These days, many countries have a water shortage and have limited access to clean water. To overcome this, a new treatment is emerging, namely, the photocatalytic processing of greywater. Photocatalytic processes to remove the organic matter from different greywater sources are critically reviewed. Their efficiency in degrading the organic matter in greywater is scrutinized along with factors that can affect the activity of photocatalysts. Modified TiO₂, ZnO and TiO₂ catalysts show great potential in degrading organic materials that are present in greywater. There are several methods that can be used to modify TiO₂ by using sol-gel, microwave and ultrasonication. Overall, the photocatalytic approach alone is not efficient in mineralizing the organic compounds, but it works well when the photocatalysis is combined with oxidants and Fe³⁺. However, factors such as pH, concentration and catalyst-loading of organic compounds can significantly affect photocatalytic efficiency.

Let $n\ge 2$ be a fixed integer. The aim of this paper is to investigate the properties of n-derivations within the framework of perfect Lie superalgebras over a commutative ring R. The main result shows that if the base ring contains ${\displaystyle \frac{1}{n-1}}$, and L is a perfect Lie superalgebra with a center equal to zero, then any n-derivation of L is necessarily a derivation. Additionally, every n-derivation of the derivation algebra $Der\left(L\right)$ is an inner derivation. Moreover, we extend the concept of n-homomorphisms to mappings between Lie superalgebras L and $L^{\prime }$ and prove that under specific assumptions, homomorphisms, anti-homomorphisms, and their combinations are all n-homomorphisms. Finally, we conclude our paper with some open problems.

For a Leibniz algebra S over a field k, we show that the space of central derivations satisfies Derz(S) ∼= Hom(S/[S, S], Z(S)). We then study the Daletskii–Takhtajan type functor DTn and prove that the induced map DTn∗ : Der(S) → Der(DTn(S)) restricts to an injective map Derz(S) ,→ Derz(DTn(S)), which becomes an isomorphism when S is DTngenerated.