Burris S. — Number theoretic density and logical limit laws :: Электронная библиотека попечительского совета мехмата МГУ

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Burris S. — Number theoretic density and logical limit laws

Burris S. — Number theoretic density and logical limit laws

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Название: Number theoretic density and logical limit laws

Автор: Burris S.

Аннотация:

From Book News, Inc.
Burris (U. of Waterloo, Canada) divides Kevin Compton's ideas into two parts, one on density in number systems, and the other on the application of number theoretic density results to obtain logical limit laws. The first part, on additive number systems, is suitable as an advanced undergraduate special topics reading text. The second, on multiplicative number systems, demonstrates the challenge and reward of becoming reasonably comfortable with Dirichlet series.Book News, Inc.®, Portland, OR

Product Description:
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.

Язык:

Рубрика: Математика/Алгебра/Комбинаторика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 289

Добавлена в каталог: 07.03.2005

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Предметный указатель

rings      145
$\Lambda$ 0—1 law      106
$\Lambda$ limit law      106
$\mathcal{K}$ -indecomposable      218
$\vec{s}$ , $\vec{Q}$ satisfy $\varphi(\vec{x}, \vec{U})$       105
0-1 law      106
Abelian groups      145
Abelian p-group      25 145
Abscissa of absolute convergence      128
Abscissa of convergence      128
Abscissa of convergence of $\mathcal{A}$       151
Absolutely convergent double series      5
Additive norm      19
Additive number system      20
Adequate class      120 218
Arity of a relation      103
Asymptotic density, global      46 106 159
Asymptotic density, local      45 106 159
Asymptotic to      13
Atomic L-formulas      104
Big O      13
Boolean algebra      145
Boolean algebra of subsets of I      218
Cauchy functional equation      178
Cauchy integral formula      97
Cauchy product      5 238 242
Chain      23
CLIQUE      22
Coefficient      19
Coefficient of       4
Coefficient, large      54
Coefficient, of $n^{-x}$       127
Coefficient, small      54
Commutative monoid      17
Component      22
Component, generalized      120
Compton's Density Theorem      92
Compton's Dirichlet Density Theorem      60
Compton's Tauberian Theorem      87
Counting function(s) of A      25 146
Counting function(s) of B      25 145
Counting function(s), local      25
CYCLE      22
Direct product of structures      218
Dirichlet convolution      128
Dirichlet density      48 160
Dirichlet polynomial      131
Dirichlet product      128
Dirichlet series      127
Dirichlet series expansion      133
Disjoint union      114
Disjunctive form      224
Domain of convergence      240
Domain of f      233
Double series      5
Ehrenfeucht — Fraisse game      109
empty function      233
Equivalence classes      22
Equivalence relation      22
Equivalent formulas      108
Euler product      147
Exponent, large      167
Exponent, small      167
Exponentially bounded function      15
Extended reals      3
Extension      234
Factorisatio Numerorum      94
Factorization      143
Fast growing      32 153
Finitely generated additive number system      39
First-order 0-1 law      217
First-order limit law      217
First-order logic      103
Formal addition      238 242
Formal multiplication      238 242
Formal scalar multiplication      238 242
Free commutative monoid      18
Free generators      18
Fteferman — Vaught sequence      219
Fteferman — Vaught Theorem      218
Fundamental identity of A      26 147
Generating series of B      25 146
Global (cumulative) count      33
Global asymptotic density      217
Global spectrum of $\mathcal{K}$       121
Graph, reflexive      22
Hua's Theorem      41
Identity function      233
Indecomposable elements      17 143
Indecomposable member of $\mathcal{K}$       120 218
Infinite Cauchy product      6
Infinite Dirichlet product      130 131
Interpretation      103
Irreflexive      22
Isomorphic to      20 144
Isomorphism      21 144
Isomorphism type      106 217
L-formula, first-order      104
L-formula, monadic second-order      104

L-structure      103
Labeled structure      xiv
Language      103
Language purely relational      103
Language relational      103
Left inverse      246
Limit law      106
Linear forest graph      23
Linear forest poset      23
Little o      13
Local asymptotic density      217
Local spectrum of $\mathcal{K}$       121
Monadic second-order logic      104
Multiplicative norm      143
Multiplicative number system      143
Neighborhood      235
nth derivative      237
Odlyzko's Product Theorem      197
Partition identity      26
Partition of $\mathcal{K}$       120 226
Partition of P      54 167
Partition set      55 167
Partition, a positive integer      26
Partition, class of $\mathcal{K}$       120 226
Permutation      22
Perron integral formula      210
Play of the game      109
Polynomial function      236
Polynomially bounded function      140
Power series      4
Power series, expansion      9
Power series, function      240
Prenex form      224
Primes of $\mathcal{A}$       143
Propositional connectives      104
Quantifier elimination      223
Quantifier rank      107
Quantifier symbols      104
Radius of convergence      5 241
Radius of convergence of $\mathcal{A}$       31
Range of f      233
Rank of      4 20 143
Rank of a free commutative monoid      18
Ratio Test      13
Real function      233
Reduced additive number system      37
Reduced form of $\mathcal{A}$       38
Refined counting function      251 256
Reflexive law      22
Regular variation at infinity of index a,      136
Restriction      234
Riemann hypothesis      152
Riemann zeta function      146
Right inverse      246
Rigid      xiv 24
Sarkoezy's Density Theorem      206
Satisfies      104
Schur's Tauberian Theorem      62
Sentence      104
Set of additives of an element      51
Set of isomorphism types      106 217
Set of multiples of an element      164
Simple partition set      67 188
Single valued      233
Slowly growing      32 153
Slowly varying at infinity      136
Stabilize      131
Standard assumption      51 164
Stewart's Sum Theorem      83
Subexponential function      15
Subpolynomial function      140
Subsystem      20 144
Subsystem, generated by      20 144
Subsystem, proper      20 144
Subsystem, trivial      20 144
Successful for k rounds      110
Sum of additive number systems      83
Superpolynomial growth of a function      43
Support of a function      37
Symbol, constant      103
Symbol, equality      103
Symbol, function      103
Symbol, relation      103
Symmetric law      22
Tauberian theorem      64
Transitive Law      22
Truncation      7 131 243
Two variable, fundamental identity of $\mathcal{A}$       251 256
Two variable, generating series      251 256
Two-colored linear forest graph      24
Two-colored linear forest poset      24
Universe      103
Unlabeled structure      xiv
Variable, bound      104
Variable, first-order      104
Variable, free      104
Variable, monadic second-order      104
Winning strategy      110
Wins the play      110

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