The combinatorial complexity of the region lit up from a point light source inside a simple n-gon with perfectly reeecting edges, after at most k specular reeections was established as O(n 2k) 2]. A lower bound of ((n=k) 2k) was also established, matching the upper bound for any xed k. In a real situation, surfaces may not be perfect mirrors; indeed most surfaces may be non-shiny or rough, caus...

A spherical n-gon is a bordered surface homeomorphic to a closed disk, with n distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these polygons and enumerate them in the case that two angles at the corners are ...

A k-triangulation of the n-gon is a maximal set of diagonals of the n-gon containing no subset of k + 1 mutually crossing diagonals. The number of k-triangulations of the n-gon, determined by Jakob Jonsson, is equal to a k×k Hankel determinant of Catalan numbers. This determinant is also equal to the number of k non-crossing Dyck paths of semi-length n− 2k. This brings up the problem of finding...

In this paper we study the geometry of Minkowski plane and obtain some results. We focus on the curve theory in Minkowski plane and prove that the total curvature of any simple closed curve equals to the total Landsberg angle. As the result, the sum of oriented exterior Landsberg angles of any polygon is also equal to the total Landsberg angle, and when the Minkowski plane is reversible, the su...

What is the minimum perimeter of a convex lattice n-gon? This question was answered by Jarńık in 1926. We solve the same question in the case when perimeter is measured by a (not necessarily symmetric) norm.

We study the maximum possible number f(k, l) of intersections of the boundaries of a simple k-gon with a simple l-gon in the plane for k, l ≥ 3. To determine the number f(k, l) is quite easy and known when k or l is even but still remains open for k and l both odd. We improve (for k ≤ l) the easy upper bound kl − l to kl − ⌈k/6⌉ − l and obtain exact bounds for k = 5 (f(5, l) = 4l − 2) in this c...

Let K n denote a regular n-gon with unit side length. For two regular polygons K n and K m let the relative Newton number N(K n ; K m) of K n and K m be the maximum number of congruent copies of K m that can touch K n without having interior points in common. In this paper we determine the exact value of N(K n ; K m) for various pairs (n; m).

OBJECTIVES To compare the optic nerve head (ONH) structure between compressive optic neuropathy (CON) and glaucomatous optic neuropathy (GON), and to determine whether selected ONH quantitative parameters effectively discriminate between GON and CON, especially CON cases presenting with a glaucoma-like disc. METHODS We prospectively assessed 34 patients with CON, 34 age-matched patients with ...

The gonad of the Caenorhabditis elegans hermaphrodite is generated by the postembryonic divisions of two somatic precursors, Z1 and Z4, and two germline precursors, Z2 and Z3. These cells begin division midway through the first larval stage. By the end of the fourth larval stage, Z1 and Z4 produce 143 descendants, while Z2 and Z3 give rise to approximately 1000 descendants. The divisions of Z2 ...

PURPOSE To examine the prevalence of glaucomatous optic neuropathy (GON) in a medium myopic to highly myopic group of patients and its association with parapapillary gamma zone and parapapillary delta zone. METHODS The retrospective observational hospital-based study included patients who had attended the Tokyo High Myopia Clinics within January 2012 and December 2012 and for whom fundus phot...