Let z1,z2 be complex numbers with $|z_1|=|z_2|=1$. Prove that $|z_1+1|+|z_2+1|+|z_1z_2+1| \ge 2$.  My attempt: I started with $z_1=e^{i\theta_1}$ and $z_2=e^{i\theta_2}$ and then plugged them into the equations given but as it came to $\theta_1+\theta_2$ in the last modulus I got pretty confused regarding how to proceed.   Please help me out.
If $|z_1|=1,|z_2|=1$, how can one prove $|1+z_1|+|1+z_2|+|1+z_1z_2|\ge2$
Replacing ArcPy with ArcGIS API for Python? Today I found out about the . I was wondering if you could use this to (eventually) replace ArcPy?  Recently I have been doing analyses that require cloud based solutions. When using AWS, Docker and Jupyter it is very convenient to stick with a Unix based OS such as Ubuntu. However, the Unix alternatives for ArcGIS have their own limitations as well such as the always hard to install python bindings for the  GDAL library. ArcGIS for Python operates cross platform.  Perhaps there are the things I should and should not expect from the ArcGIS API for Python.  Perhaps Arcpy is going to disappear altogether. How can I do the following in Git?  My current branch is branch1 and I have made some local changes. However, I now realize that I actually meant to be applying these changes to branch2. Is there a way to apply/merge these changes so that they become local changes on branch2 without committing them on branch1? Trying to find a story about an unborn telepathic baby who can save a spaceship from doom I know that I can use something like cat test.txt | pr -w 80 to wrap lines to 80 characters  wide, but that puts a lot of space on the top and bottom of the printed lines and it does not work right on some systems  What's the best way to force a text file with long lines to be wrapped at a certain width?   Bonus points if you can keep it from breaking words. How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$? Does the universe have an edge/boundary/barrier? If yes, what is at the edge? My question related kind of to asking what the shape of the universe is. Say a hypothetical alien civilization built an faster-than-light spaceship. If they keep flying would they end up where they began? ie space itself is curved...or would they hit some invisible barrier? How to replace an old version of Ubuntu with a newer one from USB/DVD? I dual booted ubuntu 11.04/11.10 a while ago with a USB stick and i want to uninstall it.   I would like to completely uninstall the old version and get 12.04.  I have a 64 bit laptop. Please help! I have an issue with my Minecraft launcher - I downloaded modpack for minecraft then the launcher started an update for the modpack version which it's 1.6.2, however then I started to get the issue      [15:40:59 ERROR]: Gave up trying to download      for job 'Version &amp; Libraries'      [15:40:59 ERROR]: Job 'Version &amp; Libraries' finished with 26   failure(s)! (took 0:01:06.362)   At the same time I had just downloaded Avira antivirus and ran it, but unfortunately  removing it and trying again didn't help. I've tried to restore the system to the point before downloading the antvirus and the mod pack a 2 days ago, I've tried turning off the firewall and I have tried removing the whole .minecraft folder all with no luck at all. I can't figure it out, all the servers are working and I have good internet connection but some of the files download fine and other don't.  The full log is here:  [15:40:59 WARN]: Couldn't download https ://libraries.minecraft.net/net/sf/jopt-simple/jopt-simple/4.5/jopt-simple-4.5.jar for job 'Version &amp; Libraries' java.net.SocketTimeoutException: connect timed out         at java.net.TwoStacksPlainSocketImpl.socketConnect(Native Method) ~[?:1.7.0_45]         at java.net.AbstractPlainSocketImpl.doConnect(Unknown Source) ~[?:1.7.0_45]         at java.net.AbstractPlainSocketImpl.connectToAddress(Unknown Source) ~[?:1.7.0_45]         at java.net.AbstractPlainSocketImpl.connect(Unknown Source) ~[?:1.7.0_45]         at java.net.PlainSocketImpl.connect(Unknown Source) ~[?:1.7.0_45]         at java.net.SocksSocketImpl.connect(Unknown Source) ~[?:1.7.0_45]         at java.net.Socket.connect(Unknown Source) ~[?:1.7.0_45]         at sun.security.ssl.SSLSocketImpl.connect(Unknown Source) ~[?:1.7.0_45]         at sun.net.NetworkClient.doConnect(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.http.HttpClient.openServer(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.http.HttpClient.openServer(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.HttpsClient.&lt;init&gt;(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.HttpsClient.New(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.AbstractDelegateHttpsURLConnection.getNewHttpClient(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.http.HttpURLConnection.plainConnect(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.AbstractDelegateHttpsURLConnection.connect(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.http.HttpURLConnection.getInputStream(Unknown Source) ~[?:1.7.0_45]         at java.net.HttpURLConnection.getResponseCode(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.HttpsURLConnectionImpl.getResponseCode(Unknown Source) ~[?:1.7.0_45]         at net.minecraft.launcher.updater.download.ChecksummedDownloadable.download(ChecksummedDownloadable.java:49) ~[launcher.jar:?]         at net.minecraft.launcher.updater.download.DownloadJob.popAndDownload(DownloadJob.java:108) [launcher.jar:?]         at net.minecraft.launcher.updater.download.DownloadJob.access$000(DownloadJob.java:12) [launcher.jar:?]         at net.minecraft.launcher.updater.download.DownloadJob$1.run(DownloadJob.java:89) [launcher.jar:?]         at java.util.concurrent.Executors$RunnableAdapter.call(Unknown Source) [?:1.7.0_45]         at java.util.concurrent.FutureTask.run(Unknown Source) [?:1.7.0_45]         at java.util.concurrent.ThreadPoolExecutor.runWorker(Unknown Source) [?:1.7.0_45]         at java.util.concurrent.ThreadPoolExecutor$Worker.run(Unknown Source) [?:1.7.0_45]         at java.lang.Thread.run(Unknown Source) [?:1.7.0_45] [15:40:59 ERROR]: Gave up trying to download https ://libraries.minecraft.net/net/sf/jopt-simple/jopt-simple/4.5/jopt-simple-4.5.jar for job 'Version &amp; Libraries' [15:40:59 ERROR]: Job 'Version &amp; Libraries' finished with 26 failure(s)! 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By Krull's principal ideal theorem, every minimal prime ideal containing $f\mod I$ has codimension one. $X$ having dimension at least one implies that there are homogenous prime prime ideals $\mathfrak{q}\supsetneq \mathfrak{r}\supset I$. This implies that $(x_o,\dots x_n)$ has codimension at least two. Thus there are prime ideals containing $f\mod I$ that are contained in $(x_o,\dots x_n)$. However, I'd need to show that there are homogenous such prime ideals, right? I'd rather get hints than full solutions. Why does this PNG image on twitter look different when clicked? This morning on twitter there was a tweet with the text "click on this picture".  When picture is clicked, it's opened in wider viewer, but looks completely different.  I tried to analyze this picture in ACDSee, and I also tried uploading on Facebook, but it's happening only when embedded on twitter. I don't have much knowledge about graphics so please explain this to me. Why does this PNG image change only when clicked on Twitter?  Screenshots (left = before clicking; right = after clicking):     Here's the original . In other words, are $\emptyset$ and $\mathbb{R}$ the only open and closed sets in $\mathbb{R}$? Why/Why not?  I tried by assuming a set is equal to its interior points and contains its limit points.  A bounded set will not do since stuff like $[1,4]$ and $\{5\}$ will not work, though that is not really proof. Help please?  Anyway, it must then be unbounded.  If $a$ is a real number then $(a,\infty)$, $(-\infty,a)$, $[a,\infty)$ and $(-\infty,a]$ don't seem to cut it so it must be $\mathbb{R}$. How do I delete Windows.old in Windows? I used to run Windows 7 32 bit, but installed the 64 bit version because of a RAM upgrade. During the installation I got this message:      The partition you selected might contain files from a previous Windows installation. If it does, these files and folders will be moved to a folder named Windows.old. You will be able to access the information in Windows.old, but you will not be able to use your previous version of Windows.   The C:\ drive now has two (2) Windows folders:   Windows (15.3 GB) Windows.old (15.7 GB)     Don't see why I need the Windows.old taking over 15GB of space on my hard drive so I tried to delete it. It didn't work.  How can I safely delete this folder? What's the difference between the five masses: inertial mass, gravitational mass, rest mass, invariant mass and relativistic mass? Different PDF output with same source files (NeurIPS 2021 package) An element of a group has the same order as its inverse If $a$ is a group element, prove that $a$ and $a^{-1}$ have the same order.   I tried doing this by contradiction.    Assume $|a|\neq|a^{-1}|$  Let $a^n=e$ for some $n\in \mathbb{Z}$ and $(a^{-1})^m=e$ for some $m\in \mathbb{Z}$, and we can assume that $m &lt; n$.  Then $e= e*e = (a^n)((a^{-1})^m) = a^{n-m}$. 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