[14] **viXra:1708.0445 [pdf]**
*replaced on 2019-04-12 06:04:26*

**Authors:** Victor Christianto, Florentin Smarandache

**Comments:** 12 Pages. This paper has been published by Octogon Mathematical Magazine, 2018. Your comments are welcome

While it is known that Euclid’s five axioms include a proposition that a line consists at least of two points, modern geometry avoid consistently any discussion on the precise definition of point, line, etc. It is our aim to clarify one of notorious question in Euclidean geometry: how many points are there in a line segment? – from discrete-cellular space (DCS) viewpoint. In retrospect, it may offer an alternative of quantum gravity, i.e. by exploring discrete gravitational theories. To elucidate our propositions, in the last section we will discuss some implications of discrete cellular-space model in several areas of interest: (a) cell biology, (b) cellular computing, (c) Maxwell equations, (d) low energy fusion, and (e) cosmology modelling

**Category:** Mathematical Physics

[13] **viXra:1708.0424 [pdf]**
*submitted on 2017-08-29 02:39:21*

**Authors:** Vu B Ho

**Comments:** 5 Pages.

In this work, we discuss the possibility to formulate a covariant Ricci flow so that it satisfies the principle of relativity and therefore can be applied to all coordinate systems defined on a Riemannian manifold. Since the investigation may be considered to be in the domain of pure mathematics, which is outside our field of physical investigations, therefore there may be errors in mathematical arguments that we are unable to foresee.

**Category:** Mathematical Physics

[12] **viXra:1708.0406 [pdf]**
*submitted on 2017-08-28 04:39:26*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 6 Pages. In French.

From the equations of the problem of $n$ body, we consider that $t$ is a function of the variables $(x_k,y_k,z_k)_{k=1,n}$. We write a new formulation of the equations of the $n$ body problem.

**Category:** Mathematical Physics

[11] **viXra:1708.0328 [pdf]**
*replaced on 2017-10-29 11:13:19*

**Authors:** Spiros Konstantogiannis

**Comments:** 17 Pages.

Making use of the Bethe ansatz, we introduce a quotient polynomial and we show that the presence of intermediate terms in it, i.e. terms other than the constant and the leading one, constitutes a non-solvability condition of the respective potential. In this context, both the exact solvability of the quantum harmonic oscillator and the quasi-exact solvability of the sextic anharmonic oscillator stem naturally from the quotient polynomial, as in the first case, it is an energy-dependent constant, while in the second case, it is a second-degree binomial with no linear term. In all other cases, the quotient polynomial has at least one intermediate term, the presence of which makes the respective potential non-solvable.

**Category:** Mathematical Physics

[10] **viXra:1708.0254 [pdf]**
*replaced on 2019-01-25 14:11:13*

**Authors:** Robert B. Easter

**Comments:** 27 pages. Extended paper, extending the 10-page conference paper Double Conformal Space-Time Algebra (ICNPAA 2016; DOI:10.1063/1.4972658).

The G(4,8) Double Conformal Space-Time Algebra (DCSTA) is a high-dimensional 12D Geometric Algebra that extends the concepts introduced with the G(8,2) Double Conformal / Darboux Cyclide Geometric Algebra (DCGA) with entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) in spacetime with a new boost operator. The base algebra in which spacetime geometry is modeled is the G(1,3) Space-Time Algebra (STA). Two G(2,4) Conformal Space-Time subalgebras (CSTA) provide spacetime entities for points, hypercones, hyperplanes, hyperpseudospheres (and their intersections) and a complete set of versors for their spacetime transformations that includes rotation, translation, isotropic dilation, hyperbolic rotation (boost), planar reflection, and (pseudo)spherical inversion. G(4,8) DCSTA is a doubling product of two orthogonal G(2,4) CSTA subalgebras that inherits doubled CSTA entities and versors from CSTA and adds new 2-vector entities for general (pseudo)quadrics and Darboux (pseudo)cyclides in spacetime that are also transformed by the doubled versors. The "pseudo" surface entities are spacetime surface entities that use the time axis as a pseudospatial dimension. The (pseudo)cyclides are the inversions of (pseudo)quadrics in hyperpseudospheres. An operation for the directed non-uniform scaling (anisotropic dilation) of the 2-vector general quadric entities is defined using the boost operator and a spatial projection. Quadric surface entities can be boosted into moving surfaces with constant velocities that display the Thomas-Wigner rotation and length contraction of special relativity. DCSTA is an algebra for computing with general quadrics and their inversive geometry in spacetime. For applications or testing, G(4,8) DCSTA can be computed using various software packages, such as the symbolic computer algebra system SymPy with the GAlgebra module.

**Category:** Mathematical Physics

[9] **viXra:1708.0226 [pdf]**
*replaced on 2018-04-20 05:33:43*

**Authors:** Vu B Ho

**Comments:** 6 Pages.

Investigations into the nature of the principle of least action have shown that there is an intrinsic relationship between geometrical and topological methods and the variational principle in classical mechanics. In this work, we follow and extend this kind of mathematical analysis into the domain of quantum mechanics. First, we show that the identification of the momentum of a quantum particle with the de Broglie wavelength in 2-dimensional space would lead to an interesting feature; namely the action principle δS=0 would be satisfied not only by the stationary path, corresponding to the classical motion, but also by any path. Thereupon the Bohr quantum condition possesses a topological character in the sense that the principal quantum number n is identified with the winding number, which is used to represent the fundamental group of paths. We extend our discussions into 3-dimensional space and show that the charge of a particle also possesses a topological character and is quantised and classified by the homotopy group of closed surfaces. We then discuss the possibility to extend our discussions into spaces with higher dimensions and show that there exist physical quantities that can be quantised by the higher homotopy groups. Finally we note that if Einstein’s field equations of general relativity are derived from Hilbert’s action through the principle of least action then for the case of n=2 the field equations are satisfied by any metric if the energy-momentum tensor is identified with the metric tensor, similar to the case when the momentum of a particle is identified with the curvature of the particle’s path.

**Category:** Mathematical Physics

[8] **viXra:1708.0198 [pdf]**
*submitted on 2017-08-17 01:42:13*

**Authors:** Vu B Ho

**Comments:** 24 Pages.

In this work we discuss the possibility of reconciling quantum mechanics with classical mechanics by formulating a temporal dynamics, which is a dynamics caused by the rate of change of time with respect to distance. First, we show that a temporal dynamics can be derived from the time dilation formula in Einstein’s theory of special relativity. Then we show that a short-lived time-dependent force derived from a dynamical equation that is obtained from the temporal dynamics in a 1-dimensional temporal manifold can be used to describe Bohr’s postulates of quantum radiation and quantum transition between stable orbits in terms of classical dynamics and differential geometry. We extend our discussions on formulating a temporal dynamics to a 3-dimensional temporal manifold. With this generalisation we are able to demonstrate that a sub-quantum dynamics is a classical dynamics.

**Category:** Mathematical Physics

[7] **viXra:1708.0197 [pdf]**
*submitted on 2017-08-17 01:50:44*

**Authors:** Vu B Ho

**Comments:** 10 Pages.

In this work we discuss the possibility of combining the Coulomb potential with the Yukawa’s potential to form a mixed potential and then investigate whether this combination can be used to explain why the electron does not radiate when it manifests in the form of circular motions around the nucleus. We show that the mixed Coulomb-Yukawa potential can yield stationary orbits with zero net force, therefore if the electron moves around the nucleus in these orbits it will not radiate according to classical electrodynamics. We also show that in these stationary orbits, the kinetic energy of the electron is converted into potential energy, therefore the radiation process of a hydrogen-like atom does not related to the transition of the electron as a classical particle between the energy levels. The radial distribution functions of the wave equation determine the energy density rather than the electron density at a distance r along a given direction from the nucleus. It is shown in the appendix that the mixed potential used in this work can be derived from Einstein’s general theory of relativity by choosing a suitable energy-momentum tensor. Even though such derivation is not essential in our discussions, it shows that there is a possible connection between general relativity and quantum physics at the quantum level.

**Category:** Mathematical Physics

[6] **viXra:1708.0196 [pdf]**
*submitted on 2017-08-17 01:54:09*

**Authors:** Vu B Ho

**Comments:** 15 Pages.

In this work we develop a theory of temporal relativity, which includes a temporal special relativity and a temporal general relativity, on the basis of a generalised Newtonian temporal dynamics. We then show that a temporal relativity can be used to study the dynamics of quantum radiation of an elementary particle from a quantum system.

**Category:** Mathematical Physics

[5] **viXra:1708.0192 [pdf]**
*replaced on 2018-07-19 18:41:23*

**Authors:** Vu B Ho

**Comments:** 11 Pages. This paper has been published in International Journal of Physics

In this work first we show that the three main formulations of physics, namely, Newton’s second law of motion, Maxwell field equations of electromagnetism and Einstein field equations of gravitation can be formulated in similar covariant forms so that the formulations differ only by the nature of the geometrical objects that represent the corresponding physical entities. We show that Newton’s law can be represented by a scalar, the electromagnetic field by a symmetric affine connection or a dual vector, and the gravitational field by a symmetric metric tensor. Then with the covariant formulation for the gravitational field we can derive differential equations that can be used to construct the spacetime structures for short-lived and stable quantum particles. We show that geometric objects, such as the Ricci scalare curvature and Gaussian curvature, exhibit probabilistic characteristics. In particular, we also show that Schrödinger wavefunctions can be used to construct spacetime structures for the quantum states of a quantum system, such as the hydrogen atom. Even though our discussions in this work are focused on the microscopic objects, the results obtained can be applied equally to the macroscopic phenomena.

**Category:** Mathematical Physics

[4] **viXra:1708.0166 [pdf]**
*submitted on 2017-08-15 06:49:15*

**Authors:** Spiros Konstantogiannis

**Comments:** 50 Pages.

Exactly solvable rational extensions of the harmonic oscillator have been constructed as supersymmetric partner potentials of the harmonic oscillator [1] as well as using the so-called prepotential approach [2]. In this work, we use the factorization property of the energy eigenfunctions of the harmonic oscillator and a simple integrability condition to construct and examine series of regular and singular rational extensions of the harmonic oscillator with two known eigenstates, one of which is the ground state. Special emphasis is given to the interrelation between the special zeros of the wave function, the poles of the potential, and the excitation of the non-ground state. In the last section, we analyze specific examples.

**Category:** Mathematical Physics

[3] **viXra:1708.0147 [pdf]**
*replaced on 2017-08-28 07:33:28*

**Authors:** Harry Watson

**Comments:** 2 Pages.

Consider the product (4pi)(4pi-1/pi)(4pi-2/pi)(4pi-2/pi)(4pi-4/pi). The product of the first three terms is 1836.15. The product of the last two terms is 134.72. The mass ratio of the proton to the electron is 1836.15. We may sharpen the result by letting the last two terms be (4pi-3/pi)(4pi-4/pi) = 131.13. harry.watson@att.net

**Category:** Mathematical Physics

[2] **viXra:1708.0031 [pdf]**
*replaced on 2018-05-26 01:00:37*

**Authors:** Hideki Mutoh

**Comments:** 6 Pages.

Dirac equation includes the 4 x 4 complex differential operator matrix, which is one of square roots of d'Alembertian with spin of half integer. We found another 4 x 4 complex differential matrix as a square root of d'Alembertian for bosons, which we call diamond operator. The extended Maxwell's equations with charge creation-annihilation field and the linear gravitational field equations with energy creation-annihilation field can be simply written by using the diamond operator. It is shown that the linear gravitational field equations derive Newton's second law of motion, Klein-Gordon equation, time independent Schrödinger equation, and the principle of quantum mechanics.

**Category:** Mathematical Physics

[1] **viXra:1708.0011 [pdf]**
*replaced on 2019-04-02 07:05:42*

**Authors:** Hong Lai Zhu

**Comments:** 53 Pages.

In this paper, using proposed three new transformation methods we have solved general solutions and exact solutions of the problems of definite solutions of the Laplace equation, Poisson equation, Schrödinger equation, the homogeneous and non-homogeneous wave equations, Helmholtz equation and heat equation. In the process of solving, we find that in the more universal case, general solutions of partial differential equations have various forms such as basic general solution, series general solution, transformational general solution, generalized series general solution and so on.

**Category:** Mathematical Physics