Monopole clusters and critical dynamics in four-dimensional U(1)

We investigate monopoles in four-dimensional compact U(1) with Wilson action. We focus our attention on monopole clusters as they can be identified unambiguously contrary to monopole loops. We locate the clusters and determine their properties near the U(1) phase transition. The Coulomb phase is characterized by several small clusters, whereas in the confined phase the small clusters coalesce to one large cluster filling up the whole system. We find that clusters winding around the periodic lattice are absent within both phases and during the transition. However, within the confined phase, we observe periodically closed monopole loops if cooling is applied.Comment: 3 pages, Wuppertal preprint WUB 93-3

Distribution of interstitial stem cells in Hydra

The distribution of interstitial stem cells along the Hydra body column was determined using a simplified cloning assay. The assay measures stem cells as clone-forming units (CFU) in aggregates of nitrogen mustard inactivated Hydra tissue. The concentration of stem cells in the gastric region was uniform at about 0.02 CFU/epithelial cell. In both the hypostome and basal disk the concentration was 20-fold lower. A decrease in the ratio of stem cells to committed nerve and nematocyte precursors was correlated with the decrease in stem cell concentration in both hypostome and basal disk. The ratio of stem cells to committed precursors is a sensitive indicator of the rate of self-renewal in the stem cell population. From the ratio it can be estimated that <10% of stem cells self-renew in the hypostome and basal disk compared to 60% in the gastric region. Thus, the results provide an explanation for the observed depletion of stem cells in these regions. The results also suggest that differentiation and self-renewal compete for the same stem cell population

Crossing numbers of composite knots and spatial graphs

We study the minimal crossing number $c(K_{1}\# K_{2})$ of composite knots $K_{1}\# K_{2}$, where $K_1$ and $K_2$ are prime, by relating it to the minimal crossing number of spatial graphs, in particular the $2n$-theta curve $\theta_{K_{1},K_{2}}^n$ that results from tying $n$ of the edges of the planar embedding of the $2n$-theta graph into $K_1$ and the remaining $n$ edges into $K_2$. We prove that for large enough $n$ we have $c(\theta_{K_1,K_2}^n)=n(c(K_1)+c(K_2))$. We also formulate additional relations between the crossing numbers of certain spatial graphs that, if satisfied, imply the additivity of the crossing number or at least give a lower bound for $c(K_1\# K_2)$.Comment: 20 pages, 11 figures, changes from version1: added Lemma 5.2 and corrected mistake in Proposition 5.3, improved quality of figure