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Continued fractions /
This book is the first authoritative and up-to-date survey of the history of Iraq from earliest times to the present in any language. It presents a concise narrative of the rich and varied history of this land, drawing on political, social, economic, artistic, technological, and intellectual materia...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Hackensack, N.J. :
World Scientific,
©2006.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Preface
- 1. Introduction. 1.1. The additive subgroup of the integers generated by a and b. 1.2. Continuants. 1.3. The continued fraction expansion of a real number. 1.4. Quadratic irrationals. 1.5. The tree of continued fraction expansions. 1.6. Diophantine approximation. 1.7. Other known continued fraction expansions
- 2. Generalizations of the gcd and the Euclidean algorithm. 2.1. Other gcd's. 2.2. Continued fraction expansions for complex numbers. 2.3. The lattice reduction algorithm of Gauss
- 3. Continued fractions with small partial quotients. 3.1. The sequence ({n[symbol]}) of multiples of a number. 3.2. Discrepancy. 3.3. The sum of {n[symbol]} from 1 to N
- 4. Ergodic theory. 4.1. Ergodic maps. 4.2. Terminology. 4.3. Nair's proof. 4.4. Generalization to E[symbol]. 4.5. A Natural extension of the dynamic system (E[symbol], [symbol], T)
- 5. Complex continued fractions. 5.1. The Schmidt regular chains algorithm. 5.2. The Hurwitz complex continued fraction. 5.3. Notation. 5.4. Growth of |qn| and the quality of the Hurwitz approximations. 5.5. Distribution of the remainders. 5.6. A class of algebraic approximants with atypical Hurwitz continued fraction expansions. 5.7. The Gauss-Kuz'min density for the Hurwitz algorithm
- 6. Multidimensional diophantine approximation. 6.1. The Hermite approximations to a real number. 6.2. The Lagarias algorithm in higher dimensions. 6.3. Convexity of expansion domains in the Lagarias algorithm
- 7. Powers of an algebraic integer. 7.1. Introduction. 7.2. Outline and plan of proof. 7.3. Proof of the existence of a unit [symbol][symbol][symbol]([symbol]) oF degree n. 7.4. The sequence v[k] of units with comparable conjugates. 7.5. Good units and good denominators. 7.6. Ratios of consecutive good q. 7.7. The surfaces associated with the scaled errors. 7.8. The general case of algebraic numbers in Q([symbol])
- 8. Marshall Hall's theorem. 8.1. The binary trees of E[symbol]. 8.2. Sums of bridges covering [[symbol], [symbol]]. 8.3. The Lagrange and Markoff spectra
- 9. Functional-analytic techniques. 9.1. Continued fraction cantor sets. 9.2. Spaces and operators. 9.3. Positive operators. 9.4. An integral representation of g[symbol]. 9.5. A Hilbert space structure for G when s = [symbol] is real. 9.6. The uniform spectral gap. 9.7. Log convexity of [symbol][symbol]
- 10. The generating function method. 10.1. Entropy. 10.2. Notation. 10.3. A sampling of results
- 11. Conformal iterated function systems
- 12. Convergence of continued fractions. 12.1. Some general results and techniques. 12.2. Special analytic continued fractions.