Bezier曲線與蚶線間之關聯性的探討與推廣
在這篇報告中，我們以貝斯曲線的做圖原理建立出一種新的曲線－環狀貝斯曲線，進而得到不少有趣的結果。我們發現有名的古典曲線－蚶線，也是屬於二次環狀貝斯曲線。軌跡方程式為：，此時，係數恰符合二項式定理。之後我們推廣至n次環狀貝斯曲線的軌跡方程式：，也符合二項式定理。
在複數平面上，給定z0、z1、z2三點，我們定義出一個二次變換 ，若，，可映射成蚶線的圖形；若z∈實數，則可映射成拋物線。利用此結果類推我們找到一個複數平面上由 z0、z1、...、zn 所決定的n次變換將以原點為圓心的單位圓，映射成n次環狀Bezier曲線。
In this essay, we use the method of forming a Bezier Curve to establish a new curve, circular Bezier Curve, and find a lot of interesting results. We discover the famous classical curve ＂limacon＂, which belongs to the Quadratic Circular Bezier Curve. The locus of Quadratic Circular Bezier Curve is, where. Its coefficients match the binomial theorem. Then we apply it to the locus of nth-circular Bezier Curve:, and it also matches the binomial theorem.On the complex plane, we define a quadratic transformation corresponding to three points—z0,z1 and z2 as .If , where , a limacon is mapped. If z is a real number, a parabola is mapped. With this result, we will find a nth transformation defined by z0、z1、...、zn on the complex plane. It will form a nth-circular Bezier Curve with unit circle centering on the origin.
會變色的金屬—神奇的奈米科技
本報告研究內容，是利用電化學氧化還原方法合成金、銀、銅三種奈米粒子，以及探討電流是否會影響電解合成奈米粒子，在前半部成功地利用控制電解的部份條件，如界面活性劑、以及電流值大小，而合成出金、銀、銅三種奈米粒子，利用UV-VIS的光譜分析，鑑定其三種奈米粒子不同的吸收波長，其光譜出現吸收的現象是因為金屬表面特殊的表面電漿共振吸收現象而產生的。但是在本實驗中發現在UV-VIS的光譜中，電壓值的大小對金奈米粒子吸收波長並沒有關係，這些奈米粒子在水溶液中藉由界面活性劑的包覆，而溶解的相當好。 The content of thesis focuses on using electrochemistry oxidation-reduction reaction to synthesis gold, silver, and copper nanoparticles. We confer whether current of the electrolysis is an influence for the synthesis of nanoparticles. We succeed in synthesizing nanoparticle by controlling some terms of the electrolysis, like the micelle concentration, and current value. Using UV-VIS spectrum to analyse wavelength of three kinds of nanoparticles. The special phenomenon of absorption spectra is appeared because the surface plasma resonance on the surface of metal. From the UV-Vis spectra, we didn’t find the exact relationship between the potential value and the absorption spectra of gold nanoparticles. Finally, we also obtained good results in spectra observation, which meant that these nanoparticles encapsulated with surfactants were well solved in the solution.
基因突變對線蟲(Caenorhabdits elegans)之神經系統退化變異株的搜尋以及對其性??
This research is mainly in observation with Caenorhabditis elegans ’s genetic mutation caused via nervous system abnormal character. In the study, I the sample have been cultivated purified and add some chemical material EMS to speed up C.elegans mutation. Then based on the character to further analysis what causeof gene deal with mutation and observe the effects in heredity. The research has two stages, on the first stage of study the mainly target is to both search and purify the mutation of C.elegans. The second stage is based on the exploration of mutation’s searching andpurifying. Because the certain mutation bodies aren’t easy to find out, the project is still on progress at the beginning of second stage, and we conclude some heredity special cases in preliminary of study. 這個實驗主要是觀察並針對線蟲因為基因的突變所產生的神經系統異常的變異性狀，在實驗中我先將樣品線蟲培養並純化至一定數量，並加入適當藥劑EMS造成其突變，經篩選並分析此性狀，進而找出造成其突變之基因，以及觀察此性狀對遺傳表現所造成的影響。 該計畫分成兩階段，第一階段的實驗重點是在突變株的搜尋以及純化上，第二階段則是在突變基因的探討上，由於特定突變株的搜尋並非容易，所以目前計畫只進展至第二階段的遺傳實驗初期，對於其遺傳特徵與突變形式上已有了初步的分析，但尚未定位出該基因的位置。
蜘蛛數
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。
死亡巧克力—切切割割好計謀
三角形的邊上取任意多個點，我們可以把這塊大三角形沿著切割線切割成較小塊的三角形，但切割線必須是點（或頂點）和點的連線，而且必須切割三角形，同時可以切任意大小的三角形，如圖（1）與圖（2）。但不可以一開始就取走整個三角形。定義拿到最後一塊三角形的人獲勝，而在多邊型中的玩法與在三角形中相同。 我們分A、B、C三種規則來討論，其中A規則即是上面提到的玩法，B規則大部分的玩法和A規則都相同，唯一不同的地方在於：A規則中，只要有一方取到剩下的圖形為三角形，另一方就可以直接取走剩下的三角形，而B規則規定即使剩下的圖形已經是三角形，也必須取到剩下的圖形邊上都沒有分點為止。C規則是限制玩家一次所能取的三角形數來進行遊戲。 我們完成了A、B、C規則中三角形與多邊形的必勝策略，並找出必勝策略之間的關聯。 ;Given any numbers of points on the sides of a triangle, the players can cut this triangle into pieces. Each cutting line has to be one, linked between two points given from two different sides. And the player can’t have to cut smaller triangles out of the original triangle. The out-cut triangles can be chosen randomly without any restriction in size, just like what’s shown in picture（1）and（2）. Meanwhile the first player can’t cut the original triangle exactly all out in the very beginning process. We define the player as the winner, who gets the last triangle. And the above way we play can be applies to any multi-side shapes. We discussed the question respectively in three rules, A, B, and C. Rule A is what we mention above. Rule B is generally the same as rule A, except for the only difference：The rule A , if there is any triangle left , the next player can get it directly, but while in rule B, the every next player has to cut out smaller triangles until no point is left on sides. Rule C proceeds on conditions that there is a limitation to a certain number of triangles cut out at a time. We has finished the winning tactic respectively in rule A, B, and C in the games with a triangle and multi-side shapes. Furthermore, we find the connection between the winning tactives.
蟹狀星雲的擴張
By comparing eight different epoch images of the crab nebula taken through 1942 to 2004, we have calculated the expansion velocity of 27 optical bubble features and 60 filaments. The mean expansion velocity of bubble features and filaments is 0.173 arcsec/yr and 0.15 arcsec/yr, respectively. We also estimated the maximum radial velocity of the expansion by analyzing the emission spectrum of the nebula. The maximum radial velocity is 1385.5 km/s. Combining these measurements indicates that the crab nebular is approximately 5870 light year away. In addition, if we assume that the nebula has been expanding at a constant rate, our expansion velocity projected backward indicates the mean date of the supernova event as A.D 1124, more than 70 yrs later than the accepted date of 1054. The result confirms the well-known acceleration in the crab's expansion. Although we have analyzed eight images with a 62 yr baseline, the acceleration still can't be derived from this study. 透過量測由1942年到2004年之間八張不同年代的蟹狀星雲中爆炸後殘骸的位置變化，可以計算出蟹狀星雲爆發的擴張速度。本研究選定了27個包狀物和60個纖狀物，計算出的擴張速度分別為0.173 arcsec/yr.和0.150 arcsec/yr。再透過分析蟹狀星雲的光譜所計算出的徑向速度（radial velocity）為1385.5 km/yr，進而推得蟹狀星雲的距離分別為5430光年和6370光年，平均值為5870光年。 另外，如果假設擴張速度是等速運動，那麼把求得的擴張速度倒推出的爆發日期是在西元1124年，這比中國紀錄中超新星爆發的1054年晚了70年。這顯示出蟹狀星雲的確非等速擴張而是有加速度的狀態，才會造成以等速倒推發生日期時，晚了70年。雖然本研究中分析了相差62年之久的八張影像，仍然無法分析出星雲的擴張的加速度情形。
變形的橢圓—從距離及距離和談起
給定一平面E，A為平面上一點。取r>0，則我們知道到其距離為定值的點形成一圓，而A為此圓圓心。如果把A改成一平面圖形，則到其距離為定值的點形成的集合會是什麼樣子？類似地，給定平面上兩焦點F1及F2在平面上，則到其距離和為定值的點形成橢圓。同樣的，若把F1及F2改成平面圖形，其圖形會是什麼樣子？藉著GSP的輔助，到目前為止，我們得到了以下的結果： \r 1. 給定一平面E及此平面上的一個凸多邊形, 我們描繪出在此平面上到此凸多邊形之距離為定值的點所形成的圖形。\r 2. 設F1和F2分別為平面E上之點或線段或多邊形（未必是凸多邊形），我們利用包絡線描繪出所有滿足d(P,F1)+d(P,F2)=k（k夠大）的點所形成的圖形。 \r 3. 設C1,C2為平面E上之兩圓，我們討論所有滿足 d(P,C1)+d(P,C2)=k\r （k夠大）的點形成的圖形並討論其性質。 \r 4. 設L1和L2分別為平面E上之兩線段，我們討論所有滿足d(P,L1)+d(P,L2)=k（k夠大）的點形成的圖形並討論其性質。 \r 5. 設A為平面E上之一點，Γ為平面上一凸多邊形，我們討論所有滿足d(P,A)+D(P,Γ)=k（k夠大）的點形成的集合並討論其特性。 \r 6. 藉由和圓作比較，我們研究了變形圓的光學性質；而對變形橢圓也做類似的討論。\r Let E be a plane and A a fixed point on E. Given , it is known that all of the points on E with distance to 0r>rA form a circle and the point A is called the center of this circle. What is the corresponding graph if we replace the point A with a set (for example,a segament or a polygon) contained in FE? Similarly, what is the case when we modify the two focuses and in the definition of an ellcpse to sets and (or example,two segments or two polygons) contained in 1F2F1F2FE ? Taking advantages of GSP and analytic geomety, we research related situations and so far we have obtained the following results:\r 1. Let Γ?E be a segment, a convex polygon or a circle , etc. and r>0 be fixed. We sketch the graph of points on E with distance r to Γ and study properties of such graphs.\r 2. Let F1 and F2 be singletons, line segments , polygons(may not be convex), or circles,etc., on E Taking advantage of envelopes, we sketch the graph of those points P on E satisfying d(P,F1)=k(K>0 is large enough).\r 3. Let C1 and C2 be circles on 1C2CE. We sketch the graph of the points P on E that satisfiy d(P,C1)6d(P,C2)=k (k>0 is large enough) and study properties of this graph.\r 4. Let L1 and L2 be two line segments on E and be a large enough constant. We sketch the graph of points P on E that satisfy d(P,L1)+d(P,L2)=k(k >0is large enough) and research properties of this graph. 0k>\r 5. Let A?E and be a convex polygon on ΓE. We sketch the graph of points on E that satisfy d(P,L1)+d(P,L2)=k(k>0 is large enough) and research properties of this graph.\r 6.We compare the optical properties of metamorphic circles with circles and we deal with metamorphic ellipses similiarly.
完全圖立方乘積之最小控制
完全圖Kn是指一個圖中有n個點，且任意一個點都跟其它的點有邊相連。兩個圖G和H的卡氏乘積G□H的點集V(G□H)＝{(g,h)| g∈V(G),h∈V(H)}，兩個點(g1,h1)和(g2,h2)有邊相連若且為若g1=g2 且h1~h2，或g1~g2且h1=h2。
三個完全圖Ka、Kb、Kc 的立方乘積是指Ka□Kb□Kc。一個圖G中的一點v所連的其它點稱為這個點v的鄰居，也就是N(v)={x | x~v}。一個點集S中的所有點的鄰居的聯集稱為這個點集的鄰居，也就是N(S)=∪v∈S N(v)。如果一個點集S和它的鄰居N(S)包含了一個圖G的所有的點，也就是S∪N(S)＝V(G)稱這個點集S是這個圖G的一個控制集。我們把圖G的所有控制集中點數最少的稱為最小控制集，並定最小控制集的點數為最小控制數γ(G)，也就是γ(G)＝min { | S |, S是G的控制}。
本文的目的在於研究完全圖立方乘積的最小控制，也就是要給γ(Ka□Kb□Kc)一個上界。特別當 a = b = c = n時，γ(Ka□Kb□Kc) = 。
A complete graph Kn is a graph with n vertices, which any vertex is adjacency to every other vertices. The Cartesian product of two graph G and H which is denoted G□H is define as follow: the vertex set V(G□H)＝{(g,h)| g∈V(G),h∈V(H)}，and two vertices (g1,h1) and (g2,h2) is adjacent if and only if g1=g2 and h1~h2 or g1~g2 and h1=h2. The Cartesian product of three complete graph Ka,Kb,Kc is Ka□Kb□Kc,which is the same with (Ka□Kb)□Kc.
In a graph G, the neighbor of a vertex v N(v) is the set of the vertices adjacent to the vertex v, that is N(v)={x | x~v}。 The neighbor of a vertex set S is N(S), which is the union of the neighbors of vertex v over S, that is N(S)=∪v∈SN(v). For a graph G, if a vertex set S unions its neighbor N(S) equal to the vertex set of G, that is S∪N(S)＝V(G), we say that S is a dominating set of G. The domination number of a graph G will be denoted as γ(G), which is the minimum size of all dominating set of G..
We give an upper bound to γ(Ka□Kb□Kc). And when a=b=c, γ(Ka□Kb□Kc) ≦
終端速度
液體中之球體運動與液體的黏滯性有關，本實驗找出球體半徑與終端速度之間的關係。利用攝錄機作為紀錄工具，拍攝三種材質（壓克力、玻璃、水晶）的球體在沙拉油中的自由落體過程。使用電腦影像處理軟體將影像分解成幅影像，時間的解析度為1/30秒。測量球體的高度與時間，分析高度與時間的變化情形，發現終端速度與球體半徑之間的關係。
流體中之運動方程Fdrag = -k1V，無法符合實驗結果。我們的實驗結果顯示油中的自由落體的運動方程應該是Fdrag = -（k1V+ k2V2）。由不同材質的壓克力球（~1.18g/cm3）、玻璃珠（~2.47g/cm3）與水晶球（~2.66g/cm3）所獲得的終端速度（Vt）與球體半徑（a）的關係為a3(ρ-ρ') = 0.00003（a Vt）2 + 0.00021（a Vt） + 0.00575，其中ρ與ρ'分別為球體密度與沙拉油密度（0.90 g/cm3）。
再者，在相同半徑的條件下，密度越大的球體終端速度越大，在靜止下落後，越久達到終端速度。
The motion of a sphere which is falling through a fluid is subject to the fluid viscosity. In this study, we find out the relation between the radius of a sphere and the terminal velocity. We used a digital camera to record the sphere's descent in the oil. The three kinds of sphere we choose are acrylic（~1.18g/cm3）, glass （~2.47g/cm3）and crystal （~2.66g/cm3）. Frame-by-frame analysis of the video footage yielded rough estimates of the sphere's location within 1/30 seconds accuracy for statistically consistent results. By measuring the location and time and analyzing them, we find out the relation between the radius of a sphere and the terminal velocity. The expression of the drag force,Fdrag = -k1V, is not cosistent with our results. The study indicates the expression of the drag force should be Fdrag = -（k1V+ k2V2）. The expression for the terminal velocities of the three kinds of sphere is of the form：a3(ρ-ρ') = 0.00003（a Vt）2 + 0.00021（a Vt）+ 0.00575, where a is the radius, ρ is the sphere density and ρ' is the oil density（0.90 g/cm3）. In addition, if the radius is the same, the terminal velocity of a denser sphere is higher and the time to approches the terminal velocity is longer.