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Spectra, Pseudospectra, and Localization for Random Bidiagonal Matrices
 Comm. Pure Appl. Math
"... There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random nonhermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a \bubble with wings" in the complex plane, and the correspon ..."
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There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random nonhermitian periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a \bubble with wings" in the complex plane, and the corresponding eigenvectors are localized in the wings, delocalized in the bubble. Here, in addition to eigenvalues, pseudospectra are analyzed, making it possible to treat the nonperiodic analogues of these random matrix problems. Inside the bubble, the resolvent norm grows exponentially with the dimension. Outside, it grows subexponentially in a bounded region that is the spectrum of the in nitedimensional operator. Localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase, respectively, with distance from the diagonal. This article presents theorems that characterize the spectra, pseudospectra, and numerical range for the four cases of nite bidiagonal matrices, in nite bidiagonal matrices (\stochastic Toeplitz operators"), nite periodic matrices, and doubly in nite bidiagonal matrices (\stochastic Laurent operators").
A Survey of Regularization Methods for FirstKind Volterra Equations
 SURVEYS ON SOLUTION METHODS FOR INVERSE PROBLEMS
, 2000
"... We survey continuous and discrete regularization methods for firstkind Volterra problems with continuous kernels. Classical regularization methods tend to destroy the nonanticipatory (or causal) nature of the original Volterra problem because such methods typically rely on computation of the V ..."
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Cited by 15 (5 self)
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We survey continuous and discrete regularization methods for firstkind Volterra problems with continuous kernels. Classical regularization methods tend to destroy the nonanticipatory (or causal) nature of the original Volterra problem because such methods typically rely on computation of the Volterra adjoint operator, an anticipatory operator. In this survey we pay special attention to particular regularization methods, both classical and nontraditional, which tend to retain the Volterra structure of the original problem. Our attention will primarily be focused on linear problems, although extensions of methods to nonlinear and integrooperator Volterra equations are mentioned when known.
Optimal Approximations Of Fractional Derivatives
, 1998
"... . In this paper we consider the following fractional differentiation problem: given noisy data f ffi 2 L 2 (IR) to f , determine the fractional derivative u = D fi f 2 L 2 (IR) for fi ? 0, which is the solution of the integral equation of first kind (A fi u)(x) = 1 \Gamma(fi ) R x \Gamma1 ..."
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. In this paper we consider the following fractional differentiation problem: given noisy data f ffi 2 L 2 (IR) to f , determine the fractional derivative u = D fi f 2 L 2 (IR) for fi ? 0, which is the solution of the integral equation of first kind (A fi u)(x) = 1 \Gamma(fi ) R x \Gamma1 u(t) dt (x\Gammat) 1\Gammafi = f(x). Assuming kf \Gamma f ffi k L 2 (IR) ffi and kuk p E (where k \Delta k p denotes the usual Sobolev norm of order p ? 0) we answer the question concerning the best possible accuracy for identifying u from the noisy data f ffi . Furthermore, we discuss special regularization methods which realize this best possible accuracy. 1 Introduction Fractional differentiation problems arise in several contexts and have important applications in science and engineering (cf., e.g., [6]), and various aspects of it have been treated in the literature of which we cannot give here an exhaustive survey, but let us quote [3, 4, 5, 8, 9, 14, 15, 16, 23]. In this pape...