This article presents a unified approach to the abstract notions of partial convolution and involution in $L^p$-function spaces over semi-direct product of locally compact groups. Let $H$ and $K$ be locally compact groups and $tau:Hto Aut(K)$ be a continuous homomorphism.  Let $G_tau=Hltimes_tau K$ be the semi-direct product of $H$ and $K$ with respect to $tau$. We define left and right $tau$-c...

A generalized Stirling numbers of the second kind $ S_{a,b}\left(p,k\right) $, involved in expansion \left(an+b\right)^{p} = \sum_{k 0}^{p}k!S_{a,b}\left(p,k\right) \binom{n}{k} where a \neq 0, b are complex numbers, have studied [16]. In this paper, we show that Bernoulli polynomials B_{p}(x) can be written terms S_{1,x}\left(p,k\right) and then use known results for to obtain several new expl...

A well-known result of Ajtai Komlós, Pintz, Spencer, and Szemerédi (J. Combin. Theory Ser. 32 (1982), 321–335) states that every k $$ -graph H on n vertices, with girth at least five, average degree t − 1 {t}^{k-1} contains an independent set size c ( log ) / cn{\left(\log t\right)}^{1/\left(k-1\right)}/t for some > 0 c>0 . In this paper we show the same can be found under weaker conditions all...

Recently, Ben Green proved that the two-color van der Waerden number ω(3, k) is bounded from below by $${k^{{b_0}\left( k \right)}}$$ where $${b_0}\left( \right) = {c_0}{\left( {{{\log \,k} \over {\log \log \,k}}} \right)^{1/3}}$$ . We prove a new lower bound of kb(k) with $$b\left( {{c\log \,k}}$$ This done modifying Green’s argument, replacing complicated result about random quadratic forms a...

Let $d(n)$ and $d^*(n)$ be, respectively, the number of divisors unitary an integer $n\geq 1.$ A divisor $d$ is to be said if it prime over $\frac{n}{d}.$ In this paper, we study mean value function $D(n)=\frac{d(n)}{d^*(n)}$ in arithmetic progressions $ \left\lbrace l+mk \mid m\in\mathbb{N}^* \text{ } (l, k)=1 \right\rbrace;$ leads back real $x\mapsto S(x;k,l)=\underset{n\equiv l[k]}{\sum\limi...

this article presents a unified approach to the abstract notions of partial convolution and involution in $l^p$-function spaces over semi-direct product of locally compact groups. let $h$ and $k$ be locally compact groups and $tau:hto aut(k)$ be a continuous homomorphism.  let $g_tau=hltimes_tau k$ be the semi-direct product of $h$ and $k$ with respect to $tau$. we define left and right $tau$-c...