Mathematical Foundations Research Articles

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.

Predicting treatment plan approval probability for high-dose-rate brachytherapy of cervical cancer using adversarial deep learning

Objective. Predicting the probability of having the plan approved by the physician is important for automatic treatment planning. Driven by the mathematical foundation of deep learning that can use a deep neural network to represent functions accurately and flexibly, we developed a deep-learning framework that learns the probability of plan approval for cervical cancer high-dose-rate brachytherapy (HDRBT). Approach. The system consisted of a dose prediction network (DPN) and a plan-approval probability network (PPN). DPN predicts organs at risk (OAR) D 2cc and CTV D 90% of the current fraction from the patient’s current anatomy and prescription dose of HDRBT. PPN outputs the probability of a given plan being acceptable to the physician based on the patients anatomy and the total dose combining HDRBT and external beam radiotherapy sessions. Training of the networks was achieved by first training them separately for a good initialization, and then jointly via an adversarial process. We collected approved treatment plans of 248 treatment fractions from 63 patients. Among them, 216 plans from 54 patients were employed in a four-fold cross validation study, and the remaining 32 plans from other 9 patients were saved for independent testing. Main results. DPN predicted equivalent dose of 2 Gy for bladder, rectum, sigmoid D 2cc and CTV D 90% with a relative error of 11.51% ± 6.92%, 8.23% ± 5.75%, 7.12% ± 6.00%, and 10.16% ± 10.42%, respectively. In a task that differentiates clinically approved plans and disapproved plans generated by perturbing doses in ground truth approved plans by 20%, PPN achieved accuracy, sensitivity, specificity, and area under the curve 0.70, 0.74, 0.65, and 0.74. Significance. We demonstrated the feasibility of developing a novel deep-learning framework that predicts a probability of plan approval for HDRBT of cervical cancer, which is an essential component in automatic treatment planning.

Foundations of the wald space for phylogenetic trees

AbstractEvolutionary relationships between species are represented by phylogenetic trees, but these relationships are subject to uncertainty due to the random nature of evolution. A geometry for the space of phylogenetic trees is necessary in order to properly quantify this uncertainty during the statistical analysis of collections of possible evolutionary trees inferred from biological data. Recently, the wald space has been introduced: a length space for trees which is a certain subset of the manifold of symmetric positive definite matrices. In this work, the wald space is introduced formally and its topology and structure is studied in detail. In particular, we show that wald space has the topology of a disjoint union of open cubes, it is contractible, and by careful characterisation of cube boundaries, we demonstrate that wald space is a Whitney stratified space of type (A). Imposing the metric induced by the affine invariant metric on symmetric positive definite matrices, we prove that wald space is a geodesic Riemann stratified space. A new numerical method is proposed and investigated for construction of geodesics, computation of Fréchet means and calculation of curvature in wald space. This work is intended to serve as a mathematical foundation for further geometric and statistical research on this space.

Mathematical Foundations of Digital Twins (NSF)

Mathematical Foundations of Digital Twins (NSF)

A Practical Study on Enhancing Students’ Mathematical Literacy by “Integration of Doing, Learning and Teaching”

In face of the mathematical foundation of students in higher vocational colleges, Tao Xingzhi’s educational thought is of great guiding significance. “Integration of doing, learning and teaching” is not only the theoretical basis of Tao Xingzhi’s educational thought, but also his methodology and action guideline. Tao Xingzhi emphasizes the unity of knowledge and action, among which action is greater than knowledge. This concept is isomorphic with the mathematical culture curriculum. Drawing on the idea of “integration of doing, learning and teaching”, we explore the math culture curriculum and expand the four forms of teaching and learning in math culture: theoretical training, practical exploration, research-based learning, and reflective learning. In the implementation of the school-based curriculum, local teaching resources should be fully exploited, constructing the mathematics teaching mode of “Integration of doing, learning and teaching”, effective teaching methods and classroom evaluation algorithms, ensuring the relevance of vocational education and improving mathematical literacy.

Correlated Decision Fusion Accompanied with Quality Information on a Multi-Band Pixel Basis for Land Cover Classification.

Decision fusion plays a crucial role in achieving a cohesive and unified outcome by merging diverse perspectives. Within the realm of remote sensing classification, these methodologies become indispensable when synthesizing data from multiple sensors to arrive at conclusive decisions. In our study, we leverage fully Polarimetric Synthetic Aperture Radar (PolSAR) and thermal infrared data to establish distinct decisions for each pixel pertaining to its land cover classification. To enhance the classification process, we employ Pauli's decomposition components and land surface temperature as features. This approach facilitates the extraction of local decisions for each pixel, which are subsequently integrated through majority voting to form a comprehensive global decision for each land cover type. Furthermore, we investigate the correlation between corresponding pixels in the data from each sensor, aiming to achieve pixel-level correlated decision fusion at the fusion center. Our methodology entails a thorough exploration of the employed classifiers, coupled with the mathematical foundations necessary for the fusion of correlated decisions. Quality information is integrated into the decision fusion process, ensuring a comprehensive and robust classification outcome. The novelty of the method is its simplicity in the number of features used as well as the simple way of fusing decisions.

Enough Model Theory to Understand Skolem’s “Paradox”

This article seeks to describe some of the basic foundations of mathematics from a model-theoretic perspective with the goal of obtaining the “Downward Löwenheim–Skolem” theorem. After the theorem is introduced the seemingly paradoxical nature of it is discussed.

Mathematical Foundations of the Non-Hermitian Skin Effect

We study the skin effect in a one-dimensional system of finitely many subwavelength resonators with a non-Hermitian imaginary gauge potential. Using Toeplitz matrix theory, we prove the condensation of bulk eigenmodes at one of the edges of the system. By introducing a generalised (complex) Brillouin zone, we can compute spectral bands of the associated infinitely periodic structure and prove that this is the limit of the spectra of the finite structures with arbitrarily large size. Finally, we contrast the non-Hermitian systems with imaginary gauge potentials considered here with systems where the non-Hermiticity arises due to complex material parameters, showing that the two systems are fundamentally distinct.

Correcting for collider effects and sample selection bias in psychological research.

Colliders, variables that serve as a common outcome of an independent and dependent variable, pose a major challenge in psychological research. Collider variables can induce bias in the estimation of a population relationship of interest when (a) the composition of a research sample is restricted by scores on a collider variable or (b) researchers adjust for a collider variable in their statistical analyses, as they might do for confounder variables. Both cases interfere with the accuracy and generalizability of statistical results. Despite their importance, however, collider effects remain relatively unknown in psychology. This tutorial article summarizes both the conceptual and the mathematical foundation for collider effects and their relevance to psychological research, and then proposes a method to correct for collider bias in cases of restrictive sample selection based on Thorndike's Case III adjustment (1982). Two simulation studies demonstrated Thorndike's correction as a viable solution for correcting collider bias in research studies, even when restriction on the collider variable was extreme and the selected sample size was as low as N = 100. Bias and relative bias results are reported to evaluate how well the correction equation approximates targeted population correlations under a variety of parameter conditions. We illustrate the application of the correction method to a hypothetical study of intelligence and conscientiousness, discuss the applicability of the method to more complex statistical models as a means of detection for collider bias, and provide code for researchers to apply to their own research. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Тождество теоретического и экспериментального подходов при прототипировании региональных проектов в сфере здравоохранения.

The formalization of processes for finding the parameters of a prototype model of a medical organization implementing a regional project is based on the unity of theoretical and experimental approaches. The purpose of the study was an attempt to create a unified and harmonious theoretical foundation for conducting an experiment in order to find a prototype of a medical organization implementing projects at the regional level. In order to reveal the potential for solving technological problems facing project stakeholders, formalization of approaches was carried out for experimentally finding the parameters of a prototype project for a medical organization implementing a regional project. Materials and methods. The research materials were the works of Russian and foreign scientists related to the issues of mathematical modeling and prototyping. In order to study the research question, more than 30 literature sources were analyzed, directly or indirectly addressing the problems of practical application of object prototyping methods. Methods of system analysis, comparison, systemic and integrated approaches, as well as mathematical modeling methods were used. Results. Complex, multi-level systems when implementing projects in the regions are poorly structured and, accordingly, not fully formalized, so the experimental approach is especially relevant when searching for prototyping tools. The degree of formalization of approaches to searching for a prototype of a project object depends on the completeness (saturation) of processes, their structure and the degree of correlation of process data in a medical organization to process performers, operations, and management objects. Prototyping is designed to optimize the procedures for planning, analyzing and processing the results of research data based on a mathematical foundation. The prototyping technique is primarily aimed at saving resources (labor, financial, time) as much as possible, therefore, justification of the need and mastery of mathematical foundations in formalizing approaches to prototyping is a fundamental factor and the purpose of this discussion. Findings. System analysis using the unity of theoretical and experimental approaches, in the author’s opinion, will allow for a deeper synthesis of knowledge about the potential possibilities for forming decisions through a prototype of a medical organization implementing projects in the healthcare industry.

الفضاءات القابلة للتفكك والفضاءات غير القابلة للتفكك عبر فضاء القرب

اهتم كثيرا من الباحثين في دراسة اغلب المشكلات بعد تحويلها الى صيغ رياضية داخل فضاء القرب لسهولة توظيف هذاالفضاء مع المشكلات الحياتية المختلفة التي تواجهنا. لذلك يصب اهتمام البحث في ايجاد صفة جديدة الى فضاءات القرب باستخدام خواص القرب وخواص العناقيد وربطها معا للحصول على نقاط تراكمية جديدة اطلقنا عليها اسم نقاط كثه ومجموعة كل هذه النقاط اطلقنا عليها اسم مجموعة كثه حيث تم تكوين اساس رياضي متماسك الى الصيغة التبلوجية التي نبحث عنها والتي اطلقنا عليها التفكك واذا كانالفضاء لا يحمل هذه الصفة نطلق عليه فضاء غير قابل للتفكك.

Fractional Reasoning with Representation: Insights from Malaysia

Objective - Fractional reasoning is a crucial aspect of mathematical understanding fundamental in various mathematical concepts, real-world applications, and higher-level mathematical skills. Comprehending and working with fractions through various strategies, such as representation, is essential for students to develop a solid foundation in mathematics. However, fractional reasoning remains challenging in classroom teaching and learning since it requires deep understanding. Methodology/Technique – The current issue is a more comprehensive and conceptually grounded approach to foster a deeper acquisition of fractional reasoning strategies. Hence, this study aims to investigate to what extent primary school pupils develop fractional reasoning strategies to solve related problems, specifically for fractions of an area and fractions of a set of objects. A case study was conducted to interview eight primary school pupils from Perak (in Malaysia) for the data collection. Finding – The participants' solutions were observed to triangulate the interview data. In the content analysis, the identification of codes was carried out. Their findings revealed that the participants relied on strategies of representation methods of enactive and symbolic representations when working on fractions of an area. Novelty – This study introduces a novel perspective by emphasising that the identified fractional reasoning strategies are not isolated skills. The primary school pupils predominantly employed enactive and symbolic representations for fractions of an area, while favouring symbolic representations when reasoning fractions for a set of objects. These insights offer valuable guidance to educators, suggesting that a varied instructional approach, incorporating real-world contexts, can contribute to a more profound and versatile comprehension of fractions across diverse mathematical scenarios. Type of Paper: Empirical JEL Classification: I26, I29 Keywords: Representation, Enactive, Symbolic, Fractions, Fractional Reasoning Reference to this paper should be referred to as follows: Kamaruddin, N.I.H; Hoon, T.S; Hong, J.B.Z. (2024). Fractional Reasoning with Representation: Insights from Malaysia, GATR-Global J. Bus. Soc. Sci. Review, 12(1), 01–14. https://doi.org/10.35609/gjbssr.2024.12.1(1)

Set Theory and Third Mathematical Crisis: A Series of Events Reflected Upon the Mathematical Foundation

Set Theory and Third Mathematical Crisis: A Series of Events Reflected Upon the Mathematical Foundation

The Application of Knowledge Engineering via the Use of a Biomimetic Digital Twin Ecosystem, Phenotype-Driven Variant Analysis, and Exome Sequencing to Understand the Molecular Mechanisms of Disease

Applied Artificial Intelligence, particularly Large Language Models, in biomedical research is accelerating, but effective discovery and validation requires a toolset without limitations or bias. On January 30, 2023, the National Academies of Sciences, Engineering, and Medicine (NAS) appointed an ad hoc committee to identify needs and opportunities to advance the mathematical, statistical, and computational foundations of digital twins in applications across science, medicine, engineering, and society. On December 15, 2023, the NAS released a 164 page report, "Foundational Research Gaps and Future Directions for Digital Twins". This report described the importance of using digital twins in biomedical research. We developed an innovative method that incorporated phenotype ranking algorithms with knowledge engineering via a biomimetic digital twin ecosystem. This ecosystem applied real-world reasoning principles to non-normalized, raw data to identify hidden or "dark data". We performed a clinical exome sequencing study on patients with endometriosis and were able to identify four VUSs potentially associated with endometriosis-related disorders in nearly all patients analyzed. One VUS was identified in all patient samples and could be a biomarker for diagnostics. To the best of our knowledge, this is the first study to incorporate the recomandations of the NAS to biomedical research. This method can be used to understand the mechanisms of any disease, for virtual clinical trials, and to identify effective new therapies.

Lognormal distribution of daily travel time and a utility model for its emergence

Understanding the nature of human movement is important in many fields, such as transportation management, urban planning, and epidemiology. In this study, we focus on travelers who are grouped by their travel modes in the Tokyo Metropolitan Area. Notably, for each group of travelers, the distribution of daily travel time follows a lognormal distribution. For any travel mode, the distributions of normalized travel time for different years (1968–2008) and distances to the city center converge to a curve, although the average travel time and share of travel modes vary. By employing the concept of utility, we developed a disaggregate model wherein an individual maximizes utility from traveling (=benefit - cost) by modifying daily travel time. This model theoretically explained the observed distributions of travel time. The model was further used to demonstrate travel mode choices. The model parameters were estimated using the distribution of travel time, travel mode share, and manageable assumptions. These results provided interesting interpretations of the perceptual utility of the daily travel time of a population. Moreover, the obtained curves of utility versus time are similar to the shapes of the overall benefits assumed in previous studies. Overall benefit is a concept that has been developed and examined since the 1980s, and our study provides a mathematical foundation for this idea.