Exploring the intersection of mathematics and reality, this book delves into how mathematical models capture the complexities of the world around us. It highlights the analytical processes that lead to surprising predictions and phenomena, emphasizing the transformative power of mathematical insights in understanding and interpreting real-world scenarios.
The new edition enhances the study of applied analysis by incorporating essential materials from both applied and theoretical sciences relevant to contemporary society. It covers mathematical modeling with ODEs, Riemann geometry, and tensor analysis in continuum mechanics, alongside detailed discussions on optimization and real analysis in PDE studies. Advanced topics include integration of ODEs, inverse Sturm-Liouville problems, and diffusion geometry, with corrections and extensions to the analysis of the Smoluchowski-Poisson equation, focusing on quantized blowup mechanisms.
Incorporating Marshallian concepts like external increasing returns and monopolistic competition, this revised edition enhances the general equilibrium framework rooted in Walrasian tradition. It features new chapters and sections that deepen the analysis of production and increasing returns, providing a comprehensive exploration of these economic theories. This updated work aims to bridge classical and modern economic thought, making it a valuable resource for scholars and students alike.
Mean field approximation has been adopted to describe macroscopic phenomena from microscopic overviews. It is still in progress; fluid mechanics, gauge theory, plasma physics, quantum chemistry, mathematical oncology, non-equilibirum thermodynamics. spite of such a wide range of scientific areas that are concerned with the mean field theory, a unified study of its mathematical structure has not been discussed explicitly in the open literature. The benefit of this point of view on nonlinear problems should have significant impact on future research, as will be seen from the underlying features of self-assembly or bottom-up self-organization which is to be illustrated in a unified way. The aim of this book is to formulate the variational and hierarchical aspects of the equations that arise in the mean field theory from macroscopic profiles to microscopic principles, from dynamics to equilibrium, and from biological models to models that arise from chemistry and physics.
Focusing on the historical evolution of engines, this book provides valuable insights into the lessons learned throughout their development. Originally published in Japanese, the English translation makes this essential reading for engine designers, enthusiasts, and automotive historians alike, highlighting the significance of engines in technological advancement.