Open Access
Access granted
Subscription Access
Vol 9, No 1 (2017)
- Year: 2017
- Articles: 8
- URL: https://journals.rcsi.science/2070-0466/issue/view/12505
Research Articles
1-14
The p-Adic order of the k-Fibonacci and k-Lucas numbers
Abstract
Let (Fk,n)n and (Lk,n)n be the k-Fibonacci and k-Lucas sequence, respectively, which satisfies the same recursive relation an+1 = kan + an−1 with initial values Fk,0 = 0, Fk,1 = 1, Lk,0 = 2 and Lk,1 = k. In this paper, we characterize the p-adic orders νp(Fk,n) and νp(Lk,n) for all primes p and all positive integers k.
15-21
The comb representation of compact ultrametric spaces
Abstract
We call a comb a map f: I → [0,∞), where I is a compact interval, such that {f ≥ ε} is finite for any ε > 0. A comb induces a (pseudo)-distance \({\overline d _f}\) on {f = 0} defined by \({\overline d _f}\left( {s,t} \right) = {\max _{\left( {s \wedge t,s \vee t} \right)}}f\). We describe the completion \(\overline I \) of {f = 0} for this metric, which is a compact ultrametric space called the comb metric space.
Conversely, we prove that any compact, ultrametric space (U, d) without isolated points is isometric to a comb metric space. We show various examples of the comb representation of well-known ultrametric spaces: the Kingman coalescent, infinite sequences of a finite alphabet, the p-adic field and spheres of locally compact real trees. In particular, for a rooted, locally compact real tree defined from its contour process h, the comb isometric to the sphere of radius T centered at the root can be extracted from h as the depths of its excursions away from T.
22-38
Self-adjoint approximations of the degenerate Schrödinger operator
Abstract
The problem of construction a quantum mechanical evolution for the Schrödinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have selfadjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of these terms are corresponded to the unitary and shift components of Wold’s decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.
39-52
Generalized fractional integrals in p-adic morrey and Herz spaces
Abstract
For Riesz potential Iβ(f) on p-adic linear space Qpn and its modification \(\widetilde{I^\beta }(f)\) we give sufficient conditions of their boundedness from radialMorrey space to anotherMorrey or Campanato space. Also we study the boundedness of modified Riesz potential \(\widetilde{I^\beta }(f)\) from Herz space to special Campanato spaces.
53-61
Limit theorems for p-adic valued asymmetric semistable laws and processes
Abstract
Limit distributions of scaled sums of p-adic valued i.i.d. are characterized as semistable laws, and a condition to assure the weak convergence of a scaled sum is verified. The limit supremum of the norm of the weakly convergent scaled sum is divergent in fact, and the exact growth rate of the sum is given. It is also shown that, if a scaled sum including a time parameter in the number of the added i.i.d. is considered, the semigroup of the limit distributions corresponds to a p-adic valued Markov process having right continuous sample paths with left limits. These are generalizations of the former results for rotation-symmetric i.i.d., with some necessary modifications.
62-77
Short Communications
On the topological structure of a mathematical model of human unconscious
Abstract
On the basis of two our previous works, in this paper, following Jacques Lacan psychoanalytic theory, we wish to outline some further remarks on the topological structure of a mathematical model of human unconscious.
78-81
82-85