Mathematics

Mathematics (11)

Sunday, 03 December 2017 19:26

Towards a Universal Probability Theory, Part I

Written by

Introduction
Andrei Khrennikov's Contextual Probability and the Two Slit Experiment

Quoted Khrennikov:

“One of problems was of purely mathematical character. The standard probabilistic formalism based on Kolmogorov’s axiomatics, 1933, was a fixed context formalism. This conventional probabilistic formalism does not provide rules of operating with probabilities calculated for different contexts. However, in quantum theory we have to operate with statistical data obtained for different complexes of physical conditions, contexts. In fact, this context dependence of probabilities as the origin of the superposition principle was already discussed by W. Heisenberg; unfortunately, only in quite general and rather philosophic framework.”1

Contents

  1. Khrennikov's contextualism
  2. What's next?
  3. Feynman and the two slit experiment
  4. Appendix

____________________________

1Andrei Khrennikov, Contextual viewpoint to quantum stochastics”; https://arxiv.org/pdf/hep-th/0112076.pdf

After registration, you may ask for a translation of the full text by email.

 

Last modified on Sunday, 03 December 2017 19:33
Wednesday, 14 June 2017 21:17

Geometric Algebras of Spacetime

Written by

The Ambiguity of Geometric Algebra in the Models of Spacetime

Contents
1.  Memo
1.1. Multivectors in GA3
1.2. Conjugations in GA3
1.2.1. Clifford conjugation
1.2.2. Pauli conjugation
1.2.3. The role of complex imaginary unit in the Clifford and Pauli conjugations
2. Minkowski spacetime generated in GA1,3
2.1. Bivektor algebra
2.2. Lorentz transformation
3. Spacetime model in classic GA3: the scalar element as a time representation
4. Summary
1. Appendix
Pauli algebra and matrices
2. Appendix
Gamma or Dirac matrices, Dirac algebra
3. Appendix
Lorentz transformation
a. Einstein description with coordinates
b. Applying hyperbolic functions
c. Usage of hyperbolic numbers to describe space time
References

After registration, you may ask for a translation of the full text by email.

 

Last modified on Thursday, 15 June 2017 15:39

Conjugation concepts in GA3

Contents

  1. Memo
  2. Involutions: reversion and conjugation in GA3
    1. Reversion
    2. Magnitude or modulus
    3. Conjugation
  3. Comments

Appendix
Pauli algebra

References

  1. David Hestenes, Space–Time Algebra”
  2. David Hestenes, „New Foundations for Classical Mechanics”
  3. Chris Doran & Anthony Lasenby, Geometric Algebra for Physicist”
  4. Stephen Gull, Anthony Lasenby, Chris Doran, „Imaginary Numbers are not Real — the Geometric Algebra of Spacetime” http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf

After registration, you may ask for a translation of the full text by email.

Last modified on Monday, 01 May 2017 16:08
Thursday, 06 April 2017 15:41

The Dimensions of the Geometric Algebra

Written by

Subjective and unfinished thoughts on GA dimensions

In everyday life we ordinarily use the term dimension as spatial size. In sciences, than the mathematics and physics, the term of the dimension is used in a diverse, different sense.

Contents

  1. Dimension and grade in GA
  2. Unit of measurement as dimension
  3. Qualitative infinite as dimension

After registration, you may ask for a translation of the full text by email.

Last modified on Thursday, 06 April 2017 15:45
Sunday, 05 February 2017 20:39

CA, GA and the Two-element Numbers – Secondly

Written by

With Exciting Dénouement

Contents

  1. Introduction
  2. Geometric Algebra of the Plane
  3. The Algebra of 3-Space
  4. Conclusions

Abstract by a quotation
„These considerations all indicate that our present thinking about quantum mechanics is infested with the deepest misconceptions. We believe, with David Hestenes, that geometric algebra is an essential ingredient in unravelling these misconceptions.”1

__________________________

1 See Stephen Gull, Anthony Lasenby, Chris Doran, „Imaginary Numbers are not Real — the Geometric Algebra of Spacetime”;
http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/ImagNumbersArentReal.pdf

After registration, you may ask for a translation of the full text by email.

 

Last modified on Thursday, 06 April 2017 15:44

C Doran A Lasenby

Abstract

The geometric algebra (GA) is a Clifford algebra (CA) of a vector space over the field of real numbers. A similar concept of complex numbers can be derived from the GA. It is very instructive to compare, as the various authors deduce one of the two-element numbers from CA, or from GA.

Contents

  1. David Hestenes
  2. Garret Sobczyk
  3. Doran-Lasenby
  4. Stefan Ulrich
  5. Summary

After registration, you may ask for a translation of the full text by email.

 

Last modified on Friday, 06 January 2017 19:10
Wednesday, 23 November 2016 19:22

Number systems – what's next?

Written by

Desiderata

Content

  • The notion of number
  • Desiderata to the extension of the numbers
  • Summary

Abstract

Although the numbers are yet considered as the simplest abstractions of mathematics, it is increasingly taking shape is a picture in which the numbers will the most complex and exciting concept in mathematics. The geometric algebra is a probable candidate for the new number system, but desiderata of the author is unlike the criteria of the geometric algebra in three points:

  • The number line of real numbers is orthogonal to the space-dimensions according to the definition of scalar product.
  • As the fundamental operation of arithmetic the product is commutative.
  • The space-dimensions evolve from time-dimension which is represented by number line of real numbers.

 

After registration, you may ask for a translation of the full text by email.

 

Last modified on Wednesday, 23 November 2016 19:26

P1050632 kv

Two-element numbers as space-time elements of probability theory

Abstract

  1. Introductory
  2. The probability theory and the space-time
  3. Different interpretations and problems of the concept of probability
    1. The Bertrand paradox and his explanation
    2. Relative frequency concept and the paradoxes
    3. Kolmogorov’s probability theory
  4. Epilog

 

After registration, you may ask for a translation of the full text by email.

Last modified on Wednesday, 23 November 2016 19:22

The role of the concept of computability in physics

“It would require infinite precision for the coordinates of a phase-space point- i.e. all the decimal places!- in order for it to make sense to say that the point is non-computable. (A number described by a finite decimal is always computable.) A finite portion of .a decimal expansion of a number tells us nothing about the computability of the entire expansion of that number. But all physical measurements have a definite limitation on how accurately they can be performed, and can only give information about a finite number of decimal places. Does this nullify the whole concept of 'computable number' as applied to physical measurements?” (Roger Penrose, The Emperor’s New Mind – Concerning Computers, Minds, and The Laws of Physics, Chapter 5; The classical World, Phase space)

Contents

1. The natural numbers and the qualitative characteristic of infinite
2. The positional numeral systems and the two-element numbers
3. “Actualization” of the potentially infinite – examples in physics
4. Epilogue

Appendix

 

After registration, you may ask for a translation of the full text by email.

Content

The infinity and the two-element numbers

Infinity differently – a heuristic approach

Infinity described with homogeneous coordinates

The infinity and the geometric algebra

The complex and hyperbolic numbers as two-dimensional Clifford algebras - examples from the literature

The hiding infinity

Abstract
Types of geometric algebra generated by vectors of 1-dimensional vector space are not at all exhibit poor structure. These geometric algebras have very exciting basic structure as they are represented by two-element-numbers which are the complex, the parabolic (dual) and the hyperbolic numbers. On the other hand, special types of infinities – related to the continuum hypothesis – are modeled by the two-element numbers. So it can be said that vectors of 1-dimensional vector space, which generates the above mentioned geometric algebras are nothing more than the infinite extensions of scalars of the geometric algebras.

 

After registration, you may ask for a translation of the full text by email.