for all $a,b>0$, the following two optimal inequalities are presented: $h^{alpha}(a,b)l^{1-alpha}(a,b)geq m_{frac{1-4alpha}{3}}(a,b)$ for $alphain[frac{1}{4},1)$, and $ h^{alpha}(a,b)l^{1-alpha}(a,b)leq m_{frac{1-4alpha}{3}}(a,b)$ for $alphain(0,frac{3sqrt{5}-5}{40}]$. here, $h(a,b)$, $l(a,b)$, and $m_p(a,b)$ denote the harmonic, logarithmic, and power means of order $p$ of two positive numbers...

LetX be a complex analytic manifold, D ⊂ X a Koszul free divisor with jacobian ideal of linear type (e.g. a locally quasi-homogeneous free divisor), j : U = X −D →֒ X the corresponding open inclusion, E an integrable logarithmic connection with respect to D and L the local system of the horizontal sections of E on U . In this paper we prove that the canonical morphisms ΩX(logD)(E(kD)) −→ Rj∗L, j...

We show that the monotone analogue of logspace computation is more powerful than monotone log-depth circuits: monotone bounded fanin circuits for a certain function in monotone logspace require depth (lg 2 n).

We explain how the field of logarithmic-exponential series constructed in [DMM1] and [DMM2] embeds as an exponential field in any field of exponential-logarithmic series (constructed in [KK1], [K] and [KS]). On the other hand, we explain why no field of exponential-logarithmic series embeds in the field of logarithmic-exponential series. This clarifies why the two constructions are intrinsicall...