Is the Collatz function $x\mapsto\dfrac{3x+1}{2^{\nu_2(3x+1)}}$ continuous on $\Bbb Z_2^\times$? Let $\Bbb Z_2^\times$ be the 2-adic units. Then e.g. $3\mapsto\dfrac{10}2=5$ Attempt Here's what I know:  The function above can be thought of as a map $x\mapsto3x+2^{\nu_2(x)}$ on all of $\Bbb Z_2\to\Bbb Z_2$ which descends to the quotient map $x\mapsto\{2^ix\in\Bbb Z_2:i\in\Bbb Z\}$. Since $x\mapsto3x+2^{\nu_2(x)}$ is continuous and the function $x\mapsto\dfrac{3x+1}{2^{\nu_2(3x+1)}}$ descends to the quotient, it must too be continuous on $\Bbb Z_2/{\sim}$ However, I think $\Bbb Z_2/{\sim}$ isn't homeomorphic to $\Bbb Z_2^\times$.  If it were, I could answer in the positive.  There's an obvious bijection between them but I think $\Bbb Z_2/{\sim}$ has the trivial topology because every element is arbitrarily close to $0$.  So the trivial continuity in $\Bbb Z_2/{\sim}$ says nothing of continuity in $\Bbb Z_2^\times$.
How do I show a function on 2-adic units is continuous (using the 2-adic metric)?   I'd be happy to learn the general rule or definition. But in particular I need to show that $f(x)=\dfrac{3x+1}{2^{v_2(3x+1)}}$ is continuous at all odd numbers.  Since the function is isometric in $\lvert\cdot\rvert_2$, i.e. since $\lvert f(x)\rvert_2=\lvert x\rvert_2$ every orbit of the function is duplicated, multiplied by any power of $2$.  This can therefore be formulated in various ways; as a function though the odd numbers, through the dyadic rationals, but the following seems good to work with.  If we let $n$ be a positive integer and $k$ be the power of $2$ we can define:  $f(2^k(2n+1))=(3(2n+1)+1)\cdot2^k\cdot\lvert3(2n+1)+1\rvert_2$  So $f(2^k(2n+1)=(3n+2)\times2^k\times\lvert3n+2\rvert_2$  Here is what I have so far:  I think to prove continuity in the 2-adic metric I need $\lvert x_n-x\rvert_2=0\implies\lvert f(x_n)-f(x)\rvert_2=0$  I think $k$ and $n$ are independent so I think I can examine them independently.  Taking $k$ to infinity brings both $x$ and $f(x)$ to zero so that seems to satisfies the continuity requirement.  Moving on to $n$; this seems to be an exercise in proving convergence within odd integers which are in a sense a subset of the 2-adic units.  I know all Cauchy sequences in these converge to 2-adic units but not much more than that.  However I do have a little insight into this particular function.  For example if we examine the inputs $x$ which map to any given output of $f(x)$, it can fairly easily be shown that these $x$'s take the form of a set $\left\{4^mp+\dfrac{4^m-1}{3}:m,n\in\mathbb{N}\right\}$ so, at least for any given output $f(x)$ the inputs always converge to $x=\frac{-1}{3}$ as the intervals between them become large powers of $2$.  In fact the orbit of the function $a(2^kx)=2^k(4x+1)$ on variation of $x$ and holding $k$ fixed is in a sense orthogonal to $f(x)$; which is a restatement of the above except not in closed form.
This might be a stupid question, but I just don't get it. In Hamiltonian mechanics when examining conditions for a $(\boldsymbol{q},\boldsymbol{p})\rightarrow(\boldsymbol{Q},\boldsymbol{P})$ transformation to be canonical one starts with $$ \dot{q}_ip^i-H(\boldsymbol{q},\boldsymbol{p},t)= \dot{Q}_iP^i-\bar{H}(\boldsymbol{Q},\boldsymbol{P},t)+\frac{d}{dt}W(\boldsymbol{q},\boldsymbol{Q},t)$$ where $\bar{H}$ is the transformed Hamiltonian, and $W$ is the generating function (now a function of $\boldsymbol{q}$ and $\boldsymbol{Q}$). This term shouldn't break Hamilton's principle, since $$ \delta\int_{t_1}^{t_2} dt\frac{d}{dt}W(\boldsymbol{q},\boldsymbol{Q},t)=\delta W(\boldsymbol{q},\boldsymbol{Q},t)|_{t_2}-\delta W(\boldsymbol{q},\boldsymbol{Q},t)|_{t_1}=0-0=0 .$$ But I don't see why the variation of $W$ should disappear at the endpoints (say at $t_1$). Expanding leads to: $$ \delta W(\boldsymbol{q},\boldsymbol{Q},t)|_{t_1}=\left(\frac{\partial W}{\partial q_i}\right)_{t_1}\underbrace{\delta q_i(t_1)}_{=0}+ \left(\frac{\partial W}{\partial Q_i}\right)_{t_1}\delta Q_i(t_1)=\left(\frac{\partial W}{\partial Q_i}\right)_{t_1}\delta Q_i(t_1).$$ $\boldsymbol{Q}$ is itself a function of $\boldsymbol{q}$ and $\boldsymbol{p}$, so $$ \delta Q_i(t_1)=\left(\frac{\partial Q_i}{\partial q_k}\right)_{t_1}\underbrace{\delta q_k(t_1)}_{=0}+\left(\frac{\partial Q_i}{\partial p_k}\right)_{t_1}\delta p_k(t_1)=\left(\frac{\partial Q_i}{\partial p_k}\right)_{t_1}\delta p_k(t_1). $$ It seems as if we also needed the variation of $\boldsymbol{p}$ to vanish at the endpoints, and I don't get this because (at least in cartesian coordinates) $\boldsymbol{p}=m\dot{\boldsymbol{q}}$ and the velocity can be different along the original and the varied orbitals even at the endpoints (they can point in totally different directions), so in general $\delta \dot{\boldsymbol{q}}(t_1)\neq 0$. What am I doing wrong? Can someone help me with this, please? Why do eukaryotic organisms have introns in their DNA? We touched on introns and exons in my bio class, but unfortunately we didn't really talk about why Eukaryotes have introns. It would seem they would have to have some purpose since prokaryotes do not have them and they evolved first chronologically, but I could easily be wrong. Did the junk sections of DNA just evolve there by some sort of randomness or necessity as opposed to an actual evolutionary advantage? Why hasn't evolution stopped us from having introns since they seem to be a 'waste' of time and DNA? Why do prokaryotes not have introns? Explicitly accept change for PPA 'Label' When I run apt-get update on my ubuntu 17.10 I receive the following error:  E: Repository 'http://ppa.launchpad.net/ondrej/php/ubuntu artful InRelease' changed its 'Label' value from '***** The main PPA for PHP (5.6, 7.0, 7.1) with many PECL extensions *****' to '***** The main PPA for supported PHP versions with many PECL extensions *****' N: This must be accepted explicitly before updates for this repository can be applied. See apt-secure(8) manpage for details."   The man page doesn't really point to a solution.  My question is, how do I explicitly accept the change? Sort thebibliography by citation order Inverse of a diagonal matrix plus a constant I am looking for an efficient solution for inverting a matrix of the following form:  $$D+aP$$   where $D$ is a (full-rank) diagonal matrix, $a$ is a constant, and $P$ is an all-ones matrix.   gives a solution to the special case where all diagonal entries of $D$ are the same.  The  formula is also capable of providing a solution to this problem by setting appropriate $u$ and $v$, but it loses efficiency ($O(n^3)$ time complexity to compute) at high dimension. I am hoping to get a result in the same form so the space and time complexity are both $O(n)$. AlwaysOn AG: Current Host Server - Owner Node - Primary Instance I'm building up a SQL Server AlwaysOn AG and I'm on the step "failover", manual and automatic:  I noticed that in the Windows Server Failover Cluster a "Current Host Server" exists. My knowledge says that this is the "master" of the cluster that the cluster hosts. But now I noticed that the "Owner Node", which hosts the Availability Group(-role) a other node could be. (The owner node is also the primary instance of the AG, right?)  Now for me there exist two questions for the normal operating of an AlwaysOn AG:  Q1: Should the "Current Host Server" the same as the Owner Node of the AlwaysOn AG?  Q2: Imagine if I want to maintain a host. Shall I first failover the AG (via SSMS or with ALTER AVAILABILITY GROUP...) to the second node/another node and at second a failover of the WSFC-Current Host?  Maybe someone could help me with this questions and give me some advice.  Thanks Installing "firmware-b43-installer" offline I was given a laptop with a totally dead display. I do not want to pay to replace the broken display, but I would like to use the laptop as a server and connect to it via SSH to control it.  I have pulled out the internal hard drive from the broken laptop, and I want to install Debian GNU/Linux and the SSH server onto this HDD using a working Debian laptop.  However, the hardware in the two laptops is very different, and I am wondering what kind of problems I would run into when I plugged the drive back in to the laptop with the broken display because of this?  How would I make this work? What is the proper way for me to install a bootable copy of Linux onto a HDD that will be used by a computer with different hardware? Is there a way to draw a sort of connection between subplots in TiKz/pgfplot, like in this ? I'm looking to achieve something like this:    But I would need to draw the "x 2000" line between the plots.  EDIT: So right now I have something like this:    Coming from my main.tex:  \input{aipcheck}  \documentclass[final]{aipproc}  \layoutstyle{8x11single} \usepackage[utf8]{inputenc} \usepackage{algorithm2e} \usepackage{color} \usepackage[usenames,dvipsnames]{xcolor} \usepackage{cleveref} \usepackage{pgfplots}  \pgfplotsset{ tick label style={font=\tiny}, %axis numbers label style={font=\small}, %axis labels legend style={font=\tiny} }  \begin{document}  (...)  \begin{figure}     \centering     \newlength\figureheight      \newlength\figurewidth      \setlength\figureheight{5cm}      \setlength\figurewidth{5cm}     \input{figs/_ddmp2d.tikz} \label{fig2D} \end{figure}   with _ddmp2d.tikz, originally created from matlab2tikz, but already modified, containing:  % This file was created by matlab2tikz v0.3.3. % Copyright (c) 2008--2013, Nico Schlömer &lt;nico.schloemer@gmail.com&gt; % All rights reserved. %  % The latest updates can be retrieved from %   http://www.mathworks.com/matlabcentral/fileexchange/22022-matlab2tikz % where you can also make suggestions and rate matlab2tikz. %  %  %   \begin{tikzpicture}  \begin{axis}[% width=\figurewidth, height=\figureheight, scale only axis, xmin=0, xmax=15, xlabel={Number of iterations}, ymin=0, ymax=16000000000000, name=plot1, after end axis/.append code={\coordinate (a) at (axis description cs:1,0);}]  \addplot [ thick, color=red, dash pattern=on 1pt off 3pt on 3pt off 3pt, forget plot ] table[row sep=crcr]{ 1 15841000000000\\ (...) 15 25160000000\\ }; \addplot [ thick, color=black, dashed, forget plot ] table[row sep=crcr]{ 1 15841000000000\\ (...) 15 52501000000\\ }; \addplot [ thick, color=blue, dotted, forget plot ] table[row sep=crcr]{ 1 15841000000000\\ (...) 15 5324200000\\ }; \addplot [ thick, color=green, solid, forget plot ] table[row sep=crcr]{ 1 15841000000000\\ (...) 15 551920000\\ }; \addplot [ thick, color=green, solid, forget plot ] table[row sep=crcr]{ 3 0\\ 3 16000000000000\\ };  \addplot [ thick, color=blue, dotted, forget plot ] table[row sep=crcr]{ 4 0\\ 4 16000000000000\\ }; \addplot [ thick, color=black, dashed, forget plot ] table[row sep=crcr]{ 5 0\\ 5 16000000000000\\ }; \addplot [ thick, color=red, dash pattern=on 1pt off 3pt on 3pt off 3pt, forget plot ] table[row sep=crcr]{ 8 0\\ 8 16000000000000\\ }; \end{axis}  \begin{scope}[xshift=5cm] \begin{axis}[% width=0.35\figurewidth, height=\figureheight, scale only axis, xmin=995, xmax=1000, xtick={997,998,999,1000}, xticklabels={$997$,$998$,$999$,$1000$}, ymin=0, ymax=8000000000, at=(plot1.right of south east), anchor=left of south west, yticklabel pos=right, axis x discontinuity=crunch, after end axis/.append code={\coordinate (b) at (axis description cs:0,1);} ] \addplot [ thick, color=red, dash pattern=on 1pt off 3pt on 3pt off 3pt ] table[row sep=crcr]{ 995 6093100000\\ 996 6093100000\\ 997 6093100000\\ 998 6093100000\\ 999 6093100000\\ 1000 6093100000\\ };  \addplot [ thick, color=black, dashed ] table[row sep=crcr]{ 995 1577100000\\ 996 1598000000\\ 997 1598500000\\ 998 1628600000\\ 999 1631000000\\ 1000 1595900000\\ };  \addplot [ thick, color=blue, dotted ] table[row sep=crcr]{ 995 2592000000\\ 996 2591900000\\ 997 2591700000\\ 998 2591600000\\ 999 2591500000\\ 1000 2591400000\\ };  \addplot [ thick, color=green, solid ] table[row sep=crcr]{ 995 391810000\\ 996 391810000\\ 997 391810000\\ 998 391810000\\ 999 391810000\\ 1000 391810000\\ };  \end{axis} \draw[gray] (a) -- (b) node[pos=0.5,sloped,fill=white] {$\times$2000}; \end{scope}  \end{tikzpicture}% How to remove edit from edit queue? I asked a question on SO:  and got an answer that, while good, I thought could be improved with a minor edit:   After a little while, a comment on the answer showed me that I was wrong to have edited the answer.  How do I undo it? Puppet client not connection to master - SSL connect Error I installed the puppet master and agent on the same machine. When the client is started, I got the following error message.  puppet agent --server=agent.com --no-daemonize --debug    err: Could not retrieve catalog from remote server: SSL_connect returned=1 errno=0 state=SSLv3 read server certificate B: certificate verify failed.  This is often because the time is out of sync on the server or client Are the contents of a loot box determined at the time it is awarded or when it is opened?  For example, if you collect multiple loot boxes before a special event but opened them during the special event, would they have the potential of spawning rewards from that event? How to style text in hyperref \url? I would like to install Office 2015/365 on Ubuntu 15.04. I tried to install Office using Wine, but that did not work. Please note that I do not want virtualisation unless it has a -like behaviour, where application windows are treated like regular Ubuntu application windows by the system. "a while" vs "awhile" Is there a difference between "awhile" and "a while"? If there is, what is it? I've been wondering this for awhile, but now I actually need the answer. Why is $a^n - b^n$ divisible by $a-b$? Recover data from SD card I have a 2&nbsp;GB Kingston  card which is about three years old. I put it in a reader today in my Windows Vista computer, wrote a 32&nbsp;MB file onto it, safely removed it, and then tried to read it elsewhere. Nothing. Putting it back in the Windows Vista computer it now says:     You need to format the disk in drive F: before you can use it.   What should I do? I have access to many computers and OSes if your recommendations need that. I would be very sad if I lost all the contents of the card. Most of the data is backed up, but there are a few things that aren't. :(  Doing a   # dd if=/dev/sdg of=~/tmp/sd.bin   gives me a 2&nbsp;GB file, and  the file it seems like lots of my data is still there. How can I put it back together? Why is the node editor showing different kinds of nodes? Regex for parsing directory and filename I'm trying to write a regex that will parse out the directory and filename of a fully qualified path using matching groups.  so...  /var/log/xyz/10032008.log   would recognize group 1 to be "/var/log/xyz" and group 2 to be "10032008.log"  Seems simple but I can't get the matching groups to work for the life of me.  NOTE:  As pointed out by some of the respondents this is probably not a good use of regular expressions.  Generally I'd prefer to use the file API of the language I was using.  What I'm actually trying to do is a little more complicated than this but would have been much more difficult to explain, so I chose a domain that everyone would be familiar with in order to most succinctly describe the root problem. Given $m$ rows with $n$ variables where $m &lt; n$, and given that there exists a combination of some of the rows that gives a zero vector. Is it promised when computing the reduced-row echelon form, with Gaussian elimination to find a zero vector?  Update: I clearly don't know, how to pick up the right words to ask my question. So let me give you the full context and please help me say it right, so the question will make sense. I am solving step 3 in the quadratic sieve algorithm, here is a  description of it.  Suppose the composite number N is being factored. Bound $B$ is chosen, and the factor base is identified (which is called $P$), the set of all primes less than or equal to $B$. Next, positive integers $z$ are sought such that $z_2$ mod $N$ is B-smooth. Therefore it can be written, for suitable exponents $a_i$, $${\displaystyle z^{2}{\text{ mod }}N=\prod _{p_{i}\in &gt; P}p_{i}^{a_{i}}}$$ When enough of these relations have been generated (it is generally sufficient that the number of relations be a few more than the size of P), the methods of linear algebra can be used (for example, Gaussian elimination) to multiply together these various relations in such a way that the exponents of the primes on the right-hand side are all even: $${\displaystyle {z_{1}^{2}z_{2}^{2}\cdots z_{k}^{2}\equiv \prod &gt; _{p_{i}\in P}p_{i}^{a_{i,1}+a_{i,2}+\cdots +a_{i,k}}\ {\pmod {N}}\quad ({\text{where }}a_{i,1}+a_{i,2}+\cdots +a_{i,k}\equiv 0{\pmod &gt; {2}})}}$$  Now, let's say I found $m$ relations and the number of primes in $P$ is $n$, such that $m&lt;n$. It is given that there exists a combination of some of the $m$ relations that their product is a square number. My question is: When I perform the Gaussian elimination over F2 to find a null space combination, is it promised that I will find one? I successfully implemented the algorithm and the matrix part of my implementation works. But what I see is that if I found a null space(a combination of making a null space) and latter I add more relations(still $m &lt; n$), I lose the null space solution and not able to solve the matrix again. Each time I add a new relation I start solving it from the beginning, I don't remember any previous state. So what I want to know, could it be that there is a limitation with the Gaussian elimination algorithm, that it does not promise to find the null space if $m &lt; n$?