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The paper analyzes how to improve the teaching of Operational Research, focusing on distance learning courses where professors and students are separated through space and time. This case study was done in a public Industrial Engineering undergraduate course, and the work structure is divided into three main parts: an exploratory‐descriptive documentary analysis, application of free software, and evaluation of learning. The authors showed the feasibility of using the SageMath tool in the teaching‐learning process. The study revealed the importance of developing alternative solutions to educational realities marked by economic and financial constraints, where structure teaching with free software is a ruling factor once Engineering education is not a similar global event everywhere. Distance learning is a phenomenon that has been growing over the last 20 years in Brazil, but this was one of the first experiences with the use of SageMath. So, it can serve as a reference for countries with socioeconomic conditions similar to Brazil. Also, this case study can help other professors to enhance their teaching in a distance learning context even in scarcity scenarios of educational resources. The software implementation would be justified in part of the groups studied. [ABSTRACT FROM AUTHOR]

Technological and methodological aspects of using freeware software, such as GeoGebra, Wolfram|Alpha, Maxima, SageMath and GRAN1, for solving tasks with parameters, are presented in the article. Criteria were defined for selection of computer mathematics system (CMS) to solve tasks with parameters, including plotting a graph of a function given in explicit and implicit forms, using a parameter in a function’s analytical definition, and automatically changing the graph of a function depending on the parameter value; ability to changing the parameter step change; plotting of a tangent and a normal to a curve at a point; ability to change the scale; determination of the coordinates of the intersection of graphs of functions; obtaining an analytical solution. In the article, some examples were presented for graphic and analytical tasks that used CMS parameters. GRAN1 and GeoGebra are recommended to use for plotting and analyzing of the graphs

Abstract The paper analyzes how to improve the teaching of Operational Research, focusing on distance learning courses where professors and students are separated through space and time. This case study was done in a public Industrial Engineering undergraduate course, and the work structure is divided into three main parts: an exploratory‐descriptive documentary analysis, application of free software, and evaluation of learning. The authors showed the feasibility of using the SageMath tool in the teaching‐learning process. The study revealed the importance of developing alternative solutions to educational realities marked by economic and financial constraints, where structure teaching with free software is a ruling factor once Engineering education is not a similar global event everywhere. Distance learning is a phenomenon that has been growing over the last 20 years in Brazil, but this was one of the first experiences with the use of SageMath. So, it can serve as a reference for countries with socioeconomic conditions similar to Brazil. Also, this case study can help other professors to enhance their teaching in a distance learning context even in scarcity scenarios of educational resources. The software implementation would be justified in part of the groups studied.

This present paper is inspired by one of the questions posed by Okeke (Results Math 78(96):1-30, 2023, see Remark 2.10b). In particular, we aim to develop a robust computer code based on the theoretical results obtained in Okeke (2023), which determines the polynomial solutions of the following functional equation, 0.1 ∑ i = 1 n γ i F (a i x + b i y) = ∑ j = 1 m (α j x + β j y) f (c j x + d j y) , for all x , y ∈ R , γ i , α j , β j ∈ R , and a i , b i , c j , d j ∈ Q , and their special forms. The primary motivation for writing such a computer code is that solving even simple equations belonging to class (0.1) needs long and tiresome calculations. Therefore, one of the advantages of such a computer code is that it allows us to solve complicated problems quickly, easily, and efficiently. Additionally, the computer code will significantly improve the level of accuracy in calculations. Along with that, there is also the factor of speed. We point out that the computer code will operate with symbolic calculations provided by the programming language Python, which means that it does not contain any numerical or approximate methods, and it yields the exact solutions of the equations considered. [ABSTRACT FROM AUTHOR]

We explored some common challenges faced by undergraduate students when studying linear algebra, particularly when dealing with algorithmic thinking skills required for topics such as matrix factorization, focusing on (orthogonal) diagonalization and singular value decomposition (SVD). To address these challenges, we introduced SageMath, a Python-based open-source computer algebra system, as a supportive tool for students performing computational tasks despite its static output nature. We further examined the potential of dynamic ChatGPT, an AI-based chatbot, by requesting examples or problem-solving assistance related to (orthogonal) diagonalization or the SVD of a specific matrix. By reinforcing essential concepts in linear algebra and enhancing computational skills through effective practice, mastering these topics can become more accessible while minimizing mistakes. Although static in nature, SageMath proved valuable for confirming calculations and handling tedious computations because of its easy-to-understand syntax and accurate solutions. However, although dynamic ChatGPT may not be fully reliable for solving linear algebra problems, the errors it produces can serve as a valuable resource for improving critical thinking skills.

We present an implementation of the Atiyah-Bott residue formula for M ‾ 0 , m (P n , d). We use this implementation to compute a large number of Gromov-Witten invariants of genus 0, including intersection numbers of rational curves on general complete intersections. We also compute some numbers of rational contact curves satisfying suitable Schubert conditions. Our computations confirm known predictions made by Mirror Symmetry. The code we developed for these problems is publicly available and can also be used for other types of computations. [ABSTRACT FROM AUTHOR]

ABSTRACT: In this study a historical perspective of the mass continuity equation for fully developed laminar flow in the lumen side of a hollow-fiber membrane contactor is presented with respect to the lumen-wall boundary condition (BC). It is shown that the constant wall concentration case (Dirichlet boundary condition) imposed by the Graetz-Lévêque postulations is a sub-case of the mixed Neumann-Dirichlet linear BC largely overestimating the performance of such contactors. For the linear BC the analytical solution derived by the separation of variables method is revisited proving that it is very accurate and practical even in the region very close to the entrance of the computational domain. The analysis is extended by incorporating and solving nonlinear lumen-wall BCs with the method-of-lines approach by discretizing the radial domain using the Gauss-Jacobi orthogonal collocation and integrating the resulting initial-value differential-algebraic system. The analytical solution, the derivation of the collocation matrices and the numerical solution are presented with the aid of the open-source SageMath and the commercial package Maple.

Runge-Kutta schemes play a very important role in solving ordinary differential equations numerically. At first we want to present the Sage routine for calculation of Butcher matrix, we call it an rk package. We tested our Sage routine in several numerical experiments with standard and symplectic schemes and verified our result by corporation with results of the calculations made by hand.Second, in Sage there are the excellent tools for investigation of algebraic sets, based on Grbner basis technique. As we all known, the choice of parameters in Runge- Kutta scheme is free. By the help of these tools we study the algebraic properties of the manifolds in affine space, coordinates of whose are Butcher coefficients in Runge-Kutta scheme. Results are given both for explicit Runge-Kutta scheme and implicit Runge-Kutta scheme by using our rk package. Examples are carried out to justify our results. All calculation are executed in the computer algebra system Sage.

We consider the problem of integrating a given differential equation in algebraic functions, which arose together with the integral calculus, but still is not completely resolved in finite form. The difficulties that modern systems of computer algebra face in solving it are examined using Maple as an example. Its solution according to the method of Lagutinskis determinants and its implementation in the form of a Sagemath package are presented. Necessary conditions for the existence of an integral of contracting derivation are given. A derivation of the ring will be called contracting, if such basis B= {m1, m2, } exists in which Dmi= cimi+o (mi). We prove that a contracting derivation of a polynomial ring admits a general integral only if among the indices c1, c2, there are equal ones. This theorem is convenient for applying to the problem of finding an algebraic integral of Briot-Bouquet equation and differential equations with symbolic parameters. A number of necessary criteria for the existence of an integral are obtained, including those for differential equations of the Briot and Bouquet. New necessary conditions for the existence of a rational integral concerning a fixed singular point are given and realized in Sage.

Using barycentric coordinates, Cartesian coordinates with some help from SageMath -- a free and open-source mathematical software system, we prove some properties of the t--pedal triangles and two generalizations of the Lester circle. [ABSTRACT FROM AUTHOR]

We give an algorithmic generalisation of Dickson's method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson's idea is to formulate from Hermite's criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for $d>6$. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order $q>8$. Such PPs exist if and only if $q\in \{11,13,19,23,27,29,31\}$ and are explicitly listed in normalised form. [ABSTRACT FROM AUTHOR]

There are many computational applications and engines used in mathematics, with some of the best-known arguably being Maple, Mathematica, MATLAB, and Mathcad. However, although they are very complete and powerful, they demand the use of commercial licences, which can be a problem for some education institutions or in cases where students desire to use the software on an unlimited number of devices or to access it from several of them simultaneously. In this contribution, we show how GeoGebra, WolframAlpha, Python, and SageMath can be applied to the teaching of different mathematical courses in engineering studies, as they are some of the most interesting representatives of free (and mostly open source) mathematical software. As the best way to show a topic in mathematics is by providing examples, this article explains how to make calculations for some of the main topics associated with Calculus, Algebra, and Coding theories. In addition to this, we provide some results associated with the usage of Mathematica in different graded activities. Moreover, the comparison between the results from students that use Mathematica and students that participate in a “traditional” course, solving problems and attending to master classes, is shown.

Up to linear transformations, a classification of all permutation polynomials (PPs) of degree 7 over F q is obtained for any odd prime power q > 7. Since all exceptional polynomials of degree 7 are precisely known up to linear transformations, it suffices to work on non-exceptional PPs of degree 7, which exist only when q ⩽ 409 by our previous result. This can be exhausted by the SageMath software running on a personal computer. To facilitate the computation, some requirements after linear transformations and equations by Hermite's criterion are provided for the polynomial coefficients. The main result is that a non-exceptional PP f of degree 7 exists over a finite field F q of odd order q > 7 if and only if q ∈ { 9 , 11 , 13 , 17 , 19 , 23 , 25 , 27 , 31 , 49 } , and f is explicitly listed up to linear transformations. [ABSTRACT FROM AUTHOR]

During the last two decades, many kinds of periodic sequences with good pseudorandom properties have been constructed from classical and generalized cyclotomic classes, and used as keystreams for stream ciphers and secure communications. Among them are a family DH-GCSd of generalized cyclotomic sequences on the basis of Ding and Helleseth's generalized cyclotomy, of length pq and order d = gcd (p − 1 , q − 1) for distinct odd primes p and q. The linear complexity (or linear span), as a valuable measure of unpredictability, is precisely determined for DH-GCS8 in this paper. Our approach is based on Edemskiy and Antonova's computation method with the help of explicit expressions of Gaussian classical cyclotomic numbers of order 8. Our result for d = 8 is compatible with Yan's low bound (pq − 1)/2 on the linear complexity for any order d, which is high enough to resist attacks of the Berlekamp–Massey algorithm. Finally, we include SageMath codes to illustrate the validity of our result by examples. [ABSTRACT FROM AUTHOR]

We consider the compilation of complete (nonextensible) dictionaries of pairwise nonoverlapping codons containing the maximum possible number of such codons. This includes three subcases: (a) codons that do not allow overlapping by two letters, but allowing overlapping by one letter (e.g. , ATC and CTG); (b) on the contrary, codons that do not allow only overlapping by one letter, but allow overlapping by two letters (e.g. , ATC and TCG); (c) when there can be no overlap (as ATC and GTC). The last subcase is also an example of the nonoverlapping code that is used in coding theory and automata theory. Herein, we use graph theory and the SageMath computer algebra system as working tools. [ABSTRACT FROM AUTHOR]

Effective homology techniques allow us to compute homology groups of a wide family of topological spaces. By the Whitehead tower method, this can also be used to compute higher homotopy groups. However, some of these techniques (in particular, the Whitehead tower) rely on the assumption that the starting space is simply connected. For some applications, this problem could be circumvented by replacing the space by its universal cover, which is a simply connected space that shares the higher homotopy groups of the initial space. In this paper, we formalize a simplicial construction for the universal cover, and represent it as a twisted Cartesian product. As we show with some examples, the universal cover of a space with effective homology does not necessarily have effective homology in general. We show two independent sufficient conditions that can ensure it: one is based on a nilpotency property of the fundamental group, and the other one on discrete vector fields. Some examples showing our implementation of these constructions in both SageMath and Kenzo are shown, together with an approach to compute the homology of the universal cover when the group is Abelian even in some cases where there is no effective homology, using the twisted homology of the space. [ABSTRACT FROM AUTHOR]

Escape rooms are an increasingly popular puzzle game, and educators have started implementing them in classrooms. This paper will describe the motivations and implementation of escape-room-like puzzles in an undergraduate cryptography course, including how the open-source mathematics software system SageMath is used. In addition, there is a discussion about generalizations of the activity to other courses. [ABSTRACT FROM AUTHOR]

The investigation of the electromagnetic field in a regular waveguide filled with a homogeneous substance reduces to the study of two independent boundary value problems for the Helmholtz equation. In the case of a waveguide filled with an inhomogeneous substance, a relationship arises between the modes of these two problems, which in numerical experiments can not always be fully taken into account. In this paper, we will show how to rewrite the Helmholtz equations in the “Hamiltonian form” to express this connection explicitly. In this case, the problem of finding monochromatic waves in a waveguide with arbitrary filling will be reduced to an infinite system of ordinary differential equations in a properly constructed Hilbert space. The results of numerical experiments on finding normal waves, realized in the computer algebra system Sage, are presented. [ABSTRACT FROM AUTHOR]

We explored some common challenges faced by undergraduate students when studying linear algebra, particularly when dealing with algorithmic thinking skills required for topics such as matrix factorization, focusing on (orthogonal) diagonalization and singular value decomposition (SVD). To address these challenges, we introduced SageMath, a Python-based open-source computer algebra system, as a supportive tool for students performing computational tasks despite its static output nature. We further examined the potential of dynamic ChatGPT, an AI-based chatbot, by requesting examples or problem-solving assistance related to (orthogonal) diagonalization or the SVD of a specific matrix. By reinforcing essential concepts in linear algebra and enhancing computational skills through effective practice, mastering these topics can become more accessible while minimizing mistakes. Although static in nature, SageMath proved valuable for confirming calculations and handling tedious computations because of its easy-to-understand syntax and accurate solutions. However, although dynamic ChatGPT may not be fully reliable for solving linear algebra problems, the errors it produces can serve as a valuable resource for improving critical thinking skills.

In this paper, we introduce free mathematical software SageMath with ten examples, which illustrate ways of using it for teaching math and encourage its use. [ABSTRACT FROM AUTHOR]

There are many computational applications and engines used in mathematics, with some of the best-known arguably being Maple, Mathematica, MATLAB, and Mathcad. However, although they are very complete and powerful, they demand the use of commercial licences, which can be a problem for some education institutions or in cases where students desire to use the software on an unlimited number of devices or to access it from several of them simultaneously. In this contribution, we show how GeoGebra, WolframAlpha, Python, and SageMath can be applied to the teaching of different mathematical courses in engineering studies, as they are some of the most interesting representatives of free (and mostly open source) mathematical software. As the best way to show a topic in mathematics is by providing examples, this article explains how to make calculations for some of the main topics associated with Calculus, Algebra, and Coding theories. In addition to this, we provide some results associated with the usage of Mathematica in different graded activities. Moreover, the comparison between the results from students that use Mathematica and students that participate in a "traditional" course, solving problems and attending to master classes, is shown. [ABSTRACT FROM AUTHOR]

Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree 8 over F 2 r with r > 3. By Bartoli et al. (2017) [1] , a polynomial f of degree 8 over F 2 r is exceptional if and only if f − f (0) is a linearized PP, which has already been classified. So it suffices to search for non-exceptional PPs of degree 8 over F 2 r , which exist only when r ⩽ 9 by a previous result. This can be exhausted by the SageMath software running on a personal computer. To facilitate the computation, some requirements after linear transformations and explicit equations by Hermite's criterion are provided for the polynomial coefficients. The main result is that a non-exceptional PP f of degree 8 over F 2 r (with r > 3) exists if and only if r ∈ { 4 , 5 , 6 } , and such f is explicitly listed up to linear transformations. [ABSTRACT FROM AUTHOR]