This is sort of a multi-part question that adds up to one larger question:   Can Sauron get to the Undying Lands? ( seems to have produced the answer to this - "No") Is the One Ring still dangerous even if Sauron can't possibly get to it? Is Elrond strong enough to resist the Ring? Could the Ring be taken to the Undying Lands, perhaps by Frodo, Bilbo, or Elrond?   Assuming the answers to these questions are 1. no, 2. not really, 3. yes, and 4. yes, then wouldn't it be feasible for someone to just bring the Ring to the Undying Lands and give/send it to someone who could safely hold on to it?  Say, perhaps, Eru Iluvatar, Manwe, Elbereth, Cirdan, etc?  This wouldn't require the person in question to actively intercede in worldly affairs - they would merely have to accept a gift and keep it forever.  Since humans (and/or elves and/or hobbits or whoever took the initiative in getting the Ring there, it would be more or less the same as bringing it to Mordor and destroying it, but less suffering and loss of life would be required in the process.  So basically, the overall question is:  Why not send the Ring to the Undying Lands?  *Note - since sea monsters never seem to attack ships carrying Elves, Valar, or Maiar to the Undying Lands, despite Sauron's hatred of them, the sea monster issue is irrelevant.  In any case, this question isn't about whether the ship would get there safely if passage were granted - it is about whether passage WOULD be granted and whether it would solve the problem.   *Note 2 - granted, this wouldn't get rid of Sauron, but Aragorn was free to set the terms of the Army of the Dead's deployment.  If he said "You must serve for a thousand years" instead of "You have to fight one battle", then the Army of the Dead could have roamed through Middle-earth and Mordor killing off every Orc and goblin Sauron enlisted.  Sauron would still exist, but he would be a more or less powerless eye stuck on top of a tower, with no hope of raising and army for a thousand years. If Aragorn said "You must serve forever", Sauron would be deprived of an army forever.
At the end of ‘The Return of the King’, Bilbo, Frodo, Gandalf and others boarded a boat and left Middle-earth for Valinor.  Is there any canon explanation why they couldn’t just take the ring there in the first place and leave Middle-earth altogether? Would Sauron have gone after them? Couldn’t the elves living there safeguard it?  I do realize that without Frodo taking the ring to Mount Doom there would be no story at all, but I wonder whether or not there is any canon explanation for this.
Determinant of block matrix with commuting blocks I know that given a $2N \times 2N$ block matrix with $N \times N$ blocks like   $$\mathbf{S} = \begin{pmatrix} A &amp; B\\  C &amp; D \end{pmatrix}$$  we can calculate $$\det(\mathbf{S})=\det(AD-BD^{-1}CD)$$ and so clearly if $D$ and $C$ commute this reduces to $\det(AD-BC)$, which is a very nice property.   My question is, for a general $nN\times nN$ matrix with $N\times N$ blocks where all of the blocks commute with each other, can we find the determinant in a similar way? That is, by first finding the determinant treating the blocks like scalars and then taking the determinant of the resulting $N\times N$ matrix.  Thanks in advance! Intuitive Understanding of the constant "$e$" Let me start this with : Very long "About me" text squeezes both the left side (flair) and the right side (the section with location, GitHub link, etc.)  Now, the reverse thing has happened:         put a long text in their "location", and it causes the right section to be overly extended, effectively squeezing the About Me section.  When will the squeezing game end? I have my CIFS setup in fstab and they are working as they are supposed to on boot. They mount as they should and work for a while. Out of nowhere it seems (could be after unlocking machine etc) I get "Host is down" error trying to access it. I have multiple and they are all down. They are also shared from the same server. At this time i check on a windows computer and an outdated 14.04 machine and they are up and functioning as they are supposed to. After clicking around on the shares in nautilus and getting repeat errors they will just start working again. To access a share that is "down" takes about 2-3min of randomly clicking different mounts and going back to the first one when automagically it shows the data in the mount point.  I do not have this problem on 14.04 machines that have not been updated in a while. All of those machines are fully functional and the CIFS never go "down". On 16.04 they were not a problem until more recently.  I have made sure to update every other day and have cleaned old linux headers (in hind sight i probably should have reverted). I do this because im begging for a fix to just appear but its been weeks of battling CIFS mounts without any solution. Why didn't the Order of the Phoenix assist with the trio (i.e.  Harry Potter, Ron Weasley and Hermione Granger) in the Horcrux hunt? Does anyone know how to get your business map location displayed in the right hand sidebar of the Google search results page? As in the screenshot below:      I already added my business to this link: .  Is adding detail in above link enough or do I need to change my website coding? I will be getting a point-and-shoot camera for taking portraits and only portraits.  Specifically portraits with a shallow depth of field.   Groups portraits sometimes.  It's just too much work to add lens blur at post-prod.  I understand that setting a P&amp;S on macro mode gives a shallow focus and DOF.    How do I read P&amp;S camera specs to shortlist those with the shallowest depth of field?   Added: By point-and-shoot, I mean a camera that is easy to use -- Not necessarily compact, inexpensive, or current.  Also, if there are two or three factors/specs to get a shallow depth of field, then which spec contributes the most to bokeh? which is second most contributing? the third, etc.? Is there an off/disable control anywhere? I realize that the INFO button toggles it off, but it comes right back on as soon as you tap the shutter, and that it a problem in low light situations where the LCD keeps blasting me in the eye. I have a follow up question to this question:   My question is: when should you add a comma when starting a letter/email with     Hello John,      It was fun to see ...   I am pretty sure that a comma goes after the name, but lately I have seen several people (academics) write      Hello, John,       It was fun to see ...   with a comma after the Hello.  I am wondering when a comma there is appropriate.    I see this question  but I am wondering about the situation where you have a comma after the name also. Exterior power "commutes" with direct sum I know that for vector spaces $V, W$ over a field $K$, we have the following identity :  $$ \bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \simeq \Lambda^n(V \oplus W) $$ which holds in a canonical way : for $0 \le k \le n$, the maps  $$ \Lambda^k(V) \times \Lambda^{n-k}(W) \to \Lambda^n(V \oplus W), \\ (v_1 \wedge\cdots \wedge v_k,w_{k+1} \wedge \cdots \wedge w_n) \mapsto (v_1 \wedge \cdots \wedge v_k) \wedge (w_{k+1} \wedge \cdots \wedge w_n) $$ factors through the tensor product's universal properties, which gives us a map $\varphi_k : \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \to \Lambda^n(V \oplus W)$, and since the category of $K$-modules has biproducts, we get a canonical map  $$ \varphi = \bigoplus_{k=0}^n \varphi_k : \bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \longrightarrow \Lambda^n(V \oplus W) $$  What if I replace $K$ by a (commutative ring with $1$) ring $A$? Does this map have a kernel? I know it is surjective (it suffices to "look at it", i.e generators of the codomain are obviously all hit by $\varphi$). I don't know if it helps, but I am working in integral domains. In the field case, this is obvious if $V$ and $W$ are finitely generated, since one can use dimension arguments. I wouldn't mind if the above identity held only in the finitely generated case either. If it worked in general it would be cool though.   Note that when trying to compute the inverse map with the universal property, there seems to be a problem... my guess would have been $$ (v_1 + w_1,\cdots,v_n + w_n) \mapsto \bigoplus_{k=0}^n (v_1 \wedge \cdots \wedge v_k) \otimes (w_{k+1} \wedge \cdots \wedge w_n) $$ and this map is obviously $A$-multilinear, but there is a problem with the alternating property ; I get some terms left. I'm guessing this is the wrong map... if there is one. Is osmosis a kind of thermal machine? The hyperphysics article defines osmosis as a process driven by internal energy and internal energy is defined as energy associated with random,disordered motion of molecules. Osmosis develops pressure and lifts weight so it has the output of a machine.  But what drives osmosis? Is osmosis driven at the expense of thermal energy although there is no temperature gradient? Prove that $1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+.........=\sqrt{3}$ Prove that $$1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+.........=\sqrt{3}$$   $\bf{My\; Try::}$ Using Binomial expansion of  $$(1-x)^{-n} = 1+nx+\frac{n(n+1)}{2}x^2+\frac{n(n+1)(n+2)}{6}x^3+.......$$  So we get $$nx=\frac{1}{3}$$ and $$\frac{nx(nx+x)}{2}=\frac{1}{3}\cdot \frac{3}{6}$$  We get $$\frac{1}{3}\left(\frac{1}{3}+x\right)=\frac{1}{3}\Rightarrow x=\frac{2}{3}$$  So we get $$n=\frac{1}{2}$$  So our series sum is $$(1-x)^{-n} = \left(1-\frac{2}{3}\right)^{-\frac{1}{2}} = \sqrt{3}$$  Although I know that this is the simplest proof, can we solve it any other way someting like defining $a_{n}$ and then use Telescopic sum.  Thanks. Are there wizarding schools in the rest of the "magical" world? Are there other schools for witches and wizards in the rest of the world? In the books they only speak of 3 schools, but what about Asian or American people?  And even more, were they affected by the events happening in Europe? Does a correlation matrix of two variables always have the same eigenvectors? Enabling CORS in GeoServer (jetty)? Let $f(z)$ be an entire function whose real part is bounded by a polynomial in $|z|$. Does it follow that $f(z)$ is a polynomial?  Or, without loss of generality and more suggestively $$(f(z)=\sum_{k=0}^\infty a_k z^k \quad \land \quad \operatorname{Re}(f(z))\leq |z|^n)\quad \forall z\in \mathbb{C} \qquad \Rightarrow \qquad f(z)=\sum_{k=0}^n a_k z^k \quad\forall z\in \mathbb{C}$$ count vs length vs size in a collection From using a number of programming languages and libraries I have noticed various terms used for the total number of elements in a collection.  The most common seem to be length, count, and size.  eg.  array.length vector.size() collection.count   Is there any preferred term to be used? Does it depend on what type of collection it is? ie. mutable/immutable  Is there a preference for it being a property instead of a method? Should +1 and thanks comments be a declined flag? As a moderator for GD.SE I was always led to believe our role is to clean old comments as they are intended to be a temporary item depending on the level of context and if they add any benefit to an answer.  On SO I haven't had any issues or declines when I read old Q&amp;As and see +1 and thanks and flag.    However, on other sites when I read an old Q&amp;A and I see thanks or +1 I try to be courteous and flag them as I am aware moderators do get measured by their cleanup and I'm accustomed in doing so on SO but these flags are declined.    So this leads me to ask:   Should +1 and thanks comments be flagged and is the intended role of a comment still to be considered as temporary? What is the proper classification for a thanks or +1 comment if it should be flagged?   I didn't see any Q&amp;As on this topic after browsing the tags:   and searching led me to:  but that was several years old. I found several plugins (like  and ) to get the ability to attach a picture to a WP user. But these plugins are not maintained any more.  Since there's apparently no plugin that does it today, is there an easy way to do it manually? Calibre stopped working recently. Tried uninstalling and reinstalling. When I open a terminal and call $calibre I get the following error message: Traceback (most recent call last):   File &quot;/usr/bin/calibre&quot;, line 20, in &lt;module&gt;     sys.exit(calibre())   File &quot;/usr/lib/calibre/calibre/gui_launch.py&quot;, line 73, in calibre     main(args)   File &quot;/usr/lib/calibre/calibre/gui2/main.py&quot;, line 543, in main     listener = create_listener()   File &quot;/usr/lib/calibre/calibre/gui2/main.py&quot;, line 514, in create_listener     return Listener(address=gui_socket_address())   File &quot;/usr/lib/calibre/calibre/utils/ipc/server.py&quot;, line 110, in __init__     self._listener._unlink.cancel() AttributeError: 'NoneType' object has no attribute 'cancel'  Do I need to install some dependency that the sudo apt install didn't get? I tried installing via command line and the software installer/updater GUI. I'm on version Ubuntu 20.04.1 LTS. I made sure that my software in general is up to date also.