Scaling of the Critical Function for the Standard Map: Some Numerical Results

The behavior of the critical function for the breakdown of the homotopically non-trivial invariant (KAM) curves for the standard map, as the rotation number tends to a rational number, is investigated using a version of Greene's residue criterion. The results are compared to the analogous ones for the radius of convergence of the Lindstedt series, in which case rigorous theorems have been proved. The conjectured interpolation of the critical function in terms of the Bryuno function is discussed.Comment: 26 pages, 3 figures, 13 table

Scaling law in the Standard Map critical function. Interpolating hamiltonian and frequency map analysis

We study the behaviour of the Standard map critical function in a neighbourhood of a fixed resonance, that is the scaling law at the fixed resonance. We prove that for the fundamental resonance the scaling law is linear. We show numerical evidence that for the other resonances $p/q$, $q \geq 2$, $p \neq 0$ and $p$ and $q$ relatively prime, the scaling law follows a power--law with exponent $1/q$.Comment: AMS-LaTeX2e, 29 pages with 8 figures, submitted to Nonlinearit

Perturbative analysis of disordered Ising models close to criticality

We consider a two-dimensional Ising model with random i.i.d. nearest-neighbor ferromagnetic couplings and no external magnetic field. We show that, if the probability of supercritical couplings is small enough, the system admits a convergent cluster expansion with probability one. The associated polymers are defined on a sequence of increasing scales; in particular the convergence of the above expansion implies the infinite differentiability of the free energy but not its analyticity. The basic tools in the proof are a general theory of graded cluster expansions and a stochastic domination of the disorder

Chaotic systems in complex phase space

This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviors of these two PT-symmetric dynamical models in complex phase space exhibit strong qualitative similarities.Comment: 22 page, 16 figure

Scaling of Self-Avoiding Walks in High Dimensions

We examine self-avoiding walks in dimensions 4 to 8 using high-precision Monte-Carlo simulations up to length N=16384, providing the first such results in dimensions $d > 4$ on which we concentrate our analysis. We analyse the scaling behaviour of the partition function and the statistics of nearest-neighbour contacts, as well as the average geometric size of the walks, and compare our results to $1/d$-expansions and to excellent rigorous bounds that exist. In particular, we obtain precise values for the connective constants, $\mu_5=8.838544(3)$, $\mu_6=10.878094(4)$, $\mu_7=12.902817(3)$, $\mu_8=14.919257(2)$ and give a revised estimate of $\mu_4=6.774043(5)$. All of these are by at least one order of magnitude more accurate than those previously given (from other approaches in $d>4$ and all approaches in $d=4$). Our results are consistent with most theoretical predictions, though in $d=5$ we find clear evidence of anomalous $N^{-1/2}$-corrections for the scaling of the geometric size of the walks, which we understand as a non-analytic correction to scaling of the general form $N^{(4-d)/2}$ (not present in pure Gaussian random walks).Comment: 14 pages, 2 figure

Determination of the exponent gamma for SAWs on the two-dimensional Manhattan lattice

We present a high-statistics Monte Carlo determination of the exponent gamma for self-avoiding walks on a Manhattan lattice in two dimensions. A conservative estimate is \gamma \gtapprox 1.3425(3), in agreement with the universal value 43/32 on regular lattices, but in conflict with predictions from conformal field theory and with a recent estimate from exact enumerations. We find strong corrections to scaling that seem to indicate the presence of a non-analytic exponent Delta < 1. If we assume Delta = 11/16 we find gamma = 1.3436(3), where the error is purely statistical.Comment: 24 pages, LaTeX2e, 4 figure

Fully Automatic Expression-Invariant Face Correspondence

We consider the problem of computing accurate point-to-point correspondences among a set of human face scans with varying expressions. Our fully automatic approach does not require any manually placed markers on the scan. Instead, the approach learns the locations of a set of landmarks present in a database and uses this knowledge to automatically predict the locations of these landmarks on a newly available scan. The predicted landmarks are then used to compute point-to-point correspondences between a template model and the newly available scan. To accurately fit the expression of the template to the expression of the scan, we use as template a blendshape model. Our algorithm was tested on a database of human faces of different ethnic groups with strongly varying expressions. Experimental results show that the obtained point-to-point correspondence is both highly accurate and consistent for most of the tested 3D face models

Two centuries of masting data for European beech and Norway spruce across the European continent

Tree masting is one of the most intensively studied ecological processes. It affects nutrient fluxes of trees, regeneration dynamics in forests, animal population densities, and ultimately influences ecosystem services. Despite a large volume of research focused on masting, its evolutionary ecology, spatial and temporal variability and environmental drivers are still matter of debate. Understanding the proximate and ultimate causes of masting at broad spatial and temporal scales will enable us to predict tree reproductive strategies and their response to changing environment. Here we provide broad spatial (distribution range-wide) and temporal (century) masting data for the two main masting tree species in Europe, European beech (Fagus sylvatica L.) and Norway spruce (Picea abies (L.) H. Karst.). We collected masting data from a total of 359 sources through an extensive literature review and from unpublished surveys. The dataset has a total of 1747 series and 18348 yearly observations from 28 countries and covering a time span of years 1677-2016 and 1791-2016 for beech and spruce, respectively. For each record, the following information is available: identification code; species; year of observation; proxy of masting (flower, pollen, fruit, seed, dendrochronological reconstructions); statistical data type (ordinal, continuous); data value; unit of measurement (only in case of continuous data); geographical location (country, Nomenclature of Units for Territorial Statistics NUTS-1 level, municipality, coordinates); first and last record year and related length; type of data source (field survey, peer reviewed scientific literature, grey literature, personal observation); source identification code; date when data were added to the database; comments. To provide a ready-to-use masting index we harmonized ordinal data into five classes. Furthermore, we computed an additional field where continuous series with length >4 years where converted into a five classes ordinal index. To our knowledge, this is the most comprehensive published database on species-specific masting behaviour. It is useful to study spatial and temporal patterns of masting and its proximate and ultimate causes, to refine studies based on tree-ring chronologies, to understand dynamics of animal species and pests vectored by these animals affecting human health, and it may serve as calibration-validation data for dynamic forest models.The paper was partly funded by the “Fondo di Ricerca Locale 2015-2016” of the University of Torino and by the Stiftelsen Stina Werners fond (grant SSWF 10-1/29-3 to I.D.)