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Probabilistic norms on the homeomorphisms of a group

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Abstract

In this paper, a probabilistic metric on the set of all auto-homeomorphisms of a group is presented. We show that the defined probabilistic group metric is right invariant and it implies a probabilistic group norm. In addition, by the probabilistic norm admissibility condition, we study the uniform continuity of homeomorphisms, and finally, we prove some theorems about topologically equivalent probabilistic norms.

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References

  • Alsina C, Schweizer B, Sklar A (1993) On the definition of a probabilistic normed space. Aequ Math 46:91–98

    Article  MathSciNet  Google Scholar 

  • Bingham NH, Ostaszewski AJ (2010a) Normed versus topological groups: dichotomy and duality, Dissertationes Mathematicae, Institute of Mathematics, Polish Academy of Sciences, pp 1–138

  • Bingham NH, Ostaszewski AJ (2010b) Topological regular variation: I. Slow variation. Topol Appl 157:1999–2013

    Article  MathSciNet  Google Scholar 

  • Birkhoff G (1936) A note on topological groups. Compos Math 3:427–430

    MathSciNet  MATH  Google Scholar 

  • Farkas DR (1990) The algebra of norms and expanding maps on groups. J Algebra 133(2):386–403

    Article  MathSciNet  Google Scholar 

  • Kakutani S (1986) Über die Metrisation der topologischen Gruppen, Proc. Imp. Acad. Tokyo 12 (1936) 82-84 (also in Selected Papers, Vol. 1, ed. Robert R. Kallman, Birkhäuser. 60-62)

  • Klee VL (1952) Invariant metrics in groups (solution of a problem of Banach). Proc Am Math Soc 3:484–487

    Article  MathSciNet  Google Scholar 

  • Klement E, R Mesiar, Pap E (2000) Triangular norms, vol 8. Trends in logic—studia logica library. Kluwer Academic Publishers, Dordrecht

    Book  Google Scholar 

  • Menger K (1942) Statistical metrics. Proc Nat Acad Sci USA 28:535–537

    Article  MathSciNet  Google Scholar 

  • Nourouzi K, Pourmoslemi AR (2017) Probabilistic normed groups. Iran J Fuzzy Syst 14(1):99–113

    MathSciNet  MATH  Google Scholar 

  • Pavlov AA (2003) Normed groups and their applications in noncommutative differential geometry. J Math Sci 113:675–682

    Article  MathSciNet  Google Scholar 

  • Pettis BJ (1950) On continuity and openness of homomorphisms in topological groups. Ann Math 2:293–308

    Article  MathSciNet  Google Scholar 

  • Pourmoslemi A, Salimi M (2009) Probabilistic n-normed spaces, D-n-compact sets and D-n-bounded sets. Chaos Solitons Fractals 42(5):2729–2734

    Article  MathSciNet  Google Scholar 

  • Schweizer B (1975) Multiplication on the space of probability distribution functions. Aequ Math 12:156–183

    Article  MathSciNet  Google Scholar 

  • Schweizer B (1983) Sklar a.: probabilistie metrie spaces. Elsevier North Holland, New York

    Google Scholar 

  • Sempi C (1982) On the space of distribution functions. Riv Mat Univ Parma 8:243–250

    MathSciNet  MATH  Google Scholar 

  • Šerstnev AN (1963) On the motion of a random normed space. Dokl Akad Nauk SSSR 149:280–283

    MathSciNet  Google Scholar 

  • Sibley DA (1971) A metric for weak convergence of distribution functions. Rocky Mt J Math 1:427–430

    Article  MathSciNet  Google Scholar 

  • Taylor MD (1985) New metrics for weak convergence of distribution functions. Stochastica 9:5–17

    MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Payame Noor University, POB 19395-4697, Tehran, Iran

    Alireza Pourmoslemi

  2. Department of Law, Economics and Human Sciences and Decisions Lab, University Mediterranea of Reggio Calabria, Reggio Calabria, Italy

    Massimiliano Ferrara & Bruno Antonio Pansera

  3. ICRIOS, The Invernizzi Center for Research on Innovation, Organization, Strategy and Entrepreneurship, Bocconi University, Milan, Italy

    Massimiliano Ferrara

  4. Center for Dynamics and Institute for Analysis, Department of Mathematics, Technische Universität Dresden, Dresden, Germany

    Mehdi Salimi

Authors
  1. Alireza Pourmoslemi
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  2. Massimiliano Ferrara
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  3. Bruno Antonio Pansera
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  4. Mehdi Salimi
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Corresponding author

Correspondence to Mehdi Salimi.

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The authors declare that they have no conflict of interest.

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Communicated by A. Di Nola.

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Pourmoslemi, A., Ferrara, M., Pansera, B.A. et al. Probabilistic norms on the homeomorphisms of a group. Soft Comput 24, 7021–7028 (2020). https://doi.org/10.1007/s00500-020-04818-7

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  • DOI: https://doi.org/10.1007/s00500-020-04818-7

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