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In this paper we studied the personal mathematical work of 14 students in initial teacher training, in their first year of studies at a public university in Chile, based on two tasks on complex numbers. These tasks were set in a Computer Aided Assessment System (CAA) based on Moodle and Wiris and it was proposed to solve them using different Computer Algebra Systems (CAS), such as GeoGebra, Symbolab, Photomath and Wolfram Alpha among others. Difficulties and potentialities were observed in the mathematical work of the students. In the difficulties, it was observed that the students had problems interpreting the information from the CAS and the CAA feedback because their previous mathematical knowledge was not solid enough to do so. In the potentialities, it was observed that different characteristics of the task, together with the articulation of two or more artifacts allowed students to give meaning to the mathematical objects involved. [ABSTRACT FROM AUTHOR]

In this paper, we present a derivation of a numerical scheme to solve fifth order initial value problems using the open source computer algebra system MAXIMA. [ABSTRACT FROM AUTHOR]

In Victoria, Australia, senior secondary mathematics students are expected to use technology and thus need to make decisions about using pen-and-paper (P&P) or technology when solving mathematics problems. The predominant technology is a Computer Algebra System (CAS). This study investigated the beliefs about CAS held by twelve Year 11 students as they learnt to use CAS and whether these beliefs were stable over time. These students held a range of beliefs related to the usefulness of CAS, speed of CAS compared to P&P, whether CAS is proper mathematics, choice of CAS or P&P, ease of use, the correctness of answers and solving problems in Mathematical Methods (i.e. the mathematics subject studied). Beliefs are often described as being stable (e.g. McLeod, 1992), but some researchers stress stability needs to be determined empirically rather than being seen as a characteristic of beliefs (e.g. Liljedahl et al., 2012). For this sample of students, stability (rather than instability) is a feature of students' beliefs about CAS. [ABSTRACT FROM AUTHOR]

In this paper we review selected significant developments in the use of digital technology in the teaching and learning of mathematics over the last five years. We focus on a number of important topics in this field, including the evolvement of STEAM and critical making as well as the process of redefining learning spaces in the transformation of the mathematics classroom. We also address the increasing use of computer algebra systems and dynamic geometry packages; and the issue of student collaboration online, especially using learning environments and social media. We briefly touch on artificial intelligence systems, including hyper-personalisation of learning, multimodality and videos. We include a brief discussion on the impact of COVID-19 on mathematics education, and lastly on the more theoretical perspective of the epistemology of digital technology and the construct of humans-with-media. We conclude the discussion with some possible concerns and mentioning some possible new topics for research in the field. [ABSTRACT FROM AUTHOR]

We found a lower bound on the rank of a square matrix where every entry in the leading diagonal is neither zero nor one and every entry outside the leading diagonal is either zero or one. The rank of this matrix is at least half its order. Under an additional condition, the lower bound is higher by one. This condition means that some auxiliary system of linear equations has no binary solution. Some examples are provided that show that the lower bound can be achieved. This lower bound on the matrix rank allows the problem of finding a binary solution to a system of linear equations with a sufficiently large number of linearly independent equations to be reduced to a similar problem in a smaller number of variables. Restrictions on the existence of a large set of solutions are found, each differing from the binary one by the value of one variable. In addition, we discuss the possibility of certifying the absence of a binary solution to a large system of linear algebraic equations. Estimates of the time required for calculating the matrix rank in the SymPy computer algebra system are also provided. It is shown that the rank of a matrix over the field of residues modulo prime number is calculated faster than it generally takes to calculate the rank of a matrix of the same order over the field of rational numbers. [ABSTRACT FROM AUTHOR]

In this article, we consider weakly singular Volterra integral equation (WSVIE) with the kernel combining the power-law and logarithmic singularities f (ζ) = Au (ζ) + ∫ 0 ζ log (ζ − η) u (η) (ζ − η) β d η , ζ ∈ [ 0 , 1 ] , β ∈ [ 0 , 1) , where, f (ζ) is a smooth function. Equations of this kind, introduced by V. Volterra himself within the framework of the theory of functional compositions, are also directly related to the modern problems of fractional dynamics. Our approach is based on employing the operational matrix technique with the shifted Legendre polynomials (SLP) as a basis. This approach reduces WSVIE to a system of linear algebraic equations with coefficients, which contain closed-form analytical expressions. This allows the use of computer algebra systems for obtaining approximate solutions based on the finite terms truncation of series. We provided a number of examples highlighting such a workflow in addition to establishing the error bound, convergence analysis and stability analysis. [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.

Studies across a variety of educational fields have shown the efficacy of feedback on student performance and learning. Web-based homework is a common feature of secondary and collegiate mathematics courses to provide such feedback. While web-based homework provides often instantaneous feedback to students as they complete assignments, the feedback is limited to the correctness of the final answer or to a limited number of pre-programmed common mistakes. Intelligent Tutoring Systems have the ability to provide step-by-step feedback and guidance tailored to individual students, however authoring content for such systems is generally considered a huge barrier to their wider use. This paper presents an alternative system, called SANYMS (Show and Name Your Math Steps), which combines the wide-scale usability of web-based homework with a Constraint-Based Intelligent Tutoring System Expert Model based on the open-source computer algebra system Maxima. With the integration of Maxima, SANYMS can provide step-by-step feedback to students constructing their own solutions to a wide variety of algebraic equations of their choice without the need for extensive content authoring inherent in many Intelligent Tutoring Systems. [ABSTRACT FROM AUTHOR]

Based on the binary product described by any Latin square, the Hadamard quasigroup product is introduced in this paper as a natural generalization of the classical Hadamard product of matrices. The successive iteration of this new product is endowed with a cyclic behaviour that enables one to define a pair of new isomorphism invariants of Latin squares. Of particular interest is the set of Latin squares for which this iteration preserves the Latin square property, which requires the existence of successive localized Latin transversals within the Latin square under consideration. In order to enumerate and classify, up to isomorphism, these Latin squares, we propose a computational algebraic geometry approach based on the computation of reduced Gröbner bases. To illustrate this point, we obtain the classification of the sought Latin squares, for order up to six, by using the open computer algebra system for polynomial computations Singular.

Visually impaired individuals face challenges in pursuing higher education, particularly in technical courses related to engineering. The lack of specialized tools and resources that allow for proper development in academic activities is a significant barrier to their admission, retention, and graduation from higher education institutions. By considering this, this paper presents the development of a blind-friendly Computer Algebra System (CAS) called IrisMath. This system enables visually impaired people to perform mathematical operations commonly used in engineering. IrisMath is a web application inspired by Jupyter Notebooks and developed using a Layered Architecture to provide modularity. It offers a variety of output formats, including LaTex, CMathML, JSON, and audio formats. Our CAS has undergone an extensive assessment of its functional, non-functional, and usability requirements, demonstrating its potential as a tool for engineering students with visual disabilities.

The authors were surprised by the number of bibliographic citations to the computer algebra system DERIVE long after it was discontinued and developed a small bibliographic study about its evolution. They later carried out a similar one about the successful dynamic geometry system GeoGebra. Finally, they presented at Maple Conference 2022 a comparative small bibliographic study about Maple and other computer algebra systems, which can be found here. [ABSTRACT FROM AUTHOR]

Based on the binary product described by any Latin square, the Hadamard quasigroup product is introduced in this paper as a natural generalization of the classical Hadamard product of matrices. The successive iteration of this new product is endowed with a cyclic behaviour that enables one to define a pair of new isomorphism invariants of Latin squares. Of particular interest is the set of Latin squares for which this iteration preserves the Latin square property, which requires the existence of successive localized Latin transversals within the Latin square under consideration. In order to enumerate and classify, up to isomorphism, these Latin squares, we propose a computational algebraic geometry approach based on the computation of reduced Gröbner bases. To illustrate this point, we obtain the classification of the sought Latin squares, for order up to six, by using the open computer algebra system for polynomial computations Singular. [ABSTRACT FROM AUTHOR]

The authors were surprised by the number of articles that used or cited the computer algebra system DERIVE more than 10 years after it was discontinued and developed a small bibliographic study about it, published in 2019. Now they address in a similar way the very successful dynamic geometry system GeoGebra that, although created 20 years ago, later than the other great dynamic geometry systems (Cabri Geometry II, The Geometer’s Sketchpad and Cinderella), has now dozens of millions of users around the world. Not surprisingly, the cites to GeoGebra in the well known bibliographic databases Scopus, Web of Science and Google Scholar show an impressive growth. [ABSTRACT FROM AUTHOR]

In this paper, we propose two Maple procedures and some related utilities to determine the maximum curvature of a cubic Bézier-spline curve that interpolates an ordered set of points in R 2 or R 3 . The procedures are designed from closed-form formulas for such open and closed curves. [ABSTRACT FROM AUTHOR]

We present the possibilities provided by the MathPartner service of calculating definite and indefinite integrals. MathPartner contains software implementation of the Risch algorithm and provides users with the ability to compute antiderivatives for elementary functions. Certain integrals, including improper integrals, can be calculated using numerical algorithms. In this case, every user has the ability to indicate the required accuracy with which he needs to know the numerical value of the integral. We highlight special functions allowing us to calculate complete elliptic integrals. These include functions for calculating the arithmetic-geometric mean and the geometric-harmonic mean, which allow us to calculate the complete elliptic integrals of the first kind. The set also includes the modified arithmetic-geometric mean, proposed by Semjon Adlaj, which allows us to calculate the complete elliptic integrals of the second kind as well as the circumference of an ellipse. The Lagutinski algorithm is of particular interest. For given differentiation in the field of bivariate rational functions, one can decide whether there exists a rational integral. The algorithm is based on calculating the Lagutinski determinant. This year we are celebrating 150th anniversary of Mikhail Lagutinski.

The author recently designed, developed and implemented in Maple a package based on the use of digraphs that analyses the connectivity of an overtaking station on a double-track line. It was used to propose an alternative track layout for this kind of station, with advantages over the track layouts usually adopted. However, that package could only deal with "standard" turnouts (but neither with crossings nor with "special" turnouts, such as "single slip turnouts" or "scissors crossings"). This new article presents an improved version of the package. It uses a trick consisting in including virtual vertices in the associated digraph that are dead ends. This way it is possible to include the "special" turnouts in the track layout. It makes it possible to evaluate different alternative track layouts, including "special" turnouts; and to finally find one track layout that has advantages over the standard one and over the one proposed in the previous article. Let us observe that the design of the track layout is key for the exploitation of the infrastructure. In fact, the Spanish infrastructure administrator is nowadays remodelling the track layout of some of its main railway stations, as well as other smaller facilities. [ABSTRACT FROM AUTHOR]

Using the symbolic calculation program Mathematica and based on the power series expansions of the common latitude with geodetic latitude as a variable, power series expansions of the common latitude with geocentric latitude as the variable are derived. The coefficients of the two groups of formulas are based on the ellipsoid eccentricity e and the ellipsoid third flattening n, which make the expansions more uniform. Taking the CGCS2000 as an example, numerical analysis is applied to verify the accuracy and reliability of the derived power series expansions. By analyzing and calculating the truncation error of the common latitude based on ellipsoidal eccentricity e and the third flattening n expansion to different orders, we obtain simplified, practical formulas for the common latitude that satisfy the requirement of geodesic accuracy. Moreover, we show that the practical formula derived has higher calculation efficiency and easier dissemination, enriches the theory of map projection, and provides a basis for better display of remote sensing images. [ABSTRACT FROM AUTHOR]

In this paper we present an algorithm for efficiently counting fixed points in a finite monoid M under a conjugacy-like action. We then prove a formula for the character table of M in terms of fixed points, which allows for the effective computation of both the character table of M other a field of null characteristic, as well as its Cartan matrix, using a formula from [Thiéry '12], again in terms of fixed points. We discuss the implementation details of the resulting algorithms and provide benchmarks of their performance. [ABSTRACT FROM AUTHOR]

Objectives: The Earth's radial diameter and mean radius of curvature are the basic parameters commonly used in measurement and Earth science calculation. According to the requirements of Earth science and space science and some requirements, the Earth's radial diameter, mean radius of curvature, mean radius of sphere, equal distance radius of sphere, equal area radius of sphere and equal volume radius of sphere are commonly used. The concept of integral mean values of ellipsoid radius vector and mean radius of curvature is introduced. With the application and development of space technology and computer technology in geodesy and cartography, it is of more important practical value to study the relationship between the Earth's radial diameter and the common Earth radius. Methods: The symbolic expressions of them are deduced by computer algebraic system and are expressed as the power series of eccentricity. We use the method that comparing the radial vector integral mean and the radius integral mean with the four common sphere radius respectively. Results: The results shows that the difference between the radial integral mean and the four common sphere radii is smaller. Since the Earth is a rotating ellipsoid, the radial and the radius of curvature deviate. When the radial vector is the largest, the radius of curvature is the smallest. When the radial is the smallest, the radius of curvature is the largest. Conclusions: The radius of curvature considered by traditional thinking cannot accurately represent the radius of the earth average values, to a certain extent, which means that integral mean value of ellipsoid radius vector is more representative of the average of the Earth's radius. These research results can provide theoretical basis for Earth science, space science and navigation and positioning. [ABSTRACT FROM AUTHOR]

Copyright of Hydrographic Surveying & Charting / Haiyang Cehui is the property of Hydrographic Surveying & Charting Editorial Board and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The article discusses the kinematic support, which allows reducing the horizontal dynamic effects on the building during earthquakes. The model of a seismic isolation support is considered from the point of view of classical mechanics, that is, we assume that the support is absolutely solid, oscillating in a vertical plane above a fixed horizontal solid plate. This approach allows a more adequate description of the interaction of the support with the soil and the base plate of the building. The paper describes the procedure for reducing the complete system of equations of motion of a massive rigid body on a fixed horizontal perfectly smooth plane to a form suitable for applying the finite difference method and its implementation in the Sage computer algebra system. The numerical calculations by the Euler method for grids with different number of elements are carried out and a mathematical model of the support as a perfectly rigid body in the Sage computer algebra system is implemented. The article presents the intermediate results of numerical experiments performed in Sage and gives a brief analysis (description) of the results.

Tujuan dari penelitian ini adalah untuk mengetahui adakah pengaruh website eLearning Moodle berbantuan Computer Algebra System (CAS) Maxima terhadap motivasi dan prestasi belajar pada pokok bahasan matriks pada siswa Madrasah Aliyah Negeri 1 Yogyakarta. Proses pengumpulan data dilaksanakan melalui dua siklus. Hasil penelitian menunjukkan website eLearning Moodle berbantuan CAS Maxima mampu meningkatkan motivasi dan prestasi belajar siswa ditunjukkan dengan sebelum menggunakan media website eLearning Moodle berbantuan CAS Maxima menunjukkan hasil yang kurang memuaskan hanya sebesar 39% yang memenuhi KKM, siklus I ketuntasan mencapai 84,375%, siklus II ketuntasan mencapai 100%. Motivasi belajar pada tindakan pra siklus memperoleh persentase 18% atau sangat rendah pada indikator rasa ingin tahu, tekun dan ulet dalam menghadapi kesulitan memperoleh persentase 26% atau kategori rendah, tidak memerlukan dorongan dari luar untuk berprestasi memperoleh persentase 20% atau kategori sangat rendah dan senang dan rajin belajar 30% berada pada kategori rendah. Pada siklus 1 terjadi peningkatan motivasi belajar yaitu semua indikator berada pada kategori sedang. Sedangkan pada siklus 2 terjadi peningkatan motivasi belajar semua indikator berada pada kategori tinggi. Â Abstract The purpose of this study was to determine whether the influence of Maxima's Computer Algebra System (CAS) eLearning Moodle website on motivation and learning achievement on the subject matter of the matrix on students of Madrasah Aliyah Negeri 1 Yogyakarta. The data collection process is carried out through two cycles. The results showed the Moodle eLearning website assisted by CAS Maxima able to improve students' motivation and learning achievement indicated by using the CAS Maxima-assisted eLearning website media showed unsatisfactory results of only 39% that met the KKM, the first cycle of completeness reached 84.375%, the second cycle of completeness reached 100% Motivation to learn in pre-cycle actions get a percentage of 18% or very low on indicators of curiosity, perseverance and resilience in the face of difficulty obtaining a percentage of 26% or low category, does not require outside encouragement to achieve achievement percentage of 20% or very low and happy categories and 30% study diligently in the low category. In cycle 1 an increase in motivation to learn that all indicators are in the medium category. Whereas in cycle 2 an increase in motivation to learn all the indicators are in the high category.

In this paper, we propose two Maple procedures and some related utilities to determine the maximum curvature of a cubic Bézier-spline curve that interpolates an ordered set of points in R2 or R3. The procedures are designed from closed-form formulas for such open and closed curves.

Let H ≤ S n be an intransitive group with orbits Ω 1 , Ω 2 , ... , Ω k. Then certainly H is a subdirect product of the direct product of its projections onto each orbit, H | Ω 1 × H | Ω 2 × ... × H | Ω k . Here we provide a polynomial time algorithm for computing the finest partition P of the H -orbits such that H = ∏ c ∈ P H | c and we demonstrate its usefulness in some applications. [ABSTRACT FROM AUTHOR]

Compared with complex and lengthy Gauss-Krüger projection series expansions and real number expressions, we improve the complex function representation of Gauss-Krüger projections and rewrite them into the “multiple Angle form”, “exponential form”, and “double Angle form”. The coefficients were expanded in the power series based on the first eccentricity e and the third flattening n, respectively, and the truncation difference was analyzed when expanded to different orders to obtain the simplified practical formulas for each form on the premise of meeting the accuracy requirements of geodesy. Through numerical analysis, the computational efficiency of the forward and inverse solutions of the Gauss-Krüger projection is analyzed, which shows the superiority of the “double Angle form”. Through the above measures, the expressions for forward and inverse solutions of the Gauss-Krüger projection are obtained, meeting the accuracy requirements with a higher computational efficiency and a more concise form.

The article is devoted to the research of a problem of the application of computer algebra systems in the course of skill training of future mathematics teachers. The possibility of study organization for students of pedagogical institutions of higher education receiving "014 secondary education (mathematics)" speciality with the use of computer algebra systems is considered. The term "computer algebra system" is understood as the software which is intended for the solving of various mathematical tasks by means of a computer with a possibility of visualization of all stages of calculations. Proceeding from the fact that the computer algebra systems are actively used in educational process all over the world and informational educational environments in the course of teaching mathematical disciplines are created on their basis, it is possible to make a conclusion that the computer algebra systems are very important components of the process of training future experts. The current state of the problem of introduction and application of the computer algebra systems in the training of future mathematics teachers is investigated. The role of the computer algebra systems in teaching mathematics subjects and general natural sciences in pedagogical establishments of higher education is defined. Such computer algebra systems as MathCAD, MatLab, Maple, Mathematica, Maxima, SAGE, and Axiom are marked out among the most modern and successfully used systems aimed at solving mathematical tasks. Arguments in favour of the use of the free computer algebra system Maxima in educational and scientific activity of students are adduced in the article. Its main advantages and disadvantages are presented. The main functional of this system is considered which allows to optimize the time of the solution of various mathematical tasks, to adjust a conclusion of the received result, to perform staged verification of correctness of calculations.

Using the symbolic calculation program Mathematica and based on the power series expansions of the common latitude with geodetic latitude as a variable, power series expansions of the common latitude with geocentric latitude as the variable are derived. The coefficients of the two groups of formulas are based on the ellipsoid eccentricity e and the ellipsoid third flattening n, which make the expansions more uniform. Taking the CGCS2000 as an example, numerical analysis is applied to verify the accuracy and reliability of the derived power series expansions. By analyzing and calculating the truncation error of the common latitude based on ellipsoidal eccentricity e and the third flattening n expansion to different orders, we obtain simplified, practical formulas for the common latitude that satisfy the requirement of geodesic accuracy. Moreover, we show that the practical formula derived has higher calculation efficiency and easier dissemination, enriches the theory of map projection, and provides a basis for better display of remote sensing images.

Copyright of Journal of East China University of Science & Technology is the property of Journal of East China University of Science & Technology Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The requirements for the training of mathematics and computer science teachers to use specialized mathematical software in professional activities are substantiated in the article. Among them: the ability to creatively choose the forms and methods of teaching for the use of specialized software; ability to find non-standard or creative mathematical problems; ability to make an informed choice of specialized software; ability to see possible ways to check the result obtained by the pupil; the ability to eliminate common mistakes when pupils use computer tools, etc. The specialized mathematical software, which is used today in mathematical training of mathematics and computer science teachers in Ukraine, has been specified. The specialized mathematical software which is used today for teaching mathematics in schools of Ukraine is given. The analysis of computer tools used by the teacher in the most common mathematical software is carried out. The most urgent problems faced by mathematics teachers in their professional activities are highlighted. The experiment with ninth-grade pupils proved the positive impact of using dynamic geometry software and appropriate mathematical tools on the level of pupils’ mathematical training, which is reason to talk about the importance of computer science and mathematics pre-service teacher training to involve such tools in professional activities. Prospects for further research in the direction of developing methods for using computer mathematical tools in a research-based learning environment are presented.

The article looks at the means of forming the research skills of students of pedagogical universities. Considerable attention is paid to the formation of students' creative thinking in the process of solving problems with parameters. A large number of applied problems, economic, physical, chemical, biological, technical, medical, etc., involve solving problems with parameters (within the framework of the constructed mathematical model). Tasks with parameters require more general research than ordinary equations, inequalities and their systems. In the article, solving problems with parameters is accompanied by graphical visualization followed by the use of the analytical method. CAS Maxima is used for graphical visualization (animation). The choice of tasks with parameters is due to the fact that mathematical, logical thinking and the ability to analyze, compare, synthesize, develop research skills evolve in the course of their solution. All this should lead to the search and introduction of new forms of pedagogy and technologies in education. The main component of the methodology here is computer-oriented, namely the use of a computer mathematics system. Using systems of computer mathematics, the student can improve the programming technique and ability to focus on the analysis of methods, immerse in the features of such concepts as the conditionality of the problem, the stability of the method, evaluation of the results of calculations. In this paper, we used the CAS Maxima animation tools with the wxMaxima graphical shell to investigate the problems with the parameters. The choice of the Maxima system is due to the following reasons: it is a freely distributed system, distributed under the GNU/GPL license; there are implementations under various operating systems, including Windows, Linux, MacOS; it offers intuitive interface and is easy and reliable in operation. The results of the study show that training of modern specialists, the development of their professional potential can only be effective provided that students are involved in research activities throughout their course of study.