In this article, we consider PAIR frames in Banach spaces and introduce Banach PAIR frames. Some various concepts in the frame theory such as frames, Schauder frames, Banach frames and atomic decompositions are considered as special kinds of (Banach) PAIR frames. Some frame-like inequalities for (Banach) PAIR frames are presented. The elements that participant in the construction of (Banach) PAIR frames are characterized. It is shown that a Banach space X has a Banach PAIR frame with respect to a Banach scalar sequence space `, when it is precisely isomorphic to a complemented subspace of `. It is shown that if we are allowed to choose the scalar sequence space, PAIR frames and Banach PAIR frames with respect to the chosen scalar sequence space denote the same concept.

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In this paper we investigate the PAIR difference cordial labelingbehaviour of diamond ladder graph,lattitude ladder, octopus graph,pagodagraph, planter graph and semi jahangir graph. Prime labeling behaviour of plantergraph, duplication of planter graph, fusion of planter graph,switching of plantergraph, joining of two copies of planter graph were studied by A.Edward samueland S.Kalaivani[4]. Dafik, Riniatul Nur Wahidah hve been examined the RainbowAntimagic coloring of special graphs like volcano, sandat graph, sunflower,octpus,semijahangir [3]. Prihandini,R M., at.el have been studied the elegant labeling ofshackle graphs and diamond ladder graphs [16]. Classical meanness of some graphssuch as one-side step graph,double-sided step graph,grid,slanding ladder,diamondladder,lattitude ladder was studied by Alanazi et. al [1]. In [17] Yeni Susanti et.al studied the edge odd geaceful labeling behaviour of prism, antiprism, cartesianproduct graphs. The notion of PAIR diference cordial labeling of a graph was introduced in [7].

This paper presents a modified nine switch inverter with two inputs and two Zsource networks. This inverter has two DC inputs and two AC outputs. Input DC voltages can be boosted to the required level. Amplitude, frequency and phase of AC output voltages can be controlled, independently. The proposed converter can be used in applications with two unregulated DC sources, which require feeding two independent loads. Compared to the conventional structure, the proposed converter requires reduced number of semiconductor switches hence improved converter reliability and less volume. While the authors have already presented a similar inverter (single z-source network DUAL-input DUAL-output inverter) with the above features, new inverter has some differences in the topology, control strategy and operation. In this paper, a new control strategy of space vector modulation for the new inverter is proposed. Also a new switching method utilizing carrier-based modulation concept is presented for both DUAL (proposed) and single z-source network DUAL-input DUAL-output inverters. Performance of the proposed inverter is verified by simulation and experimental results.

Let $(G,M)$ be a PAIR of groups, in which $M$ is a normal subgroup of a group $G.$ We study some properties of $n$-isoclinism of PAIR of groups. In fact, we show that the subgroups and quotient groups of two $n$-isoclinism PAIR of groups are $m$-isoclinic for all $m\le n.$ Moreover, the properties of $\pi$-PAIR and supersolvable PAIR of groups which are invariant under $n$-isoclinism has be studied.