Parabolic extensions of the Hecke ring of the general linear group. II

The Hecke rings of arbitrary parabolic subgroups of the general linear group are investigated. Decompositions of standard and skew-symmetric polynomials in parabolic extensions are constructed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova Akademii Nauk SSSR, Vol. 183, pp. 56–76, 1990.     

On the splitting of groups of central group extensions

It is proved that if G and V are abelian groups, then the group Ext(G, V) of all abelian extensions of the group V with the aid of the group G is extracted by the direct component in the group Opext(G, V) of all central extensions of V with the aid of G.Translated from Matematicheskie Zametki, Vol. 6, No. 2, pp. 197–200, August, 1969.     

Conformally natural extensions of continuous circle maps: II. The general case

We introduce a procedure for extending continuous circle maps in a conformally natural way to continuous maps from the closed disk, bounded by the circle to itself which generalizes Douady-Earle’s method of constructing conformally natural extensions of circle homeomorphisms. We also provide a criterion for the extensions to be surjective maps from the closed disk to itself.     

On the general theory of exact controllability for skew symmetric operators

We introduce a general formalism for linear evolution equations with skew adjoint operators. We make explicit the controllability operator as an expansion with respect to eigenfunctions. Using the fact that the eigenvalues are purely imaginary, we give sufficient controllability conditions. This approach is convenient for studying the asymptotic behaviour of the optimal control.     

A group of central locally splitting extensions

Translated from Matematicheskie Zametki, Vol. 47, No. 4, pp. 100–105, April, 1990.     

On a class of nonlinear Schrödinger equations. II. Scattering theory,general case

This is the second of a series of papers devoted to the study of a class of non linear Schrödinger equations of the form $\text{i(}\text{du}\text{dt}\text{) = (?Δ + m)u + f(u)}$ in $\text{R}$ⁿ where m is a real constant and f a complex valued non linear function. Here we study the scattering theory for the pair of equations that consists of the previous one and of the equation $\text{i(}\text{du}\text{dt}\text{) = (?Δ + m)u}\text{for}\text{n ? 2}$. Under suitable assumptions of f we prove the existence of the wave operators and asymptotic completeness for a class of repulsive interactions. The assumptions of f that ensure asymptotic completeness cover the case of a single power $\text{f(u) = λ ¦ u ¦}{}^{\mathrm{p?1}}\text{u}\text{where}\text{λ ? 0}\text{and}\text{(n + 4)}\text{n}\text{< p <}\text{(n + 2)}\text{(n ? 2)}$.     

On the homology spectral sequence for topological Hochschild homology

Marcel Bökstedt has computed the homotopy type of the topological Hochschild homology of using his definition of topological Hochschild homology for a functor with smash product. Here we show that easy conceptual proofs of his main technical result of are possible in the context of the homotopy theory of -algebras as introduced by Elmendorf, Kriz, Mandell and May. We give algebraic arguments based on naturality properties of the topological Hochschild homology spectral sequence. In the process we demonstrate the utility of the unstable ``lower' notation for the Dyer-Lashof algebra.