Atomicity Through Fractal Measure Theory: Mathematical and Physical Fundamentals with Applications

ISBN-13: 9783030295950 / Angielski / Miękka / 2020 / 184 str.

Alina Gavriluţ; Ioan Mercheş; Maricel Agop

Recenzje

"The book is addressed to a wide range of researchers in physics and mathematics and related areas using multivalued analysis." (Ivan Podvigin, zbMATH 1445.81004, 2020)

"The book is intended by the authors 'for graduate and postgraduate students, teachers, and all researchers in physics and mathematics'. ... I would recommend this book, much more narrowly, to mathematicians working in measure theory and to physicists who are already quantum field theory practitioners wishing to think on their work in measure-theoretic terms." (Vladimir García-Morales, Mathematical Reviews, September, 2020)

Spis treści

Preface.- 1. Short hypertopologies. A short overview.- 2. A Mathematical-physical approach on regularity in hit-and-miss hypertologies for fuzzy set multifunctions.- 3. Non-atomic set multifunctions.- 4. Non-atomicity and the Darboux property for fuzzy and non-fuzzy Borel/Baire multivalued set functions.- 5. Atoms and pseudo-atoms for set multifunctions.- 6. Gould integrability on atoms for set multifunctions.- 7. Continuity properties and the Alexandroff theorem in Vietoris topology.- 8. Approximation theorems for fuzzy set multifunctions in Vietoris topology. Physical implications of regularity- 9. Atomicity via regularity for non-additive set malfunctions.- 10. Extended atomicity through non-differentiability and its physical implications.- 11. On a multifractal theory of motion in a non-differentiable space. Toward a possible multifractal theory of measure.- List of symbols.- Index.

O autorze

Alina Gavrilut, PhD, is lecturer at the Faculty of  Mathematics, Alexandru Ioan Cuza University of Iasi,  Romania. Fields of competence: Mathematical Analysis – especially the Measure Theory, Nonlinear Dynamics and Quantum Physics.

Ioan Mercheş is an emeritus professor of theoretical physics at the Alexandru Ioan Cuza University of Iasi, Romania. He has co-authored 2 books published with Springer:  Mechanics: An Intensive Course (2012) and Electrodynamics: An Intensive Course (2016).

Maricel Agop is  professor of physics at the Gheorghe Asachi Technical University of Lasi, Romania.  He was awarded the title  “Doctor Honoris Causa” of the University “Vasile Alecsandri”, Bacau, Romania  (2013). Since 2018, he has been a correspondent member of the Academy of Romanian Scientists.

Blurb

This book presents an exhaustive study of atomicity from a mathematics perspective in the framework of multi-valued non-additive measure theory.  Applications to quantum physics and, more generally, to the fractal theory of the motion, are highlighted. The study details the atomicity problem through key concepts, such as the atom/pseudoatom, atomic/nonatomic measures, and different types of non-additive set-valued multifunctions. Additionally, applications of these concepts are brought to light in the study of the dynamics of complex systems.

The first chapter prepares the basics for the next chapters. In the last chapter, applications of atomicity in quantum physics are developed and new concepts, such as the fractal atom are introduced. The mathematical perspective is presented first and the discussion moves on to connect measure theory and quantum physics through quantum measure theory. New avenues of research, such as fractal/multifractal measure theory with potential applications in life sciences, are opened.