Mathematical Foundations Research Articles

An Analytical Reliability Performance Assessment of Cell-Free Massive MIMO Systems for 5G and Beyond Cellular Networks

In today‟s realm, the 5G cellular networks offer a new era of fully digital environment by connecting people anywhere anytime. This era requires inspiring demands to the networks in terms of spectral and energy efficiencies, low-latency and ultra-reliable communications. A major development to deal with these unparalleled demands is the introduction of Cell- Free Massive MIMO (CF mMIMO) systems to the cellular network. In recent years, mathematical foundations have been laid for a CF mMIMO communication system to evaluate the spectral and energy efficiencies given the number of access points. Various existing literatures have adopted mathematical models to formulate closed-form expressions for these efficiencies using uncorrelated Rayleigh fading for simple linear processing detectors. In this paper, we try to address a compelling and interesting question which has not received much attention lately. The question is how to evaluate the reliability of a connected user terminal in both a CF mMIMO and conventional mMIMO system? We answer this question by first analyzing the signal models for both systems assuming exact channel estimates and consider an uncorrelated Rayleigh fading channel model. The exact Bit Error Rate, as a reliability assessment, in closed-form are then derived based on strong mathematical foundations, which are simple to numerically evaluate in Matlab R2022, for MPSK and MQAM modulation schemes. The mathematical models presented in this work can be used in 5G and Beyond 5G networks to evaluate the distributions of error rates for a user given the number of Access points or Base Station antennas and modulation scheme. Machine Learning algorithms can be applied to the analytical expressions with a view to predict the error performance of a user terminal given the fading characteristic of the propagation environmen

Representing Infinite Periodic Hyperbolic Delaunay Triangulations Using Finitely Many Dirichlet Domains

The Delaunay triangulation of a set of points P on a hyperbolic surface is the projection of the Delaunay triangulation of the set P~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{P}$$\\end{document} of lifted points in the hyperbolic plane. Since P~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{P}$$\\end{document} is infinite, the algorithms to compute Delaunay triangulations in the plane do not generalize naturally. Using a Dirichlet domain, we exhibit a finite set of points that captures the full triangulation. We prove that an edge of a Delaunay triangulation has a combinatorial length (a notion we define in the paper) smaller than 12g-6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$12g-6$$\\end{document} with respect to a Dirichlet domain. To achieve this, we introduce new tools, of intrinsic interest, that capture the properties of length-minimizing curves in the context of closed curves. We then use these to derive structural results on Delaunay triangulations and exhibit certain distance minimizing properties of both the edges of a Delaunay triangulation and of a Dirichlet domain. The bounds produced in this paper depend only on the topology of the surface. They provide mathematical foundations for hyperbolic analogs of the algorithms to compute periodic Delaunay triangulations in Euclidean space.

Hilbert transform view of water-wave theory

A general rethinking of the mathematical foundations of water surface waves from the perspective of the Hilbert transform uncovers shortcomings of the standard multiple-scale approach as well as elucidates the interplay of non-local and dispersive effects. Application of the Hilbert transforms to planar and cylindrical settings allows us to deduce new weakly nonlinear models, including an alternative to Zakharov's equation and an envelope equation for cylindrical waves on deep water, as well as to highlight the crucial differences between these geometries.

The density–density response function in time-dependent density functional theory: Mathematical foundations and pole shifting

We establish existence and uniqueness of the solution to the Dyson equation for the density–density response function in time-dependent density functional theory (TDDFT) in the random phase approximation (RPA). We show that the poles of the RPA density–density response function are forward shifted with respect to those of the noninteracting response function, thereby explaining mathematically the well-known empirical fact that the noninteracting poles (given by the spectral gaps of the time-independent Kohn–Sham equations) underestimate the true transition frequencies. Moreover, we show that the RPA poles are solutions to an eigenvalue problem, justifying the approach commonly used in the physics community to compute these poles.

Cutting-edge tool wear monitoring in AISI4140 steel hard turning using least square-support vector machine

ABSTRACT This research addresses the critical challenge of tool wear monitoring in AISI4140 steel hard turning through the innovative application of a Least Square-Support Vector Machine (LS-SVM) prediction model. While acknowledging the growing significance of hard turning over conventional grinding techniques, the background also recognizes the challenges posed by accelerated tool wear and decreased productivity. The main goal is to create a reliable tool wear prediction model that is tailored to harsh turning circumstances. Material selection, cutting force measurement, acceleration measurement, and accurate tool wear measurement are all included in the experimental procedures. Using time and frequency domain analyses, the research methodically examines how cutting force and acceleration affect tool wear. For feature selection, non-linear regression is used to determine which parameters have the greatest influence on wear. A comprehensive training and validation procedure, structural clarification, and a detailed mathematical foundation are used in the development of the LS-SVM-based tool wear prediction model. The presentation of the findings and related discussions explores the impact of acceleration and cutting force on wear. The LS-SVM model’s successful application for precise tool wear prediction in AISI4140 steel hard turning is highlighted in the conclusions, demonstrating the model’s potential to improve tool lifespan and manufacturing efficiency.

LINEAR ALGEBRA AS THE MATHEMATICAL FOUNDATION OF ARTIFICIAL INTELLIGENCE: CONCEPTS, APPLICATIONS, AND FUTURE PROSPECTS

Linear algebra, a fundamental tool in the field of mathematics, plays an essential part in the mathematical foundation of the groundbreaking arena of Artificial Intelligence (AI). This mathematics branch offers immense possibilities for capturing complex relationships within datasets through matrices and vectors. These mathematical tools facilitate the concise representation of large volumes of data, contributing to efficient data analysis between them, bringing immense value to AI algorithms and models. This research paper investigates the pivotal role of linear algebra as the fundamental mathematical framework underpinning artificial intelligence. This paper also explores the foundational concepts of linear algebra and its direct applications in various AI algorithms, demonstrating how matrices, vectors, and transformations provide the essential mathematical scaffolding for the development and optimization of intelligent systems.

Empowering Students’ Comprehension in Calculus with eCALculator

Calculus is a fundamental subject in STEM (Science, Technology, Engineering, and Mathematics) study programs. Despite the introduction of basic calculus concepts as early as secondary school, many undergraduate students still encounter difficulties in comprehending and solving differentiation and integration problems. As calculus is fundamental to fields like science and engineering, possessing a solid mathematical and calculus foundation becomes essential for STEM students. Struggling to grasp these fundamental concepts not only impacts performance in calculus but also poses challenges when tackling advanced related topics. Recognizing this challenge, the eCALculator is introduced as a comprehensive and accessible solution. This innovative tool takes the form of a step-by-step e-manual, guiding students in determining a function’s derivatives and integrals using a calculator. The e-manual can be accessed via a flipbook, encompasses instructional videos and exercises. The eCALculator was tested to 40 engineering students enrolled in Calculus courses at UiTM Cawangan Pulau Pinang. Survey results indicate that the eCALculator not only helps improving students’ comprehension and accuracy in calculus problem-solving but also increases motivation and interest in learning, particularly among low-achieving mathematics students who require additional support.

Random walk theory and application

This project presents an overview of Random Walk Theory and its applications, as discussed in the provided project work. Random Walk Theory posits that changes in elements like stock prices follow a distribution independent of past movements, making future predictions challenging. Originating from the work of French mathematician Louis Bachelier and later popularized by economist Burton Markiel, the theory finds extensive applications beyond finance, spanning fields such as psychology, economics, and physics. The project delves into various types of random walks, including symmetric random walks, and explores their implications in different spaces, from graphs to higher-dimensional vector spaces. It provides definitions, examples, and graphical representations to elucidate random walk concepts, highlighting their relevance in practical scenarios like particle movement and stock price fluctuations. Key concepts such as the reflection principle and the main lemma are discussed to provide a comprehensive understanding of random walks and their properties. Through examples and lemmas, the project elucidates the mathematical foundations of random walks, offering insights into their behavior and applications across diverse disciplines. In summary, this project contributes to a deeper comprehension of Random Walk Theory, serving as a fundamental framework for understanding stochastic processes and their real-world implications.

Utilizing Cubic B-Spline Collocation Technique for Solving Linear and Nonlinear Fractional Integro-Differential Equations of Volterra and Fredholm Types

Fractional integro-differential equations (FIDEs) of both Volterra and Fredholm types present considerable challenges in numerical analysis and scientific computing due to their complex structures. This paper introduces a novel approach to address such equations by employing a Cubic B-spline collocation method. This method offers a robust and systematic framework for approximating solutions to the FIDEs, facilitating precise representations of complex phenomena. Within this research, we establish the mathematical foundations of the proposed scheme, elucidate its advantages over existing methods, and demonstrate its practical utility through numerical examples. We adopt the Caputo definition for fractional derivatives and conduct a stability analysis to validate the accuracy of the method. The findings showcase the precision and efficiency of the scheme in solving FIDEs, highlighting its potential as a valuable tool for addressing a wide array of practical problems.

Entropy, Deterministic Chaos, and New Forms of Intelligibility: A Shared Frame of Reference for Physics and Psychology

Prigogine’s theory of dissipative structures provides a general account of entropy-driven self-organized transitions through hierarchies of structures separated by discontinuities. The theory encompasses a wide range of evolving systems throughout nature and culture. Possibilities for operationalizing a new collective rationality spanning physics and psychology emerge from Prigogine’s emphases on two distinct senses of probability, on the concept of the sufficient statistic, and on the role and limitations of the Poisson distribution in formulating a “nonlinear master equation.” Unnoted by Prigogine are correspondences of all three of these issues in the mathematical foundations of statistics and measurement established in the works of Ronald Fisher and his student, Georg Rasch. The three areas of correspondence inform models enabling specifically metrological approaches to quality-assured quantification across the sciences. Prigogine’s sense of “deterministic chaos” is re-expressed in measurement terms as stochastic invariance and the need for “a supplementary parameter” augmenting the Poisson distribution is related to a rating scale model of measurement. In light of these connections, this paper proposes that what Prigogine anticipates as a “new intelligibility” and a new science of “collective rationality” could be pragmatically operationalized in a new metrological infrastructure made coherent by the generality of entropy-driven nonequilibrium processes.

Exploring nine different proofs of a famous logarithmic inequality

This article explores and analyzes nine different proofs of a fundamental logarithmic inequality, say "log(1+x)/x ≥ 1/ (1+x/2) for x>-1". Some proofs are already published; others are new or consider new approaches or new angles of research. They are based on various techniques, such as differentiation, series expansion, and the application of well-known inequalities such as the primitive, Cauchy-Schwarz, logarithmic mean, hyperbolic tangent function, Jensen, and Hermite-Hadamard inequalities. A graphical work illustrates some results. Therefore, our study clarifies the different mathematical foundations of this seemingly straightforward but important inequality and goes beyond it with some new material.

Real space iterative reconstruction for vector tomography (RESIRE-V)

Tomography has had an important impact on the physical, biological, and medical sciences. To date, most tomographic applications have been focused on 3D scalar reconstructions. However, in some crucial applications, vector tomography is required to reconstruct 3D vector fields such as the electric and magnetic fields. Over the years, several vector tomography methods have been developed. Here, we present the mathematical foundation and algorithmic implementation of REal Space Iterative REconstruction for Vector tomography, termed RESIRE-V. RESIRE-V uses multiple tilt series of projections and iterates between the projections and a 3D reconstruction. Each iteration consists of a forward step using the Radon transform and a backward step using its transpose, then updates the object via gradient descent. Incorporating with a 3D support constraint, the algorithm iteratively minimizes an error metric, defined as the difference between the measured and calculated projections. The algorithm can also be used to refine the tilt angles and further improve the 3D reconstruction. To validate RESIRE-V, we first apply it to a simulated data set of the 3D magnetization vector field, consisting of two orthogonal tilt series, each with a missing wedge. Our quantitative analysis shows that the three components of the reconstructed magnetization vector field agree well with the ground-truth counterparts. We then use RESIRE-V to reconstruct the 3D magnetization vector field of a ferromagnetic meta-lattice consisting of three tilt series. Our 3D vector reconstruction reveals the existence of topological magnetic defects with positive and negative charges. We expect that RESIRE-V can be incorporated into different imaging modalities as a general vector tomography method. To make the algorithm accessible to a broad user community, we have made our RESIRE-V MATLAB source codes and the data freely available at https://github.com/minhpham0309/RESIRE-V.

METHODOLOGICAL FEATURES OF TEACHING RANDOM VARIABLES AT THE COMPREHENSIVE SECONDARY SCHOOL

In the evolving landscape of educational methodologies, the integration of complex mathematical concepts such as random variables into secondary school curriculums poses significant challenges and opportunities. As the global educational framework increasingly emphasizes the importance of a robust mathematical foundation, understanding and teaching stochastic elements within the school system has become crucial. This article delves into the nuanced methodological features essential for imparting knowledge of random variables effectively to secondary school students. We explore the historical context of mathematical education reforms in Armenia, highlighting the recent inclusion of stochastic topics in the school programs since 2004. The reformed educational standards introduced in 2020 and implemented in 2023 have further emphasized the need for a structured approach to teaching random variables, alongside other advanced mathematical topics. Through this lens, we examine the preparedness of educators, the adequacy of existing teaching methods, and the potential cognitive hurdles faced by students. Our analysis is rooted in both qualitative assessments and a theoretical framework that aims to bridge the gap between abstract mathematical theories and practical understanding, ensuring that students not only learn but also appreciate the significance of probability theory in everyday contexts .

Experimental Observation of the Yang-Lee Quantum Criticality in Open Quantum Systems.

The Yang-Lee edge singularity was originally studied from the standpoint of mathematical foundations of phase transitions. However, direct observation of anomalous scaling with the negative scaling dimension has remained elusive due to an imaginary magnetic field required for the nonunitary criticality. We experimentally implement an imaginary magnetic field with an open quantum system of heralded single photons, directly measure the partition function, and demonstrate the Yang-Lee edge singularity via the quantum-classical correspondence. We also demonstrate unconventional scaling laws for finite-temperature quantum dynamics.

Utilization of Teacher-Made Learning Activity Sheets as a Tool in Improving Learners’ Numeracy Skills: An Action Research

Developing learners’ numeracy skills has been a pressing issue in the field of education, as it is one of the battlecries of the agency along with literacy. Studies revealed that addressing these learning gaps varies widely across schools, classrooms, and learners. Moreover, students learn better through the activities that their teachers provide to them. Thus, under the Mathematics Foundation Fluency Program, a numeracy intervention program of Langkaan II National High School, teacher-made learning activity sheets were used as learning materials of the learners under numeracy intervention. The objective of this study is to determine the effectiveness of utilizing teacher-made learning activity sheets in improving the numeracy skills of the learners particularly in Langkaan II National High School. The findings of the study will serve as a help in improving the numeracy intervention program of the school. The results revealed that there was a marked improvement in the numeracy skills of the learners as manifested in the improved scores of the learners during the post-test. Moreover, no non-numerates were reported during the end-of-the-school year numeracy assessment. Thus, using the teacher-made learning activity sheets were found to be an effective tool in improving the numeracy skills of the learners including number sense, number operations, measurement, geometry, patterns and algebra, and statistics and probability. It helped the students to cope up certain topic/lessons in basic mathematics that develops their numeracy skills. These materials can contribute to a more comprehensive program to improve the numeracy skills of the learners. Further studies may also be conducted as to other aspects relevant to the implementation of the whole program. Validation or modification of these learning materials may also be conducted for further enhancement.

Teachers Difficulty and Coping Strategies in Physics

This study investigates the difficulties encountered by physics educators and their coping strategies. Physics teaching presents various challenges such as addressing student misconceptions and integrating real-life examples. Understanding these challenges is crucial for developing effective coping strategies and enhancing physics education quality. Research suggests that addressing these challenges can improve teacher competency and student engagement. Investigating teachers' difficulties and coping strategies is vital for advancing the field and achieving successful learning outcomes. Ten teachers from public and private schools in Sultan Kudarat province participated in the study and were selected through purposive sampling. Data were collected using semi-structured interviews and focus group discussions with careful adherence to ethical standards. Results revealed challenges including teacher competency, students' weak mathematical foundations, and low retention. Coping mechanisms involved instructional preparation, peer mentoring, and real-life application of physics concepts. Despite the hurdles, educators exhibit resilience through various coping mechanisms, emphasizing the importance of targeted interventions and support systems. By fostering collaboration, emphasizing real-world applications, and providing ongoing professional development, educators can enhance the quality of physics education and facilitate meaningful learning experiences for students.

Image Morpher

The "Image Morphing with the Beier-Neely Method" research paper extensively explores image transformation techniques, with a special focus on the Beier- Neely approach. Image morphing, the art of seamlessly transforming one image into another, has garnered significant attention for its applications in animation, filmmaking, and scientific visualization. The research paper delves into the historical context and importance of image morphing, particularly emphasizing the role of the Beier-Neely method in addressing the limitations of earlier techniques. Classical image morphing methods, characterized by pixel-level interpolation, suffer from visual artifacts, manual correspondence definition, and sluggishness. These shortcomings set the stage for the introduction of the feature-based Beier-Neely method, which revolutionized image morphing by allowing control points on features, facilitating smooth transitions and natural transformations. This research papers objectives are clearly articulated, with the primary aim of implementing the Beier- Neely method to produce high-quality image morphs. The research paper proposes a system architecture that underscores the use of feature-based control points, Delaunay triangulation, affine transformations, and the significance of visual feedback. Additionally, it provides a detailed analysis of the mathematical foundations and algorithmic intricacies of the Beier-Neely method, elucidating the theoretical framework driving its effectiveness. Overall, the research paper provides a comprehensive exploration of image morphing techniques, highlighting the pivotal role played by the Beier-Neely method in advancing the field.. Key Words: Image Morphing, Beier-Neely Method, Seamless Transitions, Feature-based Control Points, System Architecture, Mathematical Foundations

Iterative kernel regression with preconditioning

Kernel methods are popular in nonlinear and nonparametric regression due to their solid mathematical foundations and optimal statistical properties. However, scalability remains the primary bottleneck in applying kernel methods to large-scale data regression analysis. This paper aims to improve the scalability of kernel methods. We combine Nyström subsampling and the preconditioned conjugate gradient method to solve regularized kernel regression. Our theoretical analysis indicates that achieving optimal convergence rates requires only [Formula: see text] memory and [Formula: see text] time (up to logarithmic factors). Numerical experiments show that our algorithm outperforms existing methods in time efficiency and prediction accuracy on large-scale datasets. Notably, compared to the FALKON algorithm [A. Rudi, L. Carratino and L. Rosasco, Falkon: An optimal large scale kernel method, in Advances in Neural Information Processing Systems (Curran Associates, 2017), pp. 3891–3901], which is known as the optimal large-scale kernel method, our method is more flexible (applicable to non-positive definite kernel functions) and has a lower algorithmic complexity. Additionally, our established theoretical analysis further relaxes the restrictive conditions on hyperparameters previously imposed in convergence analyses.

Transferability of Learned Knowledge in Neural Networks: Impact of Trained Weights on Untrained Networks

This research explores the intricate dynamics of neural networks (NNs) with a focus on understanding the profound implications of the training process on their performance. In a departure from conventional transfer learning, this study delves into the manual initialization of untrained neural networks with weights and biases extracted from trained networks of identical architecture. Despite their plain congruence, we meticulously investigate the nuanced distinctions between trained and untrained networks and analyze the disparities in their performance. This study introduces mathematical foundations, empirical findings, and implications that underscore the significance of the training process in the realm of neural networks. Applying the ideas of weight extraction, this research is inspired by the unique process of visual learning and “mirroring” seen in humans; the ability to mimic something simply by seeing the methods involved.

An Integrative Review of Portfolio Management

This paper delves into the evolution of portfolio theories, tracing their development from the seminal work of Markowitz's traditional portfolio theory to the contemporary landscape of modern portfolio management. Emphasizing the relevance for real-world investors, the study scrutinizes the advantages, applications, and limitations of these investment theories. The exploration commences with a comprehensive analysis of return, risk, and various factors integral to portfolio theory. Leveraging qualitative analysis and real-world examples, the paper elucidates the mathematical model underpinning the theory, furnishing investors with a structured investment strategy. The focal points of the paper are the VaR (Value at Risk) model and the beta coefficient. Through an in-depth examination of their mathematical foundations and theoretical underpinnings, the research elucidates the symbiotic relationship between their economic significance and mathematical intricacies. Furthermore, the paper expounds on their collective contribution to the overarching portfolio theory, shedding light on their role as guiding principles for risk-averse investors. By weaving together theoretical frameworks and practical implications, this paper seeks to provide a nuanced understanding of portfolio theories, equipping investors with valuable insights for informed decision-making in the dynamic landscape of financial markets.