Borel circle squaring

We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If $k \geq 1$ and $A, B \subseteq \mathbb{R}^k$ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $k$, then $A$ and $B$ are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of $\mathbb{Z}^d$.Comment: Minor typos correcte

Measurable realizations of abstract systems of congruences

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the $2$-sphere. This answers a question of Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $\mathsf{PSL}_2(\mathbb{Z})$ on $\mathsf{P}^1(\mathbb{R})$. The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.Comment: minor correction

Comparing the information content of coral reef geomorphological and biological habitat maps, Amirantes Archipelago (Seychelles), Western Indian Ocean

Increasing the use of geomorphological map products in marine spatial planning has the potential to greatly enhance return on mapping investment as they are commonly two orders of magnitude cheaper to produce than biologically-focussed maps of benthic communities and shallow substrates. The efficacy of geomorphological maps derived from remotely sensed imagery as surrogates for habitat diversity is explored by comparing two map sets of the platform reefs and atolls of the Amirantes Archipelago (Seychelles), Western Indian Ocean. One mapping campaign utilised Compact Airborne Spectrographic Imagery (19 wavebands, 1 m spatial resolution) to classify 11 islands and associated reefs into 25 biological habitat classes while the other campaign used Landsat 7 þ ETM imagery (7 bands, 30 m spatial resolution) to generate maps of 14 geomorphic classes. The maps were compared across a range of characteristics, including habitat richness (number of classes mapped), diversity (ShannoneWeiner statistic) and thematic content (Cramer’s V statistic). Between maps, a strong relationship was revealed for habitat richness (R2 ¼ 0.76), a moderate relationship for class diversity and evenness (R2 ¼ 0.63) and a variable relationship for thematic content, dependent on site complexity (V range 0.43 e0.93). Geomorphic maps emerged as robust predictors of the habitat richness in the Amirantes. Such maps therefore demonstrate high potential value for informing coastal management activities and conservation planning by drawing on information beyond their own thematic content and thus maximizing the return on mapping investment

Recent ν\nus from IceCube

IceCube is a 1 km$^3$ neutrino detector now being built at the South Pole. Its 4800 optical modules will detect Cherenkov radiation from charged particles produced in neutrino interactions. IceCube will search for neutrinos of astrophysical origin, with energies from 100 GeV up to $10^{19}$ eV. It will be able to separate $\nu_e$, $\nu_\mu$ and $\nu_\tau$. In addition to detecting astrophysical neutrinos, IceCube will also search for neutrinos from WIMP annihilation in the Sun and the Earth, look for low-energy (10 MeV) neutrinos from supernovae, and search for a host of exotic signatures. With the associated IceTop surface air shower array, it will study cosmic-ray air showers. IceCube construction is now 50% complete. After presenting preliminary results from the partial detector, I will discuss IceCube's future plans.Comment: Invited talk presented at Neutrino 2008; 7 page

Genome scan of Diabrotica virgifera virgifera for genetic variation associated with crop rotation tolerance

Crop rotation has been a valuable technique for control of Diabrotica virgifera virgifera for almost a century. However, during the last two decades, crop rotation has ceased to be effective in an expanding area of the US corn belt. This failure appears to be due to a change in the insect's oviposition behaviour, which, in all probability, has an underlying genetic basis. A preliminary genome scan using 253 amplified fragment-length polymorphism (AFLP) markers sought to identify genetic variation associated with the circumvention of crop rotation. Samples of D. v. virgifera from east-central Illinois, where crop rotation is ineffective, were compared with samples from Iowa at locations that the behavioural variant has yet to reach. A single AFLP marker showed signs of having been influenced by selection for the circumvention of crop rotation. However, this marker was not diagnostic. The lack of markers strongly associated with the trait may be due to an insufficient density of marker coverage throughout the genome. A weak but significant general heterogeneity was observed between the Illinois and Iowa samples at microsatellite loci and AFLP markers. This has not been detected in previous population genetic studies of D. v. virgifera and may indicate a reduction in gene flow between variant and wild-type beetles

Predicting ecosystem shifts requires new approaches that integrate the effects of climate change across entire systems.

Most studies that forecast the ecological consequences of climate change target a single species and a single life stage. Depending on climatic impacts on other life stages and on interacting species, however, the results from simple experiments may not translate into accurate predictions of future ecological change. Research needs to move beyond simple experimental studies and environmental envelope projections for single species towards identifying where ecosystem change is likely to occur and the drivers for this change. For this to happen, we advocate research directions that (i) identify the critical species within the target ecosystem, and the life stage(s) most susceptible to changing conditions and (ii) the key interactions between these species and components of their broader ecosystem. A combined approach using macroecology, experimentally derived data and modelling that incorporates energy budgets in life cycle models may identify critical abiotic conditions that disproportionately alter important ecological processes under forecasted climates

Phase Diagram for Anderson Disorder: beyond Single-Parameter Scaling

The Anderson model for independent electrons in a disordered potential is transformed analytically and exactly to a basis of random extended states leading to a variant of augmented space. In addition to the widely-accepted phase diagrams in all physical dimensions, a plethora of additional, weaker Anderson transitions are found, characterized by the long-distance behavior of states. Critical disorders are found for Anderson transitions at which the asymptotically dominant sector of augmented space changes for all states at the same disorder. At fixed disorder, critical energies are also found at which the localization properties of states are singular. Under the approximation of single-parameter scaling, this phase diagram reduces to the widely-accepted one in 1, 2 and 3 dimensions. In two dimensions, in addition to the Anderson transition at infinitesimal disorder, there is a transition between two localized states, characterized by a change in the nature of wave function decay.Comment: 51 pages including 4 figures, revised 30 November 200