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L. Gǎvruţa (2012) introduced a special kind of frames, named K-frames, where K is an operator, in Hilbert spaces, which is significant in frame theory and has many applications. In this paper, first of all, we have introduced the notion of approximative K-atomic decomposition in Banach spaces. We gave two characterizations regarding the existence of approximative K-atomic decompositions in Banach spaces. Also some results on the existence of approximative K-atomic decompositions are obtained. We discuss several methods to construct approximative K-atomic decomposition for Banach Spaces. Further, approximative _d-frame and approximative _d-Bessel sequence are introduced and studied. Two necessary conditions are given under which an approximative _d-Bessel sequence and approximative _d-frame give rise to a bounded operator with respect to which there is an approximative K-atomic decomposition. Example and counter example are provided to support our concept. Finally, a possible application is given.

The Protection of Consumers (Trade Descriptions) bill which, owing to the General Election, did not quite make the Statute Book in the last Parliament, is, at the moment of writing, passing through its readings, with every likelihood of becoming law in the near future. It has been criticised for the extent of the control to be exercised over general trading and that in “coddling the customer” it will place unreasonable responsibilities upon retailers. In fact, it is impossible to foresee just how far its provisions may extend, but there will be few who will disagree that new and more searching requirements are long overdue.

Renewable energy alternatives and nanoscale materials have gained huge attention in recent years due to the problems associated with fossil fuels. The recyclable battery is one of the recent developments to address the energy requirement issues. In this work, the development of nanoscale materials is focused on using green synthesis methods to address the energy requirements of hybrid electric vehicles.

The current research focuses on developing metal oxide nanoscale materials (NANO-SMs). The Zno-Aloe vera NANO-SM is prepared using the green synthesis method. The developed nanoscale materials are characterized using analysis methods like FESEM, TEM, XRD and FTIR.

The average size of ZnO-Aloe vera mono-crystalline was recorded as 60–70 nm/Hexagonal shape. The nanoscale materials are used for the detection of LPG gases. The sensitivity observed was 48%. The response time and recovery time were recorded as 8–10 s and 230–250 s, respectively. The average size of SnO₂-green papaya leaves poly-crystalline was recorded as 10–20 nm/powder form.

Nanoscale materials are developed using green synthesis methods for hybrid vehicle applications. The nanoscale materials are used for the detection of harmful gases in hybrid vehicles.