Asymptotic Behavior of Automorphisms of Graphs

Abstract

Electron heating (AT e /T e <30%, An e /n e < 100%) was produced in tokamak-like discharges (n e =2X 10 n cm" 3 , T e s 15 eV, n e /nH° = 20%) in the General Atomic Octopole by application of lower hybrid wave power. The heating efficiency was 50% for v^/vj = 1.7 and 17% for V0/v T = 3.4.

Kernels of van den Dries–Wilkie Homomorphisms and o-Automorphisms of Graphs

“…Then, for an arbitrary finitely generated group G, the intersection, by all nonprincipal ultrafilters U on , of kernels of the homomorphisms l R U contains o G . It can be deduced from Theorem and Remark 1 of [4] that this intersection and o G coincide for any finitely generated group G if and only if there exists a positive integer c such that r i+1 < cr i for all i ∈ .…”

“…By Proposition 3.1, for any nonprincipal ultrafilter U on , there exists a countable set R j j ∈ of positive infinite elements of the corresponding nonstandard extension * of such that j∈ ker l R j = o G (Moreover, by the proof of Proposition 3.1, we can take R j = i → r i j /U where r i j ∈ and r i+1 j > r i j for all i j ∈ .) At the same time, it can be deduced from Theorem and Remark 1 of [4] that, for any ultrafilter U, the set R j j ∈ cannot be chosen independently of G.…”

“…By the above, for an arbitrary finitely generated group G there exists a countable set U j j/ ∈ of nonprincipal ultrafilters on with j∈ ker l R j = o G where R j = i → r i /U j for j ∈ . It can be deduced from Theorem and Remark 1 of [4] that for any such sequence i → r i the set U j j ∈ can not be chosen independently of G.…”

On the action of a group on a graph

“…The nature of the functions realized as ~r,~(g, • ) for connected locally finite vertexsymmetric graphs F, x E V(F), and g E Aut F is studied in [1]. b) either f(n) >>.…”

“…In fact, it can be shown that the intersection of the kernels of these representations coincides with the subgroup o(Aut F) considered there. In turn the research conducted in [1] on the asymptotic behavior of automorphisms of graphs has enabled us to give a positive answer to the question of van den Dries and Wilkie.…”