arXiv subjects

Mahmoud Annaby

Publications and source records attributed to Mahmoud Annaby.

3 recordsLinked to original sources

arXiv2025

On the Zeros of $q$-Hankel Transform by Using P\'{o}lya-Hurwitz Partial Fraction Method

The technique of P\'{o}lya-Hurwitz of partial fractions is implemented to investigate the zeros of finite $q$-Hankel transforms, which are defined in terms of the third $q$-Bessel function of Jackson. The new approach, which is a $q$-counterpart of P\'{o}lya-Hurwitz technique relaxes the restrictive conditions imposed on $q$ in the previously obtained results. In the present study, we use the $q$-type sampling theorems of the $q$-Hankel transforms, which lead directly to $q$-partial fractions. Various experimental examples are established.

math.NT↗

arXiv2024

On Kakeya's Geometric Proof of Eneström-Kakeya's Theorem

This paper is devoted to demonstrate Kakeya's geometric proof of his theorem (1912), independently established earlier by Eneström (1893). By calculating centers and radii of the interlacing circles of Kakeya's method, we prove Kakeya's geometric structure, which has not been previously established. We give an equivalent proof, which is based on the construction of internally interlacing circles, which has been geometrically considered by Tomic (1948).

math.CV↗

arXiv2023

On the Extraction of Amicable Pairs Between Ibn Sina-al-Baghdadi and al-Kashi

In this note, we briefly analyse the works on the extractions of amicable pairs in some medieval Arabic literature. In this note, we briefly analyse the works Ibn Sina (c.980-1037), al-Baghdadi (c.980-1037) and al-Kashi (d. c. 1429) on the extraction of amicable pairs. We compare these works in the view of the well-known Thabit ibn Qurra's rule. Contrary to the belief that these authors are merely stating variations of the famous Thabit ibn Qurra's (d. 901) rule in different manners, we show that their statements are different. We prove that the statements of both ibn Sina and al-Baghdadi led to an unsolved conjecture, while al-Kashi's statement led to the wrong amicable pair (2024, 2296). The conjecture implied by ibn Sina and al-Baghdadi statements requires testing big primes corresponding to the known 51 Mersenne primes, and we show that no counterexamples yet exist.

math.HO↗