Narrow Results

Let $R$ be a commutative local ring with residue field $\overline{R}$. Suppose $n>1$ is an even natural number, $\overline{R}$ contains at least $n+3$ elements and $q(x)=x^2-cx+1$ is a quadratic polynomial in $R[x]$. The main goal of this paper is to prove that the covering number ${\mathrm{\text{SL}}}_n(R)$ with respect to the set of $q(x)$-quadratic matrices in ${\mathrm{\text{SL}}}_n(R)$ is equal to or less than $5$. Some corollaries are also presented.

In this paper, the concepts of nilpotent and Engel orthomodular lattices, similar with the solvable orthomodular lattices are defined and their properties are investigated. The notion of $n$-Engel orthomodular lattices as a natural generalization of distributive orthomodular lattices is introduced, and we discuss Engel orthomodular lattices, which is defined by left and right normed commutators.

The commutator length of a Hamiltonian diffeomorphism f ∈ Ham(M,ω) of a closed symplectic manifold (M,ω) is by definition the minimal k such that f can be written as a product of k commutators in Ham(M,ω). We introduce a new invariant for Hamiltonian diffeomorphisms, called the k₊-area, which measures the "distance", in a certain sense, to the subspace of all products of k commutators. Therefore, this invariant can be seen as the obstruction to writing a given Hamiltonian diffeomorphism as a product of k commutators. We also consider an infinitesimal version of the commutator problem: what is the obstruction to writing a Hamiltonian vector field as a linear combination of k Lie brackets of Hamiltonian vector fields? A natural problem related to this question is to describe explicitly, for every fixed k, the set of linear combinations of k such Lie brackets. The problem can be obviously reformulated in terms of Hamiltonians and Poisson brackets. For a given Morse function f on a symplectic Riemann surface M (verifying a weak genericity condition) we describe the linear space of commutators of the form {f, g}, with .

In this paper, we prove that if the composition symbols φ and ψ are linear fractional non-automorphisms of 𝔻 such that φ(ζ) and ψ(ζ) belong to ∂𝔻 for some ζ ∈ ∂𝔻 and u, v ∈ H^∞ are continuous on ∂𝔻 with u(ζ)v(ζ) ≠ 0, then is compact on H² if and only if ζ is the common boundary fixed point of φ and ψ and one of the following statements holds: (i) both φ and ψ are parabolic; (ii) both φ and ψ are hyperbolic and another fixed point of φ is where w is the fixed point of ψ other than ζ. We also study the commutant of a weighted composition operator on H². We verify that if φ is an analytic self-map of 𝔻 with Denjoy–Wolff point b ∈ 𝔻 and u ∈ H^∞\{0}, then every weighted composition operator in the commutant {W_{u, φ}}′ has {f ∈ H² : f(b) = 0} as its nontrivial invariant subspace.

The YOLOv8 model has high detection efficiency and classification accuracy in detecting commutator surface defects, aimed at the problem of low working efficiency of a commutator, caused by commutator surface defects. First, the theoretical framework of Region-based Convolutional Neural Networks (R-CNN), spatial pyramid pooling (SPP)-net, Fast R-CNN, and Faster R-CNN is introduced, and the detection principle and process are described in detail. Secondly, the principle of the YOLOv8 network structure, head structure, neck structure, and C2f module are explained, and the loss function is described. The average precision of the proposed algorithm for detecting cracks and small points is more than 98%, and the frames per second (FPS) is 27. The detection results are mapped to the original image, and the visualization of the commutator surface defect detection is obtained, which has a higher robustness, accuracy, and real-time performance than the R-CNN, SPP-net, Fast R-CNN, and Faster R-CNN algorithms.

Let G be a group and G′ be its commutator subgroup. Denote by c(G) the minimal number such that every element of G′ can be expressed as a product of at most c(G) commutators. We find suitable bounds for c(G) when G is a free nilpotent by abelian group. Then we prove that c(G) is finite if G is a n-generator solvable group. And G has a nilpotent by abelian normal subgroup K of finite index. Moreover we have c(G) ≤ s(s + 1)/2 + 72n² + 47n, where s is the number of generators of K. We also prove that in a solvable group of finite Pruffer rank s every element of its commutator subgroup is equal to a product of at most s(s + 1)/2 + 72s² + 47s. And finally as a corollary of the above results we show that if A is a normal subgroup of a solvable group G such that G/A is a d-generator finite group. And A has finite Pruffer rank s. Then c(G) ≤ s(s + 1)/2 + 72(s² + n²) + 47(s + n). The bounds we find are independent of the solvability length of the groups.

We improve previous results by showing that a finitely generated soluble group G is finite-by-nilpotent if and only if for all a, b ∈ G, there exists a positive integer n such that [a, _nb] belongs to γ_n+2(<a, b>).<a^<a,b>>′.

Let $A$ be a unital associative ring and let $T^{(k)}$ be the two-sided ideal of $A$ generated by all commutators $[a_1,a_2,\dots ,a_k]$$(a_i\in A)$ where $[a_1,a_2]=a_1{a}_2-a_2{a}_1$, $[a_1,\dots ,a_{k-1},a_k]=[[a_1,\dots ,a_{k-1}],a_k]$$(k>2)$. It has been known that, if either $m$ or $n$ is odd, then

We prove that the class of algebras embeddable into Leibniz algebras with respect to the commutator product is not a variety. It is shown that every commutative metabelain algebra is embeddable into Leibniz algebras with respect to the anti-commutator. Furthermore, we study polynomial identities satisfied by the commutator in every Leibniz algebra. We extend the result of Dzhumadil’daev in [A. S. Dzhumadil’daev, q-Leibniz algebras, Serdica Math. J. 34(2) (2008) 415–440]. to identities up to degree 7 and give a conjecture on identities of higher degrees. As a consequence, we obtain an example of a non-Spechtian variety of anticommutative algebras.

We show that, when restricted to the class of varieties that have a Taylor term, several commutator properties are definable by Maltsev conditions.

In this paper, we investigate properties of varieties of algebras described by a novel concept of equation that we call commutator equation. A commutator equation is a relaxation of the standard term equality obtained substituting the equality relation with the commutator relation. Namely, an algebra $A$ satisfies the commutator equation $p{\approx }_{C}q$ if for each congruence $𝜃$ in $Con(A)$ and for each substitution $p^A,q^A$ of elements in the same $𝜃$-class, we have $(p^A,q^A)\in [𝜃,𝜃]$. This notion of equation draws inspiration from the definition of a weak difference term and allows for further generalization of it. Furthermore, we present an algorithm that establishes a connection between congruence equations valid in the variety generated by the abelian algebras of the idempotent reduct of a given variety and congruence equations that hold in the entire variety. Additionally, we provide a proof that if the variety generated by the abelian algebras of the idempotent reduct of a variety satisfies a nontrivial idempotent Mal’cev condition, then also the entire variety satisfies a nontrivial idempotent Mal’cev condition, a statement that follows also from [12, Theorem 3.13].

We show that a suitably chosen position-momentum commutator can elegantly describe many features of gravity, including the IR/UV correspondence and dimensional reduction ("holography"). Using the most simplistic example based on dimensional analysis of black holes, we construct a commutator which qualitatively exhibits these novel properties of gravity. Dimensional reduction occurs because the quanta size grow quickly with momenta, and thus cannot be "packed together" as densely as naively expected. We conjecture that a more precise form of this commutator should be able to quantitatively reproduce all of these features.

Let $L=-\mathrm{\text{div}}(A\nabla )$ be a second-order divergence form elliptic operator and $A$ an accretive, $n\times n$ matrix with bounded measurable complex coefficients in ${ℝ}^n.$ In this paper, we establish $L^p$ theory for the commutators generated by the fractional differential operators related to $L$ and bounded mean oscillation (BMO)–Sobolev functions.

Let $R$ be a noncommutative division ring with center $Z$, which is algebraic, that is, $R$ is an algebraic algebra over the field $Z$. Let $f$ be an antiautomorphism of $R$ such that (i) ${[f(x),x^{m(x)}]}_{n(x)}=0$, all $x\in R$, where $m(x)$ and $n(x)$ are positive integers depending on $x$. If, further, $f$ has finite order, it was shown in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] that $f$ is commuting, that is, $[f(x),x]=0$, all $x\in R$. Posed in [M. Chacron, Antiautomorphisms with quasi-generalised Engel condition, J. Algebra Appl. 17(8) (2018) 1850145 (19 pages)] is the question which asks as to whether the finite order requirement on $f$ can be dropped. We provide here an affirmative answer to the question. The second major result of this paper is concerned with a nonnecessarily algebraic division ring $R$ with an antiautomorphism $f$ satisfying the stronger condition (ii) ${[f(x),x^m]}_n=0$, all $x\in R$, where $m$ and $n$ are fixed positive integers. It was shown in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036] that if, further, $f$ has finite order then $f$ is commuting. We show here, that again the finite order assumption on $f$ can be lifted answering thus in the affirmative the open question (see Question 2.11 in [T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45(9) (2017) 4030–4036]).

The purpose of this paper is to introduce a concept of commutator in a multiplicative Lie algebra which will be termed as a Lie commutator. Consequently, we develop the theory of Lie solvability and Lie nilpotency of multiplicative Lie algebras.

In this paper, we present a characterization of groups $G$ of order $p^7$, $p$ prime, in which not all elements of the commutator subgroup ${\gamma }_2(G)$ of $G$ are commutators in $G$. In the way, we obtain several structural results on groups of order $p^7$.

This paper introduces the notion of commutator-inversion invariance and studies several algebraic properties of commutator-inversion invariant groups. Then, a characterization of 2-Engel groups is given, and it is shown that any group whose central quotient is commutator-inversion invariant gives rise to a non-associative structure called a gyrogroup. This method yields three non-degenerate gyrogroups of order 16 as concrete examples.

We show that, making use of multiplicative lattices and idempotent endomorphisms of an algebraic structure $A$, it is possible to derive several notions concerning $A$ in a natural way. The multiplicative lattice necessary here is the complete lattice of congruences of $A$ with multiplication given by commutator of congruences. Our main application is to the study of some notions concerning left skew braces.

Let $\lambda \in (0,\infty )$ and ${△}_{\lambda }:=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$ be the Bessel operator on ${ℝ}_{+}:=(0,\infty )$. In this paper, the authors show that $b\in \mathrm{\text{BMO}}({ℝ}_{+},d{m}_{\lambda })$ (or $\mathrm{\text{CMO}}({ℝ}_{+},d{m}_{\lambda })$, respectively) if and only if the Riesz transform commutator $[b,R_{{△}_{\lambda }}]$ is bounded (or compact, respectively) on Morrey spaces $L^{p,\kappa }({ℝ}_{+},d{m}_{\lambda })$, where $d{m}_{\lambda }(x):=x^{2\lambda }dx$, $p\in (1,\infty )$ and $\kappa \in (0,1)$. A weak factorization theorem for functions belonging to the Hardy space $H^1({ℝ}_{+},d{m}_{\lambda })$ in the sense of Coifman–Rochberg–Weiss in Bessel setting, via $R_{{△}_{\lambda }}$ and its adjoint, is also obtained.

Let $\alpha \in (0,1]$, $\beta \in [0,n)$ and $T_{\mathrm{\Omega },\beta }$ be a singular or fractional integral operator with homogeneous kernel $\mathrm{\Omega }$. In this paper, a CMO type space ${\mathrm{\text{CMO}}}_{\alpha }({ℝ}^n)$ is introduced and studied. In particular, the relationship between ${\mathrm{\text{CMO}}}_{\alpha }({ℝ}^n)$ and the Lipchitz space ${\mathrm{\text{Lip}}}_{\alpha }({ℝ}^n)$ is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator ${(T_{\mathrm{\Omega },\beta })}_b^m$ on weighted Lebesgue spaces via functions in $\mathrm{\text{L}}i{p}_{\alpha }({ℝ}^n)$, and an equivalent characterization of the compactness for ${(T_{\mathrm{\Omega },\beta })}_b^m$ via functions in ${\mathrm{\text{CMO}}}_{\alpha }({ℝ}^n)$ are obtained. Some results are new even in the unweighted setting for the first-order commutators.