Algebraic Concepts Research Articles

An unbounded operator theory approach to lower frame and Riesz-Fischer sequences

Frames and orthonormal bases are important concepts in functional analysis and linear algebra. They are naturally linked to bounded operators. To describe unbounded operators those sequences might not be well suited. This has already been noted by von Neumann in the 1920ies. But modern frame theory also investigates other sequences, including those that are not naturally linked to bounded operators. The focus of this manuscript will be two such kind of sequences: lower frame and Riesz-Fischer sequences. We will discuss the inter-relation of those sequences. We will fill a hole existing in the literature regarding the classification of these sequences by their synthesis operator. We will use the idea of generalized frame operator and Gram matrix and extend it. We will use that to show properties for canonical duals for lower frame sequences, such as a minimality condition regarding its coefficients. We will also show that other results that are known for frames can be generalized to lower frame sequences. Finally, we show that the converse of a well-known result is true, i.e. that minimal lower frame sequences are equivalent to complete Riesz-Fischer sequences, without any further assumptions.To be able to tackle these tasks, we had to revisit the concept of invertibility (in particular for non-closed operators). In addition, we are able to define a particular adjoint, which is uniquely defined for any operator.

An Application Approach to Teaching Linear Algebra

This paper shares insights and experiences in developing and teaching a first-year applied linear algebra course. In addition to incorporating innovative pedagogical techniques like active and inquiry-based teaching and growth grading systems, this course implements modules and other activities which connect linear algebra concepts to applications such as computer graphics and sports analytics. These activities enhance students' experiences by providing them with clear applications of linear algebra concepts and the opportunity to learn accessible modeling techniques used by researchers. These application-centered activities have been incorporated as 50-minute lab modules or as short in-class activities and have been used across multiple modalities including asynchronous and synchronous online classes. This paper provides accessible examples and resources which can help instructors incorporate interesting applications and activities into a linear algebra course. We also provide suggestions and tips for incorporating these activities and pedagogical techniques.

Exploring group concepts in abstract algebra through ChatGPT

Traditional methods often struggle to engage students and effectively communicate the intricacies of abstract algebra. This study aims to explore the future of mathematics teaching, with a focus on integrating group theory concepts into Abstract Algebra using ChatGPT, an advanced language processing AI. Chat transcripts edited with ChatGPT were analyzed to gain insights into the evolving landscape of mathematics education, specifically concerning group theory. Group theory is a fundamental branch of Abstract Algebra poised to shape the future of mathematics instruction. ChatGPT facilitates interactive discussions and problem-solving tasks related to group operations, properties, and theorems, offering personalized explanations and guidance tailored to individual student needs. The study highlights the potential of AI-driven platforms like ChatGPT to revolutionize mathematics education by providing dynamic and engaging learning experiences centered around group theory. Furthermore, the study emphasizes the importance of incorporating group theory into mathematics curricula to foster critical thinking, problem-solving skills, and interdisciplinary connections. Educators are urged to leverage innovative tools like ChatGPT alongside traditional pedagogical approaches to ensure effective teaching and learning. Through thoughtful implementation and ongoing professional development, educators can harness AI's power to create inclusive, equitable, and enriching learning environments, empowering students in abstract mathematical reasoning and analysis.

Prior knowledge of a calculus course: The impact of prior knowledge on students’ errors

We conducted a descriptive study among first-year engineering students at the Anton de Kom University of Suriname. We analyzed students’ errors regarding necessary prior knowledge in a calculus A exam. We found that the stage of the solution in which prior knowledge is required impacts the importance of prior knowledge. We also found that many errors concerned basic algebra and trigonometry concepts and skills. We concluded that even though the required prior knowledge concerns basic algebra and trigonometry, the stage of the solution in which prior knowledge is needed is of great importance.

Album of animated drawings as an innovative means of teaching algebra in the digital environment of bilingual students

Introduction. Digitalization of education has opened up new opportunities for the development and application of the didactic potential of digital technologies in teaching algebra in the context of bilingualism. This determined the purpose of this work, which is to create a methodological system for teaching algebra to bilingual students and an album of animated drawings in a digital environment. Scientific and pedagogical research focuses on the problem of mathematical training of schoolchildren in the context of bilingualism. From the point of view of improving the quality of education, it is advisable to rely on visual-figurative and associative thinking developed by bilingual students. Materials and methods. The experimental study involved 50 students of the seventh grade of gymnasium No. 5 in the city of Kyzyl, Republic of Tyva (Russian Federation). The methods of comparative analysis of scientific, pedagogical and methodological sources, empirical and statistical methods (Student's t-test) are applied. Results. A methodological system for teaching algebra to bilingual students using an album of animated drawings in a digital environment has been developed. An album of animated drawings is presented, implemented in the form of an electronic training course in the electronic learning system of Tuvan State University and the results of its application in teaching algebra to bilingual students. As a result of the experiment, it was revealed that in the absence of statistically significant differences in the control and experimental groups at the time of the initial control, since tэмп = 2,00 < tкр = 2,01, during the final control, tэмп = 2,80 > tкр = 2,01 at the significance level p < 0,05. This indicates that the compared groups are significantly different and the experimental hypothesis about the effectiveness of using an album of animated drawings as an innovative means of teaching seventh grade algebra in the digital environment of bilingual students is confirmed. Conclusion. The use of an album of animated drawings as an innovative means of teaching algebra in the digital environment of bilingual students of the seventh grade of the Republic of Tyva increases the effectiveness of learning by increasing the clarity of algebraic concepts and statements, reducing the amount of non-essential calculations, as well as the use of mechanisms for rapid assessment of learning outcomes using a developed system of bilingual prompts. The proposed learning tool forms and develops an experimental research style of learning.

How to Support The Algebraic Thinking Skills of Elementary School Students Using The Generative Multi-Representation Learning Model Modification Schema-Based Instruction?

Elementary school teachers still do not fully utilize effective learning models, especially in teaching algebraic concepts. Lack of understanding of algebraic concepts has caused students to have difficulty mastering algebra, including calculation, representation, and mathematical modeling, as well as recognizing algebraic symbols and variables. The purpose of this study is to answer the research question (RQ) How does the Generative Multi Representation Learning Model Modified Schema Based Instruction (MGMRM-SBI) affect the algebraic thinking skills of elementary school students? In this study, the experimental method was used with a posttest-only control group design, where the sample was selected using Cluster Random Sampling with a total sample of 128 students. The research instrument used was an algebraic thinking ability test in the form of description test questions, and data analysis was carried out using a t-test with the help of SPSS statistical software. The results of the t-test showed a significant difference between the algebraic thinking skills of students in the experimental and control classes, with significance values (2 sides) = 0.000 < sig. = 0.05. Based on the results of the study, it can be concluded that the application of the Generative Multi Representation Learning model Modified Schema Based Instruction has a significant effect on the algebraic thinking ability of elementary school students, and is more effective than the expository model.

Exploration of mathematical concepts in prayer and Mushofahah movements

Knowledge of integration between mathematics and religion is necessary to increase a Muslim's mastery of scientific treasures. This article describes several mathematical concepts contained in prayer movements and Mushofahah. This research is a qualitative descriptive research type of literature review. Data/information collection methods are carried out by observation and documentation. These two methods are used to collect information about mathematical concepts contained in prayer movements and Mushofahah, obtained from several primary and secondary sources in the form of the al-Qur’an, al-Hadith, books, and scientific journals as well as documentation of prayer movements and Mushofahah (shaking hands). The data analysis technique used is content analysis. The research results showed many daily life activities, including prayer and Mushofahah, concrete examples of the integration of mathematics and Islamic religion. These concepts include: In the prayer movement, there were number concepts, which included the addition and multiplication of integers, algebraic concepts in the form of sets, arithmetic sequences, and geometric concepts in the form of lines and angles. Meanwhile, in Mushofahah, there were factorial concepts, permutation concepts, and opportunities.

Profil Miskonsepsi Siswa pada Materi Bentuk Aljabar di Sekolah Menengah Pertama Negeri Kota Banda Aceh

Learning loss has an impact on various problems, including misconceptions by students when recognizing and learning algebraic operations. This research aims to identify the profile of students' misconceptions in algebraic concepts as a consequence of learning loss due to the Covid-19 pandemic in public junior high school in Banda Aceh. The study adopts a quantitative approach. The population of the research consists of all eighth-grade students in public junior high school in Banda Aceh and the sample comprises 27 students from one of these schools. The sample was obtained through purposive sampling, selecting students from the top-ranked junior high school in Banda Aceh, known for having the highest average scores. Misconceptions were assessed using an online-based cognitive diagnostic test on algebraic concepts. Data analysis was conducted by examining the composition of misconceptions based on their types. The research findings indicate that the majority of students experience misconceptions, with 25.20% experiencing misconceptions in notation, 20.37% in generalization, 8.98% in understanding symbols, 9.97% in applying rules, 1.14% in notation and generalization, 0.57% in notation and understanding symbols, and 2% in generalization and applying rules. This study indicates a necessity to an analysis of the causes of misconceptions and efforts to minimize students' misconceptions in algebraic concepts, such as implementing cognitive conflict learning or remedial teaching.

Understanding of Linear Algebra Concepts through the TAPPS (Thinking Aloud Pair Problem Solving) Learning Model at the Faculty of Computer Science, Dharma AUB University of Surakarta

Education aims to improve the quality of human life which is technically through lectures. A good lecture produces a positive influence on students' ability. The interaction between lecturers and students in the class is an important role in achieving the goals. Linear Algebra is an important subject in the Mathematic. Many students think that Linear Algebra are difficult. Linear Algebra at the Faculty of Computer Science, Dharma AUB University of Surakarta have not been able to improve students' understanding of Linear Algebra concepts. Because of class that are centered on lecturers. Based on the description above, the author applies the TAPPS (Thinking Aloud Pair Problem Solving) cooperative learning model to increase concept understanding of Linear Algebra. Formulation of the problem in this research: "Is there an increase concept understanding of Linear Algebra after using the TAPPS learning model?". This research is a qualitative descriptive group with the type of Classroom Action Research. The data used are field notes and documentation. Based on the research results, it can be concluded that there is an increase in understanding of the concept of Linear Algebra through the TAPPS (Thinking Aloud Pair Problem Solving) model.

A Quantum Harmonic Analysis Approach to Segal Algebras

In this article, we study a commutative Banach algebra structure on the space L1(R2n)⊕T1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1(\\mathbb {R}^{2n})\\oplus {\\mathcal {T}}^1$$\\end{document}, where the T1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}^1$$\\end{document} denotes the trace class operators on L2(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\mathbb {R}^{n})$$\\end{document}. The product of this space is given by the convolutions in quantum harmonic analysis. Towards this goal, we study the closed ideals of this space, and in particular its Gelfand theory. We additionally develop the concept of quantum Segal algebras as an analogue of Segal algebras. We prove that many of the properties of Segal algebras have transfers to quantum Segal algebras. However, it should be noted that in contrast to Segal algebras, quantum Segal algebras are not ideals of the ambient space. We also give examples of different constructions that yield quantum Segal algebras.

Abstract algebra students’ conceptual metaphors for isomorphism and homomorphism

Group isomorphism and homomorphism are core concepts in abstract algebra, and student understanding of isomorphism has received extensive attention in line with the centrality of this topic. However, limited work has directly examined student conceptions of homomorphism or what metaphors students use to express their thought processes while problem solving. Based on interviews with four students, we contrast two students who used predominantly formal definition and mapping-centered metaphors for homomorphism with two who additionally used sameness-centered metaphors and note that the usage or non-usage of sameness-centered metaphors was not indicative of successful problem solving. Implications include the alignment between students’ metaphors and those used in instruction, indicating the importance of attending to metaphors when teaching, and the importance of discussing what is intended by some sameness-based metaphors, such as operation-preservation.

Classifying large N limits of multiscalar theories by algebra

We develop a new approach to RG flows and show that one-loop flows in multiscalar theories can be described by commutative but non-associative algebras. As an example related to D-brane field theories and tensor models, we study the algebra of a theory with M SU(N) adjoint scalars and its large N limits. The algebraic concepts of idempotents and Peirce numbers/Kowalevski exponents are used to characterise the RG flows. We classify and describe all large N limits of algebras of multiadjoint scalar models: the standard ‘t Hooft matrix theory limit, a ‘multi-matrix’ limit, each with one free parameter, and an intermediate case with extra symmetry and no free parameter of the algebra, but an emergent free parameter from a line of one-loop fixed points. The algebra identifies these limits without diagrammatic or combinatorial analysis.

Local modules in braided monoidal 2-categories

Given an algebra in a monoidal 2-category, one can construct a 2-category of right modules. Given a braided algebra in a braided monoidal 2-category, it is possible to refine the notion of right module to that of a local module. Under mild assumptions, we prove that the 2-category of local modules admits a braided monoidal structure. In addition, if the braided monoidal 2-category has duals, we go on to show that the 2-category of local modules also has duals. Furthermore, if it is a braided fusion 2-category, we establish that the 2-category of local modules is a braided multifusion 2-category. We examine various examples. For instance, working within the 2-category of 2-vector spaces, we find that the notion of local module recovers that of braided module 1-category. Finally, we examine the concept of a Lagrangian algebra, that is a braided algebra with trivial 2-category of local modules. In particular, we completely describe Lagrangian algebras in the Drinfeld centers of fusion 2-categories, and we discuss how this result is related to the classifications of topological boundaries of (3 + 1)d topological phases of matter.

TEACHING MATHEMATICS WITH VIRTUAL ROBOTS TO STUDENTS

The study introduces the case of development of mathematics oriented 20-h robotics course for students (n=50) with educational program of “6B01501 - Mathematics”. The research aimed at exploring the impact of robotics on students’ motivation and engagement by learning math using robotics activities. These activities include controlling virtual robots while learning Algebra concepts. Overall, the aim of practicing math using robots is to investigate the effectiveness of this approach to teaching math, and to identify ways to improve math education using technology. During the study, the researchers examined the global application of robotics and conducted a review of academic databases to collect relevant literature. The findings revealed a dearth of knowledge in the academic literature concerning this particular subject. The study was carried out for three months on the basis of NJSC "Zhetysu University named after Ilyas Zhansugurov" (Taldykorgan), during which the training program "Learning Mathematics with the help of robotics" was developed and tested. Example of lessons at roboblocky.com implemented by students presented in this article. Roboblocky is an educational platform that integrates robotics and programming to facilitate the instruction of mathematical and scientific concepts to students. The platform employs a block-based programming approach, which enhances the accessibility and comprehension of these concepts among students.Based on the available research, it can be concluded that teaching math with Roboblocky is a promising approach. Studies on using robotics to teach mathematics have shown that it can improve student engagement, achievement, problem-solving skills, and spatial reasoning. Additionally, the use of virtual robots in the classroom can promote positive attitudes towards learning among students

Algebraic Thinking Ability of Junior High School Students in Solving Linear Equations Problems

Algebraic thinking ability are important for students to master and use algebraic concepts in various situations. Students with good algebraic thinking ability will find it easier to solve real-world problems and understand more abstract mathematical concepts. This study aims to describe students' algebraic thinking ability on linear equation material. Thus, the type of research used is phenomenology with a qualitative descriptive approach. The participants in this study consisted of 37 students; six students with heterogeneous abilities (high, medium, low) were then selected. The research subjects were determined by purposive sampling based on the pattern of answers representing the phenomenon that occurred. Data were collected using algebraic thinking ability tests and interviews to fulfill the validity of the data. The results showed that: 1) students with high algebraic thinking ability can use all indicators of algebraic thinking well. 2) students with medium algebraic thinking ability have difficulty in the indicators of analytical thinking, modeling, and dynamic thinking. 3) students with low algebraic thinking ability have difficulty in all indicators of algebraic thinking, namely generalization, modeling, abstraction, analytical thinking, dynamic thinking, and organization. So, teacher efforts are needed to provide students with an understanding of algebraic thinking so that students can solve math problems well.

Creative thinking for learning algebra: Year 10 students’ problem solving and problem posing with quadratic figural patterns

There is increasing focus on giving all students opportunities to develop their creative thinking in mathematics. In a fine-grained qualitative case study, Australian Year 10 (15–16-year-old) students experienced problem-solving and problem-posing tasks in a learning sequence designed to elicit creative thinking through visualization and with algebraic (functional) concepts. This article shares findings on two pairs’ creative processes and work products for generalizing and creating quadratic figural growing patterns, related to the cognitive, affective, and aesthetic dimensions of learning. Post-sequence data on the class's preferences for types of creative thinking tasks are also shared. Students were found to exhibit different creativity aspects in their problem solving and varying levels of meta-representational competency in their problem posing. The students evidenced diverse affective responses to the different types of tasks, with proportionally more students preferring problem solving to problem posing. Some implications for engaging secondary mathematics students in creative thinking tasks are shared along with recommendations for future research.

Analytical modelling and design of linear controlled dynamic systems

A generalized canonical mathematical model of multidimensional controlled mining transportation complexes and other technical systems in form by A.M. Letov is considered. A principle of symmetry and an algebraic concept are in the basis of the developed analytical methods of design. A principle of symmetry is realized on a set of indexes of roots of a characteristic equation of the system and on a set of indexes of phase coordinates of the mathematical model. The problem of quality of dynamic processes in time is reduced to an algebraic problem of distribution of roots of a corresponding characteristic equation in a complex plane. An analogy in a procedure of transformation of a characteristic determinant into a polynomial and a structure of elementary symmetrical polynomials of roots is established. A new formulation of an analytical representation of change of phase coordinates in time in a form of ordered determinants with respect to indexes of roots and indexes of phase coordinates is obtained based on residue theorem. An illustration of the developed analytical method of design is performed on a special case of a well-known controlled technical system of the fourth order.

LINEAR ALGEBRA AS THE MATHEMATICAL FOUNDATION OF ARTIFICIAL INTELLIGENCE: CONCEPTS, APPLICATIONS, AND FUTURE PROSPECTS

Linear algebra, a fundamental tool in the field of mathematics, plays an essential part in the mathematical foundation of the groundbreaking arena of Artificial Intelligence (AI). This mathematics branch offers immense possibilities for capturing complex relationships within datasets through matrices and vectors. These mathematical tools facilitate the concise representation of large volumes of data, contributing to efficient data analysis between them, bringing immense value to AI algorithms and models. This research paper investigates the pivotal role of linear algebra as the fundamental mathematical framework underpinning artificial intelligence. This paper also explores the foundational concepts of linear algebra and its direct applications in various AI algorithms, demonstrating how matrices, vectors, and transformations provide the essential mathematical scaffolding for the development and optimization of intelligent systems.

Methods of teaching secondary school students to solve algebraic equations and inequalities and their systems

Relevance. The relevance of this study is conditioned by the integral role of algebraic equations and inequalities in mathematics education, as well as the need for effective methods and approaches to their teaching at the secondary school level. Purpose. The study aims to develop and appropriate methods for teaching secondary school students to solve algebraic equations and inequalities, including their systems, taking into account modern educational requirements and pedagogical approaches. Methodology. In the course of the study, methods such as experimental research, questionnaire surveys, and statistical analysis of data were used. Results. Because of the study, it was revealed that the innovative methodology for teaching the solution of algebraic equations and inequalities in secondary school had a positive impact on students' mastery of the material. The participants demonstrated a deeper understanding of algebraic concepts, increased independence in problem-solving, and developed critical thinking skills. They became more confident in analysing and interpreting problem conditions, identifying relationships between variables, and proposing adequate solutions. The new approach also improved students� motivation to study algebra and their academic performance, with increased interest and passion for the subject due to the interactive methods and practical tasks provided. Conclusions. The study confirms the effectiveness of the innovative methodology in the contemporary educational context of mainstream schools. The practical significance of the study lies in providing mainstream schoolteachers with tools to successfully teach students algebraic equations and inequalities, contributing to the development of analytical thinking and mathematical competence. Keywords: interactive lessons; visualization of mathematical concepts; individual progress; skill development

Symbol Sense Ability of Students to Solve Linear, Quadratic, and Rational Equations

In learning algebra, one of the skills needed to be acquired by students is solving equations, such as linear equations in one variable and quadratic equations. This study aims to analyze the symbol sense ability of students by investigating strategies used to solve algebraic equations that consist of linear, quadratic, and rational equations. A qualitative descriptive method was conducted. The instrument included a test with one linear equation, one quadratic equation, and three rational equations (involving linear or quadratic factors). The test was administered to 12 students from 12th-grade high school in Bandung. Each student was asked to solve it in two different ways of answers if possible. The results showed that students were able to use symbol sense strategies for solving problems. However, there were still mistakes in solving algebraic squared value concepts, affecting their final results of the answers. We conclude that even if students can use their symbol sense abilities properly, they need to pay attention to their mathematics concepts. Keywords: Symbol Sense Ability, Solve Linear, Quadratic, Rational Equations