Complexity of numerical integration over spherical caps in a Sobolev space setting

Nonlinear Numerical Integration Scheme in Strain Space Plasticity

Strains are applied to the integration procedure in nonlinear increments todecrease the errors arising from the linearization of plastic equations. Two deformationvectors are used to achieve this. The first vector is based on the deformations obtained bythe first iteration of the equilibrium step, and the second is acquired from the sum of thesucceeding iterations. By applying these vectors and...

Codes in spherical caps

We consider bounds on codes in spherical caps and related problems in geometry and coding theory. An extension of the Delsarte method is presented that relates upper bounds on the size of spherical codes to upper bounds on codes in caps. Several new upper bounds on codes in caps are derived. Applications of these bounds to estimates of the kissing numbers and one-sided kissing numbers are consi...

Numerical Integration Over a Sphere*

Spherical Designs, Discrepancy and Numerical Integration

A spherical design is a point configuration on the sphere, which yields exact equal-weight quadrature formulae for polynomials up to a given degree. Until now only very specific constructions for spherical designs are known. We establish connections to spherical cap discrepancy and show some general discrepancy bounds. Furthermore, we reformulate the problem of constructing designs as an optimi...

A characteristic property of spherical caps

It is proved that given H ≥ 0 and an embedded compact orientable constant mean curvature H surface M included in the half space z ≥ 0, not everywhere tangent to z = 0 along its boundary ∂M = γ ⊂ {z = 0}, the inequality H ≤ (min κ) ( min √ 1 − (κg/κ) ) is satisfied, where κ and κg are the geodesic curvatures of γ on z = 0 and on the surface M , respectively, if and only if M is a spherical cap o...