Description 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.
Contents Preface. 0.Introduction. 1. Concepts of Power Geometry. 2. Historical remarks. 3. A brief survey of the book. 1. The linear inequalitites.
1. Principal definitions and properties. 2. The normal and tangent cones. 3. Graphical solution of Problem 1. 4. The Motzkin-Burger algorithm.
5. Algorithmic solution of Problem 1. 6. Cone of the problem. 7. About the computer program. 8. An infinite set S. 9. Coherent boundary
subsets. 10. Comparison with the Bugaev-Sintsov method. 11. Linear transformations.2. Singularities of algebraic equations. 1. Implicit
function. 2. Newton polyhedron. 3. Power transformations. 4. Asymptotic solution of an algebraic equation. 5. Implicit functions. 6.
Truncated systems of equations. 7. Linear transformations of power exponents. 8. Asymptotic solution of a system of equations. 9. Positional
functions of mechanisms. 10. Historical and bibliographical remarks. 3. Asymptotics of solutions to a system of ODE. 1. Local theorems
of existence. 2. The power transformation. 3. The generalized power transformations. 4. Truncated systems. 5. The power asymptotics.
6. Logarithmic asymptotics. 7. The simplex systems. 8. A big example. 9. Remarks. 4. Hamiltonian truncations. 1. The theory. 2. The generalized
Henon-Heiles system. 3. The Sokol'skii cases of zero frequencies. 4. The restricted three-body problem. 5. Local analysis of an ODE system.
1. Introduction. 2. Normal form of a linear system. 3. The Newton polyhedron. 4. The reduction of System (3.10) 5. The classification
of System (4.2) 6. The normal form of a nonlinear system. 7. Cases I and γ1. 8. System (4.2) in Cases II and IV. 9. The non-resonant
case III. 10. The normal form in the resonant Case III. 11. The resonances of higher order. 12. The resonance 1:3 in Case III. 13. The
resonance 1:2 in Case III. 14. The normal form in Case γ2. 15. The normal form in Cases γ0 and γ3. 16. The review of the
results for System (4.2). 17. The transference of results to the original system. 18. The comparison with the Hamiltonian normal form.
19. The case μ=0. 20. The Belitskii normal form. 21. The problem of surface waves. 22. On the supernormal form. 6. Systems of
arbitrary equations. 1. Truncated systems. 2. Power transformations. 3. The logarithmic transformation. 4. A big example. 5. One partial
differential equation. 6. The viscous fluid flow around a plate. 7. Self-similar solutions. 1. Supports of a function. 2. Supports of
a differential polynomial. 3. The Lie operators. 4. Self-similar solutions. 5. The power transformation. 6. The logarithmic transformation.
7. The ordinary differential equation. 8. The system of equations. 8. On complexity of problems of Power Geometry. 1. The levels of
complexity. 2. The linear equalities. 3. The linear transformations. 4. Linear inequalities. 5. On applications of Power Geometry. 6.
Historical remarks. Bibliography. Subject index.
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