Mathematical Geometry Research Articles

Pure Rickart Modules

This paper focuses on the study of pure Rickart modules (or PR-modules for short), which are a class of modules over a commutative ring with identity. The main objective is to investigate the properties and characterizations of these modules, as well as their relationship with other classes of modules such as free, projective, and flat modules. This paper also explores the connections between PR-modules and various algebraic structures such as rings. The results obtained in this study provide a deeper understanding of the structure and behavior of PR-modules, which can have important applications in algebraic geometry, representation theory, and other areas of mathematics. Some results about PR-module have been investigated in this paper, for example we demonstrate a module is PR-module if and only if for every , is pure submodule of (or for short). Also, some kind of generalization of these rings have been constructed and demonstrated in term of PR-modules.

Federal fractals: review of Stephen Tierney, The Federal Contract

Stephen Tierney’s new book presents an exciting new theory of federal constitutionalism. His core argument is that each federal constitution is founded on a ‘federal contract’, which is an agreement among a plurality of territorial communities to form a federal political community while retaining their distinct constitutional identity as constituent states. This offers an alternative to the ‘social contact’ of liberal political theory in which the isolated individual is the sole basic unit of the constitutional order. The explanatory power of Tierney’s thesis is impressive. He shows how the idea of the federal contract can be manifested in many varied forms but in a manner that recognises the constitutional status and autonomy of the constituent states, formed into an association under which they agree to act cooperatively with each other and with the federation as a whole, under a mutual commitment of reciprocity that underwrites inter-governmental cooperation and solidarity, while also maintaining inter-governmental distinctiveness and autonomy. Under the auspices of this simple and elegant idea, Tierney offers an account of federal constitutionalism that resembles the fractal patterns of mathematical geometry insofar as each federal system reflects a particular federating logic that drives its initial creation and ongoing evolution. This opens up a promising area of further research involving an ever-closer examination of federal systems to see whether the fractal-like patterns he has identified continue to be apparent at progressively higher levels of resolution.

Editorial Foreword

Non-Euclidean Geometry in Modern Physics and Mathematics is a series of biennial conferences initiated by the Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine in 1997, dedicated to the heritage of J´anos Bolyai, Carl Friedrich Gauss, and Nikolai Ivanovich Lobachevski (BGL, alphabetically), and held at various sites connected with the founders of the new geometry; for history see: https://indico.cern.ch/event/586799/page/8964-former-blg-conferences.

Research on multibeam line layout based on geometric mathematical model and regression analysis

A multibeam sounding system is a system that emits acoustic pulse waves through a mechanical device to map the seafloor topography in real time. It can accurately and rapidly measure the size, shape and elevation changes of underwater targets. In this study, the mathematical model of the multibeam sounding system is proposed, and the optimized design of the survey line is carried out according to different seabed geomorphology. Firstly, relevant formulas are deduced by mathematical geometry and trigonometric functions in plane space, then a spatial right-angled coordinate system is established and the direction of the survey line is projected to the x-axis, which transforms the survey line problem in three-dimensional space into a planar type, and finally, according to the mathematical definitions, the shape of the multibeam irradiation on the sloping surface is judged to be an ellipse and the projection model is constructed. The bivariate model is constructed using regression analysis, and the line measurement that meets the requirements is determined by solving the optimal solution. The conclusion shows that the optimal solution for the angle between the direction of the survey line and the projection of the normal direction of the seabed slope on the horizontal plane is 45°, and the survey line that meets the requirements of the shortest measurement length, covering the entire sea area to be surveyed, and the overlap rate of 10-20% can be obtained.

Geometric trust-based secure localization technique for resiliency of GPS outage and location error in vehicular cyber-physical systems (VCPS)

Management of vehicle traffic is a challenging task as it is non-deterministic by nature. Vehicular Cyber-Physical Systems (VCPS) is the emerging field of dynamics of vehicle management. Vehicle localization is considered an important task in VCPS. Many researchers proposed methodologies for this based on the Global Positioning System (GPS) which poses few location identification errors. Also, there are more vulnerabilities to the existing vehicular positioning system due to Zig-Zag attacks and bad-mouth attacks. In this work, an error-free and secure environment for communication between dynamically moving vehicle models has been proposed. In our proposed model a localization technique based on mathematical geometry which is capable of GPS outages and encompasses the dynamism of vehicle and on-road trajectory has been developed. The proposed model includes Extended Kalman filter-based routing to predict the neighbouring vehicle position. To avoid vulnerabilities created by the malicious nodes, a trust-based computation is performed by each node on its neighbours perceiving the authenticity of received messages. To validate the methodology, NS2 tool has been used to simulate the VCPS and to test the efficiency with different scenarios such as erroneous location, GPS outage, and malicious attack. The result shows that the proposed approach is more optimal and secure than the existing methodologies.

Interpretation of the Mathematical Geometry of the Roti Canai Structure in North Sumatra

This research was aimed at finding out the role of mathematical geometry in the process of making delicious Roti Canai. This research uses a literature review method or what is usually called library research which uses library materials such as books, magazines, documents, and journal articles. The results of this research show that there is a role of mathematical geometry in making Roti Canai, which includes a circular shape (π), structure, weight, and temperature measurement. From all the involvement of geometry in the process of making Roti Canai, delicious Roti Canai is created. So, it is concluded that the role of geometry is very important in the process of making Roti Canai.

Exploration of Ethnomathematics Houses Malay ‘Limas Potong’ The Batam Islands of Riau

Ethnomathematics is the study of how a group of individuals in a particular culture understand, communicate, and use cultural ideas and practices that researchers identify as mathematics. To return cultural heritage of traditional house in Batam, Riau Islands, and to further explore the idea of mathematical geometry inherent in the traditional house, so this research aims to explore the ethnomathematics in the Malay Traditional House 'Limas Potong'. This study applied a qualitative approach that was conducted in the Malay Traditional House 'Limas Potong', specifically in the Kampung Melayu neighborhood of Batam City's Nongsa District in the Riau Archipelago. The data used in this research was obtained from these explorations or already-published articles. Observation, interviews, and documentation were used as data collection methods. In this research, data analysis was carried out in two stages: data analysis in the field and data combining. According to the research findings, each decorative and structural part of the Malay Traditional House 'Limas Potong' incorporates mathematical ideas in geometrical shapes and components. Measurement, architectural design, mathematics learning, and application of learning are all related to ethnomathematics in the Malay Traditional House 'Limas Potong'. The construction of the Malay Traditional House 'Limas Potong' can be utilized as a learning resource for students, especially in geometry material, according to some mathematical ideas contained in the structure of the traditional house. Ethnomathematics exploration will inspire teachers to make this a new alternative to developing creativity in students' mathematics learning.

The 2n Geometric Space Model of the Earth's Atmosphere and the Periodic Table of Vertical Changes the Laws of Mathematical Geometry Distribution of Atmospheric Sphere

The 2n Geometric Space Model of the Earth's Atmosphere and the Periodic Table of Vertical Changes the Laws of Mathematical Geometry Distribution of Atmospheric Sphere

Improved multi-search strategy A* algorithm to solve three-dimensional pipe routing design

Pipe routing design (PRD) has a significant impact on the safety and operational efficiency of the system. PRD is involved in multi-disciplinary knowledge such as mechanical engineering, computer graphics and mathematical geometry, and it can be abstracted as a complex combined optimization problem. A* algorithm is one of the most commonly employed approaches to solve two-dimensional (2-D) path planning. However, due to the central idea of A* algorithm is all-traversal search, the search process considers all possibilities and requires a blind search one by one, it is difficult to solve PRD problems in complex three-dimensional (3-D) space environment. In this study, an improved multi-search strategy A* algorithm (MSSA*) is presented to overcome the disadvantages of the traditional A* algorithm in solving 3-D PRD problems. The MSSA* algorithm contains four novel mechanisms which are the node directional discrimination rule, the double-layer domain extension search strategy, the multi-factor heuristic evaluation function, and the dynamic adaptive weighting strategy. Then, a range of experiments are conducted to prove the validity of the presented mechanisms. Subsequently, the MSSA* is compared with several existing algorithms for solving 3-D PRD to further verify the superiority of MSSA* in terms of high-efficiency and practicality. Finally, the MSSA* is applied to solve the 3-D PRD problems for offshore production platforms in oil and gas treatment system.

Fuzzy Intuitionistic Alpha Cut of B-Spline Curve Interpolation Modeling for Shoreline Island Data

Traditional approaches are unable to handle the uncertainty data issue, which leads to inaccurate data analysis and prediction. Data with uncertainty are frequently collected during the data collection phase but cannot be directly used to create geometric models. Therefore, using intuitionistic alpha cuts for the uncertainty data, this paper discusses B-Spline curve interpolation modeling. To resolve the uncertain data and produce the mathematical model, fuzzy set theory, intuitionistic fuzzy sets, and geometry modeling are combined. Three main procedures are used in detail, the first of which is the application of fuzzy set theory to defined uncertainty data, followed by the use of an intuitionistic fuzzy set to take into account the membership, non-membership, and indeterminacy values of the alpha, and finally the fuzzification and defuzzification procedures. The B-spline curve interpolation function is used in geometric modeling to create mathematical geometry in the form of curves. As a result, several numerical examples are provided, along with their algorithms for producing the desired curve.

The teaching of geometry in mathematics and sciences applying symmetry with folding techniques

This work offers students the use of geometry in a simpler and more practical way, providing the teaching of Symmetry and based on this assumption, the use of folding techniques, combining this work with the practical and logical field of use of concepts, providing students with a effective understanding of the geometric proposal used, in addition to putting the student in front of numerous questions required in the ENEM test, which are often seen in this exam. This activity was applied to a seventh grade class of elementary school II in a Federal Public School in the metropolitan region of Belém, showing its effectiveness and harmonization with the content of Plane Geometry, corroborating for a better teaching with an effective learning of the contents of the proposed teaching.

Eksplorasi Etnomatematika Bentuk Geometri Peninggalan Syekh Ahmad Muttamakkin

Syekh Ahmad Muttamakkin was one of the great scholars of the archipelago in the 18th century and became a role model for the community in Kajen, Pati. The history of Syekh Ahmad Muttamakkin must be preserved so that it does not disappear over time. In this way, researchers link the history of Syekh Ahmad Muttamakkin's legacy with the material of mathematical geometry. In this case the researcher associates local culture with mathematical activity. This research uses descriptive qualitative research with an ethnographic approach. The object of this study is ethnomathematics on the legacy of Syekh Ahmad Muttamakkin which includes mosque buildings, ornaments, and objects left by Syekh Ahmad Muttamakkin. The data in this study were obtained from observations, interviews, documentation, and literature from journals and internet articles. The purpose of this study is to find out the ethnomathematics studies contained in the legacy of Syekh Ahmad Muttamakkin and to dig up philosophical information about these objects. The result of this study is that the researcher found a link between the mathematical concept of recognizing geometric shapes and the legacy of Syekh Ahmad Muttamakkin. In an ethnomathematics exploratory study, in terms of the shape of mosque buildings, ornaments, and relics of Syekh Ahmad Muttamakkin, it shows that there is a connection with mathematical material, namely regarding geometry and shapes.

Stimulating mathematical communication with SPECOMATSO technology development based on digital literacy

Mathematical communication is crucial in inclusive mathematics teaching and learning to convey mathematical symbols. However, teaching mathematical symbols to deaf students in inclusive classrooms has been a persistent challenge due to the lack of specialized sign language for mathematical expressions. Media, such as SPECOMATSO technology, can serve as a bridge to address this issue. This research focused on developing valid and practical SPECOMATSO technology to strengthen students’ literacy and facilitate the delivery of mathematical symbols, which previously posed difficulties for teachers in inclusive settings. The research combined the Alessi & Trollip and V-waterfall models, encompassing seven development stages: analysis, design, implementation, unit testing, integration testing, system testing, and acceptance testing. The research instruments included observation and interview guidelines, a product validation questionnaire, and a student response questionnaire. This study produces SPECOMATSO technology that is valid and practical for learning plane geometry and angles in mathematics. Although this development still needs improvement in fixative abilities, such as storage and editing functions, it is expected to strengthen digital literacy and stimulate mathematical communication among deaf students who face challenges related to their limited auditory capabilities.

Analysis of students’ mathematical reasoning between different genders

The main purpose of this study was to examine existing studies related to mathematical reasoning skills based on gender differences using Systematic Literature Review (SLR). The following studies identify reasoning abilities based on gender differences. The review was carried out by carrying out several stages, namely the process of identification, screening, feasibility, inclusion, and data analysis on two livelihood machines, namely SCOPUS and ERIC. In reporting this study, the guidelines used were PRISMA guidelines. The data used was sorted by the specified criteria and this resulted in a total of 8 articles. Our findings identify that the study was conducted in three places: Indonesia, Sweden, and Turkey. Based on the findings of the study using three research designs, namely qualitative, mixed method, and quasi-experimental. The topics of the material used in this study are topics of geometry, algebra, and other branches of mathematics. The SLR also includes characteristics of reasoning abilities found by gender as well as the distribution of studies in terms of trends and countries.

Kinds of Mathematical Thinking Addressed in Geometry Research in Schools: A Systematic Review

Geometry is one of the content of mathematics which in many studies is associated with students' thinking abilities, such as critical thinking and reasoning abilities or others..This study aims to conduct a systematic review of the geometry research in school for identifying the types of mathematical thinking and their interconnections. We searched the Scopus database for articles published from 2003 to 2023 using relevant keywords. We applied the PRISMA method to select and evaluate the studies or articles based on the empirical data. We retrieved and evaluated data from the studies on the various styles of mathematical thinking evolved. Out of 166 titles that were initially obtained, only 10 titles passed the five stages of the systematic review protocol process. We identified 10 types of mathematical thinking that were discussed in the context of learning geometry at school: Creative Mathematical Reasoning (CMR), Computational thinking, Geometric reasoning, Geometric thinking van hiele theory, Geometric thinking (3D geometric thinking with representations), 3D geometry thinking, Visuo spatial reasoning, Geometry Spatial Reasoning, mathematical creative reasoning (MCR), and Inductive reasoning. We also found some connections of literature between these types of mathematical thinking, such as CMR and MCR, Geometric reasoning and Geometric thinking, and Visuo spatial reasoning and Geometry Spatial Reasoning. This systematic review provides an overview of the current state of research on geometry and reasoning in school mathematics and reveals some gaps and directions for future study. It also has implications for teachers who want to enhance their students’ mathematical thinking skills in geometry by exposing them to different types of mathematical thinking and their connections.

STRUCTURAL CORRELATION BETWEEN MATHEMATICS AND ART IN THE CLASSROOM EDUCATION

The correlation-integration system, that is, the methodological model, is based on the principle of interdisciplinarity of the sciences, so it establishes a system of intersubject connection, contrary to the traditional isolation of teaching subjects. Thismodel of teaching gives the most answers to students because it enables them to see subjects, phenomena and processes, relationships and connections between them as a part of the overall reality, that is, to structure knowledge from different subject areas into a common whole. The aim of the paper is to point out the importance of the application of structural cross-curricular correlation of certain teaching contents, i.e. to determine whether students of lower years of elementary school can connect knowledge from the field of geometry in mathematics and geometric abstraction in fine arts, in accordance with the current curriculum. Students of the third and fourth years of elementary school in the Tuzla area, in the second semester of the school year 2020/21, participated in the research. The research was conducted with a questionnaire, and the questions were divided into three categories: general data, recognition and naming of mathematical and artistic elements in the analysis of abstract works of art, andreduction of a figurative work of art to basic geometric shapes. The results of the correlation analysis showed, among other things, that children who achieved better results at the end of the previous year, achieved better results at the end of the semester in both subjects (mathematics and fine arts), and were also more successful at observing artistic and mathematical concepts in completely abstract art works. Given that a large percentage of students (65.6%) achieved exceptional achievement, it can be concluded that teachers are engaged when it comes to connecting teaching content, because students can mostly recognize artistic, compositional and mathematical elements in the analysis of abstract works of art. Key words: structural correlation, art and mathematics, geometric abstraction.

Mathematical Geometry and Groups for Low-Symmetry Metal Complex Systems.

Since chemistry, materials science, and crystallography deal with three-dimensional structures, they use mathematics such as geometry and symmetry. In recent years, the application of topology and mathematics to material design has yielded remarkable results. It can also be seen that differential geometry has been applied to various fields of chemistry for a relatively long time. There is also the possibility of using new mathematics, such as the crystal structure database, which represents big data, for computational chemistry (Hirshfeld surface analysis). On the other hand, group theory (space group and point group) is useful for crystal structures, including determining their electronic properties and the symmetries of molecules with relatively high symmetry. However, these strengths are not exhibited in the low-symmetry molecules that are actually handled. A new use of mathematics for chemical research is required that is suitable for the age when computational chemistry and artificial intelligence can be used.

Systematic Literature Review: Effectiveness of a Teaching Aid Based on the Achievement and Interest of Students in the Field of Measurement and Geometry

<p style="text-align:justify">The purpose of this systematic literature review (SLR) is to identify: (a) the topic of the study, (b) the research methods used, and (c) the results of research on Mathematics education in Malaysia. This study discusses the use of teaching aid (TA) in the field of syllabus and geometry for Form 2 students. The use of TA is considered highly successful and relevant for educators to improve the quality of the teacher’s instructions and students’ understanding. Therefore, using the rules of optional reporting items for systematic review and meta-analysis (PRISMA) by Moher et al. (2015), a review system was carried out to determine the appropriate strategies and variables for the field. Four stages constitute the PRISMA paradigm used in this study: identification, screening, qualification, and admission. Using criteria opted by researchers from multiple searches, including Google Scholar, Researchgate, Scopus, and Emerald, over 20 papers were identified for additional investigation. The data were then analysed quantitatively to describe the research's findings. From the results, two main research themes were found, namely (a) learning to use TA; and (b) the field of measurement and geometry of Mathematics. The results of the article analysis indicate that Mathematics education in Malaysia is currently at a moderate level and is ineffective at fostering students' understanding and interest. These results are anticipated to serve as the foundation for teachers, students, schools, and the Ministry of Education to undertake more engaging and interactive learning, particularly in the subject areas of mathematics and geometry.</p>

Grape-Bunch Identification and Location of Picking Points on Occluded Fruit Axis Based on YOLOv5-GAP

Due to the short fruit axis, many leaves, and complex background of grapes, most grape cluster axes are blocked from view, which increases robot positioning difficulty in harvesting. This study discussed the location method for picking points in the case of partial occlusion and proposed a grape cluster-detection algorithm “You Only Look Once v5-GAP” based on “You Only Look Once v5”. First, the Conv layer of the first layer of the YOLOv5 algorithm Backbone was changed to the Focus layer, then a convolution attention operation was performed on the first three C3 structures, the C3 structure layer was changed, and the Transformer in the Bottleneck module of the last layer of the C3 structure was used to reduce the computational amount and execute a better extraction of global feature information. Second, on the basis of bidirectional feature fusion, jump links were added and variable weights were used to strengthen the fusion of feature information for different resolutions. Then, the adaptive activation function was used to learn and decide whether neurons needed to be activated, such that the dynamic control of the network nonlinear degree was realized. Finally, the combination of a digital image processing algorithm and mathematical geometry was used to segment grape bunches identified by YOLOv5-GAP, and picking points were determined after finding centroid coordinates. Experimental results showed that the average precision of YOLOv5-GAP was 95.13%, which was 16.13%, 4.34%, and 2.35% higher than YOLOv4, YOLOv5, and YOLOv7 algorithms, respectively. The average positioning pixel error of the point was 6.3 pixels, which verified that the algorithm effectively detected grapes quickly and accurately.

Predictive validity of mathematics basic education certificate examination as correlate of mathematics geometry achievement among SS 2 students in Delta state

This study examined predictive validity of mathematics basic education certificate examination as correlate of mathematics geometry achievement among SS 2 students in Delta State. Two research questions and null hypotheses guided the study and Correlational research design was employed. The population of the study consisted of 105, 848 students who sat for BECE Mathematics in 2022 and SS 11 students in year 2024 in Delta State. The sample size of the study was 480 while the instruments used for this study was mathematics BECE scores and Geometry Achievement test (GAT). The internal consistency of the GAT yielded an index of 0.62. The Cronbach alpha method of estimating reliability was employed to obtain the reliability index of 0.84 for the mathematics BECE Proforma. The Pearson’s Product Moment Correlation Coefficient of Determination was used to answer the research questions while regression and fisher-z statistics were used to test the hypotheses at 0.05 level of significance. Based on this finding it was concluded that BECE Mathematics is a predictor of mathematics geometry achievement. However, the magnitude of relationship and the percentage of prediction of mathematics geometry achievement from BECE Mathematics performance were very low and weak. Based on the findings it was recommended among others that BECE Mathematics scheme, contents and curriculum should be reviewed in conformity with mathematics geometry to enhance its predictive validity, BECE that are conducted by NECO should be compulsorily used by all schools to ensure uniformity.