Prove that every planar graph is the union of at most $5$ acyclic   graphs.      Reminder: As union of two graphs $G_1(V_1, E_1)$ and $G_2(V_2, E_2)$ we consider the graph $G(V_1\cup V_2,E_1\cup E_2)$   I have tried induction on the number of vertices, but I seem to miss something
How can I prove that every planar graph can be expressed as a union of (maximum) five edge-disjoint forests?  Only thing I can think about is going somehow like the $5$ color theorem but I get stuck in the middle. Or with induction, but still something doesn't fit. Can you help?
I installed nginx a while ago using the ubuntu PPA repository.  Now I want to compile nginx from source so I can compile in and use the geo-ip module.  I know how to compile the new nginx from source.  How can I remove the nginx binary which was installed by the package?  I want to leave all the configs and logs and upstart stuff etc untouched.  I really just want to remove the old binary.  How can I do this? How to change default viewport shading in 2.8? Is it possible to change the default mesh color/shading in 2.8 to be more like 2.79? I'm finding it hard to see finer detail in 2.8. On the left is 2.8 and the right is 2.79 Shared email on 2 Facebook accounts is stopping access to both accounts My husband wanted me to set up a Facebook account for him and I accidentally used our shared email address, which is set against my own account.  Since then, neither of us has been able to log in, so I can't alter any settings.  Can anyone advise how to resolve this please? Notify question OP (and those who marked it a favourite) if an answer has been modified Seriously, I had not noticed e.g. Jeff's edit  had he not also retagged my question. Dimensional Tolerance for US Standard NEMA 1-15 Plugs I am trying to find the dimensional tolerance on US electrical plugs. I have tried looking online, but I have only been able to find that they are called , and have found the picture below showing length and height for the male connector (but not width of the male prong), and I haven't been able to find dimensional tolerances anywhere.     I ask here because I'm assuming there's a US code that governs electrical plugs, but I can't seem to find the standard anywhere. I'm using a  freely downloadable latex class file and the style is set to \bibliographystyle{ieeetr}. But when I cite something with \cite{} I get it inside ().   For example I get it as abc et al. (1) but what I want to get is abc et al. [1].   Please tell me how to get my citations inside [ ] ? What is the difference between Python's list methods append and extend? What's the difference between the list methods append() and extend()? Find Total number of ways out of N Number taking K numbers every M interval I have been stuck in a problem, that has thrown my brain out of the coding. This problem is at very high priority and I need the solution as early as possible. Problem is as :  There are exactly N advertising boards on the highway. Now a company want to advertise on some of these advertising boards(each advertising board costs some money) .  Company strategy is that, they want at least 'K' advertisement should be there among M consecutive advertising boards. But at the same time Company want to pay minimum for its advertisement .  Now, what is the total number of ways, different ways Company can advertise meeting its minimum cost strategy.  As for Example: N= 3, M = 2, K=1 ==> there is only one way for minimum cost, ie. 0C0 , where '0' denotes No company advertisement, and 'C' denotes company advertisement board.  Similarly, for N =4, M =2, K =1 ==> there is 3 possible ways, ie. C0C0, 0C0C, 0CC0.  Hope to get a reply from someone soon. Thanks. Crash or "segmentation fault" when data is copied/scanned/read to an uninitialized pointer What to do when the correct answer to a question changes over time? My question concerns . I asked the question, and at the time selected the correct answer. However, time passes and libraries get updated, and now a different answer is actually correct.   I should probably select this new answer as the correct answer, but I feel bad for the person that will lose 15 reputation points even though is answer was the a good answer at that given point in time.  Any suggestions? Why isn’t singular ‘they’ used with 3Sg verb forms? There are many (duplicate) questions about the acceptance, popularity and history of singular they (and their, them and themself) around here, it even got a tag of its own. If I didn’t miss anything, however, the proper verb form hasn’t been questioned yet.  As we all know, English third person singular pronouns (it, she, he and one, +body), names (Alice, Bob, …) and nouns (student, teacher, …) demand the +(e)s suffix be added to the finite verb form in simple present, where some auxiliary verbs have a “special” form (is &lt; *bes / *ares and has &lt; *haves). All other subjects don’t, including plural third person pronoun they.  When the plural you replaced thou (with thee, thy / thine), the other marked verb form that had remained in English – i.e. suffix +(e)st or +t – vanished, too. The first and second persons only ever appear as pronouns (I, we; you), not names or nouns, so there was no strong inclination to keep the verbal inflection.  The second person precedent would suggest that singular they be used with uninflected verb forms which is how it’s usually encountered in the wild. Assuming that they someday replaces he and she (and maybe it) it would lead to disagreement with the words the pronoun stands in for:        Alice         goes to her   place  by herself.   Bob           goes to his   place  by himself.   Alice and Bob go   to their place  by themselves. – (common)   Alice and Bob go   to their places by themselves. – (separate)                  She  goes to her   place  by herself.   He   goes to his   place  by himself.   They go   to their place  by themselves.   They go   to their places by themselves.                  They ?goes to their place  by themself.   ditto   They  go   to their place  by themselves.   They  go   to their places by themselves.                  They ?go to their place  by ?themselves.   ditto   They  go to their place  by  themselves.   They  go to their places by  themselves.      So why doesn’t singular they afford s on present-tense verbs like all other third person singular subjects do?   I learned about singular they only long after I had been taught English as a second language in school. That’s why it’s still a conscious decision to use it and hence I could easily adapt to use s forms with it, but would that sound and look funny / strange / wrong to native speakers? Where can I ask questions for website recommendations? Is there any Stack Exchange site where I can ask others to recommend me good web sites for solving specific problems?  For example I would like to know if there is any URL shortening website that can allow me to define the url.  For example instead of http://bit.ly/1gX2dD0 I would like it to be http://bit.ly/superflow - if "superflow" is not available then I will invent another word.  I can imagine that such services are paid but I don't need the link to last more than one day, and maybe there is such a website. When exactly can a caster stop concentrating on a spell? Relative log file matching with linux? Why is torque not measured in Joules? Recently, I was doing my homework and I found out that Torque can be calculated using $\tau = rF$. This means the units of torque are Newton meters. Work &amp; Energy are also measured in Newton meters which are Joules.  However, torque isn't a measure of energy. I am really confused as why it isn't measured in Joules. Proof that the relation $5 \mid (a + 4b)$ is symmetric and transitive Take the relation $R$ to be defined on the set of integers:  $$aRb \iff 5 \mid (a + 4b)$$  As part of a larger proof, I have to show that $R$ is both symmetric and transitive. I'm lost.  I see the first steps, but I can't find how to progress further. Here's what I have at this point:    Proof of Symmetry  We have to prove that if $5 \mid (a + 4b)$, then $5 \mid (b + 4a)$. Clearly, this is true if $a = b$, but apart from that, it's unclear in my mind.  Proof of Transitivity  We have to prove that if $5 \mid (a + 4x)$ and $5 \mid (x + 4b)$, then $5 \mid (a + 4b)$.    I've fiddled around with sample values, but I still don't see it. I'm pretty lost here. Thoughts? https:// search results appearing on Google for purely http:// site I started weeding through my site's search results from Google today, using a site: search, to determine if there are any links that cause 404s and thus need redirecting. To my amazement I noticed numerous https:// results relating to various pages. My site doesn't have a SSL certificate, doesn't serve such pages, doesn't internally link to https:// pages, doesn't include any such files in its sitemap.xml and, for all of these, never has.  I decided to do a Google search for https://&lt;my site&gt; and found one site that incorrectly refers to the root of my site with a https:// prefix - I will try to contact them to get them to correct this.  I'm not sure however how Googlebot managed to index the non-root files as https://. I can't find any external links to them and surely, without certification, Googlebot should have stalled at the first request?  I've just added the following lines to the site's .htaccess (although the surfer still has to navigate through the browser's "This site is a security risk. Abandon hope all ye who enter here!" message(s) first to get there):  RewriteEngine On RewriteCond %{HTTPS} on RewriteRule ^(.*)$ http://www.&lt;my site&gt;.org/$1 [R=301,L]    replacing &lt;my site&gt; with my domain name.   My big question is this though - I would like to use the Google Webmaster Tools Remove URLs feature to remove the https:// pages from the index. Can I be guaranteed that this will only remove the https:// versions of each relevant page and not the valid http:// versions?  My thanks to anyone who can help me out with this particular question and the issue in general. Is there an advantage to use a Synchronized Method instead of a Synchronized Block? Diagonalisation and characteristic polynomial Let $A\in M_3(\Bbb R)$ which is not a diagonal matrix. Let $P$ be a polynomial (in one variable), with real coefficients and of degree 3 such that $P(A) = 0$. Pick out the true statements:  a. $P = cP_A$ where $c \in\Bbb R$ and $P_A$ is the characteristic polynomial of $A$;  b. if $P$ has a complex root (i.e. a root with non-zero imaginary part), then $P = cP_A$, with $c$ and $P_A$ as above;  c. if $P$ has a complex root, then $A$ is diagonalizable over $\Bbb C$.  I guess b and c are true although I am unable to disprove a. Because A could satisfy $(X-A)$ but this is not a polynomial. Now if $P$ has a complex root, then its conjugate too is its root as the polynomial has  real coefficients. As we all know An odd degree polynomial has always a real root, what we have is three distinct roots. Since $A$ satisfies the polynomial the only difference between the characteristic polynomial and $P$ has to be a constant. How does $e^{i x}$ produce rotation around the imaginary unit circle? Euler’s formula states that $e^{i  x} = \cos(x) + i \sin(x)$.  I can see from the MacLaurin Expansion that this is indeed true; however, I don’t intuitively understand how raising $e$ to the power of $ix$ produces rotation.  Can anyone give me an intuitive understanding?