LetX be a compact, strictly convex C-hypersurface in the (n+1)-dimensional Euclidean space R. The Gauss map ofX maps the hypersurface one-to-one and onto the unit n-sphere S. One may parametrize X by the inverse of the Gauss map. Consequently, the Gauss curvature can be regarded as a function on S. The classical Minkowski problem asks conversely when a positive function K on S is the Gauss curv...

1. Introduction. Transversal theory is a branch of combinatorial mathematics which is only just beginning to emerge as a reasonably connected and coherent subject. Whether this is yet rich enough or mature enough to be called a 'theory' may be a matter for debate; indeed, it is by no means certain that this part of mathematics may not finally be classified under some broader, more comprehensive...

A bstract We obtain a Lagrangian for lightlike dimensional reduction of the nonabelian M5 brane theory in six dimension. assume that six-manifold has at least one conformal Killing spinor and two commuting vector fields, we perform along these directions.

We study the behavior of the Chern classes of graph hypersurfaces under the operation of deletion-contraction of an edge of the corresponding graph. We obtain an explicit formula when the edge satisfies two technical conditions, and prove that both these conditions hold when the edge is multiple in the graph. This leads to recursions for the Chern classes of graph hypersurfaces for graphs obtai...

The problem considered in this article is: ‘for given finite system of convex polygons in the plane which have no transversal, find such homothety transformations of polygons (having fixed centres inside given polygons) with minimal similarity ratio c > 1 that the transformed system has a transversal’. We prove that in this ‘minimal configuration’, we can always find three polygons (two of them...

The public key cryptosystem MST1 has been introduced in [9]. Its security relies on the hardness of factoring with respect to wild logarithmic signatures. To identify ‘wild-like’ logarithmic signatures, the criterion of being totally-non-transversal has been proposed. We give tame totally-non-transversal logarithmic signatures for the alternating and symmetric groups of degree ≥ 5. Hence, basin...

We prove Helly-type theorems for line transversals to disjoint unit balls in R. In particular, we show that a family of n > 2d disjoint unit balls in R has a line transversal if, for some ordering ≺ of the balls, any subfamily of 2d balls admits a line transversal consistent with ≺. We also prove that a family of n > 4d − 1 disjoint unit balls in R admits a line transversal if any subfamily of ...

Let S be an ordered set of disjoint unit spheres in R. We show that if every subset of at most six spheres from S admits a line transversal respecting the ordering, then the entire family has a line transversal. Without the order condition, we show that the existence of a line transversal for every subset of at most 11 spheres from S implies the existence of a line transversal for S.