We introduce a new g-frame (singleton g-frame), g-orthonormal basis and g-Riesz basis and study corresponding notions in some other generalizations of frames.Also, we investigate duality  for some kinds of g-frames. Finally, we illustrate an example which provides a  suitable translation from discrete frames to Sun's g-frames.

Broomcorn seed (Sorghum vulgare) was used as raw material for bioethanol production. Optimum conditions were obtained from response surface method. Broomcorn seed flour (45 g/l) was treated by alkaline treatment and dual enzymatic hydrolysis (0.7 g/l of α- amylase and 0.42 g/l of amyloglucosidase). The hydrolyzed total sugar of 25.5 g/L was used in conventional bioethanol production (8.1 g/l) u...

Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X\{u})| + d(u) ≥ n − 1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C7 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph (Kr ∪ Ks ∪ Kt) ∨ K2, 1 ≤ r ...

This function generalizes many particular functions. For f( k) = k! one gets the Smarandache function, while for f(k) = k(k: 1) one has the Pseudo-Smarandache function Z (see [1], [4-5]). In the above paper [3] we have defined also dual arithmetic functions as follows: Let 9 : N* -+ N* be a function having the property that for each n 2:: 1 there exists at least a k 2:: 1 such that g(k)ln. Let ...

In a recent paper [11] we answered to the negative a question raised in the book by Eklof and Mekler [8, p. 455, Problem 12] under the set theoretical hypothesis of ♦א1 which holds in many models of set theory. The Problem 12 in [8] reads as follows: If A is a dual (abelian) group of infinite rank, is A ∼= A⊕Z? The set theoretic hypothesis we made is the axiom ♦א1 which holds in particular in G...

We present an overview of the first combinatorial results for the so-called geometric RAC simultaneous drawing problem (or GRacSim drawing problem, for short), i.e., a combination of problems on geometric RAC drawings [3] and geometric simultaneous graph drawings [2]. According to this problem, we are given two planar graphs G1 = (V,E1) and G2 = (V,E2) that share a common vertex set but have di...

‎we present an application of the dual gabor frames to image‎ ‎processing‎. ‎our algorithm is based on finding some dual gabor‎ ‎frame generators which reconstructs accurately the elements of the‎ ‎underlying hilbert space‎. ‎the advantages of these duals‎ ‎constructed by a polynomial of gabor frame generators are compared‎ ‎with their canonical dual‎.