三等獎

本研究旨在研究費馬多邊形數定理(任意非負整數必可表成k個k邊形數的和)的一般化情況，也就是說，任意非負整數是否能表成給定的二次多項式數列中所選取的γ項和。以數學模型敘述，就是探討對一個已知的二次多項式an2+bn+c，是否可找到一正整數γ，滿足∀x∈N∪{0},∃α1,α2,…αγ,使得x=∑γi=1(aαi2+bαi+c)。 本作品主要探討若此探究模型存在，那麼數列〈an 〉的一般式an2+bn+c與γ值之間會存在什麼關係，並期望能運用一個簡潔明瞭又一般化的數學式表示。本文亦提供另一個數學模型，探討γ值與某些特殊係數a,b,c之間的關聯性。而本文探尋[a/2]n2+[b/2]n+1,a∈N,b∈Z,a+b≡0(mod 2)(此為本文主要探討的二次式)，求得此二次式所對應之γ值的方法為先令p=[2a/(a+b)]+2，再藉由所建立的模型二，求出[(p-2)/2] n2+[(4-p)/2] n的γ值，接著再用所建立的模型一來求得[a/2]n2+[b/2]n+1的γ值，進而依循此方法最後得出任意形如[a/2]n2+[b/2]n+1的二次式之γ值。

平原菟絲子對於多數植物皆會寄生，實驗中發現不易寄生於空心菜、地瓜葉、朝天椒，可證明具有寄生的選擇性。為了抑制種子的發芽，實驗無意間發現密閉中麵團黴可以完全覆蓋種子，但木瓜黴、柳丁黴、麵包黴卻沒有這種現象。麵團為穀類製品，易受黃麴毒素汙染，花生更易受黃麴毒素汙染，故選擇花生再次實驗，發現花生也能有效抑制發芽的效果。 103年5月觀察世博館旁與南寮漁港的菟絲子生長情形，從夏天到冬天都有菟絲子生長，104年8月經歷蘇迪勒Soudelor颱風和10月杜鵑Dujuan颱風過後，實地觀察發現，南寮漁港堤防外整片菟絲子全部消失，堤防內的依然存在。根據文獻顯示海水中含有35‰的氯化鈉，設計實驗發現泡過15‰以上的食鹽水後菟絲子就不會再發芽，另外，我們運用海水成分第二高的氯化鎂進行實驗，亦有抑制種子發芽的效果。

Backpropagation neural networks were used to forecast daily rainfall with minimal error for Metro Manila in order to have an inexpensive way of accurately predicting weather. Calamities brought on by heavy rainfall have caused great economic, infrastructure and human loses. Neural networks have the ability to discern complex patterns in noisy data; this makes it a viable method for weather forecasting. Daily precipitation, humidity, rain indication, sea level pressure, temperature and maximum sustained wind speed for January 2000 to December 2010 were acquired from the Philippine Atmospheric Geophysical and Astrological Services Administration. The neural network made use of Python 2.7.2 and the backpropagation program by Neil Schemenauer (python.org). It considered different neural network architectures with a total of 2844 data sets for training and 708 data sets for testing. Each neural network’s accuracy was measured with a graph of the actual and predicted values, correlation coefficient, and root mean square error. It was observed that the neural network with architecture 5-8-1 yielded the most accurate results as it had the highest correlation coefficient of 0.48599 and smallest root mean square error of 14.84. It was also observed that the trends of the predicted values followed that of the target values. This suggests that it is possible to create a neural network with a moderate correlation given daily weather data. It is recommended that further researches make use of hourly data instead of daily data for more accurate results. Other variables, which might affect rainfall, not in this study should also be considered. This research could aid in the anticipation of calamities and the decision making involved in shipping, fishing and aviation industries.

Protein C deficiency (PCD) is found in 1 out of 200 to 500 persons in the general global population which is also one of the common conditions of Inherited thrombophilia, it’s characterized by an increased tendency of blood to clot in human blood vessels. It is caused by several factors including mutations in the genes involved in thrombin binding, protein c activation and numerous clotting factors. This includes F5 (Factor 5 Leiden) gene on chromosome 1q24.2, F7 (Prothrombin) gene on chromosome 13q34, SERPINC1 (serpin peptidase inhibitor C) on chromosome 1q25.2, SERPIND1 (serpin peptidase inhibitor D) on chromosome 22q11.21, HRG (Histidine Rich Glycoprotein) on chromosome 3q27.3, PLAT (Plasminogen Activator) on chromosome 8q11.21 and THBD (Thrombomodulin) gene on chromosome 20p11.21. In the current study, a three Saudi families with inherited thrombophilia has been recruited to identify the underlying cause of this special condition. Whole exome sequencing, targeting all coding exons of the human genome, was performed using Illumina Nextera library preparation kits followed by paired-end sequencing on Illumina NextSeq500 instrument. Reads quality control was performed and reads were aligned to the reference genome using BWA software. Variants calling and annotation was performed using GATK. All known genes involved in causing inherited thrombophilia All known genes involved in causing PCD were excluded by whole exome sequencing. The genes that were previously reported to be involved in inherited thrombophilia were checked for any causative variant. No mutation has been identified in known genes. identifying a novel gene underlying PCD. The Result of this study will hopefully pave the way to better understanding the disease pathophysiology and help in developing DNA based diagnosis, carrier screening and somatic gene therapy.

本研究探討將方格填入Young Diagram之排列結構，考慮填入的順序並計算其方法數。已知Hool Length公式得以計算特定形狀的Young Tableau方法數，而本研究欲討論在給定各相連方格形式與其數量的情形下，一般化的排列圖形之性質及數量關係。首先發現骨牌的排列與「缺一格」形式之數量相同，進而深入觀察排列的結構以尋找兩者間的關聯性，並藉此對應關係(bijective relations)解釋這個特別的現象。接著，進一步的利用路徑表現此結構的相關性質，嘗試探討一般化的缺格情形，藉由路徑的對應，得知該排列方法數均為「廣義Motzkin數列」的線性組合。除此之外，更擴展至廣義的r-Ribbons「多格相連」的排列結構，發現r-Ribbons方格排列圖形在完整與「缺一格」的情況下可一一對應，若將n個r-Ribbons填入高度為H的Young Diagram中，當H不超過r時，其總方法數為Hn，以各列編號相異的方格數量進行分類，其結果為在H為空間下的Pascal分佈。

以製作適合高中生的英文篇章難易度自動分級為初衷，本研究採高中英文課文為語料，針對「如何分級」，意即從文章萃取哪些特徵、利用何工具或語料協助萃取特徵、以何工具分級等因素，進行研究與實驗，並建立一套新方法。首先進行前處理，再嘗試以單字、句型的數量或比例、句長、音節長、整合以上分析等各式特徵，支持向量機(Support Vector Machines)、隨機森林分類器(Random Forest Classifier)、決策樹分類器(Decision Tree Classifier)、卷積神經網路句分類器(Convolutional Neural Networks for Sentence Classification)等工具，進行將篇章分為高中一、二、三年級等三個難易度等級的測試，建立自動分級模型。最後製作成可供大眾使用的自動分級網頁。各項測試之中，最佳分類效能為整合各項特徵時得到的分類正確率65.04%，經模擬得知，此效能較過去研究，已有所提升。

We all do it – walk along with a cup in hand, and carelessly spill it. While it’s usually more annoying than anything else, it happens to affect almost all of us, and little is done to minimise the likelihood of it occurring. So my aim was to explain the physics behind why we spill drinks when we walk, and to investigate how we can minimise the likelihood of this occurring. I broke this investigation into two distinct parts, explaining the system of the cup, and explaining the effect of walking. From initial observations, it was clear that the cup was a resonating system. Like any resonating system, the cup has a natural frequency. When the cup is oscillated – moved back and forth – at near this frequency, the size of the liquid oscillations is very large. This is because the acceleration is in phase with the motion of the liquid, so in each cycle maximum energy is input into the system. In my investigation I experimentally measured this natural frequency, and created a mathematical model to explain this frequency. It was also found that as the size of liquid oscillations in the cup increases, so does distortion of the fluid surface, possibly enabling spilling. To systematically analyse the effect of walking, I had subjects walk on a treadmill, so walking surface and speed were controlled. However, I also needed an accurate way of measuring the motion of a carried cup. Firstly, I tried to use video analysis; however I found this far too imprecise for measuring small changes in velocity of a cup. In the end I used a smartphone to record the acceleration of a carried cup, as acceleration is what causes the movement of liquid in a cup. This allowed surprisingly accurate measurements to be made, and allowed both the size and frequency of the acceleration to be recorded. In order to relate the system of the cup and the oscillation provided whilst walking I conducted a qualitative experiment into the effect of stride frequency on the likelihood of spilling. When stride frequency was very close to the natural frequency of the cup, spilling occurred almost instantly, while it did not occur if stride frequency was much higher or lower. In the end, my research showed that to minimise the likelihood of spilling your drink walk slowly, use a narrow cup, focus on walking smoothly, and fill the cup well below the rim. Despite this, some people happen to be much smoother cup carriers than others, likely due to their individual biomechanics. And, if you really don’t want to spill your drink, you can always use a lid.

In this project, we defined a mid-tangent point with respect to a fixed point X and a tangent at a point Y on a planar curve C as a point on the tangent that is equidistant from X and Y. We studied the locus of mid-tangent points of conic sections. We found that the locus of mid-tangent points of most conic sections are non-linear curves. However, we observed and proved by using Euclidean geometry that the locus of mid-tangent points of circles are straight lines. The mapping defined by mid-tangent points was studied further. The similarity between a mid-tangent mapping and a stereographic projection was displayed as a one – to – one correspondence function. We also extended the concept of mid-tangent points to three dimensional space and found that the similarity with the stereographic projection was retained in higher dimensions. Finally, we studied the locus of mid-tangent points of a sphere to create a mapping of the sphere to a plane.

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。

近年來場址效應(Site Effect)的研究主要為探討地質造成之場址效應，不過現代，人類大多在建築物內活動，而建築物之場址效應也應該被受重視，本研究利用模擬波形探討建築物場址效應造成之放大效應，發現放大倍率會隨著不同樓層而跟著放大1到2倍。 中央氣象局致力研究地震預警系統，希望藉由預報來減少地震造成之損失，而其用於預估最大加速度之衰減式(Attenuation Relationship)考慮震源效應(Source Effect)與路徑效應(Path Effect)，而場址效應只利用經驗觀測值與預估值之比值表示。本研究以學校建築物做為研究區域，利用二階段衰減式作為模型提出三階段迴歸衰減式，加以考慮建築物造成之場址效應，在第三階段乘入利用模擬波形所得到之經驗場址效應放大倍率，最後利用線性迴歸修正，期望可有效預估建築物內之最大加速度，未來可用以建立建築物地震預警系統。