is paid on conditioning problems.     

Cancellation of projective modules over polynomial extensions over a two-dimensional ring

Let R be a commutative Noetherian ring of dimension two with $1/2\in R$ and let $A=R[X_1,?,X_n]$. Let P be a projective A-module of rank 2. In this article, we prove that P is cancellative if ${\wedge }^2(P)\oplus A$ is cancellative.     

SomeL-functions of a polynomial ring over a finite field

An L-function involved in the enumeration of certain families of irreducible polynomials over a finite field is studied. For a number of cases, we confirm the hypothesis on zeros of that function, which is an analog of the hypothesis of Riemann and Weyl postulated for algebraic varieties over finite fields. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 476–495, July–August, 1996.     

Numerical integration over polygons using an eight-node quadrilateral spline finite element

In this paper, a cubature formula over polygons is proposed and analysed. It is based on an eight-node quadrilateral spline finite element [C.-J. Li, R.-H. Wang, A new 8-node quadrilateral spline finite element, J. Comp. Appl. Math. 195 (2006) 54–65] and is exact for quadratic polynomials on arbitrary convex quadrangulations and for cubic polynomials on rectangular partitions. The convergence of sequences of the above cubatures is proved for continuous integrand functions and error bounds are derived. Some numerical examples are given, by comparisons with other known cubatures.