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Interpolation of Scattered Data: Distance Matrices and Conditionally Positive Definite Functions
 CONSTRUCTIVE APPROXIMATION
, 1986
"... Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke. ..."
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Cited by 328 (3 self)
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Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke.
Discrete orthogonal polynomial ensembles and the Plancherel measure
, 2001
"... We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble i ..."
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Cited by 152 (8 self)
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We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble is related to a twodimensional directed growth model, and the Charlier ensemble is related to the lengths of weakly increasing subsequences in random words. The Krawtchouk ensemble occurs in connection with zigzag paths in random domino tilings of the Aztec diamond, and also in a certain simplified directed firstpassage percolation model. We use the Charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of Tracy and Widom. As a limit of the Meixner ensemble or the Charlier ensemble we obtain the Plancherel measure on partitions, and using this we prove a conjecture of Baik, Deift and Johansson that under the Plancherel measure, the distribution of the lengths of the first k rows in the partition, appropriately scaled, converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the Gaussian Unitary Ensemble. In this problem a certain discrete kernel, which we call the discrete Bessel kernel, plays an important role.
Asymptotics of Plancherel measures for symmetric groups
 J. Amer. Math. Soc
, 2000
"... 1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G ∧ of irreducible representations of G which assigns to a representation π ∈ G ∧ the weight (dim π) 2 /G. For the symmetric group S(n), the set S(n) ∧ is the set o ..."
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Cited by 150 (36 self)
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1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G ∧ of irreducible representations of G which assigns to a representation π ∈ G ∧ the weight (dim π) 2 /G. For the symmetric group S(n), the set S(n) ∧ is the set of partitions λ of the number
Boussinesq equations and other systems for smallamplitude long waves in nonlinear dispersive media II: Nonlinear theory, preprint
, 2002
"... Communicated by G. Iooss Summary. Considered herein are a number of variants of the classical Boussinesq system and their higherorder generalizations. Such equations were first derived by Boussinesq to describe the twoway propagation of smallamplitude, long wavelength, gravity waves on the surfac ..."
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Cited by 94 (22 self)
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Communicated by G. Iooss Summary. Considered herein are a number of variants of the classical Boussinesq system and their higherorder generalizations. Such equations were first derived by Boussinesq to describe the twoway propagation of smallamplitude, long wavelength, gravity waves on the surface of water in a canal. These systems arise also when modeling the propagation of longcrested waves on large lakes or the ocean and in other contexts. Depending on the modeling of dispersion, the resulting system may or may not have a linearization about the rest state which is well posed. Even when well posed, the linearized system may exhibit a lack of conservation of energy that is at odds with its status as an approximation to the Euler equations. In the present script, we derive a fourparameter family of Boussinesq systems from the twodimensional Euler equations for freesurface flow and formulate criteria to help decide which of these equations one might choose in a given modeling situation. The analysis of the systems according to these criteria is initiated.
Generalization Performance of Regularization Networks and Support . . .
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2001
"... We derive new bounds for the generalization error of kernel machines, such as support vector machines and related regularization networks by obtaining new bounds on their covering numbers. The proofs make use of a viewpoint that is apparently novel in the field of statistical learning theory. The hy ..."
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Cited by 77 (17 self)
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We derive new bounds for the generalization error of kernel machines, such as support vector machines and related regularization networks by obtaining new bounds on their covering numbers. The proofs make use of a viewpoint that is apparently novel in the field of statistical learning theory. The hypothesis class is described in terms of a linear operator mapping from a possibly infinitedimensional unit ball in feature space into a finitedimensional space. The covering numbers of the class are then determined via the entropy numbers of the operator. These numbers, which characterize the degree of compactness of the operator, can be bounded in terms of the eigenvalues of an integral operator induced by the kernel function used by the machine. As a consequence, we are able to theoretically explain the effect of the choice of kernel function on the generalization performance of support vector machines.
Error Estimates for Interpolation By Compactly Supported Radial Basis Functions of Minimal Degree
, 1997
"... We consider error estimates for the interpolation by a special class of compactly supported radial basis functions. These functions consist of a univariate polynomial within their support and are of minimal degree depending on space dimension and smoothness. Their associated "native" Hilb ..."
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Cited by 60 (7 self)
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We consider error estimates for the interpolation by a special class of compactly supported radial basis functions. These functions consist of a univariate polynomial within their support and are of minimal degree depending on space dimension and smoothness. Their associated "native" Hilbert spaces are shown to be normequivalent to Sobolev spaces. Thus we can derive approximation orders for functions from Sobolev spaces which are comparable to those of thinplatespline interpolation. Finally, we investigate the numerical stability of the interpolation process.
Lévy Processes in Finance: Theory, Numerics, and Empirical Facts
, 2000
"... Lévy processes are an excellent tool for modelling price processes in mathematical finance. On the one hand, they are very flexible, since for any time increment ∆t any infinitely divisible distribution can be chosen as the increment distribution over periods of time ∆t. On the other hand, they have ..."
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Cited by 54 (2 self)
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Lévy processes are an excellent tool for modelling price processes in mathematical finance. On the one hand, they are very flexible, since for any time increment ∆t any infinitely divisible distribution can be chosen as the increment distribution over periods of time ∆t. On the other hand, they have a simple structure in comparison with general semimartingales. Thus stochastic models based on Lévy processes often allow for analytically or numerically tractable formulas. This is a key factor for practical applications. This thesis is divided into two parts. The first, consisting of Chapters 1, 2, and 3, is devoted to the study of stock price models involving exponential Lévy processes. In the second part, we study term structure models driven by Lévy processes. This part is a continuation of the research that started with the author's diploma thesis Raible (1996) and the article Eberlein and Raible (1999). The content of the chapters is as follows. In Chapter 1, we study a general stock price model where the price of a single stock follows an exponential Lévy process. Chapter 2 is devoted to the study of the Lévy measure of infinitely divisible distributions, in particular of generalized hyperbolic distributions. This yields information about what changes in the distribution of a generalized hyperbolic Lévy motion can be achieved by a locally equivalent change of the underlying probability measure. Implications for
Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger
 Operators, Amer. Math. Soc
, 2009
"... Abstract. This manuscript provides a selfcontained introduction to mathematical methods in quantum mechanics (spectral theory) with applications to Schrödinger operators. The first part covers mathematical foundations of quantum mechanics from selfadjointness, the spectral theorem, quantum dynamic ..."
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Cited by 54 (31 self)
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Abstract. This manuscript provides a selfcontained introduction to mathematical methods in quantum mechanics (spectral theory) with applications to Schrödinger operators. The first part covers mathematical foundations of quantum mechanics from selfadjointness, the spectral theorem, quantum dynamics (including Stone’s and the RAGE theorem) to perturbation theory for selfadjoint operators. The second part starts with a detailed study of the free Schrödinger operator respectively position, momentum and angular momentum operators. Then we develop WeylTitchmarsh theory for SturmLiouville operators and apply it to spherically symmetric problems, in particular to the hydrogen atom. Next we investigate selfadjointness of atomic Schrödinger operators and their essential spectrum, in particular the HVZ theorem. Finally we have a look at scattering theory and prove asymptotic completeness in the short range case.
Random maps, coalescing saddles, singularity analysis, and Airy phenomena
 Random Structures & Algorithms
, 2001
"... A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that i ..."
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Cited by 51 (6 self)
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A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.