What can one learn about Self-Organized Criticality from Dynamical Systems theory ?

We develop a dynamical system approach for the Zhang's model of Self-Organized Criticality, for which the dynamics can be described either in terms of Iterated Function Systems, or as a piecewise hyperbolic dynamical system of skew-product type. In this setting we describe the SOC attractor, and discuss its fractal structure. We show how the Lyapunov exponents, the Hausdorff dimensions, and the system size are related to the probability distribution of the avalanche size, via the Ledrappier-Young formula.Comment: 23 pages, 8 figures. to appear in Jour. of Stat. Phy

A relaxed approach for curve matching with elastic metrics

In this paper we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions. It extends previous works on this topic in several important ways. The model and resulting matching algorithm integrate within one common setting both the family of $H^2$-metrics with constant coefficients and scale-invariant $H^2$-metrics on both open and closed immersed curves. These families include as particular cases the class of first-order elastic metrics. An essential difference with prior approaches is the way that boundary constraints are dealt with. By leveraging varifold-based similarity metrics we propose a relaxed variational formulation for the matching problem that avoids the necessity of optimizing over the reparametrization group. Furthermore, we show that we can also quotient out finite-dimensional similarity groups such as translation, rotation and scaling groups. The different properties and advantages are illustrated through numerical examples in which we also provide a comparison with related diffeomorphic methods used in shape registration.Comment: 27 page

Efficient Computation of Invariant Tori in Volume-Preserving Maps

In this paper we implement a numerical algorithm to compute codimension-one tori in three-dimensional, volume-preserving maps. A torus is defined by its conjugacy to rigid rotation, which is in turn given by its Fourier series. The algorithm employs a quasi-Newton scheme to find the Fourier coefficients of a truncation of the series. This technique is based upon the theory developed in the accompanying article by Blass and de la Llave. It is guaranteed to converge assuming the torus exists, the initial estimate is suitably close, and the map satisfies certain nondegeneracy conditions. We demonstrate that the growth of the largest singular value of the derivative of the conjugacy predicts the threshold for the destruction of the torus. We use these singular values to examine the mechanics of the breakup of the tori, making comparisons to Aubry-Mather and anti-integrability theory when possible

Two-phase incremental kernel PCA for learning massive or online datasets

As a powerful nonlinear feature extractor, kernel principal component analysis (KPCA) has been widely adopted in many machine learning applications. However, KPCA is usually performed in a batch mode, leading to some potential problems when handling massive or online datasets. To overcome this drawback of KPCA, in this paper, we propose a two-phase incremental KPCA (TP-IKPCA) algorithm which can incorporate data into KPCA in an incremental fashion. In the first phase, an incremental algorithm is developed to explicitly express the data in the kernel space. In the second phase, we extend an incremental principal component analysis (IPCA) to estimate the kernel principal components. Extensive experimental results on both synthesized and real datasets showed that the proposed TP-IKPCA produces similar principal components as conventional batch-based KPCA but is computationally faster than KPCA and its several incremental variants. Therefore, our algorithm can be applied to massive or online datasets where the batch method is not available

Combinatorial cobordism maps in hat Heegaard Floer theory

In a previous paper, Sarkar and the third author gave a combinatorial description of the hat version of Heegaard Floer homology for three-manifolds. Given a cobordism between two connected three-manifolds, there is an induced map between their Heegaard Floer homologies. Assume that the first homology group of each boundary component surjects onto the first homology group of the cobordism (modulo torsion). Under this assumption, we present a procedure for finding the rank of the induced Heegaard Floer map combinatorially, in the hat version.Comment: 34 pages, 31 figures; discussion now limited to cobordisms satisfying a homological assumption; Section 4 completely rewritten; various other revisions; this version to appear in Duke Math.