PubMed · 5232445
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E E Cureton. 1966. On correlation coefficients.. https://doi.org/10.1007/bf02289528
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There are relationships among the phases and magnitudes of the structure factors that have not been extensively studied regarding their potential for enhancing procedures for structure determination. These relationships arise from a special way of writing the determinantal inequalities that form the necessary and sufficient conditions for a Fourier series to be non-negative. This particular form also lends itself readily to the development of probability measures by the use of the central limit theorem. Higher-order determinants are of interest. The relationships among the phases and magnitudes of the structure factors are algebraic relationships and the focus is on those which retain their reliability, even though the magnitudes of the structure factors contain experimental errors. The future utility of the algebraic relationships depends upon the development of suitable algorithms for solving them to obtain values for the unknown phases. One approach concerns a method for extending the range of least-squares calculations by modifying the defining equations without changing the global minima and by further altering the nature of the minimization function from time to time during the course of the least-squares calculation, while still preserving the global minima. The objective is to smooth the minimization function and alter the remaining false minima from time to time so that the minimization function is not trapped in a false minimum. Some calculations have been made that indicate the nature of the algebraic relationships among the phases and magnitudes and how the results of the calculations are benefited by having large values for the structure-factor magnitudes in the determinants.(ABSTRACT TRUNCATED AT 250 WORDS)
Dose perturbations at tissue interfaces have been of significant concern since the beginning of this century. However, comprehensive studies related to the backscatter perturbation in kilovoltage beams are still limited. The dose perturbation depends on various parameters, including beam energy, field size, and the thickness, width, position, and atomic number, Z, of the inhomogeneity creating the interface with soft tissue. Using a thin window parallel plate ion chamber having relatively flat response at low energies, the dose perturbation was measured as backscatter dose perturbation factor, BSDF, at various interfaces in kilovoltage x-ray beams. The BSDF is defined as the ratio of doses with and without an interface for identical setup conditions. Results indicate that the BSDF is strongly dependent on beam energy, like the backscatter factor. Contrary to its behavior in megavoltage beams, BSDF in kilovoltage beams does depend on the field size, suggesting a contribution from scattered photons and fluorescent radiation, originating in the high-Z material. The thickness of the high-Z medium is not critical, since a fraction of a millimeter is sufficient to provide full backscatter. The interface effect with wide inhomogeneity has two distinct regions: the high dose region (BSDF > 1.0), which is very localized and disappears within a fraction of a millimeter, and the low dose region (BSDF < 1.0), which is observed up to 10 cm. The dependence of BSDF is neither a quadratic function of Z nor a cube root of beam energy, indicating that the interface effect is complex and not predominantly due to photoelectron transport.
In a recent paper, Poggio and Girosi (1990) proposed a class of neural networks obtained from the theory of regularization. Regularized networks are capable of approximating arbitrarily well any continuous function on a compactum. In this paper we consider in detail the learning problem for the one-dimensional case. We show that in the case of output data observed with noise, regularized networks are capable of learning and approximating (on compacta) elements of certain classes of Sobolev spaces, known as reproducing kernel Hilbert spaces (RKHS), at a nonparametric rate that optimally exploits the smoothness properties of the unknown mapping. In particular we show that the total squared error, given by the sum of the squared bias and the variance, will approach zero at a rate of n(-2m)/(2m+1), where m denotes the order of differentiability of the true unknown function. On the other hand, if the unknown mapping is a continuous function but does not belong to an RKHS, then there still exists a unique regularized solution, but this is no longer guaranteed to converge in mean square to a well-defined limit. Further, even if such a solution converges, the total squared error is bounded away from zero for all n sufficiently large.