How do I find the RGB representation of a Color object?
Get value of R, G, B by color
The Higgs boson and gluons have no electric charge and photons couple to charge, so there is no tree level interaction between them and photons.  But what prevents higher order diagrams from contributing a non-zero mass term to the photon, for instance    where a photon couples to some fermion (say an electron, or a top quark) which can interact with the Higgs field.   Or consider that same diagram but with quarks and a gluon interacting between them? Or any higher diagram with even more loops?  I have heard charge conservation depends on gauge invariance, which in turn depends on photons being massless.  So it appears the photon has no mass, and these diagrams must all cancel somehow.  So I'm hoping there is a very nice symmetry explanation for why they all disappear, but if I say "because of gauge invariance" that would be circular logic, so there must be another symmetry at stake here?     What prevents photons from obtaining a mass from high-order self-energy loop diagrams? How to prevent water from freezing How do I speed up my design work? I've been a designer for 6 months, having previously had a 1st class honours degree in Graphic Communication. My issue is that I am a Very good designer, but I work too slow. It's beginning to drive a rift between myself and the creative director who desperately wants me to work quicker. I've always been a workaholic even in university, I put in 110% all the time, really work my ass off, but I'm still not fast enough to complete projects quickly, my work is good, but it's not efficient.  Do you guys have any tips to help me speed up? I am getting the following error in the top menu bar:     An error occurred.Please run the Package Manger form the right-click menu or apt-get in a terminal to see what is wrong.The error message was:'Error:BrokenCount>0'.This usually means that your installed packages have unmet dependencies. Do flight attendants prefer I press the button or walk to the back? (Long-haul flights) Let $x&gt;0$, $y&gt;0$ and $z&gt;0$. Show that   $$x^3y+y^3z+z^3x\ge xyz(x+y+z).$$   I known we can't WLOG: $x\ge y\ge z$, if this, I can use rearrangement inequality, But other I can't it. Thanks? How can I fix "unbalanced brackets" in my data tag? Which is correct?     Everyone were convinced that he would go to the game.   Everyone was convinced that he would go to the game.   I think it's "was", because "everyone" is singular, but I just wanted to check. How to prove  $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$ Background: One of my co-authors faced problem of compiling the paper that I had built using TexStudio and MikteX 2.9 on his Mac Machine running TexLive.   ShareLaTeX Role: So, one solution is to use ShareLaTeX. But the first problem pops in mind is how safe is uploading unpublished article to a publicly accessible portal?  What I did: I googled security and data breach of ShareLaTex and found nothing except some words from the ShareLaTex community about their word to ensure that our data are safe; perhaps this would not be enough.  My 1st Question: To what extent can I trust ShareLaTeX and put my ready-to-publish content there?   My 2nd Question: Is there any alternate way of sharing LaTeX with others around the globe without changing the underlying platform? EM algorithm gaussian mixtures- derivation I'm trying to make sense of a derivation I'm following from the lecture notes of Stanford's ML course.  Specifically the notes are here:   I'm particular I'm interested in the step where they are optimizing the lower bound of the log-likelihood by finding the derivative of that function w.r.t the mean (top of page 7). I'm a bit stuck on going from line 1 to line 2 in that sequence.  To copy here (using abbreviated notation for space):  $$ \nabla \mu_s \sum_{i=1}^{m}\sum_{j=1}^{k} w_j^{(i)}\log\frac{\alpha_j \exp\psi_{j}^{(i)}\phi_j}{w_j^{(i)}}  \qquad(1) $$ $$ \nabla \mu_s \sum_{i=1}^{m}\sum_{j=1}^{k} w_j^{(i)}\psi_{j}^{(i)}\qquad(2) $$ where $\psi_{j}^{(i)} =-\frac{1}{2}(x^{(i)}-\mu_j)^T\Sigma^{-1}_{j}(x^{(i)}-\mu_j)$ and $\alpha_j=\frac{1}{(2\pi)^{n/2}|\Sigma_j|^{1/2}}$.  The quantity $\mu_s$ is the mean (of a multivariate gaussian distribution) for class $s$.    In short, I'm not sure how they get from (1) to (2).  From just changing the log of (1) I see:  $$ \nabla \mu_s \sum_{i=1}^{m}\sum_{j=1}^{k} w_j^{(i)}\log \left(\alpha_j \exp\psi_{j}^{(i)}\phi_j\right) - w_j^{(i)}\log w_j^{(i)} \qquad(1) $$  To get to their form (2) my impression is that they take pre-emptively discard terms that would contain derivatives of $w_j^{(i)}$ (w.r.t. $\mu_s$).  At least if you assume that much, you get to (2).  However, as they show earlier in that PDF,  $$ w_j^{(i)} = p(z^{(i)}=j | x^{(i)}) =\frac{p(x^{(i)}|z^{(i)}=j)p(z^{(i)}=j) }{\sum_j p(x^{(i)},z^{(i)}=j) } $$ i.e. this quantity is the posterior distribution of the class labels.  Intuitively, it seems like $\partial w_j^{(i)}/ \partial \mu_s$ should be non-zero.  After all, changing the location of the mean $\mu_s$ would change your posterior probability for a particular class given the same data.  (Right?).  Just from a mechanical point of view, $\partial w_j^{(i)}/\partial \mu_j$ would seemingly have some non-zero terms since $p(x^{(i)}|z^{(i)}=j) \sim \mathcal{N}(\mu_j, \Sigma_j)$.  If I work through the algebra, I get  $$ \frac{\partial w_j^{(i)}}{\partial \mu_j} = w_j^{(i)}(1-w_j^{(i)})\frac{\partial \psi^{(i)}_{j}}{\partial \mu_j} $$  What am I missing here?  If I take the log-likelihood (without even thinking about EM algorithm),    $$ \ell = \sum_{i=1}^{m}\log \sum_{j=1}^{k}p(x^{(i)}|z^{(i)}=j)p(z^{(i)}=j) $$ and differentiate w.r.t $\mu_s$, I correctly get  $$ \mu_s^* = \frac{\sum_{i=1}^{m}w_s^{(i)}x^{(i)}}{\sum_{i=1}^{m}w_s^{(i)}} $$  so I'm comfortable with the eventual result, just not the intermediate steps. libpng problems for QGIS I'm using ubuntu 18.04 and if I download QGIS 2.18 and I try to run it, I get this:  libpng warning: Application jmp_buf size changed Segmentation fault (core dumped)   If I try with QGIS 3.2, the error is:   libpng warning: Application was compiled with png.h from libpng-1.6.34 libpng warning: Application  is  running with png.c from libpng-1.2.56 libpng error: Incompatible libpng version in application and library   So it's clear that the problem is with libpng.   I purged libpng-dev and reinstall again but nothing changed.   I tried to purge libpng16-16 but I get:  sudo apt purge libpng16-16 Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  libgdk-pixbuf2.0-0 : Depends: libpng16-16 (&gt;= 1.6.2-1) but it is not going to be installed                       Recommends: libgdk-pixbuf2.0-bin  libgtk-3-0 : Depends: libcairo-gobject2 (&gt;= 1.10.0) but it is not going to be installed               Depends: libcairo2 (&gt;= 1.14.0) but it is not going to be installed               Depends: libfontconfig1 (&gt;= 2.12) but it is not going to be installed               Depends: libpango-1.0-0 (&gt;= 1.40.5) but it is not going to be installed               Depends: libpangocairo-1.0-0 (&gt;= 1.40.5) but it is not going to be installed               Depends: libpangoft2-1.0-0 (&gt;= 1.40.5) but it is not going to be installed               Depends: adwaita-icon-theme               Recommends: libgtk-3-bin E: Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.   and then:  ~$ sudo apt install libgdk-pixbuf2.0-bin libgtk-3-bin Reading package lists... Done Building dependency tree        Reading state information... Done libgdk-pixbuf2.0-bin is already the newest version (2.36.11-2). libgtk-3-bin is already the newest version (3.22.30-1ubuntu1). 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. ~$ sudo apt install libgtk-3-0 libgdk-pixbuf2.0-0 Reading package lists... Done Building dependency tree        Reading state information... Done libgdk-pixbuf2.0-0 is already the newest version (2.36.11-2). libgtk-3-0 is already the newest version (3.22.30-1ubuntu1). 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.   At this point I'm afraid of breaking something if I continue trying things, so I come here for advice before.  I deleted and reinstalled ubuntu. I installed first thing qgis and it was working. Afterwards I compiled gdal, from source to have it from python as stand alone. It might be this, but I had done this previously without any problems. counting triangles in a graph or its complement Given a simple graph $G$ [no loops or parallel edges] on six vertices,  let $G^c$ denote its complement. It is known that either $G$ or $G^c$ must contain a triangle  $T$ in it. [An example of a Ramsey number I think.]   My question came up when I tried to find such a graph on six vertices  which had only one $T$ in it or its complement.  I believe if I did things right there are none of these. This I tried to check by drawing a copy of each graph $G$  on six vertices which has a $T$ in it,  and then looking at the complements to see if any had  no $T$'s in them. I found none.  One way to pose my question for a general number $n$ of vertices  is then as follows: Consider the set of all graphs $G$ on $n$ vertices,  and for each one construct $G^c.$  Count the number of $T$ in $G$ and add that to the number of $T$ in $G^c.$  Finally find the minimal such sum for the given $n$  and call that say $M(n).$ So my above six vertex conjecture  is that $M(6)=2.$   By considering a cycle graph on five vertices we find $M(5)=0.$   So going further one could ask if anything is known  about this minimum function $M(n)$ for larger $n$ [such as bounds, etc.] And does it have a name?  I would also like to see a simpler way to handle the six vertex case, one not involving drawing all the graphs. How to find out when an Oracle table was updated the last time Consider numbers of the type $n=2^m+1$. Prove that such an n is prime only if $n=F_K$ for some $ k \in N$, where $F_k$ is a Fermat Prime. Consider numbers of the type $n=2^m+1$. Prove that such an $n$ is prime only if $n=F_K$ for some $ k \in N$, where $F_k$ is a Fermat Prime. Consider numbers of the type $n=a^m-1$ where $a&gt;1$ is a natural number. Prove that such an $n$ is prime only if $n=M_p$ for p-prime, where $M_p$ is a Mersennt Prime.   I do not understand what the question is asking me for. Can I get some clarification? "It looks straightforward, but actually it isn't" I want to determine $$\lim\limits_{(x,y) \to (0,0)} \frac{x^2·y^3}{x^4+y^6}$$  I'm sure the limit exists, it's zero because I tried to find other different limits in line and parabola points ($(x,mx)$ and $(x,mx^2)$) but every time I got zero. So, I have to use the definition or sequences to confirm. Using url package without hyperref makes link hit boxes erratic I've currently got a url link in a footnote in my document. With the hyperref package, everything works correctly. Because the IEEE PDF Express tool doesn't like having the hyperref package, I've removed it and only left in the url package.  However, some strange behavior occurs as a result of this. Instead of just the link being clickable, entire paragraphs of text before/after end up being clickable and going to the same link. This occurs for both Skim and Preview in Mac OSX.  Does anyone have an inkling of what might be causing this? This is a follow-up to an earlier question regarding .  Having achieved the left-alignment goal, I would now like the caption of the subfigure to be bottom-aligned, while having the subfigure itself be top-aligned. As an example, in the figure below, subfigures (a) and (b) would share an uppermost edge, and their captions would share a lowermost edge, and similarly for subfigures (c) and (d). This is most appropriate for the overall layout style of my document and also the content of the subfigures themselves.    \documentclass{article} \usepackage[font=footnotesize]{subcaption} \usepackage[draft]{graphicx}  \begin{document} \begin{figure}[!ht] \captionsetup[subfigure]{justification=justified,singlelinecheck=false} \centering \begin{subfigure}[b]{0.3\textwidth} \includegraphics[scale=0.8]{fig/workflow-S} \caption{workflow $S$} \end{subfigure} \begin{subfigure}[b]{0.5\textwidth} \includegraphics[scale=0.8]{fig/workflow-S-run} \caption{run of $S$} \end{subfigure} \vskip 8pt \begin{subfigure}[b]{0.3\textwidth} \includegraphics[scale=0.8]{fig/workflow-S'} \caption{workflow $S'$} \end{subfigure} \begin{subfigure}[b]{0.5\textwidth} \includegraphics[scale=0.8]{fig/workflow-S'-run} \caption{run of $S'$} \end{subfigure} \caption{Here's a figure} \end{figure} \end{document}   I've experimented with the minipage settings for subfigure, specifically using [t] instead of [b], but it seems to have no effect. I'm unsure as to whether a position of [t] would produce the right effect anyway, or whether it would shift the captions up as well.  Why is [t] not working here, and what's the right way to produce the layout I need?   The solution should generalise to an arbitrary number of subfigures per row, subfigures which are text or tables rather than images, and should not require any hard-coded sizes. I have a significant 2x2 interaction in a between-subjects design.  I would normally report omega-squared as my effect-size measure, but I've been asked to provide Cohen's d instead?  Can Cohen's d be computed for an interaction?  If so, how?