On groups acting in phase space

The problem of the motion of a material point in a central field of general type is considered. It is shown that in the infinite-dimensional group of canonical transformations which leave the Hamiltonian function invariant there are no finite-dimensional subgroups which are significantly larger than the three-dimensional group of rotations (exact formulations in Sec. 3 and Sec. 5).Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 55–61, January, 1969.I am indebted to A. A. Kirillov for helpful discussions.     

Stability of linear almost-Hamiltonian periodic systems

A linear Hamiltonian system with periodic coefficients is subject to a small “dissipative” perturbation that makes it asymptotically stable. The conditions under which the perturbation remains dissipative for all Hamiltonian systems sufficiently close to the original one are discussed.     

Functions of two variables continuous along straigt lines

For a functionf(x, y), the setsJ _a of all its discontinuity points with a jump ofa or more (that is, such that the oscillation of the function in the neighborhood of any point fromJ _a is not smaller thana) are studied. Two cases are considered: (1)f is continuous along any straight line; (2)f is continuous along lines parallel to thex- andy-axes. In the first case, conditions that must be met by the setJ _a are given. In the second case, it is shown that a (closed) setF can be the setJ _a for a certain function if and only if the projections ofF on the coordinate axes nowhere dense. Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 306–311, August, 1997. Translated by V. N. Dubrovsky     

An Extremal Problem about Probability Distributions

A given probability distribution density is multiplied by all positive functions with a fixed ratio of upper and lower bounds. The products are normed so as to obtain probability densities again. The value of the variance in the class of probability distributions obtained by this sort of modification of the given distribution is studied. It is shown that the upper bound of the variance is attained for a piecewise constant modifying function shaped as a rectangular trough. A similar statement holds for the minimal variance. It is shown that the distribution with maximal variance is unique in the class in question.