Rearrange dataframe to a table, the opposite of "melt"
How to reshape data from long to wide format
Application of Liouville's Theorem Let $f(z)$ be an entire function such that $$|f(z)|&lt;\frac{1}{|\text{Im}(z)|},\qquad z\in\Bbb C-\Bbb R.$$ The question asked me to prove that $f(z)=0$. At least looking at it, it really seems to have an application of Liouville's theorem lurking around somewhere, but I haven't found it.   My thoughts first led me to think about doing this by contradiction and using Picard's little theorem. So, I've considered a strip containing the real axis (say of width $2$ for simplicity). This will imply the complement maps into the unit disk, which implies that this strip has to map almost everywhere else on the complex plane. Now, on the imaginary axis, I know that $f(z)$ will vanish as $z\to\infty$, and this seems like it might be useful. I'm sort of under the impression $f(z)$ might have an essential singularity at $\infty$, which would mean $f(1/z)$ has an essential singularity at $0$. Either way, it of course can't have a pole at $\infty$ because of $f(z)$ vanishing on the imaginary axis, and if it is a removable singularity, it must be $0$, which still gives the solution. Therefore, I think it must have something to do with $f(z)$ having an essential singularity at $\infty$.   I was hoping to combine this with the above inequality and deduce a contradiction, but I haven't thought of one yet.   Any hints or suggestions? I'm studying for a prelim. exam, so this isn't homework.  Update: So, in the spirit of searching methods using the essential singularity at $z=0$, we have $$\left|f\left(\frac{1}{z}\right)\right|\leq\frac{1}{\left|\text{Im}\left(\frac{1}{z}\right)\right|}=\frac{|z|^2}{|y|}.$$ Thus, for $|y|&gt;1$, we have $$\left|f\left(\frac{1}{z}\right)\right|&lt;|y|\left|f\left(\frac{1}{z}\right)\right|&lt;|z|^2.$$ Thus, if I can show that $|f(1/z)|$ is a polynomial, we'll know it can't have an essential singularity at the origin, concluding my proof. How do I install additional drivers? I've just installed Ubuntu and I need either video, wireless, or some sort of other manufacturer (non-opensource) driver. How can I run the tool that will fetch all of these for me? I'd like graphical instructions.  Related: I accidentally posted a question without being logged in. Is there a way to associate that question with my account?   For more information, visit &quot;&quot; in the .   a.k.a. How can you link a registered account to a cookie-based account? a.k.a. How do I merge an unregistered account with a registered account a.k.a. Merging users a.k.a. How do I associate anonymous questions I’ve asked with my registered account? a.k.a. Is there a way to claim an unregistered user? (in case anyone searches with those terms) Prove that $\mathrm{Res}[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$ I need to prove that if $f$ and $g$ are analytic in $D_r(z_0)$ and $g$ has a simple zero at $z_0$, then $$\mathrm{Res}[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$$   When $f(z_0)\neq 0$ and since $1/g$ has a simple pole at $z_0$ $$\mathrm{Res}[f/g,z_0]=\lim_{z\to z_0}(z-z_0)\frac{f(z)}{g(z)}=\frac{f(z_0)}{g'(z_0)}$$ But what do I need to do when $f(z_0)=0$? Any help will be appreciated thanks While building things I want to see the overlays from other modes so I can make better decisions. Can I lock an overlay somehow?  For example: When building roads I want to see the natural resources overlay. When zoning I want to see the wind overlay At  I've read that being $\sin(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$ by using  we get $\sin'(x)=\sum_{n=0}^{\infty}(2n+1) \frac{(-1)^n}{(2n+1)!}x^{2n}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}=\cos(x)$ but the page  says that if $f(x)=\sum_{n=0}^{\infty}c_n(x-a)^n$ then inside the radius of convergence of the power series we have $f'(x)=\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}$ so shouldn't it be $\sin'(x)=\sum_{n=1}^{\infty}(2n+1)\frac{(-1)^n}{(2n+1)!}x^{2n+1-1}=\sum_{n=1}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}$?  Now I understand that this last result has a missing term due to the index of the series being $1$ and not $0$ but being the $1$ mandatory because of the above theorem and since trying to shift index doesn't help (we get $\sin'(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{(2n+2)!}x^{2n+2}$) how can the result that ProofWiki shows be rigorously justified?   (I mean, I think we can't just say "oh, there's a missing term!" after having applied the theorem and magically shift the index of the series to obtain the desired result) -> 683  -> 678  ???  (Doesn't really bother me too much but figured I'd report the bug...) What is *.pkinfo file? What is a *.pkinfo file?   Google Search results talk about Package Info files and how they contain map documents and layers, so I don't see how this is different from a .mpk file.    I also have not found anything about how to create a *.pkinfo file. Let mods see the names of the flaggers for chat offensive flags There have been    (not to pick on the Bridge, just that these are the ones I found) occurrences where users abuse chat flags to cause drama. Random stuff gets flagged, and users from other rooms are needlessly summoned into the room. The issue is, while mods can tell the room to stop flagging random things, there's no way to enforce this. Spam/offensive flags in chat are anonymous, and can be abused with impunity without the mods finding out. I don't think there's anything stopping a user from letting out an endless stream of flags, annoying everyone. Can mods have the ability to see who flagged a message in chat? Evil thoughts without actions? Is there a device that could measure the speed of time? Is there (or can there be) a device that could measure the speed and acceleration of time? Does an electrolytic capacitor degrade each time it receives reverse voltage? In the following question  the notation $c\in ]a,b[$ is used. What does this mean? I have never seen it before. Prove that $||x|-|y||\le |x-y|$ I've seen the full proof of the Triangle Inequality  \begin{equation*} |x+y|\le|x|+|y|.  \end{equation*} However, I haven't seen the proof of the reverse triangle inequality:  \begin{equation*} ||x|-|y||\le|x-y|. \end{equation*} Would you please prove this using only the Triangle Inequality above?  Thank you very much. How to make sense out of the $\epsilon-\delta$ definition of a limit? The informal intuition for the limit of a function is this:     What is the value of the function $f$ as $x$ gets infinitely close to $c$?   How on earth does this monster  $$ \lim_{x \to c} f(x) = L  \iff  (\forall \varepsilon &gt; 0)(\exists \ \delta &gt; 0) (\forall x \in D)(0 &lt; |x - c | &lt; \delta \ \Rightarrow \ |f(x) - L| &lt; \varepsilon)$$ capture our intuition?  I want the answer to explain and motivate every bit of this formal statement. Remember: I am able to parse the statement, I just don't see how it captures our informal intuition. Should I do anything when I see several references that are not used in the text? In a linguistics paper I read, I noticed that several references in the bibliography are not actually used in the text. This is for about 10-15% of the references the case. Should I do anything with this? I always thought that all references must be used in the text, because even if they are just suggestions for further reading you would want some explanation why it is relevant and what you can find where.   I only know the author through their work and do not know the editors. The paper is from 2011. The references are rather general. They are relevant to the paper, but you would want page numbers with them unless you are familiar with them. They are all from different authors and different institutes. It is likely that the references were used in previous versions of the text. wp_customize_image_control default value I am trying to create a custom theme using Wordpress 3.4's customization options. I would like to create an option to change the theme's logo, but I would like to also display a default logo image.   I am using the following code in my function.php page:  $defaultbranding = "get_bloginfo('template_directory') . '/images/logo.png";      $wp_customize-&gt;add_setting( 'change_branding', array(         'default' =&gt; get_bloginfo('template_directory') . '/images/logo.png',     ) );      $wp_customize-&gt;add_control( new WP_Customize_Image_Control( $wp_customize, 'change_branding', array(         'label'   =&gt; 'Image Control',         'section' =&gt; 'theme_settings',         'settings'   =&gt; 'change_branding',     ) ) );   And in the theme's header.php page:  src='&lt;?php echo get_theme_mod( 'change_branding' , 'default' ) ?&gt;'   I would like the 'default' value on the front end to be:  get_bloginfo('template_directory') . '/images/logo.png'   Is there some way I can achieve this?  Thanks in advance for any help. Why bother with the dual problem when fitting SVM? Equivalent Definitions of the Operator Norm How do you prove that these four definitions of the operator norm are equivalent? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &amp;= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\ &amp;=\sup\{ \lVert Av\rVert\;\colon\; v\in V\text{ with }\lVert v\rVert\leq 1\}\\ &amp;=\sup\{\lVert Av\rVert\;\colon\; v\in V\text{ with }\lVert v\rVert = 1 \}\\ &amp;=\sup\left\{ \frac{\lVert Av\rVert}{\lVert v\rVert}\;\colon\; v\in V\text{ with }v\neq 0\right\}. \end{align*}$$ When there are X New Questions, the views don't update untill you refresh the page