Hyperseries in the non-Archimedean ring of Colombeau generalized numbers

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 ∑ n = 0 + ∞ a n of generalized numbers is convergent if and only if a n → 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.

Supremum, infimum and hyperlimits in the non-Archimedean ring of Colombeau generalized numbers

It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers R ~ does not generalize classical results. E.g. the sequence 1 n ↛ 0 and a sequence ( x n ) n ∈ N converges if and only if x n + 1 - x n → 0 . 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 R ~ 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.

Reconstruction of bremsstrahlung spectra from attenuation data using generalized simulated annealing

A generalized simulated annealing algorithm, combined with a suitable smoothing regularization function is used to solve the inverse problem of X-ray spectrum reconstruction from attenuation data. The approach is to set the initial acceptance and visitation temperatures and to standardize the terms of objective function to automate the algorithm to accommodate different spectra ranges. Experiments with both numerical and measured attenuation data are presented. Results show that the algorithm reconstructs spectra shapes accurately. It should be noted that in this algorithm, the regularization function was formulated to guarantee a smooth spectrum, thus, the presented technique does not apply to X-ray spectrum where characteristic radiation are present.