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131
Critical Phenomena in Gravitational Collapse
 Adv. Theor. Math. Phys
, 1997
"... Lecture given at the workshop "Mathematical aspects of theories of gravitation", Stefan Banach International Mathematical Centre, 7 March 1996. A miniintroduction to critical phenomena in gravitational collapse is combined with a more detailed discussion of the regularity of the " ..."
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Cited by 74 (2 self)
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Lecture given at the workshop "Mathematical aspects of theories of gravitation", Stefan Banach International Mathematical Centre, 7 March 1996. A miniintroduction to critical phenomena in gravitational collapse is combined with a more detailed discussion of the regularity of the "critical spacetimes" dominating these phenomena. 1 Critical phenomena in gravitational collapse Most of the material here is contained also in the paper [1], to which I refer the reader for details. Here I'll give detail only in the calculation of the critical exponent, and on how gravity regularizes selfsimilar solutions, where I hope it is of more than technical interest. Initial data for general relativity (GR) may or may not form a black hole. In a handwaving way one might compare this to a phase transition, and there is a "critical surface" in superspace (the phase space of GR) separating the two kinds of initial data. Choptuik [2] has explored this surface in a systematic way. For simplicity he...
Computational mechanics: Pattern and prediction, structure and simplicity
 Journal of Statistical Physics
, 1999
"... Computational mechanics, an approach to structural complexity, defines a process’s causal states and gives a procedure for finding them. We show that the causalstate representation—an Emachine—is the minimal one consistent with ..."
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Cited by 66 (10 self)
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Computational mechanics, an approach to structural complexity, defines a process’s causal states and gives a procedure for finding them. We show that the causalstate representation—an Emachine—is the minimal one consistent with
Statistical Mechanics: Entropy, Order Parameters, and Complexity
 Oxford Master Series in Physic
, 2006
"... The author provides this version of this manuscript with the primary intention of making the text accessible electronically—through web searches and for browsing and study on computers. Oxford University Press retains ownership of the copyright. Hardcopy printing, in particular, is subject to the s ..."
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Cited by 61 (2 self)
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The author provides this version of this manuscript with the primary intention of making the text accessible electronically—through web searches and for browsing and study on computers. Oxford University Press retains ownership of the copyright. Hardcopy printing, in particular, is subject to the same copyright rules as they would be for a printed book. CLARENDON PRESS. OXFORD
Crackling Noise
 Nature
, 2001
"... this paper, we will provide an overview of the renormalizationgroup '54 many researchers use to understand crackling noise. Briefly, the renormalization group discusses how the effective evolution laws of our system change as we measure on longer and longer length scales. (It works by gene ..."
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Cited by 34 (4 self)
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this paper, we will provide an overview of the renormalizationgroup '54 many researchers use to understand crackling noise. Briefly, the renormalization group discusses how the effective evolution laws of our system change as we measure on longer and longer length scales. (It works by generating a coarsegraining mapping in system space, the abstract space of all possible evolution laws.) The broad range of event sizes will be attributed to a selfsimilarity, where the evolution laws look the same under different length scales. Using this selfsimilarity, we are led to a method for scaling experimental data. In the simplest case this yields power laws and fractal structures, but more generally it leads to universal scaling functions  where we argue the real predictive power lies. We will only touch upon the dauntingly complex analytical methods used in this field, but we believe we can explain faithfully and fully both what our tools are useful for, and how to apply them in practice. The renormalization group is perhaps the most impressive use of abstraction in science
Belief Propagation On Partially Ordered Sets
 Mathematical Systems Theory in Biology, Communications, Computation, and Finance
, 2002
"... In this paper, which is based on the important recent work of Yedidia, Freeman, and Weiss, we present a generalized form of belief propagation, viz. belie) e propagation on a partially ordered set (PBP). PBP is an iterative messagepassing algorithm for solving, either exactly or approximately, the ..."
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Cited by 25 (0 self)
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In this paper, which is based on the important recent work of Yedidia, Freeman, and Weiss, we present a generalized form of belief propagation, viz. belie) e propagation on a partially ordered set (PBP). PBP is an iterative messagepassing algorithm for solving, either exactly or approximately, the marginalized product density problem, which is a general computational problem of wide applicability. We will show that PBP can be thought of as an algorithm for minimizing a certain "free energy" function, and by exploiting this interpretation, we will exhibit a onetoone correspondence between the fixed points of PBP and the stationary points of the free energy.
2003), Anomalous stress diffusion in earthquake triggering: Correlation length, time dependence, and directionality
"... dependence, and directionality ..."
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Convex relaxation methods for graphical models: Lagrangian and maximum entropy approaches
, 2008
"... Graphical models provide compact representations of complex probability distributions of many random variables through a collection of potential functions defined on small subsets of these variables. This representation is defined with respect to a graph in which nodes represent random variables and ..."
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Cited by 17 (2 self)
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Graphical models provide compact representations of complex probability distributions of many random variables through a collection of potential functions defined on small subsets of these variables. This representation is defined with respect to a graph in which nodes represent random variables and edges represent the interactions among those random variables. Graphical models provide a powerful and flexible approach to many problems in science and engineering, but also present serious challenges owing to the intractability of optimal inference and estimation over general graphs. In this thesis, we consider convex optimization methods to address two central problems that commonly arise for graphical models. First, we consider the problem of determining the most probable configuration—also known as the maximum a posteriori (MAP) estimate—of all variables in a graphical model, conditioned on (possibly noisy) measurements of some variables. This general problem is intractable, so we consider a Lagrangian relaxation (LR) approach to obtain a tractable dual problem. This involves using the Lagrangian decomposition technique
The role of clustering on the emergence of efficient social conventions
 In IJCAI
, 2005
"... Multiagent models of the emergence of social conventions have demonstrated that global conventions can arise from local coordination processes without a central authority. We further develop and extend previous work to address how and under what conditions emerging conventions are also socially effi ..."
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Cited by 16 (0 self)
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Multiagent models of the emergence of social conventions have demonstrated that global conventions can arise from local coordination processes without a central authority. We further develop and extend previous work to address how and under what conditions emerging conventions are also socially efficient, i.e. better for all agents than potential alternative conventions. We show with computational experiments that the clustering coefficient of the networks within which agents interact is an important condition for efficiency. We also develop an analytical approximation of the simulation model that sheds some light to the original model behavior. Finally, we combine two decision mechanisms, local optimization and imitation, to study the competition between efficient and attractive actions. Our main result is that in clustered networks a society converges to an efficient convention and is stable against invasion of suboptimal conventions under a much larger range of conditions than in a nonclustered network. On the contrary, in nonclustered networks the convention finally established heavily depends on its initial support. 1
Topologies of Social Interactions
, 2001
"... The paper extends the BrockDurlauf model of interactive discrete choice, where individuals’ decisions are influenced by the decisions of others, to richer social structures. Social structure is modelled by a graph, with individuals as vertices and interaction between individuals as edges. The paper ..."
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Cited by 13 (3 self)
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The paper extends the BrockDurlauf model of interactive discrete choice, where individuals’ decisions are influenced by the decisions of others, to richer social structures. Social structure is modelled by a graph, with individuals as vertices and interaction between individuals as edges. The paper extends the mean field case to stylized interaction topologies like the Walrasian star, the cycle and the onedimensional lattice (or path) and compares the properties of Nash equilibria when agents act on the basis of expectations over their neighbors ’ decisions versus actual knowledge of neighbors ’ decisions. It links links social interactions theory with the econometric theory of systems of simultaneous equations modelling discrete decisions. The paper obtains general results for the dynamics of adjustment towards steady states and shows that they combine spectral properties of the adjacency matrix with those associated with the nonlinearity of the reaction functions that lead to multiplicity of steady states. When all agents have the same number of neighbors the dynamics of adjustment exhibit relative persistence. Cyclical interaction is associated with endogenous and generally transient spatial oscillations that take the form of islands of conformity, but multiplicity of equilibria leads to permanent effects of initial conditions. The paper also analyzes stochastic dynamics for general interaction topologies, when agents acts with knowledge of their neighbors ’ actual decisions, which involve networked Markov chains in sample spaces of very high dimensionality.