The subject of rainfall-runoff modeling involves a wide spectrum of topics. Fundamental to each topic is the problem of accurately computing runoff at a point given rainfall data at another point. The fact that there is currently no one universally accepted approach to computing runoff, given rainfall data, indicates that a purely deter ministic solution to the problem has not yet been found. The technology employed in the modern rainfall-runoff models has evolved substantially over the last two decades, with computer models becoming increasingly more complex in their detail of describing the hydrologic and hydraulic processes which occur in the catchment. But despite the advances in including this additional detail, the level of error in runoff estimates (given rainfall) does not seem to be significantly changed with increasing model complexity; in fact it is not uncommon for the model's level of accuracy to deteriorate with increasing complexity. In a latter section of this chapter, a literature review of the state-of-the-art in rainfall-runoff modeling is compiled which includes many of the concerns noted by rainfall-runoff modelers. The review indicates that there is still no deterministic solution to the rainfall-runoff modeling problem, and that the error in runoff estimates produced from rainfall-runoff models is of such magnitude that they should not be simply ignored.

This book showcases a subclass of hereditary systems, that is, systems with behaviour depending not only on their current state but also on their past history; it is an introduction to the mathematical theory of optimal control for stochastic difference Volterra equations of neutral type. As such, it will be of much interest to researchers interested in modelling processes in physics, mechanics, automatic regulation, economics and finance, biology, sociology and medicine for all of which such equations are very popular tools. The text deals with problems of optimal control such as meeting given performance criteria, and stabilization, extending them to neutral stochastic difference Volterra equations. In particular, it contrasts the difference analogues of solutions to optimal control and optimal estimation problems for stochastic integral Volterra equations with optimal solutions for corresponding problems in stochastic difference Volterra equations. Optimal Control of Stochastic Difference Volterra Equations commences with an historical introduction to the emergence of this type of equation with some additional mathematical preliminaries. It then deals with the necessary conditions for optimality in the control of the equations and constructs a feedback control scheme. The approximation of stochastic quasilinear Volterra equations with quadratic performance functionals is then considered. Optimal stabilization is discussed and the filtering problem formulated. Finally, two methods of solving the optimal control problem for partly observable linear stochastic processes, also with quadratic performance functionals, are developed. Integrating the author’s own research within the context of the current state-of-the-art of research in difference equations, hereditary systems theory and optimal control, this book is addressed to specialists in mathematical optimal control theory and to graduate students in pure and applied mathematics and control engineering.

This volume contains current work at the frontiers of research in quantum probability, infinite dimensional stochastic analysis, quantum information and statistics. It presents a carefully chosen collection of articles by experts to highlight the latest developments in those fields. Included in this volume are expository papers which will help increase communication between researchers working in these areas. The tools and techniques presented here will be of great value to research mathematicians, graduate students and applied mathematicians.

This volume contains current work at the frontiers of research in quantum probability, infinite dimensional stochastic analysis, quantum information and statistics. It presents a carefully chosen collection of articles by experts to highlight the latest developments in those fields. Included in this volume are expository papers which will help increase communication between researchers working in these areas. The tools and techniques presented here will be of great value to research mathematicians, graduate students and applied mathematicians.