全國中小學科展

2012年

給我三個點 來畫平行多邊形

任意給定不共線相異三點A1、A2、A3,想要利用邊與其對應對角線平行的方式依序作得A4、A5、…、Am,並且再作Am+1、Am+2、Am+3重合於三點A1、A2、A3,這是一件不簡單的事!我們把這樣的多邊形稱之為平行多邊形。 為了方便探討,首先固定邊與其對應對角線長度比值固定為 1t,在複數平面上,寫成遞迴式zn+3-zn=t(zn+2-zn+1),可求得zn=pαn+qβn+s,以此可以作出任意平行m邊形,並證明平行m邊形會內接於一橢圓或圓。 其次,探討邊與其對應對角線長度比值不為定值的平行m邊形。我們求得圓內接平行m邊形,邊與其對應對角線長度比值:若m為偶數,則比值為1-cot2πmcotα1+cot2πmcotα 及cotαsin4πm-cos4πmcotα;若m為奇數,則只有2cos2kπm+1一種比例。接著將任意給定的三點對直線作伸縮變換成特定的共圓狀態,作出圓內接平行m邊形的m個頂點,再反變換回平行m邊形。 對於順序三點已有解決方式,我們再利用線性組合的性質,使任意三點只要給定邊數、不共線三點之序數及其值,便可以利用zn=pβn+qβn+s和zn=pβknαtn+qβknαtn+s兩式分別求出等比例和不等比例的平行多邊形。 最後,探討橢圓內接面積最大的m邊形必為邊與其對應對角線長度比值固定的平行m邊形。

A Backpropagation Neural Network Model on Precipitation Forecasting in the Philippines

Backpropagation neural networks were used to forecast daily rainfall with minimal error for Metro Manila in order to have an inexpensive way of accurately predicting weather. Calamities brought on by heavy rainfall have caused great economic, infrastructure and human loses. Neural networks have the ability to discern complex patterns in noisy data; this makes it a viable method for weather forecasting. Daily precipitation, humidity, rain indication, sea level pressure, temperature and maximum sustained wind speed for January 2000 to December 2010 were acquired from the Philippine Atmospheric Geophysical and Astrological Services Administration. The neural network made use of Python 2.7.2 and the backpropagation program by Neil Schemenauer (python.org). It considered different neural network architectures with a total of 2844 data sets for training and 708 data sets for testing. Each neural network’s accuracy was measured with a graph of the actual and predicted values, correlation coefficient, and root mean square error. It was observed that the neural network with architecture 5-8-1 yielded the most accurate results as it had the highest correlation coefficient of 0.48599 and smallest root mean square error of 14.84. It was also observed that the trends of the predicted values followed that of the target values. This suggests that it is possible to create a neural network with a moderate correlation given daily weather data. It is recommended that further researches make use of hourly data instead of daily data for more accurate results. Other variables, which might affect rainfall, not in this study should also be considered. This research could aid in the anticipation of calamities and the decision making involved in shipping, fishing and aviation industries.

利用全球定位系統觀測電離層海嘯現象

2011年3月11日,日本發生規模9.0的地震,同時引發海嘯。太空科學家發現海嘯會引起電離層中總電子含量TEC(Total Electron Content)的擾動。雖然海嘯引起之電離層TEC變化以及海嘯模擬結果已被公布,但兩者之關係尚未被探討。因此,我們比對海嘯模擬與當天電離層TEC變化,確認此電離層TEC擾動是由海嘯所引起的。我們進一步研究電離層TEC的擾動大小和海嘯高度的關係,發現兩者之間具有極高之相關性,且到達的時間差約為±10分鐘,而1公尺高的海嘯約會造成12.8TECu(即TEC之單位,1TECu=1016el/m2)之總電子含量變化。希望將來能利用地面GPS觀測到的電離層TEC資料,在地震後迅速確認海嘯之發生與否,並於到達岸邊前訂定其位置,同時推估高度。藉此,建立海嘯預警輔助系統,以降低海嘯對人類生命和財產的威脅與損害。

三角形同向切割線之性質推廣

William Johnston 和Joe Kennedy首先以切割重組的方式提出了任意三角形的面積為第 組同向切割線所得到中央三角形面積的7倍(即 )。本研究將此方法性質加以推廣,我們證明了任意三角形其 ,而在推廣至任意四邊形時,由於公式十分龐大複雜,無法化簡,因此我們退而求其次,證明了任意梯形第 組與第 組同向切割線所切割出的面積與線段比例特性。最後,我們也分析探討任意梯形當 時(即平行四邊形)以及任意正 邊形其第 組同向切割線之性質。

Fig Preservation

Figs have become an expanding industry here in New Zealand and are a current export fruit which could potentially provide a large amount of profit to both growers and the New Zealand market as a whole. Nicola’s family has about 10 acres of fig trees. They sell the figs locally and as an export. They generally sell for about $13 per kilogram here in New Zealand and $26 in the USA. However, figs only have a shelf life of about 7 days. This is because at present there is no proven pre or post-harvest treatment or method of storage that helps to decrease the rate of decay of the fig fruit. After researching post-harvest treatments for figs, Nicola found a report which claimed to have developed treatments that increased the shelf life of figs by about 5 weeks. With this kind of increase, it would be possible to transport, store and export figs over longer periods of time without running the risk of losing large amounts of produce, or delivering unsatisfactory fruit to customers. Nicola developed 7 different post-harvest treatments based on the ones that had shown promise in earlier research. These were hot-water baths of different temperatures, both with and without different bleach concentrations. To test these on the fruit she set up four experiments – a dry matter test, a firmness test (using a penetrometer), a colour test and observation of detrimental features of the fig. She tested these treatments at 0, 7, 14, 21 and 28 days from harvest. Nicola found that after 7 days, the firmness of all of the figs that had been treated had decreased to a large degree. The only figs that did not have a massive decrease were the untreated fruit. However after about 14 days, the firmness of all of the fruit became about the same and after this 14 day mark, she would not have considered any of the figs to be edible. However, in the appearance tests, it seemed that the treated figs that had the least amount of mould and rot were the ones that had been treated with higher levels of bleach such as the 55 degree Celsius water bath with 0.003L of bleach to every litre of water, and the 35 degree Celsius water bath with the same concentration of bleach. Overall, Nicola’s results showed that the hot water bath, and hot water bath and bleach post-harvest treatments did not slow the decay of the fruit in the earlier weeks after picking. In effect, Nicola’s research showed that the information she had relied on to help plan her study had claimed too much and that the treatments were less effective than had been stated. More research will be needed to find a more reliable way to improve the shelf life of figs.

The use of Square shaped wheels in ship harbouring using an inverted catenary surface

Riding around on a flat tire is no fun. It feels really bumpy. But a square wheel may be the ultimate flat tire. There's no way it can roll over a flat, smooth road without jolting the rider again and again. Here, I have constructed a bicycle with square wheels. It's a weird contraption, but you can ride it perfectly smoothly. My secret is the shape of the road over which the wheels roll. A square wheel can roll smoothly, keeping the axle moving in a straight line and at a constant velocity, if it travels over evenly spaced bumps of just the right shape. This special shape is called an inverted catenary. A catenary is the curve describing a rope or chain hanging loosely between two supports. Turn the curve upside down, and you get an inverted catenary--just like one of the bumps in my road. Make the road out of a whole bunch of those bumps all in a row, and you can take your square-wheeled bike for a quick spin. Just as a square rides smoothly across a roadbed of linked inverted catenaries, other regular polygons, including pentagons and hexagons, also ride smoothly over curves made up of appropriately selected pieces of inverted catenaries. As the number of a polygon's sides increases, these catenary segments get shorter and flatter. Ultimately, for an infinite number of sides (in effect, a circle), the curve becomes a straight, horizontal line. In the end, I conclude with possible enhancements in the project that might take us to a whole new world.

Project Motion in Sports

A projectile refers to any body that is thrown in space and falls under the influence of gravity and the motion of such a body is called projectile motion. In this context we will ignore the effects of air resistance to make calculations easier. Through the usage of trigonometric ratios and vectors it is possible to accurately predict the position of a body after a certain time, the maximum height attained by it and the horizontal distance it covers from the point of projection. Horizontal displacement or range of a projectile is the main index of performance in many cases of projectile motion. If air resistance is negligible, there is no net force in the horizontal direction (ΣF = 0; ax = 0) Through this topic we aim to explain the science behind the performed actions and movements in sports such as Golf, Football, Basketball and Javelin throw. Factors Affecting Distance traveled by a projectile: 1. Relative height of release 2. Speed of Release 3. Angle of release Projectile Motion: Theory v/s reality Theoretically optimal angle is about 45° however taking air resistance into consideration the angle reduces to about 42°. Long jumpers use angles of 17-23°. This is because when traveling at ~10 m/s, there is not enough time to generate a large takeoff angle. The game of Golf is based on the trajectory followed by the golf ball as it moves through the air and in this sport we have addressed issues such as the required club face angle and swing speed for the ball to go in the hole. For instance if we have a ten degree driver it will carry the ball lower than a 60 degree wedge and hence it can be deduced from the above statement that a greater angle of the club face launches the ball at a greater angle. Effects of Air resistance can be very large in case of golf. Therefore, the golf ball has dimples on its surface to negate the effect of air resistance. To depict the application of projectile motion in football, we have shot a video on our school’s football field showing the trajectory followed by a football and have addressed issues like horizontal and vertical velocity required depending on the nature of the kick. In the sport of Basketball we shot a video showing a student shooting a 3 pointer. Furthermore with the help of charts, we have calculated the velocity required for a basketball to go inside the hoop at different angles of projection such as 30, 45 and 60 degree. Finally we have included a question to determine whether a ball hit by Sachin Tendulkar will be a six or not using kinematical equations as well as equations related to projectile motion. Hence by shedding light on this wonderful topic we attempt to reveal how an athlete’s brain functions and through years and years of practice and hardwork he is able to accurately predict distances and achieve his goals.

光動力化合物的合成及應用

光動力療法為一個新興癌症治療方式,包含三大要件:光、光敏劑、氧氣。其概念為利用合成的錯合物嵌入細胞核DNA,藉由特定光激發後,將能量傳遞給氧及水分子形成自由基,進而毒殺癌細胞,達到治療惡性腫瘤的目的。 本研究利用NNO-Schiff base,成功合成出一系列的二價銅錯合物,針對不同的配位基及共配位基、濃度、光波長利用DNA裂解、細胞毒性測試,探討結構活性的關係與作用機制,以利研發相關藥物的研究,另外,錯化合物吸收波長可延伸至可見光─藍光,進而大大提升其應用的價值。

由正2n邊形與一點所衍生的三角形面積比值問題

本次研究的靈感來自於能力競賽中,一道證明由六邊形內部一點P所導致的面積比值為1的問題(詳見第6頁)。這份報告主要的目的,在探討推廣到正2n邊形內部時的情況,求出將P點移至外部時任一點的面積比值,並進一步討論面積多次方之問題。 這次研究利用GSP協助了解圖形的性質,並善用解析幾何和輔助線作圖。報告中的許多證明,可以用「各個三角形的高相加」的觀念搭配三角函數運算做出結論。 文中結果顯示,除了證明:當P點在正2n邊形內部時,必滿足面積比值為1外,並提供了當P點在正2n邊形外部時,面積比值的各種可能性。其證明方法,可作為解決正2n邊形面積問題的重要參考。

過平面上n定點作正n邊形問題與其對偶命題

本為主要探索「平面上任三點不共線的n個點是否存在n條線使其各線恰經過上述n點中之一點,並交出一正n邊形。」再則探究其對偶命題「任兩線不平行,任三線不共點的n條直線上,是否存在各線恰有一點,使其各點為一正n邊形之頂點。」 文中論及解的存在性及其一般解,並推廣至 個同心圓及 條平行線的情形。