The paper explores the history of set theory and its philosophies, with special focus on the early work of Georg Cantor. The revolutionary change of thought in the 19th century introduced set theory as a rigorous method to study infinite sets, continuity, and the structure of mathematical objects. Cantor’s introduction of transfinite numbers, cardinality, and the diagonal argument established the distinction between countable and uncountable infinity, fundamentally altering the concept of mathematical infinity. The study examines mathematics as an applied system, a philosophical investigation, and a reflection of human cognition. It traces the history of mathematical abstraction in the form of conditional notions, as opposed to formal systems based on axioms, including Zermelo–Fraenkel set theory (ZF) and Zermelo–Fraenkel with the Axiom of Choice (ZFC), through an analysis of Cantor’s work in relation to number theory, Fourier series, and the modeling of infinite sets. It also addresses historical paradoxes, such as the Galileo paradox and the Russell paradox, which set theory resolved, establishing a consistent foundation for modern mathematics. Furthermore, the paper investigates the impact of Cantor’s work on modern mathematical disciplines including topology, analysis, algebra, and logic, and emphasizes the ongoing applicability of his contributions in both theoretical and applied mathematics. The study highlights the philosophical implications of the absence of limits and the aesthetic virtue of mathematics and set theory, central to the development of meta-thinking in human cognition. Overall, this study contributes to the historical and conceptual framework of Cantor’s works and explains how set theory was born, ultimately revitalizing the principles of modern mathematics.

Background The domain of mathematics is relevant for future innovations. However, we see a decline of the number of students with exceptional mathematical competencies. Although there are numerous programs aimed at mathematically gifted secondary school students, studies report insufficient evidence regarding their design. We aimed to synthesize current findings on design features of gifted education in mathematics for secondary school students and point out research gaps. Methods For this systematic literature review, we searched ERIC, Fachportal Pädagogik, Scopus, and Web of Science Core Collection on May 22, 2024, manually via the respective interface. We included empirical studies dealing with the design features of educational programs for mathematically gifted secondary school students. We excluded studies that were either not available in English or German, or that are not published as a journal article, book chapter, or book. Design features were identified using content structuring content analysis with deductive-inductive coding based on components of a learning environment. Results Of 3,380 records identified, 49 were eligible. We identified 32 design features that were categorized regarding the six components of learning environments: tasks, content, methods, media, partners, and classroom structure. Most results of the eligible studies were consistent, but in a few cases, contradictory results were also identified. Discussion The identified design features of gifted education mostly match with principles of high quality mathematics teaching. However, not all of the five principles are covered by the design features. Features that do not align with the domain-specific principles may be considered elements of student support, which is a core component of teaching quality. The results of this systematic review are limited by the eligible studies' differences of the overall quality and contextualization of mathematical giftedness. Future research should examine the principles of high-quality mathematics teaching in gifted education in greater detail and focus on the effect of single design features.

This study aimed to improve students’ academic achievement through the implementation of the discovery learning model. The application of this model provides students with direct learning experiences by engaging them in experimental activities that facilitate the independent discovery of mathematical concepts and principles. The research problem focused on how the implementation of discovery learning could enhance mathematics learning achievement of Grade VI A students at SDN 1 Kekeri. This study described the implementation of the discovery learning model in improving students’ mathematics learning outcomes. The research employed Classroom Action Research (CAR) involving Grade VI A students of SDN 1 Kekeri, Gunung Sari District, West Lombok Regency. The data analysis revealed that students’ learning activities at the classical level reached 61.86% in Cycle I and increased to 74.99% in Cycle II. Furthermore, students’ learning achievement showed a significant improvement of 30.30%, increasing from 60.60% in Cycle I to 90.90% in Cycle II. Based on these results, it can be concluded that the implementation of the discovery learning model effectively improved students’ mathematics learning outcomes and achieved learning mastery.

This review article examines the ideas and analyses put forth by Josip Slisko in his 2026 book on matchstick puzzles, which provide a basis for projecting them onto domains of study such as math cognition with implications for math education. The book is a truly significant one bridging these two domains, showing how an apparently simple puzzle form enfolds deep mathematical ideas and principles that, when fleshed out, put on display what fundamental mathematics is all about. Above all else, Slisko’s book has specific important implications for math education, which will be highlighted in this review article.

Background This paper reports on the measures taken to develop a structured and organized approach to understanding the various design elements used in the energy interface. The approach taken in this study involves a decision-making process that selects energy interface elements from both the consumer's and developer's perspectives. Methods The decision-making process utilized the Analytic Hierarchy Process (AHP) methodology, which integrates mathematical and psychological principles. A panel of experts was invited to take part in the workshop activity to choose the best option by quantifying subjective judgments and synthesizing the opinions into a clear result using the AHP framework. Results The evaluation rounds with participants revealed key insights into points such as affordability and usability, which were crucial in energy consumers' priorities. Also, consumers prefer advanced technology features, particularly in mobile devices and smart home integration. Conclusions The recommendations from the decision-making process, which include stressing the distinct needs of various energy consumer groups, can help future product development and marketing strategies.

Quality education is highlighted in UNESCO’s Sustainable Goal 4 for Sub-Saharan Africa, with one of the chief areas of focus being equity of access for learners of diverse backgrounds and abilities. Currently, South African learner performance in Mathematics is of grave concern. Hence, this paper examines whether the ethnomathematical indigenous game of Kgati improves learner understanding of Mathematics. The theory of Culturally Responsive Teaching is the chosen theoretical framework for the paper. A qualitative case study of two teachers at separate schools, both of whom teach Intermediate Phase learners in rural KwaZulu-Natal, was undertaken to explore the phenomenon through semi-structured observations. The findings show that Kgati may be used to teach numerous Mathematics concepts, and that the game generates a high level of learner engagement; however, teachers require training to effectively draw out more mathematical principles embedded in the game. The authors recommend that teachers throughout the country be trained in the use of ethnomathematical indigenous games to teach Intermediate Phase Mathematics and that ethnomathematics be mandated in curriculum policy. The paper is significant in that it highlights the value of basing lessons on cultural artefacts and experiences, which can enrich Mathematics lessons, aligning with curriculum policy guidelines.

Electron paramagnetic resonance (EPR) is a technique for studying the microscopic structures by detecting the transitions of the spin magnetic moments of unpaired electrons. Since its discovery by E. K. Zavoisky in 1944, EPR has evolved from a tool for analyzing atomic structures in physics into a core characterization method in the fields of chemistry, biology, and materials science. In surface chemistry, due to its high sensitivity to the local environment, EPR has become a unique technique for elucidating surface-active sites, free radical intermediates, and defect structures. However, for many chemists, EPR testing and analysis, which are based on mathematical and physical principles, is not an easy field to engage in due to its high level of specialization. Some introductory textbooks provide excellent and comprehensive explanations of the basic knowledge, and recent work reports have also demonstrated the continuously developing magnetic resonance spectroscopy methods. Nevertheless, they are not intended to provide a brief and clear overview through a wide range of examples. To bridge the knowledge gap between EPR spectroscopists and chemists unfamiliar with EPR, this work reviews the progress in the application of EPR in surface chemistry, discussing its principles, applications, innovative cases and future challenges. It is hoped that nonprofessionals would gain certain knowledge and technical accumulation from this work, thereby promoting the development of surface chemistry.

This paper focused on “qualitative study on application of Mathematical principles in architectural design in Nigeria”. Mathematical principles have played a vital role in architectural design such as the foundation, the columns, beams, slabs, footings, stairs and the roofing. It was recommended that Mathematic curriculum should be design to embrace practical principles to architectural designs.

This article explores the acoustic and musical properties of aluminum stroke rods installed in Badia Tedalda, Italy, and their physical and conceptual connection to the medieval musical advances of the theorist Guido d’Arezzo. The stroke rods—aluminum rods with changing lengths supported at their midpoint harmonic node—create distinctive tonal qualities when stimulated by methods. The modern acoustic system establishes a bridge between contemporary sound exploration and medieval music theory. From the 11th century, Guido d’Arezzo’s advanced music pedagogy with a revolutionary hexachord system that furnished a methodical approach to pitch organization.1 This study illustrates, by analyzing the physical properties of the stroke rods and the mathematical principles underlying their acoustic characteristics and the pedagogical advances of Guido d’Arezzo, how both innovations represent milestone approaches to understanding and organizing sound. In what ways does the geographic placement of the stroke rods installation in the province of Arezzo create meaningful connections between medieval music pedagogy and contemporary acoustic exploration? The installation of the stroke rods at Monte Botolino in Badia Tedalda provides both a scientific and artistic demonstration of the region’s musical tradition, connecting the possible source of modern musical notation with contemporary acoustic experimentation. This investigation adds to our understanding of how mathematical relationships in sound production have supplied musical theory across centuries while giving prominence to the cultural significance of this distinctive installation in the province of Arezzo.

This study aims to identify and describe the ethnomathematical elements found in the bedug, a traditional musical instrument used by the community in Loura Subdistrict, Southwest Sumba Regency. The bedug not only functions as a musical instrument, but also has historical, philosophical, and visual value that reflects the cultural practices of the local community. This study uses a qualitative approach with a descriptive ethnographic research type. Data were collected through direct observation, interviews with traditional leaders and local communities, and documentation of the shape and structure of the bedug. The results show that the bedug contains various mathematical concepts, particularly the concept of spatial geometry, such as circles and cylinders, which are evident in the shape of the base and body of the bedug. These findings indicate that the Sumba community has intuitively applied mathematical principles in cultural practices and the creation of traditional artefacts. This study contributes to the development of ethnomathematics-based mathematics learning by integrating local wisdom, thereby making learning more contextual, relevant, and meaningful for students.

This study aims to explore the ethnomathematical concept embedded in the decorative patterns of Balinese Endek woven fabric through an analysis of its motifs, production processes, and the mathematical meanings represented by the artisans. The research employed a qualitative ethnographic approach conducted at Endek Sekar Jepun in Denpasar. Data were collected through participatory observation, in-depth interviews, and visual documentation of motifs and weaving processes using traditional non-machine looms (ATBM). The results show that Endek decorative patterns are composed of basic geometric shapes such as rhombuses, triangles, squares, and organic curves resembling bamboo shoots. These motifs apply mathematical concepts including symmetry, pattern repetition, proportion, and scale, which reflect order, balance, and the symbolic meanings of Balinese culture. The artisans’ weaving practices demonstrate intuitive applications of mathematical principles, serving as concrete examples of ethnomathematical practice. These findings have implications for culturally grounded mathematics education, the development of contextual learning materials, and the preservation of Balinese cultural heritage through systematic documentation of Endek motifs

This study addresses the gap in existing Titanic survival prediction research—where algorithm application dominates while mathematical underpinnings are overlooked—by integrating applied mathematics into three mainstream machine learning models: Logistic Regression, Random Forest, and K-Nearest Neighbors (KNN). Mathematical principles guided the entire workflow: Bayesian probability interpreted how features (e.g., gender, cabin class) contribute to survival likelihood; Gini impurity quantified decision-making logic in tree-based models; statistical distribution theory justified median imputation for missing values (e.g., 19.8% missing in the Age feature); feature engineering further embedded mathematical logic, including constructing Age and Class interaction terms to capture nonlinear dependencies between variables and binning continuous features (Age to AgeBand, Fare to FareBand) to simplify probability distributions for better model adaptability. Experimental results showed Random Forest outperformed the other models, achieving 82.68% accuracy, compared to 81.56% for KNN and 81.01% for Logistic Regression; feature importance analysis identified gender as the most critical predictor (significance score ~0.3), followed by fare (~0.25) and age (~0.24), while survival rates by title (e.g., 79.37% for Mrs, 15.67% for Mr) further reinforced gender’s dominant role in survival prediction. This work enhances the interpretability and methodological rigor of Titanic survival prediction, demonstrating how mathematical theory can ground machine learning applications in theoretical validity.

Integrating machine learning (ML) with Statistical Process Control (SPC) is important for Industry 4.0 environments. Contemporary manufacturing data exhibit high-dimensionality, autocorrelation, non-stationarity, and class imbalance, which challenge classical SPC assumptions. This systematic review, conducted following the PRISMA 2020 guidelines, provides a problem-driven synthesis that links these data challenges to corresponding methodological families in ML-based SPC. Specifically, we review approaches for (1) high-dimensional and redundant data (dimensionality reduction and feature selection), (2) autocorrelated and dynamic processes (time-series and state-space models), and (3) data scarcity and imbalance (cost-sensitive learning, generative modeling, and transfer learning). Nonlinearity is treated as a cross-cutting property within each category. For each, we outline the mathematical rationale of representative algorithms and illustrate their use with industrial examples. We also summarize open issues in interpretability, thresholding, and real-time deployment. This review offers structured guidance for selecting ML techniques suited to complex manufacturing data and for designing reliable online monitoring pipelines.

Abstract. As a result of the growth of cities and the increase in migration from rural to urban areas, the urban population has grown remarkably. Building extraction is important in many practical and strategic areas. Automatic detection of buildings from images is important in terms of both speed and preventing interpretation errors arising from the differences in experience of experts. One of the main difficulties encountered in studies on automatic building detection is the generalization problem originating from the difference in the characteristics of roofs in complex environments. Recently, software and hardware systems have gained great importance due to the development of new technologies. As a result of these innovations, studies on deep learning architecture have increased. The training of deep learning architectures aims to minimize the loss function during learning. There are many optimization algorithms based on various mathematical principles; however, an optimization algorithm that can be generalized to all problems and is optimal for all conditions is still not fully defined. Therefore, the studies in literature continue to be experimental. In this study, the effects of optimization techniques on the automatic detection of buildings with different roof types from aerial images using the U-Net architecture are analyzed. In this study, Adam, Nadam, and RMSprop optimization techniques were used. The effects of optimization techniques on classification performance were investigated by examining computational costs and performance metrics.

The principles of the Polytope Formalism - first developed for stereoisomerism - are here extended to molecular constitution (including constitutional isomerism), highlighting a deep connection between these two aspects of structure and opening the way toward a unified description of all isomerism. A key feature of this development is that it is based solely on atom connectivity, with explicit inclusion of subvalent and hypervalent species. The resulting complete sets of possible species include traditional isomers and their interconversion intermediates (transition states, higher-order saddle points, etc.), providing a powerful tool for elucidating isomerisation mechanisms. This is demonstrated through increasingly complex examples of H-tautomerism. The corresponding networks of species and interconversion pathways map directly onto their associated potential-energy surfaces and thus function as a "discretised encoding" of them. Because this framework accommodates a multidimensional implementation of transition-state theory, it describes and organises the behaviour of the chemical system. Beyond stereoisomerism and molecular constitution, the same mathematical and conceptual principles may be applied to the quantum chemical aspects (nuclear, electronic, and rovibrational states), yielding a fully discretised and physically grounded representation of molecular systems. In doing so, the Polytope Formalism provides a universal framework for the automated exploration of chemical behaviour and Chemical Space - integrating rigorous theory, digital representation, and data-driven discovery within a single coherent framework.

This paper introduces quantum-inspired fractal sustainability optimization (QIFSO), a comprehensive methodology for sustainable biosensor design that transcends conventional linear assessment frameworks. By integrating mathematical principles from quantum information theory with multifractal analysis, QIFSO enables multidimensional sustainability assessment specifically calibrated for complex biosensing technologies. The framework mathematically transforms 15 sustainability parameters into a three-dimensional state space characterized by parameter resilience (PR), sustainability momentum (SM), and criticality coefficient (CC), capturing complex interdependencies that traditional approaches overlook. Hierarchical clustering analysis using optimized k-means algorithms (1500 iterations, 10 replicates) reveals four statistically distinct sustainability regimes that occur universally across biosensor applications: resilient performers, rapid evolvers, critical constraints, and steady optimizers (Davies-Bouldin index = 1.24, Calinski-Harabasz criterion = 186.3). Multifractal analysis demonstrates that this parameter space exhibits non-integer dimensionality (Dq = 2.69 ± 0.05, p < 0.01), mathematically explaining why traditional linear frameworks consistently fail to capture complex parameter behaviors. A robust power law relationship between parameter resilience and criticality coefficient (CC = 0.45 × PR-1.68 + 0.19, R2 = 0.84, p < 0.001) provides a predictive foundation for strategic optimization. We validate this approach through comprehensive in silico case studies across four biosensor categories, including wearable sensors, implantable devices, point-of-care diagnostics, and environmental monitors, drawing on the authors' domain knowledge and prior experience in the field. These analyses indicate potential sustainability improvements ranging from 18 to 52 percent. It should be emphasized that these efforts are intended solely to illustrate the framework's potential and do not represent definitive or experimentally verified outcomes. Comparative evaluation demonstrates that QIFSO-guided optimization reduces development timelines by 60% compared to conventional approaches (mean cycle: 7.3 vs. 18.2 months, p < 0.001) while significantly improving biocompatibility, sensor longevity, and environmental performance. The framework's adaptation across 14 diverse research organizations (implementation success rate = 92%) confirms its broad applicability for accelerating sustainable innovation in biosensing technologies.

Introduction. A comprehensive review of existing literature on the role of masticatory muscle parameters in the development of pathological tooth wearing has highlighted the need for a deeper exploration of this issue. This study aims to shed light on the relationship between masticatory muscle characteristics and the progression of pathological tooth abrasion, focusing on the principles of D. Bernoulli and using advanced imaging techniques such as magnetic resonance tomography. By analyzing electromyographic data, the study will evaluate the impact of two key parameters “speed” and “force” on the onset and progression of pathological tooth wearing. Objective: to explore the masticatory muscle parameters in patients with varying degrees of pathological tooth wearing, compared to individuals without abrasion. Using D. Bernoulli’s principle, magnetic resonance tomography, and electromyography, the study seeks to determine the role of each parameter (speed and force) in the development of pathological abrasion of hard dental tissues. Additionally, the study will examine the functional activity of the masticatory muscles in detail to identify which specific parameters contribute to the onset and intensification of pathological tooth abrasion. Participants and methods. Magnetic resonance tomography was employed to assess the thickness, length, and width of the masticatory muscles. Based on D. Bernoulli's mathematical principle, the study applied a coefficient to investigate the biomechanical properties specifically the “speed” and “force” of the masticatory muscles. Electromyographic techniques were used to study muscle activity. A total of 45 patients with pathological tooth abrasion and 17 healthy controls participated in the study. The findings revealed a direct correlation between the bioelectric activity of the masticatory muscles and the ratio of muscle length to cross-sectional area. “Fast” masticatory muscles exhibited a larger ratio of muscle length to cross-sectional area, with higher peak activity amplitudes, shorter durations of bioelectric activity, and faster contraction speeds over greater distances. In contrast, “strong” masticatory muscles demonstrated a lower ratio of muscle length to cross-sectional area. These muscles exhibited longer excitation and relaxation times, resulting in prolonged pressure on the teeth. “Strong” muscles were found to be more strongly associated with the development of pathological abrasion of the tooth enamel. This study highlights the significant role of masticatory muscle parameters, particularly “speed” and “force”, in the pathogenesis of pathological tooth abrasion. Understanding the biomechanical properties of masticatory muscles can provide valuable insights into the prevention and management of this condition, emphasizing the importance of both muscle activity and muscle morphology in the preservation of dental health.

The limited availability of educational and stimulating physical play spaces in household environments and dense urban communities in Indonesia presents obstacles to optimizing hands-on learning for preschool and elementary school children (5–8 years old). This problem is exacerbated by the high level of parental concern about gadget dependence that triggers a decline in children's motor and logic skills. This study aims to design an STE(A)M-based Open Play Toy System as a non-digital intervention product solution that effectively addresses these space and behavioral issues. The design concept emphasizes modularity (components can be transformed into various forms and functions) and portability, which intelligently 'creates' functional play spaces in limited land. This toy focuses on optimizing hands-on learning through the integration of STE(A)M (Science, Technology, Engineering, Arts, Mathematics) principles, in line with children's interests in construction and arts/crafts. The design methodology is based on the ergonomics of elementary school children and an in-depth analysis of user needs in Indonesia regarding space efficiency and resistance to boredom. The resulting design is an intuitive educational toy that is flexible for indoor and outdoor use, and offers a cognitively challenging play environment. This product is expected to become a tangible intervention tool, providing quality play experiences essential for the growth and development of Indonesian children.

This study aims to explore the integration between mathematics and Islamic prayer through the conceptual pillars of patterns, numbers, and abstraction, framed within a value-based learning approach. Mathematics, as a discipline concerned with numerical operations, structural patterns, and abstract reasoning, provides a logical framework that can be meaningfully connected to spiritual practices. Prayer, as an act of devotion and direct communication with Allah SWT, embodies structural regularity, numerical precision, and abstract spiritual intention, making it a fertile context for contextualized mathematics instruction. Using a qualitative-descriptive approach with a literature study method, this research analyzes Qur’anic verses, classical tafsir, and scholarly works on mathematics in Islamic tradition. The findings reveal that the sequence and symmetry of prayer movements, the fixed counts in rakʿah and dhikr, and the unseen dimensions of intention and focus in prayer parallel mathematical principles of pattern recognition, cardinality, and abstraction. The integration of these elements into mathematics lessons can enhance cognitive skills (pattern recognition, calculation, abstraction), foster positive affective engagement by linking learning to meaningful religious contexts, and cultivate spiritual values such as discipline, mindfulness, and faith. This study concludes that mathematics and prayer are complementary pathways to understanding order in both the physical and spiritual realms, supporting the holistic objectives of Islamic education.

This study aimed to investigate the Comparative Effects of Laboratory-Based and discussion-based instruction on junior secondary students' performance and Retention in Plane Geometry. Two research questions, two null hypotheses, and two aims and objectives made up the investigation. The research was quasi-experimental in nature. For the study, intact classrooms were employed. In Rivers State, Nigeria, a sample of 119 public junior secondary class two students was selected from a total of 6,132 students. Data collection involved the use of the Plane Geometry Achievement Test (PGAT). With validation, PGAT's reliability index was 0.81. While the control group received education using a discussion-based approach, the experimental group received practical laboratory-based instruction in plane geometry. Analyses were conducted using the mean, standard deviation, and Analysis of Covariance (ANCOVA) at the 0.05 level of significance. For testing hypotheses, a significant level of 0.05 was employed. The results demonstrated that students who received instruction through laboratory-based based performed better and retained information, with a mean gain performance of 22.95, SD=13.58, than their counterparts in the control group, who had a mean gain performance of 10.74, SD=7.35, which was statistically different. It was suggested that teachers should employ laboratory-based instruction to teach mathematical principles and practical plane geometry.