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# Superposition principle

About: Superposition principle is a(n) research topic. Over the lifetime, 10750 publication(s) have been published within this topic receiving 175964 citation(s).

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01 Aug 1969Abstract: The output of an array of sansors is considered to be a homogeneous random field. In this case there is a spectral representation for this field, similar to that for stationary random processes, which consists of a superposition of traveling waves. The frequency-wavenumber power spectral density provides the mean-square value for the amplitudes of these waves and is of considerable importance in the analysis of propagating waves by means of an array of sensors. The conventional method of frequency-wavenumber power spectral density estimation uses a fixed-wavenumber window and its resolution is determined essentially by the beam pattern of the array of sensors. A high-resolution method of estimation is introduced which employs a wavenumber window whose shape changes and is a function of the wavenumber at which an estimate is obtained. It is shown that the wavenumber resolution of this method is considerably better than that of the conventional method. Application of these results is given to seismic data obtained from the large aperture seismic array located in eastern Montana. In addition, the application of the high-resolution method to other areas, such as radar, sonar, and radio astronomy, is indicated.

5,069 citations

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TL;DR: It is proved that if S is representable as a highly sparse superposition of atoms from this time-frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the l/sup 1/ norm of the coefficients among all decompositions.

Abstract: Suppose a discrete-time signal S(t), 0/spl les/t

2,096 citations

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01 Jan 1968

Abstract: In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A striking instance of such a procedure discovery by Gardner, Miura and Kruskal that the eigenvalues of the Schrodinger operator are integrals of the Korteweg-de Vries equation.
In Section 2 we prove the simplest case of a conjecture of Kruskal and Zabusky concerning the existence of double wave solutions of the Korteweg-de Vries equation, i.e., of solutions which for |I| large behave as the superposition of two solitary waves travelling at different speeds. The main tool used is the first of remarkable series of integrals discovered by Kruskal and Zabusky.

1,933 citations

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01 Jan 1975

Abstract: Part 1 Single-degree-of-freedom systems: overview of structural dynamics analysis of free vibrations response to harmonic loading response to periodic loading response to impulse loading responses to general dynamic loading - step by step methods, superposition methods generalized single degree-of-freedom systems. Part 2 Multi-degree-of-freedom systems: formulation of the MDOF equations of motion evaluation of structural-property matrices undamped free vibrations analysis of dynamic response using superposition vibration analysis by matrix iteration selection of dynamic degrees of freedom analysis of MDOF dynamic response - step by step methods variational formulation of the equations of motion. Part 3 Distributed parameter systems: partial differential equations of motion analysis of undamped free vibrations analysis if dynamic response. Part 4 Random vibrations: probability theory random processes stochastic response of linear SDOF systems stochastic response of non-linear MDOF systems. Part 5 Earthquake engineering: seismological background free-field surface ground motions deterministic structural response - including soil-structure interaction stochastic structural response.

1,627 citations

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Abstract: In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.

1,347 citations