Showing papers on "Linear approximation published in 1989"
TL;DR: In this article, the propagation of light rays in a clumpy universe constructed by cosmological version of the post-Newtonian approximation was investigated and it was shown that linear approximation to the propagation equations is valid in the region where zeta is approximately less than 1 even if the density contrast is much larger than unity.
Abstract: The propagation of light rays in a clumpy universe constructed by cosmological version of the post-Newtonian approximation was investigated. It is shown that linear approximation to the propagation equations is valid in the region where zeta is approximately less than 1 even if the density contrast is much larger than unity. Based on a gerneral order-of-magnitude statistical consideration, it is argued that the linear approximation is still valid where zeta is approximately greater than 1. A general formula for the distance-redshift relation in a clumpy universe is given. An explicit expression is derived for a simplified situation in which the effect of the gravitational potential of inhomogeneities dominates. In the light of the derived relation, the validity of the Dyer-Roeder distance is discussed. Also, statistical properties of light rays are investigated for a simple model of an inhomogeneous universe. The result of this example supports the validity of the linear approximation.
83 citations
TL;DR: In this paper, a linearized non-rigid MHD consistent displacement model is used to study the vertical stability of a plasma in an air core or iron core tokamak.
Abstract: A linearized non-rigid MHD consistent displacement model is shown to be adequate to the study of the vertical stability of a plasma in an air core or iron core tokamak. This simple but accurate approach, which can be extended to the study of shape and radial position control, can be used in determining the limits of validity of the linear approximation and simulating the behaviour of a detection system based on a set of flux measurements. The method provides a basis for the optimization of position measurements and feedback parameters. The features of the model are illustrated, with reference to the study of the vertical stability of a NET configuration, showing the significance of the choice of the input signal.
71 citations
TL;DR: In this article, the authors derived the error committed in representing the satellite ground track by a bias, tilt, and curvature, and showed that the fit is better with more parameters and as the length of the segment increases, the approximation gets worse.
Abstract: From simulations, the orbit error can be assumed to be a slowly varying sine wave with a predominant wavelength comparable to the Earth's circumference. Thus, one can derive analytically the error committed in representing the orbit error along a segment of the satellite ground track by a bias; by a bias and tilt (linear approximation); or by a bias, tilt, and curvature (quadratic approximation). The result clearly agrees with what is obvious intuitively, i.e., (1) the fit is better with more parameters, and (2) as the length of the segment increases, the approximation gets worse. But more importantly, it provides a quantitative basis to evaluate the accuracy of past results and, in the future, to select the best approximation according to the required precision and the efficiency of various approximations.
53 citations
TL;DR: A three-parameter F approximation to the distribution of a positive linear combination of central chi-squared variables is described in this paper, which is about as easy to implement as the Satterthwaite-Welsh and Hall-Buckley-Eagleson approximations.
Abstract: A three-parameter F approximation to the distribution of a positive linear combination of central chi-squared variables is described. It is about as easy to implement as the Satterthwaite-Welsh and Hall-Buckley-Eagleson approximations. Some reassuring properties of the F approximation are derived, and numerical results are presented. The numerical results indicate that the new approximation is superior to the Satterthwaite approximation and, for some purposes, better than the Hall-Buckley-Eagleson approximation. It is not quite as good as the Gamma-Weibull approximation due to Solomon and Stephens, but is easier to implement because iterative methods are not required.
51 citations
13 Dec 1989
TL;DR: In this article, an approach for the approximate input-output linearization of nonlinear systems, particularly those for which relative degree is not well defined, is presented, and the authors demonstrate some properties of the accuracy of the approximation and, in the context of the ball and beam example, show it to be far superior to the Jacobian approximation.
Abstract: The authors present an approach for the approximate input-output linearization of nonlinear systems, particularly those for which relative degree is not well defined. They show that there is a great deal of freedom in the selection of an approximation and that, by designing a tracking controller based on the approximating system, tracking of reasonable trajectories can be achieved with small error. The approximating system is itself a nonlinear system, with the difference that it is input-output linearizable by state feedback. The authors demonstrate some properties of the accuracy of the approximation and, in the context of the ball and beam example, show it to be far superior to the Jacobian approximation. The results are focused on finding regular SISO systems which are close to systems which are not regular and controlling these approximate regular systems. >
44 citations
TL;DR: In this article, the Langer theory of spinodal decomposition in binary solutions is extended to the case of ternary alloys, and a system of diffusion equations in the Cahn-Hilliard approximation and equations of motion for the three independent partial structure functions are derived.
34 citations
TL;DR: The present algorithm consistently has the shortest execution time of the program, and it tends most to require as few critical points as the optimum algorithm that was tested.
Abstract: A new algorithm is presented for the piecewise linear approximation of two-dimensional digitized curves against a square grid. The algorithm utilizes an adaptive reduction procedure in two approximation phases to select the critical points of a digitized curve such that the deviation, from the digitized curve to its final approximated curve, is bounded by a uniform error tolerance. The time complexity of this algorithm is O(m/sup 2/) rather than O(n/sup 2/) on the theoretical plane. In the experiments of fixing the initial and the final processing points, the performance of the algorithm has been compared to those of three prominent other algorithms regarding the required number of critical points and the total execution time of the program. Of the four algorithms compared, the present algorithm consistently has the shortest execution time of the program, and it tends most to require as few critical points as the optimum algorithm that was tested. >
32 citations
06 Sep 1989
TL;DR: The architecture of an integrated processor that is capable of performing addition and subtraction of 30-b numbers with 20 fractional bits in the logarithmic number system is described and compression is chosen to minimize ROM size and obtains a further reduction of 55%.
Abstract: A description is given of the architecture of an integrated processor that is capable of performing addition and subtraction of 30-b numbers with 20 fractional bits in the logarithmic number system. Previous techniques would require 70 Mb of ROM to implement this processor, which is not feasible for a single-chip implementation. The techniques presented here use a factor of 275 less memory. The key to this is the use of a linear approximation of the nonlinear functions stored in the lookup tables. The functions involved are highly nonlinear in some regions, so variable size regions are used for the approximation. The use of linear approximation alone would still require over 565 kb of ROM. Further compression is obtained by using linear approximation with differential coding of each table. The compression is chosen to minimize ROM size and obtains a further reduction of 55%. A total of 260 kb of ROM is required to implement the processor. >
29 citations
TL;DR: In this paper, the authors proposed a new technique for solving a constrained non-linear inversion, which allows to use a priori information not only in terms of soft bounds, but also hard bounds as well (usually giving more stable and accurate solutions).
Abstract: SUMMARY
Seismic data are non-linearly related to model parameters such as seismic velocities However, seismic inversion is usually considered in a linear approximation Such techniques as the Born inversion were recently applied to seismic data
Non-linear inversion is more complicated and involves extensive calculations Non-linear inversion was developed in the frame work of an unconstrained optimization procedure It uses as a priori information an initial model and probability distribution functions in the data and model spaces (This a priori information is called ‘soft’ bounds)
In this paper, we propose a new technique for solving a constrained non-linear inversion This technique will allow us to use a priori information not only in terms of ‘soft’ bounds, but ‘hard’ bounds as well (usually giving more stable and accurate solutions)
Non-linear inversion is considered as an iterative procedure which involves a dual transform at each iteration A dual transform allows for considering the problem in terms of the Lagrangian multipliers The number of Lagrangian multipliers is equal to the number of available data and thus, significantly reduces the dimension of the problem (this is true for underdetermined problems only) However, the most important property of the dual transform is that it allows us to consider a constrained problem as an unconstrained problem
Another important property is that proper constraints incorporate small-wave numbers in the generalized inversion It is shown that conventional (unconstrained non-linear inversion) is a special case of the constrained non-linear inversion developed in this paper if the truncation operator is represented by the identity matrix
25 citations
TL;DR: In this article, the plane wave propagation in nonlinear micropolar solids is asymptotically investigated, and it is shown that wave motion is governed by coupled Modified Korteweg-de Vries equations.
25 citations
TL;DR: Based on computational experience with a series of moderate and large-scale sector and risk models, it is concluded that direct nonlinear solution is more efficient than using linear approximations for sector andrisk models having objective function nonlinearities.
Abstract: Linear approximation techniques have often been applied to nonlinear mathematical programming models for computational efficiency reasons. Price-endogenous agricultural sector models and risk models have found numerous applications. This article addresses the issue of approximation efficiency. Based on computational experience with a series of moderate and large-scale sector and risk models, it is concluded that direct nonlinear solution is more efficient than using linear approximations for sector and risk models having objective function nonlinearities. On the other hand, the experimental results indicate that approximation should continue in case of models with nonlinearities in their constraints.
TL;DR: A primal approach is presented along with evidence that for random problems primal methods are to be preferred, and it is shown that the method generally considered to be the best algorithm for discrete linear approximation in the linear norm actually owes much of its efficiency to effective choice of a starting point.
Abstract: It is generally believed that methods based upon solving the dual of the linear programming formulation of the linear $l_\infty $ problem are superior to methods based upon the primal formulation.A primal approach is presented along with evidence that for random problems primal methods are to be preferred. It is then shown that the method generally considered to be the best algorithm for discrete linear approximation in the $l_\infty $-norm actually owes much of its efficiency to effective choice of a starting point. This phenomenon is explained with the aid of results from classical minimax theory, which are then used to determine a suitable initial point for a primal method.With this enhancement it would appear that the primal approach is also preferable for problems arising from function approximations.
TL;DR: In this article, a linear-time algorithm for approximating a set of n points by a linear function, or a line, that minimizes the L 1 norm is presented, which is optimal within a constant factor and enables to use linearL 1 approximation of many points in practice.
Abstract: In this paper we present a linear-time algorithm for approximating a set ofn points by a linear function, or a line, that minimizes theL1 norm. The algorithmic complexity of this problem appears not to have been investigated, although anO(n3) naive algorithm can be easily obtained based on some simple characteristics of an optimumL1 solution. Our linear-time algorithm is optimal within a constant factor and enables us to use linearL1 approximation of many points in practice. The complexity ofL1 linear approximation of a piecewise linear function is also touched upon.
08 May 1989
TL;DR: In this article, it was shown that singular value decomposition (SVD) can be applied along with 1D finite-impulse-response (FIR) techniques to design linear-phase 2D filters with arbitrary prescribed amplitude responses that are symmetrical with respect to the origin of the ( omega /sub 1/ omega/sub 2/) plane.
Abstract: It is shown that singular-value decomposition (SVD) can be applied along with 1-D finite-impulse-response (FIR) techniques to design linear-phase 2-D filters with arbitrary prescribed amplitude responses that are symmetrical with respect to the origin of the ( omega /sub 1/, omega /sub 2/) plane. the balanced approximation method is applied to 2-D linear-phase FIR filters of the type obtained by the SVD method. The method leads to economical and computationally efficient filters, usually IIR filters, which have prescribed amplitude responses and whose phase responses are approximately linear. >
TL;DR: In this article, a computer simulation based on the Monte Carlo and the surface charge methods is carried out, and a wave form of backscattered electrons (BEs), which hardly suffer from the charging phenomenon, is precisely analyzed using this simulation.
Abstract: For studying the technology of the critical dimension measurement using an electron beam (EB), a computer simulation based on the Monte Carlo and the surface charge methods is carried out, and a wave form of backscattered electrons (BE’s), which hardly suffer from the charging phenomenon, is precisely analyzed using this simulation. First, line widths obtained individually by threshold, maximum slope, and linear approximation methods are examined about various incident‐EB conditions such as an accelerating voltage and a beam diameter. The dependencies of pattern structures, such as dimension, side‐wall angle, and materials of the patterns, on measured line width are evaluated next. As a result of these calculations, it is found that the variation of the incident‐EB conditions doesn’t affect practically the results of the line‐width measurement in BE‐detection systems, while secondary‐electron signals are easily deformed by the charging phenomenon. Although the measured width much depends on the pattern structure, the linear approximation method is comparatively suitable for the purpose of obtaining the width of bottom edges in BE‐detection systems.
TL;DR: In this article, an individual machine energy function is developed which is based on the classical generator model and synchronous reference frame and is shown to be path independent once a linear approximation for the magnetic term is incorporated in the energy function.
Abstract: This investigation introduces two new tools for the determination of power system transient stability. First, an individual machine energy function is developed which is based on the classical generator model and synchronous reference frame. The synchronous reference individual machine energy function is shown to be path independent once a linear approximation for the magnetic term is incorporated in the energy function. Second, a description of the region of stability is determined based on the individual machine energy function. This region of stability unifies the Potential Energy Boundary Surface (PEBS) individual machine energy function methods [1, 10, 18] and the critical cutset methods [11, 13] of describing the region of stability.
01 Jan 1989
TL;DR: In this article, the limits of inertialess Stokes flow existence are defined; the general asymptotic relations typical for inertialness flows with high Hartmann number are obtained.
Abstract: The experimental results on the inertialess MHD-flows in a strong magnetic field are presented. Complex shaped duct flows with regular roughless walls and free-surface MHD-flows are investigated. The limits of inertialess Stokes flow existence are defined; the general asymptotic relations typical for inertialess flows with high Hartmann number are obtained.
TL;DR: In this article, the effect of thermal conduction on the temporal and spatial behavior of disturbances on a radiating fluid in thermodynamical equilibrium is studied by linear approximation in the quasi-static limit, making use of the Eddington approximation.
Abstract: The effect of thermal conduction on the temporal and spatial behavior of disturbances on a radiating fluid in thermodynamical equilibrium is studied by linear approximation in the quasi-static limit, making use of the Eddington approximation. The damping rate and frequency for the thermal and acoustic modes which result are calculated as functions of the dimensionless wavenumber and two dimensionless parameters: the conductive parameter and the Boltzmann number. Analytical expressions are obtained in the extreme cases of a nonradiating medium and a nonconductive medium. Otherwise they are calculated numerically. In the spatial analysis for the three resulting wave modes (sound, radiation diffusion, and thermal conducting wave), the phase velocity and the characteristic scale for damping are obtained as functions of the optical thickness of the fluctuation, the Boltzmann number, and the ratio of conductive to enthalpy flux. 15 references.
TL;DR: In this article, a class of linear operators generated by certain power series is studied for uniform approximation of functions of a real or a complex variable by a set of linear functions generated by a power series.
Patent•
14 Apr 1989
TL;DR: In this article, the authors propose to show an approximate straight line when a sample point set out of a position range is inputted by using a means which calculates previously the position range where conditions are satisfied for linear components even with addition of one point following a partial string of a sampling point series of characters.
Abstract: PURPOSE:To show an approximate straight line when a sample point set out of a position range is inputted by using a means which calculates previously the position range where conditions are satisfied for linear components even with addition of one point following a partial string of a sampling point series of characters, etc CONSTITUTION:Coordinate values given from a coordinate input device are successively inputted to an input terminal 100 A vector forming means 104 outputs a displacement vector ranging from a start point to an end point via delay circuits 101 and 103 While the start and end points are also inputted to a linearizing range calculating means 106 for definition of a key line for linear approximation Then a direction range is calculated for decision whether the following input point is equal to a position serving as the same linear component or not This decision is carried out by an out-range deciding means 105 If so, the calculated range is added the end of the string of values held by an FIFO 108 A switch 107 outputs an O-input direction range in an in-range state and then an I-input direction range in an out-range state respectively
TL;DR: The local constant of strong uniqueness for nonlinear approximation with respect to a norm is the local constant for approximation in the associated problem of linear approximation by the tangent space as discussed by the authors.
Abstract: The local constant of strong uniqueness for nonlinear approximation with respect to a norm is the local constant of strong uniqueness for approximation in the associated problem of linear approximation by the tangent space.
Patent•
06 Nov 1989
TL;DR: In this paper, the feature point at the time of approximating the shape of the graphic with the straight line is executed by the maximum distance between the virtual straight line drawn between both edges of the graphics to be approximated, the number of crosses of the virtual line and the graphic or the area prepared by the virtual linear line and graphic.
Abstract: PURPOSE:To obtain a linear approximation result to be visually more favorable by executing the extraction of a feature point in case of linear-approximating a graphic with a maximum distance between a virtual straight line drawn between both edges of the graphic to be approximated and the graphic and the number of crosses of the virtual straight line and graphic or an area. CONSTITUTION:The extraction of the feature point at the time of approximating the shape of the graphic with the straight line is executed by the maximum distance between the virtual straight line drawn between both edges of the graphic to be approximated, the number of crosses of the virtual straight line and graphic or the area prepared by the virtual straight line and graphic. Thus, the evaluation of the approximation is executed in consideration of the dispersion of graphic data to a displacement and an approximate straight line, a conspicuous point can be correctly approximated, an inconspicuous point can be roughly approximated, and a problem that the change of the approximation result cannot be dealt with even when it is necessary in case of the dispersions of the graphic data to the approximate straight line even in case of the graphics whose shapes are similar are different can be solved.
TL;DR: In this article, an inner product and norm are used to numerically approximate the linear operator equation by use of generalized Fourier series, and the resulting approximation is the "best" approximation in that a least squares error is minimized simultaneously for both fitting the problem's boundary conditions and satisfying the linear operation relationship (the governing equations).
Patent•
19 Dec 1989
TL;DR: In this article, a non-linear analysis of a material is presented to shorten CPU processing time and obtain the analysis accuracy of the same degree as a linear analysis by converting a stress-strain curve to a linear approximate expression consisting of plural lines, utilizing this data and a linear solver.
Abstract: PURPOSE:To shorten CPU processing time and to obtain the analysis accuracy of the same degree as a linear analysis by converting a stress-strain curve to a linear approximate expression consisting of plural lines, utilizing this data and a linear solver and executing a non-linear analysis of a material. CONSTITUTION:Plural points a1 to a6 on a stress-strain curve S-E derived by a test are set as nodal points, connected by straight lines 21a to 21f and a linear approximation is executed. By subtracting from a load and a displacement quantity, a new load or displacement quantity is obtained, and the foregoing and characteristic data obtained from an approximate straight line of the next section become input data. A linear analyzing part 4 executes an analysis from a Young's modulus derived from an approximate straight line 21a and the load or the displacement, and when the maximum stress value is larger than the stress value a1, the load or the displacement corresponding to a1 is derived, and the analysis is executed until the value becomes smaller than a1. Subsequently, the load or the displacement derived by the first analysis is subtracted from analysis input data, and by giving a Young's modulus derived from an approximate straight line 21b, the analysis is executed. Thereafter, when the analysis is executed in the same way, stress values of each nodal point are all added and become analysis output data.
Patent•
29 Nov 1989
TL;DR: In this paper, the authors propose to set a parameter in case a line pattern is evaluated by its curvature and displacement for execution of the linear approximation by changing the displacement error value in response to the linear segment length and deciding the presence or absence of a feature point based on the result of comparison between said displacement and error value.
Abstract: PURPOSE:To freely set a parameter in case a line pattern is evaluated by its curvature and displacement for execution of the linear approximation by changing the displacement error value in response to the linear segment length and deciding the presence or absence of a feature point based on the result of comparison between said displacement and error value. CONSTITUTION:When the linear approximation is applied to a line pattern, the comparison is carried out between the displacement (d) of the original data from an approximate straight line and a parameter (Thd) as well as the inter- feature point distance (l) and a parameter (Thl) for decision of the presence or absence of a feature point. In other words, an evaluation parameter required for the visually preferable linear approximation is set by changing the error value (Thd) of the displacement (d) in accordance with the linear segment length (l). Then the presence or absence of a feature point is decided by comparing the displacement (d) with the value (Thd). Thus a parameter having a large freedom degree can be set in case a line pattern obtained from extraction of contours in a picture processing job is evaluated by the curvature and the displacement of said pattern for execution of the linear approximation.
TL;DR: In this paper, the authors examined the dynamics of the approach to steady-state solutions in relation to the system of equations of equilibrium multicomponent two-phase flow through porous media.
Abstract: The question of the dynamics of the approach to steady-state solutions is examined in relation to the system of equations of equilibrium multicomponent two-phase flow through porous media [1, 2]. The equations are analyzed in the linear approximation for the case of small fluctuations. An expression is obtained for the characteristic transition time. Numerical modeling is carried out for substantial deviations from the steady-state solution. The problem of the plane linear displacement of a ten-component gas-condensate mixture by an enriched gas with subsequent transition to the extraction regime is solved. It is shown that the change in phase compositions and pressure proceeds with characteristic times of the order of the time required to create a spatially nonuniform distribution of the mixture properties during injection. However, the change in the saturations and overall composition takes place in times approximately 200 times greater than the injection time. The question of the existence of discontinuous steady-state solutions of the system of equations investigated is considered, and in the case of a binary mixture it is shown that such solutions cannot be realized.
01 Jan 1989
TL;DR: In this paper, the authors show how classical estimates in the theory of linear elliptic equations can be used to obtain parameter estimation convergence of Galerkin approximations of Dirichlet problems.
Abstract: In this section we show how classical estimates in the theory of linear elliptic equations can be used to obtain parameter estimation convergence of Galerkin approximations. We develop the results in the context of a Dirichlet problem but different boundary conditions can be treated with similar methods.
01 Jan 1989
TL;DR: In this paper, the Fokker-Planck equation was used as an approximation for the collision term in a more restricted sense, where the collision terms were first order in time and second order in space.
Abstract: Often one calls kinetic differential equations that are first order in time and second order in some other variable Fokker-Planck equations [36.1,2]. We are going to consider the Fokker-Planck equation in a more restricted sense as an approximation for the collision term.
TL;DR: In this article, the thermodynamic stability of an elastic material with a constant concentration gradient is analyzed in linear approximation within the framework of the Ginzburg-Landau theory including an external inhomogeneous field.
Abstract: The thermodynamic stability of an elastic material with a constant concentration gradient is analysed in linear approximation within the framework of the Ginzburg-Landau theory including an external inhomogeneous field. The result suggests ways to produce stress-free, paraboloidically curved finite size crystals.