What is the antonym of the word destination. Here is some context:  Every train has a destination and a ____?  I was thinking it might be origin but I've never encountered that word used in this context so I'm not sure - there might be a better word.
This question arises when I try to describe my itinerary to ticket service in the UK. Take for example, I plan to buy a ticket from A to B. B is the terminal. How to say its counter part A? I cannot find a proper term. I have to say "I will depart from A" or "I plan to go to B from A". Or I even use "departure place" to refer to B.
The ring $ℤ/nℤ$ is a field if and only if $n$ is prime Let $n \in ℕ$. Show that the ring $ℤ/nℤ$ is a field if and only if $n$   is prime.   Let $n$ prime. I need to show that if $\bar{a} \neq 0$ then $∃\bar b: \bar{a} \cdot \bar{b} = \bar{1}$. Any hints for this ?  Suppose $ℤ/nℤ$ is a field. Therefore: for every $\bar{a} \neq 0$ $∃\bar b: \bar{a}\cdot \bar{b}=1$. How can I show that $n$ must be prime ? I am trying to make a callout to AWS S3 using named credentials and Apex  The named credential is set up as follows  Label: AWS S3 Name: AWS_S3 URL: https://bucket-name.s3.amazonaws.com - Certificate: Identity Type: Named Principal Authentication Protocol: AWS Signature Version 4 AWS Access Key: AKIAxxxxxx AWS Secret Access Key: xxxxxx AWS Region: ap-southeast-2 AWS Service: s3 - Generate Authorization header: true Merge fields in header: false Merge fields in body: false   The Apex class is very simple  HttpRequest req = new HttpRequest(); req.setEndpoint('callout:AWS_S3'); req.setMethod('GET');                        Http http = new Http(); HTTPResponse res = http.send(req);                 System.debug(res.getBody()); return res.getBody();   The response body contains the following:     SignatureDoesNotMatchThe request signature we calculated does not match the signature you provided. Check your key and signing method.AKIAxxxxx   I have used the same bucket, endpoint and credentials using Postman and am able to successfully return the list of objects in the bucket so it is not an IAM permissions issue  There have been a couple of similar questions on stackexchange but unfortunately they do not provide any resolution or assistance in this case:    I've tried looking for code samples or further pointers in the Apex documentation but it doesn't really give any further guidance on how to proceed or correctly set this up so I'm a bit stuck 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 $f$? After MiKTeX update: suddenly all fonts (incl. existing path-referenced ttf files) are not found by fontspec / LuaLaTeX anymore Evaluating the sum $\Sigma \, \frac{1}{k(k+p)}$ What is a tty, and how do I access a tty? What are the best practices for battery longevity? How do you maximize the cycle count of your battery?  There seems to be a debate between:   Leave plugged in 24/7 but do a full cycle once a month Charge to ~100%, drain to ~10%, repeat.   NOTE: Taking your   Which method is better and why? How much of a difference does it really make?  Here are some of the better sources I've found about battery behavior so far. Even still I feel they don't conclusively answer the question above. See my elaborations below:           Elaborating on Option 1  Now IF Apple engineers were smart and optimized for AC draw when plugged in to not use up your charge cycles, then it seem logical to assume that you are using effectively none of your 1000 charge cycles while plugged in. If that's the case, it would seem that the LiPo drop from holding at 100% would be vastly outweighed by the fact you're not using any charge cycles.  However, this is a purely speculative assumption and I could find no evidence to either confirm or deny.  If this is NOT true, and the AC adapter only goes to charging the battery, then there would be no difference. Your battery would drain to 99%, then back up to 100%, then back down to 99%, etc. Those micro-charge cycles would add up at the same rate as 100% -> 0% -> 100%, and you would get no gain. In this case the negative effect of holding LiPo batteries at 100% would outweigh everything else.  Elaborating on Option 2  There are lots of good reasons why Option 2 is the way it is:   Because of known LiPo chemistry issues, holding a 100% charge for a long time causes the battery to degrade Apple specifically recommends not to leave it plugged in all the time  Draining a battery all the way to zero all the time is bad for the cell (Which is why ~10% is used) Heat is a killer, and when plugged in AND charging, you get extra heat that causes damage.   There are a couple reasons why I'm challenging Option 2:   It seems silly that the most intuitive use case for thousands of people that use their macs as primary computers is wrong. If the hypothesis in option 1 is true, you're just burning unnecessarily through your finite charge cycles. It's a pain to have to remember to keep plugging in and plugging out and be worried about whether or not your device is plugged in. Does the lens I just bought have a fungus? After upgrading to 12.04 from 10.04, can't login to PHPMYADMIN Proof that a sequence is convergent I'm asked to prove the convergence of the sequence $$X_n=\left(1+\frac12\right)\left(1+\frac14\right)\left(1+\frac18\right)\cdots\left(1+\frac{1}{2^n}\right)$$   I proved that it is increasing through the ratio test and then I want to prove that it is bounded.  My question is the following : is it enough to say that $X_n&lt;\left(\frac32\right)^n$ for $n&gt;1$ or not ? Namely, can we say that a sequence is bounded if $\forall$  $n$ ,  $\exists$ a $N$ such that $X_n&lt;N$ or do I have to find a N such that $\forall$  $n$ , $X_n&lt;N$?  In other words, can a bound depend on $n$ ?  Thanks for your help. Acne underneath skin Introduction  Okay, I've almost won my war. The battle, its almost won... However, I must still claim victory in my final battle. The battle against the little enemies, the ones that have fought with my blade of benzoyl peroxide every single day.  Problem  Through spot treatments I have been able to fell every single one of these spots, but only the ones on my face. I've been using a spot treatment with benzoyl peroxide in, which has worked really well for me. However, my issue comes here:     There are spots underneath my skin, I think. The areas where they come through are a bit red.   So, because there is still slightly red skin on my face, and I think this is causing my acne to constantly return. I need to stop this, and end my war once and for all.  What I've tried   Washing my face with a rag and sponge every day, three times a day Changing my pillow case every day Using a benzoyl peroxide spot treatment Minimizing how much I touch my face Not interfering with the spots with my fingers, not picking at them, not playing with them. Just letting spot creams eradicate them.   Conclusion  Seriously, this driving me up the wall. I spot cream in the evening, by the time tomorrow has ended and its time for a shower, they've all come back. Its like they're coming in waves, damn waves! I must put an end to this.  So, how can I prevent acne from coming back within the space of a single school day? How can I get rid of these once and for all? What types of measures could I try? I have no reason to visit my doctor, its only acne, but I have no idea what I can try now.  Is there even a point in me trying? I acknowledge that I am a teenager, but no others that I know seem to have acne/spots like I do. Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$ Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$.    Challenge: Prove  $$\sum_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}.$$   I have two proofs, one of which is partially combinatorial.    I'm posing this problem partly because I think some folks on this site would be interested in working on it and partly because I would like to see a purely combinatorial proof.  (But please post any proofs; I would be interested in noncombinatorial ones, too.  I've learned a lot on this site by reading alternative proofs of results I already know.)  I'll wait a few days to give others a chance to respond before posting my proofs.  EDIT: The two proofs in full are now given among the answers. Show that $\forall (x,y)$ in the first quadrant: $\frac {x^2+y^2}{4}\leq e^{x+y-2}$ In Drush 8, my site aliases are loaded correctly from ~/.drush/mysite.aliases.drushrc.php.  E.g.  $aliases['dev'] = array(   'root' =&gt; '/var/www/html/mysite.dev/docroot',   'uri' =&gt; 'mysite.prod.acquia-sites.com',   'remote-host' =&gt; 'mysite.ssh.acquia-sites.com',   'remote-user' =&gt; 'mysite.dev',   'path-aliases' =&gt; array(     '%drush-script' =&gt; 'drush' . $drush_major_version,   ), );   However these aliases aren't loaded in Drush 9.  $ drush site-alias {  }   How I should define site aliases in Drush 9? Is there any migration process? Expanding GCDs Reasoning: $\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$ I'm just wondering the reason behind being able to do this:  $$\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$$  Is there an elementary proof to the reasoning behind this? System.Exception: Too many SOQL queries: 101 Allow moderators to refresh a user's network profile Currently, moderators have the ability to reset a user's username and/or profile text, and propagate that to the rest of the network, resetting it everywhere. However, often, the user's stackexchange.com network profile will also need to be reset, and this is currently only possible to do by either the user in question or staff.  Instead of bothering staff every time, moderators should have something available similar to chat, where moderators can change a user's parent site and have access to a "refresh profile" button, which will fetch updated information such as username and avatar from the parent site.  Currently, the only thing that moderators can do on stackexchange.com is to see hidden accounts. It would make sense for moderators to be able to refresh profiles there, like chat, and avoid pinging CMs every time this comes up. Want \section* to change the page header In a book, I have several numbered sections followed by an unnumbered section. I have placed an \addcontentsline following the \section* that introduces the unnumbered section, so that it will appear (unnumbered) in the ToC; that works fine. However, the page headers for the unnumbered section are continued from the last numbered section.  I am using fancyhdr with the following setup:  \pagestyle{fancy} \fancyheadoffset{0in} \renewcommand{\chaptermark}[1]{\markboth{#1}{}} \renewcommand{\sectionmark}[1]{\markright{\thesection\ #1}}  \fancyhf{} \fancyhead[LE,RO]{\thepage} \fancyhead[C]{\bfseries \draftinfo} \fancyhead[LO]{\sf\rightmark} \fancyhead[RE]{\sf\leftmark} \fancyfoot[C]{\bfseries \draftinfo} \fancypagestyle{plain}{%   \fancyhead{} % get rid of headers   \renewcommand{\headrulewidth}{0pt} % and the line } Doppler redshift of distant galaxies gave first hint that the universe is expanding. I am curious to know how this redshift is actually measured and interpreted from observation. Suppose I observe some  visible line say red with wavelength \lambda_1.  This is not enough to tell that this line is redshifted, i.e., the  original line emitted from that galaxy had a smaller wavelength. How do I know the value of that particular line when it was emitted from that galaxy? It appears that mere observation of a line is not enough. So how do we proceed? How can I avoid name mangling in C++?