Pure Mathematics Research Articles

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

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.

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.

Stable STFT Phase Retrieval and Poincaré Inequalities

Abstract In recent work [P. Grohs and M. Rathmair. Stable Gabor Phase Retrieval and Spectral Clustering. Communications on Pure and Applied Mathematics (2018) and P. Grohs and M. Rathmair. Stable Gabor phase retrieval for multivariate functions. Journal of the European Mathematical Society (2021)], the instabilities of Gabor phase retrieval problem, that is, reconstructing $ f\in L^{2}(\mathbb{R})$ from its spectrogram, $|\mathcal{V}_{g} f|$ where $$ \begin{align*} & \mathcal{V}_g f(x,\xi) = \int_{\mathbb{R}} f(t)\overline{g(t-x)}e^{-2\pi i \xi t}\,\mbox{d}t, \end{align*} $$ have been classified in terms of the connectivity of the measurements. These findings were however crucially restricted to the case where the window $g(t)=e^{-\pi t^{2}}$ is Gaussian. In this work we establish a corresponding result for a number of other window functions including the one-sided exponential $g(t)=e^{-t}{1\kern-3.4pt1}_{[0,\infty )}(t)$ and $g(t)=\exp (t-e^{t})$. As a by-product we establish a modified version of Poincaré’s inequality, which can be applied to non-differentiable functions and may be of independent interest.

Qualitative (pure) mathematics as an alternative to measurement.

This paper focuses on the possible usage of qualitative mathematics in psychology. Qualitative mathematics is understood to be equivalent to pure mathematics. First, it is explained that mathematics is a discipline studying patterns in reproducible mental objects. Qualitative mathematics is presented as an alternative to measurement, potentially offering the same level of exactness, clarity, and rigor. This perspective might lead psychologists to explore connections between a phenomenon and any kind of mathematical structure, regardless of whether the structure is quantitative. Usage of (any) mathematical structures might require scholars who are familiar with them. Consequently, changes in mathematics education may also be needed. Introducing non-numerical structures into mathematics education-thereby partially revisiting the New Math Movement-could train individuals more prepared for a creative approach to the use of structures and less inclined to view everything as quantitative.

An analysis of learners' mental constructions in learning and solving trigonometric equations

Learners' construction of mathematical concepts is an issue of global concern. As a result, scholars advocate that mathematical topics identified as problematic for learners should be analyzed through research. This paper reports on an APOS (action, processes, objects, and schema) analysis of Grade 11 learners' mental constructions of trigonometric equations. The aim is to reveal the nature of the learners' mental constructions rather than provide a statistical analysis of their performance. This qualitative study was conducted with 17 purposively selected Grade 11 learners doing pure mathematics. The data used in this study were generated from learners' written responses and semi-structured interviews. Using the constructs of APOS theory, our analysis of learners' mental constructions revealed that most learners are operating at the action stage, which means that their knowledge construction of trigonometric equations is at the developmental stage. We further observed that learners needed external cues to trigger their thinking of how they should go about solving the trigonometric equation. All the procedural steps were explicitly written down, meaning they did not internalized the process of solving trigonometric equations. Furthermore, our analysis revealed that while learners can carry out procedures, meaning they can perform step-by-step procedures, they struggle to integrate concepts. The struggle to integrate concepts emanates from not constructing the concept image of trigonometric equations but rather constructing the concepts as isolated facts. Based on the findings, we argue that for learners to develop a schema for trigonometric equations, instructional strategies must focus on helping learners construct the concept image of trigonometric equations. Construction of the concept image will help learners start to internalize the procedures, thus leading to the interiorization of action to process and then encapsulate the process to an object stage.

Generic reasoning: A programmatic sketch

Abstract A single significant instance may support general conclusions, with possible exceptions being tolerated. This is the case in practical human reasoning (e.g. moral and legal normativity: general rules tolerating exceptions), in theoretical human reasoning engaging with external reality (e.g. empirical and social sciences: the use of case studies and model organisms) and in abstract domains (possibly mind-unrelated, e.g. pure mathematics: the use of arbitrary objects). While this has been recognized in modern times, such a process is not captured by current models of supporting general conclusions. This paper articulates the thesis that there is a kind of reasoning, generic reasoning, previously unrecognized as an independent type of reasoning. A theory of generic reasoning explains how a single significant instance may support general conclusions, with possible exceptions being tolerated. This paper will adopt, as a working hypothesis, that generic reasoning is irreducible to currently recognized kinds of ‘pure’ reasoning. The aim is to understand generic reasoning, both theoretically and in its applications.

AI-driven research in pure mathematics and theoretical physics

AI-driven research in pure mathematics and theoretical physics

Comparison of Vector Tasks in Indonesian and Singaporean Mathematics Textbooks

Vector is one of the most important subjects in mathematics, but many students face difficulties in learning vector. To find out the cause, this study is conducted to find the differences and similarities in learning opportunities presented in the vector tasks from Indonesian and Singaporean mathematics textbooks. Therefore, vector tasks from the two books are compared with the aspects of representations, contextuality, and response types. Firstly, this study finds out that the chosen Indonesian textbook provides fewer learning opportunities than the chosen Singaporean textbook in terms of vector tasks. Secondly, this study also finds out that the Indonesian textbook tends to present tasks in visual representation, while the Singaporean textbook tends to present tasks in pure mathematics form. Thirdly, this study also finds out that Indonesian textbook usually uses applicative tasks that correlate with real-life situations, while the Singaporean textbook does the otherwise. Lastly, this study also finds out that both textbooks mostly use close-ended tasks.

Mentoring Undergraduate Research with Institutional Datasets: Growth Mindset for Pure Mathematicians

Mentoring Undergraduate Research with Institutional Datasets: Growth Mindset for Pure Mathematicians

Passages

Richard W. Hamming's The Art of Doing Science and Engineering: Learning to Learn is one of the true essentials, a book to be read by every scientist or engineer (and certainly software engineer) interested in the big picture; it may be safely skipped by those who aim only at a life and career where one has "merely survived and amused" oneself, as Hamming himself puts it in his final, summative, chapter. The essence of this book is a combination of anecdote and aphorism drawn from a mature, active mind, reflecting on a great career spent at great places (Los Alamos, and then Bell Labs in its heyday, to be precise). There is also a great deal of mathematics, but the aphorism, backed by well-chosen anecdote(s), is fundamental; you do not need to know or care much about differential equations (and there are many here) to benefit from this book. It is hard to convey the value of the anecdotes, in a short review. The aphorisms, however, can be more than mentioned; what follows is, in reverse order, the entire set of Hamming's bolded, inset, aphorisms, by chapter (combining multiple chapters on the same topic, e.g., simulation). I have truncated one long item, which offers up a description of pure mathematics; you will have to read the book to get that one. The focus on aphorism is appropriate; I think Hamming himself would have agreed that this entire wonderful book is a kind of elaboration and proof of Pasteur's "Luck favors the prepared mind", which Hamming quotes not once, not twice, but seven times.

Utilizing Game Development Life Cycle Method to Develop an Educational Game for Basic Mathematics Using Unity 2D Game Engine

The rapid advancement of technology has profoundly impacted various aspects of human life, including education. While technology offers numerous advantages, such as fostering interactive and innovative learning experiences, the shift to online learning during the Covid-19 pandemic has highlighted some challenges, including a decline in students' motivation and engagement, exacerbated by distractions like video games. However, amidst these challenges, Digital Game-Based Learning (DGBL) has emerged as a promising approach to engage students and enhance learning outcomes. This research aims to develop a basic mathematical game using the Unity game engine to enrich children's learning experiences. The educational game seeks to support educators in fostering active engagement among children, specifically in fundamental mathematics. The game is developed utilizing the Game Development Life Cycle (GDLC) methodology.

Applying Pure Mathematics: IMPA and the Entanglements of Mathematical Economics in Brazil

This article studies the entanglements of mathematical economics in Brazil by delving into the emergence of a scientific community identified with the subject at the Instituto de Matemática Pura e Aplicada (IMPA). It traces how networks built since IMPA's early years in the 1950s made economics a viable alternative to applied mathematics in an institute mostly dedicated to research in pure mathematics. The article examines how interests in mathematical economics remained active during IMPA's expansion beginning in the late 1960s. This process culminated with the creation of the mathematical economics graduate program at IMPA in 1980 by Aloisio Araujo. The article follows how the program could build strong connections to the Escola de Pós-Graduação em Economia (EPGE), an important department of economics in Brazil, helping to shape recent developments in the center. The article thus aims to reveal how historical connections and entwinements of economics and mathematics groomed at IMPA have affected recent developments in the Brazilian economics community.

Perfect Codes over Non-Prime Power Alphabets: An Approach Based on Diophantine Equations

Perfect error-correcting codes allow for an optimal transmission of information while guaranteeing error correction. For this reason, proving their existence has been a classical problem in both pure mathematics and information theory. Indeed, the classification of the parameters of e-error correcting perfect codes over q-ary alphabets was a very active topic of research in the late 20th century. Consequently, all parameters of perfect e-error-correcting codes were found if e≥3, and it was conjectured that no perfect 2-error-correcting codes exist over any q-ary alphabet, where q>3. In the 1970s, this was proved for q a prime power, for q=2r3s and for only seven other values of q. Almost 50 years later, it is surprising to note that there have been no new results in this regard and the classification of 2-error-correcting codes over non-prime power alphabets remains an open problem. In this paper, we use techniques from the resolution of the generalised Ramanujan–Nagell equation and from modern computational number theory to show that perfect 2-error-correcting codes do not exist for 172 new values of q which are not prime powers, substantially increasing the values of q which are now classified. In addition, we prove that, for any fixed value of q, there can be at most finitely many perfect 2-error-correcting codes over an alphabet of size q.