Abstract:
We prove that for any prime number $p$, every finite non-abelian $p$-group $G$ of class 2 has a noninner automorphism of order $p$ leaving either the Frattini subgroup $\Phi(G)$ or $\Omega_1(Z(G))$ elementwise fixed.

Abstract:
A 22 year-old woman presented with gradual visual loss in her right eye since 1990. Medical and family histories were unremarkable. Her visual acuity was 20/80 and 20/20 in right and left eyes respectively. Slit lamp examination was quite normal with no relative afferent papillary defect.Fundi of both eyes revealed subretinal lesion with optic nerve head involvement and subretinal fluid in papillomacular bundle with macular pucker in right eye. Whole body MRI revealed a large hemangioma in the spinal canal. In 1998 the patient ahsd significant visual loss at both eyes. This is the first report of Von hippel disease with bilateral optic nerve hemangioma in Iran.

Abstract:
Let ()=,∈, for some positive integer and the composition operator on the Dirichlet space induced by . In this paper, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators ？,？ and self-commutators of , which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.

Abstract:
Let $R_n(G)$ denotes the set of all right $n$-Engel elements of a group $G$. We show that in any group $G$ whose 5th term of lower central series has no element of order 2, $R_3(G)$ is a subgroup. Furthermore we prove that $R_4(G)$ is a subgroup for locally nilpotent groups $G$ without elements of orders 2, 3 or 5; and in this case the normal closure $^G$ is nilpotent of class at most 7 for each $x\in R_4(G)$. Using a group constructed by Newman and Nickel we also show that, for each $n\geq 5$, there exists a nilpotent group of class $n+2$ containing a right $n$-Engel element $x$ and an element $a\in G$ such that both $[x^{-1},_n a]$ and $[x^{k},_n a]$ are of infinite order for all integers $k\geq 2$. We finish the paper by proving that at least one of the following happens: (1) There is an infinite finitely generated $k$-Engel group of exponent $n$ for some positive integer $k$ and some 2-power number $n$. (2) There is a group generated by finitely many bounded left Engel elements which is not an Engel group.

Abstract:
We prove that the set of right 4-Engel elements of a group $G$ is a subgroup for locally nilpotent groups $G$ without elements of orders 2, 3 or 5; and in this case the normal closure $^G$ is nilpotent of class at most 7 for each right 4-Engel elements $x$ of $G$.

Abstract:
In this paper we show that every g-frame for an \linebreak infinite dimensional Hilbert space $\mathcal{H}$ can be written as a sum of three g-orthonormal bases for $\mathcal{H}$. Also, we prove that every g-frame can be represented as a linear combination of two g-orthonormal bases if and only if it is a g-Riesz basis. Further, we show each g-Bessel multiplier is a Bessel multiplier and investigate the inversion of g-frame multipliers. Finally, we introduce the concept of controlled g-frames and weighted g-frames and show that the sequence induced by each controlled g-frame (resp. weighted g-frame) is a controlled frame (resp. weighted frame).

Abstract:
Experiments were conducted during 1996-1997 in Nough of Rafsanjan to evaluate double pollination on fruit set and development of pistachio nut. In the first experiment, Owhadi cultivar was pollinated by a combination of pollen from Beneh (P. mutica F&M), Atlantica (P. atlantica Desf) and Soltani (P.vera L.). In the second experiment, Owhadi cultivar was pollinated by pollen from Beneh, Atlantica and Khinjuk (P. Khinjuk). The results showed that in the double pollination experiment, the nut, kernel and fruit set were affected more by the first pollen than by the second one. Pollen from the wild pistachio species reduced kernel weight, number of split nuts but increased percentage of the deformity and blank nuts in Owhadi. It was concluded that the effectiveness of the first pollen on fruit set, nut and kernel development was independent of the second one. The pollen of P. vera proved to be the best pollen source for pistachio cultivars.

Abstract:
We associate a graph $\Gamma_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G | \left \text{is cyclic for all} y\in G\}$, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of $\Gamma_G$ is finite if and only if $\Gamma_G$ has no infinite clique. We prove that if $G$ is a finite nilpotent group and $H$ is a group with $\Gamma_G\cong\Gamma_H$ and $|Cyc(G)|=|Cyc(H)|=1$, then $H$ is a finite nilpotent group. We give some examples of groups $G$ whose non-cyclic graphs are ``unique'', i.e., if $\Gamma_G\cong \Gamma_H$ for some group $H$, then $G\cong H$. In view of these examples, we conjecture that every finite non-abelian simple group has a unique non-cyclic graph. Also we give some examples of finite non-cyclic groups $G$ with the property that if $\Gamma_G \cong \Gamma_H$ for some group $H$, then $|G|=|H|$. These suggest the question whether the latter property holds for all finite non-cyclic groups.

Abstract:
We associate a graph $\mathcal{C}_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G | < x,y> \text{is cyclic for all} y\in G\}$ is called the cyclicizer of $G$, and join two vertices if they do not generate a cyclic subgroup. For a simple graph $\Gamma$, $w(\Gamma)$ denotes the clique number of $\Gamma$, which is the maximum size (if it exists) of a complete subgraph of $\Gamma$. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group $G$ is solvable whenever $w(\mathcal{C}_G)<31$ and the equality for a non-solvable group $G$ holds if and only if $G/Cyc(G)\cong A_5$ or $S_5$.