Showing papers on "Linear approximation published in 1971"
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01 Jan 1971
TL;DR: In this paper, the Hahn-Banach theorem and dual extremal problems are applied to the problem of extremal problem in homogeneous Banach spaces, and the interpolation formula and Gaussian quadrature are used.
Abstract: Best uniform approximation.- The interpolation formula and Gaussian quadrature.- Best approximation and extremal problems in other norms.- Applications of the Hahn-Banach theorem and dual extremal problems.- Approximation theory and extremal problems in Hilbert spaces.- Minimal extrapolation of Fourier transforms.- General aspects of "Degree of approximation".- Approximation theory in homogeneous Banach spaces.
224 citations
TL;DR: In this article, matched asymptotic expansions were used for radiating systems in the near zone and wave zone. But their results do not depend upon any definition of gravitational field energy.
Abstract: This paper treats the slow‐motion approximation for radiating systems as a problem in singular perturbations. By using the method of matched asymptotic expansions, we can construct approximations valid both in the near zone and the wave zone. The outgoing‐wave boundary condition applied to the wave‐zone expansion leads, by matching, to a unique and easily calculable radiation resistance in the near zone. The method is developed and illustrated with model problems from mechanics and electromagnetism; these should form a useful and accessible introduction to the method of matched asymptotic expansions. The method is then applied to the general relativistic problem of gravitational radiation from gravitationally bound systems, where a significant part of the radiation can be attributed to nonlinear terms in the expansion of the metric. This analysis shows that the formulas derived from the standard linear approximation remain valid for gravitationally bound systems. In particular, it shows that, according to general relativity, bodies in free‐fall motion do indeed radiate. These results do not depend upon any definition of gravitational field energy.
153 citations
70 citations
31 citations
TL;DR: Chebyshev approximation by nonlinear families on a general compact space is studied and the existence of a minimal set on which a locally best approximation is locally best is shown.
Abstract: Chebyshev approximation by nonlinear families on a general compact space is studied. Attention is restricted to approximants satisfying a local Haar condition. A necessary and sufficient condition for the approximant to be locally best is given. A linear approximation problem is given which is equivalent to the nonlinear problem of locally best approximation. The existence of a minimal set on which a locally best approximation is locally best is shown. An alternation result is given for approximation on an interval.
17 citations
TL;DR: In this article, a modification of the Remes-algorithm is given which is always practicable and the convergence of which is proved, too, from a geometrical interpretation.
Abstract: In order to solve the linear approximation problem the Remes-algorithm is quite suitable in case of a Haar subspace. Without this assumption one must modify the algorithm even to ensure the practicability (cf. [8]). In this paper, a modification, in some respect similar to that in [8], will be given which is always practicable and the convergence of which is proved, too. The relation to the Remes-algorithm will be evident from a geometrical interpretation.
7 citations
TL;DR: A simple theory based on linear approximation of the transient waveform and on the superposition theorem permits the prediction of waveforms and amplitude of the ripple voltage at the receiving end of a line which transmits pulses with finite rise time and which is mismatched at both ends.
Abstract: A simple theory based on linear approximation of the transient waveform and on the superposition theorem permits the prediction of waveforms and amplitude of the ripple voltage at the receiving end of a line which transmits pulses with finite rise time and which is mismatched at both ends. The influence of every connection in a digital system can be determined in this way.
7 citations
TL;DR: In this article, a new approach of defining the electrical parameters of a dc machine is shown, which provides dynamic resistance and inductance characteristics which are shown to be very much off the usual values taken from a linear approximation.
Abstract: Since in many applications of dc machines an accurate knowledge of the dynamic mathematical model is necessary, it is desirable to have simple measurement methods to obtain the different parameters required. A new approach of defining the electrical parameters of a dc machine is shown. The measurement techniques provide dynamic resistance and inductance characteristics which are shown to be very much off the usual values taken from a linear approximation. Resistance values are measured by modified voltage drop methods, and inductance values are measured by load current variations. In case of transient behavior, like sudden short circuits or sudden load variations, the linear model cannot be applied, and the dynamic resistance and characteristics used in the nonlinear differential equations describing the system provide a good model within experimental error. The results show in particular how the armature inductance is dependent on the load as well as on the speed of rotation.
7 citations
01 Jan 1971
5 citations
TL;DR: In this paper, it was shown that the outgoing multipole electromagnetic waves of the linear approximation to the Einstein-Maxwell equations produce wave tails in the second approximation which, after the end of the source vibration, constitute incoming multipole waves.
Abstract: With a finite oscillating linear coherent distribution of electric charge chosen as the source of electromagnetic radiation, it is shown that the outgoing multipole electromagnetic waves of the linear approximation to the Einstein-Maxwell equations produce wave tails in the second approximation which, after the end of the source vibration, constitute incoming multipole waves.
TL;DR: In this article, it is shown that sensitivity functions are discontinuous at the discontinuous points and relations between elements of sensitivity functions at these points are linear for a given nominal solution of the system.
Abstract: : The well known method of obtaining sensitivity functions is usually restricted by the condition of continuity imposed on the functions of the coordinates in system equations. For the case of discontinuous functions, more elaborate procedures are required to give a good linear approximation. It is shown that sensitivity functions are discontinuous at the discontinuous points. Also, relations between elements of sensitivity functions at these points are shown to be linear for a given nominal solution of the system. The switching time of the desired Bang-Bang control can be estimated if variations in initial conditions are known. The changes in terminal states and cost function due to deviations in initial conditions can be determined, to first order, by the use of sensitivity functions. Bounds on the deviations in initial conditions can be found by a worst case approach so that the desired terminal conditions are satisfied within given tolerances. The fundamental importance of these techniques in a number of areas of application, for instance, guidance and control of aerospace vehicles, is well known. (Author)
TL;DR: A computer program for determining the exact equiripple approximation of a linear phase characteristic is described and a comparison with the Chebyshev type of approximation of an constant-delay characteristic is given.
Abstract: A computer program for determining the exact equiripple approximation of a linear phase characteristic is described. The program is based on a recently introduced method for the approximation of a linear phase characteristic in which only linear equations are involved. A comparison with the Chebyshev type of approximation of a constant-delay characteristic is also given.
01 Jan 1971
TL;DR: In this article, a nongeodetic particle represented by a time-like world line in the linearized field theory was developed for a non-geodesic particle in Riemannian space.
Abstract: Attempt to develop a gauge-invariant theory of the motion of singularities in an n-dimensional Riemannian space. A gauge-invariant theory is developed for a nongeodetic particle represented by a time-like world line in the linearized field theory. The solution of the basic algebraic and differential equations is to be constructed from the source line, the future null cones emanating from it, a scale factor, and nothing else, and the leading term of the solution is the Schwarzschild field. A class of possible solutions is examined, and it is shown that this class contains just one acceptable member - namely, a particle in constant acceleration.
01 Nov 1971
TL;DR: In this article, a method for finding all-pole transfer functions that approximate a linear phase characteristic in such a way as to make the phase error equal to zero at the maximum possible number of points is presented.
Abstract: A method is presented for finding all-pole transfer functions that approximate a linear phase characteristic in such a way as to make the phase error equal to zero at the maximum possible number of points. The solution is obtained in the closed form.
TL;DR: In this article, the problem of optimal profiling of the contours of plane and axisymmetric bodies in supersonic nonequilibrium flow without the formation of a shock wave is reduced to the boundary value problem for a hyperbolic system of equations, which includes the flow equations and the equations for the Lagrange multipliers.
Abstract: In [1] the problem of optimal profiling of the contours of plane and axisymmetric bodies in supersonic nonequilibrium flow without the formation of a shock wave (these bodies include, in particular, the contours of base sections and nozzles) is reduced to the boundary value problem for a hyperbolic system of equations, which includes the flow equations and the equations for the Lagrange multipliers (there is an error in Eq. (4.5) of [1]; there should be a minus sign in front of the third term in the braces). In view of the solution complexity, in [2] the construction of the optimum nozzle contour is based on the one-dimensional approximation. Although this approach does permit establishing the order of the possible gain, the conclusions concerning the contour shape which result from this approach are basically qualitative. In the following the construction of thin plane bodies of minimal wave drag in a nonequilibrium supersonic flow is carried out in the linear approximation, which leads to a more complete picture of the form of the optimum contours. Numerous examples of the use of linear theory for optimizing body shape in supersonic perfect gas flow are given in [3].
TL;DR: In this article, the authors investigated the effect of nonlinearity on the nature of Cˇerenkov radiation of particles, and showed that under certain conditions non-linearity of medium initiates radiation in media, not subjected to the radiation in linear approximation.
TL;DR: Results suggest that the nonlinear properties play a leading part in memory and association in the neuronal networks.
TL;DR: In this paper, a linear generalized polynomial in a system of independent interatomic functions is developed to extract the periodic vector set from the Patterson function, which is a relatively simple problem in linear approximation.
Abstract: Crystal-structure analysis via the Patterson function may be considered as consisting of two distinct steps. In the first step, the weighted periodic vector set is determined by establishing the location of each peak in the Patterson function. In the second step, the weighted periodic vector set is analysed to determine the crystal structure. The second step apparently offers little difficulty, since existing procedures for the analysis of periodic vector sets appear to be capable of dealing with complex structures, provided of course that the vector set is accurately determined. Unfortunately, a general and powerful method for the location of peaks in the Patterson function has not yet been developed and therefore it is the first step in the solution process which now prevents the formulation of a general method of structure analysis via the Patterson function. Such a method would be extremely useful, since the Patterson function is not restricted to centrosymmetric structures. In the present paper a way of representing the Patterson function as a linear generalized polynomial in a system of independent interatomic functions is developed. The coefficients of this polynomial determine the weighted periodic vector set. This approach, therefore, reduces the problem of extracting the periodic vector set from the Patterson function to a relatively simple problem in linear approximation, namely the determination of the coefficients of a generalized polynomial.
TL;DR: In this article, a non-linear system is examined with the help of the asymptotic method and it is shown that phase and amplitude variations with time, in a rather broad range of physical and technical problems, might be found explicitly, and that the solutions obtained reduce smoothly to those of the linear approximation when the absolute difference of the system eigenfrequency and the external perturbation becomes large.
Abstract: Oscillations externally excited in a conservative non-linear system are examined with the help of the asymptotic method. It is shown that phase and amplitude variations with time, in a rather broad range of physical and technical problems, might be found explicitly, and that the solutions obtained reduce smoothly to those of the linear approximation, when the absolute difference of the system eigenfrequency and the external perturbation becomes large.
TL;DR: In this article, a linear approximation to the Fermi-Dirac distribution function is proposed which might be of interest in practical applications, and three different ways of choosing its parameters are suggested.
Abstract: A linear approximation to the Fermi-Dirac distribution function is proposed which might be of interest in practical applications. Three different ways of choosing its parameters are suggested. In order to assess the goodness of the different choices, the specific heat curves for a closed parabolic band are calculated within this approximation and compared with the numerically computed, exact specific heat.