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Том 9, № 2 (2017)
- Год: 2017
- Статей: 6
- URL: https://journals.rcsi.science/2070-0466/issue/view/12511
Review Articles
p-Adic mathematical physics: the first 30 years
Аннотация
p-Adic mathematical physics is a branch of modern mathematical physics based on the application of p-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of p-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.
87-121
Research Articles
Positive operators on a free Banach space over the complex Levi-Civita field
Аннотация
Let C be the complex Levi-Civita field and let c0(C) or, simply, c0 denote the space of all null sequences of elements of C. A non-Archimedean norm is defined naturally on c0 with respect to which c0 is a Banach space. In this paper, we study the properties of positive operators on c0 which are similar to those of positive operators in classical functional analysis; however the proofs of many of the results are nonclassical. Then we use our study of positive operators to introduce a partial order on the set of compact and self-adjoint operators on c0 and study the properties of that partial order.
122-137
Density of characters of bounded p-adic analytic functions in the topological dual
Аннотация
Let K be an ultrametric complete algebraically closed field, let D be a disk {x ∈ K ‖x| < R} (with R in the set of absolute values of K) and let A be the Banach algebra of bounded analytic functions in D. The vector space generated by the set of characters of A is dense in the topological dual of A if and only if K is not spherically complete. Let H(D) be the Banach algebra of analytic elements in D. The vector space generated by the set of characters of H(D) is never dense in the topological dual of H(D).
138-143
On stationary points of multivalued strongly contractive mappings in partially ordered ultrametric spaces and non-Archimedean
Аннотация
The purpose of this paper is to present a fixed point theorem for multivalued strongly contractive mappings in partially ordered ultrametric spaces. Also a consequence is obtained for these mappings in partially ordered non-Archimedean normed spaces. Finally, two supporting examples of these results are given.
144-150
On solutions of the mixed Dirichlet-Steklov problem for the biharmonic equation in exterior domains
Аннотация
We study the unique solvability of the mixed Dirichlet-Steklov problem for the biharmonic equation in exterior domains under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight |x|a. Depending on the value of the parameter a, we prove uniqueness theorem or present exact formulas for the dimension of the solution space of the mixed Dirichlet-Steklov problem in the exterior of a compact set.
151-157
An analogue of the Titchmarsh theorem for the Fourier transform on the group of p-adic numbers
Аннотация
In this paper, for functions on the group Qp, we prove an analogue of the classical Titchmarsh theorem on description of the image under the Fourier transform of a set of functions satisfying the Lipschitz condition in L2.
158-164