Problem: Lets $X$ and $Y$ normed vector spaces $T:X\rightarrow Y$ a aplication such that $\Vert Tx-Ty \Vert = \Vert x-y \Vert$ for all $x,y \in X$ and $T(0)=0$ then T is a linear aplication.  My attempt: If I evaluate in $0$: $$\Vert Tx \Vert = \Vert x\Vert \quad \mbox{for all $x \in X$}$$ Then, $$\Vert T(x+y) \Vert = \Vert x+y\Vert \leq \Vert x\Vert + \Vert y\Vert = \Vert Tx\Vert+\Vert Ty\Vert$$ But, I do not know how to continue.
Suppose $f$ is an isometric (i.e., distance preserving) function on $\mathbb{E}^2$ such that $f(0,0) = (0,0)$.  Then I want to show that $f$ is necessarily linear.  Now $f$ is linear iff $f$ is both additive and homogenous.  The following is an attempted proof for the homogeneity of f (missing the last step); still more, I have no idea how to argue for the additivity of $f$.  Any ideas?  Let $x \in \mathbb{E}^2$ and $\alpha \in \mathbb{R}$   We know that $\forall x \in \mathbb{E}^2$, $\Vert x - 0 \Vert = \Vert f(x) - f(0)\Vert = \Vert f(x) - 0 \Vert$ so that $\Vert x\Vert = \Vert f(x)\Vert$.  From this we immediately have the following facts:  $\Vert x \Vert = \Vert f(x) \Vert$  $\Vert \alpha x \Vert = \Vert f(\alpha x) \Vert$  We can then argue that since $\Vert \alpha x\Vert = |\alpha| \Vert x \Vert = |\alpha| \Vert f(x) \Vert = \Vert \alpha f(x) \Vert$, we also have that $\Vert f(\alpha x) \Vert = \Vert \alpha f(x)\Vert$.  Finally, we have that  $\Vert \alpha x - x \Vert = \Vert \alpha f(x) - f(x) \Vert$   iff $ \Vert (\alpha - 1)x \Vert = \Vert (\alpha - 1) f(x) \Vert$   iff $|\alpha - 1| \Vert x \Vert = |\alpha - 1| \Vert f(x) \Vert$  Since the last of these statements is in fact true, we now have $\Vert \alpha x - x \Vert = \Vert \alpha f(x) - f(x) \Vert$ as desired.  Now at this point it seems like I have all of the facts required to assert that $f(\alpha x) = \alpha f(x)$, but I can't figure out how to formally state why without illegally appealing to visual intuition.  Any ideas?
How can I increase my disk space when Ubuntu is installed inside Windows? Before upgrading to macOS Sierra, I really liked the Character Accent Menu by doing a long press on a key (for example, the character e) and being able to add an accent on it.   However, this functionality seems to not work anymore, and so far, I can't find out why?   Also, before, doing a long press on the delete button, would repeatly delete characters. Now, it is doing only one character deletion disregarding how long I'm pressing the key. So if I want to erase 10 characters, I need to press the delete button 10 times...  Also, I am not able to log off or shut down. When doing it, all applications are closed, status bar disappears but nothing happens. The dock remains here. I have no way to shut down or log out without doing a long press on the power button.  Is someone facing the same issue?  EDIT : Logging with a new user account or guest account fix all the problems. But I still would like to know what, in my user account, became corrupted during the upgrade. A very quick question:   Is it correct to say "In case I won't be able to watch that"?  If it's acceptable to say that, what type of sentence is it?  UPDATE:  As a clarification, I want to say "Can you give me that in case I won't be able to have it later". Probably, it's more a supposition than a conditional. UV coordinates on a deformed plane Is transferring to another university an option for an unhappy PhD student? I've heard of people transferring between graduate schools because of their advisor moving to another school and taking their students with them.   Do students ever transfer for other reasons? If a student isn't happy at the school they choose, could transferring be an option? Presumably they would frame it as something positive such as "turns out I really want to work with Professor X" rather than "I hate my school", but does this ever happen? Is there a certain timeframe in which this is possible? (Assume it's a 5 year PhD program.)  If the answer varies by subject please mention the subject in your answer.  (Context: many friends have told me that if it turns out I'm not happy with my choice, I can always transfer. My impression is that this is actually difficult since many schools don't accept transfer students and in any case not being able to stick with a program looks bad -- but I don't have evidence either way so wanted to ask people who are in academia. Of course the ideal would be to pick a school one is happy with in the first place!) Prove that $\mathbb{Z}[\omega]$ is an integral domain Luggage, connecting flights and customs question for USA I'm flying to the USA, from Australia, through LAX and catching a connecting flight 6 hours later, think it's the same ticket, booked all at once. I've never traveled before, and I'm wondering what happens with customs.   Do I collect my bags at LAX so they can be checked? Do I have to check in at every airport or is it all done when I catch the first flight? There are 4 flights on my itinerary. Regex to match only letters The Unix environment has some excellent tools for parsing text in various forms. However, of late, the data is not in the traditional (historical) formats (CSV, TSV, record-based or some other delimiter-based) it used to be before. Data these days is exchanged in structured formats like XML/JSON.  I know there are some good tools like sed, awk and Perl which can chew down nearly any form of data out there. However, to work with this sort of structured data, often one has to write a complete program, and, given the little time available to extract information, one has to sit down and figure out the whole logic of what one wants to query and put it down programmatically. Sometimes this is not OK – basically because the information extracted from those files acts as inputs for further work; also because of the time it takes to search for the appropriate solution and code it up. A command line tool is needed with sufficient switches to find, query and dump data.  I'm looking for tools that take a XML/JSON or other forms of structured data and dump it into other formats like csv, etc., so that from there one could use other commands to get any information out of it.  Are there any command line utilities you know of which do this kind of a job? Are there already awk/Perl scripts available to this? Centering \vdots in a system of many equations I have a system of equations with an arbitrary number of equations (k). I'd like to use \vdots to compactly describe the system, like so:  \begin{align*}   R(-1) &amp;= \sum_{i=1}^m A(i)R(i-1) \\   R(-2) &amp;= \sum_{i=1}^m A(i)R(i-2) \\         &amp;\vdots                    \\   R(-k) &amp;= \sum_{i=1}^m A(i)R(i-k) \end{align*}     I'd like the dots, however, to be centered with = or the entire equation. Is there an elegant way of centering a column or row with AMSMath? I'm currently using an unholy combination of whitespace operators (\; \, etc.) to get the job done. Let's say when I do ls -li inside a directory, I get this: 12353538 -rw-r--r-- 6 me me 1650 2013-01-10 16:33 fun.txt  As the output shows, the file fun.txt has 6 hard links; and the inode number is 12353538. How do I find all the hard links for the file i.e. files with the same inode number? Where is a good place to learn classical optics for high school competitions? How to fix infinite bash loop (bashrc + bash_profile) when ssh-ing into an ec2 server? I mistakenly have source .bash_profile in the bashrc file and vice versa. Now when I tried to ssh into the machine (ec2), it will stuck at loading bash and get connection closed in a second. Is there a way I could fix it? Could I mount the disk to another ec2 instance to fix the bash files?  Update 1: I tried the following: %  ssh -i &quot;my-pem.pem&quot; -t ubuntu@&lt;server_address&gt;.amazonaws.com &quot;/bin/bash --noprofile --norc&quot; Connection to &lt;server_address&gt;.amazonaws.com closed.  Nothing else showed up. Do you have any idea on what was going wrong? For sanity check, if I do ssh -i &quot;my-pem.pem&quot;ubuntu@&lt;server_address&gt;.amazonaws.com, the message will be ...  28 packages can be updated. 0 of these updates are security updates. To see these additional updates run: apt list --upgradable  New release '20.04.2 LTS' available. Run 'do-release-upgrade' to upgrade to it.   Last login: Mon Feb 22 23:17:41 2021 from ip Connection to &lt;server_address&gt;.amazonaws.com closed.  Solution Just ssh onto the machine and immediately do Ctrlc and fix the bash files. Full-wave bridge rectifier with capacitor filter and ripple voltage According to various sources, the ripple voltage of a full-wave bridge rectifier is   $$ V_r = \frac{V_s}{2}fCR $$  where \$V_s\$ is the peak of the input voltage source, \$f\$ is the frequency, \$C\$ is the capacitor and \$R\$ the resistor.  I have simulated this circuit on PSPICE, circuitlab, and built it in real life:   \$R\$ is \$4.7K\$ and \$C\$ is \$10 uF\$. Voltage source is sine wave with \$0 V\$ offset, \$5V\$ amplitude and \$40 Hz\$ frequency.  All 3 experiments give me a ripple voltage of approximately \$0.65V\$, but the formula gives me \$1.330 V\$ (twice as much).  What am I missing? gpgkeys: key C927EBE00F1B0520 not found on keyserver Possible Duplicate:    I am using 10.04 LTS - Lucid Lynx. I shutdown the computer while a software update was running. Then after I restarted my computer and while the update manager was running, I got this error:  Could not download all repository indexes The repository may no longer be available or could not be contacted because of network problems. If available an older version of the failed index will be used. Otherwise the repository will be ignored. Check your network connection and ensure the repository address in the preferences is correct. GPG error: http://toolbelt.herokuapp.com ./ Release: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY C927EBE00F1B0520 Failed to fetch  http://ppa.launchpad.net/rabbitvcs/ppa/ubuntu/dists/**DISTRIBUTION**/main/binary-i386/Packages.gz 404  Not Found   I went through the answers to  and I tried this $ sudo launchpad-getkeys  Please wait... launchpad-getkeys is running an update so it can detect the missing GPG keys Trying to import all the missing keys gpg: requesting key 0F1B0520 from hkp server keyserver.ubuntu.com gpgkeys: key C927EBE00F1B0520 not found on keyserver gpg: no valid OpenPGP data found. gpg: Total number processed: 0 launchpad-getkeys has finished importing all missing GPG keys.  Try running sudo apt-get update - you shouldn't see any key  errors anymore  But when I run sudo apt-get update again, it again shows that the key was not found on the keyserver. How to solve this problem? Given: $\operatorname{gcf}(x,36)=9$ and $\operatorname{lcm}(x,36)=108$. Find $x$.   p.s I tried $54$ but although the $\operatorname{lcm}$ of $54$ and $36$ is $108$ the GCF is $18$ and it has to be $9$. Help me please. Verify that $4(29!)+5!$ is divisible by $31$. Berry naming scheme list of LaTeX font families If $e^{itx_n}$ converges for every $t\in\mathbb R$, then does $x_n$ converge? creating a diagonal matrix with a variable amount of parameters