By Vladimir I. Arnold
Arnold's difficulties includes mathematical difficulties pointed out by way of Vladimir Arnold in his well-known seminar at Moscow nation college over a number of a long time. furthermore, there are difficulties released in his a variety of papers and books.
The invariable peculiarity of those difficulties was once that Arnold didn't think of arithmetic a video game with deductive reasoning and logos, yet part of average technological know-how (especially of physics), i.e. an experimental technological know-how. a lot of those difficulties are nonetheless on the frontier of analysis this present day and are nonetheless open, or even those who are generally solved continue stimulating new examine, showing each year in journals around the globe.
The moment a part of the ebook is a suite of commentaries, regularly through Arnold's former scholars, at the present growth within the difficulties' options (featuring a bibliography encouraged via them).
This e-book can be of serious curiosity to researchers and graduate scholars in arithmetic and mathematical physics.
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For non-conic singularities (e. ) one may try to find traces of the discrete invariants of the base (e. ) in the local algebra of the singularity. What algebraic objects are encountered in this way? What happens to the characteristic classes and numbers? 1975-1 The Problems 21 1975-1. Every interesting discrete invariant of a generic singularity with Newton polyhedron F is an interesting function of the polyhedron. Study: the signature, the number of moduli, the singularity index, the integral monodromy, the variation, the Bernstein polynomial, and (i,, (for generic sections).
1981-1 43 The Problems 1981-1. Lanner schemes (Coxeter schemes for the groups generated by reflections in the walls of simplices in the Lobachevskian space). o o Simplices in the Lobachevskian space: a) the series on the plane, b) 9 simplices in the three-dimensional space, c) 5 simplices in the four-dimensional space. Find applications of these schemes in singularity theory. 1981-2. Calculate the worst Nekhoroshev indices for generic Hamiltonians with n degrees of freedom (or at least the asymptotics of these indices for n large).
Another example: apply this to the classification of the equations y" — F(x,y,y'). 1982-21. What happens to the Givental triads when the quadraticity condition is violated (generically)? 1982-22. Can a divergence-free vector field tangent to the layers of Rieb's foliation have an exponential repulsion of trajectories? 50 The Problems 1982-23 1982-23. Investigate the singularities in the problem of bypassing an obstacle when the latter is not a hypersurface in the ambient space (e. , for curves in '.
Arnold's Problems by Vladimir I. Arnold