A graph F is said to be locally primitive if, for each vertex a, the stabilizer in Aut F of a induces a primitive permutation group on the set of vertices adjacent to a. In 1978, Richard Weiss conjectured that for a finite vertex-transitive locally primitive graph F, the number of automorphisms fixing a given vertex is bounded above by some function of the valency of F. In this paper we prove t...

This paper illustrated the essential and fundamental primitive of group signature, by providing strong and formal definitions for the core requirements of anonymity and traceability. Then the authors showed that this superior definitions implied the existing informal requirements in literature, so the primitive of group signature can be unified and simplified. At last, this paper designed a con...

The classification of the primitive transitive and effective actions of Lie groups on manifolds is a problem dating back to Lie. The classification of the infinite dimensional infinitesimal actions was originally done by Cartan [3] and was made rigorous by some joint work of Guillemin, Quillen and Sternberg [8], whose proof was further simplified by Guillemin [7] recently by using some results ...

The class of all connected vertex-transitive graphs with finite valency forms a metric space under a natural combinatorially defined metric. We prove some basic properties of this metric space and discuss the structure of graphs which are limit points of the subset consisting of all finite graphs that admit a vertex-primitive group of automorphisms. A description of these limit graphs would pro...

Let $d$ be an odd square-free integer and $\zeta_8$ a primitive $8$-th root of unity. The purpose this paper is to investigate the rank $2$-class group fields $L_d=\mathbb{Q}(\zeta_8,\sqrt{d})$.

It is shown that a dynamical quantum group arising from a vertex-IRF transformation has a second realization with untwisted dynamical multiplication but nontrivial bigrading. Applied to the SL(2;C) dynamical quantum group, the second realization is naturally described in terms of Koornwinder’s twisted primitive elements. This leads to an intrinsic explanation why harmonic analysis on the “class...

Let ⌦ be a set of cardinality n, G a permutation group on ⌦, and f : ⌦ ! ⌦ a map which is not a permutation. We say that G synchronizes f if the semigroup hG, fi contains a constant map. The first author has conjectured that a primitive group synchronizes any map whose kernel is non-uniform. Rystsov proved one instance of this conjecture, namely, degree n primitive groups synchronize maps of ra...