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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]

In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called ϕ -u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On ϕ -exact sequence and ϕ -projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of ϕ -flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On ϕ -Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on ϕ -Dedekind rings and ϕ -Prüfer rings, preprint (2022), arXiv:2103.08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly ϕ -flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https://arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the ϕ -injective dimension, ϕ -projective dimension and ϕ -flat dimension for modules, and also the ϕ -(weak) global dimension of rings. Then, using these dimensions, we characterize several ϕ -rings (ϕ -Prüfer, ϕ -chained, ϕ -von Neumann, etc). Finally, we study the ϕ -(weak) global dimension of the trivial ring extensions defined by some conditions. [ABSTRACT FROM AUTHOR]

In the present paper corrected versions of the statements in the paper "Description of finite-dimensional inner Rickart and Baer Jordan algebras" by F.N. Arzikulov and U.I. Khakimov are given. In particular, it is shown that for any Jordan algebra J with an idempotent p and an associative degenerate radical D such that J = F p + ̇ D , J is an inner RJ-algebra if and only if, for any nonzero a ∈ D , a 2 = 0 and p(pa) = pa. Also, other equivalent conditions when a Jordan algebra J is an inner RJ-algebra are given. As for finite-dimensional nilpotent Jordan algebras, there is not a nilpotent inner RJ-algebra (and hence inner BJ-algebra) except the finite-dimensional Jordan algebra the square of each element of which is zero. [ABSTRACT FROM AUTHOR]

In a previous paper we have compared the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two idempotents. In the present paper, we compare the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two arrows. [ABSTRACT FROM AUTHOR]

Quantum logic (QL) and fuzzy logic (FL) have been gaining attention nowadays due to its potential to be used in quantum computing and information theory. This paper aims to provide a comprehensive overview of QL models of fuzzy representations viz. fuzzy sets (FSs), interval valued fuzzy sets (IVFSs), intuitionistic fuzzy sets (IFSs) and interval valued intuitionistic fuzzy sets (IVIFSs). These QL models can be used to analyze the behavior of quantum logical systems in a fuzzy environment, which is particularly useful for dealing with uncertainty and imprecision in quantum environment. Furthermore, this paper explores the concept of effect algebras (EAs), which are algebraic structures that provide a natural framework for studying FL. Specifically, it is shown that the family of FS, IVFS, IFS and IVIFS can be organized into an EA if the Lukasiewicz operations are considered. This result is significant because EAs can be used to model a wide range of physical and mathematical systems, including quantum systems, and provide a useful tool for analyzing the properties of FL. [ABSTRACT FROM AUTHOR]

In a recent paper, the author defined an operation of tensor product for a large class of 2-representations of U + , the positive half of the 2-category associated to sl 2. In this paper, we prove that the operation extends to give an operation of tensor product for 2-representations of the full 2-category U : when the inputs are 2-representations of the full U , the 2-product is also a 2-representation of the full U. As in the previous paper, the 2-product is given for a simple 2-representation L (1) and an abelian 2-representation V taken from the 2-category of algebras. This is the first construction of an operation of tensor product for higher representations of a full Lie algebra in the abelian setting. [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]

This work is concerned with the consistency conditions for the equations in max plus algebra. The three classes of the max plus linear equations presented in the paper Shinzawa [Ultra discrete permanent and the consistency of max plus linear equations, Linear Algebra Appl. 506 (2016) 445–477] can be equivalently converted into the corresponding system of inequalities. The equivalence relation between ultra discrete permanent and maximum cycle mean is suggested. Thus, the necessary and sufficient conditions for solvability of the equations are obtained using the maximum cycle mean in this note. [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]

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]

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 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 discuss a generalization of Vieta theorem (Vieta's formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta formulas with the ordinary Vieta formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand–Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. We introduce the notion of a simple basis‐free formula for a determinant in geometric algebra and prove that a formula of this type exists in the case of arbitrary dimension. Using this notion, we present and prove generalized Vieta theorem in geometric algebra of arbitrary dimension. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering. [ABSTRACT FROM AUTHOR]

In this paper, we develop the Hermitian refinement of symplectic Clifford analysis, by introducing a complex structure 핁 on the canonical symplectic manifold (ℝ2n,ω0)$$ \left({\mathrm{\mathbb{R}}}^{2n},{\omega}_0\right) $$. This gives rise to two symplectic Dirac operators Ds$$ {D}_s $$ and Dt$$ {D}_t $$ (in the sense of Habermann), leading to a u(n)$$ \mathfrak{u}(n) $$‐invariant system of equations on ℝ2n$$ {\mathrm{\mathbb{R}}}^{2n} $$. We discuss the solution space for this system, culminating in a Fischer decomposition for the space of (harmonic) polynomials on ℝ2n$$ {\mathrm{\mathbb{R}}}^{2n} $$ with values in the symplectic spinors. To make this decomposition explicit, we will construct the associated embedding factors using a transvector algebra. [ABSTRACT FROM AUTHOR]

In this paper, we introduce multi-layer quivers and show how to construct an (n + 1) -slice algebra of infinite type from an n -slice algebra of infinite type using the bound quivers. This leads to constructing (n + 1) -slice algebras of infinite type as matrix algebra and as tensor algebra of an n -slice algebra and equivalences of their module categories as the module categories of diagram of some quiver of type A ̃ n + 1 . [ABSTRACT FROM AUTHOR]

Co-Gorenstein algebras were introduced by Beligiannis in [A. Beligiannis, The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and co-stabilization, Comm. Algebra28(10) (2000) 4547–4596]. In [S. Kvamme and R. Marczinzik, Co-Gorenstein algebras, Appl. Categorical Struct.27(3) (2019) 277–287], the authors propose the following conjecture (co-GC): if Ω n (m o d A) is extension closed for all n ≤ 1 , then A is right co-Gorenstein, and they prove that the generalized Nakayama conjecture implies the co-GC, also that the co-GC implies the Nakayama conjecture. In this paper, we characterize the subcategory Ω ∞ (m o d A) for algebras of Ω n -finite representation type. As a consequence, we characterize when a truncated path algebra is a co-Gorenstein algebra in terms of its associated quiver. We also study the behavior of Artin algebras of Ω ∞ -infinite representation type. Finally, an example of a non-Gorenstein algebra of Ω ∞ -infinite representation type and an example of a finite dimensional algebra with infinite ϕ -dimension are presented. [ABSTRACT FROM AUTHOR]

In this paper, we prove that an affine cellular algebra A is semisimple if and only if the scheme associated to A is reduced and 0-dimensional, and the bilinear forms with respect to all layers of A are invertible. Moreover, if the ground ring is a perfect field, then A is semisimple if and only if it is separable. We also give a sufficient condition for an affine cellular algebra being Jacobson semisimple. [ABSTRACT FROM AUTHOR]

In this paper we studied the personal mathematical work of 14 students in initial teacher training, in their first year of studies at a public university in Chile, based on two tasks on complex numbers. These tasks were set in a Computer Aided Assessment System (CAA) based on Moodle and Wiris and it was proposed to solve them using different Computer Algebra Systems (CAS), such as GeoGebra, Symbolab, Photomath and Wolfram Alpha among others. Difficulties and potentialities were observed in the mathematical work of the students. In the difficulties, it was observed that the students had problems interpreting the information from the CAS and the CAA feedback because their previous mathematical knowledge was not solid enough to do so. In the potentialities, it was observed that different characteristics of the task, together with the articulation of two or more artifacts allowed students to give meaning to the mathematical objects involved. [ABSTRACT FROM AUTHOR]

Let A be the Steenrod algebra over the finite field k : = F 2 and G(q) be the general linear group of rank q over k. A well-known open problem in algebraic topology is the explicit determination of the cohomology groups of the Steenrod algebra, Ext A q , ∗ (k , k) , for all homological degrees q ⩾ 0. The Singer algebraic transfer of rank q, formulated by William Singer in 1989, serves as a valuable method for describing that Ext groups. This transfer maps from the coinvariants of a certain representation of G(q) to Ext A q , ∗ (k , k). Singer predicted that the algebraic transfer is always injective, but this has gone unanswered for all q ⩾ 4. This paper establishes Singer's conjecture for rank four in the generic degrees n = 2 s + t + 1 + 2 s + 1 - 3 whenever t ≠ 3 and s ⩾ 1 , and n = 2 s + t + 2 s - 2 whenever t ≠ 2 , 3 , 4 and s ⩾ 1. In conjunction with our previous results, this completes the proof of the Singer conjecture for rank four. [ABSTRACT FROM AUTHOR]

Let \Gamma \subseteq \mathbb {N} be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of \Gamma which depends only on the width of \Gamma, that is, the difference between the largest and the smallest generator of \Gamma. In this way, we make progress towards a conjecture of Herzog and Stamate [J. Algebra 418 (2014), pp. 8–28]. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width. [ABSTRACT FROM AUTHOR]

Equivariant monoids are very important objects in many branches of mathematics: they combine the notion of multiplication and the concept of a group action. In this paper we will construct categories which encode the structure borne by monoids with a group action by combining the theory of product and permutation categories (PROPs) and product and braid categories (PROBs) with the theory of crossed simplicial groups. PROPs and PROBs are categories used to encode structures borne by objects in symmetric and braided monoidal categories respectively, whilst crossed simplicial groups are categories which encode a unital, associative multiplication and a compatible group action. We will produce PROPs and PROBs whose categories of algebras are equivalent to the categories of monoids, comonoids and bimonoids with group action using extensions of the symmetric and braid crossed simplicial groups. We will extend this theory to balanced braided monoidal categories using the ribbon braid crossed simplicial group. Finally, we will use the hyperoctahedral crossed simplicial group to encode the structure of an involutive monoid with a compatible group action. [ABSTRACT FROM AUTHOR]

In this paper, the author gives a complete set of simple Yetter–Drinfeld modules over the Suzuki algebra A N 2 n + 1 μ λ and investigates the Nichols algebras over those irreducible Yetter–Drinfeld modules. The finite-dimensional Nichols algebras of diagonal type are of Cartan type A 1 , A 1 × A 1 , A 2 , Super type A 2 (q ; 2) and the Nichols algebra (8). And the involved finite-dimensional Nichols algebras of non-diagonal type are 1 2 , 4 m and m 2 -dimensional. The left three unsolved cases are set as open problems. In particular, all finite-dimensional Nichols algebras are given for simple Yetter–Dinfeld modules over  A 1 3 + λ . [ABSTRACT FROM AUTHOR]

For a rooted cluster algebra (Q) over a valued quiver Q , a symmetric cluster variable is any cluster variable belonging to a cluster associated with a quiver σ (Q) , for some permutation σ. The subalgebra of (Q) generated by all symmetric cluster variables, is called the symmetric mutation subalgebra and is denoted by ℬ (Q). In this paper, we identify the class of cluster algebras that satisfy ℬ (Q) = (Q) , which contains almost every quiver of finite mutation type. In the process of proving the main result, we provide a classification of quivers mutations classes that relates their maximum weights to the shapes of the initial quivers. Furthermore, some properties of symmetric mutation subalgebras are given. [ABSTRACT FROM AUTHOR]

As a quantum affinization, the quantum toroidal algebra U q , q ¯ (gl ¨ n) is defined in terms of its "left" and "right" halves, which both admit shuffle algebra presentations (Enriquez in Transform Groups 5(2):111–120, 2000; Feigin and Odesskii in Am Math Soc Transl Ser 2:185, 1998). In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the "top" and "bottom" halves instead, starting from the evaluation representation U q (gl ˙ n) ↷ C n (z) and its usual R-matrix R (z) ∈ End (C n ⊗ C n) (z) (see Faddeev et al. in Leningrad Math J 1:193–226, 1990). An upshot of this construction is a new topological coproduct on U q , q ¯ (gl ¨ n) which extends the Drinfeld–Jimbo coproduct on the horizontal subalgebra U q (gl ˙ n) ⊂ U q , q ¯ (gl ¨ n) . [ABSTRACT FROM AUTHOR]

In this paper, we show that higher Hochschild homology groups do not vanish for a class of Hochschild extension algebras of truncated quiver algebras by the standard duality module. Consequently, this result extends the result of the trivial extension algebra by the standard duality module for truncated quiver algebras. Moreover, as an application, we show that for any Hochschild extension algebra of selfinjective Nakayama algebras by the standard duality module, its Hochschild homology dimension is infinite. [ABSTRACT FROM AUTHOR]

Let D be an F-central division algebra. In this paper, we investigated the exponent of the group G (D) = D * / Nrd D (D *) D ′ , where D * is the group of units of D, Nrd D (D *) is the image of D * under the reduced norm map and D ′ is the commutator subgroup of D * . We show that if exp (G (D)) < ind (D) , then D and F satisfy strong conditions. In particular, we observe that if D is a sum cyclic algebras in Br (F) , then exp (G (D)) < ind (D) if and only if F is euclidean and D is a tensor product of an ordinary quaternion algebra and a division algebra of odd index. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate some properties of relative Rota-Baxter operators on Leibniz conformal algebras with respect to representations and their connections with Leibniz dendriform conformal algebras. Next, we introduce the notion of a twisted relative Rota-Baxter operator and construct a conformal NS-Leibniz algebra structure related to twisted relative Rota-Baxter operators. Furthermore, we define the cohomology of a twisted relative Rota-Baxter operator with coefficients in a suitable representation. Finally, we consider formal deformations of twisted relative Rota-Baxter operators from cohomological points of view. [ABSTRACT FROM AUTHOR]

In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra g with dim [ g , g ] = 1 is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra l 2 n + 1 J 1 , where J1 is the n × n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks. [ABSTRACT FROM AUTHOR]

Motivated by recent work on compatible mock-Lie algebras such that any linear combination of the two mock-Lie structures is a mock-Lie structure, in this paper we introduce compatible mock-Lie bialgebras as an analogue of compatible Lie bialgebras. They can also be regarded as a "compatible version" of mock-Lie bialgebras. Constructions of compatible mock-Lie bialgebras are presented via Manin triples and matched pairs. [ABSTRACT FROM AUTHOR]

This paper is dedicated to defining and studying for the first time a two-fold algebra over the neutrosophic real number ring by merging the fuzzy set mapping with the algebraic operations of the neutrosophic real number ring. Also, we study the elementary algebraic properties of the defined two-fold algebra through its algebraic operations and substructures such as homomorphisms and ideals. [ABSTRACT FROM AUTHOR]

This paper is dedicated to defining and studying for the first time the concept of fuzzy metric spaces based on two-fold fuzzy algebras, where the elementary properties of this new concept will be studied and presented by many theorems and related examples that explain the validity of our work. Also, many different types of open and closed balls will be discussed, as well as the relationships between these metric substructures. Keywords: fuzzy metric space, two-fold algebra, open ball, closed ball, torus Introduction and basic concepts The applications of neutrosophic sets and fuzzy sets are very wide and open research areas. In the literature, we can find many neutrosophic and fuzzy algebraic structures with deep connection with applied mathematics and number theory [5-11]. The concept of two-fold algebra was presented by Smarandache in [4], where many suggestions for the algebraic structure related to this algebra were defined and presented. This new idea has been used in [1] to study the two-fold algebra based on the standard fuzzy number theoretical system [3]. In [2], Hatip et.al. proposed the two-fold vector space and two-fold algebraic module based on fuzzy mappings, where they have studied the elementary properties of these new generalizations with many interesting examples. This work is motivated by the modern idea of two-fold algebra, and metric spaces, where we can combine those to different structures in one algebraic structure called two-fold. [ABSTRACT FROM AUTHOR]

For the purposes of our analysis, we shall work with the Steenrod algebra 풜 over the field of two elements, 픽2. Our paper focuses on expanding the outcomes from our preceding study, which serves as Part I. Precisely, we will initially compute the dimension of the “cohit” space 픽2 ⊗풜Ps, where Ps is defined as the graded polynomial ring 픽2[x1,x2,…,xs :,deg(xi) = 1,i = 1, 2,…,s], for the specific case of s = 5. This computation will be carried out explicitly for the generic degree 21 ⋅ 2t − 5, where t is a positive integer. We have developed a novel algorithm implemented in SageMath to compute the case where t = 2. We subsequently study the dimension of 픽2 ⊗풜Ps in degrees s + 5 for 8 ≤ s ≤ 9. One of our findings is the correction of some erroneous results in the previous work by Moetele and Mothebe. We, additionally, give an explicit formula for the dimension of 픽2 ⊗풜Ps in degree fourteen for all s > 0 and in degree fifteen for all s > 0,s≠10. We also describe a computational method implemented in SageMath to explicitly account for the number of spikes of degree 2s−1 − s in the 풜-module Ps, applicable for all positive integers s > 1. In application, we examine W. Singer’s conjecture on the algebraic transfer of rank 5 in degrees 21 ⋅ 2t − 5 for all t ≥ 0 and of ranks s > 0 in internal degrees d ranging from 13 to 15. Considering higher homological degrees, we analyze the behavior of the algebraic transfer of rank 7 in the generic degrees 23 ⋅ 2t − 7 for t = 0 and ℓ ⋅ 2t − 7 for ℓ ∈{9, 16},t ≤ 3. We then demonstrate that the (in)decomposable elements Pc0 ∈Ext풜7,23⋅20(픽2, 픽2), k0 = k ∈Ext풜7,9⋅22(픽2, 픽2), h6D2 ∈Ext풜7,27(픽2, 픽2), Q2(0) ∈Ext풜7,26(픽2, 픽2) and Q2(1) ∈Ext풜7,27(픽2, 픽2) are not in the image of the Singer transfer. [ABSTRACT FROM AUTHOR]

In this paper, we study Whittaker modules over the N = 2 Neveu–Schwarz algebra 픤. We first classify all simple finite-dimensional modules over the positive part of 픤. Motivated by this classification, we define Whittaker modules and investigate their simplicity. Finally, we also study the simplicity of the quotient modules of the universal Whittaker module if it is not simple. [ABSTRACT FROM AUTHOR]

In this paper, we study automorphisms of finitely generated free metabelian Novikov algebras and show that every tame automorphism of a two-generated free right nilpotent Novikov algebra of index 3 is simple reducible. We offer a method on recognizing automorphisms of finitely generated free metabelian Novikov algebras by using the theory of Gröbner–Shirshov basis. [ABSTRACT FROM AUTHOR]

In this survey paper, we collect properties of one-relator algebras in varieties of linear algebras. The main attention is paid to Schreier varieties of algebras. A variety of linear algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free. [ABSTRACT FROM AUTHOR]

In this paper, numerical characteristics of identities with involution of nonassociative algebras are studied. We construct a series of examples of algebras with overexponential growth of codimensions with involution. [ABSTRACT FROM AUTHOR]

This paper presents variants of the topological Jacobson radical of algebras in quasi-regular and modular versions of the definition, which use the ideas of the construction of the Brown–McCoy radical and the descriptions of the Jacobson radical of alternative algebras. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a spinor algebra of k-balancing numbers referred to as the k-balancing and k-Lucas-balancing spinors. First, we give k-balancing quaternions and their some algebraic properties. Then we introduce a spinor family of k-balancing numbers by defining a linear and injective correspondence between k-balancing and k-Lucas-balancing quaternions to spinors. Here, we give various algebraic properties of this spinor such as the Binet form, Catalan’s identity, d’Ocagne identity, generating and exponential generating functions. Moreover, we obtain various partial sum formulae for these sequences in closed form and also give matrix representations. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we use the Balinsky-Novikov construction to naturally realize the super Heisenberg-Virasoro algebra g . We then proceed to compute the second cohomology of g with trivial coefficients and determine its universal central extension. This yields two odd 2-cocycles in addition to the Virasoro 2-cocycle. We also calculate the first cohomology group of g in the adjoint module to classify all derivations of g . Finally, we classify all automorphisms of g and describe its automorphism group. [ABSTRACT FROM AUTHOR]