Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

We omit Gompertz’ method of fitting, which the reader will find in Makeham, below. Also omitted are Gompertz’ mortality tables, and his discussion of life expectancy and annuities under his Law. The hyperbolic logarithms in Art. 5 are the natural logs; we are not able to follow completely his integration, which is by Newton's method of fluxions.

On the Nature of the Function Expressive of the Law of Human Mortality

To detect multiple sclerosis (MS) diseases early, we proposed a novel method on the hardware of magnetic resonance imaging, and on the software of three successful methods: biorthogonal wavelet transform, kernel principal component analysis, and logistic regression. The materials were 676 MR slices containing plaques from 38 MS patients, and 880 MR slices from 34 healthy controls. The statistical analysis showed our method achieved a sensitivity of 97.12±.14%, a specificity of 98.25±0.16%, and an accuracy of 97.76±0.10%. Our method is superior to five state-of-the-art approaches in MS detection.

Multiple Sclerosis Detection Based on Biorthogonal Wavelet Transform, RBF Kernel Principal Component Analysis, and Logistic Regression

Research on convolutional neural network based on improved Relu piecewise activation function

On the approximation by single hidden layer feedforward neural networks with fixed weights.

In this note we prove more precise estimates for the approximation of the step function by sigmoidal logistic functions. Numerical examples, illustrating our results are given, too.

On the Hausdorff distance between the Heaviside step function and Verhulst logistic function

On the approximation of the step function by some sigmoid functions

We study the uniform approximation of the sigmoid cut function by smooth sigmoid functions such as the logistic and the Gompertz functions. The limiting case of the interval-valued step function is discussed  using Hausdorff metric.  Various expressions for the error estimates of the corresponding uniform and Hausdorff approximations are obtained. Numerical examples are presented using CAS MATHEMATICA.

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On the Approximation of the Cut and Step Functions by Logistic and Gompertz Functions

We  focus on some computational, modelling and approximation issues related to the logistic sigmoidal function and to Heaviside step function. The Hausdorff approximation of the Heaviside interval step function by sigmoidal functions is discussed from various computational and modelling aspects. Some relations between Verhulst model and certain biochemical reaction equations are discussed and analyzed. Numerical examples are presented using CAS Mathematica.

/pdf/sigmoidal-functions-some-computational-and-modelling-aspects-2a6eth4dik.pdf

Sigmoidal Functions: Some Computational and Modelling Aspects

Computation of polynomial zeros generalized root iteration recursive generated iterative methods two-Sides and multi-point methods factorization of a polynomial on some methods for the determination of all roots on the zeros of polynomials contraction of the SOR Weierstrass method on the critical points of Aberth's method. Supplement: a note on the Le Verrier-Fadeev method.

Nikolay Kyurkchiev

Papers

On the Hausdorff distance between the Heaviside step function and Verhulst logistic function

On the approximation of the step function by some sigmoid functions

On the Approximation of the Cut and Step Functions by Logistic and Gompertz Functions

Sigmoidal Functions: Some Computational and Modelling Aspects

Initial Approximations and Root Finding Methods