Mathematical Physics

Particle Mass Ratios

Authors: D.T. Froedge
Comments: 11 pages 20 equations 240kb

Based on the developments in a previous paper, this paper presents straightforward explanation of particle mass ratios, and the specific values for some well known particles. The additional nuclear modes postulated, are similar to the Schrödinger modes in the atom, and, though speculative, the mass ratios calculated for elementary particles are very close the observed mass ratios.

A Multiple Particle System Equation Underlying the Klein-Gordon-Dirac-Schrödinger Equations

Authors: D.T. Froedge
Comments: 16 pages 38 equations 98kb

The purpose of this paper is to illustrate a fundamental, multiple particle, system equation for which the Klein-Gordon-Dirac-Schrödinger equations are single particle special cases. In the same manner that eigenvalues of the Schrödinger equation represents energy levels of an interacting atomic system, eigenvalues represent particle energies in an interacting system of particles. An equation is proposed that has vector solutions defined in Dirac, or Clifford algebra, that treats a collection of particles as a single system..The proposed solution is a descriptor of a symmetric, light speed expanding group of interacting particles having real, as well as the familiar QM constituents.

Universal Theory of Dimensionless Constants of Nature

Authors: Mark A. Thomas
Comments: 15 pages.

An analytic structural approach to calculating dimensionless constants of Nature is presented which shows a direct relation between physics, and a number theoretic form involving transcendental forms and the Monster group. A development inherent in the consistency of the calculation leads to the deduction of the fine structure constant from pure mathematical structure. The fine structure constant is calculated as pure number in transcendental form 0.00729735256884151851344... This lies within the margin of error of Gabrielse?s 2008 experimentally determined value 0.007297352569(5). An argument is made that the naturalness of the mathematical structure explains away the hierarchical mass scale problem of physics and that the structure could be core to the Standard Model with the inclusion of the gravitational gauge force. The universality of the approach is presented to show its reaching domain. If the forms and calculation are true the fine tuning argument of physics may be overturned.

A New Generic Class of Beltrami "Force-Free" Fields. Part-i: Theoretical Considerations

Authors: T. E. Raptis
Comments: 13 pages.

We report on a new general class of solutions of the Beltrami equation, with special characteristics. We also provide examples of solutions that also satisfy Maxwell equations. A subset of these solutions can be isolated which corresponds to "gauge" fields. A special projective geometry of vacuum fields is also revealed and discussed.

Causal Set Theory and the Origin of Mass-Ratio

Authors: Carey R Carlson
Comments: 16 pages.

Quantum theory is reconstructed using standalone causal sets. The frequency ratios inherent in causal sets are used to define energy-ratios, implicating the causal link as the quantum of action. Space-time and its particle-like sequences are then constructed from causal links. A 4-D time-lattice structure is defined and then used to model neutrinos and electron clouds, which together constitute a 4-D manifold. A 6-D time-lattice is used to model the nucleons. The integration of the nucleus with its electron cloud affords calculation of the mass-ratio of the proton (or the neutron) with respect to the electron. Arrow diagrams, along with several ball-and-stick models, are used to streamline the presentation.

Handbook of Functions Errata

Authors: Fredy Zypman
Comments: 2 pages

Formulas connecting toroidal functions and elliptical functions are useful in various areas of physics. In solving a problem in electrostatics we run across an error in the Handbook of mathematical functions of Abramowitz and Stegun. In this paper we report the details.

The Geometry of CP₂ and Its Relationship to Standard Model

Authors: Matti Pitkänen
Comments: 13 Pages.

This appendix contains basic facts about CP₂ as a symmetric space and Kähler manifold. The coding of the standard model symmetries to the geometry of CP₂, the physical interpretation of the induced spinor connection in terms of electro-weak gauge potentials, and basic facts about induced gauge fields are discussed

Could the Dynamics of Kähler Action Predict the Hierarchy of Planck Constants?

Authors: Matti Pitkänen
Comments: 5 Pages.

The original justification for the hierarchy of Planck constants came from the indications that Planck constant could have large values in both astrophysical systems involving dark matter and also in biology. The realization of the hierarchy in terms of the singular coverings and possibly also factor spaces of CD and CP₂ emerged from consistency conditions. It however seems that TGD actually predicts this hierarchy of covering spaces. The extreme non-linearity of the field equations defined by Kähler action means that the correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates is 1-to-many. This leads naturally to the introduction of the covering space of CD x CP₂, where CD denotes causal diamond defined as intersection of future and past directed light-cones.

Weak Form of Electric-Magnetic Duality, Electroweak Massivation, and Color Confinement

Authors: Matti Pitkänen
Comments: 13 Pages.

The notion of electric magnetic duality emerged already two decades ago in the attempts to formulate the Kähler geometry of the "world of classical worlds". Quite recently a considerable step of progress took place in the understanding of this notion. This concept leads to the identification of the physical particles as string like objects defined by magnetic charged wormhole throats connected by magnetic ux tubes. The second end of the string contains particle having electroweak isospin neutralizing that of elementary fermion and the size scale of the string is electro-weak scale would be in question. Hence the screening of electro-weak force takes place via weak confinement. This picture generalizes to magnetic color confinement. Electric-magnetic duality leads also to a detailed understanding of how TGD reduces to almost topological quantum field theory. A surprising outcome is the necessity to replace CP₂ Kähler form in Kähler action with its sum with S² Kähler form.

How to Define Generalized Feynman Diagrams?

Authors: Matti Pitkänen
Comments: 16 Pages.

Generalized Feynman diagrams have become the central notion of quantum TGD and one might even say that space-time surfaces can be identified as generalized Feynman diagrams. The challenge is to assign a precise mathematical content for this notion, show their mathematical existence, and develop a machinery for calculating them. Zero energy ontology has led to a dramatic progress in the understanding of generalized Feynman diagrams at the level of fermionic degrees of freedom. In particular, manifest finiteness in these degrees of freedom follows trivially from the basic identifications as does also unitarity and non-trivial coupling constant evolution. There are however several formidable looking challenges left.

1. One should perform the functional integral over WCW degrees of freedom for fixed values of on mass shell momenta appearing in the internal lines. After this one must perform integral or summation over loop momenta.
2. One must define the functional integral also in the p-adic context. p-Adic Fourier analysis relying on algebraic continuation raises hopes in this respect. p-Adicity suggests strongly that the loop momenta are discretized and ZEO predicts this kind of discretization naturally.

Physics as Generalized Number Theory: Infinite Primes

Authors: Matti Pitkänen
Comments: 37 Pages.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields, and infinite primes discussed in this article. The construction of infinite primes is formally analogous to a repeated second quantization of an arithmetic quantum field theory by taking the many particle states of previous level elementary particles at the new level. Besides free many particle states also the analogs of bound states appear. In the representation in terms of polynomials the free states correspond to products of first order polynomials with rational zeros. Bound states correspond to nth order polynomials with non-rational but algebraic zeros. The construction can be generalized to classical number fields and their complexifications obtained by adding a commuting imaginary unit. Special class corresponds to hyper-octonionic primes for which the imaginary part of ordinary octonion is multiplied by the commuting imaginary unit so that one obtains a sub-space M⁸ with Minkowski signature of metric. Also in this case the basic construction reduces to that for rational or complex rational primes and more complex primes are obtained by acting using elements of the octonionic automorphism group which preserve the complex octonionic integer property. Can one map infinite primes/integers/rationals to quantum states? Do they have space-time surfaces as correlates? Quantum classical correspondence realized in terms of modified Dirac operator implies that if infinite rationals can be mapped to quantum states then the mapping of quantum states to space-time surfaces automatically gives the map to space-time surfaces. The question is therefore whether the mapping to quantum states defined by WCW spinor fields is possible. A natural hypothesis is that number theoretic fermions can be mapped to real fermions and number theoretic bosons to WCW ("world of classical worlds") Hamiltonians. The crucial observation is that one can construct infinite hierarchy of hyper-octonionic units by forming ratios of infinite integers such that their ratio equals to one in real sense: the integers have interpretation as positive and negative energy parts of zero energy states. One can construct also sums of these units with complex coefficients using commuting imaginary unit and these sums can be normalized to unity and have interpretation as states in Hilbert space. These units can be assumed to possess well defined standard model quantum numbers. It is possible to map the quantum number combinations of WCW spinor fields to these states. Hence the points of M⁸ can be said to have infinitely complex number theoretic anatomy so that quantum states of the universe can be mapped to this anatomy. One could talk about algebraic holography or number theoretic Brahman=Atman identity. One can also ask how infinite primes relate to the p-adicization program and to the hierarchy of Planck constants. The key observation is that infinite primes are in one-one correspondence with rational numbers at the lower level of hierarchy. At the first level of hierarchy the p-adic norm with respect to p-adic prime for this rational gives power p^-n so that one has two powers of p - pⁿ⁺ and p^n- since two infinite primes corresponding to fermionic vacua X±1, where X is the product of all primes at given level of hierarchy, characterize the partonic 2-surface. The proposal inspired by the p-adicization program is that Δφ = 2π/pⁿ defines angle measurement resolution crucial in the construction of p-adic variants of WCW ("world of classical world") as a union of symmetric coset spaces by starting from discrete variants of the real counterpart of symmetric space having common points with tis p-adic variant. The two measurement resolutions correspond to CD and CP₂ degrees of freedom. The hierarchy of Planck constants generalizes imbedding space to a book like structure with pages identified in terms of singular coverings and factor spaces of CD and CP₂. There are good arguments suggesting that only coverings characterized by integers n_a and n_bare realized. The identifications na = n₊ and nb = n- lead to highly non-trivial physical predictions and allow sharpen the view about the hierarchy of Planck constants. Therefore the notion of finite measurement resolution becomes the common denominator for the three threads of the number theoretic vision and give also a connection with the physics as infinite-dimensional geometry program and with the inclusions of hyper-finite factors defined by WCW spinor fields and proposed to characterize finite measurement resolution at quantum level.

Physics as Generalized Number Theory: Classical Number Fields

Authors: Matti Pitkänen
Comments: 28 Pages.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields discussed in this article, and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article the connection between standard model symmetries and classical number fields is discussed. The basis vision is that the geometry of the infinite-dimensional WCW ("world of classical worlds") is unique from its mere existence. This leads to its identification as union of symmetric spaces whose Kähler geometries are fixed by generalized conformal symmetries. This fixes space-time dimension and the decomposition M⁴ x S and the idea is that the symmetries of the Kähler manifold S make it somehow unique. The motivating observations are that the dimensions of classical number fields are the dimensions of partonic 2-surfaces, space-time surfaces, and imbedding space and M⁸ can be identified as hyper-octonions- a sub-space of complexified octonions obtained by adding a commuting imaginary unit. This stimulates some questions. Could one understand S = CP₂ number theoretically in the sense that M⁸ and H = M⁴ x CP₂ be in some deep sense equivalent ("number theoretical compactification" or M⁸ - H duality)? Could associativity define the fundamental dynamical principle so that space-time surfaces could be regarded as associative or co-associative (defined properly) sub-manifolds of M⁸ or equivalently of H. One can indeed define the associativite (co-associative) 4-surfaces using octonionic representation of gamma matrices of 8-D spaces as surfaces for which the modified gamma matrices span an associate (co-associative) sub-space at each point of space-time surface. Also M⁸ - H duality holds true if one assumes that this associative sub-space at each point contains preferred plane of M⁸ identifiable as a preferred commutative or co-commutative plane (this condition generalizes to an integral distribution of commutative planes in M⁸). These planes are parametrized by CP₂ and this leads to M⁸ - H duality. WCW itself can be identified as the space of 4-D local sub-algebras of the local Clifford algebra of M⁸ or H which are associative or co-associative. An open conjecture is that this characterization of the space-time surfaces is equivalent with the preferred extremal property of Kähler action with preferred extremal identified as a critical extremal allowing infinite-dimensional algebra of vanishing second variations.

Physics as Generalized Number Theory: P-Adic Physics and Number Theoretic Universality

Authors: Matti Pitkänen
Comments: 51 Pages.

Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article p-adic physics and the technical problems relates to the fusion of p-adic physics and real physics to a larger structure are discussed. The basic technical problems relate to the notion of definite integral both at space-time level, imbedding space level and the level of WCW (the "world of classical worlds"). The expressibility of WCW as a union of symmetric spacesleads to a proposal that harmonic analysis of symmetric spaces can be used to define various integrals as sums over Fourier components. This leads to the proposal the p-adic variant of symmetric space is obtained by a algebraic continuation through a common intersection of these spaces, which basically reduces to an algebraic variant of coset space involving algebraic extension of rationals by roots of unity. This brings in the notion of angle measurement resolution coming as Δφ = 2π/pⁿ for given p-adic prime p. Also a proposal how one can complete the discrete version of symmetric space to a continuous p-adic versions emerges and means that each point is effectively replaced with the p-adic variant of the symmetric space identifiable as a p-adic counterpart of the real discretization volume so that a fractal p-adic variant of symmetric space results. If the Kähler geometry of WCW is expressible in terms of rational or algebraic functions, it can in principle be continued the p-adic context. One can however consider the possibility that that the integrals over partonic 2-surfaces defining ux Hamiltonians exist p-adically as Riemann sums. This requires that the geometries of the partonic 2-surfaces effectively reduce to finite sub-manifold geometries in the discretized version of δM₊⁴. If Kähler action is required to exist p-adically same kind of condition applies to the space-time surfaces themselves. These strong conditions might make sense in the intersection of the real and p-adic worlds assumed to characterized living matter.

Construction of Configuration Space Spinor Structure

Authors: Matti Pitkänen
Comments: 34 Pages.

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach discussed in this article relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. This implies a geometrization of fermionic statistics. The basic philosophy is that at fundamental level the construction of WCW geometry reduces to the second quantization of the induced spinor fields using Dirac action. This assumption is parallel with the bosonic emergence stating that all gauge bosons are pairs of fermion and antifermion at opposite throats of wormhole contact. Vacuum function is identified as Dirac determinant and the conjecture is that it reduces to the exponent of Kähler function. In order to achieve internal consistency induced gamma matrices appearing in Dirac operator must be replaced by the modified gamma matrices defined uniquely by Kähler action and one must also assume that extremals of Kähler action are in question so that the classical space-time dynamics reduces to a consistency condition. This implies also super-symmetries and the fermionic oscillator algebra at partonic 2-surfaces has intepretation as N = 1 generalization of space-time supersymmetry algebra different however from standard SUSY algebra in that Majorana spinors are not needed. This algebra serves as a building brick of various super-conformal algebras involved. The requirement that there exist deformations giving rise to conserved Noether charges requires that the preferred extremals are critical in the sense that the second variation of the Kähler action vanishes for these deformations. Thus Bohr orbit property could correspond to criticality or at least involve it. Quantum classical correspondence demands that quantum numbers are coded to the properties of the preferred extremals given by the Dirac determinant and this requires a linear coupling to the conserved quantum charges in Cartan algebra. Effective 2-dimensionality allows a measurement interaction term only in 3-D Chern-Simons Dirac action assignable to the wormhole throats and the ends of the space-time surfaces at the boundaries of CD. This allows also to have physical propagators reducing to Dirac propagator not possible without the measurement interaction term. An essential point is that the measurement interaction corresponds formally to a gauge transformation for the induced Kähler gauge potential. If one accepts the weak form of electric-magnetic duality Kähler function reduces to a generalized Chern-Simons term and the effect of measurement interaction term to Kähler function reduces effectively to the same gauge transformation. The basic vision is that WCW gamma matrices are expressible as super-symplectic charges at the boundaries of CD. The basic building brick of WCW is the product of infinite-D symmetric spaces assignable to the ends of the propagator line of the generalized Feynman diagram. WCW Kähler metric has in this case "kinetic" parts associated with the ends and "interaction" part between the ends. General expressions for the super-counterparts of WCW ux Hamiltoniansand for the matrix elements of WCW metric in terms of their anticommutators are proposed on basis of this picture.

Construction of Configuration Space Geometry from Symmetry Principles

Authors: Matti Pitkänen
Comments: 26 Pages.

There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first one relies on a direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure assuming that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for the second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of spinor structure. In this article the construction of Kähler form and metric based on symmetries is discussed. The basic vision is that WCW can be regarded as the space of generalized Feynman diagrams with lines thickned to light-like 3-surfaces and vertices identified as partonic 2-surfaces. In zero energy ontology the strong form of General Coordinate Invariance (GCI) implies effective 2-dimensionality and the basic objects are pairs partonic 2-surfaces X² at opposite light-like boundaries of causal diamonds (CDs). The hypothesis is that WCW can be regarded as a union of infinite-dimensional symmetric spaces G/H labeled by zero modes having an interpretation as classical, non-quantum uctuating variables. A crucial role is played by the metric 2-dimensionality of the light-cone boundary δM₊⁴ + and of light-like 3-surfaces implying a generalization of conformal invariance. The group G acting as isometries of WCW is tentatively identified as the symplectic group of δM₊⁴ x CP₂ localized with respect to X². H is identified as Kac-Moody type group associated with isometries of H = M₊⁴ x CP₂ acting on light-like 3-surfaces and thus on X². An explicit construction for the Hamiltonians of WCW isometry algebra as so called ux Hamiltonians is proposed and also the elements of Kähler form can be constructed in terms of these. Explicit expressions for WCW ux Hamiltonians as functionals of complex coordinates of the Cartesisian product of the infinite-dimensional symmetric spaces having as points the partonic 2-surfaces defining the ends of the the light 3-surface (line of generalized Feynman diagram) are proposed.

Identification of the Configuration Space Kähler Function

Authors: Matti Pitkänen
Comments: 38 Pages.

There are two basic approaches to quantum TGD. The first approach, which is discussed in this article, is a generalization of Einstein's geometrization program of physics to an infinitedimensional context. Second approach is based on the identification of physics as a generalized number theory. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" (WCW) identified as the space of 3-surfaces in in certain 8-dimensional space. There are three separate approaches to the challenge of constructing WCW Kähler geometry and spinor structure. The first approach relies on direct guess of Kähler function. Second approach relies on the construction of Kähler form and metric utilizing the huge symmetries of the geometry needed to guarantee the mathematical existence of Riemann connection. The third approach relies on the construction of spinor structure based on the hypothesis that complexified WCW gamma matrices are representable as linear combinations of fermionic oscillator operator for second quantized free spinor fields at space-time surface and on the geometrization of super-conformal symmetries in terms of WCW spinor structure. In this article the proposal for Kähler function based on the requirement of 4-dimensional General Coordinate Invariance implying that its definition must assign to a given 3-surface a unique space-time surface. Quantum classical correspondence requires that this surface is a preferred extremal of some some general coordinate invariant action, and so called Kähler action is a unique candidate in this respect. The preferred extremal has intepretation as an analog of Bohr orbit so that classical physics becomes and exact part of WCW geometry and therefore also quantum physics. The basic challenge is the explicit identification of WCW Kähler function K. Two assumptions lead to the identification of K as a sum of Chern-Simons type terms associated with the ends of causal diamond and with the light-like wormhole throats at which the signature of the induced metric changes. The first assumption is the weak form of electric magnetic duality. Second assumption is that the Kähler current for preferred extremals satisfies the condition jK ^ djK = 0 implying that the ow parameter of the ow lines of jK defines a global space-time coordinate. This would mean that the vision about reduction to almost topological QFT would be realized. Second challenge is the understanding of the space-time correlates of quantum criticality. Electric-magnetic duality helps considerably here. The realization that the hierarchy of Planck constant realized in terms of coverings of the imbedding space follows from basic quantum TGD leads to a further understanding. The extreme non-linearity of canonical momentum densities as functions of time derivatives of the imbedding space coordinates implies that the correspondence between these two variables is not 1-1 so that it is natural to introduce coverings of CD x CP₂. This leads also to a precise geometric characterization of the criticality of the preferred extremals.

Physics as Infinite-Dimensional Geometry and Generalized Number Theory: Basic Visions

Authors: Matti Pitkänen
Comments: 32 Pages.

There are two basic approaches to the construction of quantum TGD. The first approach relies on the vision of quantum physics as infinite-dimensional Kähler geometry for the "world of classical worlds" identified as the space of 3-surfaces in in certain 8-dimensional space. Essentially a generalization of the Einstein's geometrization of physics program is in question. The second vision is the identification of physics as a generalized number theory. This program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous to a repeated second quantization of an arithmetic quantum field theory. In this article brief summaries of physics as infinite-dimensional geometry and generalized number theory are given to be followed by more detailed articles.

Unfolding the Labyrinth: Open Problems in Physics, Mathematics, Astrophysics, and Other Areas of Science

Authors: Florentin Smarandache, V. Christianto, Fu Yuhua, Radi I. Khrapko, J. Hutchison
Comments: 147 pages

The reader will find herein a collection of unsolved problems in mathematics and the physical sciences. Theoretical and experimental domains have each been given consideration. The authors have taken a liberal approach in their selection of problems and questions, and have not shied away from what might otherwise be called speculative, in order to enhance the opportunities for scientific discovery. Progress and development in our knowledge of the structure, form and function of the Universe, in the true sense of the word, its beauty and power, and its timeless presence and mystery, before which even the greatest intellect is awed and humbled, can spring forth only from an unshackled mind combined with a willingness to imagine beyond the boundaries imposed by that ossified authority by which science inevitably becomes, as history teaches us, barren and decrepit. Revealing the secrets of Nature, so that we truly see 'the sunlit plains extended, and at night the wondrous glory of the everlasting stars', requires far more than mere technical ability and mechanical dexterity learnt form books and consensus. The dustbin of scientific history is replete with discredited consensus and the grand reputations of erudite reactionaries. Only by boldly asking questions, fearlessly, despite opposition, and searching for answers where most have not looked for want of courage and independence of thought, can one hope to discover for one's self. From nothing else can creativity blossom and grow, and without which the garden of science can only aspire to an overpopulation of weeds.

Formulas of Vector Analysis

Authors: Andreï V. Serghienko
Comments: 3 pages.

We derive the formulas for the sine and the cosine of the sum, not using the notions of scalar and vector products, and using only the definitions of the sine and the cosine. We derive the formulas for the gradient operator, the divergence and the Laplace operator in different orthogonal coordinate systems, not using any additional constructions like Lamé coefficients, and using only the definitions of the sine and the cosine.

Advanced Topics in Information Dynamics

Authors: Chris Goddard
Comments: 201 pages

This work is sequel to the book "A Treatise in Information Geometry", submitted to vixra in late 2009. The aim of this dissertation is to continue the development of fractal geometry initiated in the former volume. This culminates in the construction of first order self-referential geometry, which is a special form of 8-tensor construction on a differential manifold with nice properties. The associated information theory has many powerful and interesting consequences. Additionally within this treatise, various themes in modern mathematics are surveyed- Galois theory, Category theory, K-theory, and Sieve theory, and various connections between these structures and information theory investigated. In particular it is demonstrated that the exotic geometric analogues of these constructions - save for Category theory, which is foundational - form special cases of the self referential calculus.

A 'planck-Like' Characterization of Exponential Function

Authors: Constantinos Ragazas
Comments: 8 pages

We derive a characterization of simple exponential functions that has the exact mathematical form to Planck's Formula for blackbody radiation in Quantum Physics.

Mathematical Model of Information

Authors: Elemér E Rosinger
Comments: 17 pages.

A simple and rather general mathematical model of the phenomenon of information is presented, followed by several examples and comments.

Invariants Relative to Change of Value of the Independent Variable and Their Role in the Physics.

Authors: Vladimir I. Smirnov
Comments: 34 pages, Russian and English versions included

It is identified the new class of invariants which values are constant at change of value of an inde-pendent variable. Their properties and a deriving method are shown on already known and still un-known instances, concerning to various areas of physics. In particular, new invariants (50), (55) and (57) for the straight lines intersected in one point on a plane have been discovered. Besides, the re-quest for detection of the third (not dependent on two already known) an invariant (31) electromag-netic fields for a special case of the special theory of relativity is made.

Mathematical and Phenomenological Elements of the Twin-Tori Model of Physics and Cosmology.

Authors: Chris Forbes
Comments: 10 pages

In this, a follow up to a previous paper 'A Short Article On A Newly Proposed Model Of Cosmology' (viXra:0909.0005), some of the basic mathematical structures to be used in the formulation of the model are shown, and several advantages are discussed. The paper then takes a more phenomenological approach and several simple (1+1) dimensional models are explored.

A First Order Singular Perturbation Solution to a Simple One-Phase Stefan Problem with Finite Neumann Boundary Conditions

Authors: Bruce Rout
Comments: 13 pages

This paper examines the difference between infinite and finite domains of a Stefan Problem. It is pointed out that attributes of solutions to the Diffusion Equation suggest assumptions of an infinite domain are invalid during initial times for finite domain Stefan Problems. The paper provides a solution for initial and early times from an analytical approach using a perturbation. This solution can then easily be applied to numerical models for later times. The differences of the two domains are examined and discussed.

A Treatise on Information Geometry

Authors: Chris Goddard
Comments: 292 pages

In early 1999, Professor Frieden of the University of Arizona published a book through Cambridge University Press titled "Physics from Fisher Information". It is the purpose of this dissertation to further develop some of his ideas, as well as explore various exotic differentiable structures and their relationship to physics. In addition to the original component of this work, a series of survey chapters are provided, in the interest of keeping the treatise self-contained. The first summarises the main preliminary results on the existence of non-standard structures on manifolds from the Milnor-Steenrod school. The second is a standard introduction to semi-riemannian geometry. The third introduces the language of geometric measure theory, which is important in justifying the existence of smooth solutions to variational problems with smooth structures and smooth integrands. The fourth is a short remark on PDE existence theory, which is needed for the fifth, which is essentially a typeset version of a series of lectures given by Ben Andrews and Gerhard Huisken on the Hamilton-Perelman program for proving the Geometrisation Conjecture of Bill Thurston.

Cylindrical Wave, Wave Equation, and Method of Separation of Variables

Authors: Hamid V. Ansari
Comments: 7 pages

It is shown that the wave equation cannot be solved for the general spreading of the cylindrical wave using the method of separation of variables. But an equation is presented in case of its solving the above act will have occurred. Also using this equation the above-mentioned general spreading of the cylindrical wave for large distances is obtained which contrary to what is believed consists of arbitrary functions.

New Calculuses

Authors: Vladislav Konovalov
Comments: 12 pages

In the article the new calculuses are offered similar differential and integral, but differing, that in them the analysis of the previous and subsequent values of a function is made. The new calculuses allow to decide problems, the solution which one with usage customary differential and integral calculus is impossible.

3x3 Unitary to Magic Matrix Transformations

Authors: Philip Gibbs
Comments: 5 pages, published in Prespacetime Journal, V5

We prove that any 3x3 unitary matrix can be transformed to a magic matrix by multiplying its rows and columns by phase factors. A magic matrix is defined as one for which the sum of the elements in any row or column add to the same value. This result is relevant to recent observations on particle mixing matrices.

Topological Maxwell Field Theory and Symmetry Breaking.

Authors: R. M. Kiehn
Comments: recovered from sciprint.org

Finally, I have found time to think about, and the incentive to study, how the field theory of Topological thermodynamics, electrodynamics, and hydrodynamics can be compared to field theory concepts that have been developed by Lagrangian methods, for both the classic and quantum mechanical varieties. For more than 30 years I have known that Cartan's topological methods could be applied to dissipative systems; the methods based on diffeomorphic-invariant Lagrangian field theories can not. The incentive came when I realized that the topological methods of Cartan gave dynamical results that can explain "symmetry breaking" and quantization in terms of continuous topological evolution.

Universal Theory of Dimensionless Constants of Nature

Authors: Mark A. Thomas
Comments: 16 pages.

An analytic structural approach to calculating dimensionless constants of Nature is presented which shows a direct relation between physics, and a number theoretic form involving transcendental forms and the Monster group. A development inherent in the consistency of the calculation leads to the deduction of the fine structure constant from pure mathematical structure. The fine structure constant is calculated as pure number in transcendental form 0.00729735256884151851344.... This lies within the margin of error of Gabrielse?s 2008 experimentally determined value 0.007297352569(5). An argument is made that the naturalness of the mathematical structure explains away the hierarchical mass scale problem of physics and that the structure could be core to the Standard Model with the inclusion of the gravitational gauge force. The universality of the approach is presented to show its reaching domain. If the forms and calculation are true the fine tuning argument of physics may be overturned.

Universal Theory of Dimensionless Constants of Nature

Authors: Mark A. Thomas
Comments: 15 pages.

An analytic structural approach to calculating dimensionless constants of Nature is presented which shows a direct relation between physics, and a number theoretic form involving transcendental forms and the Monster group. A development inherent in the consistency of the calculation leads to the deduction of the fine structure constant from pure mathematical structure. The fine structure constant is calculated as pure number in transcendental form 0.00729735256884151851344.... This lies within the margin of error of Gabrielse?s 2008 experimentally determined value 0.007297352569(5). An argument is made that the naturalness of the mathematical structure explains away the hierarchical mass scale problem of physics and that the structure could be core to the Standard Model with the inclusion of the gravitational gauge force. The universality of the approach is presented to show its reaching domain. If the forms and calculation are true the fine tuning argument of physics may be overturned.

A Treatise on Information Geometry

Authors: Chris Goddard
Comments: 300 pages,In this revision, I primarily review the status of the chapter on the Turbulent Geometry, and emphasise that there are three different forms of fractal structures possible from the "twisting" of two Riemannian metrics via appropriate pre-geometric operators (and only three; the underlying reason for this is due to the combinatorics of 4-tensor constructions - in particular establishing the uniqueness of particular forms of affine connections). Originally I had the misapprehension that there was only really one sensible operator - the turbulent derivative $\partial^{*}$, b ut there is also a natural transcendental or "reverb"/"lattice vibration" operator $\wedge$ that acts in a radically different manner, as well as the viscoplastic operator $\star$, the latter of which essentially provides a differentiable manifold with the structure of a geometric field. I have also slightly expanded the acknowledgments to take into account the earlier work on the idea of scale free / unparticle physics / physics subject to fractal dynamics (Schroer in the early 60s) and also to take into account the earlier work by Francis Ysidro Edgeworth (1908) on the inferential measure that was later expanded upon by R. Fisher. Finally I have reviewed the last chapter, making some attempt to clean up the later arguments.

In early 1999, Professor Frieden of the University of Arizona published a book through Cambridge University Press titled "Physics from Fisher Information". It is the purpose of this dissertation to further develop some of his ideas, as well as explore various exotic differentiable structures and their relationship to physics. In addition to the original component of this work, a series of survey chapters are provided, in the interest of keeping the treatise self-contained. The first summarises the main preliminary results on the existence of non-standard structures on manifolds from the Milnor-Steenrod school. The second is a standard introduction to semi-riemannian geometry. The third introduces the language of geometric measure theory, which is important in justifying the existence of smooth solutions to variational problems with smooth structures and smooth integrands. The fourth is a short remark on PDE existence theory, which is needed for the fifth, which is essentially a typeset version of a series of lectures given by Ben Andrews and Gerhard Huisken on the Hamilton-Perelman program for proving the Geometrisation Conjecture of Bill Thurston.

A Treatise on Information Geometry

Authors: Chris Goddard
Comments: 294 pages

In early 1999, Professor Frieden of the University of Arizona published a book through Cambridge University Press titled "Physics from Fisher Information". It is the purpose of this dissertation to further develop some of his ideas, as well as explore various exotic differentiable structures and their relationship to physics. In addition to the original component of this work, a series of survey chapters are provided, in the interest of keeping the treatise self-contained. The first summarises the main preliminary results on the existence of non-standard structures on manifolds from the Milnor-Steenrod school. The second is a standard introduction to semi-riemannian geometry. The third introduces the language of geometric measure theory, which is important in justifying the existence of smooth solutions to variational problems with smooth structures and smooth integrands. The fourth is a short remark on PDE existence theory, which is needed for the fifth, which is essentially a typeset version of a series of lectures given by Ben Andrews and Gerhard Huisken on the Hamilton-Perelman program for proving the Geometrisation Conjecture of Bill Thurston.

3x3 Unitary to Magic Matrix Transformations

Authors: Philip Gibbs
Comments: 5 pages

We prove that any 3x3 unitary matrix can be transformed to a magic matrix by multiplying its rows and columns by phase factors. A magic matrix is defined as one for which the sum of the elements in any row or column add to the same value. This result is relevant to recent observations on particle mixing matrices.