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operators and completely integrable nonlinear lattices
 Mathematical Surveys and Monographs
, 2000
"... to post this online edition! This version is for personal use only! If you like this book and want to support the idea of online versions, please consider buying this book: ..."
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Cited by 149 (43 self)
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to post this online edition! This version is for personal use only! If you like this book and want to support the idea of online versions, please consider buying this book:
KPCA plus LDA: a complete kernel Fisher discriminant framework for feature extraction and recognition
 IEEE Transactions on Pattern Analysis and Machine Intelligence
"... Abstract—This paper examines the theory of kernel Fisher discriminant analysis (KFD) in a Hilbert space and develops a twophase KFD framework, i.e., kernel principal component analysis (KPCA) plus Fisher linear discriminant analysis (LDA). This framework provides novel insights into the nature of K ..."
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Cited by 54 (4 self)
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Abstract—This paper examines the theory of kernel Fisher discriminant analysis (KFD) in a Hilbert space and develops a twophase KFD framework, i.e., kernel principal component analysis (KPCA) plus Fisher linear discriminant analysis (LDA). This framework provides novel insights into the nature of KFD. Based on this framework, the authors propose a complete kernel Fisher discriminant analysis (CKFD) algorithm. CKFD can be used to carry out discriminant analysis in “double discriminant subspaces. ” The fact that, it can make full use of two kinds of discriminant information, regular and irregular, makes CKFD a more powerful discriminator. The proposed algorithm was tested and evaluated using the FERET face database and the CENPARMI handwritten numeral database. The experimental results show that CKFD outperforms other KFD algorithms. Index Terms—Kernelbased methods, subspace methods, principal component analysis (PCA), Fisher linear discriminant analysis (LDA or FLD), feature extraction, machine learning, face recognition, handwritten digit recognition. æ 1
A characterization of the Anderson metalinsulator transport transition
 Duke Math. J
"... We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong... ..."
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Cited by 41 (17 self)
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We investigate the Anderson metalinsulator transition for random Schrödinger operators. We define the strong...
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 35 (25 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.
Optimized TensorProduct Approximation Spaces
"... . This paper is concerned with the construction of optimized grids and approximation spaces for elliptic differential and integral equations. The main result is the analysis of the approximation of the embedding of the intersection of classes of functions with bounded mixed derivatives in standard S ..."
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Cited by 25 (15 self)
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. This paper is concerned with the construction of optimized grids and approximation spaces for elliptic differential and integral equations. The main result is the analysis of the approximation of the embedding of the intersection of classes of functions with bounded mixed derivatives in standard Sobolev spaces. Based on the framework of tensorproduct biorthogonal wavelet bases and stable subspace splittings, the problem is reduced to diagonal mappings between Hilbert sequence spaces. We construct operator adapted finiteelement subspaces with a lower dimension than the standard fullgrid spaces. These new approximation spaces preserve the approximation order of the standard fullgrid spaces, provided that certain additional regularity assumptions are fulfilled. The form of the approximation spaces is governed by the ratios of the smoothness exponents of the considered classes of functions. We show in which cases the so called curse of dimensionality can be broken. The theory covers e...
Sparse Grids for Boundary Integral Equations
, 1998
"... The potential of sparse grid discretizations for solving boundary integral equations is studied for the screen problem on a square in IR 3 . Theoretical and numerical results on approximation rates, preconditioning, adaptivity and compression for piecewise constant and linear sparse grid spaces ar ..."
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Cited by 24 (16 self)
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The potential of sparse grid discretizations for solving boundary integral equations is studied for the screen problem on a square in IR 3 . Theoretical and numerical results on approximation rates, preconditioning, adaptivity and compression for piecewise constant and linear sparse grid spaces are obtained. Classification: 45L10, 65N38, 65R20, 65Y20 Keywords: boundary element method, sparse grids, adaptivity, prewavelets, matrix compression 1 Introduction This is a case study for some special boundary integral equations on a twodimensional manifold \Gamma in IR 3 (screen problems). We will focus on the example of a twodimensional unit square in IR 2 embedded into IR 3 where \Gamma = fx : (x 1 ; x 2 ) 2 [0; 1] 2 ; x 3 = 0g : (1) In general, d\Gamma x stands for the surface Lebesgue measure with respect to the variable x, jxj 2 denotes the Euclidean norm of x, and n x is the vector field of normal vectors associated with \Gamma. We specifically have in mind the single lay...
Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials, Rev
 Math. Phys
"... The object of the present study is the integrated density of states of a quantum particle in multidimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic rando ..."
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Cited by 23 (7 self)
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The object of the present study is the integrated density of states of a quantum particle in multidimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic random potential satisfying a simple moment condition, we give a detailed proof that the infinitevolume limits of spatial eigenvalue concentrations of finitevolume operators with different boundary conditions exist almost surely. Since all these limits are shown to coincide with the expectation of the trace of the spatially localized spectral family of the infinitevolume operator, the integrated density of states is almost surely nonrandom and independent of the chosen boundary condition. Our proof of the
Ordinary Differential Equations and Dynamical Systems, Universität Wien. Available online: www.mat.univie.ac.at/∼gerald/ftp/bookode/ode.pdf
, 2009
"... Abstract. This book provides an introduction to ordinary differential equations and dynamical systems. We start with some simple examples of explicitly solvable equations. Then we prove the fundamental results concerning the initial value problem: existence, uniqueness, extensibility, dependence on ..."
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Cited by 23 (1 self)
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Abstract. This book provides an introduction to ordinary differential equations and dynamical systems. We start with some simple examples of explicitly solvable equations. Then we prove the fundamental results concerning the initial value problem: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore we consider linear equations, the Floquet theorem, and the autonomous linear flow. Then we establish the Frobenius method for linear equations in the complex domain and investigate Sturm–Liouville type boundary value problems including oscillation theory. Next we introduce the concept of a dynamical system and discuss stability including the stable manifold and the Hartman–Grobman theorem for both continuous and discrete systems. We prove the Poincaré–Bendixson theorem and investigate several examples of planar systems from classical mechanics, ecology, and electrical engineering. Moreover, attractors, Hamiltonian systems, the KAM theorem,
Oscillation theory and renormalized oscillation theory for Jacobi operators
 J. Diff. Eqs
, 1996
"... Abstract. We provide a comprehensive treatment of oscillation theory for Jacobi operators with separated boundary conditions. Our main results are as follows: If u solves the Jacobi equation (Hu)(n) = a(n)u(n + 1) + a(n − 1)u(n − 1) − b(n)u(n) = λu(n), λ ∈ R (in the weak sense) on an arbitrary in ..."
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Cited by 22 (21 self)
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Abstract. We provide a comprehensive treatment of oscillation theory for Jacobi operators with separated boundary conditions. Our main results are as follows: If u solves the Jacobi equation (Hu)(n) = a(n)u(n + 1) + a(n − 1)u(n − 1) − b(n)u(n) = λu(n), λ ∈ R (in the weak sense) on an arbitrary interval and satisfies the boundary condition on the left or right, then the dimension of the spectral projection P (−∞,λ)(H) of H equals the number of nodes (i.e., sign flips if a(n) < 0) of u. Moreover, we present a reformulation of oscillation theory in terms of Wronskians of solutions, thereby extending the range of applicability for this theory; if λ1,2 ∈ R and if u1,2 solve the Jacobi equation Huj = λjuj, j = 1, 2 and respectively satisfy the boundary condition on the left/right, then the dimension of the spectral projection P (λ1,λ2)(H) equals the number of nodes of the Wronskian of u1 and u2. Furthermore, these results are applied to establish the finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Jacobi operators. 1.