Related papers: Generalized Voronoi Diagrams and Lie Sphere Geomet…
The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [4] by introducing a vast hierarchy of…
Homotopy Lie algebras are a generalization of differential graded Lie algebras encoding both the kinematics and dynamics of a given field theory. Focusing on kinematics, we show that these algebras provide a natural framework for the…
Starting with some motivating examples (classical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the "virtual structure" of its orbit space, the…
Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the…
We present an edge labeling of order-$k$ Voronoi diagrams, $V_k(S)$, of point sets $S$ in the plane, and study properties of the regions defined by them. Among them, we show that $V_k(S)$ has a small orientable cycle and path double cover,…
We present a generalized Poincar\'e sphere (G sphere) and generalized Stokes parameters (G parameters), as a geometric representation, which unifies the descriptors of a variety of vector fields. Unlike the standard Poincar\'e sphere, the…
Angular Voronoi diagram was introduced by Asano et al. as fundamental research for a mesh generation. In an angular Voronoi diagram, the edges are curves of degree three. From view of computational robustness we need to treat the curves…
It is shown that the area of Voronoi cells for a generalized Archimedean spiral lattice converges under some scale normalization, if the angle parameter is badly approximable.
We study sheaves of Lie-Rinehart algebras over locally ringed spaces. We introduce morphisms and comorphisms of such sheaves and prove factorization theorems for each kind of morphism. Using this notion of morphism, we obtain (higher)…
The characterization of systems of differential equations admitting a superposition function allowing us to write the general solution in terms of any fundamental set of particular solutions is discussed. These systems are shown to be…
All results concern characteristic 2. Two procedures that to every simple Lie algebra assign simple Lie superalgebras, most of the latter new, are offered. We prove that every simple finite-dimensional Lie superalgebra is obtained as the…
In this thesis we study sets of points in the plane and their Voronoi diagrams, in particular when the points coincide. We bring together two ways of studying point sets that have received a lot of attention in recent years: Voronoi…
Voronoi tessellations have been used to model the geometric arrangement of cells in morphogenetic or cancerous tissues, however so far only with flat hypersurfaces as cell-cell contact borders. In order to reproduce the experimentally…
We present a thorough study of the differential geometry of weightings and develop the theory of weightings for vector bundles, Lie groupoids, and Lie algebroids. We begin by extending the work of Loizides and Meinrenken on weighted…
In this paper, we introduce the category of blueprints, which is a category of algebraic objects that include both commutative (semi)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp.\…
The aim of this paper is to explain, mostly through examples, what groupoids are and how they describe symmetry. We will begin with elementary examples, with discrete symmetry, and end with examples in the differentiable setting which…
Voronoi tessellation, also known as Voronoi diagram, is an important computational geometry technique that has applications in various scientific disciplines. It involves dividing a given space into regions based on the proximity to a set…
We show that in the Klein projective ball model of hyperbolic space, the hyperbolic Voronoi diagram is affine and amounts to clip a corresponding power diagram, requiring however algebraic arithmetic. By considering the lesser-known…
The conditions for a cuspidal edge, swallowtail and other fundamental singularities are given in the context of Lie sphere geometry. We then use these conditions to study the Lie sphere transformations of a surface.
We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation on a smooth vector bundle, i.e., one whose kernel consists only of ineffective arrows. As an application,…