Related papers: Napoleonic triangles on the sphere

A geometric description is given for the Sp(2) covariant version of the field-antifield quantization of general constrained systems in the Lagrangian formalism. We develop differential geometry on manifolds in which a basic set of…

Various packing problems and simulations of hard and soft interacting particles, such as microscopic models of nematic liquid crystals, reduce to calculations of intersections and pair interactions between ellipsoids. When constrained to a…

The Clifford spectrum is a form of joint spectrum for noncommuting matrices. This theory has been applied in photonics, condensed matter and string theory. In applications, the Clifford spectrum can be efficiently approximated using…

We formulate a type B extended nilHecke algebra, following the type A construction of Naisse and Vaz. We describe an action of this algebra on extended polynomials and describe some results on the structure on the extended symmetric…

Clifford's geometric algebra has enjoyed phenomenal development over the last 60 years by mathematicians, theoretical physicists, engineers and computer scientists in robotics, artificial intelligence and data analysis, introducing a myriad…

We investigate triangulations of the two-dimensional sphere and torus with the faces properly colored white and black. We focus on matchings between white triangles and incident vertices. On the torus our objects are perfect pairings,…

The goal of this paper is to study how the symmetry of the spherical domain influences solutions of elliptic equations on such domain. The method pursued is a variant of the moving plane method, discovered by Alexandrov (1962) and used for…

We investigate several topics of triangle geometry in the elliptic and in the extended hyperbolic plane, such as: centers based on orthogonality, centers related to circumcircles and incircles, radical centers and centers of similitude,…

Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a…

This is a survey paper on various results relates to the following theorem first proved by A.D. Alexandrov: \textit{Let $S$ be an analytic convex sphere-homeomorphic surface in $\mathbb R^3$ and let $k_1(\boldsymbol{x})\leqslant…

A problem that is simple to state in the context of spherical geometry, and that seems rather interesting, appears to have been unexamined to date in the mathematical literature. The problem can also be recast as a problem in the real…

We classify the Lagrangian orientable surfaces in complex space forms with the property that the ellipse of curvature is always a circle. As a consequence, we obtain new characterizations of the Clifford torus of the complex projective…

We extend the Poincar\'{e}--Borel lemma to a weak approximation of a Brownian motion via simple functionals of uniform distributions on n-spheres in the Skorokhod space $D([0,1])$. This approach is used to simplify the proof of the…

Normed division and Clifford algebras have been extensively used in the past as a mathematical framework to accommodate the structures of the standard model and grand unified theories. Less discussed has been the question of why such…

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which…

Classes of simple polynomial and simple trigonometric splines given by Fourier series are considered. It is shown that the class of simple trigonometric splines includes the class of simple polynomial splines. For some parameter values, the…

A Circumconic passes through a triangle's vertices. We define the Circumbilliard, a circumellipse to a generic triangle for which the latter is a 3-periodic. We study its properties and associated loci.

We give a characterization of all three points in $\mathbb R^4$ with integer coordinates which are at the same Euclidean distance apart. In three dimension the problem is characterized in terms of solutions of the Diophantine equations…

We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally…

Every graph G can be embedded in a Euclidean space as a two-distance set. The Euclidean representation number of G is the smallest dimension in which G is representable by such an embedding. We consider spherical and J-spherical…