本作品為環球數學城市競賽的考題之推廣（ International Mathematics Tournament of the Towns, Senior A-Level Paper, Fall 2009, No. 7 ），然而此題目本身比較接近Scientific American （ Feb 1979 ）中 Martin Gardner 的文章 The Rotating Table 所提出的問題。 而此問題的多邊形版已被數學家解決(Ted Lewis & Stephen Willard, 1980, The Rotating Table,Mathematics Magazine, 53,Page 174-175.)。然而此份作品採用簡潔的初等數學歸納法證明了原本的多邊形情形，且專注於探討條件改變為m=n-1時的多邊形情形。 m=n-1時的情形是前人所沒有做出結果的，且m=n-1的情形遠遠比m=n 的情況來得複雜多變，其中估計上下界所使用的許多組合技巧相當特別。然而其中當n為3 和5的倍數時有比較特殊的情況，這一個部分僅能給出目前得出的k值上界。而其他情況則能夠找出所有的k值。

An increase in blood cholesterol contributes to cardiovascular diseases, the number one cause of death worldwide. Statins are currently the most effective in reducing cholesterol levels and treating patients with high cholesterol. However, these pharmaceutical agents have been shown to cause several side effects, prompting the need for a more natural solution to increasing cholesterol levels. Hence, a study was conducted to investigate the ability of lactic acid bacteria in the removal of cholesterol, explore the mechanism for the removal of cholesterol by lactic acid bacteria, and examine the effectiveness of kidney beans and sunflower seeds in inhibiting HMG-CoA reductase in the cholesterol biosynthesis pathway. Results showed that Lactobacillus plantarum was the most effective in reducing cholesterol levels and that the mechanism for cholesterol removal included both the binding to cell wall and active uptake into cells. Sunflower seeds and kidney beans were also shown to be effective in inhibiting HMG-CoA reductase, with sunflower seeds having 100% inhibition of the enzyme, similar to pravastatin, a commercial cholesterol reducing drug, and kidney beans having comparable percentage inhibition of the enzyme compared to pravastatin.

This study demonstrates that microbial film power generation is a potentially viable source of alternative energy. This research occurred over a period of two years. In the first year (2016) I tested a new method of generating renewable energy, referred to as microbial film power generation. I showed that electricity could be captured from microbial decomposition using solid graphite plates (29cm x 20cm) placed in lightly decomposed muskeg (collected in northern British Columbia). In the second year (2017) the purpose was to increase the power output of the fuel cell, while also compacting the setup. Certain changes were made to the experimental set up, namely the use of spongy graphite felt in place of solid graphite plates, thus providing a larger surface area for microbial activity to occur. The new fuel cells made produced about twice as much power. Not only was the power output greater, but it was produced from a much smaller area: 7.82 mWh/cm2 on graphite felt, compared to 0.21 mWh/cm2 on graphite plates. In other words, graphite felt produced 37 times more power per unit area than graphite plates. Furthermore, it would appear that by removing the load from the fuel cell for approximately 24 hours, the fuel cell could essentially recharge. This may be due to microbial activity releasing more electrons onto the anode permitting a new cycle to take place. This would suggest that the system could naturally recharge itself.

CNC machines use vector graphics or vector image programs that take time and effort on hobbyists. Therefore, it is important to provide accurate techniques for converting ordinary images available on the Internet or can be designed with easy programs. In order to have precise drill paths read by CNC machines directly and produce a product that does not contain rattles at the edges. This depends on the accuracy of processing the extracted paths. The development of algorithms has been completed Transforms Pixel image into Paths with XYZ extension, which is used to drill material and cut it through CNC machines. And the algorithms are based on transfer images with low quality. And Its Advantage that it can create high Paths with as few points as possible. The program can convert the pixel image into paths, and then converted into g-code, and use it in CNC machines directly.

In recent years, the ticks have reached the valley of Poschiavo and so far no study has been done to determine their diffusion. Recently, this presence has become a much discussed topic as these ticks can be carriers of pathogenes and represent a danger to humans. The goal of this work is to analyze the current situation in the valley of Poschiavo to understand in which areas the ticks are widespread, if they are carriers of pathogens and which factors could have an influence on their expansion. Several methods have been used for data collection. Ticks were found on ungulates killed during the high hunt in autumn 2016. In spring 2017, ticks were collected in various areas of the valley using the flag method that involves dragging a cotton cloth onto the ground. Some of the collected ticks were sent to a laboratory to identify the presence of the Borrelia burgdorferi, the pathogen responsible for Lyme borreliosis. To understand the evolution of the presence of ticks in the valley, the doctors and veterinarians were interviewed. Finally, to identify any climate changes related to the diffusion of ticks, the evolution of the tem-perature and relative humidity measured by two meteorological stations in the valley of Poschiavo since 1980 have been analyzed. Thanks to this study it was possible to highlight for the first time the presence in the valley of Poschiavo of ticks wich are bearer of the Borrelia burgdorferi. In fact, the bacterium was present in 26% of the analyzed ticks. Currently, the thicks populate the southern part of the valley, from the lake of Poschiavo to Campocologno, a small area in the central part of the valley and the area around Poschiavo and San Carlo. The interviews carried out showed that in recent years the ticks in the valley have increased and that the climate change could be a possible cause. In fact, since 1980 the temperature measured on the bottom of the valley has increased on average by 1.5 ° C and also the relative humidity has risen slightly. These changes could affect the diffusion of ticks in the valley of Poschiavo. In the future the temperatures will rise further and consequently the climate of the Poschiavo valley will most likely be more suited to the life of the ticks favoring their in-crease.

本研究以蒙日定理「平面上三圓彼此的外公切線交點共線」及其對偶定理「平面上三圓彼此的內公切線交點與另一圓的圓心的連線共點」出發，探討三圓更多由內、外公切線所產生的共點共線性質，進而探討四圓以上的情形，以及正多邊形、圓錐曲線等位似圖形，並推廣至空間中的球體。正如本研究作品名稱，我們將鮮少人研究的蒙日定理萌發出新枝，在日夜中茁壯，甚至最後有驚人的發現「在空間中n個外離的球，任意1個球的球心與另n-1個球的蒙日點連線會共交一點，此點稱之為n球的蒙日點」，此「點」發現，讓人不禁對宇宙中星體之間的關係產生更多無限的想像。

虎紋三角渦蟲(Girardia tigrina)是台灣本土常見的渦蟲，常用於再生之研究。然而其黏液功能與成分以及捕食蚊幼蟲機制皆尚未清楚。本研究中，從解剖顯微鏡觀察得知渦蟲爬行時分泌的黏液能將蚊幼蟲纏住，且發現渦蟲偏好從蚊幼蟲腹部末端及肛進行攻擊，並於體外行物理機制初步消化後，將蚊幼蟲組織以咽運動產生的負壓吸入體內，再行化學分解。以Congo Red、CBR250及PAS染色，發現渦蟲黏液中具有多醣、蛋白質及醣蛋白，以SDS-PAGE分離渦蟲黏液蛋白後，用銀染法分析發現渦蟲黏液包含多種蛋白質，主要為蛋白質單體15kDa，以蛋白質N端定序及LC MS/MS交叉比對分析得知渦蟲黏液中可能含有titin、calcium-binding protein等蛋白質，相關實驗還在進行中。我們也發現渦蟲在有無蚊幼蟲環境不影響其黏液蛋白分泌量，然而黏液蛋白成分組成是否有差異，需再繼續研究。本研究結果可能對應用生物防治法抑制病媒蚊有所助益。

同類相食行為廣泛發生於動物界，兩種可能原因：個體的特殊營養需求和族群的汰弱留強策略。觀察雜食性的高音符絲鱉甲蝸牛(Macrochlamys hippocastaneum)，初步發現同類相食行為與蝸殼破裂與否有關，殼完整的個體間不會發生相食行為(100%)，且當破殼程度愈大愈易發生。食性實驗中攝取過植物性養分的個體傾向選擇同類相食(χ2=22.04,***P<0.001)，同時提供同、異類蝸牛內臟時則未顯著選擇同類(χ2=3.76, P=0.053)，因此未能充分支持特殊營養需求。然而，此種蝸牛會追蹤破殼同類分泌之特殊黏液且具顯著趨向性(χ2=13.06, ***P<0.001)，與距離呈負相關(χ2=6.53, *P<0.05)。觀察腹足顯微構造中，發現內有特殊的疏鬆組織與大量分泌細胞，且通往尾端分泌特殊黏液的開口部位。結果證實破殼受傷個體產生的特殊黏液會主動引起同類相食，支持汰弱留強的族群策略，且物質誘發同類相食模式為動物行為研究中首次發現。

會議室圓桌上有n個座位，順時針依序放有編號①、②、③、 、n，共n張名牌。將參與這場會議的人也編碼，依序為1、2、3、 、 ，假設編號1的人一定會先抵達並坐到了名牌②的位置，剩下的人則依亂序到來，先找到自己名牌的座位，如果位置是空的就坐下，如果位置被佔了就往順時針方向找到下一個空位坐下（例如3到達時，若③、④均被坐了，而⑤位置是空的，則3就坐到名牌⑤的位置)。等到一個人坐定後，另一個人再進入會議室，依此規則，最後此n人到底會有幾種不同的坐法？ 我們以遞迴手法以外的方法解決了這個的問題，並深入探討： 1.能否從入座的順序立刻得知最後坐定的位置？ 2. 能否從最後坐定的位置，找出進入會議室的順序？同時得知有幾種不同進入會議室的順序會符合同一種坐法？ 3. 每個人坐錯位置的機率為何？恰k人坐錯位置的機率為何？坐錯位置人數的期望值為何？

The purpose of this project are 1)To study the relationship between global warming and acid rain with chemical model and mathematics model from temperature changing and pH of carbonic acid. 2) To create a pH measurement tool of carbonic acid in gaseous state.3) To study the impact of human activities in Loei province that affect to the relationship between global warming and acid rain. The procedures are 1)Do an experiment for studying the relationship between temperature changing and pH of carbonic acid. 2) Proof the mathematics model by using the result of experiment, the chemical reaction equation of carbonic acid solution. 3)Create a pH measurement tool of Carbonic acid by using Arduino and sensor with new formula in the computer program. 4) Using a pH measurement tool of Carbonic acid for studying impact of human activities in Loei province including industrial area, agricultural area, tourism area and forest area. The result of the mathematical model of the relationship between temperature changing and pH of carbonic acid is in form of Cubic equation in Equilibrium state and STP state. (Standard condition for Temperature and Pressure) So, we found that in this state has pH of carbonic acid is about 5.644. When the temperature rises up the effect of rainfall has a lower pH of carbonic acid solution. We also proof the new formula that create a pH measurement tool of Carbonic acid in gaseous state. The impact of human activities in Loei province found that the areas most affected by acid rain are the industrial areas, agricultural areas, tourism areas and forest areas respectively. In conclusion, when the temperature rises, it will dissolve acid solutions in the water on the earth. The loss of [H+] made the pH increases and the greenhouse gases become more atmospheric. These gases are more likely to react with atmospheric vapor. When these vapor form a cloud and condensation falls as rain, the rainfall has a lower pH, that is, global warming can result in the phenomenon of acid rain is greater.