Topic

# Linear approximation

About: Linear approximation is a(n) research topic. Over the lifetime, 3901 publication(s) have been published within this topic receiving 74764 citation(s).

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TL;DR: The approximation rate and the parsimony of the parameterization of the networks are shown to be advantageous in high-dimensional settings and the integrated squared approximation error cannot be made smaller than order 1/n/sup 2/d/ uniformly for functions satisfying the same smoothness assumption.

Abstract: Approximation properties of a class of artificial neural networks are established. It is shown that feedforward networks with one layer of sigmoidal nonlinearities achieve integrated squared error of order O(1/n), where n is the number of nodes. The approximated function is assumed to have a bound on the first moment of the magnitude distribution of the Fourier transform. The nonlinear parameters associated with the sigmoidal nodes, as well as the parameters of linear combination, are adjusted in the approximation. In contrast, it is shown that for series expansions with n terms, in which only the parameters of linear combination are adjusted, the integrated squared approximation error cannot be made smaller than order 1/n/sup 2/d/ uniformly for functions satisfying the same smoothness assumption, where d is the dimension of the input to the function. For the class of functions examined, the approximation rate and the parsimony of the parameterization of the networks are shown to be advantageous in high-dimensional settings. >

2,519 citations

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TL;DR: In this paper, a quantitative theory is presented which describes the kinetics of coagulation of colloidal systems containing more than one dispersed species, using the linear (Debye-Huckel) approximation for low surface potentials.

Abstract: A quantitative theory is presented which describes the kinetics of coagulation of colloidal systems containing more than one dispersed species. A general expression has been derived to describe the potential energy of interaction between dissimilar spherical colloidal particles, using the linear (Debye-Huckel) approximation for low surface potentials. An overall stability ratio has been defined which takes into account the possibility of interactions between like, as well as unlike, particles in the system. The errors introduced by the use of the linear approximation have been assessed in terms of their effects on the stability ratio, and found to be quite small. The theory has been used to describe the behaviour of a hypothetical system under various conditions.

1,659 citations

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TL;DR: A theoretical analysis of linear enzymatic chains is presented and three cardinal terms are proposed for the quantitative description of enzyme systems, including the effector strength, which defines the dependence of the velocity of an enzyme on the concentration of an effector.

Abstract: A theoretical analysis of linear enzymatic chains is presented By linear approximation simple analytical solutions can be obtained for the metabolite concentrations and the flux through the chain for steady-state conditions The equations are greatly simplified if the common kinetic constants are expressed as functions of two parameters, ie the thermodynamic equilibrium constant and the “characteristic time” Three cardinal terms are proposed for the quantitative description of enzyme systems The first two are the control strength and the control matrix; these indicate the dependence of the flux and the metabolite concentrations, respectively, on the kinetic properties of a given enzyme The third is the effector strength, which defines the dependence of the velocity of an enzyme on the concentration of an effector; it expresses the importance of an effector By linear approximation simple analytical expressions were derived for the control strength, the control matrix and the mass-action ratios The effector strength was calculated for two cases: for a competitive inhibitor and for allosteric effectors according to the Monod (1965) model The influence of an effector on the concentrations of the metabolites was considered

1,294 citations

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TL;DR: In this article, a review of analytical methods in electromagnetic scattering theory (i.e., geometrical and physical optics, perturbation, iteration, and integral-equation) which are applicable to the problems of remote sensing of the ocean is presented.

Abstract: This paper reviews analytical methods in electromagnetic scattering theory (i.e., geometrical and physical optics, perturbation, iteration, and integral-equation) which are applicable to the problems of remote sensing of the ocean. In dealing with Earth's surface (in this case, the weakly non-linear ocean), it is not possible to have a complete and exact description of its spatial and temporal statistics. Only the first few moments are generally available; and in the linear approximation the statistics are assumed homogeneous, stationary and Gaussian. For this case, the high-frequency methods (geometrical and physical optics) and perturbation (Rayleigh-Rice), or a combination of them, provide tractable analytical results (i.e., the specular-point, the slightly-rough Bragg scattering and the composite-surface models). The applicability and limitations of these models are discussed. At grazing incidence and for higher frequencies, other scattering mechanisms become significant; and shadowing, diffraction and trapping must be considered. The more exact methods (integral-equation and Green's function) have not been as successful in yielding tractable analytical solutions, although they have the potential to provide improved theoretical scattering results in the future.

952 citations