Mathematical Model Of Problem Research Articles

Excitation-based fully connected network for precise NIR-II fluorescence molecular tomography.

Fluorescence molecular tomography (FMT) is a novel imaging modality to obtain fluorescence biomarkers' three-dimensional (3D) distribution. However, the simplified mathematical model and complicated inverse problem limit it to achieving precise results. In this study, the second near-infrared (NIR-II) fluorescence imaging was adopted to mitigate tissue scattering and reduce noise interference. An excitation-based fully connected network was proposed to model the inverse process of NIR-II photon propagation and directly obtain the 3D distribution of the light source. An excitation block was embedded in the network allowing it to autonomously pay more attention to neurons related to the light source. The barycenter error was added to the loss function to improve the localization accuracy of the light source. Both numerical simulation and in vivo experiments showed the superiority of the novel NIR-II FMT reconstruction strategy over the baseline methods. This strategy was expected to facilitate the application of machine learning in biomedical research.

On fixed point theorems in modular spaces characterized by $C^*$-class functions

Generally, finding a solution to a theoretical mathematical modelling problem is equivalent to finding a fixed point for a suitable operator. Accordingly, fixed point theory is therefore a very important and crucial in many areas, such as mathematics, sciences, and engineering. A very popular and important fixed point theory is those formulated by Stefan Banach in 1922. The theory is related to a complete normed space and known as the Banach fixed point theory. Recently there have been numerous generalization of the Banach fixed point theory. One of them is a fixed point theory in modular spaces. In this paper, we will formulate some fixed point theorems in modular spaces by using $C^*$-class functions. The obtained results generalize and improve some results in [Supama, On Some Common Fixed Point Theorems inModulared Spaces, International Mathematical Forum, Vol. 7, no. 52, (2012), 2571 - 2579.].

Flow and thermal analysis of Oldroyd 8 constant fluid in a porous channel

AbstractThe present research is based on the thermal and flow properties of the viscoelastic Oldroyd 8 constant fluid in an upright microchannel. The energy and momentum equations were solved with the support of temperature Jump and velocity slip boundary conditions. To measure the irreversibility rate of the flow system, the acquired results of velocity and thermal equations were used. To crack the current mathematical model problem, the numerical Runge–Kutta–Fehlberg method was used. With the aid of graphs, the effect of physical parameters such as thermal radiation, thermal‐dependent heat source, Joule heating, fluid parameters, velocity slip, and temperature Jump parameters on the fluid flow, thermal energy, and system entropy generation was discussed. Fluid parameters have different effects on the velocity profile. The Grashof and Hartmann numbers demonstrate opposite effects on the momentum field. The thermal energy of the system reduces with thermal radiation and temperature Jump factor. The thermal radiation, Hartmann number, and temperature Jump parameters reduce the system's irreversibility rate. With the Brinkman number and temperature Jump parameter, the irreversibility ratio increases.

A rubric for assessing mathematical modelling problems in a scientific-engineering context

Abstract Mathematics modelling is a vital competency for students of all ages. In this study, we aim to fill the research gap about valid and reliable tools for assessing and grading mathematical modeling problems, particularly those reflecting multiple steps of the modelling cycle. We present in this paper the design of a reliable and valid assessment tool aimed at gauging the level of mathematical modelling associated with real-world modeling problems in a scientific-engineering context. The study defines and bases the central modelling processes on the proficiency levels identified in PISA Mathematics. A two-dimensional rubric was developed, reflecting the combined assessment of the type and level of a modelling process. We identified criteria that enable a clear comparison and differentiation among the different levels across each of the modelling processes. These criteria allow for concrete theoretical definitions for the various modelling processes, introducing a well-defined mathematical modelling framework from a didactical viewpoint, which can potentially contribute to promoting modelling competencies or the understanding of modelling by teachers and students. Theoretical, methodological and practical implications are discussed.

STUDY OF THE INFLUENCE OF THE PASSAGE OF HEAVY TRAINS ON THE THROUGHPUT AND CARRYING CAPACITY OF RAILWAY SECTIONS WITH ELECTRIC TRAIN TRACTION

The criterion for the effectiveness of the organization of heavy traffic on the sections of the Russian railway network is the presence of large correspondences of bulk goods presented by the consignor for transportation. An obstacle to the development of a promising cargo traffic are areas with limited throughput and carrying capacity, the overcoming of which is the organization of heavy traffic. All countries that have railways, at a certain stage of development, face capacity constraints. Summarizing the world experience, we can conclude that the priority in the development of increasing volumes of “heavy cargo” is increasing the weight norms of freight trains. The problem of passing heavy trains for Russian Railways is urgent, because the capacity of some sections has exhausted itself completely. The very timely appearance of heavy traffic becomes one of the options for solving the problem of increasing the carrying capacity of sections and directions and, as a result, creating a reserve of carrying capacity. In this article, a study of the influence of their movement on the carrying capacity and carrying capacity has been carried out.
 Aim. To consider the effectiveness of the formation and passage of heavy trains. The influence of the organization of the movement of such trains on the throughput and carrying capacity of railway sections.
 Methods. To solve the problem, graphic and analytical methods of mathematical modeling, scientific methods of collecting and processing statistical data, modern achievements in terms of general principles and methods of risk management was used.
 Results. The authors have developed a formula for determining the rate of removal of freight trains. The analysis of the influence of the values of the inter-train intervals of freight trains and heavy trains on the removal rate is given.
 Practical relevance. As the main alternative to increasing the size of traffic on the main cargo-intensive directions, with a shortage of traffic capacity, it is proposed to use the technology of heavy traffic. In modern market conditions, an increase in the weight and length of a freight train is one of the main reserves for increasing throughput and carrying capacity. By passing heavy trains, we get an increase in carrying capacity. In the article, we prove the consistency of these measures. The increase in carrying and carrying capacity is considered in the context of increasing unified weight standards.

MATHEMATICAL MODEL OF THE MARKET OF ONE PRODUCT WITH OPTIMAL DELIVERY TO THE MARKET UNDER CONDITIONS OF DELAY

MATHEMATICAL MODEL OF THE MARKET OF ONE PRODUCT WITH OPTIMAL DELIVERY TO THE MARKET UNDER CONDITIONS OF DELAY

MODELING OF SEISMIC WAVES STRESSES IN A HALF-PLANE WITH A VERTICAL CAVITY FILLED WITH WATER (THE RATIO OF WIDTH TO HEIGHT IS ONE TO TEN)

The problem of mathematical modeling of unsteady seismic waves in an elastic half-plane with a ver-tical rectangular cavity filled with water is considered. The problem of modeling problems of the transition peri-od is an actual scientific problem. A quasi-regular approach is proposed to solve a system of linear ordinary dif-ferential equations of the second order in displacements with initial conditions and to approximate the region un-der study. The method is based on the schemes: a point, a line and a plane. An algorithm and a set of programs for solving flat (two-dimensional) problems that allow obtaining a stress-strain state in complex objects have been developed. To assess the reliability of the developed methodology, algorithm and software package, the problem of the effect of a plane longitudinal wave in the form of a Heaviside function on an elastic half-plane was solved. The numerical solution corresponds quantitatively to the analytical solution. The problem of mathe-matical modeling of unsteady elastic stress waves in a half-plane with a cavity filled with water (the ratio of width to height is one to ten) under seismic influence is solved. A system of equations consisting of 8016008 un-knowns is solved. Contour stresses and components of the stress tensor are obtained in the characteristic areas of the problem under study. A cavity filled with water, with a width-to-height ratio of one to ten, reduces the amount of elastic contour stress.

Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems

Modeling complex hydrodynamic processes in coastal systems is an important problem of mathematical modeling that cannot be solved analytically. The approximation of convective terms is difficult from the point of view of error reduction. This paper proposes a difference scheme based on a linear combination of the Upwind Leapfrog scheme with 2/3 weight coefficient, and the Standard Leapfrog scheme with 1/3 weight coefficient. The weight coefficients are obtained as a result of solving the problem of minimizing the approximation error. Numerical experiments show the advantage of the developed scheme in comparison with other modifications of the Upwind Leapfrog scheme in the case when the convective transport prevails over the diffusion one. The proposed difference scheme solves transport problems more effectively than classical difference schemes in the case when the Péclet number falls in the range from 2 to 20. It follows that the considered difference scheme allows hydrodynamic problems to be solved in regions of complex shape effectively.

Mathematical Modeling Learning Design with PISA Framework on Grade X Function

Mathematical Modeling is the cornerstone and central aspect of the PISA framework. However, students still encounter difficulties in dealing with problems related to mathematical modeling. This study aims to produce a learning trajectory of mathematical modeling on functions material using the context of an online taxibike. Mathematical modeling activities on functions material are designed using the mathematical modeling cycle in the PISA framework. The subjects in this study were students of class X Bina Ilmi IT Senior High School Palembang. The research method used is validation studies. The research consisted of three stages, namely the preparation stage, the experimental design stage consisting of two cycles (cycle I: pilot experiment and cycle II: teaching experiment) and the retrospective analysis stage. Data was collected through observation, interviews and written test (students' work on the activity sheet and the results of pre-test and post-test). The results showed that learning activities that have been designed can assist students in solving mathematical modeling problems on functions. It begins with the formulate, employed, interprete/evaluate process. Through these mathematical modeling processes, students have a better understanding of functions and become accustomed to using mathematical modeling processes in solving contextual problems related to function material.

Kemampuan Pemecahan Masalah Matematis Siswa SMP pada Materi Sistem Persamaan Linear Dua Variabel

Mathematical problem-solving ability is one of the important skills to be developed and owned by students. Several studies show that students' mathematical problem solving is still low. The purpose of this study was to analyze the difficulty of students' mathematical problem-solving abilities in the variable system of linear equations of two materials. This research was conducted at SMP IT Al-Fadhilah Kp. Pasirkiamis Village. Pasirkiamis Kec. Pasirwangi District. arrowroot. This study uses an approach with a descriptive analysis method involving 3 students as a sample (simple random sampling technique). The test instrument for students' mathematical problem-solving abilities used as many as 4 items about the description. Based on the results of research and discussion, it can be said that the ease with which most students do is at the stage of making mathematical modeling and applying strategies to solve mathematical modeling problems. Students haven't done the steps to complete the problem completely, because students have not applied the principles of solving mathematical problems to the material for systems of linear equations of two variables and are not accustomed to re-examining answers carefully.

The Optimization of Mathematics Teaching Models in Colleges and Universities Based on Nonlinear Differential Equations

Abstract Nonlinear fractional differential equations are an important part of advanced mathematics teaching. The existence and uniqueness of its positive solution have always been a hot topic of academic discussion. This article uses differential inclusion theory and the Lyapunov stability method to analyze the finite-time stabilization control problem of the discontinuous mathematical adjustment model. The article uses a modified decomposition method and convergence acceleration technology in the application of fractional differential equations. The method gives an analytical approximate solution sequence that is easy to calculate, verify, and quickly converge. Finally, examples of Lyapunov stability and the construction of the V function can inspire students to understand ordinary differential equations and increase their interest in learning.

Contextual Mathematical Modelling: Problem-Solving Characterization and Feasibility

The current study investigates contextual mathematical modelling (MM) problems that were retrieved from authentic workplace situations and simplified for formal secondary school math lessons. First, the study aims to characterize contextual MM problems according to Schoenfeld’s framework of problem-solving (PS). Second, it aims to investigate the perceptions of two stakeholder groups: (1) math experts and policymakers and (2) math teachers with respect to the characteristics of the contextual MM problems and their feasibility regarding implementation in secondary school education. Based on the Delphi methodology, we employed two phases for our analysis: an open-ended questionnaire to interview ten stakeholders and, subsequently, a Likert-type questionnaire to collect data from 122 stakeholders. The main results indicate that the contextual MM problems are characterized by PS. A similar view was expressed by different stakeholder groups, and no differences were caused by various background variables, such as educational level or STEM background. Additionally, the findings revealed that both stakeholder groups perceived that it is highly feasible for these problems to be integrated into secondary school education. This study contributes theoretically to the interrelationship between MM and PS frameworks, and provides practical recommendations for the implementation of contextual MM problems in secondary schools by applying PS skills.

Model of knowledge representation in the system of information support for decision making

Objectives. The problem of mathematical modeling of the knowledge in the information support system for decision-making is being solved. The approach is based on the formalization of semantic relations in the language. The formalization of such relations provides an adequate representation of knowledge and serves as a theoretical basis for the algorithmization of user interaction with the system.When solving the problem of knowledge modeling, three main goals are pursued: the construction of a mathematical model of semantic relations, the study of the properties of marginal syntagmas and the construction of algorithms for the formation of a dictionary of syntagmatic structures.Methods. Methods of set theory, graph theory and mathematical linguistics are used.Results. A mathematical model of knowledge representation in the information decision support system has been developed. The concepts of semantic relations in the language are formalized and the properties of marginal syntagmas are investigated. The proven properties of such structures provide the algorithmization of information processes in the system under consideration.Conclusion. As an implementation of the model proposed in the article, algorithms for creating dictionaries of syntagmas and syntagmatic structures have been developed. When forming a set of syntagmas, the assertions about proved in the article marginal syntagmas are used. A set of syntagmatic structures is built taking into account their typology. Syntagmatic structures in the dictionary acquire the status of communicative fragment, i.e. stable phrases to synthesize the information system language.

Property analysis of multivariate conditional linear random processes in the problems of mathematical modelling of signals

The object of research is the process of mathematical modelling of a multidimensional random signal, which in the structure of its generation is the sum of a large number of random impulses that occur at random times. Examples of stochastic signals of this type can be, in particular, electroencephalographic and cardiographic signals, photoplethysmography signals, resource consumption processes (electricity, gas, water consumption), radar signals, vibrations of bearings of electric machines and others. A common mathematical model (especially in the multidimensional case) of this type of signal is a linear random process that allows the signal to be represented as the sum of a large number of stochastically independent random impulses that occur at Poisson moments. If the impulses are stochastically dependent (or the moments of time of their appearance are not Poisson), then the mathematical model is a conditional linear random process. The definition and analysis of the probabilistic properties of such processes for the multidimensional case have not been conducted. The paper defines a multidimensional conditional linear random process, each component of which is represented as a stochastic integral of a random kernel driven by a process with independent increments. Expressions for the characteristic function and moment functions of the specified process are obtained. The approach used was to use the mathematical apparatus of conditional characteristic functions, as well as the known representation of an infinitely divisible characteristic function of a linear random process as a functional of a process with independent increments. The obtained results provide a possibility for theoretical analysis of probabilistic properties of multichannel stochastic signals, the mathematical model of which is a multidimensional conditional linear random process. Justification of their properties of stationarity or cyclostationarity, which are the consequence of corresponding properties of the kernel and process with independent increments, can be carried out.

Mathematical Model of a Local Grid with Small Modular Reactor NPPs

The technical and economic processes of the Local Grid, regional electricity system comprising small modular reactor nuclear power plants, renewable electricity generators, and electricity storage systems are investigated. Smart grid technologies are used in the management of such a system. In relation to the electricity system, the Local Grid acts as a consumer or producer of electricity depending on the price situation in the wholesale market and its own ability to balance the volume of electricity generation-consumption.
 The Local Grid market is considered to be perfect, otherwise the wholesale electricity market is characterized by imperfect competition. The mathematical model of the Local Grid is offered. The presented model adequately reproduces the characteristics of the interaction between the local market and the wholesale market. The mathematical model is generated in the form of the commitment problem for small modular reactors and electricity storage systems. The model reflects the technological limitations for the loading modes of small modular reactors and units of the energy storage system, which operate in the modes of direct and reverse energy conversion. In particular, the model reflects the shunting modes and start-shutdown modes of small modular reactors and charge-discharge modes of energy storage system units, which together determine flexibility of the operation modes for the entire power system. The commitment problem of small modular reactors of a nuclear power plant, units of the electricity storage system, and transmission lines connecting the Local Grid with the power system is a mixed integer programming problem with an objective function of minimized system costs.
 Cluster integer functions are used to reduce the dimensionality of the problems of mathematical modeling the loading modes of the Local Grid in the descriptions for the sets of the same type small modular reactors, as well as the same type energy storage units. The Local Grid model reproduces the modes of its disconnection from the system operator's network to increase nuclear safety in the event of a natural disaster or hostilities. The results of computational experiments on modeling the commitment modes of the local electrical grid are presented.
 The experiments used standard data on the volume of electricity consumption and its production from renewable energy sources. The results of the computational experiment confirm the adequacy of the suggested mathematical model for the Local Grid. The presented model is suitable for the use both independently – to analyze Local Grid functioning - and in combination with the models of electric power industry - to determine the impact of the Local Grids on the electricity system.
 The article is developed based on the results of the research carried out under target program “Support of State-Priority Research and Scientific and Technical (Experimental) Solutions of the Department of Physical and Technical Problems of Energy of the NAS of Ukraine for 2022 – 2023” with program expenditure classification code 6541230 (applied research).

MODELING RISK FACTORS INTERACTION AND RISK ESTIMATION WITH COPULAS

Context. Various risks are inherent to practically all types of human activities. Usually the risks are characterized by availability of multiple risk factors, uncertainties, incompleteness and low quality of data available. The problem of mathematical modeling of risks is very popular with taking into consideration possible uncertainties and interaction of risk factors. Such models are required for solving the problems of loss forecasting and making appropriate managerial decisions. Objective. The purpose of the study is in development of multivariate risk modeling method using specialized copula functions.The models are developed in the form of multivariate distributions. Method. The modeling methodology is based upon exploring the special features of various copula functions that are helpful to construct appropriate multivariate distributions for the risk factors selected. The study contains formal description of selected copulas, analysis of their specific features and possibilities for practical applications in the risk management area. Examples of practical applications of the copula based approach to constructing multivariate distributions using generated and actual statistical data are provided. Results. The results achieved will be useful for further theoretical studies as well as for practical applications in the area of risk management. The distributions constructed with copula create a ground for solving the problems of forecasting possible loss and making appropriate decision regarding risk management. Conclusions. Thus the problem of constructing multivariate distributions for multiple risk factors can be solved successfully using special copula functions.

DISCRETE-TIME CONDITIONAL LINEAR RANDOM PROCESSES AND THEIR PROPERTIES

Continuous-time conditional linear random process is represented as a stochastic integral of a random kernel driven by a process with independent increments. Such processes are used in the problems of mathematical modelling, computer simulation, and processing of stochastic signals, the physical nature of which generates them to be represented as the sum of many random impulses that occur at Poisson moments. Impulses are stochastically dependent functions, in contrast to another well-known mathematical model which is a linear random process, that has a similar structure but is represented as the sum of a large amount of independent random impulses that occur at Poisson moments of time. The application areas of these models are mathematical modelling, computer simulation, and processing of electroencephalographic signals, cardio signals, resource consumption processes (such as electricity consumption, water consumption, gas consumption), radar signals, etc. A discrete-time conditional linear random process has been defined in the paper, the relationships with corresponding continuous-time model has been shown. According to the given definition the discrete-time conditional linear random process can be considered as an output of linear digital filter with random parameters on the input of the white noise which is infinitely divisible distributed. Moment functions of first and second order have been analyzed. In particular, the expressions for mathematical expectation, variance and covariance function have been obtained. The results can be utilized to study the probabilistic characteristics of the investigated information stochastic signals, which will depend on the properties of the corresponding kernel and white noise. In particular, the conditions for the process to be wide-sense stationary have been represented.

Experimental and Mathematical Investigation of the Thermophysical Properties of Coal–Water Slurries Based on Lignite

The burning of organic fuel is one of the main factors influencing the greenhouse effect on the climate of the planet. The article examines the influence of the properties of lignite and coal–water slurries on the amount of carbon dioxide molecules formed as a result of chemical reactions. The authors give an overview of the results of other researchers in recent years and give the results of their research and development in the field of burning lignite and coal–water slurries in industrial recycling plants. The authors present the results of experimental studies of the thermophysical properties of coal and a coal–water mixture. The results obtained were compared with the results of calculations using a mathematical model and the results of numerical modeling in Ansys. New methods of approximation of step functions were used for the mathematical model. These methods make it possible to reduce errors in the approximation of the functions of the thermal properties of coal. The proposed methods do not have the disadvantages of traditional decompositions of step functions into Fourier series and can be used in problems of mathematical modeling of a wide class of processes and systems. In particular, when determining the coefficient of kinematic viscosity, ash content, and humidity by the method of approximation of the obtained data, the use of new mathematical methods makes it possible to reduce the error in calculations. In addition, data on numerical modeling of hydraulic transport and combustion processes are provided, and a data validation procedure is carried out. The data convergence shown and their location in the selected band of uncertainties satisfy the requirements for verification of experimental data adopted in the European Union and in the Eurasian Union.

Power Minimization in Dual-Hop Underlay Cooperative Cognitive Radio Relay Networks for Optimal Resource Allocation

In this paper, we investigate a multi-user one-way dual-hop multi-relay cognitive radio underlay spectrum sharing network. .The joint optimal resource allocation of the bandwidth and power allocation in underlay Cooperative Cognitive Radio Relay Network (CCRRN) was studied and a convex optimization analytical framework is presented. A combined optimal power and bandwidth allocation (COPBA) scheme for the minimization of the total network power and blocking probability of the call admission control stage of the CCRRN is investigated. Our goal is to jointly optimize the bandwidth and power allocation such that the total transmission power in the CCRRN is minimized without compromising the Quality of Service (QoS) demands of the secondary users (SUs) and the interference constraint thresholds of the primary users (PUs) in the primary networks. The mathematical model and convex optimization problem with the aim of minimizing the total transmission power of the CCRRN was formulated. The resulting optimization problem was solved in the standard computer based numerical optimization solver package CONOPT in MATLAB/TOMLAB environment. Simulation results obtained is compared the equal-bandwidth optimal power allocation scheme. Keywords—Radio resource allocations; Cognitive radio Networks; Convex optimization; Cooperative relay networks, Transmit power minimization

Transportation of Raw Materials, Recondite Question in Manufacturing Enterprise

Based on the optimization theory, this paper studies the mathematical modeling problem related to the ordering and transportation of raw materials. The paper starts with reasonable assumptions, based on which the following factors are used as index: “average supply”, “order completion rate”, “average order value” and “standard deviation of supply” of each supplier, and the weight of each index is determined by using entropy weight method. On the premise of reasonable weight, the distance method of good-bad solutions (TOPSIS) is used to evaluate and rank each supplier, and finally, the most important supplier is selected. Then, a 0-1 programming model is established, in which “minimum order price” and “minimum loss” are taken as the objective functions, and the two-week inventory of the enterprise is combined with factors such as the loss rate of the forwarder, using MATLAB programming TOPSIS algorithm to solve the model to obtain the optimal ordering and transshipment plan. Considering the possibility that the enterprise needs low utilization of raw materials A and C, a bi-objective programming model is established to determine the weekly ordering and transshipment plan within the target weeks. On the other hand, when the number of suppliers is sufficient, the capacity of the forwarders limits the expansion of the capacity of the enterprise. Therefore, combined with the existing data, a linear programming model with the maximum production capacity as the objective function is established to obtain the maximum production capacity and formulate ordering and transshipment schemes. Finally, the solution process and results are summarized and analyzed.