Mathematical Calculations Research Articles

LLM and RAG Powered Chatbot for the College of Computer Science and Mathematics at the University of Mosul

This paper presents a proposal for the design and implementation of a chatbot for the website of the College of Computer Science and Mathematics at the University of Mosul. The chatbot uses a Large Language Model (LLM) supported by Retrieval Augmented Generation (RAG) technology, the first of its kind at the university. This innovative approach to designing the chatbot combines the strengths of pre-trained language models and RAG technology, enabling the chatbot to generate informative and accurate responses to a wide range of user inquiries. The chatbot is designed to provide immediate support and act as a virtual assistant for students, faculty, staff and visitors to the college website. It covers various topics related to the College of Computer Science and Mathematics at the University of Mosul, including faculty profiles, research activities, events, academic programs, admission requirements and other college-related matters.

Developing Mathematical Models to Study Changes in Groundwater Levels and Salt Concentration

Understanding how groundwater and salt concentrations change when freshwater supplies are monitored is crucial. In this work, groundwater fluxes and salt migration were considered when studying groundwater level changes and mineralisation processes. To study salt and water level changes, a more realistic mathematical model was developed to reflect the object’s basic features. To monitor geofiltration and geomigration using mathematical and numerical modelling to provide recommendations, nonlinear differential equations were applied. Scientific research on mathematical and numerical modelling and computer experimentation of the issue is briefly reviewed. Also developed were a mathematical model with two integer-valued boundary conditions and an efficient finite difference numerical approach. Changing groundwater and pressure water levels, filtration permeability, water loss coefficient, and filtration rate connected to water level all affect the environment when building a mathematical model for salt migration filtration. The article predicts groundwater, pore water, and salt concentrations in high- and low- permeability layers. Scientific research on mathematical and numerical modelling and computer experimentation of the issue is briefly reviewed. Also developed were a mathematical model with two integer-valued boundary conditions and an efficient finite difference numerical approach. Changing groundwater and pressure water levels, filtration permeability, water loss coefficient, and filtration rate connected to water level all affect the environment when building a mathematical model for salt migration filtration. This study considered soil density, active porosity, and third-order boundary conditions Open, unlike earlier studies. Since the problem is represented by nonlinear differential equations with free variables, analytical solutions are unattainable. An effective numerical method and a stable implicit system with high-accuracy approximation, finite difference approach, and forward and backward sweep approaches were given to overcome these challenges.

Evaluating the forces involved in bubble management in DMEK surgery: mathematical and computational model with clinical implications.

To model postoperative forces involved in Descemet membrane endothelial keratoplasty (DMEK) tissue adherence and bubble management, including the impact of surface tension on graft support, with a view towards clinical applications. Tennent Institute of Ophthalmology, Glasgow, and James Watt School of Engineering, University of Glasgow, Glasgow, United Kingdom. Mathematical modelling and computer simulation. Theoretical modelling of biphasic flow and interaction of gas, liquid and tissue within the anterior chamber for static horizontal scenario A (adherent DMEK with mobile bubble) and dynamic vertical scenario B (release of bubble due to pupil block following DMEK). The model assumed incompressibility for both fluids within realistically achievable pressure ranges. Cahn-Hilliard Navier-Stokes equations were discretised through the application of the Finite Element Method. Mathematical modelling and computer simulation showed bubble size, corneal curvature and force intensity influences surface tension support for DMEK tissue in scenario A. Scenario B demonstrated complex, uneven distribution of surface pressure on the DMEK graft during uncontrolled bubble release. Uneven pressure concentration can cause local tissue warping, with air/fluid displacement via capillary waves generated on the fluid-air interface adversely impacting DMEK support. We have quantitatively and qualitatively modelled the forces involved in DMEK adherence in normal circumstances. We have shown releasing air/gas can abruptly reduce DMEK tissue support via generation of large pressure gradients at the liquid/bubble/graft interfaces, creating negative local forces. Surgeons should consider these principles to reduce DMEK graft dislocation rates via optimised bubble size to graft size, longer acting bubble support and avoiding rapid decompression where possible.

Reliability of different methods of measurement of working length in endodontics

They compared measurements using radiographic images and mathematical calculations, computed tomography and electronic foraminal locator in vivo, comparing them with the real measurement determined by digital calipers after tooth extraction in 12 dental elements of patients of both sexes indicated. for tooth extraction due to advanced periodontal disease. The patients underwent Cone Beam Computed Tomography examination to determine the real length of the tooth. Afterwards, periapical radiographs were taken to establish the apparent length of the tooth. Subsequently, isolation of the operative field, the teeth had their canals accessed, emptied and subjected to conventional measurement by subtracting 3 millimeters from the radiographic vertex in this radiograph. Afterwards, a file was introduced inside the root canal, taking a new radiograph, measuring on this radiograph the distance between the tip of the instrument and the radiographic vertex, adding this measurement to the actual size of the instrument minus 0.5 millimeter as safety margin determined for the working length. Simultaneously, the canal was measured with an electronic foraminal locator. Next, the teeth were sealed and extracted, placed in containers with saline solution. The extracted teeth were measured using a digital caliper and these measurements were recorded in tables. It was concluded that odontometry using computed tomography represented the method that came closest to the real values of the dental elements, constituting the electronic locator with little significant difference in relation to the tomographic examination. Conventional dentistry, especially in anterior elements, showed a considerable degree of distortion.

Use of Scaffolding Techniques in Mathematical Problem Solving in Students Who Have Mathematics Anxiety

Solving mathematical problems is important in mathematics. This research aims to determine the mathematical problem solving abilities of students who have moderate levels of mathematics anxiety by using scaffolding techniques. This type of research is qualitative with a case study approach. The research subjects were 3 class VIII students at a junior high school in Purworejo, Central Java. The research instruments used included problem-solving ability test questions, mathematics anxiety questionnaires, and interview guides. Data is analyzed by reducing data, presenting data, and drawing conclusions. The results of this study indicate that subjects who have mathematics anxiety are having difficulty determining mathematical models and have difficulty with mathematical calculations. The scaffolding technique provided is level 2 in explaining, reviewing, and restructuring. After getting the scaffolding technique, students with mathematics anxiety are able to solve mathematical problems.

Simple Calculator Application with Client-Server Connection Using Java Socket

This research aims to develop a simple calculator application using a client-server connection based on Java Socket. The research method involves the software development process that integrates a client-server architecture to enable users to perform basic mathematical calculations over a network. The main findings of this research are the successful implementation that enables effective communication between the client and server, facilitating the exchange of data for simple mathematical operations. The conclusion of this research indicates that the use of Java Socket in a calculator application can provide reliable and efficient data transmission performance. The contribution of this research lies in the development of a practical application that demonstrates the benefits of implementing a client-server connection using Java Socket in the context of mathematical calculations.

МЕТОДИКА РОЗРАХУНКУ ОСНОВНИХ ПАРАМЕТРІВ ОСКОЛКОВИХ ВИБУХОВИХ ПРИСТРОЇВ НАПРАВЛЕНОЇ ДІЇ ДЛЯ УРАЖЕННЯ ПОВІТРЯНИХ ЦІЛЕЙ ТИПУ БПЛА

Based on the experience of the Russian-Ukrainian war, there is a tendency to increase the number of unmanned aerial vehicles used to destroy equipment, personnel, and infrastructure. For this purpose, unmanned aerial vehicles such as barrage munitions, kamikaze, drop drones, etc. are used. The share of losses of weapons and equipment, as well as personnel, caused by such weapons has become the largest compared to other means of destruction. To counteract such means, various anti-aircraft missile systems, electronic warfare equipment and mobile fire groups are used. The most effective way to defeat unmanned aerial vehicles is to hit them with a large number of destructive elements (fragments). Taking into account the ratio of the cost of unmanned aerial vehicles and means of their destruction, in particular anti-aircraft missile munitions, the paper proposes the use of fragmentation explosive devices of directed action to defeat attack unmanned aerial vehicles according to the "effectiveness-cost" criterion. The article presents an analysis of studies of the effectiveness of the use of directed-action fragmentation munitions. A study of the feasibility of using directed-action fragmentation engineering ammunition of the MON series mines is carried out. On the basis of the mathematical model of the flight of a fragment, taking into account its shape, a methodology for calculating the main parameters of fragmentation explosive devices of directed action for the destruction of air targets such as unmanned aerial vehicles is proposed. Mathematical calculations are presented to determine the main characteristics of fragmentation explosive devices of directed action for defeating unmanned aerial vehicles, taking into account the external trajectory measurements of the target and the factors that affect the formation of the fragmentation field (zone). Schemes of deployment of fragmentation explosive devices of directed action for the protection of objects are proposed.

Impact of Artificial Intelligence on the Job Landscape in the Indian IT Sector

This research paper evaluates the impact of Artificial Intelligence (AI) on the job market within India’s Information Technology (IT) sector using literature review, mathematical calculations, graphical representation, and qualitative assessment. Given the IT sector’s significant contribution to India’s GDP and AI’s leading role in the global technology revolution, this study holds particular relevance. Job roles are changing as automation follows a path from routine, data-intensive tasks to potentially displacing conventionally "smart" roles like software testing . On the other hand, roles that involve more complex cognitive tasks such as coding or data analysis, are being augmented by AI, which has positive effects on productivity and job satisfaction. In addition, new jobs roles like AI ethics officers and AI architects are also coming up, which calls for reskilling/upskilling. Nevertheless, this paper suggests that AI has a dual-faceted impact on the IT job market in India — it threatens the obsolescence of some roles but also focuses on creating more skilled jobs. It also suggests that the workforce must be equipped with the required skills and knowledge to enable such initiatives and ensure viable future growth and innovation within the industry.

Computational Mathematics and Modeling

Computational Mathematics and Modeling

Systems of quaternionic linear matrix equations: solution, computation, algorithm, and applications

<p>In applied and computational mathematics, quaternions are fundamental in representing three-dimensional rotations. However, specific types of quaternionic linear matrix equations remain few explored. This study introduces new quaternionic linear matrix equations and their necessary and sufficient conditions for solvability. We employ a methodology involving lemmas and ranks of coefficient matrices to develop a novel algorithm. This algorithm is validated through numerical examples, showing its applications in advanced fields. In control theory, these equations are used for analyzing control systems, particularly for spacecraft attitude control in aerospace engineering and for control of arms in robotics. In quantum computing, quaternionic equations model quantum gates and transformations, which are important for algorithms and error correction, contributing to the development of fault-tolerant quantum computers. In signal processing, these equations enhance multidimensional signal filtering and noise reduction, with applications in color image processing and radar signal analysis. We extend our study to include cases of $ \eta $-Hermitian and i-Hermitian solutions. Our work represents an advancement in applied mathematics, providing computational methods for solving quaternionic matrix equations and expanding their practical applications.</p>

Local convergence study of tenth-order iterative method in Banach spaces with basin of attraction

<abstract><p>Many applications from computational mathematics can be identified for a system of non-linear equations in more generalized Banach spaces. Analytical methods do not exist for solving these type of equations, and so we solve these equations using iterative methods. We introduced a new numerical technique for finding the roots of non-linear equations in Banach space. The method is tenth-order and it is an extension of the fifth-order method which is developed by Arroyo et.al. <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. We provided a convergence analysis to demonstrate that the method exhibits tenth-order convergence. Also, we discussed the local convergence properties of the suggested method which depends on the fundamental supposition that the first-order Fréchet derivative of the involved function $ \Upsilon $ satisfies the Lipschitz conditions. This new approach is not only an extension of prior research, but also establishes a theoretical concept of the radius of convergence. We validated the efficacy of our method through various numerical examples. Our method is comparable with the methods of Tao Y et al. <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>. We also compared it with higher-order iterative methods, and we observed that it either performs similarly or better for the numerical examples. We also gave the basin of attraction to demonstrate the behaviour in the complex plane.</p></abstract>

Design and Fabrication of PVC Pipe Cutting Machine with Hot Wire Mechanism

The utilization of cutting tools presents inherent limitations, encompassing sluggish production rates, inconsistent cutting precision, and elevated manufacturing expenses in the cutting of PVC (Poly-Vinyl Chloride) pipes. Addressing these challenges requires a systematic design approach to engineers to build sustainable new mechanism for PVC pipe cutting apparatus capable of streamlined operation. This study endeavours to conceptualize and develop a unique mechanism comprised of Ni-Chrome heated wire as a replacement of traditional cutter. The design focus is aimed at enhancing the cutting operation within low cost and enhanced sustainability. The methodology for machine design encompasses a complete machine design approach for each single component if a machine utilizing Solidworks 2023, complemented by rigorous mathematical design calculations. The outcomes of this study manifest in the creation of a prototype of Ni-Chrome heated wire machine capable of cutting the pipes within thickness range from 20-35mm and more based on the heated capability of wire. Noteworthy advantages of this machine design include expedited manufacturing processes, reduced energy consumption, and enhanced labour efficiency, thereby augmenting overall productivity. Future research avenues are suggested to encompass an analysis of operator safety considerations, establishment of periodic maintenance protocols, automation, and optimization of spare parts accessibility to further refine the efficacy and longevity of the proposed system.

Анализ потерь в защищенных и неизолированных проводах в стационарных режимах с учетом погодных условий

In overhead lines of power supply systems at voltages of 6–110 kV, the use of protected wires is becoming increasingly popular, which have a number of advantages over traditional non-insulated aluminum wires. On the other hand, the presence of insulation requires the development of more complex mathematical models to adequately describe their behavior and study the characteristics of new wires during operation. The results of using the developed mathematical models and calculation programs for comparative analysis of protected and non-insulated wires under conditions of changing load currents and weather conditions are presented. The dependences of temperature and loss of active power on insulation thickness are investigated. It is shown that the presence of insulation allows, under certain conditions, to reduce temperature and losses compared to wires without insulation, all other things being equal. The results obtained can be used to increase the capacity and reliability of power transmission lines.

Fault Current Unified Calculation Method for Whole Process Fault Ride-Through of DFIG-Based Wind Farms

The present inaccurate fault current calculation, due to the incomplete consideration of whole process operation stages of fault ride-through (FRT) of DFIG-based wind farms (WFs), has a distinct impact on the safe operation of local power grid. To tackle this challenge, a fault current unified calculation (FCUC) method is proposed to achieve high-accuracy and high-speed mathematical analytical calculations of DFIGs output fault current during FRT. The proposed FCUC method mainly includes: 1) Whole process unified state space model, which contains transition stage, crowbar protection stage, demagnetization stage, reactive current compensation stage, and dc-chopper protection stage, is established to thoroughly consider FRTs different internal protection circuits and control strategies; 2) Stator output fault current is conducted by the unified state space model based on the stator residual voltage and the stator flux linkage; 3) Grid side converter (GSC) output fault current, which considers both injection mode and absorption mode, is developed based on the grid residual voltage and the unified state space model; 4) Fault current contribution mechanism, which is aggregated from both stator and grid side converter, is revealed based on Laplace transform from time-frequency-time domain, and is first expressed as an unified fault current expression. Extensive simulation/experimental/calculation results under different voltage sags have validated the correctness and effectiveness of the proposed FCUC method.

DYNAMICS AND DEVELOPMENT TRENDS OF THE INTERNATIONAL MONEY TRANSFERS IN THE REPUBLIC OF ARMENIA

The importance of the study of international remittances is justified by the fact that they are important in revealing cause-and-effect relationships in the context of evaluating the development trends of the RA economy, so the article aims to comprehensively analyze the dynamics of international remittances in Armenia. Within the framework of the issues comming from the purpose of the research, the essence of the concept of "international money transfer" was presented in the article, the main goals of international money transfers have been distinguished, the factors that influence the choice of the option of making an international money transfer have been referred to, the dynamics of the international money transfers of the Aremenian commercial banks have been studied by analyzing cause-effect relationships, the index of the ratio of remittance inflow to GDP has been calculated, the factors determining the development trends of international remittances have been presented. For the research, the methods based on mathematical formula calculations, comparison of statistical deviations and logical reasoning have been used in the scientific paper. At the end of the article, main points have been listed as a conclusion, which can be basis for making professional justified judgments and processing development scenarios for international remittance within theorical and practical researches.

Mathematical modeling of the combined heat supply system of the solar house

Today, improving energy efficiency in residential heat supply systems, saving fuel and energy resources, and increasing the efficiency of using devices based on renewable energy sources is an urgent issue. The aim of the article is to develop a mathematical model of the heat balance of one-story country houses based on the use of solar energy in a non-stationary mode and conduct theoretical research. The heat balance scheme of the solar house with autonomous heat supply and the electric-heat scheme of the physical model are proposed. Based on the proposed schemes, a mathematical model and calculation algorithm of the heat balance of a one-story rural house of the research object in non-stationary mode were developed, based on the mathematical modeling, the influence of the heat capacity of the wall structure on the temperature regime of the building was studied. The main temperature characteristics, which analyzed the change of the indoor air temperature of the building depending on the ambient temperature, were built on the basis of the MATLAB/SIMULINK program. The mathematical model of the building’s heat balance in dynamic mode and the obtained calculation results are recommended for use in the development of energy-efficient solar houses.

К моделированию задачи осаждения твердой частицы в вязкой несжимаемой жидкости методом гидродинамики сглаженных частиц (SPH)

In this paper was formulates the settling velocity determining problem for a single particle. To solve it, an original version of the smoothed particle method (SPH) is proposed where settling particle affects on surrounding fluid particles movement. Herewith the calculation of forces acting on particle (except for inertial forces which takes attached mass effect) is performed according by analytical mechanics formulas for a material point. This mathematical formulation and calculation algorithm was verificated by using open source code «SPH_Lab2d». Dependences for the particle settling velocity on its diameter are obtained for various cases of fluid volume discretization. The results demonstrate a good conformance this property to values, that determined by the experimental data and known phenomenological dependences for quartz sand.

РЕКУПЕРАЦИЯ НЕФТЕСОДЕРЖАЩИХ ПРОМЫШЛЕННЫХ ОТХОДОВ КАК ФАКТОР СНИЖЕНИЯ УГРОЗ ТЕХНОСФЕРНОЙ ОПАСНОСТИ

The relevance of the study is predetermined by the unresolved problems of disposal of oil-containing waste in the Yaroslavl region. The purpose of the study was to analyze the system for preventing technospheric hazards caused by the formation of industrial waste and their negative impact on the environment. The object of the study was industrial waste, highly hazardous to the environment, characteristic of a number of enterprises in the region: acid tar. The subject of the study was the process of optimizing the composition of a bitumen binder based on acid tar in terms of the content of galvanic sludge, captax and sulfur for subsequent use in a modified form as a secondary oil-containing raw material for the production of bitumen emulsions, pastes, and lubricants. Such raw materials, technologically brought to a low-hazard level, can be successfully and effectively used in various production operations of construction, reconstruction, operation, and repair of facilities. The optimization process was carried out using statistical research methods using, during modeling, the obtained dependence of the withdrawal properties on the factors of a second-degree polynomial with two unknowns. Calculations were performed using an Excel spreadsheet package with built-in mathematical local calculation programs. The research results in the technological part have been patented and implemented in the practical activities of enterprises in the region. The value of scientific and practical developments lies in reducing the volume of formation and storage of hazardous oil-containing acid tar waste and, as a consequence, preventing and reducing the level of technospheric danger and potential man-made emergency situations.

DESIGN AND ANALYSIS OF THE OVERHEAD CRANE WITH SIX VERTICAL COLUMNS

This study delves into the comprehensive examination of an overhead crane’s construction, focusing on its frame, columns, and beams, while considering factors such as strength, stability, and stiffness. Through an in-depth review in the design domain, it is proposed a specific structural configuration for the overhead crane. This design comprises six vertical columns supporting two longitudinal beams equipped with tracks for the trolley’s longitudinal movement. Additionally, cross beams featuring winches are mounted on the trolley’s cross beams. The crane’s columns are securely mounted on a foundation, and struts are employed to attach the crane to the load-bearing wall of the building, ensuring longitudinal and transverse stability. The inclusion of cross truss structures with longitudinal struts further enhances the crane’s overall stiffness, with additional vertical struts provided to augment the left side’s longitudinal stiffness. The study also encompasses the analysis of the crane frame’s construction, complete with the development of an appropriate calculation scheme and the computation of static reactions in supports. Further calculations involve determining the cross-sections of vertical columns and longitudinal beams, ensuring compliance with strength, rigidity, and stability requirements. The selected cross-section for the columns, in the form of a square profile pipe (100×100×3 mm), is meticulously chosen to meet these criteria. Simulation modeling of load scenarios on the crane frame elements in SolidWorks software validates their strength, stiffness, and stability. Mathematical models and calculations provided the optimal parameters and characteristics of each crane component, ensuring a superior level of safety and operational efficiency. These results provide valuable insights for future research in mechanical engineering and the design of industrial mechanisms.

Machine learning for forecasting some stock market index

In this paper, we evaluate the QMLKF algorithm, designed in the previous paper [Benmoumen M. Numerical optimization of the likelihood function based on Kalman Filter in the GARCH models. Mathematical Modeling and Computing. 9 (3), 599–606 (2022)] for parameter estimation of GARCH models, by transposing it to real data and then present our machine learning for forecasting the returns of some stock indices.