By Yu. E. Gliklikh (auth.), Yuri G. Borisovich, Yuri E. Gliklikh, A. M. Vershik (eds.)

This quantity (a sequel to LNM 1108, 1214, 1334 and 1453) maintains the presentation to English talking readers of the Voronezh collage press sequence on international research and Its purposes. The papers are chosen fromtwo Russian matters entitled "Algebraic questions of study and Topology" and "Nonlinear Operators in international Analysis". CONTENTS: YuE. Gliklikh: Stochastic research, teams of diffeomorphisms and Lagrangian description of viscous incompressible fluid.- A.Ya. Helemskii: From topological homology: algebras with various houses of homological triviality.- V.V. Lychagin, L.V. Zil'bergleit: Duality in good Spencer cohomologies.- O.R. Musin: On a few difficulties of computational geometry and topology.- V.E. Nazaikinskii, B.Yu. Sternin, V.E.Shatalov: creation to Maslov's operational procedure (non-commutative research and differential equations).- Yu.B. Rudyak: the matter of attention of homology periods from Poincare as much as the present.- V.G. Zvyagin, N.M. Ratiner: orientated measure of Fredholm maps of non-negativeindex and its purposes to worldwide bifurcation of solutions.- A.A. Bolibruch: Fuchsian structures with reducible monodromy and the Riemann-Hilbert problem.- I.V. Bronstein, A.Ya. Kopanskii: Finitely delicate general sorts of vector fields within the neighborhood of a leisure point.- B.D. Gel'man: Generalized measure of multi-valued mappings.- G.N. Khimshiashvili: On Fredholmian elements of linear transmission problems.- A.S. Mishchenko: desk bound recommendations of nonlinear stochastic equations.- B.Yu. Sternin, V.E. Shatalov: Continuation of ideas to elliptic equations and localisation of singularities.- V.G. Zvyagin, V.T. Dmitrienko: Properness of nonlinear elliptic differential operators in H|lder spaces.

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As to the contractability, one can easily deduce it from Theorem 22 ((I) and (II)). The proof in the amenability case is somewhat more complicated (see Johnson [B, n o and also [B, no 204] 7 On the contrary, the following class of algebras differs from the already mentioned ones. Definition ii. A Banach algebra A is called biprojective, if it is projective as an A-bimodule. The following theorem served as the initial stimulus for the study of non-unital (that is, non-contractible; jective alge_bras.

There are hundreds (or even thousands) of works dealing with this subject. In topology and mathematical physics the partition of unity is used for the proof of various existence theo- 59 rems. In ~ i it is shown that this is an effective method for the construction of algorithms of computational geometry (cf. 6~ ). A qualitative investigation of ordinary differential equations and the search for singular points and separatrices of the corres- pending vector field are rather effective methods of the modern mathematical physics.

II) A yon Neumann algebra A is amenable-after-Connes jective (or, equivalnetly, hyperfinite). iff it is iff it is in- 39 The second assertion was proved before the first one, in 1978. l14~ (however, under the assumption of A being separable, which was removed by Elliot [B, no. 222 ] ). l. Now we, "grown wise with experience", see that (I) is an immediate corollary of (II) combined with Theorem 23 aud the theorem of Choi and Effros mentioned above. Nevertheless, actually the whole Theorem 31 was completed earlier than the simple proof of "if" part of Theorem 23 became known.

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