Speed of different colours of light
Why do prisms work (why is refraction frequency dependent)?
Complex analysis : If $z =re^{i\theta}$, then prove that $|e^{iz}| =e^{-r\sin\theta}$ Problem :   If $z =re^{i\theta}$, then prove that $|e^{iz} | =e^{-r\sin\theta}$   My working :   $z = re^{i\theta} = r(\cos\theta + i\sin\theta)$   $\Rightarrow iz = ir(\cos\theta +\sin\theta) $ = $-r\sin\theta +ir\cos\theta $  $\Rightarrow e^{iz} =e^{(-r\sin\theta +ir\cos\theta)} = e^{-\sin\theta} e^{ir\cos\theta} $   $\Rightarrow |e^{iz}| = |e^{-r\sin\theta}||e^{ri\cos\theta}|$   Now please guide how to proceed further to get the result... thanks. Adding PDF (especially GeoPDF) to ArcMap as data? I receive a lot of PDF maps (vector, image, with georeference and without) and I am having to convert them to an image file before I can add them to ArcMap and georeference.    I can't see any obvious out-of-the-box utility that would allow me to add and georeference a PDF directly without having to convert to an image file first.   Any ideas?  I would prefer to add a PDF to map directly without any prior "pdf to image" conversion Say we have the following sample and we are trying to estimate the variance of the sample mean of the population.  X = [0, -1, 2, 10, -3]   If I take an increasing number of bootstrap (e.g. 100, 1000, 10,000, 100,0000), the bootstrap distribution of the sample mean gets narrower and narrower. In other words, I can make the bootstrap distribution arbitrarily narrow.   Given the above, what's the point of using a different number of bootstrap samples here? And wouldn't this mean that I could make the standard error of the mean arbitrarily small by just taking more and more bootstrap samples? What am I missing?  Please note that I am doing a proper bootstrap sample in the sense that each bootstrap sample has the same exact size as the original sample (input array). Can newsletters take into account a user's favorite tags on that site? Stack Overflow is huge and varied. Wouldn't it be great if I could just subscribe to a newsletter with a list of my favorite tags? If I love dealing with , I think it would make sense for my version of the newsletter to focus more on Android questions.  Or, if I sign up for the  newsletter, but don't own a Mac, it would be nice if I my newsletter would focus on the part of the site I do follow (i.e., iOS).  So I would like to suggest that the newsletters for each site (optionally) take into account the user's favorited and ignored tags for that site.  I know that you can receive emails about specific filters, but I don't think that's a good solution for users who already receive the newsletter. Most of the questions it highlights cover topics that I have no interest in. It would be much better if I could filter out all the noise and get down to my likely favorites. Missing new file in Ubuntu 13.04 The New file operation is missing in Ubuntu 13.04. How can I fix it?  I have only &quot;New Folder&quot;. How can I add the New File option again? How do you backup all your emails, by copying the 'Outlook' folder? (Outlook 2013) [...] All versions of Outlook use pst files, but the disk location has changed over the years. Right-click your email account on the left in Outlook and select Data File Properties. Click Advanced to see the filename. Only the pst file is essential, but copy the whole folder to back up Outlook’s settings and preferences too. Just copy it back to restore it, or click File, Open, Open Outlook Data File. Outlook can work with any pst file stored anywhere.  Backing up your Outlook data file(s) is as easy as copying your .OST or .PST files over to another hard drive, cloud server, thumb drive, or some other storage medium. You remember how to locate your data files? Open the Mail control panel and click the “Data Files” button. Click “Open File Location…” and you can then back up your data files. [I don't display  here)  After following the instructions above, I arrive at:  C:\Users\[my username]\AppData\Local\Microsoft\Outlook  However, all Outlook Data Files here are .ost, and NOT .pst. In  File &gt; Open &amp; Export &gt; Open Outlook Data File, 'Open Outlook Data File' only works for .pst. What is wrong here? Where's the folder with all the .pst files? How can I backup all my Outlook email accounts simultaneously? What techniques can be used to define a class in JavaScript, and what are their trade-offs? I prefer to use OOP in large scale projects like the one I'm working on right now. I need to create several classes in JavaScript but, if I'm not mistaken, there are at least a couple of ways to go about doing that. What would be the syntax and why would it be done in that way?  I would like to avoid using third-party libraries - at least at first. Looking for other answers, I found the article  that discusses object-oriented programming in JavaScript. Is there a better way to do inheritance? Resize (enlarge) an image without resampling (in favor of quality loss)? How to resize an image or selection without resampling the pixels? I have some pixel art that I want to enlarge without losing the hard, non-anti-aliased, pixel edges. In other words, 1px should become 4px of the same color— fake big pixels. Is there any way to achieve this procedurally?  Example (of what I don't want):  before resize:    after resize:    Open to other tools/methods beyond photoshop, thanks in advance. Flag Options Missing - Is this a bug? This is normally what I see when flagging a post:   Which then provides the following options:    I have come across this problem before. Where I have wanted to flag a question, as not belonging here, but only have limited options.    When this happens, I use a custom flag and write in the reason, but am not happy doing so.  Is this a bug?   What is the best thing to do? During Winterbash 2014, there was a hat called , which was earned by giving an answer scoring 5 or better to question that had a score of −3 or worse at the time of answering and later rose to a score of +3 or better.  I’ve already shared the . Those cases give me a great deal of confidence that this would be a challenging badge trigger that would produce good behavior with a minor tweak. Here are the Red Barons earned on other sites:    (Logan Maingi) Mathematics  (Mike Miller) Mathematics  (Hugo) English Language and Usage  (tchrist) English Language and Usage  (Mr. Shiny and New 安宇) English Language and Usage  (Darth Satan) Science Fiction and Fantasy  (DVK) Science Fiction and Fantasy  (rolfl) Code Review  (Heslacher and Hosch250) Code Review  (Edward) Code Review  (janos and Manny Meng) Code Review  (Mark Mayo, Rory Alsop and MeNoTalk) Travel      (DJClayworth) Christianity  (McMagister) Puzzling   Looking through the timelines, I’m convinced that the answerer’s votes should be excluded from the pre-answer question score calculation. Doing that would have reduced the number of Red Barons to 16 across the network. However, the cost of making a badge harder to obtain is worth the benefit of avoiding giving people an incentive to manipulate question scores¹.  To wrap up, the hat seemed to have encouraged a number of great answers and rescued some questions that would otherwise have been lost to time. My outstanding questions are:   Were there any problems on questions where people were attempting to earn the hat, but failed? (Or any other problems I didn’t observe.) If not, should we  or create a new badge? If we create a new badge, what should it be called?   I’m especially interested in feedback that counters my own bias in favor of this idea. If I’m missing any problems, I’d like to know about them now.    ¹ Several questions were downvoted, answered, edited (sometimes trivially)  and upvoted by the user who earned the hat. My book (Concepts of Physics by H.C. Verma) writes:  It has been reported () that the force between two masses may be better represented by $$F = \frac{G_{\infty} m_{1} m_{2}}{r^2} \left[1 + \left(1 + \frac{r}{\lambda} \right) \alpha e^{-\frac{r}{\lambda}}\right]$$ where $\alpha \approx - 0.007$ and $\lambda \approx 200~\mathrm{m}$.  What is this? Such a horrendous formula! So, what about Newton's? And what's the difference between $G$ &amp; $G_{\infty}$? \mathbb{Z} yields undefined control sequence error Why use a dedicated light meter instead of the one built into the camera? Prove that every number ending in a $3$ has a multiple which consists only of ones. Prove that every number ending in a $3$ has a multiple which consists only of ones.      Eg. $3$ has $111$, $13$ has $111111$.   Also, is their any direct way (without repetitive multiplication and checking) of obtaining such  multiple of any given number( ending with $3$) ? Is a join optimized to a where clause at runtime? For each positive integer $n$, let $x_n=1/(n+1)+1/(n+2)+\cdots+1/(2n)$. Prove that the sequence $(x_n)$ converges.   This is what I have so far..  Let $\epsilon &gt; 0$ be given to us. We must show that there exists an $N \in \mathbb{N}$  such that $n\ge N \implies \left|\frac{1}{2n} - 1\right| &lt; \epsilon$. Choose $N$ to be any positive integer which is larger than $\frac{1}{\epsilon}$. (So $N&gt;\frac{1}{\epsilon}$.) Then $n\ge N \implies \left|\frac{1}{2n} - 1\right| = |\frac{1-2n}{2n}|$ When is a cellular automaton "bidirectional"? How to stop event propagation with inline onclick attribute? Consider the following:  &lt;div onclick="alert('you clicked the header')" class="header"&gt;   &lt;span onclick="alert('you clicked inside the header');"&gt;something inside the header&lt;/span&gt; &lt;/div&gt;   How can I make it so that when the user clicks the span, it does not fire the div's click event? Install App Image such as beaker browser and pin it to dock (favorites) in Ubuntu 20.04? Differences in the behaviour of pinching a garden hose and closing a tap Let's say you have a garden hose connected to an ordinary water tap which is opened fully. If you pinch the end of the hose, water leaves the hose at a higher speed (and this can be useful while watering plants, to reach pots which are further away). However when a tap (with no hose connected) is opened only slightly, water flows out at a low speed, possibly even in drops.  The actions of pinching the end of a hose and of almost-closing an open tap seem similar, so why the difference in behaviour?