Hardcover ISBN:  9780821826669 
Product Code:  SURV/86 
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eBook ISBN:  9781470413132 
Product Code:  SURV/86.E 
List Price:  $125.00 
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Hardcover ISBN:  9780821826669 
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Product Code:  SURV/86.B 
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Hardcover ISBN:  9780821826669 
Product Code:  SURV/86 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470413132 
Product Code:  SURV/86.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821826669 
eBook ISBN:  9781470413132 
Product Code:  SURV/86.B 
List Price:  $254.00$191.50 
MAA Member Price:  $228.60$172.35 
AMS Member Price:  $203.20$153.20 

Book DetailsMathematical Surveys and MonographsVolume: 86; 2001; 289 ppMSC: Primary 03; 05; 11;
This book shows how a study of generating series (power series in the additive case and Dirichlet series in the multiplicative case), combined with structure theorems for the finite models of a sentence, lead to general and powerful results on limit laws, including \(0  1\) laws. The book is unique in its approach to giving a combined treatment of topics from additive as well as from multiplicative number theory, in the setting of abstract number systems, emphasizing the remarkable parallels in the two subjects. Much evidence is collected to support the thesis that local results in additive systems lift to global results in multiplicative systems.
All necessary material is given to understand thoroughly the method of Compton for proving logical limit laws, including a full treatment of EhrenfeuchtFraissé games, the FefermanVaught Theorem, and Skolem's quantifier elimination for finite Boolean algebras. An intriguing aspect of the book is to see so many interesting tools from elementary mathematics pull together to answer the question: What is the probability that a randomly chosen structure has a given property? Prerequisites are undergraduate analysis and some exposure to abstract systems.ReadershipGraduate students and research mathematicians interested in combinatorics, number theory and logic.

Table of Contents

Chapters

1. Background from analysis

2. Counting functions and fundamental identities

3. Density and partition sets

4. The case $\rho = 1$

5. The case $0 < \rho < 1$

6. Monadic secondorder limit laws

7. Background from analysis

8. Counting functions and fundamental identities

9. Density and partition sets

10. The case $\alpha = 0$

11. The case $0 < \alpha < \infty $

12. Firstorder limit laws


Additional Material

Reviews

Shows an exciting connection between combinatorics, number theory and logic, and certainly deserves to be more widely known. The book gives a very clear account of it and it is easily readable.
Zentralblatt MATH 
This book is a lucid, selfcontained introduction to a fascinating interaction between analysis, combinatorics, logic and number theory … accessible to an undergraduate and gives interesting examples to illustrate the concepts.
Mathematical Reviews


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This book shows how a study of generating series (power series in the additive case and Dirichlet series in the multiplicative case), combined with structure theorems for the finite models of a sentence, lead to general and powerful results on limit laws, including \(0  1\) laws. The book is unique in its approach to giving a combined treatment of topics from additive as well as from multiplicative number theory, in the setting of abstract number systems, emphasizing the remarkable parallels in the two subjects. Much evidence is collected to support the thesis that local results in additive systems lift to global results in multiplicative systems.
All necessary material is given to understand thoroughly the method of Compton for proving logical limit laws, including a full treatment of EhrenfeuchtFraissé games, the FefermanVaught Theorem, and Skolem's quantifier elimination for finite Boolean algebras. An intriguing aspect of the book is to see so many interesting tools from elementary mathematics pull together to answer the question: What is the probability that a randomly chosen structure has a given property? Prerequisites are undergraduate analysis and some exposure to abstract systems.
Graduate students and research mathematicians interested in combinatorics, number theory and logic.

Chapters

1. Background from analysis

2. Counting functions and fundamental identities

3. Density and partition sets

4. The case $\rho = 1$

5. The case $0 < \rho < 1$

6. Monadic secondorder limit laws

7. Background from analysis

8. Counting functions and fundamental identities

9. Density and partition sets

10. The case $\alpha = 0$

11. The case $0 < \alpha < \infty $

12. Firstorder limit laws

Shows an exciting connection between combinatorics, number theory and logic, and certainly deserves to be more widely known. The book gives a very clear account of it and it is easily readable.
Zentralblatt MATH 
This book is a lucid, selfcontained introduction to a fascinating interaction between analysis, combinatorics, logic and number theory … accessible to an undergraduate and gives interesting examples to illustrate the concepts.
Mathematical Reviews