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Motivated by the consideration of integrable systems in three spatial dimensions in Euclidean space with integrals quadratic in the momenta we classify three-dimensional Abelian subalgebras of quadratic elements in the universal enveloping algebra of the Euclidean algebra under the assumption that the Casimir invariant p â†' â‹... l â†' vanishes in the relevant representation. We show by means of an explicit example that in the presence of magnetic field, i.e. terms linear in the momenta in the Hamiltonian, this classification allows for pairs of commuting integrals whose leading order terms cannot be written in the famous classical form of Makarov et al [17]. We consider limits simplifying the structure of the magnetic field in this example and corresponding reductions of integrals, demonstrating that singularities in the integrals may arise, forcing structural changes of the leading order terms. [ABSTRACT FROM AUTHOR]

"Real Normed Algebras Revisited," the last paper of the late Gadi Moran, attempts to reconstruct the discovery of the complex numbers, the quaternions, and the octonions, as well as proofs of their properties, using only what was known to 19th-century mathematicians. In his research, Gadi had discovered simple and elegant proofs of the above-mentioned classical results using only basic properties of the geometry of Euclidean spaces and tools from high school geometry. His reconstructions underline an interesting connection between Euclidean geometry and these algebras, and avoid the advanced machinery used in previous derivations of these results. The goal of this article is to present Gadi's derivations in a way that is accessible to a wide audience of readers. [ABSTRACT FROM AUTHOR]

Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and to utilize the representation theory of the symmetric group. But this technique is unavailable in characteristic p > 0. An alternative method, which succeeds, is the process of "hiking" a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial in which Capelli polynomial is embedded, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht's conjecture for affine algebras over fields of positive characteristic. In this paper, we develop hiking further to nonhomogeneous polynomials, to apply to the "representability question." Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable [A. Belov, Local finite basis property and local representability of varieties of associative rings, Izv. Russian Acad. Sci. (1) (2010) 3–134. English Translation Izv. Math. 74(1) (2010) 1–126]. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht's conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In Appendix A, we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems. [ABSTRACT FROM AUTHOR]

Let k be a field, let n ≥ 2 be a nonsquarefree integer not divisible by the characteristic of k. Assume that all roots of unity of degree n are contained in k. In the first part of the paper we consider pairs of symbol algebras over k with common slots D 1 ≃ (e , x) n ≃ (r , u) n , D 2 ≃ (e , y) n ≃ (r , v) n , exp D 1 = exp D 2 = n , and show that in general (e , x , y) n ≠ (r , u , v) n. As a consequence we prove that in general it is impossible to connect the pair { (e , x) n ; (e , y) n } and the pair { (r , u) n ; (r , v) n } by a chain of pairs of symbol algebras with a common slot and isomorphic to (D 1 ; D 2) in such a way that any two neighboring pairs in the chain are obtained from one another by a "natural" transformation. In the second part of the paper we prove that in contrast to the case n = 2 for any n divisible by 4 there exist symbol algebras D 1 , D 2 with deg ⁡ D 1 = deg ⁡ D 2 = n and exp D 1 = exp D 2 = n without common slot such that i D 1 + j D 2 is a symbol algebra of degree n for any i , j ∈ Z. [ABSTRACT FROM AUTHOR]

Let X be a finite connected poset, K a field of characteristic zero and I (X , K) the incidence algebra of X over K seen as a Lie algebra under the commutator product. In the first part of the paper we show that any 1 2 -derivation of I (X , K) decomposes into the sum of a central-valued 1 2 -derivation, an inner 1 2 -derivation and a 1 2 -derivation associated with a map σ : X < 2 → K that is constant on chains and cycles in X. In the second part of the paper we use this result to prove that any transposed Poisson structure on I (X , K) is the sum of a structure of Poisson type, a mutational structure and a structure determined by λ : X e 2 → K , where X e 2 is the set of (x , y) ∈ X 2 such that x < y is a maximal chain not contained in a cycle. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic 0. We study some cohomological properties of Lie subalgebras of the Witt algebra W = Der (k [ t , t − 1 ]) and the one-sided Witt algebra W ≥ − 1 = Der (k [ t ]). In the first part of the paper, we consider finite codimension subalgebras of W ≥ − 1. We compute derivations and one-dimensional extensions of such subalgebras. These correspond to Ext U (L) 1 (M , L) , where L is a subalgebra of W ≥ − 1 and M is a one-dimensional representation of L. We find that these subalgebras exhibit a kind of rigidity: their derivations and extensions are controlled by the full one-sided Witt algebra. As an application of these computations, we prove that any isomorphism between finite codimension subalgebras of W ≥ − 1 extends to an automorphism of W ≥ − 1. The second part of the paper is devoted to explaining the observed rigidity. We define a notion of "completely non-split extension" and prove that W ≥ − 1 is the universal completely non-split extension of any of its subalgebras of finite codimension. In some sense, this means that even when studying subalgebras of W ≥ − 1 as abstract Lie algebras, they remember that they are contained in W ≥ − 1. We also consider subalgebras of infinite codimension, explaining the similarities and differences between the finite and infinite codimension situations. Almost all of the results above are also true for subalgebras of the Witt algebra. We summarise results for W at the end of the paper. [ABSTRACT FROM AUTHOR]

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 the first part of the paper, we propose a different viewpoint on the theory of higher Airy structures (or Airy ideals), which may shed light on its origin. We define Airy ideals in the ħ -adic completion of the Rees Weyl algebra and show that Airy ideals are defined exactly such that they are always related to the canonical left ideal generated by derivatives by automorphisms of the Rees Weyl algebra of a simple type, which we call transvections. The standard existence and uniqueness result in the theory of Airy structures then follow immediately. In the second part of the paper, we construct Airy ideals generated by the nonnegative modes of the strong generators of the principal W -algebra of sp 2 N at level - N - 1 / 2 , following the approach developed in Borot et al. (Mem Am Math Soc, 2021). This provides an example of an Airy ideal in the Heisenberg algebra that requires realizing the zero modes as derivatives instead of variables, which leads to an interesting interpretation for the resulting partition function. [ABSTRACT FROM AUTHOR]

Bhargava and the first-named author of this paper introduced a functorial Galois closure operation for finite-rank ring extensions, generalizing constructions of Grothendieck and Katz–Mazur. In this paper, we generalize Galois closures and apply them to construct a new infinite family of irreducible components of Hilbert schemes of points. We show that these components are elementary, in the sense that they parametrize algebras supported at a point. Furthermore, we produce secondary families of elementary components obtained from Galois closures by modding out by suitable socle elements. [ABSTRACT FROM AUTHOR]

This paper considers theoretical foundations and mathematical methods of data analysis under the conditions of the Rice statistical distribution. The problem involves joint estimation of the signal and noise parameters. It is shown that this estimation requires the solution of a complex system of essentially nonlinear equations with two unknown variables, which implies significant computational costs. This study is aimed at mathematical optimization of computer algebra methods for numerical solution of the problem of Rician data analysis. As a result of the optimization, the solution of the system of two nonlinear equations is reduced to the solution of one equation with one unknown variable, which significantly simplifies algorithms for the numerical solution of the problem, reduces the amount of necessary computational resources, and enables the use of advanced methods for parameter estimation in information systems with priority of real-time operation. Results of numerical experiments carried out using Wolfram Mathematica confirm the effectiveness of the developed methods for two-parameter analysis of Rician data. The data analysis methods considered in this paper are useful for solving many scientific and applied problems that involve analysis of data described by the Rice statistical model. [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]

Two groups of mathematics faculty, one from a four-year college and one from a two-year college, redesigned their respective introductory college mathematics courses following presentation of observational data regarding how faculty had been teaching the courses. This presentation emphasised how infrequently faculty teaching introductory college mathematics employed recommended pedagogical practices. This work was part of a multi-year federal grant project designed to increase the numbers of underrepresented students majoring in STEM disciplines. While the two teams developed very different redesigned course activities, in both instances the primary motivation for initiating the work was the information provided to faculty in a professional development workshop regarding how they had previously been observed teaching the mathematics content and how infrequently they utilised the pedagogical practices recommended by STEM education experts. The paper also highlights faculty resistance to curricular reform and enumerates some ways of addressing this resistance. The paper also discusses why faculty from two-year and four-year institutions resisted working together on course redesign and provides recommendations for future efforts addressing course redesign. [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]

This document is an editorial summarizing the Applied Geometric Algebras in Computer Science and Engineering (AGACSE) conference held in Brno, Czech Republic in September 2021. The conference aimed to promote the use of geometric algebra in fields such as image processing, robotics, and quantum computing. The conference proceedings were published in the journal Mathematical Methods in the Applied Sciences. The editorial provides a list of accepted papers, covering topics such as applied geometry, technological applications, algebra, and quantum phenomena. One specific paper explores the use of geometric algebra in teaching rotations through neural networks. The document is a compilation of research papers showcasing the applications of geometric algebra in various fields, including robotics, control systems, image processing, cryptography, and physics. Each paper presents a specific problem or application and proposes a unique approach or solution using geometric algebra. The authors compare their methods with existing techniques and provide mathematical analysis to support their claims. Overall, the papers demonstrate the versatility and effectiveness of geometric algebra in different domains. [Extracted from the article]

In this paper, we present an extended dissipaton equation of motion for studying the dynamics of electronic impurity systems. Compared with the original theoretical formalism, the quadratic couplings are introduced into the Hamiltonian accounting for the interaction between the impurity and its surrounding environment. By exploiting the quadratic fermionic dissipaton algebra, the proposed extended dissipaton equation of motion offers a powerful tool for studying the dynamical behaviors of electronic impurity systems, particularly in situations where nonequilibrium and strongly correlated effects play significant roles. Numerical demonstrations are carried out to investigate the temperature dependence of the Kondo resonance in the Kondo impurity model. [ABSTRACT FROM AUTHOR]

In this paper, we classify simple Harish-Chandra modules over simple generalized Witt algebras. [ABSTRACT FROM AUTHOR]

The goal of this paper is to generalize a statement by Drinfeld, asserting that Yangians can be constructed as limit forms of the quantum loop algebras, to the super case. We establish a connection between quantum loop superalgebra and super Yangian of the general linear Lie superalgebra gl M | N in RTT type presentation. In particular, we derive the Poincaré-Birkhoff-Witt(PBW) theorem for the quantum loop superalgebra U q (Lgl M | N). Additionally, we investigate the application of the same argument to twisted super Yangian of the ortho-symplectic Lie superalgebra. For this purpose, we introduce the twisted quantum loop superalgebra as a one-sided coideal of U q (Lgl M | 2 n) with respect to the comultiplication. [ABSTRACT FROM AUTHOR]

For a ∈ C ⁎ there are two natural evaluation maps ev a and ev a from the affine Hecke algebra H ▵ (r) C to the Hecke algebra H (r) C. The maps ev a and ev a induce evaluation maps ev ˜ a and ev ˜ a from the affine quantum Schur algebra S ▵ (n , r) C to the quantum Schur algebra S (n , r) C , respectively. In this paper we prove that the evaluation map ev ˜ a (resp. ev ˜ a) is compatible with the evaluation map Ev a (resp. Ev (− 1) n a q n ) for quantum affine sl n. Furthermore we compute the Drinfeld polynomials associated with the simple S ▵ (n , r) C -modules which come from the simple S (n , r) C -modules via the evaluation maps ev ˜ a. Then we characterize finite-dimensional irreducible S ▵ (n , r) C -modules which are irreducible as S (n , r) C -modules for n > r. As an application, we characterize finite-dimensional irreducible modules for the affine Hecke algebra H ▵ (r) C which are irreducible as modules for the Hecke algebra H (r) C. [ABSTRACT FROM AUTHOR]

Let F$F$ be a field of characteristic zero and G$G$ a finite abelian group. In this paper, we prove that an affine variety of G$G$‐graded PI‐algebras is minimal if and only if it is generated by a graded algebra UT(A1,⋯,Am;γ)$UT(A_1,\dots,A_m;\gamma)$ of upper block triangular matrices where A1,⋯,Am$A_1,\dots,A_m$ are finite‐dimensional G$G$‐simple algebras. [ABSTRACT FROM AUTHOR]

Let $ n \geq ~2 $ n ≥ 2 be a natural number, and denote by $ \mathcal {M}_{n} $ M n the algebra of all $ n \times n $ n × n matrices over an algebraically closed field $ \mathbb {F} $ F of zero characteristic. Let also $ K_{1} $ K 1 and $ K_{2} $ K 2 be two non-empty proper subsets of $ \mathbb {F} $ F . In this paper, we characterize linear maps φ on $ \mathcal {M}_{n} $ M n having the property that, for every $ T \in \mathcal {M}_{n} $ T ∈ M n , the spectrum of T is a subset of $ K_1 $ K 1 if and only if the spectrum of $ \varphi (T) $ φ (T) is a subset of $ K_2 $ K 2 . We obtain a similar caracterization for the case when $ K_{1} $ K 1 and $ K_{2} $ K 2 have both at most n elements, working with the equality of the spectrum to the fixed subsets instead of the inclusion. [ABSTRACT FROM AUTHOR]

Assume that is a unital algebra over a commutative unital ring ℛ and is an -bimodule. A trivial extension algebra ⋉ is defined as an ℛ -algebra with usual operations of ℛ -module × and the multiplication defined by (u 1 , s 1) (u 2 , s 2) = (u 1 u 2 , u 1 s 2 + s 1 u 2) for all u 1 , u 2 ∈ , s 1 , s 2 ∈. In this paper, we prove that under certain conditions every Jordan n -derivation Δ on ⋉ can be expressed as Δ = d + δ , where d is a derivation and δ is both a singular Jordan derivation and an antiderivation. As applications, we characterize Jordan n -derivations on triangular algebras and generalized matrix algebras. [ABSTRACT FROM AUTHOR]

In this paper, we shall study three types of Galois extensions B which are Azumaya algebras over its center C. (1) 1 , the class of Azumaya–Galois extensions; (2) 2 , the class of Galois extensions B of B G with Galois group G such that B G is a separable C G -algebra; and (3) 3 , the class of Galois extensions which are Azumaya algebras over its center C. Clearly, 1 ⊆ 2 ⊆ 3 . Examples are given to show the proper inclusion relationship 1 ⊂ 2 ⊂ 3 , and equivalent conditions are given under which a B ∈ 3 is in 1 and a B ∈ 2 is in 1 . [ABSTRACT FROM AUTHOR]

Monounary algebras are the most simple type of an algebraic structure. Oriented graphs with one outgoing arrow from every vertex represent them. The aim of this paper is to point out the interdisciplinary relationships concerning this structure. Bernoulli shift is a paradigmatic mapping in dynamical systems. It is also called dyadic, bit shift, doubling or sawtooth. We offer a look at the properties of this mapping via monounary algebras. [ABSTRACT FROM AUTHOR]

In 2022, Bantaojai et al. [3] introduced an algebra structure called Bd-algebras. In this paper, we define a new notion called fuzzy Bd-ideals of Bd-algebras and study some of its basic properties. Moreover, we characterize fuzzy Bd-ideals by the different types of their level subsets in Bd-algebras. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new eight-dimensional algebra callked KAH-Octo. Moreover, we study subalgebras aand give some properties of the algebra such as division, isomorphism, simple, semi-simple, Jordan, Malcev, among others. Furthermore, we give some applications to some interesting areas of mathematics such as cryptography. [ABSTRACT FROM AUTHOR]

In this paper, we focus on terms with fixed variables count, terms under which the total numbers of occurrences of variables in each position are equal. Moreover, we determine conditions for which the set of terms with fixed variables count is closed under the generalized superposition. Furthermore, we form the partial algebras of such terms satisfying certain axioms as weak identities. [ABSTRACT FROM AUTHOR]

The modified RSA provides high efficiency against attacks and, as a result, it is considered the ideal choice for many applications. In this paper, we introduce an alternative to the modified RSA key encryption system called TPRSA, based on Tri-Cartesian algebra and polynomials, by modifying the mathematical structure of text encryption and decryption keys to obtain a high level of security. [ABSTRACT FROM AUTHOR]

One of the questions investigated in deformation theory is to determine to which algebras can a given associative algebra be deformed. In this paper we investigate a different but related question, namely: for a given associative finite-dimensional |${\mathbb{C}}$| -algebra |$A$|⁠ , find algebras |$N$|⁠ , which can be deformed to |$A$|⁠. We develop a simple method that produces associative and flat deformations to investigate this question. As an application of this method we answer a question of Michael Wemyss about deformations of contraction algebras. [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]

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, 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]

Frieze groups are discrete subgroups of the full group of isometries of a flat strip. We investigate here the dynamics of specific architected materials generated by acting with a frieze group on a collection of self-coupling seed resonators. We demonstrate that, under unrestricted reconfigurations of the internal structures of the seed resonators, the dynamical matrices of the materials generate the full self-adjoint sector of the stabilized group C*-algebra of the frieze group. As a consequence, in applications where the positions, orientations and internal structures of the seed resonators are adiabatically modified, the spectral bands of the dynamical matrices carry a complete set of topological invariants that are fully accounted by the K-theory of the mentioned algebra. By resolving the generators of the K-theory, we produce the model dynamical matrices that carry the elementary topological charges, which we implement with systems of plate resonators to showcase several applications in spectral engineering. The paper is written in an expository style. [ABSTRACT FROM AUTHOR]

We consider the quantum Gelfand invariants which first appeared in a landmark paper by Reshetikhin et al. [Algebra Anal. 1(1), 178–206 (1989)]. We calculate the eigenvalues of the invariants acting in irreducible highest weight representations of the quantized enveloping algebra for g l n . The calculation is based on Liouville-type formulas relating two families of central elements in the quantum affine algebras of type A. [ABSTRACT FROM AUTHOR]

A permutation statistic st is said to be shuffle-compatible if the distribution of st over the set of shuffles of two disjoint permutations π and σ depends only on st π , st σ , and the lengths of π and σ . Shuffle-compatibility is implicit in Stanley's early work on P-partitions, and was first explicitly studied by Gessel and Zhuang, who developed an algebraic framework for shuffle-compatibility centered around their notion of the shuffle algebra of a shuffle-compatible statistic. For a family of statistics called descent statistics, these shuffle algebras are isomorphic to quotients of the algebra of quasisymmetric functions. Recently, Domagalski, Liang, Minnich, Sagan, Schmidt, and Sietsema defined a version of shuffle-compatibility for statistics on cyclic permutations, and studied cyclic shuffle-compatibility through purely combinatorial means. In this paper, we define the cyclic shuffle algebra of a cyclic shuffle-compatible statistic, and develop an algebraic framework for cyclic shuffle-compatibility in which the role of quasisymmetric functions is replaced by the cyclic quasisymmetric functions recently introduced by Adin, Gessel, Reiner, and Roichman. We use our theory to provide explicit descriptions for the cyclic shuffle algebras of various cyclic permutation statistics, which in turn gives algebraic proofs for their cyclic shuffle-compatibility. [ABSTRACT FROM AUTHOR]

This paper is to look for bi-Frobenius algebra structures on quantum complete intersections over field k. We find a class of comultiplications, such that if − 1 ∈ k , then a quantum complete intersection becomes a bi-Frobenius algebra with comultiplication of this form if and only if all the parameters qij = ±1. Also, it is proved that if − 1 ∈ k then a quantum exterior algebra in two variables admits a bi-Frobenius algebra structure if and only if the parameter q = ±1. While if − 1 ∉ k , then the exterior algebra with two variables admits no bi-Frobenius algebra structures. We prove that the quantum complete intersections admit a bialgebra structure if and only if it admits a Hopf algebra structure, if and only if it is commutative, the characteristic of k is a prime p, and every ai a power of p. This also provides a large class of examples of bi-Frobenius algebras which are not bialgebras (and hence not Hopf algebras). In commutative case, other two comultiplications on complete intersection rings are given, such that they admit non-isomorphic bi-Frobenius algebra structures. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the hyperfocal subalgebra of a block with an abelian defect group and a cyclic hyperfocal subgroup is Rickard equivalent to the group algebra of the semidirect of the hyperfocal subgroup by the inertial quotient of the block. In particular, the hyperfocal subalgebra is a Brauer tree algebra, which is analogous to the structure of blocks with cyclic defect groups. As a consequence, we show that Broué's abelian defect group conjecture holds for blocks with cyclic hyperfocal subgroups. [ABSTRACT FROM AUTHOR]

A set of position-dependent noncommutative algebra in two dimension (2D) that describes the space near the Planck scale had been introduced [J. Phys. A: Math. Theor. 53 (2020) 115303]. This algebra predicted the existence of maximal length of graviton measurable at low energy. From this algebra, we deduce in the present paper, a new noncommutative algebra that is compatible with the deformed algebra proposed by Costa Filho et al. [Phys. Rev. A84 (2011) 050102] to describe the Position-Dependent Mass (PDM) system in 1D. To this aim, we derive from the momentum operators, the Schrödinger-like equation which describes PDM system in null quantum well potential. The spectrum of this system is asymetrically deformed and exhibits different behaviours from the one obtained by Costa Filho et al. Thus, we observe that by increasing the PDM, the energy decreases and this decrease is more pronounced as the quantum number increases. Finally, we evaluate the thermodynamic quantities within the canonical ensemble and we show that these results are consistent with the ones recently obtained by Bensalem and Bouaziz [Phys. A Stat. Mech. Appl.523 (2019) 583–592]. [ABSTRACT FROM AUTHOR]

In this paper, a symplectic structure on a Leibniz algebra is defined to be a symmetric nondegenerate bilinear form satisfying certain compatibility condition, and a phase space of a Leibniz algebra is defined to be a symplectic Leibniz algebra satisfying certain conditions. We show that a Leibniz algebra has a phase space if and only if there is a compatible Leibniz-dendriform algebra, and phase spaces of Leibniz algebras are one-to-one corresponds to Manin triples of Leibniz-dendriform algebras. Product (paracomplex) structures and complex structures on Leibniz algebras are studied in terms of decompositions of Leibniz algebras. A para-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a paracomplex structure satisfying a compatibility condition. We show that a symplectic Leibniz algebra admits a para-Kähler structure if and only if the Leibniz algebra is the direct sum of two Lagrangian subalgebras as vector spaces. A complex product structure on a Leibniz algebra consists of a complex structure and a product structure satisfying a compatibility condition. A pseudo-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a complex structure satisfying a compatibility condition. Various properties and relations of complex product structures and pseudo-Kähler structures are studied. In particular, Leibniz-dendriform algebras give rise to complex product structures and pseudo-Kähler structures naturally. [ABSTRACT FROM AUTHOR]

In this paper we mainly consider the notion of Drinfeld double for two weak multiplier Hopf (⁎-)algebras which are paired with each other. Then we show that the Drinfeld double is again a weak multiplier Hopf (⁎-)algebra. Furthermore, we study integrals on the Drinfeld double. Finally, we establish the correspondence between modules over a Drinfeld double D (A) and Yetter-Drinfeld modules over a weak algebraic quantum group A. [ABSTRACT FROM AUTHOR]

Let R be a (semi-) prime ring with extended centroid C, let f(X1, ... , Xk) be a multilinear polynomial over C in k noncommutative indeterminates which is not central-valued on R and let g be a generalized derivation of R. In this paper, we completely characterize the form of g and the structure of R such that (g(f(x1, ... , xk))m − γf(x1, ... , xk)n)s = 0 for all x1, ... , xk ∈ R, where γ ∈ C and m, n, s are fixed positive integers. Our results naturally improve and generalize the theorems obtained by Huang and Davvaz in [Generalized derivations of rings and Banach algebras, Comm. Algebra (2013); 43, 1188–1194] and the theorems recently obtained by De Filippis et al. in [Generalized derivations with nilpotent, power-central and invertible values in prime and semiprime rings, Comm. Algebra (2019); 47, 3025–3039]. Moreover, we describe a revised version of the theorem obtained by Huang in [On generalized derivations of prime and semiprime rings, Taiwanese J. Math. (2012); 16, 771–776.] [ABSTRACT FROM AUTHOR]

This paper introduces a novel method for solving the resection problem in two and three dimensions based on conformal geometric algebra (CGA). Advantage is taken because of the characteristics of CGA, which enables the representation of points, lines, planes, and volumes in a unified mathematical framework and offers a more intuitive and geometric understanding of the problem, in contrast to existing purely algebraic methods. Several numerical examples are presented to demonstrate the efficacy of the proposed method and to compare its validity with established techniques in the field. Numerical simulations indicate that our vector geometric algebra implementation is faster than the best-known algorithms to date, suggesting that the proposed GA-based methods can provide a more efficient and comprehensible solution to the two- and three-dimensional resection problem, paving the way for further applications and advances in geodesy research. Furthermore, the method's emphasis on graphical and geometric representation makes it particularly suitable for educational purposes, allowing the reader to grasp the concepts and principles of resection more effectively. The proposed method has potential applications in a wide range of other fields, including surveying, robotics, computer vision, or navigation. [ABSTRACT FROM AUTHOR]

Studies of issues related to computability and computational complexity involve the use of a model of computation. Central in such a model are computational processes. Processes of this kind can be described using an imperative process algebra based on ACP (Algebra of Communicating Processes). In this paper, it is investigated whether the imperative process algebra concerned can play a role in the field of models of computation. It is demonstrated that the process algebra is suitable to describe in a mathematically precise way models of computation corresponding to existing models based on sequential, asynchronous parallel, and synchronous parallel random access machines as well as time and work complexity measures for those models. [ABSTRACT FROM AUTHOR]

Let 퓟n := H*((ℝP∞)n) ≅ ℤ2[x1, x2, ..., xn] be the graded polynomial algebra over ℤ2, where ℤ2 denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra 퓟n, viewed as a graded left module over the mod-2 Steenrod algebra, 퓐. For n > 4, this problem is still unsolved, even in the case of n = 5 with the help of computers. In this article, we study the hit problem for the case n = 6 in the generic degree dr = 6(2r − 1) + 4.2r with r an arbitrary non-negative integer. By considering ℤ2 as a trivial 퓐-module, then the hit problem is equivalent to the problem of finding a basis of ℤ2-vector space ℤ2 ⊗퓐퓟n. The main goal of the current article is to explicitly determine an admissible monomial basis of the ℤ2 vector space ℤ2 ⊗퓐퓟6 in some degrees. As an application, the behavior of the sixth Singer algebraic transfer in the degree 6(2r − 1) + 4.2r is also discussed at the end of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we define and study the variety of tense modal pseudocomplemented De Morgan algebras. This variety is a proper subvariety of the variety of tense tetravalent modal algebras. A tense modal pseudocomplemented De Morgan algebra is a modal pseudocomplemented De Morgan algebra endowed with two tense operators G and H satisfying additional conditions. Also, the variety of tense modal pseudocomplemented De Morgan algebras is intimately connected with some well-known varieties of De Morgan algebras with tense operators. [ABSTRACT FROM AUTHOR]

We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies Λ ⁢ (X m , n) ⊂ E of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial. In particular, we obtain a formula for ∇ m s λ which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one. [ABSTRACT FROM AUTHOR]