Advertisement

Algorithmics of Checking whether a Mapping Is Injective, Surjective, and/or Bijective

  • E. Cabral BalreiraEmail author
  • Olga Kosheleva
  • Vladik Kreinovich
Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 539)

Abstract

In many situations, we would like to check whether an algorithmically given mapping f:A → B is injective, surjective, and/or bijective. These properties have a practical meaning: injectivity means that the events of the action f can be, in principle, reversed, while surjectivity means that every state b ∈ B can appear as a result of the corresponding action. In this paper, we discuss when algorithms are possible for checking these properties.

Keywords

Polynomial Mapping Integer Solution Computable Mapping Mathematical Term Hilbert Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Balreira, E.C.: Foliations and global inversion. Commentarii Mathematici Helvetici 85(1), 73–93 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Balreira, E.C.: Incompressibility and global inversion. Topological Methods in Nonlinear Analysis 35(1), 69–76 (2010)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Balreira, E.C., Radulescu, M., Radulescu, S.: A generalization of the Fujisawa-Kuh Global Inversion Theorem. Journal of Mathematical Analysis and Applications 385(2), 559–564 (2011)MathSciNetGoogle Scholar
  4. 4.
    Bass, H., Connell, E.H., Wright, D.: The Jacobian Conjecture: reduction of degree and formal expansion of the inverse. Bull. Amer. Math. Soc. 7(2), 287–330 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer, Berlin (2006)zbMATHGoogle Scholar
  6. 6.
    Bishop, E., Bridges, D.S.: Constructive Analysis. Springer, N.Y (1985)CrossRefzbMATHGoogle Scholar
  7. 7.
    Keller, O.-H.: Ganze Cremona-Transformationen. Monatshefte für Mathematik und Physik 47(1), 299–306 (1939)Google Scholar
  8. 8.
    Kreinovich, V., Lakeyev, A., Rohn, J., Kahl, P.: Computational complexity and feasibility of data processing and interval computations. Kluwer, Dordrecht (1998)CrossRefzbMATHGoogle Scholar
  9. 9.
    Mishra, B.: Computational real algebraic geometry. In: Handbook on Discreet and Computational Geometry. CRC Press, Boca Raton (1997)Google Scholar
  10. 10.
    Papadimitriou, C.H.: Computational Complexity. Addison Wesley, San Diego (1994)zbMATHGoogle Scholar
  11. 11.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Berlin (1989)CrossRefzbMATHGoogle Scholar
  12. 12.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. Berkeley, Los Angeles (1951)zbMATHGoogle Scholar
  13. 13.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • E. Cabral Balreira
    • 1
    Email author
  • Olga Kosheleva
    • 2
  • Vladik Kreinovich
    • 2
  1. 1.Department of MathematicsTrinity UniversitySan AntonioUSA
  2. 2.University of Texas at El PasoEl PasoUSA

Personalised recommendations