# Power Geometry as a New Calculus

• Alexander D. Bruno
Chapter
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 10)

## Abstract

Power Geometry develops Differential Calculus and aims at nonlinear problems. The algorithms of Power Geometry allow to simplify equations,to resolve their singularities, to isolate their first approximations, and to find either their solutions or the asymptotics of the solutions. This approach allows to compute also the asymptotic and the local expansions of solutions. Algorithms of Power Geometry are applicable to equations of various types: algebraic, ordinary differential and partial differential, and also to systems of such equations. Power Geometry is an alternative to Algebraic Geometry, Group Analysis, Nonstandard Analysis, Microlocal Analysis etc.

## Keywords

Normal Cone Differential Calculus Nonstandard Analysis Power Expansion Newton Polyhedron
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
Bruno, A.D.: The asymptotic behavior of solutions to nonlinear systems of differential equations. DAN SSSR 143:4 (1962), 763–766 (Russian) = Soviet Math. Dokl. 3 (1962), 464–467.Google Scholar
2. [2]
Bruno, A.D.: Local Methods in Nonlinear Differential Equations. Nauka, Moscow, 1979 ( Russian) = Springer, Berlin, 1989.Google Scholar
3. [3]
Bruno, A.D.: Power Geometry in Algebraic and Differential Equations. Fizmatlit, Moscow, 1998 ( Russian) = Elsevier Science, Amsterdam, 2000.Google Scholar
4. [4]
Bruno, A.D.: Self-similar solutions and power geometry. Uspekhi Mat. Nauk 55:1 (2000) 3–44 (Russian) = Russian Math. Surveys 55: 1 (2000), 1–42.Google Scholar
5. [5]
Bruno, A.D.: Power expansions of solutions to a single algebraic or differential equation. DAN 380:2 (2001), 155–159 (Russian) = Doklady Mathematics 64: 2 (2001), 160–164.Google Scholar
6. [6]
Bruno, A.D.: Power expansions of solutions to a system of algebraic and differential equations. DAN 380:3 (2001) (Russian) 298–304 = Doklady Mathematics 64: 2 (2001), 180–186.Google Scholar
7. [7]
Bruno, A.D., Varin, V.P.: The limit problems for the equation of oscillations of a satellite. Celestial Mechanics and Dynamical Astronomy 61: 1 (1997), 1–40.
8. [8]
Denk, R., Menniken, R., Volevich, L.: The Newton polygon and elliptic problems with parameter. Mathematische Nachrichten 192 (1998), 125–157.
9. [9]
Bruno, A.D.: Power expansions of solutions to one algebraic or differential equation. Preprint No. 63 of Inst. Appl. Math., Moscow, 2000 ( Russian).Google Scholar
10. [10]
Bruno, A.D.: Power expansions of solutions to a system of algebraic and differential equations. Preprint No. 68 of Inst. Appl. Math., Moscow, 2000 ( Russian).Google Scholar
11. [11]
Aranson, A.B.: Computation and applications of the Newton polyhedrons. Mathematics and Computers in Simulation 57: 3 (2001), 155–160.
12. [12]
Vasiliev, M.M.: Obtaining the self-similar asymptotics of solutions to the NavierStokes equations by Power Geometry. Proceedings of the 3rd ISAAC Congress, Subsection I-2. World Scientific, Singapore, to appear.Google Scholar
13. [13]
Bruno, A.D., Lunev, V.V.: The modified system of equations, describing motions of a rigid body. Preprint No. 49 of Inst. Appl. Math., Moscow, 2001 ( Russian).Google Scholar
14. [14]
Bruno, A.D., Lunev, V.V.: Local expansions of modified motions of a rigid body. Preprint No. 73 of Inst. Appl. Math., Moscow, 2001 ( Russian).Google Scholar
15. [15]
Bruno, A.D., Lunev, V.V.: Asymptotical expansions of modified motions of a rigid body. Preprint No. 90 of Inst. Appl. Math., Moscow, 2001 ( Russian).Google Scholar
16. [16]
Bruno, A.D., Lunev, V.V.: Properties of expansions of modified motions of a rigid body. Preprint No. 23 of Inst. Appl. Math., Moscow, 2002 ( Russian).Google Scholar