Try finding languages such that L1⊆L2⊆L3 where L1,L3∉ RE and L2∈ R "RE" means "recursively enumerable" and "R" means "recursive.  i am looking for the simplest solution- using a known languages such that do not demand another formal proof.
Finding languages such that $L_{1} \subseteq L_{2} \subseteq L_{3}$ where $L_{1}, L_{3} \notin \mathbb{R}$, $L_{2} \in \mathbb{R}$ I am struggling to find such languages $L_{1}$, $L_{2}$, and $L_{3}$ such that  $$ L_{1} \subseteq L_{2} \subseteq L_{3} $$ where $L_{1}, L_{3} \notin  \mathbb{R}$ and $L_{2} \in  \mathbb{R}$.  I know they exist, I need help finding them.
Logit or arcsine transformation for proportion in meta-analysis In Fringe's season 1, the episode  (1x13) shows how  travels on VertusAir flight 718. We witness how he develops some strange symptoms, withdraws into a lavatory and transforms into a porcupine-like monster.  At the end of season 3, Peter "switches" the timeline (changing the past and making the present continue in a different timeline henceforth) from blue- to orange-verse. Various statements throughout the 4th season indicate time (the date) was not reset; Olivia and Walter have been working on Fringe cases for some three years etc. - hence it is safe to say that season 4 plays 3 years after season 1.  Then, in the episode  (4x16), the events from 1x13 happen again, up to the point of Bowman's transformation, which this time doesn't happen (not aboard the plane, anyway). As far as I can tell, great care was taken to exactly recreate the events from the season 1 episode, including hiring the same actors to play the flight attendants once again ( and ), so obviously this is supposed to be the same event with slight changes.  Peter, as well as Olivia - who is in the process of regaining her memories from the blue-verse -, remember the events from 1x13 and note the difference that Bowman's transformation happened later this time, implying that in the orange-verse, everything related to flight 718 played out the same way, except for the small difference that the transformation happened a bit later.  BUT ... all of this happened 3 years later!! Is there any explanation for this?  How unlikely is it that in the orange-verse timeline, Marshall Bowman is on the very same flight with the very same flight attendants 3 years after the same would have happened in the blue-verse? Mind you, this is not a "small change" in the course of events while mostly everything else stays the same.  So, why did Peter and Olivia realize the transformation happened a tad later in the orange-verse timeline, but were not baffled by the fact that essentially the same events as in the blue-verse happened, but three years later?  EDIT: Not only the flight and the flight attendants were the same in both episodes. The passenger next to Bowman was the same, an elderly lady played by  both times, as well. Moreover, the transcripts of  and  reveal that the dialogue between Bowman, the passenger next to him, and the cabin crew is the same down to the lines of text until the moment of transformation.  Interestingly, the announcement by the captain in the beginning seems to be different - starting the inflight movie + selling headsets vs. circling before beginning the final descent + selling snacks. On the other hand, this could be presumed to be a slight retcon, as in 4x16, it was crucial for the plot that the plane would be close to landing, and starting the inflight movie would not make much sense right before landing. Convert .mov to .mp4 I want to convert .mov to .mp4! I saved the video from iMovie but it is in .mov format! How do I convert it to .mp4?  I'm using a MacBook and I want to share it to my phone, but my phone can't open .mov format video. "Above"/"below" before/after a noun I have seen sentences similar to the following:  (1) See the reference above. (2) See the reference below.  And,  (3) See the above reference.  But not,  (4) See the below reference.  Are all these forms acceptable? Which is/are preferred in formal writing? Maximum sum sublist? How can I get distribution name and version number in a simple shell script? I'm working on a simple bash script that should be able to run on Ubuntu and CentOS distributions (support for Debian and Fedora/RHEL would be a plus) and I need to know the name and version of the distribution the script is running (in order to trigger specific actions, for instance the creation of repositories). So far what I've got is this:  OS=$(awk '/DISTRIB_ID=/' /etc/*-release | sed 's/DISTRIB_ID=//' | tr '[:upper:]' '[:lower:]') ARCH=$(uname -m | sed 's/x86_//;s/i[3-6]86/32/') VERSION=$(awk '/DISTRIB_RELEASE=/' /etc/*-release | sed 's/DISTRIB_RELEASE=//' | sed 's/[.]0/./')  if [ -z "$OS" ]; then     OS=$(awk '{print $1}' /etc/*-release | tr '[:upper:]' '[:lower:]') fi  if [ -z "$VERSION" ]; then     VERSION=$(awk '{print $3}' /etc/*-release) fi  echo $OS echo $ARCH echo $VERSION   This seems to work, returning ubuntu or centos (I haven't tried others) as the release name. However, I have a feeling that there must be an easier, more reliable way of finding this out -- is that true?  It doesn't work for RedHat. /etc/redhat-release contains : Redhat Linux Entreprise release 5.5  So, the version is not the third word, you'd better use :  OS_MAJOR_VERSION=`sed -rn 's/.*([0-9])\.[0-9].*/\1/p' /etc/redhat-release` OS_MINOR_VERSION=`sed -rn 's/.*[0-9].([0-9]).*/\1/p' /etc/redhat-release` echo "RedHat/CentOS $OS_MAJOR_VERSION.$OS_MINOR_VERSION" Determining if a game is CPU- or GPU-limited What is a reliable, but not too time-consuming way to determine if a certain game is limited by the graphics card or by the processor speed?  I have some preconceptions about whether I'm mostly GPU or CPU limited, but I'd like to verify if I'm right about that. I have some ideas how I would go around and test that but performing accurate benchmarks is notoriously tricky and I don't have much experience with it.   So I'm wondering what would be an easy way to determine the bottleneck, having only one computer with a certain configuration available? What tools would I use for that purpose?  It would also be interesting to determine if the amount of VRAM on my graphics card is a limiting factor. $ df -kh Filesystem      Size  Used Avail Use% Mounted on udev            7,8G     0  7,8G   0% /dev tmpfs           1,6G   19M  1,6G   2% /run /dev/sda2       622G  227G  364G  39% / tmpfs           7,8G  1,1G  6,7G  14% /dev/shm tmpfs           5,0M  4,0K  5,0M   1% /run/lock tmpfs           7,8G     0  7,8G   0% /sys/fs/cgroup /dev/loop1      143M  143M     0 100% /snap/inkscape/2527 /dev/loop3       79M   79M     0 100% /snap/core/1441 /dev/loop0       79M   79M     0 100% /snap/core/1577 /dev/loop4       79M   79M     0 100% /snap/core/1512 /dev/loop2       62M   62M     0 100% /snap/pencilsheep/5 tmpfs           1,6G   92K  1,6G   1% /run/user/1000   What are these /dev/loop device files? My RAM is always full. Is this related? Limiting sessions to one IP at a time On a site with premium user account subscriptions, I'd like to be able to limit logins to one computer at a time. The most straightforward way to accomplish this would be to limit by IP, but I haven't had any luck finding a plugin to accomplish this. Does anybody know of one I can use to get this functionality?  Related: Often in fantasies involving magic casting spells requires a certain physical or mental toll on the caster. I don't recall any mention of spells being physically or mentally taxing in the Harry Potter universe.  Does this mean they can cast an unlimited number of spells in a given day without any negative effect to themselves? Display View counts of Questions in Mobile version I often use Stack Exchange sites from my smartphone. It works great, but I always have to switch to full version to see view counts of a question. Can you please add it? There's plenty of space on question page for it to place. I request community to suggest a location. Suggestions for a batch audio converter? Any suggestions for a good, free audio converter for Mac?  I used to use Max.app but it has long become abandonware and is now super buggy and crashy. I just need something I drop a couple (or 20 thousand) Apple Lossless files into and let it crank away making me some AAC compressed files or some MP3 or whatever. What's the best way to write Mathematical Equations on the Web? How to influence the position of float environments like figure and table in LaTeX?   This is a general question and should collect useful answers for all users. I hope we can use this as a reference What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters? What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters?  I believe the answer for this is 6. As we can write the group elements as below   (a)(b)(C) (ab)(c) (ac)(b) (bc)(a) (abc) (bac)   Can we generalize that for any $S_n$ onto $n$ the number of automorphisms will be $n!$ Also I cannot find this anywhere in my text, can we have a permutation $S_n$ onto $m$ where $n\neq m$? (for eg: is it possible to have a permutation group say $S_5$ on say 2 elements?) How to reconnect a logically disconnected USB device? Using the itemize environment inside a caption The following code does not compile. However, if I remove the itemize environment it does. I googled for this problem, and found suggestions that using the caption package would solve the problem, though I did not find any clear description of why the problem occurred in the first place. See for example    I'm using Debian squeeze with TeX Live 2009-11. The included version of caption has   \ProvidesPackage{caption}[2009/10/09 v3.1k Customizing captions (AR)]   I also tried downloading and using the caption package from     but this doesn't change anything. I put the sty files in my current directory. I assume that overrides the system files. This one corresponds to  \ProvidesPackage{caption}[2010/01/09 v3.1m Customizing captions (AR)]   Error message and example file follow.  *******************************************     ERROR: Argument of \@caption has an extra }.  --- TeX said --- &lt;inserted text&gt;                  \par  l.13     }  --- HELP --- From the .log file...  I've run across a `}' that doesn't seem to match anything. For example, `\def\a#1{...}' and `\a}' would produce this error. If you simply proceed now, the `\par' that I've just inserted will cause me to report a runaway argument that might be the root of the problem. But if your `}' was spurious, just type `2' and it will go away. ******************************************************  \documentclass{article} \usepackage{caption} \begin{document}  \begin{figure}     \captionsetup{singlelinecheck=off}     \caption     {       foo       \begin{itemize}       \item bar       \end{itemize}     } \end{figure} \end{document} How to align prices in columns when both 2 digit and 4 digit variations of a price exist? I am trying to display data similar to coinmarketcap with prices for coins, some above 1$ in which case they are rounded to 2 zeroes, some below 1$ in which case they have 4 digits following the decimal. I saw a bunch of posts here where people have suggested to use right alignment for numeric applications/prices  Approach 1 Decimal aligned   1. $13,700.05  2.      $0.0048   Approach 2 Right aligned   1. $13700.05  2.   $0.0048   Which of these approaches do you suggest? Thank you for your time Promote tag wiki creation for new tags Why is the gravitational force always attractive?