\begin{equation} \begin{aligned} \frac{\partial f_j}{\partial x_j}(\textbf{y},t-\frac{\textbf{|x|}}{c_0} + \frac{\textbf{x} \cdot \textbf{y}}{c_0 |\textbf{x}|}) &amp; = \frac{\partial f_j}{\partial t}(\textbf{y},t-\frac{\textbf{|x|}}{c_0} + \frac{\textbf{x} \cdot \textbf{y}}{c_0 |\textbf{x}|})\frac{\partial}{\partial x_j}(\textbf{y},t-\frac{\textbf{|x|}}{c_0} + \frac{\textbf{x} \cdot \textbf{y}}{c_0 |\textbf{x}|}) \\                &amp; = \frac{\partial f_j}{\partial t}(\textbf{y},t-\frac{\textbf{|x|}}{c_0} + \frac{\textbf{x} \cdot \textbf{y}}{c_0 |\textbf{x}|}) (-\frac{x_j}{c_0 |\textbf{x}|} + \frac{y_j}{c_0 |\textbf{x}|} - \frac{\textbf{x} \cdot \textbf{y}}{c_0|\textbf{x}|} )\\                 \label{3_7}                &amp; \approx -\frac{x_j}{c_0 |\textbf{x}|} \frac{\partial f_j}{\partial t} (\textbf{y},t-\frac{|x|}{c_0} + \frac{\textbf{x} \cdot \textbf{y}}{c_0 |\textbf{x}|}) , \ \ \ \  |\textbf{x}| \rightarrow \infty.                  \end{aligned} \end{equation}   I'd like the reference (3.7) to be located at the end of last equation. Could someone help? I have tried align instead of aligned but it doesn't work in amsmath
I have an equation like this:     \begin{align}       P(A \cup B) = \sum_{\omega \in A \cup B} P(\omega) = \sum_{\omega \in A} P(\omega) + \sum_{\omega \in B} P(\omega)       = P(A) + P(B)     \end{align}   For some reason, LaTeX is automatically labeling the equation. How do I turn that off? Secondly how do I write something in LaTeX on this site. I tried enclosing in '$' but it doesn't work.
One can show that $\mathbb{P}^n\times \mathbb{P}^m$ is birational to $\mathbb{P}^{n+m}$ by making note of the Zariski topologies and the canonical isomorphism between affine spaces $\mathbb{A}^n\times \mathbb{A}^m\to \mathbb{A}^{n+m}$.   Question: Does it follow that there is isomorphism $\mathbb{P}^n\times \mathbb{P}^m\to \mathbb{P}^{n+m}$? If so, what is an example of such an isomorphism?   Unfortunately I am having a hard time coming up with one, and haven't seen too many examples thus far. Adjoint pairs, triplets and quadruplets Often we have adjoint pairs $(A, B)$ (meaning $A$ is left adjoint to $B$). Sometimes we have adjoint triplets $(A,B,C)$ (meaning $A$ is left adjoint to $B$ and $B$ is left adjoint to $C$. No adjoint relation between $A$ and $C$ obviously, since they have the same source and target).  So the first question is: can we have quadruplets $(A,B,C,D)$ ? (meaning that $C$ is also left adjoint to $D$).  Second question: It would be then possible that we have $(A,D)$ besides $(A,B)$. But I think not, because I think that it is not possible to have a functor $A$ with two different right adjoints, $B$ and $D$. Is this correct? Objects not showing in final render When I switch to render mode, they show up, and they show when pressing F12, but when clicking the 'Animation' button, nothing shows. If anyone knows how to help, please do.  Edit: I have enabled rendering using the cameras, still nothing.  Edit #2: What the, some how transferring all of the objects from layer 1-2 does the trick. I guess my camera was set up for the second layer even though it was physically in the 1st. Anyone like to explain?  For anyone who's really curious, :  Also, it may seem like having the animation length of 1 frame seem bad, but I've done that for other files and they work fine. What is the difference between `grep`, `egrep`, and `fgrep`? Can anyone tell me the technical differences between grep, egrep, and fgrep and provide suitable examples?  When do I need to use grep over egrep and vice versa? What is the difference between "Have you seen this?" and "Did you see this?" What's the difference between these two phrases? If a character who was under the effect of a growth potion (double their height and eight times their weight) and weighed 2,400 lbs. was able to use Dimension Door to teleport 400 feet or more into the air, directly above a Huge-sized monster, and fell on it, how much damage would the falling creature - and the monster - take?  I would assume both would take the 20d6 max for the falling over 200 feet, but is there a estimation on additional damage for the falling creature's size?  According to the splat calculator, at 500 feet you'd be falling at 196 km/hr, and it would expend 1.6 million joules of energy, the equivalent of over 3,000 mid-sized cars hitting an object at 60 km/hr.  A 2,400-lb. barbarian falling 500 feet is like dropping a mid-sized Toyota Corolla off a 50 story building onto a monster - it should do some damage. How do I work with the Card Layout in the NetBeans GUI builder? Proof that Epicycloids are Algebraic Curves? Epicycloids are most commonly described by the parametric equations,   $x(t) = (R + a)\cos(t) – a \cos \left(\frac{R + a}{a} t \right),$  $y(t) = (R + a)\sin(t) – a \sin \left(\frac{R + a}{a} t \right).$  Where $R$ is the radius of the fixed circle and $a$ is the radius of the rolling circle.   With $R = ka$ we also have,   $x(t) = a[(k + 1)\cos(t) - \cos((k + 1)t)],$  $y(t) = a[(k + 1)\sin(t) - \sin((k + 1)t)].$  Several books discussing epicycloids mention that if the ratio of the radii of the circles $\left( \frac{R}{a} = k \right)$ is rational, then they are algebraic curves. However, I’ve only been able to find the Cartesian equations for the cardioid, nephroid and ranunculoid. With the cardioid being,   $(ax + x^2 + y^2)^2 = a^2(x^2 + y^2).$   The nephroid,   $(-4a^2 + x^2 + y^2)^3 = 108a^4y^2.$  And the ranunculoid,   $-52521875a^{12} – 1286250a^{10} (x^2 + y^2) – 32025a^8 (x^2 + y^2)^2 + 93312a^7 (x^5 – 10x^3y^2 + 5xy^4) – 812a^6 (x^2 + y^2)^3 – 21a^4 (x^2 + y^2)^4 – 42a^2(x^2 + y^2)^5 + (x^2 + y^2)^6 = 0.$  Clearly this doesn’t cover all epicycloids where $\frac{R}{a}$ is rational.   What is the proof that shows that epicycloids, where the ratio of the radii are rational, are algebraic curves? Can a Sorcerer protect their friend from Web by using Careful Spell? With the metamagic "Careful Spell" does the caster protect his friend from a Web spell for its full duration or just for the first save? If the friend stays in the area or re-enters the area does he continue to automatically save against the restraining effect? Are you allowed to use numbers as table names in MySQL? I'm thinking of having a program that dynamically creates new tables as the need arises. Can I have tables named with just numbers in MySQL? "Who wants ice-cream?" — Should I say "(not) I" or "(not) me"? I've recently updated to Ubuntu 20.04. When English is not the chosen language all shortcuts don't work (such as Ctrl + C, Ctrl + A, etc). It was working fine in the previous versions. What happened and What can I do to fix it? How to read value of variable, where the name of variable is the value of another variable Reading order for Cosmere series by Brandon Sanderson My entry into Brandon Sanderson's works is Mistborn Trilogy, I finished and loved it. Then I got to know that many of his series are connected and happen in the same universe - Cosmere. Next I started with the Way of Kings, but am now kind of worried if I might read the series in the wrong order. So what is the right order to read them? is it the publication order..?   Elantris (2005)  Mistborn Trilogy Warbreaker (2009) The Stormlight Archive: The Way of Kings (2010) Mistborn: The Alloy of Law (2011) The Emperor's Soul (2012, novella) Shadows for Silence in the Forests of Hell (2013, short story published in Dangerous Women) The Stormlight Archive: Words of Radiance (2014)    or I have seen people suggest the following order too,   Mistborn Trilogy Elantris and the Emperor's Soul The Way of Kings Warbreaker Words of Radiance The Alloy of Law Shadows of Self   I have also read posts where they say it doesn't matter which order you read it in. So am confused, since I have already started the Way Of Kings (Only a few chapters in), should I stop it and go back to Elantris, and follow an order, or should I just continue finish this and pick other books in an order. I've been modeling an F-15, but when I rotate the flaps on Y-Axis the Flaps offset from the wing and they look like they are about to fall out, so I end up correcting it by rotating in the X and Z-axis to align them correctly.  However when I rotate them back from the Y-Axis they look distorted again because I did rotations on the X and Z axis. I tried riding it, no help. I saw something about custom orientation to fix it but that was not clearly explained and it was done on an ideal plane that was (0,0,0); mine is pre-rotated on all axis.   I'll take scripts if needed and I'm using Blender 2.8.  When I rotate it on the Y-Axis:    Fixing it using X and Z rotation:    Rotating it back on the Y distorted it again:    Here's the wing with ailerons and flaps: If we pick randomly two points inside a circle centred at $O$ with radius $R$, and draw two circles centred at the two points with radius equal to the distance between them, what is the expected area of the intersection of the two cirlces that contain the origin $O$. Where to locate "sockets" I am new to this, so sorry if its obvious. I am running a Debian server and installing the likes of UWSGI, NGinx etc on there. The configurations keep talking about pointing to "sockets".  In the build options I seem to be able to specify where the sockets for each program go. By default it looks like most of them go in /tmp (not all of them).  Is this a good place for them to go? I'm trying to keep things as organized as possible but just bunging them in /tmp doesn't seem like the best option. So, I understand that by using 'tee' I can redirect the output of a command such as 'ping' to stdout as well as a file.  For example:  &gt; ping google.com | tee somefile   This would display the ping statistics on terminal and write them to 'somefile'.  Now, if I wish to modify the output from 'ping', I can use 'cut' in such a way:  &gt; ping google.com | cut -d' ' -f 1   But, if I wish to use all the three commands together, I get a no output on stdout and an empty file.  &gt; ping google.com | cut -d' ' -f 1 | tee somefile   What am I doing wrong? Is there a better way to do this? Somehow I feel I am not using 'tee' properly. Any insight would be appreciated.  I'm using bash shell, if this is relevant. Doesn't the end of Looper create a time paradox? I guess this is more a question about the nature of time travel more than anything else.  At the end of the movie,      Joe shoots himself in a field and changes the course of history.   Does this create a paradox as it changes the history that led up to that point?      If his future self had never existed he'd never have gone on the mission he did and thus he never would have ended up in the field.   Once he did what he did, he could no longer have done it, and so theoretically, time and space should have just folded in on itself and popped out of existence.  Is there something I'm missing that would explain how the movie could have ended where it did without the events at the beginning to lead it there? What characters/symbols are allowed within the CSS class selectors? I know that the following characters are invalid, but what characters are valid?  ~ ! @ $ % ^ &amp; * ( ) + = , . / ' ; : " ? &gt; &lt; [ ] \ { } | ` #