display – Triple external monitor on 13″ Macbook Pro 2018

I’ve recently been given a 13″ Macbook Pro 2018 by work so that I can work from home. Just to note I am completely new to MacOS, but have a lot of sysadmin experience on other platforms.

I have a triple monitor set up with my Windows PC and I was hoping to use this with the new Macbook. But I looked online and it seems that this model is officially limited to two external monitors. However The limit is for two 4k monitors, and I want three 1080p which is a lower total pixel count than even one 4k screen so the MBP definitely has the power required for this.

Also I don’t need the built-in display as the MBP will be tucked away. Clearly the MBP can do three displays if we include the built-in one, so is there a way to redirect that to an external screen? Is there an official or unofficial way to make this work? Would using a thunderbolt based adapter allow more bandwidth so more screens? I’ve seen reports of USB3 (not type C) display adapters that can bypass this restriction. Or there’s this https://github.com/FD-/RPiPlay that would involve using a Raspberry Pi as an airplay receiver and using that as the third screen.

arithmetic – Can (q,p) and (p-q,p) be legs of a Pythagorean Triple

I’m searching to prove that there’s no (or to find an example of) $p$ and $q$ coprimes, and $n,m$ integers such that:

$q^2 + p^2 = n^2$

$(p-q)^2 + p^2 = m^2$

I was only able to prove that in this case $p$ must be even: since there’s at least one number that is even in a pythagorean triple, it must be $p$, else there’ll be a contradiction with the parity of $q$ and $p-q$.

I conjecture that this case is impossible. Do you have any idea of how to prove or disprove it ?

rt.representation theory – Simple quotients of a triple tensor product

Let $mathcal{H}$ be a Hopf algebra over $mathbb{C}$. Let also $V_1, V_2, V_3$ finite-dimensional simple modules over $mathcal{H}$ and $Q$ be a simple quotient of $V_1otimes V_2otimes V_3$. Is it possible to show that one of the following statements is true? Is there any counterexample?

i] $Q$ is a quotient of $Notimes V_3$, for some simple quotient $N$ of $V_1otimes V_2$;

ii] $Q$ is a quotient of $V_1otimes P$, for some simple quotient $P$ of $V_2otimes V_3$.

Triple integral, any suggestions


$$iiint_Sfrac{z}{sqrt{x^2+y^2}}d V$$
Where $S:$ $x^2+y^2+z^2leq 1$, $z^2leq x^2+y^2$, $y^2leq 2xz$.

any suggestion to be able to solve it!

javascript – Largest Triple Products without using sort?

I implemented the Largest Triple Products algorithm, but I use sort which makes my time complexity O(nlogn). Is there a way to implement it without a temporary sorted array?

The problem:
You’re given a list of n integers arr(0..(n-1)). You must compute a list output(0..(n-1)) such that, for each index i (between 0 and n-1, inclusive), output(i) is equal to the product of the three largest elements out of arr(0..i) (or equal to -1 if i < 2, as arr(0..i) then includes fewer than three elements).
Note that the three largest elements used to form any product may have the same values as one another, but they must be at different indices in arr.


var arr_2 = (2, 4, 7, 1, 5, 3);
var expected_2 = (-1, -1, 56, 56, 140, 140);

My solution:

function findMaxProduct(arr) {
  // Write your code here
  if(!arr || arr.length === 0)  return ();
  let helper = arr.slice();
  helper.sort((a,b)=>a-b);   // THIS IS THE SORT
  let ans = ();
  let prod = 1;
  for(let i=0; i<arr.length; i++) {
    if(i < 2) {
      prod *= arr(i);
    else {
      if(i === 3) {
        prod *= arr(i);
      } else if(arr(i) < helper(0)) {
      } else {
        const min = helper.shift();
        prod /= min;
        prod *= arr(i);
  return ans;

Triple Integral with region enclosed by planes 4 and parabolic cylinder

where D is the region in the first octant enclosed by the planes x=0, z=0, y=0, y=6
and the parabolic cylinder z=2−x^2.

software – 1980 Mt St Helens triple exposure

You cannot undo multi exposure on film nor separate the exposures from one another.

But why not keep the triple exposure image as it is? I mean, it has an artistic value, the photo tells a story. The moment the big catastrophy happened, you experienced a smaller ‘catastrophy’ with your camera. I think that’s an interesting connection. Maybe give the photo a fitting title.

Triple boot Macbook (Arch Linux and Ubuntu) from an external SSD

I have a 2015 MacBook Air and plan to triple boot MacOS, Arch Linux and Ubuntu by installing Arch and Ubuntu onto my SSD. I have looked at this thread and read about installing rEFInd onto the SSD so that the default macOS bootloader is left unchanged. I have also read through this article and plan to do something similar.

From the article I have read that you must install Windows first before installing Ubuntu. I completely understand how to install Ubuntu after installing Windows.

The part that I am confused about is how to do this but install Arch Linux instead of Windows. If possible I would like full list of steps of how to install Arch Linux on the SSD.

Thanks in advance!

TL;DR: Trying to triple boot Arch and Ubuntu from an external SSD using this article but I am unsure how to install Arch Linux first instead of Windows.

Note: I am fine using other methods of triple-booting Arch and Ubuntu as long as it doesn’t make changes to my mac.

MacBook Info

Model: 2015 MBA

macOS Version: Catalina (10.15.3)

macos – Triple boot OS-X Catalina10.15.5, Kali-Linux 2020-2, Ubuntu 20.04 on MacBook Pro

This question is the next event of this post.

After going through the nightmare for a couple of days to retrieve my Mac partition, finally, I succeeded. But during that process, something happened with my ubuntu and it started taking 2-3min to boot(earlier hardly 15sec). So I decided to do a fresh install and hence, normalized all my partitions.

Now I want to do a triple boot as mentioned in the title. And I don’t want to mess things up anymore. so, Can someone please guide me out in this regard? I was able to find a few posts involving windows but not sure whether I can apply the same in my case.

Installing Ubuntu on Mac with macOS and Windows already installed

How can I triple boot MacOS Catalina , Ubuntu 18.04, and Windows 10

My Plan:

  1. I want to make 15GB Linux-swap, 65GB Ubuntu, and 50GB Kali.
  2. I want to make the Ubuntu default as I’ll be using it more often.
  3. I want to access the other two OS’s from the Ubuntu grub screen when needed.

My Mac-book pro is the early-2015 Model and this is the current condition of my disk.

$ diskutil list
/dev/disk0 (internal, physical):
   #:                       TYPE NAME                    SIZE       IDENTIFIER
   0:      GUID_partition_scheme                        *251.0 GB   disk0
   1:                        EFI EFI                     209.7 MB   disk0s1
   2:                 Apple_APFS Container disk1         250.8 GB   disk0s2

/dev/disk1 (synthesized):
   #:                       TYPE NAME                    SIZE       IDENTIFIER
   0:      APFS Container Scheme -                      +250.8 GB   disk1
                                 Physical Store disk0s2
   1:                APFS Volume Macintosh HD - Data     84.7 GB    disk1s1
   2:                APFS Volume Preboot                 126.6 MB   disk1s2
   3:                APFS Volume Recovery                1.6 GB     disk1s3
   4:                APFS Volume VM                      1.1 GB     disk1s4
   5:                APFS Volume Macintosh HD            11.2 GB    disk1s5
$ sudo gpt -r show /dev/disk0
        start       size  index  contents
          0          1         PMBR
          1          1         Pri GPT header
          2         32         Pri GPT table
         34          6         
         40     409600      1  GPT part - C12A7328-F81F-11D2-BA4B-00A0C93EC93B
     409640  489825072      2  GPT part - 7C3457EF-0000-11AA-AA11-00306543ECAC
  490234712          7         
  490234719         32         Sec GPT table
  490234751          1         Sec GPT header

$ sudo fdisk /dev/disk0
Disk: /dev/disk0    geometry: 30515/255/63 (490234752 sectors)
Signature: 0xAA55
         Starting       Ending
 #: id  cyl  hd sec -  cyl  hd sec (     start -       size)
 1: EE 1023 254  63 - 1023 254  63 (         1 -  490234751) <Unknown ID>
 2: 00    0   0   0 -    0   0   0 (         0 -          0) unused      
 3: 00    0   0   0 -    0   0   0 (         0 -          0) unused      
 4: 00    0   0   0 -    0   0   0 (         0 -          0) unused      

This is my boot partition

$ ls /Volumes/EFI/EFI
APPLE   refind  tools
$ ls refind 
BOOT.CSV        icons-backup        refind.conf-sample
drivers_x64     keys            refind_x64.efi
icons           refind.conf
$ ls tools 

windows 10 – Need help booting into Mac OS. Lost during triple boot MAC OS, WIN 10, UBUNTU 20.04

Excuse for the long post. Just trying to be as clear as possible. Initially I used bootcamp for installing windows 10 alongside Mac OS. And while restart it would automatically boot me into Mac OS and whenever I needed to boot into windows, I used to hold option key while starting and that gave me option to boot into either windows or continue with Mac OS.

After using it for a while, I decided to try out Ubuntu 20.04. So I partitioned my Mac OS Volume and installed Ubuntu manually. Even though I was successful, I realized that I could no longer boot into windows anymore. Auto boot was changed to Ubuntu and if I held option key during restart, I used to get option for booting into Mac (working) and Windows (this too was not working).

Here is where I actually messed up. Today I decided to retain the unused Windows partition space and hence, booted into Mac OS and deleted the windows partition using bootcamp and did a restart. Now I’m unable to log into Mac OS. Auto boot is still set to Ubuntu and no option is shown when using option key.

I know that I can do a complete reinstall but just wanted to know if there is any way to regain my Mac OS boot option without a complete restore? Currently Ubuntu is working fine.

TL;DR: Initially installed Windows 10 using bootcamp in Macbook pro. Decided to triple install Ubuntu. Succeeded but lost access to windows partition. So decided to free that space with bootcamp and now lost access to Mac OS. Is there any chance to recover the boot option for Mac OS without complete restore?

(Ubuntu Boo(Mac OS partition(Ubuntu swap(ubuntu rootWindows partitiont)2