The friendship paradox is the phenomenon first observed by the sociologist Scott L. Feld in 1991 that most people have fewer friends than their friends have, on average.^[1] It can be explained as a form of sampling bias in which people with greater numbers of friends have an increased likelihood of being observed among one’s own friends. In contradiction to this, most people believe that they have more friends than their friends have.^[2]

The same observation can be applied more generally to social networks defined by other relations than friendship: for instance, most people’s sexual partners have had (on the average) a greater number of sexual partners than they have.^[3]^[4]

─── 維基百科《Friendship paradox》

已知複數 $z = x + i \cdot y$ 是二元數，當知 $h(z)$ 可描述的現象範圍小於 $h(x,y)$ ，故而亦小於 $h(z, \bar z)$ 也。所謂 $\frac {\partial }{\partial \bar z} h(z, \bar z) = 0$ 能得全純者，將須考慮全偏之別乎？

若從任意二元微分形式恐或難知其確解來說︰

An inexact differential equation is a differential equation of the form

M(x,y)\,dx+N(x,y)\,dy=0,{\text{ where }}{\frac {\partial M}{\partial y}}\neq {\frac {\partial N}{\partial x}}.

The solution to such equations came with the invention of the integrating factor by Leonhard Euler in 1739.^[1]

Solution method

In order to solve the equation, we need to transform it into an exact differential equation. In order to do that, we need to find an integrating factor $\mu$ to multiply the equation by. We’ll start with the equation itself. $M\,dx+N\,dy=0$ , so we get $\mu M\,dx+\mu N\,dy=0$ . We will require $\mu$ to satisfy ${\frac {\partial \mu M}{\partial y}}={\frac {\partial \mu N}{\partial x}}$ . We get ${\frac {\partial \mu }{\partial y}}M+{\frac {\partial M}{\partial y}}\mu ={\frac {\partial \mu }{\partial x}}N+{\frac {\partial N}{\partial x}}\mu$ . After simplifying we get $M\mu _{y}-N\mu _{x}+(M_{y}-N_{x})\mu =0$ . Since this is a partial differential equation, it is mostly extremely hard to solve, however in most cases we will get either $\mu (x,y)=\mu (x)$ or $\mu (x,y)=\mu (y)$ , in which case we only need to find $\mu$ with a first-order linear differential equation or a separable differential equation, and as such either $\mu (y)=e^{-\int {{\frac {M_{y}-N_{x}}{M}}\,dy}}$ or $\mu (x)=e^{\int {{\frac {M_{y}-N_{x}}{N}}\,dx}}$ .

自己終需思考耶！

且先舉例講講縮放吧︰

Scaling (geometry)

In Euclidean geometry, uniform scaling (or isotropic scaling^[1]) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions. The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc.

More general is scaling with a separate scale factor for each axis direction. Non-uniform scaling (anisotropic scaling) is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling or stretching (in one direction). Non-uniform scaling changes the shape of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to the scaling axes (the angles between lines parallel to the axes are preserved, but not all angles). It occurs, for example, when a faraway billboard is viewed from an oblique angle, or when the shadow of a flat object falls on a surface that is not parallel to it.

When the scale factor is larger than 1, (uniform or non-uniform) scaling is sometimes also called dilation or enlargement. When the scale factor is a positive number smaller than 1, scaling is sometimes also called contraction.

In the most general sense, a scaling includes the case in which the directions of scaling are not perpendicular. It also includes the case in which one or more scale factors are equal to zero (projection), and the case of one or more negative scale factors (a directional scaling by -1 is equivalent to a reflection).

Scaling is a linear transformation, and a special case of homothetic transformation. In most cases, the homothetic transformations are non-linear transformations.

就從不同方向可以大小不同談起︰

Matrix representation

A scaling can be represented by a scaling matrix. To scale an object by a vector v = (v_x, v_y, v_z), each point p = (p_x, p_y, p_z) would need to be multiplied with this scaling matrix:

S_{v}={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}.

As shown below, the multiplication will give the expected result:

S_{v}p={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\end{bmatrix}}.

Such a scaling changes the diameter of an object by a factor between the scale factors, the area by a factor between the smallest and the largest product of two scale factors, and the volume by the product of all three.

The scaling is uniform if and only if the scaling factors are equal (v_x = v_y = v_z). If all except one of the scale factors are equal to 1, we have directional scaling.

In the case where v_x = v_y = v_z = k, scaling increases the area of any surface by a factor of k² and the volume of any solid object by a factor of k³.

Scaling in arbitrary dimensions

In $n$ -dimensional space $\mathbb {R} ^{n}$ , uniform scaling by a factor $v$ is accomplished by scalar multiplication with $v$ , that is, multiplying each coordinate of each point by $v$ . As a special case of linear transformation, it can be achieved also by multiplying each point (viewed as a column vector) with a diagonal matrix whose entries on the diagonal are all equal to $vI$ .

Non-uniform scaling is accomplished by multiplication with any symmetric matrix. The eigenvalues of the matrix are the scale factors, and the corresponding eigenvectors are the axes along which each scale factor applies. A special case is a diagonal matrix, with arbitrary numbers $v_{1},v_{2},\ldots v_{n}$ along the diagonal: the axes of scaling are then the coordinate axes, and the transformation scales along each axis $i$ by the factor $v_{i}$

In uniform scaling with a non-zero scale factor, all non-zero vectors retain their direction (as seen from the origin), or all have the direction reversed, depending on the sign of the scaling factor. In non-uniform scaling only the vectors that belong to an eigenspace will retain their direction. A vector that is the sum of two or more non-zero vectors belonging to different eigenspaces will be tilted towards the eigenspace with largest eigenvalue.

假設

$\left( \begin{array}{cc} x^{'} \\ y^{'} \end{array} \right) = \left( \begin{array}{cc} a & 0 \\ 0 & b \end{array} \right) \left( \begin{array}{cc} x \\ y \end{array} \right)$ ，那麼

$z^{'} = x^{'} + i \cdot y^{'} = a \cdot x + i \cdot b \cdot y$

$= a \cdot \frac{z + \bar z}{2} + i \cdot b \cdot \frac{z - \bar z}{2 i}$

$= \frac{a+b}{2} \cdot z + \frac{a-b}{2} \cdot \bar z$ 。因此

$\frac {\partial }{\partial \bar z} z^{'} = \frac{a-b}{2}$ 。

故複平面上縮放變換欲得全純者，不得不 $a=b$ 矣◎

樹莓派、樹莓派之學習、樹莓派之教育

GoPiGo 小汽車︰格點圖像算術《投影幾何》【五‧線性代數】《導引八》觀察者《變換‧D 》

2017-09-29 懸鉤子

近年來根據美國方言學會 ADS American Dialect Society Stephen Goranson 的研究，一八七七年的一次工程學會會議上 Alfred Holt 的報告上提出︰

It is found that anything that can go wrong at sea generally does go wrong sooner or later, so it is not to be wondered that owners prefer the safe to the scientific …. Sufficient stress can hardly be laid on the advantages of simplicity. The human factor cannot be safely neglected in planning machinery. If attention is to be obtained, the engine must be such that the engineer will be disposed to attend to it.

，由此看來，說不定是 De Morgan 錯記『莫非』Murphy 的了︰

Mathematician Augustus De Morgan wrote on June 23, 1866: “The first experiment already illustrates a truth of the theory, well confirmed by practice, what-ever can happen will happen if we make trials enough.” In later publications “whatever can happen will happen” occasionally is termed “Murphy’s law,” which raises the possibility — if something went wrong — that “Murphy” is “De Morgan” misremembered (an option, among others, raised by Goranson on American Dialect Society list).

這個大名鼎鼎的『莫非定律』說︰

凡事要可能出錯，必定會出錯！！

從科學和演算法方面來講，它和『最糟情境』worst-case scenario 分析同義，然而就文化層面而言，它代表著一種反諷式的幽默，也許能排解日常生活中諸多遭遇的不滿。

那人們該如何設想『莫非之機率』的呢？一九零九年時法國數學家埃米爾‧博雷爾 Félix-Édouard-Justin-Émile Borel 在一本機率書中介紹了一個『打字猴子』的概念︰

讓一隻猴子在打字機上隨機地按鍵，當這樣作的時間趨近無窮時，似乎必然能夠打出任何指定的文本，比如說整套莎士比亞的著作。

他用這隻猴子來比喻一種能夠產生無窮的隨機語詞字串之『抽象設備』。這個『無限猴子定理』理論是說︰把一個很大但有限的數看成無限的推論是錯誤的。猴子能否完全無誤的敲打出一部莎士比亞的哈姆雷特，縱使它發生的機率非常之小然而絕非是零！就像戰國時期的列禦寇在《列子‧湯問》中寫到︰

愚公移山

太行、王屋二山，方七百里，高萬仞，本在冀州之南，河陽之北。

北山愚公者，年且九十，面山而居。懲山北之塞，出入之迂也。聚室而謀曰：「吾與汝畢力平險，指通豫南，達於漢陰，可乎？」雜然相許。其妻獻疑曰：「以君之力，曾不能損魁父之丘，如太行、王屋何？且焉置土石？」雜曰：「投諸渤海之尾，隱土之北。」遂率子孫荷擔者三夫，叩石墾壤，箕畚運於渤海之尾。鄰人京城氏之孀妻有遺男，始齔，跳往助之。寒暑易節，始一反焉。

河曲智叟笑而止之曰：「甚矣，汝之不惠。以殘年餘力，曾不能毀山之一毛，其如土石何？」北山愚公長息曰：「汝心之固，固不可徹，曾不若孀妻弱子。雖我之死，有子存焉；子又生孫，孫又生子；子又有子，子又有孫；子子孫孫無窮匱也，而山不加增，何苦而不平？」河曲智叟亡以應。

操蛇之神聞之，懼其不已也，告之於帝。帝感其誠，命誇娥氏二子負二山，一厝朔東，一厝雍南。自此，冀之南，漢之陰，無隴斷焉。

即使無需神助，應該也是移的了山的吧！！

─── 《布林代數》

該怎麼看待有兩個部份 $x,y$ ，又似一個整體 $x+iy$ 的複數呢？莫非歐拉的上帝遣之來 $e^{i \pi} + 1 = 0$ ，不得不全純乎？？

全純函數（holomorphic function）是複分析研究的中心對象；它們是定義在複平面C的開子集上的，在複平面C中取值的，在每點上皆複可微的函數。這是比實可微強得多的條件，暗示著此函數無窮可微並可以用泰勒級數來描述。

解析函數（analytic function）一詞經常可以和「全純函數」互相交換使用，雖然前者有幾個其他含義。

全純函數有時稱為正則函數。在整個複平面上都全純的函數稱為整函數（entire function）。「在一點a全純」不僅表示在a可微，而且表示在某個中心為a的複平面的開鄰域上可微。雙全純（biholomorphic）表示一個有全純逆函數的全純函數。

Terminology

The word “holomorphic” was introduced by two of Cauchy‘s students, Briot (1817–1882) and Bouquet (1819–1895), and derives from the Greek ὅλος (holos) meaning “entire”, and μορφή (morphē) meaning “form” or “appearance”.^[9]

Today, the term “holomorphic function” is sometimes preferred to “analytic function”, as the latter is a more general concept. This is also because an important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow directly from the definitions. The term “analytic” is however also in wide use.

故而必得條條道路皆通羅馬也耶！

Cauchy–Riemann equations

Indeed, following Rudin (1966), suppose f is a complex function defined in an open set Ω ⊂ C. Then, writing z = x + iy for every z ∈ Ω, one can also regard Ω as an open subset of R², and f as a function of two real variables x and y, which maps Ω ⊂ R² to C. We consider the Cauchy–Riemann equations at z = z₀. So assume f is differentiable at z₀, as a function of two real variables from Ω to C. This is equivalent to the existence of the following linear approximation

f(z_{0}+\Delta z)-f(z_{0})=f_{x}\,\Delta x+f_{y}\,\Delta y+\eta (\Delta z)\,\Delta z\,

where z = x + iy and η(Δz) → 0 as Δz → 0. Since $\Delta z+\Delta {\bar {z}}=2\,\Delta x$ and $\Delta z-\Delta {\bar {z}}=2i\,\Delta y$ , the above can be re-written as

\Delta f(z_{0})={\frac {f_{x}-if_{y}}{2}}\,\Delta z+{\frac {f_{x}+if_{y}}{2}}\,\Delta {\bar {z}}+\eta (\Delta z)\,\Delta z\,

Defining the two Wirtinger derivatives as

{\frac {\partial }{\partial z}}={\frac {1}{2}}{\Bigl (}{\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}{\Bigr )},\;\;\;{\frac {\partial }{\partial {\bar {z}}}}={\frac {1}{2}}{\Bigl (}{\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}{\Bigr )},

in the limit $\Delta z\rightarrow 0,\Delta {\bar {z}}\rightarrow 0$ the above equality can be written as

\left.{\frac {df}{dz}}\right|_{z=z_{0}}=\left.{\frac {\partial f}{\partial z}}\right|_{z=z_{0}}+\left.{\frac {\partial f}{\partial {\bar {z}}}}\right|_{z=z_{0}}\cdot {\frac {\bar {dz}}{dz}}+\eta (\Delta z),\;\;\;\;(\Delta z\neq 0).

For real values of z, we have ${\bar {dz}}/dz=1$ and for purely imaginary z we have ${\bar {dz}}/dz=-1$ . Similarly, when approaching z₀ from different directions in the complex plane, the value of ${\bar {dz}}/dz$ is different. But since for complex differentiability the derivative should be the same, approaching from any direction, hence f is complex differentiable at z₀ if and only if $(\partial f/\partial {\bar {z}})=0$ at $z=z_{0}$ . But this is exactly the Cauchy–Riemann equations, thus f is differentiable at z₀ if and only if the Cauchy–Riemann equations hold at z₀.

Independence of the complex conjugate

The above proof suggests another interpretation of the Cauchy–Riemann equations. The complex conjugate of z, denoted ${\bar {z}}$ , is defined by

{\overline {x+iy}}:=x-iy

for real x and y. The Cauchy–Riemann equations can then be written as a single equation

(3)

{\dfrac {\partial f}{\partial {\bar {z}}}}=0

by using the Wirtinger derivative with respect to the conjugate variable. In this form, the Cauchy–Riemann equations can be interpreted as the statement that f is independent of the variable ${\bar {z}}$ . As such, we can view analytic functions as true functions of one complex variable as opposed to complex functions of two real variables.

Physical interpretation

A standard physical interpretation of the Cauchy–Riemann equations going back to Riemann’s work on function theory (see Klein 1893) is that u represents a velocity potential of an incompressible steady fluid flow in the plane, and v is its stream function. Suppose that the pair of (twice continuously differentiable) functions $u,v$ satisfies the Cauchy–Riemann equations. We will take u to be a velocity potential, meaning that we imagine a flow of fluid in the plane such that the velocity vector of the fluid at each point of the plane is equal to the gradient of u, defined by

\nabla u={\frac {\partial u}{\partial x}}\mathbf {i} +{\frac {\partial u}{\partial y}}\mathbf {j}

By differentiating the Cauchy–Riemann equations a second time, one shows that u solves Laplace’s equation:

{\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0.

That is, u is a harmonic function. This means that the divergence of the gradient is zero, and so the fluid is incompressible.

The function v also satisfies the Laplace equation, by a similar analysis. Also, the Cauchy–Riemann equations imply that the dot product $\nabla u\cdot \nabla v=0$ . This implies that the gradient of u must point along the $v={\text{const}}$ curves; so these are the streamlines of the flow. The $u={\text{const}}$ curves are the equipotential curves of the flow.

A holomorphic function can therefore be visualized by plotting the two families of level curves $u={\text{const}}$ and $v={\text{const}}$ . Near points where the gradient of u (or, equivalently, v) is not zero, these families form an orthogonal family of curves. At the points where $\nabla u=0$ , the stationary points of the flow, the equipotential curves of $u={\text{const}}$ intersect. The streamlines also intersect at the same point, bisecting the angles formed by the equipotential curves.

Contour plot of a pair u and v satisfying the Cauchy–Riemann equations. Streamlines (v = const, red) are perpendicular to equipotentials (u = const, blue). The point (0,0) is a stationary point of the potential flow, with six streamlines meeting, and six equipotentials also meeting and bisecting the angles formed by the streamlines.

縱知平面上任一全純函數 $f(z)$ 都可滿足拉普拉斯方程式

Laplace’s equation

In mathematics, Laplace’s equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:

\nabla ^{2}\varphi =0\qquad {\mbox{or}}\qquad \Delta \varphi =0

where ∆ = ∇² is the Laplace operator^[1] and $\varphi$ is a scalar function.

Laplace’s equation and Poisson’s equation are the simplest examples of elliptic partial differential equations. The general theory of solutions to Laplace’s equation is known as potential theory. The solutions of Laplace’s equation are the harmonic functions, which are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. In the study of heat conduction, the Laplace equation is the steady-state heat equation.

卻是如何符合邊界條件實難解矣！！

僅假借線性算子

Operator (mathematics)

In mathematics, an operator is generally a mapping that acts on the elements of a space to produce other elements of the same space. The most common operators are linear maps, which act on vector spaces. However, when using “linear operator” instead of “linear map”, mathematicians often mean actions on vector spaces of functions, which also preserve other properties, such as continuity. For example, differentiation and indefinite integration are linear operators; operators that are built from them are called differential operators, integral operators or integro-differential operators.

Operator is also used for denoting the symbol of a mathematical operation. This is related with the meaning of “operator” in computer programming, see operator (computer programming).

引用全微分概念

全微分（英語：total derivative）是微積分學的一個概念，指多元函數的全增量 $\Delta z$ 的線性主部，記為 $\operatorname dz$ 。例如，對於二元函數 $z=f(x,\ y)$ ，設f在點 $P_{0}(x_{0},\ y_{0})$ 的某個鄰域內有定義， $P(x_{0}+\Delta x,\ y_{0}+\Delta y)$ 為該鄰域內的任意一點，則該函數在點 $P_{0}(x_{0},\ y_{0})$ 的全增量可表示為

\Delta z=A\Delta x+B\Delta y+o(\rho )

，

其中 $A$ ， $B$ 僅與 $x$ ， $y$ 有關，而與 $\Delta x$ ， $\Delta y$ 無關， $\rho ={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}}$ 。若 $o(\rho )$ 是當 $\rho \rightarrow 0$ 時的高階無窮小，則稱此函數 $z=f(x,\ y)$ 在點 $(x,\ y)$ 可微分， $A\Delta x+B\Delta y$ 即為函數 $z=f(x,\ y)$ 在點 $P_{0}(x_{0},\ y_{0})$ 的全微分，記作

\operatorname dz|_{{x=x_{0},\ y=y_{0}}}=A\Delta x+B\Delta y

或 $\operatorname df(x_{0},y_{0})=A\Delta x+B\Delta y$ 。

略作些分疏於此。

因任意 $h(x,y)$ 可以看成 $h(\frac{z + \bar{z}}{2}, \frac{z - \bar{z}}{2i}) = h(z,\bar{z})$ 的形式，可設想 $\frac {\partial }{\partial z}$ 、 $\frac {\partial }{\partial \bar{z}}$ 都能表示為 $a \cdot \frac {\partial }{\partial x} + b \cdot \frac {\partial }{\partial y}$ 算符也。此處係數 $a,b$ 之定，只需援用 $z, \overline{z}$ 之獨立性 $\frac{\partial \bar z}{\partial z} = 0$ 、 $\frac{\partial z}{\partial z} = 1$ ； $\frac{\partial z}{\partial \bar{z}} = 0$ 、 $\frac{\partial \bar z}{\partial \bar{z}} = 1$ 就得矣◎

樹莓派、樹莓派之學習、樹莓派之教育

GoPiGo 小汽車︰格點圖像算術《投影幾何》【五‧線性代數】《導引八》觀察者《變換‧C 》

2017-09-28 懸鉤子

由於現今尚未有『未來』之『名目』，即使 W!o+ 願說想說，怕也是『說不得』的吧！！

Horizontal eye movement in reading. Left-to-right movement may be seen as “upstairs”, and right-to-left saccades 【掃視】are clear.

A diagram demonstrating the acuity 【清晰度】of foveal vision in reading

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

東西方人看待『事物』的『視線』 line-of-sight 常取不同的『方向』，如今『中文橫寫』已經成為主流，但是我們並不知道『去經存緯』的改變，是否會產生什麼樣的『讀‧解』變化？畢竟『閱讀中的眼睛』 Eye movement in reading 之『運動』，可是與『認知』活動息息相關的ㄚ！

要是【明月當空】 W!o 是指那千古一遇之『曌』── 一代女皇武則天獨創之字 ──，意味著『日月明空曌』，那麼

【朋比翼鳥】是『鵬』；

【社神廢祀】為『坤】；

這可是『鯤鵬巨變』的呀！一旦發生就『回不去』了！！難怪得斷之以

【有始無終】。

那其它的『辭意』呢？或『有意』，或『無意』，或『自生其意』的耶？？

『會‧通』的講︰ W!o 說︰效法

《莊子‧逍遙游》

北冥有魚，其名為鯤。鯤之大，不知其幾千里也。化而為鳥，其名為鵬。鵬之背，不知其幾千里也；怒而飛，其翼若垂天之雲。是鳥也，海運則將徙於南冥。南冥者，天池也。

齊諧者，志怪者也。諧之言曰：「鵬之徙於南冥也，水擊三千里，摶扶搖而上者九萬里，去以六月息者也。」野馬也，塵埃也，生物之以息相吹也。天之蒼蒼，其正色邪？其遠而無所至極邪？其視下也，亦若是則已矣。

且夫水之積也不厚，則其負大舟也無力。覆杯水於坳堂之上，則芥為之舟；置杯焉則膠，水淺而舟大也。風之積也不厚，則其負大翼也無力。故九萬里，則風斯在下矣，而後乃今培風，背負青天而莫之夭閼者，而後乃今將圖南。

蜩與學鳩笑之曰：「我決起而飛，搶榆枋，時則不至而控於地而已矣，奚以之九萬里而南為？」適莽蒼者，三飡而反，腹猶果然；適百里者，宿舂糧；適千里者，三月聚糧。之二蟲又何知？

小知不及大知，小年不及大年。奚以知其然也？朝菌不知晦朔，蟪蛄不知春秋，此小年也。楚之南有冥靈者，以五百歲為春，五百歲為秋；上古有大椿者，以八千歲為春，八千歲為秋。此大年也。而彭祖乃今以久特聞，眾人匹之，不亦悲乎！

……

，愛上『派生』 Python ，自然創生『 M♪o 踢呦ㄊㄜˋ』！！！

─── 《怎樣詮釋 W!o+ 之信息？？下》

視線在其方向上無遠弗屆，故難論距離矣。若以共點言，可否保角耶？

共形映射

數學上，共形變換（英語：Conformal map）或稱保角變換，來自於流體力學和幾何學的概念，是一個保持角度不變的映射。

更正式的說，一個映射

w=f(z)\,

稱為在 $z_{0}\,$ 共形(或者保角)，如果它保持穿過 $z_{0}\,$ 的曲線間的定向角度，以及它們的取向也就是說方向。共形變換保持了角度以及無窮小物體的形狀，但是不一定保持它們的尺寸。

共形的性質可以用坐標變換的導數矩陣雅可比矩陣的術語來表述。如果變換的雅可比矩陣處處都是一個純量乘以一個旋轉矩陣，則變換是共形的。

直角網格(頂部)和它在共形映射 f 下的像(底部)。可看出 f 把以 90°相交的成對的線映射成仍以 90°相交的成對曲線。

設使角都不可保，那麼怎有透視，交比可論乎？？此誠複變分析中之美妙也！

Complex analysis

“Complex analytic” redirects here. For the class of functions often called “complex analytic”, see Holomorphic function.

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

As a differentiable function of a complex variable is equal to the sum of its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions).

Plot of the function $f (x) = (x 2 - 1)(x + 2 - i) 2$ $/ (x 2 + 2 - 2 i)$ . The hue represents the function argument, while the brightness represents the magnitude.

History

Complex analysis is one of the classical branches in mathematics, with roots in the 19th century and just prior. Important mathematicians associated with complex analysis include Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number theory. In modern times, it has become very popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions. Another important application of complex analysis is in string theory which studies conformal invariants in quantum field theory.

The Mandelbrot set, a fractal.

Holomorphic functions

Holomorphic functions are complex functions, defined on an open subset of the complex plane, that are differentiable. In the context of complex analysis, the derivative of $f$ at $z_{0}$ is defined to be

$f'(z_{0})=\lim _{z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}},z\in \mathbb {C}$ .

Although superficially similar to the derivative of a real function, the behavior of complex derivatives and differentiable functions is significantly different. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach $z_{0}$ in the complex plane. Consequently, complex differentiability has much stronger consequences than usual (real) differentiability. For instance, holomorphic functions are infinitely differentiable, whereas most real differentiable functions are not. Furthermore, all holomorphic functions satisfy the stronger condition of analyticity, meaning that the function is, at every point in its domain, locally given by a convergent power series. In essence, this means that holomorphic functions can be approximated arbitrarily well by polynomials in some neighborhood of every point in its domain. This stands in sharp contrast to differentiable real functions; even infinitely differentiable real functions can be nowhere analytic.

Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, extended appropriately to complex arguments as functions $\mathbb {C} \to \mathbb {C}$ , are holomorphic over the entire complex plane, making them entire functions, while rational functions $p/q$ , where p and q are polynomials, are holomorphic on domains that exclude points where q is zero. Such functions that are holomorphic everywhere except a set of isolated points are known as meromorphic functions. On the other hand, the functions $z\mapsto \Re (z)$ , $z\mapsto |z|$ , and $z\mapsto {\bar {z}}$ are not holomorphic anywhere on the complex plane, as can be shown by their failure to satisfy the Cauchy-Riemann conditions (see below).

An important property that characterizes holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the Cauchy-Riemann conditions. If $f:\mathbb {C} \to \mathbb {C}$ , defined by $f(z)=f(x+iy)=u(x,y)+iv(x,y)$ , where $x,y,u(x,y),v(x,y)\in \mathbb {R}$ , is holomorphic on a region $\Omega$ , then $(\partial f/\partial {\bar {z}})(z_{0})=0$ must hold for all $z_{0}\in \Omega$ . Here, the differential operator $\partial /\partial {\bar {z}}$ is defined as $(1/2)(\partial /\partial x+i\partial /\partial y)$ . In terms of the real and imaginary parts of the function, u and v, this is equivalent to the pair of equations $u_{x}=v_{y}$ and $u_{y}=-v_{x}$ , where the subscripts indicate partial differentiation. However, it is important to note that functions satisfying the Cauchy-Riemann conditions are not necessarily holomorphic, unless additional continuity conditions are met (see Looman-Menchoff Theorem for a discussion).

Holomorphic functions exhibit some remarkable features. For instance, Picard’s theorem asserts that the range of an entire function can only take three possible forms: $\mathbb {C}$ , $\mathbb {C} \setminus \{z_{0}\}$ , or $\{z_{0}\}$ for some $z_{0}\in {\mathbb {C}}$ . In other words, if two distinct complex numbers $z$ and $w$ are not in the range of entire function $f$ , then $f$ is a constant function. Moreover, given a holomorphic function $f$ defined on an open set $U$ , the analytical extension of $f$ to a larger open set $V\supset U$ is unique. Thus, the value of a holomorphic function over a relatively small region in fact determines the value of the function everywhere to which it can be extended.

然既無法長篇大論，只好邀請讀者自己閱讀吧！！

LECTURE NOTES OF WILLIAM CHEN

INTRODUCTION TO COMPLEX ANALYSIS

This set of notes has been organized in such a way to create a single volume suitable for an introduction to some of the basic ideas in complex analysis. The material in Chapters 1 – 11 and 16 were used in various forms between 1981 and 1990 by the author at Imperial College, University of London. Chapters 12 – 15 were added in Sydney in 1996.

To read the notes, click the links below for connection to the appropriate PDF files.

The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.

樹莓派、樹莓派之學習、樹莓派之教育

GoPiGo 小汽車︰格點圖像算術《投影幾何》【五‧線性代數】《導引八》觀察者《變換‧B 》

2017-09-27 懸鉤子

有人說︰

天下『一切』事情，都不過是個『分類』。

『有人』就『有事』，天下何得『无事』？天下能得『無人』乎？

有人想︰

酸葡萄『沒有』想『要』，甜檸檬『有』卻想『不要』；

都是一種『心理』。

有人分︰

有所謂『好』與『壞』？分別著『對』或『錯』，

『是』其所『非』，『非』其所『是』。

有人問︰

『蜜蜂』為何不見了？

只因『益』和『害』！！

─── 《『蜜蜂』為何不見了？》

即便講︰識物辨名始於分門別類。重要的還是門類之界定性徵。若從完備無餘來看 $A \cdot z + B$ 變換形式，僅止於平面上的縮放、旋轉、平移爾，實未得仿射變換之全也。

In the plane

Affine transformations in two real dimensions include:

pure translations,
scaling in a given direction, with respect to a line in another direction (not necessarily perpendicular), combined with translation that is not purely in the direction of scaling; taking “scaling” in a generalized sense it includes the cases that the scale factor is zero (projection) or negative; the latter includes reflection, and combined with translation it includes glide reflection,
rotation combined with a homothety and a translation,
shear mapping combined with a homothety and a translation, or
squeeze mapping combined with a homothety and a translation.

To visualise the general affine transformation of the Euclidean plane, take labelled parallelograms ABCD and A′B′C′D′. Whatever the choices of points, there is an affine transformation T of the plane taking A to A′, and each vertex similarly. Supposing we exclude the degenerate case where ABCD has zero area, there is a unique such affine transformation T. Drawing out a whole grid of parallelograms based on ABCD, the image T(P) of any point P is determined by noting that T(A) = A′, T applied to the line segment AB is A′B′, T applied to the line segment AC is A′C′, and T respects scalar multiples of vectors based at A. [If A, E, F are collinear then the ratio length(AF)/length(AE) is equal to length(A′F′)/length(A′E′).] Geometrically T transforms the grid based on ABCD to that based in A′B′C′D′.

Affine transformations do not respect lengths or angles; they multiply area by a constant factor

area of A′B′C′D′ / area of ABCD.

A given T may either be direct (respect orientation), or indirect (reverse orientation), and this may be determined by its effect on signed areas (as defined, for example, by the cross product of vectors).

A central dilation. The triangles A1B1Z, A1C1Z, and B1C1Z get mapped to A2B2Z, A2C2Z, and B2C2Z, respectively.

Effect of applying various 2D affine transformation matrices on a unit square. Note that the reflection matrices are special cases of the scaling matrix.

然以立論例釋而言足矣哉。何況此本乎莫比烏斯變換

$\left( \begin{array}{cc} z^{'} \\ 1 \end{array} \right) = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \left( \begin{array}{cc} z \\ 1 \end{array} \right)$

，在 $c = 0$ 條件下所得。

恰宜說點分類基本也。

為什麼 $A \cdot z + B$ 是個平移，必得 $A = 1$ 呢？因既為平移，平面上無點能不動。假使 $A \neq 1$ ，那麼 $z = A \cdot z + B$ 有定點 $z = \frac{B}{1-A}$ ，故語意矛盾也。再由複數的極座標表示法 $r \cdot e^{i \theta}$ ，可知 $r=1$ 為旋轉，而 $r$ 就是縮放比的了。

自可解讀定點的重要性︰

Determining the fixed points

The fixed points of the transformation

f(z)={\frac {az+b}{cz+d}}

are obtained by solving the fixed point equation f(γ) = γ. For c ≠ 0, this has two roots obtained by expanding this equation to

c\gamma ^{2}-(a-d)\gamma -b=0\ ,

and applying the quadratic formula. The roots are

\gamma _{{1,2}}={\frac {(a-d)\pm {\sqrt {(a-d)^{2}+4bc}}}{2c}}={\frac {(a-d)\pm {\sqrt {(a+d)^{2}-4(ad-bc)}}}{2c}}.

Note that for parabolic transformations, which satisfy (a+d)² = 4(ad−bc), the fixed points coincide. Note also that the discriminant is

(a-d)^{2}+4cb=(a-d)^{2}+4ad-4=(a+d)^{2}-4=\operatorname {tr}^{2}{\mathfrak {H}}-4.

When c = 0, the quadratic equation degenerates into a linear equation. This corresponds to the situation that one of the fixed points is the point at infinity. When a ≠ d the second fixed point is finite and is given by

\gamma =-{\frac {b}{a-d}}.

In this case the transformation will be a simple transformation composed of translations, rotations, and dilations:

z\mapsto \alpha z+\beta .\,

If c = 0 and a = d, then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation:

z\mapsto z+\beta .

明白投影線會定

$x + \infty = \infty, if \ x \neq \infty$ 的吧◎

樹莓派、樹莓派之學習、樹莓派之教育

GoPiGo 小汽車︰格點圖像算術《投影幾何》【五‧線性代數】《導引八》觀察者《變換‧A 》

2017-09-26 懸鉤子

假使我們只說︰如果 $c = 0$ ，一個莫比烏斯變換

$\left( \begin{array}{cc} z^{'} \\ 1 \end{array} \right) = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \left( \begin{array}{cc} z \\ 1 \end{array} \right)$

就是一種仿射變換

Affine transformation

In geometry, an affine transformation, affine map^[1] or an affinity (from the Latin, affinis, “connected with”) is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence.

If $X$ and $Y$ are affine spaces, then every affine transformation $f\colon X\to Y$ is of the form $x\mapsto Mx+b$ , where $M$ is a linear transformation on $X$ and $b$ is a vector in $Y$ . Unlike a purely linear transformation, an affine map need not preserve the zero point in a linear space. Thus, every linear transformation is affine, but not every affine transformation is linear.

All Euclidean spaces are affine, but there are affine spaces that are non-Euclidean. In affine coordinates, which include Cartesian coordinates in Euclidean spaces, each output coordinate of an affine map is a linear function (in the sense of calculus) of all input coordinates. Another way to deal with affine transformations systematically is to select a point as the origin; then, any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation.

。那麼我們能知道 $c = 0$ 的條件嗎？恐生疑惑乎？？

雖說『眼見為憑』，難保不是『錯覺』呢？

此事因平面國流傳一圖而起︰

卻納悶平面國似有『透鏡』，恍惚是『平行光轉換器』耶！然懷疑

無厚能有透鏡乎？！

─── 摘自《GoPiGo 小汽車︰格點圖像算術《投影幾何》【四‧平面國】《補丁》》

即使望圖聯想！何若解析思辨耶！！

或許『圖鑑』還有一張

表白文字幾行︰

依上圖，三角形 $\Delta {\gamma}_1 {\gamma}_2} z^{'}$ 與 $\Delta {\gamma}_1 {\gamma}_2} z$ 面積比可以兩算︰

‧ 以 $z-{\gamma}_1}$ 和 $z^{'}-{\gamma}_1}$ 為底 $= \frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}}$ ；

‧ 以 $z-{\gamma}_2}$ 和 $z^{'}-{\gamma}_2}$ 為底

$= \frac{|{\gamma}_1 {\gamma}_2| \cdot \sin(\angle \phi)}{|{\gamma}_1 {\gamma}_2| \cdot \sin(\angle \theta)} \cdot \frac{z^{'} - {\gamma}_2}{z - {\gamma}_2}} = k \cdot \frac{z^{'} - {\gamma}_2}{z - {\gamma}_2}}$

$\therefore k = \frac{ \sin(\angle \phi)}{ \sin(\angle \theta)}$

既『角不變』，『角比』能變嗎★

故而 $\frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}} = k \cdot \frac{z^{'} - {\gamma}_2}{z - {\gamma}_2}}$ ，

且將兩邊乘上 $\frac{z - {\gamma}_1}{z^{'} - {\gamma}_2}$

果非所求的嘛☆

─── 摘自《GoPiGo 小汽車︰格點圖像算術《投影幾何》【五‧線性代數】《導引七‧變換組合 VII‧B 》》

當逢 ${\gamma}_2 \to \infty$ ，光從無窮遠處來，平行同光照

$\lim \limits_{{\gamma}_2 \to \infty} \left[ \frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}} = k \cdot \frac{z^{'} - {\gamma}_2}{z - {\gamma}_2}} \right] \ \Rightarrow \ \frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}} = k$ ；

反思 ${\gamma}_1 \to \infty$ ，聚焦於無限，所見果非似焉

$\lim \limits_{{\gamma}_1 \to \infty} \left[ \frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}} = k \cdot \frac{z^{'} - {\gamma}_2}{z - {\gamma}_2}} \right] \ \Rightarrow \ k \cdot \frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}} = 1$ 。

再以投影幾何之公設︰

相異兩線必交於一點。

故爾平行線不能例外也。所以代數計算求其意義一致性，亦無法分上下左右之 $\infty$ 矣。

Line extended by a point at infinity

The projective line may be identified with the line K extended by a point at infinity. More precisely, the line K may be identified with the subset of P¹(K) given by