Basic Algebra Research Articles

Geometric approach to the proof of Fermat’s last theorem

A geometric approach to the proof of Fermat’s last theorem is proposed. Instead of integers a, b, c, Fermat’s last theorem considers a triangle with side lengths a, b, c. It is proved that in the case of right-angled and obtuse-angled triangles Fermat's equation has no solutions. When considering the case when a, b, c are sides of an acute triangle, it is proved that Fermat's equation has no entire solutions for p>2. The numbers a=k, b=k+m, c=k+n, where k, m, n are natural numbers satisfying the inequalities n>m, n<k+m, exhaust all possible variants of natural numbers a, b, c which are the sides of the triangle.
 The proof in this case is carried out by introducing a new auxiliary function f(k,p)=kp+(k+m)p–(k+n)p of two variables, which is a polynomial of degree in the variable . The study of this function necessary for the proof of the theorem is carried out. A special case of Fermat’s last theorem is proved, when the variables a, b, c take consecutive integer values. The proof of Fermat’s last theorem was carried out in two stages. Namely, all possible values of natural numbers k, m, n, p were considered, satisfying the following conditions: firstly, the number (np–mp) is odd, and secondly, this number is even, where the number (np–mp) is a free member of the function f(k, p).
 Another proof of Fermat’s last theorem is proposed, in which all possible relationships between the supposed integer solution of the equation f(k, p)=0 and the number corresponding to this supposed solution are considered.
 The proof is carried out using the mathematical apparatus of the theory of integers, elements of higher algebra and the foundations of mathematical analysis. These studies are a continuation of the author's works, in which some special cases of Fermat’s last theorem are proved

Enhancing Students’ Engagement in Simultaneous Equations Involving Indices with Cooperative Learning Strategy and Technology Use

This study explores the process of enhancing students’ engagements (SE) in simultaneous equations involving indices (SEII) with cooperative learning strategy (CLS) and technology integration (TI). The concept paper sensitively expresses the need for a learning strategy (LS) and how it can change students’ learning difficulties in SEII through CLS and GeoGebra (GG). Also, to specifically to answer the research questions related to quantitative findings via MAT-test and follow-up with qualitative approach through an interview on how students successfully engage in their procedural and conceptual understanding with SEII using the LS developed. Multimodal analysis with a special package for social sciences (SPSS) and qualitative study research (QSR) via NVivo were used for the research study analysis. The study finds that the LS is a critical blueprint for formulating and creating a new approach to learning school algebra and SEII.

Algebras of slowly growing length

We investigate the class of finite-dimensional not necessarily associative algebras that have slowly growing length, that is, for any algebra in this class its length is less than or equal to its dimension. We show that this class is considerably big, in particular, finite-dimensional Lie algebras as well as many other important classical finite-dimensional algebras belong to this class, for example, Leibniz algebras, Novikov algebras and Zinbiel algebras. The exact upper bounds for the length of these algebras is proved. To do this, we transfer the method of characteristic sequences to non-unital algebras and find certain polynomial conditions on the algebra elements that guarantee the slow growth of the length function.

Gauge transformations of two-component KP and mKP hierarchies

Based on generalized Lax equations and the Sato theory on KP and mKP hierarchies derived from the basic algebra of pseudo-differential operators, Miura- and auto-Bäcklund transformations can be derived from the gauge transformations of the Lax operators. In this paper, we study the gauge transformations of two-component KP and mKP hierarchies using the two-component generalized gauge transformations of the original operators.

Classifying torsion classes for algebras with radical square zero via sign decomposition

To study the set of torsion classes of a finite dimensional basic algebra over a field, we use a decomposition, called sign-decomposition, parameterized by elements of {±1}n where n is the number of simple modules. If A is an algebra with radical square zero, then for each ϵ∈{±1}n there is a hereditary algebra Aϵ! with radical square zero and a bijection between the set of torsion classes of A associated to ϵ and the set of faithful torsion classes of Aϵ!. Furthermore, this bijection preserves the property of being functorially finite. From a point of view of tilting theory, it implies that there is a bijection between the set of isomorphism classes of basic two-term silting complexes for A associated to ϵ and the set of isomorphism classes of basic tilting Aϵ!-modules. As an application, we prove that the number of two-term tilting complexes over Brauer line algebras (respectively, Brauer cycle algebras) having n edges is (2nn) (respectively, 22n−1 if n is odd, and ∞ if n is even).

Data Analysis Techniques for Problem-Based Learning Model Development and Rewarding Elementary Linear Algebra

The problem in this research is how the data analysis techniques are used in developing problem-based learning models and giving elementary linear algebra rewards. The purpose of this study was to describe the results of data analysis in the research conducted. The type of research used is the type of RD research with qualitative and quantitative methods. The stages of data collection carried out are: Preliminary Phase, Prototyping Phase and Assessment Phase. The Preliminary stage is processed descriptively, by clearly describing the results of the preliminary analysis in the form of needs analysis, curriculum analysis, concept analysis, and student characteristics analysis, each resulting analysis is equipped with conclusions. The prototyping stage uses instrument data that has been given to experts or validators to see the errors and accuracy of the instruments presented in the form of numbers. For the Assessment Phase, validity tests were carried out on the validity of the RPS and SAP instruments, model books, lecturer books, LKM, students' mathematical ability tests, attitude questionnaires and student learning interests. Product reliability tests were carried out on RPS and SAP instruments, model books, lecturer books, LKM, students' mathematical ability tests, attitude questionnaires and student learning interests. The product effectiveness test was obtained from the student's mathematical ability test data, attitude questionnaire and student learning interest questionnaire. Based on the analysis of the data used, it can be concluded that the product validity test results obtained high validity criteria. The results of the practicality test obtained high validity. The results of the effectiveness test are that the problem-based learning model and the provision of rewards can improve students' mathematical abilities and can encourage increased understanding of a concept and solve problems that are legitimate, efficient, and effective.

The structure of completely meet irreducible congruences in strongly Fregean algebras

A strongly Fregean algebra is an algebra such that the class of its homomorphic images is Fregean and the variety generated by this algebra is congruence modular. To understand the structure of these algebras we study the prime intervals projectivity relation in the posets of their completely meet irreducible congruences and show that its cosets have the natural structure of a Boolean group. In particular, this approach allows us to represent congruences and elements of such algebras as the subsets of upward closed subsets of these posets with some special properties.

Trotter Product Formulae for *-Automorphisms of Quantum Lattice Systems

We consider the dynamics tmapsto tau _t of an infinite quantum lattice system that is generated by a local interaction. If the interaction decomposes into a finite number of terms that are themselves local interactions, we show that tau _t can be efficiently approximated by a product of n automorphisms, each of them being an alternating product generated by the individual terms. For any integer m, we construct a product formula (in the spirit of Trotter) such that the approximation error scales as n^{-m}. Our bounds hold in norm, pointwise for algebra elements that are sufficiently well approximated by finite volume observables.

Hirzebruch-Riemann-Roch and Lefschetz type formulas for finite dimensional algebras

We introduce the dimension matrix of a bimodule complex and the trace matrix of a chain endomorphism of a bimodule complex, and give the Hirzebruch-Riemann-Roch (HRR) and Lefschetz type formulas for finite dimensional elementary algebras of finite global dimension. Moreover, we compare our HRR and Lefschetz type formulas with Shklyarov's HRR type formulas and Petit's Lefschetz type formulas for dg algebras.

A Curriculum-Based Approach to Learning Trajectories in Middle School Algebra

Our aim is to contribute to the body of research on learning trajectories (LTs) in mathematics by making visible a process for articulating a hypothetical learning trajectory implicit in a widely adopted, reform-based, middle-grades mathematics curriculum. In doing so, we highlight considerations, decisions, and challenges we faced as part of this work. By describing our LT articulation process, our aim is to highlight ways in which curriculum-specific LTs can be articulated to serve as a more proximal and instrumental tool for teachers’ instructional practice. Furthermore, to illustrate we describe how the products of the work were used in practice-based professional learning experiences with middle-grades mathematics teachers.

Footprints of Geodesics in Persistent Homology

Given a metric space X and a subspace $$A\subset X$$ , we prove that A can generate various algebraic elements in persistent homology of X. We call such elements (algebraic) footprints of A. Our results imply that footprints typically appear in dimensions above $$\dim (A)$$ . Higher dimensional persistent homology thus encodes lower dimensional geometric features of X. We pay special attention to a specific type of geodesics in a geodesic surface X called geodesic circles. We explain how they may generate non-trivial odd-dimensional and two-dimensional footprints. In particular, we can detect even some contractible geodesics using two- and three-dimensional persistent homology. This provides a link between persistent homology and the length spectrum in Riemannian geometry.

Jucys–Murphy elements and Grothendieck groups for generalized rook monoids

We consider a tower of generalized rook monoid algebras over the field \mathbb{C} of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys–Murphy elements for generalized rook monoid algebras. Over an algebraically closed field \Bbbk of positive characteristic p , utilizing Jucys–Murphy elements of rook monoid algebras, for 0\leq i\leq p-1 we define the corresponding i -restriction and i -induction functors along with two extra functors. On the direct sum \mathcal{G}_{\mathbb{C}} of the Grothendieck groups of module categories over rook monoid algebras over \Bbbk , these functors induce an action of the tensor product of the universal enveloping algebra U(\widehat{\mathfrak{sl}}_p(\mathbb{C})) and the monoid algebra \mathbb{C}[\mathcal{B}] of the bicyclic monoid \mathcal{B} . Furthermore, we prove that \mathcal{G}_{\mathbb{C}} is isomorphic to the tensor product of the basic representation of U(\widehat{\mathfrak{sl}}_{p}(\mathbb{C})) and the unique infinite-dimensional simple module over \mathbb{C}[\mathcal{B}] , and also exhibit that \mathcal{G}_{\mathbb{C}} is a bialgebra. Under some natural restrictions on the characteristic of \Bbbk , we outline the corresponding result for generalized rook monoids.

MISCONCEPTION ANALYSIS OF MATH CLASS VII USING THREE TIER-TEST

This study aims to find out what misconceptions are experienced by students in mathematics material for class VII SMP and to find out what factors cause students' misconceptions in mathematics material for class VII SMP. This type of research is qualitative research using a descriptive method. The research subjects in this study were 8 students of class VII B students of SMPN 46 Makassar with data collection techniques consisting of three tier-test diagnostic tests, interviews, and documentation. Data analysis techniques in this study consisted of data reduction, data presentation, and conclusion drawing. The results of this study indicate that based on the results of the Three Tier-Test given to class VII B students, it was identified that students experienced misconceptions in algebraic material with the categories of misconceptions and misconceptions (false positive). The majority of misconceptions occur in all algebraic materials, namely: algebraic concepts, algebraic elements, addition operations, subtraction operations, multiplication operations, and division operations. In addition, it was also found that the factors causing the misconceptions of class VII B students of SMPN 54 Makassar, namely the lack of students' ability to understand algebraic concepts in-depth, lack of interest in learning specifically for algebra students, the method often used by teachers, namely the lecture’s method, these making students difficult to understand algebraic material and pre-algebraic concepts.

The SHARK integral generation and digestion system.

In this paper, the SHARK integral generation and digestion engine is described. In essence, SHARK is based on a reformulation of the popular McMurchie/Davidson approach to molecular integrals. This reformulation leads to an efficient algorithm that is driven by BLAS level 3 operations. The algorithm is particularly efficient for high angular momentum basis functions (up to L=7 is available by default, but the algorithm is programmed for arbitrary angular momenta). SHARK features a significant number of specific programming constructs that are designed to greatly simplify the workflow in quantum chemical program development and avoid undesirable code duplication to the largest possible extent. SHARK can handle segmented, generally and partially generally contracted basis sets. It can be used to generate a host of one- and two-electron integrals over various kernels including, two-, three-, and four-index repulsion integrals, integrals over Gauge Including Atomic Orbitals (GIAOs), relativistic integrals and integrals featuring a finite nucleus model. SHARK provides routines to evaluate Fock like matrices, generate integral transformations and related tasks. SHARK is the essential engine inside the ORCA package that drives essentially all tasks that are related to integrals over basis functions in version ORCA 5.0 and higher. Since the core of SHARK is based on low-level basic linear algebra (BLAS) operations, it is expected to not only perform well on present day but also on future hardware provided that the hardware manufacturer provides a properly optimized BLAS library for matrix and vector operations. Representative timings and comparisons to the Libint library used by ORCA are reported for Intel i9 and Apple M1 max processors.

An Efficacy Study of ALEKS-Based Placement in Entry-Level College Math Courses

ABSTRACT We studied ALEKS PPL placement scores, SAT scores, and entry-course grades for a population of 1659 students who placed into College Algebra or below over a span of 5 years. In addition to representing an understudied range of placement level, the study was conducted at an HBCU with majority Black/African American students, a demographic that is also underrepresented in other studies. We find that while many of the students who are placed into a course lack full mastery of the prerequisite topics, they nonetheless perform better than students who are placed into a lower-level course and worked their way up. We also find a chance (95% confidence) that ALEKS PPL will improve a student's placement by at least 1 level. ALEKS PPL proved a better predictor of student performance on Elementary Algebra topics than SAT score, but the reverse was true for a course that focused on consumer mathematics, statistics, and probability (called Math for Liberal Arts). Finally, we find that the cut scores used in most of our courses work quite well, except for the Math for Liberal Arts course.

On the Left Connected Subalgebra of the Descent Algebra of a Coxeter Group of Classical Type

A Coxeter group of classical type A_n, B_n or D_n contains a chain of subgroups of the same type. We show that intersections of conjugates of these subgroups are again of the same type, and make precise in which sense and to what extent this property is exclusive to the classical types of Coxeter groups. As the main tool for the proof, we use Solomon’s descent algebra. Using Stirling numbers, we express certain basis elements of the descent algebra as polynomials and derive explicit multiplication formulas for a commutative subalgebra of the descent algebra.

Dirac Operators for the Dunkl Angular Momentum Algebra

We define a family of Dirac operators for the Dunkl angular momentum algebra depending on certain central elements of the group algebra of the Pin cover of the Weyl group inherent to the rational Cherednik algebra. We prove an analogue of Vogan's conjecture for this family of operators and use this to show that the Dirac cohomology, when non-zero, determines the central character of representations of the angular momentum algebra. Furthermore, interpreting this algebra in the framework of (deformed) Howe dualities, we show that the natural Dirac element we define yields, up to scalars, a square root of the angular part of the Calogero-Moser Hamiltonian.

An Orthogonalization-Free Parallelizable Framework for All-Electron Calculations in Density Functional Theory

All-electron calculations play an important role in density functional theory, in which improving computational efficiency is one of the most needed and challenging tasks. In the model formulations, both nonlinear eigenvalue problem and total energy minimization problem pursue orthogonal solutions. Most existing algorithms for solving these two models invoke orthogonalization process either explicitly or implicitly in each iteration. Their efficiency suffers from this process in view of its cubic complexity and low parallel scalability in terms of the number of electrons for large scale systems. To break through this bottleneck, we propose an orthogonalization-free algorithm framework based on the total energy minimization problem. It is shown that the desired orthogonality can be gradually achieved without invoking orthogonalization in each iteration. Moreover, this framework fully consists of Basic Linear Algebra Subprograms (BLAS) operations and thus can be naturally parallelized. The global convergence of the proposed algorithm is established. We also present a precondition technique which can dramatically accelerate the convergence of the algorithm. The numerical experiments on all-electron calculations show the efficiency and high scalability of the proposed algorithm.

Indefinite linear quadratic optimal control for continuous-time rectangular descriptor Markov jump systems: infinite-time case

The indefinite linear quadratic (ILQ) optimal control problem has many important applications in financial, economic systems, etc., which has been widely researched in stochastic systems and descriptor systems, etc. However, for ILQ problem of rectangular descriptor Markov jump systems (DMJSs), because there are impulses simultaneously in descriptor subsystems and at the switching time, it is theoretically difficult to research and there is no result yet. This paper discusses the ILQ optimal control problem for continuous-time linear rectangular DMJSs. Firstly, under some rank conditions and inequality conditions, the ILQ problem for rectangular DMJSs can be equivalently transformed into standard LQ problem for Markov jump systems (MJSs) by using elementary linear algebra method. Then based on the LQ theory of MJSs, the solvable sufficient condition of the ILQ problem for rectangular DMJSs and the non-negative optimal cost value are obtained. The optimal control can be synthesised as state feedback, and the resulting optimal closed-loop system has the stochastically stable solution. In addition, with some rank inequality assumptions, the differential subsystem of the resulting optimal closed-loop system can be ensured to have a unique solution. Finally, two numerical examples are provided to illustrate the effectiveness of the methods proposed in this paper.

A BLIS-like matrix multiplication for machine learning in the RISC-V ISA-based GAP8 processor

We address the efficient realization of matrix multiplication (gemm), with application in the convolution operator for machine learning, for the RISC-V core present in the GreenWaves GAP8 processor. Our approach leverages BLIS (Basic Linear Algebra Instantiation Software) to develop an implementation that (1) re-organizes the gemm algorithm adapting its micro-kernel to exploit the hardware-supported dot product kernel in the GAP8; (2) explicitly orchestrates the data transfers across the hierarchy of scratchpad memories via DMA (direct memory access); and (3) operates with integer arithmetic.