Table of contents

It is proved that every projective connection on an -dimensional manifold is locally defined by a system of second-order ordinary differential equations resolved with respect to the second derivatives and with right-hand sides cubic in the first derivatives, and that every differential system defines a projective connection on . The notion of equivalent differential systems is introduced and necessary and sufficient conditions are found for a system to be reducible by a change of variables to a system whose integral curves are straight lines. It is proved that the symmetry group of a differential system is a group of projective transformations in -dimensional space with the associated projective connection and has dimension . Necessary and sufficient conditions are found for a system to admit the maximal symmetry group; basis vector fields and structure equations of the maximal symmetry Lie algebra are produced. As an application a classification is given of the systems of two second-order differential equations admitting three-dimensional soluble symmetry groups.

Let be a Banach space and a non-increasing function such that as and is a Lipschitz function on . A linear operator is said to be -sectorial if there exist constants and such that the spectrum of  lies in the set

and

   there exists         such that         for  

where is the resolvent of the operator . The properties of the operator exponential and fractional powers of a -sectorial operator are analysed alongside the question of the unique solubility of the Cauchy problem for the linear differential operator with -sectorial operator-valued coefficient.

Structure results are proved for submanifolds of Euclidean spaces with semiparallel Ricci tensor under certain additional conditions. Minimal submanifolds are studied in greater detail. A geometric description of a class of normally flat semi-Einstein submanifolds with multiple principal curvature vectors is presented.

For functions in the unit disc, , it is shown that the rate of approximation of the boundary function in the metric by the generalized Riesz means , , , , is equivalent to the modulus of smoothness of fractional order . This is a known result in the case of positive integer .

One considers the spectral problem with boundary conditions , , for functions  on . It is assumed that is a linear bounded operator from the Hölder space  , , into  and the are bounded linear functionals on with . Let be the linear span of the root functions of the problem , , , corresponding to the eigenvalues   with , and let . An estimate of is obtained in terms of the -functional for (the direct theorem) and an estimate of this -functional is obtained in terms of for (the inverse theorem). In several cases two-sided bounds of the -functional are found in terms of appropriate moduli of continuity, and then the direct and the inverse theorems are stated in terms of moduli of continuity. For the spectral problem with periodic boundary conditions these results coincide with Jackson's and Bernstein's direct and inverse theorems on the approximation of functions by a trigonometric system.

The intersection indices of a certain kind of analytic set (resultant cycles) are expressed in terms of the Newton polyhedra of the corresponding defining systems of functions, provided that the principal parts of the functions are in general position. Among special cases of resultant cycles are complete intersections and the loci of matrix rank drop. Among special cases of the intersection indices of such sets are the index of a singularity of a Poincaré-Hopf vector field and its generalizations to the case of singular varieties, the index of a system of germs of 1-forms at an isolated singularity of a Gusein-Zade-Ebeling complete intersection, and the Suwa residue of a system of germs of sections of a vector bundle. One also obtains as a consequence the well-known Kushnirenko-Oka formula for the Milnor number of the germ of a map in terms of the Newton polyhedra of its components. A generalization of the well-known equality of the above-mentioned invariants of singularities to the dimensions of certain local rings is also presented.