Showing papers on "Linear approximation published in 1972"

Iterative Linear AC Power Flow Solution for Fast Approximate Outage Studies

Abstract: A fast approximate method is presented for solving the ac power flow problem for line and generator outages. The method is significantly more accurate than any linear approximation and significantly faster than the Newton-Raphson method for an approximate solution. The method has applications in system planning and operations where approximate ac power flow solutions are acceptable. The method is applicable to system planning for rapid location of design criteria violations and it is particularly well adapted for system operation use as an on-line security monitor. Efficiency is achieved through decoupling of real and reactive power equations, sparse matrix methods, an experimentally determined iteration scheme and the use of the matrix inversion lemma to simulate the effect of branch outages.

Birch's Law: Why Is It So Good?

Abstract: Birch9s law arises in the physics of solids as a linear approximation, in a certain range of density, of a power law. For a change of chemical composition within the same crystal structure, the velocity-density relation is constant with a slope of nearly -0.5 in the first-order approximation.

The Energy-Density Functional of an Electron Gas in Locally Linear Approximation of the One-Body Potential

Abstract: Abstract For a system of independent electrons moving in a common one-body potential V (r) an integral representation of Dirac's density matrix is evaluated in the approximation that V(r) at the point r is replaced by a linear potential with a gradient equal to the gradient of V at r. The particle density ᵨ, ∇ᵨ and the kinetic-energy density εk are derived from the density matrix. After eliminating the potential and its gradient a parametric representation for εk in terms of ᵨ and y = |∇ᵨ |½ ᵨ-⅔ is obtained. Explicit analytical expressions are given in the limits y → 0 and y → ∞ and compared with the inhomogeneity corrections of Kirzhnits and v. Weizsäcker.

Minimax Optimization of Networks by Grazer Search

Abstract: A new optimization method called grazer search has been developed. This method is suitable for nonlinear minimax optimization of network and system responses. A linear programming problem using gradient information of one or more highest ripples in the response error function to produce a downhill direction followed by a linear search to find a minimum in that direction is central to the algorithm. Unlike the razor search method due to Bandler and Macdonald, the present method overcomes the problem of discontinuous derivatives characteristic of minimax objectives without using random moves. It can fully exploit the advantages of the adjoint network method of evaluating partial derivatives of the response function with respect to the variable parameters. Sufficient details are given to enable the grazer search method to be readily programmed and used. Although the method is intended for the computer-aided solution of an extremely wide range of design problems, it is largely compared with other methods on microwave network design problems, for which the solutions are known. Its reliability and efficiency on more arbitrary problems, examples of which are also included, is thereby established.

A unified approach to uniform real approximation by polynomials with linear restrictions

Abstract: Problems concerning approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions have been studied by several authors. This paper is an attempt to provide a unified approach to these problems. In particular, the notion of restricted derivatives approximation is seen to fit into the theory and includes as special cases the notions of monotone approximation and restricted range approximation. Also bounded coefficients approximation, c-interpolator approximation, and polynomial approximation with interpolation fit into our scheme.

An a Priori Bounded Model for Transportation Problems with Stochastic Demand and Integer Solutions

Abstract: A linear approximation model is developed for transportation problems with stochastic demand where integer solutions are required. The technique reduces the stochastic integer programming problem to a deterministic linear approximating problem which can be solved as a capacitated transportation problem. Either the transportation algorithm or the primal-dual algorithm may be used thereby insuring integer solutions.

Approximation by Alternating Families on Subsets

Abstract: A method of determining the best approximation by an alternating family on an interval is by approximating on finite subsets of the interval. In this note we show that this method can fail to converge, particularly in the case of polynomial rational approximation and exponential approximation when the best approximation is degenerate.

On Identifying Transfer Functions and State Equations for Linear Systems

Abstract: Two methods are established for identifying constantcoefficient C2n-type noise-free linear systems if the time response data of the input-output or of all states are known. 2n response data are required to identify an nth-order transfer function or state equation for an unknown linear system. The order of the unknown system can be identified by checking a sequence of determinants. The Z transform and its inversion are mainly used.

Construction of basis functions in the finite element method

Abstract: In many applications of the finite element method, the explicit form of the basis functions is not known. A well-known exception is that of piecewise linear approximation over a triangulation of the plane, where the basis functions are pyramid functions. In the present paper, the basis functions are displayed in closed form for piecewise polynomial approximation of degreen over a triangulation of the plane. These basis functions are expressed simply in terms of the pyramid functions for linear approximation.

Double-beam interferometry method for investigating axisymmetric configurations of dense plasma

Abstract: The limits of applicability of double-beam interferometry in investigations of hightemperature plasma are considered. An analysis is made of the techniques for calculating interferograms of axisymmetric objects with steep density gradients by the Abel integral method. It is shown that the calculation methods used so far, based on the linear approximation of the required function, may lead to large errors. A parabolic approximation method is suggested and the relevant coefficients are calculated. The optimal number of inhomogeneity zones in an exponential distribution of the density is obtained. It is shown that the proposed method reduces considerably the volume of the computations and increases the precision of the calculations.

Describing Reactions with Analogous Processes

Abstract: A generalized extrapolation relation is developed for probability amplitudes of unknown analytic form having parametric differences only. This expression enables a reaction to be described empirically using data from related processes. As an instrument for phenomenology, this comparative approach requires little theoretical knowledge of the related systems and relegates complexities due to structure to the input data. A linear approximation to the basic equation is obtained when the related systems are similar. Simple illustrations are provided by elastic scattering applications.

Vector potential and quadratic action

Abstract: Einstein's linear Lagrangian is replaced by a Lagrangian which is quadratic in the curvature quantities (gauge invariance). The hypothesis is made that the basic metrical field is highly agitated (due to periodic boundary conditions) thus establishing a submicroscopic basic lattice structure of the space-time world which, however, is macroscopically isotropic. All consequences follow from these assumptions. The “free vector” of Einstein's theory (void of physical significance and used for the normalization of the reference system) is no longer free but of physical significance. It becomes subject to the wave equation and the Lorentz condition. It can thus be identified with the electromagnetic vector potential. Although the present investigation does not go beyond the linear approximation, the contours of auniverse of maxium rationality emerge.

Uniform Approximation of Vector-Valued Functions with a Constraint

Abstract: This paper deals with existence and characterization of best approximations to vector-valued functions. The approximations are themselves vector-valued functions with components taken from a linear space, but the constraint is imposed that certain of the approximation parameters should be identical for all components. 1. Introduction. The following situation often occurs in the analysis of ex- perimental data. A set of experimental curves, all of which should roughly fit the same theoretical formula, has been obtained, and one desires to determine certain parameters in the theoretical formula. The experimental conditions have been such that, although some of the parameters should be the same for all of the given curves, others vary with the particular experiment. One would like, therefore, to be able to find the set of theoretical curves which simultaneously best approximates the set of given curves, under the constraint that some of the theoretical parameters should take on the same value for all members of the approximating set. As a simple example, one might want to fit a set of experimental curves by a set of straight lines, all with the same slope but with varying axis-intercepts. In the next section, we give a precise formulation of this problem for approximation from linear families. Following this, we provide existence and characterization theo- rems and discuss some examples.

Minimax nonlinear approximation by approximation on subsets

Abstract: A possible algorithm for minimax approximation on an infinite set X consists in choosing a sequence of finite point sets {Xk} which fill out X and taking a limit of minimax approximations on Xk as k → ∞. Such a procedure is considered by Rice [4, pp. 12-15]. In the case of linear approximation such a procedure has been shown to converge [1, pp. 84-88]. It has been claimed by Watson [5] that the procedure works for approxition by nonlinear families.

Role of Three-Body Potentials in Disordered Alloys. II

Abstract: The formulation of the statistical mechanics of disordered ternary alloys with arbitrary com­ position and arbitrary range of two- and three-body potentials is given. The high temperature expansion is worked out explicitly for the two- and three-site static correlation functions up to the term linear in (kT)- 1• As in the corresponding case for binaries, within the linear approximation, the presence of irreducible three-body potentials causes the two-site correlation function results to depend upon an effective, composition-dependent, pair-wise interaction. However, the dependence of the three-site correlation function upon the three-body potentials cannot be represented, even within the linear approximation, by an appropriately renormalized, composition-dependent, pair-wise potential, but is rather an explicit function of the three-body potentials. Consequently, an experimental measurement of the triplet correlation would, under suitable conditions, provide direct information about the three-body potentials.

Qiiasilinearlzation and primal-dual-eelations of chebyshev-approximation i.

Abstract: The present paper is concerned with the problem of approximating a given element of a linear space by a family of elements, depending on a parameter, as well as possible. Normlike convex functionals are used as measures for the quality of approximation. By means of quasilinearization of the convex approximation measure the approximation problem is transformed into a maximin. or programming problem, which is sometimes dealt with much easier. From the maximim-formulation a dual problem, replacing the primal approximation problems, is derived with the aid of a maximin-theorem of Ky Fan. New resultats on linear Chebyshev approximation with restricted parameters are obtained in this manner.

Continuity and condition of the linear approximation problem I

Abstract: The dependence of a best approximation on the domain of definition has not been studied so far for problems involving approximations by non-Chebychev spaces. Two simple examples are given which show that the approximation problem may not be well-conditioned (not even continuous) in this case. The examples also suggest that by a slight modification in the formulation of the problem the inherent ill-conditioning can be eliminated. This leads to the consideration of best e-approximations.