نشریه ریاضی و جامعه
سال هفتم شماره 1 (بهار 1401)

In this paper, by introducing well-established epidemiological models, compartmental models and Ising models, examples of their applications in describing epidemics, including some aspects of Covid-19, and several perspectives are presented. The importance of social distancing in preventing the spread of Covid-19 will also be demonstrated. We make a connection between the standard mathematical models employed in epidemics and well-established concepts in condensed matter Physics, such as the Fermi gas and the Fermi-liquid picture. Given the impossibility of making a precise forecast of the disease spread, the importance of taking into account additional factors, such as climate changes and urbanization, in the mathematical description of epidemics are considered.

The inherent characteristics of real-world data is uncertainty. If data is generated in valid experiments or collected standard, probability theory or fuzzy theory is a powerful tool for analysis it in the uncertainty conditions. But data is not always reliable; especially when it is not possible to perform multiple tests or reliable data collection. In this context, referring to the beliefs of experts in the field in question is an alternative approach and uncertainty theory is a tool by which the beliefs of experts can be mathematically incorporated into the problem-solving structure. A stable set has a wide range of applications in many fields, while in most cases its problems are without reliable data. In this paper, we investigate the finding of stable weighted sets with uncertain weights. These weights have an uncertain distribution based on the degree of belief of the field expert. For this purpose, we offer two methods. In the first method, by introducing the concept of chance constraint, we come to an integer linear programming model with definite coefficients. The second method is based on the concept of uncertain expected value. Finally, a numerical example for these two methods is presented.

This paper deals with one of the important and interesting topics in Number Theory under the title of prime number races, which is less discussed in Persian texts. All prime numbers (except 2) can be written in the form 4n+1 or 4n+3. The question that naturally arises is, which category contains more prime numbers? Certainly, the infinity of prime numbers makes it difficult for the spectator to predict the competition. The present article, which was written by two prominent mathematicians in this field, has comprehensively discussed the history of this problem, the types of prime number races, and the research work done in this field.

In this paper, we first consider 4D conformally flat homogeneous pseudo-Riemannian space with trivial isotropy, then, we investigate some geometrical properties such as being Ricci solitons and Walker on the spaces under consideration. Indeed, we give a complete classification of the cases giving rise to conformally flat homogeneous Ricci solitons or Walker spaces with trivial isotropy.

In this note we show that Hall polynomials exist for basic connected right 2-Nakayama algebras. This result prove the Ringel’s conjecture for right 2-Nakayama algebras.

Auslander‐Reiten Conjecture is one of the most important and long‐standing conjectures in representation theory of finite dimensional algebras. It is known to be in close relationship with a string of homological conjectures on top of which lies the well‐known Finitistic Dimension Conjecture. Recently, a dual version of Auslander‐Reiten conjecture has been posed. This dual statement deserves further study from the point of view that its validity implies the validity of the conjecture itself. This paper is devoted to deal with this Dual Auslander‐Reiten Conjecture. We discuss it for Gorenstein rings of Krull dimension at least 2. To this end, we firstly show that it suffices to focus only on the case where the Krull dimension is exactly 2. Then, switching to such rings, we show that this conjecture holds for modules of finite length over Gorenstein rings of Krull dimension at least 2.