HI distribution and kinematics of NGC 1569

We present WSRT observations of high sensitivity and resolution of the neutral hydrogen in the starburst dwarf galaxy NGC 1569. Assuming a distance of 2.2 Mpc, we find a total HI mass of 1.3 * 10^8 M_sun to be distributed in the form of a dense, clumpy ridge surrounded by more extended diffuse HI containing a few additional discrete features, such as a Western HI Arm and an HI bridge reaching out to a small counterrotating companion cloud. About 10% by mass of all HI in NGC 1569 is at unusually high velocities. Some of this HI may be associated with the mass outflow evident from H-alpha measurements, but some may also be associated with NGC 1569's HI companion and intervening HI bridge, in which case, infall rather than outflow might be the cause of the discrepant velocities. No indication of a large bubble structure was found in position-velocity maps of the high-velocity HI. The galaxy as a whole is in modest overall rotation, but the HI gas lacks any sign of rotation within 60'' (0.6 kpc) from the center, i.e. over most of the optical galaxy. Here, turbulent motions resulting from the starburst appear to dominate over rotation. In the outer disk, the rotational velocities reach a maximum of 35 \pm 6 km/s, but turbulent motion remains significant. Thus, starburst effects are still noticeable in the outer HI disk, although they are no longer dominant beyond 0.6 kpc. Even excluding the most extreme high-velocity HI clouds, NGC 1569 still has an unusually high mean HI velocity dispersion of sigma_v=21.3 km/s, more than double that of other dwarf galaxies.Comment: Figure 11a,b and Figure 14 separately in jpg forma

Cut distance identifying graphon parameters over weak* limits

The theory of graphons comes with the so-called cut norm and the derived cut distance. The cut norm is finer than the weak* topology. Dole\v{z}al and Hladk\'y [Cut-norm and entropy minimization over weak* limits, J. Combin. Theory Ser. B 137 (2019), 232-263] showed, that given a sequence of graphons, a cut distance accumulation graphon can be pinpointed in the set of weak* accumulation points as a minimizer of the entropy. Motivated by this, we study graphon parameters with the property that their minimizers or maximizers identify cut distance accumulation points over the set of weak* accumulation points. We call such parameters cut distance identifying. Of particular importance are cut distance identifying parameters coming from subgraph densities, t(H,*). This concept is closely related to the emerging field of graph norms, and the notions of the step Sidorenko property and the step forcing property introduced by Kr\'al, Martins, Pach and Wrochna [The step Sidorenko property and non-norming edge-transitive graphs, J. Combin. Theory Ser. A 162 (2019), 34-54]. We prove that a connected graph is weakly norming if and only if it is step Sidorenko, and that if a graph is norming then it is step forcing. Further, we study convexity properties of cut distance identifying graphon parameters, and find a way to identify cut distance limits using spectra of graphons. We also show that continuous cut distance identifying graphon parameters have the "pumping property", and thus can be used in the proof of the the Frieze-Kannan regularity lemma.Comment: 48 pages, 3 figures. Correction when treating disconnected norming graphs, and a new section 3.2 on index pumping in the regularity lemm