Quantum Variational Calculus

This Brief puts together two subjects, quantum and variational calculi by considering variational problems involving Hahn quantum operators. The main advantage of its results is that they are able to deal with nondifferentiable (even discontinuous) functions, which are important in applications.

Possible applications in economics are discussed. Economists model time as continuous or discrete. Although individual economic decisions are generally made at discrete time intervals, they may well be less than perfectly synchronized in ways discrete models postulate. On the other hand, the usual assumption that economic activity takes place continuously, is nothing else than a convenient abstraction that in many applications is far from reality. The Hahn quantum calculus helps to bridge the gap between the two families of models: continuous and discrete.

Quantum Variational Calculus is self-contained and unified in presentation. It provides an opportunity for an introduction to the quantum calculus of variations for experienced researchers but may be used as an advanced textbook by graduate students and even ambitious undergraduates as well. The explanations in the book are detailed to capture the interest of the curious reader, and complete to provide the necessary background material needed to go further into the subject and explore the rich research literature, motivating further research activity in the area.

From the book reviews:

“The aim of this book is cover the elementary calculus of variations problems in the context of the Hahn quantum calculus and the power quantum calculus. … The book may serve as an introduction to the Hahn quantum calculus or the power quantum calculus … or as a reference for the Euler-Lagrange equations for the associated calculus of variations problems with fixed or free endpoints.” (Roman Šimon Hilscher, Mathematical Reviews, October, 2014)

Delfim F. M. Torres was born August 16, 1971, in Nampula, Mozambique. He is currently an Associate Professor with Habilitation at the Department of Mathematics, University of Aveiro (UA), Portugal; Coordinator of the group on Mathematical Theory of Systems and Control of the Center for Research and Development in Mathematics and Applications (CIDMA); Director of The Doctoral Program in Mathematics and Applications (PDMA) of the Department of Mathematics of the University of Aveiro (DMAT-UA) and of the Department of Mathematics and Applications of the University of Minho (DMAT-UM), Portugal; Editor-in-Chief of the Int. J. Appl. Math. Stat. (IJAMAS) since 2008; Editor-in-Chief of the Int. J. Math. Stat. (IJMS) from 2007 to 2011; and Associate Editor for several other international journals.

Delfim F. M. Torres received his Diploma ("Licenciatura") honors degree in 1994 on Computer Science Engineering (a 5 years full time course) from University of Coimbra, Portugal. In 1998 he received his M.Sc. on Optimization and Control Theory (2 years full time course), and in 2002 his Ph.D. on Mathematics (3 years full time research), both degrees from UA.

Several M.Sc. dissertations, 5 Ph.D. theses, and 5 post-docs have been successfully finished under his supervision; 5 more Ph.D. theses and 3 post-docs are ongoing. Delfim F. M. Torres has written about 240 scientific and pedagogical publications, including research papers in first class international journals with referee, refereed conference proceedings, chapters in books, and books (as author and also as an editor).

Delfim F. M. Torres has 17 years of excellent teaching experience both to undergraduate and postgraduate students on Mathematics and Science and Engineering, in Portugal and abroad. He has held several positions of Invited Professor in Europe, Africa, and USA. He is a member of Scientific Committees and invited plenary speaker at several international conferences. His current research interests include several topics in the areas of the Calculus of Variations and Optimal Control, Fractional Calculus and Time Scales, and its connections with the Mathematical Theory of Control, Optimization, Epidemiology, and Mathematical Physics.