For a smooth projective variety X and a line bundle L on X there are various notions for measuring the local positivity of L at a point x ∈ X . Two such measures are the dimension of the k-osculating space at x and the Seshadri constat at x. The precise nature of the interplay between these two notions is in general an open question. Our main result gives a partial answer to this question for s...

The geometry of the second order osculating bundle OscM , is in many cases determined by its spray and the associated nonlinear connection. For a Banach manifold M , we firstly endow OscM with a fiber bundle structure over M . Three different concepts which are used in many finite dimensional literatures, that is the horizontal distributions, nonlinear connections and sprays are studied in deta...

Let $X\subset \PP^r$ be an integral projective variety. We study the dimensions of joins several copies osculating varieties $J(X,m)$ $X$. Our methods are general, but we give a full description in all cases only if $X$ is linearly normal embedding $\PP^1\times \PP^1$. For these embeddings \PP^1$ examples and then one copy arbitrary number

We continue the study undertaken in Efroimsky (2005a) where we explored the influence of spin-axis variations of an oblate planet on satellite orbits. Near-equatorial satellites had long been believed to keep up with the oblate primary’s equator in the cause of its spin-axis variations. As demonstrated by Efroimsky & Goldreich (2004), this opinion had stemmed from an inexact interpretation of a...

We continue the study undertaken in Efroimsky (2005a) where we explored the influence of spin-axis variations of an oblate planet on satellite orbits. Near-equatorial satellites had long been believed to keep up with the oblate primary’s equator in the cause of its spin-axis variations. As demonstrated by Efroimsky & Goldreich (2004), this opinion had stemmed from an inexact interpretation of a...

It is shown that sufficiently close outer and inner parallel sets to a ddimensional Lipschitz manifold in R with boundary have locally positive reach and the normal cycle of the Lipschitz manifold can be defined as limit of normal cycles of the parallel sets in the flat seminorms for currents, provided that the normal cycles of the parallel set have locally bounded mass. The Gauss-Bonnet formul...