Yoshida Nonlinear Science

The Challenge of Complex Systems
1. Auflage 2010
ISBN: 978-3-642-03406-0
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 218 Seiten, eBook

Modern science has abstracted, as compensation for establishing rigorousness, the complexity of the real world, and has inclined toward oversimpli?ed ?ctitious n- ratives; as a result, a disjunction has emerged between the wisdom of science and reality. Re?ecting on this, we see the need for science to recover reality; can it reveal new avenues for thought and investigation of the complexity? The study of science is the pursuit of clarity and distinctness. Physics,after Galilei placed it in the realm of mathematics, has been trying to establish clearness by mathematical logic. While physics and mathematics, respectively, have different intellectual incentives, they have intersected in history on countless occasions and have woven a ?awless system of wisdom. The core of rigorous science is always made of mathematical logic; the laws of science cannot be represented without the language of mathematics. Conversely, it is undoubtedly dif?cult to stimulate ma- ematical intellect without a reference to the interests of science that are directed to the real world. However, various criticisms have been raised against the discourses of sciences that explain the events of the real world as if they are 'governed' by mathematical laws. Sciences, being combined with technologies, have permeated, in the form of technical rationalism, the domain of life, politics, and even the psychological world. The criticisms accuse seemingly logical scienti?c narratives of being responsible for widespread destruction and emergence of crises, unprecedented suffering of hum- ity.

Professor Yoshida's contributions range from the mathematical physics to leading an experimental team of plasma physics. He has studied the self-organization of structures in plasmas using various theoretical methods such as variational principles, singular perturbation theory, operator theory, functional analysis, topological methods, etc., and, in particular, is well known for important contributions to the mathematical theory of the curl operator, which he has developed to extend and deepen the understanding of nonlinear structures in general vortex dynamics systems.

Zielgruppe

Research

Autoren/Hrsg.

Yoshida, Zensho

Inhaltsverzeichnis

1;Preface;5
2;Contents;8
3;1 What Is NONLINEAR?;11
3.1; Nature and Science;11
3.1.1; Natura Vexata;11
3.1.2; Syndrome;14
3.1.3; Déconstruction of Linear Theory;15
3.2; The Scale of Phenomenon / Theory with Scale;16
3.2.1; The Role of Scale in Scientific Revolutions;16
3.2.2; The Mathematical Recognition of Scale;18
3.3; The Territory of Linear Theory;20
3.3.1; Linear Space ---The Horizon of Mathematical Science;20
3.3.2; The Mathematical Definition of Vectors;22
3.3.3; Graphs---Geometric Representation of Laws;25
3.3.4; Exponential Law;30
3.4; Nonlinearity---Phenomenology and Structures;34
3.4.1; Nonlinear Phenomena;34
3.4.2; The Typology of Distortion;35
3.4.3; Nonlinearity Emerging in Small Scale---Singularity;37
3.4.4; Nonlinearity Escaping from Linearity---Criticality;39
3.4.5; Bifurcation (Polyvalency) and Discontinuity;41
3.5;Notes;43
3.6;Problems;51
3.7;Solutions;51
3.8;References;53
4;2 From Cosmos to Chaos;54
4.1; The Order of Nature---A Geometric View;54
4.1.1; Galileo's Natural Philosophy;54
4.1.2; Geometric Description of Events;55
4.1.3; Universality Discovered by Newton;57
4.2; Function---The Mathematical Representation of Order;61
4.2.1; Motion and Function;61
4.2.2; Nonlinear Regime;63
4.2.3; Beyond the Functional Representation of Motion;65
4.3; Decomposition---Elucidation of Order;67
4.3.1; The Mathematical Representation of Causality;67
4.3.2; Exponential Law---A Basic Form of Group;69
4.3.3; Resonance---Undecomposable Motion;71
4.3.4; Nonlinear Dynamics---An Infinite Chain of Interacting Modes;74
4.3.5; Chaos---Motion in the Infinite Period;76
4.3.6; Separability/Inseparability;78
4.4; Invariance in Dynamics;83
4.4.1; Constants of Motion;83
4.4.2; Chaos---True Evolution;89
4.4.3; Collective Order;90
4.4.4; Complete Solution---The Frame of SpaceEmbodying Order;92
4.4.5; The Difficulty of Infinity;94
4.5; Symmetry and Conservation Law;95
4.5.1; Symmetry in Dynamical System;95
4.5.2; The Deep Structure of Dynamical System;96
4.5.3; The Translation of Motion and Non-motion;100
4.5.4; Chaos---The Impossibility of Decomposition;103
4.6;Notes;106
4.7;Problems;114
4.8;Solutions;115
4.9;References;118
5;3 The Challenge of Macro-Systems;120
5.1; The Difficulty of Prediction;120
5.1.1; Chaos in Phenomenological Recognition;120
5.1.2; Stability;121
5.1.3; Attractors;125
5.1.4; Stability and Integrability;128
5.2; Randomness as Hypothetical Simplicity;130
5.2.1; Stochastic Process;130
5.2.2; Representation of Motion by Transition Probability;132
5.2.3; H-Theorem;134
5.2.4; Statistical Equilibrium;136
5.2.5; Statistically Plausible Particular Solutions;140
5.3; Collective Phenomena;141
5.3.1; Nonequilibrium and Macroscopic Dynamics;141
5.3.2; A Model of Collective Motion;142
5.3.3; A Statistical Model of Collisions;146
5.4;Notes;149
5.5;Problems;156
5.6;Solutions;157
5.7;References;159
6;4 Interactions of Micro and Macro Hierarchies;161
6.1; Structure and Scale Hierarchy;161
6.1.1; Crossing-Over Hierarchies;161
6.1.2; Connection of Scale Hierarchies---Structure;162
6.2; Topology---A System of Differences;165
6.2.1; The Topology of Geometry;165
6.2.2; Scale Hierarchy and Topology;166
6.2.3; Fractals---Aggregates of Scales;167
6.3; The Scale of Event / The Scale of Law;169
6.3.1; Scaling and Representation;169
6.3.2; Scale Separation;172
6.3.3; Spontaneous Selection of Scale by Nonlinearity;175
6.3.4; Singularity---Ideal Limit of Scale-Invariant Structure;176
6.4; Connections of Scale Hierarchies;178
6.4.1; Complexity---Structures with Multiple Aspects;178
6.4.2; Singular Perturbation;179
6.4.3; Collaborations of Nonlinearity and Singular Perturbation;182
6.4.4; Localized Structures in Space--Time;186
6.4.5; Irreducible Couplings of Multi-Scales;191
6.5;Notes;195
6.6;Problems;209
6.7;Solutions;210
6.8;References;214
7;Index;216

What is NONLINEAR?.- From Cosmos to Chaos.- The Challenge of Macro-Systems.- Interactions of Micro and Macro Hierarchies.

Über Autor(innen)

Professor Yoshida's contributions range from the mathematical physics to leading an experimental team of plasma physics. He has studied the self-organization of structures in plasmas using various theoretical methods such as variational principles, singular perturbation theory, operator theory, functional analysis, topological methods, etc., and, in particular, is well known for important contributions to the mathematical theory of the curl operator, which he has developed to extend and deepen the understanding of nonlinear structures in general vortex dynamics systems.