[12] **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

[11] **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

[10] **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

[9] **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

[8] **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

[7] **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

[6] **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

[5] **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

[4] **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

[3] **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

[2] **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

[1] **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