Mathematics Subject Research Articles

Morse Theory, Discrete Morse Theory and Applications

By employing a specific class of smooth functions to study a space, Morse theory establishes deep connections between analysis and topology. It is a classical subject of pure mathematics, originally pioneered by Marston Morse in the 1920s. In this article, we use Morse theory to present a proof of an interesting result on the knots, known as the Fry-Milnor theorem. We also discuss discrete Morse theory, a subject of applied mathematics developed by Robin Forman in the 1990s, and its application. We focus on elucidating especially the inherent similarity between classical Morse theory and discrete Morse theory

Role of Rectangular and Square Matrix in Graph Theory

Graph theory is one of the topics studied in modern mathematics and the concept of graph in mathematics was discussed from the past to the present. For example, graph relations and functions have many uses. The 18th century is the beginning of modern graph theory. From the 19th century onwards, graph theory was discussed for its application in various fields, which is a field of research at this time. Graph theory and matrix are two important and well-known topics in modern mathematics, so in this article the role of rectangular and square matrix in graph theory is discussed. The main purpose of preparing this article is to study rectangular and square matrix in graph theory. The research method in this article is library that was followed by using academic articles in libraries and the Internet. The findings obtained from this study show that the matrix adjacent to the graph is a square matrix and the occurrence matrix is ​​a rectangular matrix.

Promoting mathematical competence development through programming activities

Abstract The reasons for teaching programming in school are indeed manifold. Programming can for example be utilized as a vehicle for understanding and learning particular mathematical subject matter, or as a tool for solving mathematical problems. In this paper, however, we propose to utilize programming as a vehicle for developing mathematical competencies such as problem-solving, exploration, modeling, reasoning, representation, generalization, and abstraction. We perform a design study utilizing a systematic, iterative development process for designing teaching sessions, focusing on one mathematical competence at a time. We apply the proposed methodology on a set of teaching sessions that are developed, implemented, and tested in a range of classroom settings. Based on this development process, we conclude that programming can be used to develop mathematical competencies. Furthermore, we identify a set of teacher practices that can be used to facilitate mathematical competence development in programming activities.

Construcción conceptual en física a través de métodos didácticos inductivos

This contribution shows the development and evaluation of an educational process withmathematics teachers in training, focused on the use of teaching strategies with inductive characteristics, in Mathematics and Physics subject of the Bachelor’s Degree in Mathematics Teaching at the University of Colima, Mexico. The experience includes 19 students, who have had disciplinary training in mathematics and master the procedural knowledge and algebraic language involved in Physics at the basic levels, but they had had little access to conceptual knowledge of both classical physics and modern physics. With this principle, the course focused on reflection and problem solving. From this logic, the proposal included searching for both the procedural and conceptualdomain, this second one is the central objective of this research. The inductive methods included theuse of audiovisual materials and readings, that go more in the order of dissemination. The results of their work show that, without omitting a more traditional training process, such as problem-solving or classic physics textbooks, the incorporation of inductive strategies focused on the particularities of concepts like “motion” or “light” allows for a deeper understanding of fundamental principles,serving as a functional complement for a more comprehensive understanding.

UNCOVERING RELEVANT MATHEMATICS TOPICS FOR HOME ECONOMICS: SENIOR HIGH SCHOOL STUDENTS' VIEWS IN CAPE COAST, GHANA

Mathematics holds universal significance, playing a pivotal role across diverse disciplines in our daily lives. Its impact is crucial for societal development, with mathematics acting as a cornerstone. This study delves into the senior high school (SHS) mathematics topics that students studying home economics consider most relevant to their programme of study and future career needs. Employing an explanatory sequential mixed-methods design, data was collected through a questionnaire and a semi-structured interview guide from grades 11 and 12. Out of a population of 330 students, 286 were sampled using a multi-stage technique, and 277 students filled out the questionnaires, resulting in a 97% response rate. Quantitative data was analysed using percentages and frequency counts, while thematic analysis was employed for the qualitative interview data. Results reveal that ratio and rate, percentages, mensuration, and statistics emerge as the most impactful mathematics topics influencing home economics students' SHS programmes. The study recommends a focus on these topics, urging mathematics educators to always highlight the relevance of any topic they teach to the students' specific programmes of study. These findings underscore the importance of aligning the curriculum with real-world applications, thereby enhancing students’ analytical skills and preparing them for future careers.

Forming the value of a healthy lifestyle among students through mathematical problems

Introduction: Mathematical skills are of great importance since they foster critical thinking, problem-solving, and logical reasoning. Mathematical competence focuses on enabling students to apply mathematical concepts in real-life contexts, particularly to form the value of a healthy lifestyle. Objective: To study the possibilities of shaping the student's value of a healthy lifestyle and health-preserving competence through integrating mathematics problems into the educational process.Methods: To achieve the research's aim, experimental research design was applied. The experiment involved 23 instructors from different regions of Ukraine who taught mathematics in the 5th grade. A total of 404 students participated in the study, with 250 forming the Experimental Group and 154 forming the Control Group. The intervention phase lasted 12 weeks and included 7 mathematics topics. The method of descriptive statistics was selected for data analysis.Results: Various teaching strategies are used in the mathematics classroom for the formation of the value of a healthy lifestyle. Students consider video-based activities and problems the most exciting and engaging. At the same time, text activities are found to be less effective in the mathematics classroom. Besides, the experiment demonstrated that students remembered the educational information better when they solved mathematics problems in the classroom and were engaged in active or interactive activities. Using mathematics to form the value of a healthy lifestyle offers several advantages and challenges. Conclusions: The results have significant implications for secondary education improvement and student development

Worst value and robust value for general optimization problem under uncertainty data

In this paper we consider general form of a optimization problem in face of data uncertainty. We determine a necessary and sufficient condition ensuring the equality between the worst value and the robust value of this uncertain problem with attainment of the worst value. And we deduce robust strong duality and robust stable strong duality property. Keywords: Uncertain optimization problem; Robust counterpart; Robust value; Worst value; Robust strong duality. Mathematics Subject Classification: 90C17, 90C31, 90C46

A probabilistic method to prove AM-QM inequality

Inequality is one of the interesting topics in mathematics that attracts the attention of both students and researchers. Solving inequalities often requires creativity, making it a challenging topic for many students. The inequality of arithmetic and quadratic means, or AM-QM in short, states that the arithmetic mean of a list of non-negative real numbers is less than or equals to the square root of the quadratic mean of the same list. There are many methods of proving the AM-QM inequality. In this paper, we will present a probabilistic method to prove the weighted general AM-QM inequality and show that the classical AM-QM inequality is a special case of the generalized AM-QM inequality with equal weights.

AN INVESTIGATION ON CONTRIBUTION OF MATHEMATICIANS IN DEVELOPMENT OF MATHEMATICS

Mathematics is an international subject, and it has no geographical boundary for its development. Many countries have contributed for the development of the discipline of mathematics. The development of mathematical concepts started dates back to more than thousand years. For the development of the discipline mathematics, many countries mathematicians have contributed such as the Babylonian, Egyptian, Greek, Chinese and Indians etc. The concept of mathematics have come to us only for the need of calculations in daily life and commercial activities, measurement of land around for crop and to predict astronomical events happening around us by observing day and night. Therefore, mathematics is a science, because it is based on observation and recorded systematically in logical order. The contribution of Indian mathematicians in the field of mathematics is far more than western mathematicians. Hence, this study is suggesting that, there is a need to introduce among us a good number of Indian Mathematician contributions to develop positive attitude among the learners and to appreciate the mathematics subject and contribution of Indian mathematicians in the field of mathematics. Ancient Indian mathematicians have contributed immensely to the field of mathematics. The invention of zero is attributed to Indians and this contribution outweighs all other made by any other nation since it is the basis of the decimal number system, without which no advancement in mathematics would have been possible. The number system used today was invented by Indians and it is still called Indo-Arabic numerals because Indians invented them and the Arab merchants took them to the western world. It is also noticed that there is a distinct and inequitable neglect off the contributions of the sub-continent. Many of the developments of Indian mathematics remain almost completely ignored, or worse or attributed to the scholars of other nationalities, often European. However a few historians (mainly European) are reluctant to acknowledge the contributions of Indian mathematicians. Some of the great mathematicians are Pythagoras, Fibonacci, Euclid, Archimedes, Rene Descartes, and Aryabhata, who have contributed significant inputs in the field of mathematics in ancient times.

Beban Kognitif Siswa Dalam Pembelajaran BerdasarkanKurikulum Merdeka Sekolah Dasar

Cognitive Load is a theory that describes the amount of mental resources a person uses to complete a learning task. Mental resources are a person's cognitive abilities including mind, attention, memory and problem solving skills. Cognitive load in learning that occurs in the classroom describes the condition of students in dealing with learning tasks. The independent curriculum is the curriculum currently used at all levels of education including primary education. However, the curriculum faces problems both in terms of teachers and learners. The purpose of this study was to describe the condition of intrinsic, extrinsic and germane cognitive load of students during learning when using an independent curriculum. This research uses a descriptive qualitative approach. The data collection technique used purposive sampling as many as 6 informants consisting of 2 teachers and 4 students. The results of this study are the intrinsic cognitive load of students found in IPAS, PPKN, Mathematics, English and Javanese subjects. In addition, the intrinsic cognitive load is triggered by giving assignments and motivation or interest in learning from the students themselves. Extrinsic cognitive load was triggered by the lecture learning method used by the instructor or teacher due to the use of high vocabulary. Extrinsic cognitive load is also triggered by the condition of the learning environment, namely the noisy or crowded atmosphere inside and outside the classroom. The germane cognitive load is triggered when the teacher or lecturer presents the subject matter.

Analisis Implementasi Kurikulum Merdeka dalam Pembelajaran Matematika Siswa Kelas 1 Madrasah Ibtidaiyah Bojonegoro

This research is to find out and examine "Analysis of the Implementation of the Independent Curriculum in Mathematics Learning for Madrasah Ibtidaiyah Bojonegoro Students. This type of research is qualitative research which looks and hears in closer and more detailed explanations and individual understanding of their experiences. The research was carried out at MI Mafatihul Huda Bojonegoro . The population used was grade 1 students at MI Mafatihul Huda Bojonegoro. The research instrument was a questionnaire regarding the application of the Independent Learning Curriculum to mathematics subjects The overall aspect of the independent learning curriculum applied to grade 1 students at MI Mafatihul Huda is said to be high with a percentage of knowledge aspects of counting, counting and the profile of strengthening Pancasila of 80%. Utilization and retrieval aspects of 75%. The counting and counting skills aspect is 78%.

The Book Review Column

Foundation Mathematics for Computer Science: A Visual Approach, 4th edition (Springer, 2023) by John Vince (Bournemouth University, UK) is a comprehensive collection of discrete and continuous mathematical topics that are covered in most undergraduate programs in computer science. The subtitle refers to the author's use of colored graphs and tables to illustrate the concepts. Online Algorithms (Cambridge University Press, 2023) by Rahul Vaze (Tata Institute of Fundamental Research, India) is an accessible but rigorous introduction to the area aimed at advanced undergraduates and beginning graduate students. The book covers the basic as well as applied online problems with a preference of elegant analysis over performance. Privacy-preserving Computing for Big Data Analytics and AI (Cambridge University Press, 2023) by Kai Chen and Qiang Yang (Hong Kong University of Science and Technology) is a systematic examination of the history, theories, techniques, applications, and future of the field. Prize-winning neuroscientist Terrence Sejnowski (University of California at San Diego) explains the technology and mathematics behind large language models such as ChatGPT and explores the debate on their so-called comprehension of language in ChatGPT and the Future of AI: The Deep Language Revolution (The MIT Press, 2024).

Родительские отзывы об онлайн-школах: недостатки образования

The article presents the analysis of critical judgments in 52 parent reviews of online schools on the IRecommend website. Seven topics were identified using thematic modeling. The authors of the texts consider the management of educational institutions, failures of the online learning platform and the educational process as the main disadvantages of distance learning. Notably, dissatisfaction is particularly pronounced regarding the quality of instruction in mathematical subjects, physics, and chemistry. Subsequent thematic motifs reveal an excessive workload placed on parents due to their involvement in their children's online education, as well as a significant lack of social interaction among students. Conversely, concerns regarding the high cost of online education and the potential health risks associated with prolonged screen time received comparatively less attention. The results obtained in the study may be in demand in monitoring the quality of school education in Russia, as well as for the development of programs in the field of digitalization of education.

Analysis of Students' Learning Difficulties in Mathematics Subjects for Class V MIM Al-Akbar Pandeyan

Grade V MIM Al-Akbar Pandeyan students generally find it difficult to learn mathematics because they consider it a difficult and scary subject, even though mathematics has an important role in everyday life. This study aims to determine students' learning difficulties and the factors causing students' learning difficulties in mathematics subjects for grade V at MIM Al-Akbar Pandeyan, Ngemplak, Boyolali, in the 2022/2023 Academic Year. This study uses a quantitative descriptive research type. Data collection techniques use tests, questionnaires, and documentation. The results of the study show that: 1) The learning difficulties faced by students in mathematics subjects for grade V consist of two, namely students' difficulties in understanding the concept of material on nets of geometric shapes (cubes and cuboids) and difficulties in problem-solving, namely in arithmetic operations, solving story problems, and changing problem sentences into mathematical sentences; 2) The factors of learning difficulties faced by students in mathematics subjects of grade V at MIM Al-Akbar Pandeyan consist of two factors, namely internal factors (attitude in learning 54.80%, learning motivation 58.97%, physical health 44.87%, and sensory ability 45.19%) and external factors (teacher factors 39.74%, use of learning media 55.13%, school facilities and infrastructure 43.27%, and family environment 46.92%). The results of this study are expected to help teachers design more adaptive learning strategies based on student needs

Learning Media Design for Flat-Sided Space Based on Articulate Storyline for Class VII SMP

Mathematics is a subject with quite abstract material. So, it is not uncommon to be seen as difficult by some students. The low achievement of students in understanding learning materials is because students tend to understand learning Mathematics materials procedurally not conceptually. When learning material is presented in a different form, students have difficulty in understanding the material. With the help of learning media, students have a positive attitude towards learning Mathematics. Learning Mathematics using interactive multimedia needs to be a concern for teachers to improve the learning process in the future. Therefore, the purpose of designing this learning media is to help students in understanding mathematics learning, especially the material of flat-sided space building. In addition, so that this learning media can be used in the learning process. The research was conducted with a descriptive research design. The design of interactive multimedia learning media is followed by media validation, so that the media can be used in the math learning process. There are 3 design activities carried out, namely: designing learning media in mathematics subjects, designing interactive multimedia teaching material content, designing articulate storyline-based interactive multimedia on flat-sided space building material. The design of articulate storyline-based interactive multimedia can increase the involvement of students in the learning process and increase students' understanding of the material of flat-sided spaces. From the results obtained, it can be concluded that the articulate storyline-based interactive multimedia learning media on flat-sided space material that has been designed is feasible to be utilized and developed in mathematics lessons.

A Sustainable Supply Chain Model on the Impact of Carbon Emission with Pre-Payment Based Partial Free Transportation

Objectives: To analyze and justify the importance of Green technology investment for both production and distribution processes in the supply chain along with a Pre-Payment Based Partial Free Transportation strategy. Methods: A non-linear sustainable supply chain model considering a single manufacturer and multi-retailer is formulated with an incorporation of green technology investment for both production and transportation processes along with partial pre-payment based free transportation. Lagrangian method is used to obtain the optimal solution of the problem. Findings: The proposed model has been solved with the prescribed method and it gives the average profit of the manufacturing firm with green technology investment is $15537.360 and the average profit of the manufacturing firm without green technology investment is $9864.8816. This shows that the model with green technology performs better than the other one. Novelty: Green technology investment has been incorporated for both production and transportation processes, along with partially pre-paid, free transportation, whereas earlier research focused only on green investment for production. 2020 Mathematics Subject Classification: 90B05, 90B06. Keywords: Green technology, Carbon emission, Carbon cap, Carbon tax, Prepayment transportation

Multi-Objective Electric Vehicle Transshipment Problem with Battery Swapping Technology Under Fuzzy Environment

Objectives: To evaluate the efficiency of the multi-objective battery-swapping electric vehicle transshipment problem by analyzing the total cost and time involved in the entire distribution process. Methods: A multi-objective battery-swapping electric vehicle transshipment model is formulated and a case study on a food grain company is considered to demonstrate the pertinent nature of this model. Due to real-life uncertainty, Fermatean fuzzy parameters are considered in this model. Existing neutrosophic fuzzy programming and linear weighted sum method have been used to obtain the solutions for the multi-objective transshipment model. Findings: A comparative study has been made for both multi-objective electric vehicle transshipment and multi-objective electric vehicle transportation problems with battery swapping as well as charging technology. Optimum Solutions obtained for the case study by using these two prescribed methods reveal that the multi-objective transshipment problem with battery swapping gives the minimum transportation cost and time than the charging technology. Obtained solutions for multi-objective electric vehicle transshipment problems with battery swapping show a reduction of 7.8% in cost and 11.19% in the charging technology. Meanwhile, obtained solutions for multi-objective electric vehicle transshipment problems show a reduction of 14% in cost and 14.4% in time than the multi-objective transportation problem. Novelty: The efficiency of multi-objective electric vehicle transshipment problems with battery swapping technology under the Fermatean fuzzy environment has not yet investigated in the literature. 2020 Mathematics Subject Classification: 90B06. Keywords: Transshipment problem, Electric vehicle, Battery swapping technology, Fuzzy environment, Multi­objective transshipment problem

Incorporation of RFID Technology in the Inventory Management System to Optimize the Inventory Cost – ‘C’ Programming Approach

Objectives: To examine and justify the significance of Radio Frequency Identification technology in the inventory management system with scheduled shortages through a C programming approach. Methods: Radio Frequency Identification technology is examined in the inventory and analyzed between the two strategies namely i) non- Radio Frequency Identification technology and ii) Radio Frequency Identification investment in both ordering and Just In Time efficiency. The proposed methods were implemented and solved using C programming to evaluate their effectiveness and operational performance in managing inventory and reducing costs. Findings: An integrated investment evaluation model is established in this work and a comparative study has been made to analyze the assessment model with scheduled shortages in the absence and presence of Radio Frequency Identification technology by assigning the parameters using a numerical example performed by C programming. Results show that the total cost under non Radio Frequency Identification technology is 200547722.557505 whereas the total cost under Radio Frequency Identification technology investment is 100860759.476361. The number of orders placed each year under non Radio Frequency Identification technology and Radio Frequency Identification technology are 182 and 214 respectively. The results imply that the incorporation of Radio Frequency Identification technology in inventory management leads to improved efficiency and accuracy, resulting in increased profitability for an organization. Novelty: Radio Frequency Identification technology investment is incorporated into an inventory system to optimize inventory costs and annual order frequency using C programming, which remains unexplored in existing literature. 2020 Mathematics Subject Classification: 90B05. Keywords: Radio Frequency Identification technology, Investment evaluation, Inventory cost, Shortages, Economic Order Quantity

Status Sombor Indices of Unitary Cayley Graph

Objectives: The objective of the study is to obtain the values of status Sombor indices for the unitary Cayley graph. Methods: The unitary Cayley graph 𝑋𝑛 has vertex set 𝑍𝑛 = {0, 1, 2, ...𝑛 − 1}. Vertices a, and b are adjacent, if 𝑔𝑐𝑑(𝑎 − 𝑏, 𝑛) = 1. The status 𝜎 (𝑢) of a vertex u in a graph, G is the sum of distances of all other vertices from u in 𝐺. In this paper, we obtain the values of the Status Sombor index, Modified Status Sombor index, and their corresponding exponentials and compute exact formulae for unitary Cayley graph 𝑋𝑛. Findings: Different values of status Sombor indices have been obtained for unitary Cayley graphs of various orders. Novelty: The values of status Sombor indices have been extensively studied by various researchers recently and most of the research has been done for different types of graphs in chemical graph theory. The present work reveals that the unitary Cayley graph is one of the graphs that is extensively studied in both the areas of group theory and graph theory. We derive the values of status Sombor indices that are dependent on the status and degree of the vertices of the unitary Cayley graph. Mathematics Subject Classification: 05C09, 05C25, 05C50. Keywords: Unitary Cayley graphs; Status Sombor index; Modified status Sombor index; Status Sombor exponential; Modified status Sombor exponential

Sámi Body-Based Measuring in the Mathematics Classroom—The Development of a Model for Teaching

According to Norway’s Core Curriculum all students are to learn about Sámi issues, and the subject of mathematics is not exempt from it. Central to the Sámi tradition is non-standardized individual body-based measuring. Two Sámi pre-service teachers taught Sámi traditional measuring in an urban school where most students are non-Sámi. Their idea was for students to experience the usefulness of using their own bodies as measuring tools. Afterwards, a teacher educator interviewed the pre-service teachers about their experiences. The teaching unit was planned as if it were to be taught in a Sámi school, except that the language of instruction was Norwegian. Analysis shows how the teaching unit aligns with the suggestions and recommendations of Norway’s Truth and Reconciliation Commission. Analysis with respect to the Theory of Didactical Situations framework, TDS, which has not been developed for Indigenous education, show that TDS contributes to improving the teaching unit’s plan. We recommend that Sámi traditional measuring should be taught by Sámi teachers to more non-Sámi students, and we present a revised teaching plan. However, because there is a significant lack of Sámi teachers, we recommend that non-Sámi teachers use our plan as a model for teaching Sámi traditional measuring.