Linear Algebra Research Articles

A Connection Between Knot Theory and Linear Algebra

In knot theory literature, a lot of references treated the topic of computing the Alexander polynomial and proved that the Alexander polynomial is a knot invariant. In this paper, we concentrate on an unconventional connection between knot theory and linear algebra in computing the Alexander polynomial and introduce an approach depending linear algebra tools to show that the Alexander polynomial is a knot invariant. Also, we connect this approach to the way of Fox coloring to suggest another way to prove that the 3-coloring is a knot invariant and confirm what is already known about the 3-coloring of the trefoil and the figure eight knots.

Tensors in High-Dimensional Data Analysis: Methodological Opportunities and Theoretical Challenges

Large amounts of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity of such data provides vast new opportunities for modeling and analysis, but efficiently extracting information content from them, both statistically and computationally, presents unique and fundamental challenges. Addressing these challenges requires an interdisciplinary approach that brings together tools and insights from statistics, optimization, and numerical linear algebra, among other fields. Despite these hurdles, significant progress has been made in the past decade. This review seeks to examine some of the key advancements and identify common threads among them, under a number of different statistical settings.

On Saturation of Zero-Dimensional Ideals

In this paper, we explore the effective computation as well as arithmetic complexity of saturating a zero-dimensional polynomial ideal with respect to a given polynomial when a Gröbner basis of the ideal is provided. For this purpose, we first focus on studying in more detail a lesser-known algorithm due to Traverso for updating a zero-dimensional Gröbner basis by adding a new polynomial. Based on this algorithm and by applying linear algebra techniques, we propose two new algorithms for calculating the saturation of a zero-dimensional ideal with respect to a polynomial. These algorithms have been implemented in Maple and their efficiency compared to the classical method (using the Rabinowitsch trick) for computing ideal saturation is evaluated through a set of benchmark polynomial ideals.

Mental constructions for the learning of the concept of vector space

AbstractThis study contributes to the literature on linear algebra instruction by designing and researching a teaching sequence based on APOS Theory to introduce engineering students to vector spaces. The sequence offers students multiple opportunities to understand the concept. Another contribution is the evidence that introducing prerequisite concepts—such as equality, sets, and binary operations—before tackling vector space was crucial for grasping the role of proof in determining whether a set is a vector space. The findings confirm that, as other studies have shown, vector space is challenging for students. However, the results demonstrate that students were able to face this challenge. All students showed evidence of developing an understanding of the concept, with nine achieving a clear grasp of vector space and the role of proof by the end of the experience. Additionally, the progress observed—from having difficulties with symbols to successfully proving statements involving unconventional operations—underscores the effectiveness of the teaching approach.

Sketched and Truncated Polynomial Krylov Methods: Evaluation of Matrix Functions

ABSTRACTAmong randomized numerical linear algebra strategies, so‐called sketching procedures are emerging as effective reduction means to accelerate the computation of Krylov subspace methods for, for example, the solution of linear systems, eigenvalue computations, and the approximation of matrix functions. While there is plenty of experimental evidence showing that sketched Krylov solvers may dramatically improve performance over standard Krylov methods, especially when combined with a truncated orthogonalization step, many features of these schemes are still unexplored. We derive a new sketched Arnoldi‐type relation that allows us to obtain several different new theoretical results. These lead to an improvement of our understanding of sketched Krylov methods, in particular by explaining why the frequently occurring sketched Ritz values far outside the spectral region of do not negatively influence the convergence of sketched Krylov methods for . Our findings also help to identify, among several possible equivalent formulations, the most suitable sketched approximations according to their numerical stability properties. These results are also employed to analyze the error of sketched Krylov methods in the approximation of the action of matrix functions, significantly contributing to the theory available in the current literature.

Thermodynamic linear algebra

AbstractLinear algebra is central to many algorithms in engineering, science, and machine learning; hence, accelerating it would have tremendous economic impact. Quantum computing has been proposed for this purpose, although the resource requirements are far beyond current technological capabilities. We consider an alternative physics-based computing paradigm based on classical thermodynamics, to provide a near-term approach to accelerating linear algebra. At first sight, thermodynamics and linear algebra seem to be unrelated fields. Here, we connect solving linear algebra problems to sampling from the thermodynamic equilibrium distribution of a system of coupled harmonic oscillators. We present simple thermodynamic algorithms for solving linear systems of equations, computing matrix inverses, and computing matrix determinants. Under reasonable assumptions, we rigorously establish asymptotic speedups for our algorithms, relative to digital methods, that scale linearly in matrix dimension. Our algorithms exploit thermodynamic principles like ergodicity, entropy, and equilibration, highlighting the deep connection between these two seemingly distinct fields, and opening up algebraic applications for thermodynamic computers.

Circuit and graver walks and linear and integer programming

We show that a circuit walk from a given feasible point of a given linear program to an optimal point can be computed in polynomial time using only linear algebra operations and the solution of the single given linear program. We also show that a Graver walk from a given feasible point of a given integer program to an optimal point is polynomial time computable using an integer programming oracle, but without such an oracle, it is hard to compute such a walk even if an optimal solution to the given program is given as well. Combining our oracle algorithm with recent results on sparse integer programming, we also show that Graver walks from any point are polynomial time computable over matrices of bounded tree-depth and subdeterminants.

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.

Analyzing the Connection of Linear Algebra: Enhancing Visual Reasoning through Vector Spaces

Analyzing the Connection of Linear Algebra: Enhancing Visual Reasoning through Vector Spaces

Leibniz rule for the high q$$ q $$‐derivatives of a quotient of two functions

In this paper, we introduce some explicit and recursive formulas for the th ‐derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between commutative algebras and the properties of ‐differential operator. One of the formulas involves the determinant of the functions and their ‐derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, we used formulas to obtain some interesting ‐identities.

A new system of sylvester-like matrix equations with arbitrary number of equations and unknowns over the quaternion algebra

In this paper, we derive some practical necessary and sufficient conditions for the existence of a solution to a new system of coupled two-sided Sylvester-like matrix equations with arbitrary number of equations and unknowns over the quaternion algebra. As applications, we give some practical necessary and sufficient conditions for the existence of an η-Hermitian solution to a system of quaternion matrix equations in terms of ranks. We also use graphs to represent linear mappings associated with some Sylvester-type systems. As a special case of the main theorem, we prove a conjecture is correct, which was proposed in [Linear Algebra Appl. 2016;496:549–593]. The main findings of this paper widely extend almost all the known results in the literature.

The linear algebra of the generalized Pascal matrix with four variables

The linear algebra of the generalized Pascal matrix with four variables

The Impact of Using Quizizz as a Learning Media on Learning Interest of Computer Education Students

In this case, this research aims to determine the effect of using Quizizz as a learning media on the learning interest of Computer Education Students in the Linear Algebra course. The research method uses quantitative experiments. The research design used One Group Pretest-Posttest Design. The subjects of this research were 38 consisting of 4th Semester Students of the Department of Computer Education, Dehasen University, Bengkulu. The instrument used 20 items questionnaire, the data analysis techniques used normality test, homogeneity test and hypothesis test. The results of the pretest and posttest questionnaires were 40% and 85% indicating an increase in student learning interest. The results of research using paired sample testing show that the asym sig. (2-laid) of 0.000 < 0.05 where t-count is greater than t-table 24.220 > 2.037 so that H_O is rejected and H_a is accepted. It can be concluded that the use of Quizizz as a learning media (X) has a significant effect on the learning interest of computer education students at Dehasen University, Bengkulu.

The number of spanning trees in K n -chain and ring graphs

The problem of counting spanning trees of graphs or networks is a fundamental and crucial area of research in combinatorics, while has numerous important applications in statistical physics, network theory and theoretical computer science. Very recently, Kosar, Zaman, Ali and Ullah obtained a nice formula on the number of spanning trees of a K 5-chain network K5ℓ constructed by connecting ℓcopys of complete graphs K 5. They made extensive use of matrix theory and spectral graph theory, especially the normalized Laplacian of graphs. In this paper, by using a rather simple and more physical treatment (the mesh-star transformation in electrical network) without any linear algebra, we generalize their result to K n -chain graphs and K n -ring graphs. The results show that there is a simple relation between the number of spanning trees of the K n -chain graph Lnℓ and the K n -ring graph Cnℓ . We also calculate the corresponding tree entropy (or so called ‘the asymptotic growth constant’) and find that the tree entropy of the corresponding K n -chain graphs and K n -ring graphs are totally the same.

A Linear Method to Fit Equivalent Circuit Model Parameter Values to HPPC Relaxation Data From Lithium-Ion Cells

Abstract Battery management systems require mathematical models of the battery cells that they monitor and control. Commonly, equivalent circuit models are used. We would like to be able to determine the parameter values of their equations using simple tests and straightforward optimizations. Historically, it has appeared that nonlinear optimization is required to find the state-equation time constants. However, this article shows that the relaxation interval following a current or power pulse provides data that can be used to find these time constants using linear methods. After finding the time constants, the remaining parameter values can also be found via linear regression. Overall, only linear algebra is used to find all of the parameter values of the equivalent circuit model. This yields fast, robust, and simple implementations, and even enables application in an embedded system, such as a battery management system, desiring to retune its model parameter values as its cell ages.

An In-Depth Evaluation of Educational Burst Games in Relation to Learner Proficiency

Game-based learning assessments rely on educational data mining approaches such as stealth assessments and quasi mixed methods that help gather data on student learning proficiency. Rarely do we see approaches where student proficiency in learning is woven into the game’s design. Educational burst games (EBGs) represent a new approach to improving learning proficiency by designing fast-paced, short, repetitive, and skill-based games. They have the potential to be effective learning interventions both during instruction in the classroom and during after-school activities such as assignments and homework. Over five years, we have developed two EBGs aimed at improving linear algebra concepts among undergraduate students. In this study, we provide the results of an in-depth evaluation of the two EBGs developed with 45 participants that represent our target population. We discuss the role of EBGs and their design constructs, such as pace and repetition, the effect of the format (2D vs. 3D), the complexity of the levels, and the influence of prior knowledge on the learning outcomes.

On the uniqueness and computation of commuting extensions

A tuple (Z1,…,Zp) of matrices of size r×r is said to be a commuting extension of a tuple (A1,…,Ap) of matrices of size n×n if the Zi pairwise commute and each Ai sits in the upper left corner of a block decomposition of Zi (here, r and n are two arbitrary integers with n<r). This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called “quantum Zeno dynamics.” Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results:(i)Theorems on the uniqueness of commuting extensions for three matrices or more.(ii)Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to r=4n/3, and are apparently the first provably efficient algorithms for this problem applicable beyond r=n+1.(iii)A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices.

ChatGPT in Teaching Linear Algebra: Strides Forward, Steps to Go

Abstract As soon as a new technology emerges, the education community explores its affordances and the possibilities to apply it in education. In this article, we analyze sessions with ChatGPT around topics in basic linear algebra. We reflect on the affordances and changes between two versions of ChatGPT since its worldwide publication in our area of interest, namely, linear algebra. In particular, the question of whether this software can be a teaching assistant or even somehow replace the human teacher is addressed. As of the time this article is written, the answer is generally negative. For the small part where the answer can be positive, some reflections about an original instrumental genesis are given.

Applications of Laguerre transform to solve Schrödinger-type equations and differential equations of order four

The finite Laguerre transform is applied to solve differential equations problems of order higher than two and a one-dimensional steady-state Schrödinger equation, by using elementary Linear Algebra methods.

Unforgeability of Blind Schnorr in the Limited Concurrency Setting

Blind signature schemes enable a user to obtain a digital signature on a message from a signer without revealing the message itself. Among the most fundamental examples of such a scheme is blind Schnorr, but recent results show that it does not satisfy the standard notion of security against malicious users, One-More Unforgeability (OMUF), as it is vulnerable to the ROS attack. However, blind Schnorr does satisfy the weaker notion of sequential OMUF, in which only one signing session is open at a time, in the Algebraic Group Model (AGM) + Random Oracle Model (ROM), assuming the hardness of the Discrete Logarithm (DL) problem. This paper serves as a first step towards characterizing the security of blind Schnorr in the limited concurrency setting. Specifically, we show that blind Schnorr satisfies OMUF when at most two signing sessions can be concurrently open (in the AGM+ROM, assuming DL). Our argument suggests that it is plausible that blind Schnorr satisfies OMUF for up to polylogarithmically many concurrent signing sessions. Our security proof involves interesting techniques from linear algebra and combinatorics.