Algebraic Problem Research Articles

Classifying Abelian Groups Through Acyclic Matchings

AbstractThe inquiry into identifying sets of monomials that can be eliminated from a generic homogeneous polynomial via a linear change of coordinates was initiated by E. K. Wakeford. This linear algebra problem prompted C. K. Fan and J. Losonczy to introduce the notion of acyclic matchings in the additive group $$\mathbb {Z}^n$$ Z n , subsequently extended to abelian groups by the latter author. Alon, Fan, Kleitman, and Losonczy established the acyclic matching property for $$\mathbb {Z}^n$$ Z n . This note aims to classify all abelian groups with respect to the acyclic matching property.

Total Positivity and Accurate Computations Related to q-Abel Polynomials

The attainment of accurate numerical solutions of ill-conditioned linear algebraic problems involving totally positive matrices has been gathering considerable attention among researchers over the last years. In parallel, the interest of q-calculus has been steadily growing in the literature. In this work the q-analogue of the Abel polynomial basis is studied. The total positivity of the matrix of change of basis between monomial and q-Abel bases is characterized, providing its bidiagonal factorization. Moreover, well-known high relative accuracy results of Vandermonde matrices corresponding to increasing positive nodes are extended to the decreasing negative case. This further allows to solve with high relative accuracy several algebraic problems concerning collocation, Wronskian and Gramian matrices of q-Abel polynomials. Finally, a series of numerical tests support the presented theoretical results and illustrate the goodness of the method where standard approaches fail to deliver accurate solutions.

Critical buckling load analysis of Euler-Bernoulli beam on two-parameter foundations using Galerkin method

The critical buckling load determination of Euler-Bernoulli beams on two-parameter elastic foundations (EBBo2PFs) is important to avert buckling failures. The governing equation for buckling of thin beam on two-parameter elastic foundation is a homogeneous ordinary differential equation (HODE) of fourth order and constant parameters when the beam is prismatic and homogeneous. The HODE has been solved in this work by Galerkin method for simply supported, clamped and clamped-simply supported ends. One-parameter algebraic shape function formulation was used to reduce the problem to an algebraic eigenvalue problem, which is solved to find the critical buckling load for each studied case. The critical buckling load for EBBo2PF for simply supported boundary conditions was found to be closely identical to the exact solutions. The solutions for clamped-clamped edges and clamped-simple supports were found to be accurate. The merit of the Galerkin method is the simplicity and the accuracy even when one-parameter shape function has been used.

Investigating pre-tertiary students’ mistakes in solving algebraic word problems: Insights from Asutifi North District, Ghana

Algebra word problems inculcate in learners the skills and ability to think critically and look for answers to problems in society and the world. Using an explanatory sequential mixed design, the study investigated the errors made by pre-tertiary students in algebraic word problems in a selected school in the Asutifi North District of the Ahafo Region of Ghana. Whereas the Newman Error Analysis Model was used to find the types of errors students make when translating and solving algebraic word problems, protocols from topics relating to the study were carefully selected and used as interview guides alongside inspiration from the Newman Error Analysis Model. Quota and simple random sampling techniques were used to select ninety-eight (98) respondents. The Algebraic Word Problem Achievement Test (AWPAT) was given to the ninety-eight students who had studied several topics on word problems, followed by a structured interview to elicit more information from the respondents. The scores gained from the marked test items illustrated that students made more transformation errors, which happened indirectly from a lack of understanding of the concept. A careful analysis of the written responses of the five (5) students interviewed also revealed that, lack of comprehension directly translated into students’ inability to transform algebraic word problems into mathematics equations. In the same vein, the study showed that the main cause of students’ failure to translate and solve algebraic word problems was students’ lack of understanding of the concept of “word problem” among others. The study recommends that students be motivated by encouragement and praise to arouse their interest in translating and solving algebraic word problems.

The finite presentation problem for Noetherian algebras

We prove that there exists an affine Noetherian algebra over a field of characteristic zero which is not finitely presented. Specifically, we prove that the Medvedev–Passman–Resco–Small algebra, recently shown to form a counterexample to the stability problem for Noetherian algebras, is not finitely presented. This answers a question due to Bergman and Irving in characteristic zero, a case explicitly left open by Resco and Small in 1993, who gave a counterexample to this question in positive characteristic.

Extending structures for dendriform algebras

In this paper, we devote to extending structures for dendriform algebras. First, we define extending datums and unified products of dendriform algebras, and theoretically solve the extending structure problem. As an application, we consider flag datums as a special case of extending structures, and give an example of the extending structure problem. Second, we introduce matched pairs and bicrossed products of dendriform algebras and theoretically solve the factorization problem for dendriform algebras. Moreover, we also introduce cocycle semidirect products and nonabelian semidirect products as special cases of unified products. Finally, we define the deformation map on a dendriform extending structure (more general case), not necessary a matched pair, which is more practical in the classifying complements problem.

Robust Fuzzy Adaptive Fault‐Tolerant Control for a Class of Second‐Order Nonlinear Systems

ABSTRACTIn this brief, an optimal active fuzzy fault‐tolerant control (OAFFTC) scheme is proposed for a class of unknown perturbed multi‐input multi‐output (MIMO) nonlinear systems with nonaffine nonlinear actuator faults and time‐varying sensor faults. The control strategy can handle automatically (online updating) with three additives (bias, drift, and loss of accuracy) along with one multiplicative (loss of effectiveness) sensor faults and nonlinear state‐dependent actuator faults. In order to deal with uncertain system dynamics, sensor and actuator faults, and external disturbances, fuzzy systems (FSs) and backstepping techniques were combined to provide the adaptive control term as well as a robust control term. Butterworth filter is used to get rid the algebraic loop problem. The suggested robust term, can deal with approximation errors of the fuzzy systems (FSs). To automatically optimize the adaptive parameters and the starting conditions, particle swarm optimization (PSO) approach is introduced. The Lyapunov approach is employed to demonstrate the closed‐loop system's stability. To assess the efficacy and correctness of the suggested scheme, quadrotor dynamic model is performed on the simulation part.

The Borsuk–Ulam property for homotopy classes on bundles, parametrized braids groups and applications for surfaces bundles

Let M and N be fiber bundles over the same base B, where M is endowed with a free involution τ over B. A homotopy class δ∈[M,N]B (over B) is said to have the Borsuk–Ulam property with respect to τ if for every fiber-preserving map f:M→N over B which represents δ there exists a point x∈M such that f(τ(x))=f(x). In the cases that B is a K(π,1)-space and the fibers of the projections M→B and N→B are K(π,1) closed surfaces SM and SN, respectively, we show that the problem of decide if a homotopy class of a fiber-preserving map f:M→N over B has the Borsuk-Ulam property is equivalent of an algebraic problem involving the fundamental groups of M, the orbit space of M by τ and a type of generalized braid groups of N that we call parametrized braid groups. As an application, we determine the homotopy classes of fiber-preserving self maps over S1 that satisfy the Borsuk-Ulam property, with respect to all involutions τ over S1, for the torus bundles over S1 with M=N=MA and A=[1n01].

On the number of solutions to a random instance of the permuted kernel problem

The Permuted Kernel Problem (PKP) is a problem in linear algebra that was first introduced by Shamir in 1989. Roughly speaking, given an ℓ×m matrix A and an m×1 vector b over a finite field of q elements Fq, the PKP asks to find an m×m permutation matrix π such that πb belongs to the kernel of A. In recent years, several post-quantum digital signature schemes whose security can be provably reduced to the hardness of solving random instances of the PKP have been proposed. In this regard, it is important to know the expected number of solutions to a random instance of the PKP in terms of the parameters q,ℓ,m. Previous works have heuristically estimated the expected number of solutions to be m!/qℓ.We provide, and rigorously prove, exact formulas for the expected number of solutions to a random instance of the PKP and the related Inhomogeneous Permuted Kernel Problem (IPKP), considering two natural ways of generating random instances.

Analysis of students' computational thinking skills in solving algebraic problems in cultural contexts

This research aims to find out about students' computational thinking abilities in solving algebraic problems in cultural contests. This research is qualitative research carried out at Yogyakarta Middle School. The participants involved were 31 students in the even semester of grade 7. The data sources in this research were the results of students' computational thinking tests, interviews with students who were selected based on their communication skills and the results of test answers. Data instrument testing uses validity and reliability tests with data validity using triangulation techniques. Data analysis in this study used the pullulation technique between student test results and student interview results, then data reduction, data categorization, power display and conclusion drawing were carried out. The results of this research are that students in the high category can complete all indicators of computational thinking and problem-solving abilities. Students in the medium category can only complete two indicators of computational thinking and problem-solving abilities. Students in the low category have not been able to complete all indicators of computational thinking and problem-solving abilities. The study reveals a significant variance in the computational thinking abilities among junior high school students in Piyungan Development class VIII when solving algebra problems within a cultural context. High-achieving students demonstrated proficiency across all indicators of computational thinking, effectively navigating through various stages from abstraction to solution verification. In contrast, students categorized as moderate showed partial success, managing only a subset of the indicators, while struggling with the more complex stages of problem decomposition and implementation. Meanwhile, students classified in the low category were unable to achieve any indicators of computational thinking and problem-solving skills. This indicates a clear need for targeted educational interventions to enhance computational thinking among students, particularly those who are struggling, to ensure all students can develop the necessary skills to solve problems effectively.

Investigating Students' Problem Solving Achievements, the Modeling Steps reached and Modeler Types in the Process of Mathematical Modeling Activities

The aim of this study is to identify the effect of mathematical modeling activities applied as action plans on the development of students' achievement in solving algebraic verbal problems, as well as to investigate the mathematical modeling competencies they use during these mathematical modeling activities and the modeling steps they reach. Moreover, the types of mathematical modelers that the students demonstrated during this application process were examined in the study. The study was conducted as an action research with a total of 15 7th grade students in rural areas. It was aimed to reveal the mathematical modeling competencies that students achieved during mathematical modeling activities through observations, researcher and student diaries, as well as interviews conducted with students. Content analysis was implemented to analyze the data. During the study, the mathematical modeling competencies demonstrated by the students in each mathematical modeling activity and the mathematical modeling steps they reached were identified. As a result of the study, it was confirmed that students' mathematical modeling competencies were related to competencies such as general mathematical knowledge or verbal comprehension, and it was also revealed that students' mathematical knowledge competence could be examined in more detail. It was also observed that students could achieve competencies through group work and it was found that students' success in solving the verbal algebraic problems generally improved based on their mathematical modeling competencies.

Hilbert’s tenth problem for term algebras with a substitution operator

Hilbert’s tenth problem for term algebras with a substitution operator

Diagonal tensor algebra and naïve liftings

The notion of naïve lifting of differential graded (DG) modules along certain DG algebra extensions was introduced by the authors for the purpose of studying problems in homological commutative algebra that involve self-vanishing of Ext. Our goal in this paper is to deeply study naïve lifting property using the diagonal tensor algebra. Our main result provides several characterizations of naïve liftability of DG modules under certain Ext vanishing conditions. As an application, we affirmatively answer a question posed by the authors regarding a specific characterization of naïve liftability.

The contribution of working memory and spatial perception to the ability to solve geometric problems

Geometry is a branch of mathematics that deals with the properties of space, including distance, shape, size, and the relative position of figures. It is one of the oldest branches of mathematics and has applications in various fields such as science, art, architecture, and even in areas seemingly unrelated to mathematics. Studies show that working memory and spatial perception contribute to students' geometry performance. This paper presents multiple studies demonstrating the brain regions activated when solving geometric problems. Interestingly, the brain areas activated when solving algebraic problems are different from those activated when solving geometric problems. Finally, multiple studies are presented that indicate students with learning difficulties lag in geometry, as solving geometric problems requires good reading and arithmetic skills.

A local and hierarchical Koopman spectral analysis of fluid dynamics

AbstractA local and hierarchical Koopman spectral analysis is proposed to extend Koopman spectral analysis typically used in a linear system or an ergodic process to its application in general nonlinear dynamics. The continuous and analytic Koopman eigenfunctions and eigenvalues, derived from operator perturbation theory, are capable of dealing with a nonlinear transition process with mathematical rigorousness. A proliferation rule is identified to derive high‐order eigenvalues and eigenfunctions from lower‐order ones, thus various spectral patterns may be generated through recursive proliferations. The locally linear map around each state constructs base local Koopman eigenvalues and eigenfunctions from an algebraic eigenvalue problem, and high‐order ones are generated via the proliferation rule to express the systematic nonlinearity. The aforementioned hierarchy simplifies the Koopman spectral analysis and is verified by studying the development of Kármán vortex streets. The triangular chain and the lattice distribution of Koopman eigenvalues confirm the critical role of the proliferation rule and the hierarchy structure of Koopman eigenvalues. The local spectral analysis on the transition process shows that the periodic flow forms as the growth rates of the critical Koopman modes reduce to zero, and meanwhile, the Koopman modes at the same frequency superpose on each other to form the well‐known Fourier or Floquet modes, where the latter are the enhanced nonlinear motions due to the alignment of Koopman eigenvalues with the critical ones.

МАТЕМАТИКАНЫ ЖАНА АЗЫРКЫ ЗАМАНДЫ ӨНҮКТҮРҮҮ БОЮНЧА КЫРГЫЗ ЫКМАСЫ

The educational process consists of two main types of activities: the teaching activity of the teacher in educating students in knowledge, skills, and life experience, and the learning activity of the student in mastering the content taught by the teacher. Defining the learning process, teaching guides and determines learning. Therefore, the learning process is often explained as the process of educating the student. As a result, the learning process is perceived as a mechanism for the "transmission" of teaching content from teacher to student, with the teacher directly shaping the student. It formalizes the algorithm for solving problems − the problem-solving algorithm using the method of Diophantus al-Khwarizmi. This article presented a solution to text problems in algebra for grade 7 using the Kyrgyz method and a system of linear equations. It should be noted that this type of thinking was successfully used by many ancient Egyptian, Babylonian, Chinese and Indian mathematicians. The generalization of the described method can be successfully used to solve various problems.

On improving conversational interfaces in educational systems

Conversational Intelligent Tutoring Systems (CITS) have drawn increasing interest in education because of their capacity to tailor learning experiences, improve user engagement, and contribute to the effective transfer of knowledge. Conversational agents employ advanced natural language techniques to engage in a convincing human-like tutorial conversation. In solving math word problems, a significant challenge arises in enabling the system to understand user utterances and accurately map extracted entities to the essential problem quantities required for problem-solving, despite the inherent ambiguity of human natural language. In this study, we propose two possible approaches to enhance the performance of a particular CITS designed to teach learners to solve arithmetic–algebraic word problems. Firstly, we propose an ensemble approach to intent classification and entity extraction, which combines the predictions made by two distinct individual models that use constraints defined by human experts. This approach leverages the intertwined nature of the intents and entities to yield a comprehensive understanding of the user’s utterance, ultimately aiming to enhance semantic accuracy. Secondly, we introduce an adapted Term Frequency-Inverse Document Frequency technique to associate entities with problem quantity descriptions. The evaluation was conducted on the AWPS and MATH-HINTS datasets, containing conversational data and a collection of arithmetical and algebraic math problems, respectively. The results demonstrate that the proposed ensemble approach outperforms individual models, and the proposed method for entity–quantity matching surpasses the performance of typical text semantic embedding models.

PPM: A boolean optimizer for data association in multi-view pedestrian detection

To accurately localize occluded people in a crowd is a challenging problem in video surveillance. Existing end-to-end deep multi-camera detectors rely heavily on pre-training with the same multiview datasets used for testing, which compromises their real-world applications. An alternative approach presented here is to project the torso lines of the instance segmentation masks from multiple views to the ground plane and propose pedestrian candidates at the intersection points. The candidate selection process is, for the first time, formulated as a logic minimization problem in Boolean algebra. A probabilistic Petrick’s method (PPM) is proposed to seek the minimum number of candidates to account for all the foreground masks while maximizing the joint occupancy likelihoods in multiple views. Experiments on benchmark video datasets have demonstrated the much improved performance of this approach in comparison with the benchmark deep or non-deep algorithms for multiview pedestrian detection.

МАТЕМАТИКА КУРСУН ОКУТУУДА КЫЙМЫЛ МАСЕЛЕЛЕРИН ЧЫГАРУУНУН МЕТОДИКАЛЫК КӨРСӨТМӨЛӨРҮ

At present, in accordance with the program of basic general education, a special place in mathematics teaching is occupied by the development of students' independence in solving motion problems. There is no doubt that the mathematical problem of motion enables the student to formulate mathematical concepts correctly, to clarify various aspects of his environment, and to apply theoretical concepts. Solving full-scale problems defined by the program contributes to knowledge building for students. In setting motion problems, students are exposed to facts of cognitive, educational significance. Motion problems-one of the most common types of algebra problems. Simple motion problems are taught in elementary school. In grades 6-7, the solution of motion problems is reduced to a linear equation or a system of linear equations. Here we will consider motion problems related to the rational fraction equation.

Quantum Computing Techniques for Numerical Linear Algebra in Computational Mathematics

Quantum computing is a new and exciting area of computational mathematics that has the ability to solve very hard problems that traditional computing methods have not been able to solve for a long time. This abstract goes into detail about how quantum computing can be used in numerical linear algebra, which is an important part of computational mathematics that is used in many fields, such as science, engineering, and data analysis. The idea behind quantum computing is new. It uses quantum bits (qubits) and quantum gates to do calculations in a very different way, based on the rules of quantum physics. Because of this change in thinking, quantum computers might be able to solve some numerical linear algebra problems a lot faster than traditional computers. The outline then talks about some of the most important quantum computing methods and algorithms that are made for numerical linear algebra problems. There are quantum versions of standard algorithms like matrix multiplication, Gaussian elimination, and singular value decomposition. There are also new algorithms that are intended to use quantum parallelism and interference to speed up computations. For situations where full-scale quantum gear is not yet available, quantum-inspired methods are also talked about. These blend conventional and quantum techniques. The abstract talks about the difficulties and chances of using quantum algorithms for numerical linear algebra on quantum hardware systems that are available now and in the near future. It talks about things like error correction, qubit coherence times, and making efficient quantum circuit implementations. Lastly, the abstract talks about how quantum computing might change computational mathematics. It imagines that advances in quantum computing could help solve numerical linear algebra problems that were previously impossible to solve. This could have huge effects on many fields, from machine learning and optimization to quantum chemistry and cryptography. In the end, it stresses how important it is for mathematicians, computer scientists, physicists, and engineers to work together across disciplines in order to fully utilize the transformative power of quantum computing in numerical linear algebra and other areas.