攜手共解圓-扭結理論之探討
此篇研究發現在任何一個結中，都可以利用「牽手順序」和「交錯點編碼」兩種結的資訊直接看出一結化簡後的圖形。利用從「Reidemeister moves」所衍伸出的四種化簡方法{α, β, γ, δ}能更有效率的簡化結，並證明只需{α, β, γ, δ}就可化簡任何結，也利用{α, β, γ, δ}來驗證HOMFLY多項式是結不變量。由圖形及結的資訊我們發現，可使用「牽手順序」和「交錯點編碼」，搭配所討論出{α, β, γ, δ}的通式，依照步驟及通式簡化任何結。 本篇最重要的成果為：只需利用{α, β, γ, δ}即可化簡所有的結，而且比Reidemeister moves更有效率，因此可用{α, β, γ, δ}取代Reidemeister moves。 不管是一個封閉曲線或是兩個以上封閉曲線，都會遵守前述的規則，可利用{α, β, γ, δ}簡化圖形。文中也討論了較特別並具有規律的結──「星星結」，發現星星結只需使用「牽手順序」即可簡化，最後利用星星結的結論，發展出牽手遊戲中特殊的牽手遊戲情形。
Beautiful Butterfly: The Physics Behind The Colors
Even as a child, I was fascinated by the colors in nature, such as rainbows, butterflies and flowers. This fascination developed into curiosity with age, and as my school studies developed, I became particularly interested in the scientific aspects of the origin and development of colors. I wanted to answer the question: How are the different colors of the butterfly wings related to the nanostructures of scales and pigments? The color on the butterfly wings results either from the pigmentation (chemical color) or from the structure (physical color) of the wing scales. Colors such as yellow, black, red and brown are mainly created by pigments. The interaction of light and structures in and on the surface of butterfly wings, often the size of the wavelength of the light, results in physical colors. These colors are usually bright and dependent on the viewing angle (unlike chemical pigments that spread light diffusely). The colors produced here are usually golden, green, purple and blue. But, where do these colors come from and why do certain species dazzle more than others? To get to the heart of the matter, I identified two key questions: • How are the different colors of the butterfly wings related to the nanostructures of scales and to the pigments? • Using the nanostructure, can you find out the wavelength of the reflected light? In this work, I focus on the structural colors of butterflies and study the physics behind them. This includes parachuting in areas such as diffraction gratings, scattering of light, interference in thin films, and multilayer interference. In order to experience the greatest possible diversity, I selected butterflies from different species for the measurements. Using the spectrometer, I measured the light reflected from butterflies. High-resolution microscopes such as the laser microscope and the scanning electron microscope gave me the opportunity to study the detailed nanostructures of the wing. In addition, I was able to analyze and evaluate my results using existing physical models and MATLAB simulations (Maxwell equations).