Is there any reason why there are specifically Nine Riders? The Council of Elrond appointed Nine Walkers in response to the Nine, but why nine?
Is there any data in Tolkien (books/letters/etc..) to explain why there were precisely 9 Men rings and 7 Dwarf rings and 3 Elf rings? (the latter assumes that Sauron planned the number of Elvish rings to be 3 before he was thwarted by Elves making the rings themselves)  Was it because of the amount of tribes/kings he needed to deal with? Some magical/power limitations? Magical properties of numbers 9 and 7 (ala 7 Horcruxes)? A whim of Sauron's?    UPDATE: Looks like 3 and 7 are explained in the answers (3 elf lords and supposedly 7 Dwarf houses, though nobody provided canon proof).  What was NOT explained was why there were 9 Man rings?    If 9 was a meaningful amount, why? If it was merely "19-7-3" leftover, then why were there 19 total non-One rings made?
Example of non-isomorphic structures which are elementarily equivalent What happens to net acceleration in a non-inertial reference frame with cenripetal force? In a non-inertial reference frame, the centripetal force is balanced by a centrifugal force. Therefore, shouldn't the net acceleration be zero? Yet, you still feel an acceleration outwards? Why is that? How to delete pre-installed apps in Catalina? I want to delete the pre-installed apps, like Music, Books, Siri etc. from my iMac running Catalina. How do I do this?  (NOTE:  asks the same thing but was closed as a duplicate of . Annoyingly, the supposed duplicate doesn’t give an obvious answer to the question, and the original was locked can’t be answered, hence a new question. I can’t add an answer to  either) Proving that if $V$ is an inner product space and $U_1, U_2$ are subsets, then $U_1 \subset U_2$ if and only if $U^{\perp}_2 \subset U^{\perp}_1$ Why doesn't Doctor Strange remove the Infinity Gauntlet from Thanos? Doctor Strange seemingly has the ability to open portals wherever he wants. We see at the beginning of Infinity War what happens when he closes a portal and someone isn't fully through it.  Is there any reason why he can't open a portal around the left-elbow of Thanos, and then close it?  I realize that Thanos probably wouldn't allow this to happen normally,      but when he is mostly asleep, and Tony and Spider-Man are pulling on the gauntlet, Doctor Strange has ample opportunity, and plenty of space to just sever Thanos' arm.   Why doesn't he? Strange floating-point behaviour in a Java program In my program I have one array with 25 double values 0.04 When I try to sum these values in a loop I get following results:  0.0 + 0.04 = 0.04 0.04 + 0.04 = 0.08 0.08 + 0.04 = 0.12 0.12 + 0.04 = 0.16 0.16 + 0.04 = 0.2 0.2 + 0.04 = 0.24000000000000002 0.24000000000000002 + 0.04 = 0.28 0.28 + 0.04 = 0.32 0.32 + 0.04 = 0.36 0.36 + 0.04 = 0.39999999999999997 0.39999999999999997 + 0.04 = 0.43999999999999995 0.43999999999999995 + 0.04 = 0.4799999999999999 0.4799999999999999 + 0.04 = 0.5199999999999999 0.5199999999999999 + 0.04 = 0.5599999999999999 0.5599999999999999 + 0.04 = 0.6 0.6 + 0.04 = 0.64 0.64 + 0.04 = 0.68 0.68 + 0.04 = 0.7200000000000001 0.7200000000000001 + 0.04 = 0.7600000000000001 0.7600000000000001 + 0.04 = 0.8000000000000002 0.8000000000000002 + 0.04 = 0.8400000000000002 0.8400000000000002 + 0.04 = 0.8800000000000002 0.8800000000000002 + 0.04 = 0.9200000000000003 0.9200000000000003 + 0.04 = 0.9600000000000003   Why on earth could that happen?! Alien invasion movie, giant robot aliens with long legs capturing people Indirect return of all elements in an array The Bash man page describes use of ${!a} to return the contents of the variable whose name is the contents of a (a level of indirection).  I'd like to know how to return all elements in an array using this, i.e.,  a=(one two three) echo ${a[*]}   returns  one two three   I would like for:  b=a echo ${!b[*]}   to return the same.  Unfortunately, it doesn't, but returns 0 instead.  Update  Given the replies, I now realise that my example was too simple, since of course, something like:  b=("${a[@]}")   Will achieve exactly what I said I needed.  So, here's what I was trying to do:  LIST_lys=(lys1 lys2) LIST_diaspar=(diaspar1 diaspar2)  whichone=$1   # 'lys' or 'diaspar'  _LIST=LIST_$whichone LIST=${!_LIST[*]}   Of course, carefully reading the Bash man page shows that this won't work as expected because the last line simply returns the indices of the "array" $_LIST (not an array at all).  In any case, the following should do the job (as pointed out):  LIST=($(eval echo \${$_LIST[*]}))   or ... (the route that I went, eventually):  LIST_lys="lys1 lys2" ... LIST=(${!_LIST})   Assuming, of course, that elements don't contain whitespace. How to solve "permission denied" when using sudo with redirection in Bash? Is there a rule for pronouncing “th” at the beginning of a word? How to delete a file with a weird name? I wrote a program that uses POSIX memory-mapping function (mmap)  The program takes a file (a.dat) and memory-maps it for reading/writing.  Due to errors in the program, every time I run the program a file with some weird names (e.g., ?d?P?^z??d?P?^z?) is created. The error is resolved but I am not able to delete the files.  I am not able to delete it either using command line or by select/deleting from window manager.  So how should I delete it? I'm using Ubuntu 11.04. What is the logarithm of a kilometer? Is it a dimensionless number? In log-plots a quantity is plotted on a logarithmic scale. This got me thinking about what the logarithm of a unit actually is.  Suppose I have something with length $L = 1 \:\mathrm{km}$.  $\log L = \log \mathrm{km}$  It seems that the unit of $\log L$ is $\log \mathrm{km}$, but I can also say $L = 1000 \mathrm{\:m}$ and now:  $\log L = 3 + \log \mathrm{m}$  This doesn't appear to have any units at all.  This suggests that $\log \mathrm{km}$ and $\log \mathrm{m}$ are actually dimensionless numbers. But wait, I can do this with any unit! Does it actually make sense to talk about the logarithm of a unit, or some other function for that matter? Calculating $\sum_{n=1}^\infty\frac{1}{(n-1)!(n+1)}$ I want to calculate the sum:$$\sum_{n=1}^\infty\frac{1}{(n-1)!(n+1)}=$$  $$\sum_{n=1}^\infty\frac{n}{(n+1)!}=$$ $$\sum_{n=1}^\infty\frac{n+1-1}{(n+1)!}=$$ $$\sum_{n=1}^\infty\frac{n+1}{(n+1)!}-\sum_{n=1}^\infty\frac{1}{(n+1)!}=$$ $$\sum_{n=1}^\infty\frac{1}{n!}-\sum_{n=1}^\infty\frac{1}{(n+1)!}.$$  I know that $$\sum_{n=0}^\infty\frac{1}{n!}=e$$  so $$\sum_{n=1}^\infty\frac{1}{n!}=\sum_{n=0}^\infty\frac{1}{n!}-1=e-1$$  But, what can I do for $$\sum_{n=1}^\infty\frac{1}{(n+1)!}$$ ?  Am I allowed to start a sum for $n=-1$ ? How can I bring to a something similar to $$\sum_{n=0}^\infty\frac{1}{n!}$$? The ideal generated by $2$ and $ x$ in $\mathbb{Z}[x]$ Let's have 2 numbers, N and K, where K divides N. The number of K-combinations from a given set S of N elements is a well known formula. Let's concatenate N/K groups (resulting in N elements) such as the resulting set is N. How many possibilities are there, i.e. what's the formula?  For instance:    N=4, K=2, C(4, 2) = 6 {1,2},{1,3},{1,4},{2,3},{2,4},{3,4}   The 3 possibilities are:     {1,2},{3,4}   {1,3},{2,4}   {1,4},{2,3}   I generated these combinations and I think the number goes like this:  (4, 2): 3 (6, 3): 10 (6, 2): 15 (10, 5): 126 (9, 3): 280 (10, 2): 945 (14, 7): 1716 (12, 4): 5775 (15, 5): 126126 (15, 3): 1401400   Apparently, the result always divides with (N-1). What is the $n$ times iterate of $f(x)=\frac{x}{\sqrt{1+x^2}}$? Is there any way of making a Minecraft server lower than the current version? So me and my friends manged to make a server in Minecraft for us to play on and we would like to add mods. But the mods we want to add are a few versions lower than the updated version. Is there any way of making a server with a lower version of Minecraft that people can still play on? Does IBIS reduce image resolution? How does it compare to lens based IS? Prospective PhD contacts potential supervisor but receives no answer after 2-3 emails, what to do? Say a prospective PhD has a high reply rate (90-95%) to first contact emails with potential superviors. He/She has a well thought out e-mail, but a particular professor (here: USA) that is in the top preferences does not reply after two to three e-mails that were sent within a time frame of three months. I also followed the instructions on the professor personal website in the section "prospective PhD students". The student has already contacted some of the advisor students and has received positive feedback about the professor.  Should the student just give up and move on? Should the student keep trying?  Aside from the obvious "I have too many incoming e-mails to answer" or "I am out in the field", what could create a no response behavior from the professor?  I have seen the simplistic answer "move on, bad advisor, doesn't have time, etc". But this seems a contradiction in this case. The PhD students like the professor and say he/she is personable and no e-mail response after a few well written emails. On June 13th 2017 all SharePoint calculated columns with HTML markup stopped working.   Basically the fields now render text (regardless of setting the data type to number). The gist of the notice from Microsoft () is:     In Microsoft SharePoint lists and libraries, calculated fields that   are running in the classic UI mode can be used to display results   based on Excel-like formulas. This is a long standing capability, and   is documented in Calculate data in lists or libraries.      Some users have added HTML markup or script elements to calculated   fields. This is an undocumented use of the feature, and we will begin   blocking execution of custom markup in calculated fields in SharePoint   Online from June 13, 2017 onwards. We are also providing this as a   configurable option for on-premises in SharePoint Server 2016 and   SharePoint Server 2013 via the June 2017 and subsequent Public   Updates.   As we don't have on premise SharePoint (online), we have to find an immediate work around, given that this impacts our processes and thus, business.   Is anyone aware of third party applications that will allow for HTML, script markup (that works with SP online)?