We discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module. Let A be a Banach algebra and X a Banach A-module, f : X → X and g : A → A. The mappings Δ1 f,g , Δ2 f,g , Δ3 f,g , and Δ4 f,g are defined and it is proved that if ‖Δ1 f,g x, y, z,w ‖ resp., ‖Δ3 f,g x, y, z,w, α, β ‖ is dominated by φ x, y, z,w , then f is a generalized re...

The separating space of a derivation onA is a separating ideal [2, Chapter 5]; it also satisfies the same property for the left products. The following assertions are of the most famous conjectures about derivations on Banach algebras: (C1) every derivation on a Banach algebra has a nilpotent separating ideal; (C2) every derivation on a semiprime Banach algebra is continuous; (C3) every derivat...

We define four families of word-rewriting systems: the prefix/suffix systems and the left/right systems. The rewriting of prefix systems generalizes the prefix rewriting of systems: a system is prefix (suffix) if a left hand side and a right hand side are overlapping only by prefix (suffix). The rewriting of right systems generalizes the mechanism of transducers: a system is right (left) if a l...

let a be a unital r-algebra and m be a unital a-bimodule. it is shown that every jordan derivation of the trivial extension of a by m, under some conditions, is the sum of a derivation and an antiderivation.

Abstract In this article, we define the following: Let N 0 {{\mathbb{N}}}_{0} be set of all nonnegative integers and D = ( d i ) ∈ D={\left({d}_{i})}_...

Given an associative algebra satisfying the left commutativity identity abc = bac (Perm-algebra) with a derivation d, new operation a°b=ad(b) is left-symmetric (pre-Lie). We derive necessary and sufficient conditions for to be embeddable into differential Perm-algebra.

The frontal aslant tract (FAT) is a recently described major connection between the preSMA and Broca's area, whose functional role remains undefined. In this study we examined a patient presenting a morphological overregularization strategy in a verb generation task during awake surgery. This specific language deficit coincided with brain tumor resection at the level of the left FAT. During the...