Mathematical Problem Research Articles

New Generalized Derivatives for Solving Variational Inequalities Using the Nonsmooth Newton Methods

AbstractVariational inequality (VI) generalizes many mathematical programming problems and has a wide variety of applications. One class of VI solution methods is to reformulate a VI into a normal map nonsmooth equation system, which is then solved using nonsmooth equation-solving techniques. In this article, we propose a first practical approach for furnishing B-subdifferential elements of the normal map, which in turn enables solving the normal map equation system using variants of the B-subdifferential-based nonsmooth Newton method. It is shown that our new method requires less stringent conditions to achieve local convergence than some other established methods, and thus guarantees convergence in certain cases where other methods may fail. We compute a B-subdifferential element using the LD-derivative, which is a recently established generalized derivative concept. In our new approach, an LD-derivative is computed by solving a sequence of strictly convex quadratic programs, which can be terminated early under certain conditions. Numerical examples are provided to illustrate the convergence properties of our new method, based on a proof-of-concept implementation in Python.

A New Approximation for the Perimeter of an Ellipse

We consider the problem of approximating the perimeter of an ellipse, for which there is no known finite formula, in the context of high-precision performance. Ellipses are broadly used in many fields, like astronomy, manufacturing, medical imaging, and geophysics. They are applied on large and nanoscales, and while numerical integration can be used to obtain precision measurements, having a finite formula can be used for modeling. We propose an iterative symbolic regression approach, utilizing the pioneering work of Ramanujan’s second approximation introduced in 1914 and a known Padé approximation, leading to good results for both low and high eccentricities. Our proposed model is also compared with a very comprehensive historical collection of different approximations collated by Stanislav Sýkora. Compared with the best-known approximations in this centuries-old mathematical problem, our proposed model performs at both extremities while remaining consistent in mid-range eccentricities, whereas existing models excel only at one extremity.

Progress and Applications of Fully Homomorphic Encryption

In the era of digital transformation, marked by the widespread adoption of cloud-based technologies such as cloud computing and storage, ensuring data security and protecting user privacy have emerged as critical industry concerns. Fully Homomorphic Encryption (FHE) addresses these issues by allowing various computations to be performed on ciphertexts without the need for decryption, thus safeguarding data security and privacy. This paper explores the mathematical and algorithmic foundations essential for understanding FHE. It provides a detailed analysis of the advancements in FHE schemes over the past decade, focusing on various mathematical problems, including ideal lattices and integers. The discussion extends to three practical applications: cloud computing, machine learning, and electronic voting, highlighting the progress and exploring potential future research avenues and developmental strategies in the field of FHE. This comprehensive review not only underscores the significance of FHE in enhancing data security but also charts a path for its future exploration and integration into emerging technologies.

Проблеми вивчення математики в межах реалізації концепції Нової Української Школи

In this publication, we pay attention to the problems of learning mathematics in the context of the implementation of the concept of the New Ukrainian School. During preparation for publication, the following work was done. Conversations with mathematics teachers and their questionnaires were conducted. A survey of students was conducted. Sources of information in terms of informing teachers about the implementation of the concept of the New Ukrainian School and methods of teaching mathematics at the National University of Ukraine were analyzed. A comparative analysis of educational achievements of seventh-grade students who studied in experimental and control classes was made. The tasks of the final exams of high school students abroad and our graduates were analyzed.

Реализация познавательного и воспитательного потенциала рассказов М.М. Зощенко на уроках литературного чтения в начальных классах

The authors consider the possibilities of using M.M.Zoshchenko's stories in the process of forming the life experience of primary school students, reveal the essence of an integrated approach to the organization of ideological and thematic analysis of the text, methodological features of composing a mathematical problem based on the storyline of a work of art.

Evolution of the Four Fours Problem

Summary Besides being popular and entertaining, recreational mathematics problems are often of historical interest. We highlight the origin and evolution of one such simply-stated yet deep problem. For nearly three centuries the four fours problem has been generalized and expanded, providing hours of mathematical activity for enthusiasts as well as inspiring some intriguing discoveries.

<span><strong>Enhancing students’ mathematical competence in learning spatial geometry in grade 11<sup>th</sup> through practical and experiential activities</strong></span>

This study embarks on enriching 11th-grade students' mathematical competencies, particularly in spatial geometry, through experiential learning. Specifically, the study focused on the practical concepts of perpendicular and parallel relationships in space. Following the qualitative approach, we collected and analyzed the data of the students' behaviors observed within a teaching scenario, which was specifically designed for them to apply their knowledge to the task "determining the classroom's center". The scenario involved three phases: defining the "classroom center," problem-solving in the classroom, and group presentations analyzing each other's methods. The results showed that all student groups successfully solved the task, which was implemented in a serious way using their mathematical understanding in a real-life context. Thus, students enhanced their two mathematical competencies, including mathematical problem handling competency (posing and solving mathematical problems) and mathematical communication competency (communicating in, with and about mathematics). This study underscores the effectiveness of practical and experiential activities in enhancing mathematical competencies among high school students, specifically within spatial geometry. It highlights the benefits of applying mathematical concepts to practical situations, promoting more profound understanding and student engagement.

<span>Enhancing student's mathematical competency in learning the graph and properties of the sine function by Google Sheets app on smartphone</span>

This study explores the integration of the Google Sheets app on smartphones to enhance students' mathematical competency in understanding the graph and properties of the sine function. The paper was an exploratory study involving 32 11th-grade students. The teaching intervention was structured into four phases: familiarizing with the Google Sheets app, creating the sine function value table, sketching the function graph, and analyzing the graph to deduce the sine function's properties. Data were collected through worksheets and audio -video recordings of group work, focusing on manifestations of mathematical competencies. The findings indicate that students completed the tasks and demonstrated significant competencies in mathematical reasoning and using digital tools. Students were able to identify and solve mathematical problems, utilize Google Sheets effectively, and use logical reasoning to enhance their understanding of the sine function. The study concludes that integrating Google Sheets into mathematics education not only aids in visualizing and comprehending mathematical concepts but also promotes interactive and collaborative learning environments, ultimately fostering a deeper understanding and proficiency in mathematics.

PENINGKATAN KEMAMPUAN PEMECAHAN MASALAH MATEMATIS SISWA KELAS III DENGAN MEDIA POWER POINT INTERAKTIF MACAN (MATERI PECAHAN)

This research aims to improve the low level of mathematical problem solving regarding fractions using Interactive Power Point media for class III students at SD 2 Panjunan. Data collection techniques in this research used observation, interviews and documentation techniques, tests. The instruments of this research are interview sheets, observation sheets, and test question sheets. The method in this research uses the classroom action research method. The subjects in this research were class III students at SD 2 Panjunan with a total of 26 students. This research uses the Kemmis and Mc Taggart model research design, where each cycle or round consists of several stages which include, planning, action, observation, and reflection. The research results in Cycle I showed an average of 42% where the level of learning success was still low, and in cycle II after using interactive power point media it showed an average of 88% with a very good category, this shows that the application of interactive Power point in the learning process with fraction material can improve mathematics learning outcomes for class III students at SD 2 Panjunan. The conclusion of this research is that Interactive Power Point media can improve students’ mathematical problem solving abilities in fraction material.

Impact of induced electric field on the motion of a charged particle in a uniform time-dependent magnetic field that is linearly increasing with time

The classical and quantum motion of a charged particle in a uniform constant magnetic field is widely studied in the scientific literature since it is a problem that allows an exact mathematical analytical solution. On the other hand the cases of a non-uniform, time-dependent or both non-uniform and time-dependent magnetic field lead to mathematical problems that do not have general exact analytical solutions except for a few very specific cases. We present in this work, the results for the two-dimensional motion of a charged particle in a uniform time-dependent magnetic field that is linearly increasing with time for a certain finite interval and then stabilizes to a constant value afterwards. The work draws attention to certain interesting features observed for the two-dimensional motion of the charged particle as a result of the interplay between the effects of an induced electric field and the time varying magnetic field.

Variational Analysis of a Dynamic Contact Problem with Wear and Damage Involving Viscoelastic Materials with Long-Term Memory

We investigate a mathematical problem involving dynamic interaction between a viscoelastic body with long-term memory loss and an obstruction. The contact is frictional and bilateral, with a moving rigid base, resulting in wear of the contacting surface. The problem is expressed as a coupled system, with a hyperbolic quasi-variational inequality for displacement and a parabolic variational inequality for damage. We define a variational formulation for the model and demonstrate the existence of a single weak solution to the problem. The material behaviour is explained using a viscoelastic constitutive law that includes long-term memory and damage. Elastic deformations induce material deterioration, which is depicted by a parabolic inclusion. The proof is based on classical existence.

Analysis of the multi-term fractional Bateman equations in radioactive decay by means of Mikusiński algebraic calculus

Fractional Bateman equations model the concentrations of nuclear species through a radioactive decay chain, by considering non-Markovian behavior. Cruz-López and Espinosa-Paredes (Nuclear Engineering and Technology, 2022) solved the mathematical problem with a single memory coefficient, based on the Laplace-transform technique. Cruz-López, Espinosa-Paredes and François (Computer Physics Communications, 2022) extended the analysis to several fractional orders, with Laplace transforms and series expansions checked by induction. In this paper, we solve such a case of distinct memory effects, via the more general method of Mikusiński operational calculus. We do not distinguish between equal and different decay rates. Multivariate Mittag-Leffler functions appear in the closed-form expression.

Black holes, complex curves, and graph theory: Revising a conjecture by Kasner

The ratios 8/9=22/3≈0.9428 and 3/2≈0.866 appear in various contexts of black hole physics, as values of the charge-to-mass ratio Q/M or the rotation parameter a/M for Reissner-Nordström and Kerr black holes, respectively. In this work, in the Reissner-Nordström case, I relate these ratios with the quantization of the horizon area, or equivalently of the entropy. Furthermore, these ratios are related to a century-old work of Kasner, in which he conjectured that certain sequences arising from complex analysis may have a quantum interpretation. These numbers also appear in the case of Kerr black holes, but the explanation is not as straightforward. The Kasner ratio may also be relevant for understanding the random matrix and random graph approaches to black hole physics, such as fast scrambling of quantum information, via a bound related to Ramanujan graph. Intriguingly, some other pure mathematical problems in complex analysis, notably complex interpolation in the unit disk, appear to share some mathematical expressions with the black hole problem and thus also involve the Kasner ratio.

The Development of Discovery Learning Model Teaching Materials to Improve Students’ Mathematical Problem Solving Ability on Junior High School

In order to enhance students' mathematical problem-solving skills, this study looks at how teaching materials are developed using the Discovery Learning model. Since every student nowadays needs to be able to think critically, the study specifically looks at the process and outcomes. development, viability testing, feedback from students, and improving mathematical problem-solving skills. This kind of research makes use of the Research and Development (R&D) or Research-based Development research method, which is a way to create a specific product and use a model to test the product's viability. Students in class VII at MTs Al Bidayah West Bandung Regency served as the research sample for S. Thiagarajan's 4D (Define, Design, Develop, and Disseminate) study. The process and outcomes of creating educational materials lead to the conclusion that the materials, when viewed from the various stages, follow the procedures or methods outlined in 4D. Based on students' responses, which were 68.4% and 70.2% in the good category, the viability of teaching materials made possible by the Photoshop program can be concluded, along with the students' improved problem-solving skills. If this teaching material is used in a classroom setting, it is imperative that the necessary facilities be set up ahead of time because learning with it requires sufficient facilities. There are tasks and content in this teaching resource that need to be altered. so that students' learning experiences are not dull and monotonous.

Best Practice STEM-Based Differentiated Mathematics Learning for Improving Learning Outcomes and Self-efficacy of High School Students

Learning outcomes and self-efficacy are important learning factors. Mathematics learning outcomes can reflect the extent to which students have developed as learners, while self-efficacy reflects students' self-confidence in their own abilities in solving mathematical problems. This study was based on a qualitative approach, best practice. This best practice uses the STAR flow (“Situasi, Tantangan, Aksi, Refleksi”). The situation at SMAN 1 Mande, especially class XI, shows low learning outcomes and lack of self-efficacy. The challenge faced is planning and implementing differentiated learning to improve learning outcomes and student self-efficacy. Through a STEM-based differentiated learning approach, researchers designed and implemented contextual learning on the theme of global warming with integrating science, technology, engineering and mathematics. The results of this best practice indicate an increase in learning outcomes and student self-efficacy. STEM-based differentiated learning can also improve understanding of mathematical concepts at the high school level. Researcher recommend that educators embrace differentiated learning strategies by customizing their approaches to accommodate the diverse needs such as students' level of readiness, interests, and student’s learning profile. The significance of comprehending student characteristics, such as learning styles and thinking abilities is paramount, especially in the implementation of student-centered learning. Additionally, educators are advised to prioritize aligning learning objectives with the curriculum, ensuring relevance and engagement. This various approach aims to create a dynamic learning environment to resonates with students individualized level. Develop interventions to address students' lack of confidence in solving mathematical problems can improve self efficacy.

Analisis Kemampuan Mahasiswa Matematika FMIPA Unimed dalam Menyelesaikan Pertidaksamaan Nilai Mutlak dengan Berbantuan Python

This research aims to analyze the ability of Unimed FMIPA Mathematics students in solving absolute value inequalities with the help of the Python program. The problem faced is that students often have difficulty solving inequality problems manually. The purpose of this research is to see the effectiveness of using Python in helping to solve absolute value inequalities and how it affects understanding of mathematical concepts. The method used was qualitative research with a sample of 20 fifth semester students of the Mathematics Study Program, FMIPA, Medan State University. Students are given absolute value inequality problems to solve manually and with the help of Python. Data was collected through online tests and questionnaires using Google Form. The research results show that the majority of students feel that using Python is very helpful in solving absolute value inequalities. As many as 95% of students consider Python to be effective in making it easier to solve mathematical problems and increasing understanding of the concept of absolute value.

Analyzing Junior High School Students’ Mathematical Communication Ability and Self Esteem using Discovery Learning Model

Mathematical reasoning, mathematical communication, mathematical problem solving, conceptual understanding, mathematical understanding, creative thinking and critical thinking are part of the mathematical abilities that students must have. One of the goals of learning mathematics according to NCTM is learning to communicate (mathematical communication). Mathematical communication skills in mathematics learning really need to be developed. This type of research design uses a quasi-experimental research design, namely pretest and post-test with non-equivalent control group design . The research used research subjects into two groups, namely the experimental group and the control group. The subjects of this research were students of class VII I MTs As-Sa'adah Padahanten Majalengka, numbering 64 people. Subjects were selected based on experimental class VIII A, 32 people and control class VIIIB, 32 people. Walkthroughs, questionnaires, and tests are techniques used in data collection. Data processing analysis was carried out qualitatively and quantitatively. The results of this research are that there is a difference in the effectiveness of the Discovery Learning model learning approach with the ordinary (conventional) learning model in terms of the aspects of mathematical communication skills and self-esteem of MTs in statistics material. The implications of the Discovery Learning learning model in this research are that it makes it easy for students to remember the lessons learned, integrate self concept according to their initial abilities, focus, be creative in exploratory projects, and the effect of knowledge transfer on learning outcomes can last a long time.

Алгоритм для численного решения диффузионно-реакционно-дрейфового уравнения с дробной производной по времени и координате

<p>The paper is devoted to the construction and program implementation of the computational algorithm for modeling a process of diffusion-drift nature based on the fractional diffusion approach. The mathematical model is formulated as an initial-boundary value problem for the time-space fractional diffusion-drift equation in a limited domain. Time and space fractional derivatives are considered in the sense of Caputo and Riemann – Liouville, respectively. A modified implicit finite-difference scheme is constructed. The concept of the considered mathematical problem provides an example of a deterministic model of the charging process of dielectric materials. An application program has been developed that implements the constructed numerical algorithm. The results were verified using the example of solving a test problem.</p>

REVISÃO SISTEMÁTICA DE LITERATURA

This text aims to address the topic of the Systematic Literature Review (RSL), and its application to the teaching and learning of mathematics in the initial grades of elementary school. The aim is to establish a line of guidance that makes it possible to particularize and direct the child’s knowledge, in such a way that facilitates their learning, highlighting research on themes that deal with the interpretation of texts, both in reading and writing statements and mathematical problems, within the native language.

A Preliminary Study of Context Integration as A Determinant in The Effectiveness of Digital Mathematics Games

Contexts in the digital learning environment are one of the crucial components regarding students’ motivation and positive attitude development toward content. Although there is not enough guiding information in the literature about how contexts should be, previous research has shown that even digital game contexts are effective in developing students’ environmental awareness. To reveal the impact of contexts in the digital learning environment, it is necessary to consider the variables regarding the socio-economic and cultural status of the students. Therefore, it can be envisaged that the effect of digital game contexts may differ depending on students’ experiences and social environments. For this purpose, digital games were designed in two different contexts in this research. In these digital games, two opposite contexts were selected. The first context was focused on saving nature and green energy. This context includes different levels of digital games on topics such as solar energy, reforestation, and clean water sources. The second context was focused on unregulated industrial development that negatively affects nature by ignoring it. This context includes different levels of digital games such as building thermal power plants, building chemical factories, and turning forests into fields. Apart from the contexts, the other variables in both games (i.e., gameplay, artistic structure, levels of problem parameters) are the same. At each game level, students were asked to solve mathematical problems regarding ratio-proportion. In this study, the preliminary findings, which were derived from a large experimental study, were presented. The participants of this study were 15 gifted eighth-grade students from Turkey. Among the participants, eight students played the digital game regarding the context of green energy, and the remaining seven students played the digital game regarding the context of unregulated industrial development. Preliminary findings showed that students were more interested in the digital game regarding the context of unregulated industrial development as it included interesting and beneficial topics such as energy generation and economic development. The resident perception of the citizens regarding the acute need for energy generation and economic development rather than saving nature and green energy since Turkey represents a developing country may explain this finding.