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2024 | Book

Mathematical Modeling in Physical Sciences

12th IC-MSQUARE, Belgrade, Serbia, August 28–31, 2023

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About this book

This volume gathers selected papers presented at the ICMSQUARE 2023 - 12th International Conference on Mathematical Modeling in Physical Sciences held in Belgrade, Serbia from August 28–31, 2023. This proceedings offers a compilation of cutting-edge research, which aims to advance the knowledge and development of high-quality research in mathematical fields related to physics, chemistry, biology, medicine, economics, environmental sciences, and more. Annually held since 2012, the ICMSQUARE conference serves as a platform for the exchange of ideas and discussions on the latest technological trends in these fields. This book is an invaluable resource for researchers, academicians, and professionals in these areas seeking to stay up-to-date with the latest developments in mathematical modeling.

Table of Contents

Frontmatter

Mathematical Methods in Physical and Engineering Applications

Frontmatter
Single and Multi-material Topology Optimization of Continuum Structures: ABAQUS Plugin

This research addresses the need for versatile topology optimization techniques capable of optimizing both single and multi-material designs. The extended Python code incorporates the modified Bi directional Evolutionary Structural Optimization (BESO) algorithm and a material interpolation scheme to enhance its capabilities. Engineers and designers can utilize this improved approach to optimize the performance of structures, especially in the context of multi-material configurations. To increase its practicality, the code is converted into ABAQUS plugin, seamlessly integrating it with the widely used finite element analysis software. Validation examples, conducted in the ABAQUS environment, demonstrate the compatibility and accuracy of the code. This research provides an efficient and accurate solution for topology optimization, addressing the demands of multi-material designs in various engineering applications. Examples of such applications include lightweight design in automotive and aerospace industries, customized implants in biomedical engineering, and optimal material distribution in architectural structures.

Dhaval Patel, Thomas Rockenbauer, Sandra Schlögl, Margit Lang
Adaptive Filtering of Distributed Data Based on Modeling the Perception Mechanisms of Living Sensory Systems

The article is devoted to the results of the adaptive filtering synthesis in the frames of neuromorphic model inspired by biological sensory system. To adequately model the perception mechanisms the analysis and synthesis are carried out on the base of the most realistic representation of input data—external stimuli—in the form of the stream of receptor registration events. Statistical model of that stream is chosen in the form of Poisson two-dimensional point processes. On this basis, a statistical description of the input data in the form of a sampling representation is proposed. The model of neural encoding of input data is considered in the article within the framework of the receptive fields concept. A number of well-known neuro-mechanisms are implemented in the encoding model, including in particular, center/surround inhibition. Decoding issues are considered in the context of stimuli spatial contrasts restoration, which partially models the responses of so-called simple cells in the visual cortex. It is shown that the model of coupled ON-OFF-decoding admits the sharp image details restoring in the form of local edges. At the end of the article, to justify the adequacy of the synthesized encoding procedure—adaptive filtering—we demonstrate an example of image edge directed interpolation and discus the results in the terms of perceptual quality.

Viacheslav E. Antsiperov
A Method for Fast and Robust Sungular Spectrum Analysis

One of the main problems encountered in the analysis of real-world time series, is the existence of outliers originating from various sources other than the dynamics of the series. Such sources can be a momentary noise or an imperfection in the recording devices. This problem has led to the need to construct robust methods using metrics other than the usual mean squared error. In combination with non-parametric techniques such as the very popular Singular Value Decomposition (SSA), it can give iterative algorithms that deal with this problem but create huge needs in computing power, especially in problems where we need fast response of the system (for example in high frequency trading). In this paper, a technique addressing both problems is presented which exploits the peculiarity of the Singular Value Decomposition (SVD) analysis of square symmetric matrices.

Adamantia Mavrogianni
An Algorithm for Non-negative Leslie Matrices

The Leslie matrix plays a crucial role in analyzing the changes in survival and birth rates, thus determining the population’s evolution. However, when it comes to evaluating continuous solutions of population growth models, discrete solutions are necessary. The Leslie matrix model, being a discrete model based on the age stage of the population, becomes particularly relevant in this context. In this work, we present a computational mathematical tool designed be used in the Leslie matrix model, which consists of reconstructing a Leslie matrix of any order from a given set of real numbers.

Jésica Pantáz, Jonnathan Rodríguez, Luis Medina
Statistical Approach to the Gompertz Growth Model and the Underlying Timescales

The Gompertz growth model is one of the most established paradigms in biology. It is widely used to describe the growth of bacteria and tumour cell populations, as well as plants and animals. Here, a statistical approach is used to model biological growth. This approach involves the probabilistic estimation of population–environment interactions. By applying this theory, we derive stochastic equations, of which the Gompertz growth model is a limiting case. This is confirmed by numerical simulations. We examine the dynamic modes in dependence on the timescale relation. By altering the invariant measure, we suggest modified forms of the stochastic Gompertz growth model.

A. Samoletov, B. Vasiev
On 4-dimensional Hom-Lie Algebras

In this paper we obtain the classification of complex multiplicative non-Lie Hom-Lie algebras of dimension 4 associated to a particular linear map. This provides the first isomorphism classes of this type of algebras in dimension 4.

María Alejandra Alvarez, Sonia Vanesa Vera
Optimal PD-PD and State Feedback Control of Underactuated Ball and Beam System with Uncertainty and Disturbances

This research paper investigates the optimal control of the ball and beam system, a challenging fourth-degree system that is commonly used in control studies. Four different control techniques, namely cascaded PD-P, cascaded PD-PD, optimized cascaded PD-PD control using a genetic algorithm, and state feedback control, are compared in terms of their performance. The study aims to achieve specific design criteria, including low rise time, less than three percent overshoot, and less than two seconds of settling time. In addition, the impact of disturbances, uncertainty, control action, and beam angle on the robustness of the system is also examined. MATLAB software is employed to implement and evaluate the performance of the different control methods. The findings contribute to the understanding of how to achieve optimal control of complex systems like the ball and beam system.

Yaman Sahu, Bibek Gupta, Mahmoud Wael, Raafat Shalaby
Are Referees Unfair to Away Teams? Evidence from the Premier League

This study examines the contribution of the crowd effect to home advantage in the English Premier League. Using the coronavirus 2019 pandemic as a natural experiment, we test crowd effects based on the changes in the number of audiences. We find that the difference in the number of audiences has an impact on the players’ performance, leading to the home advantage. Different from previous studies, which report a referee bias induced by crowd effect, our findings suggest that the crowd effect has no direct influence on referees. Specifically, more yellow cards against away teams results from more aggressive plays by these teams, a response to performance suppression by the home crowd. Findings suggest that social pressures caused by the crowd effect influence only players, not referees.

Gahyun Choi, Kwangwon Ahn, Hanwool Jang, Daniel S. Kim
The Use of Twins in Isotopic Analysis

Twin (from TW_ice IN_terval) is a set of two nested intervals. The first interval presents an internal estimation of the calculations result, the second one presents an external estimation. The most of chemical elements has two or more stable isotopes. A sample is characterized by the different ratio of stable isotopes, or the isotopic signature. This paper deals with an application of two types of twins for analysis of stable isotopic compositions. The first type of twins is the Nesterov twin, which semantically uses external estimates. With this type of twins we can say about the presence (existence) of the chemical element in the mixture. With another type of twins we can estimate the minimum and maximum contents of element that are possible in the sample. The suggested twin approach covers a very wide application set: the cosmology, global climatology, biology, Earth sciences, forensics sciences, etc.

Tatiana Iavoruk, Alexander Bazhenov
Metaheuristic Approaches to Tune PID Controller for Ball on Plate System

This paper presents a comprehensive study on the optimization of PID controller parameters for a ball & plate system through the utilization of Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). The objective of the study is to attain precise steady-state response while shortening settling time and minimizing overshoot. The assessment of controller performance is conducted using the Integral Absolute Error (IAE) cost function. The study highlights the limitations of conventional tuning techniques and the need for metaheuristic optimization algorithms, particularly when system models and variables are incomplete or imperfect, and demonstrates that both GA and PSO can improve the performance of the controller, but GA outperforms PSO in terms of fitness function value and optimal gains. The study also highlights the importance of accurate system models and variables in achieving optimal performance. The results of this study hold significant implications for the application of PID controllers and optimization algorithms based on metaheuristic principles in various applications, particularly in process industries. The study demonstrates the potential of metaheuristic algorithms for optimizing PID controllers and highlights the need for further research in this area.

Mitterand Ekole, Omer Abdalla, Iman Shalabi, Raafat Shalaby
Frequency Criterion for the Existence of Sliding Processes in Control Systems with an Arbitrary Variable Structure

The article proposes a criterion for the existence of sliding processes according to the frequency characteristics of the control device and the control object. This approach is proposed for the first time, in contrast to the method of phase trajectories and state space equations used to analyze systems with a variable structure. This approach allows us to formulate several practical engineering provisions that are very important for the implementation of this control method in real electric drives. The main conclusions are confirmed by modeling.

V. L. Kodkin, E. A. Kuznetsova
Least Squares for Generalized Gauss–Laplace Distribution of the Error in Certain Nonlinear Regressions with Perpendicular Offsets

There are mainly two methods of calculating the parameters of the regression equations: the minimization of the squared errors (or least squares, LS) and the maximization of the likelihood. Regarding the distribution of the error of experimental observations, there are several theoretical distributions, but two of them are on the one hand better known and on the other easily generalizable into one (Gauss–Laplace, GL): the normal (or Gaussian) distribution and the double exponential (or Laplace) distribution. In the construction of the squared errors is possible to replace the classical vertical offsets (which are the sides of the squares of the errors) with perpendicular ones, more suited when all variables are equally subjected to experimental errors. In the present work, it is proposed to use an iterative algorithm for the calculation of the regression parameters using LS of perpendicular offsets under the assumption of GL distributed error. The method is exemplified on a non-linear regression model.

Carmen E. Stoenoiu, Lorentz Jäntschi
On the Influence of Absorption on the Spatial Spectrum of Radio Waves Propagation in the Polar Ionosphere

Statistical characteristics of scattered radio waves propagating in the terrestrial high-latitude conductive ionosphere are investigated considering diffraction effects applying the modify smooth perturbation method. Polarization coefficients of scattered ordinary and extraordinary waves in the polar region of the terrestrial ionosphere are obtained for the first time. Correlation function and the variances of the phase fluctuations has been derived for arbitrary correlation function of electron density fluctuations. These statistical characteristics include the polarimetric parameters, Hall’s, Pedersen and longitudinal conductivities, anisotropy parameters of elongated plasmonic structures and the velocity of a plasma flow. Numerical calculations are carried out for the anisotropic Gaussian correlation function containing anisotropy factor and the angle of inclination of these irregularities with respect to the lines of forces of a geomagnetic field using the satellite and remote sensing observation data for the polar ionosphere. Diffraction effects becomes important in proportion of distance propagating by a wave in the ionosphere. Spatial power spectrum has a maximum along the direction of an incident wave.

George Jandieri, Nika Tugushi, Irma Mindiashvili
Computational Bar Size Optimization of Single Layer Dome Structures Considering Axial Stress and Shape Disturbance

A computational method is proposed in this paper to minimize the material usage in the construction of modern spatial frame structures by prestressing a minimal number of members. The computational optimization is conducted in two steps. Firstly, a numerical model of a single-layer dome structure is used to minimize the cross-sectional area through several iterations. Different assumed ratios (r) ranging from 0.95 to 0.75 are multiplied by the designed cross-sectional area, and the optimal actual ratio (R) is determined through multiple steps using MATLAB. The selection of the optimum ratio is based on ensuring structural stability and considering various constraints. Secondly, a computational optimization is performed using the fmincon function in MATLAB, which employs an interior-point optimization algorithm to search for the minimum summation of the function. The algorithm is designed to exclude actuators with negligible actuation, thereby minimizing the number of actuators. Constraints are set on the stress of all members and the nodal displacements to maintain the desirable shape of the optimized structure. The obtained results demonstrate that the cross-sectional area of the numerical dome structure can be reduced by up to 18% by prestressing only nine members. The validity of the results is confirmed by comparing them with those obtained from MATLAB and SAP2000 software.

Ahmed Manguri, Najmadeen Saeed, Farzin Kazemi, Neda Asgarkhani, Marcin Szczepanski, Robert Jankowski
Algorithmic Feature Selection and Dimensionality Reduction in Signal Classification Tasks

This paper presents a research endeavour addressing the recognition of acoustic emission signals, aiming to enhance their utilisation in non-destructive defectoscopy and machining process control. The classification task can be accomplished through two approaches: representing signals using a suitable attribute set, or directly passing the signals in their entirety to the classification algorithm. Our primary focus was on the meticulous selection of methods and tools for automating the extraction of a comprehensive set of features from the signals, followed by dimensionality reduction techniques. Subsequently, we conducted a comprehensive performance evaluation by comparing various classifiers applied to the low-dimensional projections. Lastly, we put the feature based classification approach to the test with direct signal classification employing convolutional neural networks.

Jan Zavadil, Václav Kůs, Milan Chlada
Two-Terminal Reliability of the K4-Ladder—Revisited

The exact calculation of network reliability has been a practical, but difficult problem, solved for some periodic graphs—including ladders—using recursion and transfer matrices. We revisit the results for non-oriented K4-ladders of arbitrary length, using Markov chains (transition matrices) instead of recursion. We consider three cases, where the edges and vertices of the ladder have independent reliabilities: (i) $$p$$ p and $$1$$ 1 respectively (3 × 3 transition matrix); (ii) $$p$$ p and $$\rho$$ ρ respectively (4 × 4 transition matrix); (iii) varying throughout the ladder (6 × 6 transition matrices, indexed). To use transition matrices seems rather new in the context of graph reliability. These matrices have a clearer interpretation than their analogs, the transfer matrices, obtained from recursion: their entries represent probabilities to transit from one state to another when eliminating a K4 block. As a consequence, secondary results may be derived more easily—by modifying the initial and targeted states, or by invoking classical results in Markov chain theory.

Philippe Poulin, Simon R. Cowell, Valeriu Beiu
Comparative Analysis of Authentication Using Formant Features of Vowels and Consonants

This article discusses 3 different biometric images of the user, which are based on the formant characteristics of the sounds of the Russian language. In the 1st biometric image, the formant characteristics of vowels (including diphthongs) and consonants are used, in the 2nd—vowels and diphthongs, and in the 3rd—consonants. A cluster of algorithms is presented, consisting of the Random Forest algorithm, K Nearest Neighbors, Logistic Regression, Multilayer Perceptron and Support Vector Machines. On the basis of a cluster of algorithms, models have been developed and trained. The mechanism of creation and training of each model is considered. To compare the effectiveness of user biometric images, an experiment was conducted in which 10 people participated. In total, users pronounced 46 sounds (6 vowels, 4 diphthongs and 36 consonants). Based on the provided data, samples were formed for each biometric image. Each sample, when it enters the input of a cluster of algorithms, is divided into training and test. A comparative analysis of the authentication procedure with each biometric image was carried out. For this, the following parameters were used: the average calculation time, the average accuracy value and the average accuracy value on the test sample.

Yelena Belova
Hidden and Singular Attractors in Nonlinear Systems of Differential Equations

The concept of a “hidden” attractor is widely used in the modern literature on autonomous nonlinear chaotic systems of ordinary differential equations in the case when the system has an chaotic attractor and together with it has either no singular points or one or more stable singular points. It is shown in the paper that any hidden attractor of nonlinear system of differential equations is one of the singular attractors of the system in sense of the Feigenbaum-Sharkovsky-Magnitskii (FShM) universal bifurcation theory. That is any hidden irregular (chaotic) attractor is born during the implementation of FShM bifurcation scenario and is bounded non-periodic trajectory in phase space of finite or infinite dimension, which is the limit of the cycles of some cascade of Feigenbaum period-doubling bifurcations and contains in any of its neighborhood an infinite number of unstable periodic orbits.

Nikolai Magnitskii

Complex Systems and Complex Networks

Frontmatter
Nonlinear Features and Hybrid Optimization Algorithm for Automated Electroencephalogram Signal Analysis

In this study a hybrid approach for EEG data classification for detecting epileptic seizures was proposed. The proposed hybrid model combines Extreme Gradient Boosting (XGB), Random Forest (RF) and Light Gradient Boosted Machine (LGBM) as base models and Support Vector Machines (SVM) as final model. It is shown that the hybrid approach provides higher recognition accuracy in binary classification of epilepsy seizure detection task then the base classification models. In order to provide optimal performance and time spent in models hyperparameters optimization the AutoML genetic programming-based system TPOT was used. The dynamic characteristics of the electroencephalogram (EEG) signal, the sample entropy, the Hurst exponent, and the senior Lyapunov exponent in combination with the spectral characteristics in delta, theta, alpha, beta and gamma bands were used as informative features for the classification algorithms.

Lyudmila Egorova, Lev Kazakovtsev, Elena Vaitekunene
Properties of Model Networks Generated by Modified Small-World Scale-Free Fractal Tree Algorithm

We modified the small-world scale-free fractal tree algorithm by introducing the length distribution of branches that grow from the tree structure. The new algorithm provides a convenient tool for studying the self-organisation of small-world fractal networks, taking advantage of the simplicity of the model, where only the parameter $$\gamma $$ γ , the power-law exponent describing the degree distribution, can produce various small-world graphs with different values of the fractal dimension. The numerical result suggests the $$\gamma $$ γ -dependency of the fractal box dimension, $$d_\textrm{b} \simeq 1 + (\gamma -2)^{-0.5}$$ d b ≃ 1 + ( γ - 2 ) - 0.5 , and demonstrates that, when $$d_\textrm{b}$$ d b becomes less than 2 in response to $$\gamma $$ γ being greater than 3, it becomes difficult to maintain fractal properties while maintaining small-world properties. These results are in contrast to those for non-small-world fractal networks, where fractality can be maintained regardless of the values of $$\gamma $$ γ and $$d_\textrm{b}$$ d b . Understanding the relationship between $$\gamma $$ γ and $$d_\textrm{b}$$ d b in various types of fractal networks is important for interpreting empirical data from real networks.

Nobutoshi Ikeda
Slow-Fast Dynamical Systems with a Load Variation

We analyze a category of slow-fast dynamical systems that exhibit complex bifurcations as the dynamic parameter, representing the load on the system, undergoes variation. We establish that within such systems featuring distinct structurally stable dynamical states, a transition from the first regime to the second ensues as a consequence of load variation. We extend these concepts to investigate bifurcations in conceptual climate models, specifically those resulting from restrictions on greenhouse gas emissions.

Elena Savenkova, Sergey Vakulenko, Ivan Sudakow
The Problem of Premature Convergence in Engineering Optimization Problems

Optimization is a complex problem in mathematics and engineering applications, which has so far been solved using a variety of approaches based on heuristic methods, methods inspired by nature, methods based on the principles of fuzzy logic, and methods based on rigorous mathematical tools. The most important task is to combine various methods of modeling, optimization, design and management of complex systems within the framework of an integrated approach for a holistic description of the phenomena under study. The objective function of optimization problems combines continuous and discrete variables and various constraints, which indicates its complexity, as well as difficulties in solving such problems. Algorithms that are used in solving optimization problems may suffer from premature convergence when they stop at the optimal solution earlier than required. Based on the problem of premature convergence, a hybrid approach combining a sine–cosine algorithm (SCA) and an artificial bee colony algorithm (ABC) was proposed in the paper. In the proposed algorithm, called the hybrid sine–cosine algorithm (HSCA), both algorithms are executed alternately until the convergence criterion is satisfied.

K. A. Ponomareva, I. P. Rozhnov, L. A. Kazakovtsev
Multiple Fuzzy Soft Graphs Based on Maps and Consider Their Applications in Decision-Making

One of the most important hyper models in mathematics is the measure of fuzzy soft sets, which can handle more uncertainty associated with real-life problems. Therefore, it has a wider range of applications. The purpose of this work is to investigate and study a new model called fuzzy measure soft graphs (FMSG). Based on this model, we defined several notions like the totally regular fuzzy measure soft graph, uniform edge fuzzy measure soft graph, complement of fuzzy measure soft graph, and uniform vertex fuzzy measure soft graph. Further, using these ideas, we solved one decision-making problem for selecting the perfect parameter for universal V. In addition, several kinds of fuzzy measure topological spaces of mappings with strongly closed graphs, with a focus on $$FM\mho -\Phi$$ (resp. $$\Omega -\Psi \Phi$$ ) $$-$$ closed graphs, employing the thoughts for $$FM\mho -\Phi$$ (resp. $$\Omega -\Psi \Phi$$ ) $$-$$ open sets and $$FM\mho -\Phi$$ (resp. $$\Omega -\Psi \Phi$$ ) $$-{T}_{2}$$ . Finally, some properties of $$FM\mho -\Phi$$ (resp. $$\Omega -\Psi \Phi$$ ) $$-$$ closed graphs and almost $$\varphi -$$ minimized spaces have also been. investigated.

Shadia Majeed Noori, Shuker Khalil, Abd Ghafur Ahmad
Cross-Border Blockchain Transactions as a Tool for Adapting the National Economy of the Russian Federation to International Sanctions

In this study, we focused on the mechanisms for ensuring the sustainable development of the Russian economy in the context of correction of external financial flow owing to the adjustment of foreign economic supply chains. By constructing a system of recursive equations, it was established that a 1% reduction in volume of transnational financial flow creates the potential for a 0.45% decrease in Russia's gross domestic product. The localization of these risks can be achieved by implementing blockchain technologies in the Russian system of cross-border payments to provide a new foundation for financial flow under emerging systemic transformations in the external environment.

M. R. Safiullin, A. A. Dinmukhametova, R. T. Burganov

Sociophysics and Econophysics—Mathematical Models of Human Behaviour and Economics

Frontmatter
Remote Control of Mobile Robotic System Over Web-Based Interface for Supporting People with Physical Disabilities

The incapacity of people with physical disabilities to lead fulfilling lives is a global problem. Physical limitations lead to stress and discomfort, as may require the need for a caregiver in everyday activities. Mobile robotic systems can address and give solution to these problems, thus improving the quality of life. This paper presents an innovative method for human–robot interaction and collaboration: a concept for a mobile robotic system that aims to assist the people with physical disabilities. A unique feature is having two modes within the platform, which have separate tele-operation techniques, methods and functions. The proposed system provides direct remote control for the robot, using selectable predefined actions. The system’s feedback provides information, about the state of the robot, the camera and a visual representation of the data coming from the sensors.

Petko Stoev, Denis Chikurtev, Rosen Ficherov
Methodology for Development of Virtual Reality Based Resources for Education

The modern development of education is directly related to the development of information and communication technologies and their application in the learning process. Virtual reality is completely changing the way knowledge is transmitted and acquired. The application of technology in learning provides an opportunity for students to learn through experience. The use of virtual reality as an educational resource aims to visualize objects that in the real world are dangerous or difficult to monitor. Technology uses computer-generated objects with which the user can interact in real time. When applying the technology in the learning process, we can differentiate the following problems: need for simulated environment, need for simulated object models and need for interactive functions. This study explores systems and tools for modeling and developing educational resources designed for virtual reality. Software tools and applications for creating and processing three-dimensional models are presented. Interaction methods have been studied providing human-virtual objects interaction in a simulated environment. A methodology for optimizing the process of creating educational resources with virtual reality has been developed. The methodology presents all stages of the process of creating these resources—development of a simulated environment, development of objects and development of interactive functions for the user. The proposed methodology has the potential to improve the process of creating digital resources for virtual reality and increase their integration in learning.

Denis Chikurtev, Ava Chikurteva, Nina Bogdanova
Non-euclidean Geometries and Arts in the Beginning of the Twentieth Century: An Interdisciplinary Approach for High School Students

The action we are going to describe was carried out in a class of 27 students in the second grade of the High School (16–17 years old) IONIDIOS MODEL LYCEUM OF PIRAEUS, in the school year 2018–2019. The cooperative action of the mathematician and the philologist of the class, resulted from the common finding of a deficiency in the curricula which resulted in the students’ inability to approach holistically the changes imposed on society by the inseparable relation between art, science and technology. With this in mind, the action was designed and focused primarily on presenting trends in art and painting from Impressionism to Van Gogh in parallel with the progress and changes in science and society during that time. It was then attempted to link science, art and technology in the early twentieth century through an approach of artistic manifestations, principles and theories about mathematics and non-Euclidean geometries as they influenced architecture, sculpture, painting, the Bauhaus movement. The action was accomplished by achieving its objectives, since the artworks of high aesthetics produced by students, effectively incorporated the theory of modern mathematics and artistic avant-garde and highlighted the link between the scientific and artistic achievements of human creation.

Adamantia Mavrogianni, Georgios Kokorelis
The Impact of Sanctions Pressure on the Stability and Prospects for the Development of the National Economy of the Russian Federation

The main purpose of the study is to systematize and analyze the key parameters of the economic growth of the Russian Federation under the sanctions pressure of the 2022 model and substantiate the policy of intensification of import substitution as a key mechanism for ensuring sustainable development in the medium and long term in the new reality. The subject of the study is the sanctions restrictions imposed by a number of Western countries in relation to the Russian economy, the costs they generate and the opportunities for building a new model of economic growth. As the main results of the study, it is necessary to highlight the systematization of sanctions imposed on Russia in 2022; identified trends in the formation of key macroeconomic parameters of the Russian economy in the conditions of foreign economic perturbations, including the projection of the results of empirical analysis on theoretical models of economic dynamics (IS-LM, AD-AS).

L. A. Elshin, M. R. Gafarov, A. A. Dinmukhametova
Methodology for Assessing the Impact of Changes in Transnational Cash Flow on GDP Growth (on the Example of the Russian Federation)

The restrictions imposed by Western countries on the access of the Russian financial system to international clearing systems, if they do not localize the potential of foreign economic activity, then significantly complicate the processes of international trade, especially in terms of the growth of transaction costs. At the same time, an extremely important task is to assess the possible consequences caused by the correction of the movement of transnational cash flows in order to understand the possible risks of the sustainable development of national economic systems. This study is devoted to an overview of the emerging effects generated as a result of changes in the volumes of foreign economic supply chains. In it, within the framework of economic and mathematical modeling of perturbations in the field of correction of cash flows of the external contour, empirical estimates of a possible change in the growth rate of Russia's GDP are implemented.

M. R. Safiullin, A. Dinmukhametova, A. R. Sharapov
The Impact of Trading Environments on Commodity Futures: Evidence from Biofuel Feedstocks’ Network

This study analyzes two types of networks constructed through correlation and Granger causality, consisting of eight commodity futures that can be used as biofuel feedstock. In the correlation network, futures in emerging markets are isolated from other futures, for example, crude palm oil (CPO) and sunflower futures, whereas those in mature markets are strongly correlated. In the Granger causality network, however, CPO and sunflower futures markets are influential in the network; CPO and sunflower futures provide and receive information with high degree centrality inside the network. In particular, CPO futures is an important bridge node connecting other markets with high betweenness centrality. Findings suggest that futures markets of biofuel feedstock commodities are significantly integrated, and the role of each market cannot be ignored regardless of the trading environment.

Minhyuk Jeong, Kyohun Joo, Jinu Kim, Juyub Kim, Joohyung Kim, Kwangwon Ahn

Finite Element and Meshfree Methods

Frontmatter
Natural Convection Flow in an Inclined Wavy Porous Medium in the Presence of an Inclined Periodic Magnetic Field

In this study, unsteady natural convection flow in an inclined porous wavy cavity under the effect of an inclined periodic magnetic field (MF) is investigated. Mathematical model for the porous medium is chosen as Brinkman-extended Darcy model. Galerkin weighted residual finite element method is implemented to simulate the governing unsteady dimensionless equations. For discretization of time derivatives, Crank-Nicolson method is utilized. The obtained discrete nonlinear algebraic systems are iteratively solved by using the adaptive Newton’s method. The behavior of fluid flow and heat transfer is examined in distinct parameters as inclination angle of the cavity ( $$\theta _c=0^\circ -90^\circ $$ θ c = 0 ∘ - 90 ∘ ), amplitude ( $$A=0.025-0.2$$ A = 0.025 - 0.2 ) of wavy wall, the number of undulations ( $$N=1-4$$ N = 1 - 4 ), Hartmann number ( $$Ha=0-100$$ H a = 0 - 100 ), angle ( $$\theta _b=0^\circ -90^\circ $$ θ b = 0 ∘ - 90 ∘ ) and period ( $$\lambda $$ λ =0.1-1) of the periodic MF. The rise in Hartmann number has a dampening effect on both heat transfer and fluid flow. Period $$\lambda =1$$ λ = 1 and amplitude $$A=0.2$$ A = 0.2 has the most reducing influence on both fluid velocity and convective heat transfer. Oblique angle of periodic MF is efficient on fluid flow and heat transfer at $$\theta _b=45^\circ $$ θ b = 45 ∘ . Average Nusselt number is the largest at $$\theta _c=45^\circ $$ θ c = 45 ∘ when the periodic MF affects the system at $$\theta _b=0^\circ $$ θ b = 0 ∘ .

Shafqat Hussain, Bengisen Pekmen Geridonmez

Topics in Mathematical Physics

Frontmatter
Pseudogeometric Version of the Traveling Salesman Problem, Its Application in Quantum Physics Models and Some Heuristic Algorithms for Its Solution

The geometric version of the traveling salesman problem (TSP) has been extensively studied, leading to the development of various approaches for solving its special cases. However, these algorithms often fall short when applied to problems beyond the geometric TSP. In this paper, we explore the pseudo-geometric TSP version, a generalization of the geometric TSP, and propose an adapted geometric algorithm for solving its specific instances. We leverage the knowledge of error bounds to estimate the reconstruction error of the TSP solution even when using geometric approaches for the pseudo-geometric TSP. This allows us to achieve reliable results despite uncertainties or noise in the data. We provide a concise description of our algorithmic adaptation and present the results of computational experiments to demonstrate its effectiveness.

Boris Melnikov, Dmitrii Chaikovskii
Solution to Infinity Problem of Scattering Matrix Using Time-Evolution Operators Without Needing Renormalization

The current situation of research challenging the demanding tasks of renormalization implies that the present framework of quantum scattering theory does not offer good prospect and therefore in parallel with the development of renormalization, it is desirable to attempt to formulate a new theory able to solve the infinity problem fundamentally in a general way. Our purpose is to construct an alternative mathematical formulation capable of ensuring the convergence of the scattering matrix without relying on renormalization theory, thus preventing overlapping divergences of the scattering matrix in principle. We present alternative representations of the scattering matrix in terms of the local and global time-evolution operators which replace the Dyson series and do not need the Feynman diagram. Importantly, the obtained results clarify that substantially, there does not exist the infinity problem of the scattering matrix within the framework of our formulation. Ultimately, we draw the successful conclusion that it is possible to conceive of an alternative to the conventional scattering theory and our formalism as a new proposal can contribute to formulating a consistent theory without infinity and renormalization.

Chol Jong
Nonlinear Loading and Damping of a Single Degree of Freedom Oscillator

A single degree of freedom oscillator system subject to regular sinusoidal loading does exhibit a well-known linear solution. In real situations, we shall add damping as there in any physical system will be a damping mechanism that will limit the oscillations. In case of forced oscillations in a fluid, the loading and the damping may be nonlinear and the response may then exhibit nonlinear characteristics. A simple drag loading term proportional to the square of the oscillator's velocity is studied as an example of a system exhibiting limit cycles: When the external loading and the oscillator are in higher order resonances, where the natural period of the loading is an integer of the natural period of the system, resonances will occur. The resulting response of the system is a limit cycle oscillation. Furthermore, for oscillators with non-negligible motions, the motion will act as a nonlinear damping term, that for some situations may be interpreted as “negative damping.” In case of irregular wave loading, the system can get into higher order resonances occasionally (when the periods of the main waves are an integer of the oscillator’s periods), triggering large temporary limit cycle motions. A review of other nonlinear damping models is also given with a brief discussion of the performance of systems exposed to these damping models.

Ove Tobias Gudmestad
Correction for the Classical Conditions for a Collision in Three-Body System Using General Relativity and Machine Learning

This paper comprehensively investigates collision conditions in three-body problems, incorporating General Relativity (GR) effects. The study analyzes the initial values of the bodies to determine the collision possibility and develops a high-accuracy machine learning model for classifying collision events. The study introduces the concept of GR-effective potential energy derived from the Einstein Field Equations and solves the equation using the Schwarzschild solution for spherically symmetric gravity fields. Additionally, a code is developed to examine collisions using the GR-effective potential energy, and a machine learning model is trained accordingly. The study provides a modified equation that accurately describes the collision condition in three-body problems, accounting for GR effects. The results have significant implications in astrophysics and contribute to advancing knowledge in understanding celestial body dynamics in collision scenarios.

Hadi Salloum, Manuel Mazzara, Mohammad Reza Bahrami
A Novel Way of Calculating Scattering Integrals

The technique coined as NDIM—Negative Dimensional Integration Method by their discoverers, relies on a three-pronged basis: Gaussian integration, series expansion and analytic continuation. The technique has been successfully applied to the calculation of covariant and non covariant Feynman integrals in a generic dimensional regularization space, i.e., D-dimensional space-time for D including the negative domain values. Since the dimensionality is general, we can use specifically for one-dimensional integrals. In this work we show how this technique can be applied to tackle certain improper integrals and give an example of a particular improper integral that appears in quantum mechanical scattering process. Traditionally, improper integrals are ascribed certain values through the limiting approach or as is known, by the Cauchy principal value via residues concept technique. Here we use the NDIM approach to do the calculations and show it works fine for the improper integrals. This novel approach we believe is more straightforward and does not require to handle poles, residues, or difficult closed contours as in the traditional approach.

Alfredo Takashi Suzuki, Timothy Suzuki
Study of Linear Stability for Cylindrically Symmetrical States of Dynamic Equilibrium of Two-Component Vlasov–Poisson Plasma

We consider the linear stability problem for dynamic equilibria of two-component Vlasov–Poisson plasma in cylindrically symmetrical statement. The hydrodynamic substitution of independent variables is performed in order to transform the Vlasov–Poisson equations to an infinite system of gas-dynamic equations. It is important that exact stationary solutions to gas-dynamic equations are equivalent to exact stationary solutions to the Vlasov–Poisson equations. The sufficient condition of linear stability for exact stationary solutions to the Vlasov–Poisson equations is studied. Previously, this condition was not reversed either for small or, especially, for finite perturbations. To fulfill such reversion in the linear approximation, these gas-dynamic equations are linearized near their exact stationary solutions. The a priori exponential estimate from below is constructed for a subclass of small cylindrically symmetrical perturbations of exact stationary solutions to gas-dynamic equations, which grow over time and are described by the field of Lagrangian displacements. The countable set of sufficient conditions for linear practical instability is obtained. Thus, the Newcomb-Gardner-Rosenbluth sufficient condition for linear stability of exact stationary solutions to the Vlasov–Poisson equations is reversed. Moreover, a formal nature of this condition is revealed with respect to the considered small perturbations. As a result, by the direct Lyapunov method, an absolute instability for exact stationary solutions to the mathematical model of two-component Vlasov–Poisson plasma in relation to small cylindrically symmetrical perturbations is proved.

Yuriy G. Gubarev, Jingyue Luo
Ranking of Linear Spaces in Generating a Hierarchy Between Concepts

Human perception is composed of fundamental concepts. If we associate these with states in Hilbert spaces such that the fundamental concepts are associated with the spanning sets of the spaces, a composed concept can be defined through the superposition of states within the selected spanning sets. Under these definitions, no particular spanning set can be defined as being fundamental. Therefore, in this article, we describe the concept of a hierarchy of spaces.

Yehuda Roth

Physical Modeling Using Stochastic Differential Equations

Frontmatter
Monte Carlo Solution of Semi-linear Helmholtz Boundary Value Problem

In these work we will study a probabilistic representation of the solution of the Helmholtz boundary problem for the non-linear problem $$\begin{aligned} -\Delta u(x) + cu(x) = g \cdot f(u),\quad x \in D,\quad u{|_\Gamma } = \psi \end{aligned}$$ - Δ u ( x ) + c u ( x ) = g · f ( u ) , x ∈ D , u | Γ = ψ where, f(u) in our case could be hyperbolic functions sh(u) or ch(u). Under the assumption of the existence of a solution, an unbiased estimator is constructed on the trajectories of the proposed branching process “walk on spheres”. To do this, using Green’s formula, a special integral equation is written that connects the value of the function with its integrals over a ball and a sphere of maximum radius centered at a point and entirely contained in the region under consideration. A probabilistic representation of the solution of the problem in the form of the mathematical expectation of some random variable is obtained. In accordance with the probabilistic representation, a branching process of walk on spheres is constructed and an unbiased estimator of the solution of the problem with finite variance is constructed on its trajectories.

Abdujabbor Rasulov, Gulnora Raimova
Strongly Entangling Neural Network: Quantum-Classical Hybrid Model for Quantum Natural Language Processing

One of the most used techniques to improve a Machine Learning model is to gather more data. An interesting field in Machine Learning is Sequence Modelling, having Natural Language Processing as the peak of the field. The capabilities of Quantum Computing have been growing recently entering the novel field of Quantum Machine Learning. In this paper, we propose a Quantum Natural Language Processing classification model named Strongly Entangling Neural Network. This model leverages the quantum advantage to imitate part of the behavior of a Recurrent Neural Network to process text data into the circuit and perform the classification task. This is accomplished by representing our data in a quantum circuit that relies heavily on the entanglement property of qubits. The results of our model have very favorable metrics, particularly obtaining a $$97.70\%$$ 97.70 % of accuracy.

J. Ismael Díaz-Ortiz, Axel Villanueva, Francisco Delgado

COVID-19 and Virus Spreading Mathematical Modeling

Frontmatter
Mathematical Analysis and Optimal Strategy for a COVID-19 Pandemic Model with Intervention

The COVID-19 pandemic has spread to every corner of the globe. The virus was first spotted in Wuhan, China, in December 2019 and has since spread worldwide. It has impacted every one of us in the hardest way possible. In this paper, a mathematical model based on differential equations is proposed. This model depicts the infection patterns of COVID-19 transmission, taking asymptomatic individuals and hospitalization into account. The basic reproduction number is computed using the next-generation matrix method. This is found to be a critical signal describing the dynamics of the COVID-19 transmission. The local stability of the steady states has been investigated. The model’s global stability is demonstrated using the second method of Lyapunov and the LaSalle invariance principle. In addition, an optimal control problem is formulated to reduce fatality by considering pharmaceutical intervention options as control functions. COVID-19 transmission dynamics change when an intervention is introduced. In order to limit the number of infected individuals and reduce control costs, an appropriate objective functional has been developed and solved using Pontryagin’s maximal principle. Furthermore, extensive simulations have been conducted for various initial conditions and parameter values in order to validate the theoretical aspects.

Padma Bhushan Borah, Hemanta Kumar Sarmah
Modeling Public Fear Under the Information Environment of Emergencies as COVID-19 and Wars

Under the recent pandemic of corona-virus disease COVID-19, people have repeatedly reacted to the information concerning the spread of the epidemic, which has flooded in our society throughout the whole time. By using the statistical data obtained in Japan, a mathematical model treating the relation between the information environment and the public reaction caused by fear is proposed. The input quantities for this model are the time-varying environment of information on the COVID-19 together with the amount of public communication or the amount of information search on the COVID-19 via the SNS. Between those two quantities, we introduce three variables such as the amount of information released by the news media every day, the novelty of information, and the extent of people’s surprise to the information. Regarding the amount of information, we use the real quantity observed in Japan, whereas the novelty of information is related to the so-called oblivion function by which the extent people forget the information in the past is given. Two types of oblivion function, the power-law type and the exponential type, are introduced. On the other hand, the extent of people’s surprise by the information on COVID-19, which is determined so as to make sure the consistency of the model as a whole, has revealed to have sharp peaks at the earliest time of the pandemic, and the beginning of every emergent-state declarations over eight times. Such a reaction around the people’s surprise is due to the emotional contagion in the fearful field of information environment. To compare the time-dependent trend of those variables to the other cases of social events, we have introduced the case of Ukraine-Russia conflict reported in Japan since 1 January 2022. The people’s surprise in this case has revealed that people astonished only at the first time of the conflict, without any surprise after that even the novelty of information has been high. Discussion was made on the sources about the difference regarding the people’s surprise between the cases COVID-19 and Ukraine-Russia conflict.

Teruaki Ohnishi
Chaotic Model for Development of HIV Virus

In this paper we focus on the numerical treatment of nonlinear system of differential equations arises from biological model. Three numerical methods namely, Runge Kutta method differential transform and multistep differential transform methods are applied to solve the system. Some numerical examples are solved to show the validity and applicability of these numerical methods. Numerical results show clearly that the multistep differential transform method is one of the most efficient and accurate method in comparison with its counterparts.

Mohammad Qabaja, Jihad Asad, Rania Wannan
Stability Analysis of a COVID-19 SEIQR Model with Switching Constant Transmission Rates

In this paper, we analyze threshold conditions for a COVID-19 susceptible-exposed-infectious-quarantined-recovered (SEIQR) model with a constant recruitment rate and a piecewise constant disease transmission rate using results from the theory of switching systems. We establish that if the basic reproduction number of the system under each mode p is less than 1, then the corresponding solutions tend to the disease-free equilibrium. Further, this condition is sufficient but is not necessary to guarantee disease eradication. Simulations show that it is possible to keep the transmission of infection at bay even if the basic reproduction numbers under some modes are not less than 1.

Timothy Robin Y. Teng, Destiny S. Lutero, Mark Anthony C. Tolentino

Mathematical Modeling for Sustainable Development

Frontmatter
Modeling Air Pollution Data Using a Generalized Birnbaum-Saunders Distribution with Different Estimation Procedures

As the detrimental impact of air pollution becomes more prevalent; it is crucial to accurately model the distribution of air contamination levels. In practice, the Birnbaum-Saunders distribution is a well-known lifetime model for modeling positively skewed phenomena. Due to its relationship to the normal distribution and other desirable properties, different generalizations for the Birnbaum-Saunders distribution have been extensively studied to improve its flexibility. This article discusses inferential procedures of a specific generalization developed previously but has not received much consideration regarding this specific research area. The considered generalization has an extra shape parameter, known as Type-II generalized Birnbaum-Saunders distribution. The generalized family is more flexible than the Birnbaum-Saunders classical model, as it shares similar properties and can display both unimodality and bimodality. Eight frequentist estimation procedures are considered in this article, including the maximum likelihood and the maximum product of spacing estimation, several regression-based estimations, and goodness-of-fit estimations. Monte Carlo simulations are performed to investigate the estimation efficiency of the methods under various combinations of shape parameters, some conclusions are presented. Furthermore, air pollution concentration data are analyzed using the considered methods to illustrate their practical application. Overall, the analysis results favor the least-squares estimators according to the goodness-of-fit criteria.

Bushra Saad Alosaimi, Farouq Mohammad Alam, Hanan Mohammed Baaqeel
Improved Routing for the In-Band Network Telemetry Problem

The continuous growth of networks imposes on managers to find new technologies for better handling of traffic measurement for network wide-visibility (i.e., the ability to access most desired flow-level information per hop for all flows), performance monitoring requirements and network robustness (i.e., the ability to survive partial network failures). In-band network telemetry has emerged as one of the latest network monitoring paradigms for the mega-scale network. Previous research has focused on orchestrating the processes of collecting in-band network telemetry data from the network and addressed the scalability. Yet little has been done to route network flows efficiently to address the low network coverage and monitoring applications performance in this new network monitoring paradigm. This paper analyses two mathematical models for in-band network telemetry. The first model, proposed by [1], exploits shortest path flow routes as inputs. The second model proposed in this work, extends the first to compute flow route paths efficiently, providing a comprehensive solution to the problem.

Thierno Bocar N’Diaye, Marcelo Caggiani Luizelli, Jules Degila, Luciana Salete Buriol
House Price Prediction Using XG-Boost Grid Search and Cross-Validation Methods

The primary factors influencing the variation in housing prices in different locations are the characteristics of the housing and the conditions of the area. This study will utilize the popular machine learning technique, XGBoost, along with grid-search and cross-validation techniques in deep learning. The chosen data set is appropriate as it provides adequate representation of both the range of house prices and the diversity.

Tianlei Zhu, Yingke Yang, Shu Bao, Hassan Raza
The Effectiveness of Submerged- Emerged Breakwaters: An Analytical and Numerical Study

In this study, we investigate the wave attenuation by mangroves and submerged rigid breakwaters using a modified shallow water model. We aim to capture the interaction of waves with vegetation and breakwaters and quantify the wave height reduction. We derive the wave transmission coefficient analytically and numerically and examine the dependence of every parameter involved, from the structure and vegetation to the wave attenuation. We will show that both mangroves and submerged rigid breakwaters can effectively attenuate wave energy, with the degree of attenuation depending on various factors such as the width and length of the vegetation, the height and spacing of the breakwater, and the incident wave characteristics. Our findings have important implications for coastal management and the design of coastal protection structures.

Ikha Magdalena, Vinsensia Ferren
Validation and Numerical Simulation of a Parabolic Trough Solar Collector Plant Using an Implicit Finite-Difference Scheme

The evolving energy landscape demands reliable integration of renewable resources to ensure security, sustainability, and reduced carbon emissions. Then, a high level of understanding about the behavior of the renewable energy sources, the technologies, and their interaction is needed. Particularly, parabolic trough collector (PTC) solar plants can positively affect power systems due to their dispatchability. Since PTC harvests direct normal irradiance (DNI), complex dynamic models are required to understand how PTC behaves. This work uses a 1-D dynamic model that captures the dynamic behavior of PTC plants and is solved numerically via a three-point backward finite-difference. Through two case studies, this research highlights the dynamic behavior and performance of the PTC rows and total solar fields. The row response to varying DNI is showcased in the first case study. The model predicts heat transfer fluid (HTF) temperature profiles in 0.7 s, paving the way for effective control strategies to prevent overheating. The second case study extends the analysis to disturbances caused by clouds and shadows, reaffirming the model’s robustness. With a computation time of 2.7 s for temperature profiles over a day, the model offers both accuracy and computational efficiency for real-time applications.

Richard Rangel, Lesme Corredor, Marco Sanjuan
Transition to Rational Models of Production and Consumption in the Framework of the UN Sustainable Development Goals

The article is devoted to the problems of transition to rational models of production and consumption as the basis for the formation of a circular economy. The authors consider the evolution of theoretical approaches to the formation of the concept of circular economy. The paper determines the factors of transition to rational models of production and consumption, and proposes an economic and mathematical model for a comprehensive assessment of the impact of economic, social, demographic factors on the development of a circular economy. The national initiatives of individual countries in the field of regulating the development of the circular economy for the implementation of the UN Sustainable Development Goals were studied in the article. Indicators are proposed that evaluate the effectiveness of achieving the UN SDGs. The authors analyse best practices of implementing the circular economy model by individual companies and countries. Finally, the research identifies the problems of Russia's transition to a circular economy and proposes ways to solve them.

S. Absalyamova, L. Zulfakarova, G. Shafigullina, R. Sakhapov, M. Makhmutov
The Chemical Elements with Their Applications in Fuzzy Delta-Algebraic Systems

The preponderance of algebraic system applications has lately been generalized to many types of $${\text{fuzzy}}$$ fuzzy algebras. We created the use of a spatial class of algebras called $${\text{Delta}}\,-$$ Delta - algebras. Using $${\text{Delta}}\,-$$ Delta - algebra, we explored a new form of algebraic semi groups including, $${\text{Delta}}\,-$$ Delta - semigroup, $${\text{Delta}}\,-$$ Delta - subsemigroup, $${\text{Delta}}-{\text{ideal}}$$ Delta - ideal semigroup, $${\text{Delta}}-{\text{ideal}}$$ Delta - ideal semigroup homomorphism, and $${\text{Delta}}\,-$$ Delta - semigroup homomorphism. Some application about the atomic numbers and valency of the chemical elements are studied in $${\text{Delta}}\,-$$ Delta - algebras. Next, $${\text{fuzzy}}$$ fuzzy logic $$(FL)$$ ( F L ) terms include $${\text{fuzzy}}$$ fuzzy $${\text{Delta}}\,-$$ Delta - subsemigroup, $${\text{fuzzy}}$$ fuzzy $${\text{Delta}}-{\text{ideal}}$$ Delta - ideal semigroup, $${\text{fuzzy}}$$ fuzzy $${\text{Delta}}-{\text{ideal}}$$ Delta - ideal semigroup, bipolar $${\text{fuzzy}}$$ fuzzy $${\text{Delta}}\,-$$ Delta - subsemigroup, bipolar fuzzy $${\text{Delta}}-{\text{ideal}}$$ Delta - ideal semigroup, and bipolar fuzzy $${\text{Delta}}-{\text{ideal}}$$ Delta - ideal semigroup are introduced. In addition, specific basic aspects of our notions are studied and addressed. Any $${\text{Delta}}-{\text{ideal}}$$ Delta - ideal in a classical collection is not required to be a $${\text{Delta}}$$ Delta $${\text{par}}\,-$$ par - ideal. Therefore, in this paper, we show this matter is holed after they are generalized in $${\text{non}}-{\text{classical}}$$ non - classical sets like $${\text{fuzzy}}$$ fuzzy sets $${\text{and}}$$ and bipolar $${\text{fuzzy}}$$ fuzzy sets, and they also take a novel form in algebraic semi groups. This article investigates the $${\text{Delta}}-{\text{semigroup}}$$ Delta - semigroup homomorphism $${\text{image}}$$ image , translations, and $${\text{product}}$$ product characteristics of bipolar $${\text{fuzzy}}$$ fuzzy ( $${\text{Delta}}/\mathrm{ Delta par})-{\text{ideals}}$$ Delta / Delta par ) - ideals semigroups, and their applications are shown.

Manal Al-Labad, Shuker Khalil, Ahmed Naji Hassan
Metadata
Title
Mathematical Modeling in Physical Sciences
Editor
Dimitrios Vlachos
Copyright Year
2024
Electronic ISBN
978-3-031-52965-8
Print ISBN
978-3-031-52964-1
DOI
https://doi.org/10.1007/978-3-031-52965-8

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