‎we obtain the asymptotic expansion of the sequence with general term $frac{a_n}{g_n}$‎, ‎where $a_n$ and $g_n$ are the arithmetic and geometric means of the numbers $d(1),d(2),dots,d(n)$‎, ‎with $d(n)$ denoting the number of positive divisors of $n$‎. ‎also‎, ‎we obtain some explicit bounds concerning $g_n$ and $frac{a_n}{g_n}$.

Let \((T_t)_{t \geqslant 0}\) be a markovian (resp. submarkovian) semigroup on some \(\sigma \)-finite measure space \((\Omega ,\mu )\). We prove that its negative generator A has bounded \(H^{\infty }(\Sigma _\theta )\) calculus the weighted \(L^2(\Omega ,wd\mu as long weight \(w : \Omega \rightarrow (0,\infty finite characteristic defined by \(Q^A_2(w) = \sup _{t > 0} \left\| T_t(w) T_t \left...

The filamentous, heterocyst-forming cyanobacteria perform oxygenic photosynthesis in vegetative cells and nitrogen fixation in heterocysts, and their filaments can be hundreds of cells long. In the model heterocyst-forming cyanobacterium Anabaena sp. strain PCC 7120, the genes in the fraC-fraD-fraE operon are required for filament integrity mainly under conditions of nitrogen deprivation. The f...

Red dish mud-rich sed i ments are com mon in sev eral karst ar eas, in the form of red sur face soils and clastic cave in fill ings. The or i gin and sig nif i cance of red sur face soils have been much de bated over re cent years, while clastic mud-rich sed i ments in cave en vi ron ments have re ceived less at ten tion from geoscientists. The ge netic re la tion ship be tween these two ma te ...

Inverse Young inequality asserts that if $nu >1$, then $|zw|ge nu|z|^{frac{1}{nu}}+(1-nu)|w|^{frac{1}{1-nu}}$, for all complex numbers $z$ and $w$, and equality holds if and only if $|z|^{frac{1}{nu}}=|w|^{frac{1}{1-nu}}$. In this paper the complex representation of quaternion matrices is applied to establish the inverse Young inequality for matrices of quaternions. Moreover, a necessary and ...

background and aims. previous researches, conducted mostly on extracted teeth, have shown that efficacy of calculus removal decreases with increasing pocket depth, but there is still a lack of clinical studies concerning the correlation be-tween residual calculus and varying pocket depths. the objective of the present clinical study was to assess the efficiency of mechanical tooth depuration by...

BACKGROUND Rattan tea is a medicinal plant that has been used for many years for the treatment of inflammation, fatty liver, tumor, diabetes, and hyperlipidemia. OBJECTIVE A green and novel approach based on surfactant-mediated, ultrasonic-assisted extraction (SM-UAE) was developed for the extraction of antioxidant polyphenols from Rattan tea. A nonionic surfactant Tween-80 was selected as ex...