This study aims to provide a concrete illustration of the application of systems of linear equations in two variables (SPLDV) and two-variable linear programming in solving real-world problems commonly encountered in daily life. The background of this research is rooted in the persistently low ability of students and small business practitioners to apply linear mathematical concepts to support decision-making processes, such as determining product prices and optimizing production. Using a descriptive method with a conceptual quantitative approach, this study presents two illustrative case studies: the determination of unit prices for stationery products through SPLDV, and the optimization of the production combination of woven bags and mats in a rattan-based small enterprise through two-variable linear programming under time constraints. The results indicate that SPLDV can objectively determine product prices based on transaction data, while linear programming provides an optimal solution in the form of producing 12 bags and 6 mats, yielding a maximum profit of IDR 1,320,000. These findings demonstrate that linear mathematical models serve as effective and applicable tools to support decision-making in both microeconomic contexts and mathematics education.

In this paper, we consider the flow of a nematic liquid crystal in the domain exterior to a small spherical particle. We work within the framework of the $\unicode{x1D64C}$ -tensor model, taking into account the orientational elasticity of the medium. Under a suitable regime of physical parameters, the governing equations can be reduced to a system of linear partial differential equations. Our focus is on precise far-field asymptotics of the flow velocity with an emphasis on its anisotropic behaviour. We are able to analytically characterize the flow pattern and compare it with that of the classical isotropic Stokes flow. The expression for velocity away from the particle can be computed numerically or symbolically.

Abstract Many engineering problems involve solving large linear systems of equations. Conjugate gradient (CG) is one of the most popular iterative methods for solving such systems. CG typically requires a good preconditioner to speed up convergence, and computing these preconditioners can be challenging. In this paper, we explore whether any of the popular preconditioners can be computed today using quantum computers. We demonstrate that sparse approximate inverse (SPAI) preconditioners can be computed efficiently using quantum annealing machines. Specifically, we provide an extension of the box algorithm for computing the SPAI of very large matrices on D-Wave Advantage machines. The computation of an SPAI reduces to solving a series of quadratic unconstrained binary optimization (QUBO) problems. This is demonstrated using both well-conditioned and poorly-conditioned linear systems arising from a 2D finite difference formulation of the Poisson problem.

A frequency-independent solver for systems of linear ordinary differential equations

Meta-analyses assessing test accuracy typically require extracting true positive (TP), false negative (FN), false positive (FP), and true negative (TN) counts from each study, commonly organized in a 2 × 2 table. However, many published test accuracy studies do not report all of these counts, which can limit the ability of a meta-analysis to fully capture the available evidence on the screening or diagnostic accuracy of a given test. Fortunately, test accuracy studies often report sufficient parameters, such as sensitivity and specificity, that enable the estimation of unreported counts. The relationships between these commonly reported parameters and the unreported cell counts may be expressed mathematically and organized into a system of four linear equations. The basic principles of solving such systems using matrix methods are introduced, accompanied by examples illustrating the development and solution of linear systems with unknown TP, FN, TN, and TN counts. Approaches for handling rounding errors of reported test accuracy parameters in publications are also demonstrated. Additionally, methods for obtaining a bound solution are explored in scenarios where the solution for missing test accuracy counts results in a system with three linear equations and four unknowns, leading to non-unique solutions. Simulation studies are conducted to assess the performance of these methods, and practical guidance for their implementation is provided. The Microsoft Excel spreadsheets and SAS and R code for the examples presented in this article are available in the Supporting Information.

Students’ difficulties in understanding Systems of Linear Equations in Three Variables (SPLTV) are often caused by limited visualization and a lack of contextual learning approaches. The purpose of this study is to develop a Geogebra-assisted learning media based on Problem-Based Learning that is valid and practical, thus, it can support students in learning mathematics. The method used was design research of the development studies type with Gravemeijer’s stages, which consist of three phases, but were limited to the second trial phase in the teaching experiment stage. The subjects in this study were students of Class XI-1 at SMAN 2 Cimahi and Class XI-I at SMAN 1 Batujajar. The instruments used were expert validation sheets and student and teacher response questionnaires, which were then analyzed using a Likert scale of 1 to 5. The validation results showed that the Geogebra-assisted learning media based on Problem-Based Learning met the criteria for validity in terms of both content and media aspects. The first and second trial phases indicated that the Geogebra-assisted learning media met the Very Good criteria in terms of practicality based on student and teacher responses. Therefore, this Geogebra-assisted learning media is recommended as an alternative supporting media in mathematics learning, particularly for SPLTV material, as it can help students understand mathematical concepts more clearly, interactively, and contextually.

Determinant-free, tolerance solution of a singular fuzzy linear equation system using the shifted membership function method for control and decision-making tasks

This study analyzed the trajectories of mathematical comprehension among 461 eighth-grade students using online learning log data, identified distinct learner types, and compared their characteristics across clusters. The data was collected from students who studied the topic “Linear Inequalities and Systems of Linear Equations” between April and June 2020, and Dynamic Time Warping (DTW) based K-means clustering was applied. The analysis yielded four learner clusters. Cluster 1 represented a Low-Level group characterized by consistently low achievement and short response times; Cluster 2, labeled the Stable High group, showed high achievement and sustained engagement; Cluster 3, the Gradually Declining Mid group, exhibited a gradual downward trend from a moderate level; and Cluster 4, the Rapidly Declining Lower-Mid group, demonstrated a sharp decline from a lower-mid level. Cluster 2 recorded the highest number of solved items, test-taking days, and response times, whereas Cluster 1 displayed features resembling skipping behavior. A higher proportion of male students was observed in Clusters 1 and 4, both of which also demonstrated the lowest achievement levels. Across all groups, the lowest accuracy rates appeared in Chapter 1 (Solutions to Inequalities) and 4 (Systems of Two Linear Equations in Two Variables), indicating that students experienced greater cognitive burden when acquiring new concepts. These findings provide evidence that online learning log data can precisely capture the relationship between learner behavior and achievement, and underscore the importance of identifying declining-achievement groups at an early stage to design targeted learning support strategies.

This research aims to develop a mathematics learning medium in the form of an Indonesian culture-based e-comic integrated with the educational values ​​of Imam Syafi'i on the topic of Two-Variable Linear Equation Systems (SPLDV) to improve the mathematical problem-solving skills of students at SMP IT Al Maidani. The urgency of this research is based on the low mathematical problem-solving abilities of junior high school students, as well as the lack of contextual, engaging, and Islamic character-infused learning media. The development model used was the 4D model (Define, Design, Develop, Disseminate). The research subjects consisted of 30 students in the first trial and 32 students in the second trial. Validation was conducted by media experts, material experts, religious experts, and cultural experts. The validation results indicated that the developed learning medium was valid with an average score of 4.3 (media experts), 4.2 (material experts), 4.4 (religious experts), and 4.5 (cultural experts). The first trial resulted in an increase in the average score from 76.9 (pretest) to 83.73 (posttest), with an N-Gain of 0.32 (moderate category). Based on the reflection results of the first trial, revisions were made, including improvements to the interface, the addition of sample verification questions, and additional contextual exercises. After revisions, the second trial showed a significant increase, with an average score from 77.4 to 94.25 and an N-Gain of 0.78 (high category). A total of 87.5% of students achieved classical learning completion. The results of the study demonstrate that the developed e-comic media is valid, practical, and effective in improving students' mathematical problem-solving skills and instilling Islamic character values ​​aligned with the educational values ​​of Imam Syafi'i.

Abstract: This study aims to develop an e-module containing interactive vidoes to enhance student problem-solving abilities in the topic of there-variable linear equation systems (SPLTV). The research method used was Research and Development (R&D) employing the ADDIE model, which consists of five stages: Analysis, Design, Development, Implementation, and Evaluation. The research subjects were student of class X M2 at SMA Negeri 4 Sekadau. The instruments used included material and media expert validation sheets, teacher and student response quetionnaires, and posttest items. The validation results indicated that the e-module met the criteria of being highly valid, with material validity reaching 89,12% and media validity reaching 95,11%. The practicality level was categorized as very pratical, with teacher reponse quetionnaire scoring 100% and students reponse reaching 87,30%. The module was also declared effective, as evidenced by the posttest result of 77,77%. It can therefore be concluded that the interactive video-based e-module is highly valid, highly pratical, and effective, and can be used in the learning process. Keywords: E-module, interactive video, problem-solving ability, SPLTV.

Students' poor mathematical literacy, particularly when it comes to the content of Two-Variable Linear Equation Systems (SPLDV), which calls for the ability to comprehend information, model it mathematically, and clearly communicate the solution's findings, is what spurred this study. This project intends to improve the learning process and increase students' mathematical literacy skills by using the Contextual Teaching and Learning (CTL) approach. Using the Classroom Action study (CAR) methodology created by Kemmis and McTaggart, the research was carried out in two cycles. Thirty students from SMP Negeri 4 Tambang's class during the even semester of the 2024–2025 school year served as the study subjects. Mathematical literacy competency tests and teacher and student activity observations were used to gather research data in the form of descriptions. The results of the study indicate that the application of CTL is able to improve the implementation of learning, marked by increased student activeness in asking questions, discussing, and expressing opinions, as well as a more consistent role of teachers as facilitators. In terms of learning outcomes, students' mathematical literacy skills improved from cycle I to cycle II, both in writing problem information, using appropriate concepts and procedures, and in drawing conclusions from solutions. Thus, it can be said that using CTL effectively raises the quality of mathematics instruction while also helping students become more mathematically literate while using the SPLDV content.

A formal solution has been constructed for a countable system of first-order differential equations with two small parameters in the case where the principal matrix consists of Jordan blocks of equal dimension with distinct characteristic numbers. The asymptotic behavior of this solution has been investigated.The aim of the article is to determine the conditions under which a countable system of first-order linear differential equations with two small parameters, in the case where the principal matrix consists of Jordan blocks of equal dimension and distinct characteristic numbers, admits a solution; to construct a formal solution; and to prove its asymptotic behavior.

This study aims to explore the application of teaching materials with the Math Adventure approach to improving students' Numeracy Literacy skills in the Two-Variable Linear Equation System (SPLDV) material. The method used is a pre-experimental design with a one-group pre-test-post-test design, involving 33 eighth-grade students at SMP Negeri 2 Dobo. Data were collected through a Numeracy Literacy ability test. The results of the study showed a significant increase in students' Numeracy Literacy skills after the application of teaching materials that had been explored with the Math Adventure approach, with an average score increasing from 41.58 to 83.27 and an average N-Gain value of 0.70. A total of 51.62% of students reached the Very Advanced category and 48.38% in the Advanced category. The results of the paired sample t-test showed a significant difference between the pre-test and post-test (p < 0.05). Thus, the integration of exploration of teaching materials with the Math Adventure approach has proven effective in improving students' Numeracy.

The current manuscript introduces an efficient modification scheme, based on the Adomian decomposition method, for the numerical solution of systems of two-point boundary-value problems. Indeed, this introduced method is characterized by high computational flexibility in accurately solving both the systems of linear and nonlinear differential equations with two-point boundary conditions. Moreover, to assess the robustness of these schemes, several test numerical examples, including a well-known model in fluid dynamics, have been sought and treated. Comparatively, the results from the proposed method have been compared with those of the available exact solutions and those of deployed existing numerical methods in the literature. Lastly, in the presence of the methods used for comparison, the assessment of this method turned out to be positive, with rapid convergence and high precision speedily attained.

Volterra integral equations appear when we convert an initial value problem to an integral ‎equation. The solution of the Volterra integral equation is much easier than the original initial value ‎problem. Many problems of thermodynamics, biology, chemistry, medical sciences, physics, ‎heat flow, neutron diffusion problem, electric circuit problem, transform problem mechanics, ‎and engineering can be represented mathematically in terms of Volterra integral equations of the first and ‎second kind. Numerical Method for solving Third Kind Volterra Integral Equations (VIEs) ‎using Simpson's Rule. By approximating the unknown function using Lagrange Polynomial ‎interpolation and applying Simpson's rule for numerical integration, we obtain a system of ‎linear algebraic equations. The solution of this system provides an approximation to the ‎original integral equation‎.

The authors present the outcome of a simulation experiment on elimination of systematic errors in the state levelling network using potential technology of chronometric levelling. Its results consider two algorithms. In the initial instance, the geopotential numbers of geodetic points, as derived from the chronometric levelling, are incorporated into estimation of the parameters. In the subsequent one, these numbers are regarded as supplementary measurements of the system of linear equations during the optimization of the levelling network. The experiments were conducted on the levelling network of the RF and the Republic of Kazakhstan fundamental points heights catalogue model, class I (2120 points). The modelling program enables manipulating three variables

Temporomandibular joint (TMJ) prosthesis implantation is an effective procedure for treating temporomandibular joint disorders. Traditionally, preoperative planning for TMJ surgery has been conducted manually by experienced surgeons, which often results in longer operating time and less reliable prosthesis placement. This study proposes an automated surgical planning algorithm for TMJ prosthesis implantation that calculates the optimal position for prosthesis placement. Firstly, the STL model of the patient's craniomaxillofacial structure is populated with a point cloud, and the oriented bounding box is calculated. Next, the point cloud that meets specific constraints is filtered based on the characteristics of the anatomical structures. Subsequently, the contact surfaces of the prosthesis with the zygomatic arch and the mandible are generated according to the distribution of the point cloud. Finally, a system of linear equations is established based on the geometric constraints and solved to determine the precise placement position and orientation of the prosthesis. A group of 12 patients with 24 clinical cases was utilized for automatic planning to evaluate the efficiency and quality of the system. The results demonstrated that the average time required for the automatic planning algorithm was under 30s for the entire procedure. Furthermore, the calculation of the bone-prosthesis contact area indicated that the quality of the automatic planning is comparable to that of plans created by professional surgeons. Compared to the cumbersome manual planning methods currently used for TMJ prosthesis implantation, the approach proposed in this paper facilitates efficient and accurate automated preoperative planning. This method simulates the optimal placement of TMJ prosthesis to ensure long-term stability, demonstrating significant clinical potential for future applications in TMJ prosthesis implantation surgery.

Discrete time-delay systems arise widely in engineering and scientific applications. These systems are often modelled as high-dimensional dynamical systems, and the presence of delays increases the computational complexity and cost of both simulation and analysis. To alleviate this challenge, this paper proposes a new model reduction method in a parallel manner tailored for discrete time-delay systems. The proposed method begins with the explicit difference relations of Krawtchouk polynomials and a structural analysis of the shift-transformation matrices. We expand the original system over a Krawtchouk polynomial basis, which yields a system of linear algebraic equations characterised by a block lower triangular Toeplitz matrix. Further, applying the block discrete Fourier transform to the involved block α-circulant matrices, we design a parallel strategy to efficiently construct the projection basis used for reducing discrete time-delay systems. This is the main contribution of this paper. We provide rigorous theoretical results on the invertibility of the block α-circulant matrices and present error estimation bounds to ensure the validity and stability of the reduced model. Numerical examples demonstrate that our method achieves accurate approximation while offering substantial computational speed-ups, especially for large-scale time-delay systems.

An analytical method is proposed for determining the stress-strain state in an elastic layer with a cylindrical cavity supported by linear continuous supports perpendicular to the cavity. The need for such a development is due to the fact that in aerospace and mechanical engineering, structural elements are often affected by loads and supports described by infinite functions. This complicates the calculation for spatial bodies with complex geometry and stress concentrators. The methodology is based on the generalized Fourier method within the spatial problem of elasticity theory. The model is considered as a layer with specified stresses at the outer boundaries, where the reactions of the supports are represented as applied loads. A combined approach is used to describe the geometry using a Cartesian coordinate system for the layer and a cylindrical coordinate system for the cavity. The key idea is to decompose the original problem into two simpler ones using the principle of superposition. Auxiliary problem: the stresses in a solid layer (without a cavity) are calculated to determine the stress fields at its nominal location. Main problem: a layer with a cavity is considered, on the surface of which the stresses calculated in the first step are acting but taken with the opposite sign. The complete solution is the sum of the solutions of these two problems. Each of them is reduced to an infinite system of linear algebraic equations, which is solved by the method of reduction. This approach makes it possible to calculate the stress-strain state at any point of the body with high accuracy. Numerical analysis confirmed the correctness of satisfying the boundary conditions and showed the dependence of stresses on the nature of the distributed loads. The cylindrical cavity acts as a stress concentrator, which leads to a local increase in stresses σx and σz at the upper and lower boundaries of the layer to values that exceed both the applied load by and the calculated resistance of concrete of class C25/30.

Abstract This work suggests different Monte Carlo algorithms for solving large systems of linear algebraic equations arising from the numerical solution of the Dirichlet problem for the Helmholtz equation. Approach based on boundary integral representations, vector randomization algorithm, method of fundamental solutions, stochastic projection algorithm, and randomized singular value decomposition are applied. It is shown that the use of stochastic iterative refinement and preconditioning can significantly improve the accuracy and stability of the computations. Simulation results are presented, demonstrating the effectiveness of the proposed methods.