The Study of the Relationship between Global Warming and Acid Rain
The purpose of this project are 1)To study the relationship between global warming and acid rain with chemical model and mathematics model from temperature changing and pH of carbonic acid. 2) To create a pH measurement tool of carbonic acid in gaseous state.3) To study the impact of human activities in Loei province that affect to the relationship between global warming and acid rain. The procedures are 1)Do an experiment for studying the relationship between temperature changing and pH of carbonic acid. 2) Proof the mathematics model by using the result of experiment, the chemical reaction equation of carbonic acid solution. 3)Create a pH measurement tool of Carbonic acid by using Arduino and sensor with new formula in the computer program. 4) Using a pH measurement tool of Carbonic acid for studying impact of human activities in Loei province including industrial area, agricultural area, tourism area and forest area. The result of the mathematical model of the relationship between temperature changing and pH of carbonic acid is in form of Cubic equation in Equilibrium state and STP state. (Standard condition for Temperature and Pressure) So, we found that in this state has pH of carbonic acid is about 5.644. When the temperature rises up the effect of rainfall has a lower pH of carbonic acid solution. We also proof the new formula that create a pH measurement tool of Carbonic acid in gaseous state. The impact of human activities in Loei province found that the areas most affected by acid rain are the industrial areas, agricultural areas, tourism areas and forest areas respectively. In conclusion, when the temperature rises, it will dissolve acid solutions in the water on the earth. The loss of [H+] made the pH increases and the greenhouse gases become more atmospheric. These gases are more likely to react with atmospheric vapor. When these vapor form a cloud and condensation falls as rain, the rainfall has a lower pH, that is, global warming can result in the phenomenon of acid rain is greater.
Findings of new oscillations in BR reaction
The Briggs Rauscher reaction, i. e., BR reaction, which is one of the oscillation reactions, produces iodide ion and iodine repeatedly. Continual color changes of the solution from colorless to deep blue, and vice versa, are observed during the reaction due to the so-called “iodine test” reaction. In this work, we studied the effects of the presence of the redox active indicators on the oscillation behavior of the BR reaction. To the reaction mixture of KIO3, H2SO4, H2O2, C3H4O4, MnSO4, and starch, which are used for the general BR reaction as added a redox active reagent (indicator). Then, the changes in color and voltage of the reaction solution were recorded by a photosensor of the LEGO MINDSTORMS and a voltmeter using Pt electrodes. Under general reaction conditions, the oscillation reaction continued for ca. 5 minutes, including 18 times of oscillations. When an indicator, such as BTB, was added instead of starch to the reaction solution, splits of the voltage wave were observed, which should be a kind of new oscillation. Moreover, we found that the addition of K3[Fe(CN)6], which exhibits high redox activity, in the reaction solution instead of starch made the life-time and the numbers of the oscillation in the reaction greater by 3 times (14 min.) in time and more than 4 times (81 times) in the frequency. It’s also a kind of new oscillation. These results suggested that the oxidation-reduction reactions by the addition of ferricyanate ion effectively promotes the redox process of iodine and iodide ion. The experiments we wrote above were conducted without starch. Thus, as a reference, we conducted the same experiments under the presence of starch and got interesting results. We also studied the effects of K4[Fe(CN)6], suggeting that not only redox reaction between ferricyanide and ferrocyanide ion, but also the redox reaction with BR solution should occur in these reactions.
Effect of Air Resonance by Wind Speed Difference on Falling fruit
This study completes an air vibration equation expressed wind speed slope and wind speed. First, preliminary experiments identified air vibrations when wind speed differences occurred over distance. Several air fans were connected in series and the rotational speed of the air fan was adjusted to vary the wind speed with distance. At this time, only certain pendulum oscillates during a particular wind speed slope. It was expected that the pendulum would shake because the frequency of the air due to the slope of the wind speed was equal to the natural frequency of the pendulum. In addition, relatively short pendulum swings in large wind speed slope, long pendulum swings in short wind speed slope. After calculating the natural frequency of the seasonal growth of fruit using the physical factors model, we experiment how resonant frequency was related with cone length, angular width, wind speed, velocity and secondary derivative. the actual experiment analyzed the natural frequency of the fruit and resonance from the air vibration as the linear function of the wind speed, velocity, and secondary derivative. The experiment determined that the pendulum of a specified number of frequencies resonated with a particular wind speed pattern. It is judged that the vibration of air is related to first derivative of wind speed depending on speed and distance. However, it is very difficult to express the flow of nonlinear fluids as a function of simple function, particularly the effects of air vibrations caused by wind speed second derivative, which appeared to be associated with forces. This is a task that needs to be solved through further research.
連續函數與多倍角公式推廣研究
本研究考慮的主要問題： 若非常數之連續函數f滿足∀m∈N,∃P(x)∈C[x] s.t.f(mx)=P(f(x))，其形式應為何？ (一)、若考慮函數範圍為解析函數，則f(x)的形式必為下列三者之一： (1).axn+b (2). akx^n+b (3). acos(kxn)+b ，其中a,b,k∈C、n∈N (二)、若將考慮函數範圍改為：連續函數f:[0,∞)→C，則f(x)之形式必為下列三者之一： (1).axk'+b (2). akx^n+b (3). acos(kxn)+b ，其中a,b,k,k'∈C、n∈N、Re(k' )>0 (三)、若將考慮函數範圍改為：連續函數f:(0,∞)→C，則f(x)之形式必為下列四者之一： (1).alogx+b (2).axk'+b (3). akx^n+b (4). acos(kxn)+b ，其中a,b,k,k'∈C、n∈N 在本篇的最後，我們也將N的角色以其他正實數子集取代掉以推廣結果。
圓周上跳躍回歸問題之研究
圓周上相異n個點，將圓周分割成n段弧，每次每個點沿逆時針方向變換成與下一點所成弧之中點，若某點經m次變換後回到初始點，則m的最小值以及m的所有可能值為何？我們發現，m的最小值為n+2。更進一步發現，m的充要條件為m≧n+2且m≠kn-1, kn, kn+1，其中k為正奇數。接著，我們將問題一般化，圓周上相異n個點，沿逆時針方向變換成與下一點所成弧之p:q處，若某點經m次變換後回到初始點，則m的最小值以及m的所有可能值為何？我們發現，若p, q∈N，(p,q)=1，當變換次數r足夠大時，此n個點的位置會收斂至圓周上n等分點，同時，此n個點會在變換T=n(p+q)/(n,p)次後再次收斂至相同的位置。在這篇研究中，我們推導出任意點Pi變換r次後的點之位置坐標Ai(r)的一般式，不失一般性，我們針對P0求出A0(r)的最小極端值Lr與最大極端值Ur，在變換次數r足夠大時，透過觀察Lr與Ur對應到圓周上的收斂位置所形成的區間是否涵蓋原點，可預期P0變換r次後可否回歸。此外，我們也針對n個點具特殊初始位置座標來研究其回歸性質。
Σn=1∞(n/(Cn2n))=√(x/(4-x)3) (√x(4-x) + 4sin-1(√x/2))與其相關的無窮級數
本文從一個博奕遊戲談起，探討遊戲的期望值得到一無窮級數Σn=1∞n/Cn2n 並嘗試用相關的數學概念與方法思考，首先處理問題Σn=1∞n/Cn2n 與Σn=1∞n2/Cn2n 的值，過程中利用了Σn=1∞n/Cn2n 函數與Σn=1∞n2/Cn2n 函數的性質將欲求之無窮級數轉化成積分或微分方程式的型態，再利用奧斯特洛格拉德斯基積分方法解出所求。 為了更有效率的得到相關之無窮級數，引進了微積分工具中之冪級數的概念，輔以微分方程式公式解求出了 f(x)=Σn=1∞Xn/Cn2n =√x/(4-x)3 (√x(4-x) + 4sin-1(√x/2)), x∈(-4,4), 進而推廣、延伸與其相關的一系列無窮級數，並利用導函數f'(x)求得 Σn=1∞n·2n-1/Cn2n的值。 接下來討論與f'(x)相關的無窮級數，發現可利用f(x)的高階導函數透過迭代方式得到Σn=1∞nm/Cn2n的值，其中m為任意正整數，歸納這些級數後可以應用在本文之博奕遊戲，讓獎金的選擇更富有變化性。 最後觀察f(x)與卡塔蘭數列{Cn}的倒數所構成之冪級數有所關聯，解出 Σn=1∞Xn/Cn的收斂函數後求出了Σn=1∞1/Cn的值以及{1/Cn}的偶數項與奇數項的和。