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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]

Abstract: We show that paper-folding words contain arbitrarily large abelian powers. [Copyright &y& Elsevier]

This research addresses the issue of how to support students' representational fluency—the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper‐and‐pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth‐grade algebra students, and link to results of semi‐structured interviews with students before and after the experiment. Results of analyzing the five‐week experiment include instructional supports for students' representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students' change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students' persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool‐based representations. [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]

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]

University scientific research ability is an important indicator to express the strength of universities. In this paper, the evaluation of university scientific research ability is investigated based on the output of sci-tech papers. Four university alliances from North America, UK, Australia, and China, are selected as the case study of the university scientific research evaluation. Data coming from Thomson Reuters InCites are collected to support the evaluation. The work has contributed new framework to the issue of university scientific research ability evaluation. At first, we have established a hierarchical structure to show the factors that impact the evaluation of university scientific research ability. Then, a new MCDM method called D-AHP model is used to implement the evaluation and ranking of different university alliances, in which a data-driven approach is proposed to automatically generate the D numbers preference relations. Next, a sensitivity analysis has been given to show the impact of weights of factors and sub-factors on the evaluation result. At last, the results obtained by using different methods are compared and discussed to verify the effectiveness and reasonability of this study, and some suggestions are given to promote China’s scientific research ability. [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 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]

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]

In 1993, I. P. Shestakov posed the question of the existence of a central simple finite-dimensional algebra over a field of characteristic zero whose identities are not given by a finite set of identities. In 2012, I. M. Isaev and the author of the present paper constructed an example of a seven-dimensional central simple algebra over any field that does not have a finite basis of identities. In the present paper, we construct an example of a six-dimensional central simple algebra over a field of characteristic zero which has no finite basis of identities. [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]

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]

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]

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]

This paper focuses on the study and investigation of quotient Riesz algebras, examining their properties and exploring various topologies associated with them. The aim is to gain a better understanding of the behavior and characteristics of these quotient Riesz algebras in relation to their topological structures [ABSTRACT FROM AUTHOR]

In this paper, by using the structure properties of algebras and the technique of matrix operation, we investigate the structure of local Lie triple derivations on triangular algebras. Under some mild conditions, we prove that each local Lie triple derivation of triangular algebras can be expressed as the sum of a derivation and a linear map from the triangular algebra to its center that vanishes at Lie triple products. Finally, we apply the main result to the problem of characterizing local Lie triple derivations on an upper triangular matrix algebra. [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]

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]

In the present paper, we review the progress of the project of the classification and construction of invariant differential operators for non-compact, semisimple Lie groups. Our starting point is the class of algebras which we called earlier 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this purpose, we introduced recently the new notion of a parabolic relation between two non-compact, semi-simple Lie algebras G and G ′ that have the same complexification and possess maximal parabolic subalgebras with the same complexification. [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]

A new three-dimensional Lie algebra and its loop algebra are proposed by us, whose commutator is a vector product. Based on this, a positive flow and a negative flow are obtained by introducing a new kind of spectral problem expressed by the vector product, which reduces to a generalized KdV equation, a generalized Schrödinger equation, a sine-Gordon equation, and a sinh-Gordon equation. Next, the well-known Tu scheme is generalized for generating isospectral integrable hierarchies and non-isospectral integrable hierarchies. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy presented in the paper is worded out. Finally, we further enlarge the three-dimensional loop algebra into a six-dimensional loop algebra so that a new isospectral integrable hierarchy which is a type of extended integrable model is produced whose bi-Hamiltonian structure is also derived from the vector-product trace identity. This new approach presented in the paper possesses extensive applications in the aspect of generating integrable hierarchies of evolution equations. [ABSTRACT FROM AUTHOR]

Abstract: The paper gives proofs of some results just claimed in [R. Massy, Galois averages, J. Number Theory 113 (2005) 244–275]. For instance, it is proved that for a finite non-trivial separable extension , , of Hilbertian fields finitely generated over their prime field, the quotient group , for the corresponding multiplicative groups of non-zero elements, cannot be a torsion group of finite exponent. [Copyright &y& Elsevier]

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer's cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a "cellular algebra Clifford theory" for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G(\ell,p,n) are graded skew cellular algebras. In the special case when p = 2 this implies that the Hecke algebras of type G(\ell,2,n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck's result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G(\ell,p,n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of "adjustment matrices" in positive characteristic. [ABSTRACT FROM AUTHOR]

In the present paper, we examined the extendibility of evolution subalgebras generated by idempotents of evolution algebras. The extendibility of the isomorphism of such subalgebras to the entire algebra was investigated. Moreover, the existence of an evolution algebra generated by arbitrary idempotents was also studied. Furthermore, we described the tensor product of algebras generated by arbitrary idempotents and found the conditions of the tensor decomposability of four-dimensional S-evolution algebras. This paper's findings shed light on the field of algebraic structures, particularly in studying evolution algebras. By examining the extendibility of evolution subalgebras generated by idempotents, we provide insights into the structural properties and relationships within these algebras. Understanding the isomorphism of such subalgebras and their extension allows a deeper comprehension of the overall algebraic structure and its behaviour. [ABSTRACT FROM AUTHOR]

In this paper, we study conformal points among the class of E -models. The latter are σ-models formulated in terms of a current Poisson algebra, whose Lie-theoretic definition allows for a purely algebraic description of their dynamics and their 1-loop RG-flow. We use these results to formulate a simple algebraic condition on the defining data of such a model which ensures its 1-loop conformal invariance and the decoupling of its observables into two chiral Poisson algebras, describing the classical left- and right-moving fields of the theory. In the case of so-called non-degenerate E -models, these chiral sectors form two current algebras and the model takes the form of a WZW theory once realised as a σ-model. The case of degenerate E -models, in which a subalgebra of the current algebra is gauged, is more involved: the conformal condition yields a wider class of theories, which includes gauged WZW models but also other examples, seemingly different, which however sometimes turn out to be related to gauged WZW models based on other Lie algebras. For this class, we build non-local chiral fields of parafermionic-type as well as higher-spin local ones, forming classical W -algebras. In particular, we find an explicit and efficient algorithm to build these local chiral fields. These results (and their potential generalisations discussed at the end of the paper) open the way for the quantisation of a large class of conformal E -models using the standard operator formalism of two-dimensional CFT. [ABSTRACT FROM AUTHOR]

Let H be a complex Hilbert space and let B (H) be the algebra of all bounded linear operators on H. In this paper, for T ∈ B (H) and a unit vector x ∈ H , we introduce a local version of the reduced minimum modulus of T at x, noted by γ (T , x) . Properties of this quantity are investigated. We study the relations between γ (T , x) and the Moore–Penrose inverse, spectrum of | T | and the local spectrum of | T | at x. At the end of this paper we will be interested in several problems around this quantity (preserving, continuity, local spectral theory). [ABSTRACT FROM AUTHOR]

Abstract: We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials. [Copyright &y& Elsevier]

Let A be a finite dimensional hereditary algebra over an algebraically closed field k. In this paper, we study the tilting quiver of A from the viewpoint of τ -tilting theory. First, we prove that there exists an isomorphism between the support τ -tilting quiver Q(s τ -tilt A) of A and the tilting quiver Q(tilt A ¯ ) of the duplicated algebra A ¯ . Then, we give a new method to calculate the number of arrows in the tilting quiver Q(tilt A) when A is representation-finite. Finally, we study the conjecture given by Happel and Unger, which claims that each connected component of Q(tilt A) contains only finitely many non-saturated vertices. We provide an example to show that this conjecture does not hold for some algebras whose quivers are wild with at least four vertices. [ABSTRACT FROM AUTHOR]

In this paper, we propose the notion of a transposed Poisson superalgebra.We prove that a transposed Poisson superalgebra can be constructed by means of a commutative associative superalgebra and an even degree derivation of this algebra. Making use of this, we construct two examples of the transposed Poisson superalgebra. One of them is the graded differential algebra of differential forms on a smooth finite dimensional manifold, where we use the Lie derivative as an even degree derivation. The second example is the commutative superalgebra of basic fields of the quantum Yang--Mills theory, where we use the BRST-supersymmetry as an even degree derivation to define a graded Lie bracket. We show that a transposed Poisson superalgebra has six identities that play an important role in the study of the structure of this algebra. [ABSTRACT FROM AUTHOR]

This study introduces an ample Hausdorff groupoid Aˆ ⋊ R extracted from an ample Hausdorff groupoid G and a unital commutative ring R; a Hausdorff groupoid D which is the discrete twist over Aˆ ⋊ R. In the groupoid C*-algebra perspective, when R = C there is an isomorphism between the non-twisted groupoid C*-algebra (C∗(G)) and the twisted groupoid C*-algebra (C∗(Aˆ ⋊ R;D)). However, in this paper, in a purely algebraic setting, the non-twisted Steinberg algebra (AR(G)) and the twisted Steinberg algebra (AR(D;Aˆ⋊R)) are non-isomorphic. [ABSTRACT FROM AUTHOR]

In the present paper, we apply the concept of reverse derivation in rings on the concept of d - algebra to obtain the concept called a left-right (resp. right-left) reverse derivations of d-algebra X (briefly, (l, r) resp. (r, l)-reverse derivation of d-algebra), we will also, define some concepts such as regular map, composition two maps and study the related properties. Moreover, the notions of partial ordered edge d - algebra as well as d - subalgebra and their relation to our current study are obtained. In addition, some illustrative examples and counterexamples are discussed. [ABSTRACT FROM AUTHOR]

The aim of the paper is to construct a secondary characteristic homomorphism for Lie pseudoalgebras. The case of inner product modules is under consideration. [ABSTRACT FROM AUTHOR]

In this paper we present the Hietarinta Chern–Simons supergravity theory in three space-time dimensions which extends the simplest Poincaré supergravity theory. After approaching the construction of the action using the Chern–Simons formalism, the analysis of the corresponding asymptotic symmetry algebra is considered. For this purpose, we first propose a consistent set of asymptotic boundary conditions for the aforementioned supergravity theory whose underlying symmetry corresponds to the supersymmetric extension of the Hietarinta algebra. We then show that the corresponding charge algebra contains the super- bms 3 algebra as subalgebra, and has three independent central charges. We also show that the obtained asymptotic symmetry algebra can alternatively be recovered as a vanishing cosmological constant limit of three copies of the Virasoro algebra, one of which is augmented by supersymmetry. [ABSTRACT FROM AUTHOR]

Based on the notion of full transformations with fixed set, in this paper, we present a novel concept of n-ary Fix(In,Y)-full terms. This term can be considered as a generalization of strongly full terms, permutational full terms and full terms. Together with the superposition operation, one can form a Menger algebra of rank n. The freeness of such algebra with respect to a variety of algebras of the same types is discussed. Furthermore, we apply hypersubstitution theory to define a Fix(In,Y)-full closed identity, a Fix(In,Y)-full closed variety and present some concrete examples. [ABSTRACT FROM AUTHOR]

The main goal of this paper was to develop the structure theory of Hom-Lie superalgebras in characteristic 2. We discuss their representations, semidirect product, and α k -derivations and provide a classification in low dimension. We introduce another notion of restrictedness on Hom-Lie algebras in characteristic 2, different from the one given by Guan and Chen. This definition is inspired by the process of the queerification of restricted Lie algebras in characteristic 2. We also show that any restricted Hom-Lie algebra in characteristic 2 can be queerified to give rise to a Hom-Lie superalgebra. Moreover, we developed a cohomology theory of Hom-Lie superalgebras in characteristic 2, which provides a cohomology of ordinary Lie superalgebras. Furthermore, we established a deformation theory of Hom-Lie superalgebras in characteristic 2 based on this cohomology. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of single-valued neutrosophic hyper BCKsubalgebras as a generalization and alternative of hyper BCK-algebras and on any given nonempty set constructs at least one single-valued neutrosophic hyper BCK-subalgebra and one a single-valued neutrosophic hyper BCK-ideal. In this study level subsets play the main role in the connection between singlevalued neutrosophic hyper BCK-subalgebras and hyper BCK-subalgebras and the connection between single-valued neutrosophic hyper BCK-ideals and hyper BCK-ideals. The congruence and (strongly) regular equivalence relations are the important tools for connecting hyperstructures and structures, so the major contribution of this study is to apply and introduce a (strongly) regular relation on hyper BCK-algebras and to investigate their categorical properties (quasi commutative diagram) via single-valued neutrosophic hyper BCK-ideals. Indeed, by using the single-valued neutrosophic hyper BCK-ideals, we define a congruence relation on (weak commutative) hyper BCK-algebras that under some conditions is strongly regular and the quotient of any (single-valued neutrosophic)hyper BCK-(sub)algebra via this relation is a (single-valued neutrosophic)(hyper BCKsubalgebra) BCK-(sub)algebra. [ABSTRACT FROM AUTHOR]

In this paper, we give a basic structure theorem based on the study of extreme cases for the value of ≺ (the classical precedence relation between ultrafilters), i.e., ≺ = ∅ and no isolated element in ≺. This gives rise, respectively, to the temporal varieties O and W , with the result that O generates a variety of temporal algebras. We also characterize the simple temporal algebras by means of arithmetical properties related to basical temporal operators; we conclude that the simplicity of the temporal algebra lies in being able to make 0 any element less than 1 by repeated application to it of the L operator. We then present an algebraic construction similar to a product but in which the temporal operations are not defined componentwise. This new "product" is shown to be useful in the study of algebra order and finding of bounds by means of something similar to a lifting process. Finally, we give an alternative proof of an already known result on atoms counting in free temporal algebras. [ABSTRACT FROM AUTHOR]

Similar to gravity, an infinite tower of symmetries generated by higher-spin charges has been identified in Yang-Mills theory by studying collinear limits or celestial operator products of gluons. This work aims to recover this loop symmetry in terms of charge aspects constructed on the gluonic Fock space. We propose an explicit construction for these higher spin charge aspects as operators which are polynomial in the gluonic annihilation and creation operators. The core of the paper consists of a proof that the charges we propose form a closed loop algebra to quadratic order. This closure involves using the commutator of the cubic order expansion of the charges with the linear (soft) charge. Quite remarkably, this shows that this infinite-dimensional symmetry constrains the non-linear structure of Yang-Mills theory. We provide a similar all spin proof in gravity for the so-called global quadratic (hard) charges which form the loop wedge subalgebra of w1+∞. [ABSTRACT FROM AUTHOR]

Let be an embedded projective variety. The Lie algebra defined by the tangent directions to at smooth points is an interesting algebraic invariant of . In some cases, this algebra is isomorphic to the symbol algebra of a filtered system of distributions on a Fano manifold, which plays an important role in the theory of these manifolds. In addition, algebras defined by tangent directions are interesting on their own right. In this paper, we study the Lie algebra corresponding to a variety that is the projectivization of the orbit of the lowest weight vector of an irreducible representation of a complex semisimple Lie group. We describe these algebras in terms of generators and relations. In many cases, we can describe their structure completely. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the low dimensional cohomology of metric Hom-Jacobi{Jordan algebras. We establish one-toone correspondence between the equivalence classes of abelian quadratic extensions of a Hom-Jacobi{Jordan algebra and its second cohomology group. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the notion of a mock- Lie bialgebra which is equivalent to a Manin triple of mock-Lie algebras. The study of a special case called coboundary mock-Lie bialgebra leads to introducing the mock-Lie Yang{Baxter equation on a mock-Lie algebra which is an analogue of the classical Yang{Baxter equation on a Lie algebra. Note that a skew-symmetric solution of mock-Lie Yang{Baxter equation gives a mock-Lie bialgebra. Finally, O-operators are studied to construct a skew-symmetric solution of a mock-Lie Yang{Baxter equation. [ABSTRACT FROM AUTHOR]

Hom–Lie algebras are generalizations of Lie algebras that arise naturally in the study of nonassociative algebraic structures. In this paper, the concepts of solvable and nilpotent Hom–Lie algebras are studied further. In the theory of groups, investigations of the properties of the solvable and nilpotent groups are well-developed. We establish a theory of the solvable and nilpotent Hom–Lie algebras analogous to that of the solvable and nilpotent groups. We also provide examples to illustrate our results and discuss possible directions for further research. Dedicated to Al Farouk School & Kinder garten-Irbid-Jordan [ABSTRACT FROM AUTHOR]

Let H denote the quaternion algebra. This paper investigates the generalized complementary covariance, which is the ϕ -Hermitian quaternion matrix. We give the properties of the generalized complementary covariance matrices. In addition, we explore the unitary diagonalization of the covariance and generalized complementary covariance. Moreover, we give the generalized quaternion unitary transform algorithm and test the performance by numerical simulation. [ABSTRACT FROM AUTHOR]

Let K 〈 X d 〉 be the free associative algebra of rank d ≥ 2 over a field, K. In 1936, Wolf proved that the algebra of symmetric polynomials K 〈 X d 〉 Sym (d) is infinitely generated. In 1984 Koryukin equipped the homogeneous component of degree n of K 〈 X d 〉 with the additional action of Sym (n) by permuting the positions of the variables. He proved finite generation with respect to this additional action for the algebra of invariants K 〈 X d 〉 G of every reductive group, G. In the first part of the present paper, we established that, over a field of characteristic 0 or of characteristic p > d , the algebra K 〈 X d 〉 Sym (d) with the action of Koryukin is generated by (noncommutative version of) the elementary symmetric polynomials. Now we prove that if the field, K, is of positive characteristic at most d then the algebra K 〈 X d 〉 Sym (d) , taking into account that Koryukin's action is infinitely generated, describe a minimal generating set. [ABSTRACT FROM AUTHOR]

In this paper, we present a new class of Leonardo hybrid numbers that incorporate quantum integers into their components. This advancement presents a broader generalization of the q-Leonardo hybrid numbers. We explore some fundamental properties associated with these numbers. Moreover, we study special Leonardo quaternions over finite fields. In particular, we determine the Leonardo quaternions that are zero divisors or invertible elements in the quaternion algebra over the finite field Z p for special values of prime integer p. [ABSTRACT FROM AUTHOR]

We perform a complete and systematic analysis of the solution space of six-dimensional Einstein gravity. We show that a particular subclass of solutions — those that are analytic near I + — admit a non-trivial action of the generalised Bondi-Metzner-van der Burg-Sachs (GBMS) group which contains infinite-dimensional supertranslations and superrotations. The latter consists of all smooth volume-preserving Diff×Weyl transformations of the celestial S4. Using the covariant phase space formalism and a new technique which we develop in this paper (phase space renormalization), we are able to renormalize the symplectic potential using counterterms which are local and covariant. The Hamiltonian charges corresponding to GBMS diffeomorphisms are non-integrable. We show that the integrable part of these charges faithfully represent the GBMS algebra and in doing so, settle a long-standing open question regarding the existence of infinite-dimensional asymptotic symmetries in higher even dimensional non-linear gravity. Finally, we show that the semi-classical Ward identities for supertranslations and superrotations are precisely the leading and subleading soft-graviton theorems respectively. [ABSTRACT FROM AUTHOR]

Preconcepts are basic units of knowledge that form the basis of formal concepts in formal concept analysis (FCA). This paper investigates the relations among different kinds of preconcepts, such as protoconcepts, meet and join-semiconcepts and formal concepts. The first contribution of this paper, is to present a fuzzy powerset lattice gradation, that coincides with the preconcept lattice at its 1-cut. The second and more significant contribution, is to introduce a preconcept algebra gradation that yields different algebras for protoconcepts, semiconcepts, and concepts at different cuts. This result reveals new insights into the structure and properties of the different categories of preconcepts. [ABSTRACT FROM AUTHOR]

In this paper, we construct a symbol calculus yielding short exact sequences for the dual Toeplitz algebra generated by all bounded dual Toeplitz operators on the Hardy space associated with the polydisk D n in the unitary space C n , that have been introduced and well studied in our earlier paper (Benaissa and Guediri in Taiwan J Math 19: 31–49, 2015), as well as for the C*-subalgebra generated by dual Toeplitz operators with symbols continuous on the associated hypertorus T n . [ABSTRACT FROM AUTHOR]