ziziphus spina-christi are distributed in arid and semi-arid regions of world. most of these areas are subjected to soil salinity. so, the aim of this study was to find out the effects of different nacl concentrations on z. spina-christi seedlings growth, in the presence of a number of single and dual amf inoculums, to provide some information about possible effects of amf under salinity condit...

In this paper, we introduced and characterized the controlled $g$-duals of a frame in separable Hilbert space $\mathcal{H}$ . Afterwards, obtained new $C$-controlled $g$-dual frames from given frames. addition, approximation for was defined some their properties were investigated. Finally, relationship between approximately dual $g$-dual.

Let G be a graph with adjacency matrix A and let F be a field. An F-matrix Q is a support matrix of G if A = [Q ̸=O], the zero-nonzero pattern of Q. If G has an invertible skew-symmetric support F-matrix S, the S-dual G of G is defined as the graph with adjacency matrix [S−1 ̸= O]. An analogous adjacency matrix dual, G has been examined in the literature for those bipartite graphs G with unique p...

Let {xn} be a frame for a Hilbert space H. We investigate the conditions under which there exists a dual frame for {xn} which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether {xn} can be dilated to an orthonormal basis (under an oblique projection). A necessary and sufficient condition for the existence of Parseval duals is ...

The Lagrange dual function is: g(u, v) = min x L(x, u, v) The corresponding dual problem is: maxu,v g(u, v) subject to u ≥ 0 The Lagrange dual function can be viewd as a pointwise maximization of some affine functions so it is always concave. The dual problem is always convex even if the primal problem is not convex. For any primal problem and dual problem, the weak duality always holds: f∗ ≥ g...

Some basic results on duality of infinite graphs are established and it is proven that a block has a dual graph if and only if it is planar and any two vertices are separated by a finite edge cut. Also, the graphs having predual graphs are characterized completely and it is shown that if G* is a dual and predual graph of G, then G and G* can be represented as geometric dual graphs. The uniquene...