Klein polygons and reduced regular continued fractions

Abstract

We investigated the influence of the irradiation conditions of the coherent length and the polarization of the laser beam, the incident angle i , the pulse number, the fluence, the substrate temperature and the a-Si film thickness on the meltingcrystallization of an a-Si film and the controllability of the periodic grain boundary location by using a linearly polarized laser beam. For formation of periodic grain boundary in the crystallized Si film, the linear polarization is key factor rather than the coherent length. Also, the a-Si film thickness should be around 60 nm for large optical absorption, and the high substrate temperature and the large pulse number are preferable. By using a p-polarized beam, the grain boundary width can be expanded proportional to 1=ð1 À sin i Þ roughly. In order to suppress the ablation of the film and to increase the productivity, the fluence should be adjusted precisely.

One-sided Diophantine approximations

“…From identity (9) we obtain that the numerator on the right hand side of (25) is equal to 2q n q n−1 α − 2q n−1 p n + 1. Therefore, (25) can be rewritten as…”

“…Another and very short proof was later published by Eggan and Niven [5]. Then Finkelshtein [9] studied best upper Diophantine approximations of the 2nd kind. He found their characterization in terms of so-called reduced regular continued fractions, the formalism that is described in detail in Perron's book [15] and a paper by Zurl [22].…”

On crepant resolutions of 2-parameter series of Gorenstein cyclic quotient singularities

“…We are mainly motivated by the original works of Klein [34], [35], [36], written over hundred years ago, in which his "Umrißpolygone" (also known as Kleinian polygons, cf. Finkel'shtein [14]) were used to approximate real (not necessarily rational) numbers by only regular continued fractions. From the algorithmic point of view, Kleinian approximations are more "economic" (see remarks 3.3, 3.5 and 3.8 below).…”

Средняя Длина Приведенной Регулярной Непрерывной Дроби

“…, t l ]], где t 0 = ⌈r⌉ (верхняя целая часть r), t n 2 при n 1 (основные свойства таких дробей приведены в работе Перрона [1; глава I], см. также работу Финкельшнейна [2]). Обозначим длину приведенной регулярной непрерывной дроби [[t 0 ; t 1 , t 2 , .…”

“…3γ − ζ(2) ′ (2)) 2 − ζ ′′ (2)ζ(2) 2ζ3 (2) + O(R log 5 R).Теорема доказана.При подсчете величины E(R) учитываются все дроби вида a/b, a b R, а среди них есть равные, поэтому длины некоторых приведенных регулярных непрерывных дробей считаются несколько раз. Чтобы избежать этих повторений, можно вместо величины N (R) рассмотреть величинуN * (R) = b R a b (a,b)=1 l a b и соответствующее среднее E * (R).…”