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.

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 ...

In this paper we formulate and prove some xed and common xed pointTheorems for self-mappings dened on complete lower Transversal functionalprobabilistic spaces.

Let F be a family of compact convex sets in R. We say that F has a topological ρ-transversal of index (m, k) (ρ < m, 0 < k ≤ d − m) if there are, homologically, as many transversal m-planes to F as m-planes containing a fixed ρ-plane in R. Clearly, if F has a ρ-transversal plane, then F has a topological ρ-transversal of index (m, k), for ρ < m and k ≤ d − m. The converse is not true in general...

The aim of this article is to introduce a new class SO(2)equivariant transversal maps T R(cl(Ω), ∂Ω) and to define degree theory for such maps. We define degree for SO(2)-equivariant transversal maps and prove some properties of this invariant. Moreover, we characterize SO(2)-equivariant transversal isomorphisms and derive formula for degree of such isomorphisms.