We consider Diophantine equations of the shape , where the polynomials and are elements of power sums. Using a finiteness criterion of Bilu and Tichy, we will prove that under suitable assumptions infinitely many rational solutions (, ) with a bounded denominator are only possible in trivial cases.

It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers does not generalize classical results. E.g. the sequence and a sequence converges and only if . This has several deep consequences, e.g. in the study of series, analytic generalized functions, or sigma-additivity and classical limit theorems in integration of generalized functions. The lacking of these results is also connected to the fact that is necessarily not a complete ordered set, e.g. the set of all the infinitesimals has neither supremum nor infimum. We present a solution of these problems with the introduction of the notions of hypernatural number, hypersequence, close supremum and infimum. In this way, we can generalize all the classical theorems for the hyperlimit of a hypersequence. The paper explores ideas that can be applied to other non-Archimedean settings.

This article is the natural continuation of the paper: Mukhammadiev et al. in this journal. Since the ring of Robinson-Colombeau is non-Archimedean and Cauchy complete, a classical series of generalized numbers is convergent and only if in the sharp topology. Therefore, this property does not permit us to generalize several classical results, mainly in the study of analytic generalized functions (as well as, e.g., in the study of sigma-additivity in integration of generalized functions). Introducing the notion of hyperseries, we solve this problem recovering classical examples of analytic functions as well as several classical results.

Given a finite set of primes and an -tuple of positive, distinct integers we call the -tuple -Diophantine, if for each the quantity has prime divisors coming only from the set . For a given set we give a practical algorithm to find all -Diophantine quadruples, provided that .

Consider the divisor sum for integers and . We extract an asymptotic formula for the average divisor sum in a convenient form, and provide an explicit upper bound for this sum with the correct main term. As an application we give an improvement of the maximal possible number of -quadruples.

This article is a natural continuation of the paper Tiwari, D., Giordano, P., in this journal. We study one variable hyper-power series by analyzing the notion of radius of convergence and proving classical results such as algebraic operations, composition and reciprocal of hyper-power series. We then define and study one variable generalized real analytic functions, considering their derivation, integration, a suitable formulation of the identity theorem and the characterization by local uniform upper bounds of derivatives. On the contrary with respect to the classical use of series in the theory of Colombeau real analytic functions, we can recover several classical examples in a non-infinitesimal set of convergence. The notion of generalized real analytic function reveals to be less rigid both with respect to the classical one and to Colombeau theory, e.g. including classical non-analytic smooth functions with flat points and several distributions such as the Dirac delta. On the other hand, each Colombeau real analytic function is also a generalized real analytic function.

We prove that a family of Diophantine series satisfies an approximate functional equation. It generalizes a result by Rivoal and Roques and proves an extended version of a conjecture posed in their paper. We also characterize the convergence points.

Let be a number field of degree and let be an order in . A (GNS for short) is a pair where is monic and is a complete residue system modulo (0) containing 0. If each admits a representation of the form with and then the GNS is said to have the . To a given fundamental domain of the action of on we associate a class of GNS whose digit sets are defined in terms of in a natural way. We are able to prove general results on the finiteness property of GNS in by giving an abstract version of the well-known "dominant condition" on the absolute coefficient (0) of . In particular, depending on mild conditions on the topology of we characterize the finiteness property of for fixed and large . Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.

We study the generic behavior of the method of successive approximations for set-valued mappings in separable Banach spaces. We consider the case of nonexpansive mappings with convex and compact point images and show that for the typical such mapping and typical points of its domain the sequence of successive approximations is unique and converges to a fixed point of the mapping.

Let be a valuation ring of a global field . We show that for all positive integers and there exists an integer-valued polynomial on , that is, an element of , which has precisely essentially different factorizations into irreducible elements of whose lengths are exactly . In fact, we show more, namely that the same result holds true for every discrete valuation domain with finite residue field such that the quotient field of admits a valuation ring independent of whose maximal ideal is principal or whose residue field is finite. If the quotient field of is a purely transcendental extension of an arbitrary field, this property is satisfied. This solves an open problem proposed by Cahen, Fontana, Frisch and Glaz in these cases.

The study of the well-known partition function () counting the number of solutions to with integers has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into with and some fixed . In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle-point method.

The geometry of webs has been investigated over more than a century driven by still open problems. In our paper we contribute to extending the knowledge on webs from the perspective of the geometry of webs on surfaces in three dimensional space. Our study of AGAG-webs is motivated by architectural applications of gridshell structures where four families of manufactured curves on a curved surface are realizations of asymptotic lines and geodesic lines. We describe all discrete AGAG-webs in isotropic space and propose a method to construct them. Furthermore, we prove that some sub-nets of an AGAG-web are timelike minimal surfaces in Minkowski space and can be embedded into a one-parameter family of discrete isotropic Voss nets.

N-functions and their growth and regularity properties are crucial in order to introduce and study Orlicz classes and Orlicz spaces. We consider N-functions which are given in terms of so-called associated weight functions. These functions are frequently appearing in the theory of ultradifferentiable function classes and in this setting additional information is available since associated weight functions are defined in terms of a given weight sequence. We express and characterize several known properties for N-functions purely in terms of weight sequences which allows to construct (counter-) examples. Moreover, we study how for abstractly given N-functions this framework becomes meaningful and finally we establish a connection between the complementary N-function and the recently introduced notion of the so-called dual sequence.

Niederreiter and Halton sequences are two prominent classes of higher-dimensional sequences which are widely used in practice for numerical integration methods because of their excellent distribution qualities. In this paper we show that these sequences-even though they are uniformly distributed-fail to satisfy the stronger property of Poissonian pair correlations. This extends already established results for one-dimensional sequences and confirms a conjecture of Larcher and Stockinger who hypothesized that the Halton sequences are not Poissonian. The proofs rely on a general tool which identifies a specific regularity of a sequence to be sufficient for not having Poissonian pair correlations.

We investigate the existence of solutions to a recent model for large-scale equatorial waves, derived recently by an asymptotic method driven by the thin-shell approximation of the Earth's atmosphere in rotating spherical coordinates.

We construct an explicit steady stratified purely azimuthal flow for the governing equations of geophysical fluid dynamics. These equations are considered in a setting that applies to the Antarctic Circumpolar Current, accounting for eddy viscosity and forcing terms.

We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on and its restrictions to certain cofinalities. Our main result shows that the strengthening of Martin's Maximum does not decide whether the restriction of the non-stationary ideal on to sets of ordinals of countable cofinality is -definable by formulas with parameters in . The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on and strong forcing axioms that are compatible with . Finally, we answer a question of S. Friedman, Wu and Zdomskyy by showing that the -definability of the non-stationary ideal on is compatible with arbitrary large values of the continuum function at .

Let denote the near-ring of congruence preserving functions of the group . We investigate the question "When is a ring?". We obtain information externally via the lattice structure of the normal subgroups of and internally via structural properties of the group .

We consider the family of discrete Schrödinger-type operators in -dimensional lattice , where is the discrete Laplacian and is of rank-one. We prove that there exist coupling constant thresholds such that for any the discrete spectrum of is empty and for any the discrete spectrum of is a singleton and for and for Moreover, we study the asymptotics of as and as well as The asymptotics highly depends on and

We develop a theory of -matrix Riemann-Hilbert problems for a class of jump contours and jump matrices of low regularity. Our basic assumption is that the contour is a finite union of simple closed Carleson curves in the Riemann sphere. In particular, unbounded contours with cusps, corners, and nontransversal intersections are allowed. We introduce a notion of -Riemann-Hilbert problem and establish basic uniqueness results and Fredholm properties. We also investigate the implications of Fredholmness for the unique solvability and prove a theorem on contour deformation.

In this paper we describe several new aspects of the foundations of the representation theory of the space of smooth-automorphic forms (i.e., not necessarily -finite automorphic forms) for general connected reductive groups over number fields. Our role model for this space of smooth-automorphic forms is a "smooth version" of the space of automorphic forms, whose internal structure was the topic of Franke's famous paper (Ann Sci de l'ENS 2:181-279, 1998). We prove that the important decomposition along the parabolic support, and the even finer-and structurally more important-decomposition along the cuspidal support of automorphic forms transfer in a topologized version to the larger setting of smooth-automorphic forms. In this way, we establish smooth-automorphic versions of the main results of Franke and Schwermer (Math Ann 311:765-790, 1998) and of Mœglin and Waldspurger (Spectral Decomposition and Eisenstein Series, Cambridge University Press, 1995), III.2.6.