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In this paper, the concept of centrally extended α,β-higher derivations is studied. It is shown to be additive in a ring without nonzero central ideals. Also, we prove that in semiprime rings with no nonzero central ideals, every centrally extended α,β-higher derivation is an α,β-higher derivation. Some examples are given to show that our theorems’ assumptions cannot be relaxed. The invariance problem of the center of the ring is also investigated.

In this paper, we introduce the notion of k-derivation, generalized k-derivation and k-reverse derivation on gamma semirings, and we give some commutativity conditions on γ-prime and γ-semiprime gamma semirings. Also, we give orthogonality for pairs of k-reverse derivations on gamma semirings.

In this paper, a modified load-independent $T$ -stress constraint parameter $\\tau *}$ was defined. The $\\tau *}$ of specimens with different crack-tip constraint levels at different $J$ -integrals was calculated, and its load-independence has been validated. Based on the modified constraint parameter $\\tau *}$ and the numerically calculated JaR curves by using the GursonaTvergaardaNeedleman (GTN) model for the SENB specimens with different $a/W$ , the equations of constraint-dependent JaR curves for the A508 steel were obtained. The predicted JaR curves using the equations essentially agree with the experimental and calculated JaR curves. The transferability of the constraint-dependent JaR curves to the CT, SENT and CCT specimens was validated. The results show that the modified constraint parameter $\\tau *}$ and the GTN model can be effectively used to derive the constraint-dependent JaR curves for ductile materials.

Let A be a unital * -algebra containing a nontrivial projection, N be the set of non-negative integers. Under some mild conditions on A , it is shown that any nonlinear mixed Jordan triple * -higher derivation D=dnn∈N is an additive * -higher derivation. In particular, we apply the above result to prime * -algebras and von Neumann algebras with no central summands of type I1 .

A D-derivation of a Lie algebra L is a linear map φ for which there exists a derivation D such that φ([x,y])=[φ(x),y]+[x,D(y)] for all x,y∈L. This paper presents explicit structural results concerning D-derivations in Lie algebras over arbitrary fields. It is established that the set of D-derivations forms a Lie algebra, which decomposes as the sum of derivations and centroids, intersecting precisely at the space of central derivations. For centerless Lie algebras, the inclusion chain for D-derivations within existing derivation classes is completed, resulting in a refined hierarchy. It is proven that for both perfect and centerless Lie algebras, D-derivations decompose as a direct sum of derivations and centroids. In particular, for semisimple Lie algebras, it is shown that DerD(L)=ad(L)⊕C(L), and for simple Lie algebras over an algebraically closed field of characteristic zero, DerD(L)=ad(L)⊕FidL. Furthermore, for any centerless Lie algebra, the Lie algebra of D-derivations is shown to be isomorphic to the semidirect product of the derivation and centroid algebras, with explicit descriptions provided for semisimple and solvable cases. Examples involving so(3), so(1,3), aff(1), and h3 confirm these decompositions and offer matrix realizations of their D-derivations, thereby supporting and illustrating the main theorems.

Motivated by some results on derivations on rings and the generalizations of BCK and BCI-algebras, in 2019, P. Ganesan and N. Kandaraj introduced the notion of derivations on BH - algebras. In this paper, we study the notion of Composition of f - derivations on BH – algebras and investigate simple, interesting and elegant results.

In this article, we study twisted derivations of cyclic group rings. Let R be a commutative ring with unity, G be a finite cyclic group, and ( σ , τ ) be a pair of R -algebra endomorphisms of the group algebra RG , which are R -linear extensions of the group endomorphisms of G . In this article, we give two characterizations concerning ( σ , τ ) -derivations of the group ring RG . First, we develop a necessary and sufficient condition for a ( σ , τ ) -derivation of RG to be inner. Second, we provide a necessary and sufficient condition for an R -linear map D : R G → R G with D ( 1 ) = 0 to be a ( σ , τ ) -derivation. We also illustrate our theorems with the help of examples. As a consequence of these two characterizations, we answer the well-known twisted derivation problem for RG : Under what conditions are all ( σ , τ ) -derivations of RG inner? Or is the space of outer ( σ , τ ) -derivations trivial? More precisely, we give a sufficient condition under which all ( σ , τ ) -derivations of RG are inner and a sufficient condition under which RG has non-trivial outer ( σ , τ ) -derivations. Our result helps in generating several examples of non-trivial outer ( σ , τ ) -derivations.

Let ℜ be a unital algebra over a field k with char(k)≠2, and let ϝ,ξ,ζ:ℜ→ℜ be linear mappings. We say that ϝ is a ξ,ζ-derivation if ϝ(ϑς)=ξ(ϑ)ς+ϑζ(ς)=ζ(ϑ)ς+ϑξ(ς)forallϑ,ς∈ℜ. The mapping ϝ is said to be a Lie ξ,ζ-derivation if ϝ([ϑ,ς])=[ξ(ϑ),ς]+[ϑ,ζ(ς)]forallϑ,ς∈ℜ, where [ϑ,ς]=ϑς−ςϑ denotes the Lie product. In this paper, we prove that if every Lie ξ,ζ-derivation on ℜ is necessarily a ξ,ζ-derivation, then the same property holds for the tensor product algebra ℜ⊗ℑ, where ℑ is any commutative unital algebra. Moreover, every Lie ξ,ζ-derivation of a semiprime algebra is a ξ,ζ-derivation. As a consequence, Lie derivations on tensor products of semiprime algebras with commutative algebras reduce to derivations in the classical sense.

Adsorption processes often include three important components: kinetics, isotherm, and thermodynamics. In the study of solid–liquid adsorption, “standard” thermodynamic equilibrium constant K Eq o ; dimensionless) plays an essential role in accurately calculating three thermodynamic parameters: the standard Gibbs energy change (∆G°; kJ/mol), the standard change in enthalpy (∆H°; kJ/mol), and the standard change in entropy [∆S°; J/(mol × K)] of an adsorption process. Misconception of the derivation of the K Eq o constant that can cause calculative errors in values (magnitude and sign) of the thermodynamic parameters has been intensively reflected through certain kinds of papers (i.e., letters to editor, discussions, short communications, and correspondence like comment/rebuttal). The distribution coefficient (K D) and Freundlich constant (K F) have been intensively applied for calculating the thermodynamic parameters. However, a critical question is whether K D or K F is equal to K Eq o . This paper gives (1) thorough discussion on the derivation of thermodynamic equilibrium constant of solid–liquid adsorption process, (2) reasonable explanation on the inconsistency of (direct and indirect) application of K D or K F for calculating the thermodynamic parameters based on the derivation of K Eq o , and (3) helpful suggestions for improving the quality of papers published in this field.

In this paper, we investigated the problem of describing the form of higher Jordan triple derivations on trivial extension algebras. We show that every higher Jordan triple derivation on a$ 2 $ -torsion free$ * $ -type trivial extension algebra is a sum of a higher derivation and a higher anti-derivation. As for its applications, higher Jordan triple derivations on triangular algebras are characterized.