Combinatorial Methods

Jerome K. Percus

ISBN 10: 0387900276 / ISBN 13: 9780387900278

Editore: Springer New York, 1971

Dieser Artikel ist ein Print on Demand Artikel und wird nach Ihrer Bestellung fuer Sie gedruckt. It is not a large overstatement to claim that mathematics has traditionally arisen from attempts to understand quite concrete events in the physical world. The accelerated sophistication of the mathematical community has perhaps obscured this fact, especial. Codice articolo 5911575

Riassunto:

But the mathematical sophistication of scientists has grown rapidly too, as has the scientific sophistication of many mathematicians, and the real worl- suitably defined - is once more serving its traditional role.

Contenuti: I. Counting and Enumeration on a Set.- A. Introduction.- 1. Set Generating Functions.- 2. Numerical Generating Functions.- Examples.- Fibonacci Numbers.- B. Counting with Restrictions — Techniques.- 1. Inclusion - Exclusion Principle.- The Euler Function.- Rencontres, Derangement or Montmort Problem.- The Menage Problem.- 2. Permutations with Restricted Position. The Master Theorem.- Exercises.- Example.- Rencontre Problem.- Menage Problem.- 3. Extension of the Master Theorem.- C. Partitions, Compositions and Decompositions.- 1. Permutation Counting as a Partition Problem.- a) Counting with allowed transitions.- b) Counting with prohibited transitions.- 2. Classification of Partitions.- a) Distribution of unlabeled objects: Compositions.- b) Distribution of unlabeled objects: Partitions.- 3. Ramsey’s Theorem.- Example.- 4. Distribution of Labeled Objects.- a) Distinguishable boxes.- b) Collections of pairs — graph theory.- c) Indistinguishable boxes (and labeled objects).- d) Partially labeled graphs — The Polya Theorem.- Examples.- Proof of Polya’s Theorem.- Examples.- Exercises.- e) Counting unrooted (free) unlabeled graphs.- Dissimilarity Theorem.- Example.- II. Counting and Enumeration on a Regular Lattice.- A. Random Walk on Lattices.- 1. Regular Cubic Lattices.- Examples.- 2. General Lattices.- i) Nearest neighbor random walk on a face centered cubic lattice.- ii) Nearest neighbor random walk on a body centered cubic lattice.- B. One Dimensional Lattices.- 1. The Ballot Problem.- Example.- 2. One Dimensional Lattice Gas.- C. Two Dimensional Lattices.- 1. Counting Figures on a Lattice, General Algebraic Approach.- 2. The Dimer Problem — Transfer Matrix Method.- Exercises.- 3. The Dimer Problem — Pfaffian Method.- Exercises.- 4. The Dimer Problem — First Permanent Method.- 5. The Dimer Problem — Second Permanent Method.- D. Counting Patterns on Two Dimensional Lattices.- 1. The Ice Problem — Introduction.- 2. Square Ice — The Transfer Matrix Method.- 3. Square Ice — Exact Solution.- 4. Other Hydrogen Bonded Models — Dimer Solution.- E. The Ising Model.- 1. Introduction.- 2. Estimates of the Curie Temperature.- 3. Combinatorial Solution of the Ising Model.- 4. Other Combinatorial Solutions.- 5. Spin Correlations.

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Titolo: Combinatorial Methods
Casa editrice: Springer New York
Data di pubblicazione: 1971
Legatura: Kartoniert / Broschiert
Condizione: New