On the structure of phase transition maps for three or more coexisting phases

This paper is partly based on a lecture delivered by the author at the ERC workshop "Geometric Partial Differential Equations" held in Pisa in September 2012. What is presented here is an expanded version of that lecture.Comment: 23 pages, 6 figure

More about Birkhoff's Invariant and Thorne's Hoop Conjecture for Horizons

A recent precise formulation of the hoop conjecture in four spacetime dimensions is that the Birkhoff invariant $\beta$ (the least maximal length of any sweepout or foliation by circles) of an apparent horizon of energy $E$ and area $A$ should satisfy $\beta \le 4 \pi E$. This conjecture together with the Cosmic Censorship or Isoperimetric inequality implies that the length $\ell$ of the shortest non-trivial closed geodesic satisfies $\ell^2 \le \pi A$. We have tested these conjectures on the horizons of all four-charged rotating black hole solutions of ungauged supergravity theories and find that they always hold. They continue to hold in the the presence of a negative cosmological constant, and for multi-charged rotating solutions in gauged supergravity. Surprisingly, they also hold for the Ernst-Wild static black holes immersed in a magnetic field, which are asymptotic to the Melvin solution. In five spacetime dimensions we define $\beta$ as the least maximal area of all sweepouts of the horizon by two-dimensional tori, and find in all cases examined that $\beta(g) \le \frac{16 \pi}{3} E$, which we conjecture holds quiet generally for apparent horizons. In even spacetime dimensions $D=2N+2$, we find that for sweepouts by the product $S^1 \times S^{D-4}$, $\beta$ is bounded from above by a certain dimension-dependent multiple of the energy $E$. We also find that $\ell^{D-2}$ is bounded from above by a certain dimension-dependent multiple of the horizon area $A$. Finally, we show that $\ell^{D-3}$ is bounded from above by a certain dimension-dependent multiple of the energy, for all Kerr-AdS black holes.Comment: 25 page

When is the Best Time to Sample Aquatic Macroinvertebrates in Ponds for Biodiversity Assessment?

Ponds are sites of high biodiversity and conservation value, yet there is little or no statutory monitoring of them across most of Europe. There are clear and standardized protocols for sampling aquatic macroinvertebrate communities in ponds but the most suitable time(s) to undertake the survey(s) remains poorly specified. This paper examined the aquatic macroinvertebrate communities from 95 ponds within different landuse types over three seasons (spring, summer and autumn) to determine the most appropriate time to undertake sampling to characterise biodiversity. The combined samples from all three seasons provided the most comprehensive record of the aquatic macroinvertebrate taxa recorded within ponds (alpha and gamma diversity). Samples collected during the autumn survey yielded significantly greater macroinvertebrate richness (76% of the total diversity) than either spring or summer surveys. Macroinvertebrate diversity was greatest during autumn in meadow and agricultural ponds but taxon richness among forest and urban ponds did not differ significantly temporally. The autumn survey provided the highest measures of richness for Coleoptera, Hemiptera and Odonata. However, richness of the aquatic insect order Trichoptera was highest in spring and lowest in autumn. The results illustrate that multiple surveys, covering more than one season, provide the most comprehensive representation of macroinvertebrate biodiversity. When sampling can only be undertaken on one occasion, the most appropriate time to undertake surveys to characterise the macroinvertebrate community biodiversity is during the autumn; although this may need to be modified if other floral and faunal groups need to be incorporated in to the sampling programme