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。
Satellite Modeling of Wildfire Susceptibility in California Using Artificial Neural Networking
Wildfires have become increasingly frequent and severe due to global climatic change, demanding improved methodologies for wildfire modeling. Traditionally, wildfire severities are assessed through post-event, in-situ measurements. However, developing a reliable wildfire susceptibility model has been difficult due to failures in accounting for the dynamic components of wildfires (e.g. excessive winds). This study examined the feasibility of employing satellite observation technology in conjunction with artificial neural networking to devise a wildfire susceptibility modeling technique for two regions in California. Timeframes of investigation were July 16 to August 24, 2017, and June 25 to December 8, 2017, for the Detwiler and Salmon August Complex wildfires, respectively. NASA’s MODIS imagery was utilized to compute NDVI (Normalized Difference Vegetation Index), NDWI (Normalized Difference Water Index), land surface temperature, net evapotranspiration, and elevation values. Neural network and linear regression modeling were then conducted between these variables and ∆NBR (Normalized Burn Ratio), a measure of wildfire burn severity. The neural network model generated from the Detwiler wildfire region was subsequently applied to the Salmon August Complex wildfire. Results suggest that a significant degree of variability in ∆NBR can be attributed to variation in the tested environmental factors. Neural networking also proved to be significantly superior in modeling accuracy as compared to the linear regression. Furthermore, the neural network model generated from the Detwiler data predicted ∆NBR for the Salmon August Complex with high accuracy, suggesting that if fires share similar environmental conditions, one fire’s model can be applied to others without the need for localized training.