Vol 9, No 3 (2017)

For each finite set S of prime numbers there exists a unique completion ℚ_S of ℚ, which is a second countable, locally compact and totally disconnected topological ring. This topological ring has a natural ultrametric that allows to define a pseudodifferential operator D^α and to study an abstract heat equation on the Hilbert space L²(ℚ_S). The fundamental solution of this equation is a normal transition function of a Markov process on ℚ_S. The techniques developed provides a general framework for these kind of problems on different ultrametric groups.

This paper is concerned with dynamics in general G-ultrametric spaces, hence we discuss the introduced concepts of such spaces. Also, we obtain some fixed point existing results of strongly contractive and non-expansive mappings defined on these spaces by inspiring from the theorems proved by Mustafa and Sims.

Let (F_n)_n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as

. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±2 (mod 5), then \(p{\left| {\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]} \right._F}\) for all integers a ≥ 1. In 2015, Marques and Trojovský worked on the p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all a ≥ 1 when p ≠ 5. In this paper, we shall provide the exact p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a, b ≥ 1 and for all prime number p.

In this paper we study the spectral theory for the class of linear operators A_∞ defined on the so-called non-archimedean Hilbert space E_ω by, A_∞:= D + F_∞ where D is an unbounded diagonal linear operator and F_∞:= Σ_k=1^∞u_k ⊗ v_k is an operator of infinite rank on E_ω.