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‧E 》

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

{\displaystyle 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. {\displaystyle M\,dx+N\,dy=0}, so we get {\displaystyle \mu M\,dx+\mu N\,dy=0}. We will require \mu to satisfy  {\displaystyle {\frac {\partial \mu M}{\partial y}}={\frac {\partial \mu N}{\partial x}}}. We get {\displaystyle {\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 {\displaystyle 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 {\displaystyle \mu (x,y)=\mu (x)} or  {\displaystyle \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 {\displaystyle \mu (y)=e^{-\int {{\frac {M_{y}-N_{x}}{M}}\,dy}}} or  {\displaystyle \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 = (vx, vy, vz), each point p = (px, py, pz) 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 (vx = vy = vz). If all except one of the scale factors are equal to 1, we have directional scaling.

In the case where vx = vy = vz = 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◎