**Previous months:**

2010 - 1003(11) - 1004(4) - 1005(2) - 1006(7) - 1007(2) - 1008(5) - 1009(7) - 1010(5) - 1011(1)

2011 - 1101(4) - 1102(3) - 1103(6) - 1104(18) - 1106(2) - 1107(4) - 1108(1) - 1110(2) - 1111(1)

2012 - 1201(2) - 1202(1) - 1203(2) - 1204(2) - 1205(3) - 1208(1) - 1209(1) - 1211(4)

2013 - 1301(1) - 1303(3) - 1304(1) - 1305(2) - 1306(9) - 1307(2) - 1308(2) - 1309(2) - 1310(2) - 1311(2) - 1312(5)

2014 - 1401(2) - 1404(3) - 1405(5) - 1406(2) - 1407(3) - 1408(4) - 1409(3) - 1410(2) - 1411(3)

2015 - 1502(5) - 1504(2) - 1507(2) - 1508(8) - 1510(1) - 1511(4) - 1512(4)

2016 - 1601(1) - 1602(6) - 1604(1) - 1605(4) - 1606(2) - 1607(26) - 1608(4) - 1609(2) - 1610(3) - 1611(2) - 1612(1)

2017 - 1701(1) - 1702(1) - 1703(3) - 1704(3) - 1706(3) - 1707(7) - 1708(7) - 1709(4) - 1710(7) - 1711(2) - 1712(3)

2018 - 1801(7) - 1802(8) - 1803(4) - 1804(5) - 1805(1) - 1806(1) - 1807(3)

Any replacements are listed farther down

[308] **viXra:1807.0463 [pdf]**
*submitted on 2018-07-26 06:25:03*

**Authors:** Jan Hakenberg

**Comments:** 6 Pages.

We demonstrate that curve subdivision in the special Euclidean group SE(2) allows the design of planar curves with favorable curvature. We state the non-linear formula to position a point along a geodesic in SE(2). Curve subdivision in the Lie group consists of trigonometric functions. When projected to the plane, the refinement method reproduces circles and straight lines. The limit curves are designed by intuitive placement of control points in SE(2).

**Category:** Geometry

[307] **viXra:1807.0298 [pdf]**
*submitted on 2018-07-17 17:10:18*

**Authors:** Yeray Cachón Santana

**Comments:** 10 Pages.

This paper covers a first approach study of the angles and modulo of vectors in spaces of Hilbert considering a riemannian metric where, instead of taking the usual scalar product on space of Hilbert, this will be extended by the tensor of the geometry g. As far as I know, there is no a study covering space of Hilbert with riemannian metric. It will be shown how to get the angle and modulo on Hilbert spaces with a tensor metric, as well as vector product, symmetry and rotations. A section of variationals shows a system of differential equations for a riemennian metric.

**Category:** Geometry

[306] **viXra:1807.0234 [pdf]**
*submitted on 2018-07-12 16:30:22*

**Authors:** James A. Smith

**Comments:** 18 Pages.

As a demonstration of the coherence of Geometric Algebra's (GA's) geometric and algebraic concepts of bivectors, we add three geometric bivectors according to the procedure described by Hestenes and Macdonald, then use bivector identities to determine, from the result, two bivectors whose outer product is equal to the initial sum. In this way, we show that the procedure that GA's inventors dened for adding geometric bivectors is precisely that which is needed to give results that coincide with those obtained by calculating outer products of vectors that are expressed in terms of a 3D basis. We explain that that accomplishment is no coincidence: it is a consequence of the attributes that GA's designers assigned (or didn't) to bivectors.

**Category:** Geometry

[305] **viXra:1806.0116 [pdf]**
*submitted on 2018-06-09 15:36:41*

**Authors:** Antoine Balan

**Comments:** 1 page, written in english

In the case of a manifold which is a Lie group, a Dirac operator can be defined acting over the vector fields of the Lie group instead of the spinors.

**Category:** Geometry

[304] **viXra:1805.0030 [pdf]**
*submitted on 2018-05-01 00:29:25*

**Authors:** Johan Aspegren

**Comments:** 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to denefine certain surface measures on surfaces of convex bodies in a novel and simple way.

**Category:** Geometry

[303] **viXra:1804.0397 [pdf]**
*submitted on 2018-04-27 03:22:51*

**Authors:** Zhenghan Shen, Wen Wang, Pan Zhang

**Comments:** 9 Pages.

In this paper, by the method of heat flow and the
method of exhaustion, we prove an existence theorem of Hermitian-Yang-Mills-Higgs metrics on holomorphic line bundle over a class of non-compact Gauduchon manifold.

**Category:** Geometry

[302] **viXra:1804.0363 [pdf]**
*submitted on 2018-04-24 20:34:14*

**Authors:** James A. Smith

**Comments:** 54 Pages.

Because the shortage of worked-out examples at introductory levels is an obstacle to widespread adoption of Geometric Algebra (GA), we use GA to calculate Solar azimuths and altitudes as a function of time via the heliocentric model. We begin by representing the Earth's motions in GA terms. Our representation incorporates an estimate of the time at which the Earth would have reached perihelion in 2017 if not affected by the Moon's gravity. Using the geometry of the December 2016 solstice as a starting point, we then employ GA's capacities for handling rotations to determine the orientation of a gnomon at any given latitude and longitude during the period between the December solstices of 2016 and 2017. Subsequently, we derive equations for two angles: that between the Sun's rays and the gnomon's shaft, and that between the gnomon's shadow and the direction ``north" as traced on the ground at the gnomon's location. To validate our equations, we convert those angles to Solar azimuths and altitudes for comparison with simulations made by the program Stellarium. As further validation, we analyze our equations algebraically to predict (for example) the precise timings and locations of sunrises, sunsets, and Solar zeniths on the solstices and equinoxes. We emphasize that the accuracy of the results is only to be expected, given the high accuracy of the heliocentric model itself, and that the relevance of this work is the efficiency with which that model can be implemented via GA for teaching at the introductory level. On that point, comments and debate are encouraged and welcome.

**Category:** Geometry

[301] **viXra:1804.0360 [pdf]**
*submitted on 2018-04-25 02:11:06*

**Authors:** Hiroshi Okumura

**Comments:** 2 Pages. This paper will be submitted to Sangaku Journal of Mathematics.

A problem involving an isosceles triangle with a square and three congruent circles is generalized.

**Category:** Geometry

[300] **viXra:1804.0132 [pdf]**
*submitted on 2018-04-10 11:52:50*

**Authors:** Arturo Tozzi, James Peters

**Comments:** 6 Pages.

The first definition (prior to the well-known five postulates) of Euclid describes the point as “that of which there is no part”. Here we show how the Euclidean account of manifolds is untenable in our physical realm and that the concepts of points, lines, surfaces, volumes need to be revisited, in order to allow us to be able to describe the real world. Here we show that the basic object in a physical context is a traversal of spacetime via tiny subregions of spatial regions, rather than the Euclidean point. We also elucidate the psychological issues that lead our mind to think to points and lines as really existing in our surrounding environment.

**Category:** Geometry

[299] **viXra:1804.0032 [pdf]**
*submitted on 2018-04-03 04:56:49*

**Authors:** Ryan Haddad

**Comments:** 1 Page.

This conjecture may be a tool in defining the indefinite tangent of 90 degrees, and is a (new) mathematical coincidence that is indeed strange; why would the tangent of angles near 90 degrees be equal to the angle of the radian multiplied by powers of 10? In fact, if there is no geometrical explanation in current mathematics, it may resides in metamathematics.

**Category:** Geometry

[298] **viXra:1803.0463 [pdf]**
*submitted on 2018-03-22 09:24:56*

**Authors:** Gerasimos T. Soldatos

**Comments:** Published in: FORUM GEOMETRICORUM, VOL. 18, PAGES 93-97

A doubling of the cube is attempted as a problem equivalent to the doubling of a horn torus. Both doublings are attained through the circle of Apollonius.

**Category:** Geometry

[297] **viXra:1803.0242 [pdf]**
*submitted on 2018-03-16 18:19:17*

**Authors:** Prashanth R. Rao

**Comments:** 3 Pages.

The Playfair’s axiom is considered an equivalent of Euclid’s fifth postulate or parallel postulate in Euclidean planar geometry. It states that in a given plane, with a line in the plane and a point outside the line that is also in the same plane, one and only one line passes through that point that is also parallel to the given line. Previous proofs of Euclid’s postulate or the Playfair’s axiom have unintentionally assumed parallel postulate to prove it. Also, these axioms have different results in hyperbolic and spherical geometries. We offer proof for the Playfair’s axiom for subset of cases in the context of plane Euclidean geometry and describe another subset of cases that cannot be proven by the same approach.

**Category:** Geometry

[296] **viXra:1803.0119 [pdf]**
*submitted on 2018-03-08 13:22:17*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

Following the definition of the flow of Ricci, we construct a flow of hermitian metrics for the spinors fiber bundle.

**Category:** Geometry

[295] **viXra:1803.0050 [pdf]**
*submitted on 2018-03-04 11:55:15*

**Authors:** Antoine Balan

**Comments:** 1 page, written in french

The flow of Ricci is defined for the hermitian metric of a fiber bundle.

**Category:** Geometry

[294] **viXra:1802.0196 [pdf]**
*submitted on 2018-02-15 23:59:18*

**Authors:** Hiroshi Okumura

**Comments:** 3 Pages. This is a paper considering a problem in Wasan geometry.

A problem involving a square in the curvilinear triangle made by two touching congruent circles and their common tangent is generalized.

**Category:** Geometry

[293] **viXra:1802.0123 [pdf]**
*submitted on 2018-02-10 09:10:18*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

Proposition 23 states that two parallel lines in a plane never intersect. We use this definition with first and second postulate of Euclid to prove that two distinct lines through a single point cannot be parallel.

**Category:** Geometry

[292] **viXra:1802.0092 [pdf]**
*submitted on 2018-02-08 07:24:07*

**Authors:** Jesús Álvarez Lobo

**Comments:** 1 Page. Revista Escolar de la Olimpiada Iberoamericana de Matemática. Volume 18. Spanish.

A very simple solution to a geometric problem (proposed by Alex Sierra Cardenas, Medellin, Colombia) that involves a cevian, two perpendicular bisectors and a median in an isosceles triangle.

**Category:** Geometry

[291] **viXra:1802.0091 [pdf]**
*submitted on 2018-02-08 07:47:03*

**Authors:** Jesús Álvarez Lobo

**Comments:** 47 Pages. https://arxiv.org/abs/1110.1299

This work presents for the first time a solution to the 1821 unsolved Sawa
Masayoshi's problem, giving an explicit and algebraically exact solution for
the symmetric case (particular case b = c, i.e., for ABC isosceles right-angled triangle), see (1.60) and (1.61).
Despite the isosceles triangle restriction is not necessary, in view of the complexity of the explicit algebraic solution for the symmetric case, one can guessing the impossibility of achieving an explicit relationship for the
asymmetric case (the more general case: ABC right-angled scalene triangle). For this case is given a proof of existence and uniqueness of
solution and a proof of the impossibility of getting such a relationship, even
implicitly, if the sextic equation (2.54) it isn't solvable.
Nevertheless, in (2.56) - (2.58) it is shown the way to solve the asymmetric case under the condition that (2.54) be solvable.
Furthermore, it is proved that with a slight
modification in the final set of variables (F), it is still possible to establish a relation between them, see (2.59) and (2.61), which provides a bridge that connects the primitive relationship by means of numerical methods,
for every given right-angled triangle ABC.
And as the attempt to solve Fermat's conjecture (or Fermat's last theorem), culminated more than three centuries later by Andrew Wiles, led to the development of powerful theories of more general scope, the attempt to solve
the Masayoshi's problem has led to the development of the Theory of Overlapping
Polynomials (TOP), whose application to this problem reveals a great potential
that might be extrapolated to other frameworks.

**Category:** Geometry

[290] **viXra:1802.0079 [pdf]**
*submitted on 2018-02-08 06:28:49*

**Authors:** Jesús Álvarez Lobo

**Comments:** 11 Pages.

Sacred Mathematics: Japanese Temple Geometry. Fukagawa Hidetoshi - Tony Rothman.
Still Harder Temple Geometry Problems:
Chapter 6 - Problem 3.

**Category:** Geometry

[289] **viXra:1802.0071 [pdf]**
*submitted on 2018-02-07 07:09:05*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 13 Pages. In French.

This paper gives the elements of definition of the Bonne's map projection. It was used for the ancient cartography at 1/50000 scale in Tunisia and Algeria.

**Category:** Geometry

[288] **viXra:1802.0047 [pdf]**
*submitted on 2018-02-05 08:51:37*

**Authors:** James A. Smith

**Comments:** 14 Pages.

We express a problem from visual astronomy in terms of Geometric (Clifford) Algebra, then solve the problem by deriving expressions for the sine and cosine of the angle between projections of two vectors upon a plane. Geometric Algebra enables us to do so without deriving expressions for the projections themselves.

**Category:** Geometry

[287] **viXra:1802.0036 [pdf]**
*submitted on 2018-02-03 13:17:15*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

Studying the flow of Kaehler-Ricci, a flow is defined for a manifold which is HyperKaehler.

**Category:** Geometry

[286] **viXra:1801.0347 [pdf]**
*submitted on 2018-01-25 14:50:11*

**Authors:** Antoine Balan

**Comments:** 1 page, written in french

The flow of Ricci-Schrödinger is defined from the flow of Ricci, like the Schrödinger equation is a twist of the heat equation.

**Category:** Geometry

[285] **viXra:1801.0309 [pdf]**
*submitted on 2018-01-23 19:52:26*

**Authors:** Songting Yin, Pan Zhang

**Comments:** 9 Pages.

In this paper, we give a gradient estimate of positive solution to the equation
$$\Delta u=-\lambda^2u, \ \ \lambda\geq 0$$
on a complete non-compact Finsler manifold. Then we obtain the corresponding Liouville-type theorem and Harnack inequality for the solution.
Moreover, on a complete non-compact Finsler manifold we also prove a Liouville-type theorem for a $C^2$-nonegative function $f$ satisfying
$$\Delta f\geq cf^d, c>0, d>1, $$
which improves a result obtained by Yin and He.

**Category:** Geometry

[284] **viXra:1801.0292 [pdf]**
*submitted on 2018-01-22 15:19:04*

**Authors:** Philip Gibbs

**Comments:** 21 Pages.

The universal covering problem as posed by Henri Lebesgue in 1914 seeks to find the convex planar shape of smallest area that contains a subset congruent to any point set of unit diameter in the Euclidean plane. Methods used previously to construct such a cover can be refined and extended to provide an improved upper bound for the optimal area. An upper bound of 0.8440935944 is found.

**Category:** Geometry

[283] **viXra:1801.0156 [pdf]**
*submitted on 2018-01-13 21:02:25*

**Authors:** Xu Chen

**Comments:** 13 Pages.

In this article, we will discuss a localization formulas of equlvariant cohomology about two Killing vector fields on the set of zero points
${\rm{Zero}}(X_{M}-\sqrt{-1}Y_{M})=\{x\in M \mid |Y_{M}(x)|=|X_{M}(x)|=0 \}.$ As application, we use it to get formulas about characteristic numbers and to get a Duistermaat-Heckman type formula on symplectic manifold.

**Category:** Geometry

[282] **viXra:1801.0155 [pdf]**
*submitted on 2018-01-13 21:07:54*

**Authors:** Xu Chen

**Comments:** 7 Pages.

We extend the reslut about Poincar\'e-Hopf type formula for the difference of the Chern character numbers (cf.[3]) to the non-isolated singularities, and establish a Poincar\'e-Hopf type formula for a pair of vector field with the function $h^{T_{\mathbb{C}}M}(\cdot,\cdot)$ has non-isolated zero points over a closed, oriented smooth manifold
of dimension $2n$.

**Category:** Geometry

[281] **viXra:1801.0146 [pdf]**
*submitted on 2018-01-13 01:53:56*

**Authors:** Sennimalai Kalimuthu

**Comments:** 03 Pages. Interested people may contact k,me at any time. Tha

The 5th Euclidean postulate is 2300 years old mathematical impossibility. I have worked on this problem for nearly b35 years and found a number of consistent solutions. My findings have been appeared in international peer reviewed research journals.
Generation of power freely from space, space Bombs, Lion’s Tonic and Lemurian Yoga are my ambitious scientific projects.
Interested researchers and people may contact me at +91 8508991577.
My email is math.kalimuthu@gmail.com and arutperunjothi@outlook.com

**Category:** Geometry

[280] **viXra:1801.0056 [pdf]**
*submitted on 2018-01-05 09:56:59*

**Authors:** Carlos Alejandro Chiappini

**Comments:** 7 Pages.

Leonhard Euler demostró que en un poliedro regular convexo hay tres números ue cumplen una ley, expresada en una ecuación conocida como fórmula de Euler. Son el número de caras, el número de vértices y el número de aristas.
Este documento presenta algunas fórmulas más, obtenidas por ensayo y error a partir de una tabla que contiene los datos de los 5 poliedros regulares convexos. Estas fórmulas indemostradas tienen rasgos verosímiles. Buscar el modo de demostrar la invalidez o la validez de esas fórmulas podría ser, para las personas amantes de la topología, una tarea interesante.

**Category:** Geometry

[279] **viXra:1712.0524 [pdf]**
*submitted on 2017-12-19 13:56:07*

**Authors:** James A. Smith

**Comments:** 8 Pages.

We show how to calculate the projection of a vector, from an arbitrary direction, upon a given plane whose orientation is characterized by its normal vector, and by a bivector to which the plane is parallel. The resulting solutions are tested by means of an interactive GeoGebra construction.

**Category:** Geometry

[278] **viXra:1712.0393 [pdf]**
*submitted on 2017-12-11 16:46:30*

**Authors:** James A. Smith

**Comments:** 29 Pages. Formulas and Spreadsheets for Simple, Composite, and Complex Rotations of Vectors and Bivectors in Geometric (Clifford) Algebra

We show how to express the representations of single, composite, and ``rotated" rotations in GA terms that allow rotations to be calculated conveniently via spreadsheets. Worked examples include rotation of a single vector by a bivector angle; rotation of a vector about an axis; composite rotation of a vector; rotation of a bivector; and the ``rotation of a rotation". Spreadsheets for doing the calculations are made available via live links.

**Category:** Geometry

[277] **viXra:1711.0317 [pdf]**
*submitted on 2017-11-14 17:32:57*

**Authors:** Giordano Colò

**Comments:** 22 Pages.

We try to give a formulation of Strominger-Yau-Zaslow conjecture
on mirror symmetry by studying the singularities of special Lagrangian
submanifolds of 3-dimensional Calabi-Yau manifolds. In this
paper we’ll give the description of the boundary of the moduli space
of special Lagrangian manifolds.
We do this by introducing special Lagrangian cones in the more
general Kähler manifolds. Then we can focus on the almost Calabi-
Yau manifolds. We consider the behaviour of the Lagrangian manifolds
near the conical singular points to classify them according to
the way they are approximated from the asymptotic cone. Then we
analyze their deformations in Calabi-Yau manifolds.

**Category:** Geometry

[276] **viXra:1711.0306 [pdf]**
*submitted on 2017-11-14 06:43:32*

**Authors:** Edgar Valdebenito

**Comments:** 8 Pages.

This note presents some fractals.

**Category:** Geometry

[275] **viXra:1710.0264 [pdf]**
*submitted on 2017-10-23 07:56:01*

**Authors:** Edgar Valdebenito

**Comments:** 7 Pages.

This note presents some fractals related with the function: f(z)=((1-z^5)^2/(1+z^10))-z

**Category:** Geometry

[274] **viXra:1710.0241 [pdf]**
*submitted on 2017-10-22 16:40:29*

**Authors:** Paris Samuel Miles-Brenden

**Comments:** 5 Pages. None.

None.

**Category:** Geometry

[273] **viXra:1710.0147 [pdf]**
*submitted on 2017-10-14 08:51:55*

**Authors:** James A. Smith

**Comments:** 13 Pages.

We show how to express the representation of a composite rotation in terms that allow the rotation of a vector to be calculated conveniently via a spreadsheet that uses formulas developed, previously, for a single rotation. The work presented here (which includes a sample calculation) also shows how to determine the bivector angle that produces, in a single operation, the same rotation that is effected by the composite of two rotations.

**Category:** Geometry

[272] **viXra:1710.0137 [pdf]**
*submitted on 2017-10-13 01:17:37*

**Authors:** Antoine Balan

**Comments:** 5 pages, written in french

The Dirac operator is twisted by a symmetric automorphism, the Dirac-Lichnerowicz formula is proved. An application for the Seiberg-Witten equations is proposed.

**Category:** Geometry

[271] **viXra:1710.0131 [pdf]**
*submitted on 2017-10-11 07:44:42*

**Authors:** Edgar Valdebenito

**Comments:** 8 Pages.

This note presents a fractal image for f(z)=ln(1+g(z)).

**Category:** Geometry

[270] **viXra:1710.0127 [pdf]**
*submitted on 2017-10-11 20:27:18*

**Authors:** Choe ryujin

**Comments:** 4 Pages.

Proof of happy ending problem

**Category:** Geometry

[269] **viXra:1710.0110 [pdf]**
*submitted on 2017-10-10 11:31:30*

**Authors:** Mauro Bernardini

**Comments:** 4 Pages.

This paper attempts to provide a new vision on the 4th spatial dimension starting on the known symmetries of the Euclidean geometry. It results that, the points of the 4th dimensional complex space are circumferences of variable ray. While the axis of the 4th spatial dimension, to be orthogonal to all the three 3d cartesinan axes, is a complex line made of two specular cones surfaces symmetrical on their vertexes corresponding to the common origin of both the real and complex cartesian systems.

**Category:** Geometry

[268] **viXra:1709.0439 [pdf]**
*submitted on 2017-09-30 10:53:14*

**Authors:** Bouetou Bouetou Thomas

**Comments:** 10 Pages. non

I have noticed a situation of plagiarist and want to draw the attention of the authors and readers.

**Category:** Geometry

[267] **viXra:1709.0109 [pdf]**
*submitted on 2017-09-10 05:11:19*

**Authors:** Prashanth R. Rao

**Comments:** 3 Pages.

In this paper, we generate a special hexagon with two-fold symmetry by diagonally juxtaposing two squares of different dimensions so that they share exactly one common vertex and their adjacent sides are perpendicular to one another. We connect in specific pairs, the vertices adjacent to common vertex of both squares to generate a hexagon that is symmetrical about a line connecting the unconnected vertices. We show that this special hexagon must have one square whose points lie on its sides. With suitable modifications, it may be possible to use this technique to prove the Toeplitz conjecture for a simple closed curve generated by connecting the same six vertices of this special hexagon.

**Category:** Geometry

[266] **viXra:1709.0026 [pdf]**
*submitted on 2017-09-02 14:50:05*

**Authors:** Prashanth R. Rao

**Comments:** 2 Pages.

According to Toeplitz conjecture or the inscribed square conjecture, every simple closed curve in a plane must have atleast one set of four points on it that belong to a square. This conjecture remains unsolved for a general case although it has been proven for some special cases of simple closed curves. In this paper, we prove the conjecture for a special case of a simple closed curve derived from two simple closed curves, each of which have exactly only one set of points defining a square. The Toeplitz solution squares of two parent simple closed curves have the same dimensions and share exactly one common vertex and the adjacent sides of the two squares form a right angle. The derived simple closed curve is formed by eliminating this common vertex (that belonged to the two solutions squares to begin with) and connecting other available points on the parent curves. We show that this derived simple closed curve has atleast one solution square satisfying the Toeplitz conjecture.

**Category:** Geometry

[265] **viXra:1708.0462 [pdf]**
*submitted on 2017-08-29 17:16:30*

**Authors:** James A. Smith

**Comments:** 9 Pages.

We show how to transform a "rotate a vector around a given axis" problem into one that may be solved via GA, which rotates objects with respect to bivectors. A sample problem is worked to show how to calculate the result of such a rotation conveniently via an Excel spreadsheet, to which a link is provided.

**Category:** Geometry

[264] **viXra:1708.0459 [pdf]**
*submitted on 2017-08-30 06:57:06*

**Authors:** Adham ahmed mohamed ahmed

**Comments:** 1 Page.

This is a method to calculate pi using a computer

**Category:** Geometry

[263] **viXra:1708.0420 [pdf]**
*submitted on 2017-08-28 08:14:53*

**Authors:** Edgar Valdebenito

**Comments:** 56 Pages.

This note presents some examples of Newton-Raphson fractals.

**Category:** Geometry

[262] **viXra:1708.0236 [pdf]**
*submitted on 2017-08-19 17:45:13*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

Abstract: In this paper we prove that any two points A and B in space that are at different distances from a third point C, when connected by any curve in three dimensional space, must contain points such as D that are at intermediate distances from the third point C (length DC is intermediate to length AC and length BC).

**Category:** Geometry

[261] **viXra:1708.0229 [pdf]**
*submitted on 2017-08-19 12:49:36*

**Authors:** Prashanth R. Rao

**Comments:** 1 Page.

In this paper, we give a simple proof that if there are two points A and B that are at distinct linear distances from a third point C (AC is not equal to BC), then any curve connecting the points A and B (this curve lies within the same plane containing A,B,C) must contain points such as D that lie at an intermediate distance from C, (DC is of length intermediate to AC and BC).

**Category:** Geometry

[260] **viXra:1708.0191 [pdf]**
*submitted on 2017-08-17 03:54:59*

**Authors:** Vu B Ho

**Comments:** 6 Pages.

In this work we show that by restricting the coordinate transformations to the group of time-independent coordinate transformations it is possible to derive the Ricci flow from the contracted Bianchi identities.

**Category:** Geometry

[259] **viXra:1708.0027 [pdf]**
*submitted on 2017-08-02 13:40:27*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

The pseudo-quaternionic manifolds are studied. Several tensors are defined.

**Category:** Geometry

[258] **viXra:1707.0374 [pdf]**
*submitted on 2017-07-28 10:50:06*

**Authors:** Liu Ran

**Comments:** 3 Pages.

因为圆周率是一个比值，圆周长和直径都是测量而来，只要去测量圆周长和直径，就不可避免要在空间去度量，而空间，全都是非欧空间。

**Category:** Geometry

[257] **viXra:1707.0249 [pdf]**
*submitted on 2017-07-18 12:52:38*

**Authors:** Ulrich E. Bruchholz, Horst Eckardt

**Comments:** 19 Pages.

The well known numerical method of approximating differential
quotients by quotients of differences is used in a novel context.
This method is commonly underestimated, wrongly.
The method is explained by an ordinary differential equation first.
Then it is demonstrated how this simple method
proves successful for non-linear field equations with chaotic
behaviour. Using certain discrete values of their integration constants,
a behaviour comparable with Mandelbrot sets is obtained.
Instead of solving the
differential equations directly, their convergence behaviour is analyzed.
As an example the Einstein-Maxwell equations are investigated,
where discrete
particle quantities are obtained from a continuous theory, which is possible
only by this method.
The special set of integration constants contains values identical with
particle characteristics.
Known particle values are confirmed, and unknown values can be predicted.
In this paper, supposed neutrino masses are presented.

**Category:** Geometry

[256] **viXra:1707.0242 [pdf]**
*submitted on 2017-07-17 13:20:25*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

In this note we briefly examine the curve: x^3+y^3=sqrt(2)*x*y

**Category:** Geometry

[255] **viXra:1707.0181 [pdf]**
*submitted on 2017-07-12 20:58:33*

**Authors:** Pan Zhang

**Comments:** 14 Pages.

Let $V$ be an asymptotically cylindrical K\"{a}hler manifold with asymptotic
cross-section $\mathfrak{D}$. Let $E_\mathfrak{D}$ be a stable Higgs bundle over $\mathfrak{D}$, and $E$ a Higgs bundle over $V$ which is asymptotic to
$E_\mathfrak{D}$. In this paper, using the continuity method of Uhlenbeck and Yau, we prove that there exists an asymptotically translation-invariant Hermitian projectively Hermitian Yang-Mills metric on $E$.

**Category:** Geometry

[254] **viXra:1707.0046 [pdf]**
*submitted on 2017-07-04 11:58:48*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

A Laplacian operator is defined bound to a riemannian manifold with a pseudo-complex structure.

**Category:** Geometry

[253] **viXra:1707.0011 [pdf]**
*submitted on 2017-07-01 08:14:34*

**Authors:** John Atwell Moody

**Comments:** 17 Pages. written September 2014

When a meromorphic vector field is given on the projective plane, a complete holomorphic limit cycle, because it is a closed singular submanifold of projective space, is defined by algebraic equations. Also the meromorphic vector field is an algebraic object. Poincare had asked, is there just an algebraic calculation leading from the vector field to the defining equations of the solution, without the
mysterious intermediary of the dynamical system.
The answer is yes, that there is nothing more mysterious or wonderful that happens when a complete holomorphic limit cycle is formed than could have been defined using algebra.

**Category:** Geometry

[252] **viXra:1707.0010 [pdf]**
*submitted on 2017-07-01 08:17:31*

**Authors:** John Atwell Moody

**Comments:** Pages.

Contents:

Analytic primality testing 1

Lefshetz numbers of modular curves 23

Grothendieck sections and rational points of modular curves 29

Rational points of modular curves 33

Conclusion about modular forms 41

Outline geometric proof of Mordell’s conjecture 48

Example:the Fermat curves

63 The residue calculation 69

The meaning of positive and negative 81

**Category:** Geometry

[251] **viXra:1707.0009 [pdf]**
*submitted on 2017-07-01 08:21:26*

**Authors:** John Atwell Moody

**Comments:** Pages.

Contents

Projective toric varieties

Divisors

Maps

Chow ring

Fourier series and generalizations

**Category:** Geometry

[250] **viXra:1706.0497 [pdf]**
*submitted on 2017-06-27 02:58:55*

**Authors:** Orgest ZAKA, Kristaq FILIPI

**Comments:** 4 Pages. https://www.ijsr.net/archive/v6i6/ART20174592.pdf

In this paper we present an application possibility of the affine plane of order $n$, in the planning experiment, taking samples as his point. In this case are needed $n^2$ samples. The usefulness of the support of experimental planning in a finite affine plane consists in avoiding the partial repetition combinations within a proof. Reviewed when planning cannot directly drawn over an affine plane. In this case indicated how the problem can be completed, and when completed can he, with intent to drawn on an affine plane.

**Category:** Geometry

[249] **viXra:1706.0409 [pdf]**
*submitted on 2017-06-21 01:40:42*

**Authors:** Antoine Balan

**Comments:** 5 pages, written in french

A symplectic Dirac operator is defined for a spin Kaehler manifold. The corresponding Schrödinger-Lichnerowicz formula is proved.

**Category:** Geometry

[248] **viXra:1706.0021 [pdf]**
*submitted on 2017-06-02 19:22:09*

**Authors:** Mendzina Essomba Francois, Essomba Essomba Dieudonne Gael

**Comments:** 24 Pages.

The same mathematical equation connects the circle to the square, the sphere to the cube, the hyper-sphere to the hyper-cube, another also connects the ellipse to the rectangle, the ellipsoid to a rectangular parallelepiped, the hyper-ellipsoid To the rectangular hyper-parallelepiped.
The understanding of these equations has taken us very far in a universe so familiar to mathematicians, the universe of periodic functions, and that of geometric forms with rounded ends revealing an infinity of new mathematical constants associated with them.

**Category:** Geometry

[247] **viXra:1704.0343 [pdf]**
*submitted on 2017-04-25 13:47:47*

**Authors:** Giordano Colò

**Comments:** 28 Pages.

We describe the deformations of the moduli space M of Special Lagrangian submanifolds in the compact case and we give a characterization of the topology of M by using McLean theorem. We consider Banach spaces on bundle sections and elliptical operators and we use Hodge theory to study the topology of the manifold. Starting from McLean results, for which the moduli space of compact special Lagrangian submanifolds is smooth and its tangent space can be identified with harmonic 1-forms on these submanifolds, we can analyze their deformations. Then we introduce a Riemannian metric on M, from which we obtain other important properties.

**Category:** Geometry

[246] **viXra:1704.0328 [pdf]**
*submitted on 2017-04-25 03:20:29*

**Authors:** Ion Patrascu, Florentin Smarandache

**Comments:** 3 Pages.

In this article, we highlight some properties of the Apollonius circles of rank - 1 associated with a triangle.

**Category:** Geometry

[245] **viXra:1704.0134 [pdf]**
*submitted on 2017-04-11 04:15:20*

**Authors:** Xu Chen

**Comments:** 23 Pages.

In this article, we computed the homology groups of real Grassmann manifold $G_{n,m}(\mathbb{R})$ by Witten complex.

**Category:** Geometry

[244] **viXra:1703.0273 [pdf]**
*submitted on 2017-03-28 19:09:13*

**Authors:** Arman Maesumi

**Comments:** 5 Pages.

Given a triangle ABC, the average area of an inscribed triangle RST whose vertices are uniformly distributed on BC, CA and AB, is proven to be one-fourth of the area of ABC. The average of the square of the area of RST is shown to be one-twelfth of the square of the area of ABC, and the average of the cube of the ratio of the areas is 5/144. A Monte Carlo simulation confirms the theoretical results, as well as a Maxima program which computes the exact averages.

**Category:** Geometry

[243] **viXra:1703.0267 [pdf]**
*submitted on 2017-03-28 08:30:43*

**Authors:** Jan Hakenberg, Ulrich Reif

**Comments:** 5 Pages.

The derivation of multilinear forms used to compute the moments of sets bounded by subdivision surfaces requires solving a number of systems of linear equations. As the support of the subdivision mask or the degree of the moment grows, the corresponding linear system becomes intractable to construct, let alone to solve by Gaussian elimination. In the paper, we argue that the power iteration and the geometric series are feasible methods to approximate the multilinear forms. The tensor iterations investigated in this work are shown to converge at favorable rates, achieve arbitrary numerical accuracy, and have a small memory footprint. In particular, our approach makes it possible to compute the volume, centroid, and inertia of spatial domains bounded by Catmull-Clark and Loop subdivision surfaces.

**Category:** Geometry

[242] **viXra:1703.0080 [pdf]**
*submitted on 2017-03-09 03:29:56*

**Authors:** Gaurav S. Biraris

**Comments:** 20 Pages.

The paper proposes a generalization of geometric notion of vectors concerning dimensionality of the configuration space. In certain dimensional spaces, certain types of ordered directions exist along which elements of vector spaces can be interpreted. Scalars along the ordered directions form Banach spaces. Different types of geometrical vectors are algebraically identical, the difference arises in the configuration space geometrically. In the universe four types of vectors exists. Thus any physical quantity in the universe comes in four types of vectors. Though All the types of vectors belong to different Banach spaces (& their directions can’t be compared), their magnitudes can be compared. A gross comparison between the magnitudes of the different typed geometric vectors is obtained at end of the paper.

**Category:** Geometry

[241] **viXra:1702.0049 [pdf]**
*submitted on 2017-02-03 12:34:37*

**Authors:** Gerasimos T. Soldatos

**Comments:** 3 Pages. Published in: Forum Geometricorum, 2017, Volume 17, pp. 13-15

An “Archimedean” quadrature is attempted “borrowing” π from the 3-dimensional space of a horn torus

**Category:** Geometry

[240] **viXra:1701.0576 [pdf]**
*submitted on 2017-01-23 09:11:18*

**Authors:** Dragan Turanyanin, Svetozar Jovičin

**Comments:** 14 Pages.

The aim of this article would be to show planar (whether polar, Cartesian or parametric) functions from a different, implicit viewpoint, hence the term inpolars (inpolar curves). The whole set of brand new planar curves can be seen from that perspective. Their generic mechanism is the so called inpolar transformation as well as its inpolar inversion. One entirely new geometric system is defined this way.

**Category:** Geometry

[239] **viXra:1612.0398 [pdf]**
*submitted on 2016-12-29 12:32:03*

**Authors:** Irina I. Bodrenko

**Comments:** 3 Pages.

The some properties of hypersurfaces F3 in Euclidean space E4 of nonzero
Laplacian of the second fundamental form b are studied in this report.

**Category:** Geometry

[238] **viXra:1611.0035 [pdf]**
*submitted on 2016-11-02 18:56:08*

**Authors:** Dragan Turanyanin

**Comments:** 3 Pages.

This spiral (given in polar coordinates r, theta) can be seen as a missing member of the set of known spirals.

**Category:** Geometry

[237] **viXra:1611.0003 [pdf]**
*submitted on 2016-11-01 01:37:29*

**Authors:** Brian Ekanyu

**Comments:** 3 Pages.

This paper proves a geometric theorem about the Hebrew nation that is the Patriarchs Abraham, Isaac and Jacob and the Twelve tribes of Israel. The circle represents the state of Israel.

**Category:** Geometry

[236] **viXra:1610.0367 [pdf]**
*submitted on 2016-10-30 14:48:27*

**Authors:** Dragan Turanyanin

**Comments:** 6 Pages.

The aim of this review is firstly, to present again a new family of polar curves (e.g. thurals [1]) and secondly, to introduce their so called inpolars as main objects of one original geometrical transformation [2]. Addendum is completely new with a brief analysis of s-thural.

**Category:** Geometry

[235] **viXra:1610.0076 [pdf]**
*submitted on 2016-10-06 19:47:24*

**Authors:** James A. Smith

**Comments:** 5 Pages.

To the collections of problems solved via Geometric Algebra (GA) in References 1-13, this document adds a solution, using only dot products, to the Problem of Apollonius. The solution is provided for completeness and for contrast with the GA solutions presented in Reference 3.

**Category:** Geometry

[234] **viXra:1610.0054 [pdf]**
*submitted on 2016-10-04 18:46:06*

**Authors:** James A. Smith

**Comments:** 15 Pages.

Drawing mainly upon exercises from Hestenes's New Foundations for Classical Mechanics, this document presents, explains, and discusses common solution strategies. Included are a list of formulas and a guide to nomenclature.

**Category:** Geometry

[233] **viXra:1609.0365 [pdf]**
*submitted on 2016-09-25 18:17:04*

**Authors:** James A. Smith

**Comments:** 5 Pages.

This document adds to the collection of GA solutions to plane-geometry problems, most of them dealing with tangency, that are presented in References 1-7. Reference 1 presented several ways of solving the CPP limiting case of the Problem of Apollonius. Here, we use ideas from Reference 6 to solve that case in yet another way.

**Category:** Geometry

[232] **viXra:1609.0082 [pdf]**
*submitted on 2016-09-06 19:08:36*

**Authors:** Marvin Ray Burns

**Comments:** 8 Pages. This classic paper shows the utter simplicity of the geometric description of the MRB constant (oeis.org/A037077).

The MRB constant is the upper limit point of the sequence of partial sums defined by S(x)=sum((-
1)^n*n^(1/n),n=1..x). The goal of this paper is to show that the MRB constant is geometrically
quantifiable. To “measure” the MRB constant, we will consider a set, sequence and alternating series of
the nth roots of n. Then we will compare the length of the edges of a special set of hypercubes or ncubes
which have a content of n. (The two words hypercubes and n-cubes will be used synonymously.)
Finally, we will look at the value of the MRB constant as a representation of that comparison, of the length of the edges of a special set of hypercubes, in units of dimension 1/ (units of dimension 2 times
units of dimension 3 times units of dimension 4 times etc.). For an arbitrary example we will use units of
length/ (time*mass* density*…).

**Category:** Geometry

[231] **viXra:1608.0328 [pdf]**
*submitted on 2016-08-24 18:28:40*

**Authors:** James A. Smith

**Comments:** 38 Pages.

This document adds to the collection of solved problems presented in References [1]-[6]. The solutions presented herein are not as efficient as those in [6], but they give additional insight into ways in which GA can be used to solve this problem. After reviewing, briefly, how reflections and rotations can be expressed and manipulated via GA, it solves the CLP limiting case of the Problem of Apollonius in three ways, some of which identify the the solution circles' points of tangency with the given circle, and others of which identify the solution circles' points of tangency with the given line. For comparison, the solutions that were developed in [1] are presented in an Appendix.

**Category:** Geometry

[83] **viXra:1807.0234 [pdf]**
*replaced on 2018-07-14 06:23:07*

**Authors:** James A. Smith

**Comments:** 18 Pages.

As a demonstration of the coherence of Geometric Algebra's (GA's) geometric and algebraic concepts of bivectors, we add three geometric bivectors according to the procedure described by Hestenes and Macdonald, then use bivector identities to determine, from the result, two bivectors whose outer product is equal to the initial sum. In this way, we show that the procedure that GA's inventors dened for adding geometric bivectors is precisely that which is needed to give results that coincide with those obtained by calculating outer products of vectors that are expressed in terms of a 3D basis. We explain that that accomplishment is no coincidence: it is a consequence of the attributes that GA's designers assigned (or didn't) to bivectors.

**Category:** Geometry

[82] **viXra:1805.0030 [pdf]**
*replaced on 2018-05-30 16:22:45*

**Authors:** Johan Aspegren

**Comments:** 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to denefine certain surface measures on surfaces of convex bodies in a novel and simple way.

**Category:** Geometry

[81] **viXra:1805.0030 [pdf]**
*replaced on 2018-05-22 11:28:23*

**Authors:** Johan Aspegren

**Comments:** 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with a polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to define certain surface measures on surfaces of convex bodies in a novel and simple way.

**Category:** Geometry

[80] **viXra:1805.0030 [pdf]**
*replaced on 2018-05-20 22:09:51*

**Authors:** Johan Aspegren

**Comments:** 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with a polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to define certain surface measures on surfaces of convex bodies in a novel and simple way.

**Category:** Geometry

[79] **viXra:1805.0030 [pdf]**
*replaced on 2018-05-06 01:48:26*

**Authors:** Johan Aspegren

**Comments:** 7 Pages.

One theme of this paper is to extend known results from polygons and balls to the general convex bodies in n− dimensions. An another theme stems from approximating a convex surface with a polytope surface. Our result gives a sufficient and necessary condition for an natural approximation method to succeed (in principle) in the case of surfaces of convex bodies. Thus, Schwartz`s paradox does not affect our method. This allows us to define certain surface measures on surfaces of convex bodies in a novel and simple way.

**Category:** Geometry

[78] **viXra:1804.0003 [pdf]**
*replaced on 2018-06-05 10:27:07*

**Authors:** Antoine Balan

**Comments:** 6 pages, written in french

We define here the Seiberg-Witten equations in the quaternionic case. We formulate some algebra of the Hamilton numbers and study geometric applications of the quaternions.

**Category:** Geometry

[77] **viXra:1803.0119 [pdf]**
*replaced on 2018-03-09 09:44:09*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

Following the definition of the flow of Ricci and with help of the Dirac operator, we construct a flow of hermitian metrics for the spinors fiber bundle.

**Category:** Geometry

[76] **viXra:1803.0050 [pdf]**
*replaced on 2018-03-05 11:16:01*

**Authors:** Antoine Balan

**Comments:** 1 page, written in french

The flow of Ricci is defined for the hermitian metrics of a complex fiber bundle.

**Category:** Geometry

[75] **viXra:1802.0196 [pdf]**
*replaced on 2018-02-16 15:58:29*

**Authors:** Hiroshi Okumura

**Comments:** 3 Pages. The paper is considering a problem in Wasan geometry.

A problem involving a square in the curvilinear triangle made by two touching congruent circles and their common tangent is generalized.

**Category:** Geometry

[74] **viXra:1801.0347 [pdf]**
*replaced on 2018-01-27 07:26:09*

**Authors:** Antoine Balan

**Comments:** 2 pages, written in french

The flow of Ricci-Schrödinger is defined from the Ricci flow, like the Schrödinger equation with respect to the heat equation.

**Category:** Geometry

[73] **viXra:1712.0642 [pdf]**
*replaced on 2018-04-23 22:13:50*

**Authors:** James A. Smith

**Comments:** 13 Pages.

Published times of the Earth's perihelions do not refer to the perihelions of the orbit that the Earth would follow if unaffected by other bodies such as the Moon. To estimate the timing of that ``unperturbed" perihelion, we fit an unperturbed Kepler orbit to the timings of the year 2017's equinoxes and solstices. We find that the unperturbed 2017 perihelion, defined in that way, would occur 12.93 days after the December 2016 solstice. Using that result, calculated times of the year 2017's solstices and equinoxes differ from published values by less than five minutes. That degree of accuracy is sufficient for the intended use of the result.

**Category:** Geometry

[72] **viXra:1711.0317 [pdf]**
*replaced on 2017-12-17 12:03:53*

**Authors:** Giordano Colò

**Comments:** 22 Pages.

We try to give a formulation of Strominger-Yau-Zaslow conjecture on mirror symmetry by studying the singularities of special Lagrangian submanifolds of 3-dimensional Calabi-Yau manifolds. In this paper we’ll give the description of the boundary of the moduli space of special Lagrangian manifolds. We do this by introducing special Lagrangian cones in the more general Kähler manifolds. Then we can focus on the almost Calabi-Yau manifolds. We consider the behaviour of the Lagrangian manifolds near the conical singular points to classify them according to the way they are approximated from the asymptotic cone. Then we analyze their deformations in Calabi-Yau manifolds.

**Category:** Geometry

[71] **viXra:1711.0317 [pdf]**
*replaced on 2017-11-16 11:38:05*

**Authors:** Giordano Colò

**Comments:** 21 Pages.

We try to give a formulation of Strominger-Yau-Zaslow conjecture
on mirror symmetry by studying the singularities of special Lagrangian
submanifolds of 3-dimensional Calabi-Yau manifolds. In this
paper we’ll give the description of the boundary of the moduli space
of special Lagrangian manifolds.
We do this by introducing special Lagrangian cones in the more
general Kähler manifolds. Then we can focus on the textitalmost
Calabi-Yau manifolds. We consider the behaviour of the Lagrangian
manifolds near the conical singular points to classify them according
to the way they are approximated from the asymptotic cone. Then
we analyze their deformations in Calabi-Yau manifolds.

**Category:** Geometry

[70] **viXra:1710.0110 [pdf]**
*replaced on 2017-10-13 16:26:37*

**Authors:** Mauro Bernardini

**Comments:** 4 Pages. this version corrects some accidental writing errors of the previous loaded version.

This paper attempts to provide a new vision on the 4th spatial dimension starting on the known symmetries of the Euclidean geometry. It results that, the points of the 4th dimensional complex space are circumferences of variable ray. While the axis of the 4th spatial dimension, to be orthogonal to all the three 3d cartesinan axes, is a complex line made of two specular cones surfaces symmetrical on their vertexes corresponding to the common origin of both the real and complex cartesian systems.

**Category:** Geometry

[69] **viXra:1709.0144 [pdf]**
*replaced on 2018-01-06 02:31:12*

**Authors:** Pawan Kumar Bishwakarma

**Comments:** 16 Pages.

A regular polygon is a planar geometrical structure with all sides of equal length and all angles of equal magnitude. The ratio of perimeter of any regular polygon to the length of its longest diagonal is a constant term and the ratio converges to the value of as the number of sides of the polygon increases. The result has been shown to be valid by actually calculating the ratio for each polygon by using corresponding formula and geometrical reasoning. A computational calculation of the ratio has also been presented to validate the convergence. The values have been calculated up to 30 significant digits.

**Category:** Geometry

[68] **viXra:1707.0374 [pdf]**
*replaced on 2017-07-29 09:16:08*

**Authors:** Liu Ran

**Comments:** 3 Pages.

因为圆周率是一个比值，圆周长和直径都是测量而来，只要去测量圆周长和直径，就不可避免要在空间去度量，而空间，全都是非欧空间。

**Category:** Geometry

[67] **viXra:1706.0409 [pdf]**
*replaced on 2017-06-28 01:37:22*

**Authors:** Antoine Balan

**Comments:** 4 pages, written in french

The symplectic Dirac operator is defined over a spin Kaehler manifold. The corresponding Schrödinger-Lichnerowicz formula is proved.

**Category:** Geometry

[66] **viXra:1706.0021 [pdf]**
*replaced on 2017-06-04 16:51:38*

**Authors:** Mendzina Essomba Francois, Essomba Essomba Dieudonne Gael

**Comments:** 25 Pages.

The same mathematical equation connects the circle to the square, the sphere to the cube, the hyper-sphere to the hyper-cube, another also connects the ellipse to the rectangle, the ellipsoid to a rectangular parallelepiped, the hyper-ellipsoid To the rectangular hyper-parallelepiped.
The understanding of these equations has taken us very far in a universe so familiar to mathematicians, the universe of periodic functions, and that of geometric forms with rounded ends revealing an infinity of new mathematical constants associated with them.

**Category:** Geometry

[65] **viXra:1703.0080 [pdf]**
*replaced on 2017-11-27 07:14:25*

**Authors:** Gaurav Biraris

**Comments:** 20 Pages. A citation is published/changed

The paper proposes a generalization of geometric notion of vectors concerning dimensionality of the configuration space. In certain dimensional spaces, certain types of ordered directions exist along which elements of vector spaces can be interpreted. Scalars along the ordered directions form Banach spaces. Different types of geometrical vectors are algebraically identical, the difference arises in the configuration space geometrically. In the universe four types of vectors exists. Thus any physical quantity in the universe comes in four types of vectors. Though All the types of vectors belong to different Banach spaces (& their directions can’t be compared), their magnitudes can be compared. A gross comparison between the magnitudes of the different typed geometric vectors is obtained at end of the paper.

**Category:** Geometry

[64] **viXra:1608.0328 [pdf]**
*replaced on 2016-08-27 23:19:26*

**Authors:** James A. Smith

**Comments:** 38 Pages.

This document adds to the collection of solved problems presented in References [1]-[6]. The solutions presented herein are not as efficient as those in [6], but they give additional insight into ways in which GA can be used to solve this problem. After reviewing, briefly, how reflections and rotations can be expressed and manipulated via GA, it solves the CLP limiting case of the Problem of Apollonius in three ways, some of which identify the the solution circles' points of tangency with the given circle, and others of which identify the solution circles' points of tangency with the given line. For comparison, the solutions that were developed in [1] are presented in an Appendix.

**Category:** Geometry