Let M be a weak Nobusawa -ring and γ be a non-zero element of Γ. In this paper, we introduce concept of k-reverse derivation, Jordan k-reverse derivation, generalized k-reverse derivation, and Jordan generalized k-reverse derivation of Γ-ring, and γ-homomorphism, anti-γ-homomorphism of M. Also, we give some commutattivity conditions on γ-prime Γ-ring and γ-semiprime Γ-ring .

Let M be weak Nobusawa Γ-ring and γ be non-zero element of Γ. In this paper, we study the definition of γ-semi prime Γ-ring, and introduce definition of γ-semi simple Γ-ring and γ-Jordan ideal of M. We give equivalent definitions for γ-prime Γ-ring and γ-semi prime Γ-ring. Also by using relation between Γ-rings and rings, we give some commutativity conditions on Γ-rings.

Inithis paper, wepintroduce thepconcept ofporthogonal generalizedpderivations on Γ-semirings and provepsome resultslon semiprimeΓ-semirings.

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.

This study examined the how a logarithmic intensity function could be used to develop a segmentation model for intensity and noisy homogeneity images. Whereas inhomogeneous images demand the use of local image data, it remains notable that it remains defective for noisy images. The eventuality is that active contour motions tend to be misguided by the local information. In the proposed model, the logarithmic function was able to capture minute details contained in selected images and also counter or ignore the perceived noise. As such, the model proved effective and robust, worth applying to such images. To verify the effectiveness of the model, its results were compared to the experimental outcomes previously reported after employing local Chan-Vese Model. Indeed, the proposed model exhibited superior performance in relation to the treatment of intensity and noisy inhomogeneity images.