Last updated: 10 Oct 2009

MATHEMATICS of RELATIVITY

(OBSERVER’S MATHEMATICS)

Boris Khots, Dmitriy Khots

Abstract

    Observer dependent ascending chain of embedded sets of decimal fractions and their Cartesian products is considered. For every set, arithmetic operations are defined (these operations locally coincide with standard operations), which transform every set into a local ring. The basic problems of Algebra, Geometry, Topology, and Logic are solved for this chain. Definition of Dimension of these sets is introduced. In particular, the dimension of each of these sets is greater than or equal to seven. Euclidean, Lobachevsky, and Riemannian Geometries become the particular cases of the developed Geometry, although many others are possible. For example, we proved that two lines in a plane may intersect each other in 0 (without being parallel in the usual sense), 1, 2, 10, or even 100 points. The three classical Geometries depend on on a particular neighborhood of a given line. For example, Euclidean Geometry works in sufficiently small neighborhood of the given line, but when we enlarge the neighborhood, Lobachevsky and Riemannian Geometries take over. Developed Topology gives birth to Time, and Time becomes a function of Space. Also, the Axiom of Choice becomes invalid in the new model of Mathematics. The application of the new model to Einstein's special theory of relativity is considered. The existence of the Time and Space quantums is proved. We also prove that "small" displacements may happen instantaneously, i.e. faster than the speed of light. Classical effects of "time delay", "length reduction", and "nonsimultaneous events" become invalid for "small" displacements..

News

The Math of Relativity was presented by authors on:

September 14-20, 2009
presentation at the Novosibirsk University The International Conference "Contemporary Analysis and Geometry"
Authors would like to thank Professor Sergey Vodopyanov for the presentation of their talk “Geometrical and Analytical aspects of Observer’s Mathematics”.

June 22-27, 2009
presentation at the International Conference “Geometry “in large”, topology and applications”, devoted to the 90th anniversary Alexey Vasilievich Pogorelov,
Kharkov, Ukraine

June 1-5, 2009
presentation at the “2nd Chaotic Modeling and Simulation” International Conference,
Chania Crete, Greece

June 14-18, 2009
presentation at the Vaxjo University International Conference “Quantum Theory: Reconsideration of Foundations - 5”,
Vaxjo, Sweden

August 24-27, 2008
presentation at Vaxjo University Conference “Foundations of Probability and Physics- 5”
Vaxjo, Sweden

August 21-26, 2008
presentation at Debrecen University 6th Bolyai – Gauss - Lobachevsky Conference (BGL6) “Non-Euclidean Geometry and its Applications”, Debrecen, Hungary

August 12-18, 2008
presentation at Fifth International Conference of Applied Mathematics and Computing, Plovdiv, Bulgaria

July 28 – August 1, 2008
presentation at 16th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications (16th ICFIDCAA), Dongguk University (Gyeongju), Korea
Authors would like to thank Professor Junesang Choi for the presentation of their talk.

June 11-16, 2007
presentation at Vaxjo University Conference “Quantum Theory: Reconsideration of Foundations - 4”,  Vaxjo, Sweden.

May 18-20, 2007
presentation at Midwest Geometry Conference – 2007,  The University of Iowa, Iowa City, Iowa, USA

October 14, 2006
presentation at Kazan State University, Department of Gravity and Theory or Relativity (Prof. A. Aminova), Kazan, Russia

August 22-30, 2006
presentation at International Congress of Mathematicians 2006, ICM2006, Madrid, Spain
NOTE: Authors showed that negative solution of the classical Fermat problem depends on validity of “Axiom of Choice” in standard (“naive”) Mathematics. In Mathematics of Relativity this axiom becomes invalid (authors proved that), and Fermat problem has positive solution – see below.

October 6-8, 2005
presentation at Wolfram Technology Conference 2005, http://www.wolfram.com/techconf2005, Champaign, Illinois, USA

September 30, 2005
presentation at The University of Iowa, www.math.uiowa.edu, Department of Mathematics, Topology Seminar (Prof. Jon Simon), Iowa City, Iowa, USA

August 22-27, 2005
presentation at International Symposium “Analytic Function Theory, Fractional Calculus and Their Applications”, http://www.pims.math.ca/science/2005/05hms/program.html, University of Victoria, Department of Mathematics and Statistics, Victoria, Canada

June 22 – July3, 2005
presentation at the XVIIth Summer School-Seminar VOLGA-2005, www.ksu.ras.ru , Kazan, Tatarstan, Russia

April 21, 2005
presentation at The University of Iowa, www.math.uiowa.edu, Department of Mathematics, Mathematical Biology Seminar (Prof. Herbert Hethcold), Iowa City, Iowa, USA

29-30 September, 2004
International Conference "Informatics problems in third millennium", Kazan, Russia

22 June - 3 July, 2004
THE XVIth SUMMER SCHOOL-SEMINAR VOLGA-2004, Russia, Tatarstan, Kazan

5 - 10 July, 2004
Conference on Non Standard Mathematics, Portugal, University of Aveiro

Dmitriy Khots, Boris Khots, Chaos from Observer’s Mathematics point of view, Talk at International Conference “Chaotic Modeling and Simulation”, Chania Crete, Greece, June 2009 (in print)

Dmitriy Khots, Boris Khots, Quantum Theory from Observer’s Mathematics point of view, Talk at Vaxjo University Conference “Quantum Theory: Reconsideration of Foundation - 5”, Vaxjo, Sweden, June 2009 (in print)

Dmitriy Khots, Boris Khots, Solitary Waves and Dispersive Equations from Observer’s Mathematics point of view, Talk at International Conference “Geometry “in large”, topology and applications”, Kharkov, Ukraine, June 2009 (in print)

Dmitriy Khots, Boris Khots, Physical Aspects of Observer’s Mathematics, American Institute of Physics, volume 1101, pp 311-313, Melville, New York, 2009

Dmitriy Khots, Boris Khots, Non-Euclidean Geometry in Observer’s Mathematics, Acta Physica Debrecina, tomus XLII, pp 112-119, Debrecen, Hungary, August 2008

Dmitriy Khots, Boris Khots, Data Mining in Observer’s Mathematics, International Journal of Pure and Applied Mathematics, volume 51, #2, pp 195-201, Bulgaria, 2009

Dmitriy Khots, Boris Khots, Tenth Hilbert Problem in Observer’s Mathematics, Proceedings of the 16th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications (16th ICFIDCAA), pp 81-85, Dongguk University, Gyeongju, Korea, 2008

Dmitriy Khots, Boris Khots, Fermat’s, Mersenne’s and Waring’s problems in Observer’s Mathematics, International Journal of Pure and Applied Mathematics, Bulgaria, v. 43, #3, pp 403-408 (2008)

Dmitriy Khots, Boris Khots, Quantum Theory and Observer’s Mathematics, American Institute of Physics (AIP), volume 962, pp 261-264, 2007.

Boris Khots, Dmitriy Khots, Observer’s Mathematics – Mathematics of Relativity, Applied Mathematics and Computations, volume 187, issue 1, April 2007, pp 228-238, New York.

Boris Khots, Dmitriy Khots, An Introduction to the Mathematics of Relativity, Lecture Notes in Theoretical and Mathematical Physics, Ed. A.V. Aminova, Kazan State University, v. 7, pp 269-306, 2006.

Boris Khots, Dmitriy Khots, Physical Theory of Relativity and Mathematics of Relativity, Recent Problems in Field Theory, Ed. A.V. Aminova, Kazan State University, v. 5, pp 239-242, 2006.

Dmitriy Khots, Euclidean and Lobachevsky Geometries in Mathematics of Relativity, Recent Problems in Field Theory, Ed. A.V. Aminova, Kazan State University, v. 5, pp 243-246, 2006.

Boris Khots, Dmitriy Khots, Analogy of Fermat’s last problem in Observer’s Mathematics - Mathematics of Relativity, Talk at the International Congress of Mathematicians, Madrid 2006, Proceedings of ICM2006

Full text in .pdf you can see below.

Title
Dedication and Thanks
AMS Classification
Chapter 0. Introduction
Chapter 1. Arithmetic
Chapter 2. Algebra
Chapter 3. Geometry
Chapter 4. Analysis & Topology
Chapter 5. Logic
Chapter 6. Einstein

Appendix 1 TABLE: V2 X₂ 99.99 (nonnegative elements)
Appendix 2 TABLE: V2 X₂ 99.98 (nonnegative elements)
Appendix 3 TABLE: V2 X₂ 99.97 (nonnegative elements)
Appendix 4 TABLE: V2 X₂ 99.95 (nonnegative elements)
Appendix 5 TABLE: V2 X₂ 99.92 (nonnegative elements)
Appendix 6 TABLE: V2 X₂ 99.90 (nonnegative elements)
Appendix 7 TABLE: V2 X₂ 99.53 (nonnegative elements)
Appendix 8 TABLE 1: V2 X₂ 99.99 +₂ V2 X₂ 99.98
Appendix 9 TABLE 2: ((V2 X₂ 99.99 +₂ V2 X₂ 99.98) +₂ V2 X₂ 99.97) (except data from Table 1)
Appendix 10 TABLE 3: (((V2 X₂ 99.99 +₂ V2 X₂ 99.98) +₂ V2 X₂ 99.97) +₂ V2 X₂ 99.95) (except data from Tables 1, 2)
Appendix 11 TABLE 4: ((((V2 +₂ 99.99 +₂ V2 X₂ 99.98) +₂ V2 X₂ 99.97) +₂ V2 X₂ 99.95) +₂ V2 X₂ 99.92) (except data from Tables 1,2,3)
Appendix 12 TABLE 5: (((((V2 X₂ 99.99 +₂ V2 X₂ 99.98) +₂ V2 X₂ 99.97) +₂ V2 X₂ 99.95) +₂ V2 X₂ 99.92) +2 V2 X₂ 99.90) (except data from Tables 1,2,3,4)
Appendix 13 TABLE 6: ((((((V2 X₂ 99.99 +₂ V2 X₂ 99.98) +₂ V2 X₂ 99.97) +₂ V2 X₂ 99.95) +₂ V2 X₂ 99.92) +₂ V2 X₂ 99.90) +₂ V2 X₂ 99.53 (except data from Tables 1,2,3,4,5,6)
Appendix 14 Intersecting Lines on Plane – Empty Intersection
Appendix 15 Intersecting Lines on Plane – Intersection number = 1
Appendix 16 Intersecting Lines on Plane – Intersection number = 2
Appendix 17 Intersecting Lines on Plane – Intersection number = 10
Appendix 18 Intersecting Lines on Plane – Intersection number = 100
Appendix 19 Riemannian Geometry: Intersecting Planes on Cube – Empty Intersection
Appendix 20 Riemannian Geometry: Intersecting Planes on Cube – Intersection number = 2
Appendix 21 Riemannian Geometry: Intersecting Planes on Cube – Intersection number = 4
Appendix 22 Riemannian Geometry: Intersecting Planes on Cube – Intersection number = 20
Appendix 23 Appendixes 14-22, Summary
 

© 2004 Boris Khots, Dmitriy Khots