نشریه ریاضی و جامعه
سال هفتم شماره 3 (پاییز 1401)

Translator's abstract:This is a brief description of the life of George Boole, his wife Mary Everest Boole and their scientific activities. In this article, the remarkable life of this family and the impact of its members on mathematics and society are briefly discussed

One of the most popular theories in aesthetics is the golden ratio theory in either theoretical or practical geometry. The golden ratio, the golden proportion which is called the extreme and mean ratio by Euclid, goes by other several various names over history. The golden ratio is well known in the artistic composition and architecture of buildings, statues, and sculptures, as still used today. There are a lot of myths about the golden ratio; Some attribute it to sacred numbers among the Pythagoreans, and some figure out an excellent placement in geometry and architecture for it, mentioning that particular ancient temples and structures were built on the basis of these numbers. The present study aims to examine the written background of the golden ratio through the written heritage of the Islamic period including accessible manuscripts and treatises as primary sources, besides the articles and reports on ancient Greek works, mainly in Greek, Latin, or other languages as secondary sources. In this article, we first explain the background of this issue in the case of ancient Greece, and then we review some textbooks of the Islamic period on the subject of arithmetic or geometry that addressed this theory directly or indirectly. Since this article relies on the Islamic period in medieval and early modern times, only the treatises and some of the efforts spared including the existing written works and the visual arts from the Islamic period to the present century are introduced. Moreover, the special shapes famed as sacred or golden due to their specific geometric properties are described

Let $\mathcal{C}$ be a locally support-finite $k$-category with a $G$-action. It is proved that a Galois covering functor $P: \CC \lrt \CC/G$ induces a Galois covering of categories of their torsionless modules. Using this result, we provide a version of Gabriel's theorem for categories of torsionless modules

Let $R$ be a commutative ring, Γ be a group, and Γ′ be a subgroup of finite index. In this paper, we study the class of Gorenstein flat (resp. Gorenstein cotorsion) modules over the group ring RΓ. In particular, we discuss the relationship between Gorenstein flat (resp. Gorenstein cotorsion) dimension of RΓ−modules and that of RΓ′−modules. We will prove that, for any RΓ−module $M$ of finite Gorenstein flat dimension (∞>GfdRΓM), there is an inequality GfdRΓM ⩽ GfdRM + hdRΓ, where hdRΓ denotes the homological dimension of Γ over $R$ . Analogously, we prove that if Γ is finite, there is an inequality GctdRΓM ⩽ GctdRM + cdRΓ, where GctdRΓM (GctdRM) is the Gorenstein cotorsion dimension of module $M$ over RΓ (resp. over $R$) and cdRΓ is the cohomological dimension of $Γ over $R$.

In this paper, a fractional order model is presented to study the effect of the drug therapy on the control of HIV infection. For this purpose, we calculate the reproduction number of the model with the next generation operator method and by using that, we study the stability of equilibrium point. Reproduction number $R_0$ plays an important role in spreading or non-spreading of HIV. So that, if $R_0 < 1$, the infection-free equilibrium point is locally asymptotically stable and the infection is cleared from the $CD{4^ +}T$-cell population. In the res, we analyze the relationship between the reproduction number and the therapy parameter values, for some different values of the order of model. Also, we evaluate the effect of the therapy parameter on the immune system cells and HIV virus population. Finally, we simulate the variations of one of the parameters that show the amount of virus production by actively infected cells. Adams predictor-corrector method will be used to solve the fractional order model.

In this article, we have a brief historical overview of the facts around the axiom of choice (AC) in set theory. In this regard, we will state some of the most important equivalents of AC and also some of the weaker forms of AC. We will try to mention the most important results of AC in different branches of mathematics