## 转载：斯蒂芬·沃尔夫勒姆：宇宙的本质是计算

http://songshuhui.net/archives/91355

1. “人口过少”：任何活细胞如果活邻居少于2个，则死掉。
2. “正常”：任何活细胞如果活邻居为2个或3个，则继续活。
3. “人口过多”：任何活细胞如果活邻居大于3个，则死掉。
4. “繁殖”：任何死细胞如果活邻居正好是3个，则活过来。

【”脉冲星”：它的周期为3，看起来像一颗周期爆发的星星。】

【“滑翔者”：每4个回合“它”会向右下角走一格。虽然细胞早就是不同的细胞了，但它能保持原本的形态。】

【“轻量级飞船”：它的周期是4，每2个回合会向右边走一格。】

【“滑翔者枪”：它会不停地释放出一个又一个滑翔者。】

【“繁殖者”：它会向右行进，留下一个接一个的“滑翔者枪”。动图最后一帧定格时用三种颜色区分了繁殖者本体、滑翔者枪和它们打出来的滑翔者。】

【Rule 30的全部规则，以及从一个黑点开始的迭代结果。每一次新的迭代就是新的一行，而每一个细胞的死活由且仅由它自己和它左右两侧的细胞在上一行的状态决定。之所以是30，因为按顺序排列的二进制数00011110对应的十进制就是30。】

【250次迭代后的Rule 30。左边缘看起来还有一定规律，但大部分区域看起来近乎是随机的。】

【织锦芋螺（Conus textile）的花纹神似Rule 30的结果。】

## “我们能为宇宙建立一个完整的模型吗？我的工作假说是‘能’。”

“自由意志”这个词包括很多层面，如果我们能预测我们将会做出什么事情来，那可以说我们没有自由意志。比如说，看一只蛾子反复地撞击玻璃窗，试图飞出去，我们会说它看起来没有自由意志，似乎是决定性的反射机制。但如果我们看到一个人在做一件非常复杂的行为，我们会说他看起来好像在作出选择，因为我们无法预测他将会怎么做——在这个意义上，我们拥有自由意志。

## “我对解决别人解决过的问题从来都没有兴趣”

【Ent与沃尔夫勒姆先生的合影。】

## Distances and Riemann matrices from Divergence

【Information geometry and it’s applications】

I am reading Shun-ichi Amari’s book， Information geometry and it’s applications. This sequences of blogs will be my thought and understanding derived from Amari’s book.

One central concept in information geometry is Divergence

Definition 1.1: $D[P:Q]$ is called a divergence, when it satisfies the following criteria:

1. $D[P:Q]\leq 0$
2. $D[P:Q] = 0$ if and only if $P=Q$
3. $D:M\times M\to \mathbb{R}_+$ is a smooth function.

One direct corollary of (3) is that

Corollary: If $D[P:Q]$ is a divergence, and $\xi$ is a coordinate system around $P$. Then

• $D[\xi(P),\xi(P)+d\xi] =\frac{1}{2}\sum\limits_{i,j}gr_{ij}d\xi_id\xi_j + O(|\xi|^3)$
• $D[\xi(P)+d\xi,\xi(P)] =\frac{1}{2}\sum\limits_{i,j}gl_{ij}d\xi_id\xi_j + O(|\xi|^3)$

in which $gr_{ij}, gl_{ij}$ are positive definite symmetric matrices. Therefore the average $g=\frac{1}{2} (gr + gl)$ is also a positive definite matrix.

Each of these three tensors $gl, gr, g$ defines a Riemannian metric on manifold $M$.

• $gl_{ij} = \partial_i\partial_jD[\xi,\xi']$
• $gr_{ij} = \partial_i'\partial_j'D[\xi,\xi']$
• $g_{ij} = -\frac{1}{2}\partial_i\partial_j'D[\xi,\xi']$

The last equation is obtained from $f(\xi)=D[\xi,\xi]=0$, and take second derivative of $\partial_i\partial_j f = 0$.

There are two major differences between Divergence and Rimannian matrix

• symmetric: $D[P:Q]\neq D[Q:P]$
• Local to global:
• Riemannian metric is defined point-wisely, and the distance function is obtained by integration of Riemannian matric.
• Divergence is defined globally on $M\times M:\to \mathbb{R}_+$, the corresponding Riemannian matrices $gl, gr, g$ only describe the local behavior of $D$ around the diagnal of $M\times M$.

Example:

Consider a Bregman Divergence defined by coordinate system $\xi$ and convex function $\psi$. Then it is not hard to prove

$gl = gr = g = \partial_i\partial_j\psi$

Because $\psi$ is a globally defined function, we can say that this Riemannian matrix has a global potential. In a later blog of dual flat structure, we are going to see more details of this Divergence.

## 勒让德变换的几层理解

### 勒让德变换定义:

#### 第一层：函数上境图的支撑超平面的截距

$\{(k,b): z=k^Tx + b \text{ is a supporting plane}\},$

$f(x) = \sup\limits_k\{k^Tx+b\} = \sup\limits_k\{k^Tx-f^*(k)\} = f^{**}(x)$

$\frac{\partial}{\partial x}f = y, \frac{\partial}{\partial y}f^* =x$

$\frac{\partial^2}{\partial x^2}f = \frac{\partial}{\partial x}y, \frac{\partial^2}{\partial y^2}f^* =\frac{\partial}{\partial y}x$

$\frac{\partial^2}{\partial x^2}f \frac{\partial^2}{\partial y^2}f^* = 1$

## 线性代数与射影几何的一些结论

https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem

https://terrytao.wordpress.com/2011/07/15/pappuss-theorem-and-elliptic-curves/

## 赌场里的老贝和老贫

(1/5, 1/5, 1/5, 2/15, 2/15, 2/15)*0.1 + (1/6, 1/6, 1/6, 1/6, 1/6, 1/6)*0.9

(0.17,0.17,0.17,0.163,0.163,0.163)

theta1 = (1/5, 1/5, 1/5, 2/15, 2/15, 2/15)
theta2 = (1/6, 1/6, 1/6, 1/6, 1/6, 1/6)

P(theta1|X) ~ P(X|theta1)P(theta1) = (1/5)^60*(2/15)^40*0.1 = 1.146e-77
P(theta2|X) ~ P(X|theta2)P(theta2) = (1/6)^100 = 1.531e-78

P(theta1|X) = 0.454
P(theta2|X) = 0.545

(1/5, 1/5, 1/5, 2/15, 2/15, 2/15) * 0.454 + (1/6, 1/6, 1/6, 1/6, 1/6, 1/6)* 0.545

(0.18, 0.18, 0.18, 0.15, 0.15, 0.15)

## 否极泰来

【须菩提。于意云何。如来有肉眼不。如是。世尊。如来有肉眼。须菩提。于意云何。如来有天眼不。如是。世尊。如来有天眼。须菩提。于意云何。如来有慧眼不。如是。世尊。如来有慧眼。须菩提。于意云何。如来有法眼不。如是。世尊。如来有法眼。须菩提。于意云何。如来有佛眼不。如是。世尊。如来有佛眼。须菩提。于意云何。如恒河中所有沙。佛说是沙不。如是。世尊。如来说是沙。须菩提。于意云何。如一恒河中所有沙。有如是沙等恒河。是诸恒河所有沙数佛世界。如是宁为多不。甚多。世尊。佛告须菩提。尔所国土中。所有众生。若干种心。如来悉知。何以故。如来说诸心。皆为非心。是名为心。所以者何。须菩提。过去心不可得。现在心不可得。未来心不可得。】