Pure Mathematics Research Articles

A Brief Survey on the Categorical Semantics for Geometry of Interaction In Memory of Phil Scott

Philip Scott, better known to his friends and colleagues as Phil, was a Professor of Mathematics at the University of Ottawa (Canada). Sadly, Phil passed away 1 after a long battle with cancer - which he bravely fought till the very end - on 18 December 2023 at the age of 76. Phil was born on 27 December 1947 in Leeds (UK). When Phil was still a baby, his family moved to North Carolina (USA), where he spent his childhood. He went on to study mathematics at the University of North Carolina at Chapel Hill (USA), before moving to Canada in the 1970s. Phil did his Ph.D. at the University of Waterloo (Canada), supervised by Denis Higgs, and obtained his doctorate in pure mathematics in 1976. Phil joined the Mathematics Department at the University of Ottawa in 1982, eventually becoming Professor Emeritus, and remained there until his passing.

OPTICAL ART OF A PLANAR IDEMPOTENT DIVISOR GRAPH OF COMMUTATIVE RING

The idempotent divisor graph of a commutative ring R is a graph with a vertex set in R* = R-{0}, where any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e^2 = e ∈ R, and is denoted by Л(R). Our goal in this work is to transform the planar idempotent divisor graph after coloring its regions into optical art by depending on the reflection of vertices, edges, and planes on the x or y-axes. That is, we achieve Op art solely through pure mathematics in this paper.

Evolution of the encryption and analysis of algorithm

This paper examines how cryptography has developed from the ancient kingdom to the present and examines what modern humans understand by an acceptable encryption scheme. The significance of encryption is evident; it is typically present in data, banks and governments in offline procedures, or VPNs in online ones. Additionally, the research shows how cryptography manifests itself in the world and highlights the confidentiality and universality of encryption techniques. However, the article also discusses the similarities and differences between pure mathematics and cryptography. It presents the two most widely used encryption techniques in computer science, RSA and Hush, and outlines their benefits to show why these techniques are the most often used as well as how challenging it is to comprehend and apply. It also provides some advice and recommendations to achieve the purpose of improving ability.

The Mathematical Heart of Tom Stoppard

Mathematics represents a conspicuous through-line in Tom Stoppard’s body of work. The allusions to the Riemann hypothesis in the recently produced Leopoldstadt (2020) form a notable bookend with the musings on probability in Rosencrantz & Guildenstern Are Dead (1966) that launched Stoppard’s career. In the intervening decades, the playwright’s ongoing self-education in mathematics surfaces in a series of cleverly staged debates about providence versus free will, the existence of moral absolutes, and the origins of human consciousness. Contrary to expectations, the mathematics in these plays most often serves as a catalyst for the persistence of uncertainty in a rising tide of rationalism. The decision to place a pure mathematician at the centre of Leopoldstadt, an autobiographically-inspired family epic about the Holocaust, reveals the depth of Stoppard’s fondness for mathematics and his faith in its potential to tell human stories. As it has in earlier plays, the mathematics in Leopoldstadt offers a distinctive commentary about the tension between rational order and randomness while also making a quiet case for the perseverance of beauty in the darkest of possible worlds.

The origins of frequency-difference and frequency-sum beamforming

Beamforming the signals recorded by an array enables the determination of sound source location(s) or the arrival directions of ray paths between a sound source and the receiving array. Frequency-difference and frequency-sum beamforming are beamforming techniques that provide out-of-band information from in-band signal frequencies. Interestingly, the out-of-band frequencies can be chosen by the user, within limits set by the signal recordings, to achieve desired properties of the beamformed output, such as: increased resolution, reduced sidelobes, or greater robustness to random scattering. Both techniques are general and are not limited to any particular acoustic environment, frequency range, or array geometry. Frequency-sum beamforming, generates higher-frequency information from lower frequency signal components, enhancing beamforming results in scenarios with random scattering between the source and the receivers. However, it is limited by artifacts arising from cross-terms when multiple source signals are present in the same bandwidth. Conversely, frequency-difference beamforming manufactures lower-frequency information from higher frequency signal components, effectively mitigating the impact of spatial aliasing in situations where the receiving array is sparse. This presentation delves into the origins of frequency-difference and frequency-sum beamforming, presents the fundamental mathematics underlying their algorithms, and showcases their performance via simulations and experimental results. [Work supported by ONR.]

The practical limits of packer testing

Packer testing allows estimation of the hydraulic conductivity of the ground through analysis of an induced flow rate into the ground under various controlled water pressures. Consideration of the physical and practical limits of injecting and monitoring such flows and pressures is used to determine the limits of hydraulic conductivity that can reasonably be estimated from packer tests. A design chart is presented based on these practical limits and consideration of the fundamental mathematics underpinning packer test interpretation. The chart is proposed as a useful reference when planning or reviewing field testing programs.

“Searching for order at all levels”. Antonio Lima-de-Faria (July 4, 1921 – December 27, 2023)

“Searching for order at all levels”. Antonio Lima-de-Faria (July 4, 1921 – December 27, 2023)

Impact of Intervention on Students’ Mastery Level and Analysis of Misconceptions in Operations on Integers

One of the fundamental mathematics concepts expected of a high school student to master is integers. However, many students exhibit misconceptions in operating integers which affect their performance as they advanced to the higher level of mathematics. The purpose of this study is to determine the impact of the intervention on students’ mastery level and analyze the misconceptions in operations on integers. In this study, a quasi-experimental one-group pretest-post-test design was employed. A total of fifty-six (56) grade 7 students were conveniently chosen of which twenty-six (26) were males and thirty (30) were females. Among these, twenty-nine (29) students belonged to Grade 7A, and twenty-seven (27) students belonged to Grade 7B. The data collected were analyzed using JAMOVI, an open statistical software. The Mean Percentage Score (MPS) was calculated to determine the mastery level exhibited by the students. Results revealed significant increase in the mastery level of grade 7 students after the intervention. In addition, there were no significant differences in the mastery level of the students in terms of the gender and section. Although the mastery level of the students improved, there were recurring misconceptions observed after the intervention. The researchers suggest further study evaluating the benefits of intervention in addressing the students' misconceptions in the four fundamental operations on integers.

Is deep learning a useful tool for the pure mathematician?

A personal and informal account of what a pure mathematician might expect when using tools from deep learning in their research.

Abstraction boundaries and spec driven development in pure mathematics

In this article we discuss how abstraction boundaries can help tame complexity in mathematical research with the help of an interactive theorem prover. While many of the ideas we present here have been used implicitly by mathematicians for some time, we argue that the use of an interactive theorem prover introduces additional qualitative benefits in the implementation of these ideas.

The signature of a monomial ideal

<p>The irreducible decomposition of a monomial ideal has played an important role in combinatorial commutative algebra, with applications beyond pure mathematics, such as biology. Given a monomial ideal $ I $ of a polynomial ring $ S = k[{\bf x}] $ over a field $ k $ and variables $ {\bf x} = \{x_1, \ldots, x_n\} $, its incidence matrix, is the matrix whose rows are indexed by the variables $ {\bf x} $ and whose columns are indexed by its minimal generators. The main contribution of this paper is the introduction of a novel invariant of a monomial ideal $ I $, termed its signature, which could be thought of as a type of canonical form of its incidence matrix, and the proof that two monomial ideals with the same signature have essentially the same irreducible decomposition.</p>

The Poetry of Pure Maths – and What It Means for your Business

The Poetry of Pure Maths – and What It Means for your Business

Commutativity of some Prime Nearrings using Left-Sided outer (σ, τ)-n-Derivation

In the field of mathematics, pure mathematics is very important as it gives rise to the basis for the formation of all applicable mathematical concepts in solving real-life problems. Algebra is such an integral part of pure mathematics. It consists of the Ring theory. It has been discovered that there exists some structures similar to rings with little deformity and they are called NEARRINGS. These structures do not commute mostly as they fail to satisfy distributive law. To ascertain the commutativity of nearrings, we need derivation(s). Because several papers had been presented dealing with left-nearrings, this paper aimed to consider right-nearrings which has not been done before in that respect, to the best of our knowledge. Some methods dealing with left-nearrings have been studied and modified in this work, and new derivation has been introduced to take care of right-nearrings. For Let

Neutrosophic 𝔑-Structures in Semimodules over Semirings

The study of symmetry is a fascinating and unifying subject that connects various areas of mathematics in the twenty-first century. Algebraic structures offer a framework for comprehending the symmetries of geometric objects in pure mathematics. This paper introduces new concepts in algebraic structures, concentrating on semimodules over semirings and analysing the neutrosophic structure in this context. We explore the properties of neutrosophic subsemimodules and neutrosophic ideals after defining them. We discuss, utilizing neutrosophic products, the representations of neutrosophic ideals and subsemimodules, as well as the relationship between neutrosophic products and intersections. Finally, we derive equivalent criteria in terms of neutrosophic structures for a semiring to be fully idempotent.

Optimizing Fundamental Mathematics Concepts for Middle School Students at Al Ulum Integrated Islamic School through Innovative Teaching Methods

The concept of innovative learning has become a major focus in today's education landscape. Innovative learning places an emphasis on relevant strategies and methods to address educational challenges. Two methods that play a significant role in innovative learning are Problem Based Learning (PBL) and Discovery Learning (DL). PBL centers on problem-solving based on real-life situations, while DL involves everyday experiences in finding solutions. At Al Ulum Integrated Islamic Middle School, the primary issue at hand is a high number of students receiving mathematics scores below the Minimum Mastery Criteria (KKM). Contributing factors include unengaging conventional teaching methods, leading to student boredom in mathematics classes. To address this problem, the PBL and DL methods are considered appropriate solutions. Furthermore, the implementation of this innovative learning concept is supported by information technology, both visually and non-visually, as well as the use of teaching aids to aid students in better understanding mathematical concepts. The implementation of the innovative learning concept is expected to resolve the challenges faced at Al Ulum Integrated Islamic Middle School. An evaluation will be conducted, and the outcomes will be published in Abdimas Talenta and online media. These steps are anticipated to provide sustainable benefits for both students and educators at the school.

Generalizing the eight levels theorem: a journey to Mersenne prime discoveries and new polynomial classes

Mersenne primes, renowned for their captivating form as 2 p − 1 , have intrigued mathematicians for centuries. In this paper, we embark on a captivating quest to unveil the intricate nature of Mersenne primes, seamlessly integrating methods with the Eight Levels Theorem. Initially, we extend the Eight Levels Theorem and introduce an innovative approach that harmoniously combines arithmetic and differential techniques to compute the coefficients of the polynomial expansions of x n + y n in terms of binary quadratic forms. This endeavor leads us to the genesis of novel polynomial sequences as we scrutinize the coefficients within this expansion. Our research unearths previously uncharted connections between Mersenne numbers and the derivatives of specific polynomial sequences. By forging this linkage, we not only enhance our understanding of Mersenne primes but also bridge the divide between well-established sequences in number theory and differential equations. This broadens the applicability of our findings across diverse scientific domains, revealing fresh avenues for exploration in number theory and beyond. These polynomial bridges serve as conduits between these sequences, unveiling exciting prospects for future research. This interdisciplinary exploration opens up exciting possibilities for the broader implications of Mersenne primes, extending their significance beyond the realm of pure mathematics. In our paper, we also delve into the intriguing influence of the Golden Ratio constant, which unveils segments reminiscent of the beauty found in nature, adding an unexpected dimension to the world of arithmetic. In this harmonious journey, the allure of Mersenne primes resonates through the symphony of mathematical discovery.

Algebraic Structures on Smooth Vector Fields

The aim of this work is to investigate some algebraic structures of objects which are defined and related to a manifold. Consider L to be a smooth manifold and Γ∞(TL) to be the module of smooth vector fields over the ring of smooth functions C∞(L). We prove that the module Γ∞(TL) is projective and finitely generated, but it is not semisimple. Therefore, it has a proper socle and nonzero Jacobson radical. Furthermore, we prove that this module is reflexive by showing that it is isomorphic to its bidual. Additionally, we investigate the structure of the Lie algebra of smooth vector fields. We give some questions and open problems at the end of the paper. We believe that our results are important because they link two different disciplines in modern pure mathematics.

Picture Fuzzy Subgroup

Picture fuzzy subgroup of a crisp group is established here and some properties connected to it are investigated. Also, normalized restricted picture fuzzy set, conjugate picture fuzzy subgroup, picture fuzzy coset, picture fuzzy normal subgroup and the order of picture fuzzy subgroup are defined. The order of picture fuzzy subgroup is defined using the cardinality of a special type of crisp subgroup. Some corresponding properties are established in this regard. Significant Statement. Subgroup is an important algebraic structure in the field of Pure Mathematics. Study of different properties of subgroup in fuzzy sense is an interesting fact to the readers because fuzzy sense is the extension of classical sense. Readers can easily observe how the properties of subgroup hold in fuzzy sense like classical sense. Picture fuzzy sense is the generalization of fuzzy sense. In other words, picture fuzzy sense can be treated as advanced fuzzy sense. Readers will be interested to study how the properties of subgroup hold when the number of components increases in fuzzy environment. Our study is actually the study of an important type of advanced fuzzy algebraic structure.

Understanding quantified statements in mathematics learning at the university

The objective is to provide knowledge for pure mathematics and mathematics education students to understand quantified statements so that they can easily comprehend the meaning of mathematical propositions and facilitate their proof. The method used in this research is a literature review by collecting various sources such as books or scientific writings related to the logic used to understand mathematics, specifically related to quantifiers, both universal and existential quantifiers. When mathematics and mathematics education students understand the quantified statement ∀x[Px], what they do is take any (cannot choose specific) x, and it must be proven that x satisfies property P. Then, to understand the quantified statement ∃x[Px], what is done is to choose (at least one) x, and it must be proven that x satisfies property P. When mathematics and mathematics education students understand the multi-quantifier statement ∀x∃y[P(x,y)], what is done is to take any (cannot choose specific) x, then find (at least one) y (depending on x), and x, y must satisfy property P. Then, to understand the multi-quantifier statement ∃y ∀x[P(x,y)] or ∃y [P(x,y)∀x], what is done is to choose y (independent of x) and it must be proven that y satisfies property P along with all x.

A mathematical model for simulating photoacoustic signal generation and propagation in biological tissues

Photoacoustic (PA) medical imaging is a crossbred technique relying on light-induced ultrasonic waves due to the PA effect phenomenon recorded primarily in 1880 by A. G. Bell. Numerical simulation, also known as in-silico, studies assist scientists in minimizing incorrect PA experiments in both in-vitro and in-vivo. Numerical modeling techniques help to achieve a fast simulation process in contrast to pure mathematics alone. However, if a suitable facilitated mathematical model can be established prior to applying numerical modeling, it will be of great interest to the whole numeric model. Numerous mathematical equations, theorems, and propositions have been proposed to model the whole PA signal generation and propagation process in biological media. However, most of them are complicated and difficult to be understood by researchers, especially beginners. That’s why this paper was introduced. Our paper aims to simplify the understanding of the generation and propagation process of biomedical PA waves. We have developed a facilitated mathematical model for the entire process. The introduced developed mathematical model is based on three steps: (1) pulsed laser stimulation, (2) light diffusion, and (3) PA stress wave generation and propagation. The developed mathematical model has been implemented utilizing COMSOL Multiphysics, which relies on the finite element method (FEM) numerical modeling principle. The in-silico time-dependent study's results confirmed that the proposed mathematical model is a simple, efficient, accurate, and quick starting point for researchers to simulate biomedical PA signals' generation and propagation process utilizing any suitable software such as COMSOL multiphysics.