By Jesus Araujo-gomez, Bertin Diarra, Alain Escassut

This quantity includes papers in response to lectures given on the 11th foreign convention on $p$-adic practical research, which used to be held from July 5-9, 2010, in Clermont-Ferrand, France. The articles gathered right here characteristic fresh advancements in numerous components of non-Archimedean research: Hilbert and Banach areas, finite dimensional areas, topological vector areas and operator conception, strict topologies, areas of continuing capabilities and of strictly differentiable capabilities, isomorphisms among Banach features areas, and degree and integration. different themes mentioned during this quantity contain $p$-adic differential and $q$-difference equations, rational and non-Archimedean analytic capabilities, the spectrum of a few algebras of analytic capabilities, and maximal beliefs of the ultrametric corona algebra

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**Extra info for Advances in non-Archimedean Analysis: 11th International Conference P-adic Functional Analysis July 5-9, 2010 Universite Blaise Pascal, Clermont-ferrand, France**

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Yurova, Van der Put basis and p-adic dynamics, p-Adic Numbers, Ultrametric Analysis, and Applications, 2(2), 2010, pp. 175-178 Institute for Information Security, Lomonosov Moscow State University, Michurinsky Prosp. se Contemporary Mathematics Volume 551, 2011 q-diﬀerence equations in ultrametric ﬁelds. Najet Boudjrida, Abdelbaki Boutabaa and Samia Medjerab Abstract. Let K be a complete ultrametric algebraically closed ﬁeld and let M(K) be the ﬁeld of meromorphic functions in all K. , As (X) (s ≥ 1) be elements of K(X) such that A0 (X)As (X) = 0.

Kaup and S. Vasilache) Sur quelques propri´et´es des modules Aconvexes. Ann. Mat. Pura Appl. (4) 106 (1975), 155-169. [19] On L-bases. J. Math. Anal. Appl. 53 (1976), 508-520. [20] On a class of locally convex spaces with a Schauder basis. Indag. Math. 38 (1976), 307-312. [21] Structure theorems for locally K-convex spaces. Indag. Math. 39 (1977), 11-22. B. Robinson) Compact maps and embeddings from an inﬁnite type power series space to a ﬁnite type power series space. J. Reine Angew. Math. 293/294 (1977), 52-61.

N − 1 and mn−1 = 0, then Bm = g(m) − g(m − mn−1 pn−1 ), g(m), if m ≥ p; otherwise. It worth notice also that χ(m, x) is merely a characteristic function of the ball of radius p− logp m −1 centered at m ∈ N0 . 2. Results Given a function f : Zp → Zp , represent f via van der Put series: ∞ f (x) = Bm χ(m, x). 1. The function f : Zp → Zp is compatible if and only if it can be represented as ∞ p f (x) = logp m bm χ(m, x), m=0 where bm ∈ Zp for m = 0, 1, 2, . .. In other words, the function f is compatible (that is, satisﬁes Lipschitz condition with a constant 1) if and only if |Bm |p ≤ p− logp m for all m = 0, 1, 2, .