78 results on '"Lee DeVille"'
Search Results
2. Graph Homology and Stability of Coupled Oscillator Networks.
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Jared C. Bronski, Lee DeVille, and Timothy Ferguson
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- 2016
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3. Moment Closure and Finite-Time Blowup for Piecewise Deterministic Markov Processes.
- Author
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Lee DeVille, Sairaj V. Dhople, Alejandro D. Domínguez-García, and Jiangmeng Zhang
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- 2016
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4. A maximum entropy approach to the moment closure problem for Stochastic Hybrid Systems at equilibrium.
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Jiangmeng Zhang, Lee DeVille, Sairaj V. Dhople, and Alejandro D. Domínguez-García
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- 2014
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5. Dynamical systems defined on simplicial complexes: Symmetries, conjugacies, and invariant subspaces
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Eddie Nijholt and Lee DeVille
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34D06, 55U10, 15A18, 05C65, 91D31 ,Applied Mathematics ,FOS: Physical sciences ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Dynamical Systems (math.DS) ,Nonlinear Sciences - Chaotic Dynamics ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,Chaotic Dynamics (nlin.CD) ,Mathematics - Dynamical Systems ,Mathematical Physics - Abstract
We consider the general model for dynamical systems defined on a simplicial complex. We describe the conjugacy classes of these systems and show how symmetries in a given simplicial complex manifest in the dynamics defined thereon, especially with regard to invariant subspaces in the dynamics.
- Published
- 2022
6. Stability of Distributed Algorithms in the Face of Incessant Faults.
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R. E. Lee DeVille and Sayan Mitra
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- 2009
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7. Stability of Distributed Algorithms in the Face of Incessant Faults
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Lee DeVille, Robert E., Mitra, Sayan, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Guerraoui, Rachid, editor, and Petit, Franck, editor
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- 2009
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8. Spectral Theory for Dynamics on Graphs Containing Attractive and Repulsive Interactions.
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Jared C. Bronski and Lee DeVille
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- 2014
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9. A Stochastic Hybrid Systems framework for analysis of Markov reward models.
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Sairaj V. Dhople, Lee DeVille, and Alejandro D. Domínguez-García
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- 2014
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10. Synchrony and Periodicity in Excitable Neural Networks with Multiple Subpopulations.
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Lee DeVille and Yi Zeng
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- 2014
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11. Analysis of Power System Dynamics Subject to Stochastic Power Injections.
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Sairaj V. Dhople, Yu Christine Chen, Lee DeVille, and Alejandro D. Domínguez-García
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- 2013
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12. Stability of a Stochastic Two-Dimensional Non-Hamiltonian System.
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R. E. Lee DeVille, Navaratnam Sri Namachchivaya, and Zoi Rapti
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- 2011
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13. Weighted Flow Algorithms (WFA) for stochastic particle coagulation.
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R. E. Lee DeVille, Nicole Riemer, and Matthew West 0001
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- 2011
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14. STABILITY OF A STOCHASTIC TWO-DIMENSIONAL NON-HAMILTONIAN SYSTEM
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LEE DEVILLE, R. E., NAMACHCHIVAYA, N. SRI, and RAPTI, ZOI
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- 2011
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15. Phase-locked patterns of the Kuramoto model on 3-regular graphs
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Lee DeVille and Bard Ermentrout
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- 2016
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16. Framework for analyzing ecological trait-based models in multidimensional niche spaces
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Tommaso Biancalani, Lee DeVille, and Nigel Goldenfeld
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- 2015
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17. Spectral Theory for Networks with Attractive and Repulsive Interactions
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Jared C. Bronski and Lee DeVille
- Published
- 2013
18. Synchronization conditions in the Kuramoto model and their relationship to seminorms
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Lee DeVille, Thomas E. Carty, and Jared C. Bronski
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Convex analysis ,Permutohedron ,Applied Mathematics ,Kuramoto model ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Dynamical Systems (math.DS) ,34C15, 34D06, 52A20, 60F17 ,FOS: Mathematics ,Applied mathematics ,Mathematics - Dynamical Systems ,Extreme value theory ,Mathematical Physics ,Mathematics - Abstract
In this paper we address two questions about the synchronization of coupled oscillators in the Kuramoto model with all-to-all coupling. In the first part we use some classical results in convex geometry to prove bounds on the size of the frequency set supporting the existence of stable, phase locked solutions and show that the set of such frequencies can be expressed by a seminorm which we call the Kuramoto norm. In the second part we use some ideas from extreme order statistics to compute upper and lower bounds on the probability of synchronization for very general frequency distributions. We do so by computing exactly the limiting extreme value distribution of a quantity that is equivalent to the Kuramoto norm., Keywords: Kuramoto model, convex analysis, permutahedron, extreme-value statistics
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- 2020
19. Brjuno numbers and the symbolic dynamics of the complex exponential
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Lee DeVille, R. E.
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- 2004
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20. Circulant Type Formulas for the Eigenvalues of Linear Network Maps
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Lee DeVille and Eddie Nijholt
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Numerical Analysis ,Algebra and Number Theory ,Trace (linear algebra) ,010102 general mathematics ,Multiplicative function ,Dynamical Systems (math.DS) ,010103 numerical & computational mathematics ,Disjoint sets ,Function (mathematics) ,01 natural sciences ,Combinatorics ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Geometry and Topology ,Linear independence ,0101 mathematics ,Mathematics - Dynamical Systems ,Linear combination ,Circulant matrix ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Given an admissible map γ f for a homogeneous network N , it is known that the Jacobian D γ f ( x ) around a fully synchronous point x = ( x 0 , … , x 0 ) is again an admissible map for N . Motivated by this, we study the spectra of linear admissible maps for homogeneous networks. In particular, we define so-called network multipliers. These are (relatively small) matrices that depend linearly on the coefficients of the response function, and whose eigenvalues together make up the spectrum of the corresponding admissible map. More precisely, given a network N , we define a finite set of network multipliers ( Λ l ) l = 1 k and a class of networks C containing N . This class is furthermore closed under taking quotient networks, subnetworks and disjoint unions. We then show that the eigenvalues of an admissible map for any network in C are given by those of (a subset of) the network multipliers, with fixed multiplicities ( m l ) l = 1 k and independently of the given (finite dimensional) phase space of a node. The coefficients of all the network multipliers of C are furthermore linearly independent, which implies that one may find the multiplicities ( m l ) l = 1 k by simply expressing the trace of an admissible map as a linear combination of the traces of the multipliers. In particular, we will give examples of networks where the network multipliers need not be constructed, but can be determined by looking at small networks in C . We also show that network multipliers are multiplicative with respect to composition of linear maps.
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- 2019
21. Consensus on simplicial complexes: Results on stability and synchronization
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Lee DeVille
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Pure mathematics ,Computer science ,Generalization ,Applied Mathematics ,Dimension (graph theory) ,Stability (learning theory) ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,01 natural sciences ,010305 fluids & plasmas ,0103 physical sciences ,Synchronization (computer science) ,Balanced flow ,010306 general physics ,Laplace operator ,Mathematical Physics ,Energy functional ,Network model - Abstract
We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us to formulate flows on simplices of any dimension so that it includes edge flows, triangle flows, etc. We show that the system can be represented as the gradient flow of an energy functional and use this to deduce the stability of various steady states of the model. Finally, we demonstrate that our model contains higher-dimensional analogs of structures seen in related network models.
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- 2021
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22. Moment Closure and Finite-Time Blowup for Piecewise Deterministic Markov Processes
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Alejandro D. Dominguez-Garcia, Jiangmeng Zhang, Lee DeVille, and Sairaj V. Dhople
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0209 industrial biotechnology ,Class (set theory) ,Mathematical optimization ,Markov process ,02 engineering and technology ,Interval (mathematics) ,01 natural sciences ,Time reversibility ,symbols.namesake ,020901 industrial engineering & automation ,Moment closure ,Modeling and Simulation ,Hybrid system ,0103 physical sciences ,Piecewise ,symbols ,Applied mathematics ,Variety (universal algebra) ,010306 general physics ,Analysis ,Mathematics - Abstract
We present a variety of results analyzing the behavior of a class of stochastic processes---referred to as piecewise deterministic Markov processes (PDMPs)---for the infinite-time interval and determine general conditions on when the moments of such processes will, or will not, be well-behaved. We also characterize the types of finite-time blowups that are possible for these processes, and obtain bounds on their probabilities.
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- 2016
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23. Configurational stability for the Kuramoto-Sakaguchi model
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Lee DeVille, Thomas E. Carty, and Jared C. Bronski
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0209 industrial biotechnology ,Generalization ,media_common.quotation_subject ,General Physics and Astronomy ,Frustration ,FOS: Physical sciences ,02 engineering and technology ,Dynamical Systems (math.DS) ,Pattern Formation and Solitons (nlin.PS) ,Fixed point ,Type (model theory) ,01 natural sciences ,Stability (probability) ,Instability ,010305 fluids & plasmas ,020901 industrial engineering & automation ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Dynamical Systems ,Mathematical Physics ,Mathematics ,media_common ,Applied Mathematics ,Kuramoto model ,Mathematical analysis ,Statistical and Nonlinear Physics ,Nonlinear Sciences - Pattern Formation and Solitons ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,Flow (mathematics) ,34D06, 34D20, 37G35, 05C31 ,Adaptation and Self-Organizing Systems (nlin.AO) - Abstract
The Kuramoto–Sakaguchi model is a generalization of the well-known Kuramoto model that adds a phase-lag paramater or “frustration” to a network of phase-coupled oscillators. The Kuramoto model is a flow of gradient type, but adding a phase-lag breaks the gradient structure, significantly complicating the analysis of the model. We present several results determining the stability of phase-locked configurations: the first of these gives a sufficient condition for stability, and the second a sufficient condition for instability. In fact, the instability criterion gives a count, modulo 2, of the dimension of the unstable manifold to a fixed point and having an odd count is a sufficient condition for instability of the fixed point. We also present numerical results for both small ( N≤10) and large ( N=50) collections of Kuramoto–Sakaguchi oscillators.
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- 2018
24. Synchronization and Stability for Quantum Kuramoto
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Lee DeVille
- Subjects
Computer science ,Special solution ,Kuramoto model ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Dynamical Systems (math.DS) ,Mathematical Physics (math-ph) ,Pattern Formation and Solitons (nlin.PS) ,Topology ,Network topology ,01 natural sciences ,Stability (probability) ,Nonlinear Sciences - Pattern Formation and Solitons ,Synchronization ,010305 fluids & plasmas ,Connection (mathematics) ,Nonlinear Sciences::Chaotic Dynamics ,82C10, 34D06, 58C40, 15A18 ,0103 physical sciences ,Attractor ,FOS: Mathematics ,Mathematics - Dynamical Systems ,010306 general physics ,Quantum ,Mathematical Physics - Abstract
We present and analyze a nonabelian version of the Kuramoto system, which we call the Quantum Kuramoto system. We study the stability of several classes of special solutions to this system, and show that for certain connection topologies the system supports multiple attractors. We also present estimates on the maximal possible heterogeneity in this system that can support an attractor, and study the effect of modifications analogous to phase-lag.
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- 2018
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25. The generalized distance spectrum of a graph and applications
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Lee DeVille
- Subjects
Algebra and Number Theory ,Markov chain ,Spectral graph theory ,05C50, 92D25, 60J10, 60J22, 60J27 ,Probability (math.PR) ,Populations and Evolution (q-bio.PE) ,010103 numerical & computational mathematics ,01 natural sciences ,Distance-regular graph ,Combinatorics ,Set (abstract data type) ,Matrix (mathematics) ,Distance matrix ,FOS: Biological sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,Graph (abstract data type) ,Combinatorics (math.CO) ,0101 mathematics ,Quantitative Biology - Populations and Evolution ,Mathematics - Probability ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The generalized distance matrix of a graph is the matrix whose entries depend only on the pairwise distances between vertices, and the generalized distance spectrum is the set of eigenvalues of this matrix. This framework generalizes many of the commonly studied spectra of graphs. We show that for a large class of graphs these eigenvalues can be computed explicitly. We also present the applications of our results to competition models in ecology and rapidly mixing Markov Chains.
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- 2017
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26. Stable Configurations in Social Networks
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Lee DeVille, Jared C. Bronski, Timothy Ferguson, and Michael Livesay
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Physics::Physics and Society ,Energy (esotericism) ,Connection (vector bundle) ,General Physics and Astronomy ,Dynamical Systems (math.DS) ,Topology ,01 natural sciences ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics - Dynamical Systems ,010306 general physics ,Mathematical Physics ,Bifurcation ,Mathematics ,Social network ,business.industry ,Applied Mathematics ,010102 general mathematics ,Statistical and Nonlinear Physics ,Function (mathematics) ,Computer Science::Social and Information Networks ,Monotone polygon ,Friendship graph ,Laplacian matrix ,business - Abstract
We present and analyze a model of opinion formation on an arbitrary network whose dynamics comes from a global energy function. We study the global and local minimizers of this energy, which we call stable opinion configurations, and describe the global minimizers under certain assumptions on the friendship graph. We show a surprising result that the number of stable configurations is not necessarily monotone in the strength of connection in the social network, i.e. the model sometimes supports more stable configurations when the interpersonal connections are made stronger.
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- 2017
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27. A Stochastic Hybrid Systems framework for analysis of Markov reward models
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Alejandro D. Dominguez-Garcia, Lee DeVille, and Sairaj V. Dhople
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Stochastic differential equation ,Mathematical optimization ,Markov chain ,Stochastic process ,Computer science ,Hybrid system ,Component (UML) ,State space ,Discrete-time stochastic process ,State (functional analysis) ,Safety, Risk, Reliability and Quality ,Industrial and Manufacturing Engineering - Abstract
In this paper, we propose a framework to analyze Markov reward models, which are commonly used in system performability analysis. The framework builds on a set of analytical tools developed for a class of stochastic processes referred to as Stochastic Hybrid Systems (SHS). The state space of an SHS is comprised of: (i) a discrete state that describes the possible configurations/modes that a system can adopt, which includes the nominal (non-faulty) operational mode, but also those operational modes that arise due to component faults, and (ii) a continuous state that describes the reward. Discrete state transitions are stochastic, and governed by transition rates that are (in general) a function of time and the value of the continuous state. The evolution of the continuous state is described by a stochastic differential equation and reward measures are defined as functions of the continuous state. Additionally, each transition is associated with a reset map that defines the mapping between the pre- and post-transition values of the discrete and continuous states; these mappings enable the definition of impulses and losses in the reward. The proposed SHS-based framework unifies the analysis of a variety of previously studied reward models. We illustrate the application of the framework to performability analysis via analytical and numerical examples.
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- 2014
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28. Non-meanfield deterministic limits in chemical reaction kinetics.
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Lee DeVille, R. E., Muratov, Cyrill B., and Vanden-Eijnden, Eric
- Subjects
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MONTE Carlo method , *CHEMICAL reactions , *NUMERICAL analysis , *STOCHASTIC processes , *CHEMICAL processes , *DYNAMICS - Abstract
A general mechanism is proposed by which small intrinsic fluctuations in a system far from equilibrium can result in nearly deterministic dynamical behaviors which are markedly distinct from those realized in the meanfield limit. The mechanism is demonstrated for the kinetic Monte Carlo version of the Schnakenberg reaction where we identified a scaling limit in which the global deterministic bifurcation picture is fundamentally altered by fluctuations. Numerical simulations of the model are found to be in quantitative agreement with theoretical predictions. [ABSTRACT FROM AUTHOR]
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- 2006
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29. Optimizing Gershgorin for Symmetric Matrices
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Lee DeVille
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Numerical Analysis ,Pure mathematics ,Algebra and Number Theory ,Mathematics - Spectral Theory ,Gershgorin circle theorem ,Piecewise linear function ,Matrix (mathematics) ,Quadratic equation ,Homogeneous space ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Symmetric matrix ,Geometry and Topology ,Adjacency matrix ,Spectral Theory (math.SP) ,65F15, 15A18, 15A48 ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The Gershgorin Circle Theorem is a well-known and efficient method for bounding the eigenvalues of a matrix in terms of its entries. If $A$ is a symmetric matrix, by writing $A = B + x{\bf 1}$, where ${\bf 1}$ is the matrix with unit entries, we consider the problem of choosing $x$ to give the optimal Gershgorin bound on the eigenvalues of $B$, which then leads to one-sided bounds on the eigenvalues of $A$. We show that this $x$ can be found by an efficient linear program (whose solution can in may cases be written in closed form), and we show that for large classes of matrices, this shifting method beats all existing piecewise linear or quadratic bounds on the eigenvalues. We also apply this shifting paradigm to some nonlinear estimators and show that under certain symmetries this also gives rise to a tractable linear program. As an application, we give a novel bound on the lowest eigenvalue of a adjacency matrix in terms of the "top two" or "bottom two" degrees of the corresponding graph, and study the efficacy of this method in obtaining sharp eigenvalue estimates for certain classes of matrices., Comment: 18 pages, 7 figures
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- 2016
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30. Transitions amongst synchronous solutions in the stochastic Kuramoto model
- Author
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Lee DeVille
- Subjects
Coupling ,Applied Mathematics ,Kuramoto model ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,White noise ,Noise (electronics) ,Control theory ,Phase space ,Metastability ,Path (graph theory) ,Statistical physics ,Constant (mathematics) ,Mathematical Physics ,Mathematics - Abstract
We consider the Kuramoto model of coupled oscillators with nearest-neighbour coupling and additive white noise. We show that synchronous solutions which are stable without the addition of noise become metastable and that we have transitions amongst synchronous solutions on long timescales. We compute these timescales and, moreover, compute the most likely path in phase space that transitions will follow. We show that these transition timescales do not increase as the number of oscillators in the system increases, and are roughly constant in the system size. Finally, we show that the transitions correspond to a splitting of one synchronous solution into two communities which move independently for some time and which rejoin to form a different synchronous solution.
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- 2012
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31. Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory
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R. E. Lee DeVille, Charles S. Peskin, and Joel Spencer
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Random graph ,Discrete mathematics ,Artificial neural network ,Modeling and Simulation ,Applied Mathematics ,Attractor ,Limit (mathematics) ,Statistical physics ,Function (mathematics) ,Net (mathematics) ,Giant component ,Mathematics ,Exponential function - Abstract
We analyze a stochastic neuronal network model which corresponds to an all-to-all net- work of discretized integrate-and-fire neurons where the sy napses are failure-prone. This network exhibits different phases of behavior corresponding to syn chrony and asynchrony, and we show that this is due to the limiting mean-field system possessing multiple attractors. We also show that this mean-field limit exhibits a first-order phase transitio n as a function of the connection strength — as the synapses are made more reliable, there is a sudden onset of synchronous behavior. A detailed understanding of the dynamics involves both a characterization of the size of the giant component in a certain random graph process, and control of the pathwise dynamics of the system by obtaining exponential bounds for the probabilities of events far from the mean.
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- 2010
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32. Regular Gaits and Optimal Velocities for Motor Proteins
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Eric Vanden-Eijnden and R. E. Lee DeVille
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Stochastic Processes ,Asymptotic analysis ,Stochastic process ,Molecular Motor Proteins ,Movement ,Computation ,Myosin Type V ,Biophysics ,Observable ,Biophysical Theory and Modeling ,Biology ,Models, Biological ,Gait ,Motor protein ,Adenosine Triphosphate ,Control theory ,Control parameters - Abstract
It has been observed in numerical experiments that adding a cargo to a motor protein can regularize its gait. Here we explain these results via asymptotic analysis on a general stochastic motor protein model. This analysis permits a computation of various observables (e.g., the mean velocity) of the motor protein and shows that the presence of the cargo also makes the velocity of the motor nonmonotone in certain control parameters (e.g., ATP concentration). As an example, we consider the case of a single myosin-V protein transporting a cargo and show that, at realistic concentrations of ATP, myosin-V operates in the regime which maximizes motor velocity. Our analysis also suggests an experimental regimen which can test the efficacy of any specific motor protein model to a greater degree than was heretofore possible.
- Published
- 2008
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33. Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations
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R. E. Lee DeVille, Krešimir Josić, Matt Holzer, Anthony Harkin, and Tasso J. Kaper
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Singular perturbation ,Differential equation ,Ordinary differential equation ,Mathematical analysis ,Perturbation (astronomy) ,Statistical and Nonlinear Physics ,Vector field ,Renormalization group ,Condensed Matter Physics ,Method of averaging ,Multiple-scale analysis ,Mathematics - Abstract
For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502–4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincare–Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincare–Birkhoff normal forms for these systems up to and including terms of O ( ϵ 2 ) , where ϵ is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of O ( 1 / ϵ ) .
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- 2008
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34. Synchrony and Asynchrony in a Fully Stochastic Neural Network
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Lee DeVille and Charles S. Peskin
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business.product_category ,Bistability ,Computer science ,General Mathematics ,Models, Neurological ,Immunology ,Topology ,Synaptic Transmission ,General Biochemistry, Genetics and Molecular Biology ,Synchronization ,Control theory ,Rare events ,Animals ,Computer Simulation ,Stochastic neural network ,Randomness ,General Environmental Science ,Pharmacology ,Stochastic Processes ,Artificial neural network ,General Neuroscience ,Asynchrony (computer programming) ,Computational Theory and Mathematics ,Network switch ,Nerve Net ,General Agricultural and Biological Sciences ,business ,Algorithms - Abstract
We describe and analyze a model for a stochastic pulse-coupled neural network, in which the randomness in the model corresponds to synaptic failure and random external input. We show that the network can exhibit both synchronous and asynchronous behavior, and surprisingly, that there exists a range of parameters for which the network switches spontaneously between synchrony and asynchrony. We analyze the associated mean-field model and show that the switching parameter regime corresponds to a bistability in the mean field, and that the switches themselves correspond to rare events in the stochastic system.
- Published
- 2008
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35. Self-induced stochastic resonance for Brownian ratchets under load
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R. E. Lee DeVille and Eric Vanden-Eijnden
- Subjects
Physics ,Physics::Biological Physics ,Geometric Brownian motion ,Applied Mathematics ,General Mathematics ,Ratchet ,Brownian ratchet ,Thermal fluctuations ,Context (language use) ,Brownian motor ,Quantitative Biology::Subcellular Processes ,Diffusion process ,Statistical physics ,Brownian motion - Abstract
We consider a Brownian ratchet model where the particle on the ratchet is coupled to a cargo. We show that in a distinguished limit where the diffusion coefficient of the cargo is small, and the amplitude of thermal fluctuations is small, the system becomes completely coherent: the times at which the particle jumps across the teeth of the ratchet become deterministic. We also show that the dynamics of the ratchet-cargo system do not depend on the fine structure of the Brownian ratchet. These results are relevant in the context of molecular motors transporting a load, which are often modeled as a ratchet-cargo compound. They explain the regularity of the motor gait that has been observed in numerical experiments, as well as justify the coarsening into Markov jump processes which is commonly done in the literature.
- Published
- 2007
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36. Finite-size effects and switching times for Moran process with mutation
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Lee DeVille and Meghan Galiardi
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0301 basic medicine ,Population ,01 natural sciences ,Models, Biological ,03 medical and health sciences ,Game Theory ,0103 physical sciences ,Master equation ,Moran process ,Quantitative Biology::Populations and Evolution ,Applied mathematics ,Invariant (mathematics) ,010306 general physics ,education ,Mathematics ,Population Density ,education.field_of_study ,Stochastic Processes ,Stochastic process ,Applied Mathematics ,Agricultural and Biological Sciences (miscellaneous) ,Biological Evolution ,030104 developmental biology ,Iterated function ,Modeling and Simulation ,Mutation (genetic algorithm) ,Mutation ,Deterministic system - Abstract
We consider the Moran process with two populations competing under an iterated Prisoner’s Dilemma in the presence of mutation, and concentrate on the case where there are multiple evolutionarily stable strategies. We perform a complete bifurcation analysis of the deterministic system which arises in the infinite population size. We also study the Master equation and obtain asymptotics for the invariant distribution and metastable switching times for the stochastic process in the case of large but finite population. We also show that the stochastic system has asymmetries in the form of a skew for parameter values where the deterministic limit is symmetric.
- Published
- 2015
37. Erratum: Framework for analyzing ecological trait-based models in multidimensional niche spaces [Phys. Rev. E91, 052107 (2015)]
- Author
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Tommaso Biancalani, Lee DeVille, and Nigel Goldenfeld
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Computer science ,Stochastic process ,Niche ,Trait based ,Econometrics - Published
- 2015
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38. Graph Homology and Stability of Coupled Oscillator Networks
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Lee DeVille, Jared C. Bronski, and Timothy Ferguson
- Subjects
Explicit formulae ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,Homology (mathematics) ,Fixed point ,Topology ,01 natural sciences ,010305 fluids & plasmas ,law.invention ,Mathematics - Spectral Theory ,law ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics - Dynamical Systems ,Spectral Theory (math.SP) ,Mathematics ,Spectral graph theory ,Applied Mathematics ,Kuramoto model ,Disordered Systems and Neural Networks (cond-mat.dis-nn) ,Condensed Matter - Disordered Systems and Neural Networks ,010101 applied mathematics ,Electrical network ,Laplacian matrix ,Edge space - Abstract
There are a number of models of coupled oscillator networks where the question of the stability of fixed points reduces to calculating the index of a graph Laplacian. Some examples of such models include the Kuramoto and Kuramoto--Sakaguchi equations as well as the swing equations, which govern the behavior of generators coupled in an electrical network. We show that the index calculation can be related to a dual calculation which is done on the first homology group of the graph, rather than the vertex space. We also show that this representation is computationally attractive for relatively sparse graphs, where the dimension of the first homology group is low, as is true in many applications. We also give explicit formulae for the dimension of the unstable manifold to a phase-locked solution for graphs containing one or two loops. As an application, we present some novel results for the Kuramoto model defined on a ring and compute the longest possible edge length for a stable solution., 18 pages, 2 figures, 1 table
- Published
- 2015
39. Dynamics on Networks of Manifolds
- Author
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Eugene Lerman and Lee DeVille
- Subjects
Dynamical systems theory ,Molecular Networks (q-bio.MN) ,Dynamical Systems (math.DS) ,01 natural sciences ,010305 fluids & plasmas ,Linear dynamical system ,Surjective function ,0103 physical sciences ,FOS: Mathematics ,Quantitative Biology - Molecular Networks ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematical Physics ,Mathematics ,Discrete mathematics ,010102 general mathematics ,Directed graph ,Injective function ,Graph ,Quantitative Biology - Neurons and Cognition ,Control system ,FOS: Biological sciences ,Neurons and Cognition (q-bio.NC) ,Geometry and Topology ,Random dynamical system ,Analysis ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
We propose a precise definition of a continuous time dynamical system made up of interacting open subsystems. The interconnections of subsystems are coded by directed graphs. We prove that the appropriate maps of graphs called graph fibrations give rise to maps of dynamical systems. Consequently surjective graph fibrations give rise to invariant subsystems and injective graph fibrations give rise to projections of dynamical systems.
- Published
- 2015
40. A Nontrivial Scaling Limit for Multiscale Markov Chains
- Author
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Eric Vanden-Eijnden and Lee DeVille
- Subjects
Mathematical optimization ,Scaling limit ,Markov chain ,Matching (graph theory) ,Mean field theory ,Stochastic resonance ,Event (relativity) ,Statistical and Nonlinear Physics ,Large deviations theory ,Limit (mathematics) ,Statistical physics ,Mathematical Physics ,Mathematics - Abstract
We consider Markov chains with fast and slow variables and show that in a suitable scaling limit, the dynamics becomes deterministic, yet is far away from the standard mean field approximation. This new limit is an instance of self-induced stochastic resonance which arises due to matching between a rare event timescale on the one hand and the natural timescale separation in the underlying problem on the other. Here it is illustrated on a model of a molecular motor, where it is shown to explain the regularity of the motor gait observed in some experiments.
- Published
- 2006
- Full Text
- View/download PDF
41. Wavetrain response of an excitable medium to local stochastic forcing
- Author
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Eric Vanden-Eijnden and Lee DeVille
- Subjects
Excitable medium ,Applied Mathematics ,General Physics and Astronomy ,Mean and predicted response ,Statistical and Nonlinear Physics ,Stochastic resonance (sensory neurobiology) ,Noise (electronics) ,Classical mechanics ,Stochastic forcing ,Statistical physics ,Limit (mathematics) ,Scaling ,Mathematical Physics ,Mathematics - Abstract
We consider the effect of stochastic forcing on a one-dimensional excitable reaction–diffusion system, where the noise only acts on a small subdomain of the medium. We show that there are distinguished scaling limits in which the stochastic forcing gives rise to a mean response of a periodic wavetrain of pulses which are deterministic in the limit. Thus, the system responds coherently and periodically to stochastic forcing. This is an instance of self-induced stochastic resonance, which has previously only been analysed for finite-dimensional systems.
- Published
- 2006
- Full Text
- View/download PDF
42. Nonequilibrium statistics of a reduced model for energy transfer in waves
- Author
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Esteban G. Tabak, Ricardo J. Pignol, Eric Vanden-Eijnden, R. E. Lee DeVille, and Paul A. Milewski
- Subjects
Applied Mathematics ,General Mathematics ,Gaussian ,Non-equilibrium thermodynamics ,White noise ,Dynamical system ,Symmetry (physics) ,Schrödinger equation ,Nonlinear system ,symbols.namesake ,Stochastic differential equation ,Classical mechanics ,symbols ,Mathematics - Abstract
We study energy transfer in a “resonant duet” — a resonant quartet where symmetries support a reduced subsystem with only two degrees of freedom — where one mode is forced by white noise and the other is damped. We consider a physically motivated family of nonlinear damping forms, and investigate their effect on the dynamics of the system. A variety of statistical steady-states arise in different parameter regimes, including intermittent bursting phases, non–equilibrium states highly constrained by slaving among amplitudes and phases, and Gaussian and non-Gaussian quasi–equilibrium regimes. All of this can be understood analytically using asymptotic techniques for stochastic differential equations.
- Published
- 2006
- Full Text
- View/download PDF
43. Brjuno numbers and the symbolic dynamics of the complex exponential
- Author
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R. E. Lee DeVille
- Subjects
Discrete mathematics ,Set (abstract data type) ,symbols.namesake ,Mathematics::Dynamical Systems ,Applied Mathematics ,Dynamics (mechanics) ,Euler's formula ,symbols ,Discrete Mathematics and Combinatorics ,Applied mathematics ,The Symbolic ,Continued fraction ,Mathematics - Abstract
We show that the set of itineraries for the complex exponential family λez is closely related, through the continued fraction expansion, to the set of Brjuno numbers.
- Published
- 2004
- Full Text
- View/download PDF
44. Math PhD Careers: New Opportunities Emerging Amidst Crisis
- Author
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Richard S. Laugesen, Yuliy Baryshnikov, and Lee DeVille
- Subjects
General Mathematics ,Mathematics education - Published
- 2017
- Full Text
- View/download PDF
45. Framework for analyzing ecological trait-based models in multidimensional niche spaces
- Author
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Nigel Goldenfeld, Tommaso Biancalani, and Lee DeVille
- Subjects
Computer science ,media_common.quotation_subject ,Niche ,FOS: Physical sciences ,Ecological and Environmental Phenomena ,Quantitative Biology::Other ,Competition (biology) ,Mathematics - Spectral Theory ,FOS: Mathematics ,Quantitative Biology::Populations and Evolution ,Quantitative Biology - Populations and Evolution ,Spectral Theory (math.SP) ,Condensed Matter - Statistical Mechanics ,media_common ,Ecological niche ,Stochastic Processes ,Statistical Mechanics (cond-mat.stat-mech) ,Stochastic process ,Ecology ,Populations and Evolution (q-bio.PE) ,Niche differentiation ,Linear model ,Hamming distance ,Models, Theoretical ,Phenotype ,Kernel (image processing) ,FOS: Biological sciences ,Linear Models - Abstract
We develop a theoretical framework for analyzing ecological models with a multidimensional niche space. Our approach relies on the fact that ecological niches are described by sequences of symbols, which allows us to include multiple phenotypic traits. Ecological drivers, such as competitive exclusion, are modeled by introducing the Hamming distance between two sequences. We show that a suitable transform diagonalizes the community interaction matrix of these models, making it possible to predict the conditions for niche differentiation and, close to the instability onset, the asymptotically long time population distributions of niches. We exemplify our method using the Lotka-Volterra equations with an exponential competition kernel.
- Published
- 2014
46. A maximum entropy approach to the moment closure problem for Stochastic Hybrid Systems at equilibrium
- Author
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Lee DeVille, Jiangmeng Zhang, Sairaj V. Dhople, and Alejandro D. Dominguez-Garcia
- Subjects
Mathematical optimization ,Moment closure ,Stochastic process ,Principle of maximum entropy ,Hybrid system ,Monte Carlo method ,Maximum entropy probability distribution ,Applied mathematics ,Maximum entropy spectral estimation ,Entropy rate ,Mathematics - Abstract
We study the problem that arises in a class of stochastic processes referred to as Stochastic Hybrid Systems (SHS) when computing the moments of the states using the generator of the process and Dynkin's formula. We focus on the case when the SHS is at equilibrium or approaching equilibrium. We present a family of such processes for which infinite-dimensional linear-system analysis tools are ineffective, and discuss a few differing perspectives on how to tackle such problems by assuming that the SHS state distribution is such that its entropy is maximum. We also provide a numerical algorithm that allows us to efficiently compute maximum entropy solutions, and compare results with Monte Carlo simulations for some illustrative SHS.
- Published
- 2014
- Full Text
- View/download PDF
47. Itineraries of entire functions
- Author
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R. E. Lee DeVille
- Subjects
Discrete mathematics ,Class (set theory) ,Mathematics::Dynamical Systems ,Algebra and Number Theory ,Applied Mathematics ,Entire function ,Symbolic dynamics ,Julia set ,Exponential function ,Combinatorics ,Set (abstract data type) ,symbols.namesake ,Complex dynamics ,Euler's formula ,symbols ,Analysis ,Mathematics - Abstract
we use symbolic dynamics to describe the set of allowable itineraries of orbits of the complex exponential family λexp(z). First, we show that the set of allowable itineraries is the same for every member of this family. We then show that the set of itineraries is the same for every map of finite exponential order. In addition, we study other transcendental entire functions, and show that they also have the same set of itineraries. Finally, we give an example of a class of functions with a different set of itineraries.
- Published
- 2001
- Full Text
- View/download PDF
48. Accessible points in the Julia sets of stable exponentials
- Author
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R. E. Lee DeVille, Ranjit Bhattacharjee, Monica Moreno-Rocha, Robert L. Devaney, and Krešimir Josić
- Subjects
Discrete mathematics ,Sequence ,Mathematics::Dynamical Systems ,Applied Mathematics ,Periodic point ,Fixed point ,Julia set ,Combinatorics ,Filled Julia set ,Misiurewicz point ,symbols.namesake ,Euler's formula ,symbols ,Discrete Mathematics and Combinatorics ,Point (geometry) ,Mathematics - Abstract
In this paper we consider the question of accessibility of points in the Julia sets of complex exponential functions in the case where the exponential admits an attracting cycle. In the case of an attracting fixed point it is known that the Julia set is a Cantor bouquet and that the only points accessible from the basin are the endpoints of the bouquet. In case the cycle has period two or greater, there are many more restrictions on which points in the Julia set are accessible. In this paper we give precise conditions for a point to be accessible in the periodic point case in terms of the kneading sequence for the cycle.
- Published
- 2001
- Full Text
- View/download PDF
49. Modular dynamical systems on networks
- Author
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Lee DeVille and Eugene Lerman
- Subjects
Pure mathematics ,Physics - Physics and Society ,Dynamical systems theory ,Molecular Networks (q-bio.MN) ,General Mathematics ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,Physics and Society (physics.soc-ph) ,01 natural sciences ,010305 fluids & plasmas ,Surjective function ,Conjugacy class ,0103 physical sciences ,FOS: Mathematics ,Quantitative Biology - Molecular Networks ,0101 mathematics ,Mathematics - Dynamical Systems ,Mathematics ,Modularity (networks) ,Applied Mathematics ,010102 general mathematics ,Linear subspace ,Injective function ,FOS: Biological sciences ,Quantitative Biology - Neurons and Cognition ,Neurons and Cognition (q-bio.NC) ,Bijection, injection and surjection ,Word (computer architecture) - Abstract
We propose a new framework for the study of continuous time dynamical systems on networks. We view such dynamical systems as collections of interacting control systems. We show that a class of maps between graphs called graph fibrations give rise to maps between dynamical systems on networks. This allows us to produce conjugacy between dynamical systems out of combinatorial data. In particular we show that surjective graph fibrations lead to synchrony subspaces in networks. The injective graph fibrations, on the other hand, give rise to surjective maps from large dynamical systems to smaller ones. One can view these surjections as a kind of "fast/slow" variable decompositions or as "abstractions" in the computer science sense of the word., 37 pages. Major revision of arXiv:1008.5359 [math.DS]. Following referees' suggestions we made the paper more accessible for applied dynamicists
- Published
- 2013
50. Universal Critical Dynamics in High Resolution Neuronal Avalanche Data
- Author
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Masanori Shimono, John M. Beggs, Shinya Ito, Braden A. W. Brinkman, R. E. Lee DeVille, Karin A. Dahmen, Thomas Butler, and Nir Friedman
- Subjects
Neurons ,Physics ,Phase transition ,Critical phenomena ,Models, Neurological ,Action Potentials ,General Physics and Astronomy ,Non-equilibrium thermodynamics ,Power law ,Rats ,Models of neural computation ,Critical point (thermodynamics) ,Biological neural network ,Animals ,Statistical physics ,Nerve Net ,Scaling ,Cells, Cultured - Abstract
The tasks of neural computation are remarkably diverse. To function optimally, neuronal networks have been hypothesized to operate near a nonequilibrium critical point. However, experimental evidence for critical dynamics has been inconclusive. Here, we show that the dynamics of cultured cortical networks are critical. We analyze neuronal network data collected at the individual neuron level using the framework of nonequilibrium phase transitions. Among the most striking predictions confirmed is that the mean temporal profiles of avalanches of widely varying durations are quantitatively described by a single universal scaling function. We also show that the data have three additional features predicted by critical phenomena: approximate power law distributions of avalanche sizes and durations, samples in subcritical and supercritical phases, and scaling laws between anomalous exponents.
- Published
- 2012
- Full Text
- View/download PDF
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