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 》

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

{\displaystyle 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 {\displaystyle \Delta z+\Delta {\bar {z}}=2\,\Delta x} and  {\displaystyle \Delta z-\Delta {\bar {z}}=2i\,\Delta y}, the above can be re-written as

{\displaystyle \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  {\displaystyle v={\text{const}}} curves; so these are the streamlines of the flow. The  {\displaystyle 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  {\displaystyle u={\text{const}}} and  {\displaystyle 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  {\displaystyle 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◎