2022年

自1930年代以來，考拉茲猜想(Collatz conjecture)一直是個未解之謎，其敘述如下：選定一個自然數，如果是偶數，則用2來除；如果是奇數，則乘以3再加1，經過有限次迭代，最後一定得到1。也就是說會得到1,4,2,1,4,2,…的數列，稱之為1-2-4循環。即使此猜想敘述簡單，卻是個橫跨世紀的難題，至近幾年才有一些證明方法出現。 其中一種證明考拉茲猜想的想法為證明所有不符合考拉茲猜想的狀況為假，而其中一種狀況為除了1-2-4循環還有其他組循環，即有些正整數在經過數次考拉茲猜想的計算後，會進入一組非1-2-4循環的循環。 因此，在此篇報告中我們透過討論每一個奇數在經由乘3再加1的計算後，所得到的偶數的2的冪次，再經由反證法證明除了1-2-4循環不會有其他組循環。

觀察紫葉(Oxalis triangularis)、紫花(Oxalis corymbosa)、黃花(Oxalis corniculata )三種酢漿草葉片在早上打開，晚上閉合主要是受生理時鐘影響。實驗比較發現紫葉、紫花、和黃花酢漿草在中午強光日照下，都有葉片閉合的午睡運動，且對光線的敏感度：紫葉酢漿草>紫花酢漿草>黃花酢漿草。再用遮光網遮蔭控制光度試驗中，發現光強度是影響酢漿草午睡現象的主要因子，與水分蒸散較無關。推測其目的是避免葉綠素因強光作用而產生光氧化現象(photooxidation)，導致光合作用無法進行。而在溫度試驗也發現酢漿草在遮蔭高溫下(42℃)葉片會閉合，應是水分蒸散導致膨壓下降使葉片閉合。另在實驗過程中觀察到酢漿草也有觸發運動，在強風吹襲及水平、垂直晃動下均會導致葉片閉合，推測其目的在減少葉片摩擦產生傷口，避免病菌的感染。總結酢醬草葉片開閉受生理時鐘、光線、溫度、風吹震動及水分膨壓的影響。

My thesis focuses on studying Cretaceous microfossil specimens from the excavation of former brickworks in Štíty, especially foraminifera. In the theoretical part, I have covered the structure of the Bohemian Cretaceous Basin area, especially Bystřice Lithofacial Development. I have also processed previous paleontological researches from the locality. Emphasis was placed on field research and subsequently on laboratory research of the site. I have examined the present state of the location and gathered samples of silt clay containing a wide variety of fossils. I have acquired the microfossils, determined them, and ordered them systematically. The most important part of the thesis is the systematic and palaeoecological processing of the collection of microfossils from the locality. The thesis continues the research of the last year of SOČ, where I have gathered a collection of fossil macrofauna, flora, and ichnofauna. My collection is supplemented mainly by benthic and planktonic foraminifers. I have confirmed that the specimens found are typical representatives of marine fauna belonging to the Upper Cretaceous Coniacian. The paleoecological characteristics of the locality correspond to a nutrient-rich shallow-water environment, occasionally disturbed by storm waves.

紅瓶豬籠草的捕蟲構造是由膨大後的葉柄固定在莖上，再由葉柄下凸的維管束及扁平處的維管束內縮向前端延伸發育成籠蔓，緊接著籠蔓的前端再膨大並經由細胞凋亡特化出類圓筒型、封閉的「空心葉」，最後特化出無柄腺鑲嵌在內層細胞上負責分泌、消化、吸收。無柄腺旁的原生質絲直徑較一般植物大5-10倍，幫助吸收。由於籠蓋不能閉合，下雨天時籠內物質容易傾倒，豬籠草演化出「來者不拒」的捕食策略。除了紅色食用色素和無機物以外，不管是含氮物、水溶性、酯溶性小分子皆吸收，甚至連蔗糖及大分子的澱粉、蛋白質，都以耗能的胞吞作用吸收，這與捕蠅草只吸收小分子的含氮物質及酯溶性的物質不同。而懸吊捕蟲籠的籠蔓，可以比自身重35倍的超強支撐力，維持捕蟲籠開口向上不傾倒溢出，保持較高的自然盛載量，提升捕食競爭力。

我們已經知道， 任何分數都可以拆成數個單位分數和， 也有各種不同的拆法， 我們發現在不同的拆法中會有各種不同的長度， 例如2/5可拆成1/3+1/15,1/5+1/6+1/30，等各種不同的形式。 由於1/n=1/(n+1)+1/n(n+1) 公式， 已經拆成了各種單位分數的和之後， 又可拆出各種不同的和。 因此在這份報告中， 我們想找出各種分數最短拆分的方法。 我們的策略是在拆分的過程中，先用「貪心算法」： b/a=1/(q+1)+(b-r)/a(q+1) ， 其中a=bq+r找出最短拆分長度的一個上界， 再利用Erdös-Straus猜想、或是直接解不定方程， 找出最短拆分長度。 後來我們遇到了Schinzel猜想，我們便開始著手討論有關Erdös-Straus猜想與Schinzel猜想的證明討論，也看了許多關於這兩個猜想的文章。總歸來說，我們討論Schinzel猜想的方法是將正整數n模一個適當的整數，得到一個同餘類，去討論哪些正整數n可以寫成l/n=1/(q+t)+1/y+1/z的形式，其中q=⌊n/l⌋。

農作物常遭受病毒攻擊，造成重大經濟損失，但傳統的化學藥劑無法有效抑制病毒，且易殘留汙染環境。本研究旨在利用物理性的電擊模擬外界刺激，進而探討電擊對植物啟動水楊酸主導免疫反應的影響。實驗中以10伏特的電壓電擊菸草10分鐘，能使圓葉菸草(Nicotiana benthamiana)的水楊酸標誌基因PATHOGENESIS-RELATED PROTEIN 1(PR1)表現量提升，並使菸草鑲嵌病毒 (tobacco mosaic virus ; TMV)的感染斑點數下降。顯示電擊處理可以誘導水楊酸所主導的免疫反應，並增強整體抗病毒性。另外，相較於未電擊過的菸草，電擊後的菸草系統葉在接種TMV-GFP後PR1基因表現量提升，表示電擊可能對菸草系統性水楊酸相關抗性表現產生預起效應(priming)，可以幫助植物對抗未來的病毒感染。

Double Parallelogram Lifting Mechanism (DPLM) is a compact and stable lifting mechanism with a large extension range widely adopted in robot designs. Rubber bands and springs are often installed on the DPLM to lighten the motors' load and maintain its height, yet the installation positions are often obtained through trial and error. This project aims at finding the optimal rubber band installation positions for DPLM using modeling and optimization techniques. A mathematical model which describes the forces and moments acting on all the linkages of DPLM was derived based on the conditions for the static equilibrium and verified with a 3D simulation software. A genetic algorithm (GA) was implemented to optimize rubber band installation positions, which managed to find solutions with the overall root-mean-square- error (RMSE) of the net moment less than 2 for 2 to 6 rubber bands. A further statistical analysis of 50000 random rubber band samples showed that installing rubber bands in triangles is the best solution with the overall lowest RMSE. A test was conducted with a prototype of the DPLM and the results were consistent with our model and optimization. This project derived and verified a mathematical model for the DPLM, and found the optimal way and positions to install rubber bands. The results of this project provides a theoretical basis for controlling DPLM with rubber bands, allowing it to be further adopted in industrial robots that require repetitive lifting and lowering such as inspection robots and aerial work platforms.

在這篇作品中，研究了法里數列的性質，並試圖藉由與二維法里數列有關的福特圓，將其推廣到空間中，找到空間中與法里數列相應的數。 研究分成兩個階段，第一階段是舊定義的階段(2021國際科展作品「空間中的福特球」內容)。此階段中，定了空間中有起始三個相切的球，從這三個球繼續做更小的相切球，並找到一些性質。在經過一系列的討論，發現推廣到空間後的福特球有與平面福特圓重疊的部分。因為這個發現，我們就大膽的假設這一大堆球，可以跟另一種定義等價──直接把平面的福特圓變成三維空間的球(也就是起始球是一整排無限多顆的球)，然後做相切小球可以得到一模一樣的結果。第二個階段是由新定義開始的研究，直接將平面的福特圓變成空間中福特球的起始球，使得可以藉由法里數列的性質，快速求得福特球球心表示法，同時也找到了三維空間中的法里數列。最後研究更進一步試著推廣到D維空間(D≥3)中。

A traditional helicopter requires 60 - 80% more power to hover than when in forward or lateral flight, making the manoeuvre extremely power inefficient. To maximise efficiency, industrially many properties of the helicopter and rotor have been changed and tested, for example: optimising blade shape, fuselage shape and changing weights of different helicopter components. This report in particular aims to find a relationship between power efficiency and angular velocity for a twin bladed hovering helicopter with a single rotor. The angular velocity of a blade measures the frequency of its revolution about a fixed point. A theoretical approach was first taken and then justified with empirical data. Firstly, a model for power efficiency was derived with William Froude’s momentum and blade element theory. The efficiency equations incorporated the thrust and power coefficients. Therefore, the research focused on determining values for these coefficients by manipulating equations, using industrial specifications and simulations from the XFOIL software. In order to validate the accuracy for such theoretically generated data, an experiment was conducted for a comparison. The theoretical and empirical data were concurrent, as they followed a similar trend and the empirical values overlapped within the theoretical error bars. The power efficiency for different angular velocities were then found by substituting values for the coefficients. The results demonstrated a positive relationship; where, as angular velocity increases, power efficiency increases too, then plateaus and repeats the same trend once again. The research raises many questions and could be extended by determining if a similar relationship exists for tri-copters and quadcopters.

Stochastic gradient descent (SGD) is one of the preferred online optimization algorithms. However, one of its major drawbacks is its predisposition to forgetting previous data when optimizing through a data stream, also known as catastrophic interference. In this project, we attempt to mitigate this drawback by proposing a new low-cost approach which incorporates backward queried images with SGD during online training. Under this new approach, we propose that for every new training sample through the data stream, the neural network is optimized using the corresponding backward queried image from the initial dataset. After compiling the accuracy of the proposed method and SGD under a data-stream of 50,000 training cases with 10,000 test cases and comparing our algorithm to SGD, we see substantial improvements in the performance of the neural network with two different MNIST datasets (Fashion and Kuzushiji), classifying the MNIST datasets at a high accuracy for the mean, minimum, lower quartile, median, and upper quartile, while maintaining lower standard deviation in performance, demonstrating that our proposed algorithm can be a potential alternative to online SGD.