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118
Saturation in the Hypercube and Bootstrap Percolation
, 2014
"... Let Qd denote the hypercube of dimension d. Given d ≥ m, a spanning subgraph G of Qd is said to be (Qd, Qm)saturated if it does not contain Qm as a subgraph but adding any edge of E(Qd) \E(G) creates a copy of Qm in G. Answering a question of Johnson and Pinto [27], we show that for every fixed m ≥ ..."
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≥ 2 the minimum number of edges in a (Qd, Qm)saturated graph is Θ(2 d). We also study weak saturation, which is a form of bootstrap percolation. Given graphs F and H, a spanning subgraph G of F is said to be weakly (F,H)saturated if the edges of E(F)\E(G) can be added to G one at a time so that each
LINEAR ALGEBRA AND BOOTSTRAP PERCOLATION
, 1107
"... Abstract. In Hbootstrap percolation, a set A ⊂ V(H) of initially ‘infected ’ vertices spreads by infecting vertices which are the only uninfected vertex in an edge of the hypergraph H. A particular case of this is the Hbootstrap process, in which H encodes copies of H in a graph G. We find the min ..."
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Cited by 6 (1 self)
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the minimum size of a set A that leads to complete infection when G and H are powers of complete graphs and H encodes induced copies of H in G. The proof uses linear algebra, a technique that is new in bootstrap percolation, although standard in the study of weakly saturated graphs, which are equivalent
Estimating fractal dimension
 Journal of the Optical Society of America A
, 1990
"... Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools ..."
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Cited by 120 (4 self)
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Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools for a variety of scientific disciplines, among which is chaos. Chaotic dynamical systems exhibit trajectories in their phase space that converge to a strange attractor. The fractal dimension of this attractor counts the effective number of degrees of freedom in the dynamical system and thus quantifies its complexity. In recent years, numerical methods have been developed for estimating the dimension directly from the observed behavior of the physical system. The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of fractals to nonlinear dynamical systems, and finally to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor. Confusion is a word we have invented for an order which is not understood.Henry Miller, "Interlude," Tropic of Capricorn Numerical coincidence is a common path to intellectual perdition in our quest for meaning. We delight in catalogs of disparate items united by the same number, and often feel in our gut that some unity must underlie it all.
Glassy dynamics of kinetically constrained models
 Adv. Phys
"... We review the use of kinetically constrained models (KCMs) for the study of dynamics in glassy systems. The characteristic feature of KCMs is that they have trivial, often noninteracting, equilibrium behaviour but interesting slow dynamics due to restrictions on the allowed transitions between conf ..."
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Cited by 44 (2 self)
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We review the use of kinetically constrained models (KCMs) for the study of dynamics in glassy systems. The characteristic feature of KCMs is that they have trivial, often noninteracting, equilibrium behaviour but interesting slow dynamics due to restrictions on the allowed transitions between configurations. The basic question which KCMs ask is therefore how much glassy physics can be understood without an underlying ‘equilibrium glass transition’. After a brief review of glassy phenomenology, we describe the main model classes, which include spinfacilitated (Ising) models, constrained lattice gases, models inspired by cellular structures such as soap froths, models obtained via mappings from interacting systems without constraints, and finally related models such as urn, oscillator, tiling and needle models. We then describe the broad range of techniques that have been applied to KCMs, including exact solutions, adiabatic approximations, projection and modecoupling techniques, diagrammatic approaches and mappings to quantum systems or effective models. Finally, we give a survey of the known
Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics
, 2008
"... Kinetically constrained lattice models of glasses introduced by Kob and Andersen (KA) are analyzed. It is proved that only two behaviors are possible on hypercubic lattices: either ergodicity at all densities or trivial nonergodicity, depending on the constraint parameter and the dimensionality. Bu ..."
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Cited by 2 (1 self)
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Kinetically constrained lattice models of glasses introduced by Kob and Andersen (KA) are analyzed. It is proved that only two behaviors are possible on hypercubic lattices: either ergodicity at all densities or trivial nonergodicity, depending on the constraint parameter and the dimensionality
Phase Transitions in Quantum Chromodynamics
, 1996
"... The current understanding of finite temperature phase transitions in QCD is reviewed. A critical discussion of refined phase transition criteria in numerical lattice simulations and of analytical tools going beyond the meanfield level in effective continuum models for QCD is presented. Theoretical ..."
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Cited by 7 (1 self)
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The current understanding of finite temperature phase transitions in QCD is reviewed. A critical discussion of refined phase transition criteria in numerical lattice simulations and of analytical tools going beyond the meanfield level in effective continuum models for QCD is presented. Theoretical predictions about the order of the transitions are compared with possible experimental
Derniame (Editor
 Lecture Notes in Computer Science 635: Software Process
, 1992
"... Contract No. DEAC0376SF00515This document, and the material and data contained therein, was developed under sponsorship of the United States Government. Neither the United States nor the Department of Energy, nor the Leland Stanford Junior University, nor their employees, nor their respective cont ..."
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Cited by 7 (0 self)
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Contract No. DEAC0376SF00515This document, and the material and data contained therein, was developed under sponsorship of the United States Government. Neither the United States nor the Department of Energy, nor the Leland Stanford Junior University, nor their employees, nor their respective contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any liability of responsibility for accuracy, completeness, or usefulness of any information, apparatus, product or process disclosed, or represents that its use will not infringe privately owned rights. Mention of any product, its manufacturer, or suppliers shall not, nor is it intended to, imply approval, disapproval, or fitness of any particular use. A royaltyfree, nonexclusive right to use and disseminate same for any purpose whatsoever, is expressly reserved to the United States and the University. Prepared for the Department of Energy under contract number DEAC0376S00515 by Stanford
A Normal Form Projection Algorithm for Associative Memory
 Associative Neural Memories: Theory and Implementation
, 1993
"... this paper is contained in the projection theorem, which details the associative memory capabilities of networks utilizing the normal form projection algorithm for storage of periodic attractors. The algorithm was originally designed, using dynamical systems theory, to allow learning and pattern re ..."
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Cited by 6 (1 self)
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this paper is contained in the projection theorem, which details the associative memory capabilities of networks utilizing the normal form projection algorithm for storage of periodic attractors. The algorithm was originally designed, using dynamical systems theory, to allow learning and pattern recognition with oscillatory attractors in models of olfactory cortex. Here we concentrate on mathematical analysis and engineering oriented applications of the algorithm, and briefly discuss biological models at the end. We focus attention on the storage of periodic attractors, since that is the best understood unusual capability of this system. The storage of static and chaotic attractors are discussed as variations on this theme. We hope to give intuitive discussion and geometric perspectives to compliment and clarify the formal analysis. Other approaches to oscillatory memory may be found in [26, 17, 45, 33, 37]. The normal form projection algorithm provides one solution to the problem of storing analog attractors in a recurrent neural network. Associative memory storage of analog patterns and continuous periodic sequences in the same network is analytically guaranteed. For a network with N nodes, the capacity is N
Results 1  10
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118