This question comes from this  debate at SuperUser Forum. I have asked it here too because:     It appears not to be only a technical question, but a language question too. Most of the answers in the referred thread until now seem to me most like "Everybody says 32 bit, so it must be correct" kind, instead of reason and logic based.   The question is: as long as a computer that works with, say, 64 bit(s) uses more bit(s) for computing than a computer that works with, say, 32 bit(s), should we say ... ?   32 bit operating system 32 bits operating system   What if we change "operating system" by "graphics card", "motherboard", or "version of the program"?
Why are year, pound and mile in the singular form in the phrases below?   five-year-old children  20 pound note 10 mile run   Is that because they're acting as adjectives, which are always invariable in English?  Is it incorrect to say...   five-years-old children? 20 pounds note? 10 miles run?
What's the right terminology for Unity's UI elements? I've sent a few mails to the Ayatana mailing list regarding some UI suggestions for Unity, but I'm afraid I keep confusing the right terms for the UI element, so I would like to have some clarification.  Answer Index: Simplifying adjacent polygons using QGIS Simplify Geometries tool How do I check if an element is hidden in jQuery? Is it possible to toggle the visibility of an element, using the functions .hide(), .show() or .toggle()?  How would you test if an element is visible or hidden? Creating your own Tinyurl style uid I'm writing a small article on humanly readable alternatives to Guids/UIDs, for example those used on TinyURL for the url hashes (which are often printed in magazines, so need to be short).  The simple uid I'm generating is - 6 characters: either a lowercase letter (a-z) or 0-9.   "According to my calculations captain", that's 6 mutually exclusive events, although calculating the probability of a clash gets a little harder than P(A or B) = P(A) + P(B), as obviously it includes numbers and from the code below, you can see it works out whether to use a number or letter using 50/50.  I'm interested in the clash rate and if the code below is a realistic simulation of anticipated clash rate you'd get from generating a hash. On average I get 40-50 clashes per million, however bare in mind the uid wouldn't be generated a million times at once, but probably only around 10-1000 times a minute.  What is the probability of a clash each time, and can anyone suggest a better way of doing it?  static Random _random = new Random();  public static void main() {     // Size of the key, 6     HashSet&lt;string&gt; set = new HashSet&lt;string&gt;();     int clashes = 0;     for (int n=0;n &lt; 1000000;n++)     {         StringBuilder builder = new StringBuilder();          for (int i =0;i &lt; 7;i++)         {             if (_random.NextDouble() &gt; 0.5)             {                 builder.Append((char)_random.Next(97,123));             }             else             {                 builder.Append(_random.Next(0,9).ToString());             }         }          if (set.Contains(builder.ToString()))         {             clashes++;             Console.WriteLine("clash: (" +n+ ")" +builder.ToString());         }          set.Add(builder.ToString());         _random.Next();         //Console.Write(builder.ToString());     }      Console.WriteLine("Clashes: " +clashes);     Console.ReadLine(); }   UPDATE:   from this question  I really asked two questions here so I was cheating. The answer I was after was rcar's, however Sklivvz's is also the answer to the 2nd part (an alternative). Is it possible to make a custom unique id generator in a database, or would it be client side (which would be 2 possible reads first)?  The general idea I was after was using Ids in databases or other stores that can be used by phone or printed material, not a giant 16 byte guid.  UPDATE 2: I put the formula for two mutually exclusive events above instead of 2 independent ones (as getting an 'a' the first time doesn't mean you can't get an 'a' the second time). Should've been P(A and B) = P(A) x P(B) Suppose $f,g$: $\Bbb R$ → $\Bbb R$ are continuous functions such that $f(r) = g(r)$ for all $r$ in $\Bbb Q$. That is to say, $f$ and $g$ are equal on the rational numbers.   Prove that  $f(x) = g(x)$  for all $x$ in $\Bbb R$ Diagonal straight lines in residuals vs fitted values plot for multiple regression I am observing strange patterns in residuals for my data:   [EDIT] Here are the partial regression plots for the two variables:      [EDIT2] Added the PP Plot   The distribution  seems to be doing fine (see below) but I have no clue where this straight line might be coming from. Any ideas?   [UPDATE 31.07]  It turns out you were absolutely right, I had cases where the retweet count was indeed 0 and these ~ 15 cases resulted in those strange residual patterns.   The residuals look much better now:   I've also included the partial regressions with a loess line. I have a .bash_profile in my home directory, but it isn't getting run on login. If I do the following, then things seem to be as I expect:  ssh myhost bash source ~/.bash_profile   But normally that all happens on login. Thoughts? I downloaded and installed Ubuntu from the official site. However, I don't know if I installed the 32-bit or 64-bit version.  In Windows 7 I could right click My Computer and it listed which version it was.  Is there an easy way to check in Ubuntu? How should I calculate $\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+...+n^n}{n^n}$ How should I calculate the below limit  $$\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+...+n^n}{n^n}$$ I have no idea where to start from. If object A is moving at velocity $v$ (normalized so that $c=1$) relative to a ground observer emits object B at velocity $w$ relative to A, the velocity of B relative to the ground observer is  $$ v \oplus w = \frac{v+w}{1+vw} $$  As expected, $v \oplus 1 = 1$,  as "nothing can go faster than light".  Similarly, $v \oplus -1 = -1$. (same thing in the other direction)  But what if object A is moving at the speed of light and emits object B at the speed of light in the exact opposite direction? In other words, what is the value of $$1 \oplus -1?$$ Putting the values in the formula yields the indeterminate form $\frac{0}{0}$. This invites making sense of things by taking a limit, but $$ \lim_{(v,w)\to (1,-1)} \frac{v+w}{1+vw}$$ is not well-defined, because the limit depends on the path taken.  So what would the ground observer see? Is this even a meaningful question?  Edit: I understand $1 \oplus -1$ doesn't make sense mathematically (thought I made it clear above!), I'm asking what would happen physically. I'm getting the sense that my fears were correct, it's physically a nonsensical situation. Is there an option to create and Opt-in a user in the same API call? How do I set up an account with multiple users? The identity cannot be a commutator in a Banach algebra? The Wikipedia article on  claims, without a proof or reference, that there does not exist a (unital) Banach algebra $B$ and elements $x, y \in B$ such that $xy - yx = 1$. This is surprising to me, but maybe the proof is straightforward; anyone have a proof and/or a reference?   More generally, I would have naively thought that I could embed any ring into a Banach algebra. I guess there are actually serious restrictions to doing this; are these issues discussed anywhere? how to use these hash functions in python? In this article , specifically in the section 4.1 of the initialization phase,the following cryptographic hash functions have been chosen: \begin{align} h_1&amp;:   \mathbb G\to\mathbb Z_q^*\\ h_2&amp;:  \{0,1\}^*\times\{0,1\}^*\times\mathbb G\to\mathbb Z_q^*\\ h_3&amp;:  \{0,1\}^*\to\mathbb Z_q^* \end{align} where G is an additive group with order q. Can anyone explain to me what's the meaning of their notations and how to use them for example in SHA algortihm? Or maybe give me a simple example with real inputs? USB-C to MagSafe 2 Adapter I have a 2015 MacBook Pro with a MagSafe 2 charging cable. My wife has a new MacBook Pro from 2016 with USB-C. I was wondering whether I could technically (in the matter of electronic feasability) use her charging cable on my MacBook and vice versa with some sort of adapter / dongle, and, if so, where I would get one? Transpose Ranges repeated by row I would like to convert several range of columns into rows, and keeping the association to the original col/row source  Sample source sheet  --------------------------------------------------- | Data 1 | 1/1/2018 | 1/2/2018 | 1/3/2018 | etc...365 days | --------------------------------------------------- | name 1  |    25    |   30     |    5     | etc...|  --------------------------------------------------- | name 2  |    2     |          |    4     | etc...| --------------------------------------------------- | etc ... 100 rows   Expected Results  --------------------------------------------------- | Data 1 |   Date      |  Value   |  --------------------------------------------------- | name 1  |  1/1/2018   |   25     |    --------------------------------------------------- | name 1  |  1/2/2018   |   30     |    --------------------------------------------------- | name 1  |  1/3/2018   |   5      |    --------------------------------------------------- | name 1  |  etc.. 365 days --------------------------------------------------- | name 2  |  1/1/2018   |   2      |    --------------------------------------------------- | name 2  |  1/2/2018   |          |    --------------------------------------------------- | name 2  |  1/3/2018   |   4      |    --------------------------------------------------- | name 2  |  etc.. 365 days --------------------------------------------------- | etc ... for each change of name pull all the col dates/values   Current results  --------------------------------------------------- | Data 1 |   Date      |  Value   |  --------------------------------------------------- | name 1  |  1/1/2018   |   25     |    --------------------------------------------------- |         |  1/2/2018   |   30     |    --------------------------------------------------- |         |  1/3/2018   |   5      |    --------------------------------------------------- |         |  etc.. 365 days                      I have to copy down to bring in "name 1" for each date  --------------------------------------------------- | name 2  |  1/1/2018   |   2      |             now update the formula to bring in the second item for the sequence of dates --------------------------------------------------- |         |  1/2/2018   |          |    --------------------------------------------------- |         |  1/3/2018   |   4      |    --------------------------------------------------- |         |  etc.. 365 days --------------------------------------------------- | etc ...     =Sheet1!A:A I can bring in the name from the source list, but I have to copy it down for the number of dates.  How can I have it match the number of dates?  and be dynamic to update as dates/values are added Edit: trying to use split and rept to count the number of times to repeat, then transpose.. =transpose(rept(split(arrayformula(concatenate(Sheet1!A:A&amp;"1")),"1",True, True),COUNTA(Sheet1!I1:ZZ1)))  =TRANSPOSE(Sheet1!$C$1:$Z$1) I can use transpose to get the dates/values, but it's fixed range.  How can I make it dynamic to import dates as they are added? How can you prove that the square root of two is irrational? Should I do anything when I see several references that are not used in the text? In a linguistics paper I read, I noticed that several references in the bibliography are not actually used in the text. This is for about 10-15% of the references the case. Should I do anything with this? I always thought that all references must be used in the text, because even if they are just suggestions for further reading you would want some explanation why it is relevant and what you can find where.   I only know the author through their work and do not know the editors. The paper is from 2011. The references are rather general. They are relevant to the paper, but you would want page numbers with them unless you are familiar with them. They are all from different authors and different institutes. It is likely that the references were used in previous versions of the text. Proving $A+A'B=A+B$ without truth tables How can I prove the Boolean algebraic rule $$A+A'B=A+B$$ without using a truth table?  With the truth table, it is easy to see that the two are equal, but how can I prove it using lesser Boolean identities? How can I reliably and accurately identify the passive voice in writing or speech? How can I reliably and accurately identify the passive voice in writing or speech? I'm not interested in advice about whether or not to use it yet... I just want to know for sure what it is, so that I don't look as stupid as .