In this paper, we investigate the Fourier coefficients of eta-quotients forms q λ 0 + l k 1 24 η ( τ ) − , $$ {q}^{\frac{\lambda_0+{l}^k{\lambda}_1}{24}}\eta {\left(\tau \right)}^{-{\lambda}_0}\eta {\left({l}^k\tau \right)}^{-{\lambda}_1}, where \eta \left(\tau \right) is Dedekind eta function, = 5,7 11 13 l=5,7,11,13 and 17; a positive integer, {\lambda}_0,{\lambda}_1 are arbitrary integers. W...

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...

"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...

BACKGROUND The preclinical patterns of biological markers for Alzheimer's disease (AD) in vivo need further exploration. The aim of this study was therefore to investigate CSF biomarkers, regional cerebral blood flow (rCBF) and cognitive performance in cognitively healthy older individuals. METHOD Within a 2-week period, 32 cognitively healthy older individuals underwent CSF analysis, rCBF me...

The chiral structure of supersymmetric particle couplings involving third generation Standard Model fermions depends on left-right squark and slepton mixings as well as gaugino-higgsino mixings. The shapes and intercorrelations of invariant mass distributions of a first or second generation lepton with bottoms and taus arising from adjacent branches of SUSY cascade decays are shown to be a sens...

A bstract We study the d 2 Γ /( dωd cos θ ), / and dE distributions, which are defined in terms of visible energy polar angle charged particle from τ − decay $$ b\to c\tau \left(\mu {\overline{\nu}}_{\mu }{\nu}_{\tau },{\pi \nu}_{\tau },{\rho \nu}_{\tau}\right){\overline{\nu}}_{\tau } b → cτ</mml...

Abstract I present a method for estimating the fidelity F ( μ , τ ) between preparable quantum state and classically specified pure target $$\tau =\left|\tau \right\rangle \left\langle \tau \right|$$ τ = using simple circuits on-the-fly clas...

A bstract The semitauonic $$ {B}_c^{-}\to J/\psi {\tau}^{-}{\overline{\nu}}_{\tau } B c − → J / ψ τ ν ¯...

In this article, we analyze the first order linear delay differential equation&#x0D; \begin{equation*}&#x0D; x^{\prime }(t)+p(t)x\left( \tau (t)\right) =0,\text{ }t\geq t_{0},&#x0D; \end{equation*}&#x0D; where $p,$ $\tau \in C\left( [t_{0},\infty ),\mathbb{R}^{+}\right) $ and $%&#x0D; (t)\leq t,\ \lim_{t\rightarrow \infty }\tau (t)=\infty $. Under assumption that (t)$ is not necessarily monoton...

relative to a hereditary torsion theory $tau$ we introduce a dimension for a module $m$, called {em $tau$-rank of} $m$, which coincides with the reduced rank of $m$ whenever $tau$ is the goldie torsion theory. it is shown that the $tau$-rank of $m$ is measured by the length of certain decompositions of the $tau$-injective hull of $m$. moreover, some relations between the $tau$-rank of $m$ and c...