By A.Yu. Borisovich (auth.), Yurii G. Borisovich, Yurii E. Gliklikh, A.M. Vershik (eds.)
Contents: A.Yu. Borisovich: Plateau Operator and Bifurcations of Two-Dimensional minimum Surfaces.- Yu.G. Borisovich, T.N. Fomenko: Homological equipment within the thought of Periodic and Equivariant Maps.- B.A. Dubrovin: concept of Operators and genuine Algebraic Geometry.- B.D. Gel'man: at the constitution of the Set of recommendations for Inclusions with Multivalued Operators.- I.S. Krasil'shchik: Schouten Bracket and Canonical Algebras.- Yu.I. Sapronov: Multidimensional slumbering Tops.- B.Yu. Sternin, V.E. Shatalov: Laplace-Radon crucial Operators and Singularities of options of Differential Equations on advanced Manifolds.- V.G. Zvyagin: at the variety of ideas for definite Boundary-Value Problems.- V.I. Arnol'd: touch constitution, leisure Oscillations and Singular issues of Implicit Differential Equations.- N.M. Bliznyakov: Topological Index Estimates.- Yu.G. Borisovich: sleek method of the idea of Topological features of Nonlinear Operators. I.- A.T. Fomenko: Qualitative Geometrical thought of Integrable structures. class of Isoenergetic Surfaces and Bifurcation of Liouville Tori on the serious strength Values.- Yu.E. Gliklikh: Geometrical facets of Nelson's Stochastic Quantization.- B.Y. Sternin, V.E. Shatalov: Singularities of suggestions of Differential Equations on complicated Manifolds (Characteristic Case).- A.N. Varchenko: photograph of interval Mapping for easy Singularities.- A.M. Vershik, C.Ya. Gershkovich: The Geometry of the Nonholonomic Sphere for 3-dimensional Lie crew.
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Additional resources for Global Analysis — Studies and Applications III
Novikov [ I ~ is closely connected with nonlinear equations considered above. Lemma 8. Any stationary solution U(x) of the equations of the form (17) is a potential of a finite-gap operator L(~ ) (in the sense of Definition 2). In other words, the operator L( ~ ) is a finite-gap ope- 52 rator if and only if there exists a matrix polynomial ~( ~ ) (necessarily having the form (15)) commuting with L(~ ), [~(~),M(~)] For the proof see a spectrum of the [ 2 0 ] . We s h a l l operator L(~ ), form of an algebraic =o.
_o (17) ~imdn_m the homomorphisms of which are analogous to the corresponding homomorphisms of (l~). 5) if n ~ ~ < m, then the differentials (and the indices) should not equal zero and the sequence~ O-~-Kerd~Kerd m-r Hn(~Y ' A ~ Hr+l(Xj~ ~-~--~-Kerd u - ~ - J ~ - ~ ~ - d ~ m-n+~ r-n ~ ~mam_ r Hn(Y, X) Im~n+ 2 i-- 0 (18) -~JT-mdr _n+ 2 is exact and analogous to (17) and (14). 6) if, finally, n ~ m ~ r , then it is necessary that dr_m+ 2 = 0 and ~ - n + 2 = 0 and the exact sequence of the differential dr_n+2; 0--,~Kerd~ ~ ~ - ~ - H r + l (Y, k)---~'-Hn(Y, A) ~ ~- .
Let us discuss the interpretation of some characteristics of the s ~ f a c e r in the language of the problem on eigenvalues for the operator L( ~ ) in L2( - c~o , c~o). Lemma ~. This spectral problem is self-adjoint if the condition of positivity J A ~ o (32) is satisfied. _l Prqqf. e. the condition of self-adjointness with respect to scalar product of the form 0 This completes the proof of the le~ma. The allowed bands of the spectrum (Lyapunov stability bands) on the Riemann surface ~ are defined by the condition (Here we introduced the value p( ~ ) = (i T) -I l o g ~ ( ~ ) + 2 ~ i m , cal- led a quasimomentum.