Volume 26 - Issue 2 - December 2001

Chebyshev's influence on the development of mathematics

S. N. Bernstein (translated by O. Sheynin)

On recent cable models in neurophysiology

Roman R. Poznanski


A brief synopsis of concepts in neurophysiology for non-biologists is provided, in order to explain the basic terminology used in this field. This is followed by an overview of some recent equivalent cable models of dendrites and a discussion of dendritic information processing.

Statistical modelling and prediction associated with the HIV\AIDS epidemic

P. J. Solomon and S. R. Wilson


This paper gives an overview of the fundamental mathematical modelling and statistical procedures underlying the practices widely used in industrialised countries for predicting AIDS incidence and prevalence, as well as estimating past HIV incidence. Such predictions are needed for public health and insurance planning. In particular, we consider extrapolation forecasting, prediction of trends in small groups, stochastic epidemic modelling, backcalculation and multistage modelling.

A probabilistic approach to identifying positive value cash flows

Ilan Adler and Sheldon M. Ross


We use a probabilistic argument to find sufficient conditions on a cash flow sequence a0, a1, …, an to ensure that its present value is positive for any nonincreasing sequence of nonnegative interest rates.

Birth–death processes via MATHEMATICA®

Randall J. Swift


This paper develops a method for applying the computer algebra software package MATHEMATICA® to birth–death processes. MATHEMATICA is used to solve the differential equations associated with these processes. The method is used to obtain the transient probabilities for the birth–death–immigration and birth–death–immigration–catastrophe processes. The paper includes a discussion of MATHEMATICA's command DSolve for solving differential equations, and describes examples of its application to some elementary differential equations.

Independence of unscaled basis functions and finite mappings by neural networks

Yoshifusa Ito


A three-layered neural network having n hidden-layer units can implement a mapping of n points in Rd onto R. In this paper, the activation function of hidden-layer units is extended to higher-dimensional functions, so that the sigmoid function defined on R and the radial basis function defined on Rd can be treated on a common basis. We assume that activation functions cannot be scaled. Even under this restriction, a wide class of functions can be activation functions for the mapping of a finite number of points. If the support of the Fourier transform of a slowly increasing function includes a converging sequence of points on a line, it can be an activation function for the mapping of points without scaling. This condition does not depend on the dimension of the space on which the activation function is defined; both the logistic function on R and the Gauss kernel on Rd satisfy it. The result extends the work of Ito and Saito (1996), in which the activation function is restricted to sigmoid functions.

A Lie group approach to Bernoulli's transformations

Behzad Salimi


This paper studies the invariance properties of two special cases of the special Riccati differential equations under the action of one-parameter Lie groups. Using the coordinate functions of a particular one-parameter Lie group admitted by the special Riccati equation, the specific transformations discovered by Daniel Bernoulli are derived. The transformation to the new coordinate system reduces the solution of this class of nonlinear differential equations to quadratures for certain values of the `index' n in the special Riccati equation. The application of Lie groups leads to a systematic derivation of Bernoulli's ingenious transformations. The paper concludes with a brief discussion of how Bernoulli's transformations reduce and solve certain special Riccati equations.