This unique volume presents a fruitful and beautiful mathematical world hidden in Caianiello's neuronic equations, which describe the instantaneous behavior of a model of a brain or thinking machine. The detailed analysis from a viewpoint of “dynamical systems”, even in a single neuron case, enables us to obtain amazingly good rational approximations to the Hecke–Mahler series with two variables. Some interesting numerical applications of our rational approximations are also discussed.
This book is fundamentally self-contained and many topics required in it are explained from the beginning. Each chapter contains a number of instructive and mostly original exercises at various levels.
Contents:Basics of Discrete Dynamical SystemsCaianiello's EquationsRotation NumbersClassification of BFarey SeriesFurther Investigation of ƒ ∈ B∞Limit Sets Ωƒ and ωƒ (χ)Piecewise Linear MapsOrbital and Itinerary FunctionsFarey Structureα- and β-LeavesApproximations to Hecke–Mahler Series
Readership: Graduates and researchers interested in dynamical systems, Farey series, Hecke–Mahler series, Diophantine approximation (irrationality measures, transcendental numbers).
Key Features:A new method to obtain “good” rational approximations to functions with two variables is providedA number of mostly original exercises is contained in almost all chapters, together with their detailed solutions in the “Hints and Solutions” sectionBasic and useful mathematical notions in “discrete dynamical systems”, “elementary number theory” and in “rational approximation theory” are explained carefully