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[5] viXra:2401.0152 [pdf] submitted on 2024-01-31 21:21:51
Authors: Volker W. Thürey
Comments: 4 Pages.
We generalize the famous `Chromatic Number of the Plane'. For every finite metric space we define a similar question. We show that 15 colors suffice togenerate a coloring of the plane without monochromatic distances 1 or 2.
Category: Geometry
[4] viXra:2401.0111 [pdf] submitted on 2024-01-22 10:11:56
Authors: Archan Chattopadhyay
Comments: 8 Pages.
An analytical treatment of rotations in the Euclidean plane and 3-dimensional Euclidean space, using differential equations, is presented. Fundamental geometric results, such as the linear transformation for rotations, the invariance of the Euclidean norm, a proof of the Pythagorean theorem, and the existence of a period of rotations, are derived from a set of fundamental equations. Basic Euclidean geometry is also constructed from these equations.
Category: Geometry
[3] viXra:2401.0105 [pdf] submitted on 2024-01-21 22:02:12
Authors: Yuly Shipilevsky
Comments: 4 Pages. (Note by viXra Admin: Please list scientific references in future submissions)
We consider a mapping from the set of triangles on the same plane onto its- elf, wherein each triangle is being mapped to the triangle, having vertices, which are the orthocenter, the centroid and the incenter of the parent triangle and we consider the corresponding inverse mapping as well.
Category: Geometry
[2] viXra:2401.0075 [pdf] submitted on 2024-01-16 20:25:29
Authors: Urs Frauenfelder, Joa Weber
Comments: 45 Pages. 3 Figures. Bull. Braz. Math. Soc. (N.S.) 56, 44 (2025). https://doi.org/10.1007/s00574-025-00464-5
In the local gluing one glues local neighborhoods around the critical point of the stable and unstable manifolds to gradient flow lines defined on a finite time interval [−T,T] for large T. If the Riemannian metric around the critical point is locally Euclidean, the local gluing map can be written down explicitly. In the non-Euclidean case the construction of the local gluing map requires an intricate version of the implicit function theorem.In this paper we explain a functional analytic approach how the local gluing map can be defined. For that we are working on infinite dimensional path spaces and also interpret stable and unstable manifolds as submanifolds of path spaces. The advantage of this approach is that similar functional analytical techniques can as well be generalized to infinite dimensional versions of Morse theory, for example Floer theory.A crucial ingredient is the Newton-Picard map. We work out an abstract version of it which does not involve troublesome quadratic estimates.
Category: Geometry
[1] viXra:2401.0066 [pdf] submitted on 2024-01-13 21:14:04
Authors: Ryan J. Buchanan
Comments: 17 Pages.
We propose some questions about Fukaya categories. Given a class of isomorphisms $0 sim tau$, where $tau$ represents the truth value of a particle, and $0$ is a $0$ object in a Fukaya category, what are its spectral homology theories? This is a variation on the works of P. Seidel and E. Riehl.
Category: Geometry
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