Abstract:
The spectrum of a first-order logic sentence is the set of natural numbers that are cardinalities of its finite models. In this paper we study the hierarchy of first-order spectra based on the number of variables. It has been conjectured that it collapses to three variable. We show the opposite: it forms an infinite hierarchy. However, despite the fact that more variables can express more spectra, we show that to establish whether the class of first-order spectra is closed under complement, it is sufficient to consider sentences using only three variables and binary relations.

Abstract:
The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [4] by introducing a vast hierarchy of generalizations to the notion of graph diameter. This hierarchy provides some interesting lower bounds for the usual graph diameter. After explaining the structure of the hierarchy and discussing these bounds, we focus on clearly explaining the differences and similarities among the many diameter notions of our hierarchy. Finally, we fully characterize the hierarchy in dimension two. It collapses into fewer categories, for which we exhibit the ranges of values that can be realized as diameters.

Abstract:
Arthur Neville Brown, MA, FRAS (1864-1934) was a prolific variable star observer and served for many years as Secretary of the BAA Variable Star Section. This paper discusses Brown's life and career, both as a variable star enthusiast and as a dedicated and highly respected schoolmaster.

Abstract:
It is shown that non-supersymmetric spacetime varying string vacua can lead to an exponential hierarchy between the electroweak and the gravitational scales. The hierarchy is naturally generated by a string coupling of O(1).

Abstract:
We study a restricted-height version of the one-dimensional Oslo sandpile with conserved density, using periodic boundary conditions. Each site has a limiting height which can be either two or three. When a site reaches its limiting height it becomes active and may topple, loosing two particles, which move randomly to nearest-neighbor sites. After a site topples it is randomly assigned a new limiting height. We study the model using mean-field theory and Monte Carlo simulation, focusing on the quasi-stationary state, in which the number of active sites fluctuates about a stationary value. Using finite-size scaling analysis, we determine the critical particle density and associated critical exponents.

Abstract:
I argue that star formation is controlled by supersonic turbulence, drawing for support on a number of 3D hydrodynamical and MHD simulations as well as theoretical arguments. Clustered star formation appears to be a natural result of a lack of turbulent support, while isolated star formation is a signpost of global turbulent support.

Abstract:
Parity games are combinatorial representations of closed Boolean mu-terms. By adding to them draw positions, they have been organized by Arnold and one of the authors into a mu-calculus. As done by Berwanger et al. for the propositional modal mu-calculus, it is possible to classify parity games into levels of a hierarchy according to the number of fixed-point variables. We ask whether this hierarchy collapses w.r.t. the standard interpretation of the games mu-calculus into the class of all complete lattices. We answer this question negatively by providing, for each n >= 1, a parity game Gn with these properties: it unravels to a mu-term built up with n fixed-point variables, it is semantically equivalent to no game with strictly less than n-2 fixed-point variables.

Abstract:
The variable star known as NSVS 5597754 was identified as a short period RRc field star from ground-based photometry. This star happens to fall in the Kepler field of view and has been observed for 3 years with the Kepler spacecraft. Renamed KIC 9832227, this star was reidentified as an eclipsing binary with a periodicity of 0.457970 days. The NSVS ground based photometry yielded a period solution of 0.229 days. The Kepler photometry, which sampled the light cycle every 30 minutes, shows a peculiar behavior that is not purely eclipsing superposed over the primary variation. Follow-up ground based observations were taken in 2010 with the 0.6m MSU Observatory to obtain additional information. I present a summary of the ground based data taken of this perplexing star and conclusions derived thus far.

Abstract:
We demonstrate that, for every grand unified model based on a conventional type I seesaw mechanism leading to a normal light neutrino mass hierarchy, one can easily generate a corresponding model with an inverted hierarchy which yields the same neutrino oscillation parameters. However, the latter type model has several unattractive instabilities which will disfavor any grand unified type I seesaw model, if an inverted neutrino mass hierarchy is observed experimentally. This should be contrasted with the softly-broken $L_e - L_\mu - L_\tau$ flavor symmetry models which are eliminated, if the data favors a normal mass hierarchy.

Abstract:
Until now it has been impossible to observationally measure how star cluster scale height evolves beyond 1Gyr as only small samples have been available. Here we establish a novel method to determine the scale height of a cluster sample using modelled distributions and Kolmogorov-Smirnov tests. This allows us to determine the scale height with a 25% accuracy for samples of 38 clusters or more. We apply our method to investigate the temporal evolution of cluster scale height, using homogeneously selected sub-samples of Kharchenko et al. (MWSC), Dias et al. (DAML02), WEBDA, and Froebrich et al. (FSR). We identify a linear relationship between scale height and log(age/yr) of clusters, considerably different from field stars. The scale height increases from about 40pc at 1Myr to 75pc at 1Gyr, most likely due to internal evolution and external scattering events. After 1Gyr, there is a marked change of the behaviour, with the scale height linearly increasing with log(age/yr) to about 550pc at 3.5Gyr. The most likely interpretation is that the surviving clusters are only observable because they have been scattered away from the mid-plane in their past. A detailed understanding of this observational evidence can only be achieved with numerical simulations of the evolution of cluster samples in the Galactic Disk. Furthermore, we find a weak trend of an age-independent increase in scale height with galactocentric distance. There are no significant temporal or spatial variations of the cluster distribution zero point. We determine the Sun's vertical displacement from the Galactic Plane as $Z_\odot=18.5\pm1.2$pc.