I've been using the align environment to line up equations in proofs/explanations/etc and I like to place parenthetical commentary for each step, similar to the code below:  \documentclass{article} \usepackage{amsmath} \begin{document}  \begin{align*}   f(ax + by) &amp; = f(ax) + f(by) &amp; (By Property 1) \\              &amp; = af(x) + bf(y) &amp; (By Property 2) \\ \end{align*}  \end{document}   The problem I have with this is that the commentary completely distorts the overall alignment of the equation area; far too much space is allocated to the column containing the commentary.  I would like, insofar as it is possible, for the equations themselves to be aligned in the center of the page and for the commentary to simply appear on the right without affecting the overall justification of the equations. What's the best way to achieve this?
I'm new to LaTeX and I'm trying to figure out how to correctly format a proof. Can someone show me a basic template for how to do this?   The part I am having the most trouble with is creating new columns and customizing alignment... for instance, how do I add a right-aligned annotation (i.e. by distributive property) to a line of this sample proof? Should I use eqnarray* or is there a better option?  \begin{proof}   Let $t,u \in \mathbb{R}$ where $t=xy$ and $u=zw$. So,   \begin{eqnarray*}     4xyzw &amp;=&amp; 2\cdot2tu \\     &amp;\le&amp; 2\cdot(t^2+u^2) \\     &amp;=&amp; 2\cdot((xy)^2+(zw)^2) \\     &amp;=&amp; 2\cdot(x^2y^2+z^2w^2) \\     &amp;=&amp; 2x^2y^2+2z^2w^2 \\     &amp;\le&amp; ((x^2)^2+(y^2)^2)+((z^2)^2)+(w^2)^2) \\     &amp;=&amp; x^4+y^4+z^4+w^4   \end{eqnarray*} \end{proof}
Having trouble setting up a nginx web server on Amazon EC2 How to EFI install Ubuntu? On trying to install Ubuntu 12.10 Desktop amd64 to a desktop pc using a ga a75 ud4h motherboard (which is using an AMD 3870k APU with a new 1TB HDD) following the advice in the  and  have now got to grub option screen for EFI install. Booting live dvd in EFI mode gives &quot;Secure Boot is not enabled&quot; just before grub kernel option list shows.  If the option to &quot;Install Ubuntu&quot; or &quot;Try Ubuntu without installing&quot; is selected gives &quot;can not read cd/0&quot; and &quot;the kernel must be loaded first&quot; errors. Using DVD or USB installation media gives same results and there has never been at any point an UEFI entry in bios/firmware settings or boot device list/s. It might be helpful to say that have had an alternative non-linux OS installing EFI mode on this hardware giving a GPT installation automagically using USB installation media when the hdd space was completely unallocated space. The USB device was listed under hard drives in bios/firmware one time boot list and did not have an UEFI entry. i.e. it installed EFI although no EFI or UEFI entry was assigned to the device like what is shown in many tutorials regarding UEFI installs. The only constructive information regarding this question is found at  and  but can not work out the answer. Checking md5sum and sha256 checksums on downloaded .iso and used &quot;Check disk for defects&quot; to check burned .iso when booting DVD in BIOS mode. No errors found. &quot;Check disk for defects&quot; will not run (or any other options) when DVD booted in EFI mode. Due to trying different distributions as well as older and newer flavours of Ubuntu all giving similar results in that no options from grub option screen in EFI mode are bootable (if getting that far) as well as spending time searching Ubuntu Forums have come to conclusion the firmware on this system is hardcoded to handle Microsoft EFI files only. After installing Windows EFI mode here is a  and here is a link to a thread i started at  which may be helpful to others building a dual boot system. Installing BIOS mode and using Boot-Repair to install grub-efi is not a complete solution due to the inability of using installation media EFI mode for recovery and maintenance due to the &quot;you need to load kernel first&quot; and &quot;can not read cd/files&quot; errors as described when trying to boot installation media using EFI. Secure Boot is not an issue. Using msinfo32 from windows confirms Secure Boot is not supported on this hardware. How to edit kernel or load kernel for EFI install? = How to EFI install Ubuntu? PLEASE CONSIDER: This is not a dupe!  The linked question that is indicated to having a working answer does not have a working answer to &quot;###How to edit kernel or load kernel for EFI install? = How to EFI install Ubuntu?&quot; therefor the workaround of installing using BIOS then manually install grub-efi without renaming bootloaders which works to get Ubuntu booting using EFI or using boot-repair to do this (and many unnecessary &quot;fixes&quot; if recommended repair is not changed in advanced settings) also works.  This goes some way to confirming hardcoding of firmware is not stopping grub the boot manager seeing the vmlinuz.efi files in /casper on DVD.  I would like an answer to this so would like this question to be reopened. What would be the Slater's determinant representation for an excited state? Setup  Introducing this spinorbital notation:  \begin{align} \Psi_1=\chi_{(r1)}\alpha_{(\omega1)}  = 1 \\ \Psi_1=\chi_{(r1)}\beta_{(\omega1)}  = \bar{1} \end{align}  and the Slater's determinant, for a system of two electrons, is:  \begin{align} |\Psi\rangle = \frac{1}{\sqrt{2}} \Big|  \begin{array}{cc} 1 &amp; \bar{1}  \\ 2 &amp; \bar{2}  \\ \end{array}  \Big| = \frac{|1\bar{2}\rangle-|\bar{1}2\rangle}{\sqrt2} \end{align}  Question  Based on the example, what would be the Slater's determinant representation for an excited state like $\Psi_a^r$ (i.e. with parallel spins like: $|\bar{1}\bar{2}\rangle$) and $\Psi_{ab}^{rs}$ (feel free to use any number) ? Prove Inverses not need be Unique if Associativity fails Let $M=\{x, y, z\}$ be a set and let $\circ$ be a binary operation on $M$ such that the identity element is $x$ and $y \circ y = x = y \circ z$. Prove that inverses need not be unique if associativity fails given the multiplication table below.  $x \circ x = x, x \circ y = y, x \circ z = z$  $y \circ x = y, y \circ y = x, y \circ z = x$  $z \circ x = z, z \circ y = x, z \circ z = x$ Why is DML not allowed in Constructor? I grew up learning that the comma must be placed there, but it seems like an unnecessary interruption in a phrase that isn't ever spoken that way. Where to put my code: plugin or functions.php? Is there an easy to understand scheme to decide what kind of code belongs to a plugin or the theme’s functions.php?  There    and many  about that topic, mostly because there are some misconceptions about the inner workings of WordPress. I am asking for an answer based on facts, not on opinions.  It should explain how to handle these points (and probably more):   custom post types and taxonomies contact forms shortcodes custom widgets  SEO functions like custom meta elements theme switch   There are often pros and cons for both sides. Our most popular question  got a lot of code snippets as answers that are at least debatable. We need criteria a beginner can understand, maybe a check list – with reasons.  See also the related question by Chip Bennett on our meta site:   Related: I have been trying to solve this problem for quite a while. I am still unsure of whether any of the avenues I have pursued have been of any use. Any advice will be much appreciated. Question: Let $V$ be a finite-dimensional inner product space, and let $E$ be an idempotent linear operator on $V$. Prove that if $EE^* = E^*E$, then $E$ is self-adjoint. (This is essentially exercise 5(a) in sec. 80 on p.162 of Paul R. Halmos, Finite-Dimensional Vector Spaces, but Halmos didn't assume that the dimension of $V$ is finite.) Vertex count appears incorrect after removing doubles I've only just started studying the Blender Python API. I'm writing a script (see below) to automate something I've done by hand : joining two adjacent cubes and cleaning up the resulting mesh.   After joining and removing doubles, I expect to have 12 vertices instead of 16 in the resulting object. In the Info header, this seems to be the case, but when I print to the windows terminal I still see 16. Also I do see the output from remove_double(), here's a screenshot :    So what's wrong here ? What is the right way to access an object's mesh, and count its vertices ?  Joao  import bpy  # Create two adjacent cubes and join them bpy.ops.mesh.primitive_cube_add(location=(1,0,0)) ob = bpy.context.object bpy.ops.mesh.primitive_cube_add(location=(3,0,0)) ob.select = True bpy.ops.object.join()  # Expect 16 vertices, 24 edges, 12 faces result = bpy.context.object print(str(len(result.data.vertices)))  # Go to edit mode, remove double vertices bpy.ops.object.mode_set(mode = 'EDIT') bpy.ops.mesh.remove_doubles()  # Expect 12 vertices, 20 edges, 11 faces result = bpy.context.object print(str(len(result.data.vertices))) Today, comparing processors is obviously not just looking at the frequency and the number of cores.  When choosing a new PC, how do you compare processors? Is there some resource, like a site or something that does (not necessarily super-precise) processor comparisons?  Thanks. I am really confused in Gauss law.  Why do E3 and E2 pointing up? and also E1 pointing down?  The lecture note said infer from symmetry and you will get the following but I dont really understand. Could anyone explain to me? I am new to gauss law. How does Ubuntu flavor iso differ from a desktop package? TL;DR: how does an Ubuntu flavor iso installation ( for example Lubuntu ) differ from installing a desktop package via apt-get ? (for instance sudo apt-get install lubuntu-desktop)  Context for the question:   Some time ago I answered a question on AskUbuntu , where I tested in virtual machine removing unity-desktop and installing ubuntu-gnome-desktop. That makes the boot logo ( plymouth ) change, desktop changes, different packages get installed (gnome-network-manager instead of network-manager , if I am not mistaken). It appeared that in effect I've switched to different Ubuntu flavor, even though I started out with the standard Ubuntu system. Hence the question: what differentiates the pre-made Ubuntu flavor iso from installing a package ? Is it all the same base underneath the hood or does installation from iso have specific configurations ? If it's possible to install a flavor via package manager, why then separate flavor iso's exist ? (obvious answer is user convenience, but I think there has to exist other reasons. Canonical is a business , and it makes not a lot of sense for a business to provide additional flavor iso , if a sysadmin or end-user can get the same thing via package manager by themselves). Is the notorious $n^2 + n + 41$ prime generator the last of its type? I know that there are questions asking like "intersection of a infinite collection of sets" and I can understand that the answer for that one is a null set, but I got a question here, in which all sets are finite and nonempty. Please take a look at the pic below.  It's a True/False question.  What I'm confused about is that the "infinity" sign in the interception. Since all sets are finite, does it mean that there are many sets in the chain are the same? (because the symbol used is for subsets not for proper subsets). Please give me some hints how I should think about this question.  Thank you Proof that every number ≥ $8$ can be represented by a sum of fives and threes. My sister, cousins, and other relatives speculate that my mom's grandmother was Jewish (for various reasons). I've heard people say at times things like, for example, "I'm 50% German" or "I'm 25% Swedish." This has to do with their ancestry. But, how exactly is this calculated? Say for example that my mother's paternal grandmother was indeed 100% Jewish, what would that make me and my siblings, and how is it calculated? Watching The Lord of the Rings movies I was really captivated by the   elephant riders and the men in brazen scale armor who aided the orcs   against the forces of Gondor and Rohan. From what I know they are from   the cultures that worship Sauron. I'd like to know what Tolkien wrote   about them, their religion, their way of life, and so on.   From Existence of area-preserving diffeomorphism between two sets in $\mathbb{R}^2$ Suppose that we have open sets $A, B\subset \mathbb{R}^2$ such that $A$ and $B$ are simply connected and have smooth boundaries. Furthermore, let us assume that $m(A) = m(B) &gt; 0$, where $m$ is the Lebesgue measure on $\mathbb{R}^2$. My question is as follows: when does there exist an area-preserving diffeomorphism between $A$ and $B$? In this context, an area-preserving diffeomorphism between $A$ and $B$ is a is a $C^{\infty}$-diffeomorphism $\phi : A\to B$ such that $m(S) = m(\phi(S))$ for all measurable $S\subseteq A$. If an area-preserving diffeomorphism doesn't always exist between sets $A$ and $B$ as above, are there any extra conditions we can impose on $A$ and $B$ such that there does exist an area-preserving diffeomorphism?  My thoughts thus far are that we always have a $C^{\infty}$-diffeomorphism between $A$ and $B$: this is a consequence of the . It's also obvious to me that if we can deform $A$ linearly to get $B$, then that deformation will be area-preserving. Furthermore, it seems that aside from this case, there's no reason why an area-preserving diffeomorphism should exist. However, I'm having a hard time proving that there necessarily isn't an area-preserving diffeomorphism between $A$ and $B$ if $B$ is not a linear deformation of $A$. Does anybody have any ideas? Phantom device showing on Google Play Headphones don't work after update I updated to Ubuntu 15.04 from Ubuntu 14.10 a few days ago, and now my Logitech G230 headphones don't receive any sound. How can I fix this? PulseAudio Volume Control detects my headphones, and says they're working, but no sound comes through.  EDIT: Using alsamixer worked at first, but now it doesn't, and I get frequent bursts of static through my headphones now.