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Том 9, № 4 (2017)
- Год: 2017
- Статей: 6
- URL: https://journals.rcsi.science/2070-0466/issue/view/12518
Research Articles
Trees and ultrametric Möbius structures
Аннотация
We define the concept of an ultrametric Möbius space (Z,M) and show that the boundary at infinity of a nonelementary geodesically complete tree is naturally an ultrametric Möbius space. In addition, we construct to a given ultrametric Möbius space (Z,M) a nonelementary geodesically complete tree, unique up to isometry, with (Z,M) being its boundary at infinity. This yields a one-to-one correspondence.
247-256
p-adic valuation of exponential sums associated to trinomials and some consequences
Аннотация
In this paper, we compute the p-adic valuation of exponential sums associated to trinomials \(F\left( X \right) = a{X^{{d_1}}} + b{X^{{d_2}}} + c{X^{{d_3}}}\) over Fp. As a byproduct of our results, we obtain restrictions for permutation polynomials of type \(a{X^{{d_1}}} + b{X^{{d_2}}} + c{X^{{d_3}}}\) over Fp.
257-266
Polyadic integer numbers and finite (m, n)-fields
Аннотация
The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler function, polyadic division with a remainder, etc. are introduced. Secondary congruence classes of polyadic integer numbers, which become ordinary residue classes in the “binary limit”, and the corresponding finite polyadic rings are defined. Polyadic versions of (prime) finite fields are introduced. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is possible in the binary case. It is conjectured that a finite polyadic field should contain a certain canonical prime polyadic field, defined here, as a minimal finite subfield, which can be considered as a polyadic analogue of GF (p).
267-291
On solvability of one class of nonlinear integral equations on whole line with a weak singularity at zero
Аннотация
In present work we investigate one class of nonlinear integral equations with singularity at zero and boundary value conditions at ±∞. Above mentioned class of equations has direct applications in string theory and in the case of concrete structure of the kernel it describes the dynamics of the open-closed p-adic string for the scalar tachyon field. We prove the existence of nontrivial solution in a certain weight class of functions.With an additional restriction on the kernel the uniqueness of the obtained solution is proved.
292-305
An analogue of the Titchmarsh theorem for the Fourier transform on locally compact Vilenkin groups
Аннотация
In this paper for functions on locally compact Vilenkin groups, we prove an analogue of one classical Titchmarsh theorem on the image under the Fourier transform of a set of functions satisfying the Lipschitz condition in L2.
306-313
Local zeta functions, pseudodifferential operators and Sobolev-type spaces over non-Archimedean local fields
Аннотация
In this articlewe introduce a new type of local zeta functions and study some connections with pseudodifferential operators in the framework of non-Archimedean fields. The new local zeta functions are defined by integrating complex powers of norms of polynomials multiplied by infinitely pseudo-differentiable functions. In characteristic zero, the new local zeta functions admit meromorphic continuations to the whole complex plane, but they are not rational functions. The real parts of the possible poles have a description similar to the poles of Archimedean zeta functions. But they can be irrational real numbers while in the classical case are rational numbers. We also study, in arbitrary characteristic, certain connections between local zeta functions and the existence of fundamental solutions for pseudodifferential equations.
314-335