What is the difference between "unidos" and "juntos"?

Question: These words are used in political campaigns. What is the difference? Unidos or Juntos? One is a conjunction the other one is a verb. Not sure what the proper word is.
Comment: Note in the Spanish politics there is a coalition called [_Unidos Podemos_](https://es.wikipedia.org/wiki/Unidos_Podemos) because it consists in the union of mainly two parties: _Izquierda Unida_ and _Podemos_. So they use a part of the name of both to make the coalition name. _Juntos Podemos_ would have the same meaning, only that it would then not include a reference to the name of _Izquierda Unida_ in it.
Comment: The two words are very similar in their meanings, but have subtle differences, as you can see in Joan Esteban's answer. But the last part of your question is not completely clear for me. The proper word for what? You talk about political campaigns, but depending on what you need to say you should choose one word or the other.
Answer: Yes, they are synonymous, but consider the affective difference in this sentence:

Con mis padres vivimos juntos pero no somos muy unidos.
(With my parents we live together but aren't united.)

"Juntos" refers to close, while "unidos" refers rather connected or joined.
Comment: ¡Buen ejemplo! Curioso que en España diríamos _no estamos muy unidos_, queriendo decir que no nos llevamos especialmente bien.
Answer: From my point of view, "unir" means "to join things together so that they become one," and "junto" means "together."
Examples:

Vamos a unir estas piezas.

The sentence above means to join pieces together (as by soldering, for example) or in a way that can't be separated easily.

Vamos a juntar estas piezas.

The sentence above means to put pieces together, like in a box, for instance, but without soldering them together; the pieces are together but separated from one another inside the box.
Answer: In Spanish, these words are synonyms. There are verbs for each of these words: "unir" and "juntar", which are synonyms. According to the DRAE (Spanish Royal Academy Dictionary):

Unir

Hacer que una cosa esté al lado de otra, o en contacto con ella formando un todo.

Mezclar o trabar algunas cosas entre sí, incorporándolas.

Juntar

Unir unas cosas con otras.

The difference between "unido" and "junto" is that "unido" is the participle of the "unir" while "junto" is a noun by itself:

Junto

Unido, cercano.

Que obra o que es juntamente con otra persona, a la vez o al mismo tiempo que ella.

With this in mind, you can see that "unidos" and "juntos" mean the same.
Answer: It might be helpful to consider the same comparison to their English equivalent:

Unidos: United
Juntos: Together

So, while "united" and "together" could be seen as synonyms their meaning might be slightly different, as other answers have pointed out.

United we stand, divided we fail

versus

Alone we can do so little. Together we can do so much.

Also, taking in account that these words are used in political campaigns sometimes these two are used interchangeably since they convey a message similar to:

We can't do this unless you join us.

Additionally, this message appeals to the following:

A sense of belonging
Making a difference by becoming part of a cause

As you can see "united" and "together" are words which are meant to help forming groups, which is the very base of gathering a voting base. A big reason for one party to use "united" versus "together" is to attempt to distinguish (or brand) a message from one party (or candidate) from the remaining parties (or candidates).
Latin american countries can have more than two parties, which would make it more important to brand the messages of their candidates in a short "slogan" that can be memorable and sway voters.
Answer: The short answer (from me) would be this: What we're really talking about when we discuss the differences between "unidos" and "juntos" is the difference between "united" and "together."  The answers provided by Jorge and Rodrigo touch on this a bit, but I'll put it into my own words like this:
You can be united even if you aren't physically together.
You can be together (physically) but not united.
I don't have any stats on this,* but I would imagine there is a greater preponderance of political parties with the word "unidos" in them than there are parties with "juntos."  The word itelf — unidos — means "united."  If "unidos" has already been taken, however, and a new coalition is seeking to distinguish itself, "juntos" is definitely a good second choice because through context it implies a sense of unity that goes beyond physical distance.  Even so, after taking a closer look, it appears that as popular as words like "unidos" and "juntos" may be for a political party, political organizers in Spain and Latin America have not restricted themselves to just these two words when coming up with a name for their respective political groups.
But before I whittled my thoughts down to just that — that the difference was really about the difference between "united" and "together," I took a close look at each of these words.  I thought it might be helpful for you (and other students of Spanish) to read through some of my observations, but first a correction of sorts:
----------------------------------------------
*This is a note about my comment on not having collected any stats on how many political parties have contained "unidos" in their names and how many have contained "juntos."  Although I did get curious after writing that and looked into it, I discovered, surprisingly, that there really aren't that many, not active ones anyway.  I added what I found to the grid you see below.  As you can see, there aren't that many, but, clearly, more have "unidos" than "juntos."

The political entities in the grid above are all active, to my knowledge.  I didn't purposely look for defunct political entities (of which, I am sure there have been quite a few over the years).  But one party, now defunct, that was active recently (within the past 10 years), was a Chilean party called "Juntos Podemos Más."  According to the source I've linked you to, it was founded in 2003, but hasn't been active under that name since 2011.  If you'd like more information on any of the political entities in the grid, check out Spanish Wikipedia as well as English Wikipedia.  Believe it or not, I actually found a couple on English Wikipedia that Spanish Wikipedia didn't have.
----------------------------------------------
First impressions
The first difference I notice between these two words is that "unidos" tends to be used a lot more in political messages and "juntos" seems to be more often used when talking about relationships, especially romantic ones (although its usage goes far beyond that).  Take a look at Google images for "unidos" and "juntos" to see what I mean.  This may be due to the fact that a lot of political parties and slogans tend to have the name "unidos" in them.
Overall, "juntos" is a more common word than "unidos" ... at least that is what the Google Ngram viewer will tell you:

A general search of Google pages, as well as one filtered by News (and even Books, no less!) will indicate "unidos" is more common, but in a frequency list of 50,000 Spanish words based off of translated film subtitles, "juntos" comes in at #333 and "unidos" at #1,177 (the lower the number, the more frequent the word).
All of that may be interesting, but it doesn't help a student of Spanish determine when to use "unidos" and when to use "juntos" if both can be translated in similar ways.
----------
Let's begin with the singular form for these words because sometimes they can differ from their plural forms, but even when they don't, this can help a student understand the plural better.
Unido
According to WordReference, "unido," when used as an adjective or as a past participle, means "united." Collins only lists the adjectival meaning for this word, but translates it a bit differently; it translates it as "close" (as in a close-knit family/una familia muy unida).  Some examples of both "unido" and "unidos" follow.  These examples, and others in this article, come from headlines/articles in online news publications and Reverso's online context dictionary.  Others I wrote myself to illustrate a point, but I tried to choose examples that might be useful.
el Reino Unido
the United Kingdom
los Estados Unidos
the United States
¿Qué países son los más unidos?
Which countries are the most united?
¿Cuál es el hilo común que los ha unido?
What is the common thread that has united them?
WordReference also mentions that "unido" as an adjective can also mean "connected," but I haven't found too many good examples of that.  However, I did see some examples of it translated as "joined" when used as an adjective in a passive "se" construction:
Incluso el Museo del Prado se ha unido al '10 Year Challenge'
Even the Prado Museum has joined the '10 Year Challenge'
La derecha se ha unido al independentismo.
The right has joined the independence movement.
Junto
Now for "junto."  I'm going to simplify things by just stating that "junto" means "together" when used as an "adjective," "together" when used as an adverb, and when paired with "a" or "con," "next to/by/along" or "(together) with" when used as a preposition (which is often).  Some examples of its use as an adjective — junto/junta/juntos/juntas, its use as an adverb — junto, or preposition — junto a/con follow:
as adjective
Estamos juntos.
We are together.
Ellas están juntas.
They are together.
as adverb
The word "junto" as an adverb can sometimes be a bit complicated because it doesn't always follow a typical rule of Spanish adverbs — that they don't decline.  This was first brought to my attention in this thread here:
juntos or juntas
In the thread just mentioned, I read that "juntos" acts as an adverb (rather than an adjective) when it follows the following verbs: estudiar, trabajar, vivir.  But, after looking at a few examples, I noticed that those weren't the only verbs.  I didn't do an exhaustive study of this, but I did find several declined usages of "junto" when paired with other verbs.  So you see what I mean, I've listed what I found below:
cantamos/vamos/quedamos/salimos/volvemos/tomamos/miramos/empezamos/esperamos/entramos juntas
Obviously, I used the feminine plural inflection as a test.  So, what I interpreted from the thread I just mentioned is that Collins (the source cited in the thread) considers inflected forms of "junto" — junta/juntos/juntas — as adverbs when paired with certain verbs (estudiar, trabajar, vivir).  What I discovered by trying to find examples of this is that, in practice, these inflected forms aren't just limited to pairings with those verbs and that most English speakers would consider these inflected forms of "junto" as adverbs.
As a matter of fact, the word "together" is never categorized as an adjective in English unless we're referring to its use in informal speech where sometimes it is used to describe someone who knows what they're doing and does so competently and with conviction as in "The new employee is very together."  Just to be clear, "junto" or its many inflections never takes on this meaning of "together."
To sum up this section, basically "junto" can be an adverb that declines for gender and number, and especially tends to do so when it follows a first person plural verb, but there are instances where the writer has not declined it.  I don't know if there exists any Spanish grammar rule that says you can't, but frequency of usage can often dictate what does or doesn't sound natural to native speakers of Spanish.  Below are some examples from Reverso where "junto" has not been declined.  Initially, I thought they might be good examples of standard Spanish, but I later went back to see how frequently I found them online, and now I'm not so sure.  If any native/fluent Spanish speaker/teacher would like to chime in on these examples (or any other portion of this post) please add your comment(s) below.  Here are the examples:
Montaban junto.
They ride together.
I looked at a smattering of the examples returned from a Google search for this phrase.  The vast majority were followed by a preposition (typically "a").  So, if I were to use this adverbially — they ride together (rather than next to something), I would use "montaban juntos" (or "juntas").  I saw enough examples of this in reputable sources to believe that this is standard usage.
---------------------
Quizás nosotros los podríamos tejer junto.
Maybe we could knit them together.
This being a rather unlikely thing for most to convey, I instead did a search
for "podemos hacerlo junto" and "podemos hacerlo juntos."  As with "montaban junto(s)," the same results were found.  If I found instances of the former, "junto" was followed by a preposition.  So, again, if you need to convey "together" and not "next to," go with an inflected form of "junto."
---------------------
Estamos poniendo junto un perfil geográfico.
We're putting together a geographic profile.
As with the other two examples, results indicate that "junto," when it has not been inflected likely is serving as a preposition, is typically followed by "a," and means "next to/by/along(side)."  If you mean to convey "together," inflect "junto" for gender and number — "juntos" for males or mixed groups and "juntas" for groups of females.

I'll round out all of these examples of "junto" with those having fewer complications — the ones that act as prepositions:
Alerta por una seta tóxica que abunda junto a ríos asturianos
Alert for a toxic mushroom that abounds next to Asturian rivers
¿Quieres vivir junto al mar?
Do you want to live by the sea?
navegar junto a la costa
to sail along the coast
España, junto con Bélgica y Alemania, serán los países donde más crecerá el empleo para cumplir el Acuerdo de París
Spain, together with Belgium and Germany, will be the countries where employment will grow the most in order to comply with the Paris Agreement
Soy Wolf Blitzer, junto al mejor...
I'm Wolf Blitzer, together with the best...
A note on "unidos" and "juntos" specifically
I mentioned earlier that these two words are sometimes similarly translated.  What I mean by that is translations of "unidos" are sometimes the same ones for "juntos."  (But not usually the other way around.  It'll be a strange day, indeed, if Spanish speakers start calling the states between the 25th and 49th degrees north latitude los Estados Juntos.)
If you search for both of these words in Reverso's context dictionary, you'll find that the first translation listed for both is "together."  So, if both of these words can mean "together," how do you know when to use which?  Well, for starters, that's a question that would better be answered by a native or fluent speaker of Spanish or a Spanish teacher, but I'll take a shot at answering it based off of what I see at Reverso.  To begin, you'll find more translations of "juntos" as "together" than "unidos" as "together."  Remember when I mentioned the ranking of these two words in open subtitles?  Well, in terms of frequency that means the number of instances of "juntos" was tabulated at 108,529 and 25,514 for "unidos," meaning "juntos" was four times more frequent than "unidos," regardless of translation into English.  How does that compare with the numbers at Reverso?  Well, total instances of "juntos" totaled 64,280; total instances of "unidos" totaled 4,848.  So, immediately it is plain to see that we're dealing with a slightly different database of words since this is a 13-fold difference instead of just four.
Be that as it may, let's hone in on the translations of these words as "together."  61,461 (96%) of the instances of "juntos" are translated as "together"; only 1,736 (36%) of the instances of "unidos" are translated as "together."  What does this mean to the student of Spanish?  Well, if you're unsure about whether or not you should translate "together" as "juntos" or "unidos," you'll be hedging your bet quite safely by going with "juntos."  Apart from that, the only real difference I can see in the examples is that "unidos" when translated as "together" appears to surface in what appears to be rather formal language.  I'll provide some examples of both "juntos" as "together" and "unidos" as "together" so you can see what I mean:
Examples with "juntos":
Hacemos cosas juntos, innovamos juntos, avanzamos juntos.
We make things together, we innovate together, we advance together.
Crecieron juntos, se enlistaron juntos, pelearon juntos.
They grew up together, enlisted together, fought together.
Han dormido juntos, vivido juntos, peleado juntos.
You've slept together, lived together, fought together.
Sólo si actuamos juntos sobreviviremos juntos.
Only if we act together will we survive together.
Trabajemos juntos por el bien común europeo.
Let us work together for the common European good.
Examples with "unidos":
Sé que seguiremos trabajando unidos en el futuro.
I know we will continue to work together in the future.
Es importante que trabajemos unidos para lograr una Europa más segura y más tranquila.
It is important that we work together to ensure a safer and more secure Europe.
Los talibanes constituyen una amenaza común y ambos países deben trabajar unidos para abordarla.
The Taliban poses a common threat and both countries must work together to address it.
Fue la incertidumbre lo que nos mantuvo unidos.
It was uncertainty that kept us together.
Overall, the sentences that translated "unidos" as "together" seemed a bit loose to me.  I wouldn't say they were bad translations; they just seemed to stray a bit more from the literal meanings of words than what I found in the examples of "juntos" translated as "together."  The other thing I noticed is that there is definitely a sense of "united" in the meaning of "together" when you read through those translations that have been translated from "unidos."  Taking this thought a step further, if it weren't for the fact that "united" isn't an adverb, you could definitely just as easily swap out "united" for "together" in all of the examples where "unidos" has been translated as "together."  Now that I think about it, it might be the main reason why "unidos" would be translated as "together."  Sure, for the purposes of creating something grammatically correct in English, you could stretch your translation of "unidos" to "unitedly," but "work unitedly" just doesn't sound as good as "work together" (to me, anyway) and it just isn't a common pairing of words in English.
And this brings me to my next point and something that you might have already noticed.  When "unidos" is translated as "together," it often follows some form of "trabajar."  Even so, "trabajar juntos," is far more common than "trabajar unidos" (almost 10 times more common).  That said, I get the sense that "trabajar juntos" is more of a "working together" and "trabajar unidos" is more of a "working as one (body/unit/entity)."  The hyperlinks in the last sentence will take you to Reverso translations for each.
Lastly, "junto" and its inflected forms can serve as an adjective that sometimes means "joined":
Comment: Your contributions here are always welcome Lisa and they help to build a repository of well researched articles which is the overall aim of the Stack Exchange family.
Answer: These words come up often in politics, because groupings may be in flux, with groups sometimes splitting, sometimes coming together.  Groupings can occur at a variety of levels, from mergers to coalitions to alliances to loose groupings to umbrella group.
For starters: juntar means to gather or bring together; juntarse means to join in doing something together; unirse means to unite; unirse a means to join (an organization or movement) or to unite (for a common purpose).
But the choice isn't cut and dried.  When choosing which word to use for a particular message, slogan or chant, the following may be more important than the literal meaning:

meter and rhyme
a new group wanting to avoid using the word traditionally associated with a particular group
a new group wanting to appropriate (and perhaps redefine to some extent) a word previously used by a particular group (think of the Gray Panthers, that came after the Black Panthers -- "Panthers")
a new group wanting to show its evolution from one or more previously existing groups

The most useful thing, in understanding which word is used when, would be to look at some examples.

¡El pueblo, unido, jamás será vencido!

This is a chant, and unido works well because of the meter and the rhyme.

Únete al PRD.

This is a slogan.  It comes from unirse and means "Join the (name of political party)."

Juntos Sí Podemos

A slogan, meaning "Together Yes We Can."

Entre Todos Sí Podemos

The same, but with a different rhythm.

Miles de mujeres llegan a Chiapas para luchar junto a las zapatistas por sus derechos

This is a newspaper headline description.  "Luchar junto a" has the literal meaning of "to fight alongside," and it expresses the concept "show solidarity for."

Unidos Podemos

As I understood Fedorqui's comment, this is the name of a coalition in Spanish politics.  The two primary members are the political parties Izquierda Unida and Podemos. The coalition's name shows its etymology.

junta militar [or junta for short]

This is the name of a type of government: military dictatorship.  As an example, think of three generals who get together to take over a government as a triumvirate.  The name comes from the idea that they came together -- from the verb juntarse (join together).


Are there rules how Hindus should be named?

Question: Many non-Hindus on conversion take on Sanskrit-based names connected with Hinduism.  On the other hand I know of at least one gentleman who has converted to Sri Vaishnavism and has kept his Christian name.  Do scriptures say anything about this?
Comment: This is already asked ..See this: https://hinduism.stackexchange.com/questions/28731/what-do-our-scriptures-say-about-naming-a-child
Comment: This is not just for Hindu children - this includes adult converts to Hinduism @rickross
Comment: Oh .. then after initiation u will get an initiation name (Diksha nama) but the birth name is also retained ..
Answer:
On the other hand I know of at least one gentleman who has converted
  to Sri Vaishnavism and has kept his Christian name. Do scriptures say
  anything about this?

Yes, they do. And here, that person has retained his birth name which is alright.
During Agamic initiation, the Guru gives the disciple a name. This is known as the DikshA NAma (the initiation name). This indicates that now you are in our Sampradaya and also indicates the fact that the disciple has obtained a new birth (hence a new name).
The process of how these names are chosen might be different for different sects. In this answer I have briefly outlined the method that is for Sri VidyA sect, as quoted in Nityotsava's 2nd chapter.
The process:
This happens at the end of DikshA.

Evam idam dikshAtrayam nirvartya pashchAt tasmai ishtam mantram
  dadyAt |  Tato guruh shishyashirasi swacharanau niveshya ishtamantra
  kramopayuktAn sarvAn angamantrAn tasminneva kAle kramena vA
  yathAhadhikAramupadishya swAngeshu kimpyAngam shishyam sparshayitvA
  tadangamAtrikAksharAdim dwyaksharam tryaksaram chaturaksharam vA
  AnandanAtha shavadAntam tasya nAm kritvA dashama khandoktAn
  AchArAnanushishyAt ||
From Nityotsava's 2nd chapter, called DikshA Krama's sub-section
  called IshtamantradAnam.
In this manner the Guru should complete the three initiations and give
  the disciple his Ishta Mantra. Thereafter, the Guru, placing his foot
  on the disciple's head, shall instruct him with all the related Anga
  Mantras either during that time itself or later, in succession. Now,
  the Guru should make the disciple touch one of his limbs. Noting the
  MatrikA Akshara (one of the 50 Sanskrit alphabets), NyAsa of
  which is done during MatrikAnyAsa, on that particular limb (part of
  the body) touched, the Guru shall frame a name starting with that
  alphabet, comprised of two, three or four alphabets. And, upon adding
  the the suffix "Ananda NAtha" to the name thus obtained, he thus gives
  the disciple a new name. Thereafter, he instructs the disciple with
  the conducts that are described in ParashurAma Kalpasutara's 10th
  chapter

To know how this MAtrikA NyAsa is done, you can check this page
So, we know that on the left knee, for example, the Sanskrit alphabet Da (द), is to be Nyasa-ed with. So, assume the disciple touches the Guru's left knee.
Then the Guru has to frame a name starting with Da, comprised of two, three or four alphabets. Let's assume the chosen name is "DayA" which is meeting all the criteria. So, the DikshA NAma or Guru given name for that disciple will be DayAnanda NAtha.
(Note - Here, the suffix Ananda NAtha is peculiar to the Sri VidyA sect and it will change with sect).
But, this is briefly the process of renaming the disciple.
After the initiation, the disciple, at least in the religious circle, is known by his DikshA nAma more but he can retain his birth name and use it. For example, Swami VivekAnanda was more well-known with his Guru given (SannyAsa name) than his birth name (Narendra NAth).
But, BhAskara RAya has retained his birth name while writing many treatises where as his DikshA NAma was BhAsurAnanda NAtha.
So, it can not be said that the Christian person is at fault however he must have his own Guru-given name too.
Answer:
Are there rules how Hindus should be named?

Yes, there are rules on how Hindus should be named.
From the Manusmriti:

2.30 - One should have his 'naming' (Nāmadheya) done on the tenth or the twelfth (day), on an auspicious lunar date and at an auspicious
  moment, and under a propitious lunar asterism.
2.31 - The name of the Brāhmaṇa should be auspicious, that of the Kṣatriya connected with power, that of the Vaiśya associated with
  wealth; while that of the Śūdra contemptible [lowly/mean].
2.32 - The name of the Brāhmaṇa should be expressive of 'peace,' that of the Kṣatriya, of 'protection'; that of the Vaiśya, of 'prosperity,'
  and that of the Śūdra, of 'submissiveness.'
2.33 - That of women should be easily pronouncible, not harsh, of plain meaning, heart-captivating and auspicious; it should end in a
  long vowel and contain a benedictory term.


Is firewall needed for guest OS in VM?

Question:
Possible Duplicate:
Should I enable Firewall and AntiVirus on a VirtualPC VM?

I have good firewall on host OS (WinXP), Do I need additional firewall in guest OS (Fedora)?
Answer: That will highly depend on where your firewall is placed.
I think you will still need a firewall as most applications just bridge your physical interface with your virtual ones.
If your virtual interfaces are positioned behind the firewall than you can control in and out going to and from your guest OS with one firewall.
Answer: It will not be necessary but that will depend if you have sensitive information that you want to kept confidential inside your VM. I highly suggest that you should take a look at protecting your network right from the router itself then dig on the software firewall to further enhance your control and monitor network traffic that goes in/out of your system.

Is firewall needed for guest OS in VM?


Average Distance Between Random Points on a Line Segment

Question: Suppose I have a line segment of length $L$.  I now select two points at random along the segment.  What is the expected value of the distance between the two points, and why?
Comment: $L/3$, by symmetry.
Comment: Care to elaborate, @Byron?
Comment: @David I've added more explanation in my answer below.
Comment: Can anyone explain how by symmetry we are getting the value as @user940 has stated?
Comment: Here's the symmetry argument, roughly. If you were to randomly break a line into 3 pieces, assuming you don't have any information about how the line breaks, the average length of each piece will be L/3. That feels pretty intuitive. Now imagine you keep the pieces lined up: How many break points are there? Two! These two break points are equivalent to the two random points in OP's question. So if by symmetry the average length of the pieces is L/3, and this is equivalent to the average breakpoints being at L/3 and 2L/3 respectively, their average distance is L/3.
Answer: Byron has already answered your question, but I will attempt to provide a detailed solution...
Let $X$ be a random variable uniformly distributed over $[0,L]$, i.e., the probability density function of $X$ is the following
$$f_X (x) = \begin{cases} \frac{1}{L} & \textrm{if} \quad{} x \in [0,L]\\ 0 & \textrm{otherwise}\end{cases}$$
Let us randomly pick two points in $[0,L]$ independently. Let us denote those by $X_1$ and $X_2$, which are random variables distributed according to $f_X$. The distance between the two points is a new random variable
$$Y = |X_1 - X_2|$$
Hence, we would like to find the expected value $\mathbb{E}(Y) = \mathbb{E}( |X_1 - X_2| )$. Let us introduce function $g$
$$g (x_1,x_2) = |x_1 - x_2| = \begin{cases} x_1 - x_2 & \textrm{if} \quad{} x_1 \geq x_2\\ x_2 - x_1 & \textrm{if} \quad{} x_2 \geq x_1\end{cases}$$
Since the two points are picked independently, the joint probability density function is the product of the pdf's of $X_1$ and $X_2$, i.e., $f_{X_1 X_2} (x_1, x_2) = f_{X_1} (x_1) f_{X_2} (x_2) = 1 / L^2$ in $[0,L] \times [0,L]$. Therefore, the expected value $\mathbb{E}(Y) = \mathbb{E}(g(X_1,X_2))$ is given by
$$\begin{align} \mathbb{E}(Y) &= \displaystyle\int_{0}^L\int_{0}^L g(x_1,x_2) \, f_{X_1 X_2} (x_1, x_2) \,d x_1 \, d x_2\\[6pt]
 &= \frac{1}{L^2} \int_0^L\int_0^L |x_1 - x_2|  \,d x_1 \, d x_2\\[6pt]
 &= \frac{1}{L^2} \int_0^L\int_0^{x_1} (x_1 - x_2) \,d x_2 \, d x_1 + \frac{1}{L^2} \int_0^L\int_{x_1}^L (x_2 - x_1) \,d x_2 \, d x_1\\[6pt]
 &= \frac{L^3}{6 L^2} + \frac{L^3}{6 L^2} = \frac{L}{3}\end{align}$$
Comment: Was this problem ever studied with N randomly placed points?
Comment: After going to the link in the comment by @AlexSk, I see that $\dfrac{L}{N^2-1}$ is the expected value of the $\textbf{minimum}$ distance between any two points, for $N$ randomly placed points on the line, not the average distance between $N$ randomly placed points on the line.
Comment: @xyz Yes. It's $\frac{L}{N^2 - 1}$ Check out this answer https://mathoverflow.net/questions/1294/mean-minimum-distance-for-n-random-points-on-a-one-dimensional-line
Comment: @RodCarvalho I am solving similar problem but I am asked to find pdf function of a distance between two uniformly distributed random variables on  $[0,2]$ so generalizing it to $[0,L]$ we have situation like above. So my question is if I have convolution when looking for pdf called here $g(x_1,x_2)$ how should I put limits on the inegrals to take both situations $x_1 \leq x_2$ and $x_2<x_1$ into consideration. It is tricky because if pdfs are dependent on $X_1(t)$ and $X_2(t)$ then the integral is dependent only on t.
Answer: Sorry. I posted a cryptic comment just before running off to class. What I meant was that if $X,Y$ are independent uniform $(0,1)$ random variables, then the triple
$$(A,B,C):=(\min(X,Y),\ \max(X,Y)-\min(X,Y),\ 1-\max(X,Y))$$
is an exchangeable sequence.  In particular, $\mathbb{E}(A)=\mathbb{E}(B)=\mathbb{E}(C),$ and since $A+B+C=1$ identically we must have $\mathbb{E}(B)=\mathbb{E}(\mbox{distance})={1\over 3}.$
Intuitively, the "average" configuration of two random points on a interval
looks like this:
Comment: What I like the most about this answer is that it can be used to prove the case where we have $n$ points and not only two points $x$ and $y$. Or am I missing something ?
Comment: @AJed You are not missing anything.  Add an $n+1$th point.  Place the $n+1$ points at random on a circle of circumference $L$: clearly each of the $n+1$ gaps is identically distributed (though not independent) and has expectation $\frac{L}{n+1}$.  Now split the circle at the point you added, and you have this case for $n$ points.
Answer: Without loss of generality, let the interval length be L and the sought average difference D.
The answer is seen to be D=L/3 by the following simple argument:
Select 3 random values a,b,c on the interval of length L. The probability that c is between a and b equals 1/3 since the other equiprobable alternatives are that a or b are in the middle.  Since the interval length is L, the probability must also be D/L, thus D=L/3.
Answer: Byron's answer is short and elegant.
Here's a geometric/algebraic derivation: Let $X$ and $Y$ be two independent uniform variates in $[0,1]$. Then
$$p\left(\{|X-Y|>s\}\right) = (1-s)^2 $$
as can be seen by viewing $(X,Y)$ as a uniform variate in $[0,1]^2$, where the set $\{|X-Y|>s\}$ occupies the top left and bottom right triangles of the square box $[0,1]^2$. See figure (a). Now we have $p\left(\{|X-Y|<s\}\right) = 1-(1-s)^2$ and therefore
$$p\left(\{|X-Y|=s\}\right) = {d\over ds}p\left(\{|X-Y|<s\}\right) = 2(1-s) .$$
The result follows immediately: $\mathbb{E}\,|X-Y| = \int_0^1 2s(1-s)\,ds = {1\over 3}.$
Sidenote: The discrete case is analogous. Let $(X,Y)$ be independent uniform variates in $\left\{1,2,\dots,N\right\}^2$. Then
$$ p\left(|X-Y|=s\right) = 2(N-s), \quad s>0$$
as can be seen from mlohbiehler's plot. (Note that the equation is not valid for $n=0$.) The expectation is
$$\mathbb E |X-Y| = {1\over N^2}\sum_{s=1}^N s\,2\,(N-s) = \frac{N^2-1}{3 N} .$$
Asymptotically for $N\to\infty$ this is ${N\over3}$.
Figure (a) and (b)
Answer: Let $X_1, X_2$ be independent uniformly distributed on $[0,1]$.
Assume $X_1=x_1$. Then $P(X_2<x_1) = x_1$.
Moreover $E(X_2|X_2<x_1)=\frac{x_1}2$ and $E(X_2|X_2>x-1) = x_1+\frac{1-x_1}2=\frac{1+x_1}2$, hence $E(|X_2-x_1|) = x_1\cdot\frac{x_1}2+(1-x_1)\cdot \frac{1-x_1}2=\frac12-x_1+x_1^2$.
Finally $$E(|X_2-X_1|) = \int_0^1E(|X_2-x|)dx = \left[\frac12x-\frac12x^2+\frac13x^3\right]_0^1=\frac13.$$
Hence with an interval of length $L$ isntead of $1$, the answer is $\frac L 3$.
Comment: In the fourth sentence ("Moreover..") there are two places where on the RHS you wrote $x_2$ when you meant $x_1$, if I am not mistaken.
Comment: Yes, I did. Thanks
Answer: Let $X_1$ and $X_2$ be independent identically distributed random variables, with $f_X(x) = [0<x<1]$. It is well known that $X \stackrel{d}{=} 1-X$.
For simplicity assume $L=1$.
Therefore $|X_1-X_2| \stackrel{d}{=} |X_1+X_2-1|$. Random variable $D = X_1+X_2-1$ follows symmetric triangular distribution on $(-1,1)$, being a special case of Irwin-Hall distribution. We immediately have:
$$
   f_{|D|}(\ell) = 2 (1-\ell)[0<\ell<1]
$$
Immediately yielding the expectation:
$$
    \mathbb{E}(|D|) = \int_0^1  2 \ell(1-\ell) \mathrm{d} \ell = \frac{1}{3}
$$
Comment: What does $\buildrel d\over=$ mean?
Comment: @MJD Symbol $\stackrel{d}{=}$ stands for "[equality in distribution](http://en.wikipedia.org/wiki/Random_variable#Equality_in_distribution)".
Answer: You can picture this problem from a discrete approach, then extend it to the real number line.
For the first L natural numbers, the difference between any two of them may range from 1 through L-1. Exactly N pairs of numbers from our set will be L-N units apart. Taking that into account we sum:
Sum of distances =
 sum (0 < x < L, x*(L-x))
Which turns out to be (L-1)(L)(L+1)/6
We then divide by combinations of L in 2 and get the average distance (L+1)/3, which becomes L/3 as we allow for infinitely many more numbers in the [0;L] interval.
Answer: It is possible to solve this using limits however the solution is somewhat inelegant.
Split the line L into $(k+1)$ equal segments by placing points $p_1$, $p_2$, ... $p_{k}$ on the line. The length of each segment is $\dfrac L{k+1}$. Let $S_i$ be the sum of distances from point $p_i$ to all other points.
$S_i$ = $ \dfrac L{k+1} \sum_{j=1}^{i-1} j + \dfrac L{k+1} \sum_{j=1}^{k-i} j $
$S_i$ =
$\dfrac L{2(k+1)} ( (k-i) (k-i+1) + (i-1) (i) )$
$S_i$ =
$\dfrac L{2(k+1)} ( 2i^2 - 2i(k+1) + k(k+1) )$
Now, the sum of all distances from every point to every other point is $\sum_{i=1}^{i=k} S_i$. Also, the number of pairs of points is $k^2$. Therefore the average $A_k$ is $\dfrac{\sum_{i=1}^{i=k} S_i}{k^2}$
$\sum_{i=1}^{i=k} S_i = \dfrac L{2(k+1)} \sum_{i=1}^{i=k} ( 2i^2 - 2i(k+1) + k(k+1) ) $
$A_k = \dfrac L{2(k+1)k^2} \sum_{i=1}^{i=k} ( 2i^2 - 2i(k+1) + k(k+1) ) $
$A_k = \dfrac {2L}{2(k+1)k^2} \dfrac{k(k+1)(2k+1)}6 - \dfrac {2L(k+1)}{2(k+1)k^2} \dfrac{k(k+1)}2 + \dfrac {L(k(k+1))}{2(k+1)k^2}k $
$A_k = \dfrac {L(2k+1)}{6k} - \dfrac {L(k+1)}{2k} + \dfrac {L}{2} $
$A_k = \dfrac {L}{6k} (2k+1 - 3k-3 + 3k) $
$A_k = \dfrac {L}{3k} (k-1) $
$A_k = \dfrac L3 - \dfrac L{3k} $
Now, we just need to make the line segments infinitely small, i.e infinitely large $k$ and we're done.
$\lim_{k \to +\infty} A_k$ = $ \dfrac L3 - \lim_{k \to +\infty} \dfrac {L}{3k} = \dfrac L3 \space\blacksquare $
Answer: I get a close but different result when I approach this with brute force and ignorance. Consider an example: if my random numbers are between 5 and 14 (L=10) and uniformly distributed, I can expect after sufficient iteration to have equally observed each number for both values. Thus I can simply consider a matrix of the differences between each value, and calculate the average of the sum of the values in the matrix.
(Please excuse my formatting. I'm new here.)
<code>     5  6  7  8  9 10 11 12 13 14
    -----------------------------
 5 | 0  1  2  3  4  5  6  7  8  9
 6 | 1  0  1  2  3  4  5  6  7  8
 7 | 2  1  0  1  2  3  4  5  6  7
 8 | 3  2  1  0  1  2  3  4  5  6
 9 | 4  3  2  1  0  1  2  3  4  5
10 | 5  4  3  2  1  0  1  2  3  4
11 | 6  5  4  3  2  1  0  1  2  3
12 | 7  6  5  4  3  2  1  0  1  2
13 | 8  7  6  5  4  3  2  1  0  1
14 | 9  8  7  6  5  4  3  2  1  0
</code>
The sum of the differences in this matrix is 330, the count 100, giving an average of 3.3, not 10/3 as expected in other answers.
I don't see anything wrong with this except that i can't see anything wrong with the other answers. Can anyone explain the difference?
Comment: I think you're just seeing the difference between discrete and continuous.  I suspect if you increased $L$ without bound, the average would indeed approach $L/3$.
Comment: $10/3=3.333\ldots$
Comment: Your approach is generally correct, but the result is not absolutely accurate. In your experiment you broke the line into 10 parts. What if it was only 2 parts? Or only 1 part? (with 1 part your answer will be 0 - very inaccurate). Breaking into two parts will give you 0.25. Breaking into 10 parts gives 0.33. Why do you think 0.33 is correct, why not 0.25? Of course, more parts you take, more accurate your answer will be.
Comment: Which is not 3.3. Like I said, close but different.
Comment: @lesnik, mcmeyer provided the answer below, referencing Byron's work. In discrete cases such as mine and your examples, the answer is what it is, and it is correct. With only one part (i.e. a point) the distance between points will always be zero, so the answer of 0 will in fact be quite accurate. However, as N approaches infinity, the average approaches N/3.
Answer: Differences represented by a line from 0-15
I took the brute force approach as noted above and thought about letting the size of the line (and hence the square with the differences) range from 0 to infinity. Taking the total absolute differences for squares ranging from 0 to n then the total number of integer points that can be chosen is n+1. The total count of the number of differences calculated is (n+1)^2. The total of all the differences calculated is twice the triangular numbers which is 2(n(n+1)(n+2))/6. Therefore the average distance between two points is 2(n(n+1)(n+2))/6 all divided by (n+1)^2. To normalise this to the equivalent of a unit line we divide again by (n+1).
For the 0-15 table as above n=15 and the average distance is 0.33203 for n=20 the average distance is 0.33258 and for n=25 the distance is 0.33284. Clearly getting closer to 1/3.
If we let n get larger towards infinity then what happens to 2(n(n+1)(n+2))/6(n+1)(n+1)^2?
This becomes 2/6 or 1/3.
Hopefully this makes sense and is another way to look at the problem.
R
Answer: The pdf of the distance is highest closer to zero. This means that distance is more likely to be close to zero than any other positive value. This seems to be puzzling. Put another way, let us divide the segment into equal sized bucket, say 100 buckets .01 length each and number 0 to 99 with bucket i from i*.01 to (i+1)*.01. If you do the experiment for a large number of pairs of points and divide them into buckets, then bucket 0 will contain the largest number of samples. This seems counter intuitive at first. But it makes sense, since after selecting a point closer to the edge makes it twice more likely to select a second point closer to the first point than one farther from it. The closer point can be on either side of the original point but the farther point can only be on one side.


Is there an online program that can type pinyin for traditional Chinese?

Question: I studied pinyin so I naturally type simplified Chinese but now I want to type traditional Chinese. Is there a program available where I can type traditional Chinese using Pinyin method? Thanks
Comment: If you are looking for an __online__ application: https://translate.google.com Set the input language to Chinese and then select 'Chinese - Traditional, Taiwan - Pinyin' in the input tool dropdown. But I guess you are better off with a desktop IME. If you have Windows, you can setup the IME very easily, it's already on your computer you just need to google that. Or perhaps you have another OS, just seach for 'Chinese Traditional IME Pinyin + [your OS]'
Comment: It seems you're not actually asking for an online service, but rather something you can download? If you're actually asking for an online IME, try this one: http://www.pinyinput.com/?lang=en
Comment: Hi, essentially I am looking for a direct input method IME where i can skip the google translate step.  However, it seems google may be the safest bet.
Answer: I use Sogou input, there's a preference setting for it. And I believe many Chinese input have it too.
Answer: OK, again in mac os x, go to system preferences⋯keyboard⋯input sources⋯press +⋯choose "chinese-traditional" from the left⋯then choose "pinyin - traditional", voila :)
remember to check the "show input menu in menu bar", then you can choose the pinyin as input method, and type "out" traditional chinese characters.
have fun :)
Comment: I am not a mac user.
Comment: then, consider to change :)

the built-in functions of mac os x are excellent; for learning cantonese, and/r traditional chinese.
Comment: yes thank you, I will definitely consider using a mac as well as my pc. Thanks
Comment: @Genevieve I will write an answer for Windows for you.
Comment: @Genevieve Done. Can't believe it took me 40 minutes to write this seemingly easy answer.
Answer: Solution for Windows 8.1

Go to Control Panel\Clock, Language, and Region\Language (Alternatively, click the downward arrow on the language bar (on the taskbar) to open
the menu and click Settings...)
Click Add a language
Select Chinese (Simplified)
Select Chinese (Simplified, China)
Once 中文(中华人民共和国) is available as a language, click Options
following it.
Under Input, Microsoft Pinyin should appear by default. If it is not there, click Add an input method and select Microsoft Pinyin.
Once Microsoft Pinyin is available as an input method, click
Options following it.
Under Choose character set, select Traditional Chinese.
Congratulations! You can now type traditional Chinese with Pinyin. You can also press shift to switch between English and Chinese when Chinese (Simplified, China) is selected.

Note

This answer only considers what you can achieve in native Windows system (i.e. no third-party apps). In fact, the lack of native support for Pinyin as an input method for traditional Chinese has been repeatedly complained by users.
There is no easy method to switch between simplified Chinese and
traditional Chinese using this method. You have to go through the
same process again to change from one character set to another (you can keep the "Options" window open for quick switching).
It is possible to add tradition Chinese as a language and then select "Microsoft Bopomofo" as an input method. But note that although "Microsoft Bopomofo" is based on Pinyin, it is different from "Microsoft Pinyin" and is rather clunky to use.
Some characters outputted by "Microsoft Pinyin" using traditional Chinese character set is incorrect. For example, "只 (only)" is written the same way in both simplified Chinese and traditional Chinese. But "Microsoft Pinyin" will outputted it as "衹". There is no easy way around this problem and you have to spot it by yourself.
A fail-proof way is to type in simplified Chinese and then copy the result into Google Translate to translate them into traditional Chinese. This comes handy when you only need to type a small number of characters (note that Google Translate will mess up your formatting), or when you need to be absolutely sure that what you are typing is 100% correct.
Comment: Thank you very much. I am translating chunks of English to Chinese, I will type in pinyin and translate via google translate to change into traditional.  Many kind thanks again.
Comment: @Genevieve Glad I could help :)
Answer: Google Pinyin is another option for you.
In Windows, switch your input method to Google Pinyin, and then use Ctrl+Shift+T to switch to traditional Chinese mode. Now you can type pinyin to get traditional Chinese characters.

You can easily switch back-and-forth between simplified and traditional characters by using the same keyboard combination above.
Answer: just remembered to remind you, there're differences between fonts for simplified and traditional chinese. since you're not mac user, i would suggest downloading google font: "not sans cjk":
https://www.google.com/get/noto/help/cjk/
or, https://www.google.com/get/noto/#sans-hant
it's free, have fun :)
Answer: there is an app for typing Chinese made for Chinese people. search Sogou pinyin in google and it will take you to the home page of the app, download and install(un tick the Sogou browser because you probably don't want it, it's practically a Chinese version for Chrome) don't worry about it being illegal, it's not. after installing, you can just press ctrl + shift + f to change into traditional Chinese and same keys to change back. there are a lot of other cool settings and skins that comes with the app.
I recommend this app to all people who type Chinese it is really handy.
http://pinyin.sogou.com/
this is the website, go in click the button that says 立即下载 and it will start downloading. and just install it with your own preferred setting and you're good to go.
Hope this helps.
Comment: Sadly this only support Windows and XP, sorry Mac people
Answer: Online input method

Mandarin pinyin → Traditional Chinese: http://www.pinyinput.com/
Cantonese pinyin → Traditional Chinese: http://www.cantoneseinput.com/


iTunes help switching songs from one user to another

Question: I'm sharing a computer with my sister and she has her iTunes completely disorganized. I created my own user account on the computer, and am wondering how I can move the music from the iTunes to my user account. I go to file, library, export library and save the file but that's not working.
Answer: Go to Home > Music and copy the "iTunes" folder to an external drive.
If you don't have an external drive big enough, you can move it to Macintosh HD > Users > (Your User Name) > Public > Dropbox. (You can copy it directly to your music file from another user, as those folders are only editable when you are in your account).
Login to your user, and move the "iTunes" folder from wherever it is to Home > Music. Open iTunes: your sister's library should be loaded.
PSS: I by no means am advocating the copy of copyrighted content with this post. I imply your music is legally downloaded, and you are moving the library with the intent of organizing it and not duplicating it in any way. Don't do illegal stuff!
Answer: I would copy the files over using external media (hard drive, flash drive, etc.).


Why is double time speed different on iPod vs iPhone?

Question: I have a 6th gen iPod Nano (software version 1.2) and an iPhone 4S (iOS 6.0.1) . On the iPod, I have options to listen to podcasts at ½x, 1x, and 2x time. On the iPhone Podcasts app version 1.1.2 (209), I have options for ½x, 1x, 1.5x, 2x, and 3x time. If I listen to a podcast at 1x speed on both devices, it unsurprisingly sounds the same. However, if I listen to a podcast at 2x speed on both devices, it sounds more sped up on the iPhone. The voices sound more natural on the iPod but more like chipmunks on the iPhone. Is there a reason for this difference? Why do these two Apple devices handle audio speeds differently?
Answer: I can only offer a shot in the dark, and that is that the iPod has a weaker processor and when it tries to double the speed, the time it takes for the processor to do some calculations slows it down a little.


Saved-for-later then reheated potato mixture sparks and catches fire in microwave

Question: I have a bird who gets medicine twice a day.  I hide it in a mashed potato fruit mixture.  I make enough that the potato mixture lasts several days and add the medicine as I need it.  I take a small amount and microwave it for about 8 seconds (just enough to soften it).  Then I normally add a couple blueberries to the mixture for a little more moisture and flavor.  At that point, if I feel like I've made too much for that "serving" I'll put some back in the container for next time.  When I reheat the saved-for-later mixture, it always sparks and catches fire (about the size of a lighter flame) in the microwave (it always goes out as soon as I stop the microwave).  Of course, I never use the caught fire serving and always start over.  This only happens with the saved-for-later bits and not the original mixture.  I read the "Why did my banana catch fire in the microwave?" (Why did my banana catch fire in the microwave?) thread and that might answer it if it were all of the mixtures that caught fire, but it's only the parts that I tried not to waste and save for the next medicine-hiding dose.  Any ideas what causes just the left-over bits to catch fire?
Answer: Try putting a mug of water next to the food in the microwave.
I've had this same problem with baker's chocolate and that did the trick.   I think it's because there isn't enough water to absorb the microwaves so the excess energy causes sparking.
Comment: I'm not sure that makes sense. Sparking happens when you have a conductor. Microwaves are just EM waves so you need a conductor to get elections to move and spark. And without water, microwaves will not heat anything up. You need free water molecules (as opposed to ice which won't work). In this case, if there is no high metal content, the mix has dehydrated and surface got hot enough to catch light. You can try this with bread. I used to microwave bread about 1 minute on each side and this would dehydrate the bread enough to be crunchy!
Comment: I actually have been added a mug of water and haven't had any problems since.  I started doing this after one of the servings that wasn't a "saved-for-later" overage caught fire.  There have been no more fires!  :)


Transit airside in London Luton?

Question: Is it possible to remain completely airside in London Luton airport when changing between two Easyjet flights?  I will only have hand baggage and will already have my boarding pass printed out before going to the airport.
Answer: No it's not
All international arrivals are escorted through an isolated corridor to immigration, with no transit corridor available.
Thus, all transit passengers at Luton must comply with the UK's entry (or at least landside transit) requirements.
The UK airports that do offer airside transit are Heathrow, Gatwick and Manchester.
Comment: To add insult to injury, you still need to go through security even at those places , don't you? And it's UK security so if you refuse the full body scanner as you should then they manually inspect every item of yours and take you into a side room for patdown. It's annoying but oh well. You should still opt out.

Transit airside in London Luton?


Can an IT shop that is 'all novell' support Mac OS X?

Question: I don't have the specifics on our company's network; but all the Windows PCs (Windows XP) currently in my new company's environment use novell client to sign in and use Novell's groupwise for email.  Is it true that a Mac can't 'work' in this kind of environment?  It just seems like Novell would make an effort to support it.  Is IT right or lazy?  Can a Mac exist in such a corporate environment without causing too much of a headache?
Answer: Novell has a Groupwise client for Mac, and it works very well (but is java based, and has some font issues.) eDirectory, which is what Novell uses for logins, is basically LDAP and Mac can authenticate against it.
The only thing I'm not sure you can do without your administrator's help is to mount file shares. I generally use the web access to our novell shares, but I think that's a separate product that we paid for.
I'm in an all-Novell shop and use my personal Mac on a daily basis. 3 of our developers use Macs and are very happy. Our boss has a Mac Pro and is mostly happy with it, but the font issues frustrate him and Novell doesn't seem to be in a hurry to fix them.
Answer: Novell products are usually quite cross platform.
GroupWise has a Mac client as noted.  It shares the code base with the Linux/Unix cross platform client and is not 100% as feature complete as the Windows GW client.
However WebAccess gets better with each release and much of what you need can be done in WebAccess.  Additionally it supports IMAP and POP for email access and calendar publishing, if you want to use something like Thunderbird for reading mail, or Apple's Mail app.
For authentication if you extend your eDirectory schema and add posix (rfc2307 I think) attributes to users, then you can fairly easily authenticate to eDirectory via the Apple login process.  (You need a uid, gig, homeDirectory, etc set of attributes).
If your administrators are using OES on Linux, then they likely have those added already.  (Novell calls it LUM (Linux User Management) which is needed to let Linux boxes login properly).
There are some additional products to make life easier, like Kanaka from Condrey Consulting in case they are still all Netware, to allow for Mac support in a much simpler fashion.
If they are saying they cannot support it, there is an element of laziness, in all likelyhood.


Does USPTO accept a sequential diagram?

Question: I used to use a sequential diagram for a process flow between multiple components, but have never seen it in patent documents. Is it not allowed?
Below is just an example of sequential diagram
Comment: I don't know for sure so I'll leave it to someone else to answer. That said, I can't see why you can't use such a diagram assuming the specification explains it sufficiently. It is best if the diagram is readable entirely in black and white.
Answer: I agree Eric Shain. In fact, this kind of diagrams are quite popular en communication protocol related patents, where interactions between sender and receiver have a sequential order. For example, figures 3, 4 & 5 in Amazon patent US6675196; figure 2 in ATEN patent US7716404B2; or figure 2 in patent application US2004/0100945 A1.
Sometimes, you can even find directed graphs with multiple paths to show the flow of communication, as figures 2, 3a, 3b, 4a, 4b in Hughes Aircraft Company patent US5485464.
Answer: I would expect that a drawing that conforms to USPTO drawing requirements (black and white comes to mind, exceptions for color is accepted only under documented conditions). A search of patents.google.com would produce the examples provided by user=David.
If you suspect that something is prohibited, searching the relevant USPTO requirements (in this case drawings) is the 'do it yourself' method of determining eligibility.  One has to be careful using precedence in the sense that the rules are changing: in theory changes in rules will be documented in the updated requirements.  This response is not to discourage one from answering the question.  It is intended to generalize the process and approach solving this class of problem.
Answer: Diagrams, illustrations, photos and graphs are all allowed in a patent application. They are referred to as drawings under 35 U.S.C 113:

The applicant shall furnish a drawing where necessary for the
  understanding of the subject matter sought to be patented.

Note from the U.S.C clause above that the purpose of any drawing is only to facilitate the understanding of the subject matter that one is trying to patent.
Within the context of an utility patent, only the Claims are protected under the U.S. Patent Act.
Therefore, if the said diagram illustrates the method/process that one is trying to patent, the sequential steps of the diagram must be incorporated in the Claims themselves.
FiG.2 of US20120287823A1 shows a similar diagram.
Comment: @PanuLogic Let me attempt to explain to the best of my ability without obfuscating the issue. Firstly, diagrams and/or drawings are allowed. They are helpful but not obligatory. My response was to draw minime's attention to the following: 1) only the Claims are protected under the U.S. Patent Act. In most cases, 2)the diagrams/drawings are furnished for the understanding on the subject matter. In another words, 3) Diagrams/drawings are included only to help examiners and readers to understand the claims. Please do seek patent counsel for further understanding. I am speaking from experience.
Comment: What do you mean by "steps of the diagram must be incorporated into the Claims"? Is it not rather that they must be mentioned in the detailed description -section?  I think there are many patents like https://patents.google.com/patent/US6675196B1/en?oq=US6675196 where the content of the claims is rather small compared to the content of the diagrams. In that patent for example the claims do not refer to elements including the "steps" of the diagrams

Does USPTO accept a sequential diagram?


What kind of camera do I need to record a video with the quality that is in the linked YouTube video?

Question:

Please provide camera models if possible. Not knowing this is driving me crazy haha
Comment: @HappyTurtle - the point the posters and commenters are making is that it doesn't matter which camera (within reason) so making a recommendation on a specific camera is not helpful.
Comment: @DrMayhem It would be very helpful. DrMayhem, It's been over 2 weeks since I posted this question and ive yet to receive any useful information. Please tell me the cheapest camera model you know of that is capable of shooting a video of this quality. Sad that on StackExchange of all places I cannot get my question answered.
Comment: That is not what this site is for - it explicitly tells you that in the welcome pages. We are not a shopping recommendation site. So it's not sad. It's deliberate.
Comment: Comments are not for extended discussion; this conversation has been [moved to chat](https://chat.stackexchange.com/rooms/77754/discussion-on-question-by-happyturtle-what-kind-of-camera-do-i-need-to-record-a).
Answer: Netflix, the BBC, Amazon, and others that commission and/or broadcast content maintain a set of professional standards that specify everything from acceptable cameras to file naming conventions.  The Netflix list of approved cameras is here.  I would offer that by studying these lists (which talk about not only resolution, but also bit depth, chroma sampling, bit rate, compression rates, etc) you can see where the professionals draw the line between claims made by prosumer camera models and what is standard industry practice in 2018.
There are people who are shooting real feature-length productions for theatrical distribution on iPhones.  Because that's the director prerogative: their movie, their aesthetic.  But I would say that the approved camera lists give a good idea of what sort of camera produces the mainstream image quality you see in professional-quality productions, which that YouTube video seems to demonstrate.  And those cameras typically more than $2,000.
And, as noted before, you could have a $80,000 RED Monstro with a $45,000 Zeiss Master Prime lens and you won't get that quality without proper lighting, set design, etc.  But if you have a great set with great lighting, then you can do a lot with cameras in the $2,500-$45,000 range (and lenses in the $2,000-$20,000 range), which you can find on the Netflix approved camera list, or elsewhere (for other broadcast/streaming specifications).
Answer: Pretty much anything modern from Grass Valley, Sony Broadcast or Panasonics Broadcast division would be well capable of this, if combined with decent glass and a skilled camera and rack operator. It is after all only HD, so nothing particularly state of the art.
All of that kit is well north of £2k however (Maybe 10 - 20 times that), even before you consider the glass which is just a bit important, and the skill of the operators for which there really is no substitute.
I would bet that that crew could shoot something that looked substantially the same on a modern DSLR with a few trimmings (Matte box, some rails, some decent video glass, and a case of filters).
For many of those shots there is clearly plenty of light and given the shallow depth of field the apertures are clearly wide open so sensor noise should not be a big issue, this is not that big an ask of a camera chain.
Comment: You would have me advise on products that I have only played with a little? Far better to find someone who knows that end of the game and what the tradeoffs are. There is more to a camera then just the sensor, weight, balance, ability to tie into standard batteries, availability of an SDI feed for preview monitoring, viewfinder modes, can you rack the camera remotely, can you pull focus remotely, how about aperture? can you stripe Timecode, can you generate it? Does the thing have a tally input? How about an input for preview loopback from the switcher? What matters for your application?
Comment: Thank you for giving me a straight forward answer. I appreciate it. when you say "I would bet that that crew could shoot something that looked substantially the same on a modern DSLR with a few trimmings (Matte box, some rails, some decent video glass, and a case of filters)." Could you give me an example of the DSLR camera model you were referring to? That would help. Ex: Canon 80D, Sony a6300, etc.. Being specific helps :)
Comment: I don't really do much with the DSLR end of the game, being as they are not something you tend to see in OB Trucks and studio galleries (Which is where my gear goes), but from what I have seen I doubt that many of the current DSLR for video camera chains would have much trouble with the right glass and in the right hands. My point really is that given the right crew and DP there is nothing there that is particularly difficult from a camera chain perspective. I will leave product recommendations to someone who plays in that space.
Comment: That''s upsetting that you cannot provide a product recommendation, seeing as how you seem to really know your stuff, but I appreciate your input so far.
Comment: "What matters for your application?" Having a camera capable of shooting this quality of video. That's it. Let me worry about all the other intrinsics. I appreciate your input, even if it didn't help. Thanks for trying


What does "RAM R/W test failed" mean?

Question: I had problems installing an operating system and booting my computer, so I disabled the system's "Intel® Rapid Boot BIOS" feature. Now, whenever I start the computer, I get a message telling me that my "RAM R/W test failed." What does that mean? Do I have to do anything about it?
Answer: I think its because your RAM is faulty ;)
Answer: Rapid BIOS boot only performs a very limited number of hardware tests. By disabling it, you have allowed the BIOS to perform all of the tests (POST) which would include a more thorough testing of your memory. You now need to look into sorting out your memory fault before you attempt to install an OS.
Answer: @soandos is probably correct. Ignoring the problem does not make the problem go away. This is like the low oil level warning light in your car; you can put black tape over it, but that doesn't raise your oil level.
RAM these days is dirt cheap. You might want to consider just replacing it.
Your PC may seem to be running okay, but who knows what spurious bits are being erroneously written, possibly corrupting your data? Is you peace of mind worth it?
If you don't replace it, be sure to keep good backups.

What does "RAM R/W test failed" mean?


How many KITTs were destroyed during production of the '80s Knight Rider series?

Question: I'm re-watching the original '80s Knight Rider series. I've noticed that numerous times, after taking a big jump, KITT comes down at a pretty severe angle--but the shot always cuts away before you see the result (The Hoff is always unharmed). Is it known (roughly) how many KITTs were wrecked beyond repair during production?
Comment: I just finished watching season 1 episode 11 and literally watched KITT's front end crumble after a hard landing. Easy to catch those little fails on HDTV's. Still my favorite show.
Comment: I remember seeing an article published in the '80s that claimed they wrote off four or five per episode.  But they were just shells fired from an air cannon.
Comment: @Chenmunka: Wow.  I remember reading once that on *Dukes of Hazzard* they got a new General Lee every 1-2 episodes.  4 or 5 cars per episode sounds downright insane!
Comment: @MasonWheeler - I was riding just N of I-5 on the 14 freeway as a kid when we passed a truck load of sheriff cars that were blue rather than LASD black and whites. Then we passed a second truck load of **General Lees**.
Answer: This one is actually surprisingly hard to answer, mainly because the show actually had four different grades of cars that were used in production; "Hero cars", "Stunt cars", "Jump cars" and "Shells".
Stunt / Jump cars
According to the book "Knight Rider Legacy" (referenced here in '15 facts you might not know about Knight Rider'), the show went through an average of just under one real car every two to three shows. These stunt cars were the ones that were seen driving around corners, high-speed acceleration and performing handbrake turns.

The stunt work was hard on the cars, and the show typically ruined
four to nine each season. The Trans Ams were sold by GM to the
producers for $1 each. Each one cost about $18,000 to modify into
K.I.T.T.

This interview with Bob Ewing, the show's Associate Producer indicates that the show had two dedicated "jump cars" that could perform jump stunts without being destroyed as well as 12 other stunt cars that were dedicated to other kinds of driving, jump and landing stunts (as well as being shot at or having parts ripped off)

We have two jump cars," says Ewing. These cars look identical to
K.I.T.T., but are made of lightweight fiberglass and contain
high-powered engines. A stunt driver races this fiberglass car at high
speed toward a hidden ramp. That ramp sends the car up into the air,
then down to another hidden ramp angled for a safe landing. The car
leaps over real trains and trucks. The stunts are timed to the
split-second to make certain no one is in danger.
"We've never had an accident," Ewing says proudly. If a stunt is too
dangerous, we don't do it."
How does the driver survive? Jump cars can be specially built to
survive leaps - but why doesn't the driver get thrown out of the car
by the impact? That's easy, says Ewing: "Our stuntman is tied into
that car every which way but Sunday."

Shells
I've seen numerous references on the internet that they also managed to trash at least 2-3 "shells" (mostly old trans-ams that the producers would buy from local car dealerships and modify by replacing panels (to create the unique shape) and removing the engines to lighten them enough to fire them from an air-ram) per show but I've struggled to find a formal interview confirming it.
Hero Props
There were also a set of around six "Hero" cars that were used for close-up filming during the various seasons. According to this interview with Hotrod.com, the three original "hero" cars supplied by Pontiac were supposed to be destroyed at the end of the first season but somehow managed to survive the crusher and ended up in private hands.
Comment: @immibis - You might not be aware of this, but pretty much every family show made since 1974 is an attempt to sell toys to children. In this case the toys were slightly larger and the children slightly older...
Comment: @immibis - I'm assuming GM was more interested in the marketing they would get for their Trans Ams than getting money the show runners.
Comment: @SystemDown - Yup. Every father in the US was forced to ingest a 30 minute long Pontiac advert **every week for nearly 5 years**.
Comment: One dollar each?
Comment: @Richard And once enough of those cars had been sold *Airwolf* was released?
Comment: @HagenvonEitzen - That show sold a gazillion air-fix kits.


Why Use Adobe Camera Raw?

Question: I prefer to edit photos in Photoshop. However, my photos nowadays are all raw. So, when I open them, Photoshop forces me first into Adobe Camera Raw. It automatically adjusts various aspects of the image. For example, the tint slider is never at 0. For the image I'm working on now, it's set to +16. Does that mean Camera Raw is adding a red tint to the image? Or does it mean the original image has a red tint to it? Likewise, it always defaults to  sharpening the image. I'm thinking of just setting all the sliders to 0 and then clicking "Open image" to do my editing in Photoshop. Is there any disadvantage to doing that?
Answer:
However, my photos nowadays are all raw. So, when I open them, Photoshop forces me first into Adobe Camera Raw.

That's because the information in a raw image file must be interpreted before it can be converted to an image that is anything like what you expect to see on your screen.

Why Use Adobe Camera Raw?

There is no such thing as "The" raw image, there are only possible images that can be produced using the information in the raw file, which is a set of single linear, monochromatic luminance values for each of the photosites (a/k/a sensels, pixel wells, etc.) on the sensor.
Demosaicing to account for the differences due to color filters in the Bayer mask (even when producing a B&W image - remember how color filters altered the brightness of different things by different amounts when we used them on our cameras with B&W film?), gamma correction to convert the linear values to the logarithmic ones we need so that the image does not look more or less like solid black, color temperature, white balance correction, etc. must all be applied to the raw sensor data before it begins to look anything like the thing we took a picture of.
That is what ACR, or any other raw conversion application must do: convert the filtered monochrome luminance values for each sensel into a viewable image by applying color demosaicing, gamma correction, color temperature and white balance correction for various types of light sources, etc. It has to use some value for each of those setting to do so.
If you center everything to zero, you're basically telling Adobe Camera Raw that the photo was shot and properly exposed under more or less ideal conditions: Sunlight with a color temperature of around 5200K to 5500K, no harsh shadows, etc. ACR will apply default gamma curves, color channel multipliers, contrast adjustments, etc. optimized for those assumed conditions. If your light source(s) was(were) significantly different than what the default assumes, then those setting will not be correct to accurately reproduce the color of the scene.
Comment: Thanks for the reply! That makes sense. Am I losing anything by setting the sliders to zero and correcting the image inside Photoshop instead of Camera Raw? It appears that all of the sliders in Camera Raw have an equivalent in Photoshop. I SEEM to get better results by doing my color and light corrections in Photoshop rather than Camera Raw, but that may be just because I am not proficient with Camera Raw. Should I become proficient in Camera Raw and make corrections in there before finishing the image in Photoshop?
Comment: Another technique that SEEMS to work ok for me is to open the raw files in Nikon's ViewNX 2 and then export them from there as 16-bit TIFFs without changing them. Then, I open the TIFFs in Photoshop and make my corrections. The images always need corrections, but I've always worried that Photoshop is "flying blind" and isn't able to make adjustments as precise as Camera Raw because Camera Raw is able to make a more-informed decision. I've been using Photoshop since 2004, so I feel much more comfortable with that program than Camera Raw.
Comment: @user1177071 The thing you have to remember is that even when you use ViewNX "without changing" them, you're just using the settings that were active in the camera at the time the image was captured for things such as color temperature, white balance, contrast, etc. That works fine for a wide variety of photos, but sometimes more control than what is offered in-camera may be desired for optimizing the image to be the best it can. But the interpretation of the raw image data that is based on the in camera settings is just one of many possible interpretations of the data from the sensor.
Answer: That case means you are applying +16 Tint (which is Magenta instead of Red).
I was not aware that Auto would ever adjust White Balance though. Just the Tonal things, I think. But Adobe Daylight or Cloudy choice does set +10 Tint.
You can double click the button slider on the controls, and they return to default, usually 0.
If you don't want the auto adjustments, you do have an option in Adobe Bridge, menu Edit - Camera Raw Preferences.
Uncheck Apply Auto Tone Adjustments.
We really ought to see the Before case, to know what exposure we are doing with the camera.
But then if desired, you can still do this Auto stuff for the current picture in ACR by clicking the Auto link.
Frankly, IMO, most adjustments in ACR, especially White Balance and Exposure, are much better and much easier than in Photoshop.
Answer:
For the image I'm working on now, it's set to +16. Does that mean Camera Raw is adding a red tint to the image?

This means that the captured data's white balance is shifted towards the magenta.  Any image you open will have both temperature and tint at the settings they were at when the data was captured (notice I'm not saying image--  See Michael Clark's answer for why).
As far as I'm aware, the only thing that Adobe changes automatically in Camera Raw is the Sharpening that you mentioned.  If you want this to be turned off, you can change the Camera RAW defaults.  Usually, I don't mind the minimal sharpening that is applied, but that is a personal decision.

Why Use Adobe Camera Raw?


What kind of guitar for big hands?

Question: For someone with big hands, what kind of guitar is suitable?
What are the advantages and disadvantages of big hands for playing guitar?
Comment: Hi - what kind of guitar ? electric or acoustic ?
Answer: Your big hands are not some imaginary pair of generic big hands. Like any other new guitar purchaser, you're going to have to go try some guitars and find one (or more) that feels right to you. There is no one true answer here.
Get out there and put your hands on some guitars. Don't limit yourself to what you think you want or what someone else has told you is good, right, or best. I went shopping for an Epiphone SG and ended up in love with an Ibanez SZ520 - because the neck on the Ibanez felt better than anything else I touched.
Go to a guitar store (one that carries real guitars - stay away from Wal-Mart, etc.,) and start trying some. You don't have to know how to play anything. You don't have to impress the salesman or the other people in the store. You just have to trust yourself to know what feels good and what doesn't.
If you want to get rigorous about it, bring a small notebook and a pen. When you find a model that feels good, write it down. Then you can ask the salesman (Or, frankly ask Google when you get home - for every incredible guitar salesman out there with a huge font of knowledge, there are 150 kids who know less about guitars than they do about Nintendo.) specifics - fret size, radius, neck shape, etc.
Here's the truth: A guitar that feels good will be played more than one that doesn't. Feeling good is more important than sounding good. Sound is malleable. If you want to play rock or blues, you can get a decent approximation of any sound out of almost any guitar. Obviously, you can't play metal with a nylon-string classical, but within reason, you can make a guitar that feels right into a guitar that sounds right.
But if playing feels bad, you'll never get good enough to care how you sound anyhow.
Answer: Advantages of big hands - potentially your reach, although this depends on whether your fingers are long, or just big. And for guitars with wide necks big hands can help.
Disadvantages of big hands - you might find it tricky to play certain chords high up the neck if you can't get your fingers into a small space.
Realistically though, I have seen people with all sorts of hands play perfectly well on short scale guitars and large basses, so I'm not sure you should worry too much about this. If you do find you have trouble up at the 22nd fret, play more down the other end. If the neck feels too thin, buy a wider necked guitar.
Comment: An extreme solution to getting a guitar with a wide next would be to play a classical guitar (they have wide necks) or get a 12-string and string it with 6 strings (bad for the neck, but lotsa room). A guy I currently play with has large hands and plays mandolin quite skillfully, though, so I'm inclined to think this is mostly a matter of practice unless your hands are truly ginormous.
Answer: Honestly, I think the advantages of having larger hands far outweigh any disadvantages -- I have larger than average hands, and love being able to wrap my thumb around the neck to fret the 6th string.
I also play mandolin, which has tiny fret spacing, and only have minimal problems high up the neck.
I'd say that if you have really big hands, you would want to stick to a full size guitar.  Classical guitars have wider string spacing, which may help too.
Comment: I agree with this. I have small hands, and altough there are some advantages (I don't feel my fingers are crowded while fretting a chord), the disadvantages far outweigh the advantages. I have seen videos of people playing riffs without moving their hand on the neck while I had to move my hand to play the same riff due to a very short reach. Similarly, playing "Hendrix" chords with the thumb is extremely difficult.
Answer: Advantage of having big hands is that you can do crazy stretches which people with small hands cannot. This will add some spice in your guitar playing and will separate you from the others.
As far as Guitar recommendation is concerned, i would recommend Ibanez metal fat fret guitars which are very nice for shredding, sweep picking and performing stretched legatos.
Then at the end of the day it is you who should be comfortable with your guitar because it becomes a part of your personality.
Answer: The radius and profile of the neck, and width of the fingerboard can make a big difference in the feel for different sized hands. I have long hands - my palm and fingers are long, so I favor a wider fingerboard and a thinner profile neck.
Ray Benson, the lead guitarist for Asleep At The Wheel, has big hands, and Guitar Player Magazine interviewed him years ago and specifically asked about whether there were pluses or minuses to this; his comment was basically they let him reach further than most people can.
The only downside to having longer fingers is when I'm playing on a narrower fingerboard by the nut. Sometimes it's hard to jam my fingers together for a D chord, but for rock it's seldom a problem, because I cheat and drop the F#, or switch to a guitar with a bit wider neck at the nut.
Answer: From the finger-picking perspective:
I have been playing for decades and only recently come to realize that a wide neck is right for my large hands. I was just working on the intro to The Boxer, and yes, ability to reach is significant. But what really helps me with the wider neck is left-fingers fretting without bumping the adjacent strings. I unstrung 6 from my 60's Epi 12 with a 2 inch nut and have a new best friend!
I don't really have fat fingers but they sure feel that way. I've seen guys play even really skinny necks and make it work. We all have our own limitations to work with & there's always someone who makes it look so easy, darn it!
Answer: I have large hands and play a Telecaster. Very squeezy at the open chord end of the board but great for A scale blues lead.
Can bend a D to a G note on the G string without too much trouble.
I struggle with intricate lead around the 12th fret or above however.
As Clint said: "A man's gotta' know his limitations".
Answer: I have very large hands.My ring finger is a size 13 on my fret hand.I have struggled for years to find guitars that fit me.Salesmen always steered me away from "baseball bat" necks because "no one" likes them.As a huge Zakk Wylde fan who has gone through lots of "great playing" guitars that just hung on my wall because they hurt to play,I went against the norm and tried a Wylde Audio guitar.That baseball bat felt great in my hand,I just did not care for active pickups.Six months later I ran accross a Ritchie Kotzen signature tele.Felt great and sounded great too.That "baseball bat" went home with me that night.I sold my other 3 electrics because they never got played anymore and have not looked back.An advantage to big hands.......my friends are always amazed by my vibrato and bending techniques,and fat thumbs allow me to grip a pick just right for pinch harmonics almost naturally.

What kind of guitar for big hands?


First known computer-found error in expert chess analysis

Question: What is the first known example of published expert chess analysis that was disproven by a computer search, with the winning player completely opposite? That is to say, a widely known published analysis by a chess grandmaster of a position with the conclusion that it was winning for one player, but later a computer search proved that actually it was winning for the other player. This comes close, but I am curious to see examples with win-to-lose reversal of opinion. To clarify, I am looking for the first time we discovered that the previously accepted analysis was wrong, and our discovery arose from a computer search.
Comment: @Evargalo: Sorry for the ambiguity. I meant "the earliest time a computer evaluation reversed a preivously published analysis". The idea is that it would be the first time that we are aware of a serious mistake that may not have been revealed to us without the computer analysis. I've edited my question to clarify that, thanks!
Comment: Are you interested in the oldest assessment that has been proven to be false by a computer (whenever the comp's evaluation has been revealed), or in the earliest time a comp evaluation reversed a preivously published analysis (even if that analysis was recent) ?    What I mean is : A chess expert published an analysis showing that some position is winning in year X. A posterior analysis with computer refutes that evaluation and proves the position is losing in year Y. Are you looking for the earliest X or the earliest Y ?
Answer: In the early eighties, the evaluation of the two bishops vs knight endgame was changed thanks to computer-assisted analysis.
Previously, that endgame was thought to be a draw, but now we know the bishops can force a win.
I don't know if this is the earliest instance, but it's definiely a quite old one
Comment: It sure is mentionned in Alain Villeneuve's *Les Finales*. I just don't know since when ! The first edition was in 1982, but I only have had the ~1997 in the hands.
Comment: Can you add a source ? I thought the reassessment of KBBKN happened in the early *nineties*, but I might be wrong.
Comment: No. I can't add a source. I'm talking from the top of my head. I think I read it in a chess endgame book some years ago
Comment: I plugged the endgame into tablebase (with the pieces all relatively centered) and it says White has a forced win but in over 50 moves (usually 75-100ish). So technically this means the endgame is a theoretical draw.
Comment: I thought I was quite explicit in my question asking for a **win-to-lose reversal** of opinion due to computer analysis.
Answer: Partial answers because this is an intrinsically difficult question, especially to prove it was the "first".
Endgame tablebases have changed the evaluations of some endgames. E.g.,

Queen versus two bishops. This was thought to be a draw due to the existence of a drawing fortress position, but the queen can win most of the time by preventing the bishops from getting to the fortress. However, it can take up to 71 moves to force a win (Nunn 2002:290ff).

The question also asks about grandmaster analysis being incorrect, with the actual result being the opposite. This certainly has happened before simply because grandmasters are fallible humans. Example by Tim Krabbe:
<code>3r2k1/4qpbp/Q4P2/8/2N5/4p3/PP4PP/5RK1 w - - 0 1
</code>
Black to move. With the powerful passed pawn on e3, Black is clearly winning. 26...e2 27. fxe7 Bd4+ and White Resigns.
This game was published as a brilliancy, and it was months before someone spotted the fatal flaw in the combination: the position after 28...e2?? is actually winning for White. I won't spoil it here, but if you're interested you can find the solution on Tim Krabbe's website. Of course, in today's world, errors like this will never get published because an engine will spot the winning move in a fraction of a second.
Comment: Your example by Tim Krabbe would count **if** it had not been discovered until a computer analysis. But Tim's website says "a reader of the Observer chess column, which was one of the many where this combination was published, asked what happens if White plays 28.Ne3.". So, this doesn't count for the purposes of my question.
Comment: Yeah, I know. Same goes for the first bullet point, which didn't flip the evaluation from win to loss or vice versa. That's why I said partial answer.
Comment: @user21820 PS, maybe edit out the winning move for the next reader =)
Comment: Sorry, editing can only be done for the first 5 min. And thanks for your answer anyway!
Comment: @user21820 delete the comment and repost the relevant parts!


How to Get Rid of Smell in a Refrigerator?

Question: How do you get rid of the mixture of smells in a refrigerator (Not Unplugged) so that other items don't inherit their nasty smell? I have heard of baking soda, but are there other techniques?
Comment: Why can't you use baking soda?
Answer: Activated charcoal works better than baking soda. This is the experiment done to find the efficiency of activated charcoal vs Baking soda.
Similar to charcoal, activated carbon also helps to absorb unpleasant odors inside the refrigerator.
activated charcoal
activated carbon
Answer:
Cleaning with sponge and mixture of water and baking soda.
Cleaning with white vinegar.
Leave a package of backing soda opened in the fridge for 24 hours.
Leave a cup of white vinegar opened in the fridge for 24 hours.
Answer: Some methods are:

Baking soda. Just leave it in there, baking soda leeches odours from the air.
Throw away rotting food(Seriously). Any rotting food will make a odour, but sometimes the rotting food is in small particles.
Comment: Are there anything besides baking soda?
Answer: Lime and baking soda.
It makes your fridge smell great. Just be careful because lime is acid and you should know what happens when you mix acid with baking soda.
(HINT: Baking soda and vinegar)
Answer: This article shows a lot of ideas that I use:

Lemon juice. This is a also cutting a lemon in half, this removes the odour.

Remove refrigerator odors with ease. Dab lemon juice on a cotton ball
  or sponge and leave it in the fridge for several hours. Make sure to
  toss out any malodorous items that might be causing the bad smell.

Cotton Balls. These can also be used with a variety of scented oils. I don't know, about their lasting length, as I have only used them for a day or 2 to give a nice odour after something smelly was there.

Sometimes the refrigerator just doesn't smell fresh. Dampen a cotton
  ball with vanilla extract and place it on a shelf. You'll find it acts
  as a deodorizer, offering its own pleasant scent.

Real Extract is what I use. But the fake stuff should work.

Tomato Juice. Never used it, but it is suppose to remove skunk smell.

Did a power failure cause the food to spoil and become malodorous in
  your fridge? Get rid of spoiled-food smells in your refrigerator and
  freezer with the help of some tomato juice. After disposing of the bad
  food that caused the smell, thoroughly wipe the insides of the fridge
  and freezer with a sponge or washcloth doused in undiluted tomato
  juice. Rinse with warm, soapy water and wipe dry. If traces of the
  smell remain, repeat the procedure or substitute vinegar for the
  tomato juice.

How to Get Rid of Bad Smells in Your Fridge:
You should always clean the things that reside in your refrigerator before putting them back in. Watching the things you put in your refrigerator also helps. Cups of vinegar and trays of baking soda also helps, but there is also:

Use oats as an odor-absorber. Leave a bowl of oats out in your fridge
  and this will suck in the bad smells.

I've never used this, but oats are really absorbent and pleasant smelling, besides dusty.
Coffee grounds and charcoal are some good odour removers. But make sure that the charcoal is safe.

Let activated charcoal absorb bad odors. Similar to the coffee grounds
  method, place charcoal briquettes -- do not use briquettes with
  flammable additives as these are poisonous -- in a couple of shallow
  bowls or on baking sheets and put them on different shelves in the
  fridge. Set the refrigerator temperature to low and leave in with the
  door closed for several days. The smells in question should go away
  within a few days. Activated charcoal can be purchased from pet stores
  or drugstores. Charcoal can be refreshed by placing in the oven at
  350ºF/180C for 20 minutes.

I use clay cat litter, but you may use others. This may be pricey if you don't own a animal.

Use unscented chlorophyll cat litter. Spread in a shallow pan and
  leave in the refrigerator until the smell disappears.

This article said to unplug the refrigerator and while it is advised to unplug electric devices before cleaning them you should be fine to leave it on. I advise you greatly to consult your owners manual.
These methods work only if the refrigerator is cleaned of the rotting food, mould, or other problem that caused the smell. They can take away the smell, but resolving the odour is more hygienic.
sheknows.com has some helpful stages.

Clean

Even if you didn't have to purge the contents your fridge, this is as
  good a time as any to give it a good cleaning. Use your favorite
  antibacterial cleaning spray and a dishcloth or paper towels to wipe
  down all the surfaces.

Absorb

Next, you're going to want to put something in the refrigerator that
  will absorb the smell. Baking soda is the classic choice, but ground
  coffee or even tea bags are great alternatives. Leave them in open
  containers in the fridge for as long as there is an odor (many people
  opt to leave an open box of baking soda in the refrigerator at all
  times as a preventative measure).

Perfume

Once you have the smell mostly gone, you can start to mask it just a
  little with a few natural kitchen "perfumes." Rub a vanilla
  extract-soaked cotton ball on the shelves for a fresh vanilla scent or
  spread a few lemon slices throughout the shelves (just don't forget
  about them and let them get moldy — you'll be back at square one!).

And remember to wrap all the food up in airtight containers, and sealable jars and bags. This will make sure the odours doesn't affect other food, as well.

If the smell persists, give it another thorough clean. That, and some
  time, should be enough to rid your fridge of even the most persistent
  smells. To prevent your food from absorbing the odors, be sure
  everything is tightly sealed in jars or plastic containers. Ice cube
  trays with covers are great if you can find them.

So wear a CAP when cleaning the refrigerator.
If the refrigerator is still smelly after all of this stuff, check its vents. It may have moulded or have severe water damage. Using several methods, always works better than one.

Ways to prevent odours

Use airtight containers.
Clean spills and messes.
Watch for spoiling food and remove appropriately. Having a paper on the door of the fridge that says when food was bought, opened and its expiration date should help.
Having regular maintenance. This means cleaning any pans or vents and calling someone to fix it. Adding some deodorants to the pan works amazing.
Storing food properly.

Cleaning a unplugged refrigerator and a plugged one are pretty much the same only the methods may be a bit less noxious.


How was the composition and structure of water determined?

Question: Water is a pretty basic molecule that people used since early science. What were the steps in figuring out what water is made of and what is its structure?
Answer:
Composition (i.e. water is not an element):

1783: The Scottish inventor James Watt published a paper on the composition of water in 1783; Cavendish had performed the experiments first but published second. Controversy about priority ensued. (source: Wikipedia - Cavendish)
1783: Lavoisier began a series of experiments on the composition of water which were to prove an important capstone to his combustion theory and win many converts to it. Many investigators had been experimenting with the combination of Henry Cavendish's inflammable air, which Lavoisier termed hydrogen (Greek for "water-former"), with dephlogisticated air (oxygen) by electrically sparking mixtures of the gases. All of the researchers noted the production of water, but all interpreted the reaction in varying ways within the framework of the phlogiston theory. In cooperation with mathematician Pierre Simon de Laplace, Lavoisier synthesized water by burning jets of hydrogen and oxygen in a bell jar over mercury. (source: Wikipedia - Lavoisier)

Ratio of oxygen to hydrogen:

1805: Gay-Lussac and Humboldt discover that water is formed by two parts of hydrogen and one part of oxygen (by volume). (source: Wikipedia - Gay Lussac)
1842: Dumas fixed the ratio H:O as 2:16. (source: Atomistry - Atomic weight of hydrogen history)
Comment: It'd be cool if you explained a bit the setup of Watt's and/or Cavendish's experiments. +1
Answer: I am pretty sure that Lavoisier was the the first to experimentally prove water a compound by running it down super heated gun barrels. The oxygen bound to the iron, and hydrogen came out as a gas. By careful weighing he determined that the barrel had gained mass, indicating something from the water had bound to the metal in the barrel, and hence that the water had two components.
Cavendish and Watt came at the problem from another direction, producing hydrogen first from acids and metals and only later connecting hydrogen with water, thus inferring it was a compound.
It's actually kind of hard to tell as no journals existed at the time and most scientific discoveries were conveyed by chain letters. Nationalism played a great role as well, particularly with the English and French each claiming priority for this or that.
Lavoisier also spent most of the 1770s and 1780s perfecting French methods for producing stable and reliable gunpowder (which played a major role in the American revolution) so much of his work during that time was considered state secrets. Many of his papers were destroyed when he was murdered in the French revolution. As a result, we don't have a clear picture of exactly when he did what.
All in all though, it was a surprisingly small group of individuals, really a few dozen at most, that jump started chemistry in the late 1700s. They all communicated and built on each other's works making a collective snowball of progress, so determining priority down the wire is really more of a game than useful history.