Algebraic Problem Research Articles

Dedicated Hardware Accelerators for Processing of Sparse Matrices and Vectors: A Survey

Performance in scientific and engineering applications such as computational physics, algebraic graph problems or Convolutional Neural Networks (CNN), is dominated by the manipulation of large sparse matrices—matrices with a large number of zero elements. Specialized software using data formats for sparse matrices has been optimized for the main kernels of interest: SpMV and SpMSpM matrix multiplications, but due to the indirect memory accesses, the performance is still limited by the memory hierarchy of conventional computers. Recent work shows that specific hardware accelerators can reduce memory traffic and improve the execution time of sparse matrix multiplication, compared to the best software implementations. The performance of these sparse hardware accelerators depends on the choice of the sparse format, COO , CSR , etc, the algorithm, inner-product , outer-product , Gustavson , and many hardware design choices. In this article, we propose a systematic survey which identifies the design choices of state-of-the-art accelerators for sparse matrix multiplication kernels. We introduce the necessary concepts and then present, compare, and classify the main sparse accelerators in the literature, using consistent notations. Finally, we propose a taxonomy for these accelerators to help future designers make the best choices depending on their objectives.

Fast multi-level iteration schemes with compression technique for eigen-problems of compact integral operators

In this paper, we present multi-level iteration schemes to solve the eigen-problems of compact integral operators based on the multiscale Galerkin methods. By using the compression technique, the wavelet coefficient matrix is truncated into sparse. This technique results in a fast algorithm. To further minimize computational complexity, we establish Jacobi, Gauss-Seidel and L-H iteration schemes for solving the final algebraic eigenvalue problems obtained from discretizing the integral equation using fast multiscale Galerkin methods. We show that the proposed schemes require only linear computational complexity and obtain the desired convergence order for the approximate eigenvalues and eigenvectors. Numerical experiments are conducted and analyzed, with comparisons to the standard routine and the power iteration algorithm included. Ultimately, our multi-level iteration schemes yield faster solutions with fewer iterations, which is independent of the dimension of approximate space.

ANALYSIS OF STUDENTS' DIFFICULTIES IN SOLVING ALGEBRAIC OPERATION PROBLEMS BASED ON PROBLEM-SOLVING SKILLS IN GRADE VII AT SMP NEGERI 8 TAMBUN SELATAN

This study aims to analyze the difficulties faced by students when solving algebraic operation problems based on the criteria of problem solving ability. This study was conducted to understand the factors that master students' difficulties in solving algebraic problems. This research used descriptive method with qualitative approach. The research subjects consisted of 35 students of class VII-G at SMP N 8 Tambun Selatan. Data were collected through observation, tests, questionnaires, and documentation. Data analysis used the Miles and Huberman data analysis model. Data validity was checked through method tringulation. The results showed that of the 35 students studied: 1) 61% of students had difficulty in understanding the problem, 2) 57% of students had difficulty in planning the solution, 3) 73% of students had difficulty in carrying out the plan, and 4) 86% of students had difficulty in checking back. The main factors that influence these difficulties are students' understanding of basic algebraic concepts, difficulties in applying mathematical concepts to the algebraic context and underdeveloped problem solving skills. In understanding math problems and formulating the steps needed to solve them, students often face difficulties.

METHODOLOGICAL FEATURES OF TEACHING THE VECTOR METHOD IN A SCHOOL GEOMETRY COURSE

The article describes the main methodological features of teaching “vector methods” in a school geometry course. In order to increase the efficiency of teaching students the vector method, the connections between the vector method and coordinate methods are analyzed. Examples of solving algebraic and geometric problems using the vector method were shown, showing the scope of vector studies covered in the school course. A brief overview of the history of the development of the vector method is given. The article is recommended for students, school teachers, methodologists and all interested parties.

Analisis Kesalahan Siswa MTs dalam Menyelesaikan Soal-Soal Aljabar Menggunakan Teori Newman

This research was conducted to analyze students' errors in solving junior high school algebra problems and the causes of these errors using Newman's theory. This research uses a quantitative descriptive approach. The research subjects were 31 class VII students. The instrument of this research is written formative test questions on algebra topics which are analyzed using Newman error analysis. Based on the results of data analysis, it was concluded that students made (1) mistakes in reading the questions; (2) mistakes in understanding the questions; (3) process skills errors, and (4) errors in writing final answers. Based on the research results, it can be concluded that based on Newman's theory, there are four mistakes made by junior high school students in solving algebra problems Keywords: student errors, Newman Theory, algebra

Analysis of Student Errors in Solving Equations and Inequalities Problems in Algebra and Trigonometry Based on Newman Procedure

The purpose of this study is to examine how students make mistakes when attempting to solve equality and inequality problems in trigonometry and algebra using the Newman method. A qualitative descriptive research methodology is employed in this study. Students enrolled in Manado State University's 2023–2024 Mathematics Education Study Program during their first semester served as the research subjects. Test and interview techniques are used in data collection. Three steps made up the data analysis in this study: reduction, data display, and data verification. Data analysis utilizing the Newman approach yielded the following conclusions: 5.6% encoding errors, 4.8% reading errors, 12% comprehension errors, 8.4% transformation errors, and 18.8% process skill errors were made when solving equation and inequality problems. The inability of pupils to comprehend the concept of the subject being explained, their lack of willingness to study, and their carelessness when reading and completing questions are the causes contributing to these blunders.

Students’ Algebraic Proficiency and Attitude towards Learning Algebra

Learning algebra is a difficult process that necessitates a positive attitude in order to maintain interest and master key concepts. Previous research found that students' attitudes influenced their algebra learning. As a result, more input is required to observe the current influence of attitudes on algebra learning. This study aims to investigate secondary school students’ levels of proficiency and attitudes towards learning algebra. A case study was employed to collect data using an algebra test, a questionnaire on attitude, and task-based interviews. The subjects of this study were Form Three and Form Four students who were in the Dual Language Programme (DLP) in a school in Malaysia. A total of 93 students volunteered to participate in this study, and three of them were interviewed for insights into solving algebra problems. This study revealed that (1) there was no significant difference in algebra achievement between the Form Three and Form Four students; (2) there was a negative correlation between the achievement and attitudes; and (3) the students displayed rather low proficiency in answering the test questions, which could be attributed to difficulties with comprehension of problems in algebra and inadequate reasoning when applying problem solving strategies. These findings have implications for developing positive attitudes towards the learning of algebra. It is recommended that educators look into these areas to equip students with the right attitudes and knowledge for advanced learning at university.

Evaluating the Impact of the WEI4S Instructional Approach on Middle School Students’ Algebraic Problem-Solving Skills

This study investigates the effectiveness of an instructional approach called “Worked-out Examples Involving 4 Stages” (WEI4S) for enhancing middle school students’ algebraic problem-solving skills and reducing CL. Based on cognitive load theory (CLT), WEI4S combines four stages (analyse, explain, investigate, and apply) to facilitate deep comprehension. This study involved 36 novice seventh-grade students identified as having challenges solving algebraic word problems. Participants were randomly assigned to different groups, including the experimental group that received the WEI4S intervention and a control group that participated in problem-solving activities. Pre- and post-tests, as well as a transfer test, were administered to measure algebra performance and CL, and students’ opinions about the intervention were collected. The findings revealed that students exposed to WEI4S showed significant improvements in algebra performance, decreased intrinsic CL, and no negative changes in terms of extrinsic and germane load compared to the control group. Overall, the WEI4S approach offers a promising avenue for effective mathematics instruction by providing a structured framework that encourages active engagement and problem-solving skills.

The variational principle for probabilistic measure and Hashin–Shtrikman bounds

The paper is a review of Hashin–Shtrikman type bounds for effective moduli of conductivity and elasticity of polycrystals and composites written from the perspective of the variational principle for probabilistic measure. The results for such bounds are rederived in probabilistic terms. Remarkably, in probabilistic terms the Hashin–Shtrikman approach gets especially simple form. Besides, a clear distinction arises between the basic assumption, the choice of the trial field, and the simplifying assumptions, like geometrical isotropy, physical isotropy, texture isotropy, etc. We filled out several gaps. First, we derive an integral equation to be solved to get the bounds when the simplifying assumptions do not hold. Second, we extend the bounds for polycrystals with the cubic symmetry of crystallites to all thermodynamically possible crystallites; previously such bounds were found for crystallites with special elastic properties. One practical outcome considered is the derivation of approximate formulae for the temperature dependence of effective elastic moduli. Third, for crystallites with non-cubic symmetries, we formulated algebraic variational problems to be solved numerically to obtain the bounds, and solved these problems for several materials.

Operator norm localization property for equi-approximable families of projections

The rigidity problem for uniform Roe algebras was recently positively solved. Before its solution was found, there were positive solutions under the assumption of certain technical geometric conditions. In this paper, we introduce weaker versions of the operator norm localization property (ONL) which turn out to characterize those technical geometric conditions. We use this to obtain new rigidity results for nonmetrizable coarse spaces. As an application, we provide a novel partial answer to a question of White and Willett about Cartan subalgebras of uniform Roe algebras. We also study embeddings between uniform Roe algebras.

The comparison of general tips for mathematical problem solving generated by generative AI with those generated by human teachers

ABSTRACT In designing an intelligent tutoring system, a core area of the application of AI in education, tips from the system or virtual tutors are crucial in helping students solve difficult questions in disciplines like mathematics. Traditionally, the manual design of general tips by teachers is time-consuming and error-prone. Generative AI, like ChatGPT, presents a new channel for designing general tips. This study utilized prompt engineering and Chain of Thought to summarize general tips for given mathematical problems (one geometry problem and one algebra problem) and their solutions. A Turing test was conducted to compare ChatGPT-generated general tips with human-designed ones. Results from 121 human evaluators, each assessing 6 ChatGPT-generated and 6 human-designed general tips for each of two mathematical problems, showed that the average score for ChatGPT-generated tips is less than that of human-designed tips at a statistically significant level (p < 0.05), and Zero-Shot CoT achieved the best score. However, no evaluator could distinguish the tip types exactly. The average precision, recall and F-value of all ChatGPT-generated tips are less than 40%. AI-generated general tips can serve as a valuable reference for teachers to enhance efficiency and students’ mathematical learning.

Praxeological Differences in Institutional Transition: The Case of School Algebra

The transition from lower secondary to upper secondary school is a challenging time for many students with algebra as a focal topic. In this paper, we present a new approach to this problem, based on the anthropological theory of the didactic, particularly on what we call praxeological differences between two connected institutions. The methodology involves the construction of a praxeological reference model for school algebra based on documents such as textbooks and evaluation instruments, like national exams and screening tests, from these two institutions. To illustrate this approach, the Danish transition problem in algebra between the lower and upper secondary school is examined as a case study. The results obtained by the students from these evaluation instruments are also a part of the data, to focus on knowledge actually obtained. The results from this case indicate that praxeological difference is chiefly concentrated on rules for rewriting an algebraic model.

Effects of Case-based Strategy on Academic Achievement in Mathematics Algebraic Word Problems among Junior Secondary School Students in Ibadan Metropolis, Oyo State, Nigeria

This study investigated the Effect of Case-based Strategy on Academic Achievement in Mathematics Word Problems among Junior Secondary School Students in Ibadan Metropolis, Oyo State, Nigeria. The research focused on evaluating learning outcomes, knowledge retention, and overall students’ performance when exposed to this teaching approach based on constructivism and behaviorism theories of learning. The study employed a quasi-experimental research design. The population of the study consisted of all 20244 Junior Secondary School students in Oyo State, while two intact classes of 68 JSS2 students participated in the study. Two lesson plans and a validated Mathematics Achievement Test (MAT) were used as instruments for data collection. The MAT was tested for reliability having a value of 0.85 and the data generated was statistically analyzed using frequency counts, percentages, standard deviation and Analysis of Co-variance (ANCOVA) at a significance level of 0.05. The findings revealed that the effect of case-based strategy on students’ academic achievement was significant (F(1,65) = 72.771, p&lt; 0.05) and was found to be more effective in enhancing students’ academic achievement in algebraic word problems. It was recommended that educators should incorporate case-based strategies in teaching Mathematics Algebraic word problem at the Junior Secondary School for effective outcome.

Quantum Walk Computing: Theory, Implementation, and Application

The classical random walk formalism plays an important role in a wide range of applications. Its quantum counterpart, the quantum walk, is proposed as an important theoretical model for quantum computing. By exploiting quantum effects such as superposition, interference, and entanglement, quantum walks and their variations have been extensively studied for achieving computing power beyond that of classical computing and have been broadly used in designing quantum algorithms for algebraic and optimization problems, graph and network analysis, and quantum Hamiltonian and biochemical process simulations. Moreover, quantum walk models have been proven capable of universal quantum computation. Unlike conventional quantum circuit models, quantum walks provide a feasible path for implementing application-specific quantum computing, particularly in the noisy intermediate-scale quantum era. Recently, remarkable progress has been achieved in implementing a wide variety of quantum walks and quantum walk applications, which demonstrates the great potential of quantum walks. In this review, we provide a thorough summary of quantum walks and quantum walk computing, including theories and characteristics, physical implementations, and applications. We also discuss the challenges facing quantum walk computing, which aims to realize a practical quantum computer in the near future.

INTEGRAL TRANSFORM SOLUTION OF HEAT CONDUCTION IN ANISOTROPIC HETEROGENEOUS MEDIA

The generalized integral transform technique (GITT) is advanced to deal with conduction heat transfer in anisotropic heterogeneous media. A formal solution for exact integral transformation of conduction in anisotropic media is extended to account for heterogeneities expressed as space variable equation coefficients and source terms. The proposed eigenfunction expansion is based on biorthogonal eigenvalue problems, which results in an exact integral transformation for a class of linear problems and in a coupled transformed ordinary differential system for nonlinear situations. An algorithm is proposed for the associated eigenvalue problems, also handled through the GITT, by considering simpler biorthogonal eigenvalue problems of known analytical solution, leading to transformed algebraic eigenvalue problems. A single domain reformulation strategy is adopted to merge the information from multiple regions and materials, either isotropic or anisotropic, into one single diffusion equation. A two-dimensional transient test case is considered that presents an abrupt transition between isotropic and anisotropic materials yielding a marked change in thermal behavior in a defined region of interest formed by the anisotropic inclusion. Convergence behavior of the integral transform solution is illustrated, and the fully converged results are employed as a benchmark to inspect the accuracy of a commercial finite element code for automatically defined mesh refinement levels.

Floquet-based analysis on three-dimensional non-axisymmetric instabilities in oscillatory-driven Taylor–Couette flows and their low-frequency asymptotic behavior using Wentzel–Kramers–Brillouin method

This paper revisits the linear stability analysis of oscillatory-driven flows between two oscillating cylinders against non-axisymmetric disturbances. This study is motivated by the lack of a sufficiently reliable theoretical analysis giving insight into the experimentally observed spiral-like non-axisymmetric patterns when the cylinders are counter-oscillating. A new generalized time-dependent algebraic eigenvalue problem is constructed from the linearized set of the three-dimensional Navier–Stokes equations around the purely azimuthal basic state. Numerical evaluation of the critical eigenvalues combining both Floquet theory and spectral method reveals the existence of frequency ranges where this basic state becomes unstable against three-dimensional non-axisymmetric disturbances before it does so for two-dimensional axisymmetric ones. Indeed, as the oscillation frequency of the cylinders increases, the azimuthal wave number of the critical eigensolution is found to change from 0 to 2 to 1 and then back to 0. The primary bifurcation exchange between two instability modes with different azimuthal wave numbers occurs via different types of codimension-2 bifurcation points giving rise to discontinuities in the critical axial wave number where reversing and non-reversing non-axisymmetric Taylor vortex flows are identified. In addition, by extending our numerical calculations to the co-oscillating case, we show that the axisymmetric disturbances are the most unstable confirming thus existing experimental findings. Furthermore, a Wentzel–Kramers–Brillouin (WKB) analysis is performed to shed light on the asymptotic behavior of these time-dependent flows in the low-frequency limit when the cylinders are slowly oscillating.

Evaluating ChatGPT-Generated Linear Algebra Formative Assessments

This research explored Large Language Models potential uses on formative assessment for mathematical problem-solving process. The study provides a conceptual analysis of feedback and how the use of these models is related in the context of formative assessment for Linear Algebra problems. Particularly, the performance of a popular model known as ChatGPT in mathematical problems fails on reasoning, proofs, model construction, among others. Formative assessment is a process used by teachers and students during instruction that provides feedback to adjust ongoing teaching and learning to improve student’s achievement of intended instructional outcomes. The study analyzed and evaluated feedback provided to engineering students in their solutions, from both, instructors and ChatGPT, against fine-grained criteria of a formative feedback model that includes affective aspects. Considering preliminary outputs, and to improve performance of feedback from both agents’ instructors and ChatGPT, we developed a framework for formative assessment in mathematical problemsolving using a Large Language Model (LLM). We designed a framework to generate prompts, supported by common Linear Algebra mistakes within the context of concept development and problem-solving strategies. In this framework, the instructor acts as an agent to verify tasks in a math problem assigned to students, establishing a virtuous cycle of learning of queries supported by ChatGPT. Results revealed potentialities and challenges on how to improve feedback on graduate-level math problems, by which both educators and students adapt teaching and learning strategies.

릴의 방법과 지오지브라, 파이썬을 이용한 다항방정식 풀이

A major problem in algebra was to find a formula for the roots of polynomial equations of degree 5 or more, and it was proved by Abel and Ruffini that there was no formula for the roots. Lill developed the ‘graphical method’ by exploring the roots of polynomial equations using a geometric method rather than an algebraic method. To solve arbitrary polynomial equations, certain conditions of the ‘graphical method’ must be satisfied, but Lill did not provide a specific method for this. Beloch expressed quadratic and cubic equations using Lill's method and then discovered a method of satisfying specific conditions using origami. However, when the degree is 4 or more, several conditions must be satisfied simultaneously to solve it with origami, so there has been no research on solving the roots of arbitrary polynomial equations. Therefore, this study selected the following research questions. First, we develop a program that automatically solves quadratic and cubic equations using Lill's method, geogebra, and python. Second, we develop a program that automatically solves arbitrary polynomial equations of degree 4 or higher using Lill's method and python. As a result of the study, we were able to manually find the paths of quadratic and cubic equations using Geogebra and automatically find them using Python and develop a program to solve them using Lill's method. To find the path of an arbitrary nth degree polynomial equation of degree 4 or higher, we used object-oriented programming from the Pygame library to solve it. The significance of this research is that it visualizes the process of approximating the roots of an arbitrary polynomial equation, approaches it with a geometric method rather than an algebraic method, and develops a program to solve arbitrary polynomial equations in a generalized way.

10.47435/jtmt.v4i02.1975 Analysis Of Student Errors On Algebraic Form Material Based On Kastolan

Algebraic calculation operations are basic material that is the basis for studying other mathematical material, but there are still many students who still make mistakes in solving algebraic calculation operation questions. Therefore, it is necessary to analyze the errors made by students in working on algebraic calculation operations questions. This research aims to describe the errors made by students when completing algebraic operations according to Kastolan theory in solving algebraic arithmetic operation problems. This type of research is qualitative descriptive research. The data collection technique used was a test. The data analysis techniques used were: examining the results of students' answers by identifying and analyzing the types of errors made by students in answering questions related to the Kastolan error type; 1) Calculate the percentage of error for each type of error, 2) The results of the percentage level are classified based on the percentage of error, 3) The data that has been obtained is then calculated and will be carried out using descriptive analysis, 4) Make conclusions from the results of the analysis of students' errors in working on form operation questions algebra based on the Kastolan error type. Based on the research data and discussions obtained, 69% of conceptual errors were in the high category, 58% of procedural errors were in the sufficient category, and 59% of technical errors were in the sufficient category. It can be concluded that the mistakes that many students make when working on test questions are conceptual errors, namely errors in understanding the questions. Factors that cause students to make mistakes are not understanding the concept and not being able to apply the concept

Profile of Student Metacognition in Solving Elementary Linear Algebra Problems Viewed from Tempo Conceptual Cognitive Style

Metacognition is one of the factors that influences students' success in solving mathematical problems. On the other hand, the reality in the field is that students experience difficulties in solving mathematical problems related to Elementary Linear Algebra because they contain a lot of repetitive calculations, so they require high levels of perseverance and accuracy. This research aims to describe students' metacognitive profiles in solving Elementary Linear Algebra problems in terms of cognitive style. The cognitive style chosen is the conceptual cognitive style of pacing through Matching Familiar Figures Tests with research subjects being mathematics students at Nahdlatul Ulama University, Purwokerto. The resulting metacognition profile was obtained from Elementary Linear Algebra problem-solving test data and interviews, which were analyzed and concluded through a series of method triangulation processes. The research results show that students are reflective and impulsive in carrying out all activities that are indicators of metacognitive awareness and regulation, as well as several indicators from metacognitive evaluation. Fast, accurate students carry out all activities that become indicators of metacognitive awareness, regulation, and evaluation. Meanwhile, slowly, inaccurate students carry out all activities that are indicators of metacognitive awareness and several indicators of metacognitive regulation but still need to be able to carry out activities of metacognitive evaluation.