Did Bilbo see Gandalf in between The Hobbit and The Lord of the Rings?
How often did Gandalf return to Hobbiton after the events of The Hobbit?
Why has it become an industry standard for logos to redirect a user to a website's homepage?   Where was this first seen? Considering the irreducible (by condition) polynomial $f=X^4-2aX^2+b\in\mathbb Q[X]$ and the field extension $K=\mathbb Q(\sqrt{a^2-b})$ of $\mathbb Q$, I want to show the following result:    For $[L:\mathbb Q]=4$ we have $\sqrt b\in K$ where $L$ is the splitting field of $f$.   My thoughts so far: The four roots of $f$ are $\pm\sqrt{a\pm\sqrt{a^2-b}}$. If $L$ is of degree $4$ it must be primitive since $f$ is irreducible, so $L=\mathbb Q\left(\sqrt{a+\sqrt{a^2-b}}\right)$. Then, by multiplying the two roots $\sqrt{a+\sqrt{a^{2}-b}}\cdot\sqrt{a-\sqrt{a^{2}-b}}=\sqrt{b}$ we have that $\sqrt{b}\in L$. Now I want to look at $$\mathbb Q\subset K\subset K(\sqrt b)\subset L$$ and differentiate some cases. The case where $[L:K]=1$ is trivial and the case $[L:K]=4$ can't happen because if we square any root and subtract $a$, we get $\pm\sqrt{a^2-b}\in K$ which means $[L:K]\leq 2$. So let's look at the case where $[L:K]=2$. As I see it, we get two subcases:   For $[L:K(\sqrt b)]=2$ we have $K(\sqrt b)=K$ which finishes the proof. For $[L:K(\sqrt b)]=1$ we have $K(\sqrt b)=L$. This means that $L$ is also the splitting field of $(x^2-b)(x^2-(a^2-b))$. Is this a contradiction? How do I finish this case? In Age of Ultron, Thor seems to know something he possibly shouldn't? Sorry for the cryptic title, but I realise the film has only just been released in the UK and Europe and is yet to be released in the US.  I'll hide all substantial parts of my question, just in case. I don't want to spoil anything.     At the end of Age of Ultron, Thor states that 4 infinity stones have now shown up in the last few years.   So,     By my count we have the Tesseract, from several of the first few films. The Aether, from Thor 2. The Chitauri Scepter, again from several films. And finally, we have The Orb, from Guardians.    Now, to my knowledge     Thor, and indeed none of the Avengers, should know about The Orb. I don't remember Thor visiting the Collector between GoTG and the AoU. So, how did Thor know that The Orb had appeared? Or is he referencing a different infinity stone?   I'm looking for an in universe answer, ideally. Double and triple substitution in bash and zsh Follow-up to the background part in .  In bash I can use ${!FOO} for double substitution, in zsh ${(P)FOO}. In both, the old-school (hack-y) eval \$$FOO works.  So, the smartest and most logical thing for me would be ${${FOO}}, ${${${FOO}}}… for double/triple/n substitution. Why doesn’t this work as expected?  Second: What does the \ do in the eval statement? I reckon it’s an escape, making something like eval \$$$FOO impossible. How to do a triple/n substitution with that that works in every shell? I'm hoping there's something in the same conceptual space as the old VB6 IsNumeric() function? Website with a collection of sci fi stories set in the same universe; powerful AIs being regarded as Gods, one named Gaia, and a "technopocalypse" There is a site with lots of stories all set in a future with super powerful AIs regarded as Gods, the most powerful being matrioshka brains. They have a specific terminology but I can't remember their names. Earth has been converted into a similiar godlike computer called "Gaia" and has expelled all humans creating a dystopian exodus across the solar system. And something about rogue Von Neumann machines causing a huge crisis dubbed the "technopocalypse". SEO (or other) problems with a 404? Probability of complete coupon set in $K$ boxes where a box has $N$ distinct coupons? I am trying to find the largest file in a directory recursively. If there is a subdirectory inside of that directory the function needs to go inside that directory and check to see if the largest file is there. Once the largest file is found the output is displayed with the relative path name and the name and size of the largest file.  EX:  dude@shell2 (~...assignment/solutions) % bash maxfile.sh ~/test class/asn dude.h.gch: 9481628   This is what I have:  #!/bin/sh clear  recursiveS() {     for d in *; do         if [ -d $d ]; then             (cd $d; echo $(pwd)/$line; du -a; recursiveS;)         fi     done } recursiveS   I have been stuck for a while now. I cannot implement this by pipelining a number of existing Unix tools. Any ideas would be nice! A question about E12 series resistors I'm trying to calculate E12 resistor series myself.  Above is an illustration of the idea where such equation can be written:  x = r3 + r3 * %tol = r4 - r4 * %tol  form that we can derive:  r4 = r3* [(1 + %tol)/(1 - %tol)]  hence in generalized form we can write (%tol = 0.1 for E12 series);   r(i+1) = ri* [(1 + %tol)/(1 - %tol)]  But when I iterate the above formula in a for loop I obtain the following results for the first resistor values:  1.0000    1.2222    1.4938    1.8258    2.2315    2.7274    3.3335    4.0743    4.9797    6.0863    7.4388  Then I take [(1 + %tol)/(1 - %tol)] = 1.2 rounded value instead of 1.222222 and write the formula as:  r(i+1) = r(i)*1.2  But again when I iterate the above formula in a for loop I obtain the following results for the first resistor values:  1.0000    1.2000    1.4400    1.7280    2.0736    2.4883    2.9860    3.5832    4.2998    5.1598    6.1917  But the real values for E12 should be as:  1.0, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8  What am I doing wrong here? What is the percentage increase to any value when initial value is 0? This is a very simple maths problem, which doesn’t seems simple to me when I just put some extra thoughts on it. So need to borrow some extra brains for this.   Question: If cost of coffee in my office was Dollar 0 till yesterday and today it is Dollar 2.5 then what is the percentage increase in the cost of coffee?  Initially it seemed like 2.5% increase, simply by distributing number from 0 to 100, but this is not the case.   Some suggested it is infinitely more than the previous value, because we are calculating percentage by dividing by zero. But what about this if the cost of coffee was $0.25? Isn’t still infinitely more! Something is wrong here.   PS: Yes, coffee were free in our office but not anymore, and we are still recovering from this shocker. 😥 listing files with pattern with starting and ending I want to list all files in a directory that start with 'abc' and end with '.zip'  I'm trying to use ls.  The directory has a lot of zip files starting with abc&lt;date&gt; and xvz&lt;date&gt;. I only want to get the list of the abc&lt;date&gt;.zip Make an object invisible to a specific camera? (Cycles) I need help to find the Galois Group of $x^5 +1$. I know that it has a 5-cycle and a 4 cycle and is not a subgroup of $A_5$. Thanks! How can a question banned user fix their questions if they're deleted? If a user is question banned  and the , how can the user fix this on their own?  Until recently, advice has been to improve their questions, but since they can't see the questions on their question screen, they can't improve them:     Feature request: Allow a question banned user to see their own deleted questions, or warn them that deleting questions could lead to a question ban. What are the “spam” and “rude or abusive” (offensive) flags, and how do they work? When flagging, there are two special case options: spam and rude or abusive.   What is spam, and when should I flag content as such? What is considered offensive, rude or abusive content? How does the spam flag differ from the rude or abusive flag? What is the effect of these special flags? When are these flags removed? Transit Visa in Hong Kong for 15 hour layover between India and USA? iPhone 7 Plus takes days to turn on showing Apple logo or black screen I’ve had this issue for months but in the last couple of days it’s got very very bad.  When my phone runs out of battery it needs like 2–3 days to turn back on. At the beginning of this problem the phone would need about 1h. Every time when my phone goes off again the time to switching on goes up.  What does this all look like? Well, when my phone died I put it on the charger, then the logo appears and goes off. When it goes off, a black screen pops out with a little white line on in it, then it turns on back to Apple logo. It keeps doing this until it turns on which is 1 or 2 even now 3 days.  I tried to restore it and also update it but nothing happens, literally nothing happened. Why is compactness so important? I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel theorem, and a proof continuous functions on R from closed subintervals of R are bounded. It seems like such a strange thing to define; why would the fact every open cover admits a finite refinement be so useful? Especially as stating "for every" open cover makes compactness a concept that must be very difficult thing to prove in general - what makes it worth the effort?  If it helps answering, I am about to enter my third year of my undergraduate degree, and came to wonder this upon preliminary reading of introductory topology, where I first found the definition of compactness. Where to get the full offline-capable installer for macOS Mojave (.1 update)?