Showing papers on "Linear approximation published in 1969"
22 citations
21 citations
TL;DR: In this paper, a generalization of the projection methods developed in the previous paper is used to obtain kinetic equations for systems in strong fields, with arbitrary time dependence, and it is demonstrated that a linear approximation for small (but time dependent) outside fields can not be expected to be valid for arbitrary long times.
15 citations
TL;DR: A method for reducing the complexity of the class of linear, time-varying, dynamic control systems is developed, where some results from the theory of widths of sets in Banach space are used to obtain bounds on the optimal approximation error.
Abstract: A method for reducing the complexity of the class of linear, time-varying, dynamic control systems is developed where the approach taken is that of uniform approximation (that is, modeling for a region of initial conditions). The objective of the modeling procedure is to choose a linear time-invariant system of given dimension, that minimizes a “worst-case” type of error criterion. Some results from the theory of widths of sets in Banach space are used to obtain bounds on the optimal approximation error as a function of the dimension of the approximating system. The use of these bounds in choosing the order of the approximation is discussed. An example illustrates the use of the derived results.
12 citations
7 citations
TL;DR: In this article, a systematic approach has been made to theoretical estimation of the basic parameters characterizing the different types of spectrum stabilizers, as also of their control performance, and the mean square errors due to random disturbance and statistical fluctuations in the counting rates have been treated analytically with use made of linear approximation of the transfer functions.
Abstract: A systematic approach has been made to theoretical estimation of the basic parameters characterizing the different types of spectrum stabilizers, as also of their control performance. The transient responses to regular disturbances, and the mean square errors due to random disturbance and statistical fluctuations in the counting rates have been treated analytically with use made of linear approximation of the transfer functions. The results obtained are valid only in the limit of small gain shifts in comparison with the half width of the reference peak, but nevertheless they still provide a measure for predicting the requirements for optimum control as well as an indication of the limitations to their range of applicability.
4 citations
2 citations
TL;DR: The linear approximation of Einstein's equations for weak gravitation fields together with the linear approximation for MOLLER's condition for the frames of reference can be interpreted as a linear approximation to the field equations in a tetrad theory of gravitation.
Abstract: The linear approximation of EINSTEIN's equations for weak gravitation fields together with the linear approximation of MOLLER's condition for the frames of reference can be interpreted as a linear approximation of the field equations in a tetrad theory of gravitation, too.
2 citations
01 Sep 1969
TL;DR: This article shows the tracking and acquisition behavior of the loop for different initial phase offsets and for different ratios of initial frequency offset D to loop gain B and compares the transient and steady-state responses with the exact solution and the linear approximation solution.
Abstract: The purpose of this article is to solve the nonlinear differential equation of injection locking. Using the method of Riccati’s equation, an exact solution has been obtained which is much simpler and more explicit than that shown by Mackey. This article shows the tracking and acquisition behavior of the loop for different initial phase offsets and for different ratios of initial frequency offset D to loop gain B. It also compares the transient and steady-state responses with the exact solution and the linear approximation solution.
1 citations
TL;DR: In this paper, the effect of non-linear terms on the interaction of spin-waves with conduction electrons has been studied and it was shown that for small spin-wave amplitudes, the nonlinear terms introduce only small corrections to the linear approximation.
TL;DR: The use of operational calculus allows exact expressions for the resonance integral and the intermediate resonance parameters to be obtained as discussed by the authors, and these expressions are obtained for the more general case of an heterogeneous system of one resonance absorber with N admixed moderators and M moderators.