Pure Mathematics Research Articles

Literacy Skills of Digital-Age Primary School Students in Mandailing Natal Regency

This research aimed to investigate literacy skills, the underlying causes of low literacy levels, and the implications for primary school students in the digital age. The researcher employed a qualitative field research methodology, collecting data through participatory observation, in-depth interviews, and the utilization of Google Forms. The data analysis involved systematic reduction, presentation of findings, and the formulation of conclusions, all while employing coding techniques and categorizing themes. The findings indicate that the literacy skills of elementary school students in Indonesia remain significantly underdeveloped. A considerable number of students encounter difficulties in comprehending simple texts, organizing their thoughts in writing, and understanding the material presented by teachers. Additionally, they face challenges in grasping fundamental mathematics and science concepts, as well as effectively utilizing digital tools in their educational experiences. Several factors contribute to the observed low literacy levels, including a lack of supportive familial environments, inadequate integration of technology in educational settings, ineffective instructional models, and limited access to digital resources and training. The repercussions of insufficient literacy skills are evident in students’ difficulties in solving problems, accessing information, and articulating their thoughts, both verbally and in written form. To mitigate this issue, it is recommended that educators integrate digital tools into their pedagogical strategies. This can be achieved through the implementation of project-based learning and the use of educational podcasts disseminated via online platforms.

This is a Lot Less Obvious Than it Seems; Maths

The main point in this month�s column: Some of the things we take for granted and work with every day are not as obvious as we have come to think they are. I started out with a list of quantitative finance cases but discovered that pure maths examples could easily fill a column. So, maths now, quantitative finance later.

Effect System and Reference Frames in Rubik’s Cube and Megaminx

The two basic puzzles in Rubik’s cube can be summarized as recognizing solvable configurations and finding a common solution for valid configurations. Indeed, in the process of solving the first puzzle, we are required to come up with a common solution. The mathematical structure of moves in Rubik’s cube is also famous. This paper clarifies the difference between process and essence through an original concept: effect, and redefines the operation group G in a strict way. It also corrects a widespread mistake in pinpointing the mapping type between the operation group G and permutation groups, which appears in some former papers. The effect system grasps the essence of operations, introduces reference frames in physics through pure mathematics, and specifically facilitates the solution for the basic puzzles in Rubik’s cube and megaminx. The effect system has the potential to be utilized in physically symmetric structures with moves that permute mathematically analyzable configurations.

Applications and Related Concepts of Orbit-Stabilizer Theorem

As the essential part of abstract algebra, group theory takes a critical place on the field of pure mathematics. Approaching the study of it from the aspect of group action is a great starting point. This paper will firstly expound several related concepts and theorems, such as stabilizer, orbit, fixator, conjugacy class, center of group and centralizer, and the Lagrange’s Theorem. Based on that, the paper investigates three essential theorems themselves and their proofs, which are Orbit-Stabilizer Theorem, Class Equation, and Burnside’s Lemma. This paper specifically provides wonderful insights about the applications of the Orbit-Stabilizer Theorem. The theorem is directly utilized in the derivations of Burnside’s Lemma and the Class Equation, also in the deduction of the general formula of the order of symmetric group . There are always some connections between the concepts in these theorems and examples with the orbit and stabilizer, thus Orbit-Stabilizer Theorem can perform brilliantly. This paper provides wonderful insights that effectively connect the Orbit-Stabilizer Theorem with other parts of Group Theory, also brings some new ideas to solving problems and prove theorems.

Homotopical Algebra and Higher Structures

Homotopical algebra and higher category theory play an increasingly important role in pure mathematics, and higher methods have seen tremendous development in the last couple of decades. The talks delivered at the workshop described some of the latest progress in this area and applications to various problems of algebra, geometry, and combinatorics.

Holographic foliations: Self-similar quasicrystals from hyperbolic honeycombs

Discrete geometries in hyperbolic space are of longstanding interest in pure mathematics and have come to recent attention in holography, quantum information, and condensed matter physics. Working at a purely geometric level, we describe how any regular tessellation of (d+1)-dimensional hyperbolic space naturally admits a d-dimensional boundary geometry with self-similar “quasicrystalline” properties. In particular, the boundary geometry is described by a local, invertible, self-similar substitution tiling, that discretizes conformal geometry. We greatly refine an earlier description of these local substitution rules that appear in the 1D/2D example and use the refinement to give the first extension to higher dimensional bulks; including a detailed account for all regular 3D hyperbolic tessellations. We comment on global issues, including the reconstruction of bulk geometries from boundary data, and introduce the notion of a “holographic foliation”: a foliation by a stack of self-similar quasicrystals, where the full geometry of the bulk (and of the foliation itself) is encoded in any single leaf in a local, invertible way. In the {3,5,3} tessellation of 3D hyperbolic space by regular icosahedra, we find a 2D boundary quasicrystal admitting points of 5-fold symmetry which is not the Penrose tiling, and record and comment on a related conjecture of William Thurston. We end with a large list of open questions for future analytic and numerical studies. Published by the American Physical Society 2025

Morse Theory, Discrete Morse Theory and Applications

By employing a specific class of smooth functions to study a space, Morse theory establishes deep connections between analysis and topology. It is a classical subject of pure mathematics, originally pioneered by Marston Morse in the 1920s. In this article, we use Morse theory to present a proof of an interesting result on the knots, known as the Fry-Milnor theorem. We also discuss discrete Morse theory, a subject of applied mathematics developed by Robin Forman in the 1990s, and its application. We focus on elucidating especially the inherent similarity between classical Morse theory and discrete Morse theory

Series and Rational Solutions of the Second Kind Painlevé Equations by Using the Quantum Pseudospectral Method

The Painlevé equations and their series and rational solutions are essential in applied, pure mathematics and theoretical physics. Recently, quantum algorithms have helped to implement numerical algorithms more easily by performing linear algebra in our working. This article uses a hybrid of quantum computing schemes and spectral methods for the second Painlevé equation. Two approaches are investigated: first, a series solution is obtained, and then the rational solutions. The successive linearization method is used for the linearization of the Painlevé‐II equation. In the computer implementation, the solution value of the Painlevé‐II equation is considered as a final quantum state. In each iterative scheme, by adding the current quantum state, we can compute the final quantum state. We need different quantum models for each approach, series, and rational solutions. Numerical examples illustrate the efficiency of this method, and reasonable solutions are obtained for a wide range of parameter values.

SSCE and UTME as Correlates of Undergraduate Students’ Academic Achievement in Mathematics

This study determined students’ SSCE grades and UTME scores in Mathematics as correlates of the first year mean achievement scores of the students in undergraduate Mathematics. The study used correlational research design. It was carried out in Ebonyi State and was guided by three research questions, and three hypotheses. The study used a sample of 120 students out of the population of 146 students drawn from the 2014/2015, 2015/2016, and 2016/2017 academic sessions in both the Pure and Applied Mathematics and Mathematics Education courses using both purposive sampling and proportionate random sampling techniques. The data were collected using the researcher-made instrument which was face validated by three specialists in Measurement and Evaluation and one in Mathematics Education. The data were analysed using Multiple Linear Regression. The findings of the study showed a low positive relationship between students’ SSCE grades and the students’ achievement; and a moderate positive relationship between students’ UTME scores and the first year mean achievement scores.

A Novel Structure to Enhance Second Harmonic Generation in Plasmonic Waveguide

The graphene plasmonic nano-cavity grating is a novel plasmonic structure that has been proven to significantly boost nonlinear optical second-harmonic production. In this article, we discuss how this structure works (SHG). In the suggested structure, metal niobate is positioned such that it is sandwiched between two distinct metals and a thin sheet of grating-patterned graphene. The combination of two distinct metals in a conductor has the potential to greatly amplify the nonlinear state of the conductor, which will lead to an increase in the specific heat capacity of the conductor. Graphene gratings connect the pump beam to two SPP waves that may cancel each other out, which results in the formation of a stationary SPP wave in the region between the gratings due to the mutual interference that occurs. The distance in between the two gratings will have its distance between them fine-tuned in order to improve the formation of second harmonics. It will be shown that field sweetening in proposed waveguides may result in significant improvements in SHG by optimizing the pure mathematics of the desired structure and using different metals. These two factors are meant to be done in conjunction with one another.

Noncommutative Geometry and Spacetime: A Historical Reconstruction

Abstract Noncommutative geometry (NCG) is a branch of pure mathematics with a wide range of applications to spacetime physics. Stemming from the divergence problem in QFT, modern contributions conjecture that the fundamental structure of spacetime is noncommutative. This seemingly homogeneous picture is the result of almost a century of discontinuous interest in noncommutative spacetime (NCST). In this paper, I present a three-phase division of the development of NCST approaches. The initial phase (1930s–1950s) introduced noncommutativity as a means of addressing the divergence problem while maintaining Lorentz-invariance. The second phase (1950s–early 1990s) initially dismissed noncominutativity and focused on the pressing problem of localisation at high energies. In this context, independent alternative approaches to QG identified the value of a fundamental length scale, which ultimately contributed to a reconsideration of the conjecture of spacetime noncommutativity. This was despite the fact that it undermined the original operationalist methodology. Finally, a third phase (1990s–today) has seen a renewed interest in NCST. I argue that this resurgence can be attributed to the discovery of new mathematics in the 1980s. Furthermore, I demonstrate how the third phase builds upon and continues instances from the previous attempts, but redirects the research question. In light of this, I finally argue that the history of NCG in physics can be understood as an example of “interlaced convergence.” This instance of theory convergence may prove usefill as a case study for the more general problem of theory building in QG.

Some Fixed Point Results in Fuzzy Strong b-Metric Space

The theory of fixed points in pure mathematics is the most dynamics and active area of research activities. Also, the fuzzy metric space has been one of the important generalizations of usual metric space. As an extension of fuzzy metric space, the notion of fuzzy strong b-metric space has been introduced by Oner and Sostak in 2020. The purpose of this paper is to study the notions of fuzzy strong b-metric spaces and to establish some fixed point results in complete fuzzy strong b-metric space with examples. Our results extend and improve some well-known results in literature.

Visualization as an Intuitive Process in Mathematical Practice

In the field of the philosophy of mathematics, in recent years, there has been a resurgence of two processes: intuition and visualization. History has shown us that great mathematicians in their inventions have used these processes to arrive at their most brilliant proofs, theories and concepts. In this article, we want to defend that both intuition and visualization can be understood as processes that contribute to the development of mathematical knowledge as evidenced in the history of mathematics. Like intuition, visualization does not have a definition, and its role has become more prominent both in pure mathematics and in educational research. For us, both visualization and intuition are processes that start from the real world of those who “intuit” or “visualize,” require experience and knowledge of concepts and theories (the more expertise in the subject, the more profound the results will be) and must, in the end, be validated by the specialized academic community. In this article, we defend the importance of visualization in mathematical practice and its role in the advances of great scientists (Euclid, Euler, Galileo, Descartes, Newton, Maxwell, Riemann, Einstein, Feynman, among others) as an alternative and valuable way to symbolic thinking, which has “reigned” in the academic and scientific community.

The Proof of the Fermar's Last Theorem, Mersenne's Prime Conjecture and Poincare Conjecture in Euclidean Geometry

In order to strictly prove from the point of view of pure mathematics Goldbach's 1742 Goldbach conjecture and Hilbert's twinned prime conjecture in question 8 of his report to the International Congress of Mathematicians in 1900, and the French scholar Alfond de Polignac's 1849 Polignac conjecture, By using Euclid's principle of infinite primes, equivalent transformation principle, and the idea of normalization of set element operation, this paper proves that Goldbach's conjecture, twin primes conjecture and Polignac conjecture are completely correct. In order to strictly prove a conjecture about the solution of positive integers of indefinite equations proposed by French scholar Ferma around 1637 (usually called Ferma's last theorem) from the perspective of pure mathematics, this paper uses the general solution principle of functional equations and the idea of symmetric substitution, as well as the inverse method. It proves that Fermar's last theorem is completely correct.

Exploring determinants of engagement with voluntary mathematics assignments in economics education: a correlational study

Abstract This article uses data collections from a German university to explore determinants influencing students’ engagement with voluntary weekly mathematics homework assignments tailored for economics. Demographical, social and affective variables were collected from approximately 800 students across several years. Unlike engineering or pure mathematics disciplines in Germany, where homework assignments are often mandatory, homework in economics remains optional. The study aims to provide an exploratory insight into which students opt to tackle these voluntary tasks. For this purpose, the collected data were clustered and analyzed using regression analyses. It was found, among other things, that demographic variables such as gender and age influence the decision to engage in voluntary exercise tasks, but, for example, prior knowledge in mathematics inhibits motivation for optional task. The compulsory nature of homework assignments in other subjects made them unsuitable for comparison due to a lack of heterogeneity. Therefore, this research contributes to the Mathematics for Economists research, underscoring economics students’ unique dynamics and decision-making processes.

An Overview of the Basic Theory of Probability Theory and its Commercial Application

Abstract. Probability theory is a key field of mathematics that focus on developing theoretical models and mathematical frameworks to describe and analyze random events or processes. As a part of basic science, probability theory holds a central place in pure mathematics while also serving a crucial function across various domains, including natural science, social science and engineering technology. This paper summarizes the main concepts, basic theories and commercial applications of probability theory. Probability theory encompasses ideas such as random experiment, sample space, event, probability, conditional probability, independence, Bayes Theorem, random variable, probability distribution, expected value and variance, law of large numbers and central limit theorem. This paper provides an overview of key concepts and theorems from probability theory, along with their practical applications in everyday life, in order to make people deeply recognize the broad and profound contents and research fields of probability theory and its wide application in various fields as an academic tool, which has a great relation with people.

Mathematical Foundations of Real Numbers and its Application in Computation

The study of Real Numbers (R) is a fundamental pillar of Mathematical Analysis, serving as the cornerstone for a broad spectrum of Mathematical principles and practical applications. Real Numbers are examined both in theory and in practice, impacting fields like pure Mathematics, physics, engineering, and economics. This paper thoroughly explores Real Numbers, looking at their properties, how they are represented, and their significant role in Mathematical research and various applications. We highlight the importance of Real Numbers in advancing our understanding of Mathematics. This research article provides a comprehensive overview of the Mathematical foundations on Real Numbers and its application in computation. It explores historical developments, theoretical frameworks, Mathematical proofs, comparative analyses, applications, and future directions in the study of Real Numbers. The study aims to synthesize existing knowledge, highlight key contributions, theoretical framework and application in computation.

Preface to “Geometry and Topology with Applications”

Geometry is a very active research field in pure mathematics, with a history and tradition going back to the antiquity [...]

NOVEL APPROACHES FOR ADDRESSING EPISTEMIC UNCERTAINTY IN FRACTURE MECHANICS

This paper presents a novel optimisation approach for analysing cracked structural systems, consisting of four key steps. The first step (called the Golden Derivative) derives an equation at the discontinuity specific to cracked systems. The cracked system is transformed into an equivalent intact system for analysis in the second step. The third step converts the unique cracked system equation into a finite element equation, revealing epistemic uncertainty that leads to the formulation of Persian curves (PCs). These curves are developed using the following three methods: numerical experimentation, equivalent springs and logical reasoning, with consistent outcomes validating their effectiveness. The uncertainty is linked to the crack effect equation from fracture mechanics in the final step. It is shown by applying mathematical differentiation that this crack effect and classical fracture mechanics inherently involve epistemic uncertainty. A proposed remedy addresses this issue. The derivation of PC is based on pure mathematics and logical reasoning (bridging probabilistic and deterministic methods) making the results applicable across various fields for analysing real-world data.

Spinor–Vector Duality and Mirror Symmetry

Mirror symmetry was first observed in worldsheet string constructions, and was shown to have profound implications in the Effective Field Theory (EFT) limit of string compactifications, and for the properties of Calabi–Yau manifolds. It opened up a new field in pure mathematics, and was utilised in the area of enumerative geometry. Spinor–Vector Duality (SVD) is an extension of mirror symmetry. This can be readily understood in terms of the moduli of toroidal compactification of the Heterotic String, which includes the metric the antisymmetric tensor field and the Wilson line moduli. In terms of the toroidal moduli, mirror symmetry corresponds to mappings of the internal space moduli, whereas Spinor–Vector Duality corresponds to maps of the Wilson line moduli. In the past few of years, we demonstrated the existence of Spinor–Vector Duality in the effective field theory compactifications of string theories. This was achieved by starting with a worldsheet orbifold construction that exhibited Spinor–Vector Duality and resolving the orbifold singularities, hence generating a smooth, effective field theory limit with an imprint of the Spinor–Vector Duality. Just like mirror symmetry, the Spinor–Vector Duality can be used to study the properties of complex manifolds with vector bundles. Spinor–Vector Duality offers a top-down approach to the “Swampland” program, by exploring the imprint of the symmetries of the ultra-violet complete worldsheet string constructions in the effective field theory limit. The SVD suggests a demarcation line between (2,0) EFTs that possess an ultra-violet complete embedding versus those that do not.