四等獎

本研究中，我們將提出一些新穎結果，著重討論其在三角中的應用；同時，找出其遞迴關係式，得出三角展開式與其所對應之多項式分解式，進而討論出多種的規律性及所涵蓋的內容及推廣性質，我得到很多高中數學公式無法推導出在【4】和【8】中的漂亮公式及創新的結果，且這些等式都是由我們不太瞭解的無理數所構成的。 主要是討論我們在【7】中所得到的收穫與經驗；複數是三角、幾何、代數互動的橋樑，我是以不同的角度及嶄新的方法來綜合探討在【6】中相關的應用。提出關於正整數平方的倒數和公式更為精簡且基本的證明，將 sin−2 x 表示成級數形式的部分分式，進而應用在(a,b) = 1的機率問題上；並研究相關的等式，直接透過三角與代數來研究關於 2p 次方的倒數之求和問題，得出級數 之和的有用遞迴公式，並與最重要的常數扯上關係。 For one thing, we present diverse methods to evaluate finite trigonometric summation and related sums. Trigonometric summations over the angles equally divided on the upper half plane are investigated systematically. Several related trigonometric identities are also exhibited. What is more, we use methods of calculus, and make several surprising and unexpected transformations. A useful recursive formula for obtaining the infinite sums of even order harmonic series, infinite sums of a few even order harmonic series, which are calculated using the recursive formulas, are tabulated for easy references. Furthermore, is there any interesting results and applications？ Finally, the purpose of this paper is to develop a new proof of and related identities, but their derivations are more complicated. The following studies are completed under the instruction of the professor.

Tea is the most widely accepted and consumed beverage worldwide due to its characteristic aroma and taste. Recent studies have provided the strong scientific basis for understanding the health promoting effects and cancer preventive actions of tea. The components of tea especial the catechins are varied with the conditions of making tea. To understand and determine the chemical composition of tea is very important. Some investigations of the parameters on the storage and making of tea were carried on in this study. The kinds of tea studied were including black tea, oolong tea, green tea, and instant tea bags. A high performance liquid chromatograph combined with UV detector and mass spectrometer used to analyze the components of tea. The results showed that the composition of tea solution is dependant of the exposure of light. One of the components of tea, (-)-epicatechin methylgallate (ECMG) was oxidized to quino form. The concentrations of (-)-catechin gallate (ECG) decreased and one of new compound (M.W. 442) produced with increasing the storage time. From the results show tea made with cold water is better than that made with hot water. The ingredients in green tea were changed faster than those in fermentative black tea and oolong tea. The components of tea can be kept unchanged for a long time at low temperature. In refrigerator, the time can be extended to overnight. 茶由於其具有特殊的芳香氣味及口味，廣泛地被世人用為飲料，近年來的研究證據顯示茶具有促進身體健康和防癌功效，但茶中成分之變化，尤其兒茶素隨茶的種類、茶沖泡保存方式及置放時間而有所不同，因此對於茶中成分及沖泡方式的認知是一門重要的課題。 本實驗主要是探討茶沖泡時間及貯存條件，對於茶液所含兒茶素變化的影響，探討的茶包含紅茶、烏龍茶、綠茶及其茶包。並使用質譜儀及高效能液相層析質譜儀配備紫外光偵測器分析茶液中之成分，實驗結果顯示成分變化速率以未發酵的綠茶較發酵的紅茶和烏龍茶快；低溫儲存時，亦可延緩茶液成分的變化，例如加蓋並存放於冰箱，則茶置放至隔夜其成分均未改變；同時結果顯示成分產生變化者，主要為兒茶素氧化，如將多酚類氫氧基經氧化變成?（quino） ，其次為斷裂再產生聚合其中綠茶以不照光變化較大。本研究並經實驗發現如果以冷開水沖泡綠茶20~30分鐘，咖啡因的溶出量雖略多於沖泡2分鐘的熱茶，但各種兒茶素的溶出量卻遠高之，尤其置放至隔夜茶液成分仍無氧化現象。

流行病的傳染過程如同一個無尺度網路，但較一般無尺度網路有著更多的變數而明顯差異，因此無法直接應用一般的無尺度網路模式來描述其傳染途徑。我建立一個新模式「無尺度流行病模式」，經由比較模擬結果與疾病管制局的數據，證實此「無尺度流行病模式」是正確與確切可用，且適用於短期暴發性傳染病與長期流行病。SARS案例研究結果，顯示影響SARS疾病傳染因子的大小是：ψ>m>γ。其中降低ψ值可使SARS確定病例至5月31日止降為143人(減少確定病例190人，相當於減少死亡21人)；僅提高防疫使5=γ，亦可使確定病例減至307人(減少確定病例26人，相當於減少死亡3人)。因此強化隔離措施以減少傳染天數最為重要，且可以有效控制每日SARS新增病例，避免發生高侵襲率的現象。HIV/AIDS案例研究結果，獲知採用ψ值來進行月份模擬，則至 2005年12月HIV(+)與AIDS分別為可減少2,715與285人。而進行年度模擬結果，則至 2014年底HIV(+)與AIDS分別為可減少41,936與5,328人。無尺度流行病模式可以協助所需警戒的程度與政策決定的計畫結果。因此無尺度流行病模式在幫助政府評估社會經濟成本與健康憂慮上的有用之工具。當面臨一個全然無知的新病毒的侵襲時，如何減少死亡與傷害人數？是本研究之最終目的。因此，本研究結合了流行病、無尺度網路與灰預測，建立面對病毒侵襲，一個確切可行的對抗無尺度流行病傳染新方法，並詳細說明運作流程。\r \r \r The course of epidemic infections resembles a scale-free network. However, they are different due to more variables in the epidemic infection. Therefore, the model of scale-free networks is not enough to satisfy the reality epidemic infections. In this study, I propose a new the Scale-Free Epidemic Model. Comparison of the simulation results with Taiwan CDC report data for SARS and HIV/AIDS cases show that the Scale-Free Epidemic Model is accurate and useful. This model can be used in the short-term outbreak of infectious diseases and for the longer-term epidemics. In the SARS case study, the results show that the sequence of effect of the epidemic factors was: ψ>m>γ. The SARS confirmed cases would decrease to 143 cases (reduced 190 confirmed cases or 3 death cases) calculated to May 31, 2003, if the average infection time was reduced to two days (an optimum value of ψ). Therefore, vigorous action in isolation quarantine and treatment for SARS cases is most effective policy; the number of new cases and the attack rate would also decrease. In the HIV/AIDS case study, the simulation results of the Scale-Free Model indicates that the reduced numbers of HIV(+) and AIDS in the monthly simulation calculated to December 2005 are 2,310 and 361 and the annual simulation by December 2014 are 27,161 and 3,710. The Scale-Free Epidemic model can help determine the level of caution needed and the projected results of policy decisions. Therefore it is a useful tool in assisting the government to balance socio-economic and health concerns. The fight against a new epidemic and how to reduce the number of deaths is the main purpose of this study. So, a new method to fight against epidemics is proposed. Detailed procedures of this method are explained.

本實驗是控制奈米碳管的成長條件，研究奈米碳管的成長機制、電子場發射特性。奈米碳管的物理性質會因捲曲度(helicity)、管壁厚度(單層或多層)、管徑所影響，而不同的捲曲度，會形成不同電性(金屬性或半導體性)的奈米碳管。由此可知，探討奈米碳管的成長機制，以控制奈米碳管的穩定成長，及如何長出電性或光學性質相近的奈米碳管，是極為重要之事。為了研究奈米碳管的成長機制，我們使用微波電漿輔助化學氣相沈積法 (Microwave plasma ehanced chemical vapor deposition) ，在市售的縫衣針針尖上，成長奈米碳管，這個技術能夠經由吸收微波達到局部加熱針尖的方式，再透過控制微波功率、通入氣體的種類、壓力的變化以及催化劑的使用，能夠達到有效控制奈米碳管的成長條件。我們改變三個參數：催化劑的使用、氣體種類(Ar+CH4 或N2+CH4)及壓力，之後利用SEM 觀察奈米碳管的成長情形；再測量場發射(測量電性)，討論不同成長的條件與場發射電流關係。

本實驗利用「閃光攝影術」，由拍攝桌球碰撞球拍拍面瞬間照片，希望能測量桌球與拍面的接觸時距、摩擦力、恢復係數。近年來，數位相機已經具有單眼相機所有功能，所有攝影參數可記錄調整，並且可以馬上看，作影像處理，大幅降低拍攝費用，若配合高速閃光燈，可以拍到重複曝光照片，所有的數據由照片測量，是計算碰撞問題的新的構想，適用於所有球類的碰撞。

Objective To design build and run 5 Mouse mobiles – vehicles which use mousetraps as their only source of energy. Targets- • Go up a ramp • Raise a flag • Hit something and set it free • Deliver something • Catapult Details These mouse mobiles are original creations which include some commercially produced parts. They are powered by single spring driven standard wooden mousetraps. They are 5 in number each being used for the respective tasks. Basic structure Each mousetrap consists of a 2 parallel wooden planks which has 2 holes drilled on either sides for the rear and front axles. A rod is used as the axle and CD disks are used as wheels. The mousetrap is attached to the planks. Movement and Locomotion To move the mouse mobile a string is connected to the rear axle. The string runs through 2 pulleys which change the direction of the applied force and are attached to the hammer of the spring. When the vehicle is pulled backwards on the floor tension is created in the string (as the hammer is stretched) which gets stored in the form of spring energy. When the wheels are released on the floor the rear axle starts rotating hence propelling the mouse mobile forward. Completion of objectives 1) To go up a ramp For the required task a high amount of torque would be needed .That will be sufficiently provided by the spring energy. 2) Raise a flag For the required task a rod is placed on the wooden platform. The flag is placed at the bottom of the rod; it is connected to the hammer of the mousetrap using a string. The spring and the rod are placed in the opposite directions. When the hammer is released after being stretched it pulls the string and raises the flag. 3) Deliver something For the required task 2 mouse mobiles are used. A platform of any hard and light material is attached to the rear of a mouse mobile. After reaching the desired location, when the hammer of the primary spring is fully relaxed it hits the catch of another stationary mouse mobile. The hammer of the respective catch is connected to the platform using a string. Upon activation the hammer is released, the platform is pulled with extremely high speed and the materials on the platform, due to their moment of inertia fall down. 4) Hit something and set it free. For the required task 2 mouse mobiles are used. A small mousetrap is attached to the front of the mouse mobile which is in contact to the required object. The other mouse mobile on reaching the above vehicle hits its catch using its hammer. The hammer of the stationary mouse mobile is released and the object is hit and set free. 5) Catapult For the required task 2 mouse mobiles are used. A mousetrap is attached to the front of stationary mouse mobile which is attached to a strainer and the object to be catapulted is placed inside it .Another mouse mobile approaches is and hits the catch hence activating the hammer and the object is catapulted.

機械加工過程中往往遇到形狀複雜工件，無法用一般虎鉗夾持進行加工。若需加工複雜工件時，需使用V 形槽、壓枕……等等夾具加以輔助，但有些夾具根本無法夾持。若用特殊夾具需拆除原有之虎鉗，而且還必須校正，工作繁雜又浪費很多時間。 本設計之優點為不需更換虎鉗，直接放在虎鉗鉗口即可夾持不規則的物體，利用正向力的作用夾持而不打滑，輕易達到夾持時之穩定和足夠之夾持力，以達迅速、不需使用特殊夾具、不需再校正、可當平行塊之多功能夾具，使複雜形狀之工件加工簡單化、迅速化之設計。;When handling workpieces in complicated and irregular shape in the mechanical process, users are unable to make it with ordinary vises. V-block and clamping block might help, while some others do not work at all. In such cases, the user has to tear the vise apart and then do some correction, which is complicated and time-wasting. The strength of this design is that there is no need to replace the vise. The user just puts this device on the vise clamp to clamp the irregular object. The vertical clamping force makes the piece at work stable and allows no slipping. With this device, no special fixture or further correction is needed. It can also be used for a parallel block if necessary. In other words, as a fixture of multiple functions, the device makes the processing work simpler and more efficient than ever.

經食道超音波 (TEE) 是診斷心血管疾病的重要方法之一，目前一般的診斷中，都是由心血管超音波影像的空間分布，藉以了解心血管的狀況或是阻塞的情況；以及用超音波都卜勒影像，觀察血液在血管中的流動。在這個研究中，我們嘗試用主動脈超音波影像圖形隨時間的變化，作為新的診斷方法，藉以定量地診斷出主動脈硬化的情形。研究過程中，我們將主動脈模擬為受到心臟之週期性壓力驅動之阻尼振盪器，其中主動脈之硬度變化即相當於阻尼因子。由心血管超音波圖形隨時間變化的分析中，測量出 “主動脈擴張到最大時” 相對於 “心電圖中的R-尖峰” 之間的延遲時間，理論上，此一延遲時間即可反應出主動脈硬化的程度。我們由10個樣品的資料中，對每個樣品進行大約50至100次心跳的分析。初步的研究成功地發現：上述之主動脈擴張延遲時間超過0.17秒時，即明確地顯示出樣品具有主動脈硬化之現象，而且此一方法也可由數據的分佈發覺心血管之其他病徵。未來仍需應用此一方法對較多樣品進行研究以確定此診斷方法之可靠性，並且由體外攝取主動脈超音波影像進行分析，藉以取代經食道超音波的方法，增加診斷的方便性以及減少病患之痛苦。 Echocardiography is a very important tool for the diagnosis of cardiovascular diseases. Important information about the intracardiac blood flow, shunt direction and cardiovascular system function can be obtained by echocardiography due to its high spatial resolution capability. However, seldom message is known about the aorta stiffness. This research investigated a new method to quantitatively analyze the aorta stiffness. The aorta was modeled as a periodic-force-driven damping oscillator, in which the stiffness of aorta was simulated as the damping factor. From the temporal analysis of the echocardiographic images, the delay time of the maximal aorta distention relative to the R-peak of electrocardiographic trace was measured to study the damping and stiffness of the aorta. The preliminary study successively found that a delay time greater than 0.17 sec could be a criterion to diagnose that the aorta is quite stiff. This method could also clearly discern some abnormal cardiac performance. A large scale study with this method should be conducted in the future.

流量計在實驗室與工業領域裡是重要的儀器，如今已經有數十種依不同物理原理而發展出來的型式，可以配合多變的環境需求與測量條件而使用。然而，各種流量計所適用的範圍備受侷限。本研究主要目的在發展一種熱線式的渦流流量計，供給氣體之流量量測之用。透過自行製作儀器與設備：熱線測速儀(包括探針、探棒及電子處理器)和渦旋產生器(管道中含一三角形截面之鈍體，當流體通過時，在後方尾流產生週期性渦旋逸放)。由於熱線測速儀擁有偵測流體運動時高頻動態變化的能力(約為20000 Hz 以內)，因此結合熱線測速儀與渦旋產生器，經適當的設計與調校，可以測得在不同流體流速時渦旋產生器的三角截面鈍體後方渦旋逸放的頻率。由於渦旋產生器的截面面積為固定值，因此可以從而計算出流量與渦旋逸放頻率的關係。經由嚴格的校準與驗證步驟，本研究的結果顯示自製的熱線測速儀擁有極佳的渦旋頻率偵測能力，所量測到的校準曲線顯示渦旋產生器的三角形截面柱所產生的渦旋逸放頻率與流量成線性關係。為了降低誤差，建議在0 ~ 40 CMM 之量測範圍內分成三條方程式來代表不同範圍內的校準曲線，最大誤差僅在5%以下。若需使用在不同的流量範圍時，僅需改變渦流產生器和幾何尺寸，以使渦旋逸放頻率適合於熱線測速儀的動態響應範圍即可。倘若商品化之後，可以實際應用於風扇流量量測、引擎進氣埠流量的測量等等應用。熱線測速儀本身也可作為風速計，適用於各種場合之風速量測。Flow meter is a instrument that is vital to the laboratory as well as the industrial related field. Based on different physical principles, tens of models that work in harmony with the diverse environmental demands and measurement conditions are developed to date. However, the application of varied flow meters is still under severe restriction. The purpose of this study is to develop a hot-wire type of vortex shedding flow meter for the use of flow rate measurement. Through the home-made apparatus and device, the hot-wire anemometer (includes probe, stem and electronic processor) and the vortex generator. (duct that contains triangle’s section of the bluff body. When fluid passes through, the wake behind produces periodical vortex shedding.) The ability of hot-wire anemometer when it detects the fluid moving changes of high-frequent movement is within 2000Hz, after appropriate design and adjustment, the combination of hot-wire anemometer and vortex generator may investigate the frequency of different flow rate that generated from the vortex shedding behind the bluff body of triangle section. The section area of vortex generator is constant value, thus it can calculate the relationship of flow rate and the frequency of vortex shedding. By means of strict calibration and test procedure, the results reveal that home-made hot-wire anemometer has excellent ability to detect the frequency of vortex shedding. The calibration curve indicates a linear relationship between the frequency of vortex shedding and flow rate. In order to reduce inaccuracy, it is suggested to classify three formulas to represent the flow rate that ranges from 0 ~ 40 CMM. The greatest inaccuracy is under 5%. When applied to different flow rate range, it only has to change the size of vortex generator only if the response frequency of hot-wire anemometer suit for the range of frequency of vortex generator. After commercialization, it can be applied to measure the flow rate of fans, flow rate of intake valve of engine, etc. Hot-wire anemometer also served as anemometer, which can be applied to wind velocity measurement in any situation.

The special life cycle of the Nymphoides indica: The veins and petioles of the pleustonic leaves can asexually reproduce adventitious buds. Pleustonic adventitious buds will change the density of the plants and sink into water. When the root of submersible plants attaches to the bottom of the pool, they will extend stems to the water surface and resume the life cycle. Reticulate sigmons and astral sclerenchyma appears in the process of transformation. Astral sclerenchyma merely appear in the species of which its leaves float above the water surface, and Nymphoides sp. are the most obvious. The cell wall thickness of astral sclerenchyma is even. The cells are hollow and full of water, and they will utilize water to change turgor between axes and spines. This is also proved in imitating experiments. The reticulate sigmons are frequently seen among aguatic plants, and the horizontal structures prevent air space from collapsing. These two structures make the thin stems of Nymphoides indica sustain the leaves instead of being crushed by the floating water. 印度莕菜 ( Nymphoides indica )獨特的生長週期：浮水型葉的葉脈和葉柄可以無性繁殖不定芽，浮水型的不定芽會改變植株的密度而沉入水中，當沉水型植株的根系在池底固著後，便會以莖延伸出水面，再次展開浮水型葉的成長週期。轉變過程中會出現網狀的橫膈和星狀厚壁細胞。星狀厚壁細胞僅出現於葉片需由水中浮出水面生長的植物種類，而莕菜屬植物更明顯。星狀厚壁細胞其細胞壁厚度均勻，細胞中空且充滿水分，會利用水分在中軸與突起的棘間移動來改變膨壓變化，在模擬的實驗中也得到證實。網狀的橫膈在水生植物中是較常見到的構造，水平方向的支撐可以讓氣室不會塌陷，兩種構造的作用下使得印度莕菜細長的莖能夠支撐葉片，不會被流水沖斷。