"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.
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.
Open local folder from link What constitutes an observation/measurement in QM? Prefix or adjective meaning "one and a half" Is there a prefix or adjective that means "one and a half", as "tri-" or "triple" is for "three"?  The exact usage I have is to describe "18" in terms of a dozen. Where I live they've started making 18-egg cartons, and the local grocer was wondering what name to give them in his computerised cash register, hoping for a single word.  Even slang, informal or invented term would do if there isn't such a term already. Could you help me to point out the differences in the following sentences. for me both sentences imply same meaning.      The Restaurant , offers a broad range of locally caught fresh seafood,   prime steak and meat dishes      The Restaurant , offers a wide range of locally caught fresh seafood,   prime steak and meat dishes If $q$ is coprime to $a$ then $a\mid nq-1,\,$ so $q$ is invertible mod $a$ I think the following is true:  If $q$ is coprime to $a$ then $a\mid (nq-1)$ for some $n\in\mathbb Z$.  I have started to first reducing $n\in\mathbb Z$ to $n\in\{1,\ldots,a-1\}$ and then considering all the cases $q = ka + j$ for $j=1,2,3,\ldots,a-1$. I stopped at $j=3$ and found that this is kinda messy. Does anyone have an elegant proof for this (maybe using some results from number theory)?  [Note: question has been generalized from prime $q\nmid a$ to $\,q$ coprime to $a$] Send POST data using XMLHttpRequest What's the easiest way to use a linked list in python? In scheme, a linked list is defined simply by '(1 2 3 4 5). Python's lists, [1, 2, 3, 4, 5], and tuples, (1, 2, 3, 4, 5), are not, in fact, linked lists, and linked lists have some nice properties such as constant-time concatenation, and being able to reference separate parts of them. Make them immutable and they are really easy to work with! So my teacher gave a pretty tough (for me) problem in class today: Does the series $$ \sum_{n=1}^{\infty}\frac {\sin{\frac 1 n}} {\sqrt n} $$ converge or diverge?  So far I've thought of trying the Comparison Test and Ratio Test but couldn't really get anywhere. Intuitively I think it should converge, but how do I actually show this? This problem occurred to me when I came across a similar problem where the radii were taken over only the primes. That question was unanswered, but it seems to me infinitely many circles of radius $1/2, 1/3, 1/4...$ can fit into a unit disk. The area of all those circles would be $\pi \sum_2^\infty 1/n^2 = \pi^3/6 -\pi$, which is less than the area of the unit disk $\pi$. But can the circles actually be packed with no overlaps? Prove inequality $n<3^n$ using mathematical induction Prove that $n&lt;3^n$ where $n \in \mathbb N$,  when $ n=1 $, I have proved it's true. And assumed when $n = p$ , $p&lt;3^p$ is true. Can any body help me in showing that it is true for $n =p+1$ \colorbox height too long \documentclass{article} \usepackage[utf8]{inputenc} \usepackage{xcolor} \usepackage{minted} \usepackage{parskip}  \definecolor{codegray}{gray}{0.9} \newcommand{\code}[1]{\colorbox{codegray}{\texttt{#1}}}  \begin{document}  \code{test code}\\ anotherline  \end{document}   This is working good on sharelatex.com and looks like I expect and want it:   but on my local project using this command gives me this:   any idea what could be causing this colorbox height going to the next line? Novel about interstellar war: Both sides win, both sides think they lost Press space to continue How do I stop a bash script until a user has pressed Space?  I would like to have the question in my script     Press space to continue or CTRL+C to exit   and then the script should stop and wait until Space is pressed. When can I use a forward declaration? What is the Big-O of a nested loop, where number of iterations in the inner loop is determined by the current iteration of the outer loop? What is the Big-O time complexity of the following nested loops: for (int i = 0; i &lt; N; i++) {     for (int j = i + 1; j &lt; N; j++) {         System.out.println(&quot;i = &quot; + i + &quot; j = &quot; + j);     } }  Would it be O(N^2) still? From what I know, it is unknown whether $x_n=x_{n-1}^2 + 1$ has a closed form. Is there a recurrence relation which is known to have no closed form with a proof of inexistence?  Assuming a closed form is a non recursive description using the elementary operations of addition multiplication and power, or, assuming any other good definition of "closed form".  Edit  There are similar questions out there, but the answers are a little bit going around the question.   So is there one with proof or is it unknown? And if it depends on the definition of a closed form, then what are the (or some) options? How strong can a closed form definition be to still have a recursive relation that can be proved to not being able to have its form?  Update  Because it is a big question, and I don't want to open a new small one, I think a good representative small question about this subject is:  Is there a recurrence relation with a domain of the natural numbers, described by elementary functions, which has no closed form defined as:  A closed form of a recurrence relation with a domain of natural numbers is a function $f$ that can be described as a composition of a constant number of elementary functions without reccursion. i.e, $f(n) = f_1(f_2( \dots (f_k(n))) \dots) $ where $f_i$ is elementary for all $1 \leq i \leq k$   I think this definition is the most intuitive and basic.  For example, the relation $x_n = x_{n-1}+1$ has a closed form in this sense: $$f(n) = f_1(f_2(f_3(f_4(n))))$$  Where $f_1$ is division, $f_2(n)=(n,2)$, $f_3$ is multiplication and $f_4(n) = (n,n+1)$   Is there a recurrence relation which has no closed form in this sense? ViewExpiredException in Firefox and IE after migrating to Jboss 7.0 AS From yesterday, openweather extention on the top panel does not show the weather. Its shows "Please wait Loading".  I tried to find the solution. There was a post of creating account on openweather website and use API id. I found that it is a subscription base plan. There was a free plan which allows 60 calls free for weather data.  I thought that the extention is open source. And may be I deleted any package that is required to work the extention so I reinstall the OS. But there is no difference of doing that.    What could possibly go wrong any guidance? How do I find out what version of Linux I'm running? Is there a way to determine what version (distribution &amp; kernel version, I suppose) of Linux is running (from the command-line), that works on any Linux system? Mcrypt extension is missing in 14.04 server for mysql The php mcrypt extension on ubuntu 14.04 shows that it is installed properly, but phpmyadmin says:   The mcrypt extension is missing. Please check your PHP configuration.   My current php config is . If you need anymore information, please contact me through the comments.  $ sudo apt-get install php5-mcrypt     Reading package lists... Done     Building dependency tree            Reading state information... Done     php5-mcrypt is already the newest version.     0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.