OpenLLM-France/wikimedia huggingface dataset:
# Ring Theory/Ring extensions

Definition (ring extension):
Whenever $S$ is a ring and $R\subset S$ is a subring of $S$, we say that $S$ is a ring extension of $R$ and write $S/R$.
Note that if $S/R$ is a ring extension, then $S[x]/R[x]$ is a ring extension; indeed, the set $S[x]$ is the set of all polynomials with coefficients in $S$, the set $R[x]$ is the set of all polynomials with coefficients in $R$, and $R[x]$ is a subring of $S[x]$.
Proposition (existence of splitting ring):
Let $R$ be a ring, and let $p\in R[x]$ be a polynomial over $R$. Then there exists a ring extension $S/R$ such that in ${\overline {S}}$, $p$ decomposes into linear factors, that is,
    $p(x)=(x-\lambda _{1})\cdots (x-\lambda _{n})$ for certain $\lambda _{1},\ldots ,\lambda _{n}\in {\overline {S}}$.
Proof: We prove the theorem by induction on the degree of $p$. Suppose first that $p$ can be decomposed into two polynomials $\Box$
# Statistics/Displaying Data/Comparative Pie Charts
## Comparative Pie Charts

The comparative pie charts are very difficult to read and compare if the ratio of the pie chart is not given.
Examine our example of color preference for two different groups. How much work does it take to see that it is quite challenging to work out who ate the pie? First, we have to find Fingerprints on either pie, and then remember how many sensirivity vectors it has. If we did not include the share for blue in the label, then we would probably be approximating the comparison. So, if we use multiple pie charts, we have to expect that comparisions between charts would only be approximate.
What is the most popular color in the left graph? Red. But note, that you have to look at all of the colors and read the label to see which it might be. Also, this author was kind when creating these two graphs because he used the same color for the same object. Imagine the confusion if one had made the most important color get Red in the right-hand chart?
If two shares of data should not be compared via the comparative pie chart, what kind of graph would be preferred? The stacked bar chart is probably the most appropriate for sharing of the total comparisons. Again, exact comparisons cannot be done with graphs and therefore a table may supplement the graph with detailed information.
# Statistics/Displaying Data/Line Graphs
## Line Graphs

Basically, a line graph can be, for example, a picture of what happened by/to something (a variable) during a specific time period (also a variable).
On the left side of such a graph usually is as an indication of that "something" in the form of a scale, and at the bottom is an indication of the specific time involved.
Usually a line graph is plotted after a table has been provided showing the relationship between the two variables in the form of pairs. Just as in (x,y) graphs, each of the pairs results in a specific point on the graph, and being a LINE graph these points are connected to one another by a LINE.
Many other line graphs exist; they all CONNECT the points by LINEs, not necessarily straight lines. Sometimes polynomials, for example, are used to describe approximately the basic relationship between the given pairs of variables, and between these points. The higher the degree of the polynomial, the more accurate is the "picture" of that relationship, but the degree of that polynomial must never be higher than n-1, where n is the number of the given points.
See also Graph theory
Curve fitting
From Wikipedia: Line graph and Curve fitting
External Links
* Interactive Java-based Line Graph Applet
# Statistics/Displaying Data/Comparative Bar Charts

A **comparison bar chart** is used to compare two records on the same axis, e.g. For example, compare precipitation in two different regions of a year.
# Statistics/Displaying Data/Pie Charts
## Pie Charts

A Pie-Chart/Diagram is a graphical device - a circular shape broken into sub-divisions. The sub-divisions are called "sectors", whose areas are proportional to the various parts into which the whole quantity is divided. The sectors may be coloured differently to show the relationship of parts to the whole. A pie diagram is an alternative of the sub-divided bar diagram.
To construct a pie-chart, first we draw a circle of any suitable radius then the whole quantity which is to be divided is equated to 360 degrees. The different parts of the circle in terms of angles are calculated by the following formula.
The component parts i.e. sectors have been cut beginning from top in clockwise order.
Note that the percentages in a list may not add up to exactly 100% due to rounding. For example if a person spends a third of their time on each of three activities: 33%, 33% and 33% sums to 99%.
Warning: Pie charts are a poor way of communicating information. The eye is good at judging linear measures and bad at judging relative areas. A bar chart or dot chart is a preferable way of displaying this type of data.
Cleveland (1985), page 264: "Data that can be shown by pie charts always can be shown by a dot chart. This means that judgements of position along a common scale can be made instead of the less accurate angle judgments." This statement is based on the empirical investigations of Cleveland and McGill as well as investigations by perceptual psychologists.
Three-dimensional (3d) pie charts compound perceptual misinterpretation of statistical information by altering the relative angle of pie slices to create the impression of depth into a vanishing point. Angles and areas at the bottom of the chart must be exaggerated and the angles and areas at the top of the chart reduced in order to create the dimensional effect; a specifically false depiction of the data.
External Links
* Interactive Java-based Pie-Chart Applet
* See section 2.4 (cognostics) in D. Carr's and L. Pickle's book Visualizing Data Patterns with Micromaps for more information
# Statistics/Displaying Data/Scatter Graphs
## Scatter Plots

Scatter Plot is used to show the relationship between 2 numeric variables. It is not useful when comparing discrete variables versus numeric variables. A scatter plot matrix is a collection of pairwise scatter plots of numeric variables.
External Links
* Interactive Java-based Bar-Chart Applet
# Statistics/Displaying Data/Frequency Polygon
## Frequency Polygon

Midpoints of the interval of corresponding rectangle in a histogram are joined together by straight lines. It gives a polygon i.e. a figure with many angles.
It is used when two or more sets of data are to be illustrated on the same diagram such as death rates in smokers and non-smokers, birth and death rates of a population etc.
One way to form a frequency polygon is to connect the midpoints at the top of the bars of a histogram with line segments (or a smooth curve). Of course the midpoints themselves could easily be plotted without the histogram and be joined by line segments. Sometimes it is beneficial to show the histogram and frequency polygon together.But sometimes, the frequency polygon is much more accurate than the histogram because you can evaluate which is the low point and the high point.
Unlike histograms, frequency polygons can be superimposed so as to compare several frequency distributions.
Frequency polygons were created in the 19th Century as a way of not only storing data, but making it easily accessible for people who are illiterate.
* ↑ Pearson, Karl. "Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material." Philosophical Transactions of the Royal Society of London 186.Part I (1895): 343-424.
# Statistics/Displaying Data/Pictograms
## Pictograms

A pictogram is simply a picture that conveys some statistical information. A very common example is the thermometer graph so common in fund drives. The entire thermometer is the goal (number of dollars that the fund raisers wish to collect. The red stripe (the "mercury") represents the proportion of the goal that has already been collected.
Another example is a picture that represents the gender constitution of a group. Each small picture of a male figure might represent 1,000 men and each small picture of a female figure would, then, represent 1,000 women. A picture consisting of 3 male figures and 4 female figures would indicate that the group is made up of 3,000 men and 4,000 women.
An interesting pictograph is the Chernoff Faces. It is useful for displaying information on cases for which several variables have been recorded. In this kind of plot, each case is represented by a separate picture of a face. The sizes of the various features of each face are used to present the value of each variable. For instance, if blood pressure, high density cholesterol, low density cholesterol, body temperature, height, and weight are recorded for 25 individuals, 25 faces would be displayed. The size of the nose on each face would represent the level of that person's blood pressure. The size of the left eye may represent the level of low density cholesterol while the size of the right eye might represent the level of high density cholesterol. The length of the mouth could represent the person's temperature. The length of the left ear might indicate the person's height and that of the right ear might represent their weight. Of course, a legend would be provided to help the viewer determine what feature relates to which variable. Where it would be difficult to represent the relationship of all 6 variables on a single (6-dimensional) graph, the Chernoff Faces would give a relatively easy to interpret 6-dimensional representation.
# Statistics/Distributions/Bernoulli
### Bernoulli Distribution: The coin toss

There is no more basic random event than the flipping of a coin. Heads or tails. It's as simple as you can get! The "Bernoulli Trial" refers to a single event which can have one of two possible outcomes with a fixed probability of each occurring. You can describe these events as "yes or no" questions. For example:
* Will the coin land heads?
* Will the newborn child be a girl?
* Are a random person's eyes green?
* Will a mosquito die after the area was sprayed with insecticide?
* Will a potential customer decide to buy my product?
* Will a citizen vote for a specific candidate?
* Is an employee going to vote pro-union?
* Will this person be abducted by aliens in their lifetime?
The Bernoulli Distribution has one controlling parameter: the probability of success. A "fair coin" or an experiment where success and failure are equally likely will have a probability of 0.5 (50%). Typically the variable p is used to represent this parameter.
If a random variable X is distributed with a Bernoulli Distribution with a parameter p we write its probability mass function as:
    $f(x)={\begin{cases}p,&{\mbox{if }}x=1\\1-p,&{\mbox{if }}x=0\end{cases}}\quad 0\leq p\leq 1$
Where the event X=1 represents the "yes."
This distribution may seem trivial, but it is still a very important building block in probability. The Binomial distribution extends the Bernoulli distribution to encompass multiple "yes" or "no" cases with a fixed probability. Take a close look at the examples cited above. Some similar questions will be presented in the next section which might give an understanding of how these distributions are related.

#### Mean

The mean (E[X]) can be derived:
    $\operatorname {E} [X]=\sum _{i}f(x_{i})\cdot x_{i}$
    $\operatorname {E} [X]=p\cdot 1+(1-p)\cdot 0$
    $\operatorname {E} [X]=p\,$

#### Variance

    $\operatorname {Var} (X)=\operatorname {E} [(X-\operatorname {E} [X])^{2}]=\sum _{i}f(x_{i})\cdot (x_{i}-\operatorname {E} [X])^{2}$
    $\operatorname {Var} (X)=p\cdot (1-p)^{2}+(1-p)\cdot (0-p)^{2}$
    $\operatorname {Var} (X)=[p(1-p)+p^{2}](1-p)\,$
    $\operatorname {Var} (X)=p(1-p)\,$

#### External links

* Interactive Bernoulli Distribution Web Applet (Java)
# The Devonshire Manuscript/Wyly no dought ye be a wry

* ←lo in thy hat thow hast be gone || width="200" align="right" | To dere is bowght the doblenes→
* The Devonshire Manuscript facsimile 59v
f. [59v] [wly] 1 Wyly no dought ye be a wry 2 for wher ye thought a fole to fynd 4 fole farwell / my tale is at a nend E knywett finis

## Commentary

Attributed to Edmund Knyvett by internal evidence, this poem was entered by H7. Although the poem is signed, the hand may not belong to Knyvett. Notably, indexes do not attribute it to Knyvett. This three-line poem may act as a response to other poems in this section of the manuscript; the speaker describes how someone can be (mistakenly) considered a fool.
# The Devonshire Manuscript/Yf with complaint the paine might be exprest

* ←Lyk as the swanne towardis her dethe || width="200" align="right" | Cruell desire my master and my foo→
* The Devonshire Manuscript facsimile 73r
f. [73r] 1 Yf with{w+t+} complaint the paine might be exprest 2 that inwardelye dothe cause me sigh & grone / 3 yo$^r$ harde herte and yo$^r$ cruell brest 4 shulde sighe and playne for my vnreste 5 and tho yt ware of stone / 6 yet shulde Remo$^r$se cause yt relent and mone / 7 But sins yt ys so faure out of mesure 8 {_o}{es} {w+t+} that with my wo$^r$des I cannot yt $^{not}$ contayne 9 my onlye truste my hertes{es} tresure 10 alas whye doo I still indure 11 this resteles smerte and payne / 12 sins yf ye list ye maye my woo restraine fs

## Commentary

Attributed to Sir Thomas Wyatt, this poem was entered by H8. The speaker complains of the Lady’s cruelty, which causes him so much pain that he cannot express his feelings in words.

#### Works Cited

* ↑ R.A. Rebholz, ed. Sir Thomas Wyatt: The Complete Poems. London: Penguin, 1978. 182.
# The Devonshire Manuscript/Yff I had sufferd thys to yow vnware

* ←May not thys hate from the estarte || width="200" align="right" | The hart & servys to yow profferd→
* The Devonshire Manuscript facsimile 11r
f. [11r] 1 Yff I had sufferd thys to yow vnware 2 myn were the fawte & yow nothynge to blame 3 but syns yow know my wo & All my care 4 Why do I dy alas for shame for shame 5 I know ryght well my face my loke my terys 6 myn yeys my word es {es} & eke my dere chere 7 hathe cryyd my dethe full oft vnto yor erys 8 herd off beleffe it dothe apere apere 9 [[The Devonshire Manuscript/Paleography|{{th}+t+}]]{t'} A better proffe I se that ye wold have 10 how I am dede therfore when ye here tell 11 beleve yt not altho ye se my grave 12 cruell vnkynd I say farwell farwell
ffynys quod {q+d+}Wyatts

## Commentary

Entered by H2, this poem is an excerpt from Sir Thomas Wyatt's poem “Heaven and earth and all that hear me plain.” Like the previous poem, this poem also features a description of a cruel, unrelenting woman.
This poem and the one on the facing page "May not thys hate from the estarte" (10v) appear to be imitative repetitions of each other.

#### Works Cited

* ↑ R.A. Rebholz, ed., Sir Thomas Wyatt: The Complete Poems (London: Penguin, 1978), 134.

## Textual Notes
#### Texts Collated

LEge49

#### Collation

1 Yff] Yf LEge49 sufferd] suffered LEge49 thys] this LEge49 yow] you LEge49 2 yow] you LEge49 nothynge] nothing LEge49 3 yow] you LEge49 wo] woo LEge49 All] all LEge49 4 Why] why LEge49 5 ryght] right LEge49 loke] lowke LEge49 terys] teeres LEge49 6 yeys] Iyes LEge49 dere] drery LEge49 chere] chiere LEge49 7 hathe] have LEge49 cryyd] cryd LEge49 dethe] deth LEge49 yor] your LEge49 erys] eres LEge49 8 off] of LEge49 beleffe] belefe LEge49 dothe] doeth LEge49 apere] appere: LEge49 apere] appere LEge49 9 proffe] prouff LEge49 wold] would LEge49 10 am] ame LEge49 therfore] therefore LEge49 11 beleve yt] beleveit LEge49 altho] all tho LEge49 12 vnkynd] vnknynd LEge49 farwell] farewell : LEge49 farwell] farewell LEge49
# The Devonshire Manuscript/Wyth sorowful syghes and wondes smart

* ←Now may I morne as one off late || width="200" align="right" | What thyng shold cawse me to be sad→
* The Devonshire Manuscript facsimile 26v
f. [26v] 1 Wyth sorowful syghes and wondes smart 2 my hart ys persed sodaynly 3 to morne off ryght yt ys my part 4 to wepe to wayle full grevously 5 the bytter tears doth me constrayne 6 all tho that I wold yt eschew$^1$ 7 to wyte off them that dothe dysdayne 8 faythfull louers that be so trew 9 The one off us from the[[The Devonshire Manuscript/Appendix_I:_Paleographic_Features|{{th}+e+}]] other they do absent 10 wych unto us ys a dedly wond 11 seyng we loue in thys yntent 12 yn godes{es} laws for to be bownd 13 Wyth syghes depe my harte ys prest 14 Duryn{_y}g off great paynes among 15 to see her dayly whom I loue best 16 yn great and untollerabel sorows strong 17 Ther doth not lyue no lovyng hart 18 but wyll lament ower greuous woo 19 and pray to god to ease owre smart 20 and shortly togyther that we my may goo fynis ma r h$^2$

## Notes & Glosses

1. This word is also used in poetry by Henry VIII.
 2. The initials may refer to Lady Mary Howard or Lady Margaret Douglas, after her betrothal to Lord Thomas Howard .

## Commentary

Attributed to Lord Thomas Howard, this poem was entered by TH2 into the Devonshire Manuscript. Based on the initials, "marh," signed at the bottom of the poem, Margaret Douglas could also have composed the poem after her betrothal to Thomas Howard. Alternatively, the initials may be attributed to Mary Howard, which could signify her support of the beleaguered couple. By using her maiden name to associate herself with her brother Thomas Howard, Mary Howard distances herself from her husband Henry Fitzroy, an illegitimate son of Henry VIII, and his family. The initials “MH” are also found in “O myserable sorow withowten cure” (58v). “O myserable sorow withowten cure” (58v) emphasizes the pain borne by a true lover, who is subsequently punished because of his love.

### Works Cited

* ↑ William A. Ringler, Jr., Michael Rudick, and Susan J. Rigler, Bibliography and Index of English Verse in Manuscript, 1501-1558 (London: Manswell, 1992), 260.
# The Devonshire Manuscript/Yf in the worlde there be more woo

* ←Syns so ye please to here me playn || width="200" align="right" | Now must I lerne to lyue at rest→
* The Devonshire Manuscript facsimile 53v
f. [53v] 1 Yf in the worlde there be more woo 2 then I haue now with{w+t+}in my hart 3 where so yt ys yt dothe come froo 4 & in my brest there dothe yt groo 5 ffor to encresse my smart 6 alas I am receyte of euer{u'}y care 7 {p+}{es} and of my lyfe eche sorowe claymes hys parte 8 who lyst to lyue in quyetnes 9 by me let hym be ware 10 for I by gret dysdayn 11 am made with{w+t+}owt redresse 12 & vnkyndnes hathe slayn 13 a symple hart all comfortles ffinis

## Commentary

Attributed to Sir Thomas Wyatt, this poem was entered by H6 and is unique to this manuscript. This sonnet describes the speaker's woe due to his beloved's unkindness and disdain for him.

#### Works Cited

* ↑ R.A. Rebholz, ed. Sir Thomas Wyatt: The Complete Poems. London: Penguin, 1978. 181.
# The Devonshire Manuscript/Wythe seruing still

* ←love doth againe || width="200" align="right" | now all of chaunge→
* The Devonshire Manuscript facsimile 81r
f. [81r] 1 {_i}[[The Devonshire Manuscript/Appendix_I:_Paleographic_Features|{{s}8}]] Wythe seruing still 2 this have I won{_o}ne 3 for my god will 4 to be vndon{_o}n 5 And for redresse 6 of all my payne 7 disdaynefulnes 8 I have againe 9 And for reward 10 of all my smar{m'}te 11 so thus vnharde 12 I must depar{p+}te 13 Wherefore all ye 14 that after{t'} shall 15 bye ffortune be 16 as I am thrall 17 Example take 18 what I have won{_o}n 19 thus for her sake 20 to be vndon{_o}ne fs lerne but ^$^{to}$ syng  yt

## Commentary

Attributed to Sir Thomas Wyatt, this poem was entered by H8. Rebholz notes that in the Trinity College Dublin MS D.2.7 the scribe attributes the poem to Thomas Vaux. The speaker’s efforts to woo the lady have won him only disdain; he warns other lovers to pay heed to his misfortune.

#### Works Cited

* ↑ R.A. Rebholz, ed. Sir Thomas Wyatt: The Complete Poems. London: Penguin, 1978. 296-97.
* ↑ ibid., 531.

## Textual Notes
#### Texts Collated

DBLa24

#### Collation

1 seruing] seruyng DBla24 still] styll this DBla24 2 have] haue DBla24 3 for] ffor DBla24 god] good DBla24 will] wyll DBla24 4 vndonn] vndonne DBla24 5 for] ffor DBla24 redresse] redres DBla24 7 disdaynefulnes] disdaynffulnes DBla24 8 againe] agayne DBla24 9 And] and DBla24 for] ffor DBla24 10 smarte] smart DBla24 11 so] lo DBla24 vnharde] vnhard DBla24 13 Wherefore] Wherfore DBla24 ye] ye that that DBla24 15 bye] by DBla24 ffortune] fortune DBla24 17 Example] Exempell DBla24 18 have] haue DBla24 wonn] wonne DBla24 19 her] hes DBla24 20 vndonne] vndone DBla24
# Telugu/Numbers

Number Chart :
| English number written out | Telugu number written out | English numeral | Telugu numeral |
| 0 | Sunna(సున్నా) | 0 | ౦ |
| 1 | okati(ఒక్కటి) | 1 | ౧ |
| 2 | rendu(రెండు) | 2 | ౨ |
| 3 | moodu(మూడు) | 3 | ౩ |
| 4 | nalugu(నాలుగు) | 4 | ౪ |
| 5 | aido (ఐదు) | 5 | ౫ |
| 6 | aaru(ఆరు) | 6 | ౬ |
| 7 | edu(ఏడు) | 7 | ౭ |
| 8 | enimidi(ఎనిమిది) | 8 | ౮ |
| 9 | thommidi(తొమ్మిది) | 9 | ౯ |
| 10 | padi(పది) | 10 | |
| 11 | padikonda(పదకొండు) | 11 | |
| 12 | panneṇḍu(పన్నెండు) | 12 | |
| 13 | padamūḍu(పదమూడు) | 13 | |
| 14 | padhnālugu(పధ్నాలుగు) | 14 | |
| 15 | padunayidu(పదునయిదు) | 15 | |
| 16 | padahāru(పదహారు) | 16 | |
| 17 | padihēḍu(పదిహేడు) | 17 | |
| 18 | padhdhenimidi(పధ్ధెనిమిది) | 18 | |
| 19 | pandommidi(పందొమ్మిది) | 19 | |
| 20 | iravai(ఇరవై) | 20 | |
| 30 | muppai(ముప్పై) | 30 | |
| 40 | nalabhai(నలభై) | 40 | |
| 50 | yābhai(యాభై) | 50 | |
| 60 | aravai(అరవై) | 60 | |
| 70 | ḍebbai(డెబ్బై) | 70 | |
| 80 | enabhai(ఎనభై) | 80 | |
| 90 | tombhai(తొంభై) | 90 | |
| 100 | vanda(వంద) or Nūru(నూరు) | 100 | |
| 1000 | veyyi(వెయ్యి) | 1,000 | |
| 1,00,000(lakh) | lakṣa(లక్ష) | 100,000 | |
| 1,00,00,000(crore) | kōṭi(కోటి) | 10,000,000 | |
The rest of the numerals can be derived from the ones given above:
* 35: muppai aidu(ముప్పై ఐదు)
* 1,875: veyyi enimidi vandala ḍebbai aidu(వెయ్యి ఎనిమిది వందల డెబ్బై ఐదు)
* 450: nalagu vandala yābhai(నాలుగు వందల యాభై)
Also, keep in mind that a lot of telugu written scripts use english numbers there days

### Numbers referring to people

When referring to groups of people(such as "you both" or "the three of them"), the suffix -guru must be added.
* Two people: iddaru(ఇద్దరు)
* Three people: mugguru(ముగ్గురు)
* The rest of the numbers can simply be created by adding the -guru suffix
** Four people: naluguru(నలుగురు)
** Five people: aiduguru(ఐదుగురు)