Skip to main content
SpringerLink
Log in

Kolmogorov’s Axioms for Probabilities with Values in Hyperbolic Numbers

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

We introduce the notion of a probabilistic measure which takes values in hyperbolic numbers and which satisfies the system of axioms generalizing directly Kolmogorov’s system of axioms. We show that this new measure verifies the usual properties of a probability; in particular, we treat the conditional hyperbolic probability and we prove the hyperbolic analogues of the multiplication theorem, of the law of total probability and of Bayes’ theorem. Our probability may take values which are zero-divisors and we discuss carefully this peculiarity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Accardi, L.: Quantum Probability: An Introduction to Some Basic Ideas and Trends. Stochastic Models, II. In: Hernandez, D., Lopez-Mimbela, J.A., Quezada, R. (eds.) Aportaciones Mat. Investig., vol. 16. Soc. Mat. Mexicana, Mexico, pp. 1–128 (2001)

  2. Alpay, D., Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C.: Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis. Springer Briefs in Mathematics, vol. XV (2014)

  3. Luna-Elizarrarás M.E., Perez-Regalado C.O., Shapiro M.: On linear functionals and Hahn–Banach theorems for hyperbolic and bicomplex modules. Adv. Appl. Clifford Algebras 23(4), 1105–1129 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Luna-Elizarrarás M.E., Shapiro M., Struppa D.C., Vajiac A.: Bicomplex numbers and their elementary functions. Cubo A Math. J. 14(2), 61–80 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Luna-Elizarrarás M.E., Shapiro M., Struppa D.C., Vajiac A.: Complex Laplacian and derivatives of bicomplex functions. Complex Anal. Oper. Theory 7(5), 1675–1711 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.: Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers, Frontiers in Mathematics. Birkhäuser, Basel (2015)

  7. Meyer, P.A.: Quantum Probability for Probabilists. Lecture Notes in Mathematics. Springer, Berlin (1993)

  8. Parthasarathy, K.R.: An Introduction to Quantum Stochastic Calculus. Monographs in Mathematics, vol. 85. Birkhäuser, Basel (1992)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Chapman University, von Neumann Hall, Orange, CA, 92866, USA

    Daniel Alpay

  2. Escuela Superior de Fisica y Matemáticas, Instituto Politécnico Nacional, Mexico City, Mexico

    M. Elena Luna-Elizarrarás & Michael Shapiro

Authors
  1. Daniel Alpay
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Elena Luna-Elizarrarás
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael Shapiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Alpay.

Additional information

M. Elena Luna-Elizarrarás and Michael Shapiro–On leave from Escuela Superior de Fisica y Matemáticas, Instituto Politécnico Nacional.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alpay, D., Luna-Elizarrarás, M.E. & Shapiro, M. Kolmogorov’s Axioms for Probabilities with Values in Hyperbolic Numbers. Adv. Appl. Clifford Algebras 27, 913–929 (2017). https://doi.org/10.1007/s00006-016-0706-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00006-016-0706-6

Search

Navigation