Basic Algebra Research Articles

On Superization of Nonlinear Integrable Dynamical Systems

We study an interesting superization problem of integrable nonlinear dynamical systems on functional manifolds. As an example, we considered a quantum many-particle Schrödinger–Davydov model on the axis, whose quasi-classical reduction proved to be a completely integrable Hamiltonian system on a smooth functional manifold. We checked that the so-called "naive" approach, based on the superization of the related phase space variables via extending the corresponding Poisson brackets upon the related functional supermanifold, fails to retain the dynamical system super-integrability. Moreover, we demonstrated that there exists a wide class of classical Lax-type integrable nonlinear dynamical systems on axes in relation to which a superization scheme consists in a reasonable superization of the related Lax-type representation by means of passing from the basic algebra of pseudo-differential operators on the axis to the corresponding superalgebra of super-pseudodifferential operators on the superaxis.

Elliptic hyperlogarithms

Abstract Let ${\mathcal {E}}$ be a complex elliptic curve and S be a non-empty finite subset of ${\mathcal {E}}$ . We show that the functions $\tilde {\Gamma }$ introduced in [BDDT] out of string theory motivations give rise to a basis (as a vector space) of the minimal algebra $A_{{\mathcal {E}}{\smallsetminus } S}$ of holomorphic multivalued functions on ${\mathcal {E}}{\smallsetminus } S$ which is stable under integration, introduced in [EZ]; this basis is alternative to the basis of $A_{{\mathcal {E}}{\smallsetminus } S}$ constructed in loc. cit. using elliptic analogs of the hyperlogarithm functions.

Six-dimensional Supermultiplets from Bundles on Projective Spaces

The projective variety of square-zero elements in the six-dimensional minimal supersymmetry algebra is isomorphic to P1×P3. We use this fact, together with the pure spinor superfield formalism, to study supermultiplets in six dimensions, starting from vector bundles on projective spaces. We classify all multiplets whose derived invariants for the supertranslation algebra form a line bundle over the nilpotence variety; one can think of such multiplets as being those whose holomorphic twists have rank one over Dolbeault forms on spacetime. In addition, we explicitly construct multiplets associated to natural higher-rank equivariant vector bundles, including the tangent and normal bundles as well as their duals. Among the multiplets constructed are the vector multiplet and hypermultiplet, the family of O(n)-multiplets, and the supergravity and gravitino multiplets. Along the way, we tackle various theoretical problems within the pure spinor superfield formalism. In particular, we give some general discussion about the relation of the projective nilpotence variety to multiplets and prove general results on short exact sequences and dualities of sheaves in the context of the pure spinor superfield formalism.

Octonion Polynomials for a More Secure RSA Public Key Cryptosystem

In this paper, we design a new version of the RSA public key cryptosystem which utilizes the octonion polynomials that are elements of an octonion algebra with eight-dimensions. In the proposed version, which is called OP-RSA cryptosystem, the keys are created as the octonion polynomials. The plaintext is chosen as an octonion polynomial and the ciphertext for it is an element in an octonion algebra (OA) which is an octonion polynomial as well. The decryption process is proved mathematically based on the octonion polynomials. The security issues are determined. The proposed OP-RSA cryptosystem is more secure in comparison with other presented versions.

Algebra Problems in Middle School Textbooks from Six Countries

This study compared the algebra content of middle school mathematics textbooks from Finland, the United States, Singapore, China, Taiwan, and Nepal. Content analysis was employed to investigate five aspects of the algebra problems: the total number of algebra problems, representational forms, contextualization, response types, and cognitive demand. The results revealed that the six mathematics textbook series differed in all five aspects, indicating uneven opportunities to learn middle school algebra across countries. The insights gained can be valuable in designing mathematics textbooks and curricular materials. We provide recommendations for stakeholders and curriculum developers based on the results.

Symmetric Functions and Rings of Multinumbers Associated with Finite Groups

In this paper, we introduce ωn-symmetric polynomials associated with the finite group ωn, which consists of roots of unity, and groups of permutations acting on the Cartesian product of Banach spaces ℓ1. These polynomials extend the classical notions of symmetric and supersymmetric polynomials on ℓ1. We explore algebraic bases in the algebra of ωn-symmetric polynomials and derive corresponding generating functions. Building on this foundation, we construct rings of multisets (multinumbers), defined as equivalence classes on the underlying space under the action of ωn-symmetric polynomials, and investigate their fundamental properties. Furthermore, we examine the ring of integer multinumbers associated with the group ωn, proving that it forms an integral domain when n is prime or n=4.

Generalized Pentagon Equations

AbstractDrinfeld defined the Knizhnik–Zamolodchikov (KZ) associator $$\Phi _{\textrm{KZ}}$$ Φ KZ by considering the regularized holonomy of the KZ connection along the droit chemin [0, 1]. The KZ associator is a group-like element of the free associative algebra with two generators, and it satisfies the pentagon equation. In this paper, we consider paths on $${\mathbb {C}}\backslash \{ z_1, \dots , z_n\}$$ C \ { z 1 , ⋯ , z n } which start and end at tangential base points. These paths are not necessarily straight, and they may have a finite number of transversal self-intersections. We show that the regularized holonomy H of the KZ connection associated with such a path satisfies a generalization of Drinfeld’s pentagon equation. In this equation, we encounter H, $$\Phi _{\textrm{KZ}}$$ Φ KZ , and new factors associated with self-intersections, tangential base points, and the rotation number of the path.

Clebsch-Gordan Coefficients for Macdonald Polynomials

AbstractIn this paper we use the double affine Hecke algebra to compute the Macdonald polynomial products $$E_\ell P_m$$ E ℓ P m and $$P_\ell P_m$$ P ℓ P m for type $$SL_2$$ S L 2 and type $$GL_2$$ G L 2 Macdonald polynomials. Our method follows the ideas of Martha Yip but executes a compression to reduce the sum from $$2\cdot 3^{\ell -1}$$ 2 · 3 ℓ - 1 signed terms to $$2\ell $$ 2 ℓ positive terms. We show that our rule for $$P_\ell P_m$$ P ℓ P m is equivalent to a special case of the Pieri rule of Macdonald. Our method shows that computing $$E_\ell {\textbf {1}}_0$$ E ℓ 1 0 and $${\textbf {1}}_0 E_\ell {\textbf {1}}_0$$ 1 0 E ℓ 1 0 in terms of a special basis of the double affine Hecke algebra provides universal compressed formulas for multiplication by $$E_\ell $$ E ℓ and $$P_\ell $$ P ℓ . The formulas for a specific products $$E_\ell P_m$$ E ℓ P m and $$P_\ell P_m$$ P ℓ P m are obtained by evaluating the universal formulas at $$t^{-\frac{1}{2}}q^{-\frac{m}{2}}$$ t - 1 2 q - m 2 .

Algebraic structures formalizing the logic of effect algebras incorporating time dimension

Abstract Effect algebras were introduced in order to describe the structure of effects, i.e. events in quantum mechanics. They are partial algebras describing the logic behind the corresponding events. It is natural to ask how to introduce the logical connective implication in effect algebras. For lattice-ordered effect algebras this task was already solved by several authors, including the present ones. We concentrate on effect algebras that need not be lattice-ordered since these can better describe the events occurring in the quantum physical system. Although an effect algebra is only partial, we try to find a logical connective implication which is everywhere defined. But such a connective can be “unsharp” or “inexact” because its outputs for given pairs of entries need not be elements of the underlying effect algebra, but may be subsets of (mutually incomparable) maximal elements. We introduce such an implication together with its adjoint functor representing conjunction. Then we consider so-called tense operators on effect algebras. Of course, also these operators turn out to be “unsharp” in the aforementioned sense, but they are in a certain relation with the operators implication and conjunction. Finally, for given tense operators and given time set T, we describe two methods how to construct a time preference relation R on T such that the given tense operators are either comparable with or equivalent to those induced by the time frame (T, R).

Mathematics of Machine Learning: An Introduction

Over the course of just a few decades, machine learning has grown into a force which shapes modern life perhaps as much as the combustion engine or wireless communication. As we drive to work, machine learning algorithms extract license plate numbers from images captured by automatic cameras at busy intersections. At work, they measure our productivity and govern our supply chains. In our personal lives, they power product recommendations on online shopping sites and suggestions on social media. In our homes and on our devices, they recognize our voices and our faces. They process our loan applications and evaluate our medical images. More globally, they stabilize power grids and assist in planning flight routes. There is not a space in our public and personal lives which machine learning has not at least begun to affect. For a field as ubiquitous, it is fairly poorly represented in our common knowledge. This article aims at giving a high level introduction to some core tasks, ideas and methods of machine learning for readers who are familiar with at least some undergraduate mathematics. Advanced readers may get more from certain sections, but our goal is to present a self-contained picture which requires little knowledge beyond calculus in multiple variables and elementary linear algebra. For readers who have not taken many of the advanced classes, this may also motivate why certain fields of study are of interest in applications. One big omission in this article are deep neural networks, i.e. the models which underly 'deep learning.' Due to their special importance, they will be discussed in a separate companion article.

A simple algebraic derivation of the equation for a line of best fit

Abstract We present a simple algebraic derivation of the equation for a line of best fit. Unlike conventional derivations, which require multi-variable calculus, our argument relies on elementary algebra alone, and avoids more advanced notation (such as the sigma summation symbol) by exploiting routine data averages. In this way, our approach demystifies the process of computing a linear regression, and is likely to be of interest to students and instructors alike. We illustrate the main results with a worked example taken from an introductory physics context.

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.

Exploring fuzzy filters and their relationships on Sheffer stroke basic algebras

Exploring fuzzy filters and their relationships on Sheffer stroke basic algebras

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.

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.

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.