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

Description

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. Front Matter ....Pages i-xiii Several hypertopologies: A short overview (Alina Gavriluţ, Ioan Mercheş, Maricel Agop)....Pages 1-7 A mathematical-physical approach on regularity in hit-and-miss hypertopologies for fuzzy set multifunctions (Alina Gavriluţ, Ioan Mercheş, Maricel Agop)....Pages 9-19 Non-atomic set multifunctions (Alina Gavriluţ, Ioan Mercheş, Maricel Agop)....Pages 21-35 Non-atomicity and the Darboux property for fuzzy and non-fuzzy Borel/Baire multivalued set functions (Alina Gavriluţ, Ioan Mercheş, Maricel Agop)....Pages 37-50 Atoms and pseudo-atoms for set multifunctions (Alina Gavriluţ, Ioan Mercheş, Maricel Agop)....Pages 51-69 Gould integrability on atoms for set multifunctions (Alina Gavriluţ, Ioan Mercheş, Maricel Agop)....Pages 71-82 Continuity properties and Alexandroff theorem in Vietoris topology (Alina Gavriluţ, Ioan Mercheş, Maricel Agop)....Pages 83-103 Approximation theorems for fuzzy set multifunctions in Vietoris topology: Physical implications of regularity (Alina Gavriluţ, Ioan Mercheş, Maricel Agop)....Pages 105-122 Atomicity via regularity for non-additive set multifunctions (Alina Gavriluţ, Ioan Mercheş, Maricel Agop)....Pages 123-131 Extended atomicity through non-differentiability and its physical implications (Alina Gavriluţ, Ioan Mercheş, Maricel Agop)....Pages 133-162 On a multifractal theory of motion in a non-differentiable space: Toward a possible multifractal theory of measure (Alina Gavriluţ, Ioan Mercheş, Maricel Agop)....Pages 163-179 Back Matter ....Pages 181-184