Coherent structure

The mental picture of turbulence structure had progressed from the hypothesis that it mimics molecular motion to the belief that turbulence is highly organized. The history of turbulence structure is briefly reviewed

The Cooling of Coronal Plasmas. iv: Catastrophic Cooling of Loops

We examine the radiative cooling of coronal loops and demonstrate that the recently identified catastrophic cooling (Reale and Landi, 2012) is due to the inability of a loop to sustain radiative / enthalpy cooling below a critical temperature, which can be > 1 MK in flares, 0.5 - 1 MK in active regions and 0.1 MK in long tenuous loops. Catastrophic cooling is characterised by a rapid fall in coronal temperature while the coronal density changes by a small amount. Analytic expressions for the critical temperature are derived and show good agreement with numerical results. This effect limits very considerably the lifetime of coronal plasmas below the critical temperature

Ethnography, education and on-line research

This paper is an attempt to establish the methodological basis for carrying out ethnographies of online education communities, in particular in the Continuing Professional Development VITAL project co-ordinated by the Faculty of Education and Language Studies at The Open University www.vital.ac.uk/ A much shorter earlier draft version of this paper was given at the Qualitative Research For Web 2.0/3.0: The Next Leap! 25 & 26 March 2010 in Berlin. Organised by Merlien. The arguments and references in this paper are almost all to be found in two books 'one authored and one edited – by Professor Christine Hine of Surrey University, UK (Hine 2000; Hine 2005)

Fractional colorings of partial tt-trees with no large clique

Dvo\v{r}\'ak and Kawarabayashi [European Journal of Combinatorics, 2017] asked, what is the largest chromatic number attainable by a graph of treewidth $t$ with no $K_r$ subgraph? In this paper, we consider the fractional version of this question. We prove that if $G$ has treewidth $t$ and clique number $2 \leq \omega \leq t$, then $\chi_f(G) \leq t + \frac{\omega - 1}{t}$, and we show that this bound is tight for $\omega = t$. We also show that for each value $0 < c < \frac{1}{2}$, there exists a graph $G$ of a large treewidth $t$ and clique number $\omega = \lfloor (1 - c)t \rfloor$ satisfying $\chi_f(G) \geq t + 1 + \log(1-2c) + o(1)$.Comment: 9 page

Graph colorings with restricted bicolored subgraphs: I. Acyclic, star, and treewidth colorings

We show that for any fixed integer $m \geq 1$, a graph of maximum degree $\Delta$ has a coloring with $O(\Delta^{(m+1)/m})$ colors in which every connected bicolored subgraph contains at most $m$ edges. This result unifies previously known upper bounds on the number of colors sufficient for certain types of graph colorings, including star colorings, for which $O(\Delta^{3/2})$ colors suffice, and acyclic colorings, for which $O(\Delta^{4/3})$ colors suffice. Our proof uses a probabilistic method of Alon, McDiarmid, and Reed. This result also gives previously unknown upper bounds, including the fact that a graph of maximum degree $\Delta$ has a proper coloring with $O(\Delta^{9/8})$ colors in which every bicolored subgraph is planar, as well as a proper coloring with $O(\Delta^{13/12})$ colors in which every bicolored subgraph has treewidth at most $3$.Comment: 6 page

Rainbow spanning trees in random edge-colored graphs

A well known result of Erd\H{o}s and R\'enyi states that if $p = \frac{c \log n}{n}$ and $G$ is a random graph constructed from $G(n,p)$, $G$ is a.a.s. disconnected when $c 1$. When $c > 1$, we may equivalently say that $G$ a.a.s. contains a spanning tree. We find analogous thresholds in the setting of random edge-colored graphs. Specifically, we consider a family $\mathcal G$ of $n-1$ graphs on a common set $X$ of $n$ vertices, each of a different color, and each randomly chosen from $G(n,p)$, with $p = \frac{c \log n}{n^2}$. We show that when $c > 2$, there a.a.s. exists a spanning tree on $X$ using exactly one edge of each color, and we show that such a spanning tree a.a.s. does not exist when $c < 2$.Comment: It was discovered that the main result follows from Frieze, McKay, "Multicolored trees in random graphs," RSA, 199