and Applied Analysis 3 From 2.3 we obtain c 1 b b 1 2b 1 b 1 ⇒ b 1 b 1 2b 1 3b 2 2b 1 , 2.4 which implies b2 − b − 1 0. Hence b x1 or b x2. From this and 2.3 it follows that a b c x1 or a b c x2, from which the result follows. Theorem 2.3. Equation 1.1 has no prime period-four solutions. Proof. Let xn n −2 be a prime period-four solution of 1.1 and x−2 a, x−1 b, x0 c. Then we have x1 x0x−2 − 1 ...

Let $G$ be a finite group and $pi(G)$ be the set of all prime divisors of $|G|$. The prime graph of $G$ is a simple graph $Gamma(G)$ with vertex set $pi(G)$ and two distinct vertices $p$ and $q$ in $pi(G)$ are adjacent by an edge if an only if $G$ has an element of order $pq$. In this case, we write $psim q$. Let $|G= p_1^{alpha_1}cdot p_2^{alpha_2}cdots p_k^{alpha_k}$, where $p_1

let $g$ be a finite group. we construct the prime graph of $ g $,which is denoted by $ gamma(g) $ as follows: the vertex set of thisgraph is the prime divisors of $ |g| $ and two distinct vertices $ p$ and $ q $ are joined by an edge if and only if $ g $ contains anelement of order $ pq $.in this paper, we determine finite groups $ g $ with $ gamma(g) =gamma(l_3(q)) $, $2 leq q < 100 $ and prov...

Let $G$ be a finite group. The prime graph of $G$ is a graph $Gamma(G)$ with vertex set $pi(G)$, the set of all prime divisors of $|G|$, and two distinct vertices $p$ and $q$ are adjacent by an edge if $G$ has an element of order $pq$. In this paper we prove that if $Gamma(G)=Gamma(G_2(5))$, then $G$ has a normal subgroup $N$ such that $pi(N)subseteq{2,3,5}$ and $G/Nequiv G_2(5)$.

‎let $r$ be a domain with quotiont field $k$‎, ‎and‎ ‎let $n$ be a submodule of an $r$-module $m$‎. ‎we say that $n$ is‎ ‎powerful (strongly primary) if $x,yin k$ and‎ ‎$xymsubseteq n$‎, ‎then $xin r$ or $yin r$ ($xmsubseteq n$‎ ‎or $y^nmsubseteq n$ for some $ngeq1$)‎. ‎we show that a submodule‎ ‎with either of these properties is comparable to every prime‎ ‎submodule of $m$‎, ‎also we show tha...

let $n$ be a submodule of a module $m$ and a minimal primary decomposition of $n$ is known‎. ‎a formula to compute baer&apos;s lower nilradical of $n$ is given‎. ‎the relations between classical prime submodules and their nilradicals are investigated‎. ‎some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

Families of pseudorandom sequences with low cross correlation have important applications in communications and cryptography. Among several known constructions of sequences with low cross correlations, interleaved constructions proposed by Gong uses two sequences of the same period with two-level autocorrelation. In this paper, we study the balance property and the cross correlation of interlea...

This Case describes two similar outsourced IS projects that took place in Wellington, New Zealand (NZ) during the period 1992-99, with emphasis on the reasons why one project failed (the prime contractor repudiated the contract and the project was abandoned) and why the other succeeded. The same prime contractor was used for both projects and there were a number of other similarities, yet the o...

Abstract We present the generalized optical theorem and its applications with special emphasis on roles of bound states. First, we prove which gives a necessary sufficient condition for function $\langle {\bf {\it k}^{\prime}} | T k}} \rangle$ two variables $\ k}}^{\prime }$ ${\bf k}}$ to be physically acceptable as half-on-shell T-matrix, i.e., have an underlying Hermitian potential V. Second,...