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Том 11, № 3 (2019)
- Год: 2019
- Статей: 6
- URL: https://journals.rcsi.science/2070-0466/issue/view/12540
Research Articles
Finite Ultrametric Balls
Аннотация
The necessary and sufficient conditions under which a given family ℱ of subsets of finite set X coincides with the family BX of all balls generated by some ultrametric d on X are found. It is shown that the representing tree of the ultrametric space (BX, dH) with the Hausdorff distance dH can be obtained from the representing tree TX of ultrametric space (X, d) by adding a leaf to every internal vertex of TX.
177-191
p-Adic Multiwavelet Sets
Аннотация
This paper contains a brief review of p-adic MRA theory along with the introduction of p-adic multiwavelet set. Here we have studied various properties of p-adic multiwavelet sets such as disjointness of their dilates, counting formula for the elements in a wavelet set etc. and proved some results analogous to real setting, in special cases.
192-204
Axioms of Soft Logic
Аннотация
In this paper, we develop the foundation of a new mathematical language, which we term “Soft Logic”. This language enables us to present an extension of the number 0 from a singular point to a continuous line. We create a distinction between −0 and +0 and generate a new type of numbers, which we call ‘Bridge Numbers’ (BN):
where a, b are real numbers, “a” is the value on the \(\overline {\bf{0}} \) axis, and “b” is the value on the \(\overline {\bf{1}} \) axis. We proceed by defining arithmetic and algebraic operations on the Bridge Numbers, investigate their properties, and conclude by defining goals for further research. In the Attachment, we continue comparing our results with existing mathematical work on Infinitesimals, Dual numbers, and Nonstandard analysis. The research is a part of “Digital living 2030” project with Stanford University.
205-215
Some Results on Arithmetic Properties of p-Adic Liouville Numbers
Аннотация
In 1906, Clark defined and studied the set of p-adic Liouville numbers and, in 1985, Schikhof also studied this set in his book Ultrametric Calculus. In this paper, we introduce the set of weak p-adic Liouville numbers, which is a set of transcendental p-adic numbers that contains the p-adic Liouville numbers, and we show some properties about these numbers. In particular, we shall prove an analogous result to a classic theorem of Maillet about Liouville numbers.
216-222
Adinkras, Dessins, Origami, and Supersymmetry Spectral Triples
Аннотация
We investigate the spectral geometry and spectral action functionals associated to 1D Supersymmetry Algebras, using the classification of these superalgebras in terms of Adinkra graphs and the construction of associated dessin d’enfant and origami curves. The resulting spectral action functionals are computed in terms of the Selberg (super) trace formula and of a Poisson summation formula, respectively.
223-247
Periodic Jacobi-Perron Algorithms in Cubic Fields and Fundamental Units
Аннотация
In this paper, we study the Jacobi-Perron algorithm of (α, α2) and (1/α, 1/α2) where α is the unique real root of monic cubic irreducible polynomials in certain infinite families. We also investigate the associated Hasse-Bernstein units, along with when they are fundamental units in ℤ[α] and ℚ(α).
248-254