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) ≦
On Course Line Management
The Online Course Management system was developed in 2012 by George Moon to address the issue of creating course books at Burnside High School in Christchurch, New Zealand. The course books are designed to inform students, staff and parents of the many courses that are available for students to choose for their next year of study. In the past, the system that the school used consisted of large amounts of paperwork and duplication. Not only did this system require a lot of effort from staff, but the course book cost the school thousands of dollars to produce, as it had to be sent off to be published into a large book that would be read by students for a week, then likely thrown out. This year the school decided to digitise the course book, so that students would look at their courses online. Earlier this year, the school believed that the new School Management System (SMS) ‘KAMAR’ would be able to handle all of the necessary information, however this was not the case. Because of this, they needed a simple solution that would collate all of the course data, and then output it as a course book. I developed my project to do this. It is a web based program that is accessible by staff on their computers which enables them to enter in all of the course and assessment data for their departments. As it is all securely stored on a central database, it reduces duplication and staff workload, as well as the added environmental bonus of less paper being used. The program also outputs data in a number of ways including as a coursebook PDF (digital document which can be uploaded or printed), an Excel spreadsheet and a webpage for easy viewing. It can be sorted or printed by different categories (such as level, faculty, department), which proved to be a very useful feature. Following some research on areas such as design principles, browser compatibility and screen resolution (computer screen size), the program was designed to make best use of this this information. For example, most of the computers that staff would access the website on were of a similar size screen, so I made sure that my website worked well for them. I also used my research on design principles to try and create a simple, clean interface that users with limited computer skills would easily be able to navigate around. The outcome was real, as it was used by the school to generate their coursebook this year. Following a 95% student completion rate of course selection many months earlier than previous years, the system (although it had some issues) was pronounced a success, and the school is looking to use it in the years to come. There are a number of steps I am looking to take in the future with this program including the potential sale to other schools, so they can take advantage of the features it has to offer.
「膠」流電-黏度及外加電壓對電解質溶液離子暫穩態通道之影響
在本次實驗中，我發現膠狀電解質溶液中的帶電離子，會因為離子團的熱運動，和電偶極的庫倫吸引力 (electric dipole) 的交互作用下，使溶液的I-V curve (電流-電壓曲線)，具有類似磁滯曲線(Hysteresis curve) 的效果；而膠狀溶液之濃度越高，電解起始點的對應I-V 值也越大。此外，白金電極與銅箔電極的距離若改變，也會使溶液的I-V curve 變的不一樣。另一方面，我也發現，在給予膠狀電解質溶液一緩慢外加的電壓或衝擊電壓並持續維持此一定額外加電壓時，會因為該溶液的黏度持續增高、帶電離子濃度增高且反應不斷變化下，而使該溶液的對應電壓，形成一重複出現「先降-後升-再降」的震盪現象，且電壓值節節升高。最後，我利用掃描式電子顯微鏡（SEM）及能量分散光譜儀（EDS）觀察銅箔電極之表面變化並分析其上之化學組成，藉此嘗試解釋上述這些有趣的現象。In this experiment, with the interaction of the heating action of ionic atmosphere and electric dipole, I find that ions in the gel make the I-V curve in the colloid electrolyte liquor show up with the effect similar to Hysteresis curve. The higher concentration of the colloidal solution, the bigger value of I-V at the initial electrolysis reaction was found. Furthermore, the shape of I-V curve is dependent on the distance between platinum electrode and cupper electrode. On the other hand, I find that when I apply a gradual extra-voltage or a fast extra-voltage to the colloidal electrolyte solutions and then maintain to a fixed value, this will make a unique ‘two peaks’ state oscillation of corresponding voltage. The reason is owing to the climbing viscosity and ion concentration in the solution. With the methods of scanning electron microscope (SEM) and energy dispersive spectrometer (EDS), I observe the change and analyze the components of chemicals on the surface of the cupper electrodes. Finally, I present the interesting results and try to interpret these phenomena.
明察秋毫-動態測微器
The purpose of this research is to create a device that is able to precisely measure small dynamic changes which cannot be recognized by the human eyes. The Vernier Caliper and the screw micrometer are common tools used to precisely measure lengths of objects. However, things which are measured by the Vernier Caliper or the screw micrometer have to be in a solid state, and the shape cannot be changed. By applying the light lever principle on Lego bricks, this research uses the LabVIEW graphical programming system to design a device which is able to automatically measure small dynamic changes. The precision of this device is higher than that of the Vernier Caliper and the screw micrometer. Moreover, this device is able to precisely detect the small dynamic changes of solids and liquids as well. Through numerous tests, the least count of the device can reach the level of 10-3cm. Also, this device has successfully measured small changes, such as the height of the liquid surface by one drop of water, the evaporation of water in one minute, and the growth of a plant in one hour. By popularizing this device, people will be able to precisely measure small dynamic changes which are difficult to be measured in a short time.