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90
Geometrically Induced Spectrum in Curved Leaky Wires
 J. Phys. A34
, 2001
"... Introduction The aim of the present paper is to elucidate some geometrically induced spectral properties for the Laplacian in L 2 (R 2 ) perturbed by a negative multiple of the Dirac measure of an infinite curve \Gamma in the plane. This problem has at least two motivations. On the physics side ..."
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Cited by 32 (13 self)
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Introduction The aim of the present paper is to elucidate some geometrically induced spectral properties for the Laplacian in L 2 (R 2 ) perturbed by a negative multiple of the Dirac measure of an infinite curve \Gamma in the plane. This problem has at least two motivations. On the physics side we note that quantum mechanics of electrons confined to narrow tubelike regions has attracted a considerable interest, because such systems represent a natural model for semiconductor "quantum wires". In some examples the region in question is a strip or tube with hard walls  see, e.g., [DE] and references therein  while other treatments assume even stronger localization to a curve 1 or a graph  a rich bibliography to such models can be found in [KS]. Various interesting spectral effects were found in such a setting related to the geometry and topology of the underlying restricted configuration space. One of th
FINITE SAMPLE APPROXIMATION RESULTS FOR PRINCIPAL COMPONENT ANALYSIS: A MATRIX PERTURBATION APPROACH
"... Principal Component Analysis (PCA) is a standard tool for dimensional reduction of a set of n observations (samples), each with p variables. In this paper, using a matrix perturbation approach, we study the nonasymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite ..."
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Cited by 24 (11 self)
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Principal Component Analysis (PCA) is a standard tool for dimensional reduction of a set of n observations (samples), each with p variables. In this paper, using a matrix perturbation approach, we study the nonasymptotic relation between the eigenvalues and eigenvectors of PCA computed on a finite sample of size n, to those of the limiting population PCA as n → ∞. As in machine learning, we present a finite sample theorem which holds with high probability for the closeness between the leading eigenvalue and eigenvector of sample PCA and population PCA under a spiked covariance model. In addition, we also consider the relation between finite sample PCA and the asymptotic results in the joint limit p, n → ∞, with p/n = c. We present a matrix perturbation view of the “phase transition phenomenon”, and a simple linearalgebra based derivation of the eigenvalue and eigenvector overlap in this asymptotic limit. Moreover, our analysis also applies for finite p, n where we show that although there is no sharp phase transition as in the infinite case, either as a function of noise level or as a function of sample size n, the eigenvector of sample PCA may exhibit a sharp ”loss of tracking”, suddenly losing its relation to the (true) eigenvector of the population PCA matrix. This occurs due to a crossover between the eigenvalue due to the signal and the largest eigenvalue due to noise, whose eigenvector points in a random direction.
Numerical Exterior Algebra and the Compound Matrix Method
 Numer. Math
, 2000
"... The compound matrix method, which has been proposed by Ng & Reid for numerically integrating systems of dierential equations in hydrodynamic stability on k = 2; 3 dimensional subspaces, is reformulated in terms of exterior algebra. This formulation leads to a general framework for deriving the in ..."
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Cited by 19 (1 self)
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The compound matrix method, which has been proposed by Ng & Reid for numerically integrating systems of dierential equations in hydrodynamic stability on k = 2; 3 dimensional subspaces, is reformulated in terms of exterior algebra. This formulation leads to a general framework for deriving the induced systems, and leads to several new results including: the role of Hodge duality in constructing systems, adjoints and boundary conditions, the role of analyticity for systems on unbounded domains, general formulation of induced boundary conditions, and the role of geometric integrators for preserving the manifold of k dimensional subspaces. The formulation is presented for kdimensional subspaces of systems on C with k and n arbitrary, and detailed examples are given for the case k = 2 and n = 4, with an indication of implementation details for systems of larger dimension. The theory is then applied to two examples: 2D boundarylayer ow past a compliant surface and the instability of jetlike pro les.
Gauge theoretic invariants of Dehn surgeries on knots
 Geom. Topol
"... The goal of this article is to develop new methods for computing a variety of gauge theoretic invariants for 3manifolds obtained by Dehn surgery on knots. These invariants include the ChernSimons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible SU( ..."
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Cited by 14 (6 self)
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The goal of this article is to develop new methods for computing a variety of gauge theoretic invariants for 3manifolds obtained by Dehn surgery on knots. These invariants include the ChernSimons invariants, the spectral flow of the odd signature operator, and the rho invariants of irreducible SU(2) representations. The rho invariants and spectral flow
Asymptotics of repeated interaction quantum systems
"... Prépublication de l’Institut Fourier n o 682 (2005) wwwfourier.ujfgrenoble.fr/prepublications.html A quantum system S interacts in a successive way with elements E of a chain of identical independent quantum subsystems. Each interaction lasts for a duration τ and is governed by a fixed coupling be ..."
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Cited by 14 (4 self)
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Prépublication de l’Institut Fourier n o 682 (2005) wwwfourier.ujfgrenoble.fr/prepublications.html A quantum system S interacts in a successive way with elements E of a chain of identical independent quantum subsystems. Each interaction lasts for a duration τ and is governed by a fixed coupling between S and E. We show that the system, initially in any state close to a reference state, approaches a repeated interaction asymptotic state in the limit of large times. This state is τ–periodic in time and does not depend on the initial state. If the reference state is chosen so that S and E are individually in equilibrium at positive temperatures, then the repeated interaction asymptotic state satisfies an average second law of thermodynamics.
The longtime dynamics of Dirac particles in the KerrNewman black hole geometry
 Adv. Theor. Math. Phys
, 2003
"... We consider the Cauchy problem for the massive Dirac equation in the nonextreme KerrNewman geometry outside the event horizon. We derive an integral representation for the Dirac propagator involving the solutions of the ODEs which arise in Chandrasekhar’s separation of variables. It is proved that ..."
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Cited by 13 (7 self)
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We consider the Cauchy problem for the massive Dirac equation in the nonextreme KerrNewman geometry outside the event horizon. We derive an integral representation for the Dirac propagator involving the solutions of the ODEs which arise in Chandrasekhar’s separation of variables. It is proved that for initial data in L ∞ loc near the event horizon with L 2 decay at infinity, the probability of the Dirac particle to be in any compact region of space tends to zero as t goes to infinity. This means that the Dirac particle must either disappear in the black hole or escape to infinity. 1
An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry
, 2004
"... We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation ..."
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Cited by 12 (7 self)
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We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. In particular, we prove completeness of the solutions of the separated ODEs. This integral representation is a suitable starting point for a detailed analysis of
Hankel Operators and Gramians for Nonlinear Systems
, 1998
"... In the theory for continuoustime linear systems, the system Hankel operator plays an important role in a number of realization problems ranging from providing an abstract notion of state to yielding tests for state space minimality and algorithms for model reduction. But in the case of continuoust ..."
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Cited by 10 (9 self)
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In the theory for continuoustime linear systems, the system Hankel operator plays an important role in a number of realization problems ranging from providing an abstract notion of state to yielding tests for state space minimality and algorithms for model reduction. But in the case of continuoustime nonlinear systems, Hankel theory is considerably less developed beyond a well known Hankel mapping introduced by Fliess in 1974. In this paper, a definition of a system Hankel operator is developed for causal L 2 stable inputoutput systems. If a generating series representation of the inputoutput system is given then an explicit representation of the corresponding Hankel operator is possible. If, in addition, an a#ne state space model is available with certain stability properties then a unique factorization of the Hankel operator can be constructed with direct connections to well known and new nonlinear Gramian extensions. 1. Introduction In the theory of continuoustime linear syst...
Band gap of the Schrödinger operator with a strong deltainteraction on a periodic curve
"... this paper we study the operator H = ( ) in L ), where is a smooth periodic curve in R . We obtain the asymptotic form of the band spectrum of H as tends to in nity. Furthermore, we prove the existence of the band gap of (H ) for suciently large > 0. Finally, we also derive the spectral b ..."
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Cited by 10 (2 self)
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this paper we study the operator H = ( ) in L ), where is a smooth periodic curve in R . We obtain the asymptotic form of the band spectrum of H as tends to in nity. Furthermore, we prove the existence of the band gap of (H ) for suciently large > 0. Finally, we also derive the spectral behaviour for ! 1 in the case when is nonperiodic and asymptotically straight. 1