Terms Of Spectral Decomposition Research Articles

Non-local Solvable Birth\u2013Death Processes

In this paper, we study strong solutions of some non-local difference–differential equations linked to a class of birth–death processes arising as discrete approximations of Pearson diffusions by means of a spectral decomposition in terms of orthogonal polynomials and eigenfunctions of some non-local derivatives. Moreover, we give a stochastic representation of such solutions in terms of time-changed birth–death processes and study their invariant and their limit distribution. Finally, we describe the correlation structure of the aforementioned time-changed birth–death processes.

Limit behaviors of solutions to the pseudo-relativistic Lane-Emden equation

In this paper, we study the pseudo-relativistic Lane-Emden equation(−Δ+m2−m)u=|u|p−1uinΩ, with the Dirichlet boundary condition u=0 on ∂Ω. Here, Ω⊂Rn is bounded and the operator (−Δ+m2−m) is defined in terms of spectral decomposition. In the authors' previous paper [8], it is shown the existence and nonexistence results for the least energy solutions, depending on the range of parameter p. This paper is devoted to study the asymptotic behavior of nontrivial solutions when the mass parameter m in the equation either goes to 0+ or diverges to infinity.

De Sitter-invariant states from holography

A class of invariant states under de Sitter isometries is constructed in d-dimensional conformal field theories (CFTs) from the universal sector of AdS/CFT dualities. These states extend the Mottola–Allen α-vacua to theories containing scalar primary operators of arbitrary scaling dimensions, and are proven to explicitly break conformal symmetry. Two-point correlators in these states are shown to satisfy a generalized spectral decomposition in terms of free massive propagators. It is also pointed out that the spectral decomposition of generic Euclidean correlators is known in the mathematics literature as Olevsky index transform, of which computational use can be made.

Semilinear elliptic equations with the pseudo-relativistic operator on a bounded domain

We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, (−Δ+m2−m)u=|u|p−1uinΩ,with the Dirichlet boundary condition u=0 on ∂Ω. Here, p∈(1,∞) and the operator (−Δ+m2−m) is defined in terms of spectral decomposition. In this paper, we investigate existence and nonexistence of a nontrivial solution, depending on the choice of p, m and Ω. Precisely, we show that (i) if p is not H1 subcritical (p≥n+2n−2) and Ω is star-shaped, the equation has no nontrivial solution for all m>0; (ii) if p is not H1∕2 supercritical (1<p≤n+1n−1), then there exists a least energy solution for all m>0 and any bounded domain Ω; (iii) finally, in the intermediate range (n+1n−1<p<n+2n−2), the problem has a nontrivial solution, provided that m is sufficiently large and the problem (0.1)−Δu=|u|p−1uinΩ,u=0on∂Ωadmits a non-degenerate nontrivial solution, for example, when Ω is a ball or an annulus.

Spectral decomposition of nonlinear systems with memory.

We present an alternative approach to the analysis of nonlinear systems with long-term memory that is based on the Koopman operator and a Lévy transformation in time. Memory effects are considered to be the result of interactions between a system and its surrounding environment. The analysis leads to the decomposition of a nonlinear system with memory into modes whose temporal behavior is anomalous and lacks a characteristic scale. On average, the time evolution of a mode follows a Mittag-Leffler function, and the system can be described using the fractional calculus. The general theory is demonstrated on the fractional linear harmonic oscillator and the fractional nonlinear logistic equation. When analyzing data from an ill-defined (black-box) system, the spectral decomposition in terms of Mittag-Leffler functions that we propose may uncover inherent memory effects through identification of a small set of dynamically relevant structures that would otherwise be obscured by conventional spectral methods. Consequently, the theoretical concepts we present may be useful for developing more general methods for numerical modeling that are able to determine whether observables of a dynamical system are better represented by memoryless operators, or operators with long-term memory in time, when model details are unknown.

Identification of the Spectrum Signature of Air-Bearing Slider Dynamics

Characterization of the motion of air-bearing slider with sub-10-nm clearance is becoming a critical aspect of developing advanced head–disk interfaces (HDI) in hard disk drives to achieve higher areal data densities. In this article, the response of sub-10-nm clearance air-bearing slider induced by a bump contact is recorded using laser Doppler vibrometer (LDV). To identify system dynamics in terms of spectral decomposition, the slider response is studied using FFT, power spectrum density, spectrogram, and Hilbert instantaneous spectrum analysis. The results demonstrate that the response of air-bearing slider in instantaneous contact exhibits nonstationary and nonlinear properties which can be accurately identified using Hilbert instantaneous spectrum. The interpretation and spectrum identification based on Fourier analysis and its extension in time–frequency domain could lead to inaccurate results due to their limitation in resolution and linearity assumption.

Gravitational collapse of scalar fields via spectral methods

In this paper we present a new numerical code based on the Galerkin method to integrate the field equations for the spherical collapse of massive and massless scalar fields. By using a spectral decomposition in terms of the radial coordinate, the field equations were reduced to a finite set of ordinary differential equations in the space of modes associated with the Galerkin expansion of the scalar field, together with algebraic sets of equations connecting modes associated with the metric functions. The set of ordinary differential equations with respect to the null coordinate is then integrated using an eighth-order Runge-Kutta method. The numerical tests have confirmed the high accuracy and fast convergence of the code. As an application we have evaluated the whole spectrum of black hole masses which ranges from infinitesimal to large values obtained after varying the amplitude of the initial scalar field distribution. We have found strong numerical evidence that this spectrum is described by a nonextensive distribution law.

The fundamental solutions for irrotational and rotational Stokes flow in spheroidal geometry

AbstractThe Stokes operator E2 governs the irrotational axisymmetric Stokes flow and its square governs the corresponding rotational flow. In spheroidal coordinates the elements of the solution space ker E2 enjoy a spectral decomposition into separable eignefunction, while the elements of the ker E4 accept a spectral decomposition in terms of semiseparable eigensolutions involving 3D-by-3D eigenfunctions of the Gegenbauer operator. These spectral characteristics are utilized to construct the fundamental solutions for both the E2 and the E4 operators in spheroidal geometry. The fundamental solution for E2 is expressed in terms of the elements of the irrotational space ker E2, while the fundamental solution for E4 is expressed in terms of the corresponding generalized eigenfunctions alone.

Methodology and convergence rates for functional linear regression

In functional linear regression, the slope parameter is a function. Therefore, in a nonparametric context, it is determined by an infinite number of unknowns. Its estimation involves solving an ill-posed problem and has points of contact with a range of methodologies, including statistical smoothing and deconvolution. The standard approach to estimating the slope function is based explicitly on functional principal components analysis and, consequently, on spectral decomposition in terms of eigenvalues and eigen-functions. We discuss this approach in detail and show that in certain circumstances, optimal convergence rates are achieved by the PCA technique. An alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view.

Uncertainty propagation in CFD using polynomial chaos decomposition

Uncertainty quantification in CFD computations is receiving increased interest, due in large part to the increasing complexity of physical models, and the inherent introduction of random model data. This paper focuses on recent application of PC methods for uncertainty representation and propagation in CFD computations. The fundamental concept on which polynomial chaos (PC) representations are based is to regard uncertainty as generating a new set of dimensions, and the solution as being dependent on these dimensions. A spectral decomposition in terms of orthogonal basis functions is used, the evolution of the basis coefficients providing quantitative estimates of the effect of random model data. A general overview of PC applications in CFD is provided, focusing exclusively on applications involving the unreduced Navier–Stokes equations. Included in the present review are an exposition of the mechanics of PC decompositions, an illustration of various means of implementing these representations, and a perspective on the applicability of the corresponding techniques to propagate and quantify uncertainty in Navier–Stokes computations.

Computation of isotropic tensor functions

AbstractAn explicit formulation and a procedure for the computation of isotropic tensor‐valued tensor functions is discussed. The formulation is based on a spectral decomposition in terms of second‐order eigenvalue bases, which avoids the costly computation of eigenvectors. As an important result a compact structure of the fourth‐order derivatives of general second‐order isotropic tensor functions is presented.

Universal notions characterizing spectral decompositions

In this note we characterize certain types of spectral decomposition in terms of “universal” notions valid for any operator on a Banach space. To be precise, let X be a complex Banach space and let T be a bounded linear operator on X. If F is a closed set in the plane C, let X(T, F) consist of all y ∈ X satisfying thes identitywhere f:C\F → X is analytic. It is then easy to see that X(T, F) is a T-invariant linear manifold in X. Moreover, if y ∈ X thenis a compact subset of the spectrum σ(T). Our aim is to give necessary and sufficient conditions for a decomposable or strongly decomposable operator in terms of X(T, F) and γ(y, T). Recall that T is decomposable if whenever G1G2 are open and cover C there exist T-invariant closed linear manifolds M1, M2 with X= M1 + M2 and σ(T | M1) ⊂ Gi(i = 1,2) (equivalently, σ(T | Mi)⊂ Ḡi, see [4, p. 57]). In this case, X(T, F) is norm closed if Fis closed and each y in X has a unique maximally defined local resolvent satisfying (1.1) on C\Fy; Fy is called the local spectrum σ(y, T) and coincides with γ(y, T). Hence T has the single valued extension property (SVEP); i.e., zero is the only analytic function f:V → X satisfying (z − T)f(z) = 0 on V. If T is decomposable and the restriction T | X(T, F) is also decomposable for each closed F, then T is called strongly decomposable. We point out that Albrecht [2] has shown by example that not every decomposable operator is strongly decomposable, while Eschmeier [6]has given a simpler construction to show that this phenomenon occurs even in Hilbert space.

The probability distribution of S+ gyrospeeds in the Io torus

We report a measured probability distribution for S+ gyrospeeds in the Io torus derived from [S II] 6716 Å, 6731 Å emission line profiles recorded at axial distance 5.9 RJ from Jupiter. This ion species is not in kinetic thermodynamic equilibrium. An artificial spectral decomposition in terms of three Maxwellian distributions finds perpendicular temperatures T⊥ = 2.7, 28, and 192 eV with weights 0.28, 0.50, and 0.22, respectively. The average temeprature is about 60 eV, which would be seriously underestimated by a simple measurement of the line width. We find upper limits of ∼5% or ∼20% to the mixing ratios of S+ ions having the pickup speed of ionized ballistic atoms, assuming a flat or isotropic pitch angle distribution, respectively. Due to the low S+ density and charge, Coulomb collisional power transfer from the observed S+ population to the ambient electrons delivers only 0.01 eV cm−3 s−1, or 1/40 the UV emission power requirement. Existing [S III] linewidth temperatures indicate S2+ is hotter than S+, indicating that kinetic thermodynamic equilibrium exists neither between nor within species in the hot, outer plasma torus.