Let $T:\Bbb R^4\longrightarrow\Bbb R^4$ be a linear transformation satisfying $T^3+3T^2=4I$, where $I$ is the identity transformation. Then the linear transformations $S=T^4+3T^3-4I$ is a). one-one but not onto. b). onto but not one-one. c). Invertible. d). Non-invertible.  on solving i get $S=4T-4I$, but how does this suggest invertibilty? Also, Logically thinking option $a,b$, and $d$ are interlinked. So if only one choice is correct then i could simply say that yes option $c$ is correct. But can someone give the general aproach?
Let $T,S$ be linear transformations, $T:\mathbb R^4 \rightarrow \mathbb R^4$, such that $T^3+3T^2=4I, S=T^4+3T^3-4I$. Then S is:   one-one but not onto onto but not one one invertible non-invertible   (One or more correct options)  My attempt: $S=T^4+3T^3-4I=T(T^3+3T^2)-4I=T(4I)-4I=4T-4I$  How do I go about proving or disproving my options?   My thoughts on options # 1. and 2.: I'm guessing $S$ is one-one if $Ker(S)=\theta$ is one way to go, but how do I obtain the kernel when I don't know what the transformation is? $Ker(S)=\theta$ if $T=I$, which does satisfy $T^3+3T^2=4I$. But so can other $T$s.   My thoughts on options # 3. and 4.: $S$ is invertible if $det(S)\neq0$, which is possible $det(T-I)\neq0$, i.e. $T\neq I$. How do I show that?  Please help!
Simple PHP Pagination script wp_delete_auto_drafts() deletes links in menus Since yesterday the wp_scheduled_auto_draft_delete cron job somehow deletes all page entries from our menus. All entries in the table wp_postmeta with meta_key = '_menu_item_object' and meta_value = 'page' get deleted in the process. By tracing back the DELETE query I found out that the menu entries are deleted by a function called _wp_delete_post_menu_item. I got the following stacktrace for that function:  #0  _wp_delete_post_menu_item(3244) #1  call_user_func_array(_wp_delete_post_menu_item, Array ([0] =&gt; 3244)) called at [/srv/www/htdocs/biva/wp-includes/plugin.php:470] #2  do_action(delete_post, 3244) called at [/srv/www/htdocs/biva/wp-includes/post.php:2483] #3  wp_delete_post(3244, 1) called at [/srv/www/htdocs/biva/wp-includes/post.php:5645] #4  wp_delete_auto_drafts() #5  call_user_func_array(wp_delete_auto_drafts, Array ()) called at [/srv/www/htdocs/biva/wp-includes/plugin.php:546] #6  do_action_ref_array(wp_scheduled_auto_draft_delete, Array ()) called at [/srv/www/htdocs/biva/wp-cron.php:100]   But I'm stuck here, I'm not a wordpress developer and I have no clue why deleting old drafts results in deleting attached menu entries although the main pages still exist.  Workaround: As a workaround we added the following to the top of wp-config, wich will disable wp-cron:  /** Disable wp-cron to prevent menu from getting deleted. Detailed Bug description: http://wordpress.stackexchange.com/questions/145953 **/ define('DISABLE_WP_CRON', true);   If you are experiencing similar difficulties you might try locking the rows in the table wp_postmeta with meta_key = '_menu_item_object' and meta_value = 'page' and then tracing back the error message when WP trys to delete these rows, that's how we nailed down our error.  Reproduce the Error: It is definitely the stealth publish plugin that is causing our error, one can easily reproduce it, by doing the following:   Install a fresh Wordpress 3.9.1. Install the stealth publish plugin . Create a few test pages and add them to a new menu. Create a new page, with some dummy content, but do not save it!. Leave the page without saving. Change the system time 8 days into the future. The wp_scheduled_auto_draft_delete cron job will run because it runs daily and it hasn't run for 7 days as soon as you visit your page. It will then find your never saved dummy page delete it as it is older than 7 days and the menu will vanish. You might need to refresh the page a few times till the effect becomes visible.   I've recorded a screencast on how to reproduce the error (but I took not the time to comment it): Initially, my question was: is "focussed" or "focused" the correct past tense of "focus", but since this applies to a lot of words, I would like to generalize and ask: is there supposed to be a rule when to double the consonant? Below is experimental data for the ratio $$R=\frac{\sigma(e^+e^-\rightarrow hadrons)}{\sigma(e^+e^-\rightarrow\mu^+\mu^-)}$$ as a function of the centre of mass energy $\sqrt s$. I am interested in the peak at around $100GeV$ which corresponds to the resonance of the $Z^0$ boson. There are two ways of looking at this that I have in mind:  We can effectively ignore the electromagnetic process around this energy, and so each process should have it's own Breit-Wigner peak centred on the mass energy of the $Z^0$. The ratio of these is just a constant. Thinking in terms of Feynman rules and again ignoring the electromagnetic process around this region, each process has the same propagator and vertex factors (roughly - ignore quark mixing) and there are some extra factors due to different quarks being possible and colour degeneracy, but still the ratio should be a constant (one possible issue here is interference between Feynman diagrams which I have neglected?).  So my question is why does the peak exist in the data? Thunderbolt 2 docks have been quite expensive and they will be obsolete once I purchase the new Macbooks with USB-C ports. I currently use a 2014 MacBook Pro with Thunderbolt 2 ports, and I want to futureproof my purchase by acquiring a cheaper USB-C dock now.  The cheapest Thunderbolt 2 dock costs around $200 while USB-C docks cost around $30. Being able to use a USB-C dock with older Thunderbolt 2 Macbook Pros would not only be cheaper but also allows me to use the same USB-C dock if I decided to upgrade to a future Macbook.  For that to work I need a Thunderbolt 2 (male) to USB-C female adaptor, after some searching it doesn't seem to exist. Is there a reason why something like this hasn't been made? Or is the USB-C specification designed such that it is impossible? (hence businesses can still continue to sell expensive Thunderbolt 2 docks...)  I will accept an answer that meets the following requirements:   If there is such an adapter please give a link and some form of evidence that these would have 100% compatibility with USB-C docks. If there is no such adapter please explain:   Is it impossible to produce? If not, why hasn't a vendor created one yet? I've explained that its a better financial investment than buying a TB2 dock... surely there's a market for those still using TB2 MBPs wanting to upgrade in the next couple of years. Not only docks but it would open a gateway to cheaper alternatives to those hideously expensive TB2 peripherals. Defining font color inside highlight fails Since I wanted a better readability of highlights of a would be black &amp; white-printed document, I needed to change both the highlighting color (gray) AND the font color (white).  Depending which command you implement inside the other will compile or not: \hl{\textcolor{white}{Fail}} does not work, compiling returns Package xcolor Error: Undefined color '{white}'.  Here's the mwe:  \documentclass{article} \usepackage{xcolor} \usepackage{color} \usepackage{soul} % Enables highlighting \sethlcolor{gray}  \begin{document}      \textcolor{white}{\hl{Success}}     %% Error:     %% Package xcolor Error: Undefined color `{white}'.     \hl{\textcolor{white}{Fail}}  \end{document}   So, in the end, the problem is sorta solved but I thought you might have an answer as to what happens. Real Time LaTeX? Possible Duplicate:          I'm looking for an application in which I can write pure LaTeX in one half of the screen and on the other half to see the compiled document in real time. This seems really like an obvious way to do LaTeX authoring but it doesn't look like any software is able to do it. LyX is not what I'm looking for. I want to edit the actual LaTeX and see it rendered in real time. Why isn't this already being done somewhere? I want players to be able to click text to run an OP command even if that player is not an OP. For example something like:  Console: Do you want to enable Fly? [If you do, Please click here]  *Player1 clicks that text*  Console: /fly Player1   Is something like this possible? lib32bz2-1.0 missing on 16.04 I have tried installing this package (needed for android studio).  This is the error i'm getting  E: Unable to locate package lib32bz2-1.0 E: Couldn't find any package by glob 'lib32bz2-1.0' E: Couldn't find any package by regex 'lib32bz2-1.0'   I have already tried:  sudo dpkg --add-architecture i386 sudo apt-get update sudo apt-get install lib32bz2-1.0   &amp;  sudo dpkg --add-architecture i386 sudo apt-get update sudo apt-get install lib32bz2-1.0:i386   I also searched on  and did not find lib32bz2-1.0 for Ubuntu 16.04. So how do i get this package which is available for trusty and precise? Crop jpeg into circular tikz node \documentclass[demo]{article} \usepackage{graphicx} \usepackage{tikz} \begin{document} \begin{tikzpicture} \node[circle,draw,inner sep=2cm]{\includegraphics{myimage.jpg}}; \end{tikzpicture} \end{document}   How do I modify the above code so that the included image is adapted to the circle shape and cropped of the exceeding parts?  I want to specify a shape (a circle) and then include an image respecting the given dimension and cropped. Can someone give me a short description and comparison of the different road cycling pedal / shoe systems? I am looking at buying a pedal/shoe system as the clips and straps I am using are hurting my feet again.  It's been a long time since I have bought a road pedal / shoe system, and I have forgotten the differences between SPDs, Look pedals and all the others.    Can someone give a short comparison of each pedal system with pros and cons of each? My usage is/will be mainly city street cycling and then road cycling (recreation, not racing).  I'd prefer if people would give the pros and cons of each and not their opinion on the "pedal system war".  :) However, "I use them because I find XYZ and it helps in ABC ways" is very useful. How do I backup all files? (.dbus and dconf backup problems) What are /home/user/.cache/dconf and /home/user/.dbus? Why does their backup fail?  I tried following  (see code below) but while it allowed me to backup the above files, now I am told it couldn't backup /home/[user]/.dbus/session-bus.    $ man chown $ sudo chown user .cache/dconf   [sudo] password for user:  $ sudo chown user .dbus $ man chgrp $ sudo chgrp user .cache/dconf $ sudo chgrp user .dbus   How do I fix it so it will not take my time closing an error message after each backup? Should I exclude these folders ? Why are they not needed? Why would someone want to back them up? I don't understand philosophy very well, and so I am wondering whether you can "prove" anything in philosophy. It always seems you can go a layer down, and find another question, almost endlessly until you get to the question of "why/how/are we here?"  Can you prove, absolutely, and 100% that something is true, in the field of philosophy? How can I create launchers on my desktop? In older versions it was easy to create a launcher on my desktop. All I had to do is right click on my desktop and select the "create launcher" option.  How can I create such launchers now? Creating intuition about Laplace & Fourier transforms Is there a problem with Beamer in the latest MiKTeX update? Image of concave mirror when object is farther than the focal point I have a set of concave/convex mirrors/lenses, and I have drawn out all of the ray tracing diagrams for each possible combination and type of image each optical element can produce. When I look at the ray tracing diagrams then reproduce the situation in the real world with myself as the &quot;object&quot;, all the scenarios make sense to me except one: when I as the object stand beyond the focal point of a concave mirror.  The image ends up as &quot;real&quot; and appears to be in front of the mirror. Now I understand that in the situation above, I as the object would stand at 6m and this would result in the image appearing at 12m. So if I placed a wall or screen at the 12m mark, the image would form on that surface (just like a movie theater screen).... at least I think I interpreted that correctly. But I still see myself when I look in the mirror?! If the image is supposed to form and come into focus behind me, what in the world am I seeing when looking at the surface of the mirror? I still see myself, just inverted. The surface of the concave mirror still has a reflection on it. It doesn't look quite like a normal flat mirror, in that the image doesn't really look like it's &quot;inside&quot; the mirror, more like it's right on the surface of it. It's really frustrating because every other possible arrangement of optical elements makes total sense to me, except this one. Any suggestions on how to think about it? Thank you for your time! How to add texture to fill colors in ggplot2 How to configure Micromax 352g USB Modem? I want the driver software for my Micromax 352g USB modem for Ubuntu 12.04 LTS. On Ubuntu 10.04 sakis3g was working well, but it is not working on Ubuntu 12.04 LTS. Display Animated GIF