Distinguishing labeling of group actions

AbstractSuppose Γ is a group acting on a set X. An r-labeling f:X→{1,2,…,r} of X is distinguishing (with respect to Γ) if the only label preserving permutation of X in Γ is the identity. The distinguishing number, DΓ(X), of the action of Γ on X is the minimum r for which there is an r-labeling which is distinguishing. This paper investigates the relation between the cardinality of a set X and the distinguishing numbers of group actions on X. For a positive integer n, let D(n) be the set of distinguishing numbers of transitive group actions on a set X of cardinality n, i.e., D(n)={DΓ(X):|X|=n and Γ acts transitively on X}. We prove that |D(n)|=O(n). Then we consider the problem of an arbitrary fixed group Γ acting on a large set. We prove that if for any action of Γ on a set Y, for each proper normal subgroup H of Γ, DH(Y)≤2, then there is an integer n such that for any set X with |X|≥n, for any action of Γ on X with no fixed points, DΓ(X)≤2

Application of optical single-sideband laser in Raman atom interferometry

A frequency doubled I/Q modulator based optical single-sideband (OSSB) laser system is demonstrated for atomic physics research, specifically for atom interferometry where the presence of additional sidebands causes parasitic transitions. The performance of the OSSB technique and the spectrum after second harmonic generation are measured and analyzed. The additional sidebands are removed with better than 20 dB suppression, and the influence of parasitic transitions upon stimulated Raman transitions at varying spatial positions is shown to be removed beneath experimental noise. This technique will facilitate the development of compact atom interferometry based sensors with improved accuracy and reduced complexity

Weighted-1-antimagic graphs of prime power order

AbstractSuppose G is a graph, k is a non-negative integer. We say G is weighted-k-antimagic if for any vertex weight function w:V→N, there is an injection f:E→{1,2,…,∣E∣+k} such that for any two distinct vertices u and v, ∑e∈E(v)f(e)+w(v)≠∑e∈E(u)f(e)+w(u). There are connected graphs G≠K2 which are not weighted-1-antimagic. It was asked in Wong and Zhu (in press) [13] whether every connected graph other than K2 is weighted-2-antimagic, and whether every connected graph on an odd number of vertices is weighted-1-antimagic. It was proved in Wong and Zhu (in press) [13] that if a connected graph G has a universal vertex, then G is weighted-2-antimagic, and moreover if G has an odd number of vertices, then G is weighted-1-antimagic. In this paper, by restricting to graphs of odd prime power order, we improve this result in two directions: if G has odd prime power order pz and has total domination number 2 with the degree of one vertex in the total dominating set not a multiple of p, then G is weighted-1-antimagic. If G has odd prime power order pz, p≠3 and has maximum degree at least ∣V(G)∣−3, then G is weighted-1-antimagic