扭轉圓筒的摺痕與幾何機制

本論文包含了兩個工作：（一）扭轉圓筒的摺痕與幾何機制（二）語言統計學的縮放結構。兩者皆是跨領域的研究，前者包含了摺紙數學還有彈塑性力學，後者涉及統計學、語言學、自然語言處理與物理。論文的第一部分，討論在不同幾何限制下，薄膜圓筒被扭轉時因挫屈（buckling）產生的規律摺痕。發現圓筒上的摺痕數量N只要透過半徑R與寬度w的比例就能被唯一決定，即N = N(R/w)，是一個無關厚度的二維參數。從簡單的幾何關係、能量最小化與一對可以直接描述摺痕形成的新材料參數，我們理論推導出的解析解，可以成功擬合實驗事實，在摺痕力學的描述上比舊有的彈塑性力學參數表現更加簡單。除了摺痕數量，R/w還能用來描述摺痕的型態：斷裂摺痕、規律摺痕與不規律摺痕三種型態。本文還會探討如何從這一簡單、可解析的結果出發，建立一座串聯小形變與大形變問題的橋樑，不需要透過複雜的von Kármán–Donnell equations求數值解。除了最簡單的圓筒，我們還將問題拓展到截錐、多邊形管、斜切管、圓球等不同幾何結構，試圖在材料和幾何上統一這類問題。根據尺度縮放不影響摺痕形態的這一特性，我們將此結果應用在解釋地球板塊裂痕與山脈形成的機制上，提出因扭轉產生板塊形變的機制。論文的第二部分，延續了前任學長蔡孟學在詞（word）頻和字/音節（character/ syllable）頻排名的統計關係研究，深入探討「縮放結構」（scaling structure）的來源：將詞的出現次數排名x與其組成字的出現次數排名y畫成y

關鍵字

摺痕；扭轉；彈塑性力學；摺紙；斷詞；語言學

並列摘要

This thesis contain two work: 1. Geometric Mechanism of Crease Formation on Twisting Cylindrical Shell, and 2. Scaling Pattern: A Fundamental Structure in Statistical Linguistics. Both of them are interdisciplinary research. The former contain mathematics of paper folding and elastic-plastic mechanics, the latter involved statistics, linguistics, natural language processing and physics. The first part of thesis discuss the mechanism of regular creases’ pattern formation due to twisting buckling under different geometry constraints. Discovering the numbr of creases N can unique determined by the ratio of radius R and width w of cylindrical shell, a 2D parameter, N = N(R/w). We get analytic solutions that fits the experimental data from simple geometry, energy minimization and two new material properties. These new properties display more concise than ordinary material properties used in elastic-plastic mechanics. In addition to N, R/w can be used to depict three types of creases: fracture, regular and irregular. This thesis also allow us to build a bridge between small and large deformation problem form simple and analytic results instead of numerical solutions of von Kármán–Donnell equations which are complicated. Beside the simplest case, cylindrical shell, we extend this problem to truncated cone, polygon cylindrical shell, beveled end pipe and ball, trying to unify the influence of material and geometry. According to the scaling property of types of creases, we apply the result to explain of creases of earth’s plates, the formation of mountains, and propose the twisting buckling mechanism of deformation of earth’s plates. The second part of this thesis continues, the work of Meng-Xue Tsai in our 13 lab, the reaserch of statistical relationship between frequency rank of words and that of characters/syllables and investigate the origin of “scaling structure”: ploting word rank x and character/syllables ranky, you will get a stepped layer structure. This structure shown in different natrual language corpus, writing styles and languages. That is to say, we discover a stastistics law which is complementary to Zipf’s law. Furthermore we describe the completeness of scaling structure with three quantities: SP vaule, Collocation function and Link function. These index can be used to judge accuracy of word segmentation, and does not dependent on large data or prior knowledge of the language. We popose a new word segmentation algorithm which can improve its ability by self-scoring mechanism, Due to its feature, this algorithm can be applied to unknown language or language with small data base such as animal language or baby language, or finding out the basic statistics unit and structure in a field. On top of that, by finding out conditions ruining scaling structure, we prove this structure is not arbitrary , but has statistically significant.