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Vol 10, No 2 (2018)
- Year: 2018
- Articles: 6
- URL: https://journals.rcsi.science/2070-0466/issue/view/12524
Research Articles
C-Algebras on some Free-Banach Spaces
Abstract
The main goal of this work is to study the Gelfand spaces of some commutative Banach algebras with unit within the space of bounded linear operators. We will also show, under special condition, that this algebra is isometrically isomorphic to some space of continuous functions defined over a compact. Such isometries preserve idempotent elements. This fact will allow us to define the respective associated measure which is known as spectral measure. Let us also notice that this measure is obtained by restriction of the reciprocal of the Gelfand transform to the set of characteristic functions of clopen subsets of the spectrum of above algebra. We will finish this work showing that each element of such algebras described above can be represented as an integral of some continuous function, where the integral has been defined through the spectral measure.
81-89
Solvability of a Class of Nonlinear Pseudo-Differential Equations in ℝn
Abstract
In the present work the existence of continuous and bounded solutions for a class of nonlinear pseudo-differential equations on ℝn is proved. The monotonicity, asymptotic behavior and other properties for obtained solutions are also presented. Mentioned class of equations arises in p-adic string theory.
90-99
Tauberian Theorems for the (\(\bar N\) , pm,n) and (M, λm,n) Methods for Double Series over Ultrametric Fields
Abstract
In this paper, K denotes a complete, non-trivially valued, non-archimedean (or ultrametric) field. Entries of double sequences, double series and 4-dimensional infinite matrices are in K.We prove Tauberian theorems for the Weighted Mean and (M,λm,n) methods for double series.
100-107
Weak Similarities of Finite Ultrametric and Semimetric Spaces
Abstract
Weak similarities form a special class of mappings between semimetric spaces. Two semimetric spaces X and Y are weakly similar if there exists a weak similarity Φ: X → Y. We find a structural characteristic of finite ultrametric spaces for which the isomorphism of its representing trees implies a weak similarity of the spaces. We also find conditions under which the Hasse diagrams of balleans of finite semimetric spaces are isomorphic.
108-117
Some Problems in the Theory of Approximation of Functions on the Group of p-Adic Numbers
Abstract
Some problems in the theory of approximation of complex-valued functions on the group Qp in the metric of Lρ, 1 ≤ ρ≤∞ by functions with bounded spectrum, are investigated. A description of certain function spaces in terms of the best approximations are obtained and some imbedding theorems are proved.
118-129
Model of p-Adic Random Walk in a Potential
Abstract
We consider the p-adic random walk model in a potential which can be viewed as a generalization of p-adic random walk models used for describing protein conformational dynamics. This model is based on the Kolmogorov-Feller equations for the distribution function defined on the field of p-adic numbers in which the transition rate depends on ultrametric distance between the transition points as well as on function of potential violating the spatial homogeneity of a random process. This equation which will be called the equation of p-adic random walk in a potential, is equivalent to the equation of p-adic random walk with modified measure and reaction source. With a special choice of a power-law potential the last equation is shown to have an exact analytic solution. We find the analytic solution of the Cauchy problem for such equation with an initial condition, whose support lies in the ring of integer p-adic numbers.We also examine the asymptotic behaviour of the distribution function for large times. It is shown that in the limit t→∞ the distribution function tends to the equilibrium solution according to the law, which is bounded from above and below by power laws with the same exponent. Our principal conclusion is that the introduction of a potential in the model of p-adic random walk conserves the power-law behaviour of relaxation curves for large times.
130-150