每日彙整: 2017-01-09

樹莓派、樹莓派之學習、樹莓派之教育

L4K ︰ Python Turtle《十》

道德經‧首起

道可道，非常道。名可名，非常名。無名天地之始；有名萬物之母 。故常無欲，以觀其妙；常有欲，以觀其徼。此兩者，同出而異名 ，同謂之玄。玄之又玄，衆妙之門。

 

恍兮惚兮，小海龜領悟『微分』和『差分』之不同耶！

150px-Pierre_Francois_Verhulst

Logistic-curve.svg

P(t) = \frac{1}{1 + \mathrm e^{-t}}

350px-Logit.svg

\operatorname{logit}(p)=\log\left( \frac{p}{1-p} \right)

220px-Linear_regression.svg

Y \approx F(X, \Box)

Maple_logistic_plot_small

x_{n+1} = r x_n(1 - x_n)

Logistic_map_animation

Logistic_map_phase_plot_of_x-n+1--x-n-_vs_x-n-

相圖

512px-LogisticMap_BifurcationDiagram

Logistic_map

Logistic_map_scatterplots_large

LogisticCobwebChaos
定點‧震盪‧混沌

200px-Ganzhi001

300px-NewtonIteration_Ani

一八三八年，比利時數學家 Pierre François Verhulst 發表了一個『人口成長』方程式，

\frac{dN}{dt} = r N \left(1 - \frac {N}{K} \right)

，此處 N(t) 是某時的人口數，r 是自然成長率， K 是環境承載力。求解後得到

N(t) = \frac{K}{1+ C K e^{-rt}}

，此處 C = \frac{1}{N(0)} - \frac{1}{K} 是初始條件。 Verhulst 將這個函數稱作『logistic function』，於是那個微分方程式也就叫做『 logistic equation』。假使用 P = \frac{N}{K} 改寫成 \frac{dP}{dt} = r P \left(1 - P \right)，將它『標準化』，取 CK = 1 與 r = 1，從左圖的解答來看， 0 < P <1，也就是講人口數成長不可能超過環境承載力的啊！

如果求 P(t) 的反函數，得到 t = \ln{\frac {1 -P}{P}}，這個反函數被稱之為『Logit』函數，定義為

\operatorname{logit}(p)=\log\left( \frac{p}{1-p} \right) , \ 0 < p < 1

，一般常用於『二元選擇』，比方說『To Be or Not To Be』的『機率分佈』，也用於『迴歸分析』 Regression Analysis 來看看兩個『變量』在統計上是『相干』還是『無干』的ㄡ！假使試著用『無窮小』 數來看 \log\left( \frac{\delta p}{1-\delta p} \right) = \log(\delta p) \approx - \infty 和 \log\left( \frac{1-\delta p} {\delta p}\right) = \log(\frac{1}{\delta p}) = \log(H) \approx \infty，或許更能體會『兩極性』的吧！！

一九七六年，澳洲科學家 Robert McCredie May 發表了一篇《Simple mathematical models with very complicated dynamics》文章，提出了一個『單峰映象』 logistic map 遞迴關係式 x_{n+1} = r x_n(1 - x_n), \ 0\leq x_n <1。這個遞迴關係式很像是『差分版』的『 logistic equation』，竟然是產生『混沌現象』的經典範例。假使說一個『遞迴關係式』有『極限值』x_{\infty} = x_H 的話，此時 x_H = r x_H(1-x_H)，可以得到 r{x_H}^2 = (r - 1) x_H，於是 x_H \approx 0 或者 x_H \approx \frac{r - 1}{r}。在 r < 1 之時，『單峰映象』或快或慢的收斂到『零』； 當 1 < r < 2 之時，它很快的逼近 \frac{r - 1}{r}；於 2 < r < 3 之時，線性的上下震盪趨近 \frac{r - 1}{r}；雖然 r=3 也收斂到 \frac{r - 1}{r}，然而已經是很緩慢而且不是線性的了；當 r > 1 + \sqrt{6} \approx 3.45 時，對幾乎各個『初始條件』而言，系統開始發生兩值『震盪現象』，而後變成四值、八值、十六值…等等的『持續震盪』；最終於大約 r = 3.5699 時，這個震盪現象消失了，系統就步入了所謂的『混沌狀態』的了！！

『連續的』微分方程式沒有『混沌性』，『離散的』差分方程式反倒發生了『混沌現象』，那麼這個『量子』的『宇宙』到底是不是『混沌』的呢？？回想之前『λ 運算』裡的『遞迴函式』，與數學中的『定點』定義，『單峰映象』可以看成函數 f(x) = r \cdot x(1 - x) 的『迭代求值』︰x_1 = f(x_0), x_2 = f(x_1), \cdots x_{k+1} = f(x_k) \cdots。當 f^{(p)} (x_f) = f \cdots p -2 times f \cdots f(x_f) = x_f，這個 x_f 就是『定點』，左圖中顯示出不同的 r 值的求解現象，從有『定點』向『震盪』到『混沌』。如果我們將『 logistic equation』 改寫成 \Delta P(t) = P(t + \Delta t) - P(t) = \left( r P(t) \left[ 1 - P(t) \right] \right) \cdot \Delta t，假使取 t = n \Delta t, \Delta t = 1，可以得到 P(n + 1) - P(n) = r P(n) \left[ 1 - P(n) \right]，它的『極限值』 P(H) \approx 0, 1，根本與 r 沒有關係，這也就說明了兩者的『根源』是不同的啊！然而這卻建議著一種『時間序列』的觀點，如將 x_n 看成 x(n \Delta t), \ \Delta t = 1，這樣 \frac{x[(n+1) \Delta t] - x[n \Delta t]}{\Delta t} = x_{n+1} - x_n 就說是『速度』的了，於是 (x_n, x_{n+1} - x_n) 便構成了假想的『相空間』，這可就把一個『遞迴關係式』轉譯成了一種『符號動力學』的了！！

在某些特定的 r 值，這個『遞迴關係式』有『正確解』 exact solution，比方說 r=2 時，x_n = \frac{1}{2} - \frac{1}{2}(1-2x_0)^{2^{n}}，因為 x_0 \in [0,1)，所以 (1-2x_0)\in (-1,1)，於是 n \approx \infty \Longrightarrow (1-2x_0)^{2^{n}} \approx 0，因此 x_H \approx \frac{1}{2}。再者由於『指數項』 2^n 是『偶數』，所以此『符號動力系統』不等速 ── 非線性 ── 而且不震盪的逼近『極限值』的啊。

─── 《【Sonic π】電路學之補充《四》無窮小算術‧中下上》

 

將如之何『道可道、名可名』乎？

Paul’s Online Math Notes

Welcome!

Welcome to my online math tutorials and notes. The intent of this site is to provide a complete set of free online (and downloadable) notes and/or tutorials for classes that I teach at Lamar University. I’ve tried to write the notes/tutorials in such a way that they should be accessible to anyone wanting to learn the subject regardless of whether you are in my classes or not. In other words, they do not assume you’ve got any prior knowledge other than the standard set of prerequisite material needed for that class. In other words, it is assumed that you know Algebra and Trig prior to reading the Calculus I notes, know Calculus I prior to reading the Calculus II notes, etc. The assumptions about your background that I’ve made are given with each description below.

I’d like to thank Fred J., Mike K. and David A. for all the typos that they’ve found and sent my way! I’ve tried to proof read these pages and catch as many typos as I could, however it just isn’t possible to catch all of them when you are also the person who wrote the material. Fred, Mike and David have caught quite a few typos that I’d missed and been nice enough to send them my way. Thanks again Fred, Mike and David!

If you are one of my current students and are here looking for homework assignments I’ve got a set of links that will get you to the right pages listed here.

At present I’ve gotten the notes/tutorials for my Algebra (Math 1314), Calculus I (Math 2413), Calculus II (Math 2414), Calculus III (Math 3435) and Differential Equations (Math 3301) class online. I’ve also got a couple of Review/Extras available as well. Among the reviews/extras that I’ve got are an Algebra/Trig review for my Calculus Students, a Complex Number primer, a set of Common Math Errors, and some tips on How to Study Math.

I’ve made most of the pages on this site available for download as well. These downloadable versions are in pdf format. Each subject on this site is available as a complete download and in the case of very large documents I’ve also split them up into smaller portions that mostly correspond to each of the individual topics. Near the top of each page you will see one or two download buttons depending on whether the subject is available as only as a complete document or is also available in pieces. You can see a complete listing of all the available downloads by selecting the Downloads option in the menu.

Classes & Extras

Here is a complete listing of all the subjects that are currently available on this site as well as brief descriptions of each.

Cheat Sheets & Tables

Algebra Cheat Sheet – This is as many common algebra facts, properties, formulas, and functions that I could think of. There is also a page of common algebra errors included. Currently the cheat sheet is four pages long.

Algebra Cheat Sheet (Reduced) – This is the same cheat sheet as above except it has been reduced so that it will fit onto the front and back of a single piece of paper. It contains all the information that the normal sized cheat sheet does.

Trig Cheat Sheet – Here is a set of common trig facts, properties and formulas. A unit circle (completely filled out) is also included. Currently this cheat sheet is four pages long.

Trig Cheat Sheet (Reduced) – My standard trig cheat sheet reduced to fit onto the front and back of a single piece of paper. It contains all the information that the normal sized cheat sheet does.

Calculus Cheat Sheets – These are a series of Calculus Cheat Sheets that covers most of a standard Calculus I course and a few topics from a Calculus II course.

Common Derivatives and Integrals – Here is a set of common derivatives and integrals that are used somewhat regularly in a Calculus I or Calculus II class. Also included are reminders on several integration techniques. Currently this cheat sheet is four pages long.

Common Derivatives and Integrals (Reduced) – My common derivatives and integrals table reduced to fit onto the front and back of a single piece of paper. It contains all the information that the normal sized table does.

Table of Laplace Transforms – Here is a list of Laplace transforms for a differential equations class. This table gives many of the commonly used Laplace transforms and formulas.

 

Calculus II – Notes

Tangents with Parametric Equations