Unexpected behavior of subtraction with approximate numbers
Dr. StrangeNumbers or: How I Learned to Stop Worrying and Love Floating Point Arithmetic
I wanted to align a set of equations to left. I found on the Internet that you can do it easily with align* environment.   I found it here:   So I did it:  \begin{align*}  f(b)-f(a)=&amp;f((b_1,b_2,\dots,b_k))-f((a_1,a_2,\dots,a_k))=\ &amp;=\sum_{i=1}^{k}[f((a_1,\dots,a_{i-1},b_i,\dots, b_k))-f((a_1,\dots,a_i,b_{i+1},\dots, b_k))  \end{align*}   Unfortunately, I got bunch of red errors saying that there are missing $s. Why? Doesn't that align* environment provide a math environment already? How to find the number of anti-symmetric relations? What's the easiest way to install a missing Perl module? I get this error:  Can't locate Foo.pm in @INC  Is there an easier way to install it than downloading, untarring, making, etc? what is the [count=\i] in \foreach for? Can someone explain what does [count=\i] in a foreach mean?  Antal S-Z's answer to  use this. It is also mentioned in , but I am not able to locate documentation on this in the TikZ-PGF manual (V2.10) What license to use while putting papers on the Arxiv? I am new to this and not sure what to make of all the options available. I am interested in putting a machine-learning-related preprint, which will potentially be submitted to places like NIPS/ICLR/ICML in the future.   arXiv.org perpetual, non-exclusive license to distribute this article (Minimal rights required by arXiv.org)  Creative Commons Attribution license (CC BY 4.0)  Creative Commons Attribution-ShareAlike license (CC BY-SA 4.0)  Creative Commons Attribution-Noncommercial-ShareAlike license (CC BY-NC-SA 4.0)  Creative Commons Public Domain Declaration (CC0 1.0) Continuously differentiable map from $\mathbb{R}^{m+n}$ to $\mathbb{R}^n$ Suppose $m,n&gt;0$, $U$ an open subset of $\mathbb{R}^{m+n}$ and let $f: U \to \mathbb{R}^n$ be continuously differentiable.  Is it possible for $f$ to be injective?    My thinking is that continuous differentiability+injectivity suggests that this is sort of getting at a converse to the Inverse Function Theorem.  We know the Jacobian is not invertible at any point in $U$, and the question is asking if this implies that $f$ cannot be invertible in any neighborhood contained in $U$. subcaption vs. subfig: Best package for referencing a subfigure Legendre's formula counts the number of positive integers less than or equal to a number $x$ which are not divisible by any of the first $a$ primes: $$\begin{align} &amp;\phi(x,a)=\lfloor x \rfloor-\sum_{p_i\le a}\left\lfloor \dfrac{ x }{(p_i)}\right\rfloor+\sum_{p_i&lt;p_j\le a}\left\lfloor\dfrac{ x}{(p_ip_j)}\right\rfloor-\sum_{p_i&lt;p_j&lt;p_k\le a}\left\lfloor \dfrac{x}{(p_ip_jp_k)}\right\rfloor+\dots \end{align}$$ then $\pi(x)=\phi\bigl(x,\pi (\sqrt{x})\bigr)+\pi (\sqrt{x})-1$ as given .  My question is, why does $\bigl\lfloor\phi(x,\lfloor\sqrt{x}\rfloor)+\sqrt{x}-1\bigr\rfloor$ also equal $\pi(x)?$ Bash/Korn shell script edited on Windows throws error '...^M: not found' Upgrade SQL Server 2000 database to 2008 R2 and enable new features I need to show that for prime numbers of the form $p=4n+1$, $x=(2n)!$ solves the congruence $x^2 \equiv-1\pmod p$. I then need to show this implies p isn't a gaussian prime.   I have started to solve this using Wilson's theorem that a number $z$ is prime iff $(z-1)!\equiv-1\pmod z$. Therefore the endpoint of my proof should be that $(p-1)!\equiv-1\pmod p$.  As $p$ is of the form $p=4n+1$, I only need to prove that $4n!$ is congruent to $-1$ modulo $p$.   Here is my working so far: Starting with the congruence $x^2 \equiv -1\pmod p$:    $$\eqalign{x^2 = -1\pmod p&amp;\implies   x^2 + 1 = kp \implies (x-i)(x+i) = kp\cr &amp;\implies ((2n)!-i)((2n!)+i))=kp \implies 4n!- 1 =kp\cr}$$   This is where I start to run out of any ideas that seem to get me anywhere. Any tips would be greatly appreciated! Does “gay” still include the meaning “merry”? Dictionary.com lists , with “merry, lively” as the first entry.  Microsoft  for listing  (a real place) as his hometown:   Xbox Live considered the term gay absolutely inappropriate in any context.     As a non-native English speaker, my question is whether in contemporary English, does gay always mean homosexual? Are British and American English the same in this respect? "Least Astonishment" and the Mutable Default Argument Is there a value that can be assigned to my time, i.e. an hourly rate, so that I can be given a receipt to remit to CRA come Income Tax Filing time?  I live in Canada. I'm currently taking Quantum Mechanics in school and I came across a table displaying the following in the given order:   $E_n (\text{energy levels}), g_n (\text{eigen functions})$   $n \ l, [m]$  $E_1, 1$   $1s, [0]$  $E_2, 4$   $2s, [0]$ $2p, [-1,0,1]$  $E_3, 9$   $3s, [0]$ $3p, [-1,0,1]$ $3d, [-2,-1,0,1,2]$    Now, as I understand $n$ defines the size of the orbital, $l$ the shape and $m$ the orientation.  My question is, how does the atoms shell model fit into this if one can have two electrons per orbital? Are $1s$ the same as the orbital $2s$, but just renamed for the sake of higher energy levels? Or do two similarly looking orbits exist for different energy levels? For example two "layers" of spherical electron clouds for an atom with $4$ electrons.  I just can't see how $2$, $8$, $18$ becomes $2$, $8$, $8$ as we were taught in Physics at high school? Am I missing something? I want to untriangulate every object in my scene with this script:  import bpy  for ob in bpy.data.objects:     bpy.ops.mesh.tris_convert_to_quads(limit=0.9, uvs=False, vcols=False, sharp=False, materials=False)   But it doesn't work in object mode, any ideas on how I can get it working? I'm trying to hide the app icon from the title bar in GNOME in Ubuntu 18.04. I know there is  asking the same thing but it's from 2011 and I can't find what it requires to find in the shell css. I'm also using the Arc-Dark custom theme.   Here are two examples of what I want to remove:      To be clear, I don't want to resize it to fit, I want to straight away remove it completely. I have built a peak detector circuit using op-amps/comparators to amplify the signal from an electret microphone, hold the peak signal and create a sharp logic edge when a peak is detected to be sent to a microcontroller.  I have built the circuit as follows setting up OA1 as a non inverting amplifier with a gain of 1000, but the voltage out from the OA1 remains fixed at about 4.5V no matter how much noise I make into the microphone. The comparator part of the circuit is outputting approximately 2V low and 5V high depending on how I set the potentiometer. All the components appear to be working.         &ndash; Schematic created using Is there an easier or more 'correct' way to open gedit as root? I have ubuntu 12.04 running in a virtual machine on virtual box.  Its existence is mainly as a 'toy' to play about with Ubuntu, so for this reason I wanted the GUI (is that what 'Dash' is in ubuntu 12.04?  Anyway...)  One thing I am currently using this machine for is a LAMP server.     I use 'gedit' to edit my files, but because I haven't correctly set permissions yet I just open it with sudo in terminal.  Just now, to try to simplify this process I created a shell script on the desktop with a very simple job - sudo gedit  I can't help thinking there must be an even more obvious way to open gedit as root from the GUI than this.  So is there?  Edit:  I just 'improved' my script by replacing sudo with gksudo.  Now I don't have to have a terminal window open while gedit is open.   So I guess this is quite convenient for now :)   It would be good, for the sake of knowledge, to know if this is anything close to how the more seasoned ubuntu users would do this type of thing. Ubuntu 16.04 multitouch gestures