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High school math projects center on solving real-world examples in a wide variety of situations that require a creative skill set. Creativity in high school mathematics takes place in a subtle form. In contrast, creativity presents itself conspicuously in art. Therefore, this paper will explore the utilization of polynomial equations in drawing various figures as part of algebra class projects. Additionally, these projects incorporate integrated digital tools in mathematics. The paper will analyze three algebraic projects: the first project involves drawing a bird, the second project focuses on drawing a butterfly, and the third project entails drawing a bat using polynomial equations with the technological tool Desmos. All three projects are products of student work that encompass figures requiring up to forty polynomial equations. The students' impressive work demonstrates the power of mathematics in pushing the boundaries of other subjects. As digital technology becomes increasingly integral, mathematical tools are positioned as universal tools for the future and are poised to rival artists in producing visual art. This raises the intriguing question: can a mathematician create better art with mathematical equations through digital technology than an artist?

The paper describes the results of a study whose main aim was to find the interrelationship between pupils' school grades in Czech language (native), mathematics, and physics and pupils' cognitive predispositions to problem-solving in science and mathematics diagnosed by the Lawson Classroom Test of Scientific Reasoning and the Culture of Problem Solving test. A total of 180 pupils from the Czech Republic aged 14-15 took part in this study. The results show that pupils with better grades in the monitored subjects achieve better results in both tests. It also turns out that there are generally statistically insignificant differences between the results of pupils assessed by grades 1 or 2 and between those assessed by grades 3 or 4. Pupils' performance in the two tests might help to strengthen the objectivization of grading at school. They might also help identify the indicators important for developing problem-solving skills. The research specifically points to the need to develop algebraic thinking, the conception of infinity, spatial imagination, geometric imagination in the plane, proportional reasoning, and the ability to control variables.

Gaming the system is a persistent problem in Computer-Based Learning Platforms. While substantial progress has been made in identifying and understanding such behaviors, effective interventions remain scarce. This study uses a method of causal moderation known as Fully Latent Principal Stratification to explore the impact of two types of interventions--gamification and manipulation of assistance access on the learning outcomes of students who tend to game the system. The results indicate that gamification does not consistently mitigate these negative behaviors. One gamified condition had a consistently positive effect on learning regardless of students' propensity to game the system, whereas the other had a negative effect on gamers. However, delaying access to hints and feedback may have a positive effect on the learning outcomes of those gaming the system. This paper also illustrates the potential for integrating detection and causal methodologies within educational data mining to evaluate effective responses to detected behaviors.

Nowadays, researchers are very interested in studying various Diophantine equations due to their importance in Cryptography, Chemistry, Knot Theory, Astronomy, Geometry, Trigonometry, Biology, Algebra, Electrical Engineering, Economics, and Astrology. The present paper is about the non-negative integer solution of the exponential Diophantine equation nx + 43y = z², where are non-negative integers, is a positive integer with nx + 43y = z², where n = 2 (mod 129) and n + 1 and is not a perfect square. The authors use the famous Catalan conjecture for this purpose. Results of the present paper indicate that 2, 3, 0, and 3 are the only required values of and respectively, that satisfy the exponential Diophantine equation, where are non-negative integers, is a positive integer with nx + 43y = z², where n = 2 (mod 129) and n + 1 and is not a perfect square. The present technique of this paper proposes a new approach to solving the Diophantine equations, which is the main scientific contribution of this study, and it is very beneficial, especially for researchers, scholars, academicians, and people interested in the same field. [ABSTRACT FROM AUTHOR]

In this paper, we discuss some extensions of the Maligranda–Orlicz lemma. It deals with the problem of constructing a norm in a subspace of the space of bounded functions, for which it becomes a normed algebra so that the norm introduced is equivalent to the initial norm of the subspace. This is done by satisfying some inequality between these norms. We show in this paper how this inequality is relevant to the study of operator equations in Banach algebras. In fact, we study how to equip a subspace of the space of bounded functions with a norm equivalent to a given one so that it is a normed algebra. We give a general condition for the construction of such norms, which allows us to easily check whether a space with a given norm is an algebra with a pointwise product and the consequences of such a choice for measures of noncompactness in such spaces. We also study quasi-normed spaces. We introduce a general property of measures of noncompactness that allows the study of quadratic operator equations, prove a fixed-point theorem suitable for such problems, and complete the whole with examples and applications. [ABSTRACT FROM AUTHOR]

Consider an infinite-dimentional Banach space denoted as X, and designate ∂(X) as the algebra of all bounded linear operators on X. Moreover, let s (A) denote the spectrum of ..., and ...(A)) indicate the boundary of s (A). A map ... is termed a ρ-spectrum if ... (A) for all A ... In this paper, we characterize all surjective maps f1 and f2 on ρ(X) satisfying ... = (AB) for all A,B(X). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the necessary and sufficient conditions for solving a dual split quaternion matrix equation A X B   =   C , and present the general solution expression when the solvability conditions are met. As an application, we delve into the necessary and sufficient conditions for the existence of a Hermitian solution to this equation by using a newly defined real representation method. Furthermore, we obtain the solutions for the dual split quaternion matrix equations A X   =   C and X B   =   C . Finally, we provide a numerical example to demonstrate the findings of this paper. [ABSTRACT FROM AUTHOR]

The notion of a transposed Poisson n -Lie algebra has been developed as a natural generalization of a transposed Poisson algebra. It was conjectured that a transposed Poisson n -Lie algebra with a derivation gives rise to a transposed Poisson (n + 1) -Lie algebra. In this paper, we focus on transposed Poisson n -Lie algebras. We have obtained a rich family of identities for these algebras. As an application of these formulas, we provide a construction of (n + 1) -Lie algebras from transposed Poisson n -Lie algebras with derivations under a certain strong condition, and we prove the conjecture in these cases. The notion of a transposed Poisson -Lie algebra has been developed as a natural generalization of a transposed Poisson algebra. It was conjectured that a transposed Poisson -Lie algebra with a derivation gives rise to a transposed Poisson -Lie algebra. In this paper, we focus on transposed Poisson -Lie algebras. We have obtained a rich family of identities for these algebras. As an application of these formulas, we provide a construction of -Lie algebras from transposed Poisson -Lie algebras with derivations under a certain strong condition, and we prove the conjecture in these cases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept and representations of modified λ-differential Hom-Lie algebras. We then develop the cohomology of modified λ-differential Hom-Lie algebras with coefficients in a suitable representation. As applications, abelian extensions and skeletal modified λ-differential Hom-Lie 2-algebras are characterized in terms of cohomology groups. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of a Quasi B-algebra, building upon the foundational work on Post algebras by G. Epstein and the investigation of semi B-algebras. We systematically explore the properties of Quasi B-algebras, establishing equivalent conditions for the transition from a semi B-algebra to a Quasi B-algebra. [ABSTRACT FROM AUTHOR]

This is a noticeably short biography and introductory paper on multiplier Hopf algebras. It delves into questions regarding the significance of this abstract construction and the motivation behind its creation. It also concerns quantum linear groups, especially the coordinate ring of Mq(n) and the observation that K [Mq(n)] is a quadratic algebra, and can be equipped with a multiplier Hopf ∗-algebra structure in the sense of quantum permutation groups developed byWang and an observation by Rollier–Vaes. In our next paper, we will propose the study of multiplier Hopf graph algebras. The current paper can be viewed as a precursor to this upcoming work, serving as a crucial intermediary bridging the gap between the abstract concept of multiplier Hopf algebras and the well-developed field of graph theory, thereby establishing connections between them! This survey review paper is dedicated to the 78th birthday anniversary of Professor Alfons Van Daele. [ABSTRACT FROM AUTHOR]

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Let K be a field and S = K[x1,..., xn] a standard polynomial ring over K. In this paper, we give new combinatorial algorithms to compute the smallest t-spread lexicographic set and the smallest t-spread strongly stable set containing a given set of t-spread monomials of S. Some technical tools allowing to compute the cardinality of t-spread strongly stable sets avoiding their construction are also presented. Such functions are also implemented in a Macaulay2 package, TSpreadIdeals, to ease the computation of well-known results about algebraic invariants for t-spread ideals. [ABSTRACT FROM AUTHOR]

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander's correspondence, classical Schur–Weyl duality and Soergel's Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as the Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, B, and module categories of endomorphism algebras of generator-relative cogenerators over B which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree n so that the endomorphism algebra of a generator-cogenerator provides an n-faithful cover, in the sense of Rouquier, of B. The degree n is known as the Hemmer–Nakano dimension of the standard modules. We prove that, in general, the Hemmer–Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra B is bounded above by the number of non-isomorphic simple modules of B. We establish methods for reducing computations of Hemmer–Nakano dimensions in the integral setup to computations of Hemmer–Nakano dimensions over finite-dimensional algebras, and vice-versa. In addition, we extend the framework to study Hemmer–Nakano dimensions of arbitrary resolving subcategories. In this setup, we find that the relative dominant dimension over (projective) Noetherian algebras is an important tool in the computation of these degrees, extending the previous work of Fang and Koenig. In particular, this theory allows us to derive results for Schur algebras and the BGG category \mathcal {O} in the integral setup from the finite-dimensional case. More precisely, we use the relative dominant dimension of Schur algebras to completely determine the Hemmer–Nakano dimension of standard modules with respect to Schur functors between module categories of Schur algebras over regular Noetherian rings and module categories of group algebras of symmetric groups over regular Noetherian rings. We exhibit several structural properties of deformations of the blocks of the Bernstein-Gelfand-Gelfand category \mathcal {O} establishing an integral version of Soergel's Struktursatz. We show that deformations of the combinatorial Soergel's functor have better homological properties than the classical one. [ABSTRACT FROM AUTHOR]

In this paper, we make use of the simultaneous decomposition of eight quaternion matrices to study the solvability conditions and general solutions to a system of two-sided coupled Sylvester-type quaternion matrix equations A i X i C i + B i X i + 1 D i = Ω i , i = 1 , 2 , 3 , 4. We design an algorithm to compute the general solution to the system and give a numerical example. Additionally, we consider the application of the system in the encryption and decryption of color images. [ABSTRACT FROM AUTHOR]

Let A be a unital A-algebra containing a non-trivial projection. In this paper, we prove that if a map Ω: A → A such that Ω([K, F]A ⊙ D) = [Ω(K), F]A ⊙ D + [K, Ω(F)]A ⊙ D + [K, F]A ⊙ Ω(D), where [K, F]A = KF - FKA and K ⊙ F = KAF + FKA for all K, F, D ∈ A, then Ω is an additive A-derivation. Furthermore, we extend its results on factor von Neumann algebras, standard operator algebras and prime A-algebras. Additionally, we provide an example illustrating the existence of such maps. [ABSTRACT FROM AUTHOR]

The goal of the present paper is to provide a cohomology theory and crossed modules of modified Rota–Baxter pre-Lie algebras. We introduce the notion of a modified Rota–Baxter pre-Lie algebra and its bimodule. We define a cohomology of modified Rota–Baxter pre-Lie algebras with coefficients in a suitable bimodule. Furthermore, we study the infinitesimal deformations and abelian extensions of modified Rota–Baxter pre-Lie algebras and relate them with the second cohomology groups. Finally, we investigate skeletal and strict modified Rota–Baxter pre-Lie 2-algebras. We show that skeletal modified Rota–Baxter pre-Lie 2-algebras can be classified into the third cohomology group, and strict modified Rota–Baxter pre-Lie 2-algebras are equivalent to the crossed modules of modified Rota–Baxter pre-Lie algebras. [ABSTRACT FROM AUTHOR]

Quantum groups occupy a significant position in both mathematics and physics, contributing to progress in these fields. It is interesting to obtain new quantum groups by the quantization of Lie bialgebras. In this paper, the quantization of the rank two Heisenberg–Virasoro algebra by Drinfel'd twists is presented, Lie bialgebra structures of which have been investigated by the authors recently. [ABSTRACT FROM AUTHOR]

In this paper, we give another definition of Ruston elements and almost Ruston elements, which is equivalent to the definitions given by Mouton and Raubenheimer in the case that the homomorphism has a closed range and Riesz property. For two homomorphisms, we consider the preserver problems of Fredholm theory and Fredholm spectrum theory. In addition, we study the spectral mapping theorems of Fredholm (Weyl, Browder, Ruston, and almost Ruston) elements relative to a homomorphism. Last but not least, the dependence of Fredholm theory on three homomorphisms is considered, and meanwhile, the transitivity of Fredholm theory relative to three homomorphisms is illustrated. Furthermore, we consider the Fredholm theory relative to more homomorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the concept of hyper-instability as a strong version of multiplicative instability. This concept provides a powerful tool to study the multiplicative instability of Banach algebras. It replaces the condition of the iterated limits in the definition of multiplicative instability with conditions that are easier to examine. In particular, special conditions are suggested for Banach algebras that admit bounded approximate identities. Moreover, these conditions are preserved under isomorphisms. This enlarges the class of studied Banach algebras. We prove that many interesting Banach algebras are hyper-unstable, such as C-algebras, Fourier algebras, and the algebra of compact operators on Banach spaces, each under certain conditions. [ABSTRACT FROM AUTHOR]

Convolution plays a pivotal role in the domains of signal processing and optics. This paper primarily focuses on studying the weighted convolution for quaternion linear canonical cosine transform (QLCcT) and its application in multiplicative filter analysis. Firstly, we propose QLCcT by combining quaternion algebra with linear canonical cosine transform (LCcT), which extends LCcT to Hamiltonian quaternion algebra. Secondly, we introduce weighted convolution and correlation operations for QLCcT, accompanied by their corresponding theorems. We also explore the properties of QLCcT. Thirdly, we utilize these proposed convolution structures to analyze multiplicative filter models that offer lower computational complexity compared to existing methods based on quaternion linear canonical transform (QLCT). Additionally, we discuss the rationale behind studying such transforms using quaternion functions as an illustrative example. [ABSTRACT FROM AUTHOR]

One of the most relevant topics in web theory is linearization. A particular class of linearizable webs is the Grassmannizable web. Akivis gave a characterization of such a web, showing that Grassmannizable webs are equivalent to isoclinic and transversally geodesic webs. The obstructions given by Akivis that characterize isoclinic and transversally geodesic webs are computed locally, and it is difficult to give them an interpretation in relation to torsion or curvature of the unique Chern connection associated with a web. In this paper, using Nagy's web formalism, Frölisher—Nejenhuis theory for derivation associated with vector differential forms, and Grifone's connection theory for tensorial algebra on the tangent bundle, we find invariants associated with almost-Grassmann structures expressed in terms of torsion, curvature, and Nagy's tensors, and we provide an interpretation in terms of these invariants for the isoclinic, transversally geodesic, Grassmannizable, and parallelizable webs. [ABSTRACT FROM AUTHOR]

In this paper, we define a cohomology theory of a modified λ -differential left-symmetric algebra. Moreover, we introduce the notion of modified λ -differential left-symmetric 2-algebras, which is the categorization of a modified λ -differential left-symmetric algebra. As applications of cohomology, we classify linear deformations and abelian extensions of modified λ -differential left-symmetric algebras using the second cohomology group and classify skeletal modified λ -differential left-symmetric 2-algebra using the third cohomology group. Finally, we show that strict modified λ -differential left-symmetric 2-algebras are equivalent to crossed modules of modified λ -differential left-symmetric algebras. [ABSTRACT FROM AUTHOR]

In this paper, we provide some properties of k-potent elements in algebras obtained by the Cayley–Dickson process over Z p . Moreover, we find a structure of nonunitary ring over Fibonacci quaternions over Z 3 and we present a method to encrypt plain texts, by using invertible elements in some of these algebras. [ABSTRACT FROM AUTHOR]

In this paper we investigate left ideals as codes in twisted skew group rings. The considered rings, which are in most cases algebras over a finite field, allow us to retrieve many of the well-known codes. The presentation, given here, unifies the concept of group codes, twisted group codes and skew group codes. [ABSTRACT FROM AUTHOR]

In this paper, by using the notion of fuzzy points and equality algebras, the notions of fuzzy point equality algebra, equality-subalgebra, and ideal were established. Some characterizations of fuzzy subalgebras were provided by using such concepts. We defined the concepts of (∈, ∈) and (∈, ∈ ∨ q)-fuzzy ideals of equality algebras, discussed some properties, and found some equivalent definitions of them. In addition, we investigated the relation between different kinds of (α, β)-fuzzy subalgebras and (α, β)-fuzzy ideals on equality algebras. Also, by using the notion of (∈, ∈)-fuzzy ideal, we defined two equivalence relations on equality algebras and we introduced an order on classes of X, and we proved that the set of all classes of X by these order is a poset. [ABSTRACT FROM AUTHOR]

Suppose that S T 1 is a class of close-to-star functions. In this paper, we investigated the estimate of Zalcman functional on the logarithmic coefficients and the third Hankel determinant for the class S T 1 with the determinant entry of logarithmic coefficients. Also, we obtained the sharp bounds of Zalcman functional J 2 , 4 f and J 3 , 3 f for the class S T 1 . [ABSTRACT FROM AUTHOR]

To elucidate the combinatorial architecture of neural codes, the neural ideal , an algebraic object, was introduced. Represented in its canonical form, provides a succinct characterization of the inherent receptive field architecture within the code. The polynomials in are also instrumental in determining the relationships among the neurons' receptive fields. Consequently, the computation of the collection of canonical forms is pivotal. In this paper, based on the study of relations between pseudo-monomials, the authors present a computationally efficient iterative algorithm for the canonical forms of the neural ideal. Additionally, we introduce a new relationship among the neurons' receptive fields, which can be characterized by if-and-only-if statements, relating both to and to a larger ideal of a code . [ABSTRACT FROM AUTHOR]

In this paper we show how the Functional Separation of Variables (FSoV) method can be applied to the problem of computing overlaps with integrable boundary states in integrable systems. We demonstrate our general method on the example of a particular boundary state, a singlet of the symmetry group, in an su 3 rational spin chain in an alternating fundamental-anti-fundamental representation. The FSoV formalism allows us to compute in determinant form not only the overlaps of the boundary state with the eigenstates of the transfer matrix, but in fact with any factorisable state. This includes off-shell Bethe states, whose overlaps with the boundary state have been out of reach with other methods. Furthermore, we also found determinant representations for insertions of so-called Principal Operators (forming a complete algebra of all observables) between the boundary and the factorisable state as well as certain types of multiple insertions of Principal Operators. Concise formulas for the matrix elements of the boundary state in the SoV basis and su N generalisations are presented. Finally, we managed to construct a complete basis of integrable boundary states by repeated action of conserved charges on the singlet state. As a result, we are also able to compute the overlaps of all of these states with integral of motion eigenstates. [ABSTRACT FROM AUTHOR]

In the literature on Kleene algebra, a number of variants have been proposed which impose additional structure specified by a theory, such as Kleene algebra with tests (KAT) and the recent Kleene algebra with observations (KAO), or make specific assumptions about certain constants, as for instance in NetKAT. Many of these variants fit within the unifying perspective offered by Kleene algebra with hypotheses, which comes with a canonical language model constructed from a given set of hypotheses. For the case of KAT, this model corresponds to the familiar interpretation of expressions as languages of guarded strings. A relevant question therefore is whether Kleene algebra together with a given set of hypotheses is complete with respect to its canonical language model. In this paper, we revisit, combine and extend existing results on this question to obtain tools for proving completeness in a modular way. We showcase these tools by giving new and modular proofs of completeness for KAT, KAO and NetKAT, and we prove completeness for new variants of KAT: KAT extended with a constant for the full relation, KAT extended with a converse operation, and a version of KAT where the collection of tests only forms a distributive lattice. [ABSTRACT FROM AUTHOR]

In this paper, we first define the pre-Lie family algebra associated to a dendriform family algebra in the case of a commutative semigroup. Then we construct a pre-Lie family algebra via typed decorated rooted trees, and we prove the freeness of this pre-Lie family algebra. We also construct pre-Lie family operad in terms of typed labeled rooted trees, and we obtain that the operad of pre-Lie family algebras is isomorphic to the operad of typed labeled rooted trees, which generalizes the result of Chapoton and Livernet. In the end, we construct Zinbiel and pre-Poisson family algebras and generalize results of Aguiar. [ABSTRACT FROM AUTHOR]

Vibration measurements can be used to evaluate the operation status of power equipment and are widely applied in equipment quality inspection and fault identification. Event-sensing technology can sense the change in surface light intensity caused by object vibration and provide a visual description of vibration behavior. Based on the analysis of the principle underlying the transformation of vibration behavior into event flow data by an event sensor, this paper proposes an algorithm to reconstruct event flow data into a relationship correlating vibration displacement and time to extract the amplitude-frequency characteristics of the vibration signal. A vibration measurement test platform is constructed, and feasibility and effectiveness tests are performed for the vibration motor and other power equipment. The results show that event-sensing technology can effectively perceive the surface vibration behavior of power and provide a wide dynamic range. Furthermore, the vibration measurement and visualization algorithm for power equipment constructed using this technology offers high measurement accuracy and efficiency. The results of this study provide a new noncontact and visual method for locating vibrations and performing amplitude-frequency analysis on power equipment. [ABSTRACT FROM AUTHOR]

It has been shown that by using a Lagrange multiplier field to ensure that the classical equations of motion are satisfied, radiative effects beyond one-loop order are eliminated. It has also been shown that through the contribution of some additional ghost fields, the effective action becomes form invariant under a redefinition of field variables, and furthermore, the usual one-loop results coincide with the quantum corrections obtained from this effective action. In this paper, we consider the consequences of a gauge invariance being present in the classical action. The resulting gauge transformations for the Lagrange multiplier field as well as for the additional ghost fields are found. These gauge transformations result in a set of Faddeev–Popov ghost fields arising in the effective action. If the gauge algebra is closed, we find the Becci–Rouet–Stora–Tyutin (BRST) transformations that leave the effective action invariant. [ABSTRACT FROM AUTHOR]

This research article introduces and investigates the concepts of quasi covered ideals and quasi bases within the context of semigroup theory a fundamental field of study in algebra. Quasi covered ideals represent a novel subset of semigroups, offering a versatile perspective that extends beyond conventional ideals, enabling a more flexible analysis of semigroup structures. In this paper, we delve into the properties and attributes of quasi covered ideals, providing a comprehensive exploration of their characteristics. Additionally, we establish intricate relationship between covered ideals, the greatest ideal, quasi covered ideals, and quasi bases, shedding light on the interconnections among these fundamental elements within the realm of semigroup theory. [ABSTRACT FROM AUTHOR]

In this paper, the  (2 + 1) -dimensional seventh-order Caudrey–Dodd–Gibbon–KP equation is investigated through the Lie group method. The Lie algebra of infinitesimal symmetries, commutative and adjoint tables, and one-dimensional optimal systems is presented. Then, the seventh-order Caudrey–Dodd–Gibbon–KP equation is reduced to nine types of  (1 + 1) -dimensional equations with the help of symmetry subalgebras. Finally, the unified algebra method is used to obtain the soliton solutions, trigonometric function solutions, and Jacobi elliptic function solutions of the seventh-order Caudrey–Dodd–Gibbon–KP equation. [ABSTRACT FROM AUTHOR]

In this paper, first, we introduce the notion of post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and the fact that there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra. A novel property is that a cocommutative post-Hopf algebra gives rise to a generalized Grossman-Larson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions to the Yang-Baxter equation. Then, we introduce the notion of relative Rota-Baxter operator on Hopf algebras. A cocommutative post-Hopf algebra gives rise to a relative Rota-Baxter operator on its subadjacent Hopf algebra. Conversely, a relative Rota-Baxter operator also induces a post-Hopf algebra. Finally, we show that relative Rota-Baxter operators give rise to matched pairs of Hopf algebras. Consequently, post-Hopf algebras and relative Rota-Baxter operators give solutions to the Yang-Baxter equation in certain cocommutative Hopf algebras. [ABSTRACT FROM AUTHOR]

Using the first cohomology from the mirror Heisenberg–Virasoro algebra to the twisted Heisenberg algebra (as the mirror Heisenberg–Virasoro algebra module), in this paper, we determined the derivations on the mirror Heisenberg–Virasoro algebra. Based on this result, we proved that any two-local derivation on the mirror Heisenberg–Virasoro algebra is a derivation. All half-derivations are described, and as corollaries, we have descriptions of transposed Poisson structures and local (two-local) half-derivations on the mirror Heisenberg–Virasoro algebra. [ABSTRACT FROM AUTHOR]

The aim of the paper was to give a description of nonlinear Jordan triple derivable mappings on trivial extension algebras. We proved that every nonlinear Jordan triple derivable mapping on a 2 -torsion free ∗ -type trivial extension algebra is a sum of an additive derivation and an additive antiderivation. As an application, nonlinear Jordan triple derivable mappings on triangular algebras were characterized. [ABSTRACT FROM AUTHOR]

In this paper, we investigated the problem of describing the form of higher Jordan triple derivations on trivial extension algebras. We show that every higher Jordan triple derivation on a -torsion free -type trivial extension algebra is a sum of a higher derivation and a higher anti-derivation. As for its applications, higher Jordan triple derivations on triangular algebras are characterized. [ABSTRACT FROM AUTHOR]

In this paper, we define R-norm entropy and conditional R-norm entropy of partitions of algebraic structures, and we establish some of their basic properties. We show that the Shannon entropy and the conditional Shannon entropy of partitions can be derived from the R-norm entropy and the conditional R-norm entropy of partitions, respectively, by letting R tend to 1. Finally, using the notion of entropy for partitions, we define the R-norm entropy of a dynamical system. We prove that the R-norm entropies of isomorphic dynamical systems are equal. [ABSTRACT FROM AUTHOR]

Most students are aware of natural numbers, integers, rational numbers, irrational numbers, real numbers, pure imaginary numbers, and complex numbers. Unfortunately, far fewer have heard of algebraic and transcendental numbers and exponentially fewer know about transcendental numbers beyond that p and e are transcendental. Although almost all real numbers are transcendental, they remain virtually unstudied at the grades 9-16 level. This paper introduces transcendental numbers and follows with a novel approach to constructing Patterned Transcendental Numbers through techniques available to high school algebra students and beyond and provides two apps that will allow readers to create patterned transcendental numbers. This paper ends with student recreational investigations regarding developing transcendental numbers. [ABSTRACT FROM AUTHOR]

Uchino initiated the investigation of twisted Rota-Baxter operators on associative algebras. Relevant studies have been extensive in recent times. In this paper, we introduce the notion of a twisted Rota-Baxter operator on a Hom-Lie algebra. By utilizing higher derived brackets, we establish an explicit L∞-algebra whose Maurer-Cartan elements are precisely twisted Rota-Baxter operators on Hom-Lie algebras. Additionally, we employ Getzler's technique of twisting L∞-algebras to establish the cohomology of twisted Rota-Baxter operators. We demonstrate that this cohomology can be regarded as the Chevalley-Eilenberg cohomology of a specific Hom-Lie algebra with coefficients in an appropriate representation. Finally, we study the linear and formal deformations of twisted Rota-Baxter operators by using the cohomology defined above. We also show that the rigidity of a twisted Rota-Baxter operator can be derived from Nijenhuis elements associated with a Hom-Lie algebra. [ABSTRACT FROM AUTHOR]

This paper was devoted to the study of the so-called nonlinear higher Lie n-derivation of triangular algebras T, where n is a nonnegative integer greater than two. Under some mild conditions, we proved that every nonlinear higher Lie n-derivation by local actions on the triangular algebras is of a standard form. As an application, we gave a characterization of higher Lie n-derivation by local actions on upper triangular matrix algebras, block upper triangular matrix algebras and nest algebras, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of algebras Hn, which are generated by three generators x, y, z, with the following relations: (1) x2n = 1, y² = xy + y, xy = yx; and (2) z² = z, xz = zx = z, zy = 2z. First, it shows that Hn is a positively based algebra. Then, all the indecomposable modules of Hn are constructed. Additionally, it shows that the dimension of each indecomposable Hn-module is at most 2. Finally, all the left (right) cells and left (right) cell modules of Hn are described, and the decompositions of the decomposable left cell modules are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we extend fractional-order derivative for the shifted Vieta-Lucas polynomial to generalized-fractional integro-differential equations involving non-local boundary conditions using Galerkin method as transformation technique and obtained N - ⌈δ⌉ + 1 system of linear algebraic equations with λi, i = 0, . . ., N unknowns, together with ⌈δ⌉ non-local boundary conditions, we obtained (N + 1)- linear equations. The accuracy and effectiveness of the scheme was tested on some selected problems from the literature. Judging from the table of results and figures, we observed that the approximate solution corresponding to the problem that has exact solution in polynomial form gives a closed form solution while problem with non-polynomial exact solution gives better accuracy compared to the existing results. [ABSTRACT FROM AUTHOR]

The novel concept called neutrosophic set was launched to take care of indeterminate factors in real-life data. The hybrid model of neutrosophic set and soft set has been widely studied in different areas of algebra, especially in associative structures such as fields, groups, rings, and modules. In this current paper, the novel concept is further introduce to a non-associative structure termed Q-neutrosophic soft quasigroup (Q-NSG) and investigate its different algebraic properties of the quasigroups. We shown the conditions for the sets of a-level cut of Q-NSG to be subquasigroups, the condition for each set of subquasigroups of a quasigroup to be Q-level cut neutrosophic soft subquasigroup were established. It was shown that Q-NSG obeys alternative property and flexible law. In addition, We defined Q-neutrosophic soft loop and investigate some of its characteristics. In particular, it was shown that Q-neutrosophic soft loop obeys inverse, weak inverse and cross inverse properties. We established the condition for a Q-neutrosophic soft loop to obey antiautomorphic inverse, semi-automorphic inverse and super anti-automorphic inverse properties. The necessary and sufficient condition for Q-neutrosophic soft set under a loop (G, ◦,/, \) to be a Q-neutrosophic soft loop was also established. [ABSTRACT FROM AUTHOR]

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 C s 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. [ABSTRACT FROM AUTHOR]