OpenLLM-France/wikimedia huggingface dataset:
# Section 6.3
## 6.3.1

Show the following equation holds for all n in the set of all counting numbers.
(I) 1+3+5+...+(2n-1)=n^2
First we prove the base case. Note that 1 = 1^2. Next, assume that for any integer k, equation (I) holds. Add (2(k+1)-1)=2k+1 to both sides of (I). This yields 1+3+5+...+(2k-1)+(2(k+1)-1)=k^2+2k+1. Note that k^2+2k+1 = (k+1)^2. Thus, 1+3+5+...+(2k-1)+(2(k+1)-1)=(k+1)^2. Therefore, if (I) holds for a given integer k, it also holds for k+1. Since it has been shown that (I) holds for k=1, by induction, (I) is necessarily true for all n in the set of all counting numbers.
(II)

## 6.3.11

trivial
# Exercise 1A
## 1.

* $9x$
* $p-13$
* $k-4m+4n$
* $0$
* $r+2s-15t$

## 2.

* $4(x+2y)$
* $3(4a+5b-6c)$
* $12(6f+3g-4h)$
* $p(p-q+r)$
* $12k(k+12m-6n)$

## 3.

* $28(x+y)$
* $7b+13c$
* $-p+24q+33r$
* $2(5l+3w-h)$
* $2(w+2v)$

## 4.

* $2ab$
* $km-mn$
* $2pq-sq$
* $4x+8$
* $-2$

## 5.

* $6x^{3}y^{2}$
* $30a^{3}b^{3}c^{4}$
* $k^{2}m^{2}n^{2}$
* $162p^{4}q^{4}r^{4}$
* $24r^{2}s^{2}t^{2}u^{2}$

## 6.

* ${b} \over {c}$
* ${e} \over {2f}$
* ${x} \over {5}$
* $2a$
* ${2} \over {pr}$
# Solutions To Mathematics Textbooks/Pure Mathematics 1/Chapter 3

The fourth term of arithmetic progression is 10 and the eleventh term of it exceeds three times the fourth term by 1 find the sum of the first 20 term of the progression
# Structural Biochemistry/Bacterial transformation technique
## Bacterial Transformation

Bacterial Transformation is a basic technique that is commonly used in a molecular biological laboratory. The technique is performed by introducing a foreign plasmid into the bacteria and the bacteria is used to amplify the foreign DNA. Bacteria can replicate the foreign DNA if the foreign DNA has an origin of replication that its DNA polymerases can recognize. The bacteria will then replicate the foreign DNA along with its own. This new DNA strand can at times provide the organism with new trait(s) that can be identified after transformation.

### Role of Heat Shock in Transformation

Heat Shock is subjecting a cell to a higher temperature than it is normally found in the organism. Heat Shock causes pores to form in the outer membrane which permits DNA to enter the cell. The heat source is then removed from the cell and the membrane reforms with the DNA inside it.

### Role of Electroporation in Transformation

The role of electroporation in transformation is the same as Heat Shock, though the method is different. In electroporation, an electric field is applied to the cell that has a significant increase in the electrical conductivity and permeability of the cell membrane which allows foreign DNA to enter the cell. Bacteria also contain a cell wall made of peptidoglycan that are naturally porous and only act as a shell that protects the bacteria. If plasmids are mixed with bacteria, plasmids can be transferred into the cell after electroporation. The electric field in electroporation is created by appliances called electroporators.

### Role of Calcium Chloride in Transformation

The addition of Calcium Chloride produces calcium ions which help the cell membrane become more permeable to DNA plasmid. This method of disrupting the cell membrane using electronics is generally used in conjunction to heat shocking. In which drastic changes in temperature causes the membrane to be more porous, enabling a greater efficiency for the desired DNA segments to transform into the targeted bacteria. This method of competence is more effective when used with plasmid DNA than linear DNA since certain enzymes exist within the bacteria that readily degrade linear DNA.
# Structural Biochemistry/Bilirubin
## Introduction

Bilirubin is the product that is formed from further breakdown of heme. As a waste product, bilirubin is secreted into bile and is also what gives urine it's yellow color. Additionally, yellow bruises obtain their color also from bilirubin in a condition called jaundice. Since bilirubin is a breakdown from hemoglobin, it is generated in large quantities as red blood cells undergo turnover. Phagocytes then absorb the dead red blood cells where it is converted into free bilirubin and then released into plasma and eventually absorbed by liver cells. Finally, bacteria in the intestine metabolizes bilirubin where it is eliminated as urine or feces, which is also one of the reasons for its brown color.

## Bilirubin Blood Test

As the name suggests, the bilirubin test measures the amount of bilirubin in a blood sample usually taken from a heel stick for a baby or from the vein for an adult. The reason for this test is because it can be used to check for liver function and watch out for early signs of liver disease, diagnose conditions that affect the destruction of red blood cells, or to find out if something blocking
Conditions that may affect the test are caffeine, which can lower bilirubin levels, and fasting, which increases bilirubin levels.

## References

"Biliary Excretion of Waste Products: Elimination of Bilirubin." About.com: Biology. N.p., n.d. Web. 5 Dec. 2012. <http://biology.about.com/library/organs/bldigestliver4.htm>.
Pope, John. "Bilirubin." WebMD. Healthwise, 13 May 2010. Web. 5 Dec. 2012. <http://www.webmd.com/digestive-disorders/bilirubin-15434>.
# Structural Biochemistry/Benadryl
## Introduction

Benadryl, also known as Diphenhydramine, is an antihistamine drug. It is usually found as a crystalline, white powder and is soluble in water and alcohol. Diphenhydramine is typically used for treating allergic reactions.
In the body, histamine is released when one suffers through an allergic reaction or even viral infections. The histamine then binds to cell receptors, triggering changes in the cell that lead to allergy symptoms such as itching and sneezing.
As an antihistamine drug, diphenhydramine competes against the histamine for the cell receptors. The antihistamine not only prevents histamines from binding and stimulating the cells but while doing so, it also binds to the receptors—they do not stimulate the cells, therefore allergy symptoms will not show.

## History

Benadryl was discovered by George Rieveschl, who took part in many experiments with focus on relieving muscle pain at University of Cincinnati. Benadryl is an antihistamine drug that used for treating people who suffer with allergy. Benadryl was first available through prescription in 1946. Later in 1980s, it was approved as over-the-counter drug.

## Content

Benadryl contains the histamine-blocker diphenhydramine. This drug will help patients to recover from common symptoms: sneezing, watery eyes, runny nose, and sore throat. One application of Benadryl is Benadryl Chesty Cough & Nasal Congestion: a drugs that is used to relieve the symptoms of chesty cough and cold.

## Side Effects

Do not use this drugs if you are currently treating with depression, heart condition, blood pressure, and behavioral disorder. As many other drugs, Benadryl does have some common side effects:
* sleepiness
* dizziness
* headache
* dry mouth
* difficulty urinating.

## Chemistry

Diphenhydramine is synthesized as shown below. The molecular formula of Diphenhydramine is C17H21NO • HCl.

## References

"Benadryl (Diphenhydramine) Drug Information". RXList. 2009. Retrieved March 6, 2009
Ohio History central: Benadryl; accessed Jan. 5, 2011
Benadryl for the Family Chesty Cough & Nasal Congestion Oral Liquid ; accessed Oct. 13, 2011
http://www.medicinenet.com/diphenhydramine/article.htm
Image from Wikipedia Commons
# Spanish/Translation Reference

These pages contain an alphabetical listing of English words and their Spanish counterparts for quick and easy translation.
* A
* B
* C
* D
* E
* F
* G
* H
* I
* J
* K
* L
* M
* N
* O
* P
* Q
* R
* S
* T
* U
* V
* W
# Spanish/Tenses/Subjunctive/Simple/Present

| | -ar | -ar | -ar | -er/-ir | -er/-ir | -er/-ir |
| | conjugation | example | translation | conjugation | example | translation |
| | | hablar | to speak | | comer | to eat |
| Yo | stem + e | hable | I speak | stem + a | coma | I eat |
| Tú | stem + es | hables | You speak | stem + as | comas | You eat |
| Él | stem + e | hable | He speaks | stem + a | coma | He eats |
| Nosotros | stem + emos | hablemos | We speak | stem + amos | comamos | We eat |
| Vosotros | stem + éis | habléis | You all speak | stem + áis | comáis | You all eat |
| Ellos | stem + en | hablen | They speak | stem + an | coman | They eat |
# Spanish/Tenses/Subjunctive
### The Subjunctive Mood

* Present
* Imperfect
* Future
* Present
* Imperfect
* Future
* Present
* Imperfect
* Future
* Present
* Imperfect
* Future
# Spanish/The Wrong Way To Learn Spanish/Part 2

^ The Wrong Way To Learn Spanish ^ - Part Two. In this section , we'll be looking at some paragraphs given below and translate them into Spanish.
Dorothy lived in Kansas with Uncle Henry and Aunt Em. Their house was small. It had one room and a cellar.
Dorothy vivia en Kansas con Tío Henry y Tía Em. Su casa era pequeña. Estuvo un habatacion y una sotano.
The land around the house was very flat. The soil was dry and the grass was dead. The sun had turned everything dull and gray.
La tierra alrededor de la casa era muy plana. El suelo era seco y la hierba era muerta. El sol had turned todo embotado y gris.
Aunt Em was once young and pretty. The sun and wind withered her. They made her sad, too. When the orphan Dorothy came, her laughter made Aunt Em scream in surprise.
Tía Em fue joven y bonita. El sol y el viento le marchitaban. La hicieron triste, demasiado. Cuando llegó la huérfana Dorothy, su risa hizo tía Em grito de sorpresa.
Uncle Henry never laughed. He worked hard from morning till night. He was sad and gray also. He looked stern and solemn, and rarely spoke.
Tío Henry nunca se rió. Trabajó duro desde la mañana hasta la noche. Estaba triste y gris también. Miró austero y solemne y raramente hablaba.
Toto made Dorothy laugh. He saved her from growing sad and gray. Toto was not gray; he was a little black dog. He was very cute. Toto and Dorothy played all day. She loved him very much.
Toto reír a Dorothy. Le salvó de creciente triste y gris. Toto no era gris; era un pequeño perro negro. Era muy guapo. Totó y Dorothy jugaron todo el día. Ella lo amaba mucho.
Today, they were not playing. Uncle Henry looked at the sky. It was very grey today. Uncle Henry was anxious. Dorothy held Toto. She looked at the sky, too. Aunt Em was washing the dishes.
Hoy en día, no estaban jugando. Tío Henry miró al cielo. Hoy ha sido muy gris. Tío Henry estaba impaciente. Dorothy sostuvo Toto. Ella miró el cielo, también. Tía Em estaba lavando los platos.
# Spanish/Third Conjugation Verb

Verb partir (to leave or to cut)
This is the typical example of a regular third conjugation verb in Spanish.

## Formas simples

| | Indicativo | Indicativo | Indicativo | Indicativo | Indicativo | Subjuntivo | Subjuntivo | Subjuntivo | Imperativo |
| | Presente | Pretérito imperfecto | Pretérito perfecto simple | Futuro | Condicional | Presente | Pretérito imperfecto | Futuro | Presente |
| yo | parto | partía | partí | partiré | partiría | parta | partiera / partiese | partiere | --- |
| tú | partes | partías | partiste | partirás | partirías | partas | partieras / partieses | partieres | parte |
| él | parte | partía | partió | partirá | partiría | parta | partiera / partiese | partiere | parta |
| nosotros | partimos | partíamos | partimos | partiremos | partiríamos | partamos | partiéramos / partiésemos | partiéremos | partamos |
| vosotros | partís | partíais | partisteis | partiréis | partiríais | partáis | partierais / partieseis | partiereis | partid |
| ellos | parten | partían | partieron | partirán | partirían | partan | partieran / partiesen | partieren | partan |

## Formas no personales

* Infinitivo: partir
* Gerundio: partiendo
* Participio: partido
# Spanish/The Passive Voice

La voz pasiva (the passive voice) is formed in Spanish with ser + participio pasado.
    El soldado en la pintura "Guernica" fue mutilado por las bombas.
    The soldier in the painting "Guernica" was mutilated by the bombs.
It is used when the agent of the verb is unknown, but can also be used (as in this example) to put more emphasis on the person or thing which receives the action as opposed to the subject.
Back to Spanish.
# Spanish/Thesaurus of Spanish Cognates/Everyday Vocabulary/Conversational Forms
## Spanish (Romance)

Note 1: The Spanish words can be mainly categorized in three groups according to what is the stressed syllable: agudas, when the stressed syllable is the last one; llanas, when the stressed syllable is the penultimate (i.e. the last syllable but one); and esdrújulas, when the stressed syllable is the antepenultimate (i.e. the last syllable but two).
* All the agudas words which end in vowel or in the consonants n or s, have a written accent on the stressed syllable (the last one).
* All the llanas words which do NOT end in vowel nor in consonants n or s, have a written accent on the stressed syllable (the penultimate one).
* And all the esdrújulas words have an written accent on the stressed syllable (the antepenultimate one).
Note 2: The s and th [θ] are not observed by all the Spanish speakers. In Latin America and some regions of Spain, people pronounce z and c before e and i (th pronunciation) as an s, this phenomenon is called seseo. On the other hand, when the s pronunciation is replaced for the th one, it is called ceceo.
| Translation | Phrase | IPA | Pronunciation | Remarks |
| Spanish | castellano español | /kasteˈʎano/ /kasteˈʝano/ /espaˈɲol/ | |
| let's go | vámonos | /'bamonos/ | | |
| hello | hola | /ˈola/ | listen | |
| good morning | buenos días | /bwenosˈð̞iːas/ | listen | lit. good days |
| goodbye | adiós | /aˈð̞jos/ | listen | from a Dios to God |
| see you (later) | hasta luego | /ˈastaˈlweɰo/ | listen | lit. until (we see) later |
| please | por favor | /por faˈβ̞or/ | listen | lit. by favour |
| thank you | gracias | /ˈgraθjas/ /ˈgrasjas/ | listen | |
| you’re welcome don’t mention it | de nada | /deˈnað̞a/ | listen | lit. (it was) of nothing |
| sorry | perdón | /perˈð̞on/ | listen | lit. (I beg your) pardon |
| pardon me | perdóneme | /pɛɾˈð̞o.ne.me/ | |
| I’m sorry | lo siento | /loˈsjɛn.to/ | listen | lit. I feel (bad) for it |
| forgive me | discúlpame disculpe (formal) | /disˈkul.pa.me/ | | |
| this one | éste (masculine) ésta (feminine) | /ˈeste/ /ˈesta/ | listen listen | |
| that one | ése (masculine) ésa (feminine) | /ˈese/ /ˈesa/ | | |
| that yonder | aquél (masculine) aquella (feminine) | /aˈkel/ /aˈkeʎa/ /aˈkeʝa/ | | |
| how much? | ¿cuánto? | /ˈkwanto/ | | |
| English | inglés | [iŋˈgles] | | |
| yes | sí | /ˈsi/ | | |
| no | no | /ˈno/ | | |
| I don’t understand | no comprendo no entiendo | /nokomˈprendo/ /noenˈtjendo/ | | |
| I don’t know | no sé | /noˈse/ | | |
| where is the bathroom? | ¿Dónde está el baño? | /dondesˈtaelˈβ̞aɲo/ | |
| bless you | salud | /saˈluð̞/ | | lit. health |
| Do you speak English? (direct polite) | ¿Habla inglés? | /ˈaβ̞laiŋˈgles/ | (listen) | polite question |
| What is your name? | ¿Cómo se llama? | | | polite question |
| hopefully, I hope (that) | ojalá (que) | /oxaˈla/ | | |
# Spanish/Tenses/Subjunctive/Simple

The present subjunctive in Spanish is formed by adding the following endings to the stems of verbs: -ar: -e, -es, -e, -emos,- éis, -en -er/-ir: -a, -as, -a, -amos, -áis, -an
Verbs that are stem changing/ boot verbs in the present indicative will do the same in the present subjunctive. If a verb has an irregular yo form in the present indicative that is the base form for the present subjunctive; hence tener, tengo, becomes: tenga, tengas, tenga, etc., or oír, oigo, becomes: oiga, oigas, oiga, etc.
Verbs that end in –uir, like huir, which have a ‘y’ between stem and ending in the boot (all forms but nosotros and vosotros) in the present indicative function as yo irregulars: all six forms in the present subjunctive have the ‘y’: huya, huyas, huya, huyamos, huyáis, huyan.
For phonetic reasons there are various spell changes. The –car, -gar, -zar, -guar yo spell changes in the preterit occur in all six forms of the present subjunctive. Sacar: saque, saques, saque, etc. Llegar: llegue, llegues, llegue, etc. Lanzar: lance, lances, lance, etc. Averiguar: averigüe, averigües, averigüe, etc. Also yo spell changes from the present indicative will occur in all six forms in the present subjunctive for: -ger/gir, consonent+-zar, -guir. Escoger: escoja, escojas, escoja, etc. Vencer: venza, venzas, venza, etc. Seguir: siga, sigas, siga, etc.
-ir stem changing verbs will have a change in the nosotros and vosotros forms, with the ‘o’ in the stem becoming a ‘u’ or the ‘e’ becoming an ‘i’. Morir: muera, mueras, muera, muramos, muráis, mueran. Sentir: sienta, sientas, sienta, sintamos, sintáis, sientan.
There are six irregular verbs in the present subjunctive: Ser: sea, seas, sea, seamos, seáis, sean Ir: vaya, vayas, vaya, vayamos, vayáis, vayan Saber: sepa, sepas, sepa, sepamos, sepáis, sepan Haber: haya, hayas, haya, hayamos, hayáis, hayan Estar: esté, estés, esté, estemos, estéis, estén Dar: dé, des, dé, demos, deis, den
The imperfect subjunctive in Spanish is formed by adding the following endings to the third person plural of the preterit after removing the ‘-ron’ -ra, -ras, -ra, ’-ramos, -rais, -ran
Hence hablar, hablaron, becomes: hablara, hablaras, hablara, habláramos, hablarais, hablaran
Or vivir, vivieron, becomes: viviera, vivieras, viviera, viviéramos, vivierais, vivieran An alternative, less used, set of endings is also available: -se, -ses, -se, ’-semos, -seis, -sen
The future subjunctive (not commonly used but an accepted form), like the imperfect subjunctive, is formed using the third person plural of the preterit after removing the ‘-ron’. The endings are: -re, -res, -re, ’-remos, -reis, -ren
# Structural Biochemistry/Immunohistochemistry

What is it?
It is a method that is utilized to detect antigens on a cellular level within a tissue or population of cells. In nature, antibodies are created from lymphocytes (mammals) and B cells, these antibodies will recognize certain antigens and bind to their epitopes and promote antigen elimination. Many times certain areas under study are usually a small and specific site, with this blocking must be applied. Blocking is usually done by applying 5% of serum that is normal, this serum contains random host antibodies and allows these random antibodies to bind to the non-specific sites so that the primary antibody will only bind to the site of study.
# Structural Biochemistry/Incorrectly Established Results

The field of biology is an experimental science, where the logical theory is only as good as its experimental background. Therefore, there comes a dilemma when accepting theories, should the scientist faithfully accept the known theory for what it is or should the scientist attempt to verify them experimentally? One of the main causes of mental inertia is due to these incorrectly established results. The first form of mental inertia is belief in generally accepted observations, oftentimes due to blind faith in the authoritative source. For example "...Aristotle thought there were eight legs ona fly and wrote it down. For centuries, scholars were conetnt to quote his authority'. This idea could be easily refuted yet due to mental inertia scientists just accepted this theory of years and years.
Another form of mental inertia is when scientists still adhere to the observations made with faulty techniques when a better technique is already available. If an incorrect observation was made before better techniques came around, then that is an mistake, not mental inertia. It is only mental inertia if better techniques came, that would allow the scientists to improve the experiment and observations, and the scientist still uses the old observations for the basis of their research.

## References

* ↑ Chase, S. (1938) The Tyranny of Words, Harcourt, Brace and Company
# Structural Biochemistry/Incorrect Understanding of a Mechanism Action
## Overview

Mechanism actions are constantly being improved and updated as new information becomes available. Its a scientists' responsibility to keep up to date with what's new, as well as think critically about what is being stated and implied. However, when scientists don't think critically and merely accepts things for they are, they fall prey to mental inertia. Mental inertia caused by an incorrect understanding of a mechanism action can be split in four categories: naming, linkage groups, scientific paradigms, and improper controls.
Mental inertia can be caused by the naming of a mechanism. While it may not seem like it, the naming of a mechanism is extremely important. A mechanism's name implies its function and the group of mechanisms it belongs to. Therefore if a mechanism is poorly named it could cause confusion or incorrect assumptions. For example, hormone-independent activation of hormone receptors were overlooked for years because these proteins were initially called 'receptors for hormones'. Mental inertia can also be caused by incorrectly looking at linkage groups-when involvement of the first mechanism is considered as evidence for involvement of the second, and likewise the lack of involvement of the first mechanism means the lack of involvement of the second mechanism. This causes faulty thinking when different mechanisms for different stages of the same process are seen as different indications of a single mechanism. Scientific paradigms causes mental inertia by hindering scientists from thinking outside the traditional paradigms. For example, a set scientific paradigm was the idea that the death of a normal mammalian cell only occurs because something is killing it. Because of this paradigm, nobody considered the idea that normal cells can commit suicide for years. Now, this idea of cell suicide is currently one of the cornerstones of biology. Lastly, mental inertia can be caused by improper controls for an experiment, when controls are merely accepted because they are commonly used. This mental inertia is self-perpetuating because as more improper controls are used, the more common they become.