Common multiples of complete graphs

A graph H is said to divide a graph G if there exists a set S of subgraphs of G, all isomorphic to H, such that the edge set of G is partitioned by the edge sets of the subgraphs in S. Thus, a graph G is a common multiple of two graphs if each of the two graphs divides G.

Common multiples of complete graphs and a 4-cycle

A graph G is a common multiple of two graphs H-1 and H-2 if there exists a decomposition of G into edge-disjoint copies of H-1 and also a decomposition of G into edge-disjoint copies of H-2. In this paper, we consider the case where H-1 is the 4-cycle C-4 and H-2 is the complete graph with n vertices K-n. We determine, for all positive integers n, the set of integers q for which there exists a common multiple of C-4 and K-n having precisely q edges. (C) 2003 Elsevier B.V. All rights reserved.

Growth hormone binding protein in normal and aneuploid pregnancy: a paradoxical decrease in trisomy 18

Objective To explore whether abnormalities in growth hormone binding protein (GHBP) may underlie the growth restriction associated with fetal aneuploidy. Design A retrospective casecontrol study. Setting Monash Medical Centre, Clayton, Victoria, Australia. Population Twenty-one trisomy 18, and 30 trisomy 21 pregnancies, and 170 chromosomally normal pregnancies at 15-18 weeks of gestation representing three to five controls per case matched for source, gestation and duration of storage. Methods GHBP was measured using a ligand immunofunctional assay. Results In the chromosomally normal pregnancies GHBP levels decreased slightly but significantly across the narrow gestational window studied. Compared with controls, levels of GHBP, expressed as median (95% CI) multiples of the median (MoM), in the trisomy 21 pregnancies were similar, 1.0 (0.92-1.39) MoM and 1.27 (1.04-1.50) MoM, respectively; P = 0.061 (Mann-Whitney CI test) but were significantly reduced in the trisomy 18 pregnancies, 0.68 (0.51-0.84) MoM; P = 0.0014 (Mann-Whitney U test). Conclusions These data suggest that decreased levels of maternal growth hormone binding protein, and by implication growth hormone receptor complement, may underlie the early severe growth restriction that is characteristic of trisomy 18.

Fluorescence spectrum of a two-level atom driven by a multiple modulated field

We investigate the fluorescence spectrum of a two-level atom driven by a multiple amplitude-modulated field. The driving held is modeled as a polychromatic field composed of a strong central (resonant) component and a large number of symmetrically detuned sideband fields displaced from the central component by integer multiples of a constant detuning. Spectra obtained here differ qualitatively from those observed for a single pair of modulating fields [B. Blind, P.R. Fontana, and P. Thomann, J. Phys. B 13, 2717 (1980)]. In the case of a small number of the modulating fields, a multipeaked spectrum is obtained with the spectral features located at fixed frequencies that are independent of the number of modulating fields and their Rabi frequencies. As the number of the modulating fields increases, the spectrum ultimately evolves to the well-known Mellow triplet with the sidebands shifted from the central component by an effective Rabi frequency whose magnitude depends on the initial relative phases of the components of the driving held. For equal relative phases, the effective Rabi frequency of the driving field can be reduced to zero resulting in the disappearance of fluorescence spectrum, i.e., the atom can stop interacting with the field. When the central component and the modulating fields are 180 degrees out of phase, the spectrum retains its triplet structure with the sidebands located at frequencies equal to the sum of the Rabi frequencies of the component of the driving field. Moreover, we shaw that the frequency of spontaneous emission can be controlled and switched from one frequency to another when the Rabi frequency or initial phase of the modulating fields are varied.