How does the mass of black holes become so much larger than the mass of an average star? Since black holes are formed from the collapse of large stars, how come the mass of black holes is of astronomical scale. (If I am  right, i heard there must be a limit to how big a star can become, beyond that it wont be stable?). so this scale of size of mass of super massive black holes does not seem to add up to me, unless they devours many stars nearby it. Any explanation?   Can there be a possibility a more than one larger black holes compete to be for center of a galaxy?
Scientists have found very bright source of light which they call  and the are found to be . So these black holes are so massive that they cannot be formed by a supernova. So how are these formed?
If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$. and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $ If  $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$ and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $. Progress We have   $\gcd(a,b)=1\implies \exists u,v\in \mathbb Z$ such that $au+bv=1\implies a^2u'+bv'=1, u',v'\in \mathbb Z$. Let $\gcd(a+b, a-b)=d$. Then $\mid (a+b)x+(a-b)y, \forall x,y\in \mathbb Z$. How to show $\gcd(a+b, a-b)=1$ or $2$ and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $. I've setup a HTTPS and FTP site in IIS but how can I add users to use the FTP site?  Does they necessarily need an account with password in the computer to be able to use it? ^@ spam in tty (but seems to be system-wide) I recently noticed some weird behavior on my GNU/Linux machine. I've been trying to narrow it down but I am unsure where to go next. My setup uses the following:   i3 gdm arch systemd grub2 pulseaudio 4.13.3-1-ARCH   Here is how it manifests iself:   when trying to resize panes in i3 (alt+right click and hold) the action will be randomly canceled (happens on key presses) alt+dragging windows meets a similar fate booting to tty periodically adds ^@s to the login prompt, I'd estimate them to be ~0.5s intervals in tty I can see those being passed to vim when I open the command bar (:) using a pager like less in tty mode and searching with / will result in  (?) being spammed htop's search (/) is instantly canceled by the periodic keypress  holding a key and trying to have it be repeated (such as holding space) will stop the other occurrences from happening (e.g. only one or two happen) any tooltip (such as chrome tooltips) get instantly closed because of key presses   Here is what I have tried:   tried switching DE temporarily (gnome, cinnamon, xfce) tried killing programs one by one until I was left with a tty session and essentially systemd services + pulseaudio tried booting on an arch livecd - problem does not manifest  overriding grub2 with init=/bin/bash drops me into a shell where the problem does not manifest I have tried pressing all keys to verify it wasn't a hardware problem, however previous steps point to it not being hardware I have made a diff of dmesg, from the init=/bin/bash and the regular boot - the main difference seems to be systemd being started in the regular boot Made a diff of lsmod from init=/bin/bash and regular boot, tried to rmmod modules that seemed related   My system is up to date, I'm not quite sure what other steps I can do to track this down. Any other debugging tips? Can “whose” refer to an inanimate object? We lit a fire whose fuel was old timber wood.   Is the word whose referring to fire, an inanimate object, correct in this sentence? Or is there a more appropriate word? Closing a Java FileInputStream From a technical perspective, what exactly is RAW and how does it differ from JPG or bitmap images? I'm not asking about the pros/cons versus another format, but rather what constitutes RAW format, how is the data stored, how does it differ from JPG (other than lack of compression), and how is it different than bitmap? What is the meaning of this sentence - "Were it not for the bodies there on the ground, it would have appeared to be a normal day at the oasis."? Possible Duplicate:          What is the meaning of this sentence - "Were it not for the bodies there on the ground, it would have appeared to be a normal day at the oasis."?  Does it mean something like this - If the bodies were not there on the ground, it would have appeared to be a normal day at the oasis.  Can anyone also explain the structure of this sentence and example of some similar sentences? Mutable or immutable class? The coefficient of kinetic friction can be calculated using $\mu_k = F_k/F_n$. A change in angle does not affect the normal force, but doesn't a change in angle affect the friction? As in if the angle is steeper, wouldn't the object have a greater propensity to move down the slope, thus there will be less friction? I have updated from Windows 7 Ultimate x64 to Windows 10 Pro x64 using Media Creation Tool.  After some weeks when I'm clicking on the start button nothing happens and the whole status bar becomes unresponsive. After trying to restart explorer.exe from Task Manager explorer.exe fails to initialize. It just shows the start button, Cortana and the search bar, TaskViewer (if that's how it is called) and the Action Centre button.  Any ideas?  I tried running:   sfc /scannow and no errors were found chkdsk /f and no errors were found   A full system restart seems to do the trick but after some time like a few hours it stops responding again.  My configuration:   CPU is Intel i5 4690K OC-d to 4.5Ghz GPU is AMD Radeon HD 7750 Saphire 1Gb GDDR5 Edition   EDIT: After some GOOGLE SEARCHING if i try to run from Task Manager C:\Windows i can actually open a window. The start bar isn't responding though and there are still no desktop icons or wallpaper. If $X$ is an orthogonal matrix, why does $X^TX = I$? It's not immediately clear to me why this is true. My notes say that putting $n$ orthonormal vectors $ v_1, ..., v_n$ in the columns of $X$ gives $X^TX = I$, and it follows from this that the rows of $X$ are orthonormal.   Can you explain this better to me? I tried the $2\times 2$ case, but it isn't clear to me why this is true. Why jquery validation is not working on appended elements? The trigonometric ratios of an angle greater than $90^\circ$ are equal to the supplementary angle's ratios.  I'm just clarifying this, but the ratios don't actually exist for angles greater than $90^\circ$ right? (since by definition these ratios are of a right angle triangle). Is it just mathematical custom to assume the ratios of an angle greater than $90^\circ$ to be equivalent to its supplementary angle's ratios? For convenience? Why is it "come" rather than "go" ? Markdown does not allow for bold italic, then bold, then bold italic I am trying in one of my posts to do this:  ***how many*** **and** ***where***   Which for some reason comes out like this:    Why is that happening?  Shouldn't the middle word be bold?  Is this  for markdown or is it (as I assume) a ? date difference in days, in Android Why is "union" an exception to the "a/an" rule? SEO scope for multiple domains hosting on the same IP address Unable to install Ubuntu on Lenovo Y500 I am totally unable to install any flavor of Linux on my lenovo Y500. TL;DR:  with EFI on -&gt; cannot even access installer with legacy - &gt; installer runs fine, but installed OS never boots  Long Version: It is going to be hard to recap what I did, because I tried so many things, but here is the gist of it: EFI mode (secure boot disabled): Trial #1  ubuntu 12.10 x64 &amp; Mint 14 x64 boot from usb see screen with choices pick &quot;try ubuntu&quot; get black screen  Trial #2  ubuntu 12.10 x64 &amp; Mint 14 x64 boot from usb see screen with choices press &quot;e&quot; to enter edit mode remove &quot;quiet splash&quot; replace with &quot;nomodeset&quot; end up on the command line with an error message from x server: &quot;no screen found&quot; try fiddling around with xorg.conf to no avail  Trial #3  Arch Linux Live USB EFI mode no boot  Trial #4  ubuntu 12.10 x64 &amp; Mint 14 x64 set &quot;e&quot; to enter edit mode in menu add acpi=off end up with &quot;(initramfs) unable to find a medium&quot; message  Trial #5  same as above set SATA mode to legacy (it was AHCI) error loop that says something along the lines of &quot;ata1.00: hard resetting link&quot;  Legacy Mode Trial #1  ubuntu 12.10 x64 &amp; Mint 14 x64 USB boot Everything seems to work Proceed to normal install (picks SSD drive by default) Everything seems to work reboot blank screen reboot with &quot;shift&quot; pressed see message &quot;grub loading.&quot; and nothing As per advised on Ubuntu's community EFI page, run boot-repair same results run boot-repair again same results  Trial #2  ubuntu 12.10 x64 &amp; Mint 14 x64 same as above, but pick 1T HDD as target install same results  Trial #3  create a /boot partition on the SSD install on SSD same results (&quot;grub loading.&quot;) run boot repair twice same results  Trial #4  create a GPT filesystem on HDD create a /boot partition on the HDD install on HDD same results as above  Trial #5  install ubuntu on HDD install boot on all drives (HDD &amp; SSD) same results  Trials #6  install Arch on SSD no boot loader found  Notes:  Every step has been tried with multiple USBs. All isos' MD5Sums have been verified LiveUSBs have been created both through ubuntu's embedded app and unetbootin (save for Arch's live USB which was created by DDing the iso as per advised in the Arch wiki) I have wiped out windows in my first trials, and I don't care, I am never going to use it. I don't want to dual-boot. I just want Ubuntu or Mint or Arch as far as I can remember (it gets hard after 50+ trials), I tried every option with sata mode IDE or normal. Not sure though.  I've also tried numerous other things that I can't recall right now. All in all I've been giving this 3 to 5 hours a day for a week and a half to absolutely no avail. I am extremely disappointed as this is my first Lenovo and word is Lenovo is Linux-friendly...I have never spent more than an hour installing linux on any machine. What's even more frustrating is that it seems to be working for , and I can't find any hint telling me how they managed. I wrote to one guy who posted a serie of youtube videos about how to get everything working perfectly on the Y500, and he told me he just installed mint without any problems. Might be I am a particular case because I have the SLI version of the Y500. If anyone has pointers, incomplete solutions, ideas, black magic, I'll be forever in debt. Thank you P.S: This question, afaik, is not a duplicate of an existing question. I've scoured all questions containing &quot;lenovo&quot; and &quot;install&quot;, and none applies to me. The following could have shown up as an exercise in a basic Abstract Algebra text, and if anyone can give me a reference, I will be most grateful.  Consider a set $X$ with an associative law of composition, not known to have an identity or inverses. Suppose that for every $g\in X$, there is a unique $x\in X$ with $gxg=g$. Show that $X$ is a group.  Note that I’m not asking for a proof (though a really short one would please me!), just some place where this has been published.