### A note on representation of positive definite binary quadratic forms by positive definite quadratic forms in 6 variables

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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 ${\mathbb{R}}^{p,q}$, the real vector space ${\mathbb{R}}^{p+q}$, furnished with the quadratic form ${x}^{\left(2\right)}=x\xb7x=-{x}_{1}^{2}-{x}_{2}^{2}-...-{x}_{p}^{2}+{x}_{p+1}^{2}+...+{x}_{p+q}^{2}$, and especially with a description of this group that involves Clifford algebras.

In this paper we show that well-known relationships connecting the Clifford algebra on negative euclidean space, Vahlen matrices, and Möbius transformations extend to connections with the Möbius loop or gyrogroup on the open unit ball $B$ in $n$-dimensional euclidean space ${\mathbb{R}}^{n}$. One notable achievement is a compact, convenient formula for the Möbius loop operation $a*b=(a+b){(1-ab)}^{-1}$, where the operations on the right are those arising from the Clifford algebra (a formula comparable to $(w+z){(1+\overline{w}z)}^{-1}$ for the Möbius loop multiplication...

This is the last of a series of three papers where we compute the unramified cohomology of quadrics in degree up to 4. Complete results were obtained in the two previous papers for quadrics of dimension $\le 4$ and $\ge 11$. Here we deal with the remaining dimensions between 5 and 10. We also prove that the unramified cohomology of Pfister quadrics with divisible coefficients always comes from the ground field, and that the same holds for their unramified Witt rings. We apply these results to real quadrics....

Let $K$ be a totally real algebraic number field whose ring of integers $R$ is a principal ideal domain. Let $f({x}_{1},{x}_{2},{x}_{3})$ be a totally definite ternary quadratic form with coefficients in $R$. We shall study representations of totally positive elements $N\in R$ by $f$. We prove a quantitative formula relating the number of representations of $N$ by different classes in the genus of $f$ to the class number of $R\left[\sqrt{-{c}_{f}N}\right]$, where ${c}_{f}\in R$ is a constant depending only on $f$. We give an algebraic proof of a classical result of H. Maass on representations...