Umfang:
Online-Ressource (XIII, 159 S)
ISBN:
9780883850244
Serie:
The Carus mathematical monographs 22
Anmerkung:
Description based upon print version of record
,
""Front Cover""; ""Random Walks and Electric Networks""; ""Copyright Page""; ""Preface""; ""Contents""; ""Part I: Random Walks on Finite Networks""; ""Chapter 1. Random Walks in One Dimension""; ""1.1. A random walk along Madison Avenue""; ""1.2. The same problem as a penny matching game""; ""1.3. The probability of winning: basic properties""; ""1.4. An electric network problem: the same problem?""; ""1.5. Harmonic functions in one dimension; the Uniqueness Principle""; ""1.6. The solution as a fair game (martingale)""; ""Chapter 2. Random Walks in Two Dimensions""; ""2.1. An example""
,
""2.2. Harmonic functions in two dimensions""""2.3. The Monte Carlo solution""; ""2.4. The original Dirichlet problem; the method of relaxations""; ""2.5. Solution by solving linear equations""; ""2.6. Solution by the method of Markov chains""; ""Chapter 3. Random Walks on More General Networks""; ""3.1. General resistor networks and reversible Markov chains""; ""3.2. Voltages for general networks; probabilistic interpretation""; ""3.3. Probabilistic interpretation of current""; ""3.4. Effective resistance and the escape probability""; ""3.5. Currents minimize energy dissipation""
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""Chapter 4. Rayleigh's Monotonicity Law""""4.1. Rayleigh's Monotonicity Law""; ""4.2. A probabilistic explanation of the Monotonicity Law""; ""4.3. A Markov chain proof of the Monotonicity Law""; ""Part II: Random Walks on Infinite Networks""; ""Chapter 5. Pólya's Recurrence Problem""; ""5.1. Random walks on lattices""; ""5.2. The question of recurrence""; ""5.3. Polya's original question""; ""5.4. Polya's Theorem: recurrence in the plane, transience in space""; ""5.5. The escape probability as a limit of escape probabilities for finite graphs""
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""5.6. Electrical formulation of the type problem""""5.7. One dimension is easy, but what about higher dimensions?""; ""5.8. Getting around the lack of rotational symmetry of the lattice""; ""5.9. Rayleigh: shorting shows recurrence in the plane, cutting shows transience in space""; ""Chapter 6. Rayleigh's Short-Cut Method""; ""6.1. Shorting and cutting""; ""6.2. The Shorting Law and the Cutting Law; Rayleigh's idea""; ""6.3. The plane is easy""; ""6.4. Space: searching for a residual network""; ""6.5. Trees are easy to analyze""; ""6.6. The full binary tree is too big""
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""6.7. NT3: a ""three-dimensional"" tree""""6.8. NT3 has finite resistance""; ""6.9. But does NT3 fit in the three-dimensional lattice?""; ""6.10. What we have done; what we will do""; ""Chapter 7. The Classical Proofs of Pólya's Theorem""; ""7.1. Recurrence is equivalent to an infinite expected number of returns""; ""7.2. Simple random walk in one dimension""; ""7.3. Simple random walk in two dimensions""; ""7.4. Simple random walk in three dimensions""; ""7.5. The probability of return in three dimensions: exact calculations""
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""7.6. Simple random walk in two dimensions is the same as two independent one-dimensional random walks""
Weitere Ausg.:
ISBN 9781614440222
Weitere Ausg.:
Druckausg. Doyle, Peter G. Random walks and electric networks Washington, DC : Mathematical Association of America, 1984 ISBN 0883850249
Sprache:
Englisch
Fachgebiete:
Mathematik
Schlagwort(e):
Elektrisches Netzwerk
;
Irrfahrtsproblem
;
Elektrisches Netzwerk
;
Irrfahrtsproblem
;
Electronic books
Mehr zum Autor:
Snell, James Laurie 1925-2011
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