For any distinct two primes \(p_1\equiv p_2\equiv 3\) \((\text {mod }4)\), let \(h(-p_1)\), \(h(-p_2)\) and \(h(p_1p_2)\) be the class numbers of quadratic fields \(\mathbb {Q}(\sqrt{-p_1})\), {Q}(\sqrt{-p_2})\) {Q}(\sqrt{p_1p_2})\), respectively. Let \(\omega _{p_1p_2}:=(1+\sqrt{p_1p_2})/2\) \(\varPsi (\omega _{p_1p_2})\) Hirzebruch sum _{p_1p_2}\). We show that \(h(-p_1)h(-p_2)\equiv h(p_1p_2...

Let $S$ be an inverse semigroup and let $E$ be its subsemigroup of idempotents. In this paper we define the $n$-th module cohomology group of Banach algebras and show that the first module cohomology group $HH^1_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is zero, for every odd $ninmathbb{N}$. Next, for a Clifford semigroup $S$ we show that $HH^2_{ell^1(E)}(ell^1(S),ell^1(S)^{(n)})$ is a Banach sp...

Aliphatic polyamides derived from odd diamine or odd dicarboxylic acid units cannot adopt a conventional sheet structure when molecular chains have an all trans conformation. However, typical fiber diffraction patterns of this sheet structure were observed in several polyamides derived from odd units such as nylons 65 and 56. Consequently, a new structure based on the establishment of intermole...