停車就是彈硬幣
在這個科展中我們要研究兩個非常有趣的問題:\r 停車場問題 與 彈硬幣遊戲.\r 停車場問題是這樣的:在一條單行道上有n個車位,編號從1到n。現在有n個司機排成一排要進入停車。但是每個司機都有怪癖,各自有最想要停的位子。他們依序將車子開進單行道,如果想要停的位子是空的,當然停在這個位子。但是如果不巧那個位子已經被停了,不得已只好找下一個空位,姑且停之。但是如果往下找都沒有空位,由於是單行道,司機就只好開走不停了。\r 比如說,如果現在有五輛車,司機的喜好分別是(3,1,2,5,2)。則五輛車都可以順利停車。但是司機的喜好如果是(3,1,4,5,4),有些車就無法停車了。\r 彈硬幣遊戲是這樣的:考慮圓內接正n+1邊形,任意兩點都連線。這正n+1邊形中有一個頂點P是特殊的,每個頂點上一開始都放有一些硬幣(各點硬幣數可以不同)。如果P以外的某個頂點上的硬幣數n個,我們可以對這個頂點進行操作:一次操作是指將這個頂點上的硬幣各分一個給每個其他頂點。點P只在其他點都無法操作時操作。我們不理會頂點P上的錢數,因此這個遊戲可以無限地玩下去。
魔術猜牌-由再生訊息延伸推展猜中比值之研究
本研究是藉由數學手法探討;如何由一疊 36 張四種花色的撲克牌中,尋找出保證可猜中最多張花色的方法。研究過程是以在適當的猜牌時機,以邏輯推理、二進位、分析與歸納 … … 等數學原理與方法,搭配巧妙的策略運用而達到目的。 猜牌方法:先約定好猜牌規則,助手將 36 張牌背圖樣相同但非對稱的撲克牌,以旋轉牌背的方向傳達訊息。在本研究中得出「經由巧妙的猜牌方法保證可以猜中不少於 26 張花色」,並得出「當總張數趨近於無窮大時,保證可以猜中不少於 81 . 07 %的牌,並且證出若僅使用獨立的訊息猜牌,無論任何猜牌方法皆無法猜中多於 87 . 37 %的牌」 · 其中一個猜中多於 80 %的例子是‘「當總張數等於 23006 張時,保證可以猜中不少於 1 8405 張牌(18405/23006 > 4/5)」 ·The study is mathematically based with reasonable explanations behind it. We are to correctly guess as many cards as possible from a deck of 36 cards, with random numbers and four different suits. We will apply mathematical methods, such as logic inference, binary system, and analytical reduction, upon right timing. Using careful arrangement of the principles and reasoning, we can reach our ultimate goal. To state guessing: Conference between the guesser and the assistant about the guessing rules, the assistant will have 36 cards with the same exact pattern on the back but not symmetrical. The pattern of the cards will be different when rotated 180o. The only communication between the two is by rotating cards. In this study, we can prove that through mathematical method, we can assure 26 or more cards can be correctly guessed. Furthermore, when the total amount of cards is close to infinity, we can assure 81.07% or more of the cards can be correctly guessed, and prove that if the cards are guessed from independent information, no more than 87.37% of the cards will be correctly guessed by any guessing methods. One of the examples, which 80% of the cards are correctly guessed, is that when the amount of the cards is 23006, 18405 or more of the cards can be correctly guessed. (18405/23006 > 4/5)
A Novel Contrast-Enhanced Brain Mimicking Hydrogel for Testing Implantable Brain Electrodes
Paralysis is a debilitating disorder that does not currently have safe and effective treatments. Implantable brain electrodes can be used to read brain waves and convert them into a corresponding motor function to restore movement in paralyzed patients. Tissue deformation induced around the implant site is believed to reduce their viability through the foreign body response. Developing electrodes that minimize deformation is challenging because the mechanical aspects of deformation are not fully understood and non-animal tissue models for testing electrodes are unavailable. Development of pre-clinical models for in vitro testing of the mechanical properties of electrodes can lead to a better understanding of this prevalent problem. The objective of this study was to construct a novel contrast-enhanced, brain mimicking hydrogel using photopolymerizable polyethylene glycol (PEG) polymer that contains alginate microspheres with enclosed gadolinium (Gd) contrast agent. 1.5% alginate microspheres were constructed with enclosed Gd-DTPA-BSA contrast agent and successively added into 10% PEG. Then, this mixture was photopolymerized using a 5 mW/cm2UV lamp to result in a successful brain mimicking hydrogel. Rheological testing showed that its elastic modulus was approximately 1.5 kPa, which is similar to that of a normal human brain. The model is valuable because the presence of the contrast agent in the hydrogel resulted in distinct bright spots on the MRI. This can facilitate the visualization of tissue deformation caused by electrode insertion via comparison of pre-insertion and post-insertion images. This brain-mimicking model has the potential to improve understanding of neural deformation from electrode implants in order to assist patients suffering from paralysis.
探討以最經濟的方式偵測低濃度溶液之金屬離子
A novel technology for detection of trace concentration of ions at ppb level was developed. The detection limit of this method can be reached to 1ppb. The Wheatstone bridge usually be used for determination the concentration of metal ion, the major disadvantage of this technique is the precision too low. Atom absorption spectroscopy is better for monitoring trace metal ion, but the cost of instrument is too expensive. In this study, the oscilloscope was used to modify the Wheatstone bridge Circuit. The difference between the voltages of two connection points of Wheatstone bridge was used to estimate the concentration of ions. The amplification system was used in this method to improve the limit to detect ion from 10 ppm to 10ppb. The parameters of measurement including types of probe, match of Wheatstone bridge resistances, parameters of input signal and electrolysis in solution were also being studied. 本研究主要是發展一個富有創新且經濟的ppb ( parts per billion ; 10-9)級,微濃度金屬離子量策技術,目前本實驗設計已可以達到 ppb的偵測極限,尤其本方法不僅能夠應用於定量,尚可做定性偵測。簡易的溶液濃度技術常使用惠司通電橋的方法,但這個方法的缺點在於精確度較低,而另外常用的儀器是原子吸收光譜儀,卻價格高昂。因此,我們回頭對惠司通電橋的技術作改良和創新技術。首先,最重要的新技術是對示波器的改良。示波器被用來量測惠司通電橋的電位,我們比較電橋上兩特定點的電位來判斷濃度,當兩個點的電位相同時代表溶液電阻和控制電阻相同。但當輸入信號的電壓太強時,示波器上並無法判別細微的電位變化,因此造成量測的精度不高。本組設計新的電路用以放大在高強度訊號的波形細微變化,成功將測量極限由 10 ppm推進到 ppb。另外關於此設計中的各種參數,包括探針類型、惠司通電路的電阻匹配、輸入訊號之強度頻率、電極間電容效應及溶液中可能發生的電解效應等,都有精確的探討。最後的設計結果將是創新且有別於以往的設計概念。