Admissible Sets and Structures

Description

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. Admissible set theory is a major source of interaction between model theory, recursion theory and set theory, and plays an important role in definability theory. In this volume, the seventh publication in the Perspectives in Logic series, Jon Barwise presents the basic facts about admissible sets and admissible ordinals in a way that makes them accessible to logic students and specialists alike. It fills the artificial gap between model theory and recursion theory and covers everything the logician should know about admissible sets. Contents Preface 1 CFD in Perspective 1.1 The Nature of CFD 1.2 Overview of the Book 1.3 Algorithm, Numerical Method, Implementation and Simulation 1.4 Models and Methods 1.5 Round-off Error 1.6 A Hierarchy of Computation 1.7 Ramification, Turbulence and the Complexity of Fluid Flows 1.8 The Development of Computer Hardware 1.9 Some Remarks on Software 2 Mappings 2.1 Numerical Methods as Mappings 2.2 Fixed Points: Stability, Instability and Superstability 2.3 Stability of Exact Solution, Implicit and Explicit Approximations 2.4 More on Mappings 2.5 Random Number Generation 2.6 Newton’s Method in the Complex Plane 2.7 Mappings and Fluid Flows 3 Ordinary Differential Equations: Initial Value Problem 3.1 Some Conventional Wisdom 3.2 Explicit Euler and Implicit Euler Schemes 3.3 Runge–Kutta Methods 3.4 Adams–Bashforth–Moulton Methods 3.5 Other Methods and Considerations 3.6 Bashforth’s Problem: Sessile Drop on a Flat Plate 3.7 Flow Due to a Collapsing Bubble 3.8 Motion of a Solid in Ideal Fluid 3.9 The Point-Vortex Equations 3.10 Vortex Sheet Roll-up 4 Spatial Discretization 4.1 Forward, Backward and Central Difference 4.2 Matrix Derivative Operators 4.3 Compact Differences 4.4 Non-uniform Discretization 4.5 Numerical Interpolation 4.6 Numerical Integration 5 Boundary Value and Eigenvalue ODEs 5.1 Linear Boundary Value Problems 5.2 Nonlinear Boundary Value Problems 5.3 Boundary Value Problems in Viscous Flow 5.4 Eigenvalue Problems 5.5 Hydrodynamic Instability 6 Methods Based on Functional Expansions 6.1 Introduction 6.2 Fourier Approximation 6.3 Polynomial Approximation 6.4 Galerkin, Tau, Collocation and Pseudo-spectral Methods 6.5 Some Examples 7 Partial Differential Equations 7.1 Definitions and Preliminaries 7.2 The Advection Equation 7.3 The Diffusion Equation 7.4 The Advection–Diffusion Equation 7.5 Godunov’s Theorem 7.6 More on Stability: Non-periodic Boundary Conditions 7.7 Burgers’ Equation 7.8 Implicit Time-differencing 7.9 Direct Solution with Matrix Representation 8 Multi-dimensional Partial Differential Equations 8.1 Multi-dimensions 8.2 Navier–Stokes Equations 8.3 Navier–Stokes Equations in Spectral Form 8.4 Finite Volume Formulation 8.5 CFD for Complex Geometries 8.6 Sharp Interface Cartesian Grid Method 8.7 Immersed Boundary Method References Index