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We study the homological properties of the factor space G/P, where G is a complex semisimple Lie group and P a parabolic subgroup of G. To this end we compare two descriptions of the cohomology of such spaces. One of these makes use of the partition of G/P into cells (Schubert cells), while the other consists in identifying the cohomology of G/P with certain polynomials on the Lie algebra of the Cartan subgroup H of G. The results obtained are used to describe the algebraic action of the Weyl group W of G on the cohomology of G/P.

CONTENTS Introduction § 1. Holomorphic continuation of functions from a product of two neighbourhoods § 2. Holomorphic continuation of functions from a product of real axes § 3. The connection between Bernstein's theorem and theorems on the thin end of the wedge § 4. Removal of the boundedness assumption for the separate continuations § 5. Some generalizations References

The vector equation of the static theory of elasticity for a homogeneous isotropic medium is

(1)

where , and is Poisson's constant, being treated as a spectral parameter. This is then the problem: to examine the spectrum of the family of operators on the left-hand side of (1) for boundary conditions of first or second kind. The problem was first posed at the end of the 19th century by Eugène and François Cosserat; it has been investigated in recent years by V. G. Maz'ya and the present author. The main results obtained are for an elastic domain , which may be finite, or infinite with a sufficiently smooth finite boundary. In the case of the first boundary-value problem the family operators of the theory of elasticity has a countable system of eigenvectors, orthogonal in the metric of the Dirichlet integral; this system is complete in each of the spaces and . The eigenvalues condense at the three points ; and are isolated eigenvalues of infinite multiplicity. Similar results are obtained also, for the second boundary-value problem. The essential difference lies in the fact that in this case the eigenvalues have one further condensation point , and examples show that need not be a point of condensation for eigenvalues of the second problem.

The matrix Riccati differential equation is discussed, from the point of view of dissipativity or conservativity of its solutions. A survey is given of results relating to analytic properties of these solutions and to the geometry of the corresponding semigroup of matrix linear fractional transformations; further, a probabilistic interpretation is given of the properties of being dissipative or conservative, and the connection between dissipativity of the solutions of the Riccati equation and the stability of the screw method is studied. Physical and technical applications of the mathematical theory are given. There are 87 references.