| Communications in Mathematics |
We introduce noncommutative rings with $DK$-property (Dubrovin-Komarnytsky's property) and investigate elementary divisor rings with such property. Mostly we pay attention to these kinds of noncommutative rings which have stable range $1$. A theory of reduction matrices over such rings is constructed. As a consequence, new families of non-commutative rings of elementary divisor rings are constructed.
Let $D$ be a division ring with infinite center $F$; $σ$ be an anti-automorphism of $D$ and $m$ be a positive integer such that $σ^m\neq \mathrm{Id}$. In this paper, we show that if $D$ satisfies a $σ^m$-GRI, then $D$ is centrally finite.
Let N be the maximal unipotent subgroup in the simple algebraic group of type Φ. It naturally acts on the space dual to the Lie algebra n of N, and this action is called coadjoint. Such orbits play the key role in the orbit method of A.A. Kirillov. In this work, we classify the orbits of this action in the case of Φ = F_4 in terms of supports of canonical forms. This means that we will present a set S of linear forms such that for any coadjoint orbit there exists a unique form from S belonging to that orbit. The set of canonical forms will be explicitly described in terms of supports.
A filtration of the morphisms of the $k$-linearization $k \mathbf{FS}$ of the category $\mathbf{FS}$ of finite sets and surjections is constructed using a natural $k \mathbf{FI}^{op}$-module structure induced by restriction, where $\mathbf{FI}$ is the category of finite sets and injections. In particular, this yields the `primitive' subcategory $ k \mathbf{FS}^0 \subset k \mathbf{FS}$ that is of independent interest; for example, the category of $k \mathbf{FS}^0$-modules is closely related to the category of $k \mathbf{FA}$-modules, where $\mathbf{FA}$ is the category of finite sets and all maps. Working over a field of characteristic zero, the subquotients of this filtration are identified as bimodules over $k \mathbf{FB}$, where $\mathbf{FB}$ is the category of finite sets and bijections, also exhibiting and exploiting additional structure. In particular, this describes the underlying $k \mathbf{FB}$-bimodule of $k \mathbf{FS}^0$.
In complex production lines, it is essential to have strict, fast-acting rules to determine whether the system is In Control (InC) or Out of Control (OutC). This study explores a bio-inspired method that digitally mimics ant colony behavior to classify InC/OutC states and forecast imminent transitions requiring maintenance. A case study on industrial potato chip frying provides the application context. During each two-minute frying cycle, sequences of eight temperature readings are collected. Each sequence is treated as a digital ant depositing virtual pheromones, generating a Base Score. New sequences, representing new ants, can either reinforce or weaken this score, leading to a Modified Base Score that reflects the system's evolving condition. Signals such as extreme temperatures, large variations within a sequence, or the detection of change-points contribute to a Threat Score, which is added to the Modified Base Score. Since pheromones naturally decay over time unless reinforced, an Environmental Score is incorporated to reflect recent system dynamics, imitating real ant behavior. This score is calculated from the Modified Base Scores collected over the past hour. The resulting Total Score, obtained as the sum of the Modified Base Score, Threat Score, and Environmental Score, is used as the main indicator for real-time system classification and forecasting of transitions from InC to OutC. This ant colony optimization-inspired approach provides an adaptive and […]
We study a nonlinear analogue of additive commutators, known as \textit{polynomial commutators}, defined by $p(ab) - p(ba)$ for a polynomial $p \in F[x]$ and elements $a, b$ in an algebra $R$ over a field $F$. Originally introduced by Laffey and West for matrices over fields, this notion is here extended to broader algebraic settings. We first show that in division rings, polynomial commutators can generate maximal subfields and even the entire ring as an algebra. In the matrix setting, we prove that matrices similar to ones with zero diagonal are polynomial commutators, and under mild assumptions, every matrix can be written as a product of at most three such commutators. Furthermore, we demonstrate that the matrix algebra can be decomposed as the sum of its center and the linear span of all polynomial commutators. Using the theory of rational identities in division rings, we also exhibit that the trace of a polynomial commutator in the matrix ring can be nonzero in noncommutative cases. Lastly, we explore the size of polynomial commutators via matrix norms.
This paper introduces and investigates some properties of algebras constructed from the algebra of polynomials via derivation and integration operators using a process presented by Dzhumadildaev in a previous work. In particular, we discover new classes of infinite-dimensional simple conservative algebras and describe derivations of these algebras of ranks $1$ and $2$.