Valerii Vasil'evich Kozlov

Invariant and quasi-invariant measures on infinite-dimensional spaces

Extensions of locally convex topological spaces are considered such that finite cylindrical measures which are not countably additive on their initial domains turn out to be countably additive on the extensions. Extensions of certain transformations of the initial spaces with respect to which the initial measures are invariant or quasi-invariant to the extensions of these spaces are described. Similar questions are considered for differentiable measures. The constructions may find applications in statistical mechanics and quantum field theory.

A summation formula for divergent continued fractions

Paradoxical formulas which have been proposed by a number of authors for evaluating divergent continued fractions are discussed. The point of the paradoxes is that limit passages (which are to be rigorously defined) result in the convergence real sequences to complex values. These formulas are refined and substantiated on the basis of the theory of uniform distribution supplemented with certain statements of complex analysis.

Symplectic geometry of a linear transformation with a quadratic invariant

A linear transformation with an invariant being a nondegenerate quadratic form is symplectic. The geometric properties of such transformations are discussed. A complete set of quadratic invariants which are pairwise in involution is explicitly specified. The structure of the isotropic cone on which all these integrals simultaneously vanish is investigated. Applications of the general results to the problem on the stability of a fixed point of a linear transformation with a quadratic invariant are discussed.

Vladimir Andreevich Steklov (1863–1926)

The article outlines the life and work of Vladimir Andreevich Steklov, a full member of the Russian Academy of Sciences, where the Mathematical Institute was named after him. The article is being released for the 150th anniversary of the scientist. The authors employ his Recollections, published in 1991 by the Leningrad Department of the Institute for the History of Science and Technology (IHST) of the Academy of Sciences of the USSR (Scientific Heritage, vol. 17), as the main source of information about Steklov’s life and work, the development of his personality, and his mathematical and public career. The article highlights the decisive role Vladimir Steklov played in the formation of the Russian Academy of Sciences as the successor of the Imperial St. Petersburg Academy of Sciences and its transformation into the Academy of Sciences of the USSR. The information about Vladimir Steklov’s contribution to this area comes from the archives of the Russian Academy of Sciences and from Science in the First Years of Soviet Rule (1917–1925), a collection of documents published by the Leningrad Department of the IHST in 1968. The article also summarizes Vladimir Steklov’s scientific achievements in mathematical physics and mechanics.