Algebraic Problem Research Articles

Rasch model assessment of algebraic word problem among year 8 Malaysian students

<p>Word problems continue to be a challenge for students today. All students must meet the prerequisites for problem solving and reasoning skills, which are important components of the critical thinking component of 21st century skills. This study is being conducted to assess students’ strategies for solving word problems with numbers, consecutives, and ages. The Rasch model is used to analyze the item difficulty level of word problems and students’ strategies for solving ten-word problems at various levels of item difficulty in a similar trait. Then, Pearson correlation analysis is used to investigate the item difficulty level in relation to linguistic, algebraic, and arithmetic factors of word problems before evaluating students’ performance on solving these word problems using various strategies. Rasch model found these algebraic word-problem questions are slightly harder for year 8 Malaysian students in relative to an international standard. Meanwhile, the item difficulty of word problems is driven by linguistic and algebra factors where students can score accurately if the word problems contained explicit information. However, the students encountered difficulties while losing their solution strategy when the questions contained implicit data that demanded critical thinking ability.</p>

Event-Triggered-Based Adaptive NN Cooperative Control of Six-Rotor UAVs With Finite-Time Prescribed Performance

This paper studies the disturbance-observer-based adaptive neural network cooperative event-triggered control problem for six-rotor unmanned aerial vehicles with finite-time prescribed performance. Six-rotor unmanned aerial vehicle systems are divided into the position subsystem and the attitude subsystem. The switching threshold event-triggered mechanism that considers the influence of event triggering on the consensus control performance is designed of each six-rotor unmanned aerial vehicle to save the transmission resources, which excludes the Zeno behavior. Neural networks are introduced for estimating the uncertainties and solving the algebraic loop problem. In addition, for the position subsystem, by combining the prescribed performance with the velocity function, a finite-time control scheme is proposed, which can guarantee that the consensus errors converge to a prespecified neighborhood of the origin, and all closed-loop signals are bounded. Based on the designed control input signal of position subsystem, an adaptive neural network event-triggered control mechanism is designed to stabilize the attitude subsystem. Finally, some verification results are given to test the rationality of the proposed control strategy. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The purpose of this paper is to design an event-triggered cooperative control scheme to save the transmission resources for six-rotor unmanned aerial vehicle systems with performance limitations and lumped disturbances. In practical applications, if the impact of the event-triggered mechanism on the system performance is not taken into account, the system performance may be significantly degraded while the information transmission resource is saved. Thus, this paper designs an event-triggered mechanism that considers the system performance, which can effectively weigh the relationship between system performance and transmission resource consumption and better meet the practical requirements. Moreover, a finite-time specified performance control strategy is introduced to avoid the problem of difficult determination of residual set and improve the transient-state and steady-state performances of unmanned aerial vehicle systems, which is more in line with the actual demand.

Bidiagonal Factorizations of Filbert and Lilbert Matrices

Extensions of Filbert and Lilbert matrices are addressed in this work. They are reciprocal Hankel matrices based on Fibonacci and Lucas numbers, respectively, and both are related to Hilbert matrices. The Neville elimination is applied to provide explicit expressions for their bidiagonal factorization. As a byproduct, formulae for the determinants of these matrices are obtained. Finally, numerical experiments show that several algebraic problems involving these matrices can be solved with outstanding accuracy, in contrast with traditional approaches.

Analysis of Mathematical Connection Ability in Minimum Competency Assessment Algebra Test Viewed From Students’ Intrapersonal and Interpersonal Intelligence

Mathematical connection is a crucial skill for students, as it reduces the need to memorize numerous concepts and procedures independently. This study explores the variance in mathematical connection abilities among students with different intelligence types, specifically intrapersonal and interpersonal, in the context of solving algebraic problems on the AKM (Minimum Competency Assessment) test. Utilizing a qualitative descriptive methodology, this research focused on four students from class VIII-B at an Islamic Junior High School in Blitar. Data were gathered through questionnaires assessing intrapersonal and interpersonal intelligence, followed by the AKM Algebra test and interviews with selected students—two from each intelligence category. The findings reveal that students with interpersonal intelligence demonstrated superior mathematical connection abilities compared to their intrapersonal counterparts. Individuals with interpersonal intelligence met all indicators of mathematical connections, whereas intrapersonal intelligence subjects varied, with some meeting only one or two. This discrepancy underscores that students of the same intelligence type can exhibit differing levels of mathematical connection abilities, influenced by various factors such as mathematical skills, development of geometric thinking, and reasoning abilities.

MODUL KEMAHIRAN BERFIKIR ALGEBRA UNTUK PELAJAR TINGKATAN SATU

Algebraic Thinking Skills (KBA) is one aspect of skills that students need to master in order to solve non-routine problems. These skills are also needed as a basis for preparing students to enter university studies and job fields that require logical and analytical thinking. However, the performance of Malaysian students in solving algebraic problems has not yet reached a satisfactory level according to the TIMSS 2019 and PISA 2018 reports. Thus, the Algebraic Thinking Skills Module (MBA) was developed to foster KBA through three constructs namely i) arithmetic generalization, ii) functions, and iii) modeling. MBA is developed based on heuristic methods through Polya absorption Model and digital bar. A bar model is a rectangular illustration constructed to represent known and unknown quantities and relationships between quantities. The digital bar model refers to the free application of the bar model at the link https://mathsbot.com/manipulatives/bar. The KBA test was constructed and administered as a pre- and post-test to 120 Form One students from a rural school in Sabah. The results of the paired sample t-test showed that there was a significant difference in the mean score between the pre-test and the post-test (t(119) = -17.553, p &lt; .05) after the intervention using MBA. This shows that MBA can improve KBA through heuristic methods. This innovation is expected to be developed in rural schools to form students who think algebraically and are literate in digital technology.

Command filter-based event-triggered fixed-time control for nonlinear systems with prescribed performance and actuator faults

This paper investigates command filter-based adaptive neural network (NN) minimal learning practical fixed-time control for stochastic nonlinear systems with prescribed performance and actuator faults. The considered system is in a high-order nonstrict-feedback stochastic structure with unknown dynamics and external disturbances. By combining NN with minimal learning parameter method, the need for prior knowledge of nonlinear functions is eliminated, and the number of weights’ updating laws for the NN is reduced to one, regardless of the system's order and number of neural nodes. This reduction significantly decreases the computational burden. Additionally, a unique property of the Gaussian basis function of NNs is applied to solve the algebraic loop problem of the nonstrict-feedback structure. A novel event-triggered control mechanism is proposed to save communication resources. In order to surmount the "explosion of complexity" and "singularity" problems, a novel fixed-time command filter is suggested, and then, a modified compensation mechanism is proposed to mitigate the errors that emerge from command filters. Furthermore, to improve the transient and steady-state performance of the tracking error, a prescribed performance function is taken into account. Via the Lyapunov stability theory, it will be shown that the developed adaptive backstepping control scheme, guarantees that the closed-loop system signals are bounded in probability in a fixed time, that the convergence time is independent of the initial value, and the tracking error remains within the decaying prescribed performance bounds all the time. Finally, the effectiveness and practicability of the theoretical results are verified by a practical simulation example.

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

The analysis of stability problems of beams on two-parameter foundations (Bo2PFs) is an important part of their design for compressive loads. This work presents novel first principles derivation of the governing differential equations of elastic stability (GDES) of thin beams resting on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models. The requirements of translational and rotational equilibrium of all the applied, reactive and internal forces on an infinitesimal segment of the Bo2PF and the laws of infinitesimal calculus were used to formulate the GDES as a fourth order ordinary differential equation (ODE) in terms of the transverse displacement function ux. The GDES is non-homogeneous in the presence of applied transverse load qx but homogeneous when qx vanishes. This study presents the Fourier series method (FSM) for solving the governing differential equation of stability (GDES) for the case of Dirichlet boundary conditions. The FSM has the advantage of amenability to differentiation, and integration due to the orthogonality properties of the sinusoidal functions. Implementation of the FSM by assuming the unknown function in the GDES as a Fourier series of infinite terms and the exploitation of orthogonalization simplifies the problem to an algebraic eigenvalue problem which is the characteristic buckling equation. The exact eigenvalues are found by algebraic solution of the buckling equation. The exact eigenvalues were used to find the exact buckling loads and the exact buckling load coefficients. The critical buckling load was found to correspond to the first buckling mode (n= 1), and is identical with previous solutions in the literature. Numerical calculations for the critical buckling load parameters Kcr were presented for the Bo2PF problem for values of the dimensionless foundation parameters k-1= 0, k-2= 0; k-1= 100, k-2= 0; k-1= 0, k-2= 1; k-1= 100, k-2= 100; k-1= 0, k-2= 2.5; k-1= 100, k-2= 2.5. The present solutions were compared with previous solutions for Kcr in the literature. The comparison shows that the present FSM results are identical with previous results obtained using various other methods such as Recursive Differentiation Method, Finite Element Method, Generalized Integral Transform Method (GITM) and Stodola-Vianello Iteration Method. The study has illustrated the effectiveness of the FSM for solving Bo2PFs.

In-Plane Vibrations of Deep Sandwich Arches with Different End Conditions Based on a Logarithmic Shear Deformation Theory

This paper deals with the vibrational frequencies of deep sandwich arches to enhance their application domain and possibly use them for energy harvesting. The circular arch with porous nanocomposite core and titanium alloy face sheets having different end conditions is numerically analyzed. The middle core of the sandwich arch is made of a six-layered porous aluminum reinforced with graphene nanoplatelets. The kinematic equations are formulated in this study based on the higher-order shear deformation theory using a logarithmic function of radius. The effective properties of the nanocomposite media are modeled by employing the Halpin–Tsai modified rule. The equations of motion are determined by applying the principle of virtual displacement. The partial differential equations are reduced using the generalized differential quadrature technique to solving an algebraic eigenvalue problem. Novel numerical results are given to show the effects of geometrical parameters, material properties, and boundary conditions on the vibrations of deep sandwich arch.

Quantum Communication Complexity of Linear Regression

Quantum computers may achieve speedups over their classical counterparts for solving linear algebra problems. However, in some cases—such as for low-rank matrices—dequantized algorithms demonstrate that there cannot be an exponential quantum speedup. In this work, we show that quantum computers have provable polynomial and exponential speedups in terms of communication complexity for some fundamental linear algebra problems if there is no restriction on the rank. We mainly focus on solving linear regression and Hamiltonian simulation. In the quantum case, the task is to prepare the quantum state of the result. To allow for a fair comparison, in the classical case, the task is to sample from the result. We investigate these two problems in two-party and multiparty models, propose near-optimal quantum protocols, and prove quantum/classical lower bounds. In this process, we propose an efficient quantum protocol for quantum singular value transformation, which is a powerful technique for designing quantum algorithms. We feel this will be helpful in developing efficient quantum protocols for many other problems.

On the Geometry and Topology of Discrete Groups: An Overview

In this paper, we provide a brief introduction to the main notions of geometric group theory and of asymptotic topology of finitely generated groups. We will start by presenting the basis of discrete groups and of the topology at infinity, then we will state some of the main theorems in these fields. Our aim is to give a sample of how the presence of a group action may affect the geometry of the underlying space and how in many cases topological methods may help the determine solutions of algebraic problems which may appear unrelated.

A note on the normal complement problem in semisimple group algebras

A note on the normal complement problem in semisimple group algebras

CONSTRUCT THE TRIPLE ZERO GRAPH OF RING Z_n USING PYTHON

Let be a commutative ring with nonzero identity and there exists such that , , , denotes the set of all triple zero elements of . The triple zero graph of , denoted by , is an undirected graph with vertex set where two distinct vertices and are adjacent if and only if , and there exists a nonzero element of such that , , and . Python is a programming language with simple and easy-to-learn code that can be used to solve problems in algebra and graphs. In this paper, we construct the triple zero graph of ring using Python. Based on the output of the program, several properties of are obtained, such as if and , then is a planar graph, if with is prime numbers, then is a complete graph , and if with is prime numbers and , then is a connected graph.

Analisis Kemampuan Literasi Matematika dalam Menyelesaikan Soal Aljabar Berbasis HOTS Ditinjau dari Kemampuan Awal Siswa

This research aims to analyze in depth the mathematical literacy abilities possessed by students at SMP Negeri 11 Sungai Ambawang which are reviewed based on the initial abilities possessed by students with three categories of initial abilities, namely high, medium and low levels. This research is a descriptive qualitative research method in the form of a case study. In this research, the case that will be studied is how mathematical literacy skills are in solving HOTS-based algebra problems. Data collection tools in this research are tests and interview instruments. The subjects of this research were 24 students in class VIII A of SMPN 11 Sungai Ambawang. In this research, it was concluded that students with a high level of initial ability were able to fulfill three indicators of mathematical literacy, namely understanding and formulating, using and applying, and interpreting, while students with a medium level of initial ability were only able to fulfill two indicators, namely understanding and formulating, using and apply but cannot interpret. Meanwhile, students with low initial abilities are unable to meet the three indicators.

Student’s Reversible Thinking Processes: An Analysis Based on Adversity Quotient Type Climbers

Reversibility thinking carried out mentally in mathematical operations has an important role in the process of understanding concepts as it involves developing a thinking process from beginning to end and from end to beginning. This qualitative research aims to describe students' reversible thinking processes in solving algebra problems, specifically in climber-type adversity quotient. A total of 2 students who have the climber type were selected as participants from 36 potential subjects involved in the research. Data collection was carried out by providing Adversity Response Profile questionnaires, Test of Thinking Reversible, and semi-structured interviews. Our participants solved problems according to polya stages and through forward and reverse processes. The results of data analysis show that in the forward process, there are 2 aspects of reversible thinking, namely negation and reciprocity, while in the reverse process, they involve 2 aspects of reversible thinking, namely the capability to return to initial data after obtaining the result and negation. The results suggest that student's ability to perform mental reversals indicates a strong reversible thought process, leading to more precise cognitive thinking.

STODOLA-VIANELLO ITERATION METHOD FOR FREE TORSIONAL VIBRATION ANALYSIS OF MONOSYMMETRIC BOX-BEAM BRIDGES

Box-beam bridges are used in large spans and wider decks due to their high strength and greater torsional and flexural stiffness. They are usually prone to vibration due to moving vehicular traffic. Their eigenfrequency analysis is a crucial aspect of their design in order to ensure that the natural frequency is not close to the excitation frequency to avert resonance failures. The free torsional vibration equation is a fourth order partial differential equation (PDE) with variable parameters. For prismatic cross-sections, homogeneous and isotropic materials the governing PDE have constant parameters. This paper explores the Stodola-Vianello iteration method (SVIM) for solving the PDE for isotropic, homogeneous prismatic box-beams. Harmonic response is assumed, decoupling the PDE to two equations, one in terms of time and the second an ordinary differential equation (ODE) in terms of space coordinates. The Stodola-Vianello method is used by the method of four successive integrations to express the ODE as an algebraic iteration problem with four constants of integration. The four boundary conditions are used to solve for the four constants of integration, thus making the problem determinate. Application of the boundary conditions results in the full determination of the iteration equations. For simply supported boundaries studied in the work, a trigonometric buckling shape function that satisfies all the boundary conditions is employed in the SVIM formula to obtain the next bucking modal shape function. The requirement for convergence is then used to establish the characteristic buckling equation from which the eigenvalues are obtained. The solution to the characteristic buckling equation gave the exact mathematical expression for the natural torsional frequency at the nth vibration mode. The frequencies are determined for the first eight vibration modes and compared with previously obtained values. It was found that the natural frequencies are identical with previously found values of the exact natural frequencies. The natural frequency obtained is exact because it satisfies the PDE and the boundary conditions at all points in the domain.

Total positivity and high relative accuracy for several classes of Hankel matrices

SummaryGramian matrices with respect to inner products defined for Hilbert spaces supported on bounded and unbounded intervals are represented through a bidiagonal factorization. It is proved that the considered matrices are strictly totally positive Hankel matrices and their catalecticant determinants are also calculated. Using the proposed representation, the numerical resolution of linear algebra problems with these matrices can be achieved to high relative accuracy. Numerical experiments are provided, and they illustrate the excellent results obtained when applying the theoretical results.

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.