Let $p\equiv 1(mod8)$ and $q\equiv 3(mod8)$ be two prime integers and let $\ell \notin \{-1,p,q\}$ be a positive odd square-free integer. Assuming that the fundamental unit of $\mathbb{Q}\left(\sqrt{2p}\right)$ has a negative norm, we investigate the unit group of the fields $\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q},\sqrt{-\ell })$.

Let $p$ and $q$ be two different prime integers such that $p\equiv q\equiv 3(mod8)$ with $(p/q)=1$, and $l$ a positive odd square-free integer relatively prime to $p$ and $q$. In this paper we investigate the unit groups of number fields $𝕃=\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q},\sqrt{-l})$.

Let $k$ be a number field with a 2-class group isomorphic to the Klein four-group. The aim of this paper is to give a characterization of capitulation types using group properties. Furthermore, as applications, we determine the structure of the second 2-class groups of some special Dirichlet fields $𝕜=\mathbb{Q}(\sqrt{d},\sqrt{-1})$, which leads to a correction of some parts in the main results of A. Azizi and A. Zekhini (2020).

Condizione necessaria e sufficiente affinché una funzione rapidamente decrescente di variabile reale sia uniformemente analitica è che per i suoi coefficienti ${\gamma }_0,{\gamma }_1,\mathrm{\dots }$ di Fourier-Hermite riesca ${\gamma }_m=0(e^{\sqrt{mt}})$ per $t>0$ abbastanza piccolo.

In a recent paper we proved that there are at most finitely many complex numbers $t\ne 0,1$ such that the points $(2,\sqrt{2(2-t)})$ and $(3,\sqrt{6(3-t)})$ are both torsion on the Legendre elliptic curve defined by $y^2=x(x-1)(x-t)$. In a sequel we gave a generalization to any two points with coordinates algebraic over the field $\mathbf{Q}\left(t\right)$ and even over $\mathbf{C}\left(t\right)$. Here we reconsider the special case $(u,\sqrt{u(u-1)(u-t)})$ and $(v,\sqrt{v(v-1)(v-t)})$ with complex numbers $u$ and $v$.

Condizione necessaria e sufficiente affinché una funzione rapidamente decrescente di variabile reale sia uniformemente analitica è che per i suoi coefficienti ${\gamma }_0,{\gamma }_1,\mathrm{\dots }$ di Fourier-Hermite riesca ${\gamma }_m=0(e^{\sqrt{mt}})$ per $t>0$ abbastanza piccolo.

The binary operation $aba$, called Jordan triple product, and its variants (such as e.g. the sequential product $\sqrt{a}b\sqrt{a}$ or the inverted Jordan triple product $a{b}^{-1}a$) appear in several branches of operator theory and matrix analysis. In this paper we briefly survey some analytic and algebraic properties of these operations, and investigate their intimate connection to Thompson type isometries in different operator algebras.

By iterating the Bolyai-Rényi transformation $T\left(x\right)={(x+1)}^2(mod1)$, almost every real number $x\in [0,1)$ can be expanded as a continued radical expression $$x=-1+\sqrt{x_1+\sqrt{x_2+\cdots +\sqrt{x_n+\cdots }}}$$ with digits $x_n\in \{0,1,2\}$ for all $n\in \mathbb{N}$. For any real number $x\in [0,1)$ and digit $i\in \{0,1,2\}$, let $r_n(x,i)$ be the maximal length of consecutive $i$’s in the first $n$ digits of the Bolyai-Rényi expansion of $x$. We study the asymptotic behavior of the run-length function $r_n(x,i)$. We prove that for any digit $i\in \{0,1,2\}$, the Lebesgue measure of the set $$D\left(i\right)=\left\{x\in [0,1):\underset{n\to \infty }{lim}\frac{r_n(x,i)}{logn}=\frac{1}{log{\theta }_i}\right\}$$ is $1$, where ${\theta }_i=1+\sqrt{4i+1}$. We also obtain that the level set $$E_{\alpha }\left(i\right)=\left\{x\in [0,1):\underset{n\to \infty }{lim}\frac{r_n(x,i)}{logn}=\alpha \right\}$$ is of full Hausdorff...

Let $𝕜=\mathbb{Q}\left(\sqrt{2},\sqrt{d}\right)$ be an imaginary bicyclic biquadratic number field, where $d$ is an odd negative square-free integer and ${𝕜}_2^{\left(2\right)}$ its second Hilbert $2$-class field. Denote by $G=\mathrm{Gal}({𝕜}_2^{\left(2\right)}/𝕜)$ the Galois group of ${𝕜}_2^{\left(2\right)}/𝕜$. The purpose of this note is to investigate the Hilbert $2$-class field tower of $𝕜$ and then deduce the structure of $G$.

Let $A$ be a closed set of $M\simeq {\mathbb{R}}^n$, whose conormai cones $x+y_x^{*}\left(A\right)$, $x\in A$, have locally empty intersection. We first show in §1 that $\text{dist}\left(x,A\right)$, $x\in M\setminus A$ is a $C^1$ function. We then represent the n microfunctions of ${\mathcal{C}}_{A|X}$, $X\simeq {\mathbb{C}}^n$, using cohomology groups of ${\mathcal{O}}_X$ of degree 1. By the results of § 1-3, we are able to prove in §4 that the sections of ${\left.{\mathcal{C}}_{A|X}\right|}_{{\dot{\pi }}^{-1}\left(x_0\right)}$, $x_0\in \partial A$, satisfy the principle of the analytic continuation in the complex integral manifolds of ${\left\{H\left({\varphi }_i^C\right)\right\}}_{i=1,\mathrm{\dots },m}$, $\left\{{\varphi }_i\right\}$ being a base for the linear hull of ${\gamma }_{x_0}^{*}\left(A\right)$ in $T_{x_0}^{*}M$; in particular we get ${\left.{\mathrm{\Gamma }}_{A\times {}_{M}T{}^{*}{}_{M}X}\left({\mathcal{C}}_{A|X}\right)\right|}_{\partial A\times {}_M\dot{T}{}^{*}{}_{M}X}=0$. When $A$is a half space with...