[7] **viXra:1302.0154 [pdf]**
*replaced on 2014-05-30 13:13:59*

**Authors:** Branko Zivlak

**Comments:** 7 Pages. 16 formulas, 1 table

In "The Code of Nature" [1] Helmut Söllinger gives Carl Friedrich von Weizsäcker's (1912-2007) the relationship between the fundamental physical constants. The first section examines the relationship between the mass of protons and electrons, and the fundamental physical constants. The second is about the Weizsacker's assumption of proportionality between the Planck length, Compton wavelength and radius of the Universe. In the third section, I try to explain, the previous relationships, in light of the attractive-repulsive forces of the Ruđer Bošković (1711-1787), the earliest founder of quantum theory. This version is written on the languages of Bošković’s father and mother.

**Category:** Mathematical Physics

[6] **viXra:1302.0132 [pdf]**
*submitted on 2013-02-19 21:26:30*

**Authors:** Zhanqiang Bai, Guowu Meng, Erxiao Wang

**Comments:** 13 Pages.

It is demonstrated that, for the recently introduced classical magnetized Kepler problems in dimension $2k+1$, the non-colliding orbits in the ``external configuration space" $\mathbb R^{2k+1}\setminus\{\mathbf 0\}$ are all conics, moreover, a conic orbit is an ellipse, a parabola, and a branch of a hyperbola according as the total energy is negative, zero, and positive. It is also demonstrated that the Lie group ${\mr {SO}}^+(1,2k+1)\times {\bb R}_+$ acts transitively on both the set of oriented elliptic orbits and the set of oriented parabolic orbits.

**Category:** Mathematical Physics

[5] **viXra:1302.0112 [pdf]**
*submitted on 2013-02-17 02:09:57*

**Authors:** M. Pitkanen

**Comments:** 7 Pages.

I have been reading two highly interesting articles by Robert Kiehn. There are very many contacts on TGD inspired vision and its open interpretational problems.
The notion of Falaco soliton has surprisingly close resemblance with K\"ahler magnetic flux tubes defining fundamental structures in TGD Universe. Fermionic strings are also fundamental structures of TGD accompanying magnetic flux tubes and this supports the vision that these string like objects could allow reduction of various condensed matter phenomena such as sound waves -usually regarded as emergent phenomena allowing only highly phenomenological description - to the fundamental microscopic level in TGD framework. This can be seen as the basic outcome of this article.
Kiehn proposed a new description for the generation of various instability patterns of hydrodynamics flows (Kelvin-Helmholtz and Rayleigh-Taylor instabilities) in terms of hyperbolic dynamics so that a connection with wave phenomena like interference and diffraction would emerge. The role of characteristic surfaces as surfaces of tangential and also normal discontinuities is central for the approach. In TGD framework the characteristic surfaces have as analogs light-like wormhole throats at which the signature of the induced 4-metric changes and these surfaces indeed define boundaries of two phases and of material objects in general. This inspires a more detailed comparison of Kiehn's approach with TGD.

**Category:** Mathematical Physics

[4] **viXra:1302.0110 [pdf]**
*submitted on 2013-02-17 02:17:26*

**Authors:** M. Pitkanen

**Comments:** 40 Pages.

The original focus of this article was p-adic icosahedron. The discussion of attempt to define this notion however leads to the challenge of defining the concept of p-adic sphere, and more generally, that of p-adic manifold, and this problem soon became the main target of attention since it is one of the key challenges of also TGD.
There exists two basic philosophies concerning the construction of both real and p-adic manifolds: algebraic and topological approach. Also in TGD these approaches have been competing: algebraic approach relates real and p-adic space-time points by identifying the common rationals. Finite pinary cutoff is however required to achieve continuity and has interpretation in terms of finite measurement resolution. Canonical identification maps p-adics to reals and vice versa in a continuous manner but is not consistent with p-adic analyticity nor field equations unless one poses a pinary cutoff. It seems that pinary cutoff reflecting the notion of finite measurement resolution is necessary in both approaches. This represents a new notion from the point of view of mathematics.
a) One can try to generalize the theory of real manifolds to p-adic context. The basic problem is that p-adic balls are either disjoint or nested so that the usual construction by gluing partially overlapping spheres fails. This leads to the notion of Berkovich disk obtained as a completion of p-adic disk having path connected topology (non-ultrametric) and containing p-adic disk as a dense subset. This plus the complexity of the construction is heavy price to be paid for path-connectedness. A related notion is Bruhat-Tits tree defining kind of skeleton making p-adic manifold path connected. The notion makes sense for the p-adic counterparts of projective spaces, which suggests that p-adic projective spaces (S^{2} and CP_{2} in TGD framework) are physically very special.
b) Second approach is algebraic and restricts the consideration to algebraic varieties for which also topological invariants have algebraic counterparts. This approach looks very natural in TGD framework - at least for imbedding space. Preferred extremals of Kähler action can be characterized purely algebraically - even in a manner independent of the action principle - so that they might make sense also p-adically.
Number theoretical universality is central element of TGD. Physical considerations force to generalize the number concept by gluing reals and various p-adic number fields along rationals and possible common algebraic numbers. This idea makes sense also at the level of space-time and of "world of classical worlds" (WCW).
Algebraic continuation between different number fields is the key notion. Algebraic continuation between real and p-adic sectors takes place along their intersection which at the level of WCW correspond to surfaces allowing interpretation both as real and p-adic surfaces for some value(s) of prime p. The algebraic continuation from the intersection of real and p-adic WCWs is not possible for all p-adic number fields. For instance, real integrals as functions of parameters need not make sense for all p-adic number fields. This apparent mathematical weakness can be however turned to physical strength: real space-time surfaces assignable to elementary particles can correspond only some particular p-adic primes. This would explain why elementary particles are characterized by preferred p-adic primes. The p-adic prime determining the mass scale of the elementary particle could be fixed number theoretically rather than by some dynamical principle formulated in real context (number theoretic anatomy of rational number does not depend smoothly on its real magnitude!).
Although Berkovich construction of p-adic disk does not look promising in TGD framework, it suggests that the difficulty posed by the total disconnectedness of p-adic topology is real. TGD in turn suggests that the difficulty could be overcome without the completion to a non-ultrametric topology. Two approaches emerge, which ought to be equivalent.
a) The TGD inspired solution to the construction of path connected effective p-adic topology is based on the notion of canonical identification mapping reals to p-adics and vice versa in a continuous manner. The trivial but striking observation was that canonical identification satisfies triangle inequality and thus defines an Archimedean norm allowing to induce real topology to p-adic context. Canonical identification with finite measurement resolution defines chart maps from p-adics to reals and vice versa and preferred extremal property allows to complete the discrete image to hopefully space-time surface unique within finite measurement resolution so that topological and algebraic approach are combined. Finite resolution would become part of the manifold theory. p-Adic manifold theory would also have interpretation in terms of cognitive representations as maps between realities and p-adicities.
b) One can ask whether the physical content of path connectedness could be also formulated as a quantum physical rather than primarily topological notion, and could boil down to the non-triviality of correlation functions for second quantized induced spinor fields essential for the formulation of WCW spinor structure. Fermion fields and their n-point functions could become part of a number theoretically universal definition of manifold in accordance with the TGD inspired vision that WCW geometry - and perhaps even space-time geometry - allow a formulation in terms of fermions. This option is a mere conjecture whereas the first one is on rigorous basis.

**Category:** Mathematical Physics

[3] **viXra:1302.0095 [pdf]**
*replaced on 2013-07-09 00:23:49*

**Authors:** Hosein Nasrolahpour

**Comments:** 5 Pages.

The fractional variational principle represents an important part of fractional calculus and has
found many applications. There are several versions of fractional variational principles and so
different kinds of fractional Euler-Lagrange equations. In this paper, we propose the fractional
sine-Gordon Lagrangian density. Then using the fractional Euler-Lagrange equations we obtain
fractional sine-Gordon equation.

**Category:** Mathematical Physics

[2] **viXra:1302.0059 [pdf]**
*replaced on 2014-12-27 16:00:06*

**Authors:** Sergey V. Ershkov

**Comments:** 8 Pages. Keywords: Helmholtz equation, paraxial approximation, Gaussian beam

A new type of exact solutions of the full 3 dimensional spatial Helmholtz equation for the case of non-paraxial Gaussian beams is presented here.
We consider appropriate representation of the solution for Gaussian beams in a spherical coordinate system by substituting it to the full 3 dimensional spatial Helmholtz Equation.
Analyzing the structure of the final equation, we obtain that governing equations for the components of our solution are represented by the proper Riccati equations of complex value, which has no analytical solution in general case.
But we find one of the possible exact solution which is proved to satisfy to such an equations for Gaussian beams.

**Category:** Mathematical Physics

[1] **viXra:1302.0020 [pdf]**
*submitted on 2013-02-04 05:20:58*

**Authors:** Chun-Xuan Jiang

**Comments:** 13 Pages.

We make the new periodic table of elements 1-92

**Category:** Mathematical Physics