Description: High-order numerical methods provide an efficient approach to simulating many physical problems. This book considers the range of mathematical, engineering, and computer science topics that form the foundation of high-order numerical methods for the simulation of incompressible fluid flows in complex domains. Introductory chapters present high-order spatial and temporal discretizations for one-dimensional problems. These are extended to multiple space dimensions with a detailed discussion of tensor-product forms, multi-domain methods, and preconditioners for iterative solution techniques. Numerous discretizations of the steady and unsteady Stokes and Navier-Stokes equations are presented, with particular attention given to enforcement of incompressibility. Advanced discretizations, implementation issues, and parallel and vector performance are considered in the closing sections. Numerous examples are provided throughout to illustrate the capabilities of high-order methods in actual applications. Computer scientists, engineers and applied mathematicians interested in developing software for solving flow problems will find this book a valuable reference.
Contents: Preface; 1. Fluid mechanics and computation: an introduction; 2. Approximation methods for elliptic problems; 3. Parabolic and hyperbolic problems; 4. Mutidimensional problems; 5. Steady Stokes and Navier-Stokes equations; 6. Unsteady Stokes and Navier-Stokes equations; 7. Domain decomposition; 8. Vector and parallel implementations; Appendix A. Preliminary mathematical concepts; Appendix B. Orthogonal polynomials and discrete transforms.
Essential Information
ISBN, Binding, : 0-521-45309-7 Hardback Pages:
528
Approximate Publication Date: 15/08/2002
Main Subject Category: Mathematics - numerical
analysis
Series: Cambridge Monographs on Applied and
Computational Mathematics, No. 9
Description: Neurons in the brain communicate by short electrical pulses, the so-called action potentials or spikes. How can we understand the process of spike generation? How can we understand information transmission by neurons? What happens if thousands of neurons are coupled together in a seemingly random network? How does the network connectivity determine the activity patterns? And, vice versa, how does the spike activity influence the connectivity pattern? These questions are addressed in this introduction to spiking neurons aimed at those taking courses in computational neuroscience, theoretical biology, biophysics, or neural networks. The approach will suit students of physics, mathematics, or computer science; it will also be useful for biologists who are interested in mathematical modelling. The text is enhanced by many worked examples and illustrations. There are no mathematical prerequisites beyond what the audience would meet as undergraduates: more advanced techniques are introduced in an elementary, concrete fashion when needed.
Contents: 1. Introduction; Part I. Single Neuron Models: 2. Detailed neuron models; 3. Two-dimensional neuron models; 4. Formal spiking neuron models; 5. Noise in spiking neuron models; Part II. Population Models: 6. Population equations; 7. Signal transmission and neuronal coding; 8. Oscillations and synchrony; 9. Spatially structured networks; Part III. Models of Synaptic Plasticity: 10. Hebbian models; 11. Learning equations; 12. Plasticity and coding; Bibliography; Index.
Essential Information
ISBN, Binding, : 0-521-81384-0 Hardback Pages:
450
Approximate Publication Date: 15/08/2002
Main Subject Category: Computer applications
Description: This book examines the development of calculus in Britain during the century following Newton. It is usually maintained that this was a period of decline in British mathematics. However, the authors research has shown that the methods used by researchers of the period yielded considerable success in laying the foundations and investigating the applications of the calculus. Even when declineEwas at its worst point, in mid-century, the foundations of the reform, which were to change the direction and nature of the mathematics community, were being laid. The book considers the importance of the work of mathematicians such as Isaac Newton, Roger Cotes, Brook Taylor, James Stirling, Abraham de Moivre, Colin Maclaurin, Thomas Bayes, John Landen and Edward Waring. It will be useful to science historians and philosophers studying the period, and to students of British history studying the teaching of mathematics.
Contents: Introduction; Overture: Newtons published work on the calculus of fluxions; Part I. The Early Period: 1. The diffusion of the calculus (1700-1730); 2. Developments in the calculus of fluxions (1714-1733); 3. The controversy on the foundations of the calculus (1734-1742); Part II. The Middle Period: 4. The textbooks on fluxions (1736-1758); 5. Some applications of the calculus (1740-1743); 6. The analytic art (1755-1785); Part III. The Reform: 7. Scotland (1785-1809); 8. The Military Schools (1773-1819); 9. Cambridge and Dublin (1790-1820); 10. Tables; Endnotes; Bibliography; Index.
Essential Information
ISBN, Binding, 0-521-52484-9 Paperback
Pages: 240
Approximate Publication Date: c.31/08/2002
Main Subject Category: Theoretical, mathematical
physics
Description: Numerical weather prediction is a problem of mathematical physics. The complex flows in the atmosphere and oceans are believed to be accurately modelled by the Navier-Stokes equations of fluid mechanics together with classical thermodynamics. However, due to the enormous complexity of these equations, meteorologists and oceanographers have constructed approximate models of the dominant, large-scale flows that control the evolution of weather systems and that describe, for example, the dynamics of cyclones and ocean eddies. The simplifications often result in models that are amenable to solution both analytically and numerically. The lectures in these volumes examine and explain why such simplifications to Newtons second law produce accurate, useful models and, just as the meteorologist seeks patterns in the weather, mathematicians seek structure in the governing equations, such as groups of transformations, Hamiltonian structure and stability. This book and its companion show how geometry and analysis facilitate solution strategies.
Contents: Introduction J. C. R. Hunt, J. Norbury and I. Roulstone; 1. A view of the equations of meteorological dynamics and various approximations A. A. White; 2. Extended-geostrophic Euler-Poincare models for mesoscale oceanographic flow J. S. Allen, D. D. Holm and P. A. Newberger; 3. Fast singular oscillating limits of stably stratified three-dimensional Euler-Boussinesq equations and ageostrophic wave fronts A. Babin, A. Mahalov and B. Nicolaenko; 4. New mathematical developments in atmosphere and ocean dynamics, and their application to computer simulations M. J. P. Cullen; 5. Rearrangements of functions with applications to meteorology and ideal fluid flow R. J. Douglas; 6. Statistical methods in atmospheric dynamics: probability metrics and discrepancy measures as a means of defining balance S. Baigent and J. Norbury.
Essential Information
ISBN, Binding, 0-521-80681-X Hardback
Pages: 320
Approximate Publication Date: 15/08/2002
Main Subject Category: Applied mathematics,
mathematical physics
Description: Numerical weather prediction is a problem of mathematical physics. The complex flows in the atmosphere and oceans are believed to be accurately modelled by the Navier-Stokes equations of fluid mechanics together with classical thermodynamics. However, due to the enormous complexity of these equations, meteorologists and oceanographers have constructed approximate models of the dominant, large-scale flows that control the evolution of weather systems and that describe, for example, the dynamics of cyclones and ocean eddies. The simplifications often result in models that are amenable to solution both analytically and numerically. The lectures in these volumes examine and explain why such simplifications to Newtons second law produce accurate, useful models and, just as the meteorologist seeks patterns in the weather, mathematicians seek structure in the governing equations, such as groups of transformations, Hamiltonian structure and stability. This book and its companion show how geometry and analysis facilitate solution strategies.
Contents: Introduction J. C. R. Hunt, J. Norbury and I. Roulstone; 1. Balanced models in geophysical fluid dynamics: Hamiltonian formulation, constraints and formal stability O. Bokhove; 2. The swinging spring: a simple model of atmospheric balance P. Lynch; 3. On the stationary spectra for an ensemble of plane weakly nonlinear internal gravity waves P. Caillol and V. Zeitlin; 4. Hamiltonian description of shear flow N. J. Balmforth and P. J. Morrison; 5. Some applications of transformation theory in mechanics M. J. Sewell; 6. Legendre-transformable semi-geostrophic theories R. J. Purser; 7. The Euler-PoincarEequations in geophysical fluid dynamics D. D. Holm, J. E. Marsden and T. Ratiu; 8. Are there higher-accuracy analogues of semi-geostrophic theory? M. E. McIntyre and I. Roulstone.
Essential Information
ISBN, Binding, 0-521-80757-3 Hardback
Pages: 390
Approximate Publication Date: 15/08/2002
Main Subject Category: Applied mathematics,
mathematical physics
Description: Macroscopic quantum phenomena are particularly important when considering the problem of Schrodinger's cat. This book contains a coherent and self-contained account of such phenomena, focusing on the central role played by macroscopic quantum tunneling. Beginning with an explanation of the nature and significance of the cat problem, Shin Takagi introduces the concept of macroscopic quantum tunneling. He deals with typical examples in detail, elucidating how quantum mechanical coherence may be lost (so-called decoherenceE or how it may be maintained despite the effects of environment and measurement processes. Recent experimental and theoretical advances are discussed, and the remaining problems described. The final chapter describes an experiment to decide between quantum mechanics and macrorealism in the light of Einsteins moon. Assuming only a knowledge of elementary quantum mechanics, this book emphasises conceptual aspects rather than technical details. It provides a firm introduction to the subject for graduate students and researchers.
Contents: 1. Introduction; 2. Overview of macroscopic quantum tunneling; 3. Some candidate systems for macroscopic quantum tunneling; 4. Environmental problems; 5. Harmonic environment; 6. Quantum resonant oscillation in the harmonic environment; 7. Quantum decay in the harmonic environment; 8. General versus harmonic environments; 9. The cat in the moonlight; 10. Appendix A. Euclidean space and Hilbert space; 11. Appendix B. Virtual ground state of a system of a single degree of freedom and its decay; 12. Appendix C. Functional derivative; 13. Appendix D. Miscellanea about spin; 14. References.
Essential Information
ISBN, Binding, 0-521-80002-1 Hardback Pages:
221
Approximate Publication Date: 15/08/2002
Main Subject Category: Physics - condensed
matter