Math and Modernity: Critical Reflections

Besides the birth of new revolutionary concepts and methods, and of new areas of research, mathematicians, logicians, and philosophers have put into question the foundations of the discipline itself and the whole meaning of “mathematical truth.” Before then, at the end of the eighteenth century, mathematics was mainly concerned with explaining the “real world” and its laws. At the beginning of the “modern era” things started to change, sometimes slowly, other times abruptly. Abstract mathematics was no longer intimately related to the real world and its description. This abstract approach, both on research and on mathematical education, generated critical reactions in the mathematical community, and some “modern” ideas were rejected or neglected after several decades of experimentation.

A Graphic Method for Detecting Multiple Roots Based on Self-Maps of the Hopf Fibration and Nullity Tolerances

The aim of this paper is to study, from a topological and geometrical point of view, the iteration map obtained by the application of iterative methods (Newton or relaxed Newton’s method) to a polynomial equation. In fact, we present a collection of algorithms that avoid the problem of overflows caused by denominators close to zero and the problem of indetermination which appears when simultaneously the numerator and denominator are equal to zero. This is solved by working with homogeneous coordinates and the iteration of self-maps of the Hopf fibration. As an application, our algorithms can be used to check the existence of multiple roots for polynomial equations as well as to give a graphical representation of the union of the basins of attraction of simple roots and the union of the basins of multiple roots. Finally, we would like to highlight that all the algorithms developed in this work have been implemented in Julia, a programming language with increasing use in the mathematical community.

The Legacy of Peter Wynn

After the death of Peter Wynn in December 2017, manuscript documents he left came to our knowledge. They concern continued fractions, rational (Padé) approximation, Thiele interpolation, orthogonal polynomials, moment problems, series, and abstract algebra. The purpose of this paper is to analyze them and to make them available to the mathematical community. Some of them are in quite good shape, almost finished, and ready to be published by anyone willing to check and complete them. Others are rough notes, and need to be reworked. Anyway, we think that these works are valuable additions to the literature on these topics and that they cannot be left unknown since they contain ideas that were never exploited. They can lead to new research and results. Two unpublished papers are also mentioned here for the first time.

A CASE STUDY ON HOW PRIMARY-SCHOOL IN-SERVICE TEACHERS CONJECTURE AND PROVE: AN APPROACH FROM THE MATHEMATICAL COMMUNITY

This paper studies how four primary-school in-service teachers develop the mathematical practices of conjecturing and proving. From the consideration of professional development as the legitimate peripheral participation in communities of practice, these teachers’ mathematical practices have been characterised by using a theoretical framework (consisting of categories of activities) that describes and explains how a research mathematician develops these two mathematical practices. This research has adopted a qualitative methodology and, in particular, a case study methodological approach. Data was collected in a working session on professional development while the four participants discussed two questions that invoked the development of the mathematical practices of conjecturing and proving. The results of this study show the significant presence of informal activities when the four participants conjecture, while few informal activities have been observed when they strive to prove a result. In addition, the use of examples (an informal activity) differs in the two practices, since examples support the conjecturing process but constitute obstacles for the proving process. Finally, the findings are contrasted with other related studies and several suggestions are presented that may be derived from this work to enhance professional development.

Mathematics in the interwar period in Central-Eastern Europe. The report on an international research project for the years 2018–2020

In the article, we will show the main important results of the international research project The impact of WWI on the formation and transformation of the scientific life of the mathematical community. It was supported by the Czech Science Foundation for the years 2018–2020 and brought together ten scientists from five countries (Czech Republic, Poland, Slovakia, USA, and Ukraine) and used the collaboration with historians of mathematics and mathematicians from many other European countries. We will discuss our motivation for the creation of the project, our methodological and professional preparations which profited from the international composition of the team and its longtime collaborations, profound specializations and experiences of the team members, and their deep and long-term studies of many archival sources and basic published works. We will present our choice of the general research trends, our definition of the scientific questions, and our determination of the main topics of our studies. We will describe our most important results (books, articles, visiting lectures, presentations at national and international conferences, seminars and book fairs, exhibitions, popularizations of the results between students, teachers, mathematicians, historians of sciences, and people who love mathematics and its history). We will analyze the new benefit that the project created for the future, for example, good platforms for future international research and cooperation, the discovery of many new interesting research questions, problems, and plans.

Analytical Parameter Estimation of the SIR Epidemic Model. Applications to the COVID-19 Pandemic

The SIR (Susceptible-Infected-Removed) model is a simple mathematical model of epidemic outbreaks, yet for decades it evaded the efforts of the mathematical community to derive an explicit solution. The present paper reports novel analytical results and numerical algorithms suitable for parametric estimation of the SIR model. Notably, a series solution of the incidence variable of the model is derived. It is proven that the explicit solution of the model requires the introduction of a new transcendental special function, describing the incidence, which is a solution of a non-elementary integral equation. The paper introduces iterative algorithms approximating the incidence variable, which allows for estimation of the model parameters from the numbers of observed cases. The approach is applied to the case study of the ongoing coronavirus disease 2019 (COVID-19) pandemic in five European countries: Belgium, Bulgaria, Germany, Italy and the Netherlands. Incidence and case fatality data obtained from the European Centre for Disease Prevention and Control (ECDC) are analysed and the model parameters are estimated and compared for the period Jan-Dec 2020.