Mathematical Theory Research Articles

On thermally driven fluid flows arising in astrophysics

We investigate a potential model for an unbounded celestial bodies of finite mass composed of a solid core and a gaseous atmosphere. The system is governed by the Navier–Stokes–Fourier–Poisson equations, incorporating no-slip boundary conditions for velocity and a specified temperature distribution on the surface of the solid core. Additionally, a positive far-field condition is imposed on the temperature. This manuscript extends the mathematical theory of open fluid systems to unbounded exterior domains addressing these physically motivated yet highly challenging combination of boundary conditions. Notably, we establish the existence of global-in-time weak solutions and demonstrate the weak-strong uniqueness principle

Pure Rickart Modules

This paper focuses on the study of pure Rickart modules (or PR-modules for short), which are a class of modules over a commutative ring with identity. The main objective is to investigate the properties and characterizations of these modules, as well as their relationship with other classes of modules such as free, projective, and flat modules. This paper also explores the connections between PR-modules and various algebraic structures such as rings. The results obtained in this study provide a deeper understanding of the structure and behavior of PR-modules, which can have important applications in algebraic geometry, representation theory, and other areas of mathematics. Some results about PR-module have been investigated in this paper, for example we demonstrate a module is PR-module if and only if for every , is pure submodule of (or for short). Also, some kind of generalization of these rings have been constructed and demonstrated in term of PR-modules.

Структурный подход к оценке работоспособности системы мониторинга инженерных конструкций вантового моста через Петровский канал в створе автомобильной дороги «Западный скоростной диаметр» в городе Санкт-Петербурге. Часть 2

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 cablestayed 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.

Проблемы поиска и преследования беспилотных летательных аппаратов с использованием теоретико-игрового подхода

Currently, the application of mathematical modeling theory is widely used in various fields and, among other things, can be used to prevent illegal actions. This study examines scenarios involving a single pursuer tracking a single evader, as well as situations involving multiple pursuers pursuing multiple evaders. The authors formulate this problem as a search and pursuit problem for four-rotor unmanned aerial vehicles (UAVs) or quadcopters. The solution to the problem is based on game theory, as it provides a mathematical framework for modeling and studying strategic interactions involving multiple decision makers. The authors consider a game where the set of strategies of the runner is the set of possible combinations of speeds and directions of his movement, and the set of strategies of the pursuer is the set of all possible permutations of the elements of his speeds. The matrix of the resulting game consists of elements that are the time of capture. A mathematical problem on the optimal assignments of searching and pursuing unmanned aerial vehicles is formulated and solved. Purpose: optimizing the effectiveness of strategies for detecting and capturing four-rotor unmanned aerial vehicles (quadcopters). Methods: methods of mathematical modeling, game theory apparatus, Hungarian method of solving the problem of assignments, decision theory, the principle of dynamic programming, the Maple package for solving examples were used. Results: In order to fulfill the conditions for solving the problem of optimal assignments, it is necessary and sufficient that it is balanced. This assignment problem can be balanced by entering the required number of fictitious boats or escaping boats. After that, it is possible to formulate and solve the dual problem of optimal appointments. The resulting game can be solved by any method of solving matrix games. In this way, it is possible to determine the policy of harassment and search between drones. Practical importance: All escaped quadcopter UAVs can be chased and successfully intercepted using the developed models. Examples of the study of mathematical models using the Maple software package are given.

Applications of Optimal Transportation

The mathematical theory of optimal transportation is constantly expanding its range of application, while applications give impulses for new research directions in the field. This workshop was specifically devoted to applications of optimal transportation in the natural sciences, and to the recent developments of the theory that have been motivated by these. The bouquet of current applications includes mathematical models for large-scale air motion, dynamics of plasmas, material design, pattern formation in fluids, collective behaviour in biology, and many more. Related theoretical developments are in the analysis of the Hellinger-Kantorovich metric for modeling reaction–diffusion processes, and in efficient numerical methods for multi-marginal optimal transport, to name two of many examples encountered in this workshop.

Behavioral swarms: A mathematical theory toward swarm intelligence

This paper presents a mathematical theory of behavioral swarms, where the state of interacting entities, which are called active particles, includes in addition to position and velocity, an internal variable, called activity, which has the ability to interact with mechanical variables thus affecting the interaction rules. In turn, the mechanical variables can modify the dynamics of activity variable. This approach is useful for describing the dynamics of living systems with a finite number of interacting entities. This paper provides a general conceptual framework that extends the pioneering work [N. Bellomo, S.-Y. Ha and N. Outada, Towards a mathematical theory of behavioral swarms, ESAIM: Control Theory Var. Calculus 26 (2020) 125, https://doi.org/10.1051/cocv/2020071 ]. The theory is firstly developed for the constant number of active particles. Then, it is considered for the case of particles tending to infinity. This theory is useful for describing the dynamics of living systems. Therefore, it provides the conceptual basis for developing a mathematical theory of behavioral swarms, which naturally lead to the study of a theory of swarm intelligence. A study of the dynamics of swarms shows how the theory can be applied to real world collective dynamics.

David Ferrier’s “complex whole”: Early traces of a “brain network” concept

ABSTRACT Currently, the idea that the brain is a complex network of interacting brain regions is hardly controversial. The rapid development of this field is often attributed to the emergence of powerful brain-imaging techniques and, around the millennium, the merging of the neuroscience of brain networks with modern mathematical graph theory. However, little is known about the historical roots of this concept. It is interesting to know when the first traces of a concept of brain networks can be found in the work of early neuroscientists, how this concept evolved over time, and what factors may have influenced this evolution. This study aims to set a first step in addressing these questions by a detailed analysis of David Ferrier’s classic study, The Functions of the Brain. From this analysis it will become clear that, in addition to a clear notion of localized functions in the brain, Ferrier speculated in several places about the need for several of these brain regions to communicate and interact in order to bring about higher brain functions. He referred to this perspective on the brain as a “complex whole,” which could be interpreted as an early precursor of the modern concept of brain networks.

Exploling the Scientific Model for Constructing a Platform for the International Communication of PingJiang Opera

Purpose: Considerable changes have occurred in many areas of science and technology as a result of the current information technologies' quick development and use. Specifically, new theories in mathematics have surfaced; they may be used to construct and refine current mathematical models of different processes and objects. Method: Regulatory and permitting decision-makers are worried about the ambiguity surrounding the possible environmental consequences arising from the introduction of these innovative devices in coastal and riverine ecosystems as research and testing of maritime energy devices progress. The Triton Initiative conducts studies and makes suggestions for environmental monitoring techniques and technology to give industry stakeholders with the information required to approve maritime energy system testing. Results: The efficient distribution of research findings is crucial in enhancing data accessibility for stakeholders who might use the information to make informed policy choices. Nevertheless, there are few frameworks available for carrying out scientific communications related to marine energy projects. Using the Triton Initiative's pilot scientific communication program as a case study, we discuss methods, resources, and strategies for creating a science communication framework for marine energy projects or other related fields of research in this article. Conclusion: This article uses the historic Pingjiang Road in Suzhou, China, as a case study to investigate how, in the era of globalization and digital media, urban spaces that have been revitalized for historic and cultural consumption may demonstrate the publicness or potential capacity for publicity.

Intelligent Method for Mutation of Input Cases with Feedback

Relevance. Fuzzing is one of the effective ways to improve the software reliability and is included in the mandatory list of research carried out at the stage of qualification testing according to national standard GOST R 56939-2016. The use of standard mutators reduces the fuzzing process to brute force, which negatively affects the time of incorrect program behavior detection. In this regard, it is important to rationalize the selection of input data, which takes into account the data corpus specifics, as well as the context describing the software response under test and allowing to determine the mutations at the next iteration of testing. Purpose of the research is to increase the efficiency of fuzzing by intellectualizing the standard mutator using neural networks, which takes into account the syntactic and semantic features of the input corpus and uses program feedback.Methods. The methods of analysis and synthesis, theory of algorithms, discrete and computational mathematics, machine learning were used.Result. The advantages and disadvantages of the standard module for AFL fuzzer’s input corpus mutation are considered. The justification of neural network choice with LSTM-architecture as a mechanism that realizes the intelligent control of input corpora’s’ generation and transformation is given. The proposed mutation method is described, which implies the integration of decision making mechanism on the amount and format of necessary mutations to increase the code coverage into the standard mutator, as well as the subsequent refinement of input data by shell-code to check the operability of the fragment that caused abnormal software’s behavior. The scheme of the mutation module is presented, which includes a component of input corporas conversion for generation of program execution traces and a component aimed at concept confirmation and re-call of abnormal software behavior using the generated shell-code.Novelty. Unlike the known ones, the proposed method uses feedback, fixing the software reaction, when forming the data mutation strategy, which determines the scientific novelty of the obtained results.Significance. The proposed solution allows reducing the program testing time while maintaining the code coverage. The results obtained in the research are universal and, in the future, can be used in white, black and gray box fuzzing methods.

Towards a mathematical foundation for music theory and composition: a theory of structure

This paper proposes mathematical foundations for music theory and composition. While mathematical methods have proven effective in music theory, a deficiency in rigorous mathematical foundations often leads to ad-hoc constructions and a reliance on intuitive notions with inexplicit definitions. The proposal introduces a comprehensive environment for encoding musical phenomena, as well as a theory of musical parameter. Drawing from the methodologies of (Mazzola, Guerino. 2002. The Topos of Music: Geometric Logic of Concepts, Theory, and Performance. Basel: Birkhäuser Verlag), the proposed framework aims to extend his theory of forms to accommodate a wider class of musical structures. Additionally, I advocate for a framework characterized by “comprehensive comprehension” (accommodating a broad class of musical objects), “explicit encoding” (capturing essential features of objects), and “limited ontological commitments” (minimizing posited ontological primitives). These three aspects aim to ensure the framework's generality, explicit representation of structure, and economic efficiency in theoretical constructions. Minimizing ontological commitments also has the benefit of facilitating easier comparisons between entities. This approach is motivated by the desire for a highly versatile framework that enables systematic and standardized construction methods without constraining the intended domain of discourse. To achieve these goals, the paper introduces a mathematical theory of structure grounded in topos theory. By offering a unified and systematic approach, this work contributes to establishing a more rigorous and standardized basis for mathematical music theory. This, in turn, fosters a deeper understanding of musical phenomena and enables more robust theoretical constructions and compositional applications.

Transdisciplinary approach as a resource for the development of critical thinking in higher mathematical education

Introduction. In a modern digital and globalized society, the higher education system is faced with the task of developing critical thinking skills in graduates that allow them to solve complex problems through the integration of knowledge from various disciplines at a new transdisciplinary level of knowledge. Such integration seems possible based on the use of modeling methodology, methods of computational and discrete mathematics, and mathematical cybernetics, which form the foundation of the language of information technology and processes. The purpose of the study is to identify the role of the transdisciplinary approach in the development of students’ critical thinking through mathematics. Materials and methods. 76 students of the Vyatka State University majoring in 06.03.01 Biology took part in the study, covering 2022–2023. The main methods for measuring critical thinking, presented by the UF/EMI Critical Thinking Disposition Instrument and The Critical Thinking Barriers Scale (CTBS), were supplemented with diagnostic tools from specially designed tasks, taking into account specifics of teaching mathematics to future biotechnologists. Statistical analysis of the obtained results was performed using the Mann-Whitney U test. Results. A number of methodological conditions have been formulated to ensure the development of students' critical thinking in the process of teaching mathematics, which corresponds to the priority goals of modern higher education. The expediency of using contextual mathematical problems containing a real data set, the method of mathematical modeling, the inclusion of modern mathematical theories and their applications in the teaching content, a joint form of work in the classroom and work with scientific articles is substantiated. A quantitative analysis of the results confirmed the significance of the differences (Uemp = 65 < Ucr = 66; р < 0,01), which indicates the effectiveness of the intervention based on the proposed methodological conditions. Conclusion. The practical significance of the study is determined by the possibilities of using its results in improving the critical thinking skills of students in the process of teaching mathematics.

Research on the influence of acoustic short circuit fault of air spring isolator on ship floating raft isolation system

Air spring isolator is the crucial component of ship floating raft isolation system. However, due to various factors such as manufacturing errors and installation conditions of air spring isolators, abnormal contacts cannot be avoided in the internal limiters of air spring isolator, resulting in the acoustic short circuit fault. This leads to a significant decrease in isolation effectiveness. To address this issue, experiments on the acoustic short circuit fault of air spring isolators are conducted, and the experimental results are analysed. Then the causes behind the phenomena and results of the acoustic short circuit experiments are further elucidated through finite element simulations and mathematical theory analysis. The research results indicate that the acoustic short circuit of air spring isolators affects the isolation effectiveness. The more severe the acoustic short circuit fault, the greater its impact on the isolation effectiveness, particularly in the low-frequency isolation component. This is because, during the acoustic short circuit of air spring isolators, the rubber part of the internal limiter comes into contact, causing an increase in the stiffness of the air spring support, thereby resulting in an increase in the modal frequency of the isolation device and consequently severely impacting the low-frequency isolation effectiveness.

Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓq(·)(Mp(·),v(·)) with Variable Exponents

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.

Research on upper bound limit analysis of slope stability in open-pit mine based on least square principle

So as to extend the application of upper bound limit analysis for slope stability in open-pit mine, this paper analyzed failure mechanism of rock mass when shear failure occurred based on plastic mechanics principles, and combined with discrete mathematical theory and least square principle to propose a visualization method of logarithmic spiral. At the same time, the visualization method was applied to engineering practice for slope stability in open-pit mine, and cycle algorithm of stability analysis for open-pit mine slope was proposed through energy balance conditions, then stability analysis results were evaluated reasonably through uniting with limit equilibrium method. The results show that visualization method of logarithmic spiral proposed in this paper is highly consistent with original curve, and can ensure correlation coefficient square is greater than 0.9995. What is more, upper bound limit analysis method has a superhigh accuracy in slope engineering. At the same time, the most dangerous sliding surface obtained by this method can satisfy the requirements of velocity separation, which solved velocity problems during the process of slope evolution. Therefore, the method proposed in this paper has a stronger application in the field of open-pit mining.

Introduction to Algebraic Deformation Theory and The Case of k-Linear Categories

Deformation theory is a branch of mathematics which studies how mathematical objects, such as algebraic varieties, schemes, algebras, or categories, can be deformed “continuously” depending on a space of parameter while preserving certain algebraic or geometric structures. Algebraic deformation theory, which was pioneered by Murray Gerstenhaber in 1960s-1970s, established its role as a cornerstone in modern mathematics and theoretical physics. This theory provides a powerful framework for understanding the subtle variations and deformations of mathematical and physical objects depending on a parameter space. Lying at the interface of algebra, geometry and topology this theory has been being studied extensively worldwide and obtained many applications in various areas of mathematics and theoretical physics, such as the study of Calabi-Yau manifolds, mirror symmetry, quantum physics. In such context, this article aims to introduce this important mathematical theory to the community of Vietnamese mathematicians in a hope to bring more attention of Vietnamese mathematicians and math students to this vibrant research area. We also expect that this topic will be taught at universities in Vietnam in the near future.

A systematic literature review on the mathematical underpinning of model‐based systems engineering

AbstractThe International Council on Systems Engineering (INCOSE) has initiated a Future of Systems Engineering (FuSE) program that includes a stream for advancing the theoretical foundations of the discipline of Systems Engineering (SE). A near‐term goal of FuSE is to assess the adequacy of current theoretical foundations of SE. The discipline of SE is converging toward model‐based practices (i.e., MBSE) that have not yet reached the maturity of model‐based practices in other engineering domains. For example, finite element analysis and computational fluid dynamics are grounded in mathematical theory, while, generally, MBSE is not. However, some attempts have been made to underpin MBSE with theoretical richness. This article presents a systematic literature study that surveyed state of the art on providing MBSE with mathematical foundations. Our protocol collected over 2000 publications that were reviewed for inclusion/exclusion, categorized, and analyzed. We provide insights to the type of mathematical theories used, domains of applications, and areas of SE to which the math was applied to, among other analysis. We also provide a synthesized discussion about the field moving forward, emphasizing positive trends along with the negatives and areas of concern. Overall, we found the field to be nascent.

Exploring the space of graphs with fixed discrete curvatures

Abstract Discrete curvatures are quantities associated to the nodes and edges of a graph that reflect the local geometry around them. These curvatures have a rich mathematical theory and they have recently found success as a tool to analyze networks across a wide range of domains. In this work, we consider the problem of constructing graphs with a prescribed set of discrete edge curvatures, and explore the space of such graphs. We address this problem in two ways: first, we develop an evolutionary algorithm to sample graphs with discrete curvatures close to a given set. We use this algorithm to explore how other network statistics vary when constrained by the discrete curvatures in the network. Second, we solve the exact reconstruction problem for the specific case of Forman–Ricci curvature. By leveraging the theory of Markov bases, we obtain a finite set of rewiring moves that connects the space of all graphs with a fixed discrete curvature.

Results from a Nonlinear Wave Equation with Acoustic and Fractional Boundary Conditions Coupling by Logarithmic Source and Delay Terms: Global Existence and Asymptotic Behavior of Solutions

The nonlinear wave equation with acoustic and fractional boundary conditions, coupled with logarithmic source and delay terms, is significant for its ability to model complex systems, its contribution to the advancement of mathematical theory, and its wide-ranging applicability to real-world problems. This paper examines the global existence and general decay of solutions to a wave equation characterized by coupling with logarithmic source and delay terms, and governed by both fractional and acoustic boundary conditions. The global existence of solutions is analyzed under a range of hypotheses, and the general decay behavior is established through the construction and application of an appropriate Lyapunov function.

Wittgenstein in Cantor's paradise

AbstractThis paper offers an evaluation of Wittgenstein's critique of Cantorian set theory, illustrating his broader philosophical stance on mathematics. By emphasizing the constructed nature of mathematical theories, Wittgenstein encourages a reflective approach to mathematics that acknowledges human agency in its development. His engagement with Cantorian set theory provides valuable insights into the philosophical and practical dimensions of mathematics, urging a reconsideration of its foundations and the nature of mathematical proofs. This perspective aligns closely with the philosophy of mathematical practice, which emphasizes the human side of mathematics by its focus on mathematical practice.

Optimal capacity configuration and operation strategy of typical industry load with energy storage in fast frequency regulation

As the potential and competent load-side resources for frequency response and control in modern power grids, typical industrial load can compensate for the deficiency of frequency response capability for new-type power systems. However, the operational flexibility is seriously enforced by the operation conditions uncertainties of industrial load. With “Online Calculation, and Real-time Matching” as the core, based on fuzzy mathematical theory, the coordinated operation strategy of typical industrial loads and energy storage systems (ESS) is proposed to finish fast frequency regulation (FFR) tasks. And an optimal capacity configuration model of industrial loads with ESSs is established to evaluate the whole economic profitability in FFR. Specially, in terms of algorithms, a novel Two-layer Parallel Particle Swarm Optimization and Genetic Algorithm (TPPSGA) is innovatively designed in this manuscript. Comparing with traditional heuristic algorithms, TPPSOGA improves computation speed by 6.84 times. Simulation results show that our proposed strategy can reasonably estimate the frequency capability of industrial load by membership degree function, and prove that the industrial load participating in FFR with the support of ESSs is able to get more revenue and stability than industrial load or ESSs separately is in regulation of FFR. The economic analyze in market price and ESS parameters can help the load agent to screen the suitable-type ESS and its capacity to serve for FFR and enhance the flexibility in load-side frequency regulation.