Grassmann image of non-isotropic surface of pseudo-Euclidean space

We consider submanifolds of non-isotropic planes of the Grassman manifold of the pseudo-Euclidean space. We prove a theorem about the unboundedness of the sectional curvature of the submanifolds of the two-dimensional non-isotropic planes of the four-dimensional pseudo-Euclidean space with the help of immersion in the six-dimensional pseudo-Euclidean space of index 3. We also introduce a concept of the indicatrix of normal curvature and study the properties of this indicatrix and the Grassman image of the non-isotropic surface of the pseudo-Euclidean space. We find a connection between the curvature of the Grassman image and the intrinsic geometry of the plane. We suggest the classification of the points of the Grassman image.     

Example of a function of two variables that cannot be an <Emphasis Type="Italic">R</Emphasis>-function

We note that the definition of R-functions depends on the choice of a certain surjection and pose the problem of the construction of a function of two variables that is not an R-function for any choice of a surjective mapping. It is shown that the function x ₁ x ₂ − 1 possesses this property. We prove a theorem according to which, in the case of finite sets, every mapping is an R-mapping for a proper choice of a surjection.     

Topologies on the n-Element Set Consistent with Topologies Close to Discrete on an (n−1)-Element Set

Ukrainian Mathematical Journal - Topologies on a finite set are described by a nondecreasing sequence of nonnegative integers (vector of topologies). We study T0-topologies on the n-element set...