Inspired by a theorem Skornjakov-Hughes-Pasini [9, 7, 8] and problem which turned up in our recent paper [13], we start study of epimorphisms with source thick generalized m-gon target thin m-gon. In this first part the series, classify cases m = 3, 4 6 when polygons are finite. Then show that infinite case is very different, construct examples strongly deviate from finite case. A number genera...

The physical and antibacterial properties of ophthalmic lenses fabricated by copolymerization with hydrogel monomers using two types graphene were measured, their usability as contact lens materials was analyzed. For polymerization, silicone monomers, including SID-OH, 3-(methacryloxy)propyl tris(trimethylsiloxy)silane, decamethylcyclopentasiloxane, used, N,N-dimethylacetamide, ethylene glycol ...

In [13], we determined all dense near 2n-gons of order (2, t) with a nice chain of subpolygons, i.e. a chain F0 ⊂ F1 ⊂ · · · ⊂ Fn of convex subpolygons satisfying (i) Fi, i ∈ {0, . . . , n}, is a near 2i-gon, and (ii) Fi, i ∈ {0, . . . , n− 1}, is big in Fi+1. In the present paper, we describe a method which can be used for classifying general dense near polygons with such a chain. We will use ...

We show that for any arrangement A of n lines in general position in the plane there exists a simple closed polygon with n edges having the property that every edge of the polygon lies on a distinct line of A.

Hypoperfusion of the optic nerve might be involved in pathogenesis normal-tension glaucomatous neuropathy (GON). Mean arterial pressure (MAP) drives ocular perfusion, but no previous studies have addressed risk GON relation to blood (BP) variability, independent BP level. In a cross-sectional study, 93 residents Maracaibo, Venezuela, underwent optical coherence tomography, visual field assessme...

We prove that n + 4 finger probes are sufficient to determine the shape of a convex n-gon from a finite collection of models, improving the previous result of 2n + 1. Further, we show that n − 1 are necessary, proving this is optimal to within an additive constant. For line probes, we show that 2n + 4 probes are sufficient and 2n − 3 necessary. The difference between these results is particular...

Let S be a set of n points distributed uniformly and independently in the unit square. Then the expected number of empty four-gons with vertices from S is Θ(n2 logn). A four-gon is empty if it contains no points of S in its interior.

In this paper we study the following problem: we are given a set of imprecise points modeled as parallel line segments, and we wish to place a point on each line segment such that the resulting point set maximizes/minimizes the size of the largest/smallest area k-gon. We first study the problem for the case k = 3. We show that for a given set of parallel line segments of equal length the larges...

We study the minimum number g(m,n) (respectively, p(m, n)) of pieces needed to dissect a regular m-gon into a regular n-gon of the same area using glass-cuts (respectively, polygonal cuts). First we study regular polygon-square dissections and show that dn/2e − 2 ≤ g(4, n) ≤ n 2 + o(n) and dn/4e ≤ g(n, 4) ≤ n 2 + o(n) hold for sufficiently large n. We also consider polygonal cuts, i.e., the min...

In this paper we generalize a result of Benson to all finite generalized polygons. In particular, given a collineation θ of a finite generalized polygon S, we obtain a relation between the parameters of S and, for various natural numbers i, the number of points x which are mapped to a point at distance i from x by θ. As a special case we consider generalized 2n-gons of order (1, t) and determin...