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P-Adic metric preserving functions and their analogues

  • Robert W. Vallin and Oleksiy A. Dovgoshey EMAIL logo
Published/Copyright: April 14, 2021
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 71 Issue 2

Abstract

The p-adic completion ℚp of the rational numbers induces a different absolute value |⋅|p than the typical | ⋅| we have on the real numbers. In this paper we compare and contrast functions f : ℝ+ → ℝ+, for which the composition with the p-adic metric dp generated by |⋅|p is still a metric on ℚp, with the usual metric preserving functions and the functions that preserve the Euclidean metric on ℝ. In particular, it is shown that f ∘ dp is still an ultrametric on ℚp if and only if there is a function g such that f ∘ dp = g ∘ dp and g ∘ d is still an ultrametric for every ultrametric d. Some general variants of the last statement are also proved.

Keywords: ultrametric space; p-adic number; ultrametric preserving function; metric preserving function; poset
MSC 2010: Primary 54E35; 26A21

The second author was partially supported in the frame of the project: Development of Mathematical Models, Numerical and Analytical Methods, and Algorithms for Solving Modern Problems of Biomedical Research. State registration number: 0117U002165.

  1. (Communicated by David Buhagiar )

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Received: 2019-12-22
Accepted: 2020-06-23
Published Online: 2021-04-14
Published in Print: 2021-04-27

© 2021 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2017-0476
Keywords for this article
ultrametric space; p-adic number; ultrametric preserving function; metric preserving function; poset

Articles in the same Issue

  1. Regular papers
  2. Prof. RNDr. Michal Fečkan, DrSc. – Sexagenarian?
  3. Tribonacci numbers with two blocks of repdigits
  4. Padovan numbers that are concatenations of two distinct repdigits
  5. On the 2-rank and 4-rank of the class group of some real pure quartic number fields
  6. A general inverse matrix series relation and associated polynomials – II
  7. Some hardy type inequalities with finsler norms
  8. Starlikeness and convexity of the product of certain multivalent functions with higher-order derivatives
  9. Block Hessenberg matrices and spectral transformations for matrix orthogonal polynomials on the unit circle
  10. How is the period of a simple pendulum growing with increasing amplitude?
  11. Fourier transforms of convolution operators on orlicz spaces
  12. Some characterizations of property of trans-Sasakian 3-manifolds
  13. P-Adic metric preserving functions and their analogues
  14. On statistical convergence of sequences of closed sets in metric spaces
  15. A characterization of the uniform convergence points set of some convergent sequence of functions
  16. A nonparametric estimation of the conditional ageing intensity function in censored data: A local linear approach
  17. Donsker’s fuzzy invariance principle under the Lindeberg condition
  18. Characterization of generalized Gamma-Lindley distribution using truncated moments of order statistics
  19. Matrix variate pareto distributions
  20. Global exponential periodicity and stability of neural network models with generalized piecewise constant delay
  21. Optimal inequalities for contact CR-submanifolds in almost contact metric manifolds
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