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...

Total Body Irradiation (TBI) is a radiological therapy mainly used for patient conditioning before bone marrow transplantation. Basically, the therapy consists in delivering a suited X-ray dose to the entire body of the patient. Achieving a uniform dose over the whole body is one of the major problems because of variations in thickness and density of patient’s tissues. This paper presents the d...