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On statistical convergence in quasi-metric spaces

İlkhan Merve, Kara Emrah Evren

Demonstratio Mathematica

2019

Vol. 52, nr 1

225--236

A quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric in general. Quasi-metrics are a subject of comprehensive investigation both in pure and applied mathematics in areas such as in functional analysis, topology and computer science. The main purpose of this paper is to extend the convergence and Cauchy conditions in a quasi-metric space by using the notion of asymptotic density. Furthermore, some results obtained are related to completeness, compactness and precompactness in this setting using statistically Cauchy sequences.

On the Asymptotic Density of Prime k-tuples and a Conjecture of Hardy and Littlewood

Tóth L.

Computational Methods in Science and Technology

2019

Vol. 25, No. 3

143--148

In 1922 Hardy and Littlewood proposed a conjecture on the asymptotic density of admissible prime k-tuples. In 2011, Wolf computed the “Skewes number” for twin primes, i.e., the first prime at which a reversal of the HardyLittlewood inequality occurs. In this paper, we find “Skewes numbers” for 8 more prime k-tuples and provide numerical data in support of the Hardy-Littlewood conjecture. Moreover, we present several algorithms to compute such numbers.