In the present paper, we study simple algebras, which do not belong to the well-known classes of algebras (associative algebras, alternative algebras, Lie algebras, Jordan algebras, etc.). The simple ...finite-dimensional algebras over a field of characteristic 0 without finite basis of identities, constructed by Kislitsin, are such algebras. In the present paper, we consider two such algebras: the simple seven-dimensional anticommutative algebra \(\mathcal{D}\) and the seven-dimensional central simple commutative algebra \(\mathcal{C}\). We prove that every local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is a derivation, and every 2-local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also a derivation. We also prove that every local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is an automorphism, and every 2-local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also an automorphism.
In this paper we define two types of implicative derivations on pseudo-BCK algebras, we investigate their properties and we give a characterization of isotone implicative derivations. The ...multiplicative derivations on BCK-algebras with product are introduced and investigated as well. At the same time, particular cases of multiplicative derivations are defined and their properties are investigated. Moreover, we prove that two good ideal multiplicative derivations coincide if and only if their corresponding sets of fixed points coincide. We also define the translation and scaling derivation operators, we study their properties and we show that these operators form a residuated pair. Finally, we prove that there exists an order preserving bijection between the fixed points sets of the two operators.
This paper proposes a review focused on exotic chemotaxis and cross-diffusion models in complex environments. The term exotic is used to denote the dynamics of models interacting with a time-evolving ...external system and, specifically, models derived with the aim of describing the dynamics of living systems. The presentation first, considers the derivation of phenomenological models of chemotaxis and cross-diffusion models with particular attention on nonlinear characteristics. Then, a variety of exotic models is presented with some hints toward the derivation of new models, by accounting for a critical analysis looking ahead to perspectives. The second part of the paper is devoted to a survey of analytical problems concerning the application of models to the study of real world dynamics. Finally, the focus shifts to research perspectives within the framework of a multiscale vision, where different paths are examined to move from the dynamics at the microscopic scale to collective behaviors at the macroscopic scale.
Let $R$ be a prime ring, $\alpha$ an automorphism of $R$ and $b$ an element of $Q$, the maximal right ring of quotients of $R$. The main purpose of this paper is to characterize skew $b$-derivations ...in prime rings which satisfy various differential identities. Further, we provide an example to show that the assumed restrictions cannot be relaxed.
Let
M
be a finite von Neumann algebra with no central summands of type I
1
. We show that each nonlinear 2-local Lie n-derivation
δ
:
M
→
M
with
n
≥
3
is of the form d + h, where
d
:
M
→
M
is a ...linear derivation and h is a homogeneous central-valued mapping which annihilates each
(
n
−
1
)
-th commutator of
M
.
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