Topic
Fourier series
About: Fourier series is a research topic. Over the lifetime, 16548 publications have been published within this topic receiving 322486 citations.
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TL;DR: The turbulent spectral properties of the dynamical equation of Hasegawa and Mima (1978) governing the evolution of the electrostatic potential in drift-wave turbulence is investigated for two formulations of the problem: (1) as a nondissipative initial value problem, with the potential represented by a truncated Fourier series with large number of terms; (2) as dissipative problem with a small viscous dissipation at very short spatial scales, and a long wavelength forcing term at longer wavelengths.
Abstract: The turbulent spectral properties of the dynamical equation of Hasegawa and Mima (1978) governing the evolution of the electrostatic potential in drift-wave turbulence is investigated for two formulations of the problem: (1) as a nondissipative initial value problem, with the potential represented by a truncated Fourier series with large number of terms, and (2) as a dissipative problem with a small viscous dissipation at very short spatial scales, and a long wavelength forcing term at longer wavelengths It is found that Hasegawa and Mima's prediction for the nondissipative, truncated initial value modal problem is accurate, but substantial differences exist for the forced dissipative case between computer results and analytical predictions based on a wave kinetic equation of Kadomtsev Much better agreement is found with a simple dual-cascade model based on Kraichnan's generalization of Kolmogorov's cascade arguments
76 citations
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TL;DR: In this article, a new plane stress model of composite beams with interlayer slips is developed by the state space method, which is obtained analytically for beams with two simply supported ends, which rigorously satisfies the governing equations and specified boundary conditions.
76 citations
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TL;DR: In this article, a modulated Fourier expansion in time is used to show long-time nearconservation of the harmonic actions associated with spatial Fourier modes along the solutions of nonlinear wave equations with small initial data.
Abstract: A modulated Fourier expansion in time is used to show long-time near-conservation of the harmonic actions associated with spatial Fourier modes along the solutions of nonlinear wave equations with small initial data. The result implies the long-time near-preservation of the Sobolev-type norm that specifies the smallness condition on the initial data.
75 citations
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TL;DR: In this paper, a kinetic equation for the collisional evolution of stable, bound, self-gravitating and slowly relaxing systems is established, which is valid when the number of constituents is very large.
Abstract: A kinetic equation for the collisional evolution of stable, bound, self-gravitating and slowly relaxing systems is established, which is valid when the number of constituents is very large. It accounts for the detailed dynamics and self-consistent dressing by collective gravitational interaction of the colliding particles, for the system's inhomogeneity and for different constituents' masses. It describes the coupled evolution of collisionally interacting populations, such as stars in a thick disc and the molecular clouds off which they scatter.
The kinetic equation derives from the BBGKY hierarchy in the limit of weak, but non-vanishing, binary correlations, an approximation which is well justified for large stellar systems. The evolution of the 1-body distribution function is described in action–angle space. The collective response is calculated using a biorthogonal basis of pairs of density–potential functions.
The collision operators are expressed in terms of the collective response function allowed by the existing distribution functions at any given time and involve particles in resonant motion. These equations are shown to satisfy an H theorem. Because of the inhomogeneous character of the system, the relaxation causes the potential as well as the orbits of the particles to secularly evolve. The changing orbits also cause the angle Fourier coefficients of the basis potentials to change with time. We derive the set of equations which describes this coupled evolution of distribution functions, potential and basis Fourier coefficients for spherically symmetric systems. In the homogeneous limit, which sacrifices the description of the evolution of the spatial structure of the system but retains the effect of collective gravitational dressing, the kinetic equation reduces to a form similar to the Balescu–Lenard equation of plasma physics.
75 citations
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TL;DR: A new theory of MR imaging is described that utilizes prior information in the form of a set of “training” images thought to be similar to the “unknown” objects to be scanned that provides the basis for developing efficient scanning and image reconstruction techniques that are “tailored” to each body part or to particular disease states.
Abstract: We describe a new theory of MR imaging that utilizes prior information in the form of a set of "training" images thought to be similar to the "unknown" objects to be scanned. First, the training images are processed to find an orthonormal series representation of these images that is more convergent than the usual Fourier series. The coefficients in this new series can be calculated from a subset of the phase-encoded signals needed to construct the Fourier image representation. The characteristics of the training images also determine exactly which phase-encoded signals should be measured in order to minimize error in the image reconstruction. The optimal phase-encodings are usually scattered nonuniformly in kappa-space. Good results were obtained when this theory was applied to imaging data from simulated objects and to experimental data from phantom scans. This theory provides the basis for developing efficient scanning and image reconstruction techniques that are "tailored" to each body part or to particular disease states.
75 citations