Xia B. Automated inequality proving and discovering (Singapore; Hackensack, London, 2016). - ОГЛАВЛЕНИЕ / CONTENTS
Навигация
Архив выставки новых поступлений | Отечественные поступления | Иностранные поступления | Сиглы
ОбложкаXia B. Automated inequality proving and discovering / B.Xia, L.Yang. - Singapore; Hackensack, London: World Scientific, 2016. - xii, 332 p.: ill. - ISBN 978-981-4759-11-3
Шифр: (И/В16-X58) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
Preface ......................................................... v

1    Basics of Elimination Method ............................... 1
1.1  Pseudo-division ............................................ 1
1.2  Resultants ................................................. 5
1.3  Subresultants .............................................. 8
     1.3.1  The Habicht Theorem ................................ 10
     1.3.2  Subresultant Chain Theorem ......................... 14
     1.3.3  Subresultant Polynomial Remainder Sequence ......... 18

2    Zero Decomposition of Polynomial System ................... 25
2.1  Notations ................................................. 25
2.2  Wu's Zero Decomposition ................................... 26
2.3  Relatively Simplicial Decomposition (RSD) ................. 29
     2.3.1  Regular Chain ...................................... 29
     2.3.2  RSD ................................................ 32
2.4  Weakly RSD ................................................ 37
     2.4.1  Concepts and Definitions ........................... 37
     2.4.2  Algorithm WRSD ..................................... 40
     2.4.3  Correctness of Algorithm WRSD ...................... 41
2.5  Generic Regular Decomposition ............................. 47
2.6  Zero Decomposition Keeping Multiplicity ................... 50
     2.6.1  Multiplicity ....................................... 52
     2.6.2  Zero Decomposition Keeping Multiplicity ............ 54

3    Triangularization of Semi-Algebraic System ................ 57
3.1  Triangular SAS ............................................ 58
3.2  Triangular Decomposition of SASs .......................... 60
     3.2.1  Generic Zero-Dimensional Case ...................... 61
     3.2.2  Positive Dimensional Case .......................... 66

4    Real Root Counting ........................................ 77
4.1  Classical Results ......................................... 77
4.2  Discrimination Systems for Polynomials .................... 83
4.3  Proof of Discrimination Theorem ........................... 92
4.4  Properties of Discrimination Matrix ....................... 97

5    Real Root Isolation ...................................... 109
5.1  Real Root Isolation for Polynomials ...................... 109
5.2  Real Root Isolation for Constant Semi-Algebraic Systems .. 112
     5.2.1  Interval Arithmetic ............................... 113
     5.2.2  Algorithm ......................................... 114
     5.2.3  Examples .......................................... 119
5.3  Real Root Counting for Constant Semi-Algebraic Systems ... 121
5.4  Termination of Linear Programs ........................... 125

6    Real Root Classification ................................. 133
6.1  Border Polynomial and Discrimination Polynomial .......... 134
6.2  Generic Zero-Dimensional Case ............................ 135
     6.2.1  Regular Zero-Dimensional TSA ...................... 135
     6.2.2  Generic Zero-Dimensional SAS ...................... 139
     6.2.3  Algorithm ......................................... 140
6.3  Positive Dimensional and Over-Determined Cases ........... 144
6.4  DISCOVERER ............................................... 148
6.5  Automated Discovering of Geometric Inequalities .......... 152
6.6  Algebraic Analysis of Biological Systems ................. 163
6.7  Program Verification Through SASs Solving ................ 169
     6.7.1  Non-linear Ranking Function Discovering ........... 170
     6.7.2  Reachability Computation .......................... 173

7    Open Weak CAD ............................................ 181
7.1  Quantifier Elimination and Cylindrical Algebraic
     Decomposition ............................................ 182
7.2  Open CAD ................................................. 186
7.3  Projection Operator Np ................................... 190
     7.3.1  An Illustrative Example ........................... 190
     7.3.2  Notations ......................................... 192
     7.3.3  Algorithm DPS ..................................... 193
     7.3.4  The Correctness of Algorithm DPS .................. 196
     7.3.5  Examples of Proving Polynomial Inequalities ....... 202
     7.3.6  Polynomial Optimization via Np .................... 204
7.4  Open Weak CAD ............................................ 211
     7.4.1  Concepts .......................................... 212
     7.4.2  Projection Operator Hp ............................ 214
     7.4.3  Computing Open Sample ............................. 224
     7.4.4  Combining Hp and Np ............................... 226
     7.4.5  Examples .......................................... 229

8    Dimension-Decreasing Algorithm ........................... 233
8.1  Inequalities with Radicals ............................... 233
8.2  Dimension-Decreasing Algorithm ........................... 235
     8.2.1  Concepts .......................................... 235
     8.2.2  Algorithm ......................................... 239
8.3  Inequalities on Triangles ................................ 240
8.4  BOTTEMA .................................................. 242
     8.4.1  Inequality Proving with BOTTEMA ................... 244
     8.4.2  Non-linear Optimization with BOTTEMA .............. 246

9    SOS Decomposition ........................................ 251
9.1  Preliminary .............................................. 253
9.2  Convex Cover Polynomial .................................. 257
9.3  Split Polynomial ......................................... 261
9.4  Algorithm ................................................ 264
9.5  Experiments .............................................. 269

10   Successive Difference Substitution ....................... 271
     10.1 Basic Idea .......................................... 271
     10.1.1 An Example ........................................ 271
     10.1.2 Difference Substitution ........................... 273
     10.1.3 Successive Difference Substitution ................ 278
     10.2 Weighted Successive Difference Substitution ......... 283
     10.2.1 Concepts .......................................... 283
     10.2.2 Geometric Meaning ................................. 284
     10.2.3 Termination ....................................... 287
10.3 Examples ................................................. 290
10.4 Polya's Theorem .......................................... 293

11   Proving Inequalities Beyond the Tarski Model ............. 297
11.1 Symmetric Forms of Degrees Less Than Five ................ 298
     11.1.1 Problem ........................................... 298
     11.1.2 Algorithm ......................................... 301
     11.1.3 Examples and Discussion ........................... 306
     11.2.4 Class of Symmetric Forms of Any Degrees ........... 307
     11.2.1 Problem ........................................... 307
     11.2.2 Algorithm ......................................... 309
     11.2.3 Examples .......................................... 312

Bibliography .................................................. 315
Index ......................................................... 327

Архив выставки новых поступлений | Отечественные поступления | Иностранные поступления | Сиглы
 

[О библиотеке | Академгородок | Новости | Выставки | Ресурсы | Библиография | Партнеры | ИнфоЛоция | Поиск | English]
  Пожелания и письма: www@prometeus.nsc.ru
© 1997-2018 Отделение ГПНТБ СО РАН (Новосибирск)
Статистика доступов: архив | текущая статистика
 

Документ изменен: Thu Apr 5 17:11:24 2018. Размер: 11,078 bytes.
Посещение N 161 c 24.10.2017