OpenLLM-France/wikimedia huggingface dataset:
# Definition of Homomorphism

Let G, K be groups with binary operations $\ast$ and $\circledast$ respectively.
$f\colon G\to K$ is homomorphism iff
   * $\forall \;g_{1},g_{2}\in G:f(g_{1}\ast g_{2})=f(g_{1})\circledast f(g_{2})$

# Definition of Kernel

Let e$_K$ be identity of K
    ${\text{kernel}}~f={\text{ker}}~f=\lbrace g\in G\;|\;f(g)={\color {OliveGreen}e_{K}}\rbrace$

# Definition of Image

    ${\text{Image}}~f={\text{im}}~f=\lbrace k\in K\;|\;\exists \;g\in G:f(g)=k\rbrace$
# Theorem

    In a group, each element only has one inverse.

# Proof

    0. Choose ${\color {OliveGreen}g}\in G$. Then, inverse g$_1$$^{−1}$ of g is also in G. 1. Assume g has a different inverse g$_2$$^{−1}$ in G 2. $({\color {BrickRed}g_{1}^{-1}}\ast {\color {OliveGreen}g})\ast {\color {Purple}g_{2}^{-1}}={\color {BrickRed}g_{1}^{-1}}\ast ({\color {OliveGreen}g}\ast {\color {Purple}g_{2}^{-1}})$ $\ast$ is associative on G 3. $e_{G}\ast {\color {Purple}g_{2}^{-1}}={\color {BrickRed}g_{1}^{-1}}\ast e_{G}$ g$_1$$^{-1}$ and g$_2$$^{-1}$ are inverses of g on G (usage 3) 4. ${\color {Purple}g_{2}^{-1}}={\color {BrickRed}g_{1}^{-1}}$, contradicting 1. e$_G$ is identity of G (usage 3)
# Theorem

Let f be a homomorphism from group G to group K. Let e$_K$ be identity of K.
    ${\text{ker}}~f={\lbrace e_{G}\rbrace }$ means f is injective.

# Proof

    0. Choose $x,y\in G$ such that $f(x)=f(y)$ 1. $f(y\ast x^{-1})=f(y)\circledast f(x^{-1})$ f is a homomorphism 2. $=f(x)\circledast f(x^{-1})$ 0. 3. $=f(x\ast x^{-1})$ f is a homomorphism 4. $=f(e_{G})=e_{K}$ homomorphism maps identity to identity 5. $y\ast x^{-1}\in {\text{ker}}~f$ 1,2,3,4. 6. $y\ast x^{-1}=e_{G}$ given ${\text{ker}}~f={\lbrace e_{G}\rbrace }$ 7. $y=x$
# Theorem

Let f be a homomorphism from group G to Group K.
Let g be any element of G.
    f(g$^{-1}$) = [f(g)]$^{-1}$

# Proof

    0. $f({\color {Blue}g})\circledast f({\color {BrickRed}g^{-1}})=f({\color {Blue}g}\ast {\color {BrickRed}g^{-1}})$ f is a homomorphism 1. $f({\color {Blue}g})\circledast f({\color {BrickRed}g^{-1}})=f({\color {Blue}e_{G}})$ definition of inverse in G . 2. $f({\color {Blue}g})\circledast f({\color {BrickRed}g^{-1}})={\color {OliveGreen}e_{K}}$ homomorphism f maps identity to identity 3. ${\color {BrickRed}f({\color {Blue}g}){\color {BrickRed}]^{-1}}\circledast f({\color {Blue}g})\circledast f({\color {BrickRed}g^{-1}})={\color {BrickRed}f({\color {Blue}g}){\color {BrickRed}]^{-1}}\circledast {\color {OliveGreen}e_{K}}$ as f(g) is in K, so is its inverse [f(g)]$^{−1}$ . 4. ${\color {OliveGreen}e_{K}}\circledast f({\color {BrickRed}g^{-1}})={\color {BrickRed}f({\color {Blue}g}){\color {BrickRed}]^{-1}}$ inverse on K, e$_K$ is identity of K 5. $f({\color {BrickRed}g^{-1}})={\color {BrickRed}f({\color {Blue}g}){\color {BrickRed}]^{-1}}$ e$_K$ is identity of K
# Abstract Algebra/Group Theory/Homomorphism/Image of a Homomorphism is a Subgroup
## Theorem

Let f be a homomorphism from group G to group K. Let e$_K$ be identity of K.
    ${\text{im}}~f=\lbrace k\in K\;|\;\exists \;g\in G:f(g)=k\rbrace$ is a subgroup of K.

## Proof
### Identity

    0. $f({\color {Blue}e_{G}})={\color {OliveGreen}e_{K}}$ homomorphism maps identity to identity 1. ${\color {OliveGreen}e_{K}}\in \lbrace k\in K\;|\;\exists \;g\in G:f(g)=k\rbrace$ 0. and ${\color {Blue}e_{G}}\in G$ 2. Choose $i\in \lbrace k\in K\;|\;\exists \;g\in G:f(g)=k\rbrace$ 3. $i\in K$ 2. 4. ${\color {OliveGreen}e_{K}}\ast i=i\ast {\color {OliveGreen}e_{K}}=i$ i is in K and e$_K$ is identity of K(usage3) 5. $\forall \;i\in {\text{im}}f:{\color {OliveGreen}e_{K}}\ast i=i\ast {\color {OliveGreen}e_{K}}=i$ 2, 3, and 4. 6. ${\color {OliveGreen}e_{K}}$ is identity of ${\text{im}}\;f$ definition of identity(usage 4)

### Inverse

    0. Choose ${\color {OliveGreen}i}\in \lbrace k\in K\;|\;\exists \;g\in G:f(g)=k\rbrace$ 1. $\exists \;{\color {OliveGreen}g}\in G:f({\color {OliveGreen}g})={\color {OliveGreen}i}$ 0. 2. $f({\color {OliveGreen}g})\circledast f({\color {BrickRed}g^{-1}})=f({\color {BrickRed}g^{-1}})\circledast f({\color {OliveGreen}g})=e_{K}$ homomorphism maps inverse to inverse between G and K 3. ${\color {OliveGreen}i}\circledast f({\color {BrickRed}g^{-1}})=f({\color {BrickRed}g^{-1}})\circledast {\color {OliveGreen}i}=e_{K}$ homomorphism maps inverse to inverse 4. i has inverse f( k$^{-1}$) in im f 2, 3, and e$_K$ is identity of im f 5. Every element of im f has an inverse.

### Closure

    0. Choose $i_{1},i_{2}\in \lbrace k\in K\;|\;\exists \;g\in G:f(g)=k\rbrace$ 1. $\exists \;g_{1},g_{2}\in G:f(g_{1})=i_{1},f(g_{2})=i_{2}$ 0. 2. $g_{1}\ast g_{2}\in G$ Closure in G 3. $f(g_{1}\ast g_{2})\in \lbrace k\in K\;|\;\exists \;g\in G:f(g)=k\rbrace$ 4. $i_{1}\circledast i_{2}=f(g_{1})\circledast f(g_{2})=f(g_{1}\ast g_{2})$ f is a homomorphism, 0. 5. $i_{1}\circledast i_{2}\in imf$ 3. and 4.

### Associativity

    0. im f is a subset of K 1. $\circledast$ is associative in K 2. $\circledast$ is associative in im f1 and 2
# Abstract Algebra/Group Theory/Group/a Cyclic Group of Order n is Isomorphic to Integer Moduluo n with Addition
## Theorem

Let C$_m$ be a cyclic group of order m generated by g with $\ast$
Let $(\mathbb {Z} /m,+)$ be the group of integers modulo m with addition
       C$_m$ is isomorphic to $(\mathbb {Z} /m,+)$

## Lemma

Let n be the minimal positive integer such that g$^n$ = e
       $g^{i}=g^{j}\leftrightarrow i=j~{\text{mod}}~n$
    Let i > j. Let i - j = sn + r where 0 ≤ r < n and s,r,n are all integers.
    and 0 ≤ r < n

## Proof

    0. Define ${\begin{aligned}f\colon C_{m}&\to \mathbb {Z} /m\\g^{i}&\mapsto i~{\text{mod}}~m\end{aligned}}$
    Lemma shows f is well defined (only has one output for each input).
    f is homomorphism:
       $f(g^{i})+f(g^{j})=i+j~{\text{mod}}~m=f(g^{i+j})=f(g^{i}\ast g^{j})$
    f is injective by lemma
    f is surjective as both $\mathbb {Z} /m$ and $C_{m}$ have m elements and f is injective
# Theorem

Let f be a homomorphism from group G to group K.
Let e$_G$ and e$_K$ be identities of G and K.
    f(e$_G$) = e$_K$

# Proof

    0. $f({\color {Blue}e_{G}})\in K$ f maps to K 1. ${\color {BrickRed}f({\color {Blue}e_{G}}){\color {BrickRed}]^{-1}}$ inverse in K . 2. $f({\color {Blue}e_{G}}\ast {\color {Blue}e_{G}})=f({\color {Blue}e_{G}})\circledast f({\color {Blue}e_{G}})$ f is a homomorphism 3. $f({\color {Blue}e_{G}})=f({\color {Blue}e_{G}})\circledast f({\color {Blue}e_{G}})$ identity e$_G$ . 4. ${\color {BrickRed}f({\color {Blue}e_{G}}){\color {BrickRed}]^{-1}}\circledast f({\color {Blue}e_{G}})={\color {BrickRed}f({\color {Blue}e_{G}}){\color {BrickRed}]^{-1}}\circledast f({\color {Blue}e_{G}})\circledast f({\color {Blue}e_{G}})$ 1. . 5. ${\color {OliveGreen}e_{K}}={\color {OliveGreen}e_{K}}\circledast f({\color {Blue}e_{G}})$ identity e$_K$, definition of inverse 6. ${\color {OliveGreen}e_{K}}=f({\color {Blue}e_{G}})$ identity e$_K$
# Cognitive Science: An Introduction/Serious Games and Crowdsourcing Methods

Some aspects of science require enormous amounts of human labor, too much for the individual scientist to do in their own lab.

## Crowdsourcing

One way around this is to hire people to do this labor. This requires coordination and a lot of grant money. It has been made less costly by recent internet applications, such as Amazon Mechanical Turk, which facilitates the hiring of people on the internet for small jobs that can be done through a web browser. The cumulative effort of laypersons working on one task is known as crowdsourcing.

## Citizen Scientists

Some crowdsourcing projects work without payment. People contribute simply because they want to. Most open-source software projects are like this. There are also science sites where laypersons can volunteer to help out. These helpers are known as citizen scientists.

## Serious Games

If a crowdsourcing project is gamified (made into a game), then some people will play the game, and help out on a scientific project, even if they don't care about the scientific aspect: They will play simply because it's fun. These are serious games. Most of the serious games related to cognitive science are more directly aimed at furthering artificial intelligence research, but some are for understanding the human mind and brain.
In the game Eyewire (2012, https://eyewire.org/explore), players trace cells of the mouse retina in images they see. This has contributed to the discovery of six neuron types. Its successor, Neo, is currently being developed.
The ESP game
Peekaboom
Quanty
* ↑ Deweerdt, S. (2019). Deep connections. Nature, 571(7766), S6-S8.
# Cognitive Science: An Introduction/Study Questions

At the end of every page there are a list of study questions and answers. They are written in a particular format so that they can be easily imported into a spaced-repetition learning system.

## Spaced-Repetition Learning Systems

Spaced-repetition learning systems (SRS) These are, basically, flash card programs. What makes these programs special is that they decide what cards are to be reviewed every day. It does this based on psychological studies of forgetting. Ideally, one uses the SRS program every day. The program gives you the cards you need to review that day, based on the notion that the best time to review a card is just before you are likely to forget it.
You see the question, try to answer it in your head, and then look at the answer. You indicate to the SRS program if you got the question correct or not, and that feeds information into the algorithm to choose when to show you the card again. If you know a fact very well, it might not ask you for a year or more.

## Available SRS Systems

See a complete list at List of flashcard software.
The author recommends using a program called Anki. When available, Anki "decks" of cards will be linked from pages in this book.

## How To Make Good SRS Flash Cards

There are several websites on how to make good cards. The author recommends this one .
# Cognitive Science: An Introduction/Senses In Non-human Animals and Robots
## Single-Celled Organism Perception

Very early on in the evolution of life, single-celled organisms had a primitive kind of perception, as single-celled organisms still do today. The surface of the cells had detectors that could identify chemicals in the environment. If the chemical was something good, like food, then the organism might move in the direction that had a higher concentration of that chemical. If the presence of a chemical is higher in some areas, and lower in others, this is known as a chemical gradient. Similarly, harmful environmental conditions, like extreme heat, cause small organisms to move away. This is the biological origin of perception. But single-celled organisms are too small to detect which direction the gradient is facing. What they do is they have a very primitive memory with one slot in it. The bacterium will swim straight ahead as long as the desired chemical magnitude is greater than that in memory. Otherwise it changes course.
Single-celled eukaryotes often have more complex perception mechanisms, including a primitive kind of light detection. They have eyespots that are sensitive to light. They will move toward or away from light depending on their energy needs in the moment.
Multicellular organisms (humans included) have much more complex sensory systems.

## Detection of Polarization of Light

Octopuses have, as a part of their visual system, the ability to detect the polarization of light. Light is made of waves, and the plane of the oscillation determines its polarization. Certain sunglasses work by only allowing certain polarizations to get through, so when you turn your head, you can sometimes see some light blocked but not other.

## Echolocation

Many bats use echolocation to navigate in their environment. With practice, human beings can do a limited version of this as well.

## Electrolocation

Platypuses and electric eels can detect electrical fields given off by other animals. They use this sense to detect prey.

## Magnetoreception

Magnetoreception is the ability to detect Earth's magnetic field, so that the individual can tell which way is north. Many bacteria have this sense, as do many fish, amphibians, reptiles, birds, and, rarely, mammals such as mice, mole-rats, and bats.
* ↑ Gazzaley, A., & Rosen, L. D. (2016). The Distracted Mind: Ancient Brains in a High-tech World. Cambridge, MA: MIT Press. Page 20.
* ↑ Godfrey-Smith, P. (2016). Other minds: The octopus, the sea, and the deep origins of consciousness. Farrar, Straus and Giroux. Pages 16--17.
* ↑ Godfrey-Smith, P. (2016). Other minds: The octopus, the sea, and the deep origins of consciousness. Farrar, Straus and Giroux. Page 17.
* 1 2 Mitchell, K. J. (2018). ‘’Innate: How the wiring of our brains shapes who we are.’’ Princeton, NJ: Princeton University Press. Page 132
# Cognitive Science: An Introduction/Statistics
## Factor Analysis

When we're trying to understand the mind, and how it works, one of the things we want to know is how we can think of the different parts of the mind. Keep in mind that this might be a different question from the different parts of the brain--there are likely functions in the mind that are not restricted to a particular location, much like a function in a company might not be strongly associated with a specific, few rooms in a building.
One popular way to try to understand the different parts of the mind is by using a statistical technique called "factor analysis" (in other fields it is known as "principle components analysis.") What factor analysis does is look for relationships between large numbers of variables (typically, scores of different kinds of questions on a written test). Suppose you had a 200 question test given to 1000 people. Some of these questions might involve questions about world history, and others might test algebra abilities. We might expect that people who did well on world history question A might also do well on world history question B. Likewise, we might not expect whether somebody got A right will also get a math question right. We might expect a factor analysis to look at all of these questions, and the patterns of correct and incorrect answers given, and find two "factors." The scientist will then look at what questions belong in which factors and notice that some are about history and some are about math, and name the factors "history" and "math." You can think of it as a way of reducing the number of variables--in this case, 200 questions to 2 factors. A factor is the weighted sum of the variables. A math question would be called a "marker" for the math factor, and "load" on that factor. In psychology, we tend to interpret factors as representations of underlying abilities.
This is what's called an "exploratory factor analysis." It's exploratory because we basically just give the data to the technique and it returns the patterns it found. We don't use our pre-conceived hypotheses about the structure in the data. A problem with exploratory factor analyses is that there are several ways to do it, and depending on which way you do it, you can get different factors, and different numbers of factors. Often the choice of technique is subjective.
If a scientist already has an idea of the structure of the data (that is, what she thinks the factors might be), then she can run a "confirmatory factor analysis," which tests whether the factors she thinks are there really are. Some have argued that this is a better way to do science, and is able to find evidence against theories in a way that exploratory factor analysis cannot.
* ↑ Hegarty, M. & Waller, D.A. (2005). Individual differences in spatial abilities. In P. Shah and A. Miyake (Eds.) The Cambridge Handbook of Visuospatial Thinking. Cambridge University Press: Cambridge. 121--169.
# Adventist Youth Honors Answer Book/Nature/Rocks & Minerals/Luster

Luster is a description of the way light interacts with the surface of a crystal, rock, or mineral. For example, a diamond is said to have an adamantine luster and pyrite is said to have a metallic luster.
Other descriptive terms used for gems include vitreous, like glass; resinous, like amber; waxy, like jade; greasy, like soapstone; pearly; and silky.
# Adventist Youth Honors Answer Book/Nature/Rivers and Streams

Rivers and Streams :
| Nature North American Division | Skill Level Unknown |
| Nature North American Division | Year of Introduction: 2011 |
* The requirements for this honor have not yet been published. The NAD has accepted the honor, but has not yet released the official requirements. The requirements listed below should be considered an advance, working copy, and they are therefore subject to change — perhaps significantly.
# Adventist Youth Honors Answer Book/Nature/Rocks & Minerals/Cleavage

Cleavage is the tendency of crystalline materials to split along definite planes, creating smooth surfaces, of which there are several named types:
* Basal cleavage: cleavage parallel to the base of a crystal, or to the plane of the lateral axes. This occurs quite easily in graphite, making the material feel slippery.
* Cubic cleavage: cleavage parallel to the faces of a cube. This is the source of the cubic shape seen in crystals of ground table salt.
* Diagonal cleavage: cleavage parallel to a diagonal plane.
* Lateral cleavage: cleavage parallel to the lateral planes.
* Octahedral, Dodecahedral, or Rhombohedral cleavage: cleavage parallel to the faces of an octahedron, dodecahedron, or rhombohedron (respectively). Octahedral cleavage is seen in common semiconductors used in electronics.
* Prismatic cleavage, cleavage parallel to a vertical prism.
Cleavage is of technical importance in the electronics industry and in the cutting of gemstones. While precious stones are generally cleaved by impact, man-made single crystals of semiconductor materials are generally sold as thin wafers which are much easier to cleave. Simply pressing a silicon wafer against a soft surface and scratching its edge with a diamond scribe is usually enough to cause cleavage.
# Adventist Youth Honors Answer Book/Nature/Rocks & Minerals/Streak

The streak (also called powder color) of a mineral is the color of the powder produced when it is dragged across an unweathered surface. Unlike the apparent color of a mineral, which for most minerals can vary considerably, the trail of finely ground powder generally has a more consistent characteristic color, and is thus an important diagnostic tool in mineral identification. If no streak seems to be made, the mineral's streak is said to be white or colorless. Streak is particularly important as a diagnostic for opaque and colored materials. It is less useful for silicates, most of which have a white streak and are too hard to powder easily.
The apparent color can vary widely because of trace impurities or a disturbed macroscopic crystal structure. Small amounts of an impurity that strongly absorbs a particular wavelength can radically change the wavelengths of light that are reflected by the specimen, and thus change the apparent color. However, when the specimen is dragged to produce a streak, it is broken into randomly oriented microscopic crystals, and small impurities do not greatly affect the absorption of light.
The surface across which the mineral is dragged is called a "streak plate," and is generally made of unglazed porcelain tile. In the absence of a streak plate, the underside of a porcelain bowl or vase, the surface of an electrical fuse, or the back of a glazed tile will work. Sometimes a streak is more easily or accurately described by comparing it with the "streak" made by another streak plate.
Because the trail left behind results from the mineral being crushed into powder, a streak can only be made of minerals softer than the streak plate, around 7 on the Mohs scale of mineral hardness. In this case, the color of the powder can be determined by filing or crushing with a hammer a small sample, which is then usually rubbed on a streak plate. Most minerals that are harder have an unhelpful white streak.
Some minerals leave a streak similar to their natural color, such as cinnabar and azurite. Other minerals leave surprising colors, such as fluorite, which always has a white streak, although it can appear in purple, blue, yellow, or green crystals. Hematite, which is black in appearance, leaves a red streak which accounts for its name, which comes from the Greek word "haima," meaning "blood." Galena, which can be similar in appearance to hematite, is easily distinguished by its gray streak.
# Adventist Youth Honors Answer Book/Nature/Scorpions

Try to draw an insect. Here are some vicous species of scorpions that you may wish to download and draw.
* Common Yellow Scorpion
* Bark Scorpion ( Centruroides sculpturatus ) Desert Hairy Scorpion ( Hadrurus arizonensis ) Bark Scorpion
* Desert Hairy Scorpion
* Death Stalker
* Spider Hunting Scorpion
* Central American Bark Scorpion
* Black Tailed Bark Scorpion
* Red Bark Scorpion
* Death Stalker Scorpion ( Leiurus quinquestriatus ) Isometroides vescus with babies Centruroides limbatus Centruroides bicolor Centruroides vittaus Brownish Bark Scorpion
# ERP Internals/Data Tables/General Ledger Budget

This table contains one record for every period-end budget for every General Ledger account. There can be one or more budgets with each having a different budget type.
Used by: General Ledger
Key: Budget Type, Account Number, Fiscal Year, Period Number
Fields:
* Budget Type
* Account Number
* Fiscal Year
* Period Number
* Budget
* Units
# ERP Internals/Data Tables/HR Benefits

This data consists of the following two tables:
* Benefit
* Benefit Link
# ERP Internals/Data Tables/Employee

This table contains one record for every employee.
Used by: Human Resources, Payroll
Key: Employee Number
Fields:
* Employee Number (a unique identifier assigned by the user)
* Type
* Active (Yes/No)
* Last Name
* First Name
* Middle Name
* Suffix
* Title (Mr/Ms/Mrs/etc.)
* Job Title
* Department (this is selected from a table unlike all the other department fields)
* Address 1
* Address 2
* City
* State/Province
* Country
* Zip/Postal Code
* Business Telephone
* Telephone Extension
* Home Telephone
* Mobile
* Pager
* E-mail
* Pay Rate
* Pay Frequency
* Vacation Accrual Rate
* Accrued Vacation
* Sick Day Accrual Rate
* Accrued Sick Days
* Picture
* Ethnic Origin (used for US government reporting)
* Military Status
* Country-Specific Tax Fields
* Hire Date
* Next Review Date
* Termination Date
* Termination Reason (ROE Code)
* Birthdate
* Gender
* Marital Status
* Number of Dependents
* Emergency Contact Name
* Emergency Telephone Number
* SSN/SIN
* Bank Name (for Direct Deposit)
* Bank Account Number (for Direct Deposit)
* Manager Employee Number
* Union
* Union Number
* GL Distribution Group (contains all the postable account numbers)
* Language
* Notes
# ERP Internals/Data Tables/Employee Transaction

This table contains one record for every employee transaction.
Used by: Human Resources, Payroll
Key: Transaction Number
Fields:
* Transaction Number (a unique number assigned by the ERP package)
* Date
* Type (Hours/Pay/Deduction/DB Adjustment/CR Adjustment)
* Hours
* Amount
* Currency
# ERP Internals/Data Tables/General Ledger Account Balance

This table contains one record for every period-end balance for every General Ledger account.
Used by: General Ledger
Key: Account Number, Fiscal Year, Period Number
Fields:
* Account Number
* Fiscal Year
* Period Number (a unique number designating the accounting period within the year)
* Balance
* Units
# ERP Internals/Data Tables/Fixed Asset

This table contains one record for every fixed asset.
Used by: Fixed Assets
Key: Fixed Asset Number
Fields:
* Fixed Asset Number (a unique identifier assigned by the user)
* Type
* Active (Yes/No)
* Description
* Purchase Price
* Purchase Date
* Depreciation Type
* Accumulated Depreciation
* Disposal Date
* Serial Number
* GL Distribution Group (contains all the postable account numbers)
* Notes
# ERP Internals/Data Tables/Employee Check

This table contains one record for every employee check.
Used by: Human Resources, Payroll
Key: Employee Number, Check Number
Fields:
* Employee Number
* Bank Account Number
* Check Number
* Amount
* Posted to General Ledger Date/Time