光的世界︰派生科學計算二‧中

2016-07-24 懸鉤子

照說一元二次方程式 $x^2 + px + q = 0$ 用『配方法』來求解，應是自然順理成章之事。若從『發現』之『邏輯』來考察這一『思路』莫非源於 ${(x - \Box)}^2 = {\bigcirc}^2$ 的『形式』之『啟發』耶？如果由二次式函數 $f(x) = x^2 + px +q$ 之觀點來看，難到不能平移『座標系』 $x = y + \alpha$ ，使之簡化為 $y \times (y - \Box)$ 之『形式』乎？？一個簡單的例子 $x^2 - 4x + 3 = 0$ 或可說明，如此尋找 $\alpha$ 以求簡化方程式的求解，實無攸利也！那個 $\alpha$ 還得滿足原方程式 ${\alpha}^2 - 4 \alpha + 3 = 0$ 矣！！

Figure q_0

Fig 1 : $f(x) = x^2 - 4x +3 = (x - 1) \times (x - 3)$

Figure q_1

Fig 2 : $f(y) = y^2 - 1 = (y - 1) \times (y + 1)$

Figure q_2

Fig 3 : $f(y) = y^2 - 2y = y \times (y - 2)$

pi@raspberrypi:~ x^2 = 1x = \pm 1x^2 + px + q = (x - \alpha)(x -\beta)r_1 = \alpha + \betar_2 = \alpha - \betaNr_2 = \alpha - \beta \beta - \alpha- r_2\alpha - \beta{(\alpha - \beta)}^2 = {(\alpha + \beta)}^2 - 4 \alpha \betar_2\alpha = \frac{r_1 + r_2}{2}\beta = \frac{r_1 - r_2}{2}\sqrt[4]{-1}\sqrt[3]{-1}\pm \sqrt{x}\sqrt[3]{x}\sqrt[n]{x}P(x) = \sum \limits_{i=0}^{i=n} c_i \cdot x^i = 0 \ =?= \prod \limits_{k=1}^{k=n} x - x_kc_ix_k\sqrt{2}\sqrt{2}\frac{p}{q}pqQ\sqrt{Q}\sqrt{Q} = \frac{p}{q}p^2 = q^2 \cdot Qp, qp = k \cdot Qk, qq^2 = k^2 \cdot Qq = m QpqQ\sqrt{Q}P(x) = x^2 -2 = 0 = (x + \sqrt{2})(x - \sqrt{2})$ 時，這裡所說的『多項式』與『方程式』是一樣的嗎？它們的內在聯繫又是什麼的呢？

── 摘自《【Sonic π】電路學之補充《四》無窮小算術‧中下下‧上》

樹莓派、樹莓派之學習、樹莓派之教育

光的世界︰派生科學計算二‧上

2016-07-23 懸鉤子

為什麼『IPython』那麼強調『互動性』 interactive 呢？因為人們的『認知』是積累的，偶或『靈感』乍現，當下若不『捕捉』，恐怕稍後不復『記得』矣！更何況今日之『計算』極其『複雜』，若是每每紙筆『推導』，或是一定得先寫好『程式』，總覺不切實際，終將誰願為的耶？？反觀則朗朗親見『互動性』伴隨『符號運算』的好處？？！！

口說無憑，且以拉格朗日之『預解形‧式』再探伽羅瓦『方程式』論的淵源，及於『SymPy』之省勞少錯方便乎！！？？

就讓我們略窺一下『伽羅瓦』的思考法吧。假使 $x_1,x_2,\cdots, x_n$ 是『多項式』 $P(x) = \sum \limits_{k=0}^{k=n} c_k x^k = 0$ 的『解』，此處係數 $c_k$ 都是『有理數』。如果我們建構一個『對稱函數』

$f(x_1,x_2,x_3,\cdots,x_n) = (x-x_1)(x-x_2)(x-x_3)\cdots(x-x_n)$

，將它展開後 $c_n f(x_1,x_2,x_3,\cdots,x_n)$ 應該就是 $P(x)$ 的吧。如果將這些『解』 $x_1,x_2,\cdots, x_n$ ，作任意的『排列』permutation $\begin{pmatrix} x_1 & x_2 & x_3 & \cdots & x_n \\ x_2 & x_n & x_4 & \cdots & x_1\end{pmatrix}$ ，此處是說上一排的『解』的『位置』用下一排的『解』來『置換』，由於 $f$ 函數的特殊『形式』，我們會得到 $f(x_1,x_2,x_3, \cdots,x_n)=f(x_2,x_n,x_4,\cdots,x_1)$ 。事實上對於任意的『置換』，都會有 $f(x_1,x_2,...,x_n)=f(x_2,x_1,\cdots,x_n)=f(x_3,x_1,\cdots,x_n,x_{n−1})$ 。所以函數 $f$ 稱之為『對稱函數』，這個『置換』的『不變性』就是『伽羅瓦』主要研究的對象。舉例來說，考慮一個二次方程式 $x^2 + A x + B = 0$ 有兩個根 $\lambda_1, \lambda_2$ ， $F(\lambda_1, \lambda_2) = (x - \lambda_1)(x - \lambda_2)$
$= x^2 -(\lambda_1 + \lambda_2) x + \lambda_1 \cdot \lambda_2$
$= x^2 + A x + B$
，比對後得到
$\lambda_1 + \lambda_2 = -A$ ，和
$\lambda_1 \cdot \lambda_2 = B$
。這兩個『代數式』對於 $\lambda_1, \lambda_2$ 來講，也是『對稱的』，如果將它們看成兩個變數的『聯立方程組』，化簡後所得到的也定然就是『對等的』二次方程式 ${\lambda_1}^2 + A \lambda_1 + B = 0$ 與 ${\lambda_2}^2 + A \lambda_2 + B = 0$ 。

這產生了很重要的結果，假使 $\lambda_1 = a + b \sqrt{Q}$ 是方程式的一個根，假設另一個根是 $\lambda_2 = c + d \sqrt{Q}$ ，由於
$\lambda_1 + \lambda_2 = -A$
$\Longrightarrow (a + b \sqrt{Q}) + (c + d \sqrt{Q}) = -A$
$\Longrightarrow (a + c +A) + (b + d) \sqrt{Q} = 0$
$\therefore a + c + A = 0, \ b + d =0$
，再由
$\lambda_1 \cdot \lambda_2 = B$
$\Longrightarrow (a + b \sqrt{Q}) \cdot (c + d \sqrt{Q}) = B$
$\Longrightarrow (a c + b d Q - B) + (a d + b c) \sqrt{Q}) =0$
$\therefore a c + b d Q = B, \ a d + b c =0$
。因此得到 $d = -b, \ c = a$ 。而且 $a = - \frac{A}{2}, \ b \sqrt{Q} = \frac{\sqrt{A^2 - 4B}}{2}$ 。也就是說這兩個根是熟悉的 $\frac{- A + \sqrt{A^2 - 4B}}{2}$ 與 $\frac{- A - \sqrt{A^2 - 4B}}{2}$ 。於是一個『對稱函數』 $f(x_1,x_2,x_3,\cdots,x_n) = (x-x_1)(x-x_2)(x-x_3)\cdots(x-x_n)$ 如果某一個根 $x_k$ 是 $a + b \sqrt{Q}$ 的形式﹐那麼必然有另一個根 $x_j$ 是 $a - b \sqrt{Q}$ 的形式，這就是由於那個『多項式』的係數是『有理數』的原故，它的『二次方根』的解，總是『成對』出現的啊！於是『二次方根』解的個數也必然是『偶數』的了！！

── 摘自《【Sonic π】電路學之補充《四》無窮小算術‧中下下‧中》

Resolvent (Galois theory)

In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root. Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois. Nowadays they are still a fundamental tool to compute Galois groups. The simplest examples of resolvents are

$X^{2}-\Delta$ where $\Delta$ is the discriminant, which is a resolvent for the alternating group. In the case of a cubic equation, this resolvent is sometimes called the quadratic resolvent; its roots appear explicitly in the formulas for the roots of a cubic equation.
The cubic resolvent of a quartic equation, which is a resolvent for the dihedral group of 8 elements.
The Cayley resolvent is a resolvent for the maximal resoluble Galois group in degree five. It is a polynomial of degree 6.

These three resolvents have the property of being always separable, which means that, if they have a multiple root, then the polynomial p is not irreducible. It is not known if there is an always separable resolvent for every group of permutations.

For every equation the roots may be expressed in terms of radicals and of a root of a resolvent for a resoluble group, because, the Galois group of the equation over the field generated by this root is resoluble.

───

即使祇以文獻上記載的歷史來說，

配方法

配方法是一種代數的計算技巧，可以用來解二次方程式、判別解析幾何中某些方程式的圖形，或者用來計算微積分中的某些積分型式。配方法最主要的目的就是將一個一元二次方程式或多項式化為一個一次式的完全平方，以便簡化計算。

將下方左邊的二次式化成右邊的形式，就是配方法的目標：

ax^{2}+bx+c=a(x-h)^{2}+k

，其中h和k是常數

其由來亦古早矣︰

求根公式的由來

中亞細亞的花拉子米

花拉子米全名是阿布·阿卜杜拉·穆罕默德·伊本·穆薩·花拉子米^[1]（Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī，約780年－約850年^[2]），他是一位波斯數學家、天文學家及地理學家，也是巴格達智慧之家的學者。

他的《代數學》（Kitab al-Jabr wa-l-Muqabala）是第一本解決一次方程及一元二次方程的系統著作，他因而被稱為代數的創造者^[3]，與丟番圖享名。十二世紀，花拉子米在印度數字方面的著作被翻譯成拉丁文，十進制因此傳入西方世界^[4]。此外，他修訂了托勒密的《地理》，並著有天文學及占星學方面的書籍。

從一些詞就可以看出他對數學的重要貢獻，「代數」（algebra）一詞出自阿拉伯文拉丁轉寫「al-jabr」^[5]，「al-jabr」是用以解決一元二次方程的兩個辦法之一。算法（Algorism、Algorithm）出自「Algoritmi」，這是花拉子米（al-Khwārizmī）的拉丁文譯名^[6]，而西班牙語「guarismo」及葡萄牙語「algarismo」亦是由此名字而來，這兩個詞語都解作數字^[7]。

……

(約780-約850) 在公元820年左右出版了《代數學》。書中給出了一元二次方程的求根公式，並把方程的未知數叫做「根」，其後譯成拉丁文radix。

我們通常把 $x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$ 稱之為 $ax^{2}+bx+c=0\,$ 的求根公式：

${\begin{aligned}ax^{2}+bx+c&=0\\x^{2}+{\frac {b}{a}}x+{\frac {c}{a}}&=0\\x^{2}+{\frac {b}{a}}x+\left({\frac {b}{2a}}\right)^{2}-\left({\frac {b}{2a}}\right)^{2}+{\frac {c}{a}}&=0\\\left(x+{\frac {b}{2a}}\right)^{2}-{\frac {b^{2}}{4a^{2}}}+{\frac {c}{a}}&=0\\\left(x+{\frac {b}{2a}}\right)^{2}&={\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}\\\left(x+{\frac {b}{2a}}\right)^{2}&={\frac {b^{2}-4ac}{4a^{2}}}\\x+{\frac {b}{2a}}&={\frac {\pm {\sqrt {b^{2}-4ac}}}{2a}}\\x&={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\end{aligned}}$

或不將 $x^{2}$ 係數化為1：

${\begin{aligned}ax^{2}+bx+c&=0\\ax^{2}+bx+\left({\frac {b}{2{\sqrt {a}}}}\right)^{2}&=\left({\frac {b}{2{\sqrt {a}}}}\right)^{2}-c\\\left(x{\sqrt {a}}+{\frac {b}{2{\sqrt {a}}}}\right)^{2}&=\left({\frac {b}{2{\sqrt {a}}}}\right)^{2}-c\\x{\sqrt {a}}+{\frac {b}{2{\sqrt {a}}}}&=\pm {\sqrt {\left({\frac {b}{2{\sqrt {a}}}}\right)^{2}-c}}\\x{\sqrt {a}}+{\frac {b}{2{\sqrt {a}}}}&=\pm {\sqrt {{\frac {b^{2}}{4a}}-c}}\\x+{\frac {b}{2a}}&=\pm {\sqrt {{\frac {b^{2}}{4a^{2}}}-{\frac {c}{a}}}}\\x+{\frac {b}{2a}}&=\pm {\sqrt {{\frac {b^{2}}{4a^{2}}}-{\frac {4ac}{4a^{2}}}}}\\x&=-{\frac {b}{2a}}\pm {\sqrt {{\frac {b^{2}-4ac}{4a^{2}}}}}\\x&={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}\end{aligned}}$

至於說用『單位圓』之

單位根

數學上， $n\,$ 次單位根是 } $n\,$ 次冪為1的複數。它們位於複平面的單位圓上，構成正n邊形的頂點，其中一個頂點是1。

定義

z^{n}=1\qquad (n=1,2,3,\cdots )

這方程的複數根 $z \,$ 為 $n\,$ 次單位根。

單位的 $n\,$ 次根有 $n\,$ 個：

e^{\frac {2\pi k{i}}{n}}\qquad (k=0,1,2,\cdots ,n-1)

。

本原根

單位的 $n\,$ 次根以乘法構成 $n$ 階循環群。它的生成元是 $n\,$ 次本原單位根。 $n\,$ 次本原單位根是 $e^{\frac {2\pi k{i}}{n}}$ ，其中 $k\,$ 和 $n\,$ 互質。 $n\,$ 次本原單位根數目為歐拉函數 $\varphi (n)$ 。

例子

一次單位根有一個 $1\,$ 。

二次單位根有兩個： $+1\,$ 和 $-1\,$ ，只有 $-1\,$ 是本原根。

三次單位根是

\left\{1,{\frac {-1+{\sqrt {3}}{i}}{2}},{\frac {-1-{\sqrt {3}}{i}}{2}}\right\},

其中 ${{\mathrm {i}}}\,$ 是虛數單位；除 $1\,$ 外都是本原根。

四次單位根是

\left\{1,+{i},-1,-{i}\right\},

其中 $+{{\mathrm {i}}}\,$ 和 $-{{\mathrm {i}}}\,$ 是本原根。

來『系統化』研究『一元多次方程式』之求解卻是拉格朗日的直覺『先見』哩！！此處僅擷維基百科之文本一段︰

Quadratic formula

In elementary algebra, the quadratic formula is the solution of the quadratic equation. There are other ways to solve the quadratic equation instead of using the quadratic formula, such as factoring, completing the square, or graphing. Using the quadratic formula is often the most convenient way.

The general quadratic equation is

ax^{2}+bx+c=0.

Here $x$ represents an unknown, while $a$ , $b$ , and $c$ are constants with $a$ not equal to 0. One can verify that the quadratic formula satisfies the quadratic equation, by inserting the former into the latter. Each of the solutions given by the quadratic formula is called a root of the quadratic equation.

Geometrically, these roots represent the $x$ values at which any parabola, explicitly given as $y = ax 2 + bx + c$ , crosses the $x$ -axis. As well as being a formula that will yield the zeros of any parabola, the quadratic equation will give the axis of symmetry of the parabola, and it can be used to immediately determine how many zeros it has.

……

By Lagrange resolvents

For more details on this topic, see Lagrange resolvents.

An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents,^[24] which is an early part of Galois theory.^[25] This method can be generalized to give the roots of cubic polynomials and quartic polynomials, and leads to Galois theory, which allows one to understand the solution of algebraic equations of any degree in terms of the symmetry group of their roots, the Galois group.

This approach focuses on the roots more than on rearranging the original equation. Given a monic quadratic polynomial

x^2+px+q,

assume that it factors as

x^2+px+q=(x-\alpha)(x-\beta),

Expanding yields

x^2+px+q=x^2-(\alpha+\beta)x+\alpha \beta,

where $p = -(α + β)$ and $q = αβ$ .

Since the order of multiplication does not matter, one can switch $α$ and $β$ and the values of $p$ and $q$ will not change: one can say that $p$ and $q$ are symmetric polynomials in $α$ and $β$ . In fact, they are the elementary symmetric polynomials – any symmetric polynomial in $α$ and $β$ can be expressed in terms of $α + β$ and $αβ$ The Galois theory approach to analyzing and solving polynomials is: given the coefficients of a polynomial, which are symmetric functions in the roots, can one “break the symmetry” and recover the roots? Thus solving a polynomial of degree $n$ is related to the ways of rearranging (“permuting”) $n$ terms, which is called the symmetric group on $n$ letters, and denoted $S n$ . For the quadratic polynomial, the only way to rearrange two terms is to swap them (“transpose” them), and thus solving a quadratic polynomial is simple.

To find the roots $α$ and $β$ , consider their sum and difference:

\text{[math]}

These are called the Lagrange resolvents of the polynomial; notice that one of these depends on the order of the roots, which is the key point. One can recover the roots from the resolvents by inverting the above equations:

\text{[math]}

Thus, solving for the resolvents gives the original roots.

Now $r 1 = α + β$ is a symmetric function in $α$ and $β$ , so it can be expressed in terms of $p$ and $q$ , and in fact $r 1 = - p$ as noted above. But $r 2 = α - β$ is not symmetric, since switching $α$ and $β$ yields $- r 2 = β - α$ (formally, this is termed a group action of the symmetric group of the roots). Since $r 2$ is not symmetric, it cannot be expressed in terms of the coefficients $p$ and $q$ , as these are symmetric in the roots and thus so is any polynomial expression involving them. Changing the order of the roots only changes $r 2$ by a factor of −1, and thus the square $r 22 = (α - β) 2$ is symmetric in the roots, and thus expressible in terms of $p$ and $q$ . Using the equation

(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta\!

yields

r_2^2 = p^2 - 4q\!

and thus

r_2 = \pm \sqrt{p^2 - 4q}.\!

If one takes the positive root, breaking symmetry, one obtains:

\text{[math]}

and thus

\text{[math]}

Thus the roots are

\textstyle{\frac{1}{2}}\left(-p \pm \sqrt{p^2 - 4q}\right)

which is the quadratic formula. Substituting $p = b / a, q = c / a$ yields the usual form for when a quadratic is not monic. The resolvents can be recognized as $r 1 / 2 = - p / 2 = - b / 2 a$ being the vertex, and $r 22 = p 2 - 4 q$ is the discriminant (of a monic polynomial).

A similar but more complicated method works for cubic equations, where one has three resolvents and a quadratic equation (the “resolving polynomial”) relating $r 2$ and $r 3$ , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved.^[24] The same method for a quintic equation yields a polynomial of degree 24, which does not simplify the problem, and in fact solutions to quintic equations in general cannot be expressed using only roots.

───

不知是否能得『配方法』之根本焉？？？

樹莓派、樹莓派之學習、樹莓派之教育

光的世界︰派生科學計算一

2016-07-22 懸鉤子

學習『工具』最簡單的方法，就是『使用』它︰

pi@raspberrypi:~ $ ipython3
Python 3.4.2 (default, Oct 19 2014, 13:31:11) 
Type "copyright", "credits" or "license" for more information.

IPython 2.3.0 -- An enhanced Interactive Python.
?         -> Introduction and overview of IPython's features.
%quickref -> Quick reference.
help      -> Python's own help system.
object?   -> Details about 'object', use 'object??' for extra details.

In [1]:

不過對此『工具』是什麼□○都尚且不知，又將如何『使用』呢？通常『工具』的『使用』始於閱讀『概觀』︰

Introduction

Overview

One of Python’s most useful features is its interactive interpreter. It allows for very fast testing of ideas without the overhead of creating test files as is typical in most programming languages. However, the interpreter supplied with the standard Python distribution is somewhat limited for extended interactive use.

The goal of IPython is to create a comprehensive environment for interactive and exploratory computing. To support this goal, IPython has three main components:

An enhanced interactive Python shell.
A decoupled two-process communication model, which allows for multiple clients to connect to a computation kernel, most notably the web-based notebook
An architecture for interactive parallel computing.

All of IPython is open source (released under the revised BSD license).

………

從『簡介』尋找，發現『上手』之處︰

Introducing IPython

You don’t need to know anything beyond Python to start using IPython – just type commands as you would at the standard Python prompt. But IPython can do much more than the standard prompt. Some key features are described here. For more information, check the tips page, or look at examples in the IPython cookbook.

If you’ve never used Python before, you might want to look at the official tutorial or an alternative, Dive into Python.

The four most helpful commands

The four most helpful commands, as well as their brief description, is shown to you in a banner, every time you start IPython:

command	description
?	Introduction and overview of IPython’s features.
%quickref	Quick reference.
help	Python’s own help system.
object?	Details about ‘object’, use ‘object??’ for extra details.

Tab completion

Tab completion, especially for attributes, is a convenient way to explore the structure of any object you’re dealing with. Simply type object_name.<TAB> to view the object’s attributes (see the readline section for more). Besides Python objects and keywords, tab completion also works on file and directory names.

………

要是還能知道『參考文件』在哪裡固然很好︰

IPython Documentation

Release:	3.2.1
Date:	September 25, 2015

Welcome to the official IPython documentation.

───

往往根本什麼說明也沒有哩！？此時亦無礙徒手上陣，嘗試錯誤，將之 TRY 出耶？！

依稀記得作者當初下的第一個指令就是『?』︰

? -> Introduction and overview of IPython’s features.

匆匆瀏覽後，隨即上路︰

IPython — An enhanced Interactive Python
=========================================

IPython offers a combination of convenient shell features, special commands and a history mechanism for both input (command history) and output (results caching, similar to Mathematica). It is intended to be a fully compatible replacement for the standard Python interpreter, while offering vastly improved functionality and flexibility.

At your system command line, type ‘ipython -h’ to see the command line options available. This document only describes interactive features.

MAIN FEATURES
————-

* Access to the standard Python help. As of Python 2.1, a help system is available with access to object docstrings and the Python manuals. Simply type ‘help’ (no quotes) to access it.

* Magic commands: type %magic for information on the magic subsystem.

* System command aliases, via the %alias command or the configuration file(s).

* Dynamic object information:

Typing ?word or word? prints detailed information about an object. If
certain strings in the object are too long (docstrings, code, etc.) they get snipped in the center for brevity.

Typing ??word or word?? gives access to the full information without
snipping long strings. Long strings are sent to the screen through the less pager if longer than the screen, printed otherwise.

The ?/?? system gives access to the full source code for any object (if
available), shows function prototypes and other useful information.

If you just want to see an object’s docstring, type ‘%pdoc object’ (without quotes, and without % if you have automagic on).

Both %pdoc and ?/?? give you access to documentation even on things which are not explicitely defined. Try for example typing {}.get? or after import os, type os.path.abspath??. The magic functions %pdef, %source and %file operate similarly.

* Completion in the local namespace, by typing TAB at the prompt.

At any time, hitting tab will complete any available python commands or variable names, and show you a list of the possible completions if there’s no unambiguous one. It will also complete filenames in the current directory.

This feature requires the readline and rlcomplete modules, so it won’t work if your Python lacks readline support (such as under Windows).
* Search previous command history in two ways (also requires readline):

– Start typing, and then use Ctrl-p (previous,up) and Ctrl-n (next,down) to search through only the history items that match what you’ve typed so far. If you use Ctrl-p/Ctrl-n at a blank prompt, they just behave like normal arrow keys.

– Hit Ctrl-r: opens a search prompt. Begin typing and the system searches your history for lines that match what you’ve typed so far, completing as much as it can.

– %hist: search history by index (this does *not* require readline).

* Persistent command history across sessions.

* Logging of input with the ability to save and restore a working session.

* System escape with !. Typing !ls will run ‘ls’ in the current directory.

* The reload command does a ‘deep’ reload of a module: changes made to the module since you imported will actually be available without having to exit.

* Verbose and colored exception traceback printouts. See the magic xmode and xcolor functions for details (just type %magic).

* Input caching system:

IPython offers numbered prompts (In/Out) with input and output caching. All input is saved and can be retrieved as variables (besides the usual arrow key recall).

The following GLOBAL variables always exist (so don’t overwrite them!):
_i: stores previous input.
_ii: next previous.
_iii: next-next previous.
_ih : a list of all input _ih[n] is the input from line n.

Additionally, global variables named _i<n> are dynamically created (<n> being the prompt counter), such that _i<n> == _ih[<n>]

For example, what you typed at prompt 14 is available as _i14 and _ih[14].

You can create macros which contain multiple input lines from this history, for later re-execution, with the %macro function.

The history function %hist allows you to see any part of your input history by printing a range of the _i variables. Note that inputs which contain magic functions (%) appear in the history with a prepended comment. This is because they aren’t really valid Python code, so you can’t exec them.

* Output caching system:
For output that is returned from actions, a system similar to the input
cache exists but using _ instead of _i. Only actions that produce a result (NOT assignments, for example) are cached. If you are familiar with Mathematica, IPython’s _ variables behave exactly like Mathematica’s % variables.

The following GLOBAL variables always exist (so don’t overwrite them!):
_ (one underscore): previous output.
__ (two underscores): next previous.
___ (three underscores): next-next previous.

Global variables named _<n> are dynamically created (<n> being the prompt counter), such that the result of output <n> is always available as _<n>.

Finally, a global dictionary named _oh exists with entries for all lines
which generated output.

* Directory history:

Your history of visited directories is kept in the global list _dh, and the
magic %cd command can be used to go to any entry in that list.

* Auto-parentheses and auto-quotes (adapted from Nathan Gray’s LazyPython)

1. Auto-parentheses

Callable objects (i.e. functions, methods, etc) can be invoked like
this (notice the commas between the arguments)::

In [1]: callable_ob arg1, arg2, arg3

and the input will be translated to this::

callable_ob(arg1, arg2, arg3)

This feature is off by default (in rare cases it can produce
undesirable side-effects), but you can activate it at the command-line
by starting IPython with `–autocall 1`, set it permanently in your
configuration file, or turn on at runtime with `%autocall 1`.

You can force auto-parentheses by using ‘/’ as the first character
of a line. For example::

In [1]: /globals # becomes ‘globals()’

Note that the ‘/’ MUST be the first character on the line! This
won’t work::

In [2]: print /globals # syntax error

In most cases the automatic algorithm should work, so you should
rarely need to explicitly invoke /. One notable exception is if you
are trying to call a function with a list of tuples as arguments (the
callable_ob(arg1, arg2, arg3)

You can force auto-parentheses by using ‘/’ as the first character
of a line. For example::

In [1]: /globals # becomes ‘globals()’

Note that the ‘/’ MUST be the first character on the line! This
won’t work::

In [2]: print /globals # syntax error

In [1]: zip (1,2,3),(4,5,6) # won’t work

but this will work::

In [2]: /zip (1,2,3),(4,5,6)
——> zip ((1,2,3),(4,5,6))
Out[2]= [(1, 4), (2, 5), (3, 6)]

IPython tells you that it has altered your command line by
displaying the new command line preceded by –>. e.g.::

In [18]: callable list
——-> callable (list)

2. Auto-Quoting

You can force auto-quoting of a function’s arguments by using ‘,’ as
the first character of a line. For example::

In [1]: ,my_function /home/me # becomes my_function(“/home/me”)

If you use ‘;’ instead, the whole argument is quoted as a single
string (while ‘,’ splits on whitespace)::

In [2]: ,my_function a b c # becomes my_function(“a”,”b”,”c”)
In [3]: ;my_function a b c # becomes my_function(“a b c”)

Note that the ‘,’ MUST be the first character on the line! This
won’t work::

In [4]: x = ,my_function /home/me # syntax error

一旦需要什麼 □□ ，或想要怎樣 ○○ ，再查詢、翻閱、谷歌……文件，仔細精讀文本一番乎！！？？果然常用之故，有時大腦恐已忘記，誰知指頭竟能自書，當真『熟能生巧』隨心所欲哩？？！！

因此『派生互動筆記』界面，作者也作如是觀矣！！！

The IPython Notebook

Introduction

The notebook extends the console-based approach to interactive computing in a qualitatively new direction, providing a web-based application suitable for capturing the whole computation process: developing, documenting, and executing code, as well as communicating the results. The IPython notebook combines two components:

A web application: a browser-based tool for interactive authoring of documents which combine explanatory text, mathematics, computations and their rich media output.

Notebook documents: a representation of all content visible in the web application, including inputs and outputs of the computations, explanatory text, mathematics, images, and rich media representations of objects.

Main features of the web application

In-browser editing for code, with automatic syntax highlighting, indentation, and tab completion/introspection.
The ability to execute code from the browser, with the results of computations attached to the code which generated them.
Displaying the result of computation using rich media representations, such as HTML, LaTeX, PNG, SVG, etc. For example, publication-quality figures rendered by the matplotlib library, can be included inline.
In-browser editing for rich text using the Markdown markup language, which can provide commentary for the code, is not limited to plain text.
The ability to easily include mathematical notation within markdown cells using LaTeX, and rendered natively by MathJax.

Notebook documents

Notebook documents contains the inputs and outputs of a interactive session as well as additional text that accompanies the code but is not meant for execution. In this way, notebook files can serve as a complete computational record of a session, interleaving executable code with explanatory text, mathematics, and rich representations of resulting objects. These documents are internally JSON files and are saved with the .ipynb extension. Since JSON is a plain text format, they can be version-controlled and shared with colleagues.

Notebooks may be exported to a range of static formats, including HTML (for example, for blog posts), reStructuredText, LaTeX, PDF, and slide shows, via the new nbconvert command.

Furthermore, any .ipynb notebook document available from a public URL can be shared via the IPython Notebook Viewer (nbviewer). This service loads the notebook document from the URL and renders it as a static web page. The results may thus be shared with a colleague, or as a public blog post, without other users needing to install IPython themselves. In effect, nbviewer is simply nbconvert as a web service, so you can do your own static conversions with nbconvert, without relying on nbviewer.

光的世界︰幾何光學八

2016-07-21 懸鉤子

在進入『矩陣光學』之前，先作個簡單的總結。相信讀者已經清楚明白維基百科

幾何光學

高斯光學

幾何光學中研究和討論光學系統理想成像性質的分支稱為高斯光學，或稱近軸光學。它通常只討論對某一軸線(即光軸)具有旋轉對稱性的光學系統。如果從物點發出的所有光線經光學系統以後都交於同一點，則稱此點是物點的完善像。

光學影像

如果物點在垂軸平面上移動時，其完善像點也在垂軸平面上作線性移動，則此光學系統成像是理想的。可以證明，非常靠近光軸的細小物體，其每個物點都以很細的、很靠近光軸的單色光束被光學系統成像時，像是完善的。這表明，任何實際的光學系統(包括單個球面、單個透鏡)的近軸區都具有理想成像的性質。

為便於一般地了解光學系統的成像性質和規律，在研究近軸區成像規律的基礎上建立起被稱為理想光學系統的光學模型。這個模型完全撇開具體的光學系統結構，僅以幾對基本點的位置以及一對基本量的大小來表征。

根據基本點的性質能方便地導出成像公式，從而可以了解任意位置的物體被此模型成像時，像的位置、大小、正倒和虛實等各種成像特性和規律。反過來也可以根據成像要求求得相應的光學模型。任何具體的光學系統都能與一個等效模型相對應，對於不同的系統，模型的差別僅在於基本點位置和焦距大小有所不同而已。

高斯光學的理論是進行光學系統的整體分析和計算有關光學參量的必要基礎。

利用光學系統的近軸區可以獲得完善成像，但沒有什麼實用價值。因為近軸區只有很小的孔徑(即成像光束的孔徑角)和很小的視場(即成像範圍)，而光學系統的功能，包括對物體細節的分辨能力、對光能量的傳遞能力以及傳遞光學資訊的多少等，正好是被這兩個因素所決定的。要使光學系統有良好的功能，其孔徑和視場要遠比近軸區所限定的為大。

當光學系統的孔徑和視場超出近軸區時，成像質量會逐漸下降。這是因為自然點發出的光束中，遠離近軸區的那些光線在系統中的傳播光路偏離理想途徑，而不再相交於高斯像點(即理想像點)之故。這時，一點的像不再是一個點，而是一個模糊的彌散斑；物平面的像不再是一個平面，而是一個曲面，而且像相對於物還失去了相似性。所有這些成像缺陷，稱為像差。

有關光學系統的一些要求

一個光學系統須滿足一系列要求，包括：放大率、物像共軛距、轉像和光軸轉折等高斯光學要求；孔徑和視場等性能要求，以及校正像差和成像質量等方面的要求。這些要求都需要在設計時予以考慮和滿足。因此，光學系統設計工作應包括：對光學系統進行整體安排，並計算和確定系統或系統的各個組成部分的有關高斯光學參量和性能參量；選取或確定系統或系統各組成部分的結構形式並計算其初始結構參量；校正和平衡像差；評價像質。

像差與光學系統結構參量(如透鏡厚度、透鏡表面曲率半徑等)之間的關係極其複雜，不可能以具體的函數.式表達出來，因而無法採用聯立方程式之類的辦法直接由像差要求計算出系統的精確結構參量。現在能做到的是求得滿足初級像差要求的解。

初級像差是實際像差的近似表示，僅在孔徑和視場較小時能反映實際的像差情況，因此，按初級像差要求求得的解只是初始的結構參量，需對其進行修改才能達到像差的進一步校正和平衡，在這一過程中，傳統的做法是根據追跡光線得到的像差數據及其在系統各面上的分布情況，進行分析、判斷，找出對像差影響大的參量，加以修改，然後再追跡光線求出新的像差數據加以訐價。如此反覆修改，直到把應該考慮的各種像差都校正和平衡到符合要求為止。這是一個極其繁複和費時很多的過程。

詞條之內容。可以知道『光線追跡』

Ray tracing (physics)

In physics, ray tracing is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, change direction, or reflect off surfaces, complicating analysis. Ray tracing solves the problem by repeatedly advancing idealized narrow beams called rays through the medium by discrete amounts. Simple problems can be analyzed by propagating a few rays using simple mathematics. More detailed analyses can be performed by using a computer to propagate many rays.

When applied to problems of electromagnetic radiation, ray tracing often relies on approximate solutions to Maxwell’s equations that are valid as long as the light waves propagate through and around objects whose dimensions are much greater than the light’s wavelength. Ray theory does not describe phenomena such as interference and diffraction, which require wave theory (involving the phase of the wave).

Technique

Ray tracing works by assuming that the particle or wave can be modeled as a large number of very narrow beams (rays), and that there exists some distance, possibly very small, over which such a ray is locally straight. The ray tracer will advance the ray over this distance, and then use a local derivative of the medium to calculate the ray’s new direction. From this location, a new ray is sent out and the process is repeated until a complete path is generated. If the simulation includes solid objects, the ray may be tested for intersection with them at each step, making adjustments to the ray’s direction if a collision is found. Other properties of the ray may be altered as the simulation advances as well, such as intensity, wavelength, or polarization. The process is repeated with as many rays as are necessary to understand the behavior of the system.

$760px-Raytrace_changing_refractive_index.svg$

Ray tracing of a beam of light passing through a medium with changing refractive index. The ray is advanced by a small amount, and then the direction is re-calculated.

Optical design

光的世界︰幾何光學七

2016-07-20 懸鉤子

不知是否能從若干帶著簡述之圖形︰

In ray transfer (ABCD) matrix analysis, an optical element (here, a thick lens) gives a transformation between $(x_{1},\theta _{1})$ at the input plane and $(x_{2},\theta _{2})$ when the ray arrives at the output plane.

300px-Optical_axis_en

Optical axis (coincides with red ray) and rays symmetrical to optical axis (pair of blue and pair of green rays) propagating through different lenses.

In 2-dimensions an archery target has circular symmetry.

120px-Surface_of_revolution_illustration

A surface of revolution has circular symmetry around an axis in 3-dimensions.

220px-DoubleCone

A double-cone is a surface of revolution, generated by a line.

外添一個模擬軟件

Geometic Optics 幾何光學 2.05 - Mozilla Firefox_089

PhET Geometic Optics 幾何光學

，人們就可以明白『幾何光學』之『座標系』 $(x , \theta)$ 選取的道理？能夠知道『物面』── 物體所在面，垂直於光軸 ── 以及『像面』── 成像所在面，也垂直於光軸 ── 可用『極座標』。因此這一『物圓』對應那一『像圈』，而且此『圓圈』僅需 $x$ 決定而已？？從此『圓圈』各向 $\theta$ 所發射的『小角光』，定然會聚焦！！

並且了解為什麼照片是『方』的耶！！？？

1280px-View_from_the_Window_at_Le_Gras,_Joseph_Nicéphore_Niépce

尼埃普斯用暗箱拍攝的照片，為現存最早的照片，拍攝年份為1826年，花了8小時曝光。這張照片一度被世人遺忘，由1898年最後一次在倫敦公開展出，至1952年被歷史學家尋回，相隔超過半世紀。

Yuanmingyuan_before_the_burning,_Beijing,_6–18_October,_1860

焚毀前的圓明園照片，由費利斯·比特攝於1860年10月6日-18日

樂觀的想，視覺闡釋周遭環境，潛意識補足推理所需，

Visual perception is the ability to interpret the surrounding environment by processing information that is contained in visible light. The resulting perception is also known as eyesight, sight, or vision (adjectival form: visual, optical, or ocular). The various physiological components involved in vision are referred to collectively as the visual system, and are the focus of much research in psychology, cognitive science, neuroscience, and molecular biology, collectively referred to as vision science.

Leonardo da Vinci (1452–1519) is believed to be the first to recognize the special optical qualities of the eye. He wrote “The function of the human eye … was described by a large number of authors in a certain way. But I found it to be completely different.” His main experimental finding was that there is only a distinct and clear vision at the line of sight—the optical line that ends at the fovea. Although he did not use these words literally he actually is the father of the modern distinction between foveal and peripheral vision.^{[citation needed]}

220px-Eye_Line_of_sight

Leonardo da Vinci: The eye has a central line and everything that reaches the eye through this central line can be seen distinctly.

※Fovea centralis

The fovea centralis (the term fovea comes from the Latin, meaning pit or pitfall) is a small, central pit composed of closely packed cones in the eye. It is located in the center of the macula lutea of the retina.^[1]^[2]

The fovea is responsible for sharp central vision (also called foveal vision), which is necessary in humans for activities where visual detail is of primary importance, such as reading and driving. The fovea is surrounded by the parafovea belt, and the perifovea outer region.^[2] The parafovea is the intermediate belt, where the ganglion cell layer is composed of more than five rows of cells, as well as the highest density of cones; the perifovea is the outermost region where the ganglion cell layer contains two to four rows of cells, and is where visual acuity is below the optimum. The perifovea contains an even more diminished density of cones, having 12 per 100 micrometres versus 50 per 100 micrometres in the most central fovea. This, in turn, is surrounded by a larger peripheral area that delivers highly compressed information of low resolution following the pattern of compression in foveated imaging.

Approximately half of the nerve fibers in the optic nerve carry information from the fovea, while the remaining half carry information from the rest of the retina. The parafovea extends to a radius of 1.25 mm from the central fovea, and the perifovea is found at a 2.75 mm radius from the fovea centralis.^[3]

Schematic diagram of the human eye, with the fovea at the bottom. It shows a horizontal section through the right eye.

Unconscious inference

Main article: Unconscious inference

Hermann von Helmholtz is often credited with the first study of visual perception in modern times. Helmholtz examined the human eye and concluded that it was, optically, rather poor. The poor-quality information gathered via the eye seemed to him to make vision impossible. He therefore concluded that vision could only be the result of some form of unconscious inferences: a matter of making assumptions and conclusions from incomplete data, based on previous experiences.^{[citation needed]}

Inference requires prior experience of the world.

Examples of well-known assumptions, based on visual experience, are:

light comes from above
objects are normally not viewed from below
faces are seen (and recognized) upright.^[6]
closer objects can block the view of more distant objects, but not vice versa
figures (i.e., foreground objects) tend to have convex borders

The study of visual illusions (cases when the inference process goes wrong) has yielded much insight into what sort of assumptions the visual system makes.

Another type of the unconscious inference hypothesis (based on probabilities) has recently been revived in so-called Bayesian studies of visual perception.^[7] Proponents of this approach consider that the visual system performs some form of Bayesian inference to derive a perception from sensory data. Models based on this idea have been used to describe various visual perceptual functions, such as the perception of motion, the perception of depth, and figure-ground perception.^[8]^[9] The “wholly empirical theory of perception” is a related and newer approach that rationalizes visual perception without explicitly invoking Bayesian formalisms.

Gestalt theory

Main article: Gestalt psychology

Gestalt psychologists working primarily in the 1930s and 1940s raised many of the research questions that are studied by vision scientists today.

The Gestalt Laws of Organization have guided the study of how people perceive visual components as organized patterns or wholes, instead of many different parts. “Gestalt” is a German word that partially translates to “configuration or pattern” along with “whole or emergent structure”. According to this theory, there are eight main factors that determine how the visual system automatically groups elements into patterns: Proximity, Similarity, Closure, Symmetry, Common Fate (i.e. common motion), Continuity as well as Good Gestalt (pattern that is regular, simple, and orderly) and Past Experience.

，恰是『視覺』『直覺』之『完備』與『不足』的乎？？！！

格式塔學派（德語：Gestalttheorie）是心理學重要流派之一，興起於20世紀初的德國，又稱為完形心理學^[1]。由馬科斯·韋特墨（1880－1943）、沃爾夫岡·苛勒（1887－1967）和科特·考夫卡（1886－1941）三位德國心理學家在研究似動現象的基礎上創立。格式塔是德文Gestalt的譯音，意即「模式、形狀、形式」等，意思是指「動態的整體（dynamic wholes）」。

格式塔學派主張人腦的運作原理是整體的，「整體不同於其部件的總和」。例如，我們對一朵花的感知，並非純粹單單從對花的形狀、顏色、大小等感官資訊而來，還包括我們對花過去的經驗和印象，加起來才是我們對一朵花的感知^[2]。

260px-Reification

260px-Invariance

具體化

具體化（Reification）是知覺的「建設性」的或「生成性的」方面，這種知覺經驗，比起其所基於的感覺刺激，包括了更多外在的空間信息。例如，圖形A可以被知覺為三角形，儘管在事實上並未畫三角形。圖形C可以被視為三維球體，事實上也沒有畫三維球體。

組織性

「組織性」（Multistability或「組織性知覺」multistable perception）是趨勢模糊知覺經驗，不穩定地在兩個或兩個以上不同解釋之間往返。例如左圖所示「內克爾立方體」和「魯賓圖/花瓶幻覺」。

恆常性

恆常性（Invariance）知覺認可的簡單幾何組件，形成獨立的旋轉，平移、大小以及其他一些變化（如彈性變形，不同的燈光和不同的組件功能）。例如圖例’A’在圖中都立即確認為相同的基本形式，立即有別於’B’的形式。在彈性變形的’C’，描繪時使用不同的圖形元素，如’D’類。產生「具體化」、「多重穩定性」、「不變性」和「不可分模塊單獨進行建模」，它們是不同方面的統一機制。

一朵活生生之『玫瑰花』是一『整體』，一枝已離株的『玫瑰花』難免凋零。不知誰說︰就算將之『解析』為根、莖、葉、…… 終究沒了『生命』★不過假使不『解析』，它果能活的長久嗎☆

FreeSandal

光的世界︰派生科學計算二‧中

光的世界︰派生科學計算二‧上

Resolvent (Galois theory)

配方法

求根公式的由來

單位根

定義

本原根

例子

Quadratic formula

By Lagrange resolvents

光的世界︰派生科學計算一

Introduction

Overview

Introducing IPython

The four most helpful commands

Tab completion

IPython Documentation

Contents

The IPython Notebook

Introduction

Main features of the web application

Notebook documents

光的世界︰幾何光學八

幾何光學

高斯光學

光學影像

有關光學系統的一些要求

Ray tracing (physics)

Technique

Optical design

光的世界︰幾何光學七

Unconscious inference

Gestalt theory

具體化

組織性

恆常性

輕。鬆。學。部落客

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7	8	9	10	11	12	13
14	15	16	17	18	19	20
21	22	23	24	25	26	27
28	29	30