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2016 - 1601(5) - 1602(10) - 1603(10) - 1604(6) - 1605(14) - 1606(11) - 1607(6) - 1608(13) - 1609(17) - 1610(18) - 1611(6)

Any replacements are listed further down

[554] **viXra:1611.0293 [pdf]**
*submitted on 2016-11-20 13:15:56*

**Authors:** Nicolas Poupart

**Comments:** 6 Pages.

It will be demonstrated in this paper that dark mass is a necessary consequence of the relativistic mechanics. This demonstration disregards the forces of physics and is therefore a purely mechanical explanation. The Tully-Fisher relation will be deduced naturally, without appealing to any new physics, and it is therefore possible to derive this law without modifying the Newtonian gravity or general relativity. In addition, a theoretical prediction on a new form of frequency shift will be carried out which makes it possible to refute or confirm this theory.

**Category:** Mathematical Physics

[553] **viXra:1611.0253 [pdf]**
*submitted on 2016-11-16 18:03:50*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 15 Pages.

A Guide Through Metaspace.

**Category:** Mathematical Physics

[552] **viXra:1611.0214 [pdf]**
*submitted on 2016-11-14 10:00:31*

**Authors:** L. H. Koudahoun, Y. J. F. Kpomahou, D. K. K. Adjaï, J. Akande, B. Rath, P. Mallick, M. D. Monsia

**Comments:** 11 pages

A mathematical model describing the nonlinear oscillations in elastic structures is proposed. The Energy Balance Method (EBM) is applied to solve the generalized nonlinear Duffing equation obtained in absence of excitation. The numerical results show an excellent agreement with the periodic solutions obtained through the Energy Balance Method. Finally the effects of different parameters on the system behavior are studied.

**Category:** Mathematical Physics

[551] **viXra:1611.0163 [pdf]**
*submitted on 2016-11-11 08:42:21*

**Authors:** M.E. Hassani

**Comments:** 8 Pages;8 References

In 1916, Johannes Droste independently found an exact (vacuum) solution to the Einstein's (gravitational) field equations in empty space. Droste's solution is quasi-comparable to Schwarzschild's one . Droste published his paper entitled “The field of a single centre in Einstein's theory of gravitation, and the motion of a particle in that fieldˮ. The paper communicated (in the meeting of May 27, 1916) by Prof. H.A. Lorentz, and published in ʻProceedings of the Royal Netherlands Academy of Arts and Science. 19 (1): 197-215 (1917)ʼ. In the present article, the Droste's solution is scrutinized and proven to be invalid purely and simply because the procedure used by Droste is mathematically questionable since he had systematically, deliberately, and without any justification ‒removed the constant coefficient ʻ2ʼ from the differential term (v'w') in Eq.(6) and added the differential term (wv'') to the same Eq.(6) in order to obtain Eq.(7) which was and is his principal objective, that is, the desired solution. Consequently, Eqs.(6,7) had clearly been falsified.

**Category:** Mathematical Physics

[550] **viXra:1611.0162 [pdf]**
*submitted on 2016-11-11 08:59:33*

**Authors:** Biswanath Rath, P. Mallick, Jean Akande, Damien K.K. Adjaï, Lucas H. Koudahoun, Y.J. Fernando Kpomahou, Marc D. Monsia

**Comments:** 11 pages

We generate a general model of Liénard type of second order differential
equation and study its classical solution. We also generate Hamiltonian
from the differential equation and study its stable eigenvalues.

**Category:** Mathematical Physics

[549] **viXra:1611.0146 [pdf]**
*submitted on 2016-11-11 00:07:25*

**Authors:** Biswanath Rath

**Comments:** 07 Pages. A new Eigenvalue mathod sing fractional matrix has been introduced .

A new member in Matrix representation has been introduced under
the name Fractional − Matrix and defined properly. Further we show
how one can address spectral analysis using this Fractional − Matrix.
Interesting examples have been considerd .

**Category:** Mathematical Physics

[548] **viXra:1610.0386 [pdf]**
*submitted on 2016-10-31 23:23:00*

**Authors:** Renzun Lian

**Comments:** 12 Pages.

An ElectroMagnetic-Power-based Characteristic Mode Theory (CMT) for Metal-Material combined objects (MM-EMP-CMT) was built by expressing the various electromagnetic powers as the functions of the line current on metal line, the surface current on metal surface, the surface current on the boundary of metal volume, and the total field in material volume, so it can be simply called as the Line-Surface-Volume formulation for the MM-EMP-CMT (LSV-MM-EMP-CMT). As a companion to the LSV-MM-EMP-CMT, a Line-Surface formulation for the MM-EMP-CMT (LS-MM-EMP-CMT) is established in this paper by expressing the various powers as the functions of the line and surface currents on metal part and the surface equivalent current on the boundary of material part.
The physical essence of LS-MM-EMP-CMT is the same as LSV-MM-EMP-CMT, i.e., to construct the various power-based Characteristic Mode (CM) sets for metal-material combined objects, but the LS-MM-EMP-CMT is more advantageous than the LSV-MM-EMP-CMT in some aspects. For example, the former saves computational resources; the former avoids to calculate the modal scattering field in source region; the field-based definitions for the impedance and admittance of metal-material combined electromagnetic systems can be easily introduced into the former.

**Category:** Mathematical Physics

[547] **viXra:1610.0354 [pdf]**
*submitted on 2016-10-29 07:32:58*

**Authors:** Renzun Lian

**Comments:** 8 Pages.

As a companion to the ElectroMagnetic-Power-based Characteristic Mode Theory (EMP-CMT) for PEC systems (PEC-EMP-CMT) and the EMP-CMT for Material bodies (Mat-EMP-CMT), an EMP-CMT for Metal-Material combined objects (MM-EMP-CMT) is established in this paper, and then some power-based Characteristic Mode (CM) sets are constructed for depicting the inherent power characteristics of metal-material combined objects. The MM-EMP-CMT is valuable for analyzing and designing the metal-material combined electromagnetic structures, such as the microstrip antennas and the Dielectric Resonant Antennas (DRAs) mounted on metal platforms etc. In addition, a variational formulation for the scattering problem of metal-material combined objects is provided based on the conservation law of energy.

**Category:** Mathematical Physics

[546] **viXra:1610.0345 [pdf]**
*submitted on 2016-10-28 22:26:32*

**Authors:** Renzun Lian

**Comments:** 13 Pages.

Recently, a volume formulation of the ElectroMagnetic-Power-based Characteristic Mode Theory (CMT) for Material bodies (Mat-EMP-CMT) is built by expressing various electromagnetic powers as the functions of the total fields in material bodies, so it can be simply called as Vol-Mat-EMP-CMT. As a companion to the Vol-Mat-EMP-CMT, several Surface formulations of the Mat-EMP-CMT (Surf-Mat-EMP-CMT) are established in this paper by expressing various electromagnetic powers as the functions of the surface equivalent sources on the boundaries of material bodies.
The physical essence of Surf-Mat-EMP-CMT is the same as the Vol-Mat-EMP-CMT, i.e., to construct the various power-based Characteristic Mode (CM) sets for material bodies, but the former is more advantageous than the latter in some aspects. For example, the former saves computational resources; the former avoids to compute the modal scattering field in source region; the field-based definitions for the impedance and admittance of material bodies can be easily introduced into the former.

**Category:** Mathematical Physics

[545] **viXra:1610.0340 [pdf]**
*submitted on 2016-10-28 05:21:45*

**Authors:** Renzun Lian

**Comments:** 15 Pages.

In this paper, an ElectroMagnetic-Power-based Characteristic Mode Theory (CMT) for Material bodies (Mat-EMP-CMT) is provided. The Mat-EMP-CMT is valid for the inhomogeneous and lossy material bodies, and it is applicable to the bodies which are placed in complex electromagnetic environments.
Under the Mat-EMP-CMT framework, a series of power-based Characteristic Mode (CM) sets are constructed, and they have abilities to depict the inherent power characteristics of material bodies from different aspects. All power-based CM sets are independent of the external electromagnetic environment and excitation.
Among the various power-based CM sets constructed in Mat-EMP-CMT, only the Input power CM (InpCM) set has the same physical essence as the CM set constructed in Mat-VIE-CMT (the Volume Integral Equation based CMT for Material bodies), and the other CM sets are completely new. However, the power characteristic of the InpCM set is more physically reasonable than the CM set derived from Mat-VIE-CMT.
In addition, not only radiative CMs and real characteristic currents but also non-radiative CMs and complex characteristic currents can be constructed under the Mat-EMP-CMT framework; the traditional characteristic quantity, Modal Significance (MS), is generalized, and some new characteristic and non-characteristic quantities are introduced to depict the modal characteristics from different aspects; a variational formulation for the scattering problem of material scatterer is established based on the conservation law of energy.

**Category:** Mathematical Physics

[544] **viXra:1610.0337 [pdf]**
*submitted on 2016-10-27 20:25:31*

**Authors:** Nicolas Poupart

**Comments:** 6 Pages.

Nous démontrerons dans cet article que la masse noire est une conséquence nécessaire de la mécanique relativiste soit de la combinaison de la mécanique classique et de la relativité restreinte. Ainsi, il faudra soit accepter ce résultat soit mettre en doute la consistance de la mécanique relativiste.

**Category:** Mathematical Physics

[543] **viXra:1610.0332 [pdf]**
*submitted on 2016-10-28 04:20:20*

**Authors:** Renzun Lian

**Comments:** 17 Pages.

In this paper, an ElectroMagnetic-Power-based Characteristic Mode Theory (CMT) for PEC systems (PEC-EMP-CMT) is built. The PEC-EMP-CMT is valid for the PEC systems which are surrounded by any electromagnetic environment, and it can construct the complex characteristic currents and non-radiative Characteristic Modes (CMs).
In this paper, some traditional concepts, such as the system input impedance and modal input impedance etc., are redefined; the traditional characteristic quantity, Modal Significance (MS), is generalized; a series of new power-based CM sets are introduced.
It is proven in this paper that various power-based CM sets of a certain objective PEC structure are independent of the external environment and excitation; the non-radiative space constituted by all non-radiative modes is identical to the interior resonance space constituted by all interior resonant modes of closed PEC structures, and the non-radiative CMs constitute a basis of the space. Based on above these, the normal Eigen-Mode Theory (EMT) for closed PEC structures is classified into the PEC-EMP-CMT framework.
In addition, a variational formulation for the external scattering problem of PEC structures is provided in this paper, based on the conservation law of energy.

**Category:** Mathematical Physics

[542] **viXra:1610.0315 [pdf]**
*submitted on 2016-10-26 08:43:06*

**Authors:** M.E.Hassani

**Comments:** 6 Pages; 4 References.

In the present paper, the Schwarzschild’s original solution (1916) is scrutinized and proven to be logically, mathematically and physically not only wrong but basically meaningless because the ʻeasy trickʼ used by Schwarzschild violated the fundamental concepts of analytic geometry (rectangular coordinates), trigonometry (triangles) and dimensional analysis (consistency and homogeneity). It seems that Schwarzschild had systematically and deliberately violated these fundamental concepts in order to avoid/break an unavoidable/unbreakable impasse (the determinant ≠ 1). Then he had mathematically cheated to have the determinant =1 in an anti-mathematical manner since he was not attached to his initial claim, viz., ‒ x1, x2, x3 and x, y, z are rectangular coordinates ‒ Thus, as scientists we should not forget one very important thing, namely, mathematics is not only an exact science, but it is the language of Science itself.

**Category:** Mathematical Physics

[541] **viXra:1610.0231 [pdf]**
*submitted on 2016-10-19 15:07:45*

**Authors:** Arturo Tozzi, James F Peters

**Comments:** 83 Pages.

This manuscript encompasses our published and unpublished topological results in physics. Topology, the mathematical branch that assesses objects and their properties preserved through deformations, stretching and twisting, allows the investigation of the most general physical systems features. In particular, the Borsuk-Ulam Theorem (BUT) states that, if a single point projects to a higher spatial dimension, it gives rise to two antipodal points with matching description. Physical counterparts of BUT and its variants allow an inquiry of physical problems. The opportunity to treat systems as topological structures makes BUT a universal principle underlying natural phenomena.

**Category:** Mathematical Physics

[540] **viXra:1610.0221 [pdf]**
*submitted on 2016-10-19 03:43:29*

**Authors:** Arturo Tozzi, James F Peters

**Comments:** 8 Pages.

The Borsuk-Ulam Theorem (BUT) states that a single point, if embedded in one spatial dimension higher, gives rise to two antipodal points with matching descriptions and similar features. Novel BUT variants allow the assessment of countless physical systems, from entropies to quantum entanglement. We argue that BUT, cast in a quantitative fashion which has the potential of being operationalized, is a universal principle underlying a number of natural phenomena.

**Category:** Mathematical Physics

[539] **viXra:1610.0198 [pdf]**
*submitted on 2016-10-17 11:30:45*

**Authors:** Taha Sochi

**Comments:** 15 Pages.

We use a generic and general numerical method to obtain solutions for the flow of generalized Newtonian fluids through circular pipes and plane slits. The method, which is simple and robust can produce highly accurate solutions which virtually match any analytical solutions. The method is based on employing the stress, as a function of the pipe radius or slit thickness dimension, combined with the rate of strain function as represented by the fluid rheological constitutive relation that correlates the rate of strain to stress. Nine types of generalized Newtonian fluids are tested in this investigation and the solutions obtained from the generic method are compared to the analytical solutions which are obtained from the Weissenberg-Rabinowitsch-Mooney-Schofield method. Very good agreement was obtained in all the investigated cases. All the required quantities of the flow which include local viscosity, rate of strain, flow velocity profile and volumetric flow rate, as well as shear stress, can be obtained from the generic method. This is an advantage as compared to some traditional methods which only produce some of these quantities. The method is also superior to the numerical meshing techniques which may be used for resolving the flow in these systems. The method is particularly useful when analytical solutions are not available or when the available analytical solutions do not yield all the flow parameters.

**Category:** Mathematical Physics

[538] **viXra:1610.0196 [pdf]**
*submitted on 2016-10-17 11:34:00*

**Authors:** Taha Sochi

**Comments:** 16 Pages.

We investigate the possibility that the spatial dependency of stress in generalized Newtonian flow systems is a function of the applied pressure field and the conduit geometry but not of the fluid rheology. This possibility is well established for the case of a one-dimensional flow through simply connected regions, specifically tubes of circular uniform cross sections and plane thin slits. If it can also be established for the more general case of generalized Newtonian flow through non-circular or multiply connected geometries, such as the two-dimensional flow through conduits of rectangular or elliptical cross sections or the flow through annular circular pipes, then analytical or semi-analytical or highly accurate numerical solutions; regarding stress, rate of strain, velocity profile and volumetric flow rate; for these geometries can be obtained from the stress function, which can be easily obtained from the Newtonian case, in combination with the constitutive rheological relation for the particular non-Newtonian fluid, as done previously for the case of the one-dimensional flow through simply connected regions.

**Category:** Mathematical Physics

[537] **viXra:1610.0195 [pdf]**
*submitted on 2016-10-17 11:36:50*

**Authors:** Taha Sochi

**Comments:** 9 Pages.

In this article we challenge the claim that the previously proposed variational method to obtain flow solutions for generalized Newtonian fluids in circular tubes and plane slits is exact only for power law fluids. We also defend the theoretical foundation and formalism of the method which is based on minimizing the total stress through the application of the Euler-Lagrange principle.

**Category:** Mathematical Physics

[536] **viXra:1610.0194 [pdf]**
*submitted on 2016-10-17 11:51:12*

**Authors:** Taha Sochi

**Comments:** 43 Pages.

In this article, the extensional flow and viscosity and the converging-diverging geometry were examined as the basis of the peculiar viscoelastic behavior in porous media. The modified Bautista-Manero model, which successfully describes shearthinning, elasticity and thixotropic time-dependency, was used for modeling the flow of viscoelastic materials which also show thixotropic attributes. An algorithm, originally proposed by Philippe Tardy, that employs this model to simulate steadystate time-dependent flow was implemented in a non-Newtonian flow simulation code using pore-scale modeling and the initial results were analyzed. The findings are encouraging for further future development.

**Category:** Mathematical Physics

[535] **viXra:1610.0193 [pdf]**
*submitted on 2016-10-17 11:53:33*

**Authors:** Taha Sochi

**Comments:** 27 Pages.

Yield-stress is a problematic and controversial non-Newtonian flow phenomenon. In this article, we investigate the flow of yield-stress substances through porous media within the framework of pore-scale network modeling. We also investigate the validity of the Minimum Threshold Path (MTP) algorithms to predict the pressure yield point of a network depicting random or regular porous media. Percolation theory as a basis for predicting the yield point of a network is briefly presented and assessed. In the course of this study, a yield-stress flow simulation model alongside several numerical algorithms related to yield-stress in porous media were developed, implemented and assessed. The general conclusion is that modeling the flow of yield-stress fluids in porous media is too difficult and problematic. More fundamental modeling strategies are required to tackle this problem in the future.

**Category:** Mathematical Physics

[534] **viXra:1610.0150 [pdf]**
*submitted on 2016-10-14 01:23:44*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 31 Pages.

A 100 year task of minimal duration and of limitless measure

**Category:** Mathematical Physics

[533] **viXra:1610.0078 [pdf]**
*submitted on 2016-10-06 18:16:19*

**Authors:** Vito R. D'Angelo

**Comments:** 2 Pages.

In the spirit of the Pythagorean school of thought, that everything in the universe can be reduced to pure numbers; utilizing the Rydberg constant, arguably the most precise value in physics, an attempt is made to bring to fruition the aforementioned tenet. The relationships of seven well known constants are utilized within the context of four equations.

**Category:** Mathematical Physics

[532] **viXra:1610.0059 [pdf]**
*submitted on 2016-10-05 01:38:20*

**Authors:** Claude Michael Cassano

**Comments:** 9 Pages.

Elementary Linear Algebra theory handles transformations between sets of n-square matrix vector bases of M-dimensions well. Considering n-square & m-square matrices of M-dimensions, further theory and techniques, shown here, may be applied to yield results; including relationships between spin matrices and components.

**Category:** Mathematical Physics

[531] **viXra:1610.0027 [pdf]**
*submitted on 2016-10-03 07:39:34*

**Authors:** Ricardo Gil

**Comments:** 1 Page.

In quantum field theory, the mass gap is the difference in energy between the vacuum and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle. The purpose of this paper is to suggest that the lowest state in a system is the entanglement which’s gravity for the Entanglement State is calculated by 2.99E12 x 1G / 9.8 m/s2 = 305102040846 G= 1/30510204086 G = 3.277592E-12 G X 1E-32m= 3.277 E-44 at 1/8.96 E20 Joules /Kg=1.11E-21 Joules/Kg energy state.

**Category:** Mathematical Physics

[530] **viXra:1609.0364 [pdf]**
*submitted on 2016-09-25 21:12:02*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 3 Pages.

At a certain point computation will initiate catastrophe. One must then impose computational factorization of variants [of stringy] in metaspace.

**Category:** Mathematical Physics

[529] **viXra:1609.0332 [pdf]**
*submitted on 2016-09-23 06:00:56*

**Authors:** Prado, Pf et al

**Comments:** 1 Page.

Addendum to hypothesis

**Category:** Mathematical Physics

[528] **viXra:1609.0302 [pdf]**
*submitted on 2016-09-20 22:49:31*

**Authors:** Lamont Williams

**Comments:** 2 Pages.

When thinking about the origin of the fine-structure constant, renowned physicist Richard Feynman speculated that it might be related to pi or the base of the natural logarithm. However, he could not envision how pi or the natural logarithm’s base could be associated with the constant. This article presents an equation that helps to address Feynman's question.

**Category:** Mathematical Physics

[527] **viXra:1609.0287 [pdf]**
*submitted on 2016-09-19 14:39:03*

**Authors:** Prado, PF et al

**Comments:** 1 Page.

Temprtive Math exploration

**Category:** Mathematical Physics

[526] **viXra:1609.0267 [pdf]**
*submitted on 2016-09-18 09:09:58*

**Authors:** J. Akande, L. H. Koudahoun, D. K. K. Adjaï, Y. J. F. Kpomahou, M. D. Monsia

**Comments:** 2 pages

The problem to solve consists of finding explicit solutions for the classical equation of motion of a particle subject to an exponential-type potential and for the quantum version.

**Category:** Mathematical Physics

[525] **viXra:1609.0261 [pdf]**
*submitted on 2016-09-17 12:07:14*

**Authors:** Vito R. D'Angelo

**Comments:** 7 Pages.

The Hertz-kilogram relationship constant, although obscure, is nonetheless a bona fide constant listed in the National Institute of Standards and Technology's (CODATA values) fundamental constants list[pg.6]. The Hertz-kilogram relationship constant is being used due to its value of (7.372497201 x10^-51 (91)) being very close to the Democritean indivisible unit, Y' (8.134865168 10^-54) It is shown that the 13/12 schematic of U-theory [2] theoretically calculates the fundamental constants within the uncertainty limits of (NIST) CODATA values [1]; utilizing twelve dimensionless constants.

**Category:** Mathematical Physics

[524] **viXra:1609.0260 [pdf]**
*submitted on 2016-09-17 08:51:16*

**Authors:** J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou, M. D. Monsia

**Comments:** 5 pages

The generalized modified Emden equation also known as the generalized second order Riccati equation, is exactly solved in terms of the periodic solution of the linear harmonic oscillator. The solutions for specific values of parameters are discussed. The conditions for isochronous oscillations are also investigated.

**Category:** Mathematical Physics

[523] **viXra:1609.0247 [pdf]**
*submitted on 2016-09-16 13:54:39*

**Authors:** Prado, PF et al

**Comments:** 4 Pages.

Offers a math framework to the Perspective: Is there a structural resonant ruler in Nature? An exploratory framework to neuroscience and fundamental physics studies. Addition to Nextex set of conjectures: Version 1.0. Concluding that the resonant phenomenon probably is just the best signaling mark to the Principal phenomenon underlayered in Nature:the physical (realistic) implementation of the Fundamental theorem of Calculus or the called Theorem of Stokes.

**Category:** Mathematical Physics

[522] **viXra:1609.0239 [pdf]**
*submitted on 2016-09-15 16:32:52*

**Authors:** D. K. K. Adjaï, Y. J. F. Kpomahou, J. Akande, L. H. Koudahoun, M. D. Monsia

**Comments:** 4 pages

In this paper the qualitative properties of a family of an anharmonic oscillator equations of motion were carried out with phase portraits.

**Category:** Mathematical Physics

[521] **viXra:1609.0211 [pdf]**
*submitted on 2016-09-13 14:54:33*

**Authors:** Prado, Pf

**Comments:** 3 Pages.

Discussion on ressonance considerations about fundamental physics

**Category:** Mathematical Physics

[520] **viXra:1609.0142 [pdf]**
*submitted on 2016-09-11 12:00:21*

**Authors:** J. Akande, L. H. Koudahoun, D. K. K. Adjaï, Y. J. F. Kpomahou, M. D. Monsia

**Comments:** 3 pages

In this paper the Schrödinger equation is derived for a dynamical system with exponential-type restoring force function.

**Category:** Mathematical Physics

[519] **viXra:1609.0055 [pdf]**
*submitted on 2016-09-05 08:44:45*

**Authors:** J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou, M. D. Monsia

**Comments:** 5 pages

This research work proposes a Lagrangian and Hamiltonian analysis for a class of exactly integrable quadratic Liénard-type harmonic nonlinear oscillator equations and its inverted version admitting a position-dependent mass dynamics.

**Category:** Mathematical Physics

[518] **viXra:1609.0037 [pdf]**
*submitted on 2016-09-04 06:28:30*

**Authors:** Y.F.J. Kpomahou, L.H. Koudahoun, D.K.K. Adjaï, J. Akande, M.D. Monsia

**Comments:** 3 pages

The main objective of this paper is to propose two simple analytical linearizing transformation for determining the exact analytical harmonic periodic solution of a class of Liénard-type position-dependent mass oscillator equations. The performed exact analysis shows that both employed analytical methods are efficient to solve this class of equations.

**Category:** Mathematical Physics

[517] **viXra:1609.0036 [pdf]**
*submitted on 2016-09-03 12:10:32*

**Authors:** M. D. Monsia, J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou

**Comments:** 3 pages

This paper purposes to show the existence of exact analytical solutions to a class of generalized Duffing-van der Pol and modified Emden type oscillator equations using nonlocal transformation.

**Category:** Mathematical Physics

[516] **viXra:1609.0029 [pdf]**
*submitted on 2016-09-02 15:17:50*

**Authors:** M. D. Monsia, J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou

**Comments:** 3 pages

The objective in this paper is to show that the generalized Duffing-van der Pol and modified Emden type equations consist of limiting cases of the exactly integrable Monsia et al.[2] nonlinear oscillator equation by expanding the exponential-type damping and restoring forces in a Taylor series.

**Category:** Mathematical Physics

[515] **viXra:1609.0024 [pdf]**
*submitted on 2016-09-02 07:47:41*

**Authors:** Miroslav Josipovic

**Comments:** 70 Pages.

Text is intended as some motivational surway of geometric algebra in 3D. This is just first (condensed) version.

**Category:** Mathematical Physics

[514] **viXra:1609.0003 [pdf]**
*submitted on 2016-09-01 08:27:12*

**Authors:** M. D. Monsia, J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou

**Comments:** 2 pages

This communication consists of additions to a previous work [1]. It presents certain Liénard-type and Duffing-type nonlinear oscillators equations with exponential-type restoring force, according to the recent theory of nonlinear differential equations of position-dependent mass oscillators formulated by Monsia et al. [1].

**Category:** Mathematical Physics

[513] **viXra:1608.0422 [pdf]**
*submitted on 2016-08-31 13:04:28*

**Authors:** Dan Visser

**Comments:** 17 Pages.

Dan Visser (DAN) creates Art from a way of thinking involving another universe. He is retired and has written down his way of thinking in articles through bypassing the institutions. “I know what dark matter is, and I know the institutions still don’t know”. I calculate the percentages of dark matter, dark energy and visible matter close to the measured Planck-satellite values. I also calculated the energy for a dark matter-force (a few years ago), which now appears to fit the experimental values performed by the decay of Berilium-8 showing a value of 16,7 MeV at the atom-scale of 10^-10 meter and a time-scale of about 10^-14 seconds. I also show dark matter co-operates with quantum-gravity in a new formula. The force of dark matter is also showing-up in an experiment revealing the proton-radius becomes 4% smaller due to a muon in orbit instead of an electron. This effect can be calculated by my new dark matter-topology and dark matter–dimensions. Both the experiments (berilium-8 decay and smaller proton-radius), as well as the used dark matter-topology and dark matter–dimensions, gave me the idea to posit a principle for an UFO-motor. Such a motor can be tuned to overcome a gravitational field like earth, or any other planet. Such motor enables to move within the hologram universe, instead of in the Big Bang space-time universe.

**Category:** Mathematical Physics

[512] **viXra:1608.0398 [pdf]**
*submitted on 2016-08-29 08:11:19*

**Authors:** M. D. Monsia, J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou

**Comments:** 2 pages

The present letter adds to the paper ’’ A Class of Position-Dependent Mass Liénard Differential Equations via a General Nonlocal Transformation’’. The purpose is to emphasize the fact that the mathematical theory of position-dependent mass nonlinear oscillator differential equations previously developed [1] provides exact analytical trigonometric periodic solutions to inverted quadratic Mathews-Lakshmanan oscillator equations.

**Category:** Mathematical Physics

[511] **viXra:1608.0368 [pdf]**
*submitted on 2016-08-26 19:09:34*

**Authors:** M. D. Monsia, J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou

**Comments:** 3 pages

This letter consists of additions to the paper ’’ A Class of Position-Dependent Mass Liénard Differential Equations via a General Nonlocal Transformation’’. The objective is to highlight the fact that the general second-order nonlinear differential equation theory of position-dependent mass oscillators developed previously has the ability to provide exact analytical periodic solutions with sinusoidal form to the class of quadratic Liénard-type equations, like the motion of a particle on a rotating parabola and Morse- type oscillator equation, under question.

**Category:** Mathematical Physics

[510] **viXra:1608.0317 [pdf]**
*submitted on 2016-08-25 03:50:33*

**Authors:** Robert G Wallace

**Comments:** 8 Pages.

An algebra for unit multivector components for a manifold of five poly-complex dimensions is presented. The algebra has many properties that suggest it may provide a basis for a grand unification theory.

**Category:** Mathematical Physics

[509] **viXra:1608.0266 [pdf]**
*submitted on 2016-08-23 09:05:27*

**Authors:** M. D. Monsia, J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou

**Comments:** 3 pages

This work aims to present some specific examples of the generalized mixed Liénard differential equation and position-dependent mass Liénard equation depicted in A Class of Position-Dependent Mass Liénard Differential Equations via a General Nonlocal Transformation.

**Category:** Mathematical Physics

[508] **viXra:1608.0244 [pdf]**
*submitted on 2016-08-22 11:03:47*

**Authors:** M.W.Kalinowski

**Comments:** 178 Pages. the paper is written in polish

The Cauchy initial value problem for the Klein-Gordon equation has been considered in a class of
tempered distributions using a notion of a section
of a distribution with a hyperplane. We consider also different linear PDE derivable from Klein-Gordon equation as Dirac, Proca ,Weyl and all the most important wave equations of relativistic quantum mechanics and quantum field theory. We consider also Maxwell equations.We consider also
classical Cauchy initial value problem for those
equations using obtained generalized results e.g.
for Maxwell equations.

**Category:** Mathematical Physics

[507] **viXra:1608.0232 [pdf]**
*submitted on 2016-08-21 14:31:45*

**Authors:** Christian Rakotonirina

**Comments:** 94 Pages. in French

Properties of tensor product of matrices have been constructed. These properties are used to study factorization by tensor product of matrices of some real Clifford algebras of square matrices. Applying these factorizations, we have found a way to get , from the Pauli matrices, twelve systems and only twelve. Each of them is formed of four matrices coefficients of a Dirac equation. We have looked for solutions of these twelve equations for free fundamental fermions. These twelve equations can be constructed by quantification of the relativistic energy-momentum relation. We have introduced a notion that we call ‘’equivalence of particles’’. Then, the equivalence between free fundamental fermions have been studied. Finally, we have proved equivalence between the Dirac equation and the Hestenes equation.

**Category:** Mathematical Physics

[506] **viXra:1608.0226 [pdf]**
*submitted on 2016-08-20 18:47:15*

**Authors:** M. D. Monsia, J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou

**Comments:** 2 pages

The objective, in this paper, consists of mapping the damped linear harmonic oscillator equation onto a class of Liénard nonlinear differential equations that incorporates the well known position dependent mass Mathews-Lakshmanan oscillator equations as specific examples through a general nonlocal transformation.

**Category:** Mathematical Physics

[505] **viXra:1608.0181 [pdf]**
*submitted on 2016-08-17 14:44:03*

**Authors:** J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou, M. D. Monsia

**Comments:** 2 pages

This letter is devoted to show the existence of a general class of integrable mixed Liénard-type equations that includes some physically important nonlinear differential equations like the generalized modified Emden-type equation (MEE) through the first integral under differentiation approach.

**Category:** Mathematical Physics

[504] **viXra:1608.0124 [pdf]**
*submitted on 2016-08-12 08:02:59*

**Authors:** J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou, M. D. Monsia

**Comments:** 7 pages

The inverted quadratic Liénard type equation is very useful in various branches of classical and quantum theories, since it admits a position dependent mass dynamics. The objective of the present work is to show that some interesting inverted nonlinear oscillator equations like the inverted version of Mathews-Lakshmanan oscillator belong to a general class of exactly solvable inverted quadratic Liénard equations. This class of equations is generated from a first integral formulated as an integro-differential equation. The obtained results may be used for the identification and integrability of a family of dynamical systems equations.

**Category:** Mathematical Physics

[503] **viXra:1608.0096 [pdf]**
*submitted on 2016-08-08 16:47:42*

**Authors:** Gary D. Simpson

**Comments:** 39 Pages.

This text develops various identities for Hamilton's quaternions. The results are presented in order of difficulty. Results are organized as Axioms, Vectors, Quaternions, and Matrices. There are also sections for Octonions and Pentuples. Axioms are presented first and are largely without rigorous proof. Subsequent identities are constructed from prior identities. When complex conjugates are discussed, the author's thinking is biased towards the original quaternion having a positive vector portion and the conjugate having a negative vector portion. To genuinely understand what is presented, it is recommended that the reader should visualize the concepts in addition to manipulating them algebraically. The algebra is certainly true, but the visual understanding is more elegant and intuitive. This text will likely be updated occasionally.

**Category:** Mathematical Physics

[502] **viXra:1608.0095 [pdf]**
*submitted on 2016-08-08 17:38:43*

**Authors:** Bing Wang

**Comments:** 10 Pages.

Effects of L{\'{e}}vy noise on self-propelled particles in a two-dimensional potential is investigated. The current reversal phenomenon appear in the system. $V$($x$ direction average velocity) changes from negative to positive with increasing asymmetry parameter $\beta$, and changes from positive to negative with increasing self-propelled velocity $v_0$. The $x$ direction average velocity $V$ has a maximum with increasing modulation constant $\lambda$.

**Category:** Mathematical Physics

[501] **viXra:1608.0059 [pdf]**
*submitted on 2016-08-05 12:30:18*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

The solution for the problem of Breakdown of Euler Equations, like the Millenium Problem for Navier-Stokes equations.

**Category:** Mathematical Physics

[500] **viXra:1607.0384 [pdf]**
*submitted on 2016-07-20 15:09:39*

**Authors:** Jerry L. Decker

**Comments:** 5 Pages. Eliminating transformations space like and time like at high speed

A method was found for constructing coordinate systems larger than four dimensions. Velocity vectors and base units were used to define a reference frame by attaching a clock to each of the velocity components. Additional meters were also attached to velocity components. All of the resulting systems can be physically constructed and tested.

**Category:** Mathematical Physics

[499] **viXra:1607.0374 [pdf]**
*submitted on 2016-07-19 18:32:01*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 3 Pages.

Why did unification work better with the Super-Yang Mills Gauge Analog than any other unification attempts?

**Category:** Mathematical Physics

[498] **viXra:1607.0199 [pdf]**
*submitted on 2016-07-17 15:09:37*

**Authors:** M. D. Monsia, J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou

**Comments:** 10 pages

The dynamics of quadratic Liénard type equations is usually investigated in the only context of periodic solutions. The problem of interest in this work is then to demonstrate the existence of a simple variable transformation generating a class of exactly integrable quadratic Liénard type equations that preserves the three distinct damped dynamical operating regimes of nonlinear oscillators. Specific examples of equation belonging to this class and their exact solutions in terms of the periodic solution to linear harmonic oscillator are provided for illustrating the developed theory.

**Category:** Mathematical Physics

[497] **viXra:1607.0148 [pdf]**
*submitted on 2016-07-12 13:43:28*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 4 Pages.

What is metaspace?

**Category:** Mathematical Physics

[496] **viXra:1607.0123 [pdf]**
*submitted on 2016-07-10 14:02:38*

**Authors:** Furkan Semih Dundar

**Comments:** 4 Pages.

$N$-scales are a generalization of time-scales that has been put forward to unify continuous and discrete analyses to higher dimensions. In this paper we investigate massive scalar field theory on $n$-scales. In a specific case of a regular 2-scale, we find that the IR energy spectrum is almost unmodified when there are enough spatial points. This is regarded as a good sign because the model reproduces the known results in the continuum approximation. Then we give field equation on a general $n$-scale. It has been seen that the field equation can only be solved via computer simulations. Lastly, we propose that $n$-scales might be a good way to model singularities encountered in the general theory of relativity.

**Category:** Mathematical Physics

[495] **viXra:1607.0096 [pdf]**
*submitted on 2016-07-08 06:09:08*

**Authors:** Zoran B. Vosika

**Comments:** 17 Pages.

Generalization of fractal density on a fractals for spaces with positive and negative fractal dimensions. Fractal-fractional generalized physics (i.e. classical or quantum physics). Generalized Hausdorff measures. Numbers and generalized functions as generalized logical values. Beyond logics and numbers. Generalized concept of physical field, i.e generalized universes and multiverses. Fractional generalization of path integrals.

**Category:** Mathematical Physics

[494] **viXra:1606.0294 [pdf]**
*submitted on 2016-06-27 10:58:21*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 16 Pages.

This note represents an attempt of solving the Navier-Stokes equations under the assumptions (A) of the problem as described by the Clay Institute (C.L. Fefferman, 2006).

**Category:** Mathematical Physics

[493] **viXra:1606.0259 [pdf]**
*submitted on 2016-06-24 10:04:44*

**Authors:** Jean-Luc Paillet, Andrew Meulenberg

**Comments:** Pages. Presented to 1st Symp. on RNBE (LENR), March 18-20 2016, Avignon, France

This work continues our previous works on electron deep orbits of the hydrogen atom. An introduction shows the importance of the deep orbits of hydrogen (H or D) for research in the LENR domain, and gives some general considerations on the Electron Deep Orbits (EDOs). In a first part we quickly recall the known criticism against the EDO and how we face it. In particular, a solution to fix all problems is to consider a modified Coulomb potential with finite value inside the nucleus. For this reason, we deeply analyzed the specific work of Maly and Va’vra on deep orbits as solutions of the Dirac equation, with such a modified Coulomb potential without singular point. Then, by using a more complete ansatz, we made numerous computations on the wavefunctions of these EDOs, allowing to confirm the approximate size of the mean radii

**Category:** Mathematical Physics

[492] **viXra:1606.0185 [pdf]**
*submitted on 2016-06-17 22:38:08*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 18 Pages.

In this research investigation, the author has presented a theory of ‘Universal
Relative Metric That Generates A Field Super-Set To The Fields Generated By
Various Distinct Relative Metrics’.

**Category:** Mathematical Physics

[491] **viXra:1606.0174 [pdf]**
*submitted on 2016-06-17 08:49:58*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 16 Pages.

In this research investigation, the author has presented a theory of ‘Universal
Holistic Beauty Primality Tree Of Any Set, Universal Growth Of Any Given Set’.

**Category:** Mathematical Physics

[490] **viXra:1606.0153 [pdf]**
*submitted on 2016-06-15 07:32:26*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 14 Pages.

In this research investigation, the author has presented a theory of ‘The Universal
Irreducible Any Field Generating Metric’.

**Category:** Mathematical Physics

[489] **viXra:1606.0134 [pdf]**
*submitted on 2016-06-14 00:59:43*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 13 Pages.

In this research investigation, the author has presented a theory of ‘The Universal Irreducible Any Field Generating Metric’.

**Category:** Mathematical Physics

[488] **viXra:1606.0089 [pdf]**
*submitted on 2016-06-09 23:29:20*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 12 Pages.

In this research investigation, the author has presented the theory of ‘Complete Recursive Sub-Sets Found To Exhaustion Of A Set’ with ‘The Example Of The Same Explaining The Quantization Scheme Of Any Universal Natural Manifestation In Holisticness’.

**Category:** Mathematical Physics

[487] **viXra:1606.0086 [pdf]**
*submitted on 2016-06-09 06:42:55*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 12 Pages.

In this research investigation, the author has presented the theory of ‘Complete Recursive Sub-Sets Found To Exhaustion Of A Set’ with ‘The Example Of The Same Explaining The Quantization Scheme Of Any Universal Natural Manifestation In Holisticness’.

**Category:** Mathematical Physics

[486] **viXra:1606.0061 [pdf]**
*submitted on 2016-06-06 15:38:04*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 3 Pages.

Is infrared divergence irrelevent to D-energy?

**Category:** Mathematical Physics

[485] **viXra:1606.0056 [pdf]**
*submitted on 2016-06-06 06:37:02*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 9 Pages.

In this research investigation, the author has presented the notion of ‘Relative Metric And Field Generation Based On The Same’.

**Category:** Mathematical Physics

[484] **viXra:1606.0037 [pdf]**
*submitted on 2016-06-03 23:03:41*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 8 Pages.

In this research investigation, the author has presented the notion of ‘Relative
Metric And Field Generation Based On The Same’.

**Category:** Mathematical Physics

[483] **viXra:1605.0292 [pdf]**
*submitted on 2016-05-29 10:43:19*

**Authors:** Ricardo Gil

**Comments:** 3 Pages.

Einstein merged Space with Time hence Spacetime. Tesla has a Unified Theory based on Fields and Electromagnetism. The logical progression is to merge the two and suggest Frequential Spacetime in which the frequency of matter dictates the shape and speed of entanglement of matter in Spacetime.

**Category:** Mathematical Physics

[482] **viXra:1605.0270 [pdf]**
*submitted on 2016-05-26 07:15:55*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 5 Pages.

In this research investigation, the author has presented a Scheme for ‘Positive And Negative Time Scaling And Time Portal Engineering’

**Category:** Mathematical Physics

[481] **viXra:1605.0255 [pdf]**
*submitted on 2016-05-24 15:15:40*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 3 Pages.

Does any unification candidate necessitate the grand unification scheme?

**Category:** Mathematical Physics

[480] **viXra:1605.0240 [pdf]**
*submitted on 2016-05-23 04:23:55*

**Authors:** I. V. Drozdov

**Comments:** 25 Pages.

The second-order approach to the entropy gradient maximization for systems with many degrees of freedom provides the dynamic equations of first order and light-like second order without additional ergodicity conditions like conservation laws.
The first order dynamics lead to the definition of the conserved kinetic energy
and potential energy. In terms of proper degrees of freedom the total energy conservation reproduces the Einstein's mass-energy relation.
The newtonian interpretation of the second order dynamic equations suggests the definition for general inertial mass and for the interaction potential.

**Category:** Mathematical Physics

[479] **viXra:1605.0199 [pdf]**
*submitted on 2016-05-19 09:42:01*

**Authors:** V. A. Budarin

**Comments:** 9 Pages. MSC 76D09

The velocity field culculation method is based on the use of two special cases of the Newtonian fluid motion equations, not including the Navier-Stokes equation. Two shear stress calculation ways are considered. The first way is the differentiation on the velocity field equation, and the second one requires the solution of the first-order differential equation. The second way provides the distribution of shear stresses for any continuous medium, including Newtonian fluid.
Culculation equations for a laminar flow in a round pipe are found. It is shown that parabolic velocity distribution along the radius is a special case of more general equation.
The factors affecting the shear stresses for the three flow models are found. Stresses are determined by the linear velocity gradients in the laminar flow. In the 3D vortex, they can be found by various equations, which include vorticity. Total stresses for the averaged turbulent flow are culculated by summing the previously found stresses.
The equations of the method are incomplete and may be used for the accurate solution of simple problems.

**Category:** Mathematical Physics

[478] **viXra:1605.0183 [pdf]**
*submitted on 2016-05-17 08:17:14*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 4 Pages.

In this research investigation, the author has detailed a scheme for 'Positive And Negative Time Scaling And Time Portal Engineering'.

**Category:** Mathematical Physics

[477] **viXra:1605.0167 [pdf]**
*submitted on 2016-05-15 03:57:58*

**Authors:** Dionysios G. Raftopoulos

**Comments:** 424 Pages.

The Theory of the Harmonicity of the Field of Light is an axiomatic theory motivated by the logical contradictions of Special Relativity Theory and based on two fundamental acceptances:
1. The adoption of the natural philosophy of Werner Heisenberg and the school of Copenhagen, according to which a consistent natural description of the Cosmos shouldn’t ignore the existence of the Observer or at least the instrument of observation and measurement and
2. The choice of the Projective Space as the Geometrical Space of its natural description. This choice is validated following the fundamental separation of the Perceptible Space, which is objective, and the Geometrical Space, that exists only in our minds. As all logically consistent Geometries are accepted in Mathematical Science, the adoption of a Geometrical Space by a Theory of Physics is free.
Further on, this theory adopts as its first fundamental hypothesis the second hypothesis of the Special Relativity Theory, properly modified. Then, during the study of the kinematics of the material point, the property of harmonicity of the field of light emerges practically automatically.
Via this theory Human Reason is restored.

**Category:** Mathematical Physics

[476] **viXra:1605.0163 [pdf]**
*submitted on 2016-05-14 21:39:22*

**Authors:** Frank H. Makinson

**Comments:** 3 Pages.

The commonly used units of measure have changed over the centuries. The numeric value for the speed of light was measured before it was known that light was an electromagnetic wave and how electromagnetic waves were produced. The scientific community adopted a set of units of measures that were originally developed for joint commercial, scientific and common use. There is an on going effort by the various metrology organizations to update them. There is no need to change the generally accepted units of measure for commercial and common use, but scientific units of measure need major improvements.

**Category:** Mathematical Physics

[475] **viXra:1605.0126 [pdf]**
*submitted on 2016-05-12 07:50:50*

**Authors:** Giuseppe Azzarello

**Comments:** 8 Pages.

You will be shown the symmetry properties of the Planck’s particle, and will be drawn to its magnetic
charge. This will unify the gravitational, electrical and magnetic forces in a single force, now known as superforce. This is possible by introducing a new constant, symmetrical coupling factor call, which allows the transformation between forces.

**Category:** Mathematical Physics

[474] **viXra:1605.0101 [pdf]**
*submitted on 2016-05-10 15:26:40*

**Authors:** J.A.J. van Leunen

**Comments:** 10 Pages. The .docx version is accessible via the author's website http://www.e-physics.eu

Quaternionic Hilbert spaces can store discrete quaternions and quaternionic continuums in the eigenspaces of operators that reside in these Hilbert spaces. The reverse bra-ket method is an extension of the bra-ket notation that was introduced by P.M. Dirac. The reverse bra-ket method can create natural parameter spaces from quaternionic number systems and can relate the combinations of functions and their parameter spaces with eigenspaces and eigenvectors of corresponding operators that reside in non-separable Hilbert spaces. This also works for separable Hilbert spaces. The defining functions relate the separable Hilbert space with its non-separable companion. In this way, the method links Hilbert space technology with function technology, differential technology and integral technology. Quaternionic number systems exist in several versions that differ in the way that they are ordered. This is applied by defining multiple types of parameter spaces in the same Hilbert space. The set of closed subspaces of a separable Hilbert space has the relational structure of an orthomodular lattice. This fact makes the Hilbert space suitable for modelling quantum physical systems. The reverse bra-ket method is a powerful tool for generating quaternionic models that help investigating quantum physical models.

**Category:** Mathematical Physics

[473] **viXra:1605.0099 [pdf]**
*submitted on 2016-05-10 12:41:46*

**Authors:** Jean-Luc Paillet, Andrew Meulenberg

**Comments:** 17 Pages. paper presented to the 11th Workshop on Anomalies in Hydrogen Loaded Metals, Airbus-Toulouse (Fr.), 15-16 Oct. 2015

This work continues our previous work [1] and in a more developed form [2]), on electron deep orbits of the hydrogen atom. An introduction shows the importance of the deep orbits of hydrogen (H or D) for research in the LENR domain, and gives some general considerations on the EDO (Electron Deep Orbits) and on other works about deep orbits.
A first part recalls the known criticism against the EDO and how we face it. At this occasion we highlight the difference of resolution of these problems between the relativistic Schrödinger equation and the Dirac equation, which leads for this latter, to consider a modified Coulomb potential with finite value inside the nucleus.
In the second part, we consider the specific work of Maly and Va’vra [3], [4]) on deep orbits as solutions of the Dirac equation, so-called Deep Dirac Levels (DDLs). As a result of some criticism about the matching conditions at the boundary, we verified their computation, but by using a more complete ansatz for the “inside” solution. We can confirm the approximate size of the mean radii

**Category:** Mathematical Physics

[472] **viXra:1605.0083 [pdf]**
*submitted on 2016-05-09 00:20:00*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 3 Pages.

Given a unification candidate which is both inefficient and non-optimal will the experimental physics have validity with all such candidates?

**Category:** Mathematical Physics

[471] **viXra:1605.0025 [pdf]**
*submitted on 2016-05-02 18:29:22*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 3 Pages.

What to do when there is a lack of bonding between D-variants?

**Category:** Mathematical Physics

[470] **viXra:1604.0284 [pdf]**
*submitted on 2016-04-19 10:21:38*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 3 Pages.

We aim to discuss some earlier unification candidates and whether such unification candidates are suitable or non-suitable for the grand unification scheme?

**Category:** Mathematical Physics

[469] **viXra:1604.0283 [pdf]**
*submitted on 2016-04-19 08:58:26*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages. In english.

We find an exact solution for the system of Euler equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Mathematical Physics

[468] **viXra:1604.0266 [pdf]**
*submitted on 2016-04-18 08:44:03*

**Authors:** Ashes Mi

**Comments:** 113 Pages.

Solves Prime Numbers using PRIMES to Count With

**Category:** Mathematical Physics

[467] **viXra:1604.0202 [pdf]**
*submitted on 2016-04-12 12:51:08*

**Authors:** Terubumi Honjou

**Comments:** 7 Pages.

The prime number is connected to the quantum-mechanical basic equation.
Mathematician Euler discovered a prime number and a connection with π (Japanese yen) for the first time. The left side of a go board of the following equation that I had only with a prime number is equal to π2/6. I transformed the following equation and had the equation of the area of Japanese yen. Then it became the equation that the prime number equation (zeta function) of the oiler assumed a prime number a radius. Here, a prime number and the correlation with what I set were provided on the top of the pulsation wave pattern of the figure of prime number, physics fusion as if I showed it to a figure of of the Lehman expectation proof that I contributed from an association between Schrodinger equation and circular motion of the elementary particle pulsation principle correlation chart in the online posting before last time. The prime number has a quantum-mechanical basic equation, the connection that are close to Schrodinger equation.

**Category:** Mathematical Physics

[466] **viXra:1604.0159 [pdf]**
*submitted on 2016-04-10 02:14:06*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 3 Pages.

Infrared divergences are prominent in the process toward grand unification. Where do we see infrared divergences in the grand unification scheme?

**Category:** Mathematical Physics

[465] **viXra:1604.0021 [pdf]**
*submitted on 2016-04-03 15:39:27*

**Authors:** M.E. Hassani

**Comments:** 18 Pages; 2 Tables; 13 References.

The causality principle is physically investigated in the framework of special relativity theory (SRT) and it is proven to be absolutely valid for subluminal, luminal and superluminal signals under any natural and/or artificial circumstances; also Einstein's thought experiment (1907), Tolman's paradox (1917), tachyonic antitelephone (1970) and Moller's thought experiment (1952) are re-examined in order to show more conclusively that the so-called causality paradoxes are in fact a pure mental construction resulted from some too-common misconceptions which mainly spring from the confusion between the concepts of (relative) velocity and (relative) speed. Thus, in light of the present work, the old and semi-persistent concern about causality violation by superluminal signals and its consequences at micro and macrophysical levels may be conceptually ruled out if one approaches the physico-mathematical formalism of SRT in an open-minded way.

**Category:** Mathematical Physics

[464] **viXra:1603.0377 [pdf]**
*submitted on 2016-03-27 23:47:52*

**Authors:** Moises Dominguez-Espinosa, Jaime Melendez-Martinez

**Comments:** 4 Pages. 3 figures, dynamical systems, quatum mechanics.

There is a paradigm in Quantum Mechanics that explains quantization through normal vibration modes called Eigenstates that arise from Schrodinger wave equation. In this contribution we propose an alternative methodology of quantization by using basic concepts of mechanics and chaos from which a Toy Model is built.

**Category:** Mathematical Physics

[463] **viXra:1603.0371 [pdf]**
*submitted on 2016-03-27 10:37:50*

**Authors:** Jonathan Tooker

**Comments:** 1 Page.

Wick rotation produces numbers that agree with experiment and yet the method is mathematically wrong and not allowed by any self-consistent rule. We explore a small slice of wiggle room in complex analysis and show that it may be possible to use QFT without reliance Wick rotations.

**Category:** Mathematical Physics

[462] **viXra:1603.0363 [pdf]**
*submitted on 2016-03-25 18:33:06*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 3 Pages.

What is required and restricted by the grand unification scheme when it comes to energy for any variant?

**Category:** Mathematical Physics

[461] **viXra:1603.0245 [pdf]**
*submitted on 2016-03-16 14:29:28*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 3 Pages.

Conceptual definition of GRS and physics.

**Category:** Mathematical Physics

[460] **viXra:1603.0229 [pdf]**
*submitted on 2016-03-15 22:31:38*

**Authors:** Victor Christianto, Yunita Umniyati

**Comments:** 8 Pages. This paper has been submitted to IJET (www.scipress.com). Your comments are welcome

This paper was at-least-partially inspired by the problem of relatively slow internet connection in our country. We believe that the same common problem has plagued other developing countries like ours, so it seems that we need a new technology to increase the internet capacity, especially the wireless network capacity. One way to do that is to look at the photon electrodynamics theory. In a series of papers, Bo Lehnert has suggested screw-shaped model of photon, inspired by his Revised Quantum Electrodynamics (RQED). Therefore in this paper we will review 4 possible methods to extend his screw-shaped photon model. In the mean time, there is recent debate concerning theoretical basis and utilization of photon orbital angular momentum (OAM), in particular as a means to increase wireless internet capacity. Promising results have been reported from laboratory experiments carried out by Bo Thide group and others too. But considering Vigier’s proposal to consider photon as soliton, in this paper we will discuss not the usual photon OAM as suggested by Thide group, instead we will consider soliton orbital angular momentum. If the proposed concept holds true, then it is possible to develop soliton radio wave based on OAM, which we call here as SOAmR (Soliton Orbital Angular Momentum Radio).

**Category:** Mathematical Physics

[459] **viXra:1603.0223 [pdf]**
*submitted on 2016-03-15 14:34:15*

**Authors:** Wei Cen, Ning Gu

**Comments:** Pages.

The bio-heat transfer equation for homogeneous material model can be easily calculated by using second order finite difference approximation to discretize the spatial derivatives and explicit finite-difference time-domain (FDTD) scheme for time domain discretization. Mr. Gandhi and colleagues solved the bio-heat equation for inhomogeneous models utilizing implicit finite-difference method. Whereas we appreciate their research, we would like to address a few issues that may help further clarify or confirm the research.

**Category:** Mathematical Physics

[458] **viXra:1603.0121 [pdf]**
*submitted on 2016-03-07 21:09:17*

**Authors:** Claude Latourre

**Comments:** 8 Pages. en francais

For more than a century, the equations of general relativity have evolved according to the observations of the universe. These changes are expressed through the cosmological constant (Λ), which was first added on the space-time part to account for a stationary universe, then removed when observed the evolution of it. More recently, the constant reappeared on the energy-momentum part to describe an accelerated expansion of the universe.
Let's see now, how the contraction of the equations of general relativity can express exactly the value of the cosmological constant: Λ = -1/4 (R + κ T) and also to deduce an equivalent reformulation the equations of general relativity: (Rµ√ -1/4 gµ√ R) = κ (Tµ√ -1/4 gµ√ T). All this, without using any physical concept: Dark energy, Quintessence…

**Category:** Mathematical Physics

[457] **viXra:1603.0115 [pdf]**
*submitted on 2016-03-07 15:34:26*

**Authors:** Robert G. Wallace

**Comments:** 26 Pages.

Philosophical considerations suggest a dimensional structure of reality based on infinite subdivision of singularity via Cayley Dickson doubling. An algebra, labelled the Kaotic algebra, is constructed based on algebras obtained by Cayley-Dickson doubling. It provides a natural basis for a non-associative geometry with links to the associative geometry of Cl(1,10). Subalgebras of the kaotic algebra have a phenomenology corresponding to that of the fermions of the standard model.

**Category:** Mathematical Physics

[456] **viXra:1603.0052 [pdf]**
*submitted on 2016-03-04 04:38:37*

**Authors:** Na Liu, Xihua Xu, Yibing Chen

**Comments:** 24 Pages. numerical method in CFD

In this paper, an arbitrary high-order compact method is developed for compressible
multi-component flows with a stiffened gas equations of state(EOS).
The main contribution is combining the high-order, conservative, compact spectral
volume scheme(SV) with the non-oscillatory kinetic scheme(NOK) to solve
the quasi-conservative extended Euler equations of compressible multi-component
flows. The new scheme consists of two parts: the conservative part and the
non-conservative part. The original high order compact SV scheme is used to
discretize the conservative part directly. In order to treat the equation of state
of the stiffened gas, the NOK scheme is utilized to compute the numerical flux.
Then, careful analysis is made to satisfy the necessary condition to avoid unphysical
oscillation near the material interfaces. After that, a high-order compact
scheme for the non-conservative part is obtained. This new scheme has the following
advantages for numerical simulations of compressible multi-component
stiffened gas: high order accuracy with compact stencil and oscillation-free near
the material interfaces. Numerical tests demonstrate the good performance and
the efficiency of the new scheme for multi-component flow simulations.

**Category:** Mathematical Physics

[455] **viXra:1602.0255 [pdf]**
*submitted on 2016-02-20 10:38:07*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 75 Pages. In French.

It is a traduction to French of the booklet of Prof. Torben Krarup " A Contribution on the Mathematical Foundation of Physical Geodesy " publication N°44 of the Institute of Geodesy of Danmark, 1969. It represents the foundation of physical geodesy after Molodensky and Hotine contributions.

**Category:** Mathematical Physics

[454] **viXra:1602.0254 [pdf]**
*submitted on 2016-02-20 10:55:25*

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 16 Pages. In French.

The papier presents an essay of the resolution of Navier-Stokes equations under the hypothesis (A) of the open problem cited by Clay Institute (C.L. Fefferman, 2006).

**Category:** Mathematical Physics

[453] **viXra:1602.0167 [pdf]**
*submitted on 2016-02-14 13:07:31*

**Authors:** R E Watson

**Comments:** 39 Pages.

Kaluza's 1921 theory of gravity and electromagnetism using a fifth wrapped-up spatial dimension is inspiration for many modern attempts to develop new physical theories. The original theory has problems which may well be overcome, and thus Kaluza theory should be looked at again: it is a natural, if not necessary, geometric unification of gravity and electromagnetism. Here a general demonstration that the Lorentz force law can be derived from a range of Kaluza theories is presented. This is investigated via non-Maxwellian kinetic definitions of charge that are divergence-free and relate Maxwellian charge to 5D components of momentum. The possible role of torsion is considered as an extension. It is shown, however, that symmetric torsion components are likely not admissible in any prospective theory. As a result Kaluza's original theory is rehabilitated and a call for deeper analysis made.

**Category:** Mathematical Physics

[452] **viXra:1602.0122 [pdf]**
*submitted on 2016-02-10 15:47:26*

**Authors:** Hans Detlef Hüttenbach

**Comments:** 4 Pages.

A simple mathematical proof reveals that time-inversion symmetry and reversibility are different concepts, which also resolves the
Loschmidt paradox.

**Category:** Mathematical Physics

[451] **viXra:1602.0114 [pdf]**
*submitted on 2016-02-09 13:34:59*

**Authors:** Robert B. Easter

**Comments:** 87 Pages.

This paper introduces the G(4,8) Double Conformal Space-Time Algebra (DCSTA). G(4,8) DCSTA is a straightforward extension of the G(8,2) Double Conformal / Darboux Cyclide Geometric Algebra (DCGA). G(4,8) DCSTA extends G(8,2) DCGA with spacetime boost operations and differential operators for differentiation with respect to the time-like w=c t direction and time t. The spacetime boost operation can implement anisotropic dilation (directed non-uniform scaling) of quadric surface entities. Quadric surface entities can be boosted into moving surfaces with constant velocities that display the length contraction effect of special relativity. To demonstrate G(4,8) DCSTA as concrete mathematics with possible applications, this paper includes sample code and example calculations using the symbolic computer algebra system SymPy.

**Category:** Mathematical Physics

[450] **viXra:1602.0095 [pdf]**
*submitted on 2016-02-08 12:08:17*

**Authors:** Branko Zivlak

**Comments:** 4 Pages. 12 formulas

After examining the relationships in the universe, is determined a hypothetical quantum temperature and then obtained temperature of cosmic microwave background radiation (CMB). It is rejected an explanation of that temperature as a result of the relic radiation from the Big Bang.

**Category:** Mathematical Physics

[449] **viXra:1602.0066 [pdf]**
*submitted on 2016-02-05 13:54:42*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 3 Pages.

Give formal definition of SUPREME and explain it's implications.

**Category:** Mathematical Physics

[448] **viXra:1602.0049 [pdf]**
*submitted on 2016-02-04 10:02:29*

**Authors:** Jean Claude Dutailly

**Comments:** 380 Pages. New on QM, RG, gravitation

This book proposes a review and, on some important points, a new interpretation of the main concepts of Theoretical Physics. Rather than offering an interpretation based on exotic physical assumptions (additional dimension, new particle, cosmological phenomenon,…) or a brand new abstract mathematical formalism, it proceeds to a systematic review of the main concepts of Physics, as Physicists have always understood them : space, time, material body, force fields, momentum, energy… and propose the right mathematical tools to deal with them, chosen among well known mathematical theories.
After a short introduction about the place of Mathematics in Physics, a new interpretation of the main axioms of Quantum Mechanics is proposed. It is proven that these axioms come actually from the way mathematical models are expressed, and this leads to theorems which validate most of the usual computations and provide safe and clear conditions for their use, as it is shown in the rest of the book.
Relativity is introduced through the construct of the Geometry of General Relativity, based on 5 propositions and the use of tetrads and fiber bundles, which provide tools to deal with practical problems, such as deformable solids.
A review of the concept of momenta leads to the introduction of spinors in the framework of Clifford algebras. It gives a clear understanding of spin and antiparticles.
The force fields are introduced through connections, in the, now well known, framework of gauge theories, which is here extended to the gravitational field. It shows that this field has actually a rotational and a transversal component, which are masked under the usual treatment by the metric and the Levy-Civita connection. A thorough attention is given to the topic of the propagation of fields with interesting results, notably to explore gravitation.
The general theory of lagrangians in the application of the Principle of Least Action is reviewed, and two general models, incorporating all particles and fields are explored, and used for the introduction of the concepts of currents and energy-momentum tensor. Precise guidelines are given to find operational solutions of the equations of the gravitational field in the most general case.
The last chapter shows that bosons can be understood as discontinuities in the fields.
In this 4th version of this book, changes have been made :
- in Relativist Geometry : the ideas are the same, but the chapter has been rewritten, notably to introduce the causal structure and explain the link with the practical measures of time and space;
- in Spinors : the relation with momenta has been introduced explicitly
- in Force fields : the section dedicated to the propagation of fields is new, and is an important addition.
- in Continuous Models : the section about currents and energy-momentum tensor are new.
- in Discontinuous Processes : the section about bosons has been rewritten and the model improved.

**Category:** Mathematical Physics

[447] **viXra:1602.0048 [pdf]**
*submitted on 2016-02-04 10:11:31*

**Authors:** Domenico Oricchio

**Comments:** 11 Pages.

I open a dusty old drawer, and I found this article, rejected by every journal, with a complete, total loss of time; a thing that I'll never make; and I change only the bibliography, that is to initially conceived .
The old idea sounds interesting, and here, on vixra, there is not rejection.
I don't remember the whole theory, and the whole programs, but it can be useful to others; so I share it with you.
It seem that without the complication of the least common divisor, the calculation is more simple, and elegant.

**Category:** Mathematical Physics

[446] **viXra:1601.0312 [pdf]**
*submitted on 2016-01-29 03:55:49*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 18 Pages. Need yet revision and complete translation for english.

A solution to the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy. We have proved that there are initial velocities u^0 (x) and forces f(x,t) such that there is no physically reasonable solution to the Navier-Stokes equations for t>0, which corresponds to the case (C) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute.

**Category:** Mathematical Physics

[445] **viXra:1601.0280 [pdf]**
*submitted on 2016-01-25 20:07:50*

**Authors:** Miguel A. Sanchez-Rey

**Comments:** 3 Pages.

Procedure will be stated.

**Category:** Mathematical Physics

[444] **viXra:1601.0134 [pdf]**
*submitted on 2016-01-12 21:34:25*

**Authors:** Sai Venkatesh Balasubramanian

**Comments:** 27 Pages.

We have obtained an equivalent circuit that explains selfheating effects and have tested the presence of nonlinearity and IMD using simulation. The significance of the approach lies in successfully modeling thermal effects in interconnects in ICs using SOI technology with al conductors. Future work: proposing of circuits and techniques that can be used for compensation of selfheating effects.

**Category:** Mathematical Physics

[443] **viXra:1601.0075 [pdf]**
*submitted on 2016-01-08 13:26:02*

**Authors:** Rami Mehrem

**Comments:** 5 Pages.

The integral $\int_0^\infty r^{2-\lambda}\,
j_0(k_1r)\,j_0(k_2r)\,j_\lambda(k_3r)\,dr$ is evaluated for any real positive $k_1$, $k_2$, $k_3$ and $\lambda \ge 0$

**Category:** Mathematical Physics

[442] **viXra:1601.0045 [pdf]**
*submitted on 2016-01-06 09:49:26*

**Authors:** Carsten S.P. Spanheimer

**Comments:** 26 Pages. in german language

Die Einführung 'quaternionischer Differentialformen' auf einer Mannigfaltigkeit ergibt ein vielversprechendes mathematisches System zur Beschreibung unserer physikalischen (3+1)-Raumzeit schon mit einem Minimum von Grundannahmen.
Hier wird die Grundlage dieses Modells einer 'Quaternionischen Raumzeit' dargestellt.

**Category:** Mathematical Physics

[441] **viXra:1512.0458 [pdf]**
*submitted on 2015-12-28 10:27:37*

**Authors:** Markos Georgallides

**Comments:** 33 Pages. The Unsolved ancient-Greek Problems , The Unsolved Special E-Problems

The Special Problems of E-geometry consist the Quantization Moulds of Euclidean Geometry in it , to become → The Basic monad , through mould of Space –Anti-space in itself , which is the material dipole in inner monad Structure which is Identical with the Electromagnetic cycloidal field → Linearly , through mould of Parallel Theorem , which are the equal distances of common Line-meter between the points of parallel and line → In Plane , through mould of Squaring the circle , where the two equal and perpendicular monads consist a Plane acquiring the common Plane-meter ,pi,→ and in Space ,(volume) , through mould of the Duplication of the Cube , where any two Un-equal and perpendicular monads acquire the common Space-meter ³√2 ,to be twice each other , as this in analytical methods explained . The article consist a provocation to all scarce today Geometers and mathematicians in order to give an answer to these Old-age standing Unsolved Problems.
All Geometrical solutions are clearly Exposed and presented on Dr-Geo machine , and unveill the pass-over-faults of Relativity . E-Geometry is proved to be the base of all natural sciences and also the reflective logic from the objective reality , which is nature , to us

**Category:** Mathematical Physics

[440] **viXra:1512.0443 [pdf]**
*submitted on 2015-12-26 14:29:46*

**Authors:** Dan Visser

**Comments:** 11 Pages.

The formulation of the Einstein-field equations are changed: Herewith I present dynamical formulations for the multiplication of Big Bang-universes embedded in an eternal rotating holographic universe. This is beyond the current assumption the universe could be a hologram; that assumption was made without detailed knowledge of shape and deeper dynamics of a holographic universe. The new formulations introduce a dynamical process of deeper dynamics by which an object or subject could be transformed to another part of the holographic universe. This may happen by recalculation. But how? The recalculation described here is based on an amount of ‘duo-bits of dark matter’, which recalculate the quantum-dynamics. The recalculation-process emerges from a domain below the Planck-scale. The ‘duo-bit recalculation’ starts for a calculated lower-limit of Planck-areas. From my calculation follows, for example: 17 x 10^53 ‘duo-bits’ can recalculate 64 x 10^6 Planck-areas. This subquantum informational process is embedded in the modified Einstein-field-equations by a geometrical ratio of quantum-gravity and Planck-areas. This ratio is correlated to an additional energy-tensor of dark matter-energy. The geometrical ratio is more refined than the Einstein cosmological constant, which is only a number. The refined ratio replaces the cosmological constant. Therefore the main-issue in this paper is, that the accelerated space-expansion in the current Big Bang-universe is based on a ‘fake-dynamic’ due to one of many possible quasi Big Bang-universes that directly exist in physical reality. Or in other words: Directly existing parallel universes are part of the holographic dynamics. According to the new formulations parallel universes do not exist outside the conservative Big Bang-universe. In an appendix a description is added for the media written in Dutch.

**Category:** Mathematical Physics

[439] **viXra:1512.0401 [pdf]**
*submitted on 2015-12-22 11:25:45*

**Authors:** Sergey V. Ershkov

**Comments:** 10 Pages. Keywords: Arnold-Beltrami-Childress (ABC) flow, helical flow, stationary points

The existence of stationary points for the dynamical system of ABC-flow is considered.
The ABC-flow, a three-parameter velocity field that provides a simple stationary solution of Euler's equations in three dimensions for incompressible, inviscid fluid flows, is the prototype for the study of turbulence (it provides a simple example of dynamical chaos).
But, nevertheless, between the chaotic trajectories of the appropriate solutions of such a system we can reveal the stationary points, the deterministic basis among the chaotic behaviour of ABC-flow dynamical system. It has been proved the existence of 1 point for two partial cases of parameters {A, B, C}: 1) A = B = 1; 2) C = 1 (A² + B² = 1). Moreover, dynamical system of ABC-flow allows 3 points of such a type, depending on the meanings of parameters {A, B, C}.

**Category:** Mathematical Physics

[438] **viXra:1512.0341 [pdf]**
*submitted on 2015-12-16 14:56:32*

**Authors:** Branko Zivlak

**Comments:** 3 Pages. 1 Formula, 1 Table

The formula connecting fundamental physical constants has been verified based on the 2010 and 2014 recommended CODATA values of physical constants.

**Category:** Mathematical Physics

[437] **viXra:1512.0334 [pdf]**
*submitted on 2015-12-16 00:15:11*

**Authors:** A. A. Frempong

**Comments:** 46 Pages. Copyright © A. A. Frempong

Over nearly a year and half ago, the Navier-Stokes (N-S) equations in 3-D for incompressible fluid flow were analytically solved by the author. However, some of the solutions contained implicit terms. In this paper, the implicit terms have been expressed explicitly in terms of x, y, z and t. The author proposed and applied a new law, the law of definite ratio for incompressible fluid flow. This law states that in incompressible fluid flow, the other terms of the fluid flow equation divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. It was mathematically shown that without gravity forces on earth, there would be no incompressible fluid flow on earth as is known, and also, there would be no magnetohydrodynamics. In addition to the usual method of solving these equations, the N-S equations were also solved by a second method in which the three equations in the system were added to produce a single equation which was then integrated. The solutions by the two methods were identical, except for the constants involved. Ratios were used to split-up the equations; and the resulting sub-equations were readily integrable, and even, the nonlinear sub-equations were readily integrated. The examples in the preliminaries show everyday examples on using ratios to divide a quantity into parts, as well as possible applications of the solution method in mathematics, science, engineering, business, economics, finance, investment and personnel management decisions. The
x-direction Navier-Stokes equation was linearized, solved, and the solution analyzed. This solution was followed by the solution of the Euler equation of fluid flow. The Euler equation represents the nonlinear part of the Navier-Stokes equation. Following the Euler solution, the Navier-Stokes equation was solved essentially by combining the solutions of the linearized equation and the Euler solution. For the Navier-Stokes equation, the linear part of the relation obtained from the integration of the linear part of the equation satisfied the linear part of the equation; and the relation from the integration of the non-linear part satisfied the non-linear part of the equation. The solutions and relations revealed the role of each term of the Navier-Stokes equations in fluid flow. The gravity term is the indispensable term in fluid flow, and it is involved in the parabolic and forward motion of fluids. The pressure gradient term is also involved in the parabolic motion. The viscosity terms are involved in the parabolic, periodic and decreasingly exponential motion. Periodicity increases with viscosity. The variable acceleration term is also involved in the periodic and decreasingly exponential motion.
The fluid flow in the Navier-Stokes solution may be characterized as follows. The x-direction solution consists of linear, parabolic, and hyperbolic terms. The first three terms characterize parabolas. If one assumes that in laminar flow, the axis of symmetry of the parabola for horizontal velocity flow profile is in the direction of fluid flow, then in turbulent flow, the axis of symmetry of the parabola would have been rotated 90 degrees from that for laminar flow. The characteristic curve for the x-nonlinear term is such a parabola whose axis of symmetry has been rotated 90 degrees from that of laminar flow. The y-nonlinear term is similar parabolically to the x-nonlinear term. The characteristic curve for the z-nonlinear term is a combination of two similar parabolas and a hyperbola. If the above x-direction flow is repeated simultaneously in the y-and z-directions, the flow is chaotic and consequently turbulent.
For a spin-off, the smooth solutions from above are specialized and extended to satisfy the requirements of the CMI Millennium Prize Problems, and thus prove the existence of smooth solutions of the Navier-Stokes equations.

**Category:** Mathematical Physics

[436] **viXra:1512.0333 [pdf]**
*submitted on 2015-12-16 00:24:04*

**Authors:** A. A. Frempong

**Comments:** 19 Pages. Copyright © A. A. Frempong

The system of magnetohydrodynamic (MHD) equations have been solved analytically in this paper. The author applied the technique used in solving the Navier-Stokes equations and applied a new law, the law of definite ratio for MHD. This law states that in MHD, the other terms of the system of equations divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. It is shown that without gravity forces on earth, there would be no magnetohydrodynamics on earth as is known. The equations in the system of equations were added to produce a single equation which was then integrated. Ratios were used to split-up this single equation into sub-equations which were readily integrated, and even, the non-linear sub-equations were readily integrated. Twenty-seven sub-equations were integrated. The linear part of the relation obtained from the integration of the linear part of the equation satisfied the linear part of the equation; and the relation from the integration of the non-linear part satisfied the non-linear part of the equation. The solutions revealed the role of each term in magnetohydrodynamics. In particular, the gravity term is the indispensable term in magnetohydrodynamics. The solutions of the MHD equations were compared with the solutions of the N-S equations, and there were similarities and dissimilarities.

**Category:** Mathematical Physics

[435] **viXra:1512.0332 [pdf]**
*submitted on 2015-12-16 00:45:22*

**Authors:** A. A. Frempong

**Comments:** 12 Pages. Copyright © A. A. Frempong

This paper covers the solutions of the Euler equations in 3-D and 4-D for incompressible fluid flow. The solutions are the spin-offs of the author's previous analytic solutions of the Navier-Stokes equations (vixra:1405.0251 of 2014). However, some of the solutions contained implicit terms. In this paper, the implicit terms have been expressed explicitly in terms of x, y, z and t.
The author applied a new law, the law of definite ratio for fluid flow. This law states that in incompressible fluid flow, the other terms of the fluid flow equation divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. This law evolved from the author's earlier solutions of the Navier-Stokes equations. In addition to the usual approach of solving these equations, the Euler equations have also been solved by a second method in which the three equations in the system are added to produce a single equation which is then integrated. The solutions by the two approaches are identical, except for the constants involved. From the experience gained in solving the linearized Navier-Stokes equations, only the equation with the gravity term as the subject of the equation was integrated. The experience was that when each of the terms of the Navier-Stokes equation was used as the subject of the equation, only the equation with the gravity term as the subject of the equation produced a solution. Ratios were used to split-up the x-direction Euler equation with the gravity term as the subject of the equation. The resulting five sub-equations were readily integrable, and even, the non-linear sub-equations were readily integrated. The integration results were combined. The combined results satisfied the corresponding equation. This equation which satisfied its corresponding equation would be defined as the driver equation; and each of the other equations which would not satisfy its corresponding equation would be called a supporter equation. A supporter equation does not satisfy its corresponding equation completely, but provides useful information which is not apparent in the solution of the driver equation. The solutions and relations revealed the role of each term of the Euler equations in fluid flow. The gravity term is the indispensable term in fluid flow, and it is involved in the forward motion of fluids. The pressure gradient term is also involved in the forward motion. The variable acceleration term is also involved in the forward motion. The fluid flow behavior in the Euler solution may be characterized as follows. The x-direction solution consists of linear, parabolic, and hyperbolic terms. If one assumes that in laminar flow, the axis of symmetry of the parabola for horizontal velocity flow profile is in the direction of fluid flow, then in turbulent flow, the axis of symmetry of the parabola would have been rotated 90 degrees from that for laminar flow. The characteristic curve for the x-nonlinear term is such a parabola whose axis of symmetry has been rotated 90 degrees from that of laminar flow. The y-nonlinear term is similar parabolically to the x-nonlinear term. The characteristic curve for the z-nonlinear term is a combination of two similar parabolas and a hyperbola. If the above x-direction flow is repeated simultaneously in the y-and z-directions, the flow is chaotic and consequently turbulent.

**Category:** Mathematical Physics

[434] **viXra:1512.0261 [pdf]**
*submitted on 2015-12-07 21:43:24*

**Authors:** Rodney Bartlett

**Comments:** 6 Pages.

The Fields Medal is an award established in 1924 by the ICM (International Congress of Mathematicians) and is restricted to mathematicians up to the age of 40. It recognizes both existing work and future promise, and is equivalent to the Nobel Prize.
A trick is shown in this article that proves 1=2. However, during the steps division by zero is used. Since this is not allowed, the conclusion is false. Or it would be unless zero could be shown not to be nothing. Zeros are something because they're paired with ones to compose the binary digits essential to the formation of everything in space-time. This means zero has been misunderstood throughout history, division by zero is indeed permitted and the conclusion that 1=2 is correct. In turn, this means that while 1+1=2, 1+1 must also equal 1. This mathematically validates the centuries-long march of physics towards unification of everything in this cosmos. Unification occurs not just in the present but the entire past and future are naturally part of this one entity/event.
The article then explores possible consequences for 1) Einstein being correct to divide by zero, 2) Hidden Variables and Determinism, 3) that zero is not nothing but actually something, 4) that zero redefines the term infinity, 5) that there really is another explanation for the origin of the universe besides the Big Bang, 6) and another explanation for black holes, 7) possibilities regarding life after death and life before conception.

**Category:** Mathematical Physics

[406] **viXra:1610.0337 [pdf]**
*replaced on 2016-11-20 13:13:40*

**Authors:** Nicolas Poupart

**Comments:** 6 Pages.

Il sera démontré dans cet article que la masse noire est une conséquence nécessaire de la mécanique relativiste. Cette démonstration fait abstraction des forces de la physique et est donc une explication purement mécanique. La relation de Tully-Fisher sera déduite naturellement, sans appel à une quelconque nouvelle physique, et il est par conséquent possible de dériver cette loi sans modifier la gravitation de Newton ou la relativité générale. De plus, une prédiction théorique sur une nouvelle forme de décalage de fréquence sera effectuée permettant de réfuter ou de confirmer cette théorie.

**Category:** Mathematical Physics

[405] **viXra:1609.0055 [pdf]**
*replaced on 2016-10-05 08:35:53*

**Authors:** J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou, M. D. Monsia

**Comments:** 5 pages

This research work proposes a Lagrangian and Hamiltonian analysis for the unique class of position-dependent mass oscillator characterized by a harmonic periodic solution and parabolic potential energy and its inverted version admitting a position-dependent mass dynamics.

**Category:** Mathematical Physics

[404] **viXra:1608.0317 [pdf]**
*replaced on 2016-08-25 22:30:14*

**Authors:** Robert G Wallace

**Comments:** 9 Pages.

An algebra for unit multivector components for a manifold of five poly-complex dimensions is presented. The algebra has many properties that suggest it may provide a basis for a grand unification theory.

**Category:** Mathematical Physics

[403] **viXra:1608.0181 [pdf]**
*replaced on 2016-09-15 07:53:55*

**Authors:** J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou, M. D. Monsia

**Comments:** 4 pages

The Lagrangian description of a dynamical system from the equation of motion consists of an inverse problem in mechanics. This problem is solved for a class of exactly integrable mixed and quadratic Liénard type oscillator equations from a given first integral of motion. The dynamics of this class of equations, which contains the generalized modified Emden equation, also known as the second-order Riccati equation, and the inverted versions of the Mathews-Lakshmanan equations, is then investigated from Hamiltonian and Lagrangian points of view.

**Category:** Mathematical Physics

[402] **viXra:1608.0059 [pdf]**
*replaced on 2016-08-27 13:33:54*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 11 Pages.

The solution for the problem of Breakdown of Euler Equations, like the Millenium Problem for Navier-Stokes equations.

**Category:** Mathematical Physics

[401] **viXra:1608.0059 [pdf]**
*replaced on 2016-08-19 06:48:23*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 11 Pages.

The solution for the problem of Breakdown of Euler Equations, like the Millenium Problem for Navier-Stokes equations.

**Category:** Mathematical Physics

[400] **viXra:1608.0059 [pdf]**
*replaced on 2016-08-15 18:59:23*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 10 Pages.

**Category:** Mathematical Physics

[399] **viXra:1608.0059 [pdf]**
*replaced on 2016-08-15 06:56:32*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages.

First date: remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Second date: verifying that for potential and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. The first date is historical only. Last date: non-uniqueness in time for incompressible and potential flows, if u≠0.

**Category:** Mathematical Physics

[398] **viXra:1608.0059 [pdf]**
*replaced on 2016-08-13 16:24:12*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 10 Pages.

**Category:** Mathematical Physics

[397] **viXra:1608.0059 [pdf]**
*replaced on 2016-08-12 13:12:29*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 10 Pages.

**Category:** Mathematical Physics

[396] **viXra:1608.0059 [pdf]**
*replaced on 2016-08-08 12:48:47*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 9 Pages.

**Category:** Mathematical Physics

[395] **viXra:1608.0059 [pdf]**
*replaced on 2016-08-08 03:21:24*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 9 Pages.

**Category:** Mathematical Physics

[394] **viXra:1608.0059 [pdf]**
*replaced on 2016-08-06 11:13:40*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 9 Pages.

**Category:** Mathematical Physics

[393] **viXra:1605.0192 [pdf]**
*replaced on 2016-09-14 11:29:59*

**Authors:** Bernd Ganter

**Comments:** 8 pages. The paper is the condensed version in English translation of the German parent document viXra:1408.0018. Comments please to bernd.ganter.fsk@gmx.de

In order to explain the strongly differing forces of the fundamental interactions, we use optimal consecutive exponentiation (power towers) as mathematical instrument for the optimization of volumes or densities, and thus for information storage in the universe. From this approach it is possible to derive and calculate the value of the fine structure constant, the value of which has remained unexplained for 100 years now, as 137.035999100.

**Category:** Mathematical Physics

[392] **viXra:1605.0101 [pdf]**
*replaced on 2016-09-28 04:34:35*

**Authors:** J.A.J. van Leunen

**Comments:** 13 Pages.

Hilbert spaces can store discrete members of division rings and continuums that are formed by members of division rings in the eigenspaces of operators that reside in these Hilbert spaces. Only three suitable division rings exist. These are the real numbers, the complex numbers and the quaternions.
The reverse bra-ket method is an extension of the bra-ket notation that was introduced by P.M. Dirac. The reverse bra-ket method can create natural parameter spaces from number systems that are division rings and can relate the combinations of functions and their parameter spaces with eigenspaces and eigenvectors of corresponding operators that reside in non-separable Hilbert spaces. This also works for separable Hilbert spaces. The defining functions relate the separable Hilbert space with its non-separable companion. In this way, the method links Hilbert space technology with function technology, differential technology and integral technology. Quaternionic number systems exist in several versions that differ in the way that they are ordered. This is applied by defining multiple types of parameter spaces in the same Hilbert space. The set of closed subspaces of a separable Hilbert space has the relational structure of an orthomodular lattice. This fact makes the Hilbert space suitable for modelling quantum physical systems. The reverse bra-ket method is a powerful tool for generating quaternionic models that help investigating quantum physical models.

**Category:** Mathematical Physics

[391] **viXra:1604.0283 [pdf]**
*replaced on 2016-08-25 21:21:42*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 10 Pages.

We find an exact solution for the system of Euler equations, supposing that there is some solution, following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, it is possible that there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Mathematical Physics

[390] **viXra:1604.0283 [pdf]**
*replaced on 2016-07-14 12:34:15*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 8 Pages.

We find an exact solution for the system of Euler equations, supposing that there is some solution, following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, it is possible that there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Mathematical Physics

[389] **viXra:1604.0283 [pdf]**
*replaced on 2016-07-07 12:37:12*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 7 Pages.

We find an exact solution for the system of Euler equations following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Mathematical Physics

[388] **viXra:1604.0283 [pdf]**
*replaced on 2016-06-24 12:16:36*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 7 Pages.

We find an exact solution for the system of Euler equations following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Mathematical Physics

[387] **viXra:1604.0283 [pdf]**
*replaced on 2016-06-23 12:07:43*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 7 Pages.

We find an exact solution for the system of Euler equations following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Mathematical Physics

[386] **viXra:1604.0283 [pdf]**
*replaced on 2016-06-20 07:03:31*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 7 Pages.

**Category:** Mathematical Physics

[385] **viXra:1604.0283 [pdf]**
*replaced on 2016-06-16 12:20:05*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages.

We find an exact solution for the system of Euler equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Mathematical Physics

[384] **viXra:1604.0283 [pdf]**
*replaced on 2016-06-15 09:26:06*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages. Sorry, this paper yet is not good.

We find an exact solution for the system of Euler equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity.

**Category:** Mathematical Physics

[383] **viXra:1604.0283 [pdf]**
*replaced on 2016-06-12 22:09:03*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages. Sorry, this paper yet is not good.

**Category:** Mathematical Physics

[382] **viXra:1604.0283 [pdf]**
*replaced on 2016-06-09 12:51:35*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** ERRATA: Page 4, between equations (12) and (13) include the word "square" before the word "module": square module.

**Category:** Mathematical Physics

[381] **viXra:1604.0283 [pdf]**
*replaced on 2016-06-04 16:06:11*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 6 Pages.

**Category:** Mathematical Physics

[380] **viXra:1604.0283 [pdf]**
*replaced on 2016-06-02 09:30:27*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 5 Pages.

**Category:** Mathematical Physics

[379] **viXra:1604.0283 [pdf]**
*replaced on 2016-05-31 07:59:27*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 5 Pages.

**Category:** Mathematical Physics

[378] **viXra:1604.0283 [pdf]**
*replaced on 2016-05-09 08:25:53*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages. ERRATA: x,y,z in equation (7) are in Lagrangian description

**Category:** Mathematical Physics

[377] **viXra:1604.0283 [pdf]**
*replaced on 2016-04-27 08:02:57*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages.

**Category:** Mathematical Physics

[376] **viXra:1604.0283 [pdf]**
*replaced on 2016-04-23 20:37:16*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages. In english.

**Category:** Mathematical Physics

[375] **viXra:1604.0283 [pdf]**
*replaced on 2016-04-22 19:03:20*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 4 Pages. Sorry. Unfortunately, x, y, z are not so free.

**Category:** Mathematical Physics

[374] **viXra:1603.0377 [pdf]**
*replaced on 2016-03-30 20:22:38*

**Authors:** Moises Dominguez-Espinosa, Jaime Melendez-Martinez

**Comments:** 4 Pages. 3 Figures, Dynamical Systems, Quantum Mechanics.

There is a paradigm in Quantum Mechanics that explains quantization through normal vibration modes called Eigenstates that arise from Schrodinger wave equation. In this contribution we propose an alternative methodology of quantization by using basic concepts of mechanics and chaos from which a Toy Model is built.

**Category:** Mathematical Physics

[373] **viXra:1603.0377 [pdf]**
*replaced on 2016-03-29 08:02:39*

**Authors:** Moises Dominguez-Espinosa, Jaime Melendez-Martinez

**Comments:** 4 Pages. 3 Figures, Dynamical Systems, Quantum Mechanics

There is a paradigm in Quantum Mechanics that explains quantization
through normal vibration modes called Eigenstates that arise from Schrödinger wave
equation. In this contribution we propose an alternative methodology of quantization by
using basic concepts of mechanics and chaos from which a Toy Model is built.

**Category:** Mathematical Physics

[372] **viXra:1603.0371 [pdf]**
*replaced on 2016-04-15 17:31:25*

**Authors:** Jonathan Tooker

**Comments:** 1 Page. nice paper

Wick rotation produces numbers that agree with experiment and yet the method is mathematically wrong and not allowed by any self-consistent rule. We explore a small slice of wiggle room in complex analysis and show that it may be possible to use QFT without reliance on Wick rotations.

**Category:** Mathematical Physics

[371] **viXra:1603.0213 [pdf]**
*replaced on 2016-05-15 22:47:47*

**Authors:** Doug Jensen

**Comments:** 2 Pages.

There are many good reasons why we should replace pi with tau, this paper focuses on one definitive reason.

**Category:** Mathematical Physics

[370] **viXra:1603.0121 [pdf]**
*replaced on 2016-03-30 20:17:57*

**Authors:** CLaude Latourre

**Comments:** 8 Pages. en français

For more than a century, the equations of general relativity have evolved according to the observations of the universe. These changes are expressed through the cosmological constant (Λ), which was first added on the space-time part to account for a stationary universe, then removed when observed the evolution of it. More recently, the constant reappeared on the energy-momentum part to describe an accelerated expansion of the universe. Let's see now, how the contraction of the equations of general relativity can express exactly the value of the cosmological constant: Λ = -1/4 (R + κ T) and also to deduce an equivalent reformulation the equations of General Relativity: (Rµ√ -1/4 gµ√ R) = κ (Tµ√ -1/4 gµ√ T). All this, without using any physical concept: dark energy, Quintessence…

**Category:** Mathematical Physics

[369] **viXra:1603.0115 [pdf]**
*replaced on 2016-05-13 02:57:59*

**Authors:** Robert G Wallace

**Comments:** 48 pages

Manifolds of any dimension and signature have an associated Clifford algebra. Division algebras can be generated by repeated application of the Cayley-Dickson construction, which can be further extended to power associative algebras such as the sedenions. It is possible to apply the Cayley-Dickson construction to Clifford algebras. The series of algebras generated this procedure are referred to in this paper as kaotic algebras, and the notation Ka^n(p,q) is proposed to designate particular kaotic algebra obtained by $n$ applications of the Cayley-Dickson construction to a matrix group isomorphic to a Clifford algebra Cl(p,q). The Cayley table for Ka^3(1,4) is generated. This table has aspects suggesting that this algebra may have applications in physics.

**Category:** Mathematical Physics

[368] **viXra:1603.0115 [pdf]**
*replaced on 2016-04-19 20:36:50*

**Authors:** Rob Wallace

**Comments:** 25 Pages.

The direct product of a right and left handed quaternion algebras generates an algebra isomorphic to the Clifford algebra Cl(3,1) which describes space-time with signature (+++-). Once a higher dimensional background is proposed as an underlying basis for reality, it becomes logical to seek an equivalent product for octonions. However, the non-associativity of octonions means that a direct product is not defined for them. In this paper, a modified Moufang loop construction is used to generate an algebra based on products of octonions which differs from that of the octo-octonions, labelled the kaotic algebra. Subalgebras of the kaotic algebra can be found that correspond to several models of particle physics proposed by others.

**Category:** Mathematical Physics

[367] **viXra:1603.0115 [pdf]**
*replaced on 2016-04-18 01:48:40*

**Authors:** Rob Wallace

**Comments:** 25 Pages.

The direct product of a right and left handed quaternion algebras generates an algebra isomorphic to the Clifford algebra Cl(3,1) which describes space-time with signature (+++-). Once a higher dimensional background is proposed as an underlying basis for reality, it becomes logical to seek an equivalent product for octonions. However, the non-associativity of octonions means that a direct product is not defined for them. In this paper, a modified Moufang loop construction is used to generate an algebra based on products of octonions which differs from that of the octo-octonions, labelled the kaotic algebra. Subalgebras of the kaotic algebra can be found that correspond to several models of particle physics proposed by others.

**Category:** Mathematical Physics

[366] **viXra:1602.0167 [pdf]**
*replaced on 2016-05-28 11:19:59*

**Authors:** Robert Watson

**Comments:** 39 Pages. Some important corrections made in v2 using dimensional analysis

Kaluza's 1921 theory of gravity and electromagnetism using a fifth wrapped-up spatial dimension is inspiration for many modern attempts to develop new physical theories. The original theory has problems which may well be overcome, and thus Kaluza theory should be looked at again: it is a natural, if not necessary, geometric unification of gravity and electromagnetism. Here a general demonstration that the Lorentz force law can be derived from a range of Kaluza theories is presented. This is investigated via non-Maxwellian kinetic definitions of charge that are divergence-free and relate Maxwellian charge to 5D components of momentum. The possible role of torsion is considered as an extension. It is shown, however, that symmetric torsion components are likely not admissible in any prospective theory. As a result Kaluza's original theory is rehabilitated and a call for deeper analysis made.

**Category:** Mathematical Physics

[365] **viXra:1602.0122 [pdf]**
*replaced on 2016-07-19 15:17:16*

**Authors:** Hans Detlef Hüttenbach

**Comments:** 4 Pages.

A simple mathematical proof reveals that time-inversion symmetry and reversibility are different concepts, which also resolves the Loschmidt paradox.

**Category:** Mathematical Physics

[364] **viXra:1602.0122 [pdf]**
*replaced on 2016-02-12 14:46:47*

**Authors:** Hans Detlef Hüttenbach

**Comments:** 4 Pages.

A simple mathematical proof reveals that time inversion symmetry and reversibility are different concepts, which also resolves the Loschmidt paradox.

**Category:** Mathematical Physics

[363] **viXra:1602.0114 [pdf]**
*replaced on 2016-10-04 04:52:04*

**Authors:** Robert B. Easter

**Comments:** 184 Pages.

This paper introduces the G(4,8) Double Conformal Space-Time Algebra (DCSTA). G(4,8) DCSTA is a straightforward extension of the G(2,8) Double Conformal Space Algebra (DCSA), which is a different form of the G(8,2) Double Conformal / Darboux Cyclide Geometric Algebra (DCGA). G(4,8) DCSTA extends G(2,8) DCSA with spacetime boost operations and differential operators for differentiation with respect to the pseudospatial time w=ct direction and time t. The spacetime boost operation can implement anisotropic dilation (directed non-uniform scaling) of quadric surface entities. DCSTA is a high-dimensional 12D embedding of the G(1,3) Space-Time Algebra (STA) and is a doubling of the G(2,4) Conformal Space-Time Algebra (CSTA). The 2-vector quadric surface entities of the DCSA subalgebra appear in DCSTA as quadric surfaces at zero velocity that can be boosted into moving surfaces with constant velocities that display the length contraction effect of special relativity. DCSTA inherits doubled forms of all CSTA entities and versors. The doubled CSTA entities (standard DCSTA entities) include points, hypercones, hyperplanes, hyperpseudospheres, and other entities formed as their intersections, such as planes, lines, spatial spheres and circles, and spacetime hyperboloids (pseudospheres) and hyperbolas (pseudocircles). The doubled CSTA versors (DCSTA versors) include rotor, hyperbolic rotor (boost), translator, dilator, and their compositions such as the translated-rotor, translated-boost, and translated-dilator. The DCSTA versors provide a complete set of spacetime transformation operators on all DCSTA entities. DCSTA inherits the DCSA 2-vector spatial entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) and gains Darboux pseudocyclides formed in spacetime with the pseudospatial time dimension. All DCSTA entities can be reflected in, and intersected with, the standard DCSTA entities. To demonstrate G(4,8) DCSTA as concrete mathematics with possible applications, this paper includes sample code and example calculations using the symbolic computer algebra system SymPy.

**Category:** Mathematical Physics

[362] **viXra:1602.0114 [pdf]**
*replaced on 2016-07-28 13:07:54*

**Authors:** Robert B. Easter

**Comments:** 185 Pages.

This paper introduces the G(4,8) Double Conformal Space-Time Algebra (DCSTA). G(4,8) DCSTA is a straightforward extension of the G(2,8) Double Conformal Space Algebra (DCSA), which is a different form of the G(8,2) Double Conformal / Darboux Cyclide Geometric Algebra (DCGA). G(4,8) DCSTA extends G(2,8) DCSA with spacetime boost operations and differential operators for differentiation with respect to the pseudospatial time w=ct direction and time t. The spacetime boost operation can implement anisotropic dilation (directed non-uniform scaling) of quadric surface entities. DCSTA is a high-dimensional 12D embedding of the G(1,3) Space-Time Algebra (STA) and is a doubling of the G(2,4) Conformal Space-Time Algebra (CSTA). The 2-vector quadric surface entities of the DCSA subalgebra appear in DCSTA as quadric surfaces at zero velocity that can be boosted into moving surfaces with constant velocities that display the length contraction effect of special relativity. DCSTA inherits doubled forms of all CSTA entities and versors. The doubled CSTA entities (standard DCSTA entities) include points, hypercones, hyperplanes, hyperpseudospheres, and other entities formed as their intersections, such as planes, lines, spatial spheres and circles, and spacetime hyperboloids (pseudospheres) and hyperbolas (pseudocircles). The doubled CSTA versors (DCSTA versors) include rotor, hyperbolic rotor (boost), translator, dilator, and their compositions such as the translated-rotor, translated-boost, and translated-dilator. The DCSTA versors provide a complete set of spacetime transformation operators on all DCSTA entities. DCSTA inherits the DCSA 2-vector spatial entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) and gains Darboux pseudocyclides formed in spacetime with the pseudospatial time dimension. All DCSTA entities can be reflected in, and intersected with, the standard DCSTA entities. To demonstrate G(4,8) DCSTA as concrete mathematics with possible applications, this paper includes sample code and example calculations using the symbolic computer algebra system SymPy.

**Category:** Mathematical Physics

[361] **viXra:1602.0114 [pdf]**
*replaced on 2016-07-06 17:36:15*

**Authors:** Robert B. Easter

**Comments:** 111 Pages.

This paper introduces the G(4,8) Double Conformal Space-Time Algebra (DCSTA). G(4,8) DCSTA is a straightforward extension of the G(2,8) Double Conformal Space Algebra (DCSA), which is a different form of the G(8,2) Double Conformal / Darboux Cyclide Geometric Algebra (DCGA). G(4,8) DCSTA extends G(2,8) DCSA with spacetime boost operations and differential operators for differentiation with respect to the pseudospatial time w=ct direction and time t. The spacetime boost operation can implement anisotropic dilation (directed non-uniform scaling) of quadric surface entities. DCSTA is a high-dimensional 12D embedding of the G(1,3) Space-Time Algebra (STA) and is a doubling of the G(2,4) Conformal Space-Time Algebra (CSTA). The 2-vector quadric surface entities of the DCSA subalgebra appear in DCSTA as quadric surfaces at zero velocity that can be boosted into moving surfaces with constant velocities that display the length contraction effect of special relativity. DCSTA inherits doubled forms of all CSTA entities and versors. The doubled CSTA entities (standard DCSTA entities) include points, hypercones, hyperplanes, hyperpseudospheres, and other entities formed as their intersections, such as planes, lines, spatial spheres and circles, and spacetime hyperboloids (pseudospheres) and hyperbolas (pseudocircles). The doubled CSTA versors (DCSTA versors) include rotor, hyperbolic rotor (boost), translator, dilator, and their compositions such as the translated-rotor, translated-boost, and translated-dilator. The DCSTA versors provide a complete set of spacetime transformation operators on all DCSTA entities. DCSTA inherits the DCSA 2-vector spatial entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) and gains Darboux pseudocyclides formed in spacetime with the pseudospatial time dimension. All DCSTA entities can be reflected in, and intersected with, the standard DCSTA entities. To demonstrate G(4,8) DCSTA as concrete mathematics with possible applications, this paper includes sample code and example calculations using the symbolic computer algebra system SymPy.

**Category:** Mathematical Physics

[360] **viXra:1602.0114 [pdf]**
*replaced on 2016-02-24 01:21:52*

**Authors:** Robert B. Easter

**Comments:** 88 Pages.

This paper introduces the G(4,8) Double Conformal Space-Time Algebra (DCSTA). G(4,8) DCSTA is a straightforward extension of the G(8,2) Double Conformal / Darboux Cyclide Geometric Algebra (DCGA). G(4,8) DCSTA extends G(8,2) DCGA with spacetime boost operations and differential operators for differentiation with respect to the time-like w=ct direction and time t. The spacetime boost operation can implement anisotropic dilation (directed non-uniform scaling) of quadric surface entities. Quadric surface entities can be boosted into moving surfaces with constant velocities that display the length contraction effect of special relativity. To demonstrate G(4,8) DCSTA as concrete mathematics with possible applications, this paper includes sample code and example calculations using the symbolic computer algebra system SymPy.

**Category:** Mathematical Physics

[359] **viXra:1602.0010 [pdf]**
*replaced on 2016-02-05 04:29:56*

**Authors:** Zhang Yunfan

**Comments:** 36 Pages.

The theory of idealiscience is an accurate theoretical model, by the model we can deduce most important laws of Physics, explain a lot of physical mysteries, even a lot of basic and important philosophical questions. we can also get the theoretical values of a lot of physical constants, even some of the constants can not be deduced by traditional physical theories, such as neutron mass and magnetic moment,Avogadro constant and so on.

**Category:** Mathematical Physics

[358] **viXra:1601.0312 [pdf]**
*replaced on 2016-02-11 04:12:01*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 16 Pages. English and portuguese versions.

A solution to the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy. We have proved that there are initial velocities u^0 (x) and forces f(x,t) such that there is no physically reasonable solution to the Navier-Stokes equations for t>0, which corresponds to the case (C) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute.

**Category:** Mathematical Physics

[357] **viXra:1601.0312 [pdf]**
*replaced on 2016-02-05 04:43:59*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 16 Pages. English translation still incomplete.

A solution to the 6th millenium problem, respect to breakdown of Navier-Stokes solutions and the bounded energy. We have proved that there are initial velocities u^0 (x) and forces f(x,t) such that there is no physically reasonable solution to the Navier-Stokes equations for t>0, which corresponds to the case (C) of the problem relating to Navier-Stokes equations available on the website of the Clay Institute.

**Category:** Mathematical Physics

[356] **viXra:1601.0312 [pdf]**
*replaced on 2016-02-03 12:12:20*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 22 Pages. Sorry... example 2 wrong again.

**Category:** Mathematical Physics

[355] **viXra:1601.0312 [pdf]**
*replaced on 2016-02-02 08:17:02*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 20 Pages. Almost done (in portuguese).

**Category:** Mathematical Physics

[354] **viXra:1601.0312 [pdf]**
*replaced on 2016-01-31 15:05:53*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 20 Pages. Need yet revision and complete translation for english.

**Category:** Mathematical Physics

[353] **viXra:1601.0045 [pdf]**
*replaced on 2016-04-17 08:36:40*

**Authors:** Carsten S.P. Spanheimer

**Comments:** 26 Pages. In german language.

Die Einführung 'quaternionischer Differentialformen' auf dem Tangentialraum einer vierdimensionalen Mannigfaltigkeit ergibt ein vielversprechendes mathematisches System zur Beschreibung unserer physikalischen (3+1)-Raumzeit
schon mit einem Minimum von Grundannahmen.
Hier wird die Grundlage dieses Modells einer 'Quaternionischen Raumzeit' dargestellt.

**Category:** Mathematical Physics

[352] **viXra:1512.0334 [pdf]**
*replaced on 2016-02-26 01:37:13*

**Authors:** A. A. Frempong

**Comments:** 47 Pages. Copyright © by A. A. Frempong

Over nearly a year and half ago, the Navier-Stokes (N-S) equations in 3-D for incompressible fluid flow were analytically solved by the author. However, some of the solutions contained implicit terms. In this paper, the implicit terms have been expressed explicitly in terms of x, y, z and t. The author proposed and applied a new law, the law of definite ratio for incompressible fluid flow. This law states that in incompressible fluid flow, the other terms of the fluid flow equation divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. It was mathematically shown that without gravity forces on earth, there would be no incompressible fluid flow on earth as is known, and also, there would be no magnetohydrodynamics. In addition to the usual method of solving these equations, the N-S equations were also solved by a second method in which the three equations in the system were added to produce a single
equation which was then integrated. The solutions by the two methods were identical, except for the constants involved. Ratios were used to split-up the equations; and the resulting sub-equations were readily integrable; and even, the nonlinear sub-equations were readily integrated. The examples in the preliminaries show everyday examples on using ratios to divide a quantity into parts, as well as possible applications of the solution method in mathematics, science, engineering, business, economics, finance, investment and personnel management decisions. The x-direction Navier-Stokes equation was linearized, solved, and the solution analyzed. This solution was followed by the solution of the Euler equation of fluid flow. The Euler equation represents the nonlinear part of the Navier-Stokes equation. Following the Euler solution, the Navier-Stokes equation was solved essentially by combining the solutions of the linearized equation and the Euler solution. For the Navier-Stokes equati
on, the linear part of the relation obtained from the integration of the linear part of the equation satisfied the linear part of the equation; and the relation from the integration of the non-linear part satisfied the non-linear part of the equation. The solutions and relations revealed the role of each term of the Navier-Stokes equations in fluid flow. The gravity term is the indispensable term in fluid flow, and it is involved in the parabolic and forward motion of fluids. The pressure gradient term is also involved in the parabolic motion. The viscosity terms are involved in the parabolic, periodic and decreasingly exponential motion. Periodicity increases with viscosity. The variable acceleration term is also involved in the periodic and decreasingly exponential motion. The fluid flow in the Navier-Stokes solution may be characterized as follows. The x-direction solution consists of linear, parabolic, and hyperbolic terms. The first three terms characterize parabolas. If one ass
umes that in laminar flow, the axis of symmetry of each parabola for horizontal velocity flow profile is in the direction of fluid flow, then in turbulent flow, the axes of symmetry of some of the parabolas would be at right angles to that for laminar flow. The characteristic curve for the integral of the x-nonlinear term is such a parabola whose axis of symmetry is at right angles that of laminar flow. The integral of the y-nonlinear term is similar, parabolically, to that of the x-nonlinear term. The characteristic curve for the integral of the z-nonlinear term is a combination of two similar parabolas and a hyperbola. If the above x-direction flow is repeated simultaneously in the y-and z-directions, the flow is chaotic and consequently turbulent.
For a spin-off, the smooth solutions from above are specialized and extended to satisfy the requirements of the CMI Millennium Prize Problems, and thus prove the existence of smooth solutions of the Navier-Stokes equations.

**Category:** Mathematical Physics

[351] **viXra:1512.0334 [pdf]**
*replaced on 2016-02-22 03:06:28*

**Authors:** A. A. Frempong

**Comments:** 47 Pages. Copyright © A. A. Frempong

Over nearly a year and half ago, the Navier-Stokes (N-S) equations in 3-D for incompressible fluid flow were analytically solved by the author. However, some of the solutions contained implicit terms. In this paper, the implicit terms have been expressed explicitly in terms of x, y, z and t. The author proposed and applied a new law, the law of definite ratio for incompressible fluid flow. This law states that in incompressible fluid flow, the other terms of the fluid flow equation divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. It was mathematically shown that without gravity forces on earth, there would be no incompressible fluid flow on earth as is known, and also, there would be no magnetohydrodynamics. In addition to the usual method of solving these equations, the N-S equations were also solved by a second method in which the three equations in the system were added to produce a single equation which was then integrated. The solutions by the two methods were identical, except for the constants involved. Ratios were used to split-up the equations; and the resulting sub-equations were readily integrable; and even, the nonlinear sub-equations were readily integrated. The examples in the preliminaries show everyday examples on using ratios to divide a quantity into parts, as well as possible applications of the solution method in mathematics, science, engineering, business, economics, finance, investment and personnel management decisions. The x-direction Navier-Stokes equation was linearized, solved, and the solution analyzed. This solution was followed by the solution of the Euler equation of fluid flow. The Euler equation represents the nonlinear part of the Navier-Stokes equation. Following the Euler solution, the Navier-Stokes equation was solved essentially by combining the solutions of the linearized equation and the Euler solution. For the Navier-Stokes equation, the linear part of the relation obtained from the integration of the linear part of the equation satisfied the linear part of the equation; and the relation from the integration of the non-linear part satisfied the non-linear part of the equation. The solutions and relations revealed the role of each term of the Navier-Stokes equations in fluid flow. The gravity term is the indispensable term in fluid flow, and it is involved in the parabolic and forward motion of fluids. The pressure gradient term is also involved in the parabolic motion. The viscosity terms are involved in the parabolic, periodic and decreasingly exponential motion. Periodicity increases with viscosity. The variable acceleration term is also involved in the periodic and decreasingly exponential motion. The fluid flow in the Navier-Stokes solution may be characterized as follows. The x-direction solution consists of linear, parabolic, and hyperbolic terms. The first three terms characterize parabolas. If one assumes that in laminar flow, the axis of symmetry of each parabola for horizontal velocity flow profile is in the direction of fluid flow, then in turbulent flow, the axes of symmetry of some of the parabolas would be at right angles to that for laminar flow. The characteristic curve for the integral of the x-nonlinear term is such a parabola whose axis of symmetry is at right angles that of laminar flow. The integral of the y-nonlinear term is similar, parabolically, to that of the x-nonlinear term. The characteristic curve for the integral of the z-nonlinear term is a combination of two similar parabolas and a hyperbola. If the above x-direction flow is repeated simultaneously in the y-and z-directions, the flow is chaotic and consequently turbulent.
For a spin-off, the smooth solutions from above are specialized and extended to satisfy the requirements of the CMI Millennium Prize Problems, and thus prove the existence of smooth solutions of the Navier-Stokes equations.

**Category:** Mathematical Physics

[350] **viXra:1512.0334 [pdf]**
*replaced on 2015-12-16 22:28:55*

**Authors:** A. A. Frempong

**Comments:** 46 Pages. Copyright © A. A. Frempong

Over nearly a year and half ago, the Navier-Stokes (N-S) equations in 3-D for incompressible fluid flow were analytically solved by the author. However, some of the solutions contained implicit terms. In this paper, the implicit terms have been expressed explicitly in terms of x, y, z and t. The author proposed and applied a new law, the law of definite ratio for incompressible fluid flow. This law states that in incompressible fluid flow, the other terms of the fluid flow equation divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. It was mathematically shown that without gravity forces on earth, there would be no incompressible fluid flow on earth as is known, and also, there would be no magnetohydrodynamics. In addition to the usual method of solving these equations, the N-S equations were also solved by a second method in which the three equations in the system were added to produce a single equation which was then integrated. The solutions by the two methods were identical, except for the constants involved. Ratios were used to split-up the equations; and the resulting sub-equations were readily integrable, and even, the nonlinear sub-equations were readily integrated. The examples in the preliminaries show everyday examples on using ratios to divide a quantity into parts, as well as possible applications of the solution method in mathematics, science, engineering, business, economics, finance, investment and personnel management decisions. The x-direction Navier-Stokes equation was linearized, solved, and the solution analyzed. This solution was followed by the solution of the Euler equation of fluid flow. The Euler equation represents the nonlinear part of the Navier-Stokes equation. Following the Euler solution, the Navier-Stokes equation was solved essentially by combining the solutions of the linearized equation and the Euler solution. For the Navier-Stokes equation, the linear part of the relation obtained from the integration of the linear part of the equation satisfied the linear part of the equation; and the relation from the integration of the non-linear part satisfied the non-linear part of the equation. The solutions and relations revealed the role of each term of the Navier-Stokes equations in fluid flow. The gravity term is the indispensable term in fluid flow, and it is involved in the parabolic and forward motion of fluids. The pressure gradient term is also involved in the parabolic motion. The viscosity terms are involved in the parabolic, periodic and decreasingly exponential motion. Periodicity increases with viscosity. The variable acceleration term is also involved in the periodic and decreasingly exponential motion. The fluid flow in the Navier-Stokes solution may be characterized as follows. The x-direction solution consists of linear, parabolic, and hyperbolic terms. The first three terms characterize parabolas. If one assumes that in laminar flow, the axis of symmetry of the parabola for horizontal velocity flow profile is in the direction of fluid flow, then in turbulent flow, the axis of symmetry of the parabola would have been rotated 90 degrees from that for laminar flow. The characteristic curve for the x-nonlinear term is such a parabola whose axis of symmetry has been rotated 90 degrees from that of laminar flow. The y-nonlinear term is similar parabolically to the x-nonlinear term. The characteristic curve for the z-nonlinear term is a combination of two similar parabolas and a hyperbola. If the above x-direction flow is repeated simultaneously in the y-and z-directions, the flow is chaotic and consequently turbulent.
For a spin-off, the smooth solutions from above are specialized and extended to satisfy the requirements of the CMI Millennium Prize Problems, and thus prove the existence of smooth solutions of the Navier-Stokes equations.

**Category:** Mathematical Physics

[349] **viXra:1512.0333 [pdf]**
*replaced on 2015-12-16 22:38:15*

**Authors:** A. A. Frempong

**Comments:** 10 Pages. Copyright © A. A. Frempong

The system of magnetohydrodynamic (MHD) equations has been solved analytically in this paper. The author applied the technique used in solving the Navier-Stokes equations and applied a new law, the law of definite ratio for MHD. This law states that in MHD, the other terms of the system of equations divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. It is shown that without gravity forces on earth, there would be no magnetohydrodynamics on earth as is known. The equations in the system of equations were added to produce a single equation which was then integrated. Ratios were used to split-up this single equation into sub-equations which were readily integrated, and even, the non-linear sub-equations were readily integrated. Twenty-seven sub-equations were integrated. The linear part of the relation obtained from the integration of the linear part of the equation satisfied the linear part of the equation; and the relation from the integration of the non-linear part satisfied the non-linear part of the equation. The solutions revealed the role of each term in magnetohydrodynamics. In particular, the gravity term is the indispensable term in magnetohydrodynamics. The solutions of the MHD equations were compared with the solutions of the N-S equations, and there were similarities and dissimilarities.

**Category:** Mathematical Physics

[348] **viXra:1512.0332 [pdf]**
*replaced on 2016-04-18 01:01:43*

**Authors:** A. A. Frempong

**Comments:** 12 Pages. Copyright © A. A. Frempong

This paper covers the solutions of the Euler equations in 3-D and 4-D for incompressible fluid flow. The solutions are the spin-offs of the author's previous analytic solutions of the Navier-Stokes equations (vixra:1405.0251 of 2014). However, some of the solutions contained implicit terms. In this paper, the implicit terms have been expressed explicitly in terms of x, y, z and t.
The author applied a new law, the law of definite ratio for fluid flow. This law states that in incompressible fluid flow, the other terms of the fluid flow equation divide the gravity term in a definite ratio, and each term utilizes gravity to function. The sum of the terms of the ratio is always unity. This law evolved from the author's earlier solutions of the Navier-Stokes equations. In addition to the usual approach of solving these equations, the Euler equations have also been solved by a second method in which the three equations in the system are added to produce a single equation which is then integrated. The solutions by the two approaches are identical, except for the constants involved. From the experience gained in solving the linearized Navier-Stokes equations, only the equation with the gravity term as the subject of the equation was integrated. The experience was that when each of the terms of the Navier-Stokes equation was used as the subject of the equation, only the equation with the gravity term as the subject of the equation produced a solution. Ratios were used to split-up the x-direction Euler equation with the gravity term as the subject of the equation. The resulting five sub-equations were readily integrable, and even, the non-linear sub-equations were readily integrated. The integration results were combined. The combined results satisfied the corresponding equation. This equation which satisfied its corresponding equation would be defined as the driver equation; and each of the other equations which would not satisfy its corresponding equation would be called a supporter equation. A supporter equation does not satisfy its corresponding equation completely, but provides useful information which is not apparent in the solution of the driver equation. The solutions and relations revealed the role of each term of the Euler equations in fluid flow. The gravity term is the indispensable term in fluid flow, and it is involved in the forward motion of fluids. The pressure gradient term is also involved in the forward motion. The variable acceleration term is also involved in the forward motion. The fluid flow behavior in the Euler solution may be characterized as follows. The x-direction solution consists of linear, parabolic, and hyperbolic terms. If one assumes that in laminar flow, the axis of symmetry of the parabola for horizontal velocity flow profile is in the direction of fluid flow, then in turbulent flow, the axis of symmetry of the parabola would have been rotated 90 degrees from that for laminar flow. The characteristic curve for the x-nonlinear term is such a parabola whose axis of symmetry has been rotated 90 degrees from that of laminar flow. The y-nonlinear term is similar parabolically to the x-nonlinear term. The characteristic curve for the z-nonlinear term is a combination of two similar parabolas and a hyperbola. If the above x-direction flow is repeated simultaneously in the y-and z-directions, the flow is chaotic and consequently turbulent.

**Category:** Mathematical Physics