Related papers: Uniqueness of non-Euclidean Mass Center System and…
The notion of center of mass, which is very useful in kinematics, proves to be very handy in geometry (see [1]-[2]). Countless applications of center of mass to geometry go back to Archimedes. Unfortunately, the center of mass cannot be…
In this article is given a simple expression for the \textit{ center of mass} for a system of material points in a two-dimensional surface of constant negative Gaussian curvature. Using basic techniques of Geometry, an expression in…
This thesis discusses the Newtonian limit of General Relativity for static isolated systems with compactly supported matter. We call these systems "geometrostatic" to underline their geometric nature. We introduce new quasi-local notions of…
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 present a new geometric approach to the study of static isolated general relativistic systems for which we suggest the name geometrostatics. After describing the setup, we introduce localized formulas for the ADM-mass and ADM/CMC-center…
We propose a definition of center of mass for asymptotically flat manifolds satisfying Regge-Teitelboim condition at infinity. This definition has a coordinate-free expression and natural properties. Furthermore, we prove that our…
We define and study a variant of the center of mass of a polygon and, more generally, of a simplicial polytope which we call the Circumcenter of Mass (CCM). The Circumcenter of Mass is an affine combination of the circumcenters of the…
In this article, I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us (the…
The concept of centre of mass of two particles in 2D spaces of constant Gaussian curvature is discussed by recalling the notion of "relativistic rule of lever" introduced by Galperin [Comm. Math. Phys. 154 (1993), 63--84] and comparing it…
In this paper we introduce a family of center of masses that complement the definition of the family of Gauss-Bonnet-Chern masses by Ge-Wang-Wu and Li-Nguyen. In order to prove the existence and the well-definedness of the center of mass,…
Suppose that to every non-degenerate simplex Delta in n-dimensional Euclidean space a `center' C(Delta) is assigned so that the following assumptions hold: (i) The map that assigns C(Delta) to Delta commutes with similarities and is…
Asymptotic normality for the natural volume measure of random polytopes generated by random points distributed uniformly in a convex body in spherical or hyperbolic spaces is proved. Also the case of Hilbert geometries is treated and…
This paper deals with the question of how to calculate the volume of a body in the three-dimensional Euclidean space when it is cut into slices perpendicular to a given curve. The answer is provided by a formula that can be considered as a…
In this work, we review the concept of center of a geometric object as an equivariant map, unifying and generalizing different approaches followed by authors such as C. Kimberling or A. Edmonds. We provide examples to illustrate that this…
The center of mass of a finite measure with respect to a radially increasing weight is shown to exist, be unique, and depend continuously on the measure.
In 1996, Huisen-Yau proved that every three-dimensional, asymptotically Schwarzschilden manifold with positive mass is uniquely foliated by stable spheres of constant mean curvature and they defined the center of mass using this…
We provide a computer-assisted proof of the exact count of classes of central configurations for five bodies for several sets of mass values that are exceptional from the point of view of the finiteness results of Albouy and Kaloshin in the…
Pappus' Involution Theorem is a powerful tool for proving theorems about non-euclidean triangles and generalized triangles in Cayley-Klein models. Its power is illustrated by proving with it some theorems about euclidean and non-euclidean…
The classical notion of center of mass for an isolated system in general relativity is derived from the Hamiltonian formulation and represented by a flux integral at infinity. In contrast to mass and linear momentum which are well-defined…
We present a method for establishing invariant manifolds for saddle--center fixed points. The method is based on cone conditions, suitably formulated to allow for application in computer assisted proofs, and does not require rigorous…