Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book examines a system of parabolic-elliptic partial differential eq- tions proposed in mathematical biology, statistical mechanics, and chemical kinetics. In the context of biology, this system of equations describes the chemotactic feature of cellular slime molds and also the capillary formation of blood vessels in angiogenesis. There are several methods to derive this system. One is the biased random walk of the individual, and another is the reinforced random walk of one particle modelled on the cellular automaton. In the context of statistical mechanics or chemical kinetics, this system of equations describes the motion of a mean ?eld of many particles, interacting under the gravitational inner force or the chemical reaction, and therefore this system is af?liated with a hierarchy of equations: Langevin, Fokker–Planck, Liouville–Gel’fand, and the gradient ?ow. All of the equations are subject to the second law of thermodynamics — the decrease of free energy. The mat- matical principle of this hierarchy, on the other hand, is referred to as the qu- tized blowup mechanism; the blowup solution of our system develops delta function singularities with the quantized mass.

Differential equations, Parabolic. --- Differential equations, Partial. --- Lattice dynamics. --- Geometry, Differential. --- Biomathematics. --- Statistical mechanics. --- Chemical kinetics. --- Biology --- Mathematics --- Chemical reaction, Kinetics of --- Chemical reaction, Rate of --- Chemical reaction, Velocity of --- Chemical reaction rate --- Chemical reaction velocity --- Kinetics, Chemical --- Rate of chemical reaction --- Reaction rate (Chemistry) --- Velocity of chemical reaction --- Chemical affinity --- Reactivity (Chemistry) --- Mechanics --- Mechanics, Analytic --- Quantum statistics --- Statistical physics --- Thermodynamics --- Differential geometry --- Dynamics, Lattice --- Crystal lattices --- Phonons --- Solids --- Partial differential equations --- Parabolic differential equations --- Parabolic partial differential equations --- Differential equations, Partial --- Mathematics. --- Differential equations, partial. --- Global analysis (Mathematics). --- Mathematical physics. --- Engineering mathematics. --- Applications of Mathematics. --- Partial Differential Equations. --- Analysis. --- Mathematical Methods in Physics. --- Mathematical and Computational Biology. --- Mathematical and Computational Engineering. --- Engineering --- Engineering analysis --- Mathematical analysis --- Physical mathematics --- Physics --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Math --- Science --- Applied mathematics. --- Partial differential equations. --- Mathematical analysis. --- Analysis (Mathematics). --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- 517.1 Mathematical analysis

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

A mathematical theory is introduced in this book to unify a large class of nonlinear partial differential equation (PDE) models for better understanding and analysis of the physical and biological phenomena they represent. The so-called mean field approximation approach is adopted to describe the macroscopic phenomena from certain microscopic principles for this unified mathematical formulation. Two key ingredients for this approach are the notions of “duality” according to the PDE weak solutions and “hierarchy” for revealing the details of the otherwise hidden secrets, such as physical mystery hidden between particle density and field concentration, quantized blow up biological mechanism sealed in chemotaxis systems, as well as multi-scale mathematical explanations of the Smoluchowski–Poisson model in non-equilibrium thermodynamics, two-dimensional turbulence theory, self-dual gauge theory, and so forth. This book shows how and why many different nonlinear problems are inter-connected in terms of the properties of duality and scaling, and the way to analyze them mathematically.

Differential equations, Partial. --- Mathematics. --- Mean field theory. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Differential equations. --- Partial differential equations. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Math --- Science --- Differential Equations. --- Differential equations, partial.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Oncologia --- Matemàtica --- Matemàtiques (Ciència) --- Ciència --- Alfabetisme matemàtic --- Àlgebra --- Anàlisi matemàtica --- Aritmètica --- Axiomes --- Biomatemàtica --- Congruències (Geometria) --- Congruències i residus --- Constants matemàtiques --- Descomposició (Matemàtica) --- Dinàmica --- Estadística --- Factorització (Matemàtica) --- Filtres (Matemàtica) --- Formes (Matemàtica) --- Formes normals (Matemàtica) --- Geometria --- Geografia matemàtica --- Inducció (Matemàtica) --- Infinit --- Lògica matemàtica --- Matemàtica aplicada --- Matemàtica japonesa --- Màxims i mínims --- Nombres --- Quarta dimensió --- Successions (Matemàtica) --- Teoria de conjunts --- Teoria de l'índex (Matemàtica) --- Teoria de la computació --- Variables (Matemàtica) --- Didàctica de la matemàtica --- Ensenyament de la matemàtica --- Història de la matemàtica --- Matemàtics --- Medicina --- Oncologia geriàtrica --- Oncologia pediàtrica --- Oncologia veterinària --- Càncer --- Infermeria oncològica --- Tumors --- Oncology --- Mathematics. --- Mathematics

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Partial differential equations --- Mathematics --- Mathematical physics --- Chemistry --- Biomathematics. Biometry. Biostatistics --- Engineering sciences. Technology --- differentiaalvergelijkingen --- analyse (wiskunde) --- toegepaste wiskunde --- biomathematica --- chemie --- biometrie --- wiskunde --- ingenieurswetenschappen --- fysica

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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.

Functional analysis --- Mathematical analysis --- Numerical methods of optimisation --- Mathematics --- Mathematical physics --- Biomathematics. Biometry. Biostatistics --- Genetics --- Human biochemistry --- medische biochemie --- analyse (wiskunde) --- biomathematica --- genetica --- wiskunde --- fysica --- kansrekening --- optimalisatie