Volume XII. Number 1. April 2022

This is number 1 of volume XII of the Journal of the year 2022. From then on, the publication of the Journal will be annual, every April a new number of Mathematical Thought will be made available to everyone.

We continue to improve with regard to Covid-19 despite the fact that other worrying fronts are opening up, such as the war in Ukraine. We hope that logic prevails and we are able to live in peace.

As always, this issue of Mathematical Thought includes a variety of interesting works distributed in each of the sections of the publication.

Título: Créditos editoriales e Índice de artículos del Número 1 (Vol. XII)
Autor: Equipo Editorial

This work presents a method of approximated numerical integration of a function defined on an equispaced interval [a, b] with n subintervals of length ∆t = (b−a)/n . Furthermore, a bound of the error between the approximation by said method and the exact value of the integral has been determined for a function whose integral is known.

This method is compared with the trapezoidal rule analyzing the approximation error using a particular function demonstrating that this error is smaller than that obtained by the trapezoidal rule

Título: Asymmetric parabolic rule and error bound for numerical integration of functions in equispaced intervals
Autor: José Manuel Recio-López

This article analyzes the process of obtaining solutions of nonlinear equations, through iterative procedures, based on the fixed point theorems. The article proposes some improvement algorithms to be implemented in an appropriate programming language.

The results are based on the impossibility of solving algebraic equations of degree higher than four using purely algebraic methods. This work can also be considered an introduction to the theory of discrete dynamical systems.

Título: Algorithm convergence improvements
Autor: Federico Ruiz López

This article aims to analyze some of the properties of the integer part function from an algebraic component, some of them are demonstrated and in others the geometric interpretation is used to obtain a better explanation of the property or to complement it. Additionally, some applications and some mathematical contexts where this function makes sense are studied, such as network points, complex numbers, tessellations, number theory, probability, among others.

Título: Integer part function and some properties
Autor: Juan Carlos Arango Parra & Yeisson Alexis Acevedo Agudelo
Teaching Experiences

This paper presents the work performed by the authors with the objective of carrying out a practical experience with the students of the second course of Computer Engineering degree in the computer room. In this experience, firstly, the students are divided into groups. Due to the construction of the exercise, each group of students needed the results of the previous group to do its part of the work. In this way, the practical session improves the ability of the students to work by teams in a sequenced way. Since team working is one of the main abilities that Engineering Companies value in their workers, this didactic experience could be an interesting opportunity to enhance this skill in future engineers.

The practice is framed in the field of Discrete Mathematics. It consists in encoding and decoding a message by applying the RSA algorithm by means of Mathematica (the math software of the Wolfram team). The algorithm was developed in the theoretical sessions within the topic dedicated to the Number Theory as an example of the real life application of classical concepts that are studied in this field, such as the prime numbers, the modular arithmetic and the Euler theorem. Therefore, besides its didactical importance, the practice also provides resources for the students to implement this application.

Título: A practice to enhance sequential team work in mathematical topics
Autor: Javier Rodrigo Hitos & Susana Merchán Rubira
History of Mathematics

Music and art constitute two of the most intimate and at the same time most universal forms of communication of the human being. Through them you can discover spiritual realities that no other language is able to express in all its magnitude.

Rhythm is the driving force behind music. Historically, it constituted, with all kinds of rudimentary instruments as a medium, the first organized musical expression of primitive man. Musically speaking, rhythm is the expression of the duration of sounds. Of course, just as not all sounds have the same height, neither do all sounds last for the same time. In addition, rhythm is the key musical element that is used to distinguish all musical styles or genres.

The use of rhythmic sequence as the basis of composition can also be applied to any other manifestation of art. It is precisely this rhythmic sequence that provides a certain personality to the work of art. Silences are also rhythms, just as they are in spoken language. It is necessary to resort to pauses or silences that help to better understand the text or any artistic manifestation.

In art it is possible to find relationships that establish measures of unity, similar to the musical compass. Thus, in the same way that the compass divides a composition into parts of equal duration, establishing a periodic and regular order, in painting or sculpture proportions and sequences can be established according to certain patterns that evolve and, sometimes, are repeated. These patterns have a geometric characterization sometimes evident and others difficult to appreciate.

This article takes a tour of different manifestations of art, whether painting, sculpture, photography, fashion, architecture, engineering, decorative arts and others, collecting their relationships with musical rhythms and their geometric characteristics.

Título: Rhythms, art and geometry
Autor: Jesús Alonso Trigueros

It is well-known that Nobel Prizes are awarded in six modalities and none of them is Mathematics. However, this is not an obstacle for many mathematicians to have been awarded that Prize. This communication shows the biographies of all those mathematicians who came to be awarded in some of the modalities in which these Awards are granted, not only scientific, such as Physics or Chemistry, but also in Economics and even in Literature, as it is the case of the prestigious Spanish writer José de Echegaray, the first mathematician awarded that distinction.

Título: Mathematicians awarded the Nobel Prize
Autor: Juan Núñez Valdés, Luis Rabasco González
Games and Mathematical Oddities

In 1965, the book «Vedic Mathematics», by Bharati Krishna, was published with a compilation of procedures that allow arithmetic calculations to be carried out mentally, without pencil or paper. The author claims that it is a compilation of the mathematical methods described in the ancient Vedas. There are serious doubts about the veracity of this statement, but the procedures described in the book offer a valuable tool that can favour the learning of mathematics among the little ones. This article explains the fundamentals of some of these mathematical algorithms.

Título: Vedic Mathematics
Autor: Santiago Higuera de Frutos
Mathematical Tales

A tale for children is presented. It is oriented for children from 6 to 9 years old (mainly for 6 years old). In the tale the basic arithmetic operations are introduced in a funny way, by means of a children’s show.

Título: The ogre Goro
Autor: Javier Rodrigo
Critics and Reviews

In this paper a report about the Collaborative Project MathTales is performed. This Project took part of the Citizen Laboratory Experimentamates, carried out in November, 2021 in the Polytechnical University of Madrid. The Project consists on the creation of a web page that stands as a repository of mathematical tales oriented to different ages as well as classified by didactic methods and contents.

Título: Collaborative Project MathTales in Experimentamates
Autor: Alejandra Fernández Sánchez, Verónica Navarro Fernández & Javier Rodrigo Hitos

In this article we interview Pablo Daniel Pajares Galeano, science communicator and teacher from Extremadura who tells us about his experience.

Título: Pedro Daniel Pajares Galeano “from monologist to teacher”
Autor: José Manuel Sánchez Muñoz

You can download the full issue here: