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Algorithms for the computation of the forward and inverse geodesic problems for an ellipsoid of revolution are derived. These are accurate to better than 15 nm when applied to the terrestrial ellipsoids. The solutions of other problems…
In [Heimann, Lehrenfeld, Preu{\ss}, SIAM J. Sci. Comp. 45(2), 2023, B139 - B165] new geometrically unfitted space-time Finite Element methods for partial differential equations posed on moving domains of higher-order accuracy in space and…
In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space $\mathbb{R}^{n+1}$ which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both…
Asymptotic results for weighted floating bodies are established and used to obtain new proofs for the existence of floating areas on the sphere and in hyperbolic space and to establish the existence of floating areas in Hilbert geometries.…
We introduce a class of examples which provide an affine generalization of the nonholonomic problem of a convex body rolling without slipping on the plane. We investigate dynamical aspects of the system such as existence of first integrals,…
A numerical method to build an orthonormal basis of properly symmetrized hyperspherical harmonic functions is developed. As a part of it, refined algorithms for calculating the transformation coefficients between hyperspherical harmonics…
A sphere is a fundamental geometric object widely used in (computer aided) geometric design. It possesses rational parameterizations but no parametric polynomial parameterization exists. The present study provides an approach to the optimal…
Affine geometry is usually regarded as a framework in which metric notions such as distance and angle are absent. However, just as projective geometry produces various metric geometries by introducing additional structures on the line at…
Working in univalent foundations, we investigate the symmetries of spheres, i.e., the types of the form $\mathbb{S}^n = \mathbb{S}^n$. The case of the circle has a slick answer: the symmetries of the circle form two copies of the circle.…
We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit…
Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from…
The reassignment method for the wavelet transform is investigated. Particularly good results are obtained if the wavelet is an extremal for the uncertainty relation of the affine group.
An alternative method is suggested for the description of the velocity and pressure fields in an unbounded incompressible viscous fluid induced by an arbitrary number of spheres moving and rotating in it. Within the framework of this…
The 0th order of affine perturbation series of the deflection angle of a ray near a photon sphere is more accurate than a deflection angle in a strong deflection limit, which is used often, because the later has hidden error terms. We…
Position uncertainty (delocalization) measures for a particle on the sphere are proposed and illustrated on several examples of states. The new measures are constructed using suitably the standard multiplication angle operator variances.…
Three papers describing different methods to solve the inverse scattering problem of the reconstruction of the shape and/or impedance of an obstacle have been chosen for analysis. This literature review consists of an evaluation of these…
The classification of solutions to semilinear partial differential equations, as well as the classification of critical points of the corresponding functionals, have wide applications in the study of partial differential equations and…
In this paper, we study strictly convex affine hypersurfaces centroaffinely congruent to their centre map, in the case when the shape operator has two distinct eigenvalues: one of multiplicity 1, and one nonzero of multiplicity n-1. We show…
The two dimensional sphere can't be approximated by finite homogeneous spaces. We describe the optimal approximation and its distance from the sphere.
An adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption. The method is based on the so-called moving mesh partial differential…