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...

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...

We study the oscillatory and nonoscillatory behavior of solutions a class forced impulsive nonlinear neutral differential systems form $$ \left\{\begin{array}{c}{\left(r(t){\left(y(t)+p(t)y\left(t-\tau \right)\right)}^{\prime}\right)}^{\prime }+q(t)G\left(y\left(t-\sigma \right)\right)=f(t),\kern1.75em t\ne {\tau}_k,\kern0.75em k\ \epsilon\ \mathbb{N},\\ {}\Delta \left(r\left({\tau}_k\right){\l...

we describe under various conditions abelian subgroups of the automorphism‎ ‎group $mathrm{aut}(t_{n})$ of the regular $n$-ary tree $t_{n}$‎, ‎which are‎ ‎normalized by the $n$-ary adding machine $tau =(e‎, ‎dots‎, ‎e,tau )sigma _{tau‎ ‎}$ where $sigma _{tau }$ is the $n$-cycle $left( 0,1‎, ‎dots‎, ‎n-1right) $‎. ‎as‎ ‎an application‎, ‎for $n=p$ a prime number‎, ‎and for $n=4$‎, ‎we prove that...

We study the transition between phases at large $R$-charge on a conformal manifold. These are characterized by behaviour of lowest operator dimension $\Delta(Q_R)$ for fixed and $Q_R$. focus, as an example, $D=3$, $\mathcal{N}=2$ Wess-Zumino model with cubic superpotential $W=XYZ+\frac{\tau}{6}(X^3+Y^3+Z^3)$, compute $\Delta(Q_R,\tau)$ using $\epsilon$-expansion in three interesting limits. In ...

Single potassium M-channels in rat sympathetic neurons have multiple voltage-dependent kinetic components in their activity: short, medium, and long closed times (tau(CS), tau(CM), and tau(CL)) and short and long open times (tau(OS) and tau(OL)). All five components can be detected in cell-attached patches, but only four of them (tau(CS), tau(CM), tau(OS), and tau(OL)) in excised patches (, J. ...

In this paper, we obtain asymptotic formula on the "hyperbolic" summation \begin{equation*} \underset{mn\leq x}{\sum }D_{k}\left( \gcd \left( m,n\right) \right) \text{ \ }\left( k\in \mathbb{Z}_{\geq 2}\right), \end{equation*} such that $D_{k}\left( n\right) = \dfrac{\tau _{k}\left( }{\tau_{k}^{\ast }$, where $\tau =\!\!\sum\limits_{n_{1}n_{2}\ldots n_{k}=n}\!\!1$ denotes Piltz divisor function...

A bstract In $$ {\Lambda}_b^0\to {\Lambda}_c^{+}\left(\to {\Lambda}^0{\pi}^{+}\right){\tau}^{-}{\overline{v}}_{\tau } ? b 0 ? c + ? ...

"The goal of this paper is to study the nonexistence nontrivial solutions following Cauchy problem $$\left\{ \begin{array}{ll} u_{tt}+(-\Delta)^{\beta/2} u+u_{t}=\displaystyle\int_{0}^{t}\left(t-\tau \right) ^{-\gamma}\left\vert u(\tau ,\cdot) \right\vert^{p}d\tau,\\ \cr u(0,x)=u_{0}(x),\quad u_t(0,x)=u_1(x),\quad x\in\mathbb{R}^n, \end{array}\right.$$ where $p>1,\ 0<\gamma <1,\,\, \be...

Left ventricular relaxation time constant, Tau, is the best index to evaluate left ventricular diastolic function, but the measurement is only available traditionally in catheter lab. In Echo lab, several methods of non-invasive measurement of Tau have been tried since 1992, however almost all the methods are still utilizing the same formula to calculate Tau as in catheter lab, which makes them...