OpenLLM-France/wikimedia huggingface dataset:
# Roman Culture/Achilles

Achilles, the great Trojan soldier, was born of the Nereid Thetis and the mortal Peleus. After his birth, his mother Thetis left him due to certain complications. His father took him to the wise Chiron, a Centaur living in Mount Pelion. Chiron educated and fed Achilles throughout his childhood years. During that time, Calchas prophesied that Trojan would soon be taken by Achilles.
However, because of this conflict, Achilles will end up dying at a very young age. Realizing the destiny that was bestowed upon her son, Thetis tried many different things to help him avoid it. One of her plans involved dressing Achilles in a female outfit and entrusting him to king Lycomedes. This did not last very long, as Odysseus foiled the plan by revealing Achilles' true identity. After this event took place, Achilles became a general at the age of 15, and the siege of Troy soon began. The purpose of the war again Troy was to restore Helen and the Spartan property which was stolen by Paris. On the way to Troy, Achilles came upon a island called Tenedos. There, Achilles killed King Tenes, who was honored by Apollo. It was said that whoever killed Tenes would be slain by Apollo himself. Thetis sent a servant to warn Achilles of the situation, but he was clouded with anger after his sweetheart Hemithea (Tenes' sister) got away with the help of her brother. Achilles' mother also warned him not to be the first to step foot on Trojan ground, for the first to step on Trojan ground is the first to die.
During the war, King Agamamnon took one of Apollo's priest daughters who set a plague on the Trojan Army. Seeing the situation, Calcas advised the king to return the daughter to save the Army. Agamamnon agreed, but was not satisfied, and so decided to take claim of Achilles' prize which was Briseis. With this, Achilles was filled with anger and sorrow which caused him to hide in depression away from the ongoing war. Without his presences on the battlefield, the Achaeans faced many defeats against the Trojan army. Agamamnon, realizing how important Achilles was, agreed to give him wealth and riches in return for his service. However, Achilles refused those gift because of his hatred towards the king.
As the Trojans came closer to the campsite of the Archaeans, Achilles ordered his best friend Patroclus to help with the offensive. Sadly, Patroclus was killed in the battle by Hector, which brought great grief upon Achilles. He felt regret and resentment of the times in which he was idle instead of helping out with the war effort. With this, he begged his mom to make him new armor which he used to destroy Hector and avenge his best friend, Patroclus. His mother agreed, and with his new armor Achilles dove into the Trojan army and destroyed many of their men. Soon, with Achilles' help, the Archaeans took the over the city of Troy.
There was no record of how Achilles died but it was said to he was slain by Paris and Apollo.

## References

* Achillies. N.p., n.d. Web. 12 Dec. 2011. <http://www.maicar.com/GML/Achilles.html>.
Checked for grammar and originality by Griffin Donohue on 12/15/11
# Role Playing Games/Types of RPGs
## Settings

One common way of differentiating RPG is by setting. Common settings are Science Fiction, Contemporary, Historical, Fantasy, or some mixture of these.
For example, Mass Effect and Cyberpunk 2020 are science fiction series. The Fallout series is primarily science fiction, with light touches of Fantasy. A game like Shadowrun is nearly half science fiction and half fantasy.
Some games take place in a setting which is either meant to mirror the real world, or at least be similar to it, such as The Escapist or to a lesser extent Heavy Rain. A game like Vampire: The Masquerade uses both a contemporary setting and a fantasy setting, and integrates separation between the two into gameplay.
Some fantasy RPGs include a typical game of Dungeons and Dragons. This genre also includes Dragon Age and The Elder Scrolls, among other series.
Next to discuss is Text Adventures.
# Roman Culture/Battle of Cannae

The Battle of Cannae The Battle of Cannae occurred in the spring of 216 BC and is the feature battle of the Second Punic War. Hannibal had led Carthage to victories over the Romans in the battles of Trebia and Trasimene, in 218 BC and 217 BC respectively and was to again show his tactical superiority over the Romans once again at Cannae in southeast Italy. Against the odds his army of approximately 46,000 infantry and 8,000 Calvary defeated the enormous Roman army that consisted of about 87,000 men.
The tactics that Hannibal put into effect were the reasons the outnumbered Carthaginians defeated the Romans with such ease. Hannibal slowly guided the front like back in a timely retreat. The infantry of Carthage bent and the African mercenaries pushed up the flanks and attacked the sides of the Roman block of men. After the Calvary of the Carthaginians had chased the Roman Calvary off, they hit the Romans from the back, closing the Romans in completely. In all the Carthaginians were able to systematically kill Romans, killing more than 50,000 Roman men in all.
EDITED: by Ariel Turpin on 12/13/2011 Checked for spelling, grammar, and plagiarism.
Sources: http://militaryhistory.about.com/od/battleswarsto1000/p/cannae.htm, http://www.youtube.com/watch?v=CQNCGqfjaBc
# Roman Culture/Dining

Although there were many different types of people in Ancient Rome, nearly all of them had the same dining rituals. Breakfast and lunch were light meals. Lunch was usually a cold meal and it was eaten around 11:00 a.m. Dinner was the most important meal of all. The dining room was called a triclinium, and it consisted of a square table surrounded by three couches. Wealthy Romans probably would have had several dining rooms to entertain more guests or they would eat outside if the weather was nice. Diners had a unique way of eating. They would lie on their sides, leaning on their left elbows while facing the table. Their servants or slaves would serve them food. The diners would then eat the food with their fingers and sometimes spoons were used. Wealthy families would usually have three courses. The appetizers, gustatio, would include eggs, shellfish or vegetables. The entrees, prima mensa, would usually be cooked vegetables and meat. The dessert, mensa secunda, would usually be fruit or pastries. Drinking wine was part of daily life as well.
If guests were invited, the dinners were much more extravagant. These dinner parties would involve many elaborate courses. Hosts would try to impress their guests with the meals, often offering unique dishes, like ostrich or flamingo. There would often be entertainment between each course, and a literary performance after dinner. Also, there was special seating arrangement for guests. Guests were seated according to their status. The best seat was on the middle couch, to the right of the host.
For most Romans, however, dinner was much more simple. Poor Romans often ate cold food due to their lack of cooking facilities. They would usually only eat porridge and bread, and would buy meat and vegetables only if they could afford them. Although dinner menus may have been different among the Roman families, dinnertime was an important part of being a Roman, no matter what their status was.
EDIT: 12-10-11 Rojen: scanned for spelling and plagiarism. added reference to docstoc due to very similar topic and phrasing.

## References

"Ancient Roman Diet - Crystalinks." Crystalinks Home Page. Web. 01 Dec. 2011. <http://www.crystalinks.com/romefood.html>.
"The Roman Empire: in the First Century. The Roman Empire. Life In Roman Times. Home Life | PBS." PBS: Public Broadcasting Service. 2006. Web. 01 Dec. 2011. <http://www.pbs.org/empires/romans/empire/home.html>.
"The Senators." Docstoc – Documents, Templates, Forms, Ebooks, Papers & Presentations. Web. 10 Dec. 2011. <http://www.docstoc.com/docs/83448234/The-Senators>.
# Real Analysis/Section 2 Exercises/Answers
## On Sequences
### Problem 1

Strictly Increasing:
* We need to establish if it's monotone, and what kind (is it strictly increasing, strictly decreasing, non-increasing, non-decreasing, what?). Given the problem, we'll assume strictly increasing.
* First, we should prove the base case. This means proving that a$_1$ < a$_2$.${\begin{aligned}a_{1}&<a_{2}\\{\frac {1}{2}}&<{\frac {2}{3}}\\3&<4\end{aligned}}$
* Next, we should prove that this works for any number n${\begin{aligned}a_{n}&<a_{n+1}\\{\frac {n}{n+1}}&<{\frac {n+1}{n+2}}\\n(n+2)&<(n+1)(n+1)\\n^{2}+2n&<n^{2}+2n+1\\0&<1\\\end{aligned}}$
* You're done! Everything checks out and is valid.

## General

For problem 2, we see that supx_n =1 since n/(n+1) gets close to 1.
# Real Analysis/Symbols

We begin with listing various sets of numbers that are important in mathematical analysis.
Sets of numbers :
| $\mathbb {N}$ or N | The natural numbers |
| $\mathbb {Z}$ or Z | The integers |
| $\mathbb {Q}$ or Q | The rational numbers |
| $\mathbb {R}$ or R | The real numbers |
| $\mathbb {C}$ or C | The complex numbers |
List of mathematical symbols :
| $\forall$ | For all |
| $\exists$ | Exists/There Exists |
| $\subseteq ,\subset$ | Subset, Proper Subset |
| $\supseteq ,\supset$ | Superset, Proper Superset |
| $\in$ | Belongs to |
| $\setminus$ | Set Subtraction |
| $\cup$ | Union |
| $\cap$ | Intersection |
| $|x|$ | Absolute value |
| $\sup$ | Supremum/Least Upper Bound |
| $\inf$ | Infimum/Greatest Lower Bound |
| $\emptyset ,\phi$ | Empty Set |
| $\wedge$ | Logical And |
The Greek Alphabet :
| $\mathrm {A} ,\alpha$ | Alpha |
| $\mathrm {B} ,\beta$ | Beta |
| $\Gamma ,\gamma$ | Gamma |
| $\Delta ,\delta$ | Delta |
| $\mathrm {E} ,\epsilon$ | Epsilon |
| $\mathrm {Z} ,\zeta$ | Zeta |
| $\mathrm {H} ,\eta$ | Eta |
| $\Theta ,\theta$ | Theta |
| $\mathrm {I} ,\iota$ | Iota |
| $\mathrm {K} ,\kappa$ | Kappa |
| $\Lambda ,\lambda$ | Lambda |
| $\mathrm {M} ,\mu$ | Mu |
| $\mathrm {N} ,\nu$ | Nu |
| $\Xi ,\xi$ | Xi |
| $O,o$ | Omicron |
| $\Pi ,\pi$ | Pi |
| $\mathrm {P} ,\rho$ | Rho |
| $\Sigma ,\sigma$ | Sigma |
| $\mathrm {T} ,\tau$ | Tau |
| $\Upsilon ,\upsilon$ | Upsilon |
| $\Phi ,\phi$ | Phi |
| $\mathrm {X} ,\chi$ | Chi |
| $\Psi ,\psi$ | Psi |
| $\Omega ,\omega$ | Omega |
# Making a Three String English Sinnet

The Three String English Sinnet is the most simple sinnet to create. You start by putting the three strings next to each other. Then you pick the left string and cross it over the middle string, making the left string the middle string and vice versa. Then, you pick the right string, and cross it over the middle string, making the right string the middle string, and vice versa. Repeat this process alternating the left and right string, until the desired length is reached.

# Description

This sinnet is a braid, and is therefore primarily used for decoration. Another use for this sinnet is creating a stronger rope by braiding several weaker ropes. A Three String English Sinnet will create a rope three times stronger than either of the three individual ropes (provided the three ropes used were of the same quality and strength).

# Other Names

* Common Braid
* Braid
* Three String Braid
* Vlecht (Dutch)
* Driestrengs Vlecht (Dutch)
* Driestrengs Engelse Platting (Dutch)

# Related Knots

* Five String English Sinnet
# Scrabble/Challenging

If a player forms any word on the board that an opponent believes is not in the agreed-upon dictionary, the opponent can challenge the play. If the play is valid, the challenger loses his or her turn. If the play is invalid, it must be removed from the board, and the person who played it loses her/his turn. In tournament rules, if player A makes a play, his or her opponent has the right to challenge until player A draws and looks at a tile.
Note: If any challenged word is found to be unacceptable, the entire play is unacceptable.
Note: The challenge penalty is not always universal, and some tournaments (including the World Scrabble Championship, and many North American CSW tournaments) utilize 5-point challenge, where incorrectly challenging a valid play penalizes the challenger 5 points per word challenged (or 5 points are added per word for the challenged player), but no loss of turn. If a player plays a phony which is challenged, the player still loses his/her turn.
These challenge rules suggest strategies:
The playing level of the opponent should be taken into account. If you are playing an expert and she/he plays an obscure 15-point play, it is probably not worth challenging such a low score. However, if you are playing a weaker opponent and he/she plays a word you have never seen before, you may want to consider challenging. You also want to assess the board position if the play is left on the board - if your opponent plays an obviously phony word that gives you a huge play, you should generally accept the play rather than challenge it off immediately.
Bluffing is occasionally implemented in tournaments. However, you should not play a word and try to draw a challenge by saying, "I'm not too sure if it's valid" or "I'm not sure if it's spelled with an A or E" after playing CALENDER. This practice is not allowed in tournament play. Bluffing is most common when the player has a very high-scoring play in which she/he is unsure of the validity, or as a desperation play where any other play leads to defeat.
Additionally, it is sometimes advisable to set up traps for a weaker opponent. For example, you can play THIO, setting up (THIO)L. An unsuspecting opponent who tacks an S onto THIO, making THIOS*, will lose her/his turn.
In club and tournament play, you can "hold" a play if you are deciding on challenging. This does not neutralize the game clock. If you are holding, the opponent cannot draw tiles for 15 seconds. After 15 seconds, the opponent may draw courtesy tiles, but must keep them separate from remaining tiles, and must show them to you if his/her play is challenged off the board. Holding a play can be useful when the decision to challenge a play is more complex.
Lastly, always challenge the final play of the game if you are the slightest bit unsure of its validity.
# Scrabble/Dealing With a Bad Rack

| A1 | E1 | E1 | E1 | I1 | I1 | I1 |
Sometimes, due to luck of the draw, you will get stuck with a poor rack such as above. Typically, such racks could consist of all vowels, all consonants, or letters that simply do not combine well together (e.g. BCFHUUV, LLLNNOO, UUUUVVW).

## Should I exchange?

You may be tempted to exchange most or all your tiles when faced with a bad rack. In some cases, this is indeed the best option. However, there are some things to consider.
In some cases, you may be able to dump your poor tiles to form a long word, such as AALII or AUREOLAE. Here, it helps to know a lot of vowel-heavy words.
If you can't play a long word, it's still quite possible to make a good score with a short word provided you land on bonus squares. For instance, if you have a rack with only consonants, but have a Q and a T, and there is an open A on the board, you can play QAT for at least twelve points, and more if you hit bonus squares. (This usually works better with consonants, as vowels do not score as much.) However, take a look at your leave as well. If you have a bad rack now and you play only two letters, the chances of having a bad rack on your next turn are pretty high - after all, you will still have five bad tiles.
Remember that you cannot exchange when there are fewer than seven tiles left in the bag.
If you do decide to exchange tiles, make sure that the tiles you keep are balanced (e.g. don't keep too many vowels or too many consonants) and likely to score well. It is generally a good idea to avoid duplicating tiles, other than E, S, or blank.

## Should I pass?

During the beginning or middle of the game, passing is almost never a good strategy, even if you cannot make a good play due to a poor rack, or if you can play a really high-scoring bingo if your opponent puts an open E. Passing signals to your opponent that you want him to play a certain tile, and as a result, a strong opponent will likely play a closed board, or exchange to improve his rack, which does you no good. Other than endgame situations where making a valid play is impossible, and some rare pre-endgame positions, passing with a poor rack is never a good idea.

## Memorize vowel-heavy words

Vowel dumps can be very handy if you are stuck with a rack full of vowels. For example, if your rack is AEEILOU, rather than exchanging, you should look for a G to play EULO(G)IAE. Vowel dumps worth memorizing are the 4-letter words with three vowels, the 5- and 6-letter words with four vowels, and the 7- and 8-letter words with five or six vowels.
# Scouting/Knots/Decorational Knots/Sinnets

Sinnets consist of 3 or more interwinded laces or ropes. The most simple and most famous sinnet is the Three String English Sinnet. Sinnets can consist of almost any number of strings although they become more and more complicated when more strings are added. Sinnets also include variants of braids.
* Three String English Sinnet or Common Braid
* Five String English Sinnet
* Four String French Sinnet
* Five String French Sinnet
* Four String Square Sinnet
* Eight String Square Sinnet
* Six String Round Sinnet
* Crown Sinnet
* Alternating Crown Sinnet
* Round Crown Sinnet
* Chain Sinnet
* Solomon Bar
* Bannister Bar
# Making a Five String English Sinnet

The Five String English Sinnet is a very simple sinnet to create, and closely related to the Three Strings version. You start by putting the five strings next to each other. Then you pick the leftmost string and cross it over the middle string. Then, you pick the rightmost string, and cross it over the middle string. Repeat this process alternating the leftmost and rightmost string, until the desired length is reached.

# Description

This sinnet is a braid, and is therefore primarily used for decoration. Another use for this sinnet is creating a stronger rope by braiding several weaker ropes. A Five String English Sinnet will create a rope five times stronger than either of the five individual ropes (provided the five ropes used were of the same quality and strength).

# Other Names

* Five String Braid
* Vijfstrengs Vlecht (Dutch)
* Vijfstrengs Engelse Platting (Dutch)

# Related Knots

* Three String English Sinnet
* English Sinnets with more strings are created the same manner. Any uneven number of strings can be used.
# UMD Probability Qualifying Exams/Aug2010Probability
## Problem 1

* Two persons, A and B, are playing a game. If A winsa a round, he gets $4 from B and wins the next round with probability 0.7. If A loses the round, he pays $5 to B and wins the next round with probability 0.5.
* (i) Write downt he transition matrix of the Markov chain with two states, {A won the current round, B won the current round} and find the stationary probabilities of the states
* (ii) Find $\lim _{n\to \infty }P(A{\text{ has more money after }}n{\text{ rounds than before the game}})$.

### Solution

(i) The Markov transition matrix will be the 2x2 matrix $Q=(q_{ij})$ where $i,j=1$ corresponds to a win for Player A and $i,j=0$ corresponds to a loss for Player A. For example, $q_{11}$ is the probability that Player A wins after winning in the previous hand; $q_{01}$ is the probability that Player A wins after losing in the previous hand; etc. This will give
$Q=\left({\begin{array}{cc}0.7&0.3\\0.5&0.5\end{array}}\right)$
The stationary distribution will be the tuple $\pi =(a,b)$ such that $\pi Q=\pi$. We can calculate this explicitly:
$(a,b)\left({\begin{array}{cc}0.7&0.3\\0.5&0.5\end{array}}\right)=(a,b)$ yields the following system of equations: $.7a+.5b=a;.3a+.5b=b$ Using the fact that $\pi$ must be a probability (i.e. $a+b=1$) we get $a={\frac {5}{8}},b={\frac {3}{8}}$.
(ii) Since $Q$ is positive, and hence ergodic, then any initial probability distribution will converge to the stationary distribution just calculated, $\pi$. Thus as $n\to \infty$ Player A will win with probability $5/8$. Can Player A expect to have more money though? For sufficiently large $n$ we can compute Player A's expected winnings in one round:
$E[{\text{winnings}}]=5/8*4+3/8*(-5)=5/8>0$
Thus Player A should expect to have more money than before the game with probability 1.

## Problem 2

* (i) Let $X$ be a random variable with zero mean and finite variance $\sigma ^{2}$. Show that for any $c>0$
* $P(X>c)\leq {\frac {\sigma ^{2}}{\sigma ^{2}+c^{2}}}$.
* (ii) Let $\{X_{n},n\geq 1\}$ be a square-integrable martingale with $E(X_{1})=0$. Show that for any $c>0$
* $P(\max _{1\leq i\leq n}X_{i}\geq c)\leq {\frac {var(X_{n})}{var(X_{n})+c^{2}}}$.

### Solution

(i) $P(X\geq c)\leq P(|X|\geq c)=P(|X-E[x]|\geq c)\leq P(|X-E[X]|\geq c+\sigma )\leq {\frac {\sigma ^{2}}{c^{2}+2c\sigma +\sigma ^{2}}}\leq {\frac {\sigma ^{2}}{c^{2}+\sigma ^{2}}}$ were the second-to-last inequality is the standard Chebyshev's inequality.

## Problem 4

* Let $X,y$ be random variables with finite expectations.
* (i) Show that $E(X|Y)=0$ implies $E(|X+Y|)\geq E(|Y|)$.
* (ii) Show that if $(X,Y)$ is identically distributed with $(Y,X)$, then
* $E(|3X-Y|)\geq E(|X+Y|).$

### Solution

(i) Let $f(x)=|x+Y|$. Easy to see that $f$ is convex.
Then by Jensen's Inequality we have
$f(E(X|Y))\leq E(f(X)|Y)$
$|E(X|Y)+Y|\leq E(|X+Y||Y)$.
Taking the expectation on both sides gives
$E(|Y|)\leq E(|X+Y|)$.
# UMD Probability Qualifying Exams/Aug2008Probability
## Problem 1

* Let $(\xi _{1},\xi _{2})$ be a Gaussian vector with zero mean and covariance matrix $D$ with entries $D_{11}=D_{22}=1,D_{12}=D_{21}=1/2$. Find $E(\xi _{1}^{2}\xi _{2}|2\xi _{1}-\xi _{2})$

### Solution
## Problem 2

* Let $X_{n},n\geq 0$ be a Markov chain on the state space $\mathrm {X} =\{1,2\}$ having transition matrix $P$ with elements $P_{11}=1/3,P_{12}=2/3,P_{21}=P_{22}=1/2$. Let $f:\mathrm {X} \to \mathbb {R}$ be the function with $f(1)=1$ and $f(2)=4$. Find a function $g:\mathrm {X} \to \mathbb {R}$ such that
* $Y_{n}=f(X_{n})-f(X_{0})-\sum _{i=0}^{n-1}g(X_{i}),n\geq 1,$
* is a martingale relative to the filtration ${\mathcal {F}}_{n}^{X}$ generated by the process $X_{n}$.

### Solution

Notice that since $f,g$ are measurable functions, then $Y_{n}$ is composed of linear combinations of ${\mathcal {F}}_{n}$-measurable functions and hence $Y_{n}$ is ${\mathcal {F}}_{n}$-adapted. Furthermore, for any $n$, $Y_{n}$ is finite everywhere, hence is $L^{1}$.
Therefore, we only need to check the conditional martingale property, i.e. we want to show $Y_{n}=E(Y_{n+1}|{\mathcal {F}}_{n}$.
That is, we want
${\begin{aligned}f(X_{n})-f(X_{0})-\sum _{i=1}^{n-1}g(X_{i})&=E[f(X_{n+1})-f(X_{0})-\sum _{i=1}^{n}g(X_{i})|{\mathcal {F}}_{n}]\\f(X_{n})&=E[f(X_{n+1})|{\mathcal {F}}_{n}]-g(X_{n})\end{aligned}}$
Therefore, if $Y_{n}$ is to be a martingale, we must have
$g(X_{n})=E[f(X_{n+1})|{\mathcal {F}}_{n}]-f(X_{n})$.
Since $\mathrm {X} =\{1,2\}$, we can compute the right hand side without too much work.
$g(1)=E[f(X_{n+1})|X_{n}=1]-f(1)=(1\cdot 1/3+4\cdot 2/3)-1=2$
$g(2)=E[f(X_{n+1})|X_{n}=2]-f(2)=(1\cdot 1/2+4\cdot 1/2)=-{\frac {3}{2}}$
This explicitly defines the function $g$ and verifies that $Y_{n}$ is a martingale.

## Problem 3

* Let $\xi _{n}$ be independent identically distributed random variables with uniform distribution on [0,1]. For which values of $\alpha >0$ does the series
* $\sum _{n=1}^{\infty }(\xi _{n}+n^{-\alpha })^{(n^{\alpha +1})}$
* converge almost surely?
# Acknowledgements

We would like to thank the following contributors for their time and effort in creating this project:
* srinig (discuss · email · contribs · logs · count) Author
* laxmi (discuss · email · contribs · logs · count) Editor
* kgrr (discuss · email · contribs · logs · count) Initial Outline
# UMD Probability Qualifying Exams

This is a compilation of problems and solutions from past Probability qualifying (PhD) exams at the University of Maryland Mathematics Department.

## Old Exams

A compilation of old exams is located here:
Probability Exams

## Probability Exams

* January 2015
* August 2014
* January 2011
* August 2010
* January 2010
* August 2009
* January 2009
* August 2008
* January 2008
* August 2007
* January 2007
* August 2006
* January 2006