R-Equivalence and A1-Connectedness in Anisotropic Groups

The graph of equivalence classes and Isoclinism of groups

‎Let $G$ be a non-abelian group and let $Gamma(G)$ be the non-commuting graph of $G$‎. ‎In this paper we define an equivalence relation $sim$ on the set of $V(Gamma(G))=Gsetminus Z(G)$ by taking $xsim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)={uin G | x textrm{ and } u textrm{ are adjacent in }Gamma(G)}$ is the open neighborhood of $x$ in $Gamma(G)$‎. ‎We introduce a new graph determined ...

Finiteness of R-equivalence Groups of Some Adjoint Classical Groups of Type

Let F be a field of charateristic different from 2. We construct families of adjoint groups G of type D3 defined over F (but not over k) such that G(F )/R is finite for various fields F which are finitely generated over their prime subfield. We also construct families of examples of such groups G for which G(F )/R ≃ Z /2Z when F = k(t), and k is (almost) arbitrary. This gives the first examples...

Sparsity Equivalence of Anisotropic Decompositions

Anisotropic decompositions using representation systems such as curvelets, contourlet, or shearlets have recently attracted significantly increased attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sp...

Weak approximation, Brauer and R-equivalence in algebraic groups over arithmetical fields

We prove some new relations between weak approximation and some rational equivalence relations (Brauer and R-equivalence) in algebraic groups over arithmetical fields. By using weak approximation and local global approach, we compute completely the group of Brauer equivalence classes of connected linear algebraic groups over number fields, and also completely compute the group of R-equivalence ...

Connectedness of Degree Graphs of Nonsolvable Groups