In this note cubature formulae of degree 5 are studied for n-dimensional integrals over the ball with constant weight function. We apply the method of reproducing kernel to show that the existence of such formulae attaining the best known lower bound is equivalent to the existence of tight spherical 5-designs. The known results concerning spherical 5-designs show that the lower bound for the integral under consideration will not be attained in general. The bound will be attained for n=2,3,7,23 and possibly for n=(2ρ+1)2−2, ρ⩾5. In all other cases the bound must be increased at least by 1, in particular, Stroud's formulae for n=4,5,6,7 are minimal.

An overview of the lower bounds for the number of points for integrals over the square and triangle is presented. This is compared with the number of points in known cubature formulae.

In this note an interactive program is presented which allows the visualization of the common roots of two-dimensional polynomials. The program is useful when constructing cubature formulas by use of orthogonal polynomials.

In this note a minimal cubature formula of degree 3 will be determined for integrals over the surface of the torus with arbitrary radii. The construction is based on a normalization of the problem, ideal theory, and computer algebra.

In a comprehensive investigation in the 1960s Krall and Sheffer [Ann. Mat. Pura Appl., 76 (1967), pp. 325–376] characterized all bivariate orthogonal polynomial systems which are generated by a second-order differential equation. Actually, they prove that these nine systems are weakly orthogonal and (positive) definite except possibly for two systems. Their paper is completed by showing that these systems are also definite and by determining all parameters for which the classical positive definite systems remain definite. The authors further derive an explicit form of the three-term recursion formulae for all systems. In addition, it is shown that for the associated cubature problem Moller's lower bound applies.

It is shown that for two classes of integrals the results of Gaussian quadrature can be extended straightforwardly to the bivariate case.For these classes Gaussian formulae of an arbitrary degree are derived.

For two classes of integrals it will be shown that the number of nodes of cubature formulae of degree 4n + 1, n > 1, will not attain Möller’s lower bound. Thus in these cases that bound has to be increased by 1.

Recently, the authors were asked by Professor Klaus Donner whether the set of matrices, the determinant of which in its absolute value is less than or equal to one, is Chebyshevian within the space of n x n matrices endowed with the / 2 -operator norm.The answer is yes.Although our proof is elementary we think that the result and its proof are of general interest.In addition, there is one further nice example of a Chebyshev set.

A cubature formula with degree of exactness ≦2k−2 has at leastk(k+1)/2 knots. The existence of such minimal formulae is equivalent to the existence of solutions of a system of quadratic equations. Each solution of such a system generates in a uniquely determined way a minimal formula. For the square [−1, 1]2 as domain of integration and for some weight-functions these systems have a simple form, but they seem to be hard to solve. In this note attention is drawn to these systems and to experiences made in order to obtain numerical solutions.

An metallische Hüftgelenkimplantate, deren Querschnitt durch die Markhöhle begrenzt ist, werden folgende Anforderungen gestellt: 1. Verträglichkeit, d.h. im Kontakt mit der Körperflüssigkeit darf auch unter mechanischer Beanspruchung keine Metallauflösung (Korrosion) und kein Abrieb auftreten, um unerwünschte Reaktionen mit dem umliegenden Gewebe und Reaktionen, die zu Systemerkrankungen führen, zu vermeiden.

AbstractWith a view to investigate the biological degradation of „Doriden”︁ some glutarimides have been synthesized which are presumptive intermediate products.

Several basically substituted lactams have been synthesized from cyclic ketones by means of the Schmidt reaction.