Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count

The nodal domains of eigenvectors of the discrete Schrodinger operator on simple, finite and connected graphs are considered. Courant's well known nodal domain theorem applies in the present case, and sets an upper bound to the number of nodal domains of eigenvectors: Arranging the spectrum as a non decreasing sequence, and denoting by $\nu_n$ the number of nodal domains of the $n$'th eigenvector, Courant's theorem guarantees that the nodal deficiency $n-\nu_n$ is non negative. (The above applies for generic eigenvectors. Special care should be exercised for eigenvectors with vanishing components.) The main result of the present work is that the nodal deficiency for generic eigenvectors equals to a Morse index of an energy functional whose value at its relevant critical points coincides with the eigenvalue. The association of the nodal deficiency to the stability of an energy functional at its critical points was recently discussed in the context of quantum graphs [arXiv:1103.1423] and Dirichlet Laplacian in bounded domains in $R^d$ [arXiv:1107.3489]. The present work adapts this result to the discrete case. The definition of the energy functional in the discrete case requires a special setting, substantially different from the one used in [arXiv:1103.1423,arXiv:1107.3489] and it is presented here in detail.Comment: 15 pages, 1 figur

Number preferences in lotteries

We explore people's preferences for numbers in large proprietary data sets from two different lottery games. We find that choice is far from uniform, and exhibits some familiar and some new tendencies and biases. Players favor personally meaningful and situationally available numbers, and are attracted towards numbers in the center of the choice form. Frequent players avoid winning numbers from recent draws, whereas infrequent players chase these. Combinations of numbers are formed with an eye for aesthetics, and players tend to spread their numbers relatively evenly across the possible range

Soil water-holding capacity and monodominance in Southern Amazon tropical forests

Background and aims: We explored the hypothesis that low soil water-holding capacity is the main factor driving the monodominance of Brosimum rubescens in a monodominant forest in Southern Amazonia. Tropical monodominant forests are rare ecosystems with low diversity and high dominance of a single tree species. The causes of this atypical condition are still poorly understood. Some studies have shown a relationship between monodominance and waterlogging or soil attributes, while others have concluded that edaphic factors have little or no explanatory value, but none has accounted for soil-moisture variation other than waterlogging. This study is the first to explicitly explore how low soil water-holding capacity influences the monodominance of tropical forests. Methods: We conducted in situ measurements of vertical soil moisture using electrical resistance collected over 1 year at 0–5; 35–40 and 75–80 cm depths in a B. rubescens monodominant forest and in an adjacent mixed-species forest in the Amazon-Cerrado transition zone, Brazil. Minimum leaf water potential (Ψmin) of the seven most common species, including B. rubescens, and soil water-holding capacity for both forests were determined. Results: The vertical soil moisture decay pattern was similar in both forests for all depths. However, the slightly higher water availability in the monodominant forest and Ψmin similarity between B. rubescens and nearby mixed forest species indicate that low water-availability does not cause the monodominance. Conclusions: We reject the hypothesis that monodominance of B. rubescens is primarily determined by low soil water-holding capacity, reinforcing the idea that monodominance in tropical forests is not determined by a single factor

The genetic architecture of the human cerebral cortex

The cerebral cortex underlies our complex cognitive capabilities, yet little is known about the specific genetic loci that influence human cortical structure. To identify genetic variants that affect cortical structure, we conducted a genome-wide association meta-analysis of brain magnetic resonance imaging data from 51,665 individuals. We analyzed the surface area and average thickness of the whole cortex and 34 regions with known functional specializations. We identified 199 significant loci and found significant enrichment for loci influencing total surface area within regulatory elements that are active during prenatal cortical development, supporting the radial unit hypothesis. Loci that affect regional surface area cluster near genes in Wnt signaling pathways, which influence progenitor expansion and areal identity. Variation in cortical structure is genetically correlated with cognitive function, Parkinson's disease, insomnia, depression, neuroticism, and attention deficit hyperactivity disorder