Mathematical Physics

1701 Submissions

[9] viXra:1701.0679 [pdf] submitted on 2017-01-30 21:21:09

Metamorphic Topological Schemes

Authors: Miguel A. Sanchez-Rey
Comments: 2 Pages.

Establish topological schemes in metamorphic space as A-scheme and B-scheme.
Category: Mathematical Physics

[8] viXra:1701.0653 [pdf] submitted on 2017-01-28 10:44:20

Advance on Electron Deep Orbits of the Hydrogen Atom

Authors: J.L.Paillet, A.Meulenberg
Comments: 12 Pages.

In previous works, we discussed arguments for and against the deep orbits, as exemplified in published solutions. So we considered the works of Maly and Va’vra on the topic, the most complete solution available and one showing an infinite family of EDO solutions. In particular, we deeply analyzed their 2nd of these papers, where they consider a finite nucleus and look for solutions with a Coulomb potential modified inside the nucleus. In the present paper, we quickly recall our analysis, verification, and extension of their results. Moreover, we answer to a recent criticism that the EDOs would represent negative energy states and therefore would not qualify as an answer to the questions posed by Cold Fusion results. We can prove, by means of a simple algebraic argument based on the solution process, that, while at the transition region, the energy of the EDOs are positive. Next, we deepen the essential role of Special Relativity as source of the EDOs, which we discussed in previous papers. But the central topic of our present study is an initial analysis of the magnetic interactions near the nucleus, with the aim of solving important physical questions: do the EDOs satisfy the Heisenberg Uncertainty relation (HUR)? Are the orbits stable? So, we examine some works related to the Vigier-Barut Model, with potentials including magnetic coupling. We also carried out approximate computations to evaluate the strength of these interactions and the possibilities of their answering some of our questions. As a first result, we can expect the HUR to be respected by EDOs, due to the high energies of the magnetic interactions near the nucleus. Present computations for stability do not yet give a plain result; we need further studies and tools based on QED to face the complexity of the near-nuclear region. For the creation of EDOs, we outline a possibility based on magnetic coupling.
Category: Mathematical Physics

[7] viXra:1701.0651 [pdf] submitted on 2017-01-28 08:06:57

Double Conformal Space-Time Algebra (ICNPAA 2016)

Authors: Robert B. Easter, Eckhard Hitzer
Comments: 10 pages. In proceedings: S. Sivasundaram (ed.), International Conference in Nonlinear Problems in Aviation and Aerospace ICNPAA 2016, AIP Conf. Proc., Vol. 1798, 020066 (2017); doi: 10.1063/1.4972658. 4 color figures.

The Double Conformal Space-Time Algebra (DCSTA) is a high-dimensional 12D Geometric Algebra G(4,8) that extends the concepts introduced with the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA) G(8,2) with entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) in spacetime with a new boost operator. The base algebra in which spacetime geometry is modeled is the Space-Time Algebra (STA) G(1,3). Two Conformal Space-Time subalgebras (CSTA) G(2,4) provide spacetime entities for points, flats (incl. worldlines), and hyperbolics, and a complete set of versors for their spacetime transformations that includes rotation, translation, isotropic dilation, hyperbolic rotation (boost), planar reflection, and (pseudo)spherical inversion in rounds or hyperbolics. The DCSTA G(4,8) is a doubling product of two G(2,4) CSTA subalgebras that inherits doubled CSTA entities and versors from CSTA and adds new bivector entities for (pseudo)quadrics and Darboux (pseudo)cyclides in spacetime that are also transformed by the doubled versors. The "pseudo" surface entities are spacetime hyperbolics or other surface entities using the time axis as a pseudospatial dimension. The (pseudo)cyclides are the inversions of (pseudo)quadrics in rounds or hyperbolics. An operation for the directed non-uniform scaling (anisotropic dilation) of the bivector general quadric entities is defined using the boost operator and a spatial projection. DCSTA allows general quadric surfaces to be transformed in spacetime by the same complete set of doubled CSTA versor (i.e., DCSTA versor) operations that are also valid on the doubled CSTA point entity (i.e., DCSTA point) and the other doubled CSTA entities. The new DCSTA bivector entities are formed by extracting values from the DCSTA point entity using specifically defined inner product extraction operators. Quadric surface entities can be boosted into moving surfaces with constant velocities that display the length contraction effect of special relativity. DCSTA is an algebra for computing with quadrics and their cyclide inversions in spacetime. For applications or testing, DCSTA G(4,8) can be computed using various software packages, such as Gaalop, the Clifford Multivector Toolbox (for MATLAB), or the symbolic computer algebra system SymPy with the GAlgebra module.
Category: Mathematical Physics

[6] viXra:1701.0540 [pdf] submitted on 2017-01-18 19:08:15

Solving Partial Differential Equations by Self-Generated Stochasticity.

Authors: Michail Zak
Comments: 7 Pages.

New physical principle for simulations of PDE has been introduced. It is based upon replacing the PDE to be solved by the system of ODE for which the PDE represents the corresponding Liouville equation. The proposed approach has a polynomial (rather than exponential) algorithmic complexity, and it is applicable to nonlinear parabolic, hyperbolic, and elliptic PDE.
Category: Mathematical Physics

[5] viXra:1701.0533 [pdf] submitted on 2017-01-18 05:02:17

A Note on the Gravitational Equations Analogous to Maxwell's Electromagnetic Equations.

Authors: J. Dunning-Davies, J. P. Dunning-Davies
Comments: 7 Pages.

Ever since Oliver Heaviside's suggestion of the possible existence of a set of equations, analogous to Maxwell's equations for the electromagnetic field, to describe the gravitational field, others have considered and built on the original notion. However, if such equations do exist and really are analogous to Maxwell's electromagnetic equations, new problems could arise related to presently accepted notions concerning special relativity. This note, as well as offering a translation of a highly relevant paper by Carstoiu, addresses these concerns in the same manner as similar concerns regarding Maxwell's equations were.
Category: Mathematical Physics

[4] viXra:1701.0523 [pdf] replaced on 2017-04-26 09:09:52

Draft Introduction to Abstract Kinematics

Authors: Grushka Ya.I.
Comments: 208 Pages. Mathematics Subject Classification: 03E75; 70A05; 83A05; 47B99. DOI: 10.13140/RG.2.2.24968.62720

This work lays the foundations of the theory of kinematic changeable sets ("abstract kinematics"). Theory of kinematic changeable sets is based on the theory of changeable sets. From an intuitive point of view, changeable sets are sets of objects which, unlike elements of ordinary (static) sets, may be in the process of continuous transformations, and which may change properties depending on the point of view on them (that is depending on the reference frame). From the philosophical and imaginative point of view the changeable sets may look like as "worlds" in which evolution obeys arbitrary laws. Kinematic changeable sets are the mathematical objects, consisting of changeable sets, equipped by different geometrical or topological structures (namely metric, topological, linear, Banach, Hilbert and other spaces). In author opinion, theories of changeable and kinematic changeable sets (in the process of their development and improvement), may become some tools of solving the sixth Hilbert problem at least for physics of macrocosm. Investigations in this direction may be interesting for astrophysics, because there exists the hypothesis, that in the large scale of Universe, physical laws (in particular, the laws of kinematics) may be different from the laws, acting in the neighborhood of our solar System. Also these investigations may be applied for the construction of mathematical foundations of tachyon kinematics. We believe, that theories of changeable and kinematic changeable sets may be interesting not only for theoretical physics but also for other fields of science as some, new, mathematical apparatus for description of evolution of complex systems.
Category: Mathematical Physics

[3] viXra:1701.0309 [pdf] replaced on 2017-02-04 13:13:24

Inversions And Invariants Of Space And Time

Authors: Hans Detlef Hüttenbach
Comments: 5 Pages.

This paper is on the mathematical structure of space, time, and gravity. It is shown that electrodynamics is neither charge inversion invariant, nor is it time inversion invariant.
Category: Mathematical Physics

[2] viXra:1701.0299 [pdf] replaced on 2017-01-15 11:21:14

A Child's Guide to Spinors

Authors: William O. Straub
Comments: 13 Pages. Finalized, with typos fixed in Equations (6.2.2) and (6.3.2)

A very elementary overview of the spinor concept, intended as a guide for undergraduates.
Category: Mathematical Physics

[1] viXra:1701.0166 [pdf] submitted on 2017-01-03 13:20:03

Prolate Spheroidal Wave Function as Exact Solution of the Schrödinger Equation

Authors: J. Akande, D. K. K. Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou, M. D. Monsia
Comments: 6 pages

In quantum mechanics, the wave function and energy are required for the complete characterization of fundamental properties of a physical system subject to a potential energy. It is proved in this work, the existence of a Schrödinger equation with position-dependent mass having the prolate spheroidal wave function as exact solution, resulting from a classical quadratic Liénard-type oscillator equation. This fact may allow the extension of the current one-dimensional model to three dimensions and increase the understanding of analytical features of quantum systems.
Category: Mathematical Physics