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74
A General Model of Web Graphs
, 2003
"... We describe a very general model of a random graph process whose proportional degree sequence obeys a power law. Such laws have recently been observed in graphs associated with the world wide web. ..."
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Cited by 82 (7 self)
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We describe a very general model of a random graph process whose proportional degree sequence obeys a power law. Such laws have recently been observed in graphs associated with the world wide web.
Efficient Solution Of Parabolic Equations By Krylov Approximation Methods
 SIAM J. Sci. Statist. Comput
, 1992
"... . In this paper we take a new look at numerical techniques for solving parabolic equations by the method of lines. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action ..."
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Cited by 49 (3 self)
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. In this paper we take a new look at numerical techniques for solving parabolic equations by the method of lines. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of very small dimension to a known vector which is, in turn, computed accurately by exploiting highorder rational Chebyshev and Pad'e approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrixbyvector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Further parallelism is introduced by expanding the rational approximations into partial fractions. Some ...
WeylTitchmarsh MFunction Asymptotics, Local Uniqueness Results, Trace Formulas, And BorgType Theorems For Dirac Operators
 Proc. London Math. Soc
, 2001
"... We explicitly determine the highenergy asymptotics for WeylTitchmarsh matrices associated with general Diractype operators on halflines and on R. We also prove new local uniqueness results for Diractype operators in terms of exponentially small differences of WeylTitchmarsh matrices. As con ..."
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Cited by 34 (17 self)
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We explicitly determine the highenergy asymptotics for WeylTitchmarsh matrices associated with general Diractype operators on halflines and on R. We also prove new local uniqueness results for Diractype operators in terms of exponentially small differences of WeylTitchmarsh matrices. As concrete applications of the asymptotic highenergy expansion we derive a trace formula for Dirac operators and use it to prove a Borgtype theorem.
Harmonic functions on multiplicative graphs and interpolation polynomials, Electron
 J. Combin. 7 (2000), Research paper
"... Abstract. We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur’ ..."
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Cited by 23 (9 self)
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Abstract. We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur’s S and P functions and with Jack symmetric functions. As a by–product, we compute certain Selberg–type integrals.
Fast Algorithms With Preprocessing for MatrixVector Multiplication Problems
, 1994
"... In this paper the problem of complexity of multiplication of a matrix with a vector is studied for Toeplitz, Hankel, Vandermonde and Cauchy matrices and for matrices connected with them (i.e. for transpose, inverse and transpose to inverse matrices). The proposed ..."
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Cited by 23 (1 self)
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In this paper the problem of complexity of multiplication of a matrix with a vector is studied for Toeplitz, Hankel, Vandermonde and Cauchy matrices and for matrices connected with them (i.e. for transpose, inverse and transpose to inverse matrices). The proposed
On Local Borg–Marchenko Uniqueness Results
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2000
"... We provide a new short proof of the following fact, first proved by one of us in 1998: If two Weyl–Titchmarsh mfunctions, mj (z), of two Schrödinger operators Hj = − d2 dx2 + qj, j = 1, 2inL2 ((0,R)),0
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Cited by 22 (12 self)
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We provide a new short proof of the following fact, first proved by one of us in 1998: If two Weyl–Titchmarsh mfunctions, mj (z), of two Schrödinger operators Hj = − d2 dx2 + qj, j = 1, 2inL2 ((0,R)),0<R≤∞, are exponentially close, that is, m1(z) − m2(z)  = z→ ∞ O(e−2Im(z1/2)a),0<a<R, then q1 = q2 a.e. on [0,a]. The result applies to any boundary conditions at x = 0 and x = R and should be considered a local version of the celebrated Borg–Marchenko uniqueness result (which is quickly recovered as a corollary to our proof). Moreover, we extend the local uniqueness result to matrixvalued Schrödinger operators.
The Spectral Theory of Multiresolution Operators and Applications
, 1994
"... this article we explore the notion of the multiresolution operator, its spectral theory, and applications. This operator (also called the transition operator) has appeared as a fundamental tool in several aspects of wavelet theory, such as the LawtonCohen theorem on wavelet orthonormal bases [2, 18 ..."
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Cited by 17 (2 self)
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this article we explore the notion of the multiresolution operator, its spectral theory, and applications. This operator (also called the transition operator) has appeared as a fundamental tool in several aspects of wavelet theory, such as the LawtonCohen theorem on wavelet orthonormal bases [2, 18, 19], and the work of Eirola [8] and others [1, 3, 4, 5, 23] on the Sobolev smoothness of wavelet scaling functions. After introducing the multiresolution operator and observing its connection with the convolution and downsampling operations of multirate signal processing, we present a review of the work of Lawton and that of Eirola. Throughout the paper we work in the setting of rank m wavelet systems (m is the integer dilation factor, not necessarily 2). This involves some generalization of previous work, and yields initial results on the differentiability of wavelet scaling functions for rank m ? 2. In particular, we find that the minimal support rank 3 scaling functions with
A General Model of Undirected Web Graphs
, 2001
"... We describe a general model of a random graph whose degree sequence obeys a power law. Such laws have recently been observed in graphs associated with the world wide web. ..."
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Cited by 17 (3 self)
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We describe a general model of a random graph whose degree sequence obeys a power law. Such laws have recently been observed in graphs associated with the world wide web.
Nonlinear elliptic equations in conformal geometry
"... This is the set of Nachdiplom lectures which I have given during AprilJuly 2001 at Zurich. In the lectures, I have focused the study on some nonlinear partial differential equations related to curvature invariants in conformal geometry. A model of such a differential equation on compact surface is ..."
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Cited by 16 (0 self)
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This is the set of Nachdiplom lectures which I have given during AprilJuly 2001 at Zurich. In the lectures, I have focused the study on some nonlinear partial differential equations related to curvature invariants in conformal geometry. A model of such a differential equation on compact surface is the the Gaussian curvature equation under conformal change of metrics. On manifolds of dimension four, an analogue of Gaussian curvature is the study of the Pffafian integrand in the GaussBonnet formula. To be more precise, on a Riemannian manifold (M, g) of dimension four, denote the WeylSchouten tensor A as Aij = Rij − R 6 gij where Rij is the Ricci tensor and R is the scalar curvature of the Riemannian metric g; denote the second elementary symmetric function of A as σ2(A) = ∑ i<j λiλj = 1 2 [(T rA)2 − A  2], where λi (1 ≤ i ≤ 4) are the eigenvalues of A; then one has the Gauss Bonnet formula