A generic rectangular partition   is a partition of a rectangle into a finite number of rectangles provided that no four of them meet at a point.  A graph $\mathcal{H}$ is called  dual of a plane graph $\mathcal{G}$ if there is one$-$to$-$one correspondence between the vertices of $\mathcal{G}$ and the regions of $\mathcal{H}$, and  two vertices of $\mathcal{G}$ are adjacent if and only if the corresponding regions of $\mathcal{H}$ are adjacent. A plane graph is a  rectangularly dualizable graph  if its dual  can be embedded as a  rectangular partition.   A rectangular dual  $\mathcal{R}$ of a plane graph $\mathcal{G}$ is a partition of a  rectangle  into $n-$rectangles such that (i) no four rectangles of $\mathcal{R}$ meet at a point, (ii)  rectangles in  $\mathcal{R}$ are mapped to vertices of $\mathcal{G}$,  and (iii)  two rectangles in $\mathcal{R}$ share a common boundary segment if and only if the corresponding vertices are adjacent in $\mathcal{G}$. In this paper, we derive a necessary and sufficient  for a rectangularly dualizable graph $\mathcal{G}$ to admit a unique rectangular dual upto combinatorial  equivalence. Further we show that $\mathcal{G}$ always admits   a slicible as well as an area$-$universal  rectangular dual.

In this paper we get some results and applications for duals and approximate duals of g-frames in Hilbert spaces. In particular, we consider the stability of duals and approximate duals under bounded operators and we study duals and approximate duals of g-frames in the direct sum of Hilbert spaces. We also obtain some results for perturbations of approximate duals.

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module‎. ‎The purpose of this paper is to introduce and investigate the submodules of an $R$-module $M$ which satisfy the dual of Property $\mathcal{A}$‎, ‎the dual of strong Property $\mathcal{A}$‎, ‎and the dual of proper strong Property $\mathcal{A}$‎. ‎Moreover‎, ‎a submodule $N$ of $M$ which satisfy Property $\mathcal{S_J(N)}$ and Property $\mathcal{I^M_J(N)}$ will be introduced and investigated‎.

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module‎. ‎The purpose of this paper is to introduce and investigate the submodules of an $R$-module $M$ which satisfy the dual of Property $\mathcal{A}$‎, ‎the dual of strong Property $\mathcal{A}$‎, ‎and the dual of proper strong Property $\mathcal{A}$‎. ‎Moreover‎, ‎a submodule $N$ of $M$ which satisfy Property $\mathcal{S_J(N)}$ and Property $\mathcal{I^M_J(N)}$ will be introduced and investigated‎.

There has been considerable progress in the study of hypergroup algebras. In some papers, we have already studied the structure of L(X), the second dual of L(X), and related problems. Here we examine the weighted case, and among other results we generalize the above results to the weighted case.

‎The sequences of the form {Embgn} m,nÎZ, where Emb is the modulation operator‎, ‎b>0 and gn is the‎ ‎window function in L2(R)‎, ‎construct Fourier-like‎ ‎systems‎. ‎We try to consider some sufficient conditions on the window‎ ‎functions of Fourier-like systems‎, ‎to make a frame and find a dual‎ ‎frame with the same structure‎. ‎We also extend the given two Bessel‎ ‎Fourier-like systems to make a pair of dual frames and prove that‎ ‎the window functions of Fourier-like Bessel sequences share the‎ ‎compactly supported property with their extensions‎. ‎But for‎ ‎polynomials windows‎, ‎a result of this type does not happen.