Related papers: A 4-Dimensional Peabody of Constant Width
We introduce the stable presentation length of a finitely presented group. The stable presentation length of the fundamental group of a 3-manifold can be considered as an analogue of the simplicial volume. We show that the stable…
The purpose of this paper is to introduce a geometric structure called pseudo-conformal quaternionic CR structure on a (4n+3)-dimensional mamnifold and then exhibit a quaternionic analogue of Chern-Moser's CR structure and uniformization.
It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.
The isoperimetric quotient of the whole family of inner and outer parallel bodies of a convex body is shown to be decreasing in the parameter of definition of parallel bodies, along with a characterization of those convex bodies for which…
In this paper, we classify $3$-dimensional complete self-shrinkers in Euclidean space $\mathbb R^{4}$ with constant squared norm of the second fundamental form $S$ and constant $f_{4}$.
We prove that complete non-locally symmetric quaternionic K\"ahler manifolds with an end of finite volume exist in all dimensions $4m\ge 4$.
The intrinsic volumes of a convex body are fundamental invariants that capture information about the average volume of the projection of the convex body onto a random subspace of fixed dimension. The intrinsic volumes also play a central…
We classify rational, irreducible quartic symmetroids in projective 3-space. They are either singular along a line or a smooth conic section, or they have a triple point or a tacnode.
We theoretically study discrete photonic lattices in more than three dimensions and point out that such systems can exist in continuous three-dimensional (3D) space. We study discrete diffraction in the linear regime, and predict the…
This paper shows the existence of a periodic orbit with singularity in the symmetric collinear four body problem. In each period of the orbit, there is a binary collision (BC) between the inner two bodies and a simultaneous binary collision…
Suppose that we have the unit Euclidean ball in $\R^n$ and construct new bodies using three operations - linear transformations, closure in the radial metric and multiplicative summation defined by $\|x\|_{K+_0L} = \sqrt{\|x\|_K\|x\|_L}.$…
We develop, in the context of general relativity, the notion of a geoid -- a surface of constant "gravitational potential". In particular, we show how this idea naturally emerges as a specific choice of a previously proposed, more general…
All four dimensional orbit spaces of compact coregular linear groups have been determined. The results are obtained through the integration of a universal differential equation, that only requires as input the number of elements of an…
In this paper we determine the cosmological constant as a topological invariant by applying certain techniques from low dimensional differential topology. We work with a small exotic $R^{4}$ which is embedded into the standard…
A variant of the flatness problem from integer programming is studied, in which one considers convex bodies in $\mathbb{R}^d$ with at most $k$ interior lattice points. The maximum lattice width of such a body is denoted by Flt(d,k) and it…
In this paper, we study $4$-dimensional complete hypersurfaces with $w$-constant mean curvature in the unit sphere. We give a lower bound of the scalar curvature for $4$-dimensional complete hypersurfaces with $w$-constant mean curvature.…
We investigate a novel setting for polytope rigidity, where a flex must preserve edge lengths and the planarity of faces, but is allowed to change the shapes of faces. For instance, the regular cube is flexible in this notion. We present…
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the image of a non-closed geodesic has 0 distance from the set of conical points.…
The present work studies the robustness of certain basic homoclinic motions in an equilateral restricted four body problem. The problem can be viewed as a two parameter family of conservative autonomous vector fields. The main tools are…
The purpose of this paper is to study convex bodies $C$ for which there exists no convex body $C^\prime\subsetneq C$ of the same lattice width. Such bodies shall be called ``lattice reduced'', and they occur naturally in the study of the…