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[6] viXra:2002.0584 [pdf] submitted on 2020-02-29 06:37:02
Authors: Jan Hakenberg
Comments: 9 Pages.
We construct biinvariant generalized barycentric coordinates for scattered sets of points in any Lie group. The coordinates are invariant under left-action, right-action, and inversion, and satisfy the Lagrange property. The construction does not utilize a metric on the Lie group, unlike inverse distance coordinates. Instead, proximity is determined in a vector space of higher dimensions than the group using the Euclidean norm. The coordinates that we propose are an inverse to the unique, biinvariant weighted average in the Lie group.
Category: Geometry
[5] viXra:2002.0551 [pdf] replaced on 2020-03-01 06:02:08
Authors: Suaib Lateef
Comments: 6 pages
I present a proof of a theorem which is a generalization of Pythagoras’ theorem. According to Wikipedia, the cosine rule is considered a general case of Pythagoras’ theorem. However, it is known that the cosine rule includes an angle. The new theorem to be presented does not include any angle.
Category: Geometry
[4] viXra:2002.0148 [pdf] submitted on 2020-02-07 10:42:35
Authors: Antoine Balan
Comments: 2 pages, written in english
We define a Dirac type operator called the spinorial Dirac operator.
Category: Geometry
[3] viXra:2002.0129 [pdf] submitted on 2020-02-07 07:01:13
Authors: Jan Hakenberg
Comments: 3 Pages.
We present meshfree generalized barycentric coordinates for scattered sets of points in d-dimensional vector space. The coordinates satisfy the Lagrange property. Our derivation is based on the projection of Shepard’s popular inverse distance weights to their best fit in the subspace of coordinates with linear reproduction. The notion of distance between a pair of points is sufficient for the construction of coordinates.
Category: Geometry
[2] viXra:2002.0061 [pdf] submitted on 2020-02-04 04:06:09
Authors: Jaykov Foukzon, Elena Men’kova, Alexander Potapov
Comments: 150 Pages.
This book is an exposition of "Singular Semi-Riemannian Geometry"- the study
of a smooth manifold furnished with a degenerate (singular) metric tensor of arbitrary
signature. This book also dealing with Colombeau extension of the Einstein field
equations using apparatus of the Colombeau generalized function and contemporary
generalization of the classical Lorentzian geometry named in literature Colombeau
distributional geometry.The regularizations of singularities present in some Colombeau
solutions of the Einstein equations is an important part of this approach. Any singularities
present in some solutions of the Einstein equations recognized only in the sense of
Colombeau generalized functions and not classically.In this paper essentially new class
Colombeau solutions to Einstein fild equations is obtained. We leave the neighborhood
of the singularity at the origin and turn to the singularity at the horizon.Using nonlinear
distributional geometry and Colombeau generalized functions it seems possible to show
that the horizon singularity is not only a coordinate singularity without leaving
Schwarzschild coordinates.However the Tolman formula for the total energy ET of a
static and asymptotically flat spacetime,gives ET m, as it should be. The vacuum
energy density of free scalar quantum field with a distributional background spacetime
also is considered.It has been widely believed that, except in very extreme situations, the
influence of gravity on quantum fields should amount to just small, sub-dominant
contributions. Here we argue that this belief is false by showing that there exist
well-behaved spacetime evolutions where the vacuum energy density of free quantum
fields is forced, by the very same background distributional spacetime such distributional
BHs, to become dominant over any classical energy density component. This
semiclassical gravity effect finds its roots in the singular behavior of quantum fields on
curved distributional spacetimes. In particular we obtain that the vacuum fluctuations
2 has a singular behavior on BHs horizon r: 2r~|r r |2. A CHALLENGE TO
THE BRIGHTNESS TEMPERATURE LIMIT OF THE QUASAR 3C273 explained
successfully.
Keywords: Colombeau nonlinear generalized funct
Category: Geometry
[1] viXra:2002.0004 [pdf] submitted on 2020-02-01 04:48:37
Authors: Antoine Balan
Comments: 1 page, written in english
We define a moduli space called the Dirac moduli space with help of the Dirac operator.
Category: Geometry
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