Basic Algebra Research Articles

Crossed modules, non-abelian extensions of associative conformal algebras and wells exact sequences

In this paper, we introduce the notions of crossed modules of associative conformal algebras, two-term strongly homotopy associative conformal algebras, and discuss the relationship between them and the third Hochschild cohomology of associative conformal algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. We show that non-abelian extensions of an associative conformal algebra can be viewed as Maurer–Cartan elements of a suitable differential graded Lie algebra, and prove that there is a one-one correspondence between the Deligne groupoid of this differential graded Lie algebra and the non-abelian cohomology. For any two associative conformal algebras [Formula: see text] and [Formula: see text], we provide the condition that a pair of automorphisms in Aut(A) × Aut(B) can be extended to an automorphisms of the non-abelian extension algebra of [Formula: see text] by [Formula: see text], and give the fundamental sequence of Wells in the context of associative conformal algebras.

Derived deformation theory of crepant curves

AbstractThis paper determines the full, derived deformation theory of certain smooth rational curves in Calabi–Yau 3‐folds, by determining all higher ‐products in its controlling DG‐algebra. This geometric setup includes very general cases where does not contract, cases where the curve neighbourhood is not rational, all known simple smooth 3‐fold flops, and all known divisorial contractions to curves. As a corollary, it is shown that the non‐commutative deformation theory of is described via a superpotential algebra derived from what we call free necklace polynomials, which are elements in the free algebra obtained via a closed formula from combinatorial gluing data. The description of these polynomials, together with the above results, establishes a suitably interpreted string theory prediction due to Ferrari (Adv. Theor. Math. Phys. 7 (2003) 619–665), Aspinwall–Katz (Comm. Math. Phys.. 264 (2006) 227–253) and Curto–Morrison (J. Algebraic Geom. 22 (2013) 599–627). Perhaps most significantly, the main results give both the language and evidence to finally formulate new contractibility conjectures for rational curves in CY 3‐folds, which lift Artin's (Amer. J. Math. 84 (1962) 485–496) celebrated results from surfaces.

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.

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.

A 4-valued logic for double Stone algebras

This paper investigates the logical structure of the 4-element chain considered as a double Stone algebra. It has been shown that any element of a double Stone algebra can be identified as monotone ordered triplet of sets. As a consequence, we obtain the 4-valued semantics for the logic LD of double Stone algebras. Furthermore, the rough set semantics of the logic LD is provided by dividing the boundary region (uncertainty) into two disjoint subregions.

Finitely generated subgroups of algebraic elements of plane Cremona groups are bounded

Finitely generated subgroups of algebraic elements of plane Cremona groups are bounded

Effects of Collaborative Team Teaching on Students' Performance in High School Algebra

Effects of Collaborative Team Teaching on Students' Performance in High School Algebra

SIMPLIFYING IRRATIONAL EXPRESSIONS AND APPLICATION OF THE MAPLE COMPUTER PACKAGE

The article is devoted to the presentation of one of the main concepts of the school algebra course: “traditional” and “non-traditional” approaches of simplifying irrational expressions, as well as a description of the simplification of irrational expressions using the computer package MAPLE. The article examines irrational expressions, containing multiple roots of the 2nd degree, and presents both “traditional” approaches of simplifying them, using full quadratic division and formula, and “non-traditional” approach of noting-squaring. Specific examples are considered in order to more clearly describe the application of each of the above approaches, point out their differences, features, advantages and provide clear methodological instructions. In addition to the above approaches for simplifying irrational expressions, the article also describes the implementation of simplifying irrational expressions using the computer package MAPLE: the simplification of an irrational expression given in the article is also implemented in the MAPLE environment, bringing methodological instructions, as a result of which it becomes possible to instantly simplify an irrational expression.

Representations of Basic Hopf Algebras of Tame Type

Representations of Basic Hopf Algebras of Tame Type

Lorentz-violating extension of Wigner function formalism and chiral kinetic theory

The quantum kinetic equation for the gauge-invariant Wigner function, constructed from spinor fields that obey the Dirac equation modified by CPT and Lorentz symmetry-violating terms, is presented. The equations for the components of the Wigner function in the Clifford algebra basis are accomplished. Focusing on the massless case, an extended semiclassical chiral kinetic theory in the presence of external electromagnetic fields is developed. We calculate the chiral currents and establish the anomalous magnetic and separation effects in a Lorentz-violating background. The chiral anomaly within the context of extended quantum electrodynamics is elucidated. Finally, we derive the semiclassical Lorentz-violating extended chiral transport equation. Published by the American Physical Society 2024

Hamiltonians for the quantised Volterra hierarchy

This paper builds upon our recent work, published in Carpentier et al (2022 Lett. Math. Phys. 112 94), where we established that the integrable Volterra lattice on a free associative algebra and the whole hierarchy of its symmetries admit a quantisation dependent on a parameter ω. We also uncovered an intriguing aspect: all odd-degree symmetries of the hierarchy admit an alternative, non-deformation quantisation, resulting in a non-commutative algebra for any choice of the quantisation parameter ω. In this study, we demonstrate that each equation within the quantum Volterra hierarchy can be expressed in the Heisenberg form. We provide explicit expressions for all quantum Hamiltonians and establish their commutativity. In the classical limit, these quantum Hamiltonians yield explicit expressions for the classical ones of the commutative Volterra hierarchy. Furthermore, we present Heisenberg equations and their Hamiltonians in the case of non-deformation quantisation. Finally, we discuss commuting first integrals, central elements of the quantum algebra, and the integrability problem for periodic reductions of the Volterra lattice in the context of both quantisations.

О перманенте многомерных матриц

The permanent of multidimensional matrices is expressed in terms of operations on elements of commutative algebra with nilpotent index 2 generators. Using a technique based on this relationship, several properties of the permanent have been proved. Various types of multidimensional permutations are considered. The permanent of multidimensional matrices is considered from the point of view of the enumeration function of multidimensional permutations.

The algebraic learning of middle school students’ evolution during a school year: a statistical large-scale study based on results in mathematics didactics

AbstractThis large-scale study characterizes the algebraic learning of a cohort of nearly 800 French students during the last year of middle school (9th grade, 14–15 years old) and examines their evolution in relation to the composition of the classes to which the students belong. Based on the Anthropological Theory of the Didactic (ATD), the study is founded both on a reference model of elementary algebra and on technological-theoretical levels characterizing the students’ reasoning and their knowledge regarding institutional expectations. These are used to code the data and define the statistical variables on which multivariate descriptive analyses are carried out. Students’ learning is analyzed through Pépite, an algebra test given at the beginning and at the end of the year. A cluster analysis resulted in three distinct clusters of similar classes and of similar students both at the beginning and end of the grade 9 school year. They are interpreted didactically and could be useful for a teacher who has to manage the heterogeneity of learning in a class. Some classes and students move from one cluster to another between the beginning and the end of the year, showing a wide variety of ways in which students’ algebra learning progresses or regresses over a school year.

Exact solutions, wave dynamics and conservation laws of a generalized geophysical Korteweg de Vries equation in ocean physics using Lie symmetry analysis

An evolution equation is a partial diferential equation that describes the time evolution of a physical system starting from given initial data. Evolution equations arise from many areas of applied and engineering sciences. To this end, this article investigates the analytical studies of a generalized geophysical Korteweg-de Vries equation in ocean physics. The examination of this model is conducted via the Lie group theory of diferential equations. In the first place, point symmetries, which are constituent elements of a four-dimensional Lie algebra, are systematically computed. Thereafter, one-parameter transformation groups for the algebra are calculated. Besides, going forward, a one-dimensional optimal system of subalgebras is derived in a procedural manner. Sequel to this, the subalgebras and combination of the achieved symmetries are invoked in the reduction proces which enables the derivation of nonlinear ordinary diferential equations associated with the generalized geophysical Korteweg-de Vries equation under study. Most of the achieved nonlinear ordinary diferential equations are further solved either via direct integration or using a power series approach. Furthermore, travelling wave solutions are initially obtained. This is attained via direct integration and the use of Jacobi elliptic function approach. These techniques enable the attainment of various exact soliton solutions, including non-topological soliton solutions as well as general periodic function solutions of note, such as cosine amplitude, sine amplitude, and delta amplitude solutions of the model. Furthermore, numerical simulations of the solutions are invoked to gain a gross knowledge of the physical phenomena represented by the under-study generalized geophysical Korteweg-de Vries equation in ocean physics. In the end, the investigation further gives attention to the calculation of conserved vectors for the model using Ibragimov's theorem for conservation laws, as well as Noether's theorem.

On the cubic Shimura lift to $\operatorname{PGL}(3)$: The fundamental lemma

The classical Shimura correspondence lifts automorphic representations on the double cover of \mathrm{SL}_{2} to automorphic representations on \mathrm{PGL}_{2} . Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of \mathrm{SL}_{3} to \mathrm{PGL}_{3} , and also characterize the image of the lift. The characterization would be through the non-vanishing of a certain global period involving a function in the space of the automorphic minimal representation \Theta_{\mathrm{SO}_8} for split \mathrm{SO}_{8}(\mathbb{A}) , consistent with a conjecture of Bump, Friedberg and Ginzburg (2001). In this paper, we first analyze a global distribution on \mathrm{PGL}_{3}(\mathbb{A}) involving this period and show that it is a sum of factorizable orbital integrals. The same is true for the Kuznetsov distribution attached to the triple cover of \mathrm{SL}_{3}(\mathbb{A}) . We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the fundamental lemma.

A note on splittable linear Lie algebras

A linear Lie algebra is splittable if it contains the semisimple and nilpotent parts of each element. It is early known that a solvable linear Lie algebra g is splittable if and only if g=a+n, where a is an abelian subalgebra of g composed of semisimple elements and n is the ideal of all nilpotent matrices of g. In this paper, using elementary linear algebra we give a direct proof of the theorem and related results. Besides, we determine the structure of linear Lie algebras composed of semisimple or nilpotent elements.

Diagnostic study of mathematical reasoning in novice university students

This paper presents the results of a study aimed at diagnosing how newly enrolled mathematics majors reason from a mathematical standpoint when they are required to perform inductive and deductive processes and validate or refute claims of a general nature. The theoretical framework underpinning this research comprises studies and investigations related to mathematical reasoning. The results of this diagnostic reveal that students lack mathematical reasoning skills that would allow them to justify statements of a general nature. In addition, they face difficulties in the development of deductive reasoning, are prone to generalize based on specific cases and lack skills in mathematical argumentation. On the other hand, they lack basic algebra skills such as developing square binomials and factoring. These findings highlight the need to design and implement a didactic proposal that fosters the development of reasoning and argumentation in students.

On Neutrosophic Transitivity and Absorbent Filters of Basic Logic Algebras

On Neutrosophic Transitivity and Absorbent Filters of Basic Logic Algebras

A direct and analytical method for inverse problems under uncertainty in energy system design: combining inverse simulation and Polynomial Chaos theory

This article introduces and formalizes a novel stochastic method that combines inverse simulation with the theory of generalized Polynomial Chaos (gPC) to solve and study inverse problems under uncertainty in energy system design applications. The method is particularly relevant to design tasks where only a deterministic forward model of a physical system is available, in which a target design quantity is an input to the model that cannot be obtained directly, but can be quantified reversely via the outputs of the model. In this scenario, the proposed method offers an analytical and direct approach to invert such system models. The method puts emphasis on user-friendliness, as it enables its users to conduct the inverse simulation under uncertainty directly in the gPC domain by redefining basic algebra operations for computations. Moreover, the method incorporates an optimization-based approach to integrate supplementary constraints on stochastic quantities. This feature enables the solution of inverse problems bounding the statistical moments of stochastic system variables. The authors exemplify the application of the proposed method with proof-of-concept tests in energy system design, specifically performing uncertainty quantification and sensitivity analysis for a Multi-Energy System (MES). The findings demonstrate the high accuracy of the method as well as clear advantages over conventional sampling-based methods when dealing with a small number of stochastic variables in a system or model. However, the case studies also highlight the current limitations of the proposed method such as slow execution speed due to the optimization-based approach and the challenges associated with, for example, the curse of dimensionality in gPC.

Gröbner bases for bipartite determinantal ideals

Nakajima’s graded quiver varieties naturally appear in the study of bases of cluster algebras. One particular family of these varieties, namely the bipartite determinantal varieties, can be defined for any bipartite quiver and gives a vast generalization of classical determinantal varieties with broad applications to algebra, geometry, combinatorics, and statistics. The ideals that define bipartite determinantal varieties are called bipartite determinantal ideals. We provide an elementary method of proof showing that the natural generators of a bipartite determinantal ideal form a Gröbner basis, using an S-polynomial construction method that relies on the Leibniz formula for determinants. This method is developed from an idea by Narasimhan and Caniglia–Guccione–Guccione. As applications, we study the connection between double determinantal ideals (which are bipartite determinantal ideals of a quiver with two vertices) and tensors, and provide an interpretation of these ideals within the context of algebraic statistics.