Applications of deferred Cesàro statistical convergence of sequences of fuzzy numbers of order (ξ, ω)

The purpose of this article is to study deferred Cesrào statistical convergence of order (ξ, ω) associated with a modulus function involving the concept of difference sequences of fuzzy numbers. The study reveals that the statistical convergence of these newly formed sequence spaces behave well for ξ ≤ ω and convergence is not possible for ξ > ω. We also define p-deferred Cesàro summability and establish several interesting results. In addition, we provide some examples which explain the validity of the theoretical results and the effectiveness of constructed sequence spaces. Finally, with the help of MATLAB software, we examine that if the sequence of fuzzy numbers is bounded and deferred Cesàro statistical convergent of order (ξ, ω) in (Δ, F, f), then it need not be strongly p-deferred Cesàro summable of order (ξ, ω) in general for 0 < ξ ≤ ω ≤ 1.

A new density-modification procedure extending the application of the recent |ρ|-based phasing algorithm to larger crystal structures

The incorporation of the new peakness-enhancing fast Fourier transform compatible ipp procedure (ipp = inner-pixel preservation) into the recently published SM algorithm based on |ρ| [Rius (2020). Acta Cryst A76, 489–493] improves its phasing efficiency for larger crystal structures with atomic resolution data. Its effectiveness is clearly demonstrated via a collection of test crystal structures (taken from the Protein Data Bank) either starting from random phase values or by using the randomly shifted modulus function (a Patterson-type synthesis) as initial ρ estimate. It has been found that in the presence of medium scatterers (e.g. S or Cl atoms) crystal structures with 1500 × c atoms in the unit cell (c = number of centerings) can be routinely solved. In the presence of strong scatterers like Fe, Cu or Zn atoms this number increases to around 5000 × c atoms. The implementation of this strengthened SM algorithm is simple, since it only includes a few easy-to-adjust parameters.

A Note on Probability Convergence Defined by Unbounded Modulus Function and $alphabeta$-Statistical Convergence

In this paper we define f − αβ-statistical convergence of order γ in probability and f − αβ-strong p-Ces`aro summability of order γ in probability for a sequence of random variables under unbounded modulus function and examine the relation between these two concepts. We show by an example that this notion of f − αβ-statistical convergence of order γ in probability is stronger than αβ-statistical convergence of order γ in probability.