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Given a locally compact abelian group G and a Quasi-inner product A-module (M, $\phi$) with $\phi$ a positive sesquilinear map on M, we introduce the Fourier transform for the Bochner integrable M-valued functions on G and investigate some of its properties including a Parseval type equality.

Quasi module is a new algebraic structure, based on module, which is composed of a semigroup structure and a partial order accompanied with an external ring multiplication. We proposed this structure in our paper [1] while we were studying the hyperspace C(M) consisting of all nonempty compact subsets of a topological module M over some topological ring R. Quasi module can be considered as a generalisation of module in some sense. In the present paper we have defi ned topological Quasi module and given some examples of it. We have shown that the Cartesian product of arbitrary family of topological Quasi modules is again a topological Quasi module over some topological unitary ring. Finally we have defi ned projective system of topological Quasi modules and projective limit of this system. We have proved various topological properties of the projective limit of a projective system.

Let A be a Banach algebra and E be a Banach A -bimodule then S=A ÅE, the l1-direct sum of A and E becomes a module extension Banach algebra when equipped with the algebras product (a, x): (a', x')=(aa', a.x'+x.a'). In this paper, we investigate D-amenability for these Banach algebras and we show that for discrete inverse semigroup S with the set of idempotents ES, the module extension Banach algebra S=l1 (ES) Å l1 (S) is D-amenable as a l1 (ES) -module if and only if l1 (ES) is amenable as Banach algebra.

In this paper we defined the concept of module amenability of Banach algebras and module connes amenability of module dual Banach algebras. Also we assert the concept of module Arens regularity that is different with [1] and investigate the relation between module amenability of Banach algebras and connes module amenability of module second dual Banach algebras. In the following we study the relation between module amenability, weak module amenability and module approximate amenability of Banach algebra. The notation of amenability of Banach algebras was introduced by B. Johnson in [7]. A Banach algebra A is amenable if every bounded derivation from A into any dual Banach A-bimodule is inner, equivalently if H(A; X) = 0 for any Banach A-bimodule X, where H(A; X) is the first Hochschild co-homology group of A with coefficient in X. Also, a Banach algebra A is weakly amenable if H(A; A) = 0. Bade, Curtis and Dales introduced the notion of weak amenability on Banach algebras in [4]. They considered this concept only for commutative Banach algebras. After that Johnson defined the weak amenability for arbitrary Banach algebras.

Using free simplicial algebras with given CW−basis, it is shown how to construct a free or totally free Quasi 3-crossed module on suitable construction data. Quasi 3-crossed complexes are introduced and similar freeness results are given for these are discussed.

In this paper we define a congruence ~ on inverse semigroup S such that amenability of S is equivalent to amenability of S/ ~. We study module amenability of semigroup algebra i1(S/ ~) when S is an inverse semigroup with idempotents E and prove that it is equivalent to module amenability of i1 (S). The main difference of this action with the more studied trivial action is that in this case the corresponding homomorphic image is a Clifford semigroup rather than a discrete group.

Let (R; m) be a Gorenstein local ring, and M and N be two  nitely-generated modules over R. Nagel proved that if M and N are in the same even liaison class, then one has Hi m(M)  = H im (N) for all i < dimM = dimN. In this paper, we provide a short proof to this result.

There is an increasing demand for low-cost single-phase DC– AC inverters in many applications such as PV systems. These systems may be used without the transformer to improve efficiency and make the whole system lighter, smaller, and easier to install. With the transformerless topology, the system efficiency may be increased by about 2%, and the related cost may be decreased by about 25%. There are large leakage currents in transformerless topologies, especially in photovoltaic systems, where safety issues and electromagnetic interference problems often occur. To overcome such disadvantages, common-ground topologies can be used as they can minimize the leakage current of the transformerless inverter ones. The transformerless semi-Quasi-Z-source inverter (SqZS) with the common-ground structure offers many advantages over conventional single-phase inverters. Leakage current elimination, higher power density, fewer components, and low-cost features make it an attractive option as a micro-inverter in photovoltaic applications. The basic topology of SqZSI is especially suitable for a PV module in low-voltage applications as a low-cost micro-inverter with high-voltage SiC switching devices. However, the unity voltage gain is one of the disadvantages of SqZS inverters, which is referred to as a drawback. In other words, the conventional structure of SqZS is not capable of stepping up the voltage, and the maximum amplitude of AC voltage that can be extracted is equal to the input DC voltage. This paper proposes a modified structure for single-phase inverters (MSqZS) to achieve voltage boost capability. The voltage boost is achieved in a single-stage conversion just by adding an additional series DC blocking capacitor to the basic inverter. It also maintains the common-ground feature. Nonlinear sinusoidal modulation (NLSPWM) is modified to allow the SqZS basic structure to achieve high voltage gain. Although the proposed inverter is modified in topology and modulation, it is not more complex than the basic SqZSI. The proposed inverter has the least number of components among the similar step-up common-ground topologies. In this paper, the closed-loop control is proposed to improve the performance of the MSqZS under variable input voltage, as well as output load and compensation for the undesired and non-ideal effect of the parasitic elements. In addition, the proposed inverter is capable of generating reactive power. Also, the design consideration for series capacitor is analysed for proper capacitor selection. The simulation and experimental results under various closed-loop and open-loop scenarios comply with the IEEE Std 1547 and verify the appropriate performance of the proposed inverter.