Mahler measures, elliptic curves, and L-functions for the free energy of the Ising model.

This work establishes links between the Ising model and elliptic curves via Mahler measures. First, we reformulate the partition function of the Ising model on the square, triangular, and honeycomb lattices in terms of the Mahler measure of a Laurent polynomial whose variety's projective closure defines an elliptic curve. Next, we obtain hypergeometric formulas for the partition functions on the triangular and honeycomb lattices and review the known series for the square lattice.

What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution.

An exact solution of the Ising model on the simple cubic lattice is one of the long-standing open problems in rigorous statistical mechanics. Indeed, it is generally believed that settling it would constitute a methodological breakthrough, fomenting great prospects for further application, similarly to what happened when Lars Onsager solved the two-dimensional model eighty years ago. Hence, there have been many attempts to find analytic expressions for the exact partition function , but all such attempts have failed due to unavoidable conceptual or mathematical obstructions.

Survival in patchy landscapes: the interplay between dispersal, habitat loss and fragmentation.

Habitat loss and fragmentation are important factors determining animal population dynamics and spatial distribution. Such landscape changes can lead to the deleterious impact of a significant drop in the number of species, caused by critically reduced survival rates for organisms. In order to obtain a deeper understanding of the threeway interplay between habitat loss, fragmentation and survival rates, we propose here a spatially explicit multi-scaled movement model of individuals that search for habitat.

Stochastic optimal foraging: tuning intensive and extensive dynamics in random searches.

Recent theoretical developments had laid down the proper mathematical means to understand how the structural complexity of search patterns may improve foraging efficiency. Under information-deprived scenarios and specific landscape configurations, Lévy walks and flights are known to lead to high search efficiencies. Based on a one-dimensional comparative analysis we show a mechanism by which, at random, a searcher can optimize the encounter with close and distant targets.

Brain complex network analysis by means of resting state fMRI and graph analysis: will it be helpful in clinical epilepsy?

Functional magnetic resonance imaging (fMRI) has just completed 20 years of existence. It currently serves as a research tool in a broad range of human brain studies in normal and pathological conditions, as is the case of epilepsy. To date, most fMRI studies aimed at characterizing brain activity in response to various active paradigms.

Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer.

The study of animal foraging behaviour is of practical ecological importance, and exemplifies the wider scientific problem of optimizing search strategies. Lévy flights are random walks, the step lengths of which come from probability distributions with heavy power-law tails, such that clusters of short steps are connected by rare long steps. Lévy flights display fractal properties, have no typical scale, and occur in physical and chemical systems.