Power Geometry in Algebraic and Differential Equations
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Table of contents
- Cover
- Contentsvii
- Prefacev
- Chapter 0. Introduction1
- 1. Concepts of Power Geometry1
- 2. Historical remarks4
- 3. A brief survey of the book5
- Chapter 1. The linear inequalities9
- 1. Principal definitions and properties9
- 2. The normal and tangent cones14
- 3. Graphical solution of Problem 117
- 4. The Motzkin–Burger algorithm19
- 5. Algorithmic solution of Problem 122
- 6. Cone of the problem32
- 7. About the computer program34
- 8. An infinite set S39
- 9. Coherent boundary subsets43
- 10. Comparison with the Bugaev–Sintsov method46
- 11. Linear transformations49
- Chapter 2. Singularities of algebraic equations55
- 1. Implicit function55
- 2. Newton polyhedron59
- 3. Power transformations63
- 4. Asymptotic solution of an algebraic equation65
- 5. Implicit functions71
- 6. Truncated systems of equations72
- 7. Linear transformations of power exponents77
- 8. Asymptotic solution of a system of equations81
- 9. Positional functions of mechanisms88
- 10. Historical and bibliographical remarks99
- Chapter 3. Asymptotics of solutions to a system of ODE105
- 1. Local theorems of existence105
- 2. The power transformation110
- 3. The generalized power transformations116
- 4. Truncated systems122
- 5. The power asymptotics128
- 6. Logarithmic asymptotics135
- 7. The simplex systems145
- 8. A big example150
- 9. Remarks158
- Chapter 4. Hamiltonian truncations161
- 1. The theory161
- 2. The generalized Henon–Heiles system171
- 3. The Sokol'skii cases of zero frequencies174
- 4. The restricted three-body problem186
- Chapter 5. Local analysis of an ODE system191
- 1. Introduction191
- 2. Normal form of a linear system195
- 3. The Newton polyhedron196
- 4. The reduction of System (3.10)203
- 5. The classification of System (4.2)205
- 6. The normal form of a nonlinear system214
- 7. Cases I and γ1219
- 8. System (4.2) in Cases II and IV221
- 9. The non-resonant case III223
- 10. The normal form in the resonant Case III228
- 11. The resonances of higher order236
- 12. The resonance 1:3 in Case III240
- 13. The resonance 1:2 in Case III244
- 14. The normal form in Case γ2247
- 15. The normal form in Cases γ0 and γ3251
- 16. The review of the results for System (4.2)259
- 17. The transference of results to the original system261
- 18. The comparison with the Hamiltonian normal form262
- 19. The case μ=0264
- 20. The Belitskii normal form264
- 21. The problem of surface waves271
- 22. On the supernormal form273
- Chapter 6. Systems of arbitrary equations277
- 1. Truncated systems277
- 2. Power transformations286
- 3. The logarithmic transformation290
- 4. A big example293
- 5. One partial differential equation298
- 6. The viscous fluid flow around a plate300
- Chapter 7. Self-similar solutions315
- 1. Supports of a function315
- 2. Supports of a differential polynomial316
- 3. The Lie operators317
- 4. Self-similar solutions319
- 5. The power transformation325
- 6. The logarithmic transformation329
- 7. The ordinary differential equation332
- 8. The system of equations336
- Chapter 8. On complexity of problems of Power Geometry341
- 1. The levels of complexity341
- 2. The linear equalities343
- 3. The linear transformations347
- 4. Linear inequalities349
- 5. On applications of Power Geometry352
- 6. Historical remarks353
- Bibliography359
- Subject index383
The geometry of power exponents includes the Newton polyhedron, normal cones of its faces, power and logarithmic transformations. On the basis of the geometry universal algorithms for simplifications of systems of nonlinear equations (algebraic, ordinary differential and partial differential) were developed.
The algorithms form a new calculus which allows to make local and asymptotical analysis of solutions to those systems.
The efficiency of the calculus is demonstrated with regard to several complicated problems from Robotics, Celestial Mechanics, Hydrodynamics and Thermodynamics. The calculus also gives classical results obtained earlier intuitively and is an alternative to Algebraic Geometry, Differential Algebra, Lie group Analysis and Nonstandard Analysis.
The algorithms form a new calculus which allows to make local and asymptotical analysis of solutions to those systems.
The efficiency of the calculus is demonstrated with regard to several complicated problems from Robotics, Celestial Mechanics, Hydrodynamics and Thermodynamics. The calculus also gives classical results obtained earlier intuitively and is an alternative to Algebraic Geometry, Differential Algebra, Lie group Analysis and Nonstandard Analysis.