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Existence and Stability of Standing Pulses in Neural Networks
 I. Existence. SIAM Journal on Applied Dynamical Systems
, 2003
"... Abstract. We analyze the stability of standing pulse solutions of a neural network integrodifferential equation. The network consists of a coarsegrained layer of neurons synaptically connected by lateral inhibition with a nonsaturating nonlinear gain function. When two standing singlepulse soluti ..."
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Cited by 37 (2 self)
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Abstract. We analyze the stability of standing pulse solutions of a neural network integrodifferential equation. The network consists of a coarsegrained layer of neurons synaptically connected by lateral inhibition with a nonsaturating nonlinear gain function. When two standing singlepulse solutions coexist, the small pulse is unstable, and the large pulse is stable. The large single pulse is bistable with the “alloff ” state. This bistable localized activity may have strong implications for the mechanism underlying working memory. We show that dimple pulses have similar stability properties to large pulses but double pulses are unstable.
Borel Subrings Of The Reals
 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
, 2002
"... A Borel (or even analytic) subring of R either has Hausdorff dimension 0 or is all of R. Extensions of the method of proof yield (among other things) that any analytic subring of C having positive Hausdorff dimension is equal to either R or C. ..."
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Cited by 14 (1 self)
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A Borel (or even analytic) subring of R either has Hausdorff dimension 0 or is all of R. Extensions of the method of proof yield (among other things) that any analytic subring of C having positive Hausdorff dimension is equal to either R or C.
Spectral properties of continuous refinement operators
 Proc. Amer. Math. Soc
, 1998
"... Abstract. This paper studies the spectrum of continuous refinement operators and relates their spectral properties with the solutions of the corresponding continuous refinement equations. 1. ..."
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Cited by 11 (3 self)
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Abstract. This paper studies the spectrum of continuous refinement operators and relates their spectral properties with the solutions of the corresponding continuous refinement equations. 1.
The distributional Denjoy integral
, 2006
"... Abstract. Let f be a distribution (generalised function) on the real line. If there is a continuous function F with real limits at infinity such that F ′ = f (distributional derivative) then the distributional integral of f is defined as ∫ ∞ f = F(∞)−F(−∞). It is shown that this simple definition g ..."
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Cited by 11 (8 self)
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Abstract. Let f be a distribution (generalised function) on the real line. If there is a continuous function F with real limits at infinity such that F ′ = f (distributional derivative) then the distributional integral of f is defined as ∫ ∞ f = F(∞)−F(−∞). It is shown that this simple definition gives an integral that includes the Lebesgue and Henstock–Kurzweil integrals. The Alexiewicz norm leads to a Banach space of integrable distributions that is isometrically isomorphic to the space of continuous functions on the extended real line with uniform norm. The dual space is identified with the functions of bounded variation. Basic properties of integrals are established using elementary properties of distributions: integration by parts, Hölder inequality, change of variables, convergence theorems, Banach lattice structure, Hake theorem, Taylor theorem, second mean value theorem. Applications are made to the half plane Poisson integral and Laplace transform. The paper includes a short history of Denjoy’s descriptive integral definitions. Distributional integrals in Euclidean spaces are discussed and a more general distributional integral that also integrates Radon measures is proposed. 2000 subject classification: 26A39, 46E15, 46F05, 46G12 1
Dynamic Legislative Policy Making
, 2007
"... We prove existence of stationary Markov perfect equilibria in an infinitehorizon model of legislative policy making in which the policy outcome in one period determines the status quo in the next. We allow for a multidimensional policy space and arbitrary smooth stage utilities. We prove that all s ..."
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Cited by 10 (1 self)
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We prove existence of stationary Markov perfect equilibria in an infinitehorizon model of legislative policy making in which the policy outcome in one period determines the status quo in the next. We allow for a multidimensional policy space and arbitrary smooth stage utilities. We prove that all such equilibria are essentially in pure strategies and that proposal strategies are differentiable almost everywhere. We establish upper hemicontinuity of the equilibrium correspondence, and we derive conditions under which each equilibrium of our model determines a unique invariant distribution characterizing long run policy outcomes. We illustrate the equilibria of the model in a numerical example of policy making in a single dimension, and we discuss extensions of our approach to accommodate much of the institutional structure observed in realworld politics.
A general approach to sparse basis selection: Majorization, concavity, and affine scaling
 IN PROCEEDINGS OF THE TWELFTH ANNUAL CONFERENCE ON COMPUTATIONAL LEARNING THEORY
, 1997
"... Measures for sparse best–basis selection are analyzed and shown to fit into a general framework based on majorization, Schurconcavity, and concavity. This framework facilitates the analysis of algorithm performance and clarifies the relationships between existing proposed concentration measures use ..."
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Cited by 10 (5 self)
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Measures for sparse best–basis selection are analyzed and shown to fit into a general framework based on majorization, Schurconcavity, and concavity. This framework facilitates the analysis of algorithm performance and clarifies the relationships between existing proposed concentration measures useful for sparse basis selection. It also allows one to define new concentration measures, and several general classes of measures are proposed and analyzed in this paper. Admissible measures are given by the Schurconcave functions, which are the class of functions consistent with the socalled Lorentz ordering (a partial ordering on vectors also known as majorization). In particular, concave functions form an important subclass of the Schurconcave functions which attain their minima at sparse solutions to the best basis selection problem. A general affine scaling optimization algorithm obtained from a special factorization of the gradient function is developed and proved to converge to a sparse solution for measures chosen from within this subclass.
Onebit distributed sensing and coding for field estimation in sensor networks
 IEEE Trans. Signal Processing
, 2008
"... This paper formulates and studies a general distributed field reconstruction problem using a dense network of noisy one–bit randomized scalar quantizers in the presence of additive observation noise of unknown distribution. A constructive quantization, coding, and field reconstruction scheme is deve ..."
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Cited by 7 (0 self)
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This paper formulates and studies a general distributed field reconstruction problem using a dense network of noisy one–bit randomized scalar quantizers in the presence of additive observation noise of unknown distribution. A constructive quantization, coding, and field reconstruction scheme is developed and an upper–bound to the associated mean squared error (MSE) at any point and any snapshot is derived in terms of the local spatio–temporal smoothness properties of the underlying field. It is shown that when the noise, sensor placement pattern, and the sensor schedule satisfy certain weak technical requirements, it is possible to drive the MSE to zero with increasing sensor density at points of field continuity while ensuring that the per–sensor bitrate and sensing–related network overhead rate simultaneously go to zero. The proposed scheme achieves the order–optimal MSE versus sensor density scaling behavior for the class of spatially constant spatio–temporal fields. I. INTRODUCTION AND OVERVIEW We study the problem of reconstructing, at a data fusion center, a temporal sequence of spatial data fields, in a bounded geographical region of interest, from finite bit–rate messages generated by a dense noncooperative network of sensors. The data–gathering sensor network is made up of noisy
Stochastic Growth: Asymptotic Distributions
, 2003
"... this paper, (1) is said to satisfy the law of large numbers if, for any Lipschitz function g : X R, N g(x)#(dx) (3) Palmost surely as N ## ..."
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Cited by 4 (2 self)
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this paper, (1) is said to satisfy the law of large numbers if, for any Lipschitz function g : X R, N g(x)#(dx) (3) Palmost surely as N ##
SPATIAL MODELS OF BOOLEAN ACTIONS AND GROUPS OF ISOMETRIES
"... Abstract. Given a Polish group G of isometries of a locally compact separable metric space, we prove that each measure preserving Boolean action by G has a spatial model or, in other words, has a point realization. This result extends both a classical theorem of Mackey and a recent theorem of Glasne ..."
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Abstract. Given a Polish group G of isometries of a locally compact separable metric space, we prove that each measure preserving Boolean action by G has a spatial model or, in other words, has a point realization. This result extends both a classical theorem of Mackey and a recent theorem of Glasner and Weiss, and it covers interesting new examples. In order to prove our result, we give a characterization of Polish groups of isometries of locally compact separable metric spaces which may be of independent interest. The solution to Hilbert’s fifth problem plays an important role in establishing this characterization. 1.