Pure Mathematics Research Articles

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 [...]

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.

Comment: Paper on the progress of pure mathematics "proof of 3x + 1 conjecture"

The unresolved problem in number theory: the 3x+1 problem, deeply loved by math enthusiasts. I saw a paper titled "Proof of 3x+1 Conjecture" in the Journal of Pure Mathematical Progress (ISSN Print: 2160-0368), and its proof was incorrect.

“Imagine, maths is used anywhere, and we don’t get to know this”—upper secondary students and the relevance of advanced mathematics

People are more motivated to put effort into learning when they know they will be able to put the learnt content to use. These relevance perceptions play a motivating role in the learning of mathematics, a subject renowned for its abstraction, hard examinations, and usefulness in many fields in society (research, industries, etc.). In this article, we describe a study on upper secondary students in an advanced mathematics course and their perception of the relevance of mathematics in future professions, in particular regarding two concepts in their curriculum (logarithms, trigonometry). We defined relevance as a connection between an object (relevance of what?), a subject (relevant for whom?), an asserter (relevant according to whom?), and a purpose (relevant to what end?). The aim of the study was to know (1) what relevance perceptions students held regarding the advanced abstract mathematical concepts, and (2) how students can develop these considering that students do not yet know exactly what future is ahead of them. We interviewed pairs of students (n = 14, 17–19 years old) in two parts. The first part of the interview revealed that students learned mathematics within a traditional school culture that emphasized practice-and-drill of pure mathematics and that did not in any way inform students about the use of mathematics in research and workplaces. After exposing students to authentic applications of logarithms and trigonometry, the second part of the interview showed the power of imagination when it connected students to future professions for which mathematics was relevant. Instrumental in effectively prompting students’ imagination were visualizations showing applications of the mathematical concepts within workplace contexts. Drawing on Leont’ev’s version of Activity Theory, we theorize students’ assertions of the relevance of mathematics through the dialectics of self and collective (relating one’s own goals and more general motives), the dialectics of use-value and exchange-value (needing mathematics for later life or for examinations), and the dialectics of mathematization and de-mathematization (while mathematics is used in many workplaces, it is hidden in instruments).

Pareto’s legacy in the Italian tradition: the case of mathematical economics

Pareto’s personality had a great influence on the evolution of economic theory in the last century at least in the Italian context. At the beginning of the twentieth century a small but combative circle of followers aimed to defend and spread the new Paretian approach based on the idea of general economic equilibrium. Paretian scholars were well recognizable in the Italian context for their contributions to mathematical and statistical aspects. The purpose of this contribution is to outline the essential features of the Italian Paretian school in the field of pure economics or as we would say today of mathematical economics. This favorable position of the Italian tradition in the field of mathematical economics was basically due to two circumstances. The first was represented by the legacy of Pareto and his school while the second was to be found in the fact that among the students of Pareto there was a pure mathematician of great value Luigi Amoroso. We see then that an important voice even if poorly valued was that of Bruno de Finetti who published a remarkable series of articles on mathematical economics in the second half of the 1930s. After World War II the Paretian tradition in this field weakened considerably and effectively disappeared internationally.

IDSUML: Intrusion Detection System Using Machine Learning

Abstract: Network and system security is essential to modern digital communication. There will be multiple successful attempts by hackers and other intruders to obtain illegal access to networks and internet services. The development of secure systems is continuously accompanied by new dangers and the corresponding responses. Using intrusion detection systems (IDS) is one choice. An intrusion detection system. Main objective is to safeguard its resources from possible dangers. These acts are categorized as either potential or conventional attacks since it is capable of analysing and forecasting user behaviour. To find network intrusions, we typically employ Support Vector Machines (SVM) and Rough Pure Mathematics (RST). RST is used to preprocess and decrease the amount of data when packets are retrieved from the network. The RST-selected options provided to the SVM model enable it to provide many views and an explanation. A good application of the approach is to reduce the density of knowledge in a certain field. By comparing the data with Principal element analysis, the experiments show how RST and SVM schema may lower the false positive rate and increase accuracy (PCA)

Research on welding seam tracking algorithm for automatic welding process of X-shaped tip of concrete piles using laser distance sensor

AbstractThe manufacturing of the X-shaped tip of prestressed centrifugal concrete piles is nowadays done half automatically by combining the manual worker and the automatic welding robot. To make this welding process full automatically, the welding seam tracking algorithm is considered. There are many types of sensors that can be used to detect the welding seam such as vision sensor, laser vision sensor, arc sensor, or touch sensor. Each type of sensor has its advantages and disadvantages. In this paper, an algorithm for welding seam tracking using laser distance sensor is proposed. Firstly, the fundamental mathematics theory of the algorithm is presented. Next, the positioning table system supports the procedure is designed and manufactured. The object of this research is the fillet joint because of the characteristics of the X-shaped tip of the concrete piles. This paper proposes a new method to determine the welding trajectory of the tip using laser distance sensor. After that, the experimental results are received to verify the proposed idea. Finally, the improved proposal of the algorithm is considered to increase the accuracy of the suggested algorithm.