The Fourth-order Bessel–type Differential Equation
TL;DR: The Bessel-type functions as discussed by the authors are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity on the point of infinity of the complex plane, which are derived by linear combinations and limit processes from the classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials.
Abstract: The Bessel-type functions, structured as extensions of the classical Bessel functions, were defined by Everitt and Markett in 1994. These special functions are derived by linear combinations and limit processes from the classical orthogonal polynomials, classical Bessel functions and the Krall Jacobi-type and Laguerre-type orthogonal polynomials. These Bessel-type functions are solutions of higher-order linear differential equations, with a regular singularity at the origin and an irregular singularity at the point of infinity of the complex plane. There is a Bessel-type differential equation for each even-order integer; the equation of order two is the classical Bessel differential equation. These even-order Bessel-type equations are not formal powers of the classical Bessel equation. When the independent variable of these equations is restricted to the positive real axis of the plane they can be written in the Lagrange symmetric (formally self-adjoint) form of the Glazman–Naimark type, with real coeffic...
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Dissertation•
08 Feb 2018
TL;DR: In this paper, the effect of plasticity on the propagation and mechanical equilibrium of blisters, although experimentally observed, had not been systematically studied to date, and the authors focused on the observation and characterization of buckling structures observed on gold films deposited on silicon substrates.
Abstract: Thin film coatings submitted to high compressive stresses may experience a simultaneous buckling and delamination phenomenon called "blistering". The mechanism of formation and propagation of blisters in the form of straight wrinkles and circular blisters has been extensively studied in the literature considering a linear elastic behavior for the film. However, the effect of plasticity on the propagation and mechanical equilibrium of such blisters, although experimentally observed, had not been systematically studied to date.In this work, we are interested in the observation and characterization of buckling structures observed on gold films deposited on silicon substrates. The effects of plasticity on the morphology or critical buckling load of buckled structures are quantitatively demonstrated using small scale surface observation techniques such as AFM, as well as mechanical testing by nanoindentation tests and stress measurement methods.A mechanical model is developed in order to model the film as a geometric nonlinear plate with elastic-plastic behavior in unilateral contact with a rigid support representing the substrate. In addition, a cohesive zone model is introduced between the plate and the support in order to take into account the delamination of the film, with a separation work depending on the mode mix of the interface loading.This model allowed us to highlight the effect of plasticity on the equilibrium profiles resulting from elastic-plastic blistering, for both straight and circular blisters morphologies. The effect on the offset of the critical buckling load has also been studied. Finally, the influence of plastic deformation on the propagation mechanism of the interfacial fracture itself has been studied. In particular, a stabilizing effect of the circular blister form, which has been observed experimentally in various studies, has been demonstrated through calculation.
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Book•
01 Jan 1993
TL;DR: In this article, the main properties of bounded and unbounded operators, adjoint operators, symmetric and self-adjoint operators in hilbert spaces are discussed, as well as the stability of self-jointness under small perturbations.
Abstract: linear operators in hilbert spaces | springerlink abstract. we recall some fundamental notions of the theory of linear operators in hilbert spaces which are required for a rigorous formulation of the rules of quantum mechanics in the one-body case. in particular, we introduce and discuss the main properties of bounded and unbounded operators, adjoint operators, symmetric and self-adjoint operators, self-adjointness criterion and stability of self-adjointness under small perturbations, spectrum, isometric and unitary operators, spectral
2,282 citations
Book•
01 Jun 1985
TL;DR: In this paper, the Laurent series is used for expanding functions in Taylor series, and the calculus of residues is used to expand functions in Laurent series volumes II, III, and IV.
Abstract: Volume I, Part 1: Basic Concepts: I.1 Introduction I.2 Complex numbers I.3 Sets and functions. Limits and continuity I.4 Connectedness. Curves and domains I.5. Infinity and stereographic projection I.6 Homeomorphisms Part 2: Differentiation. Elementary Functions: I.7 Differentiation and the Cauchy-Riemann equations I.8 Geometric interpretation of the derivative. Conformal mapping I.9 Elementary entire functions I.10 Elementary meromorphic functions I.11 Elementary multiple-valued functions Part 3: Integration. Power Series: I.12 Rectifiable curves. Complex integrals I.13 Cauchy's integral theorem I.14 Cauchy's integral and related topics I.15 Uniform convergence. Infinite products I.16 Power series: rudiments I.17 Power series: ramifications I.18 Methods for expanding functions in Taylor series Volume II, Part 1: Laurent Series. Calculus of Residues: II.1 Laurent's series. Isolated singular points II.2 The calculus of residues and its applications II.3 Inverse and implicit functions II.4 Univalent functions Part 2: Harmonic and Subharmonic Functions: II.5 Basic properties of harmonic functions II.6 Applications to fluid dynamics II.7 Subharmonic functions II.8 The Poisson-Jensen formula and related topics Part 3: Entire and Meromorphic Functions: II.9 Basic properties of entire functions II.10 Infinite product and partial fraction expansions Volume III, Part 1: Conformal Mapping. Approximation Theory: III.1 Conformal mapping: rudiments III.2 Conformal mapping: ramifications III.3 Approximation by rational functions and polynomials Part 2: Periodic and Elliptic Functions: III.4 Periodic meromorphic functions III.5 Elliptic functions: Weierstrass' theory III.6 Elliptic functions: Jacobi's theory Part 3: Riemann Surfaces. Analytic Continuation: III.7 Riemann surfaces III.8 Analytic continuation III.9 The symmetry principle and its applications Bibliography Index.
1,426 citations
Book•
05 May 1980
TL;DR: The spectral theory of self-adjoint and normal operators on L2(a, b) spaces has been studied in this article, where it has been shown that the existence and completeness of wave operators can be proved.
Abstract: 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.
1,346 citations
Book•
01 Mar 1978
TL;DR: In this paper, the authors define the following criteria for exponential and ordinary dichotomies: stability, roughness, reducibility, robustness, and robustness of an exponential dichotomy.
Abstract: Stability.- Exponential and ordinary dichotomies.- Dichotomies and functional analysis.- Roughness.- Dichotomies and reducibility.- Criteria for an exponential dichotomy.- Dichotomies and lyapunov functions.- Equations on ? and almost periodic equations.- Dichotomies and the hull of an equation.
1,151 citations