Mathematical Theory Research Articles

Tracking System for Living Beings and Objects: Integration of Accessible Mathematical Contributions and Graph Theory in Tracking System Design

This paper presents a theoretical framework for a tracking system, wherein we generalize the formulation of a tracking system de- signed for living beings and objects. Many tracking systems are typically developed within specific frameworks, either for tracking in limited or unlimited space. The latter often relies on technical tools dedicated to tracking living beings or objects. In this study, we propose a system theory that formulates the task of tracking both living beings and ob- jects. Graphical modeling is widely employed in tracking to establish correct connections between the elements to be tracked and other com- ponents in the system. However, basing a tracking system on graphs in both its theoretical and practical aspects remains the optimal method for achieving a high-performing, relevant, and adaptable system in vari- ous situations. This paper introduces a tracking system based on graph learning and hypergraphs, fully leveraging direct and indirect relations while considering the order between multiple system links. Tracking is thus formulated as a search problem on graphs and hypergraphs, with vertices representing the elements of the system (living beings or ob- jects), and edges representing the types of connections between these elements. We define a law governing the relationships between the ver- tices, managing the shared data between the elements of the system and other processes. Furthermore, examples of single and multi-context track- ing situations demonstrate that the proposed system, in its theoretical foundation, outperforms existing systems.

Thermal Runaway in Battery Cells: Guiding Principles for Continuum Scale Modeling

Lithium-ion batteries are sensitive to temperature. Low-temperature or high-temperature operation may lead to lithium plating and dendrites, SEI decomposition, and separator collapse. Such phenomena may cause internal short circuits, rapid heat generation and, subsequently, may trigger thermal runaway, resulting in fire or explosion [1][2]. Hence, accurate modeling and prediction of the thermal response of battery cells inside the pack is critical to preventing safety hazards as well as optimizing battery performance. Battery thermal management system (BTMS) plays a significant role in maintaining an optimal operation temperature and ensuring thermal safety for electric vehicles [3]. However, the large number of battery cells inside one module makes it challenging for BTMS algorithms to realize real-time simulations based on fully resolved physics-based models. The conventional lumped capacitance (LC) thermal model achieves reduced computation costs under the hypothesis that temperature is spatially uniform. Nevertheless, such a model works better for small Biot number conditions, usually under low C-rate operation and moderate cooling scenarios, and it lacks justification under a broad range of battery operating conditions and materials [4]. To overcome such limitations, rigorous multiscale models, based on mathematical coarse-graining theories, have been developed to predict the thermal response of batteries at the module scale [5] while not compromising on computational cost. In this work, we investigate the accuracy of the classical LC thermal model for twenty potential battery modules constructed with commercial batteries and packing materials. We show that the classical LC thermal model may be inaccurate under a broad range of design conditions. In the context of multiscale modeling, we identify which effective models should be instead used, and analyze the impact of SOC and the relative strength of heat insulation and heat dissipation on model selection and accuracy. This work provides guidance, based on first principles and rigorous multiscale theory, on best practices for model selection in practical applications involving thermal behavior and thermal runaway in battery cells at the module scale.

Research on the static and dynamic tensile damage mechanism of concrete based on fuzzy sets

The quantitative analysis of microscopic damage mechanisms in concrete and the formulation of damage evolution equations using mathematical theory and computerized tomography (CT) imaging are still emerging fields. The test data from CT scans have yet to be fully exploited. This study proposes a method for quantitatively characterizing mesoscale damage evolution in concrete. By segmenting CT images from static and dynamic tensile tests, we define metrics such as the aggregate rate, mortar rate, and porosity to examine the damage progression under both stress conditions. We then establish a damage evolution model based on porosity changes. Our findings reveal that crack growth in concrete is slower under static tension, with cracks navigating around aggregates towards weaker regions. Conversely, under dynamic loads, cracks propagate more rapidly and directly through aggregates, following paths of quicker energy dispersion. Static loading results in less aggregate damage than does dynamic loading, with a notable increase in porosity and a more varied change in the mortar rate. The damage to the finer components of concrete increases as the crack-adjacent region diminishes. Damage is generally more extensive under static loading than under dynamic loading. The proposed porosity-based damage evolution model effectively describes concrete damage under static tension, leveraging CT technology to enhance the universality of the research and offering a microscopic-level theoretical foundation for studying concrete mechanical properties.

The “Galenic Question”: A Solution Based on Historical Sources and a Mathematical Analysis of Texts

How many different writers authored the huge number of texts attributed to Galen of Pergamum (129~216 Anno Domini (AD)), medical doctor and philosopher, a giant in the history of medicine? The quest to find out which texts were his and which ones were written by others is known as the “Galenic Question”. We propose a “solution” to it through a multidisciplinary approach based (a) on historical research and (b) on a mathematical analysis of the Greek texts. The historical approach considers historical independent sources and anachronisms. The mathematical approach is based on a mathematical theory concerning deep language variables, rarely consciously controlled by any author, and is therefore capable of giving indications on the similarity of texts, with little or no bias. The multidisciplinary approach has convinced us that at least three authors wrote the texts attributed to Galen. The first two were very likely real historical persons: (a) a certain Galen living between the end of the I century Before Christ (BC) and the second half of the I century AD, and (b) the historical Galen of Pergamum (II–III centuries AD). We believe the third (c) to be represented by several unknown authors hiding under the name Galen, but likely living after Galen of Pergamum’s death.

Uncertainty measures: A critical survey

Classical probability is not the only mathematical theory of uncertainty, or the most general. Many authors have argued that probability theory is ill-equipped to model the ‘epistemic’, reducible uncertainty about the process generating the data. To address this, many alternative theories of uncertainty have been formulated. In this paper, we highlight how uncertainty theories can be seen as forming clusters characterised by a shared rationale, are connected to each other in an intricate but interesting way, and can be ranked according to their degree of generality. Our objective is to propose a structured, critical summary of the research landscape in uncertainty theory, and discuss its potential for wider adoption in artificial intelligence.

Simulation of the impact of an ultrasonic drill as a composition of a robotic platform for investigation of soil on celestial bodies

The article discusses the theoretical issues of developing an ultrasonic drill for drilling soil on celestial bodies as part of a robotic platform and formulates the tasks of its control. The specifics of using the drill require a wide temperature range during operation, low energy consumption, an efficient operating cycle and the requirement to ensure the preservation of the structure and composition of the soil when exposed to the tool. One of the promising areas of development is devices with vibration decoupling from the ground by ultrasound due to the transformation of ultrasonic vibrations into repeated impacts of sound frequency using a specially introduced “free” body between the ultrasonic and impact parts of the drill (bit). This allows continuous drilling regardless of the condition of the impact bit, including jamming. The purpose of the work is to select the most effective drill design and compile a mathematical model that provides simulation modeling. Materials and methods. Technical solutions of both domestic and foreign authors were considered, and a prototype was selected. Theoretical works were analyzed and it was found that the materials presented in them do not contain detailed mathematical models necessary for simulation modeling at the stage of design and optimization of the structure. In the theoretical part of the article, mathematical models are proposed, obtained on the basis of equivalent parameters found from a distributed description of the drill design elements. Methods of mathematical physics, classical mechanics and similarity theory were used. The results are that two alternative mathematical models are developed. One of the models has a continuous description with a significantly nonlinear rigidity of the “wall” upon impact, while the impact motion is considered as a type of self-oscillation. The second model is based on the theory of impact. Conclusion. A parametric study was carried out and the dependences of the oscillation frequency of a free body on the size of the impact gap, body mass and amplitude of oscillations of the concentrator, etc. were obtained. The considered models can be used for simulation modeling when developing a drill design. The tasks of controlling a robotic drilling platform are formulated from the position of ensuring the operability of the drill when drilling various soils.

Homogenization for convection-enhanced thermal transport in sea ice

Sea ice regulates heat exchange between the ocean and atmosphere in Earth’s polar regions. The thermal conductivity of sea ice governs this exchange, and is a key parameter in climate modelling. However, it is challenging to measure and predict due to its sensitive dependence on temperature, salinity and brine microstructure. Moreover, as temperature increases, sea ice becomes permeable, and fluid can flow through the porous microstructure. While models for thermal diffusion through sea ice have been obtained, advective contributions to transport have not been considered theoretically. Here, we homogenize a multiscale advection–diffusion equation that models thermal transport through porous sea ice when fluid flow is present. We consider two-dimensional models of convective flow and use an integral representation to derive bounds on the thermal conductivity as a function of the Péclet number. These bounds guarantee enhancement in the thermal conductivity due to the added flow. Further, we relate the Péclet number to temperature, making these bounds useful for global climate models. Our analytic approach offers a mathematical theory which can not only improve predictions of atmosphere–ice–ocean heat exchanges in climate models, but can provide a theoretical framework for a range of problems involving advection–diffusion processes in various fields of application.

Multiobjective optimal control problems. Stationary Navier–Stokes equations

This paper deals with the solution of some multi-objective optimal control problems for stationary Navier–Stokes equations. More precisely, we look for Pareto and Nash equilibria associated to standard cost functionals. First, we prove the existence of equilibria and we deduce appropriate optimality systems. Then, we analyse the existence and characterization of Pareto and Nash equilibria for the Navier–Stokes equations. Here, we use the formalism of Dubovitskii and Milyoutin., see [Girsanov FV. Lectures on mathematical theory of extremum problems. Berlin: Springer-Verlag; 1972. (Notes in economics and mathematical systems; vol. 67)]. Finally, we also present a finite element approximation of the bi-objective problem, we illustrate the techniques with several numerical experiments and we compare the Pareto and Nash equilibria.

High accuracy solutions for the Pochhammer–Chree equation in elastic media

This study investigates the nonlinear Pochhammer–Chree equation, a model crucial for understanding wave propagation in elastic rods, through the application of the Khater III method. The research aims to derive precise analytical solutions and validate them using He’s variational iteration method (VIM). The Pochhammer–Chree equation’s relationship to other nonlinear evolution equations, such as the Korteweg-de Vries and nonlinear Schrödinger equations, underscores its significance in the field of nonlinear wave dynamics. The methodology employs the Khater III method for deriving analytical solutions, while He’s VIM serves as a numerical validation tool, ensuring the accuracy and stability of the obtained results. This dual approach not only yields novel solutions but also provides a robust framework for analyzing complex wave phenomena in elastic media. The findings of this study have significant implications for material science and engineering applications, offering new insights into the behavior of waves in elastic rods. By bridging the gap between theoretical models and practical applications, this research contributes to the advancement of both mathematical theory and physical understanding of nonlinear wave dynamics. Situated within the domain of applied mathematics, with a focus on nonlinear wave equations, this work exemplifies the interdisciplinary nature of contemporary research in mathematical physics. The results presented herein open new avenues for future investigations in related fields and highlight the potential for innovative applications in material science and engineering.

Mini-Workshop: Standard Subspaces in Quantum Field Theory and Representation Theory

Real standard subspaces of complex Hilbert spaces are long known to provide the right language for Tomita–Takesaki modular theory of von Neumann algebras. In recent years they have also become an object of prominent interest in mathematical quantum field theory (QFT) and unitary representation theory of Lie groups. This workshop brought together mathematicians and physicists working with standard subspaces, particularly in QFT (construction of QFT models, characterization of entropy, information-theoretic aspects), nets of standard subspaces on causal homogeneous spaces and aspects of reflection positivity and euclidean models related to standard subspaces and modular theory.

Структурный подход к оценке работоспособности системы мониторинга инженерных конструкций вантового моста через Петровский канал в створе автомобильной дороги «Западный скоростной диаметр» в г. Санкт-Петербурге. Часть 1

Purpose: assessing the performance of the monitoring system for engineering structures and assessing the current state of the structures of the cable-stayed bridge across the Petrovsky Canal in the alignment of the Western High-Speed Diameter highway in the city of St. Petersburg due to the occurrence of emergency situations in which accelerometers on the pylons record values exceeding the limit. Methods: statistical data from the databases of the existing system of monitoring engineering structures is used to obtain the results. The tasks, set within the framework of the study, are accomplished by applying the theoretical methods of scientific knowledge: the analytical method, the mathematical statistics theory, induction. Results: the resulting structural approach to assessing the performance of the monitoring system for engineering structures on cable-stayed bridges makes it possible to reduce the number of false alarms of the system and assess the current state of the bridge structure. Within the framework of the III stage of the structural approach, it is proposed to develop a methodology for assessing the performance of the monitoring system for engineering structures with subsequent assessment of the current state of cable-stayed bridge structures. Practical significance: the results of the work are important for construction as they expand the understanding of the features of the mechanism of soil freezing.

A noise‐tolerant fuzzy‐type zeroing neural network for robust synchronization of chaotic systems

SummaryAs a significant research issue in control and science field, chaos synchronization has attracted wide attention in recent years. However, it is difficult for traditional control methods to realize synchronization in predefined time and resist external interference effectively. Inspired by the excellent performance of zeroing neural network (ZNN) and the wide application of fuzzy logic system (FLS), a noise‐tolerant fuzzy‐type zeroing neural network (NTFTZNN) with fuzzy time‐varying convergent parameter is proposed for the synchronization of chaotic systems in this paper. Notably the fuzzy parameter generated from FLS combined with traditional convergent parameter embedded into this NTFTZNN can adjust the convergence rate according to the synchronization errors. For the sake of emphasizing the advantages of NTFTZNN model, other three sets of contrast models (FTZNN, VPZNN, and PTZNN) are constructed for the purpose of comparison. Besides, the predefined‐time convergence and noise‐tolerant ability of NTFTZNN model are distinctly demonstrated by detailed theoretical analysis. Furthermore, synchronization simulation experiments including two chaotic systems with different dimensions are provided to verify the related mathematical theories. Finally, the schematic of NTFTZNN model for chaos synchronization is accomplished completely through Simulink, further accentuating its effectiveness and potentials in practical applications.

Development of a model for assessing the technical condition of Baghdad bridges

The research aims to develop a significant model for assessing the condition of bridges. This model may serve as a supplementary source of information in determining bridge condition ratings not only in Baghdad, but also in other regions with similar operating conditions. The authors also used the methods of mathematical statistics, probability theory, and mathematical theory of planning to conduct an experiment using the results of bridge survey. Currently, the Commission on Audit uses financial depreciation models to determine the service life of structures, with no alternative formulas available for predicting this parameter in the country. Therefore, the authors consider a new approach on the basis of available research on depreciation models. This approach is based on assessment of the condition of 30 structural elements of a bridge, as well as the results of inspections conducted by the Department of Public Works and Highway Bridges. The purpose was to determine the applicability of available data as meaningful variables in identifying a regression model to predict the condition of such bridges in Baghdad. Statistical data for determining the coefficients and dependent variables provided a relevant model for bridge condition assessment, which can be applied to identify the technical state of bridge structures.

A mathematical theory of relational generalization in transitive inference

Humans and animals routinely infer relations between different items or events and generalize these relations to novel combinations of items. This allows them to respond appropriately to radically novel circumstances and is fundamental to advanced cognition. However, how learning systems (including the brain) can implement the necessary inductive biases has been unclear. We investigated transitive inference (TI), a classic relational task paradigm in which subjects must learn a relation ([Formula: see text] and [Formula: see text]) and generalize it to new combinations of items ([Formula: see text]). Through mathematical analysis, we found that a broad range of biologically relevant learning models (e.g. gradient flow or ridge regression) perform TI successfully and recapitulate signature behavioral patterns long observed in living subjects. First, we found that models with item-wise additive representations automatically encode transitive relations. Second, for more general representations, a single scalar "conjunctivity factor" determines model behavior on TI and, further, the principle of norm minimization (a standard statistical inductive bias) enables models with fixed, partly conjunctive representations to generalize transitively. Finally, neural networks in the "rich regime," which enables representation learning and improves generalization on many tasks, unexpectedly show poor generalization and anomalous behavior on TI. We find that such networks implement a form of norm minimization (over hidden weights) that yields a local encoding mechanism lacking transitivity. Our findings show how minimal statistical learning principles give rise to a classical relational inductive bias (transitivity), explain empirically observed behaviors, and establish a formal approach to understanding the neural basis of relational abstraction.

Mathematics Learning for Students of Basic Age: Montessori Theory Applicative Review

Although previous studies have discussed about the theme of Montessori theory in learning mathematics, but the focus of studying in basic age students needs to be done in depth. Because, basic age children are seen as crucial age in teaching concrete and critical thinking skills using the Montessori method. This study aims to describe how the Montessori method in learning mathematics. The research procedure is carried out based on literature or literature studies. The results of the study found that operationally the application of the Montessori method in mathematics learning: First, the introduction of identity that can be done by making a relationship between the visual aspects of an object and its name. Second, recognition of comparison. This step is carried out to find an affirmation of the level of student understanding of the material. Third, make a variety of visualizations but allied and require students to be able to explain it. With the steps that provide attention to the development of the learning reasoning will gradually reduce the "phobia" of students to mathematics, and will have implications for increasing mathematical learning outcomes.

Patient-Specific, Mechanistic Models of Tumor Growth Incorporating Artificial Intelligence and Big Data.

Despite the remarkable advances in cancer diagnosis, treatment, and management over the past decade, malignant tumors remain a major public health problem. Further progress in combating cancer may be enabled by personalizing the delivery of therapies according to the predicted response for each individual patient. The design of personalized therapies requires the integration of patient-specific information with an appropriate mathematical model of tumor response. A fundamental barrier to realizing this paradigm is the current lack of a rigorous yet practical mathematical theory of tumor initiation, development, invasion, and response to therapy. We begin this review with an overview of different approaches to modeling tumor growth and treatment, including mechanistic as well as data-driven models based on big data and artificial intelligence. We then present illustrative examples of mathematical models manifesting their utility and discuss the limitations of stand-alone mechanistic and data-driven models. We then discuss the potential of mechanistic models for not only predicting but also optimizing response to therapy on a patient-specific basis. We describe current efforts and future possibilities to integrate mechanistic and data-driven models. We conclude by proposing five fundamental challenges that must be addressed to fully realize personalized care for cancer patients driven by computational models.

Development of Mathematics Teaching Models According to the Theory of Knowledge Creation to Promote the Mathematical Competence of Mathayom 6 Student

Background and Aims: The development of human thought is greatly aided by mathematics, which fosters creativity, and logical and systematic thought, benefits life, and raises the standard of living. (1) the first goal of this research is to create a mathematical teaching model for Mathayom 6 students that is based on the theory of knowledge creation. And (2) Examine the effects of utilizing a theory of knowledge creation-based mathematics teaching model to enhance Mathayom 6 students' mathematical proficiency. Methodology: Three research steps make up this study. (1) Phase 1: Basic information, current conditions, and conditions for teaching mathematics are studied, along with concepts, theories, and research about models of teaching mathematics based on knowledge creation theory to enhance secondary school students' mathematical competency. Year 6 by speaking with five education specialists. (2) Phase 2: Creating a teaching model for mathematics based on the theory of knowledge creation to help students in Mathematical Competency 6 and (3) Phase 3: Testing out a model of teaching mathematics based on the theory of knowledge creation to improve Mathayom 6 students' performance in the subject. Thirty Mathayom 6 students from Thesaban 2 School in Nangrong Municipality, Nangrong District, Buriram Province participated in the model's testing as a sample group during the second semester of the academic year 2022; (1) A mathematics teaching model based on the theory of knowledge creation to support Mathayom 6 students' mathematical competency is one of the instruments used to gather data. (2) Learning Satisfaction Scale; (3) Mathematics Competency Test; and (2) Achievement Test, the t-test (Dependent), mean, percentage, and standard deviation are among the statistics used in data analysis. Results: (1) A mathematical competency-promoting teaching model for Mathayom 6 students, based on the theory of knowledge creation. There are six parts. These include the model's goals, instructional strategies, social systems, response theories, and support networks. Four steps make up the teaching process: (1.1) Step of operation (Operation: O) (1.2) Step two: Action (Acting: A) (1.3) Step three: Knowledge creation (Creation: C), additionally, (1.4) Recapitulating the idea (Idea: C) and a high degree of appropriateness was indicated by the model's evaluation results, which had a mean of 4.47 and a standard deviation of 0.41. (2) The application of the theory of knowledge creation to the mathematics teaching model revealed that at a statistical significance of.05., students' academic achievement is higher than it was before they started studying. When Mathayom 6 students use the mathematics teaching model, their performance in mathematics exceeds the 70% threshold. Additionally, a high degree of student satisfaction has been observed with the mathematics teaching model. Conclusion: To emphasize mathematical competency, the study presents a comprehensive teaching model for Mathayom 6 students that is based on the theory of knowledge creation. The effectiveness of the model is demonstrated by the evaluation results, which show a significant improvement in academic achievement and student satisfaction, both of which exceed the performance threshold of 70%.

Философия микроэкономики: ключевые тенденции XXI в.

The article analyzes the current key philosophical trends in academic publications on microeconomics. The author investigates new sense that economists imply into traditional methodological grounds of Microeconomics. Based on classical marginalist microeconomic theory, contemporary Russian and World economists introduce new meanings and new interpretations into such important aspects of Microeconomics as microeconomic agents, choice and limitations for producers and consumers, market interactions and competition, time and space of microeconomic phenomena. New methodological grounds of Microeconomics in the sphere of decision-making are analysed on the ground of content-analysis. The process of economic choice in new conditions of the 21st century is based on deliberate consumption of economic goods and thoughtful limitations for time, place and number of alternatives when decision is made. Maximization of utility as the goal of behavior of a rational individual is replaced by the criterion of satisfaction and happiness which now form important elements of microeconomic analysis. New approaches to microeconomic models are formed on the ground of mathematical theories of dynamic chaos and nonlinear processes that allow to forecast more precisely microeconomic events in the era of turbulence. Stochastic concepts and differentiated games take the place of traditional statistical strict determinism of microeconomic theories and uncertainty acquires new depth as a fundamental pattern of functioning of microeconomic processes. The article specifies the motives and goals of individual and collective agents acting at micro level. Thus microeconomics of the 21st century has reconsidered its key philosophical grounds, which would contribute to a new paradigm of Microeconomic Analysis.

Statistical wave field theory.

In this paper, we introduce the foundations of the Statistical Wave Field Theory. This theory establishes the statistical laws of waves propagating in a closed bounded volume, that are mathematically implied by the boundary-value problem of the wave equation. These laws are derived from the Sturm-Liouville theory and the mathematical theory of dynamical billiards. They hold after many reflections on the boundary surface, and at high frequency. This is the first statistical theory of reverberation which provides the closed-form expression of the power distribution and the correlations of the wave field jointly over time, frequency, and space inside the bounded volume, in terms of the geometry and the specific admittance of its boundary surface. The Statistical Wave Field Theory may find applications in various science fields, including room acoustics, electromagnetic theory, and nuclear physics.

The General Mathematics Course ‘Mathematics and Life Sciences’ for Non-science and Engineering Students

In modern society, alongside the rapid advancement of science and technology, the role and importance of mathematics are increasingly being emphasized. Such advancements, including data analysis and artificial intelligence development, rely fundamentally on mathematical thinking and theory, which have become essential elements across all academic disciplines. Against this backdrop, the role of university general mathematics education has become undeniably important. It is necessary not only for students in science and engineering, but also for those in the humanities and social sciences, to have access to balanced mathematical education. To this end, universities must focus on developing a variety of general mathematics courses. This paper presents a detailed case study of the design and operation of a basic general mathematics course titled “Mathematics and Life Sciences,” offered to students in the humanities and social sciences at Y University in Seoul. The course is structured to enable students to learn essential basic mathematics theories that are crucial across various disciplines and to apply them to phenomena in life sciences. In doing so, students will gain a better understanding of certain interdisciplinary connections as well as enhance their thinking and problem-solving skills. Additionally, this paper analyzes the educational effects and positive feedback from students obtained through a decade of continuous improvement in the content and operation of this course. Based on this analysis, this paper proposes future directions for development and research tasks in the educational field. This research is expected to contribute to the exploration of the direction in which university general mathematics education should head.