The second author introduced notions of weak permutablity and commutativity between groups and proved the finiteness of a group generated by two weakly permutable finite subgroups. Two groups H, K weakly commute provided there exists a bijection f : H → K which fixes the identity and such that h commutes with its image h for all h ∈ H. The present paper gives support to conjectures about the ni...

Let G be an abelian group of finite order n, and let h be a positive integer. A subset A of G is called weakly h-incomplete if not every element of G can be written as the sum of h distinct elements of A; in particular, if A does not contain h distinct elements that add to zero, then A is called weakly h-zero-sum-free. We investigate the maximum size of weakly hincomplete and weakly h-zero-sum-...

in this paper, we first discuss about canonical dual of g-frameλp = {λip ∈ b(h, hi) : i ∈ i}, where λ = {λi ∈ b(h, hi) :i ∈ i} is a g-frame for a hilbert space h and p is the orthogonalprojection from h onto a closed subspace m. next, we provethat, if λ = {λi ∈ b(h, hi) : i ∈ i} and θ = {θi ∈ b(k, hi) :i ∈ i} be respective g-frames for non zero hilbert spaces hand k, and λ and θ are unitarily e...

Given a graph H on vertex set {1, 2, · · · , n} and a function f : [0, 1] → R, define ‖f‖H := ∣∣∣∣∣ ∫ ∏ ij∈E(H) f(xi, xj)dμ |V (H)| ∣∣∣∣∣ 1/|E(H)| , where μ is the Lebesgue measure on [0, 1]. We say that H is norming if ‖·‖H is a semi-norm. A similar notion ‖·‖r(H) is defined by ‖f‖r(H) := ‖|f |‖H and H is said to be weakly norming if ‖·‖r(H) is a norm. Classical results show that weakly normin...

Let G be an abelian group and let R be a commutative G-graded super-ring (briefly, graded super-ring) with unity 1 6= 0. We say that a ∈ h(R), where h(R) is the set of homogeneous elements in R, is weakly prime to a graded superideal I of R if 0 6= r a ∈ I , where r ∈ h(R), then r ∈ I . If ν(I ) is the set of homogeneous elements in R that are not weakly prime to I , then we define I to be weak...