Showing papers on "Linear approximation published in 1968"
TL;DR: A nonlinear programming problem will be solved by using a linear approximation, solving the linear programming problem, and then correcting the linear approximation through the use of differences between the linear and nonlinear results.
Abstract: In the planning and design of a high-voltage transmission network it is sometimes desirable to install controllable kilovar ar capacity at several locations to support bus voltages during emergencies. This problem arises when transmission line or generation outages cause bus voltage magnitudes to decrease below desirable limits. The problem of selecting where and how much kilovar capacity is required has many feasible solutions which satisfy the conditions imposed. A method for locating that solution with the minimum total installed capacity is presented. This nonlinear programming problem will be solved by using a linear approximation, solving the linear programming problem, and then correcting the linear approximation through the use of differences between the linear and nonlinear results. The method is illustrated by an application to a portion of a large high-voltage network.
40 citations
38 citations
TL;DR: In this article, a static drift-velocity field-strength characteristic is used to describe the dynamic behavior in the Gunn effect, and a reciprocity theorem between high field and low field domains is derived.
Abstract: Starting from a few basic assumptions, namely the continuity equation for the carrier current, the Poisson equation, and a static drift-velocity field-strength characteristic, systems of ordinary differential equations are derived, describing the dynamic behaviour in the Gunn-effect. Solutions are obtained and discussed in the following cases: (1) steady-state domains, allowing generalization to space-dependent doping density in some cases; (2) linear approximation, yielding a condition for sample stability; (3) time-dependent processes like domain formation and extinction. The last point especially provides some new understanding of dynamic phenomena because many results can be gained, without any numerical calculation, by geometric consideration using the characteristic, e.g. the relation between triangular and trapezoidal domains sometimes discussed, and the triggered operation are investigated in this way. Hence, a reciprocity theorem between high field and low field domains is derived, and a direct verification of the principle of minimum entropy production is obtained in some cases.
26 citations
23 citations
22 citations
TL;DR: In this article, the independent variable in a regression situation for a measured value of the dependent variable (inverse estimation) is discussed along with corresponding design problems under correct classification and misclassification of the model.
Abstract: Estimation of the independent variable in a regression situation for a measured value of the dependent variable (inverse estimation) is discussed along with the corresponding design problems under correct classification and misclassification of the model. Optimal designs are derived using a linear approximation for the particular cases when the true model is linear or quadratic. Figures are provided to enable the experimenter to locate optimal designs for various sets of the parameters. The design criterion considered in this paper is minimization of the integral over the range of x of E( – x)2 with respect to design.
22 citations
22 citations
20 citations
01 Aug 1968
TL;DR: In this article, a generalization of the direct method of Lyapunov is employed to determine the stability of neuristor waveforms, and the validity of the linear approximation for distributed systems is discussed.
Abstract: A generalization of the direct method of Lyapunov is employed to determine the stability of neuristor waveforms. The validity of the linear approximation for distributed systems is discussed. It is shown that a neuristor waveform on a RC ± G transmission line is stable if its derivative represents the minimum eigenvalue solution of the linearized perturbation equation.
11 citations
TL;DR: In this article, the authors compared I?(a:) to P(a;) for x > 2 and showed that I?[a:] appears to be a better fit than Hasting's approximation [4] for x> 2 and E(x) appeared to be superior to P[a;).
Abstract: _ xexpj-A/2) IV + 6a:4 14a:2 28] a; + 2 \\-x + ox — 20a; — 4 -i and the motivation for such a definition may be found in [2] and [3]. It is readily apparent that EOx) and F(x) have the following properties in common: 1. For x > 2, E(x) is real, positive and finite. 2. For x > 2, dE/dx is real, negative and finite. 3. Asa:-» oo,E(x) -» 0. 4. As x —* oo, x exp ix2/2) Eix) —» 1. 5. As x —> oo, dE/dx —* 0. The following table compares I?(a:) to P(a;). P(a;) is a better fit than Hasting's approximation [4] for x > 2 and E(x) appears to be superior to P(a;). The relative error, e^(a;) is the quantity selected as a basis for comparison and
10 citations
01 Jan 1968
TL;DR: In this article, the Taylor approximation of prescribed network characteristics in the neighborhood of an arbitrary frequency was studied, where the desired characteristic may represent either the phase, or the loss, or both.
Abstract: Design procedures are developed for the Taylor approximation of prescribed network characteristics in the neighborhood of an arbitrary frequency. The desired characteristic may represent either the phase, or the loss, or both.
TL;DR: In this paper, an angular diffusion coefficient expressing the rate at which the mean square deviation increases with trajectory length is evaluated from the results and shown to be given by an expression very similar to that obtained in the linear approximation, although they work in a strongly nonlinear regime.
Abstract: Calculations of the refractive scattering of radiation in a medium with randomly varying refractive index have usually been limited to the linear approximation where the scattering is weak. WORT (1966) has considered strong scattering in the non-linear regime but only for very restricted and physically unrealistic type of refractive index fluctuation. In this paper we describe a numerical experiment designed to remove this restriction and investigate a case with strong but realistic refractive index fluctuations. Examples of random surfaces are set up and ray trajectories computed through such surfaces. An angular diffusion coefficient expressing the rate at which the mean square deviation increases with trajectory length is evaluated from the results and shown to be given by an expression very similar to that obtained in the linear approximation, although we work in a strongly non-linear regime.
TL;DR: In this article, it was shown that linear momentum is carried by the waves cyclically in such a way that the center of rotation is a fixed point outside the axis of symmetry of the rod.
Abstract: Gravitational waves emitted from a spinning rod - taken as a rigid axially symmetric distribution of matter of uniform small cross section - are studied within the framework of general relativity, with special interest on transport of linear momentum from the rotating source. It is found that linear momentum is carried by the waves cyclically in such a way that the centre of rotation is a fixed point outside the axis of symmetry of the rod. Two results for the rate of momentum flux, differing in numerical content, are derived from the solution of the linear approximation, one by the use of Synge's argument involving the energy-momentum tensor, and the other by means of the pseudo-tensor. The discrepancy between the results is explained.
Patent•
28 May 1968
TL;DR: In this article, the structure for providing a multiple linear segment approximation of an arcuate segment and the method of such approximation for use in pattern tracer servocontrols or the like is disclosed.
Abstract: Structure for providing a multiple linear segment approximation of an arcuate segment and the method of such approximation for use in pattern tracer servocontrols or the like is disclosed.
TL;DR: In this paper, the flow of a nonviscous and nonheat-conducting gas through a channel of variable cross section under the influence of a transverse magnetic field is discussed.
Abstract: In this paper, we discuss the flow of a nonviscous and non-heat-conducting gas through a channel of variable cross section under the influence of a transverse magnetic field.
01 Aug 1968
TL;DR: In this paper, the approximation technique presented in previous letters is extended to linear-slope delay functions having positive slope, which is a special case of delay functions with positive slope.
Abstract: The approximation technique presented in previous letters is extended to linear-slope delay functions having positive slope.
01 Jun 1968
TL;DR: In this paper, an extension of the approximation technique, presented in previous letters, is presented, and the approximation polynomial is, again, of the form P n (s) = M(s) + kN(s).
Abstract: The approximation technique, presented in previous letters, is extended. The approximation polynomial is, again, of the form P n (s) = M(s) + kN(s). The method of optimization of constant k is considered, and it is shown that an equal percentage ripple approximation can be achieved by a very simple procedure. The technique is particularly useful when a time delay network consists of a number of sections in tandem.