四等獎

令Sn 為{1,2,…,n}任意排列所成的集合，π ? Sn 為其中的一個元素，我們記π = (π(1), π(2),…, π(n))。今給定π ? Sn ，若對所有i,1? i ? n，都有π (π (i)) = i 時，我們稱π 為involution。假設π ? Sn ，並給定σ ? Sm (m ? n)，當π 中任取m 項，其大小關係的順序都和σ 不同，我們稱π 避開σ，或稱π 是一個σ-avoiding 排列。在這篇報告中，我們主要分析了2143-avoiding involution，1234-avoiding involution，和3412-avoiding involution 中的一些統計量，給出了十數個結果與幾個猜想。Let Sn be the set of permutations on {1,2,…,n} and π ? Sn be an element in Sn. Denote π as π = (π(1), π(2),…, π(n)). We say that π is an involution if π(π(i)) = i for every i, 1? i ? n. Given π ? Sn and σ ? Sm (m ? n) , we say that π avoids σ (or π is an σ-avoiding permutation) if π does not contain any m-term subsequence in the order of σ. In this paper, we discuss some classic statistics on 2143-avoiding involutions, 1234-avoiding involutions and 3412-avoiding involutions. We get many new results in this field and give some interesting conjectures.

We understood the definition and meaning of spider number by reading〝Wonders of Numbers〞. It interested us so much. So, we took further step to study the situation of extreme value when the gap sometimes lie on the line and sometimes on the circle or even on both. That is to say, we explored the relation between spider number and the gap when the spider number is maximum or minimum. New research for the application of spider number involves several directions. First, we design a new game called〝Stepping Land Mine〞with the rule of spider number. Give you a net with several hidden gaps, trying to find the right positions of gaps. Second is the further result for a different type of net about regular n-polygon. Third is a tactic for a net with destroying of the strategy points. In this situation, the gaps amount on the circle and on the line are fixed. At the same time, consider the situation of circles and lines designing the tactic of placing the gaps to attain the maximum of the destructive effect. 在本文中我們定義一個蜘蛛網上的蜘蛛數，若在蜘蛛網中加入缺口後，會影響蜘蛛數的大小。我們探討蜘蛛網上的缺口，該如何分配才能夠得到蜘蛛數的極值(最大值及最小值)。先觀察一直線和圓上缺口如何放置蜘蛛數有極值，再探討許多條直線及圓上的情況，進而推展至許多同心圓及通過圓心的許多條放射線的缺口，該如何放置，蜘蛛數才會有極值發生。

近在台灣新發現的筆筒樹萎凋病，造成大量筆筒樹(Cyathea lepifera)枯萎死亡，對於此疾病的防治已成亟需解決的問題。先前的研究中，已確認病原菌為新屬真菌並命名為Ophiodiaporthe cyatheae sp. nov.。本研究則在於以生物交互作用方式進行防治，尋找拮抗菌，針對特性實際應用。 研究中因實驗素材與操作方法的差異分為細菌與真菌類拮抗菌兩大主軸，且皆成功自土壤中各分離出數種有應用潛力的拮抗菌。細菌方面，尋獲並經鑑定為Burkholderia gladioli，對於病原菌有良好抑制效果，然由於此菌本身安全性有疑慮，實用上仍有待商榷。真菌方面，拮抗菌以超寄生方式與快速的生長能力制伏了病原菌，期待透過比較消除病原菌的速度，篩選出其中最強大適切的拮抗菌種，運用於實際田間病殘株處理。

跳舞草(Desmodium gyrans) 屬多年生木本豆科植物，其特殊之處在於小葉會對外界的聲音有所感應。本實驗以訊號產生器固定聲音強度，發出2、4、6、8、10 KHz不同聲頻之聲波刺激跳舞草，並以每5秒為單位紀錄小葉擺動角度之變化，分析其擺動週期、擺動幅度等不同的變化。實驗結果為跳舞草小葉之擺動週期與擺動振幅是隨著聲音頻率的增加而呈現sin函數變化之圖形。Desmodium gyrans (Leguminosae) is a perennial woody plant. Acoustic waves can stimulate stipules and cause to oscillation. This experiment used the coroma to immobilize strength, emitted the frequency of 2, 4, 6, 8, 10 KHz acoustic wave to stimulate stipules and recorded the changes of oscillation angle every five seconds. We calculated the oscillation cycle、oscillation span, and analyzed experiment data. The most importance result is that the experiment graphs of oscillation angle and oscillation span with different frequency of acoustic waves display sin function metamorphic diagram.

這是一系列關於水蒸氣冷凝為極細微小水珠的長程實驗。其中可以分為下列三個階段：第一階段是基礎實驗。將水氣導入至潔淨的光滑表面上(蓋玻片)，觀察水珠冷凝的機制。第二階段是在外加磁場及電場作用下，將水氣導入至潔淨的光滑表面（蓋玻片），觀察水珠冷凝的機制。這部分的實驗推翻了一般「水分子是電中性，在電場或磁場中不受影響 」的刻板觀念！實驗所呈現出來的冷凝水珠，不但有明確的自我組成模式( Self assembly pattern)。並且發現：電場會增速凝結水珠的成長(Aggregation)，而磁場則會抑制凝結水珠的成長。第三階段是將水蒸氣導引至超聲波的環境中：我們先將超聲波訊號產生器(變頻、定頻)面向於載台旁，再讓水氣導入至潔淨的光滑表面上(蓋玻片)，觀察冷凝水珠的機制。當使用固定頻率超聲波波源，我們發現：在超聲波場中水珠的成長會受到抑制，且成長速率會隨著頻率的升高而逐漸減小。第一階段與第二階段的實驗結果與討論已分別發表於2004 年及2005 年的台灣國際科學展覽報告中，本作品將詳述第三階段。 This experiment explores the basic nature of the condensation of water vapor into droplets on the surfaces of cover glasses. This condensation occurs because of the difference in temperature between the water vapor and the cover glass. The condensation process is observed under a microscope. The growth of the droplets can be described as: nucleation, aggregation (piling up) and coalescence. The growth is irrelevant to surfaces or environments. It is found that the temperature difference of moist air over the cover glass do not affect the nucleation size of the droplets in simple plain surroundings; while the change of flow rate does. In general, the coalescence is speeded up at higher temperatures. Furthermore, the effects of electric fields 、magnetic fields and ultrasonic waves are also studied. It can be observed that the size of water droplets become smaller and grow more uniformly under magnetic fields or imposed ultrasonic waves; also, the aggregation rate is decreased by imposed magnetic fields or ultrasonic waves, and it is increased by imposed electric fields. These effects of magnetic fields 、electric fields and imposed ultrasonic waves might be related to the flow conditions and the vibration of surrounding air in the system. This experiment provides the first step in the understanding of the formation of water droplets and their self assembly mechanism in different environment.

給定下面範例：\r 058823529411764705882 +235294117647058823532\r =0588235294117647058823529411764705882353，\r 其等式結果與質數17 的倒數結果(1/17)有某種關聯(卻沒有一個決定性的證據)，意即\r 1/17=0.0588235294117647=\r 0.058823529411764705882352941176470588235294117647...... ( Len(17) =16 )\r \r 曾經在下列網站上發現過幾組數字(挑戰試題)，引起我們極大的興趣。\r http://www.domino.research.ibm.com/Comm/wwwr_pondernsf/challenges/March2000.html\r http://www.math.smsu.edu/~les/POW08_96.html\r \r \r The two examples that I have are 0588 2+23532=05882353 and 058823529411764705882+23529411764705882353 2=0588235294117647058823529411764705882353 These were found by the Canadian professor Alf van der Poorten, and he gave a talk on these identities in December at the west coast number theory conference. He was unspecific as to exactly where these identities were coming from, but they are connected with reciprocals of primes:1/17 = 0.0588235294117647= 0.058823529411764705882352941176470588235294117647 ΛΛ ( Len(17) = 16 ) Though not mentioning how to obtain these equations, Prof. Poorten demonstrated the relationship between the above examples and the reciprocal of the prime numbers 17 (1/17 ) without a definitive proof.

在拋物線上置掛正三角形看似簡單，其實不然。本篇文章研究在二次函數的各種不同情況下，可做正三角形的分佈以及其個數。 1. 在一條拋物線上時，最多只能作正三角形。 4. 在三條對稱軸相等的拋物線和共頂點開口大小不同之拋物線上，本篇文章證明一定能找出正三角形落在它們之上。但由於最多有四個分界點，要解四次方乘組過於繁複，於是本篇文章對分界點作了一些估計，找出了分界點的極限值。 5. 本篇文章證明了對於給定的正n 邊形，存在一1 元n-1 次方程式可以通過它所有頂點。 Building a regular triangle on a parabolic curve looks easy . In fact , it doesn’t . This Article researches regular triangles distributions and its numbers in different conditions. 1. On one parabolic curve can only build regular triangles , squares and other regular polygons can’t be built. 4. For three parabolic curves which has same symmetrical axis or three concurrent parabolic curves, we prove that it can build at least one regular triangle on them .But because it can have at most 4 boundary points, to solve quartic equation is to complicated. So we do some estimation of boundary points, and find out some limits. 5. This Article prove that for given regular polygons , there exists a one dimension n-1 orders equation can pass all its apexes.

The purpose of the study was to develop and create a semi-automatic multi-purpose sorting and counting machine of standard articles. Currently, there is a problem of equipping industrial enterprises as well as small trade companies and large retailers with computing sorting machines of standard products of a certain shape. We would like to fill this missing link up with a simple, compact and inexpensive device. Procedures The proposed research consisted of a consistent design of a virtual model of the device and its electronic-mechanical implementation. The virtual model is simulated by a computer program "SolidWorks" object, which graphically shows the operation of the future device. In the development of the computing sorting device standard electronic devices and their associated software have been used. The created simplified real model demonstrates the basic principles and characteristics of the proposed device. Data As an example for the implementation of the concept device a computing device for sorting coins, in circulation in Russia has been created. As a basic principle of sorting objects by their geometric and weight characteristics were used. It is important that the device is focused on the correct form of the objects of sorting (balls, rings, coins, regular polyhedrons, screws, nuts, etc). To confirm the effectiveness of the computing device of this type of sorting, a series of tests of counting of objects manually have been carried out . The effectiveness of the device is determined by comparing the time characteristics of manual and automatic sorting. Findings and conclusions As a result of research and work performed, we have concluded that: 1) The proposed device can be used in various industries. (for example, while sorting ball bearings.) 2) Such a computing sorting device may find it's application in various commercial enterprises: to assist cashiers in retail ATMs. 3) Can be used in payment terminals. 4) 4) After a certain modernization of the device it can be used for money encashment.

氫是一種非常理想的能源。不僅效率極高，且不會造成環境污染。空氣中的含量極少，常用的電解水方式又效果不彰。我們利用Ag、Ru、Cu 、C-CuPu、C 等數種電極，與H2SO4、 H3PO4 、HNO3 電解液，分別在10V 雙電極與-0.8V 三電極下做電解水實驗，研究何種電極與何種電解液能得到最多的氫氣量。在低電壓下找出最好的電極，液與太陽能電池結合，成為電池中的一部份，讓發電效果更好，以利未來燃料電池H is a kind of great power.It is not only effective,but also no environment pollution .Owing to the H in the air is very little,so it is not easy to take .Addition to,we often to take it by electro liquid,but the electrolysis effect is not ideal. Except Ag、Ru、Cu、C-CuPt、C and so on ,in theH2SO4、 H3PO4 、HNO3 electro liquid,we have a experiment in the 10V dual electrode and -0.8V triple electrode to study which electrode and which electro liquid to get the most of H.Under the low voltage,we can fund the best electrode.It is easy to bind with solar energy battery,andit can be a part of battery.Besides, it makes the generator effect better,and it is convenient to make the fuel cell commercialize in the future.

二十四節氣是中國古代對氣象觀測的智慧結晶，由於近幾年來聖\r 嬰與反聖嬰現象的影響，在透過嘉義氣象觀測站的協助之下，利用\r Excel 進行簡易的雨量和氣溫統計，並配合中央氣象局發布的環流型\r 態和農業災害，來探討二十四節氣在嘉義地區的適用性。\r 以陽曆來區分節氣，每年會誤差6 小時，加上四年一閏，剛好彌\r 補差距，農曆則需19 年。至於聖嬰現象對嘉義地區則呈現涼夏和暖\r 冬，但在反聖嬰現象中則變為加強四季之特性，即是熱夏與寒冬。兩\r 者都造成雨量集中夏秋兩季，形成冬春乾旱。至於二十四節氣則是以\r 夏、秋的氣溫和雨量相關的諺語較適合嘉義地區。\r Abstract\r The 24 Solar Terms, is the achievement of weather observation from\r ancient China. Because of the effects from El Nina and La Nina these\r years, we proceed simple statistics of rainfall and temperature by using\r Excel and the assistance from Chia-yi Weather Station. We adopt the\r Circulation Type and the damage of agriculture from Central Weather\r Bureau, to discuss if the 24 Solar Terms for in Chia-yi area.\r To divide the 24 Solar Terms according to the solar calendar, there’ll\r be a six –hour gap every year. The gap could be made up every four years.\r But with the lunar calendar, it takes nineteen years to offset. The El Nina\r shows cool summer and warm winter in Chia-yi area, but the La Nina\r emphasizes the characteristics of the four seasons. It shows hot summer\r and cold winter. Both of them rain mostly in summer and fall, and cause\r the drought in winter and spring. The saying of the 24 Solar Terms is\r suitable for the temperature and the rainfall during summer and fall in\r Chia-yi area.