[9] **viXra:1310.0235 [pdf]**
*replaced on 2014-04-04 09:01:45*

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

**Comments:** 20 Pages.

A framework for calculations in a semi-Riemannian space with the typical metric connection and curvature expressions is developed, with an emphasis on deriving them from an embedding function as a more fundamental object than the metric tensor.
The scale-invariant and 'linearizing' logarithmic nature of an 'infinitesimal embedding' of a tangent space into its neighbourhood is observed, and a composition scheme of spacetime scenarios from 'outer' non-invariant and 'inner' scale-invariant embeddings is briefly outlined.

**Category:** Mathematical Physics

[8] **viXra:1310.0234 [pdf]**
*submitted on 2013-10-26 11:40:40*

**Authors:** Solomon I. Khmelnik

**Comments:** Pages. The rules Clay Mathematics Institute written as follows: "In the case of … the Navier-Stokes problem, the SAB will consider the award of the Millennium Prize for deciding the question in either direction." I suggest my decision of this problem.

In this book we formulate and prove the variational extremum principle for viscous incompressible and compressible fluid, from which principle follows that the Navier-Stokes equations represent the extremum conditions of a certain functional. We describe the method of seeking solution for these equations, which consists in moving along the gradient to this functional extremum. We formulate the conditions of reaching this extremum, which are at the same time necessary and sufficient conditions of this functional global extremum existence.
Then we consider the so-called closed systems. We prove that for them the necessary and sufficient conditions of global extremum for the named functional always exist. Accordingly, the search for global extremum is always successful, and so the unique solution of Naviet-Stokes is found.
We contend that the systems described by Navier-Stokes equations with determined boundary solutions (pressure or speed) on all the boundaries, are closed systems. We show that such type of systems include systems bounded by impermeable walls, by free space under a known pressure, by movable walls under known pressure, by the so-called generating surfaces, through which the fluid flow passes with a known speed.
The book is supplemented by open code programs in the MATLAB system – functions realizing the calculation method and test programs. Links on test programs are given in the text of the book when the examples are described. The programs may be obtained from the author by request at solik@netvision.net.il

**Category:** Mathematical Physics

[7] **viXra:1310.0231 [pdf]**
*submitted on 2013-10-25 21:29:54*

**Authors:** Jian-zhong Zhao

**Comments:** 17 Pages.

Mathematical provability , then classification, of Saint-Venant's Principle are discussed. Beginning with the simplest case of Saint-Venant's Principle, four problems of elasticity are discussed mathematically. It is concluded that there exist two categories of elastic problems concerning Saint-Venant's Principle: Experimental Problems, whose Saint-Venant's Principle is established in virtue of supporting experiment, and Analytical Problems, whose Saint-Venant's decay is proved or disproved mathematically, based on fundamental equations of linear elasticity. The boundary-value problems whose stress boundary condition consists of Dirac measure, a "singular distribution ", can not be dealt with by the mathematics of elasticity for " proof " or "disproof " of their Saint-Venant's decay, in
terms of mathematical coverage.

**Category:** Mathematical Physics

[6] **viXra:1310.0191 [pdf]**
*replaced on 2016-02-07 07:56:52*

**Authors:** Malcolm Macleod

**Comments:** 3 Pages.

In this essay I look at a simple geometrical model for an expanding black-hole universe with a contracting white-hole universe twin. Discrete Planck unit 'drops' are transferred in integer steps from the white-hole universe to the black-hole universe thereby forcing an expansion of the black-hole universe and giving a rationale for Planck time, the arrow of time, the speed of light, dark energy and dark matter. Presuming that universe expansion will cease when the universe reaches absolute zero gives a cosmological constant Omega_u = 1.0137x10^{123} (units of Planck time). Selected related cosmic microwave background CMB parameters calculated using age in units of Planck time give a best fit for a 14.643 billion year old black hole.

**Category:** Mathematical Physics

[5] **viXra:1310.0165 [pdf]**
*submitted on 2013-10-17 04:11:02*

**Authors:** Claude Michael Cassano

**Comments:** 17 Pages.

A soliton solution of the Klein-Gordon equation, ke^{-|m⋅r|} , consistent with the mass-generalized Maxwell's equations and the preon foundation for the fermions is presented, proved and discussed.

**Category:** Mathematical Physics

[4] **viXra:1310.0146 [pdf]**
*submitted on 2013-10-16 11:16:19*

**Authors:** Marcos Georgallides

**Comments:** 15 Pages.

Lagrange equation of motion for a single point ( Primary Point A is the only Space ) , states that this point must move from the Initial Position A to another position say B . This Equilibrium for points A and B , presupposes in Mechanics the Principle of Virtual Displacements and the work done is W = ∫ P.ds = 0 , or when ds = distance AB then →
[ ds .( PA + P B ) = 0 ] ...(1).. From Equation (1) are self created all the Spaces [S] the equilibrium Anti-Spaces [AS] and the Sub-Spaces [SS] with infinite points in them and with a finite work on it . Monad dš (dipole AB) is a complex number of type [ dš = z = x+i.y ] ..(1a) representing the real part (x) , the distance AB , and imaginary parts (i.y) which is the work of …(1) . Complex number , z , the first dimentional unit AB , is such that either repeated by itself as monad ( z• = z.z.z.z. w-times ) or repeated times itself in monad ( ⁿ√z = z /ⁿ = z• , z/ⁿ.z/ⁿ.z/ⁿ….w = 1/n-times , or the nth roots of z equal to w = 1/n ) remains unaltered forming Spaces ( z•) , Anti-spaces( - z•) and the inversing Sub-spaces (ⁿ•z) , meaning that , unit circle is mapped on itself simultaneously on the two bases , 1 and n=1/w , where w.n = 1. This duality of coexistance on AB [ the w.th power and the n.th root of z where w.n =1 ] presupposes a common base ,m, which creates this unit polynomial exponentiation . Analysing this exponentiation according to one of the four basic properties of logs then becomes →
log.w(1= w.n) = log.w(w)+log.w(n=1/w) = 1+1/w = 1+ n
and it is the base of natural logarithms e and since 1= w.n
then → ( 1+ n )• = (1+1/w)• = constant = m = e ← ....(2) Since the first dimentional unit AB is a complex number with many imaginary parts (and this because of the infinite variables) then this unit has the general type of quaternion .i.e.
m^±(ª+₫.i) = q• = (Tq)•.[cos.wφ + ε.sin.wφ] ……where
m = lim(1+1/w)• for w = 1→ ∞ , q = z = ± ( x+y.i )
sinφ = y/•x²+y² , cosφ = x/•x²+y² , |z| = •x²+y²,
Tq = • x²+y1²+y2²+ ….yn² , Ty = • y1²+y2²+ ….yn²
ε = (y.i/Ty)=[y.i ] / [Ty]=(y1.a1+y2.a2+.)/(• y1²+y2²+yn²)
[PNS] ↔ quaternion ↔ [ dŝ = x+i.y ] is a Vector with two components , the one x is the only Space with Scalar Potential field Φo , which is only half lengths of Space , Anti-Space , ( the longitudinal position ) , (x) → (-x) straight line connecting Space [S] , Anti-Space [AS] in [PNS] and in it exist , the initial Work , or Impulse , bounded on points which cannot be created or destroyed which is analogous to the (x) magnitude , and the other one y is the infinite local curl fields So , due to the Spin which is the intrinsic rotation of the Space and Anti-Space . Because in [S] and [AS] forces PA - PB are acting in the same straight line so moment lever is zero ( 0 ) , therefore Primary [S] and [AS] are ir-rotational and so it is possible to express this Primary field as a scalar function (Φo). This shows that [PNS] is a Space Work or Space- Spin or < Space Energy Existence > , where Time is not existing , because Φo and So are not time-varying . The same is holding also for the infinite dipole AiBi which are also complex numbers with all their properties , that of quaternions .
Because quaternion properties are wrapped in lower and higher dimensions only by rotation , this is the property of spaces , so all dipole AnBn may have commons , which may bleed off in any Space , a very useful device for Quantum-mechanics . Geometrically states that , this property of commons allows to the dipole AnBn or to Spaces ↔ [ dŝ = xn+i.yn ] , to be also a Space -Time existence wrapped in the , Space-Energy Existence because of ,
Operation ↔ Quaternion → Notation m^±(ª+₫.i) = q• → = (Tq)•. [cos.wφ + ε.sin.wφ]
Scatters , Part or all Content of the quaternion q , in all Spaces and Sub-Spaces as q•
[ i.e. The duality of coexistence ↔ of the content of dŝ , from w.th power to the n.th root of q , where w.n =1 , is the measuring of , drag areas and other trapped accumulator , and by rotation to convert them in Spaces ]. An extend analysis of this unification follows in [23]

**Category:** Mathematical Physics

[3] **viXra:1310.0082 [pdf]**
*replaced on 2013-12-05 01:12:07*

**Authors:** Sergey V. Ershkov

**Comments:** 7 Pages. Keywords: Navier-Stokes equations, self-similar solutions, incompressible flow

A new classification of self-similar solutions of the Navier-Stokes system of equations is presented here.
We consider equations of motion for incompressible flow (of Newtonian fluids) in the curl rotating co-ordinate system. Then the equation of momentum should be split into the sub-system of 2 equations: an irrotational (curl-free) one, and a solenoidal (divergence-free) one.
The irrotational (curl-free) equation used for obtaining of the components of pressure gradient. As a term of such an equation, we used the irrotational (curl-free) vector field of flow velocity, which is given by the proper potential (besides, the continuity equation determines such a potential as a harmonic function).
As for solenoidal (divergence-free) equation, the transition from Cartesian to curl rotating co-ordinate system transforms equation of motion to the Helmholtz vector differential equation for time-dependent self-similar solutions. The Helmholtz differential equation can be solved by separation of variables in only 11 coordinate systems, so it forms a complete set of all possible cases of self-similar solutions for Navier-Stokes system of equations.

**Category:** Mathematical Physics

[2] **viXra:1310.0081 [pdf]**
*submitted on 2013-10-12 16:06:14*

**Authors:** Ali R. Hadjesfandiari

**Comments:** 39 Pages.

By examining the theory of relativity, as originally proposed by Lorentz and Poincaré, the fundamental relationship between space-time and matter is discovered, thus completing the theory of relativity and electrodynamics. As a result, the four-dimensional theory of general motion and the four-dimensional vortex theory of interaction are developed. It is seen that the electromagnetic four-vector potential and strength fields are the four-dimensional velocity and vorticity fields, respectively. Furthermore, the four-vector electric current density is proportional to the four-dimensional mean curvature of the four-vector potential field. This is the fundamental geometrical theory of electromagnetism, which determines the origin of electromagnetic interaction and clarifies some of the existing ambiguities. Interestingly, the governing geometry of motion and interaction is non-Euclidean.

**Category:** Mathematical Physics

[1] **viXra:1310.0021 [pdf]**
*submitted on 2013-10-03 14:27:31*

**Authors:** Thomas Günther

**Comments:** 3 Pages.

This article presents a new exact solution of Einstein’s equations with cosmological constant,
which includes de Sitter’s metric as a special case. The generalized solution admits a nonzero stress energy
momentum tensor. The second section is concerned with a transformation of the line element into a spherical
symmetric but anisotropic form.

**Category:** Mathematical Physics