Monatshefte fur Mathematik最新文献

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.

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 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 V be a valuation ring of a global field K. We show that for all positive integers k and $1<n_1\le \cdots \le n_k$ there exists an integer-valued polynomial on V, that is, an element of $\text{Int}\left(V\right)=\{f\in K[X]\mid f(V)\subseteq V\}$, which has precisely k essentially different factorizations into irreducible elements of $\text{Int}\left(V\right)$ whose lengths are exactly $n_1,\dots ,n_k$. In fact, we show more, namely that the same result holds true for every discrete valuation domain V with finite residue field such that the quotient field of V admits a valuation ring independent of V whose maximal ideal is principal or whose residue field is finite. If the quotient field of V 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.

This article is the natural continuation of the paper: Mukhammadiev et al. Supremum, infimum and hyperlimits of Colombeau generalized numbers in this journal. Since the ring of Robinson-Colombeau is non-Archimedean and Cauchy complete, a classical series ${\sum }_{n=0}^{+\infty }{a}_n$ of generalized numbers is convergent if and only if $a_n\to 0$ 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.

We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on ${\omega }_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening ${\mathrm{MM}}^{++}$ of Martin's Maximum does not decide whether the restriction of the non-stationary ideal on ${\omega }_2$ to sets of ordinals of countable cofinality is ${\Delta }_1$ -definable by formulas with parameters in $H\left({\omega }_3\right)$ . The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on ${\omega }_2$ and strong forcing axioms that are compatible with $\mathrm{CH}$ . Finally, we answer a question of S. Friedman, Wu and Zdomskyy by showing that the ${\Delta }_1$ -definability of the non-stationary ideal on ${\omega }_2$ is compatible with arbitrary large values of the continuum function at ${\omega }_2$ .

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