OpenLLM-France/wikimedia huggingface dataset:
# Cellular Automata/Partial Differential Equations

We traditionally describe phenomena in differential equations, but we would like to simulate and eventually predict their behaviour using numerical methods. This can be done using a traditional top-down approach or a less common bottom up approach.
An inverted method would be to construct from bottom up a cellular automaton, with the same properties as the observed phenomenon differential equation.

## The top down approach

The transformation of a differential equation into a discrete system that can be simulated on a computer is called the top-down approach. Let us try a simple example. One dimensional diffusion equation
    ${\frac {\partial C}{\partial t}}=\kappa {\frac {\partial ^{2}C}{\partial x^{2}}}$
is discretized forward in time and symmetric in space
    $c_{x}^{t}=c_{x}^{t-1}+k\left(c_{x+1}^{t-1}-2c_{x}^{t-1}+c_{x-1}^{t-1}\right)=c_{x}^{t}=k\,c_{x+1}^{t-1}+\left(1-2k\right)c_{x}^{t-1}+k\,c_{x-1}^{t-1}$
with $k:=\kappa \,{\Delta t}\,({\Delta x})^{-2}$. The produced discrete equation can be seen as a cellular automaton local transition function
    $c_{x}^{t}=f\left(c_{x+1}^{t-1}c_{x}^{t-1}c_{x-1}^{t-1}\right)$
The time and space discretization is the same for both systems but there is a crucial difference in the discretization of variables. On a common computer a variable is represented using 64-bit floating point numbers whereas a cellular automaton has variables of 2 to 8 bits.
As numerical approximation errors is an important problem using floating point numbers and would be unbearable using only a few bits. We see that there is no direct method to for transforming differential equations into cellular automata, a different approach must be used.

## The bottom-up approach

This approach begins with selecting a discrete system that inherently possesses some basic properties needed for a simplified model of the original problem. An example of such basic properties is reversibility and preservation of quantities as mass, momentum and energy. The model is then modified and trimmed till it is sufficiently close to the original problem.
The basic model is usually as simple as possible. If cellular automata are used for the basic model, their neighborhood

## References

* Joe D. Hoffman, Numerical Methods for Engineers and Scientists
# Cellular Automata/Global Dynamics
## Attractor Basin

When discussing global dynamics of cellular automata the word state is used to describe configurations.

### Graphic conventions

The following diagrams/captions are based on the original figure State-space and basins of attraction on the DDLab website. Sub-trees, basins of attraction, and the entire basin of attraction field (for cellular automata, random Boolean networks, and discrete dynamic networks in general) can be computed and drawn with the DDLab software.
The state space is made of all available states of a cellular automaton. It's size is $S=v^{n}$ where $n$ is the size of the (finite) lattice, and $v$ is the value-range or alphabet, $v=2$ for binary systems. $B$ is one of the states.
The trajectory $A\rightarrow B\rightarrow C$ is defined by the global transition function. State $A$ is a predecessor and state $C$ is the successor of state $B$.
If the global transition function is not injective, than states may have more than one predecessor (preimage). The in-degree of the state $B$ is the number of its predecessors. A state may have no predecessors, than it is called garden of Eden.
A trajectory may encounter a state occurred previously (it must if the lattice is finite), thus it has entered an attraction cycle. A point attraction cycle has the length of a single cell.
By recursively searching predecessors from a root state on the attraction cycle (the predecessor on the cycle is excluded) till all garden of Eden states are reached a transient tree is constructed. A subtree can be constructed from an arbitrary state on the tree, the root of the subtree.
By adding transient trees to each state on the attraction cycle, a basin of attraction is constructed. Some attraction cycles may have no transient trees, this is especially true for reversible cellular automata.
By grouping all the states in the state space into respective basins of attraction, a basin of attraction field is constructed.

## Injectivity, surjectivity and bijectivity

The CA rule is reversible, if the global transition function $F$ is bijective (both injective and surjective).

### Injectivity of the global transition function

$F$ is injective, there is at most one preimage for any CA configuration.
$F:X\mapsto Y\qquad \forall x_{1},x_{2}\in \Omega \;[x_{1}\neq x_{2}\Rightarrow F(x_{1})\neq F(x_{2})]$
So there are no configurations with more than one predecessor.

### Surjectivity of the global transition function

$F$ is injective, there is at least one preimage for any CA configuration.
$F:X\mapsto Y\qquad \forall y\in \Omega \;\exists x[y=F(x)]$
So there are no configurations without a predecessor (so called Garden of Eden).
It is easy to determine the existence of garden of Eden states for one dimensional CA using De Bruijn Diagrams.

### Bijectivity of the global transition function

$F$ is bijective if it is both injective and surjective. For any configuration there is exactly one predecessor.
# Cellular Automata/Information Dynamics
## Information speed

    $v={\frac {\Delta x}{\Delta t}}$
It is defined as the number

## Speed of light

Is the maximum speed information can travel inside CA.It depends on the size and shape of the neighborhood and it may depend on the direction.
For asymmetric 1D CA (left and right direction)
    $c_{L}=k_{0}$ $c_{R}=k-k_{0}-1$
For symmetric 1D CA (radious $r$)
    $c=r$

## References

* Kolmogorov Complexity
* Shanon communication theory
* Hidenosuke Nishio , Takashi Saito, Information dynamics of cellular automata I: an algebraic study, Fundamenta Informaticae, v.58 n.3-4, p.399-420, December 2003
* Hidenosuke Nishio, Takashi Saito, Information Dynamics of Cellular Automata II: Completeness, Degeneracy and Entropy, September 4, 2002
* Hideyuki Takahashi, The Maximum Invariant Set of an Automaton System, Information and Control 32, 307-354 (1976)
* Hideyuki Takahashi, Undecidable Questions about the Maximum Invariant Set, Information and Control 33, 1-34 (1977)
* Hideyuki Takahashi, Information Transmission in One-Dimensional Cellular Space and the Maximum Invariant Set, Information and Control 33, 35-55 (1977)
* E. Banks. Information Processing and Transmission in Cellular Automata. Ph.D. Thesis, MIT (1971).
# Cellular Automata/Information and Reversibility
## References

Kenichi Morita, Reversible simulation
* Kenichi Morita and Masateru Harao, Computation universality of one-dimensional reversible (injective) cellular automata, Trans. IEICE Japan, E72, 758-762, 1989. PDF
* Kenichi Morita, Computation-universality of one-dimensional one-way reversible cellular automata, Information Processing Letters, 42, 325-329, 1992.
* Kenichi Morita, Reversible simulation of one-dimensional irreversible cellular automata, Theoretical Computer Science, 148, 157-163, 1995 PS
* Tommaso Toffoli, "Computation and construction universality of reversible cellular automata", Journal of Computer and System Sciences, 15 (1977), 213-231.
* Reversible Cellular Automata (Harold V. McIntosh)
* Reversible Cellular Automata (NSK)
* Global Reversibility of Cellular Automata (MathPages)
# Chronicles of Narnia/Places/Narnia
## General Overview

Narnia is the land in which our story actually takes place. It is a magical land, occupied by the creatures of our legends and by talking animals.

## Extended Description

In Narnia's creation story, The Magician's Nephew, we learn that Narnia was created largely by an act of will by Aslan, who is thereafter effectively the God of Narnia. It is at that time that most of the talking animals are created, as are the mythical creatures, such as Fauns, Dryads, and flying horses. At the same time, as it was a person from our world who sang the song that created Narnia, dominion over that world was given to the Sons of Adam and Daughters of Eve.
If one reads the stories in the order in which they were published, one can receive the impression that the entire world is Narnia, at least through the first few books. We do hear that there are fierce Giants in the northern ranges of the land, who must be vanquished when they venture into Narnia, but one does not gather that these northern lands are outside of Narnia until the fourth book, The Silver Chair, when we actually journey through these lands. And it is in the fifth-published book, The Horse and His Boy, that we first learn that there is an entire land, Calormen, separated from Narnia by the Great Desert, and that there is an outlying kingdom, Archenland, standing guard against attack from that quarter. Narnia itself seems to be surrounded by the mountains of the Northern March, Lantern Waste to the west, the mountain range in which Archenland was founded to the south, and the ocean to the east. The talking animals and mythical creatures seem to be restricted to this area, as is the belief in the godhood of Aslan.
In The Magician's Nephew, the entire world is created, though our story involves little other than Narnia; and in The Last Battle, the entire world, including the Northern Marches and Calormen, is ended.
# Chrono Cross/Combos
## Combos

Like Chrono Trigger, Chrono Cross has many different combos. Combos involving two characters are called Double Techs. Combos involving three characters are called triple techs.

### Double Techs

| Double Tech | Character and Tech | Character and Tech | Description |
| X-Strike (Red) | Serge - Dash&Slash (Level 3) | Glenn - Dash&Gash (Level 3) | Does large damage to one enemy |
| VitalForce (White) | Radius - VitalEnergy (Level 7) | Viper - AirForce Level 5) | Does massive damage to one enemy |
| DragonSpike (Green) | Karsh - DragonRider (Level 3) | Zoah - Toss&Spike (Level 7) | Does large damage to one enemy |
| PitchBlack (Black) | Grobyc - StrongArm (Level 7) | Norris - TopShot (Level 7) | Does large damage to all enemies |
| SwordStorm (Black) | Guile - WandaSwords (Level 5) | Sneff - SwordTrick (Level 7) | Does large damage to one enemy |
| TossedSalad (Green) | Turnip - VegOut (Level 7) | NeoFio - BamBamBam (Level 7) | Does large damage to one enemy |
| DraggyRider (Yellow) | Leah - TripleKick (Level 7) | Draggy - BigBreath (Level 7) | Does large damage to one enemy |
| DoubleTake (Red) | Kid - Pilfer (Level 3) | Mel - Snatch (Level 3) | Increases the chance of stealing an item |
| Flamenco (Red) | Nikki - LimeLight (Level 7) | Miki - DanceOnAir (Level 7) | Temporarily raises all characters stats |

### Triple Techs

| Triple Tech Name | Character and Tech | Character and Tech | Character and Tech | Description |
| DeltaForce (White) | Serge - Luminaire (Level 5) | Leena - MaidenFaith (Level 7) | Razzly - Raz-Flower (Level 7) | Does Massive Damage to all enemies |
| Z-Slash (Blue) | Serge - FlyingArrow (Level 7) | Kid - RedPin (Level 5) | Sprigg [in Slash doppelgang Form] - DashSlash | Does Massive Damage to all enemies |
# Chronicles of Narnia/Introduction/Beginner
## Included Content

The Beginner reader is not expected to have fully explored the Chronicles of Narnia storyline. A Beginner reader is assumed to have read perhaps one or two of the books (maybe even none at all). Pages marked for Beginners include some short synopses of the book plots, brief introductions to some of the key characters in the series, and overviews of several important locations where many important events happen in the series.

## Excluded Content

Much of the Chronicles of Narnia series has yet to be explored. A great majority of the character list as well as all of the chapter summaries are beyond the Beginner level.

## What's Next

Once the reader has completed more of the Narnia series, the next step is the Intermediate level. The Intermediate level details much more of the storyline and is suited for a reader who has nearly completed reading the series.
# Chronicles of Narnia/Places
## Common

Places where many events happen and/or where the characters spend a great deal of time.
* Cair Paravel
* Narnia
# Chronicles of Narnia/Introduction/Intermediate
## Included Content

The Intermediate reader is expected to be rather well-versed in the storyline of the Chronicles of Narnia. Much of the plot is known to the reader and he/she wishes to expand on that with analysis and commentary. Most of the character pages and sections are open to the Intermediate reader. In addition, Intermediate readers have access to chapter summaries for referencing specific details.

## Excluded Content

Not much is kept from the Intermediate reader. Most major content is covered and filling in the gaps is all that is left. An Intermediate reader should feel knowledgeable about the topics covered in the books, but perhaps not quite grasping the full importance of scenes and character actions, referred to as the Greater Picture which is excluded until the reading of the series can be finished.

## What's Next

Once the reader has completed the entire Narnia series, the next step is the Advanced level. The Advanced level details the entire storyline. Advanced readers have access to the Greater Picture section seen on many pages that connects all the pieces together.
# English-Hanzi/Form

A form is a type of something.
类型 是 某事物 的 种类。
# English-Hanzi/Friday

Friday is the day after Thursday.
星期五 是 星期四 之后 一日。
# English-Hanzi/Frozen food

Frozen food is a food preserved by freezing for preventing spoilage. Frozen food does not need preservative.
冷冻食品 是 一种 食物，通过 冷冻 保存 以 防止 腐坏。冷冻食品 不需要 防腐剂。
# English-Hanzi/Folic acid

Special:search/folic acid prefix:English-Hanzi/
Also called: Folacin, Folate, Pteroylglutamic acid, Vitamin B9
也 称为：叶酸，叶酰谷氨酸，维生素B9
Folic acid is a B vitamin. It helps the body make healthy new cells. Everyone needs folic acid. For women who may get pregnant, it is really important. Getting enough folic acid before and during pregnancy can prevent major birth defects of her baby's brain or spine.
叶酸 是 一种 B族维生素。它 帮助 身体 制造 健康 的 新 细胞。每个人 都 需要 叶酸。对于 可能 怀孕 的 妇女，这 是 真的 很 重要。在 怀孕 前 和 怀孕 期间 获得 足够 的 叶酸 可以 防止 婴儿 的 大脑 或 脊柱 的 重大 先天性 缺陷。
Foods with folic acid in them include
* Leafy green vegetables
* Fruits
* Dried beans, peas, and nuts
* Enriched breads, cereals and other grain products
含 叶酸 的 食物 包括
* 绿叶 蔬菜
* 水果
* 干豆，豌豆，和 坚果
* 强化 的 面包，谷物 和 其他 粮食 制品
If you don't get enough folic acid from the foods you eat, you can also take it as a dietary supplement.
如果 你 没有 从 你 吃 的 食物 得到 足够 的 叶酸，你 也 可以 从 膳食 补充剂 摄取。
# English-Hanzi/Fluke worm

Fluke worm is a parasite. It lives inside the body of another animal. It can bring disease to that animal.
吸虫 是 寄生虫。它活在另一动物体内。它可给该动物带来疾病。
# English-Hanzi/Foot

The foot is the lowest part of the body. It is below the ankle.
足部 是 身体 的 最 低 部分。它 在 脚踝 下面。
# English-Hanzi/Force

Force is a power that can make a thing to move.
力量 是 一种 可以 使 某物 移动 的 动力。
# English-Hanzi/Fluid

Fluid is a substance that can flow.
流体是一种可以流动的物质。
# English-Hanzi/Food allergy

Food allergy is the immune system reacts to some foods. Usually the reaction is mild but sometimes is serious.
食物过敏是免疫系统对某些食物的反应。通常反应温和，但有时是严重的。
# English-Hanzi/France

Special:search/France prefix:English-Hanzi/
France is a country in the west of Europe. Its population is 67.2 million. Its capital is Paris.
法国 是 一个 在 欧洲 西部 的 国家。它 的 人口 是 6720万。它 的 首都 是 Paris。
# English-Hanzi/Recycling

Recycling is a process that makes useless things reusable.
循环再造 是 一个 使 无用的 东西 可重新利用的 过程。
# English-Hanzi/Risk

A risk is a danger.
风险 就是 危险。
# English-Hanzi/Rapeseed oil

Special:search/rapeseed oil prefix:English-Hanzi/
Special:search/rapeseed prefix:English-Hanzi/
Special:search/oil prefix:English-Hanzi/
Rapeseed oil is an oil obtained from rapeseed. It is rich in omega-3 and vitamin E, and low in saturated fat.
菜油是油菜籽的油，富含omega-3和vitamin E，饱和脂肪低。
# English-Hanzi/Red blood cell

Red blood cells are the red color cells in the blood. They deliver oxygen from the lungs to the tissues of the body.
红血球 是 血液 中 的 红色 细胞。它们 将 氧气 从 肺部 传递 到 身体 的 组织。
# English-Hanzi/Region

A region is an area.
一个 区域 是 一个 地区。
# English-Hanzi/Rest

A rest is a period of relaxing.
休息 是 放松 的 期间。
# English-Hanzi/Ravioli

Special:search/ravioli prefix:English-Hanzi/
Special:search/dumpling prefix:English-Hanzi/
Ravioli is an Italian pasta. It is a dumpling made with different fillings such as meat, vegetables, and cheese.
ravioli是一种Italy面食。它是一种用不同的馅料如肉类，蔬菜，和cheese制成的团子。
# English-Hanzi/Ratio

A ratio between two or more quantities is a way of measuring their sizes compared to each other.
两个 或 更多 数量 之间 的 比率 是 相互 比较 测量 它们 尺寸 的 一种 方式。
# English-Hanzi/Rate

A rate is a measurement of the speed that something changes.
速率 是 衡量 某事物 变化 的 速度。
# English-Hanzi/Rice

Special:search/rice prefix:English-Hanzi/
Rice is a cereal plant. Its seeds are processed to be white rice or brown rice. We usually eat cooked rice with meat and vegetables.
米 是 谷类 植物。其 种子 加工成 白米 或 糙米。我们 通常 用 肉类 和 蔬菜 下饭。
# Circuit Theory/All Pages

This page will contain an automatic listing of all pages in the "Circuit Theory" namespace.
# Circuit Theory/2Source Excitement/Thevenin and Norton
## Thevenin Voltage

* removing the load R$_3$
* redrawing up and down
* opening the current source (using superposition)
* shorting the voltage source (using superposition)
After opening the current source, have a series circuit. The overall current is going to be :
    $\mathbb {I} _{t}={\frac {\mathbb {V} _{s}}{X_{L1}+X_{L2}+R_{1}+X_{C2}+X_{C1}}}$
V$_{AB}$ due to the voltage source is going to be the drop across C2 and L2 which is (matlab code):
    $\mathbb {V} _{AB}=\mathbb {I} _{t}*(-4j)=0.0204-0.0344*i$
V$_{AB}$ due to the current source is a little more difficult. Put ground in between C$_2$ and L$_2$. Use node analysis at the junction of C$_1$, R$_2$ and R$_1$:
    $\mathbb {I} _{S}-{\frac {\mathbb {V} _{N}}{R_{1}+X_{L1}+X_{L2}}}-{\frac {\mathbb {V} _{N}}{X_{C1}+X_{C2}}}=0$
Solving for V$_N$:
    $\mathbb {V} _{N}=\mathbb {I} _{S}{\frac {1}{{\frac {1}{R_{1}+X_{L1}+X_{L2}}}+{\frac {1}{X_{C1}+X_{C2}}}}}$
Now can find the current in the two parallel branches. Then V$_{AB}$ is going to be the drop across C$_2$ minus the drop across L$_2$ (because the currents are going in opposite directions):
    $V_{AB}={\frac {\mathbb {V} _{N}}{X_{c1}+X_{C2}}}*X_{C2}-{\frac {\mathbb {V} _{N}}{R_{1}+X_{L1}+X_{L2}}}*X_{L2}=118.0-118.0*i$
So the Thevenin voltage is going to be the addition of the two V$_{AB}$'s (matlab code):
    $V_{th}=118.0-118.0*i$

## Norton Current

* Load resistor R$_3$ shorted, norton current identified
* Current source opened (calculate using superposition)
* Voltage source shorted (calculate using superposition)
Current in the short due to the voltage source is the total current:
    $\mathbb {I} _{N}={\frac {\mathbb {V} _{1}}{R_{1}+X_{C1}+X_{L1}}}$
Current in the short due to the current source is a little more difficult. From the first drawing:
    $\mathbb {I} _{N}=\mathbb {I} _{C}1-\mathbb {I} _{C}2$
Both I$_{C1}$ and I$_{C2}$ can be found through current dividers, so I$_N$ is:
    $\mathbb {I} _{N}=\mathbb {I} _{s}*({\frac {R1+X_{L1}}{R1+X_{L1}+X_{C1}}}-{\frac {X_{L2}}{X_{L2}+X_{C2}}})$
# Circuit Theory/Active Filters/Example90

Start with writing a node equation:
    $i_{1}-i_{2}-i_{3}-i_{0}=0$
    ${\frac {V_{in}-V_{node}}{R_{1}}}-V_{node}sC_{1}-{\frac {V_{node}-V_{o}}{R_{3}}}-{\frac {V_{node}}{R_{2}}}=0$
But the last expression for i$_o$ could have also been written using C$_2$. And we need to eliminate V$_{node}$. So our second equation is associated with the feedback resistor:
    ${\frac {V_{node}}{R_{2}}}-(0-V_{o}sC_{2})=0$
Now these two equations can be solved for V$_o$/V$_{in}$:
The transfer function is going to be the ratio of V$_{out}$/V$_{in}$ so:
Yields this:
    $-{\frac {R_{3}}{R_{1}+(C_{2}R_{1}R_{2}+C_{2}R_{1}R_{3}+C_{2}R_{2}R_{3})s+C_{1}C_{2}R_{1}R_{2}R_{3}s^{2}}}$
Which looks like a low pass filter (inverted because of the op amp).
# Circuit Theory/Active Filters

Active filters have op amps or amplifiers in them. They usually don't have inductors because they are expensive, heavy and large when compared to capacitors.

## Minimal Filter Transfer Functions

Each filter type has a characteristic transfer function:
| Filter Type | Transfer Function |
| High Pass | ${\frac {s^{2}}{aS^{2}+bs+c}}$ |
| Low Pass | ${\frac {1}{aS^{2}+bs+c}}$ |
| Band Pass | ${\frac {s}{aS^{2}+bs+c}}$ |
| Notch | ${\frac {s^{2}+d}{aS^{2}+bs+c}}$ |
First the goal is to build a transfer function from the op-amp circuit so that a bode diagram can be plotted. This is a simple matter of substituting s for derivatives in the terminal relations.

## Active Filter Components

Rather than draw an example op amp circuit for each of the above filters, it is easier to draw four op amp circuits that are more related to the desired transfer function:
| Generates | Transfer Function |
| Complex Pole | derivation $-{\frac {R_{3}}{R_{1}+(C_{2}R_{1}R_{2}+C_{2}R_{1}R_{3}+C_{2}R_{2}R_{3})s+C_{1}C_{2}R_{1}R_{2}R_{3}s^{2}}}$ |
| Zero at Origin | $-sRC$ |
| Real Pole | ${\frac {-{\frac {1}{R_{1}C}}}{s+{\frac {1}{R_{2}C}}}}$ |
| Real Zero | $-R_{2}C(s+{\frac {1}{R_{1}C}})$ |
Most filter design starts with the desired transfer function. Then the circuit is built. With the above four configurations, any transfer function can be created.
Better filters are have stepper slopes and narrower bands in their bode diagrams. This is measured with a factor called Q for quality. This is why there are so many different types of filter designs.

## Under Constrained Design

Suppose you wanted this transfer function:
    ${\frac {s+1}{(s+2)(s+3)(s+4)}}$
Then an active filter could be built like this:
* stage A $-R_{2}C(s+{\frac {1}{R_{1}C}})$
* stage B ${\frac {-{\frac {1}{R_{1}C}}}{s+{\frac {1}{R_{2}C}}}}$
* stage C ${\frac {-{\frac {1}{R_{1}C}}}{s+{\frac {1}{R_{2}C}}}}$
* stage D ${\frac {-{\frac {1}{R_{1}C}}}{s+{\frac {1}{R_{2}C}}}}$
Choosing values to match the above transfer function results in a table like this:
| Input Stage Variables | Feedback stage variables |
| $R_{1_{A}}C_{A}=1$ | $R_{2_{A}}C_{A}=1$ |
| $R_{1_{B}}C_{B}={\frac {1}{2}}$ | $R_{2_{B}}C_{B}=1$ |
| $R_{1_{C}}C_{C}={\frac {1}{3}}$ | $R_{2_{C}}C_{C}=1$ |
| $R_{1_{D}}C_{D}={\frac {1}{4}}$ | $R_{2_{D}}C_{D}=1$ |
There are 12 unknowns and 8 equations. This is called an under constrained problem. It means that one can apply other design rules. For example, capacitors are more expensive than resistors. Buying in bulk is less expensive. So ... choose one value for all the capacitors and then adjust resistor values. Choose C = 1μF Then all the resistors are 1 M&Ohm; accept R$_{1B}$, R$_{1C}$ and R$_{1D}$.
# Exercise as it relates to Disease/teaching resource/HDE template1

The templates are designed to assist you in your wikibook fact sheet assignment. A range of design features are shown so that you can copy and paste (or further edit) anything that will assist in your assignment.

## Template 1

* Will be added to soon