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ON ELLIPSOIDAL TECHNIQUES FOR REACHABILITY ANALYSIS. PART II: INTERNAL APPROXIMATIONS BOXVALUED CONSTRAINTS
, 2000
"... Following Part I, this article continues to describe the calculation of the reach sets and tubes for linear control systems with timevarying coefficients and ellipsoidal hard bounds on the controls and initial states. It deals with parametrized families of internal ellipsoidal approximations constr ..."
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Cited by 119 (6 self)
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Following Part I, this article continues to describe the calculation of the reach sets and tubes for linear control systems with timevarying coefficients and ellipsoidal hard bounds on the controls and initial states. It deals with parametrized families of internal ellipsoidal approximations constructed such that they touch the reach sets at every point of their boundary at any instant of time. The reach tubes are thus touched internally by ellipsoidal tubes along some curves. The ellipsoidal tubes are chosen here in such a way that the touching curves do not intersect and that the boundary of the reach tube would be entirely covered by such curves. This allows exact parametric representation of reach tubes through unions of tight internal ellipsoidal tubes as compared with earlier methods based on constructing one or several isolated approximating tubes. The method of external and internal ellipsoidal approximations is then propagated to systems with boxvalued hard bounds on the controls and initial states. It appears that the proposed technique may well work for nonellipsoidal, boxvalued constraints. This broadens the range of applications of the approach and opens new routes to the arrangement of efficient numerical algorithms.
What Matters in Neuronal Locking?
"... Present and permanent address: PhysikDepartment der TU Munchen Exploiting local stability we show what neuronal characteristics are essential to ensure that coherent oscillations are asymptotically stable in a spatially homogeneous network of spiking neurons. Under standard conditions, a necessa ..."
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Cited by 46 (10 self)
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Present and permanent address: PhysikDepartment der TU Munchen Exploiting local stability we show what neuronal characteristics are essential to ensure that coherent oscillations are asymptotically stable in a spatially homogeneous network of spiking neurons. Under standard conditions, a necessary and in the limit of a large number of interacting neighbors also sufficient condition is that the postsynaptic potential is increasing in time as the neurons fire. If the postsynaptic potential is decreasing, oscillations are bound to be unstable. This is a kind of locking theorem and boils down to a subtle interplay of axonal delays, postsynaptic potentials, and refractory behavior. The theorem also allows for mixtures of excitatory and inhibitory interactions. On the basis of the locking theorem we present a simple geometric method to verify existence and local stability of a coherent oscillation. 2 1
Canonical Correlation Analysis, Approximate Covariance Extension, and Identification of Stationary Time Series
 Automatica
, 1996
"... In this paper we analyze a class of statespace identification algorithms for timeseries, based on canonical correlation analysis, in the light of recent results on stochastic systems theory. In principle, these so called "subspace methods" can be described as covariance estimation followed by stoc ..."
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Cited by 37 (17 self)
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In this paper we analyze a class of statespace identification algorithms for timeseries, based on canonical correlation analysis, in the light of recent results on stochastic systems theory. In principle, these so called "subspace methods" can be described as covariance estimation followed by stochastic realization. The methods o#er the major advantage of converting the nonlinear parameter estimation phase in traditional ARMA models identification into the solution of a Riccati equation but introduce at the same time some nontrivial mathematical problems related to positivity. The reason for this is that an essential part of the problem is equivalent to the wellknown rational covariance extension problem. Therefore the usual deterministic arguments based on factorization of a Hankel matrix are not valid for generic data, something that is habitually overlooked in the literature. We demonstrate that there is no guarantee that several popular identification procedures based on the same principle will not fail to produce a positive extension, unless some rather stringent assumptions are made which, in general, are not explicitly reported. In this paper the statistical problem of stochastic modeling from estimated covariances is phrased in the geometric language of stochastic realization theory. We review the basic ideas of stochastic realization theory in the context of identification, discuss the concept of stochastic balancing and of stochastic model reduction by principal subsystem truncation. The model reduction method of Desai and Pal, based on truncated balanced stochastic realizations, is partially justified, showing that the reduced system structure has a positive covariance sequence but is in general not balanced. As a byproduct of this analysis we obtain a t...
Guaranteed Passive Balancing Transformations for Model Order Reduction
, 2002
"... The major concerns in stateoftheart model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms such as PRIMA generate guaranteedpassive models ..."
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Cited by 28 (4 self)
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The major concerns in stateoftheart model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms such as PRIMA generate guaranteedpassive models, for systems with special internal structure, using numerically stable and efficient Krylovsubspace iterations. Truncated Balanced Realization (TBR) algorithms, as used to date in the design automation community, can achieve smaller models with better error control, but do not necessarily preserve passivity. In this paper we show how to construct TBRlike methods that guarantee passive reduced models and in addition are applicable to statespace systems with arbitrary internal structure.
Random incidence matrices: moments of the spectral density
 J. Stat. Phys
, 2001
"... We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices: any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large ..."
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Cited by 18 (3 self)
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We study numerically and analytically the spectrum of incidence matrices of random labeled graphs on N vertices: any pair of vertices is connected by an edge with probability p. We give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in N and p. For large N and fixed p the spectrum contains a large eigenvalue at Np and a semicircle of “small ” eigenvalues. For large N and fixed average connectivity pN (dilute or sparse random matrices limit) we show that the spectrum always contains a discrete component. An anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed. We develop recursion relations to compute the moments as explicit polynomials in pN. Their growth is slow enough so that they determine the spectrum. The extension of our methods to the Laplacian matrix is given in Appendix.
On the Complexity of Finding Short Vector in Integer Lattices
, 1983
"... this paper we present a modified version of this algorithm n ..."
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Cited by 13 (1 self)
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this paper we present a modified version of this algorithm n
Algorithms for Matrix Groups and the Tits Alternative
 Proc. 36th IEEE FOCS
, 1999
"... l over the generators grows as c l for some constant c>1 depending on G. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals an ..."
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Cited by 11 (2 self)
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l over the generators grows as c l for some constant c>1 depending on G. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals and Babai, who give a Las Vegas algorithm for the case of finite groups, as well as recent work of Babai, Beals, Cai, Ivanyos, and Luks, who give a deterministic algorithm for the case of abelian groups. # 1999 Academic Press Article ID jcss.1998.1614, available online at http:##www.idealibrary.com on 260 00220000#99 #30.00 Copyright # 1999 by Academic Press All rights of reproduction in any form reserved. * Research conducted while visiting IAS and DIMACS and supported in part by an NSF Mathematical Sciences
Random Heuristic Search
 Theoretical Computer Science
, 1999
"... There is a developing theory of growing power which, at its current stage of development (indeed, for a number of years now), speaks to qualitative and quantitative aspects of search strategies. Although it has been specialized and applied to genetic algorithms, it's implications and applicability a ..."
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Cited by 11 (1 self)
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There is a developing theory of growing power which, at its current stage of development (indeed, for a number of years now), speaks to qualitative and quantitative aspects of search strategies. Although it has been specialized and applied to genetic algorithms, it's implications and applicability are far more general. This paper deals with the broad outlines of the theory, introducing basic principles and results rather than analyzing or specializing to particular algorithms. A few specific examples are included for illustrative purposes, but the theory's basic structure, as opposed to applications, remains the focus. Key words: Random Heuristic Search, Modeling Evolutionary Algorithms, Degenerate Royal Road Functions. 1 Introduction Vose [20] introduced a rigorous dynamical system model for the binary representation genetic algorithm with proportional selection, mutation determined by a rate, and onepoint crossover, using the simplifying assumption of an infinite population. 1 ...
Simplified discontinuous Galerkin methods for systems of conservation laws with convex extension
 in Discontinuous Galerkin Methods
, 1999
"... Abstract. Simplified forms of the spacetime discontinuous Galerkin (DG) and discontinuous Galerkin leastsquares (DGLS) finite element method are developed and analyzed. The new formulations exploit simplifying properties of entropy endowed conservation law systems while retaining the favorable ene ..."
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Cited by 9 (4 self)
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Abstract. Simplified forms of the spacetime discontinuous Galerkin (DG) and discontinuous Galerkin leastsquares (DGLS) finite element method are developed and analyzed. The new formulations exploit simplifying properties of entropy endowed conservation law systems while retaining the favorable energy properties associated with symmetric variable formulations. 1 2