We introduce a new nonconservative self-organized critical model. This model is equivalent to a quasistatic two-dimensional version of the Burridge-KnopoA spring-block model of earthquakes. Our model displays a robust power-law behavior. The exponent is not universal; rather it depends on the level of conservation. A dynamical phase transition from localized to nonlocalized behavior is seen as the level of conservation is increased. The model gives a good prediction of the Gutenberg-Richter law and an explanation to the variances in the observed b values. PACS numbers: 9 I.30.Px, 05.40.+j, 05.45.+b The dynamics of earthquake faults may provide a physical realization of the recently proposed idea of selforganized criticality (SOC). Bak, Tang, and Wiesenfeld (BTW) introduced the concept of self-organized criticality: Dynamical many-body systems reach a critical state without the need to fine-tune the system parameters [1]. BTW showed that a certain class of systems drive themselves into a statistically stationary state characterized by spatial and temporal correlation functions exhibiting power-law behavior. Hence, the system has no intrinsic length or time scale and is in this sense critical. The study of the SOC systems has to a great extent been based on simulations using cellular automaton models.The majority of these simulations have been limited to conservative models. It has been suggested that the necessary (and sufficient) condition for SOC is indeed a conservation law [2,3]. This seems to be the situation for SOC models where perturbation is done locally as in the original BTW model [4]. Recently, though, it was shown that a special nonconservative model with a global perturbation displays SOC [5].Earthquakes are probably the most relevant paradigm of self-organized criticality. In 1956 Gutenberg and Richter realized that the rate of occurrence of earthquakes of magnitude M greater than m is given by the relation log]OE =c+dm,where the parameter d is 1 and & for small and large earthquakes, respectively [8]. Thus the Gutenberg-Richter law is transformed into a power law for the number of observed earthquakes with energy greater than E, N(go & g)g (3) log~uN(M & m) =a bm . -This is the Gutenberg-Richter law [6]. The parameter b has been recorded to have a wide range of values for diff'erent faults. Findings of b from 0.80 to 1.06 for small earthquakes and 1.23 to 1.54 for large earthquakes have been reported [7]. The energy (seismic moment) E released during the earthquake is believed to increase exponentially with the earthquake magnitude, Note that B is in the same range for both small and large earthquakes, namely, 0.80-1.05. Bak and Tang indicated that the simple conservative SOC models can serve as a framework for explaining the power-law behavior, giving a 8 value of 0.2 [9]. Similar results are obtained for two-dimensional models in [10,11]. Otsuka was the first to simulate a 2D version of the Burridge-Knopoff model and he found B=0.8 [12]. Carlson and Langer proposed a 1D dynamical version of th...

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Corrections NEUROSCIENCE. For the article ''Development of 17 O NMR approach for fast imaging of cerebral metabolic rate of oxygen in rat brain at high field,'' authors note the following correction. After the publication of this article, the authors found a technical error in the setup of parameters used for acquiring the 3D 17 O magnetic resonance spectroscopic (MRS) images reported in this article. This error led to overestimations of the spatial resolution of 3D 17 O MRS images. The claimed voxel sizes of the 3D 17 O MRS images (Figs. 2 and 3) and the cerebral metabolic rate of oxygen (CMRO 2) image (Fig. 6) are 57% smaller than the actual voxel size in each spatial dimension. Therefore, the correct voxel size was 0.10 ml, and the correct field-of-view (FOV) used in the 3D 17 O MRS images was 28 28 24 mm 3. This correction should not significantly affect the major conclusions and methodology presented in this article. However, the correction could reveal that the current sensitivity of 17 O NMR and, alternatively, the spatial resolution of the 17 O MRS image achieved at 9.4 tesla may be potentially limited for determining and imaging CMRO 2 in small brain structures such as the white matter in the rat brain. www.pnas.orgcgidoi10.1073pnas.0837868100 COLLOQUIUM. For the colloquium paper ''Unified scaling law for earthquakes,'' by Kim Christensen, Leon Danon

The probability density function (PDF) of a global measure in a large class of highly correlated systems has been suggested to be of the same functional form. Here, we identify the analytical form of the PDF of one such measure, the order parameter in the low temperature phase of the 2D-XY model. We demonstrate that this function describes the fluctuations of global quantities in other correlated, equilibrium and non-equilibrium systems. These include a coupled rotor model, Ising and percolation models, models of forest fires, sand-piles, avalanches and granular media in a self organized critical state. We discuss the relationship with both Gaussian and extremal statistics.PACS numbers: 05.40, 05.65, 47.27, 68.35.Rh Self similarity is an important feature of the natural world. It arises in strongly correlated many body systems when fluctuations over all scales from a microscopic length a to a diverging correlation length ξ lead to the appearence of "anomalous dimension" [1] and fractal properties. However, even in an ideal world the divergence of of ξ must ultimately be cut off by a macroscopic length L, allowing the definition of a range of scales between a and L, over which the anomalous behaviour can occur. Such systems are found, for example, in critical phenomena, in Self-Organized Criticality [2,3] or in turbulent flow problems. By analogy with fluid mechanics we shall call these finite size critical systems "inertial systems" and the range of scales between a and L the "inertial range". One of the anomalous statistical properties of inertial systems is that, whatever their size, they can never be divided into mesoscopic regions that are statistically independent. As a result they do not satisfy the basic criterion of the central limit theorem and one should not necessarily expect global, or spatially averaged quantities to have Gaussian fluctuations about the mean value. In Ref.[4](BHP) it was demonstrated that two of these systems, a model of finite size critical behaviour and a steady state in a closed turbulent flow experiment, share the same non-Gaussian probability distribution function (PDF) for fluctuations of global quantities. Consequently it was proposed that these two systems -so utterly dissimilar in regards to their microscopic details -share the same statistics simply because they are critical. If this is the case, one should then be able to describe turbulence as a finite-size critical phenomenon, with an effective "universality class". As, however, turbulence and the magnetic model are very unlikely to share the same universality class, it was implied that the differences that separate critical phenomena into universality classes represent at most a minor perturbation on the functional form of the PDF. In this paper, to test this proposition, we determine the functional form of the BHP fluctuation spectrum and show that it indeed applies to a large class of inertial systems [5].The magnetic model studied by BHP, the spin wave limit to the two dimensional XY (2D-XY) model, is defined by ...

We have studied experimentally transport properties in a slowly driven granular system which recently was shown to display self-organized criticality [Frette et al., Nature 379, 49 (1996)]. Tracer particles were added to a pile and their transit times measured. The distribution of transit times is a constant with a crossover to a decaying power law. The average transport velocity decreases with system size. This is due to an increase in the active zone depth with system size. The relaxation processes generate coherently moving regions of grains mixed with convection. This picture is supported by considering transport in a 1D cellular automaton modeling the experiment. The avalanches that occur when grains are dropped onto a pile illustrate the spontaneous generation of complexity in simple dynamical systems [1]. When grains are dropped onto a finite base, a pile builds up. However, it cannot become infinitely high, and, eventually, the system settles in a stationary state where the outflux over the edge of the base on average equals the influx. Intermittent flow of grains down the slope of the pile (small and large avalanches) maintain the system in this state. Bak, Tang, and Wiesenfeld constructed a 2D cellular automaton of a slowly driven dynamical system. They showed, that the "pile" spontaneously evolves, or self-organizes, into a state with avalanches of all sizes distributed according to a power law, that is, there is no internal system-specific scale. Because of the lack of any characteristic avalanche size, the system is referred to as critical [1].It has been a longstanding question whether real granular systems display self-organized criticality (SOC) when slowly driven. Recently, however, an experiment on a quasi one-dimensional pile of rice has shown that the occurrence of SOC depends on details in the grain-level dissipation mechanisms [2]. With nearly spherical grains, a characteristic avalanche size appeared, inconsistent with SOC. Only with sufficiently elongated grains, avalanches with a power-law distribution occurred. We focus in this Letter on the transport that results from avalanches in the system displaying SOC. The elongated rice grains could pack in a variety of ways, and each avalanche replaced, locally or globally, one surface configuration with another. Thus a dynamically varying medium disorder (coupled to the relaxation processes) was generated. This is conceptually different from transport in media with a quenched disorder, see e.g. Refs. [3,4]. Furthermore, in SOC systems, a small perturbation may lead to arbitrarily large avalanches, and it is not clear at all, how this affects the transport properties. Thus it is quite surprising, that there are no experiments and only a few theoretical and numerical studies on transport in systems displaying SOC [5][6][7].We have measured the transit times of colored tracer particles in the rice pile. Experimentally, we find that the distribution of transit times is essentially constant for small transit times T and decays as a power law for l...

International audienceWe show how pre-averaging can be applied to the problem of measuring the ex-post covariance of financial asset returns under microstructure noise and non-synchronous trading. A pre-averaged realised covariance is proposed, and we present an asymptotic theory for this new estimator, which can be configured to possess an optimal convergence rate or to ensure positive semi-definite covariance matrix estimates. We also derive a noise-robust Hayashi-Yoshida estimator that can be implemented on the original data without prior alignment of prices. We uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high-frequency equity data

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