相关论文: Platonic configurations of points and lines
Atiyah's conjecture concerning configurations of N points in the Euclidean three-space is verified for the following nonplanar configurations: The first m points lie on a line L and the remaining n=N-m (>2) points are the vertices of a…
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…
We construct a series of examples of non--flat non--homogeneous parabolic geometries that carry a symmetry of the parabolic geometry at each point.
The stability of equilibrium configurations of galaxies or stars are time honored problems in astrophysics. We present mathematical results on these problems which have in recent years been obtained by Yan Guo and the author in the context…
In this paper we obtain presentations of fundamental groups of the complements of three quadric-line arrangements in P^2. The first arrangement is a smooth quadric Q with n tangent lines to Q, and the second one is a quadric Q with n lines…
The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, \Sigma)$…
Matrix configurations coming from matrix models comprise many important aspects of modern physics. They represent special quantum spaces and are thus strongly related to noncommutative geometry. In order to establish a semiclassical limit…
In this paper we construct several arrangements of lines and/or conics that are derived from the geometry of the Klein arrangement of $21$ lines in the complex projective plane.
We put forward a method of constructing discrete coherent states for n qubits. After establishing appropriate displacement operators, the coherent states appear as displaced versions of a fiducial vector that is fixed by imposing a number…
We show that Euclidean geometry in suitably high dimension can be expressed as a theory of orthogonality of subspaces with fixed dimensions and fixed dimension of their meet.
In this paper we review some author's results about Weingarten surfaces in Euclidean space $\r^3$ and hyperbolic space $\h^3$. We stress here in the search of examples of linear Weingarten surfaces that satisfy a certain geometric property.…
On the space $\mathbb Q_p^n$, where $p\ne 2$ and $p$ does not divide $n$, we construct a p-adic counterpart of spherical coordinates. As applications, a description of homogeneous distributions on $\mathbb Q_p^n$ and a skew product…
This paper addresses formation control of reduced attitudes in which a continuous control protocol is proposed for achieving and stabilizing all regular polyhedra (also known as Platonic solids) under a unified framework. The protocol…
We conclude the construction of the algebraic complex, consisting of spaces of differentials of Euclidean metric values, for four-dimensional piecewise-linear manifolds. Assuming that the complex is acyclic, we investigate how its torsion…
We introduce a family of reconfiguration puzzles arising from ideas in geometry and topology. We present their construction from square-tiled shapes, discuss some of the underlying mathematics and describe how they are naturally associated…
We study point-line configurations and their associated matroid and circuit varieties. We aim to find a finite set of defining equations for matroid varieties and an irreducible decomposition for circuit varieties. To solve the former…
In this paper, extending the work of Gal'perin (Comm. Math. Phys. 154: 63-84, 1993), we investigate generalizations of the concepts of centroids and static equilibrium points of a convex body in spherical, hyperbolic and normed spaces. In…
We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we introduce three distinct methods for generating polynomials within the associated ideals of…
The purpose of this paper is to prove that every finite set of points that can be constructed in the Euclidean plane by using a compass and a ruler can also be constructed by using unitary match-sticks in a non-simultaneous way and…
Planar central configurations can be seen as critical points of the reduced potential or solutions of a system of equations. By the homogeneity and invariance of the potential with respect to SO(2), it is possible to see that the…