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Let is the -algebra of entire functions defined on a complete ultrametric algebraically closed field. For a p-adic entire function and for r > 0, sup{| f (x)|:|x|= r} is denoted by | f |(r), where |⋅|(r) is a multiplicative norm on. Taking φ(r): [0, +∞) → (0, +∞) as a non-decreasing unbounded function of r, in this paper we develop some results of composite p-adic entire functions in terms of their relative (p, q) – φ order and relative (p, q) - φ lower order along with relative (p, q) - φ type and relative (p, q) - φ weak type, where p, q are two positive integers. [ABSTRACT FROM AUTHOR]

A new Nevanlinna theorem on q p-adic small functions is given. Let f, g, be two meromorphic functions on a complete ultrametric algebraically closed field K of characteristic 0, or two meromorphic functions in an open disk of K , that are not quotients of bounded analytic functions by polynomials. If f and g share 7 small meromorphic functions I.M., then f = g . Better results hold when f and g satisfy some property of growth. Particularly, if f and g have finitely many poles or finitely many zeros and share 3 small meromorphic functions I.M., then f = g . [ABSTRACT FROM AUTHOR]

In many situations, the notion of function is not sufficient and it needs to be extended. A classical way to do this is to introduce the notion of weak solution; another approach is to use generalized functions. Ultrafunctions are a particular class of generalized functions that has been previously introduced and used to define generalized solutions of stationary problems in [4,7,9,11,12] . In this paper we generalize this notion in order to study also evolution problems. In particular, we introduce the notion of Generalized Ultrafunction Solution (GUS) for a large family of PDEs, and we confront it with classical strong and weak solutions. Moreover, we prove an existence and uniqueness result of GUS's for a large family of PDEs, including the nonlinear Schroedinger equation and the nonlinear wave equation. Finally, we study in detail GUS's of Burgers' equation, proving that (in a precise sense) the GUS's of this equation provide a description of the phenomenon at microscopic level. [ABSTRACT FROM AUTHOR]

We investigate whether certain Diophantine equations have or have not solutions in entire or meromorphic functions defined on a non-Archimedean algebraically closed field of characteristic zero. We prove that there are no non-constant meromorphic functions solving the Erdös–Selfridge equation except when the corresponding curve is a conic. We also show that there are infinitely many non-constant entire solutions to the Markoff–Hurwitz equation. [ABSTRACT FROM PUBLISHER]

Letbe a complete ultrametric algebraically closed field of characteristic 0. According to thep-adic Hayman conjecture, given a transcendental meromorphic functionfin, for each,takes every valueinfinitely many times. It was proven by the second author for. Here, we prove it forby using properties of meromorphic functions having finitely many multiple poles. [ABSTRACT FROM PUBLISHER]

Using the fixed point method, we prove the Hyers-Ulam stability of an orthogonally quintic functional equation in Banach spaces and in non-Archimedean Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper we continue the development of the differential calculus started in Aragona et al. (Monatsh. Math. 144:13-29, ). Guided by the so-called sharp topology and the interpretation of Colombeau generalized functions as point functions on generalized point sets, we introduce the notion of membranes and extend the definition of integrals, given in Aragona et al. (Monatsh. Math. 144:13-29, ), to integrals defined on membranes. We use this to prove a generalized version of the Cauchy formula and to obtain the Goursat Theorem for generalized holomorphic functions. A number of results from classical differential and integral calculus, like the inverse and implicit function theorems and Green's theorem, are transferred to the generalized setting. Further, we indicate that solution formulas for transport and wave equations with generalized initial data can be obtained as well. [ABSTRACT FROM AUTHOR]

We discuss Mumford curves in the pencil on a Del Pezzo quintic surface constructed by Edge [Ed1]. The abstract group structures of the normalizer of the corresponding Schottky groups are described, which give us some knowledges on Mumford loci in moduli space of curves. [ABSTRACT FROM AUTHOR]

We study some connections between Non-Standard Analysis and algebraical theories of generalized functions. More precisely, we point out the fact that the construction of these algebras can be interpreted in terms of Non Standard asymptotic properties. For example, we show that the construction of the J. F. Colombeau'Algebra is equivalent (in a certain sense) to a monadic property. Conversely, we show that a certain galactic property (being exponentially small with respect to a parameter) leads to a new algebra. [ABSTRACT FROM AUTHOR]