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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.

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.

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.

In this paper, we introduce definitions of γ-orthogonality for two pairs of k-derivations, generalized k-derivations, and k-reverse derivations. And we present some results concerning with these notions on γ-semiprime gamma ring.

In this paper, we study derivations in group rings completed by various types of norms. The main attention is paid to the class of groups in which conjugations act in a controlled manner in some sense. Using the method of identifying derivations and characters on some category, we obtain an alternative way of proving that for this class of groups all derivations are inner.

This paper mainly records the theoretical system of gravitational waves and the mathematical derivation process. At the same time, it also mentions the working principle of the laser interference gravitational wave observatory. It belongs to the learning of the gravitational wave entry-level, and it can also help you understand and learn gravity waves faster from a professional point of view. The end of the article contains some of my discussions on the future development prospects of gravitational waves.

This work establishes a unified structural theory for non-global Lie higher derivations on triangular algebras T, without assuming the existence of a unit element. The primary contribution is the introduction of extreme non-global Lie higher derivations and the proof that every non-global Lie higher derivation on T admits a unique decomposition into three components: a higher derivation, an extreme non-global Lie higher derivation, and a central map vanishing on all commutators [x,y], where x,y∈T satisfy xy=0. This general framework is then explicitly applied to describe such derivations on two significant classes of algebras: upper triangular matrix algebras over faithful algebras and over semiprime algebras. By encompassing both unital and non-unital cases within a single characterization, the theory developed here not only generalizes numerous earlier results but also substantially expands the scope of the existing research landscape.