Linear Dynamical Systems.- Semigroups, Generators, and Resolvents.- Perturbation and Approximation of Semigroups.- Spectral Theory for Semigroups and Generators.- Asymptotics of Semigroups.- Semigroups Everywhere.- A Brief History of the Exponential Function.

https://www.springer.com/productFlyer_978-0-387-98463-6.pdf?SGWID=0-0-1297-1590487-0

One-Parameter Semigroups for Linear Evolution Equations

This article focuses on the construction, directly in physical space, of simply parametrized covariance functions for data-assimilation applications. A self-contained, rigorous mathematical summary of relevant topics from correlation theory is provided as a foundation for this construction. Covariance and correlation functions are defined, and common notions of homogeneity and isotropy are clarified. Classical results are stated, and proven where instructive. Included are smoothness properties relevant to multivariate statistical-analysis algorithms where wind/wind and wind/mass correlation models are obtained by differentiating the correlation model of a mass variable. the Convolution Theorem is introduced as the primary tool used to construct classes of covariance and cross-covariance functions on three-dimensional Euclidean space R3. Among these are classes of compactly supported functions that restrict to covariance and cross-covariance functions on the unit sphere S2, and that vanish identically on subsets of positive measure on S2. It is shown that these covariance and cross-covariance functions on S2, referred to as being space-limited, cannot be obtained using truncated spectral expansions. Compactly supported and space-limited covariance functions determine sparse covariance matrices when evaluated on a grid, thereby easing computational burdens in atmospheric data-analysis algorithms.

Convolution integrals leading to practical examples of compactly supported covariance and cross-covariance functions on R3 are reduced and evaluated. More specifically, suppose that gi and gj are radially symmetric functions defined on R3 such that gi(x) = 0 for |x| > di and gj(x) = 0 for |xv > dj, O di + dj and |x - y| > 2di, respectively, Additional covariance functions on R3 are constructed using convolutions over the real numbers R, rather than R3. Families of compactly supported approximants to standard second- and third-order autoregressive functions are constructed as illustrative examples. Compactly supported covariance functions of the form C(x,y) := Co(|x - y|), x,y ∈ R3, where the functions Co(r) for r ∈ R are 5th-order piecewise rational functions, are also constructed. These functions are used to develop space-limited product covariance functions B(x, y) C(x, y), x, y ∈ S2, approximating given covariance functions B(x, y) supported on all of S2 × S2.

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Construction of correlation functions in two and three dimensions

Over the past decade, the field of finite-dimensional variational inequality and complementarity problems has seen a rapid development in its theory of existence, uniqueness and sensitivity of solution(s), in the theory of algorithms, and in the application of these techniques to transportation planning, regional science, socio-economic analysis, energy modeling, and game theory. This paper provides a state-of-the-art review of these developments as well as a summary of some open research topics in this growing field.

Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications

This paper examines the theory of kernel Fisher discriminant analysis (KFD) in a Hilbert space and develops a two-phase 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.

/pdf/kpca-plus-lda-a-complete-kernel-fisher-discriminant-540vxq5jmz.pdf

KPCA plus LDA: a complete kernel Fisher discriminant framework for feature extraction and recognition

We introduce the Hilbert space-valued Wiener process and the corresponding stochastic integral of Ito type. This is then used together with semigroup theory to obtain existence and uniqueness of weak solutions of linear and semilinear stochastic evolution problems in Hilbert space. Finally, this abstract theory is applied to the linear heat and wave equations driven by additive noise.

/pdf/introduction-to-stochastic-partial-differential-equations-4mpb9c3p8y.pdf

Introduction to stochastic partial differential equations

1 Vector spaces with a scalar product, pre-Hilbert spaces.- 1.1 Sesquilinear forms.- 1.2 Scalar products and norms.- 2 Hilbert spaces.- 2.1 Convergence and completeness.- 2.2 Topological notions.- 3 Orthogonality.- 3.1 The projection theorem.- 3.2 Orthonormal systems and orthonormal bases.- 3.3 Existence of orthonormal bases, dimension of a Hilbert space.- 3.4 Tensor products of Hilbert spaces.- 4 Linear operators and their adjoints.- 4.1 Basic notions.- 4.2 Bounded linear operators and functionals.- 4.3 Isomorphisms, completion.- 4.4 Adjoint operator.- 4.5 The theorem of Banach-Steinhaus, strong and weak convergence.- 4.6 Orthogonal projections, isometric and unitary operators.- 5 Closed linear operators.- 5.1 Closed and closable operators, the closed graph theorem.- 5.2 The fundamentals of spectral theory.- 5.3 Symmetric and self-adjoint operators.- 5.4 Self-adjoint extensions of symmetric operators.- 5.5 Operators defined by sesquilinear forms (Friedrichs' extension).- 5.6 Normal operators.- 6 Special classes of linear operators.- 6.1 Finite rank and compact operators.- 6.2 Hilbert-Schmidt operators and Carleman operators.- 6.3 Matrix operators and integral operators.- 6.4 Differential operators on L2(a, b) with constant coefficients.- 7 The spectral theory of self-adjoint and normal operators.- 7.1 The spectral theorem for compact operators, the spaces Bp (H1H2).- 7.2 Integration with respect to a spectral family.- 7.3 The spectral theorem for self-adjoint operators.- 7.4 Spectra of self-adjoint operators.- 7.5 The spectral theorem for normal operators.- 7.6 One-parameter unitary groups.- 8 Self-adjoint extensions of symmetric operators.- 8.1 Defect indices and Cayley transforms.- 8.2 Construction of self-adjoint extensions.- 8.3 Spectra of self-adjoint extensions of a symmetric operator.- 8.4 Second order ordinary differential operators.- 8.5 Analytic vectors and tensor products of self-adjoint operators.- 9 Perturbation theory for self-adjoint operators.- 9.1 Relatively bounded perturbations.- 9.2 Relatively compact perturbations and the essential spectrum.- 9.3 Strong resolvent convergence.- 10 Differential operators on L2(?m).- 10.1 The Fourier transformation on L2(?m).- 10.2 Sobolev spaces and differential operators on L2(?m) with constant coefficients.- 10.3 Relatively bounded and relatively compact perturbations.- 10.4 Essentially self-adjoint Schrodinger operators.- 10.5 Spectra of Schrodinger operators.- 10.6 Dirac operators.- 11 Scattering theory.- 11.1 Wave operators.- 11.2 The existence and completeness of wave operators.- 11.3 Applications to differential operators on L2(?m).- A.1 Definition of the integral.- A.2 Limit theorems.- A.3 Measurable functions and sets.- A.4 The Fubini-Tonelli theorem.- A.5 The Radon-Nikodym theorem.- References.- Index of symbols.- Author and subject index.

Linear Operators in Hilbert Spaces

Formally self-adjoint differential expressions.- Appendix to section 1: The separation of the Dirac operator.- Fundamental properties and general assumptions.- Appendix to section 2: Proof of the Lagrange identity for n>2.- The minimal operator and the maximal operator.- Deficiency indices and self-adjoint extensions of T0.- The solutions of the inhomogeneous differential equation (?-?)u=f Weyl's alternative.- Limit point-limit circle criteria.- Appendix to section 6: Semi-boundedness of Sturm-Liouville type operators.- The resolvents of self-adjoint extensions of T0.- The spectral representation of self-adjoint extensions of T0.- Computation of the spectral matrix ?.- Special properties of the spectral representation, spectral multiplicities.- L2-solutions and essential spectrum.- Differential operators with periodic coefficients.- Appendix to section 12: Operators with periodic coefficients on the half-line.- Oscillation theory for regular Sturm-Liouville operators.- Oscillation theory for singular Sturm-Liouville operators.- Essential spectrum and absolutely continuous spectrum of Sturm-Liouville operators.- Oscillation theory for Dirac systems, essential spectrum and absolutely continuous spectrum.- Some explicitly solvable problems.

https://www.springer.com/productFlyer_978-3-540-17902-3.pdf?SGWID=0-0-1297-173811638-0

Joachim Weidmann

Papers

Linear Operators in Hilbert Spaces

Spectral Theory of Ordinary Differential Operators

Lineare Operatoren in Hilberträumen

Zur Spektraltheorie von Sturm-Liouville-Operatoren

On the continuous spectrum of schrödinger operators