OpenLLM-France/wikimedia huggingface dataset:
# Reinforcement Learning/Markov Decision Process

Markov Decision Process (MDP) is Markov Chain + Rewards function + Actions.
The Markov Decision Process is reduced to Markov Rewards process by choosing a "policy" that specifies the action taken given the state, $\pi (s)$.

## Definition

A Markov decision process is a 4-tuple $(S,A,P_{a},R_{a})$, where
* $S$ is a finite set of states,
* $A$ is a finite set of actions (alternatively, is the finite set of actions available from state ),
* $P_{a}(s,s')={\text{Pr}}(s_{t+1}=s'|s_{t}=s,a_{t}=a)$ is the probability that action $a$ in state $s$ at time $t$ will lead to state $s'$ at time $t+1$,
* $R_{a}(s,s')$ is the immediate reward (or expected immediate reward) received after transitioning from state $s$ to state $s'$, due to action $a$
(Note: The theory of Markov decision processes does not state that $S$ or $A$ are finite, but the basic algorithms below assume that they are finite.)

## Policy Specification

A policy if a function $\pi$ that specifies the action $a=\pi (s)$ that the decision maker will choose when it is in state $s$.
Once a Markov decision process is combined with a policy, this fixes the action for each state and the resulting combination behaves like a Markov chain$\Pr(s_{t+1}=s'\mid s_{t}=s,a_{t}=a)=\Pr(s_{t+1}=s'\mid s_{t}=s,a_{t}=\pi (s))$
The goal is to choose a policy $\pi$ that will maximize some cumulative stochastic rewards function.
Typically the expected the cumulative reward is a discounted sum over a potentially infinite horizon:
    $\mathbb {E} [\sum _{t=0}^{\infty }{\gamma ^{t}R_{a_{t}}(s_{t},s_{t+1})}]$ (where we choose $a_{t}=\pi (s_{t})$, i.e. actions given by the policy). And the expectation is taken over $s_{t+1}\sim P_{a_{t}}(s_{t},s_{t+1})$
where $\ \gamma \$ is the discount factor satisfying $0\leq \ \gamma \ \leq \ 1$, which is usually close to 1. (For example, $\gamma =1/(1+r)$ for some discount rate r.)
Because of the Markov property, the optimal policy for this particular problem can indeed be written as a function of $s$ only, as assumed above.
The discount-factor motivates the decision maker to favor taking actions early, rather not postpone them indefinitely.
# Regular ring/Positive characteristic/Tight closure trivial/Fact

Suppose that $R$ is a regular local ring of positive characteristic $p$ and of dimension $d$.
$I$
    $I^{*}=I\,.$
# Riemann integral/1 over sqrt(x) of 0 to 1/Lower integral exists, no upper integral/Exercise

Prove that for the function
    $]0,1]\longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{\sqrt {x}}},$
the lower integral exists, but the upper integral does not exist.
# Riemann integral/Step functions with equal limit/Integral/Fact/Proof/Exercise

Let $I=[a,b]$ be a compact interval and let
    $f\colon I\longrightarrow \mathbb {R}$
be a function. Consider a sequence of staircase functions ${\left(s_{n}\right)}_{n\in \mathbb {N} }$ such that $s_{n}\leq f$ and a sequence of staircase functions ${\left(t_{n}\right)}_{n\in \mathbb {N} }$ such that $t_{n}\geq f$. Assume that the two Riemann sums corresponding to the sequences converge and that their limits coincide. Prove that $f$ is Riemann-integrable and that
    $\lim _{n\rightarrow \infty }\int _{a}^{b}s_{n}(x)dx=\int _{a}^{b}f(x)dx=\lim _{n\rightarrow \infty }\int _{a}^{b}t_{n}(x)dx\,.$
# Riemann integral/x^2 from a to b/Explicit via staircase integrals/Exercise

Determine the definite integral
    $\int _{a}^{b}t^{2}dt$
as a function of $a$ and $b$ explicitly with lower and upper staircase functions.
# Riemann integrable function/Integral function/Definition

Let $I$ denote a real interval, let
    $f\colon I\longrightarrow \mathbb {R}$
denote a Riemann-integrable function, and let $a\in I$. Then the function
    $I\longrightarrow \mathbb {R} ,x\longmapsto \int _{a}^{x}f(t)\,dt,$
is called the integral function for $f$ for the starting point $a$.
# Riemann integral/-x^3+3x^2-2x+5 from -2 to 7/Explicit via staircase integrals/Exercise

Determine the definite integral
    $\int _{-2}^{7}-t^{3}+3t^{2}-2t+5dt$
explicitly with upper and lower staircase functions.
# Riemann integral/1 over x/No lower and no upper integral/Exercise

Prove that for the function
    $]0,1]\longrightarrow \mathbb {R} ,x\longmapsto {\frac {1}{x}},$
neither the lower nor the upper integral exist.
# Riemann integral/7x^3+4x from 3 to 5/Explicit via staircase integrals/Exercise

Determine the definite integral
    $\int _{3}^{5}7t^{3}+4tdt$
explicitly with upper and lower staircase functions.
# Riemann integrability/Elementary properties/Product/Fact/Proof/Exercise

Let $I=[a,b]$ be a compact interval and let $f,g\colon I\rightarrow \mathbb {R}$ be two Riemann-integrable functions. Prove that $fg$ is also Riemann-integrable.
# Riemann integral/Step functions with equal limit/Integral/Fact

Let $I$ denote a compact interval, and let
    $f\colon I\longrightarrow \mathbb {R}$
denote a function. Suppose that there exists a sequence of lower step functions ${\left(s_{n}\right)}_{n\in \mathbb {N} }$ with $s_{n}\leq f$ and a sequence of upper step functions ${\left(t_{n}\right)}_{n\in \mathbb {N} }$ with $t_{n}\geq f$. Suppose furthermore that the corresponding sequences of step integrals converge to the same real number.
$f$
Riemann-integrable, and the definite integral equals this limit, so
    $\lim _{n\rightarrow \infty }\int _{a}^{b}s_{n}(x)\,dx=\int _{a}^{b}f(x)\,dx=\lim _{n\rightarrow \infty }\int _{a}^{b}t_{n}(x)\,dx\,.$
# Student Projects/Environment
## ENVIRONMENT

An environment is the natural surrounding which helps grow and nourish life on this planet. All living things that live on this earth come under the environment. The environment also includes air, water, sunlight, plants, animals, etc. Moreover, the earth is considered the only planet in the universe that supports life.

## IMPORTANCE OF ENVIRONMENT

We truly cannot understand the real worth of the environment. It plays a vital role in keeping living things healthy in the environment. Likewise, it maintains the ecological balance that will keep check of life on earth. It provides food, shelter, air, and fulfils all the human needs whether big or small. Moreover, the entire life support of humans depends wholly on the environmental factors. In addition, it also helps in maintaining various life cycles on earth. Most importantly, our environment is the source of natural beauty and is necessary for maintaining physical and mental health. The environment gives us countless benefits that we can’t repay our entire life. As they are connected with the forest, trees, animals, water, and air. The forest and trees filter the air and absorb harmful gases. Plants purify water, reduce the chances of flood maintain natural balance and many others. Moreover, the environment keeps a close check on the environment and its functioning, It regulates the vital systems that are essential for the ecosystem. Besides, it maintains the culture and quality of life on earth. The environment regulates various natural cycles that happen daily. These cycles help in maintaining the natural balance between living things and the environment. Disturbance of these things can ultimately affect the life cycle of humans and other living beings. The environment has helped us and other living beings to flourish and grow from thousands of years. The environment provides us fertile land, water, air, livestock and many essential things for survival.

## CAUSE OF ENVIRONMENTAL DEGRADATION

Human activities are the major cause of environmental degradation. The activities of humans that causes environmental degradation is pollution, defective environmental policies, chemicals, greenhouse gases, global warming, ozone depletion, etc. All these affect the environment badly. Besides, these the overuse of natural resources will create a situation in the future there will be no resources for consumption. And the most basic necessity of living air will get so polluted that humans have to use bottled oxygen for breathing. Above all, increasing human activity is exerting more pressure on the surface of the earth which is causing many disasters in an unnatural form. Also, we are using the natural resources at a pace that within a few years they will vanish from the earth.
To conclude, we can say that it is the environment that is keeping us alive. Without the blanket of environment, we won’t be able to survive. Moreover, the environment’s contribution to life cannot be repaid. Besides, still what the environment has done for us, in return we only have damaged and degraded it.
# Student Projects/Emotional well-being

Emotional well being
The World Health Organization defined Health as “a state of complete physical, mental and social well being”. Emotions are part of our life. We go through different emotions every day. Emotional well being is related to our mental health and concerns. We experience different kind of feelings throughout our day. Happiness, sad, anger, affection these are all emotions. Emotional health can help the individual to have physically and psychologically well being and their overall health. We express our emotions in different way, but lack of control over emotions can create physiological and psychological disturbances in our personal life. People who are emotionally healthy are in control of their thoughts and behavior.
Effects of emotional disturbance
Emotional disturbances cause physiological problems such as; Sleeplessness, Eating disorder, lack of energy, Tiredness, aggression, etc
Psychological problems such as lack of interest, loneliness, Inferiority, depression, Irrational thoughts etc.
Causes for emotional disturbance
Emotional distress is likely someone with; low self-esteem, Emotionally sensitive , Very self –critical, Negative thoughts/ irrational thoughts, Impulsiveness
People with emotional disturbance criticize themselves, others and situations. They tend to be worry about future circumstances. Irrational thoughts come and interfere their daily activities, which leads to low self-esteem and mood fluctuations. Many factors influence our emotional health, such as our physiological health condition, function of glands, our thinking pattern etc.
Importance of Emotional well being
Emotional health is essential for every individual. It helps us to build physical psychological and social well being. It improves our quality of life and leads to happiness and better life. People with emotional problems can get counseling and therapies to overcome their obstacles which led to emotional disturbance. Proper psychological therapies and counseling approaches can help them to adapt healthy life style. Meditation, yoga, physical exercises, involving in various activities such as music, dance, sports can boost your overall health.
# Student Projects/Flower

== Parts of a flower: ==
* Calyx
* Corolla • Androecium • Gynoecium
Calyx and Corolla are known as accessory whorls because they help in protecting the flower in its bud stage and attract the insects for pollination. Androecium and gynoecium are known as reproductive whorls as they are necessary for the sexual reproduction.
1. Calyx: It is the outermost whorl of a flower consisting of sepals. Usually sepals are green in colour. They help in protecting the
2. Corolla: It is the second whorl consisting of petals. Usually petals are colourful. It attracts the insects for pollination. 3. Androecium: It is the third whorl of a flower composed of stamens. A stamen consist of a stalk like filament and an anther. The
4. Gynoecium: It is the innermost whorl of a flower composed of one or many carpels/pistil. Each pistil consist of a basal ovary,
# Student Projects/Environmental Pollution

Environment is the complex of physical, chemical and biotic factors that act upon an organism or a ecological community and ultimately determine its form and survival.
ENVIRONMENTAL POLLUTION :
Pollution is introduction of contaminants in natural environment that causes adverse change.
Most forms of pollution are invisible to human eye and come in variety of different forms.
There are seven major types of pollution. They are:
1.water pollution
2.soil pollution
3.Air pollution
4. Noise pollution
1. Water pollution : This type of pollution refers to the contamination of water bodies including ground water. As all living organisms depend on water to live ,the pollution of a body of water ends to affect every level of ecosystem including human health . common causes for water pollution include industrial waste. insecticides, pesticides, detergents and oil spills. These pollutants either work by killing off organisms through their toxicity (industrial waste, insecticides), reducing oxygen values in water(known as eutrophication) by blocking out sunlight (detergents, oil).
2.Soil pollution : soil can be stripped of its nutrients by a number of chemical agents and when this occurs it is known as soil pollution. Common causes of soil pollution are insecticides, pesticides, agricultural chemicals, industrial wastes, and radioactive wastes. Plants depend on the nutrients in the soil in order to grow, but many of these chemical compounds absorb the nitrogenous compounds present in the soil which the plants depend on. Aside from making an area barren ,soil pollution is a common cause of erosion, as plants and other living organisms play an important role in keeping the soil held together.
3.Air pollution: Air usually comprises 78% nitrogen,21%oxygen,0.9% oxide gases and0.1% inert gases. When the air becomes contaminated with other elements such as poisonous gases or particles, it can cause serious problems to human health. The most common causes of air pollution include partially combusted exhaust gases, poisonous gases which are a by-product of industry including sulphur dioxide and carbon monoxide ,and carcinogenic gases released through the burning of plastic, rubber and wood . Particle contamination can also occur when a substance such as friable asbestos fibres are disturbed and released into the air. Air pollution works either by poisoning living organisms which breathe it in ,or by disturbing the atmosphere and mixing with their air and clouds to cause acid rain.
4.Noise pollution: Noise pollution refers to an excess or unpleasant sounds emanating from industry, infrastructure, machinery, transportation. In humans noise pollution have detrimental effect on both mental and physical health and has been linked to high stress,heavy loss, hypertension, depression, sleep disturbances and increase in incidences of coronary artery disease.
# Psycholinguistics/Participants
### List of Participants

* Aaron J Newman
* Garrett Nicolai
* Alexandria Muise-Hennessey
* Bengisu Gonul
* Matt Orr
* Charlie Kramer
* Samantha Arseneau
* Kitt Holeman
* Laura Doane
* Elaine Toombs
* Claire Brousseau
* Chantel Deck
* Meg Smith
* Stepheny Vossen
* Angela Taylor
* Joel Sardinha
* Ju wen Ou
* Andrea VanderHeyden
* Jacqueline Vincent
* Nathanael Crawford
* John Boudreau
* Jillian Baker
* Hilary Van Loon
* Katelyn Grechuk
* Kathryn Fraser
* Ben Corrigan
* Jennifer Browning
* Taylor Dyon
* Megan J Mac Donald
* Heather Gallant
* Amie Windle
* Minzhou Sun
* K-Lee Fraser
* Anna Richter
* Eleanor Greer
* Keegan Colville
* Ben Parker
* Maged Aly
* Emily McGuire
* Michelle Jansen
* Jennifer Chetwynd
* Moragh Jang
* Ellen Fesseha
* Emma Horner
* Julien Gallant
* Suzanne Walsh
* Catherine McIntyre
* Allie DePetrillo
* Martha Schram
* Jennica Lounsbury
* Jayme Segal
* Emre Vs
# Vector space/Simple properties/Fact

Let $K$ be a field, and let $V$ be a $K$-vector space. Then the following properties hold (for $v\in V$ and $s\in K$).
* We have $0v=0$.
* We have $s0=0$.
* We have $(-1)v=-v$.
* If $s\neq 0$ and $v\neq 0$, then $sv\neq 0$.
# Vector space/Spanned linear subspace/Definition

Let $K$ be a field, and let $V$ be a $K$-vector space. For a family $v_{i}$, $i\in I$, we set
    $\langle v_{i},\,i\in I\rangle ={\left\{\sum _{i\in J}s_{i}v_{i}\mid s_{i}\in K,\,J\subseteq I{\text{ finite subset}}\right\}}\,,$
and call this the linear span of the family, or the generated linear subspace.
# Vector space/Product/Dimension/Exercise/Solution

The product space has dimension $n+m$. To proof this let $v_{1},\ldots ,v_{n}$ be a basis of $V$ and let $w_{1},\ldots ,w_{m}$ be a basis of $W$. We claim that the elements
    $(v_{j},0),\,j\in \{1,\ldots ,n\},{\text{ and }}(0,w_{i}),\,i\in \{1,\ldots ,m\},$
form a basis of $V\times W$.
Let $(v,w)\in V\times W$. Then there are representations
    $v=\sum _{j=1}^{n}a_{j}v_{j}{\text{ and }}w=\sum _{i=1}^{m}b_{i}w_{i}.$
Therefore
    ${\begin{aligned}(v,w)&=(\sum _{j=1}^{n}a_{j}v_{j},\sum _{i=1}^{m}b_{i}w_{i})\\&=(\sum _{j=1}^{n}a_{j}v_{j},0)+(0,\sum _{i=1}^{m}b_{i}w_{i})\\&=\sum _{j=1}^{n}a_{j}(v_{j},0)+\sum _{i=1}^{m}b_{i}(0,w_{i}),\end{aligned}}$
which means that we have a generating system.
To show linear independence, suppose that
    $\sum _{j=1}^{n}a_{j}(v_{j},0)+\sum _{i=1}^{m}b_{i}(0,w_{i})=0=(0,0)\,.$
The same computation shows
    $(\sum _{j=1}^{n}a_{j}v_{j},\sum _{i=1}^{m}b_{i}w_{i})=(0,0)\,$
and this means
    $\sum _{j=1}^{n}a_{j}v_{j}=0{\text{ and }}\sum _{i=1}^{m}b_{i}w_{i}=0.$
Since each family forms a basis, we conclude $a_{j}=0$ for all $j$ and $b_{i}=0$ for all $i$.
# Vector space/Set of sequences/R/Cauchy sequences as subspace/Exercise

Let
    $C={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid {\text{Cauchy sequence in }}\mathbb {R} \right\}}\,$
be the set of all real Cauchy sequences. Show that $C$ is a linear subspace of the space of all sequences
    $F={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid {\text{sequence in }}\mathbb {R} \right\}}\,.$
# Vector space/Product of two spaces/Exercise

Let $K$ be a field, and let $V$ and $W$ be vector spaces over $K$. Show that the product set
    $V\times W$
is also a $K$-vector space.
# Vector space/Simple Properties/Fact/Proof/Exercise

Let $K$ be a field and let $V$ be a $K$-vector space. Show that the following properties hold (where $\lambda \in K$ and $v\in V$).
* We have $0v=0$.
* We have $\lambda 0=0$.
* We have $(-1)v=-v$.
* If $\lambda \neq 0$ and $v\neq 0$ then $\lambda v\neq 0$.
# Software Design/Don't add a parameter that won't be used

Checklist questions:
* Somebody would ever want to change the function parameter's value?
* Somebody would ever want to change the class parameter's value?
* Somebody would ever want to change the configuration parameter's value?
Before adding a new parameter to a function or a class, or a new configuration parameter, ask yourself if any user of the function, or the class, or the software would ever need or want to change the default value to something else. If the answer is "No" or "Very unlikely", hardcode the default value and avoid the complexity of a parameter.

## Why

An extra parameter to a function or a class contributes to their size. To provide an option to not specify the extra parameter for a function there should be an overloading or an additional function with a different name. To provide an option to not specify the extra parameter for a class the latter needs to have an additional constructor, or an additional static factory function, or an additional function in the builder. These solutions increase the API size.
On the other hand, if a function or a class doesn't provide an option to omit the extra parameter there is an opportunity for the API users to make a mistake by providing a wrong value, or confusing several parameters which have the same type.
Regarding configuration, even if an extra parameter doesn't contribute to the complexity and the size of the codebase much (or it's not perceived as a problem), when software has more configuration parameters it's harder to learn and make sense of them for new users. A lot of unimportant parameters may draw the attention of users and operators away from the few important ones or may make harder to find the important parameters.
When there are many configuration parameters, it's hard to test all of them. Some untested combinations of values for different parameters may lead to degraded performance or errors. In other words, the software is less reliable.
# Software Design/Don't extend abstract class unless needed

Checklist questions:

## Why

Populating an abstract class with another implementation makes it harder to comprehend the logic which uses the abstract class: it may not be easy to understand for the reader which implementations can and can not be used from that logic.
More implementations for an abstract class make harder to change it: for example, to add a new function, or to change a signature of an existing function.
An object of the class may be mistakenly passed into a function expecting the abstract class, i. e. the robustness of such functions is reduced.
The fact that a class extends an abstract class without apparent purpose may puzzle the readers.

## Why not

If a class extends a well-known abstract class, the reader may be able to immediately know the principles of the workings of the class, without studying all its logic. For example, if a class implements Iterator abstract class core to the used programming language, the reader may be confident they understand the semantics of the code working with this Iterator implementation, even though they didn't read its code. In other words, extending an abstract class is a form of structure.
# Software Design/Don't omit function's signature (return type)

In some programming languages, the return type of a function could be omitted (and inferred by the compiler). In others, e. g. Haskell, the function's signature could be omitted entirely.

## Why

* Compatibility + robustness
* Apparency of semantics
* Reducing cognitive load

## Why not

* Software Design/Code size
* Code clarity