Basic Algebra Research Articles

Systematic Human Learning and Generalization From a Brief Tutorial With Explanatory Feedback.

We investigate human adults' ability to learn an abstract reasoning task quickly and to generalize outside of the range of training examples. Using a task based on a solution strategy in Sudoku, we provide Sudoku-naive participants with a brief instructional tutorial with explanatory feedback using a narrow range of training examples. We find that most participants who master the task do so within 10 practice trials and generalize well to puzzles outside of the training range. We also find that most of those who master the task can describe a valid solution strategy, and such participants perform better on transfer puzzles than those whose strategy descriptions are vague or incomplete. Interestingly, fewer than half of our human participants were successful in acquiring a valid solution strategy, and this ability was associated with completion of high school algebra and geometry. We consider the implications of these findings for understanding human systematic reasoning, as well as the challenges these findings pose for building computational models that capture all aspects of our findings, and we point toward a role for learning from instructions and explanations to support rapid learning and generalization.

Stochastic modelling of symmetric positive definite material tensors

Spatial symmetries and invariances play an important role in the behaviour of materials and should be respected in the description and modelling of material properties. The focus here is the class of physically symmetric and positive definite tensors, as they appear often in the description of materials, and one wants to be able to prescribe certain classes of spatial symmetries and invariances for each member of the whole ensemble, while at the same time demanding that the mean or expected value of the ensemble be subject to a possibly ‘higher’ spatial invariance class. We formulate a modelling framework which not only respects these two requirements—positive definiteness and invariance—but also allows a fine control over orientation on one hand, and strength / size on the other. As the set of positive definite tensors is not a linear space, but rather an open convex cone in the linear space of physically symmetric tensors, we consider it advantageous to widen the notion of mean to the so-called Fréchet mean on a metric space, which is based on distance measures or metrics between positive definite tensors other than the usual Euclidean one. It is shown how the random ensemble can be modelled and generated, independently in its scaling and orientational or directional aspects, with a Lie algebra representation via a memoryless transformation. The parameters which describe the elements in this Lie algebra are then to be considered as random fields on the domain of interest. As an example, a 2D and a 3D model of steady-state heat conduction in a human proximal femur, a bone with high material anisotropy, is modelled with a random thermal conductivity tensor, and the numerical results show the distinct impact of incorporating into the constitutive model different material uncertainties—scaling, orientation, and prescribed material symmetry—on the desired quantities of interest.

Integrating the Case Studies Pedagogy and the Anthropological Theory of Didactics to Enhance the Teaching and Learning of Algebra in Undergraduate STEM Education

Methodology case studies plays a pivotal role in STEM education, especially in teaching undergraduate algebra. This article emphasizes the power of real-world scenarios and active pedagogy, a student-centered approach that makes abstract algebraic ideas tangible. The Anthropological Theory of Didactics (ATD) is spotlighted as a perspective that treats mathematical knowledge as a cultural artifact, embedding mathematics concepts deeply within cultural and historical contexts. By comparing active pedagogy with ATD, the article unveils a compelling overlap that values the significance of context in learning. Students don't just engage actively but also delve into the cultural essence of mathematical problems, gaining a profound understanding. Addressing the hurdles undergraduates face in transitioning from high school algebra to university level, the article suggests activities that merge active pedagogy and ATD. This holistic approach has the potential to transform algebra education into a more engaging, relevant, and culturally resonant experience.

Vertex operators for imaginary [formula omitted] subalgebras of the Monster Lie algebra

The Monster Lie algebra m is a quotient of the physical space of the vertex algebra V=V♮⊗V1,1, where V♮ is the Moonshine module vertex operator algebra of Frenkel, Lepowsky, and Meurman, and V1,1 is the vertex algebra corresponding to the rank 2 even unimodular lattice II1,1. We construct vertex algebra elements that project to bases for subalgebras of m isomorphic to gl2, corresponding to each imaginary simple root, denoted (1,j) for j>0. Our method requires the existence of pairs of primary vectors in V♮ satisfying some natural conditions, which we prove. We show that the action of the Monster finite simple group M on the subspace of primary vectors in V♮ induces an M-action on the set of gl2 subalgebras corresponding to a fixed imaginary simple root. We use the generating function for dimensions of subspaces of primary vectors of V♮ to prove that this action is non-trivial for small values of j.

Quantum de Sitter Geometry

Quantum de Sitter geometry is discussed using elementary field operator algebras in Krein space quantization from an observer-independent point of view, i.e., ambient space formalism. In quantum geometry, the conformal sector of the metric becomes a dynamical degree of freedom, which can be written in terms of a massless minimally coupled scalar field. The elementary fields necessary for the construction of quantum geometry are introduced and classified. A complete Krein–Fock space structure for elementary fields is presented using field operator algebras. We conclude that since quantum de Sitter geometry can be constructed by elementary fields operators, the geometry quantum state is immersed in the Krein–Fock space and evolves in it. The total number of accessible quantum states in the universe is chosen as a parameter of quantum state evolution, which has a relationship with the universe’s entropy. Inspired by the Wheeler–DeWitt constraint equation in cosmology, the evolution equation of the geometry quantum state is formulated in terms of the Lagrangian density of interaction fields in ambient space formalism.

Block-Supersymmetric Polynomials on Spaces of Absolutely Convergent Series

In this paper, we consider a supersymmetric version of block-symmetric polynomials on a Banach space of two-sided absolutely summing series of vectors in Cs for some positive integer s>1. We describe some sequences of generators of the algebra of block-supersymmetric polynomials and algebraic relations between the generators for the finite-dimensional case and construct algebraic bases of block-supersymmetric polynomials in the infinite-dimensional case. Furthermore, we propose some consequences for algebras of block-supersymmetric analytic functions of bounded type and their spectra. Finally, we consider some special derivatives in algebras of block-symmetric and block-supersymmetric analytic functions and find related Appell-type sequences of polynomials.

Algebraically generated groups and their Lie algebras

AbstractThe automorphism group of an affine variety is an ind‐group. Its Lie algebra is canonically embedded into the Lie algebra of vector fields on . We study the relations between subgroups of and Lie subalgebras of . We show that a subgroup generated by a family of connected algebraic subgroups of is algebraic if and only if the Lie algebras generate a finite‐dimensional Lie subalgebra of . Extending a result by Cohen–Draisma (Transform. Groups 8 (2003), no. 1, 51–68), we prove that a locally finite Lie algebra generated by locally nilpotent vector fields is algebraic, that is, for an algebraic subgroup . Along the same lines, we prove that if a subgroup generated by finitely many connected algebraic groups is solvable, then it is an algebraic group. We also show that a unipotent algebraic subgroup has derived length . This result is based on the following triangulation theorem: Every unipotent algebraic subgroup of with a dense orbit in is conjugate to a subgroup of the de Jonquières subgroup. Furthermore, we give an example of a free subgroup generated by two algebraic elements such that the Zariski closure is a free product of two nested commutative closed unipotent ind‐subgroups. To any affine ind‐group , one can associate a canonical ideal . It is linearly generated by the tangent spaces for all algebraic subsets that are smooth in . It has the important property that for a surjective homomorphism , the induced homomorphism is surjective as well. Moreover, if is a subnormal closed ind‐subgroup of finite codimension, then has finite codimension in .

DEVELOPMENT OF MODULE ELEMENTARY LINEAR ALGEBRA INTEGRATED TO MEDIA PREZI FOR PRE-SERVICE MATHEMATICS TEACHER

This study purposed to describe the development and produce elementary linear algebra course modules on system of linear equation (SLE) and matrices integrated with Prezi learning media that are valid, practical and effective. This development research referred to the development design steps developed by Allesi & Trollip. The development design is grouped into three development procedures, which include: planning, design, and development stage. The product trial consisted of an alpha test conducted by two material experts and two media experts, as well as a beta test which was carried out through 2 stages, beta test 1 (small group trial) to 8 subjects; representatives of two students at cohort 2019/2020, 2020/2021, 2021/2022, and 2022/2023. Beta test 2 was conducted on 70 students who took the Elementary Linear Algebra course in cohort 2023/2024. Data collection used questionnaires for material experts, questionnaires for media experts, student response questionnaires, and tests. The developed module has several advantages. First, the material is prepared based on the results of research that has been done in previous studies so that it can minimize epistemological learning obstacles. Second, it is integrated with Prezi-based learning media that supports a deeper understanding of SLE content material.

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

Solving systems of homogeneous linear algebraic equations in pricing problems of an alternative financial model

Subject. This article focuses on the problems of solving systems of homogeneous linear algebraic equations in the pricing problems of an alternative financial model. Objectives. The article aims to develop a non-trivial approach to solving the problem of pricing from the scratch in the construction of a national alternative financial model, which boils down to solving large homogeneous systems of linear algebraic equations, complicated by the fuzzy setting of initial parameters (initial data) within the framework of creating a State model of a socially oriented economy, guaranteed to ensure economic, military and other types of integrated security of the Russian Federation. Methods. For the study, we used tensorial notations involving the use of elements of linear algebra, including operations with matrices and vectors, the apparatus of the fuzzy set theory (interval arithmetic), as well as methods and principles of scientific research and complex logical analysis and modeling of pricing and technological processes. Results. The article shows that an effective non-zero (different from zero, i.e. classical, trivial) solution of systems of homogeneous algebraic equations in pricing problems of an alternative financial model from the scratch is possible if the initial parameters are specified by fuzzy sets, which significantly simplifies the procedure of the computational process and the volume of arithmetic operations. Conclusions. The results obtained can be used in structures involved in the creation and testing of financial and economic models both at the federal and the Russian Federation constituent entity levels, as well as in governmental and non-governmental organizations dealing with the issues of pricing, planning and logistics support for business entities not only in the Russian Federation, but also in other countries experiencing problems in the field of financing.

BLAS Kütüphanelerinin GPU Mimarilerindeki Nicel Performans Analizi

Basic Linear Algebra Subprograms (BLAS) are a set of linear algebra routines commonly used by machine learning applications and scientific computing. BLAS libraries with optimized implementations of BLAS routines offer high performance by exploiting parallel execution units in target computing systems. With massively large number of cores, graphics processing units (GPUs) exhibit high performance for computationally-heavy workloads. Recent BLAS libraries utilize parallel cores of GPU architectures efficiently by employing inherent data parallelism. In this study, we analyze GPU-targeted functions from two BLAS libraries, cuBLAS and MAGMA, and evaluate their performance on a single-GPU NVIDIA architecture by considering architectural features and limitations. We collect architectural performance metrics and explore resource utilization characteristics. Our work aims to help researchers and programmers to understand the performance behavior and GPU resource utilization of the BLAS routines implemented by the libraries.

Game Theory Analysis of Algebraic Content Selection in Middle School Mathematics Curricula

The algebra section of middle school math textbooks is indispensable. The characteristics of the textbook and the rationality of its elements need to be algebra content of middle school math textbooks mainly consists of algebraic formulas, equations and inequalities, functions and other elements. This paper focuses on modeling the analysis based on game theory and then exploring the rationality of the algebra textbook study is conducted by reviewing the literature and constructing an analytical model based on cooperative game and Nash equilibrium theory. It is obtained that in the strategy where all three elements are used at the same time, the one that focuses on the orchestration function can maximize the reorganization of unit algebra content design is needed to achieve rational weighting of individual algebra selection of elements needs to fully reflect the articulation between the individual algebra elements. Algebraic elements are designed to spiral in difficulty to match the needs of teachers and students using them.

Construction of free quasi-idempotent differential Rota-Baxter algebras by Gröbner-Shirshov bases

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma. Communicated by P. Kolesnikov

Symmetry analysis and invariant solutions of generalized coupled Zakharov-Kuznetsov equations using optimal system of Lie subalgebra

Abstract This research focuses on the examination of nonlinear evolution equations, with a specific emphasis on the generalized coupled Zakharov-Kuznetsov (CZK) equations serving as a primary application. Given the wide application of classical Lie symmetry methods in this field, this study employs a Lie symmetry analysis to investigate the CZK equations, as detailed in this research. Our methodology involves the construction of a nine-dimensional optimal system by leveraging the fundamental elements of the Lie algebra. Subsequently, we apply similarity reductions to the equations using each subalgebra. The resulting invariant solutions find diverse applications within the realm of physics and can also be adapted to solve a broad range of related nonlinear evolution equations. We meticulously validate all these solutions through a straightforward verification process. To enhance our comprehension of the physical implications of these solutions, we employ Mathematica simulations to visually represent various solution scenarios. Additionally, to preserve conservation laws, we incorporate Ibragimov’s novel conservation law theorem as a crucial component of our analysis.

Lie structure of the Heisenberg-Weyl algebra

As an associative algebra, the Heisenberg--Weyl algebra $\HWeyl$ is generated by two elements $A$, $B$ subject to the relation $AB-BA=1$. As a Lie algebra, however, where the usual commutator serves as Lie bracket, the elements $A$ and $B$ are not able to generate the whole space $\HWeyl$. We identify a non-nilpotent but solvable Lie subalgebra $\coreLie$ of $\HWeyl$, for which, using some facts from the theory of bases for free Lie algebras, we give a presentation by generators and relations. Under this presentation, we show that, for some algebra isomorphism $\isoH:\HWeyl\into\HWeyl$, the Lie algebra $\HWeyl$ is generated by the generators of $\coreLie$, together with their images under $\isoH$, and that $\HWeyl$ is the sum of $\coreLie$, $\isoH(\coreLie)$ and $\lbrak \coreLie,\isoH(\coreLie)\rbrak$.

Modified rational six vertex model on the rectangular lattice

We consider a rational six vertex model on a rectangular lattice with boundary conditions that generalize the usual domain wall type. We find that the partition function of the inhomogeneous version of this model is given by a modified Izergin determinant. The proofs are based on the quantum inverse scattering method and its representation theory together with elementary linear algebra.

A Study on Some Topics in Algebra

This study provides a comprehensive examination of various topics in algebra, exploring its historical development, fundamental concepts, and diverse branches. The paper delves into elementary algebra, abstract algebra, linear algebra, and commutative algebra, offering detailed insights into each area. Additionally, it introduces a problem-analysis model to evaluate algebraic concepts. This exploration aims to highlight the significance of algebra in mathematical theory and its practical applications, underscoring its role as a foundational pillar in both academic research and applied sciences.

On Some Novel Generalizations of Weak Fuzzy Complex Numbers

The ring of weak fuzzy complex numbers is an extension of real numbers ring by using an algebraic element with fuzzy property. In this paper, we present two novel generalizations of weak fuzzy complex numbers, where the concepts of strong fuzzy complex numbers and split-complex weak fuzzy complex numbers will be defined for the first time with a general study of their elementary properties and special elements. On the other hand, we provide an algorithm to compute the dempotent elements in the ring of split-complex weak fuzzy complex numbers with many related examples that clarify the validity of our work.

Generalized solution of burger equation

In this paper, The existence and uniqueness of the generalized solution of the Burger equation is studied with initial conditions are distributions (elements of Colombeau algebra). then we look at the association notion in conjunction with the classic solution.

On bases of g-invariant endomorphism algebras

On bases of g-invariant endomorphism algebras