Natural Halting Probabilities, Partial Randomness, and Zeta Functions

We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin's Omega number, halting probability, and program-size complexity. A classification of Turing machines according to their zeta numbers is proposed: divergent, convergent and tuatara. We prove the existence of universal convergent and tuatara machines. Various results on (algorithmic) randomness and partial randomness are proved. For example, we show that the zeta number of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic) randomness--which cannot be naturally characterised in terms of plain complexity--asymptotic randomness admits such a characterisation.Comment: Accepted for publication in Information and Computin

Accumulation horizons and period-adding in optically injected semiconductor lasers

We study the hierarchical structuring of islands of stable periodic oscillations inside chaotic regions in phase diagrams of single-mode semiconductor lasers with optical injection. Phase diagrams display remarkable {\it accumulation horizons}: boundaries formed by the accumulation of infinite cascades of self-similar islands of periodic solutions of ever-increasing period. Each cascade follows a specific period-adding route. The riddling of chaotic laser phases by such networks of periodic solutions may compromise applications operating with chaotic signals such as e.g. secure communications.Comment: 4 pages, 4 figures, laser phase diagrams, to appear in Phys. Rev. E, vol. 7

Entropic effects in large-scale Monte Carlo simulations

The efficiency of Monte Carlo samplers is dictated not only by energetic effects, such as large barriers, but also by entropic effects that are due to the sheer volume that is sampled. The latter effects appear in the form of an entropic mismatch or divergence between the direct and reverse trial moves. We provide lower and upper bounds for the average acceptance probability in terms of the Renyi divergence of order 1/2. We show that the asymptotic finitude of the entropic divergence is the necessary and sufficient condition for non-vanishing acceptance probabilities in the limit of large dimensions. Furthermore, we demonstrate that the upper bound is reasonably tight by showing that the exponent is asymptotically exact for systems made up of a large number of independent and identically distributed subsystems. For the last statement, we provide an alternative proof that relies on the reformulation of the acceptance probability as a large deviation problem. The reformulation also leads to a class of low-variance estimators for strongly asymmetric distributions. We show that the entropy divergence causes a decay in the average displacements with the number of dimensions n that are simultaneously updated. For systems that have a well-defined thermodynamic limit, the decay is demonstrated to be n^{-1/2} for random-walk Monte Carlo and n^{-1/6} for Smart Monte Carlo (SMC). Numerical simulations of the LJ_38 cluster show that SMC is virtually as efficient as the Markov chain implementation of the Gibbs sampler, which is normally utilized for Lennard-Jones clusters. An application of the entropic inequalities to the parallel tempering method demonstrates that the number of replicas increases as the square root of the heat capacity of the system.Comment: minor corrections; the best compromise for the value of the epsilon parameter in Eq. A9 is now shown to be log(2); 13 pages, 4 figures, to appear in PR

Dynamics of Saccharomyces cerevisiae strains isolated from vine bark in vineyard: Influence of plant age and strain presence during grape must spontaneous fermentations

In this study, two vineyards of different age were chosen. During three years, a sampling campaign was performed for isolating vineyard-associated Saccharomyces cerevisiae (S. cerevisiae) strains. Bark portions and, when present, grape bunches were regularly collected from the same vine plants during the overall sampling period. Each bark portion was added to a synthetic must, while each grape bunch was manually crushed, and fermentations were run to isolate S. cerevisiae strains. All collected yeasts were identified at different species and strain levels to evaluate the genetic variability of S. cerevisiae strains in the two vineyards and strains dynamics. Moreover, bark-associated strains were compared with those isolated from spontaneous fermentations of grapes collected during the two harvests. Regarding the youngest vineyard, no S. cerevisiae was identified on bark and grape surface, highlighting the importance of vine age on yeast colonization. Results reported the isolation of S. cerevisiae from vine bark of the old vineyard at all sampling times, regardless of the presence of the grape bunch. Therefore, this environment can be considered an alternative ecological niche that permanently hosts S. cerevisiae. Bark-associated strains were not found on grape bunches and during pilot-scale vinifications, indicating no significative strain transfer from vine bark to the grape must. Commercial starters were identified as well both in vineyards and during vinifications

Reconstructing the Density of States by History-Dependent Metadynamics

We present a novel method for the calculation of the energy density of states D(E) for systems described by classical statistical mechanics. The method builds on an extension of a recently proposed strategy that allows the free energy profile of a canonical system to be recovered within a pre-assigned accuracy,[A. Laio and M. Parrinello, PNAS 2002]. The method allows a good control over the error on the recovered system entropy. This fact is exploited to obtain D(E) more efficiently by combining measurements at different temperatures. The accuracy and efficiency of the method are tested for the two-dimensional Ising model (up to size 50x50) by comparison with both exact results and previous studies. This method is a general one and should be applicable to more realistic model systems

Crystal energy functions via the charge in types A and C

The Ram-Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types A and C it can be defined on tensor products of Kashiwara-Nakashima single column crystals. In this paper we prove that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of the combinatorial R-matrix.Comment: 25 pages; 1 figur