Vol 9, No 2 (2017)

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.

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.

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.

In this paper, for functions on the group Q_p, 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.