Methods Of Discrete Mathematics Research Articles

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.

A Discrete Mathematics Approach for Understanding Risk Factors in Overactive Bladder Treatment.

Introduction Discrete mathematics, a branch of mathematics that includes graph theory, combinatorics, and logic, focuses on discrete mathematical structures. Its application in the medical field, particularly in analyzing patterns in patient data and optimizing treatment methods, is invaluable. This study, focusing on post-void residual (PVR) urine following overactive bladder (OAB) treatment, utilized discrete mathematics techniques to analyze PVR and its associated risk factors. Methods A retrospective study was conducted on 128 OAB patients who received intradetrusor onabotulinum toxin A injections between 2020 and 2022. Network graphs based on graph theory were used to analyze correlations between clinical variables, and clustering analysis was performed with PVR as the primary variable. Results The network graph analysis revealed that frailty, daytime frequency, and nocturia episodes were closely related to PVR. Clustering analysis with PVR as the primary variable divided the patients into three groups, suggesting that the group with particularly high frailty (Cluster 1) is at high risk for PVR. Moreover, significant differences in clinical indicators such as age, voiding efficiency, Overactive Bladder Symptom Score, and International Consultation on Incontinence Questionnaire-Short Form were observed in the remaining two clusters (Cluster 0 and 2). Conclusion This study demonstrates the effectiveness of discrete mathematics methods in identifying risk factors for PVR after OAB treatment and in distinguishing clinical subgroups based on patient characteristics. This approach could contribute to the formulation of individualized treatment strategies and the improvement of patient care quality. Further development and clinical application of this methodology are expected in future research.

Geometry Problems in Formalizing the Creation of Machining Processes on Machine Tools

Introduction: The main problems hindering the creation of fully automatic systems of design of parts and technologies of their production are considered. The search for and formalization of regularities and relationships between the design parameters of parts and the technology of their production is an urgent task. At the same time, it has not yet been possible to create a unified general theory of geometric construction and transformation of objects at the macro level. Purpose: To show and formulate the key problems of formalizing the design process and identify possible ways to solve them. Methods: The study uses a wide range of methods of algebra, geometry, and discrete mathematics, as well as the main provisions of the geometry of non-ideal objects. Results: The need for fundamental laws of generation of geometric configurations and their elements is a key formalization problem. The geometry of non-ideal objects allows us to formally solve these problems. A geometric object is represented in six-dimensional space (three linear and three angular dimensions) with binary non-uniform metrics. The considered geometry has a principal possibility to formally generate geometric configurations of objects and their elements. The geometrical structure of a part is unambiguously connected with a set of technological schemes providing the mutual arrangement of surfaces during machining. Rules for the formal generation of a complete set of technological structures that are guaranteed to provide the specified parameters during the machining of parts. Conclusion: Sets of surfaces and relations between them define the individual structure of geometric configuration for each specific object. This individuality can be effectively used for the formal recognition of geometric objects. The geometry of non-ideal objects can serve as a fundamental basis for the formal theory of the creation of original technological processes of their processing.

Technology of Digital Maturity Estimation for an Educational Organization. Part II

We present the developed system of estimating the digital maturity of an educational organization. The objective of estimating digital maturity has been formalized thanks to analyzing the existing approaches to validation of the digital maturity for an organizational object. The novelty of the given study is in introducing the fuzzy mechanism of integral estimation calculation of digital maturity into the digital maturity evaluation methodology. The methods of expert estimation, discrete mathematics and fuzzy logical inference were used to create the model. The utilization of the model developed is demonstrated on the example of higher education in the Russian Federation. The technique is adaptive and can be applied not only in educational organizations but also in organizations in other spheres of activity.

Ground state degeneration of frustrated antiferromagnetic Ising wheels

In this paper, a finite spin system corresponding to magnetic molecules with the wheel geometry is discussed in the Ising-like limit of the Heisenberg Hamiltonian. We put stress on the ground state degeneration, which is determined with the use of some discrete mathematics methods, especially investigating some integer sequences, including the famous Fibonacci one.

A Time Two-Mesh Finite Difference Numerical Scheme for the Symmetric Regularized Long Wave Equation

The symmetric regularized long wave (SRLW) equation is a mathematical model used in many areas of physics; the solution of the SRLW equation can accurately describe the behavior of long waves in shallow water. To approximate the solutions of the equation, a time two-mesh (TT-M) decoupled finite difference numerical scheme is proposed in this paper to improve the efficiency of solving the SRLW equation. Based on the time two-mesh technique and two time-level finite difference method, the proposed scheme can calculate the velocity u(x,t) and density ρ(x,t) in the SRLW equation simultaneously. The linearization process involves a modification similar to the Gauss-Seidel method used for linear systems to improve the accuracy of the calculation to obtain solutions. By using the discrete energy and mathematical induction methods, the convergence results with O(τC2+τF+h2) in the discrete L∞-norm for u(x,t) and in the discrete L2-norm for ρ(x,t) are proved, respectively. The stability of the scheme was also analyzed. Finally, some numerical examples, including error estimate, computational time and preservation of conservation laws, are given to verify the efficiency of the scheme. The numerical results show that the new method preserves conservation laws of four quantities successfully. Furthermore, by comparing with the original two-level nonlinear finite difference scheme, the proposed scheme can save the CPU time.

A Discrete Electrical Model for Photovoltaic Solar Cells—d1MxP

Solar cell equivalent circuit modelling is usually based on continuous I-V models, with a set of data obtained by analytical expressions. This work proposes an almost discrete novel mathematical method and correspondent electrical model, based on the I-V curve adjustment at every two adjacent points. It is based on the discretisation of any diode model behaviour, such as the 1M5P (also known as 1D5P) or the 1M7P (also known as 2D7P). For this reason, the model is named d1MxP, meaning that it is a discrete (d) model (1M) with x parameters (xP). The modelling methodology validation process uses experimental data already published in the literature. According to the presented results, the proposed method shows increased accuracy when compared to the 1M5P or 1M7P equivalent models. The accuracy on the maximum power point and fill factor determination is relevant, resulting in an improvement of up to 3.34% in the maximum power, up to 5.70% in its voltage and up to 8.20% in its current, for the analysed data. Furthermore, Fill Factor values, have variation from up to 35.98%. The temperature influence on the silicon solar cell is also analysed, to validate the results. The proposed method allows highly accurate curve fitting to the (experimental) points and consequently, to obtain a more accurate model to analyse the performance of solar cells under different conditions.

Model of Threats to the Integrity and Availability of Information Processed in Cyberspace

Depending on their motivation, offenders have different goals, and disclosure of information is not always such a goal. It often happens that the purpose of the offender is to disrupt the normal operation of the system. This can be achieved both by acting directly on the information and by acting on the elements of the system. Actions of this kind lead to a violation of integrity and availability, but not confidentiality. It follows that the process of forming a threat model for the integrity and availability of information differs from a similar process for confidentiality threats. The purpose of this study is to develop an information integrity threat model that focuses on threats disrupting the normal operation of the system. The research methodology is based on the methods of system analysis, graph theory, discrete mathematics, and automata theory. As a result of the research, we proposed a model of threats to the integrity and availability of information. The proposed threat model differs from analogues by a high level of abstraction without reference to the subject area and identification of threats to the availability of information as a subset of threats to the integrity of the information transmission channel.

High-order compact finite difference schemes for solving the regularized long-wave equation

In this paper, high-order compact finite difference schemes for the one-dimensional regularized long wave equation are proposed. Firstly, the original regularized long-wave equation is deformed in two ways. The fourth-order compact difference scheme and the fourth-order Padé scheme are used for discretization of the second and first derivatives of the two deformed equations in space direction, respectively, and the θ-weighted scheme is used to discretize the time derivative of the first deformed, the fourth-order backward difference formula is used for the competition of the time derivative of the second deformed equation. Secondly, the nonlinear terms in the two schemes are linearized by Taylor series expansion method. Thirdly, the existence, uniqueness and conservation of energy of numerical solutions are proved by discrete energy method and mathematical induction method. And the two new schemes are shown to be conserved and unconditionally stable. Since each time level involves three grid points, a tridiagonal linear system is formed, which can be solved directly using the Thomas algorithm. Finally, the accuracy and reliability of the present method are verified by numerical experiments.

Design of Verbal Models for Forming an Optimal Strategy for Sustainable Development of Service Enterprises in the Conditions of a Crisis

Introduction. The presence of a strategy and a strategic plan, necessary for determining the direction of the enterprise development and for making informed decisions, allows enterprise to achieve its goals with confidence and, at the same time, to respond flexibly to changes in the external environment, especially in times of crisis.Problem Statement. The formation of sustainable development strategies requires sufficient consultancy costs and technologies that are designed for being used by large enterprises with professionals having relevant expertise. However, most of all, during the crisis, small and medium-sized enterprises of the service sector suffer from it, which prevents them from using such consulting and technologies.Purpose. The development of a verbal model for shaping an optimal strategy for sustainable development of small and medium-sized enterprises of the service sector in times of crisis to find, to evaluate, and to support effective management decisions. Materials and Methods. Methods of formal logic, qualitative methods of verbal analysis, methods of discrete mathematics, and benchmarking method.Results. A verbal model has been developed for classifying the current state of enterprises on the basis of qualitative methods. The model is adapted for the possibility of its use in the presence of an arbitrary set of criteria and their values. A model has been developed for the automatically finding the optimal path (strategy) from a certain alternative (the current state of the enterprise) to an alternative that is guaranteed to belong either to the next better class (short-term strategy), or to the best class (long-term strategy).Conclusions. The verbal model allows us to describe and to classify the states of small and medium-sized enterprises of the service sector in any branch of business and automatically produces optimal strategies for their sustainable development. The criteria for optimal strategy may be the time and cost of achieving the required state of enterprise.

Numerical solution of the two-dimensional regularized long-wave equation with a conservative linearized high order finite difference scheme

In this article, a finite difference scheme is investigated to solve a two-dimensional regularized long-wave equation. The original equation is deformed into another formulation. The fourth order compact difference scheme and the fourth order Padé scheme are used for the treatment of the second and first derivatives of the deformation equation in space direction, and the fourth order backward difference formula is used for the discretization of the time derivative. The nonlinear term in the proposed scheme is linearized by Taylor series expansion method. The new scheme is shown to be of the fourth order accuracy in both temporal and spatial dimensions. The existence, uniqueness and conservation of energy of numerical solutions are proved by discrete energy method and mathematical induction method. And the new scheme is shown to be conservative and unconditionally stable. Finally, numerical experiments are provided to verify our theoretical analysis results of the new scheme.

Logic and the Mathematic Model of the Subject-Area Description for Corporate Information Systems

Aiming to build and receive a logic-mathematical substantiation and model of the process of constructing the description of any subject area for corporate information systems, the authors divided the subject area for building the system into constituent elements and the relationships between them. Using discrete mathematics methods to the graphs and two-seater predicates of the first order, we built an interconnected model of the subject area. Logical and mathematical models were built upon the main components of the subject area. Based on a technique used to construct its description, the use of the graphical description, combined with the application of transformation in a semantic model of the predicate type, allows us to use further the received model for constructing the normalized databases. The research reveals that constructing the qualitative description of the subject domain of the corporate automated information system requires us to use some consecutive iterations. Logic and mathematical models allow us to construct a general view of the description of the subject area of the corporate automated information system. Further, this generalized model can be transferred to absolutely any concrete subject area. The advantage of constructing the description of the subject area in the form of the predicate type’s semantic model is the integral sight at all considered subject areas, as information representation in the integral form is better perceived. When constructing such a graph, there is a possibility of completely restoring missing logical links. The purpose of the research is to describe the subject area of the corporate system using the graph form of representation. The novelty lies in developing information support model designed for managing processes of acquiring and using knowledge (this model allows to consider usefulness of information elements and didactic connections between elements) and in approach to estimation of achieving information support goals with the use of graphs describing the subject area of corporate system.

Methods and algorithm for calculating the focal parameters of a hollow conical electron beam in high-voltage glow discharge electron guns with a focusing magnetic lens

The algorithm is considered for calculating the focal distance of a hollow conical electron beam generated by high-voltage glow discharge electron guns with magnetic focusing of the beam in the drift region, as well as a method for calculating the diameter of the focal ring and its thickness for such a beam. The proposed algorithm is based on the theory of electron drift in the field of a focusing magnetic lens and is designed using the methods of discrete mathematics and the minimax analysis. The obtained simulation results made it possible to establish the influence of the magnetic lens current on the focal diameter of a hollow conical electron beam and on its focal ring thickness. It is shown that the change in the focal parameters of a hollow conical electron beam can be effectively provided through the regulation of the magnetic lens current.

Development of interaction model for air traffic controllers and pilots using discrete mathematics methods

The purpose of the article is to create a database of errors and to develop an algorithm for a situational decision-making model taking into account availability of potential errors of air traffic controllers and pilots. Air traffic controllers and pilots typical errors were compiled and analyzed, arrays of specialists errors were created, binary error relations based on methods of discrete mathematics were also compiled in this article. This decision is caused by the need to formalize the interaction of specialists, since each error of the air traffic controller can be compared with a certain set of pilot errors and vice versa. In case of further in-depth analysis, it is possible to expand the database by adding additional errors arrays of the adjacent point controller, aerodrome service, planning service, etc. The goal is formed after analyzing the features of simulator training in higher educational institutions. The peculiarity is the absence of hazardous factors during the simulator training. This training takes place according to the ideal model. Undoubtedly, this approach is aimed at developing the correct algorithm of actions in normal or abnormal flight conditions, but thus the trainee can’t work out the decision-making skills if there is an error in the ideal algorithm. At the same time, existing specialists face unintended errors every working day, so having experience in this field plays an important role in minimizing the impact of the human factor on flight safety. In our case, it is proposed to include such a dangerous factor as an unintentional error in the joint training program for air traffic controllers and pilots, which will improve the training quality of specialists.

MIMO-Terahertz in 6G Nano-Communications: Channel Modeling and Analysis

With the development of wireless mobile communication technology, the demand for wireless communication rate and frequency increases year by year. Existing wireless mobile communication frequency tends to be saturated, which demands for new solutions. Terahertz (THz) communication has great potential for the future mobile communications (Beyond 5G), and is also an important technique for the high data rate transmission in spatial information network. THz communication has great application prospects in military-civilian integration and coordinated development. In China, important breakthroughs have been achieved for the key techniques of THz high data rate communications, which is practically keeping up with the most advanced technological level in the world. Therefore, further intensifying efforts on the development of THz communication have the strategic importance for China in leading the development of future wireless communication techniques and the standardization process of Beyond 5G. This paper analyzes the performance of the MIMO channel in the Terahertz (THz) band and a discrete mathematical method is used to propose a novel channel model. Then, a channel capacity model is proposed by the combination of path loss and molecular absorption in the THz band based on the CSI at the receiver. Simulation results show that the integration of MIMO in the THz band gives better data rate and channel capacity as compared with a single channel.

The use of discrete mathematics algorithms in technological design

The article examines the possibility of the formalization of the main stages of a detail mechanical processing using discrete mathematics methods. There are examples of using the graph theory for modeling practically all the stages of the development of processing technology. For structural optimization of technological process it is offered to use the problem of the traveling salesman. The choice of technological bases may be carried out both with using the graph theory and the set theory. The article makes a conclusion about the effectiveness of discrete mathematics methods for the formalization of decision making processes during technological processing.

Development of Moscow region passenger transport system using discrete mathematics methods

Transformation of settlement systems, consolidation of existing cities to the level of agglomerations, conurbations and megalopolises require a qualitative change in the transport system. Today Moscow and the Moscow region are in the stage of active development and interaction. The most important issue is to ensure transport connectivity and territory accessibility. The high cost of infrastructure projects and their duration require high-quality justification of decisions to be made. This publication discusses the use of the theory of graphs in the development of Light-Rail-Transit (LRT) systems, which are planned to be used as an element of the new unified transport system of Moscow agglomeration. The materials for the article are data on the existing and planned development of Moscow region settlements, the basis for calculations is the mathematical apparatus of the theory of graphs. The result of the work is represented by recommendations on the use of the theory of graphs for the purposes of substantiation of Moscow region transport system development, as well as main areas for continuing research were determined.

Система целей обучения дискретной математике будущих учителей математики и информатики в условиях интегративно­модульного подхода к образованию

The article deals with the theoretical foundations, methodological features and practical problems of teaching discrete mathematics to future teachers of mathematics and computer science in terms of integrative-modular approach to the organization of the educational process. In particular, the multilevel system of purposes of continuous teaching discrete mathematics according to the scheme “school — bachelor — master” is constructed. The role of discrete mathematics in the modern world, science and education is analyzed. The necessity of introduction to the basic questions of discrete content line of schoolchildren and students is justified. The place of discipline “Discrete Mathematics” in the system of modern domestic general and higher education is revealed. Methodical expediency of development and implementation in the pedagogical practice of continuous learning methods of discrete mathematics in the system “school — bachelor — master” on the basis of integrative-modular approach to the organization of the educational process is justified. Based on the basic provisions of the competence approach, guided by the principles of fundamentality, continuity, integrativeness, professional orientation and modularity, a multi-level model of subject competencies — a system of goals for teaching discrete mathematics to future teachers of mathematics and computer sciences has been developed. The issues of practical implementation of the developed method are considered.

Проблемы и методы экспериментальной географической экологии

The author’s paradigm of experimental geographical ecology and its core-landscape ecology—has been set forth. The empirical statistical models and the ecological geographical concepts developed on their basis are described. They disclose the mechanisms of formation of regional- and local-level landscape-ecological systems, their natural and anthropogenic dynamics and evolutionary trends. New provisions on the mono- and polysystemic organization of the geographical environment have been put forward and substantiated: 1) environmental principles and mechanisms for the formation of zonal geospace and its borders; 2) the concept of polymorphism of landscape-zonal systems as a basic model for the further development of integrated physical geography; 3) a paleo-forecasting concept that describes the predicted scenarios and their paleogeographic analogues as a single system of global changes in the environment; 4) topo-ecological forecasting concept as a scientific and methodological basis of local geosystem monitoring; 5) new provisions of the theory of sustainability of natural ecosystems and methods for calculating their functional stability based on the parameters of the biological cycle. A methodology for numerical landscape-ecological forecasting developed by the author and brought to the prescription level is presented, and regional and local landscape-ecological forecasts are given based on forthcoming global climate changes. The predictive ecological analysis was performed by models of base states of the geo(eco-)systems and there climatogenic changes, with application of methods of empirical simulation techniques for the estimated prognostic situations. A strategy is proposed, and algorithms are developed for applying landscape ecology methods for quantitative forecast estimates of the biotic regulation of the carbon cycle under global climate changes. In the constructed models, geographical ecology acquires an effective formalized tool for analysis and prediction using the methods of discrete mathematics for the processing and generalization of bulk empirical data obtained from field and laboratory landscape studies.

Innovative solutions of the underground exploitation of coal deposits

The article offers the development of a tool to estimate the state of operation schedules of coal mines relying upon the analysis of actual state of an operation schedule taking into consideration hierarchic connections between the system components. Comprehensive research method has been applied. The method involves statistical analysis to determine the dependence between performance of potential index and a number of independent indices concerning production and commercial operations of a mine, and Solow-Swan model method to estimate level of potential and coal mine reception of innovations. Methods of discrete mathematics have been used in the context of network models to optimize operational parameters and to normalize a structure of manufacturing relations. Regularities, concerning formation of production and commercial operations of mines, have been determined. The regularities involve concentration level of mining, labour productivity, stope advance, and prime cost of end product. Mathematical models, describing “performance potential” index, have been developed. The index is based upon the analysis of dependences between it and mining and geological as well as technological parameters of a mine. It has been proved that simulation of optimum parameter values of operation schedule means complete implementation of economic potential of the mine, i.e. ultimately achievable (reference) level since the level correspondence makes operation schedule of a mine receptive to innovations. A model has been proposed to analyze the efficiency of operation schedules of coal mines as for the estimation of a level of potential and coal mine reception of innovations. Further representation of a structure of manufacturing relations in the form of a network model helps reduce prime cost of mining and to increase labour productivity. The approach can be applied in the process of operative and long-term planning activities by coal industry enterprises as well as enterprises belonging to the related industries, and to determined ultimately achievable rate of economic surplus value. Corresponding software has been developed to implement the methods.