Mathematical Model Of Problem Research Articles

Metacognitive Strategies for Mathematical Modeling with Engineering Groups of Students: Adaptation and Validation of a Questionnaire

A sequential exploratory mixed-methods study is implemented to develop an instrument that allows for the evaluation of the metacognitive strategies used by engineering groups of students when solving mathematical modeling problems. The findings of the qualitative study guided by observations and interviews reveal the use of metacognitive strategies of ‘planning’, ‘monitoring and, if necessary, regulation’, and ‘evaluation’. In this article, we present the final categories of the qualitative analysis and discuss how these data were shaped into a theoretical construct and items of an instrument to measure metacognitive strategies. The psychometric properties of the instrument are analyzed, and it is argued that it has a similar interpretation among males and females, as there are no significant differences in these results. The development of the present study demonstrates how the qualitative method can support the adaptation of an instrument to measure metacognitive strategies, thus contributing to validity and applicability.

Bridging Gaps: Pre-Service Mathematics Teachers’ Handling the Difficulties in Posing Real-World Mathematical Problems

Real-world mathematical problem (RWMP) solving and posing are important aspects of teaching and learning mathematical modelling, as well as developing a mathematization disposition for both teachers and students. Several researchers have explored blockages or difficulties in such modelling processes and in problem posing. However, prior research has identified difficulties that the pre-service mathematics teachers (PSMTs) encountered when they tried to pose a modelling problem by choosing the general topic themselves, as there is little known about possible obstacles that PSMTs can encounter when trying to pose a real-world problem relevant to a given mathematical topic. The current study explored the difficulties encountered by PSMTs in a RWMP-posing activity. The target group was 23 PSMTs with prior experience in mathematical modelling and mathematical problem posing. The findings showed that the PSMTs struggled with: (a) task organization, which involved selecting and understanding mathematical knowledge; (b) specialized content knowledge, which included a lack of real-world knowledge and difficulty in connecting mathematical concepts to real-world contexts; and (c) individual considerations of aptness, which encompassed authenticity, interest, complexity, language, and relevance to task organization. The PSMTs applied various strategies to complete the posing task, such as using problem-posing and solving heuristics, adapting existing problems, sharing and discussing with friends, and considering the perspective of a typical student. The implications of these findings should help in developing preparatory instructional practices for mathematics teachers.

Contributions and problems of mathematical models in COVID‐19 prevention in Japan

AbstractThis article reviews the essential role of mathematical models in understanding and combatting the pandemic of novel coronaviruses, in particular focusing the advance in the use of mathematical models in disease control in Japan. Highlighting the integral role of mathematical models in public health, the article introduces a model that factors in the heterogeneity of infectious contacts, concentrating on the effectiveness of testing and isolation, alongside a model that involves economic losses. The models exhibit how, given such heterogeneity, milder behavioral restrictions can still achieve suppression, rigorous testing and isolation can effectively curb the spread, and containment measures can mitigate economic losses. These models aid in grasping the complicated dynamics of disease transmission and optimizing interventions. The knowledge of population ecology is also considered effective for public health in statistical analysis, organizing concepts using dynamic mathematical models, which lead to policy proposals and deepen understanding. Evolution theory may help the understanding of virulence subject to change. However, effective prevention necessitates not only models but also the practical implementation of efficacious measures. The cooperation of various disciplines is particularly crucial in achieving a balance between health measures, economic interests, and human rights. Moreover, the article acknowledges the limitations of models and underscores the significance of real‐world execution. Overall, the article advocates for a broader outlook to tackle future pandemics and related challenges, underscoring the importance of ongoing academic cooperation and global governance to effectively address emerging infectious diseases and their far‐reaching implications.

A PROPOSAL FOR AN INNOVATIVE APPROACH TO EXTENDED MODELLING CYCLES INCORPORATING PROBLEM-POSING FROM A CREATIVITY PERSPECTIVE

Many studies have shown that mathematical modelling, creativity, and problem-posing activities are deeply related. However, there is a lack of research that poses mathematical modelling problems. Additionally,only few studies have conducted statistical analyses of the relationship between creativity in problem-posing and creativity in psychology. The purpose of this research is to conduct a survey to quantitatively evaluate creativity, and to propose a more effective modelling cycle based on the survey results. This study focuses on fluency a creativity factor that is measured by the number of problems posed by students. The survey results showed a moderate positive correlation between fluency in posing mathematical modelling problems and creativity in psychology (r = 0.44, p <0.01). Based on the results, an extended modelling cycle is proposed and discussed.

Математичне моделювання динамічних впливів на будівельні конструкції

The solution of the problem of mathematical modeling of dynamic influences on building structures is considered in general and in connection with important scientific and practical tasks. In modern conditions, special attention should be paid to the problems of taking into account dynamic influences, parameters of load-bearing structures of buildings that affect their structural reliability and safety. For many years, information about catastrophic destruction of buildings and structures has been appearing more often, which are the result not only of errors that arose during the construction of facilities, but also due to dynamic influences that arose during their operation. A buildable technical system will be used in advance with the technical parameters set, which is responsible for the control of the process of the preparation of structures, the installation of the equipment, during the operation and operation. The development will be carried out and the sporud will be constantly used to promote the development of future technologies, the development and the establishment of the future, as well as the development of dynamic developments in the future. In particular, it is important to understand the development of the engineering and geological minds of the spores, as they have known the front deformation of the non-essential axis of the runt foundations. Various methods are used for structural dynamics, such as modal analysis, response spectrum analysis, time analysis, and finite element analysis. These methods include mathematical modeling, numerical simulation, and computational methods to predict the behavior and response of a structure under dynamic load conditions. Knowing the frequency and form of the structure, it is possible to estimate the dynamic reaction of the design to the application. For the purpose of including the transfer of the world, the destruction of internal forces in the real parts of the construction. Analyses of the response to the additional assessment of the characteristics of the construction, the identification of critical areas and the re-conversion, which the project is able to meet the criteria.

Using different digital tools in designing and solving mathematical modelling problems

AbstractWhile existing studies acknowledge the importance of using technology in the mathematical modelling process, questions about how to integrate digital tools into mathematical modelling are not still answered. This study aims to examine pre-service mathematics teachers’ designing and solving mathematical modelling problems by using different digital tools through the collaboration of two different courses in their education program. The GeoGebra-supported modelling cycle was adopted as the conceptual framework of the study. The case study was used to design the research. The participants of the study were three volunteer pre-service secondary mathematics teachers, who were able to use and integrate digital tools in mathematics education. The pre-service teachers were asked to design a mathematical modelling problem by integrating different digital tools as the final assignment in both courses. The designed mathematical modelling problem, the video records of pre-service teachers’ explanations including the designing and solving process, and the video records of the focus group interview were the data of the study. The results showed that the pre-service teachers were able to design and solve an effective and comprehensive mathematical modelling problem using technology by combining their skills that they had learnt through different courses. In particular, they were able to use different digital tools in both designing and solving a mathematical modelling problem. Their solution also led to the extension of the adopted framework by adding new technological actions into the modelling stages.

STAGES OF ACCUMULATION OF OIL DROPLETS AND GAS BUBBLES IN THE DOME FOR CASES OF DEEP-SEA SPILLS: PART 2

In a previously published article, the problem of mathematical modeling of the operation and installation of a special device – a separator dome designed to collect hydrocarbons for cases of deep-sea leaks was given, three initial stages of modeling the installation of the device were described. This paper describes in detail the remaining stages of installing the dome: the accumulation of a layer of gas, the final stage of installing the dome, when the tubes for pumping oil and gas are connected to the device and the device begins to work in stationary mode. Let's take a closer look at each of the stages. The fourth stage of the dome installation is associated with the achievement of a given height (at which the gas bubbles have not yet passed into the hydrate state) and the beginning of the migration of gas bubbles inside the dome. First, the gas bubbles migrate into layer of liquid immiscible with water, then in the oil layer. During this process, all the gas coming from the jet completely enters the tank. At the fifth stage, gas accumulations inside the tank are considered. At the sixth stage, the study of the accumulation of layers of gas and oil continues, but at the same time the reservoir itself sinks to the bottom of the reservoir at a constant speed. This is necessary to subsequently secure the dome and prevent oil and gas from leaking out of the dome. The seventh stage is the final one. At this stage, we believe that the dome is fixed on the ocean floor, pipes for pumping oil and gas are connected to it. As a result of the created mathematical model of the dome-separator operation, dependences characterizing the process of installation and functioning of the dome, the temperature dependences of gas and oil layers accumulated inside the dome, as well as the dynamics of accumulation of the corresponding layers in the dome are obtained. The characteristic start and end times of each of the stages are described.

Моделирование работы газотурбинной установки SGT5-4000F с исследованием влияния режимных и климатических факторов на устойчивость горения в камере сгорания

When regulating gas turbine units, it is necessary to be confident in the stable combustion process in the combustion chambers, considering the dynamics of the processes and changes of external climatic factors. Many scientific studies are devoted to the problems of experimental research and mathematical modeling of processes in combustion chambers. The possibilities of flame failure, expansion of the lower limit of combustion, determination of the stable position of the flame front and other issues have been assessed. Even though the problem of design of mathematical models and their use remains relevant. The purpose of this research is to model and study the influence of operation and climatic factors on the stability of combustion in the combustion chamber of the SGT5-4000F gas turbine unit. The subject of the research is the SGT5-4000F gas turbine unit with a low-emission combustion chamber. The simulation model of the gas turbine unit has been developed in the environment of dynamic modeling of technical systems SimInTech. The research is carried out using theoretical methods of calculating fuel combustion, thermodynamic foundations of the theory of gas turbine engines, methods of mathematical and simulation modeling, as well as data from the archive of automated process control systems. A simulation model of SGT5-4000F gas turbine unit has been developed. It is distinguished by the ability to assess the proximity of the compressor operating mode to the stability limit and determine the boundaries of stable combustion procedure in the combustion chamber, considering environmental climatic factors. The authors have studied the stability of the compressor and combustion chamber. The results of the experimental studies of a gas turbine unit in the operating load range from 113 to 282 MW and in the range of outside air temperature from –12 to 30 °C have shown that the combustion process in the combustion chamber is unvarying over the entire operational stress spectrum. The results obtained are verified by comparing model values with technological parameters of a gas turbine unit taken from the archive of the power plant control system. The authors have assessed the influence of electrical load on the performance indicators of a gas turbine unit at outdoor temperatures of +30, +15 and –12 °C. According to the results of simulation modeling, the model of SGT5-4000F gas turbine unit is found to be adequate. The developed model can be used as a tool at the stage of functional and technological design of automated control systems for the development of effective automatic control systems.

Robust generation expansion planning in power grids under renewable energy penetration via honey badger algorithm

Robust reliability Generation Expansion Planning (GEP) turns out to be a crucial step for an efficient energy management system in a modern power grid, especially under renewable energy employment. The integration of all such components in a GEP model makes it a large-scale, nonlinear, and mixed-variable mathematical modeling problem. In this paper, the presence of wind energy uncertainty is analyzed. Both long and short-term uncertainties are incorporated into the proposed GEP model. The first step concerns the impact of long-term wind uncertainties through the annual variations of the capacity credit of two real sites in Egypt at Zafaranh and Shark El-ouinate. The second step deals with the short-term uncertainties of each wind site. The wind speed uncertainty of each wind site is modeled by probability distribution function. Then, wind power is estimated from the wind power curve for each wind site and Monte-Carlo Simulation is performed. Fast Gas Turbine and/or Pump Hydro Storage are incorporated to cope with short-term uncertainties. Sensitivity analysis is implemented for 3, 6, and 12 stages as short and long planning horizons to minimize the total costs with wind energy penetration and emission reduction over planning horizons. Also, a novel Honey Badger Algorithm (HBA) with model modifications such as Virtual Mapping Procedure, Penalty Factor Approach, and the Modified of Intelligent Initial Population Generation is utilized for solving the proposed GEP problem. The obtained results are compared with other algorithms to ensure the superior performance of the proposed HBA. According to the results of the applicable test systems, the proposed HBA performs better than the others, with percentage reductions over CSA, AO, BES, and PSO ranging up to 4.2, 2.72, 2.7, and 3.4%, respectively.

A Comparative Study of Mathematical Modelling Problems in High School Mathematics Textbooks—Taking People’s Education Edition A, Jiangsu Education Edition and Hubei Education Edition as Examples

A Comparative Study of Mathematical Modelling Problems in High School Mathematics Textbooks—Taking People’s Education Edition A, Jiangsu Education Edition and Hubei Education Edition as Examples

Model Oluşturma Etkinliklerinde Matematiksel İlişkilendirmelerin Belirlenmesi: Tasarımcı Olarak Matematik Öğretmen Adaylarıyla Bir Durum Çalışması

This paper aimed to explore mathematical connections that pre-service elementary mathematics teachers identified in model-eliciting activities that they designed. The study used a qualitative, in-depth exploratory case study approach conducted with a total of 42 undergraduate students selected using a purposive sampling method. Data were collected from mathematical modeling problems prepared by pre-service mathematics teachers and from written assignments in which pre-service teachers evaluated their problems in terms of mathematical connections. Data analysis utilized open coding to identify and label common patterns thematically. Results showed that most of the problems designed were consistent with the principles of the model-eliciting activities. A cross-case analysis of model-eliciting activities revealed that four types of mathematical connections were identified: (i) connections to real-world situations, (ii) connections between mathematical concepts, (iii) connections between different modes of representation, and (iv) connections to other disciplines. The mathematical connections identified in the modeling activities did not seem to be mutually exclusive, but rather supportive and sometimes overlapping. Pre-service mathematics teachers’ identification of mathematical connections in modeling problems reflects both their awareness of mathematical connections and their perception that the skills required for the mathematical modeling process are directly related to mathematical connection skills. Therefore, it is important that pre-service mathematics teachers have a comprehensive understanding of mathematical connections and believe that this is an essential element of the modeling process so that the various dimensions of mathematical connections, along with relevant mathematical, pedagogical, and curricular knowledge, can be effectively integrated into model-eliciting activities.

Open modelling problems: cognitive barriers and instructional prompts

Open mathematical modelling problems that can be solved with multiple methods and have multiple possible results are an important part of school curricula in mathematics and science. Solving open modelling problems in school should prepare students to apply their mathematical knowledge in their current and future lives. One characteristic of these problems is that information that is essential for solving the problems is missing. In the present study, we aimed to analyze students’ cognitive barriers while they solved open modelling problems, and we evaluated the effects of instructional prompts on their success in solving such problems. A quantitative experimental study (N = 263) and a qualitative study (N = 4) with secondary school students indicated that identifying unknown quantities and making numerical assumptions about these quantities are important cognitive barriers to solving open modelling problems. Task-specific instructional prompts helped students overcome these barriers and improved their solution rates. Students who were given instructional prompts included numerical assumptions in their solutions more often than students who were not given such prompts. These findings contribute to theories about solving open modelling problems by uncovering cognitive barriers and describing students’ cognitive processes as they solve these problems. In addition, the findings contribute to improving teaching practice by indicating the potential and limitations of task-specific instructional prompts that can be used to support students’ solution processes in the classroom.

The concept of a complex digital twin of a furnace for metal heat processing

A number of mathematical models, digital twins and diagnostic systems are considered, which are used for individual processes and elements of flame furnaces. An idea was put forward about the expediency of creating a complex digital twin, which simultaneously takes into account the main processes in metal heat processing (heating for rolling or heat treatment) in a flame furnace. A conceptual definition of the structure and interconnection of the elements of a complex digital twin of a furnace for metal heat processing is carried out. It is substantiated that the data to ensure the operation of the digital twin are divided into two groups: constantly or periodically measured values and data that are unchanged. The minimum set of quantities, the measurement of which is necessary for the operation of the proposed complex digital twin of the furnace, is determined. A list of objective data about the unit and the technological process, which are unchanged over a long period of operation, is also compiled. At the same time, a scheme for organization the interaction of special computing units has been developed, which allows to achieve high accuracy in solving the traditional problem of mathematical modeling of metal heat processing ‒ determining the metal temperature field. To achieve this result, computing units are used to identify the parameters of all processes that affect the solution of the problem: the value of air suction into the furnace chamber and along the smoke path, the value of the recovery coeffi-cient, the oxygen consumption for the oxidation of the metal surface, the average heat flux density absorbed by the metal surface. Special blocks are also provided for diagnosing the state of the heat exchanger, fan, smoke path, chimney and elements of the mechanical equipment of the furnace. Ultimately, top-level blocks are provided that allow, based on the processing of information from all the basic blocks of the complex digital twin, to analyze the energy efficiency of the furnace unit, obtain the technical and economic characteristics of its operation, as well as to issue recommendations to clarify the current parameters of technological processes and maintenance of equipment elements.

Comparative Evaluation of the Accuracy of Numerical Methods on Reference Solutions

When organizing a mass practical solution of computational problems of gas dynamics with the help of mathematical modeling, information on the comparative accuracy of the numerical methods used is now increasingly in demand. As a rule, calculators need information not only for a particular combination of the defining gas-dynamic parameters of the problem (characteristic Mach, Reynolds numbers, etc.), but also for the variation of these parameters in certain ranges. This work presents numerical studies devoted to a comparative assessment of the accuracy of numerical methods for a number of problems with reference solutions. The calculation of these problems is carried out for ranges of characteristic numbers using various numerical methods. The results obtained are compared with the reference solution and make it possible to estimate the error for each of the numerical methods. Calculations are carried out using the construction of a generalized computational experiment. A generalized computational experiment is a computational technology that combines the solution of mathematical modeling problems, parallel technologies and visual analytics technologies. The results of the generalized computational experiment are multidimensional arrays, where the dimension of the arrays corresponds to the defining parameters. Analysis and visual representation of the obtained results provide information on the comparative accuracy of the numerical method for the selected class of problems.

Learning statistics using realistic mathematics education: A case of fifth-grade students of elementary school

This study aims to describe realistic mathematics education (RME) in helping students' understanding of concepts in statistical material about data collection. This research uses descriptive qualitative research methods with a design research approach. The subjects of the study were 29 fifth-grade students of a public school in Pekanbaru, Riau, Indonesia. The results of this study show that RME is applied in learning with the following steps, namely first, providing and explaining contextual problems to be solved by students; second, students solve contextual problems for mathematical models in their own way; third, discuss and compare each problem solving given by students; Fourth, students conclude and summarize the learning concept. From these activities, students can develop knowledge and skills in collecting and presenting data.

Mathematical model for the dynamics of bacterial meningitis (Meningococcal meningitis): a case study of Yobe State Specialist Hospital, Damaturu, Nigeria

A model for bacterial meningitis was created by adding a class of transporters to the basic Susceptible Carrier Infected and Recovered (SCIR) model, since vaccination and treatment are the best methods of controlling the transmission of most overpowering sicknesses. Immunization assists helpless people with building either a drawn out invulnerability or transient resistance while treatment decreases the quantity of sickness actuated passing and the quantity of irresistible people locally or country. This study comprises of a mathematical model for bacterial meningitis dynamics that can be used to a wide range of mathematical modeling problems. In this exploration, a nonlinear deterministic model with time reliance controls has been proposed to depict the elements of bacterial meningitis in a populace. We discovered that the (EEP) and (DFE) are locally asymptotically stable in our study. We now advise the researcher to determine whether it is globally asymptotically stable in order to achieve optimal disease control. The presence of an endemic harmony and the calculation of the reproduction number . The mathematical arrangement shows that vaccinating vulnerable people will prompt disposal of the infection in the public and furthermore it will lessen the weight on wellbeing suppliers. Numerical simulations were presented to explain the parameters in the end path in the model were carried out by using MAPLE software. Most likeable of this research work shows that the rate at which treatment and vaccination rate increases to the higher value the recovered compartment increases to the peak point. This means treatment and vaccination has an impact on reducing the case of bacterial meningitis in a population. Now we conclude that a high infection transmission rate requires a high vaccine and treatment rate on the effect of vaccination against meningitis.

Mathematical modeling of movement of multi-rotor type aircraft

The problem of determining the features and setting the problem of mathematical modeling of multirotor aircraft (LA) is considered. Differences of their mathematical models from the classical mathematical models of aircraft and single-rotor helicopter type aircraft are considered. The analysis and substantiation of the forces and moments acting on a multirotor aircraft are carried out, taking into account the peculiarities of considering the corresponding coordinate systems necessary for studying the spatial motion of the aircraft. The problem of controlling the trajectory motion of an aircraft is formulated taking into account the rotation of its structure around the center of mass. Based on the consideration of the structural diagram of one of the most common four-rotor aircraft (quadcopter), a scheme for creating control forces and moments acting on the aircraft, under the influence of which the aircraft’s trajectory in space changes, is substantiated. The main mathematical dependences characterizing the kinematics of the motion of a multi-rotor aircraft are given. On the basis of the analysis carried out, a generalized block diagram of the control process for such an aircraft is substantiated and presented. To test the performance and adequacy of the mathematical model, a study was made of the movement of a quadrocopter in a vertical plane between given points in space in accordance with the law of forced control, which ensures the movement of an aircraft in space with maximum speed. The computer simulation of the obtained analytically mathematical dependences showed that this approach is applicable to the construction of mathematical models of the motion of multirotor type aircraft of various design layouts.

Students’ errors in solving climate change context mathematical modeling problems

This research is a descriptive study that aims to describe the errors of students of the Faculty of Education and Teacher Training (FKIP), Sriwijaya University, in solving Climate Change context mathematical modeling problems, to minimize errors made in solving mathematical modeling problems, especially in the context of climate change in the future. No one has researched student errors in solving mathematical modeling problems in the Climate Change context using the Newman error procedure. Data collection techniques used in this study included tests and semi-structured interviews. The data analysis technique used for test data is by analyzing errors using the Newman procedure. The results of this study with 52 participants were types of errors by FKIP Sriwijaya University in solving Climate Change context mathematical modeling problems as follows for question 1 and question 2, and respectively there were 23.1% and 15.4% errors in understanding the problem, 46.2% and 96.2% transformation errors, 32.7% and 96.2% writing. The causes of errors are errors in reading, errors in determining the problem and existing information, errors in making assumptions, errors in identifying variables, and errors in making mathematical models.

A comprehensive approach to double inverted pendulum modelling

The problem of mathematical modelling and indication of properties of a DIP has been investigated in this paper. The aim of this work is to aggregate the knowledge on a DIP modelling using the Euler-Lagrange formalism in the presence of external forces and friction. To indicate the main properties important for simulation, model parameters identification and control system synthesis, analytical and numerical tools have been used. The investigated properties include stability of equilibrium points, a chaos of dynamics and non-minimum phase behaviour around an upper position. The presented results refer to the model of a physical (constructed) DIP system.

Development of mathematical modeling methods filtration flows of multicolored liquids

Today the relevance of the issues of environmental consequences of technological human intervention in the world around us is beyond doubt. The rapid development of technique and technology entails the development of new ways and methods of environmental protection. At the same time, issues of theoretical research, the creation of mathematical, and then computer models of the studied phenomena, play an important role. The article considers the influence of pollution of stationary filtration flows of an incompressible multicolored liquid on the environment. The problems of mathematical modeling of the filtration flows of multicolored liquids are solved. It is assumed that pollution does not significantly change the properties of the filtered liquid, its density and viscosity. Liquids do not mix and have a clear interface, which does not have a harmful effect on the environment.