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El. knyga: Stochastic Modelling of Reaction-Diffusion Processes

4.50/5 (5 ratings by Goodreads)
(University of Oxford), (University of Oxford)
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This practical introduction covers mathematical methods for the analysis of stochastic models and their biological applications. Based on courses taught at the University of Oxford, the book can be used for self-study or as a supporting text for advanced undergraduate or beginning graduate-level courses in applied mathematics.

This practical introduction to stochastic reaction-diffusion modelling is based on courses taught at the University of Oxford. The authors discuss the essence of mathematical methods which appear (under different names) in a number of interdisciplinary scientific fields bridging mathematics and computations with biology and chemistry. The book can be used both for self-study and as a supporting text for advanced undergraduate or beginning graduate-level courses in applied mathematics. New mathematical approaches are explained using simple examples of biological models, which range in size from simulations of small biomolecules to groups of animals. The book starts with stochastic modelling of chemical reactions, introducing stochastic simulation algorithms and mathematical methods for analysis of stochastic models. Different stochastic spatio-temporal models are then studied, including models of diffusion and stochastic reaction-diffusion modelling. The methods covered include molecular dynamics, Brownian dynamics, velocity jump processes and compartment-based (lattice-based) models.

Recenzijos

'The text can be used effectively for solitary study or as a textbook for a course offered at the boundary between undergraduate and beginning graduate study This is a remarkable, even admirable, work that bears the mark of its Oxford origins. Its potential audience includes chemists and mathematicians as well as adventuresome biologists and physicists and perhaps even bright or intrepid general readers.' A. E. Viste, Choice 'This textbook is an example-driven introduction to stochastic modeling in mathematical biology Beyond serving as a course textbook, the book could serve as a good general introduction to the area of stochastic modeling in biology for researchers, particularly given the copious citations to more specialist texts.' Andrew Krause, MAA Reviews 'Erban and Chapman's Stochastic Modelling of ReactionDiffusion Processes will be valuable both as a reference for practitioners and as a textbook for a graduate course on stochastic modelling. Every chapter includes problems for the reader. The problems are well written and appropriate for most intended readers of the book. I hope that this book is widely adopted and that it becomes a standard textbook in the field.' Michael A. Salins, Mathematical Reviews/MathSciNet Review 'This book is also available at a reduced price as an e-book on Kindle. Based on the sample I viewed, all the features of the printed book have been perfectly preserved, with no loss of clarity in the layout or the mathematical symbols or the graphs and diagrams.' David Hopkins, The Mathematical Gazette

Daugiau informacijos

Practical introduction for advanced undergraduate or beginning graduate students of applied mathematics, developed at the University of Oxford.
Preface ix
1 Stochastic Simulation of Chemical Reactions
1(32)
1.1 Stochastic Simulation of Degradation
1(7)
1.2 Stochastic Simulation of Production and Degradation
8(6)
1.3 Higher-Order Chemical Reactions
14(2)
1.4 Stochastic Simulation of Dimerization
16(9)
1.5 Gillespie Algorithm
25(8)
Exercises
29(4)
2 Deterministic versus Stochastic Modelling
33(26)
2.1 Systems with Multiple Favourable States
34(3)
2.2 Self-Induced Stochastic Resonance
37(5)
2.3 Stochastic Focusing
42(7)
2.4 Designing Stochastic Chemical Systems
49(10)
Exercises
55(4)
3 Stochastic Differential Equations
59(36)
3.1 A Computational Definition of SDE
60(2)
3.2 Examples of SDEs
62(4)
3.3 Fokker-Planck Equation
66(6)
3.4 Boundary Conditions on the Fokker-Planck Equation
72(3)
3.5 Kolmogorov Backward Equation
75(1)
3.6 SDEs with Multiple Favourable States
76(4)
3.7 Chemical Fokker-Planck Equation
80(5)
3.8 Analysis of Problem from Section 2.1
85(3)
3.9 Analysis of Problem from Section 2.2
88(7)
Exercises
92(3)
4 Diffusion
95(42)
4.1 Diffusion Modelled by SDEs
96(4)
4.2 Compartment-Based Approach to Diffusion
100(7)
4.3 Diffusion and Velocity-Jump Processes
107(10)
4.4 Diffusion to Adsorbing Surfaces
117(8)
4.5 Reactive Boundary Conditions
125(5)
4.6 Einstein-Smoluchowski Relation
130(7)
Exercises
133(4)
5 Efficient Stochastic Modelling of Chemical Reactions
137(23)
5.1 A Simple Multiscale Problem
139(3)
5.2 Multiscale SSA with Partial Equilibrium Assumption
142(6)
5.3 Multiscale Modelling
148(3)
5.4 First-Reaction SSA
151(1)
5.5 Exact Efficient SSAs
152(8)
Exercises
157(3)
6 Stochastic Reaction-Diffusion Models
160(32)
6.1 A Compartment-Based Reaction-Diffusion Algorithm
161(3)
6.2 A Reaction-Diffusion SSA Based on the SDE Model of Diffusion
164(2)
6.3 Compartment-Based SSA for Higher-Order Reactions
166(3)
6.4 A Choice of Compartment Size h
169(5)
6.5 Molecular-Based Approaches for Second-Order Reactions
174(3)
6.6 Reaction Radius and Reaction Probability
177(6)
6.7 Modelling Reversible Reactions
183(3)
6.8 Biological Pattern Formation
186(6)
Exercises
190(2)
7 SSAs for Reaction-Diffusion-Advection Processes
192(34)
7.1 SSAs for Diffusion-Advection Processes
193(3)
7.2 Reaction-Diffusion-Advection SSAs
196(3)
7.3 Bacterial Chemotaxis
199(7)
7.4 Collective Behaviour of Locusts
206(5)
7.5 Ions and Ion Channels
211(5)
7.6 Metropolis-Hastings Algorithm
216(10)
Exercises
222(4)
8 Microscopic Models of Brownian Motion
226(42)
8.1 One-Particle Solvent Model
227(6)
8.2 Generalized Langevin Equation
233(9)
8.3 Solvent as Harmonic Oscillators
242(4)
8.4 Solvent as Points Colliding with the Diffusing Particle
246(6)
8.5 Forces Between Atoms and Molecules
252(5)
8.6 Molecular Dynamics
257(11)
Exercises
265(3)
9 Multiscale and Multi-Resolution Methods
268(25)
9.1 Coupling SDE-Based and Compartment-Based Models
270(8)
9.2 Coupling Molecular Dynamics with Langevin Dynamics
278(7)
9.3 Multi-Resolution Molecular and Brownian Dynamics
285(8)
Exercises
289(4)
Appendix
293(4)
Appendix A Deterministic Modelling of Chemical Reactions
293(2)
Appendix B Discrete Probability Distributions
295(1)
Appendix C Continuous Probability Distributions
296(1)
References 297(8)
Index 305
Radek Erban is Professor of Mathematics at the University of Oxford, a Fellow of Merton College, Oxford and a Royal Society University Research Fellow. S. Jonathan Chapman is Professor of Mathematics and its Applications at the University of Oxford, and a Fellow of Mansfield College, Oxford.
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