2019年

本研究旨在對費馬多邊形數定理(任意非負整數皆可表成n個n邊形數的和)進行更進一步的延伸探討，更精確地說，即是：對於給定的二次式an2+bn+c，定義一數列〈an 〉n=-1∞=〈a-1=0,an=an2+bn+c, ∀n≥0〉，而若存在一最小正整數γ，使得對於所有非負整數x，可由數列〈a_n 〉_(n=-1)^∞中取出共γ項，滿足x恰為這γ項之和。這時，我們稱正整數γ為二次式an^2+bn+c的指標值，定義函數Yi使得γ=Yi(an2+bn+c)。 在本研究中，首先先行探討函數Yi的一些基本性質，再藉由電腦以暴力法算出一些二次式指標值的下界，從這些指標值中找出規律，將其推廣至所有我們所討論的二次式，並證明之，至於再探討二次式指標值的上界的部分，我將數學家MelvynB.Nathanson證明費馬多邊形數定理的證明技巧，稍作改寫，使其能夠應用至更為一般的情況，藉此系統性的求得二次式指標值的可能上界。最後，經由不斷的優化上界與下界，即可求得二次式的指標值。

一個裝有水的玻璃杯，用喇叭撥出大聲的聲音，便能觀察到共振的現象發生，玻璃杯將會不停的震動，甚至導致破裂。本研究將會藉由傅立葉轉換(Fourier transform)來分析玻璃杯的振動的本徵頻率，並進一步探討頻譜的峰值。另外我們也研究了不同水位的高低以及液體密度對於共振頻率的影響，並以能量守恆的觀點進一步推導出玻璃杯的震動方程式。最後我們也近一步討論higher harmonic，並且發現non-linear dispersion relation的現象，有別於我們對於f∝n的觀念。

ARC Generator is a acronym for Alternating Rotational Convertor. The purpose of the generator is to convert rotational motion into an alternating current. What makes this generator unique from other generators is that it uses a combination of aspects from rotational as well as linear generators. The A.R.C Generator is a experiment to generate electricity in new ways, potentially opening new doors in the area of electrical generation. The final goals for the ARC Generator project are to: 1. Make a electrical generator that is unique compared to other types of generators. 2. Make a hydro power based generator that is simple as well as affordable for private use. The generator itself is split into four distinct parts: • The channels • The inner chamber • The core • The coils

此研究在探討「給定一圓及多個已知點，求作各邊或其延長線上恰含有前述已知點中之一點的圓內接多邊形」之作圖方法及圖形的性質。 研究當中，以反演的幾何變換結合代數方程式並透過遞迴關係，除了找到該問題之解的個數與條件外，並求得有唯一解之點的軌跡方程式，從中發現其軌跡圖形為一圓錐曲線，且隨著給定點在圓外個數的奇偶性，會有或為橢圓或為雙曲線之不同的圖形變化。同時也發現給定點本身使得滿足條件的圓內接多邊形存在之個數將決定該軌跡圖形與給定圓間之切點個數等的有趣性質。

本研究以實作字體風格轉換的生成對抗網路模型為動機，將Conditional GAN當作模型的基礎，探討pix2pix模型及其他研究的一些方法對模型會產生甚麼影響，以得出能最優化預測成效的深度學習模型。 首先進行的是前處理的步驟，將字體的truetype檔案轉換成模型輸入的jpeg檔，再以生成器（Generator）和判別器（Discriminator）建立Condional GAN的基礎模型，然後探討加入U-Net、Category Embedding等方法，以及訓練資料集大小對模型造成的影響，最後實作整合的pix2pix模型和CycleGAN模型進行比較。 經過實驗後發現，U-Net和Category Embedding都對模型的預測成果有所幫助。另外，對相似的字體而言，pix2pix的效果較好，而對兩種風格差異較大的字體則需用Category Embedding的方式，融入更多字體進行訓練以達到更好的成效。

As our planet suffers the effects of climate change, it is only a matter of time before society will have to centre all aspects of development around sustainability. In the past, clean solutions for transportation have been dismissed due to the higher cost, and lower efficiency than fossil fuels. However, in the past few decades, there has been a steep decline in solar module cost, and and a steady climb towards higher efficiency. From my findings in this project, I have concluded that we are now at a point where we can embrace the clean, renewable potential which our sun offers. I have created and tested a proof-of concept electric vehicle (Solar EV), which can run indefinitely during daylight hours, provided sunny conditions. There are several mechanical features of my project which highlight the potential that renewable energy in transportation can have. Firstly the vehicle’s 500W motor is powered by 3 100W solar modules, and 3 50W modules, for a total of 450W or power generation. This means that when driving at anything less than 90% throttle, the Solar EV can run continuously without needing to stop to charge or refuel. Another design mechanism installed in the vehicle are three 12V lead acid batteries. These batteries allow the Solar EV to be powered for over 1.5 hours, which is useful during cloudy conditions, night, and most importantly, when driving through areas of shade. A unique efficiency component designed into my vehicle is the linear actuator I installed into the module racking system. This design element allows the tilt of the modules to be altered, to maximize the efficiency of the solar module array. At early or late hours of the day, it can be heavily tilted with the press toggle switch, or kept at a relatively flat level when the sun is the highest in the sky. I ran a series of trials to figure out whether or not the theoretical data matches up with the experimental results. After my series of trials, the bike was yet to run out of power. The solar vehicle reaches speeds up to 32 km/h, however comfortably glides at around 25 km/h. The linear actuator I installed allows the solar modules’ tilt to change . During different times of day or year, the sun is at different heights in the sky, however it is very important to maximize the solar potential. With the press of a switch, the module can be actuated to account for this. Lastly, regenerative braking captures the energy from braking. Using the reversible nature of a DC motor with a specialized motor controller responding to feedback from the brake actuators allows the vehicle to reuse energy that would otherwise be wasted as heat.

本研究在於製造出一顆環保並可回收利用的電池，在顧及環保的同時，也要有更多經濟與日常的考量。 在電解液方面：藉由添加植物肥料中常見的含氮、磷元素之化合物，及金屬離子以產生氧化還原反應來獲得電動勢，如此一來，電池使用完畢後，還能將剩餘的電解液做為肥料來為植物施肥。 在電極方面：在政府教育的推廣下，使用後的乾電池都必須要進行回收以免破壞生態。據此，我們進一步從廢棄乾電池中取出電極部分並作為環保電池的電極運用，如此一來便能使廢棄乾電池重獲新生。 在外殼方面：我們利用可生物分解的甘蔗渣吸管為包覆材質。 我們已自製成功可產生約1.5 V的環保電池，後續將發完電的電池做為肥料並用來使植物茁壯成長，未來若能取代現有乾電池，那麼對於廢棄電池造成的環境汙染，將具有極大的改善效益。

本研究藉由基因誘導之第二型糖尿病小鼠（Male B6.BKS-Leprdb mice; db/db mice）做為動物模型，並利用氫質譜儀與代謝體學之平台（1H NMR-based untargeted metabolomics）了解第二型糖尿病所導致的腎病變之疾病進程中的代謝途徑變化，及白胺酸的介入對於疾病進程的影響。研究發現，三羧酸循環及支鍊胺基酸代謝途徑中的代謝物，在基因誘導之第二型糖尿病小鼠尿液中有顯著下降，此發現指出糖尿病腎病變所造成的粒線體能量代謝功能下降。在肝臟組織萃取中，脂化膽固醇（esterified cholesterol）的上升及 β-胡蘿蔔素（β-carotene）、維生素Ａ（vitamin A）及維生素Ａ酸（retinoic acid）的下降顯示肝臟產生脂肪肝及其氧化壓力之上升。白胺酸的介入對於減緩腎臟粒線體受損及降低脂肪肝與氧化壓力並無顯著功效，此現象可歸因於支鍊胺基酸啟動mTORC1代謝途徑。本研究所標定之代謝物可被應用於第二型糖尿病及其所導致之腎病變的生物指標。

Wildfires have become increasingly frequent and severe due to global climatic change, demanding improved methodologies for wildfire modeling. Traditionally, wildfire severities are assessed through post-event, in-situ measurements. However, developing a reliable wildfire susceptibility model has been difficult due to failures in accounting for the dynamic components of wildfires (e.g. excessive winds). This study examined the feasibility of employing satellite observation technology in conjunction with artificial neural networking to devise a wildfire susceptibility modeling technique for two regions in California. Timeframes of investigation were July 16 to August 24, 2017, and June 25 to December 8, 2017, for the Detwiler and Salmon August Complex wildfires, respectively. NASA’s MODIS imagery was utilized to compute NDVI (Normalized Difference Vegetation Index), NDWI (Normalized Difference Water Index), land surface temperature, net evapotranspiration, and elevation values. Neural network and linear regression modeling were then conducted between these variables and ∆NBR (Normalized Burn Ratio), a measure of wildfire burn severity. The neural network model generated from the Detwiler wildfire region was subsequently applied to the Salmon August Complex wildfire. Results suggest that a significant degree of variability in ∆NBR can be attributed to variation in the tested environmental factors. Neural networking also proved to be significantly superior in modeling accuracy as compared to the linear regression. Furthermore, the neural network model generated from the Detwiler data predicted ∆NBR for the Salmon August Complex with high accuracy, suggesting that if fires share similar environmental conditions, one fire’s model can be applied to others without the need for localized training.

本研究考慮的主要問題： 若非常數之連續函數f滿足∀m∈N,∃P(x)∈C[x] s.t.f(mx)=P(f(x))，其形式應為何？ (一)、若考慮函數範圍為解析函數，則f(x)的形式必為下列三者之一： (1).axn+b (2). akx^n+b (3). acos⁡(kxn)+b ，其中a,b,k∈C、n∈N (二)、若將考慮函數範圍改為：連續函數f:[0,∞)→C，則f(x)之形式必為下列三者之一： (1).axk'+b (2). akx^n+b (3). acos⁡(kxn)+b ，其中a,b,k,k'∈C、n∈N、Re(k' )>0 (三)、若將考慮函數範圍改為：連續函數f:(0,∞)→C，則f(x)之形式必為下列四者之一： (1).alogx+b (2).axk'+b (3). akx^n+b (4). acos⁡(kxn)+b ，其中a,b,k,k'∈C、n∈N 在本篇的最後，我們也將N的角色以其他正實數子集取代掉以推廣結果。