Export PNG image in Sketch, looks poor quality
My sketch exports seem to be pixelated even when am exporting it to 1x
Sieve of eratosthenes infinite primes I don't understand why the sieve of eratosthenes doesn't prove an infinite number of primes.  For example the number 2 eliminates 1/2 of an infinite number of numbers.  The next prime, 3, eliminates 1/3 of the remainder, not 1/3 of all the numbers and so on.  5 would eliminate 1/5 of the remaining numbers. Why doesn't that prove an infinite number of primes? Thanks How do ideal quotients behave with respect to localization? Suppose $R$ is commutative ring with unity. For ideals $I$, $J \subseteq R$, the ideal quotient $(J:I)$ is $$(J:I) := \{x\in R \, : \, xI \subseteq J\}$$ Let $S\subset R$ be a multiplicative set. When does localization at $S$ commute with taking quotients, i.e. when does the equality $$(S^{-1} J : S^{-1} I) = S^{-1} (J:I)$$ hold? More generally, when is it true that $$S^{-1} \text{Ann}_{R} (M) = \text{Ann}_{S^{-1} R} (S^{-1} M)$$ for an $R$-module $M$? Grub does not show a Windows 8 option after dual boot So, I've successfully dual-booted my Windows 8 machine with Ubuntu 12.04 . However, I still don't have a convenient method of choosing what OS to load at boot time.  After installing Ubuntu, my computer still loads Windows 8 directly. I then added grubx64.efi to the white list of my boot loader. But after that, my machine loads Ubuntu directly without even a shadow of GRUB showing up!  I used boot-repair and I got this paste.ubuntu URL: . After running boot-repair (and re-white listing the grubx64.efi file), GRUB now shows up but without any Windows 8 option!  Lastly, I ran sudo fdisk -l and it gave me this:  WARNING: GPT (GUID Partition Table) detected on '/dev/sda'! The util fdisk doesn't support GPT. Use GNU Parted.   Disk /dev/sda: 750.2 GB, 750156374016 bytes 255 heads, 63 sectors/track, 91201 cylinders, total 1465149168 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 4096 bytes I/O size (minimum/optimal): 4096 bytes / 4096 bytes Disk identifier: 0x6396389f     Device Boot      Start         End      Blocks   Id  System /dev/sda1               1  1465149167   732574583+  ee  GPT Partition 1 does not start on physical sector boundary.   I'm guessing my problem has something to do with the warning from fdisk above but I don't know what to do with it. How do I proceed now?  Edit  My etc/default/grub is as follows:  # If you change this file, run 'update-grub' afterwards to update # /boot/grub/grub.cfg. # For full documentation of the options in this file, see: #   info -f grub -n 'Simple configuration'  GRUB_DEFAULT=0 #GRUB_HIDDEN_TIMEOUT=0 GRUB_HIDDEN_TIMEOUT_QUIET=true GRUB_TIMEOUT=10 GRUB_DISTRIBUTOR=`lsb_release -i -s 2&gt; /dev/null || echo Debian` GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX=""  # Uncomment to enable BadRAM filtering, modify to suit your needs # This works with Linux (no patch required) and with any kernel that obtains # the memory map information from GRUB (GNU Mach, kernel of FreeBSD ...) #GRUB_BADRAM="0x01234567,0xfefefefe,0x89abcdef,0xefefefef"  # Uncomment to disable graphical terminal (grub-pc only) #GRUB_TERMINAL=console  # The resolution used on graphical terminal # note that you can use only modes which your graphic card supports via VBE # you can see them in real GRUB with the command `vbeinfo' #GRUB_GFXMODE=640x480  # Uncomment if you don't want GRUB to pass "root=UUID=xxx" parameter to Linux #GRUB_DISABLE_LINUX_UUID=true  # Uncomment to disable generation of recovery mode menu entries #GRUB_DISABLE_RECOVERY="true"  # Uncomment to get a beep at grub start #GRUB_INIT_TUNE="480 440 1"   Also, just for the record,  in installing my dual-boot. The tutorial (which uses, 11.10, I must note), doesn't mention anything about this problem. Is there anything there I shouldn't have done? Why do video cables (DVI, VGA) still screw in? I guess I could imagine that there was a time when accidentally disconnecting the monitor would have been catastrophic, though I don't see that being the case today.  Is it just a legacy holdover that they are still secured with screws?  Also, why is it that the video connection is really the only one that is still secured this way?  It seems like the power cable would be a better candidate to be secured with screws.  That really could be catastrophic.  In the end, the reason this question occurred to me is that I've never had a video cable come out when not screwed in, so why the significant security? What happen if I install VirtualBox Guest Additions on a host? I have a bootable USB drive with Ubuntu installed.   This morning I Installed VirtualBox and booted my Ubuntu drive in a VM. I then installed the guest additions in Ubuntu.   My question/curiosity is, what would happen if I boot the drive again outside of the VirtualBox VM, using a physical computer?  Will my Ubuntu work normally or is something going to change? Proofs of Determinants of Block matrices Based on the table Encounter XP Multipliers from the Basic Rules' DM section, Page 57, when you're building encounters in 5E, you include an XP modifier depending on the number of creatures your PCs will be fighting: 1.5x for a pair, 2x for 3-6 and so on. However, the PCs only get the base XP as a reward.     This doesn't change the actual xp reward ... just your calculations of how difficult the encounter is.   Why is this? After acknowledging that battles vs multiple opponents are more difficult, why not reward them as such? This question is in response to . The original post mentioned that they wish to retract their paper from a "fake" journal. I don't know what a fake journal is but better safe than sorry so I want to make sure what defines a fake journal. In the comments, other people also mentioned "shady" journals and "predatory" journals. This is the first time I encounter these descriptions so I would like to know more about what they entail. I would be interested in hearing about your experience with such journals as well, if any. Why am I getting different results from two servers that located in the same destination network? There's two servers with the same network IP, and the only difference between them is the last two numbers like: 37.211.15.247 and 37.211.166.178. The Hosting Provider claims that the subnet mask for both networks is 255.254.0.0  When I try the command Tracert (Traceroute) to test the ping and to see what hops does it take to reach the destination server, I get different ping times and different forward path results which is strange because they are all located in the same company host.  Is there something wrong? Shouldn’t it take the same path too?  I have added the the full IP addresses to both servers as requested by some commenters. Well the two servers aren’t “International” anyways, so as far as I know, the only thing that comes to mind is the possibility that the two are configured on different subnetworks just like what . Though it’s suspicious.  UPDATE: After doing so much research, I found out the Subnet of these two hosts. They are on a /15 (255.254.0.0) subnet mask. Unfortunately, this is what that hosting provider claimed to me which is not true, and the true Subnet they apparently use is a /17 (255.255.128.0) Subnet mask. It turns out that they were definitely on different Subnets. That's why I got different traceroutes when I tested these two IPs 37.211.15.247 and 37.211.166.178  because they are clearly on different Subnets. Thank you everyone for these good answers, especially  his answer helped me a lot. Proof that the largest eigenvalue of a stochastic matrix is $1$ The largest eigenvalue of a  (i.e. a matrix whose entries are positive and whose rows add up to $1$) is $1$.  Wikipedia marks this as a special case of the , but I wonder if there is a simpler (more direct) way to demonstrate this result. Why couldn't Harry see Thestrals at the end of Goblet of Fire? Why couldn't Harry see Thestrals at the end of Goblet of Fire after the senseless and shocking death of Cedric Diggory?   At the end of Goblet of Fire, the carriages still appeared horseless to Harry:     Hermione turned away, smiling at the horseless carriages which were now trundling towards them up the drive, as Krum, looking surprised, but gratified, signed a fragment of   parchment for Ron.   Goblet of Fire - Page 629 - British Hardcover - Adult Edition   At the beginning of Order of the Phoenix, Harry sees the Thestrals for the first time:     Here stood the hundred or so horseless stagecoaches that always took the students above first year up to the castle. Harry glanced quickly at them, turned away to keep a lookout for Ron and Hermione, then did a double-take.   The coaches were no longer horseless. There were creatures standing between the carriage shafts.   Order of the Phoenix - Page 178 - British Hardcover - Adult Edition   We know that only those who have seen death can see Thestrals:     ‘Don’ worry, it won’ hurt yeh,’ said Hagrid patiently. ‘Righ’, now, who can tell me why some o’ yeh can see ’em an’ some can’t?’   Hermione raised her hand.   ‘Go on then,’ said Hagrid, beaming at her.   ‘The only people who can see Thestrals,’ she said, ‘are people who have seen death.’   ‘Tha’s exactly right,’ said Hagrid solemnly [...]   Order of the Phoenix - Page 394 - British Hardcover - Adult Edition   A month went by between Cedric's death and the end of the spring term -- is this enough time for Harry to have truly internalized Cedric's death? Goblet of Fire indicates that Harry was in shock following Cedric's murder, but was able to relate the story of Cedric's death to Mr. and Mrs. Diggory.     When he looked back, even a month later, Harry found he had few memories of the following days. It was as though he had been through too much to take in any more. The recollections he did have were very painful. The worst, perhaps, was the meeting with the Diggorys that took place the following morning.   [...]   ‘He suffered very little, then,’ she said, when Harry had told her how Cedric had died.'   Goblet of Fire - Page 621 - British Hardcover - Adult Edition   So, then, why couldn't Harry see Thestrals at the end of Goblet of Fire? Difference between ' and " on command line (bash)? I used to use '' and "" (single and double-quotes) interchangeably on the command line, but I recently noticed that '$HOME/some/dir' is not expanded, while "$HOME/some/dir" is. I searched around a little bit and found that "" allows some protection of special characters, while '' allows full protection. So what other characters are protected by '' and not ""?  EDIT: What are practical situations in which these differences might be significant? Convergence of sequence given by $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n^2+1}$ Be $(x_n)_n\ge 1 $  such that $x_1=1$ and $x_{n+1}=x_n+\sqrt{x_n^2+1}$ for every $n\ge 1$. Prove that the sequence $y_n=(2^n/x_n)_n \ge 1$ is convergent and find it s limit.  Being positive and decreasing, $y_n$ is clearly convergent. But finding it's limit really put me intro trouble. Any help?  Thank you! Rule of thumb for choosing an implementation of a Java Collection? Anyone have a good rule of thumb for choosing between different implementations of Java Collection interfaces like List, Map, or Set?  For example, generally why or in what cases would I prefer to use a Vector or an ArrayList, a Hashtable or a HashMap? How to resolve "iTunes could not back up the iPhone because the backup was corrupt"? When I try to sync my iPhone with my computer, I get the following message:       iTunes could not back up the iPhone "iPhone Name"   because the backup was corrupt or not compatible   with the iPhone.      Delete the backup for this iPhone, then try again.   Yet, here is no option to delete the backup. Where are backups stored / how can I delete the backup?  This is iTunes version 10.5.3 running under  Mac OS X 10.6.8. How to avoid WSP deployment globally using Visual Studio 2010? How can we avoid the Globally Deployment of a WSP using Visual Studio 2010? Why is no root pass needed to access MariaDB after upgrade to Ubuntu 16.04? No sound in Ubuntu 13.04, only Dummy output device listed My sounds suddenly stopped playing on Ubuntu 13.04 64bit. I can only see Dummy output device in the Sound Settings and no sound is played during testing.  I can get the sound temporarily working by killing one of the strange processes, but I have to repeat the same after reboot  jan@jan-HP-Desktop:~$ ps -ef|grep pulseaudio 112       1405     1  0 09:38 ?        00:00:00 /usr/bin/pulseaudio --start --log-target=syslog 112       2132  1405  0 09:38 ?        00:00:00 /usr/lib/pulseaudio/pulse/gconf-helper jan       2575     1  0 09:39 ?        00:00:00 /usr/bin/pulseaudio --start --log-target=syslog jan       2597  2575  0 09:39 ?        00:00:00 /usr/lib/pulseaudio/pulse/gconf-helper jan       3160  2975  0 09:40 pts/2    00:00:00 grep --color=auto pulseaudio  jan@jan-HP-Desktop:~$ sudo kill 1405   Is there a permanent solution for this problem? Here's a screen capture from a site where I can view /site-analytics:    The orange "all posts" curve should be strictly larger than the blue "questions" curve, but they are mistakenly displaying the same data. How to find the joint PDF? Ley X,Y two random variables with density $f(x,y)=8xy \  \text{if} \   0&lt;x&lt;y&lt;1$  Find the joint distribution F(x,y).  I find that $F(x,y)=2x^2y^2-x^4\ \text{  if}  \ 0&lt;x&lt;y&lt;1$ but I dont know how to find the didtribution in other points.