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Chapter 7 describes the second-order reliability method (SORM), which employs a second-order approximation of the limit-state surface fitted at the design point in the standard normal space. Three distinct SORM approximations are presented. The classical SORM fits the second-order approximating surface to the principal curvatures of the limit-state surface at the design point. This approach requires computing the Hessian (second-derivative matrix) of the limit-state function at the design point and its eigenvalues as the principal curvatures. The second approach computes the principal curvatures iteratively in the process of finding the design point. This approach requires only first-order derivatives of the limit-state function but repeated solutions of the optimization problem for finding the design point. One advantage is that the principal curvatures are found in decreasing order of magnitude and, hence, the computations can be stopped when the curvature found is sufficiently small. The third approach fits the approximating second-order surface to fitting points in the neighborhood of the design point. This approach also avoids computing the Hessian. Furthermore, it corrects for situations where the curvature is zero but the surface is curved, e.g., when the design point is an inflection point of the surface. Results from the three methods are compared numerically.
In this paper we are interested in comparing the spectra of two elliptic operators acting on a closed minimal submanifold of the Euclidean unit sphere. Using an approach introduced by Savo in [A Savo. Index Bounds for Minimal Hypersurfaces of the Sphere. Indiana Univ. Math. J. 59 (2010), 823-837.], we are able to compare the eigenvalues of the stability operator acting on sections of the normal bundle and the Hodge Laplacian operator acting on $1$-forms. As a byproduct of the technique and under a suitable hypothesis on the Ricci curvature of the submanifold we obtain that its first Betti's number is bounded from above by a multiple of the Morse index, which provide evidence for a well-known conjecture of Schoen and Marques & Neves in the setting of higher codimension.
We show that the elements of the dual of the Euclidean distance matrix cone can be described via an inequality on a certain weighted sum of its eigenvalues.
Concentrated random variables are frequently used in representing deterministic delays in stochastic models. The squared coefficient of variation ($\mathrm {SCV}$) of the most concentrated phase-type distribution of order $N$ is $1/N$. To further reduce the $\mathrm {SCV}$, concentrated matrix exponential (CME) distributions with complex eigenvalues were investigated recently. It was obtained that the $\mathrm {SCV}$ of an order $N$ CME distribution can be less than $n^{-2.1}$ for odd $N=2n+1$ orders, and the matrix exponential distribution, which exhibits such a low $\mathrm {SCV}$ has complex eigenvalues. In this paper, we consider CME distributions with real eigenvalues (CME-R). We present efficient numerical methods for identifying a CME-R distribution with smallest SCV for a given order $n$. Our investigations show that the $\mathrm {SCV}$ of the most concentrated CME-R of order $N=2n+1$ is less than $n^{-1.85}$. We also discuss how CME-R can be used for numerical inverse Laplace transformation, which is beneficial when the Laplace transform function is impossible to evaluate at complex points.
A holomorphic endomorphism of ${{\mathbb {CP}}}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study the eigenvalues of the differential of such a map along a periodic cycle. When $n=1$, a well-known fact is that the eigenvalue along a periodic cycle of a post-critically finite rational map is either superattracting or repelling. We prove that, when $n=2$, the eigenvalues are still either superattracting or repelling. This is an improvement of a result by Mattias Jonsson [Some properties of 2-critically finite holomorphic maps of P2. Ergod. Th. & Dynam. Sys.18(1) (1998), 171–187]. When $n\geq 2$ and the cycle is outside the post-critical set, we prove that the eigenvalues are repelling. This result improves one obtained by Fornæss and Sibony [Complex dynamics in higher dimension. II. Modern Methods in Complex Analysis (Princeton, NJ, 1992) (Annals of Mathematics Studies, 137). Ed. T. Bloom, D. W. Catlin, J. P. D’Angelo and Y.-T. Siu, Princeton University Press, 1995, pp. 135–182] under a hyperbolicity assumption on the complement of the post-critical set.
The algebraic eigenproblem is the mathematical answer to the physical questions: What are the principal stresses in a solid or fluid and on what planes do they act? What are the natural frequencies of a system? Is the system stable to small disturbances? What is the best basis with respect to which to solve a system of linear algebraic equations with a real symmetric coefficient matrix? What is the best basis with respect to which to solve a system of linear ordinary differential equations? What is the best basis with respect to which to represent an experimental or numerical data set?
The eigenvalues and eigenfunctions of self-adjoint differential operators provide the basis functions with respect to which ordinary and partial differential equations can be solved. These methods are extensions of those used to solve linear systems of algebraic equations and ordinary differential equations. Eigenfunction expansions also provide the basis for advanced numerical methods, such as spectral methods, and data-reduction techniques, such as proper-orthogonal decomposition.
In the preceding chapter on rigid-body motion we took a step beyond single-particle mechanics to explore the behavior of a more complex system containing many particles bonded rigidly together. Now we will explore additional sets of many-particle systems in which the individual particles are connected by linear, Hooke’s-law springs. These have some interest in themselves, but more generally they serve as a model for a large number of coupled systems that oscillate harmonically when disturbed from their natural state of equilibrium, such as elastic solids, electric circuits, and multi-atom molecules. We will begin with the oscillations of a few coupled masses and end with the behavior of a continuum of masses described by a linear mass density. The mathematical techniques required to analyze such coupled oscillators are used throughout physics, including linear algebra and matrices, normal modes, eigenvalues and eigenvectors, and Fourier series and Fourier transforms.
Chapter 8 concerns a group of WEC units that may be realised in a more distant future, namely groups or arrays of individual WEC units and two-dimensional WEC units, which needs to be rather big structures. Firstly, a group of WEC bodies is analysed. Next a group consisting of WEC bodies as well as OWCs is analysed. Then the previous real radiation resistance needs to be replaced by a complex radiation damping matrix which is complex, but Hermitian, which means that its eigenvalues are real.
We define the resolvent and spectrum of a bounded linear operator and discuss the relationship between the spectrum and the ‘point spectrum’, which is the set of all eigenvalues. We prove some basic properties of the spectrum and the Spectral Mapping Theorem for polynomials.
For a family of elliptic operators with periodically oscillating coefficients, $-{\rm div}(A(\cdot /\varepsilon )\nabla )$ with tiny ε > 0, we comprehensively study the first-order expansions of eigenvalues and eigenfunctions (eigenspaces) for both the Dirichlet and Neumann problems in bounded, smooth and strictly convex domains (or more general domains of finite type). A new first-order correction term is introduced to derive the expansion of eigenfunctions in L2 or $H^1_{\rm loc}$. Our results rely on the recent progress on the homogenization of boundary layer problems.
Let $X$ be a compact, metric and totally disconnected space and let $f:X\rightarrow X$ be a continuous map. We relate the eigenvalues of $f_{\ast }:\check{H}_{0}(X;\mathbb{C})\rightarrow \check{H}_{0}(X;\mathbb{C})$ to dynamical properties of $f$, roughly showing that if the dynamics is complicated then every complex number of modulus different from 0, 1 is an eigenvalue. This stands in contrast with a classical inequality of Manning that bounds the entropy of $f$ below by the spectral radius of $f_{\ast }$.
We consider the Steklov eigenvalues of the Laplace operator as limiting Neumann eigenvalues in a problem of mass concentration at the boundary of a ball. We discuss the asymptotic behaviour of the Neumann eigenvalues and find explicit formulae for their derivatives in the limiting problem. We deduce that the Neumann eigenvalues have a monotone behaviour in the limit and that Steklov eigenvalues locally minimize the Neumann eigenvalues.
In this work we study the homogenisation problem for nonlinear elliptic equations involving $p$-Laplacian-type operators with sign-changing weights. We study the asymptotic behaviour of variational eigenvalues which consist of a double sequence of eigenvalues. We show that the $k$th positive eigenvalue goes to infinity when the average of the weights is nonpositive, and converges to the $k$th variational eigenvalue of the limit problem when the average is positive for any $k\geq 1$.
In this paper, we consider the dependence of eigenvalues of sixth-order boundary value problems on the boundary. We show that the eigenvalues depend not only continuously but also smoothly on boundary points, and that the derivative of the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$th eigenvalue as a function of an endpoint satisfies a first-order differential equation. In addition, we prove that as the length of the interval shrinks to zero all higher eigenvalues of such boundary value problems march off to plus infinity. This is also true for the first (that is, lowest) eigenvalue.
inf{αλ1(Ω) + βλ2(Ω) + (1 − α − β)λ3(Ω) | Ω open in ℝn, |Ω| ≤ 1},
for α, β ∈ [0, 1], α + β ≤ 1, where
λk(Ω)
is the kth
eigenvalue of the Dirichlet Laplacian acting in L2(Ω) and
|Ω| is
the Lebesgue measure of Ω. We investigate for which values of α, β every minimiser is
connected.
We consider a class of eigenvalue problems for polyharmonic operators, including
Dirichlet and buckling-type eigenvalue problems. We prove an analyticity result for the
dependence of the symmetric functions of the eigenvalues upon domain perturbations and
compute Hadamard-type formulas for the Frechét differentials. We also consider
isovolumetric domain perturbations and characterize the corresponding critical domains for
the symmetric functions of the eigenvalues. Finally, we prove that balls are critical
domains.
We consider the problem of minimising the nth-eigenvalue of the Robin
Laplacian in RN. Although for n = 1,2 and a
positive boundary parameter α it is known that the minimisers do not
depend on α, we demonstrate numerically that this will not always be the
case and illustrate how the optimiser will depend on α. We derive a
Wolf–Keller type result for this problem and show that optimal eigenvalues grow at most
with n1/N, which is in sharp contrast with
the Weyl asymptotics for a fixed domain. We further show that the gap between consecutive
eigenvalues does go to zero as n goes to infinity. Numerical results then
support the conjecture that for each n there exists a positive value of
αn such that the nth
eigenvalue is minimised by n disks for all
0 < α < αn
and, combined with analytic estimates, that this value is expected to grow with
n1/N.
Trace inequalities for sums and products of matrices are presented. Relations between the given inequalities and earlier results are discussed. Among other inequalities, it is shown that if A and B are positive semidefinite matrices then for any positive integer k.
In this paper, we are going to investigate the canonical property of solutions of systems of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm–Liouville equation with turning point. Using of the asymptotic estimates provided by Eberhard, Freiling, and Schneider for a special fundamental system of solutions of the Sturm–Liouville equation, we study the infinite product representation of solutions of the systems. Then we transform the Sturm–Liouville equation with turning point to the equation with singularity, then we study the asymptotic behavior of its solutions. Such representations are relevant to the inverse spectral problem.