在generalized Petersen graph P(n,5)中的hyper Hamiltonian
Generalized Petersen graph P(n,k)，定義為n 為不小於2 的整數以及1≤ k ≤ n−1，有頂點{ u0, u1, . . . , un−1, v0 , v1 , . . . , vn−1 }，及路徑{ uiui+1 , uivi , vivi+k：1≤ i ≤ n−1 }。在 [2] 中，我們可以知道P(n,5) 是Hamiltonian 等價於當n≠11。
在這一篇報告中，我們證明當generalized Petersen graph P(n,5) 是hyper Hamiltonian（一種Hamiltonian graph 再去掉任何一點後，仍然是Hamiltonian graph）的充要條件是n 為不等於11 的奇數且n ≥ 7。
The generalized Petersen graph P(n,k), n ≥ 2 and 1≤ k ≤ n−1, has vertex-set { u0, u1, . . . , un−1, v0 , v1 , . . . , vn−1 } and edge-set { uiui+1 , uivi , vivi+k：1≤ i ≤ n−1 with subscripts reduced modulo n}. And we can know that P(n,5) is Hamiltonian if and only if n≠11 from [2].In this paper it is proved that generalized Petersen graph P(n,5) is Hyper Hamiltonian (A Hamiltonian graph can still be a Hamiltonian graph when any one of the nodes fault) if and only if n is odd and n≠11.
蟲以食為天一線蟲覓食與攝食行為之探討
秀麗隱桿線蟲(Caenorhabditis elegans)為在科學界備受矚目的模式生物，對於其生殖及發育過程已有諸多研究，然而對於其覓食及攝食行為的相關研究卻仍闕如；每一種生物都有其獨特的行為模式，而為了解線蟲發展出何種覓食與攝食的行為以適應自然的環境，因此著手進行研究。本研究歸納出線蟲的七種一般行為：前進、探頭、偏移、擺頭、後退、轉向及拐彎行為，並探討其意義。其中，前進及探頭行為為恆常出現。偏移與轉向行為與攝食有關，在有菌環境中出現，可避免其遠離食物來源，而連續的後退及轉向行為也構成特殊行為之樹枝狀行為，推測應為幫助線蟲的消化所出現的行為。擺頭與拐彎行為與覓食有關，在無菌環境中出現。後退行為雖在有無菌環境中皆出現，但由後退後行為的不同可以了解後退行為在兩種環境下具有不同意義。在有菌環境中，後退後通常伴隨轉向行為；在無菌環境中，後退後通常伴隨拐彎行為，為逃離不適環境且藉以覓食的行為。經過研究，發現線蟲的覓食機制為嗅覺，且對嗅覺有依賴性及專一性，在其他氣味存在的環境下會大大降低覓食的效率，但最後仍然能正確地找到食物來源。線蟲於覓食的過程中所出現的特殊行為之混亂行為，由實驗結果發現為是E. coli 所釋放至培養基的化學物質所造成，化學物質濃度越濃，線蟲所表現出的混亂行為就越明顯。推測此種現象發生的原因，應為E. coli 釋放化學物質以干擾線蟲，以避免本身受到捕食，然而，線蟲也演化出適應的能力，因此在發生混亂行為後仍然能抵達菌落，此種特殊的交互作用表現出線蟲與E. coli 間的共演化現象。Cenorhabditis elegans has long been a model organism used for developmental biology studies. However, researches about its foraging and feeding behaviors are still limited. Every organism has its unique behaviors, and this research was conducted in order to understand more on what kind of foraging and feeding behaviors were developed by C. elegans and how they helped C. elegans to adapt to the environment in which it lives. The research concluded with seven general behaviors of C. elegans : advancing, probing, slanting, swaying, backing, turning and bending, and the meanings of it. The advancing and probing behaviors are constant. The slanting and turning behaviors, which appear in presence of food, are feeding-associated behaviors, which help C. elegans avoid leaving the food source. Continuous backing and turning behaviors also make up the branching behavior, one of the special behaviors, which help C. elegans’ digestion. Swaying and bending behaviors, which appear in absence of food, are foraging-associated behaviors. Although backing behavior appear both in presence and absence of food, the difference in post-backing behaviors show that there are distinct meanings of backing behavior in each environment. In presence of food, turning after backing is usually observed; in absence of food, bending after backing is usually observed, which indicates the act of escaping from unsuitable environment and searching for food. Studies show that the foraging mechanism of C. elegans is by olfaction, and C. elegans is highly olfaction-dependent and specific: efficiency of foraging decreases significantly under environment with scents other than that of E. coli, but C. elegans is still able to locate the food. According the experiments, the chaos behavior, one of the special behaviors, appears during the foraging process of C. elegans, and is caused by chemicals released to the medium by E. coli. The density of the chemicals and the extent of chaos behavior is in direct proportion. The theory is E. coli secrets chemicals to disturb the worm to avoid being preyed upon. C. elegans, however, also evolved the ability to adapt to the chemical so that it can still reaches the food source even after chaos behavior appeared. This unique interaction is a manifestation of co-evolution between C. elegans and E. coli.
「圖形板」的圖形軌跡之探討及其延伸
Starting from the problem in AMC competition of Australia, we try to find out the locus and its length when a point in a regular polygon rolls in a circle. The result is that the locus has a wonderful and regular cycle.Next, we discuss the regularity of the cycle when a regular polygon（n sides） rolls in another regular polygon. Furthermore,we discuss the the equation of the locus by changing the radius and the angle of rolling. we find out the argument function of the locus of a point inside when a a regular polygon（n sides）rolls in another regular polygon （m sides）： ， Aj is the summits of the regular polygon（m sides）, Bjcorresponds Aj when a point inside the regular polygon (n sides) rolls, ) And then, we do some moving simulation with some computer math software, such as Cabri Geometry、Mupad, etc. We discuss the regularity of the locus and its equation of a point inside when some special cycloids, like asteroids, cardioids, etc, roll in a certain condition. Moreover, with the result of research 2, we create the “plate＂ and apply for a patent on it. We hope to study math by playing games.
從澳洲AMC 競賽題出發，嘗試探討一正n 邊形中的一點在單位圓內滾動軌跡及其軌跡長度，發現該軌跡均會產生奇妙的循環規律。
接下來，推廣探討正n 邊形在其他正多邊形中滾動時循環的規律，並利用旋轉半徑及角度之間的變化深入探討其滾動軌跡方程式，發現正n 邊形繞正m 邊形滾動時其內部一點軌跡參數式為，其中， Aj 為 正m 邊形之各頂點、Bj 為正n 邊形中內部一點旋轉時對應 Aj 之點，。
進一步想嘗試使用數學電腦軟體如：Cabri Geometry、Mupad 等對以上研究去做一些動態模擬，並再探討一些特殊擺線如：星狀線、心臟線…等，在條件下相切滾動時，圖中某一點的軌跡規律性及其方程式。另外，應用研究二中的結果，創造出寓數學於遊戲的「圖形板」，並申請了新型專利。
The unknown gene interacts with dll , abdA,Ubx
We operated the misexpression screen between the EP lines and the pattern lines with the genotypes of eq1>dll, eq1>abdA, eq1>Ubx, eq1-GAL4, ey-GAL4 or dpp-GAL4. After the screening, we found that five of these 1,800 strains of filial generation had special phenotypes. It had shorter antennae and defects in the anterior equatorial region of eyes. We used plasmid rescue and IPCR to sequence the certain target gene, and found that it was escargot, abbreviated as esg. To identify when, where and how the overexpression of escargot induces such phenotype, we operated the staining of eye-antenna disc in third-instar larval period of wild type, eq> esg×UAS-GFP and eq>GFP with anti-dll, anti-caspase3 and anti-esg. The result shows that escargot cannot be detected before puparium formation. But the expression of dll, a gene controls the eye development, was reduced in the eye disc. We except the overexpression cause the defect of distal antennae and the anterior equatorial region of eyes mainly in the 3-day-long pupal life.我們用異位表現法篩選出和eq1>dll、eq1>abdA、eq1>Ubx、eq1-GAL4、ey-GAL4或dpp-GAL4 這些pattern lines有交互作用的EP lines。在這1800種的果蠅子代品系中，有五種具有特殊的性 狀。它們具有觸角短化以及複眼前緣中央區有缺刻的現象(形成心型眼)。我們使用質體救援 法以及IPCR的方法來定序這段未知基因序列，發現這是一個叫做escargot的基因(簡稱esg)。 為了了解過分表現此基因會造成何種分子影響，以至於產生此種性狀，因此我們使用 anti-dll 、anti-caspase3 和anti-esg 進行野生型、eq>esg×UAS-GFP 和eq>GFP 三齡幼蟲的 eye-antenna disc的螢光免疫染色。結果在幼蟲成蛹前都沒有偵測到esg的表現現象；不過在eye disc中，控制眼睛發育的基因dll的表現有被抑制的現象。因此我們推測過分表現esg的過程因 該是發生在為其短短三天的蛹期。也就是說，這種表型應該是在化蛹後形成。
太陽系外行星光度研究
搜尋太陽系外行星（extra-solar planet；exoplanet）是當今天文學家最重要的計畫之ㄧ；其中，當系外行星經過其主恆星碟面時，會造成光度略微下降，稱之為凌日現象（transit） 。本研究便是利用鹿林天文台SLT（Sne Little Telescope）望遠鏡來拍攝兩顆已知的系外行星HD 189733 b 和HD 209458 b在凌日時的影像。 我們得到約4000 張影像，再經過影像修正與資料分析後，我們作出凌日光變曲線圖、Bin圖及相位圖，並從中探討其特性。 經分析判斷後，HD 189733 看到一個剛開始發生和一個完整的凌日現象，分別由B濾鏡和 R 濾鏡所拍攝；而 HD 209458 則看到兩個剛開始發生，三個要結束的凌日現象，分別由B濾鏡、R 濾鏡和 Hα 濾鏡。 並由光變曲線，推得HD 189733 的基本參數，包括：行星半徑為 1.242個木星半徑、軌道傾角為88.94 度以及軌道半徑為 0.03115 天文單位。 Since 1995, more than 200 extra-solar planets have been found. Until today, astronomers still make efforts in searching extra-solar planets because it’s one of the most important projects in this century. During the crossing by the stellar disks (i.e., transits), the exoplanets will cause the brightness to decrease by a small amount. In the project, we have focused on the observations of two known transit exoplanets, HD 189733 b and HD 209458 b. The photometric observations were performed with the 40-cm SLT telescope on the Lulin Observatory. The results from the image procuring, data reduction and analysis are presented. We accrue and plot the light curve, bin and phase. We found out that HD 189733 has two transit ingresses with B, R band, and HD 209458 has two transit ingresses, two transit egresses with B, R, Hα band on phase. We figure out planetary parameters, RP is 1.242 RJ, inclination is 88.94° and orbital semi-major axis is 0.03115 AU by transit light curve.
新型空氣清淨燈具之研究與開發
本研究主要的目的是在開發同時具有空氣清淨與照明的兩種燈具。其中桌燈是基於自然對流原理，利用燈泡發熱讓氣流通過燈具上方的濾網達到過濾功能，為了尋求過濾效果與照度兼顧的最佳值，本研究並提出比較因子的概念。在吊燈方面，除了運用自然對流原理之外，還更進一步利用太陽能驅動風扇，進行強制對流，強化過濾的效果，使得本研究成果更趨於完善。 由實驗結果可得知，桌燈在四星期長期測試條件之下，其過濾效果增進率分別為39.1, 40.8與 40.1%。在吊燈四週長期實驗的結果方面，螺旋與 100W 鎢絲燈泡在自然對流的過濾效果增進率分別為49.1%與 51.4%，而100W鎢絲燈強制對流方面過濾效果增進率則為60.2%。由整個研究結果可以發現，本燈具對於空氣清淨有極佳的效果，在不增加額外耗能條件之下，能增加燈具的散熱效果與延長壽命，同時又具備空氣清淨效果，對環境空氣品質具有相當的貢獻。 The purpose of this study is to develop a novel lamp with both the functions of air-cleaning and lighting. One of it is the desk light. Basing on free convection principle, it makes the air run through the filter on the top of the lamp by its heat in order to attain the aim of air cleaning. To find the optimum value of both cleaning effect and illumination, we advanced the compare factor. The other is the droplight, though it is based on the same principle, we use the solar energy as its power to drive the fan. So that the effect of the filter can be augmented and the result of this research approach perfect. According to the experimental result, in the four-week experiment with desk light, the enhanced efficiency of filter is 39.1%, 40.8% and 40.1% respectively. On the way of droplight with four-week experiment, the enhanced efficiency of filter is 49.1% and 51.4% with helix and tungsten(100W) lamp under the condition of free convection; the enhanced efficiency of filter is 60.2% with tungsten(100W) lamp under forced convection. All these results of the research shows that the novel lamp has great performance on air cleaning and much better effect of heat sink without extra consuming of energy, also the lifespan of the lamp can be extended. Furthermore, it is capable of air cleaning and contributes to the quality of environmental air.
流體碰撞物體所產生的波形之研究及應用
當流體由圓管流下，在碰撞到物體後水流會產生類似駐波的形狀。為瞭解此現象的產生機制，及影響此現象的變因，我們改變流體的表面張力、流速及與碰撞物體間的距離，以探討各變因對波形所產生的影響，進而研究此現象的成因。由實驗結果發現波形會因流速加快、擋板距離增加、表面張力減少而有波長變短的趨勢，且可以用表面張力波的理論解釋。由理論推導的結果，可測量液體表面張力。由於圓球狀的外型使表面積增大，可增加液體之散熱的面積，因此可應用在水冷系統方面。A phenomenon similar to the standing wave, which occurs when a slow-velocity fluid jet collides with an obstacle, was observed. Because the free surface profile was observed to be stable, the phenomenon was not considered as standing wave. To understand the mechanism of this phenomenon and the factors that can affect the free surface profile, the surface tension of the fluid, jet velocity and the distance between the exit of the tube and the obstacle are varied to study their influences on the free surface profile. According to our experiment, the wave length is shortened when the jet velocity or the distance between the tube and the obstacle increases or when the surface tension decreases. The tendency of the investigated phenomenon can be explained by the capillary wave theory. Based on Bernoulli’s principle, continuity principle and surface tension\r equation, an ODE (ordinary differential equation) could be formulated. By using numerical method to solve this ODE, we predict the free surface profile which could match the experimental photo well. The tendency of the phenomenon can also be explained by the ODE. In order to measure the surface tension of the fluid, we wish to minimize the experiment apparatus. To enhance our assumption we use laser to locate the individual particle that we add in the fluid and calculate the velocity field of the flow jet.
液態導體的磁效應
本文所探討的議題為電解質溶滿通以電流後所產生的效應與機制。本實驗所採用的方法為電解與電鍍，運用這兩種方法，來比較電解液在不同狀況下所產生的結果；經過多次的實驗，累積了許多實驗結果，使我們可以得到更精確的數據 · 在此次實驗中，我們發現電解液在相同的電壓下，通以電流後的穩定性與金屬的活性有關，活性越大越不穩定；反之，活性越小越穩定。另一個發現為，只有單一極性離子移動的情形，可通過的電流，比陰陽離 r 同時移動時為大；但因通過的電流大使電解液反應劇烈，產物時時覆蓋電極使電流下降。所以就穩定性來說，是以陰陽離子同時移動為佳 · 在展望方面，希望可以發展到液態磁屏避的設備，可減少設備過重之問題 ·This is a study of how electric current effects the electrolyte solution. The experiment was conducted through two methods: electrolysis and electroplating, the results of which were compared. The experiment of the same designs hi been conducted repeatedly and, as a result, accurate data were collected and accumulated.One of the two major findings from the experiments was that, when under the same voltage, stability of the electric current varied with the change of activity of the metals; the greater the activity of the metals, the less stability of the current, and vice versa. The other major finding was that, with the movement of dipole-ion, a greater amount of current would go through the solution than that which would go through with the movement of cathode and anode; however, the greater amount of current would cause intense reaction of electrolyte solution, hence merging the electrode and reducing the current. So as long as stability is concerned, the movement of cathode and anode is preferable.It is hoped that more sophisticated experiments designed on the basis of the similar principles will eventually lead to the construction of equipment of liquid-magnetic shielding of smaller weight and size.