Reference request: linear operators into $L^\infty$ can be extended presrving the norm.
Do continuous linear functions between Banach spaces extend?
Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z}[X])$ is irreducible Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z})[X]$ is irreducible. (Hint: if it were reducible, it would either have a root or be of the form $g(X) \cdot h(x)$, where deg$g(X) = 2$ and deg$h(X) = 3$) Recall that $\bar{a}$ is shorthand for the coset $a + 2\mathbb{Z} \space $in$ \space (\mathbb{Z}/2\mathbb{Z})[X]$ (for any $a \in \mathbb{Z}$)  This question is worth $7$ marks out of a possible $75$, this is my solution..  $f(X) = X^5 + X^3 + \bar{1}$ is irreducible in $\mathbb{Z}/2\mathbb{Z}[X]$ iff it has no roots in $\mathbb{Z}/2\mathbb{Z}$. Since $f(\bar{0}) = \bar{1}$ and $f(\bar{1}) = \bar{1}$, it follows that $f(x)$ is irreducible in $\mathbb{Z}/2\mathbb{Z}[X]$.  Would this be a sufficient answer? am I missing something? Ubuntu 20.04 still supporting Python2 If $|f(z)|\le |g(z)|$, can we say $f$ is constant? Let $f$ and $g$ be entire functions and $g(z)\neq 0$ for all $z\in \mathbb{C}$. If $|f(z)|\le |g(z)|$, can we say $f$ is constant?  Liouville theorem says that an entire bounded function is constant, but g is not given bounded here. So I think it should be false. What else can we say about $f$? What is the fundamental group of $(\mathbb{C} \setminus {\{0\}})~/~\{e,a\}$, where $e$ is the identity homeomorphism and $az = -\bar{z}$?  Clearly this is homeomorphic to the half cylinder , which is homotopy equivalent to the half circle.  But what is the fundamental group of the half circle? Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$. Problem: Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.  My work: So I think I have to do a proof by induction and I just wanted some help editing my proof.  My attempt:  Let $P(n)=1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.  Then $$P(1)=1^2=\frac{1(1+1)(2+1)}{6}$$ $$1=\frac{6}{6}.$$ So $P(1)$ is true.  Next suppose that $P(k)=1^2+2^2+\cdots+k^2=\frac{k(k+1)(2k+1)}{6}$ for $k \in \mathbb{N}$. Then adding $(k+1)^2$ to both sides of $P(k)$ we obtain the following: $$1^2+2^2+\cdots+k^2+(k+1)^2=\frac{k(k+1)(2k+1)}{6}+(k+1)^2$$ $$=\frac{2k^3+3k^2+k+6(k^2+2k+1)}{6}$$ $$=\frac{2k^3+9k^2+13k+6}{6}$$ $$=\frac{(k^2+3k+2)(2k+3)}{6}$$ $$=\frac{(k+1)(k+2)(2k+3)}{6}$$ $$=\frac{(k+1)((k+1)+1)(2(k+1)+1)}{6}$$ $$=P(k+1).$$ Thus $P(k)$ is true for $k \in \mathbb{N}$. Hence by mathematical induction, $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ is true for $n \in \mathbb{N}$. Sylow Theorem Explanation The Sylow theorems say that all $p$-Sylow subgroups are conjugate.   Is that the only thing they are conjugate to? In other words could a $p$-Sylow subgroup also be conjugate to something that is not a $p$-Sylow group?   Or does the fact that every element with order dividing $p$ is in a $p$-Sylow group mean that that is never possible? Is there a proof from the first principle that the Lagrangian L = T - V? Database general error: 2006 MySQL server has gone away Same gravatar is displayed for different users in the stackexchange.com , under the Top Network Askers in the right side bar.   The red circled image's source (&lt;img src=) are from   https://www.gravatar.com/avatar/?s=18&amp;d=identicon&amp;r=PG&amp;s=30                                ^   Seems the gravatar id is missing here in the image source.  For an example from the list, the user  gravatar id should be    https://www.gravatar.com/avatar/66c66c4d97a7bc3859a85253f003e1fb?s=18&amp;d=identicon&amp;r=PG&amp;s=30   but it is mapped with   https://www.gravatar.com/avatar/?s=18&amp;d=identicon&amp;r=PG&amp;s=30   which caused the display issue. How to use ampscript in order to pull user data from a second data extension? We have several data extensions that have data relationships with each other.  How can I send an email using one data extension while pulling data from a second data extension to personalize my email with ampscript?  I've tried Lookup in many different ways, but I've only managed to pull one specific row (the same row which is not personalized to each user) or all rows at once. I haven't been able to pull personalized data for each user.  In practice, I want to pull data from a DE called "Survey". Each user has a specific ID and some users have an "AnswerID", which varies. I want to to create IF/THEN statements based on that AnswerID.  Thanks, Sylvie Finding $\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2} dx$ Last two digits of $2^{1000}$ via Chinese Remainder Theorem? I bumped across the aforementioned question in my notes while studying today and I have completely forgot how to do this. I remember using CRT to solve a problem like this on one of my tests, too bad they didn't give back my solutions :(.  Since $\gcd(100,2) = 2$, we can't use the usual Euler's theorem to solve via $\pmod {100}$. So $100 = 5^2 2^2$, and applying Euler's on the 25 gives me $2^{20} \equiv 1 \pmod {25}$. However since $\gcd(2,4) = 2$, we can't do the same for $\bmod 4$. Accordingly how do I set up the other modulo congruence so I can apply CRT? When my phone MOTOROLA G3(2015) is updated to Marshmallow, its not showing files when connected to PC. While connecting, it shows options as charing only, PTP, MTP, MIDI. When I select MTP, it shows the external card, but not showing the contents, where as internal memory is detected and all files are shown. Make account merges propagate to chat When two accounts are merged, all the posts/votes are moved over to the "master" account, and the other account ("slave"?) is deleted. The merge is propagated network-wide, so a similar thing happens on each site.  But, this does not happen to the chat user. The "slave" chat user is kept, with a 404 parent profile link. I feel that the chat user should be merged as well, re attributing all the chat messages.  Thoughts? show that a distance function is continuous I am animating a Rubik's cube, and as soon as I get to the second rotation the blocks that have previously been rotated start tripping out even though it should have been reset. I tried making the graph linear which fixed a lagging issue I had, but the blocks still start to spin instead of rotating around a single axis. My Area51 account crossed 200 rep today, but still other accounts don't reflect the change in reputation. When will the changes be applied? or is there no association bonus for Area51 account? Why does extended piston become stuck if air above it is powered? To reproduce this problem make layout like this and then power the piston. Once the piston is extended, removing the power source will not retract it:    Since piston head block is transparent, I see no explanation for this. Also, even if it wasn't transparent, it still wouldn't power a piston. For example this doesn't power the piston:    But if I place redstone and THEN place the piston, it's powered:    It also remains powered no matter what, as long as there isn't a block update event:    I suppose this behavior can be used to make a block update detector, but in my case this sucks. I wanted such setup for automatic smooth stone generator (without any clocks):    But due to this bug, it won't work - the piston will be always extended. Bash history search, partial + up-arrow I have searched, but not found anything on this. I am looking for a functionality in bash, using a terminal.  Way back when, I had a user on a debian system, and a friend set me up with a convenient history search feature (I believe I used tcsh then), where I would type the beginning of a previous command, hit up-arrow, and it would do a search, based on the partial string.  E.g. if my history is:  ./script.pl ./script.pl arg1 cat output cat output | grep yada   And I type ., and press up-arrow, it would show me: ./script.pl arg1. Press it again and it would show ./script.pl, etc.  Very much like it would perform a grep on .bash_history. Is there a way to get this functionality? I've run across the following snippet in themes from time to time:  if ( ! defined('ABSPATH')) exit('restricted access');   It's at the beginning of some (all?) PHP files in a theme and it's supposed to prevent direct access of the file by nefarious sources.  I see that this isn't included in Twenty Ten or Eleven and I've never seen it recommended in official WordPress documentation. It seems like a good idea to me, but I also don't know enough about security to judge it and can't find much with Google.  Is this something I should have in my custom themes? If so, should it be in all PHP files or just some?