Mathematical Issues Research Articles

Numerical Simulation of Nonlinear Equations by using Modified Newton Raphson Technique

Non-linear equations are a significant number of complicated issues in mathematics and related subjects that must be solved. The Newton method and its preliminary variations are among the most basic, yet insufficiently effective, methods for resolving non-linear equations. To solve non-linear equations efficiently, a method's desirable feature is to find root with fewer iterations, minimum error (usually less than precision limit), which is typically less than the precision limit. In, this work, we present two approaches (Proposed techniques I and II) for solving non-linear equations by modifying Newton Raphson's technique using first derivative's forward and central difference approximations. Two examples are used to compare performance of the suggested techniques with the existing technique (Secant technique). Finding roots of nonlinear equations under consideration required fewer iterations and a lower absolute relative approximate error while using proposed method II, which performed better than both existing methods (the Secant technique and suggested method I). The suggested approaches I and II were determined to be appropriate substitutes for solving nonlinear equations.

Contributions of Ottoman Fatwā Collections to Inheritance Law: An Analysis of Legal Opinions by the 16th-Century Chief Muftīs

Abstract This article investigates the contributions of Ottoman fatwā collections to inheritance law, focusing on the 16th-century chief muftīs’ fatāwā. It emphasizes the significance of fatawa in understanding Ottoman inheritance law and explores various aspects of the division of estate, highlighting the socio-legal dynamics and the use of fatawa by individuals seeking legal advice. Within this framework, it examines why individuals preferred obtaining fatawa from chief muftīs for the division of the estate over other options. The methodology of this study involved a comprehensive analysis of fatwā collections from 16th-century Ottoman chief muftīs, focusing on categorizing and interpreting these fatawa to understand their impact on inheritance law. The study reveals that economic considerations, complex family structures, mathematical issues in estate division, and specific legal situations significantly influenced individuals’ preference for seeking fatāwā. Through analyzing fatwā collections, the study uncovers economic motivations behind this preference, revealing how fatāwā offered a cost-effective solution for heirs, alongside addressing specific legal challenges like ‘awl and munāsakha.

Leveraging ChatGPT for Scoring Students' Subjective Tests

Assessing students' understanding of circle-related material through subjective tests is effective, though grading these tests can be challenging and often requires technological support. ChatGPT has shown promise in providing reliable and objective evaluations. Many teachers in Indonesia, however, continue to face difficulties integrating technology into assessments, highlighting the need for research on leveraging ChatGPT for scoring subjective tests. This study explores the potential of ChatGPT in mathematics education, specifically for evaluating students’ answers on subjective tests related to circles. Using a qualitative case study approach, the research investigates how ChatGPT analyzes student responses. The findings reveal that ChatGPT effectively processes the answers of 15 vocational school students, delivering results that are logical and well-structured. ChatGPT aids teachers by offering a comprehensive evaluation focused on accuracy, clarity, completeness, and technical terminology. This approach supports effective feedback and objective assessment, making ChatGPT a valuable tool for understanding and addressing students' mathematical comprehension issues.

Quantifying higher-order epistasis: beware the chimera.

Epistasis, or interactions in which alleles at one locus modify the fitness effects of alleles at other loci, plays a fundamental role in genetics, protein evolution, and many other areas of biology. Epistasis is typically quantified by computing the deviation from the expected fitness under an additive or multiplicative model using one of several formulae. However, these formulae are not all equivalent. Importantly, one widely used formula - which we call the chimeric formula - measures deviations from a multiplicative fitness model on an additive scale, thus mixing two measurement scales. We show that for pairwise interactions, the chimeric formula yields a different magnitude, but the same sign (synergistic vs. antagonistic) of epistasis compared to the multiplicative formula that measures both fitness and deviations on a multiplicative scale. However, for higher-order interactions, we show that the chimeric formula can have both different magnitude and sign compared to the multiplicative formula - thus confusing negative epistatic interactions with positive interactions, and vice versa. We resolve these inconsistencies by deriving fundamental connections between the different epistasis formulae and the parameters of the multivariate Bernoulli distribution . Our results demonstrate that the additive and multiplicative epistasis formulae are more mathematically sound than the chimeric formula. Moreover, we demonstrate that the mathematical issues with the chimeric epistasis formula lead to markedly different biological interpretations of real data. Analyzing multi-gene knockout data in yeast, multi-way drug interactions in E. coli , and deep mutational scanning (DMS) of several proteins, we find that 10 - 60% of higher-order interactions have a change in sign with the multiplicative or additive epistasis formula. These sign changes result in qualitatively different findings on functional divergence in the yeast genome, synergistic vs. antagonistic drug interactions, and and epistasis between protein mutations. In particular, in the yeast data, the more appropriate multiplicative formula identifies nearly 500 additional negative three-way interactions, thus extending the trigenic interaction network by 25%.

Improved chaotic Bat algorithm for optimal coordinated tuning of power system stabilizers for multimachine power system

Power systems exhibit nonlinearity. causing dynamic instability and complex power oscillations. This research proposes an innovative strategy using the Novel Bat Algorithm (NBA) to achieve ideal Power System Stabilizers (PSSs) in a multimachine power system. The approach shifts electromechanical modes to specific areas in the s-plane. Enhancing the multi-machine power system and establishing stabilizer parameters for dynamic performance. The study examines the designed approach aptitude for standard lead-lag PSSs configurations. In order to elevate the global search problem and transfer some static operators for the optimum optimization process. the chaos mapping. also known as CNBA. is introduced into NBA. Four different forms of chaos maps are compared in experiments to resolve unconstrained mathematical issues in order to illustrate CNBA performance. In any other case. the challenge of designing PSS under a wide range of loading situations is transformed into an optimization challenge with the damping ratio of electromechanical modes with low damping as the target function. The optimal stabilizers’ gains are gotten by employing the CNBA algorithm. Second plan. an effective technique is astutely established to delineate the PSS location and quantity using CNBA and another side using participation factor. To examine the efficacy of the proposed CNBA-based PSS on a large system; it is tested on the interconnected of New-England/New-York (16 generators and 68 buses) power grid. and verified by comparative study with NBA through eigenvalue analysis and nonlinear simulation to provide evidence the algorithmic competence of CNBA. The CNBA approach yields a minimum damping ratio of 37%. which is consistent with the its eigenvalue. In contrast, the NBA approach achieves a minimum damping ratio of 31%. The simulation results reveal the fine performance of the proposed CNBA-PSS in a convincing manner and its capacity to provide an excellent damping for inter-area and local oscillations under diverse operating cases compared to NBA-PSS then in the case of PSS location.

Pre-service teachers’ perceptions of mathematics education and social justice

Many instances have presented mathematics as a difficult subject for students, and some of these difficulties have been perceived as an injustice to those who suffer from it. Meanwhile, no child is to be left behind in building a just society if all were to experience the collective good of mathematics. For this reason, it became necessary to explore pre-service teachers’ perceptions of mathematics as social justice in their lived experiences. In doing this, the qualitative interpretive approach was adopted, where qualitative data was collected, transcribed, and analysed using the case study design. A sample of eleven pre-service teachers were involved in the study. The findings indicated that the pre-service teachers had experienced severe issues of mathematics as social justice in their previous schooling that were unfair to them. However, they now felt mathematics could be interesting if delivered in a manner that let their voices be heard in the delivery process.

Problem-based Curriculum to Conceptually Develop the Multiplication Principle for Counting

Counting exercises are difficult (e.g., Batanero et al., 1997; Hadar & Hadass, 1981). As a case in point, although multiplication is a relatively simple concept, students often use multiplication incorrectly on counting problems (Batanero et al., 1997). To address this issue, this paper develops a problem-based curriculum to help students learn about the multiplication principle (MP) for counting. In this paper, we outline the important mathematical issues and nuances associated with the MP in our proposed exercise set, and discuss how mathematics educators could implement this sequence of exercises to develop students’ conceptual understanding of the MP. We conclude by discussing how our exercise set aligns with a broader theory of learning – variation theory.

Analisis Kemampuan Representasi Matematis Ditinjau dari Gaya Kognitif Field Dependent dan Field Independent

This study focuses on using different cognitive styles to solve mathematical issues. Their cognitive styles can be observed when students work on mathematical problems involving mathematical representation skills. Furthermore, the analyst should assess the mathematical representation abilities of pupils at MTsN 1 Banda Aceh using the field-dependent (FD) and Field Independent (FI) cognitive styles. A descriptive, qualitative technique is used. This study employs purposive sampling procedures and focuses on two pupils from class VIII-2, one identified as FD and the other as FI. Instruments utilized include representation skill tests, interview guidelines, and GEFT questions. Analyzing the research data involves data reduction, presentation, triangulation, and conclusion formulation. The research findings point to many conclusions: (1) FI students find it easier to understand issues, can provide better conceptual explanations, solve problems faster, and explain their solutions accurately. (2) FD students have more difficulty solving issues because they rely on teacher directions. Additional concerns include their failure to analyze and comprehend information and retain concepts from challenges effectively. It is envisaged that students will understand their cognitive types to improve their mathematical representation skills in math and other courses.

Mathematical biases in the calculation of the Living Planet Index lead to overestimation of vertebrate population decline

The Living Planet Index (LPI) measures the average change in population size of vertebrate species over recent decades and has been repeatedly used to assess the changing state of nature. The LPI indicates that vertebrate populations have decreased by almost 70% over the last 50 years. This is in striking contrast with current studies based on the same population time series data that show that increasing and decreasing populations are balanced on average. Here, we examine the methodological pipeline of calculating the LPI to search for the source of this discrepancy. We find that the calculation of the LPI is biased by several mathematical issues which impose an imbalance between detected increasing and decreasing trends and overestimate population declines. Rather than indicating that vertebrate populations do not substantially change, our findings imply that we need better measures for providing a balanced picture of current biodiversity changes. We also show some modifications to improve the reliability of the LPI.

A Systematic Review of Teachers’ Perceptions and Beliefs Towards Gender Issues in Mathematics Learning

There is still very little concern especially, regarding teacher's perceptions towards gender differences and their influences in mathematics learning. Teachers are prominent factors in mathematics learning and the importance of perceptions and beliefs of teachers towards gendering, their long-term influences and their differential impacts on males and females cannot be ignored. The purpose of this study is to synthesize existing researches related to teachers' perceptions and beliefs towards gender in learning Mathematics. A qualitative systematic review of related literature of teachers’ perceptions and beliefs towards gendering in learning mathematics is conducted. The qualitative review has followed the PRISMA protocol that was outlined by Moher et al. Total of 10 articles (six quantitative, four qualitative including a systematic literature review) are chosen for inclusion for this study. Thematic analysis suggested by Braun and Clarke is used as a method to draw findings and conclusions by generating themes within the information. The thematic findings of the study are associated with the major themes that are related to teachers' perceptions and beliefs towards gender in mathematics learning. The findings are based on the major themes: participation and motivation; behaviors and performance; abilities and ICT use; teaching methods, curriculum and practices and career choices. Hence, the importance of exploring teachers' perceptions and beliefs towards gender issues in mathematics learning has been acknowledged in this study. The study has claimed that gender parity can be achieved when mathematics teachers believe that students (both boys and girls) are valuable potentials and they praise them for their effort and achievements to develop confidentiality in mathematical thinking and problem-solving abilities. Recommendations to the concerned stakeholders to enhance equity in mathematics learning are proposed in this study.

Economic epidemiological modelling: A progress report

Prior to the Covid-19 crisis, the integration of epidemiology and economics that is, economic epidemiology modelling (epi-econ), was relatively limited. The emergence of the Covid-19 crisis has prompted an unprecedented surge in this literature. This paper identifies and develops the main conceptual and modelling challenges involved in the expanding epi-econ stream, with a particular attention to the mathematical issues due, in particular, to the non-convex nature of epi-econ models. Recent extensions are also examined and a few future areas of research highlighted.

Analisis Kesalahan Siswa Kelas IX SMP Negeri 3 Tondano Dalam Menyelesaikan Soal Cerita Pokok Bahasan Kekongruenan dan Kesebangunan

The study aims to analyze the mistakes of students in solving stories related to the material of congruence and construction in the 9th grade of the 3rd Tondano State High School. This research is a qualitative descriptive research with research subjects being 32 students and selected 6 students as samples of interviews. Data is collected through observations, written tests, and interviews. Data analysis is done through data summaries, data presentation, and interpretation. There are three types of errors observed, i.e. I misunderstood the problem, II mismade the mathematical model, III misled the algebraic operation. The results of the study showed that Type II error was the most common error, with 32 students making this error on issue 2. The error percentage reached 100%, suggesting that this error occurred in all students tested. Type III error, also occurred at a significant number, namely 27 students on issue 3, with a rate of error of 84%. The combination of Type II and III errors on issue 4, which combines both types of errors, shows that the majority of students (27 students percent 84%) make type II mistakes, while the number of students who make type III mistakes (26 students percent 81%) is also quite high. It suggests that the combination of both types of errors is still a challenge in understanding and applying mathematical concepts. The conclusion is that students have difficulties mainly in creating mathematical models (type II errors), performing algebraic operations (typ III errors) and a combination of both types of errors in solving mathematics issues. By knowing what kinds of common mistakes occur, teachers can focus on areas that require more attention to the problems that occur so that the learning process can be optimized and students are able to master the mathematical concepts.

APPLICATION OF THERMAL ANEMOMETRY FOR TURBULENT FLOW ASSESSMENT

This paper is devoted to a review of the physical principle and effective methodology of thermal anemometry. Its uniqueness lies in the ability to realize a comprehensive high-frequency assessment of the dynamic and thermal state of the turbulent flow. It is worth noting that the American Nobel Prize winner in physics Richard Feynman called turbulence "the most important unsolved problem of classical physics", since there is no direct description of this phenomenon according to classical principles. Therefore, its physical interpretation is still considered one of the six most important mathematical issues of our time. Thus, despite its long history, thermal anemometry remains one of the leading techniques for studying of turbulent flow. Which has a significant impact on hydromechanical and heat-mass exchange processes, particularly in food technology. One of the most common types of thermal anemometry is the Hot-Wire anemometer, where a thin platinum wire is used as a sensor. The principle of operation of the anemometer is to maintain a constant heating of the wire while it is cooled by the surrounding flow of liquid or gas. Thus, the power required to compensate for the thermal state of the sensor wire correlates with the flow rate. The first part of the paper is devoted to the description of the physical principles of thermal anemometry, its advantages and limitations. In particular, special attention is paid to the mathematical interpretation of the heat transfer process between the incremental element of the sensor wire and the surrounding flow. After that, the paper provides a detailed analysis of the design features and practical application of various types of wire sensors. Finally, the last section discusses the calibration methodology and various approaches to the linearization procedure of the obtained calibration curves. Among them, the linearization method based on the Collis-Williams law deserves special attention, since it provides highly accurate interpolation of calibration data and takes into account the temperature compensation of the sensor.

Special Issue “Recent Advances of Discrete Optimization and Scheduling”

This Special Issue of the journal Mathematics is dedicated to new results on the topic of discrete optimization and scheduling [...]

직관주의 수학 및 고전 수학 관점에 따른 교과서의 귀류법 증명에서 무리수 √2의 존재성 문제

Intuitionistic mathematics philosophy, which distanced itself from classical mathematics, takes the view that mathematical objects are constructed by the human mind, like constructivism, which has been discussed in mathematics education. This paper aims to analyze the content of school mathematics and obtain educational implications based on a discussion of central issues in intuitionistic mathematics. Concerning the proof by contradiction, which can reveal the difference between intuitionistic mathematics and classical mathematics, among the expressions of the proposition presented as an example of the proof by contradiction in textbooks, in ‘√2 is not a rational number’, contrast to ‘√2 is an irrational number’, avoided a mention about its existence as an irrational number. With this background, we examined the issues of intuitionistic mathematical philosophy, focusing on the existence of mathematical objects, and based on this, we analyzed the textbook's related proofs by contradiction in terms of the existence of √2 as an irrational number. This analysis found that the perspective of intuitionistic mathematics permeates the expression and the proof which do not mention and avoid the existence of √2 as an irrational number. Based on the above, we discussed the reflection on the existence of mathematical objects in school mathematics and the educational value of mathematics from a humanistic aspect.

Harnessing the Power of Supercomputers: Solving Complex Mathematical Problems and Queueing Theory

Supercomputers represent the pinnacle of computational power, offering unparalleled capabilities to tackle complex mathematical problems across various domains. This paper provides a comprehensive overview of the ways supercomputers are utilized in solving intricate mathematical issues, ranging from understanding the nature of these problems to exploring the challenges and future directions in the field. Supercomputers play a critical role in the practical applications of queueing theory across various industries by enabling the analysis and optimization of complex systems that involve waiting lines and service processes. In telecommunications networks, for instance, supercomputers are used to model and optimize traffic flow, ensuring efficient data transmission and minimal latency. By simulating network conditions under different traffic loads and service protocols, supercomputers help identify optimal resource allocations and configurations, significantly improving network reliability and performance. This capability is crucial for managing the ever-increasing data demands of modern communication systems.

Optimization Techniques in Cooperative and Distributed MAC Protocols

The tremendous increase in wireless network application finds distributed allocation of resources allocation very useful in the network. Packet delivery ratio and delay can be improved by concentrating on payload size, mobility, and density of nodes in the network. In this article, a survey is carried out on different cooperative and distributed MAC protocols for communication and optimization algorithms for various applications and the mathematical issues related to game theory optimizations in MAC protocol. Spatial reuse of channel improved by (3-29) % and multi-channel improves throughput by 8% using distributed MAC protocol. The energy utility of individual players can be focused to get better network performance with NASH equilibrium. Fuzzy logic improves channel selection by 17% and secondary users' involvement by 8%. Jamming, interference problems can be addressed using cross layer approach in the MAC and simultaneous data, voice transmissions in IoT; WSN applications can be attained using hybrid distributed MAC protocol.

“Differential Equations of Mathematical Physics and Related Problems of Mechanics”—Editorial 2021–2023

Based on the published papers in this Special Issue of the elite scientific journal Mathematics, we herein present the Editorial for “Differential Equations of Mathematical Physics and Related Problems of Mechanics”, the main topics of which are fundamental and applied research on differential equations in mathematical physics and mechanics [...]

An overview of numerical methods for discrete dynamical systems with frictionless unilateral constraints: mathematical issues and convergence results

An overview of numerical methods for discrete dynamical systems with frictionless unilateral constraints: mathematical issues and convergence results

On the collision of railcars as an interaction of nonlinear shock wave-like perturbations

The article examines the collision of railcars based on the solution of hydrodynamic equations using shock waves during the transition from weak nonlinearity to perturbations of arbitrary amplitude. Several new mathematical issues have been set for the development of high-speed transport, which can be solved in the framework of hydrodynamics to describe the process of hydrodynamics, creating effective rolling stock dampers. which requires the improvement and development of the corresponding mathematical apparatus. In this work, we use a hydrodynamic approach to find the density distributions of matter during railcar collisions at high speeds, which is important in light of the problems of high-speed transport. In our approach, we found an analytical solution to the obtained hydrodynamic equations for the one-dimensional case. The equations under study were obtained taking into account nonequilibrium processes. To find a solution to the hydrodynamic equations, the shock wave approximation is used, similar to the soliton solutions we considered earlier. Taking into account possible deviations from the results of a one-dimensional problem is considered. Such a reduction of solutions of hydrodynamic equations to shock waves has not been considered previously and may be of interest for a wide variety of applied problems. The resulting consideration of railcar collisions is important for solving problems of transport safety and technospheric safety.