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Data assimilation is an iterative approach to the problem of estimating the state of a dynam-ical system using both current and past observations of the system together with a model for the system's time evolution. Rather than solving the problem from scratch each time new observations become available, one uses the model to " forecast " the current state,(More)
In this paper, we introduce a new, local formulation of the ensemble Kalman Filter approach for atmospheric data assimilation. Our scheme is based on the hypothesis that, when the Earth's surface is divided up into local regions of moderate size, vectors of the forecast uncertainties in such regions tend to lie in a subspace of much lower dimension than(More)
We study the transition from incoherence to coherence in large networks of coupled phase oscillators. We present various approximations that describe the behavior of an appropriately defined order parameter past the transition and generalize recent results for the critical coupling strength. We find that, under appropriate conditions, the coupling strength(More)
The largest eigenvalue of the adjacency matrix of networks is a key quantity determining several important dynamical processes on complex networks. Based on this fact, we present a quantitative, objective characterization of the dynamical importance of network nodes and links in terms of their effect on the largest eigenvalue. We show how our(More)
We consider the image of a fractal set X in a Banach space under typical linear and nonlinear projections π into R N. We prove that when N exceeds twice the box-counting dimension of X, then almost every (in the sense of prevalence) such π is one-to-one on X, and we give an explicit bound on the Hölder exponent of the inverse of the restriction of π to X.(More)
We introduce a new potential-theoretic definition of the dimension spectrum D q of a probability measure for q > 1 and explain its relation to prior definitions. We apply this definition to prove that if 1 < q 2 and µ is a Borel probability measure with compact support in R n , then under almost every linear transformation from R n to R m , the q-dimension(More)
Regions in the parameter space of chaotic systems that correspond to stable behavior are often referred to as windows. In this Letter, we elucidate the occurrence of such regions in higher dimensional chaotic systems. We describe the fundamental structure of these windows, and also indicate under what circumstances one can expect to find them. These results(More)
The largest eigenvalue of the adjacency matrix of a network plays an important role in several network processes (e.g., synchronization of oscillators, percolation on directed networks, and linear stability of equilibria of network coupled systems). In this paper we develop approximations to the largest eigenvalue of adjacency matrices and discuss the(More)
MOTIVATION Comparison of nucleic acid and protein sequences is a fundamental tool of modern bioinformatics. A dominant method of such string matching is the 'seed-and-extend' approach, in which occurrences of short subsequences called 'seeds' are used to search for potentially longer matches in a large database of sequences. Each such potential match is(More)
We study the emergence of coherence in large complex networks of interacting heterogeneous dynamical systems. We show that for a large class of dynamical systems and network topologies there is a critical coupling strength at which the systems undergo a transition from incoherent to coherent behavior. We find that the critical coupling strength at which(More)