Number theoretic density and logical limit laws / Stanley N. Burris.

Format:

Book

Author/Creator:

Burris, Stanley, author.

Series:

Mathematical surveys and monographs ; no. 86.

Mathematical surveys and monographs ; volume 86

Language:

English

Subjects (All):

Generating functions.

Limit theorems (Probability theory).

Physical Description:

1 online resource (313 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2001]

Language Note:

English

Summary:

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 Ehrenfeucht-Fraissé games, the Feferman-Vaught 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.

Contents:

""Contents""; ""Preface""; ""Overview""; ""Notation Guide""; ""Part 1. Additive Number Systems""; ""Chapter 1. Background from Analysis""; ""1.1. Series and infinite products""; ""1.2. Power series expansions""; ""1.3. Big O, little o, and ~ notation""; ""1.4. The radius of convergence and ...""; ""1.5. Growth rate of coefficients""; ""Chapter 2. Counting Functions and Fundamental Identities""; ""2.1. Defining additive number systems""; ""2.2. Examples of additive number systems""; ""2.3. Counting functions, fundamental identities""; ""2.4. Global counts""

""2.5. Alternate version of the fundamental identity""""2.6. Reduced additive number systems""; ""2.7. Finitely generated number systems""; ""2.8. a*(n) is eventually positive""; ""Chapter 3. Density and Partition Sets""; ""3.1. Asymptotic density""; ""3.2. Dirichlet density""; ""3.3. The standard assumption""; ""3.4. The set of additives of an element""; ""3.5. Partition sets""; ""3.6. Generating series of partition sets""; ""3.7. Partition sets have Dirichlet density""; ""3.8. Schur's Tauberian Theorem""; ""3.9. Simple partition sets""; ""3.10. The asymptotic density of ...""

""8.3. Counting functions, fundamental identities""""8.4. Alternate version of the fundamental identity""; ""8.5. Finitely generated multiplicative number systems""; ""Chapter 9. Density and Partition Sets""; ""9.1. Asymptotic density""; ""9.2. Dirichlet density""; ""9.3. The standard assumption""; ""9.4. The set of multiples of an element""; ""9.5. Partition sets""; ""9.6. Generating series of partition sets""; ""9.7. Partition sets have Dirichlet density""; ""9.8. Discrete multiplicative number systems""; ""9.9. When sets bA have global asymptotic density""

""9.10. The strictly multiplicative case and RV[sub(α)]""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 281-283) and index.

Description based on print version record.

ISBN:

1-4704-1313-2