Mathematical Basis Research Articles

A Survey on Ring Theory for its Mathematical Foundation and Applications

The origin of the ring theory can be dated to the early 19th century. With the rapid development of technology, more and more complex engineering problems and computer problems need to obtain support and results from more basic mathematics theory. Hence, based on this demand, the researcher discovered that ring theory is a very noteworthy math theory for solving the complicated problems above. This paper investigated the ring theory and related articles based on number theory and corresponding applications, which produces and results in a literature review in this particular direction.

Cauchy’s Residue Theorem and Its Application

One of the key theorems in complex analysis, Cauchy’s Residue Theorem, which refers to the integral value of an analytic function along any simple closed contour surrounding an isolated singularity in a certain ring domain divided by 2πi, will greatly simplify the process of computing integrals on contour surrounding singularities. In the field of basic mathematics, the Cauchy’ Residue Theorem plays a key role in the integral calculation of analytic functions, nonuniform complex functions and the argument principle, which establishes a connection between a curve’s winding number and the quantity of zeros and poles inside the curve. Furthermore, Residue Theorem can combine with various subjects, including electromagnetism. This paper gives an overview of the Cauchy’s Residue Theorem and its application. Firstly, it talks about the definition and proof of the Cauchy’ Residue Theorem, along with the definition of residue. Then two examples of using Residue Theorem to integrate functions will be given. Finally, this paper involves application of Residue Theorem in trigonometric sum identities and a remark on the method.

Derivation of Power Density Equation of Hydro Turbine: Mathematical Basis of Osmosis Energy to Rotate a Reaction Turbine

Derivation of Power Density Equation of Hydro Turbine: Mathematical Basis of Osmosis Energy to Rotate a Reaction Turbine

The importance of Method 300 to improve the mathematical foundation of first-year STEM Brazilian undergraduates

This research explores strategies to support first-year STEM Brazilian undergraduates in two programs, computer science and mathematics, through partnerships among faculty and student members and the implementation of an active methodology known as “Method 300” within an elementary mathematics course. The primary goal of these initiatives was to enhance the academic performance of freshmen. In Method 300, for students to benefit from grade improvements, they must assist each other, regularly attend team meetings, and commit to the activities that constitute the method. The method was implemented only during the first half of the course, fortunately it also implicated into the second part of the course, showing that a tenth of the students who performed poorly at the beginning achieved satisfactory and excellent performance in the last assessment, even though the content was more complex. According to the assessment system of the whole course, by monitoring students who initially had poor academic performance, it was found that half of them achieved satisfactory performance by the end of the course, which contributed to a final approval rate above 73% of all students. The condition of student heterogeneity is crucial for the effective application of Method 300; without it, this could become a potential limitation to its successful implementation. In addition to this strategy, the course was supported by voluntary extracurricular tutors, although these were underutilized by the freshmen, prompting us to consider new ways of providing support. These results suggest that future research in this field should focus on improving students’ academic performance by systematically replicating Method 300 in other courses for freshmen students.

Basic aspects of IOL calculation

IOL calculation can now be considered a solved problem. However, the problem of sharing the knowledge of the underlying physical and mathematical basics is still unsolved. Also, the necessary hard- and software tools for an IOL calculation approximating the accuracy of prescribed glasses are not yet generally available. The aim of this survey is a clarification of the basics and of some historical pitfalls that still lead to persisting misunderstandings.

Teaching Reform of Public Mathematics Basic Courses--Taking Electronic Information Engineering Major as an Example

Teaching Reform of Public Mathematics Basic Courses--Taking Electronic Information Engineering Major as an Example

Examining TPACK Enactment in Elementary Mathematics with Various Learning Technologies

As schools continue to invest more resources into learning technologies and their infrastructure, there is a need to consider ways to support teachers’ professional learning related to the appropriate uses of learning technologies. The construct of TPACK continues to provide a framework for the field to help make sense of the knowledge and skills related to effectively integrating learning technologies into teaching. The article includes three vignettes that describe research-aligned enactments of TPACK using three different learning technologies: an internet-based activity focused on lower-level fluency, an internet-based simulation/game on budgeting, and an internet-based open-ended tool. The three vignettes suggest that TPACK can be enacted regardless of the learning technologies and that the culture and context of schools greatly influences whether and how teachers integrate technology into their mathematics teaching.

Detailed Late Pliocene-Quaternary sedimentation history and diagenesis processes in the eastern part of the Norwegian-Greenland Basin based on mathematical statistical methods

As a result of univariate regression analysis, quantitative parameters on the distribution of Mn/Al and Mn/Fe ratios in the sections of the studied holes of ODP Leg 151 were obtained. On the basis of correlation analysis data with Pearson matrix construction and classical factor analysis with varimax rotation, a division of rather homogeneous Upper Pliocene-Quaternary sediments was carried out with the allocation of 3 main factors that characterize the history of sedimentation and diagenesis in the Norwegian-Greenland Basin. The boundaries between lithostratigraphic subdivisions are drawn on a strict mathematical basis. It is established that the nature and degree of intensity of diagenesis processes in this region were influenced by the following main factors: sedimentation environments in the Norwegian-Greenland Basin in the Late Pliocene-Quaternary time; grain-size and chemical composition of the studied sediments; total organic carbon content in the sediments; sedimentation rates.

The Numerically Leveled Performance Assessment Framework (NLPAF): A Literature-Based Evaluation of Acceptability

ABSTRACT The purpose of this study is to present a literature-based, initial “evaluation of acceptability” (American Educational Research Association et al., 2014, p. 7) of the Numerically Leveled Performance Assessment Framework (NLPAF). Its results enable confidence in continuing its use in the authors’ professional development work and in making it available to professional developers and researchers. The authors developed the NLPAF to guide elementary mathematics teachers enrolled in a professional development program in constructing performance assessments they then administer up to three times annually to their students. These assessments measure growth across time in mathematical proficiency relating to the domain of whole number operations most prominent in the curriculum standards for the grade level taught. Numerically leveled refers to the process teachers learn and apply to estimate the size and numerical complexity of computations to be used in designing performance tasks for their students’ unique levels.

Perception of the elementary mathematics teachers about assessment for learning: a case study of Sukkur IBA community colleges, Sindh, Pakistan

This study aims to explore the perceptions of elementary mathematics teachers regarding assessment for learning (AfL). The participants of this study included three elementary mathematics teachers, each with over 5 years of teaching experience at Sukkur IBA community colleges and schools in Sindh, Pakistan. These participants were selected using purposive sampling techniques. A qualitative exploratory case study research approach was used, with data collected through semi-structured interviews and observations. To ensure the trustworthiness of the study, the interview guide and observation checklist were reviewed by experts. The data were analyzed using thematic analysis. The findings of this study revealed that teachers had a limited understanding of the concept of assessment for learning. Based on these results, the study recommends several measures: the initiation of an assessment literacy program, an increase in class time, reducing the number of students per class, using moveable chairs to enhance classroom flexibility, and making amendments to the assessment policy.

Revisiting the fitting of the Nelson–Siegel and Svensson models

The Nelson–Siegel and the Svensson models are two widely used models for the term structure of interest rates. These models are quite simple and intuitive, but fitting them to market data is numerically challenging and various difficulties have been reported. In this paper, we provide a novel mathematical analysis of the fitting problem based on parametric optimization. We formulate the fitting problem as a separable nonlinear least-squares problem, in which the linear parameters can be eliminated. We provide a thorough discussion on the conditioning of the inner part of the reformulated problem and show that many of the reported difficulties encountered when solving it are inherent to the problem formulation itself and cannot be tackled by choosing a particular optimization algorithm. Our stability analysis provides novel insights that we use to show that some of the ill-conditioning can be avoided, and that a suitably chosen penalty approach can be used to address the remaining ill-conditioning. Numerical results indicate that this approach has the expected impact while being independent of any choice of a particular optimization algorithm. We further establish smoothness properties of the reduced objective function, putting global optimization methods for the reduced problem on a sound mathematical basis.

“Our Voyage with Knowing and Teaching Elementary Mathematics”

From the Archives highlights articles from NCTM’s legacy journals, previously discussed during MTLT Teacher Talk.

DEFINING THE ROLE OF EDUCATIONAL PLATFORMS IN MATHEMATICS IN THE EFFECTIVE TRAINING OF FUTURE MATHEMATICS TEACHERS

The present study analyzes the results of a survey of teachers and students designed to identify the preferences and needs of universities in using educational platforms in mathematics to study the subject «Elementary Mathematics». The aim of the study was to determine the preferred characteristics and functionality of mathematics learning platforms that can contribute to effectivelearning and increase interest in mathematics for future mathematics teachers. To achieve this goal, questionnaires were developed, which were distributed among mathematics teachers and first-year students in the specialty «Mathematics» when teaching the subject «Elementary Mathematics» and third and fourth-year students when teaching the subject «Methods of teaching mathematics». The questionnaires included questions about preferences in using educational platforms in mathematics, the level of comfort andperception of educational material when working with technologies, preferred learning formats and expected learning outcomes. The analysis of the obtained results allowed us to identify the key principles that should be taken into account when using educational platforms in mathematics to study the subject «Elementary Mathematics» at the university. Following these principles contributes to the use of educational platforms in mathematics, and as a result, to improving the quality of mathematical education at the university and improving the learning ability of future mathematics teachers.

Secret sharing in a special linear group

Objectives. The problem of developing the mathematical foundations of modular secret sharing in a special linear group over the ring of integers is being solved.The relevance of the problem is reduced to the fact that a large number of requirements are imposed on secret sharing schemes. These include the ideality of the scheme, the possibility of verification, changing the threshold without the participation of the dealer, the implementation of a non-threshold access structure and some others. Every secret sharing scheme developed to date does not fully satisfy all these requirements. It only has a certain configuration of these properties. The development of a scheme on a new mathematical basis is intended to expand the list of these configurations, which creates more opportunities for the user in choosing the optimal option.Methods. Group theory, modular arithmetic and theory of secret sharing schemes are used.Results. A fundamental domain with respect to the action of the main congruence subgroup by right shifts in the special linear group of second-order matrices over the ring of integers is constructed. On this basis, methods for modular secret sharing and its threshold restoration are proposed.Conclusion. A rigorous mathematical justification is given for the correctness of the algorithms for generating partial secrets and restoring the main secret in the special linear group over the ring of integers. These results will be used to study the configuration of secret sharing properties in this group.

Implementing Equitable Practice-Based Teacher Education to Advance Rural STEM Teacher Candidates’ Pedagogical and Content Knowledge

This longitudinal three-year study explored implementation of practice-based teacher preparation incorporating high-leverage teaching practices to advance PCK. Participants included teacher candidates (n = 45) who completed elementary science and mathematics methods courses at a rural Midwestern university. Students engaged in teaching rehearsals to explore STEM concepts and equitable teaching practices. Data from surveys and student reflections revealed an increase in teacher candidates' perceptions of their knowledge and competence of teaching science and math content equitably.

Implementing Equitable Practice-Based Teacher Education to Advance Rural STEM Teacher Candidates’ Pedagogical and Content Knowledge

This longitudinal three-year study explored implementation of practice-based teacher preparation incorporating high-leverage teaching practices to advance PCK. Participants included teacher candidates (n = 45) who completed elementary science and mathematics methods courses at a rural Midwestern university. Students engaged in teaching rehearsals to explore STEM concepts and equitable teaching practices. Data from surveys and student reflections revealed an increase in teacher candidates' perceptions of their knowledge and competence of teaching science and math content equitably.

OPTIMIZING MATHEMATICS LEARNING IN FIFTH GRADES: THE CRITICAL ROLE OF EVALUATION IN IMPROVING STUDENT ACHIEVEMENT AND CHARACTER

The aim of this research is to describe the role of learning evaluation in fifth grade elementary school mathematics subjects. This research was conducted on Fifth-Grades students of UPT SD Negeri 060809 Medan Kota. The research method used in this research is descriptive qualitative. The subjects in this research are Fifth-Grades students and Fifth-Grades teachers and the objects in this research are the results of interviews and observations. Data collection techniques in this research are interviews with students and teachers, conducting classroom observations, analyzing supporting documents and literature studies. The data analysis technique used in this research is content analysis. The research results show that evaluation of mathematics learning in Fifth Grades of SD Negeri 060809 Medan Kota has an important role in measuring learning achievement, providing feedback, identifying weaknesses and strengths, and directing learning improvements. The evaluation instruments used include oral tests, assignments and portfolios. Apart from that, evaluation also aims to measure academic achievement and character, foster multicultural awareness, and encourage the application of values in real contexts. Furthermore, evaluation also plays a role in developing student learning competencies by measuring student progress, identifying their learning needs, and planning appropriate interventions. It also involves providing constructive feedback, developing metacognitive skills, and encouraging reflection and continuous improvement for students.

Мәтіндік және стандартты емес есептерді шешу әдістері

One of the most important tasks of primary education is the development of logical thinking. Textual and non-standard tasks have a special place in the formation of students' thinking abilities. By mastering the solution of such problems, we form in the student such qualities as search for innovation, self-attitude, perseverance. The article discusses textual and non-standard tasks and solutions found in the course of elementary mathematics, the technology of criterion-based assessment. The concepts of "text task", "non-standard task" were defined. This article presents a methodology for criterion-based assessment of students' academic achievements in mathematics lessons. The peculiarity of this technology is that when solving a task, students create descriptors, which are subsequently used in formative and aggregate assessments. Each descriptor is a step in the algorithm that solves the problem. Particular attention is paid to the use of various methods by primary school teachers in solving text problems, the importance of observing and supporting the child's non-standard ideas. The habit of finding other ways to solve the problem, the ability to use various ways and means of solving the problem plays a huge role in the future work, scientific and creative activity of the student. As a result of the research work, the main recommendations for finding a solution to text and non-standard tasks were developed. The advantages of using the criterion-based assessment system in mathematics lessons are revealed.

Dynamic Analysis and Approximate Solution of Transient Stability Targeting Fault Process in Power Systems

Modern power systems are high-dimensional, strongly coupled nonlinear systems with complex and diverse dynamic characteristics. The polynomial model of the power system is a key focus in stability research. Therefore, this paper presents a study on the approximate transient stability solution targeting the fault process in power systems. Firstly, based on the inherent sinusoidal coupling characteristics of the power system swing equations, a generalized polynomial matrix description in perturbation form is presented using the Taylor expansion formula. Secondly, considering the staged characteristics of the stability process in power systems, the approximate solutions of the polynomial model during and after the fault are provided using coordinate transformation and regular perturbation techniques. Then, the structural characteristics of the approximate solutions are analyzed, revealing the mathematical basis of the stable motion patterns of the power grid, and a monotonicity rule of the system’s power angle oscillation amplitude is discovered. Finally, the effectiveness of the proposed methods and analyses is further validated by numerical examples of the IEEE 3-machine 9-bus system and IEEE 10-machine 39-bus system.

Signature Genes Selection and Functional Analysis of Astrocytoma Phenotypes: A Comparative Study.

Novel cancer biomarkers discoveries are driven by the application of omics technologies. The vast quantity of highly dimensional data necessitates the implementation of feature selection. The mathematical basis of different selection methods varies considerably, which may influence subsequent inferences. In the study, feature selection and classification methods were employed to identify six signature gene sets of grade 2 and 3 astrocytoma samples from the Rembrandt repository. Subsequently, the impact of these variables on classification and further discovery of biological patterns was analysed. Principal component analysis (PCA), uniform manifold approximation and projection (UMAP), and hierarchical clustering revealed that the data set (10,096 genes) exhibited a high degree of noise, feature redundancy, and lack of distinct patterns. The application of feature selection methods resulted in a reduction in the number of genes to between 28 and 128. Notably, no single gene was selected by all of the methods tested. Selection led to an increase in classification accuracy and noise reduction. Significant differences in the Gene Ontology terms were discovered, with only 13 terms overlapping. One selection method did not result in any enriched terms. KEGG pathway analysis revealed only one pathway in common (cell cycle), while the two methods did not yield any enriched pathways. The results demonstrated a significant difference in outcomes when classification-type algorithms were utilised in comparison to mixed types (selection and classification). This may result in the inadvertent omission of biological phenomena, while simultaneously achieving enhanced classification outcomes.