If $p,q,r,s$ are integers such that $ps - qr = \pm 1$, and $a,b,a',b' $ are integers such that $a' = pa +qb, b' = ra + sb$, prove that $(a, b) = (a', b')$. Here, the particular value of $ps - qr = \pm 1$ enables us to get integer values. Can there be an algebraic or intuitive explanation for that? Have two explanations. My explanation is given for both sequentially. The first explanation is offered as a geometrical / intuitive one. While the second one uses number theoretic approach, and linear system of equalities. Request vetting &amp; improvement for the first one First, needs linear algebra, and orthogonal matrices. $$\begin{bmatrix}         a'\\         b' \\         \end{bmatrix} =  \begin{bmatrix}         p &amp; q\\         r &amp; s \\         \end{bmatrix} \begin{bmatrix}         a\\         b\\         \end{bmatrix} $$ The condition $ps - qr = \pm 1$, states two things about matrix $$\begin{bmatrix}         p &amp; q\\         r &amp; s \\         \end{bmatrix} $$ #1. determinant is non-zero; hence invertible matrix. #2. Also, is an orthogonal matrix. A function is invertible iff it is bijective. A matrix is only a convenient representation of linear mapping function between two sets, wrt a basis of the domain and co-domain of the function. #1: So, for linear maps, the determinant of corresponding matrix as non-zero means the mapping is possible in both directions without collapsing of any dimension (row/column). #2: the area (min. possible contained entity, as order $2$ is the minimum dimension; similarly, for order $3$ matrix, it is volume) is unit as for orthogonal matrix of order $n$ 'that' is given by: $I^n = 1$. Hence, $(a,b) = (a', b')$. Think that need some geometrical insight in orthogonal matrices. Edit ::  in case of orthogonal matrices, they practically arise in rotation or reflection, where the determinant is $\pm 1$. Can this explanation be modified to take into account that?  Second explanation is equivalent, though needs to ignore the matrix representation. But, needs the linear system of equalities to use the Bezout's lemma. This is a number theoretic approach. $$a' = pa + qb$$ $$b' = ra + sb$$ any common divisor of $a, b$ will divide both $a', b'$, so $(a,b) ≤ (a',b')$. Solving for $a, b$ in terms of $a', b'$ gives $$a = (s/(ps-qr))a' -(q/(ps-qr))b'$$ $$b = (-r/(ps-qr))a' +(p/(ps-qr))b'$$ Since $ps-qr =\pm1$, all the coefficients in this system are integers, so $(a',b')$ is a common divisor of $a$ and of $b$, therefore, $(a',b') \le (a,b)$.
I'm having problems with an exercise from Apostol's Introduction to Analytic Number Theory.     Given $x$ and $y$, let $m=ax+by$, $n=cx+dy$, where $ad-bc= \pm 1$. Prove that $(m,n)=(x,y)$.   I've tried to give a proof, but I suspect it's wrong (or at least not very good). I would be very thankful for any hints/help/advice!  My proof:     We observe that since $ad-bc= \pm 1$, $ad=bc \pm 1$, and $(ad,bc)=1$.   Now, $(a,b) \mid m$ and $(c,d) \mid n$ but      $$(a,b) = ((d,1)a,(c,1)b)=(ad,a,bc,b)=(ad,bc,a,b)=((ad,bc),a,b)=(1,a,b)=1.$$      Similarly, we determine $(c,d)=1$. So, $1=(a,b) \mid m$ and $1=(c,d) &gt; \mid n$. But $(x,y)$ also divide $m$ and $n$. Since $(x,y) \geq (a,b)=(c,d)=1$, this implies that $(x,y)=m,n$. Hence   $(m,n)=((x,y),(x,y))=(x,y)$.
How to compute the series: $\sum_{n=0}^{\infty} (-1)^{n-1}\binom{1/2}{n}$ I've only played one placement match so far and I won it because my opponents were less than stellar.  I'm afraid that if I get similar placement matches in the future and end up winning them all I might get put into a league beyond my skill.  How many different leagues are there?  Once placed in a league, is there any way to move to a different one?  How? Remove paint from the head tube I'm planning to build a new bike from scratch.  Part of that is buying a new frame, and possibly facing/reaming it. That's actually the part I'm scared of the most.  Fortunately, the brand I'm targeting, Surly, seems to do that in the factory. But as they state, there may be some paint left where it should not be:     Since Surly frames are faced prior to painting, it is possible to   shave the paint off the head tube and BB shell ends with a carpet   knife blade.  This usually results in an acceptable result, saving you   money, but we still recommend having it done the proper way to ensure   longest life from your components.   So what is the "proper way" to remove the extra paint, if the frame is properly reamed/faced already?  Should I choose not to do it the proper way, which components are going to suffer? I suppose at least the headset and bottom bracket, but could the frame suffer as well? When I install Selenium I get the following error:  Shubham@Shubham-To-be-filled-by-O-E-M:~$ sudo apt-get update     Get:1 http://security.ubuntu.com/ubuntu xenial-security InRelease [107 kB]     Hit:2 https://repo.skype.com/deb stable InRelease                            Hit:3 http://in.archive.ubuntu.com/ubuntu xenial InRelease                          Get:4 http://in.archive.ubuntu.com/ubuntu xenial-updates InRelease [109 kB]     Get:5 http://in.archive.ubuntu.com/ubuntu xenial-backports InRelease [107 kB]       Fetched 323 kB in 8s (38.6 kB/s)                                                    Reading package lists... Done     Shubham@Shubham-To-be-filled-by-O-E-M:~$ sudo pip install selenium     Traceback (most recent call last):       File "/usr/bin/pip", line 9, in &lt;module&gt;         from pip import main     ImportError: cannot import name main   How should I proceed? Resetting Ubuntu to factory settings How to find out what init mode system is using? I was looking at this and I'm not sure how to find out which startup service is used on my machine (Ubuntu 14.04 on Digital Ocean). Can it use all of them? (upstart, system v, systemd) Use chown to set the ownership of all a folder's subfolders and files? What is the first reference to an internet of computers in science fiction? I showed up here to write a perfectly normal rant about misaligned pixels, and there's a big fat checkbox cluttering up the "Ask" page:    WHAT THE EEEEK IS THIS NEW ABOMINATION? Find $\,\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$ Basic desktop actions are not available on Ubuntu 20.04 In Ubuntu 20.04, on the desktop, there is a number of actions I wish I could do, such as, by order of importance :  use keyboard shortcuts such as select all, copy / paste. use drag &amp; drop with icons deactivate the &quot;mouse over&quot; highlight, and restrict the clicking area to only the icon, not the surrounding square  To put it simply, I'd like to have the exact same features as offered by the desktop of Ubuntu 18.04 (probably related to a specific version of GNOME, and not Ubuntu itself). Am I missing something obvious here? I actually see no reason why those actions are removed or deactivated on the desktop, since they all work in any Nautilus window. Does deleting iCloud photos affect them on devices? If I delete a photo from my iCloud will it also be removed from my devices that are connected to my account? What happens with the force of gravity when the distance between two objects is 0? so I had my first approximation to the gravity equation  \begin{equation} F=GmM/r^2 \end{equation} and some questions arose that my teacher couldnt respond: if r approximates to 0 with mM being constant, then the force should tend to infinity, right? so that being true, then if we would travel to the center of the earth, we would be crushed by gravity? if so, how can there be anything right in the center of the earth? wouldnt anything there be compressed and destroyed by the infinite force?  having that in mind, my next question is as gravity force is measured from the center of an object, any part of the same object which is not at the center would be affected by the same object's gravity? if so, how can we differentiate between two objects that are not in a vacuum? my question originated thinking in us standing on the earth, in contact with it, are we, in a gravitational sense, one object? this lead to the last question, if gravity is a property of matter, then how applying the same logic as before, how does it work between two neutrons?(chose neutrons because i assumed it would be more difficult obtaining gravitational force if there was electrostatic forces from charges inbetween) do two newtrons in contact count as 1 gravitational object? and how thats its own gravity affect the newtron from center and out? Limit of sequence in which each term is defined by the average of preceding two terms We have a sequence of numbers $x_n$ determined by the equality   $$x_n = \frac{x_{n-1} + x_{n-2}}{2}$$  The first and zeroth term are $x_1$ and $x_0$.The following limit must be expressed in terms of $x_0$ and $x_1$ $$\lim_{n\rightarrow\infty} x_n $$  The options are:   A)$\frac{x_0 + 2x_1}{3}$ B)$\frac{2x_0 + 2x_1}{3}$  C)$\frac{2x_0 + 3x_1}{3}$ D)$\frac{2x_0 - 3x_1}{3}$  Since it was a multiple choice exam I plugged $x_0=1$ and $x_1=1$. Which means that all terms of this sequence is $1$,i.e, $$x_n=1, n\in \mathbb{N} $$  From this I concluded that option A was correct.I could not find any way to solve this one hence I resorted to this trick. What is the actual method to find the sequence's limit? How to find $\lim\limits_{x\to0}\frac{e^x-1-x}{x^2}$ without using l'Hopital's rule nor any series expansion? How do I lock compiled Java classes to prevent decompilation?  I know this must be very well discussed topic on the Internet, but I could not come to any conclusion after referring them.  Many people do suggest obfuscator, but they just do renaming of classes, methods, and fields with tough-to-remember character sequences but what about sensitive constant values?  For example, you have developed the encryption and decryption component based on a password based encryption technique. Now in this case, any average Java person can use  to decompile the class file and easily retrieve the password value (defined as constant) as well as  and in turn can decrypt the data by writing small independent program!  Or should such sensitive components be built in native code (for example, VC++) and call them via ? At what stage should acknowledgements be inserted into the paper if review is not double-blind? When the review process is double-blind, I guess (although I have not read this anywhere) that acknowledgements should not be written in the submitted paper, as this might break anonimity. They should only be inserted into the camera-ready version, after the paper is accepted.  What is the custom when the review process is not double-blind? Should I write the acknowledgements in the submitted version, or wait to the camera-ready version? Who's at fault with this error concerning csquotes and LuaLaTeX? I ran into this with the package lilyglyphs but I'm not sure whose fault this error is.  When I compile the following with lualatex:  \documentclass{article} \usepackage{csquotes} \MakeOuterQuote{"} \EnableQuotes \begin{document} \directlua{print("hello")} \end{document}   I get an error saying   \directlua]:1: unexpected symbol near '\'. l.6 \directlua{print("hello")}    Now suppose that I do not directly write \directlua in my document, but I include a package that does it internally. Such as   \documentclass{article} \usepackage{fontspec, csquotes, lilyglyphs} \MakeOuterQuote{"} \EnableQuotes \begin{document} \wholeNote \end{document}   I have no knowledge why compilation fails, because the error now becomes   ! Missing number, treated as zero. &lt;to be read again&gt; \begingroup l.7 \end       {document}    From my user point of view I did nothing wrong, because I'm using the public API provided by the packages. The csquotes documentation specifically says:     Note that all characters are automatically checked for validity as they are allocated. This package will reject characters which are unsuitable as active quotes... In sum, the following   characters will be considered as reserved by this package:   A–Z a–z 0–9 . , ; : ! ? ' - # $ % &amp; ^ _ ` ~ \ @ * { } [ ]    So my question is whose fault is this error?   My fault, the user's fault, because I should have known (but how?) that even though the quote is not listed as unsuitable in the csquotes manual, it actually is? The fault of the implementation or documentation of csquotes for failing to mention " as unsuitable? The fault of the implementation of the \directlua command because it doesn't prevent LaTeX catcode changes/active characters from impacting its argument? A shortcoming of TeX itself because of its global state based architecture? How do I get past the "do not enter" traffic sign that stops me from installing "Sierra" instead of "High Sierra"?  I booted into rescue mode and deleted the original "High Sierra" installation, but booting from the USB stick with the "Sierra" installer on it still gives me the "no". How to use notify-send to notify users as root? #!/bin/bash export DISPLAY=:0 state=$(upower -i $(upower -e | grep BAT) | grep --color=never -E "state" ) stat=${state:25:32} batperct=$(upower -i $(upower -e | grep '/battery') | grep  -E "percentage:") bat=${batperct:24:26} intbat=$(echo $bat | cut -d'%' -f1) if [ "$intbat" -lt "15" -a "$stat" == "discharging" ]; then     notify-send 'battery low..! please plugin charger..... charge is only' "$intbat"     fi     The script runs fine when run as a normal user (bash script.sh). The desktop notification is displayed.  If I run it as root with sudo bash script.sh, the desktop notification is not displayed.  How can I use notify-send to notify users as root?  I am using Arch Linux.