نشریه ریاضی و جامعه
سال نهم شماره 3 (پاییز 1403)

This is the second paper from a trio which reviews some of the progress in low dimensional topology in the past century. Starting with the work of Poincare in the last years of 19th century and first few years of 20th century, we review the major steps in putting 3-manifolds and their algebraic topology in a solid mathematical framework, and the important theorems which strengthened the understanding of 3-manifolds, including the prime decomposition theorem and JSJ decomposition of 3-manifolds. Highlighting the significance of hyperbolic 3-manifolds, proving the monster theorem and formulating the geometrization conjecture by Thurston has been a turning point in 3-manifold topology. The proof of geometrization conjecture by Perelman, using Ricci flow of Hamilton, affirmed that the fundamental group is an almost perfect invariant of closed 3-manifolds. Yet, it is not clear how geometric properties are reflected in the fundamental group, and its is difficult to verify whether two group presentations give isomorphic fundamental groups or not. Alternative approaches to the study of 3-manifolds and 4-dimensional cobordisms between them using abelian groups include, in particular, the theories which are formulated as topological quantum field theories (TQFTs). These approaches are also reviewed in the paper. In particular, a theorem of the author which addresses the strength of the later invariants in distinguishing 3-manifolds from the standard 3-sphere is discussed.

Over the last decade or so, wavelets have had a growing impact on signal processing theory and practice, both because of their unifying role and their successes in applications. Filter banks, which lie at the heart of wavelet-based algorithms, have become standard signal processing operators, used routinely in applications ranging from compression to modems. The contributions of wavelets have often been in the subtle interplay between discrete-time and continuous-time signal processing. The purpose of this article is to look at recent wavelet advances from a signal processing perspective. In particular, approximation results are reviewed, and the implication on compression algorithms is discussed. New constructions and open problems are also addressed. Finding a good basis to solve aproblem dates at least back to Fourier and his investigation of the heat equation. The series proposed by Fourier has several distinguishing features: The series is able to represent any finite energy function on an interval. The basis functions are eigenfunctions of linear shift invariant systems or, in other words, Fourier series diagonalize linear, shift invariant operators.

A distinguishing coloring of a simple graph $G$ is a vertex coloring of $G$ which is preserved only by the identity automorphism of $G$. In other words, this coloring ``breaks'' all symmetries of $G$. The distinguishing number $D(G)$ of a graph $G$ is defined to be the smallest number of colors in a distinguishing coloring of $G$. This concept of “symmetry breaking” was first proposed by Babai in 1977 and after the publication of a seminal paper by Albertson in 1996, it attracted the attention of many mathematicians. In this paper, along with studying some relations between $D(G)$ and some other important graph parameters, we introduce the concept of a $(D,\alpha)$-ordinary graph which displays the comparison between $D(G)$ and the independence number $\alpha(G)$. Then we consider the classification of $(D,\alpha)$-ordinary graphs in various families of graphs such as forests, cycles, generalized Johnson graphs, Cartesian products of graphs and line graphs of connected claw-free graphs. We also propose some conjectures and discuss about some future research directions in this topic.

In this paper, quasi-rigid $C_{0}$-semigroups are introduced and their properties are investigated. Also, some sufficient conditions for quasi-rigidity are stated. It is proved that quasi-rigid $C_{0}$-semigroups are recurrent. It is stated that if a $C_{0}$-semigroup contains a quasi-rigid operator, then it is quasi-rigid. The concept of topologically quasi-rigid $C_{0}$-semigroups is introduced. It is proved that topologically quasi-rigidity and quasi-rigidity are equivalent. The direct sum of quasi-rigid $C_{0}$-semigroups is investigated. It is demonstrated that a $C_{0}$-semigroup is quasi-rigid if and only if the direct sum of the $C_{0}$-semigroup with itself is quasi-rigid. Moreover, the finite direct sum of a quasi-rigid operator with itself is recurrent. So, quasi-rigidity with respect to the recurrency is similar to the weakly-mixing with respect to the hypercyclicity.

Identification of population ratio disruption in the population structure is one of the challenges that every country faces. Population aging is a kind of demographic abnormality that lack of attention causes population problems. A timely warning about the aging of society can be useful for planning about having children on the one hand and providing suitable facilities for the elderly on the other hand. For example, the capital of Japan is a good example of an urban environment suitable for the elderly.One of the anomaly detection tools is spatial clustering using scan statistics. In the last three decades, the scan statistic method has been a very important and active field in statistical research. Identifying areas on geographic maps, where the concentration of points (elderly, sick, criminals, certain animal species, etc.) is significant, is important in many fields such as epidemiology, politics, criminology, zoology, and so on. With the help of scan statistic method, spatial clusters can be identified.In this article, we introduce the scan statistic method based on Poisson distribution. Using simulation, we investigate the efficiency of this method in identifying spatial clusters. Based on the results obtained from the simulation, the Poisson scan statistic method is a suitable method for detecting anomalies in the count spatial data. As an application of spatial clustering, we consider the population structure of the state of Georgia and identify areas where the elderly population is significantly high. These areas should be prioritized in the implementation of population reform programs.

Mathematical Challenge is one of the main components in mathematics education, which aims to improve the ability of learners in the field of mathematics. Also, creativity is a personal and social characteristic that leads to the development of people at all levels and in all historical periods. In fact, applying information and knowledge and using creativity in solving problems is very important. In between, for a mathematics teacher, in addition to having good techniques and knowledge, the use of software and theories of mathematics education is also effective on the flourishing of students' talent and creativity and helps them to solve a higher-level problem. In geometric concepts, using Euclid's principles and mastering them gives the teacher a better perspective for the problem solving process. In this article, a geometrical problem suitable for the 8th and 9th grade level has been proposed for the 7th grade in Sampad school, and while expressing the importance of Euclid's principles, both the dynamic environment of mathematics and the paper-pencil environment have been used. By using the Van Hiele’s Theory in mathematical education, by designing an educational scaffold and dialogue between the teacher and students and with the help of Variation Theory, the answer of the problem was guessed by several students and a creative student gave a complete answer. Finally, we stated the problem as if and only if. In the meantime, Dimensional Deconstruction of Shapes was also used in the dynamic environment and paper-pencil environment.