Mathematical Fundamentals Research Articles

A unified-description of curvature, torsion, and non-metricity of the metric-affine geometry with the Möbius representation

We establish the mathematical fundamentals for a unified description of curvature, torsion, and non-metricity 2-forms in the way extending the so-called Möbius representation of the affine group, which is the method to convert the semi-direct product into the ordinary matrix product, to revive the fertility of gauge theories of gravity. First of all, we illustrate the basic concepts for constructing the metric-affine geometry. Then the curvature and torsion 2-forms are described in a unified manner by using the Cartan connection of the Möbius representation of the affine group. In this unified-description, the curvature and torsion are derived by Cartan’s structure equation with respect to a common connection 1-form. After that, extending the Möbius representation, the dilation and shear 2-forms, or equivalently, the non-metricity 2-form, are introduced in the same unified manner. Based on the unified-description established in this paper, introducing a new group parametrization and applying the Inönü–Wigner group contraction to the full theory, the relationships among symmetries, geometric quantities, and geometries are investigated with respect to the three gauge groups: the metric-affine group and its extension, and an extension of the (anti)-de Sitter group in which the non-metricity exists. Finally, possible applications to theories of gravity are briefly discussed.

Practice and Exploration of Integrating Ordinary Differential Equations into Course Ideology and Politics

Currently, ordinary differential equation courses in various universities in China generally emphasize the teaching of mathematical fundamentals, while neglecting the infiltration and application of ideological and political elements. With the active development of ideological and political education in university courses, universities should deeply explore the ideological and political elements contained in different types of courses, and the same applies to ordinary and differentiated courses. Based on this, this article discusses the importance of integrating ideological and political education into ordinary differential equation courses, and proposes two practical methods for developing courses: mathematical thinking and practical life integration, in order to provide reference and assistance for integrating ideological and political education into ordinary differential equation courses.

A COMPUTATIONAL AND FRACTIONAL MATHEMATICAL MODEL FOR ASSESSING HEPATITIS B WITH OPTIMAL THERAPY

This paper presents a mathematical study applied to Hepatitis B disease utilizing fractional calculus and numerical methods. We begin by discussing mathematical fundamentals, focusing on fractional calculus concepts and relevant numerical aspects for disease modeling. We then formulate the problem, describing a nonlinear fractional differential system that models hepatitis B virus (HBV) infection and its response to drug therapy. A detailed theoretical analysis explores system properties and biological implications. We provide a qualitative analysis of the model, discussing important dynamic behaviors such as stability and parameter sensitivity. Subsequently, numerical analysis is performed to validate the model and estimate unknown parameters from patient data. Our findings include significant insights into biological parameters and their impact on therapeutic response. We conclude that the proposed fractional-order model is effective, offering optimal therapeutic strategies for hepatitis B.

SOCRATIC METHOD OF TUTORING: AN INNOVATION FOR BETTER STUDENTS’ NUMERACY SKILLS

This action research sought to determine the students numeracy skills before and after using the Socratic Method of Tutoring (SMT). This exploration also measured the effect of SMT on the students mathematical operations of numbers. Additionally, the study investigated the challenges encountered by the peer tutors in handling their peers numeracy skills and their coping mechanisms in addressing the problems they experienced. The study used sequential explanatory research. The research participants consisted of thirty (30) students from the ABM strand and eighteen (18) peer tutors from the STEM strand of Zamboanga del Sur National High School – Senior High during the school year 2022-2023. The students obtained below-normal scores before the utilization of SMT, but the performance of the students in numeracy tests significantly improved after the intervention. The drastic shift in students numeracy skills proved the effectiveness of the Socratic Method of Tutoring to the students. The Peer tutors experienced four (4) major problems or challenges: (1) the peers lack of mathematical fundamentals, (2) the distracted peers focus, (3) insufficient knowledge as tutors, and (4) management of time constraints as tutors. Further, the peer tutors articulated their four (4) coping mechanisms: (1) reviewing and reteaching the competencies to their Peers, (2) being patient towards their Peers, (3) self-preparation of the topic as tutors, and (4) proper time management as tutors. The findings of this research support the conduct of a learning program to capacitate the students in their numeracy skills and enhance the pedagogical skills of the Peer Tutors of the school. KEYWORDS: Socratic Method of Tutoring; Numeracy Skills; Peer Tutors

A IMPORTÂNCIA DO ENSINO DE SEQUÊNCIAS NUMÉRICAS MATEMÁTICAS: ESTRATÉGIAS E ABORDAGENS

This article explores the importance of teaching mathematical number sequences in the context of mathematics education. It highlights the relevance of number sequences as a fundamental tool for developing logical thinking, pattern identification skills and an understanding of the fundamentals of mathematics. It then discusses effective strategies for teaching number sequences, including practical approaches, visual and technological resources. Finally, the importance of an interactive and contextualized approach that promotes student engagement and facilitates the transfer of knowledge to real-world situations is emphasized.

Network Calculus Theory and Application: A New Graduate Course for Engineering Students

With the development of modern networks, the quality of service guarantee has become even more important than before. However, various new types of traffic and heterogeneous network architecture make it quite challenging to evaluate the quality of service, such as delay and reliability. Therefore, a novel analysis method is needed to solve this situation. Network calculus is a theory used to analyze the queueing problems in networks, which is the main content to be taught in this graduate course. Before network calculus theory is introduced, the classical queueing theory will be taught. Then, the mathematical fundamentals of network calculus, such as min-plus algebra and cumulative arrival/service process, will be introduced. Next, two branches of network calculus, i.e., deterministic network calculus and stochastic network calculus, will be introduced separately, including the definition of deterministic/stochastic arrival curve and service curve, as well as important theorems, such as delay theorem, flow aggregation theorem, server concatenation theorem. After the theoretical knowledge, practical content will be taught in two parts. First, students are encouraged to establish system models based on their graduate research topics, analyze the service guarantee of their models, and present their work in class. Second, a traffic data collection platform will be taught, and students will collect various traffic data and analyze traffic characteristics based on network calculus theory. Based on the feedback from students during the past three years, students are satisfied with this new course because it focuses on frontier technology and the teaching activities are well-organized.

Unlocking Mathematical Success: A Qualitative Case Study on Grade 10 Learners’ Challenges and Triumphs with the DAMATH Strategy

Mathematics remains a challenging subject for students, necessitating innovative teaching strategies to enhance engagement and understanding. This study investigates the role of the DAMATH strategy, a game-based learning approach, in boosting academic achievement and cognitive engagement among Grade 10 learners. Using a constructivist theoretical framework, the study explores the issues and concerns students face when implementing DAMATH, their coping mechanisms, and the insights gained. Qualitative data from semi-structured interviews with six purposively selected participants reveal three main challenges: lack of understanding and strategic knowledge, the complexity of mathematical operations, and the pressure of problem-solving to achieve victory. Effective coping strategies identified include consistent practice, focusing on mathematical fundamentals, and collaborative learning. Students reported improvements in cognitive and problem-solving abilities, as well as enhanced mathematical skills, through regular DAMATH gameplay. The study concludes that DAMATH is an effective educational tool that promotes cognitive growth, critical thinking, and mathematical proficiency. Moreover, by integrating DAMATH into the curriculum, educators can enhance students’ learning experiences, making mathematics more accessible and engaging. These insights inform future educational strategies and suggest areas for further research on game-based learning and its broader impacts on student motivation and engagement.

Community-scale models of microbiomes: Articulating metabolic modelling and metagenome sequencing.

Building models is essential for understanding the functions and dynamics of microbial communities. Metabolic models built on genome-scale metabolic network reconstructions (GENREs) are especially relevant as a means to decipher the complex interactions occurring among species. Model reconstruction increasingly relies on metagenomics, which permits direct characterisation of naturally occurring communities that may contain organisms that cannot be isolated or cultured. In this review, we provide an overview of the field of metabolic modelling and its increasing reliance on and synergy with metagenomics and bioinformatics. We survey the means of assigning functions and reconstructing metabolic networks from (meta-)genomes, and present the variety and mathematical fundamentals of metabolic models that foster the understanding of microbial dynamics. We emphasise the characterisation of interactions and the scaling of model construction to large communities, two important bottlenecks in the applicability of these models. We give an overview of the current state of the art in metagenome sequencing and bioinformatics analysis, focusing on the reconstruction of genomes in microbial communities. Metagenomics benefits tremendously from third-generation sequencing, and we discuss the opportunities of long-read sequencing, strain-level characterisation and eukaryotic metagenomics. We aim at providing algorithmic and mathematical support, together with tool and application resources, that permit bridging the gap between metagenomics and metabolic modelling.

Mathematical fundamentals of spherical kinematics of plate tectonics in terms of quaternions

To be a quantitative and testable tectonic model, plate tectonics requires spherical geometry and spherical kinematics in terms of finite rotations conveniently parametrized by their angle and axis and described by unit quaternions. In treatises on “Plate Tectonics” infinitesimal, instantaneous, and finite rotations, absolute and relative rotations are said to be applied to model the motion of tectonic plates. Even though these terms are strictly defined in mathematics, they are often casually used in geosciences. Here, their definitions are recalled and clarified as well as the terms rotation, orientation, and location on the sphere. For instance, infinitesimal rotations refer to a mathematical limit, when the angle of rotation tends to zero. Their rules do not apply to finite rotations, no matter how small their finite angles of rotation are. Mathematical approaches applying appropriate and feasible assumptions to model spherical motion of tectonic plates over geological times of hundreds of millions of years are derived including (i) sequences of incremental finite rotations, (ii) sequences of accumulating successive concatenations of finite rotations, and (iii) continuous rotations in terms of fully transient quaternions. The incremental and the accumulating approaches provide complementary views. While the relative Euler pole appears to migrate in the latter, it appears fixed in the former. Path, mean, and instantaneous velocity of the migrating Euler pole are derived as well as the angular and trajectoral velocity of the rotational motion about it. The approaches are illustrated by a geological example with actual data and a numerical yet geologically inspired example with artificial data. The former revisits the three‐plate scenario with stationary axes of two “absolute” rotations implying transient “relative” rotations about a migrating Euler pole and employs a proper plate‐circuit argument to determine them numerically without resorting to approximations. The latter applies an involved interplay of incremental and accumulating modeling inducing split–join cycles to approximate sinusoidal trajectories as reported to record plates' motion during the Gondwana breakup.

Comparison of different blade shapes of translatory moving soil tillage tools

BACKGROUND: The efficiency and energy capacity of translatory moving soil tillage tools are largely determined by the parameters of their blades. If the parameters do not match the specified operating conditions, clogging the working tools with plant residues becomes possible, and a less energy-consuming type of soil cutting — sliding — may be replaced with a more energy-consuming type — chopping cutting. In this regard, it is very important for scientists engaged in improving the designs of translatory moving soil tillage tools to have a method for assessing the shape of the blades.
 AIMS: Development of a method for comparative assessment of various blade shapes of translatory moving soil tillage tools.
 METHODS: For this study, the fundamentals of agricultural mechanics, higher mathematics, and theoretical mechanics were used. The objects of the study were the blades of various soil tillage tools, providing with sliding cutting of soil. The study was conducted at the Gorsky State Agrarian University in the period from 2022 to 2023. The calculations results were compared with the materials of the corresponding standards.
 RESULTS: The method that helps to determine the relative values and directions of the resultant force of soil cutting resistance with a translatory moving blade of a soil tillage tool having a diverse shape has been developed. An example of the application of the method for assessment the effectiveness of blades of a garden tool, the cutting part of which is described by the equation of a straight line, a circle, an Archimedes spiral, a logarithmic spiral, as well as an example of assessment of a curved blade for soil tillage tools of cultivators are given.
 CONCLUSIONS: The proposed method for assessing the influence of the shape of a translatory moving flat blade of the soil tillage tools on the value of soil cutting resistance forces is easy to use and can be the basis for further development in order to analyze the effectiveness of the knives’ blade shape performing a plane-parallel movement.

Regularized Optimal Transport Layers for Generalized Global Pooling Operations.

Global pooling is one of the most significant operations in many machine learning models and tasks, which works for information fusion and structured data (like sets and graphs) representation. However, without solid mathematical fundamentals, its practical implementations often depend on empirical mechanisms and thus lead to sub-optimal, even unsatisfactory performance. In this work, we develop a novel and generalized global pooling framework through the lens of optimal transport. The proposed framework is interpretable from the perspective of expectation-maximization. Essentially, it aims at learning an optimal transport across sample indices and feature dimensions, making the corresponding pooling operation maximize the conditional expectation of input data. We demonstrate that most existing pooling methods are equivalent to solving a regularized optimal transport (ROT) problem with different specializations, and more sophisticated pooling operations can be implemented by hierarchically solving multiple ROT problems. Making the parameters of the ROT problem learnable, we develop a family of regularized optimal transport pooling (ROTP) layers. We implement the ROTP layers as a new kind of deep implicit layer. Their model architectures correspond to different optimization algorithms. We test our ROTP layers in several representative set-level machine learning scenarios, including multi-instance learning (MIL), graph classification, graph set representation, and image classification. Experimental results show that applying our ROTP layers can reduce the difficulty of the design and selection of global pooling - our ROTP layers may either imitate some existing global pooling methods or lead to some new pooling layers fitting data better. The code is available at https://github.com/SDS-Lab/ROT-Pooling.

Dual-functional metalenses for the polarization-controlled generation of focalized vector beams in the telecom infrared

The availability of static tiny optical devices is mandatory to reduce the complexity of optical paths that typically use dynamic optical components and/or many standard elements for the generation of complex states of light, leading to unprecedented levels of miniaturization and compactness of optical systems. In particular, the design of flat and integrated optical elements capable of multiple vector beams generation with high resolution in the visible and infrared range is very attractive in many fields, from life science to information and communication technology. In this regard, we propose dual-functional transmission dielectric metalenses that act simultaneously on the dynamic and geometric phases in order to manipulate independently right-handed and left-handed circularly polarized states of light and generate focused vector beams in a compact and versatile way. In the specific, starting from the mathematical fundamentals for the compact generation of vector beams using dual-functional optical elements, we provide the numerical algorithms for the computation of metaoptics and apply those techniques to the design and fabrication of silicon metalenses which are able to generate and focus different vector beams in the telecom infrared, depending on the linear polarization state in input. This approach provides new integrated optics for applications in the fields of high-resolution microscopy, optical manipulation, and optical communications, both in the classical and single-photon regimes.

Fractional-Order Control Techniques for Renewable Energy and Energy-Storage-Integrated Power Systems: A Review

The worldwide energy revolution has accelerated the utilization of demand-side manageable energy systems such as wind turbines, photovoltaic panels, electric vehicles, and energy storage systems in order to deal with the growing energy crisis and greenhouse emissions. The control system of renewable energy units and energy storage systems has a high effect on their performance and absolutely on the efficiency of the total power network. Classical controllers are based on integer-order differentiation and integration, while the fractional-order controller has tremendous potential to change the order for better modeling and controlling the system. This paper presents a comprehensive review of the energy system of renewable energy units and energy storage devices. Various papers are evaluated, and their methods and results are presented. Moreover, the mathematical fundamentals of the fractional-order method are mentioned, and the various studies are categorized based on different parameters. Various definitions for fractional-order calculus are also explained using their mathematical formula. Different studies and numerical evaluations present appropriate efficiency and accuracy of the fractional-order techniques for estimating, controlling, and improving the performance of energy systems in various operational conditions so that the average error of the fractional-order methods is considerably lower than other ones.

Research-driven education: An introductory course to systems and synthetic biology

Systems and Synthetic Biology are complementary fields emerging side-by-side into mainstream scientific research. Whilst systems biologists focus on understanding natural systems, synthetic biologists wish to modify, adapt and re-purpose biological systems towards certain desired goals, for example enhancing efficiency and robustness of desired biological traits. In both fields, data analysis, predictive mathematical modelling, experimental design, and controlled experimentation are crucial to obtain reproducible results and understand how applications can be scaled to larger systems and processes. As such, students from Life Sciences, Engineering, and Mathematics backgrounds must be taught fundamentals in biological systems, experimental techniques, mathematics, and data analysis/statistics. In addition, students must be trained for future multidisciplinary careers, where the interaction and communication between experimental and modelling researchers is fundamental. With the acceleration of technological developments (both computational and experimental) continuing unabated, educators need to bridge the increasing gap between fundamentally-required knowledge and skills that students need to pursue future academic or industrial research projects. In this paper, we will discuss how we have re-designed an introductory course in Systems and Synthetic Biology at Wageningen University and Research (Netherlands) that is targeted simultaneously to mathematical/computational students with an interest in biology and experimental methods, and to Life Science students interested in learning how biological systems can be mathematically analysed and modelled. The course highlights the links between fundamental methodologies and recently developed technologies within the Systems and Synthetic Biology fields. The course was re-designed for the 2021/22 academic year, we report that students from biology and biotechnology programmes graded their satisfaction of the course as 4.4 out of 5. We discuss how the course can act as a gateway to advanced courses in Systems Biology-oriented curricula (comprising: data infrastructure, modelling, and experimental synthetic biology), and towards future research projects.

Determination method for structuring specialists skilled in motor transport in the region

Introduction. The existing methodology for determining quantitative indicators of training specialists for motor transport complexes of the regions, taking into account the technological needs of economic entities, does not reflect the rules for the formation of structural characteristics of security in specialists and staff of motor transport. This methodology needs to be further developed to identify structural indicators of the human resources system. The main indicator of this structure is the ratio of the number of specialists by training levels. This article fills the gap and aims to improve the structural indicators of the training of motor transport specialists and staff. This is due to the relevance of developing an appropriate methodology.Materials and methods. Classical methods of analysis, synthesis, structural modelling, provisions of the theory of motor vehicles maintenance, methods of mathematical analysis and fundamentals of algorithms are used in the work. The obtained results are applicable for their use in improving specialists and staff training in organisations and enterprises of motor transport, regardless of their departmental subordination.Results. The main result of the work is a methodology for determining the structure of specialists skilled in motor transport in the region, differing from the known ones that it takes into account the distribution of the number of the skilled staff according to the levels of vocational higher and secondary education between motor transport and car service enterprises, regardless of the departmental subordination. The method of structural ranking of the specialists skilled in motor transport enterprise management has been developed. These results represent the scientific novelty of the study. The results of the structural distribution of specialists by levels of education have been determined, the corresponding algorithm has been developed.Discussion and conclusions. A methodological tool for determining the structure of specialists with motor transport education in the region has been developed. The application of the results of the work will make it possible to increase the validity of the state task to improve the training of specialists and motor transport staff in the regions.

Architectural and mathematical fundamentals of improvement neural networks for classification of images

The article proposes ways to solve the problem of structural synthesis of neural networks architectures; based on non-traditional approaches to their analytical formalization and application of new ones data processing operations. The example of the CIFAR10 dataset proves the possibility of improving the accuracy of the solution image classification tasks within an alternative architecture with expandable input and narrowing structures of trained neural networks. The results of the application of input expanding image taken with Resizing and Conv2DTranspose layers at the input trained neural networks indicate the effectiveness of solving such classification problems by example CIFAR10 dataset. Achieved on the basis of the Xception neural network, the average classification accuracy of 10 classes CIFAR10 images are 97.3%. The side effect of pre-scaling images is leveling the accuracy of the classification of different classes, which allows us to consider such a change in size as an option data augmentation in the dataset. Tensor-matrix methods have been introduced for further development of this approach formalization of the description of neural networks on the basis of the penetrating end product of matrices and its block ones modifications. On this basis, a number of new convolution and maxpooling operations are proposed, as well as combinations on input of the narrowing segment not only symmetrically enlarged image, but also its variants obtained on the basis of a generalized penetrating product. It is about drawing images expanded by rows of pixels (horizontally) and columns (vertically), as well as a combination of several different options symmetrical image extensions on the principle of constructing a pyramidal segment of the PSPNet neural network.

A Survey of Current Balloon Trajectory Prediction Technology

High-altitude scientific balloons allow for very little control over their trajectories. Although systems exist to influence the burst altitude of the balloons, their horizontal motion is dictated solely by the vagrancies of the wind. The accurate prediction of balloon trajectories is thus a topic of great importance to flight operations personnel, since launch/no-launch decisions must be made based on the likelihood of the balloon landing in some undesirable location. Such concerns are particularly acute in areas with high population densities or significant areas of restricted airspace, both of which are prevalent in the Mid-Atlantic region. This paper presents an overview of the present state of balloon trajectory prediction. The mathematical fundamentals of balloon trajectory prediction are discussed, as are the major available software implementations.

Comprehensive Direct Georeferencing of Aerial Images for Unmanned Aerial Systems Applications.

Optical image sensors are the most common remote sensing data acquisition devices present in Unmanned Aerial Systems (UAS). In this context, assigning a location in a geographic frame of reference to the acquired image is a necessary task in the majority of the applications. This process is denominated direct georeferencing when ground control points are not used. Despite it applies simple mathematical fundamentals, the complete direct georeferencing process involves much information, such as camera sensor characteristics, mounting measurements, attitude and position of the UAS, among others. In addition, there are many rotations and translations between the different reference frames, among many other details, which makes the whole process a considerable complex operation. Another problem is that manufacturers and software tools may use different reference frames posing additional difficulty when implementing the direct georeferencing. As this information is spread among many sources, researchers may face difficulties on having a complete vision of the method. In fact, there is absolutely no paper in the literature that explain this process in a comprehensive way. In order to supply this implicit demand, this paper presents a comprehensive method for direct georeferencing of aerial images acquired by cameras mounted on UAS, where all required information, mathematical operations and implementation steps are explained in detail. Finally, in order to show the practical use of the method and to prove its accuracy, both simulated and real flights were performed, where objects of the acquired images were georeferenced.

Aplicación interactiva para el aprendizaje con evaluaciones en el área de las matemáticas

The following document presents the development process of an interactive mobile application, as a support strategy for the teaching - learning process in the area of mathematics, for students of the Educational Experience (EE), Fundamentals of Mathematics at the Faculty of Statistics and Informatics. Using User Centered Design (UCD), which proposes to put the client at the center of research and design, to obtain an application that achieves a user experience (UX), satisfactory, when interacting with the application, allowing greater speed in the completion of tasks, which is simple to learn and intuitive, this to increase its pleasure and ease of use for the beneficiary, obtaining these results with usability tests to the application. All this because currently many applications are known as support in the area of mathematics, which have usability problems obtaining a bad UX. This work follows a user-centered methodology to create usable applications and a good UX for university students in the area of mathematics.

Mathematical Approaches in Functional Safety Assessment for E/E/PE Safety-Related Software

Safety integrity level (SIL)-based functional safety assessment is widely required in designing safety functions and checking their validity of electrical/electronic/programmable electronic (E/E/PE) safety-related systems after being issued IEC 61508 in 2010. For the hardware of E/E/PE safety-related systems, quantitative functional safety assessment based on target failure measures is needed for deciding or allocating the level of SIL. On the other hand, IEC 61508 does not provide any quantitative safety assessment method for allocating SIL for the software of E/E/PE safety-related systems because the software failure is treated as a systematic failure in IEC 61508. We discuss the needfulness of quantitative safety assessment for software of E/E/PE safety-related systems and propose mathematical fundamentals for conducting quantitative SIL-based safety assessment for the software of E/E/PE safety-related systems by applying the notion of software reliability modeling and assessment technologies. We show numerical examples for explaining how to use our approaches.