Our concern is with the group of conformal transformations of a finite-dimensional real quadratic space of signature (p,q), that is one that is isomorphic to ${ℝ}^{p,q}$, the real vector space ${ℝ}^{p+q}$, furnished with the quadratic form $x^{\left(2\right)}=x·x=-x_1^2-x_2^2-...-x_p^2+x_{p+1}^2+...+x_{p+q}^2$, and especially with a description of this group that involves Clifford algebras.

A Lagrange Theorem in dimension 2 is proved in this paper, for a particular two dimensional continued fraction algorithm, with a very natural geometrical definition. Dirichlet type properties for the convergence of this algorithm are also proved. These properties proceed from a geometrical quality of the algorithm. The links between all these properties are studied. In relation with this algorithm, some references are given to the works of various authors, in the domain of multidimensional continued...

Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form $x={\sum }_{i=1}^{\infty }{c}_i{q}^{-i}$ with integer coefficients $c_i$ satisfying $0\le c_i<q$, i ≥ 1. In this case we say that $\left(c_i\right)=c₁c₂...$ is an expansion of x in base q. Let U be the set of couples (x,q) ∈ J such that x has exactly one expansion in base q. In this paper we deduce some topological and combinatorial properties of the set U. We characterize the closure of U, and we determine its Hausdorff dimension. For (x,q) ∈ J, we also prove new properties...

Let $S/R$ be a finite extension of discrete valuation rings of characteristic $p\&gt;0$, and suppose that the corresponding extension $L/K$ of fields of fractions is separable and is $H$-Galois for some $K$-Hopf algebra $H$. Let ${𝔻}_{S/R}$ be the different of $S/R$. We show that if $S/R$ is totally ramified and its degree $n$ is a power of $p$, then any element $\rho$ of $L$ with $v_L\left(\rho \right)\equiv -v_L\left({𝔻}_{S/R}\right)-1(modn)$ generates $L$ as an $H$-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois extensions.

We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $$\left\{\begin{array}{c}n=a_1+a_2+\cdots +a_{s-1},\hfill \\ a_1{a}_2\cdots a_{s-1}(a_1+a_2+\cdots +a_{s-1})=b^s\hfill \end{array}\right.$$ has positive integer or rational solutions $n$, $b$, $a_i$, $i=1,2,\cdots ,s-1$, $s\ge 3.$ Using the theory of elliptic curves, we prove that it has no positive integer solution for $s=3$, but there are infinitely many positive integers $n$ such that it has a positive integer solution for $s\ge 4$. As a corollary, for $s\ge 4$ and any positive integer $n$, the above Diophantine system has a positive rational solution. Meanwhile, we give conditions such that...