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Assign the results from a Solve to variable(s)
Show that:  If $U$ is a principal ultrafilter, then the canonical inmersion $j$ is an isomorphism between $\mathcal A$ and $Ult_{U}\mathcal A$ For equal cost, why would you select the MacBook Air over the MacBook Pro? I can see that the 13" version of MacBook Pro and MacBook Air is same $1199, but the Pro is more feature-loaded than Macbook Air. Why would anyone buy MacBook Air? What's the catch in this?    I was thinking of buying a MacBook for development purposes, and worried if there's any problem in the Pro version, because it comes with more features (like more storage).  Edit:    Thanks for the answer guys, I bought a MacBook Pro a week ago and already loving it. Why $\sum_{k=0}^m x^k $ is equal to $\dfrac{x^{m+1}-1}{x-1} $? Who can help me proving this (but you can no longer use $\sum$ in the proof):  $$\sum_{k=0}^m x^k=\dfrac{x^{m+1}-1}{x-1}$$ Every time I try update (using sudo apt-get update) my Ubuntu I get this problem:  W: The repository 'cdrom://Ubuntu 16.04 LTS _Xenial Xerus_ - Release amd64 (20160420.1) xenial Release' does not have a Release file. N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use. N: See apt-secure(8) manpage for repository creation and user configuration details. W: The repository 'ppa.launchpad.net/kirillshkrogalev/ffmpeg-next/ubuntu xenial Release' does not have a Release file. N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use. N: See apt-secure(8) manpage for repository creation and user configuration details. E: Failed to fetch cdrom://Ubuntu 16.04 LTS _Xenial Xerus_ - Release amd64 (20160420.1)/dists/xenial/main/binary-amd64/Packages  Please use apt-cdrom to make this CD-ROM recognized by APT. apt-get update cannot be used to add new CD-ROMs E: Failed to fetch cdrom://Ubuntu 16.04 LTS _Xenial Xerus_ - Release amd64 (20160420.1)/dists/xenial/main/binary-i386/Packages  Please use apt-cdrom to make this CD-ROM recognized by APT. apt-get update cannot be used to add new CD-ROMs E: Failed to fetch ppa.launchpad.net/kirillshkrogalev/ffmpeg-next/ubuntu/dists/xenial/main/binary-amd64/Packages  404  Not Found E: Failed to fetch ppa.launchpad.net/kirillshkrogalev/ffmpeg-next/ubuntu/dists/xenial/main/binary-i386/Packages  404  Not Found E: Some index files failed to download. They have been ignored, or old ones used instead.   How can I fix it? I use Ubuntu 16.04 LTS I need to print an image at a certain size. What dimensions and resolution should I use? (This question is meant to be a resource to direct others to given that this is a common question.)  Given the task of creating an image that will be printed at a certain size, what pixel dimensions and resolution should I be setting my file at? Can I use an inner tube that's too narrow for the tyre? I recently got a puncture and when I went to replace the inner tube I discovered that I'd bought the wrong size tube. My tyre is 700 x 40c, the tube is 700 x 28-35c.  What is likely to happen if I use this tube? Will the tube explode when I attempt to inflate it? "Compared with" vs "Compared to"—which is used when? Is only one of them correct? Are they used in different situations? Or are they interchangeable? How to use \graphicspath? I am using MikTex 2.9 and TeXstudio for my TeX preparation. I would like to set the path to my figures folder, which contains all the figures, plots etc. required for my work.   I used \graphicspath to set the directory, but it gives me an error message:  \documentclass[12pt]{article}  \usepackage{graphicx}  \graphicspath{D:/LATEX/Reports@IIT/figures}  \begin{document} \includegraphics[width=4.5cm,height=4cm,angle=0]{logo.jpg}  \end{document}      Error: Package pdftex.def Error: File `logo.jpg' not found   However, when I prepend the filenames of the images with the same path in the \includegraphics command, it works just fine:  \documentclass[12pt]{article}  \usepackage{graphicx}  \begin{document}  \includegraphics[width=4.5cm,height=4cm,angle=0]{D:/LATEX/Reports@IIT/figures/logo.jpg}  \end{document}   Why is \graphicspath not setting the path? In Nginx, how can I rewrite all http requests to https while maintaining sub-domain? I want to rewrite all http requests on my web server to be https requests, I started with the following:   server {     listen      80;      location / {       rewrite     ^(.*)   https://mysite.com$1 permanent;     } ...    One Problem is that this strips away any subdomain information (e.g., node1.mysite.com/folder), how could I rewrite the above to reroute everything to https and maintain the sub-domain? What's the difference between $\equiv$ and $\leftrightarrow$ in a formal proof? How to get width of the video and scale another video to that width using ffmpeg? I'm using ffmpeg to stack two videos vertically. They might be different size so I need to scale them to same size. I have found following commands.  First one works for different sizes (but won't scale to proper size):  ffmpeg -i input1 -i input2 -filter_complex '[0:v]pad=iw:ih*2[int];[int][1:v]overlay=0:H/2[vid]' -map [vid] -c:v libx264 -crf 23 -preset veryfast output   Second one works if videos have same width and same pixel format:  ffmpeg -i input1 -i input2 -filter_complex vstack output   How to get width of first input inside complex filter argument so second input could be scaled to same width? What would be the command to do so? I would prefer vstack filter since it should be faster than combination of pad and overlay. How to derive variance-covariance matrix of coefficients in linear regression I am reading a book on linear regression and have some trouble understanding the variance-covariance matrix of $\mathbf{b}$:    The diagonal items are easy enough, but the off-diagonal ones are a bit more difficult, what puzzles me is that  $$ \sigma(b_0, b_1) = E(b_0 b_1) - E(b_0)E(b_1) = E(b_0 b_1) - \beta_0 \beta_1 $$  but there is no trace of $\beta_0$ and $\beta_1$ here. Segmentation fault on large array sizes The cardinality of $\{f\subseteq\mathbb R^2\mid f\colon\mathbb{R\to R}\text{ is a function}\}$ is strictly greater than the cardinality of $\mathbb R$ Texmaker generated pdf file not showing the position of cursor I am compiling LaTeX files with Texmaker on Ubuntu 12.04 system.   When I run pdflatex and then view the pdf file (internal), it does not show me the exact place where I am editing on Texmaker i.e. the generated pdf file does not show me the position of the cursor.   But some days before it was working fine. I don't know what causes this problem.  I already tried uninstalling and then reinstalling, I did update and upgrade too. But the problem still persists. Ubuntu 16.04.1 has appeared as .rar file. How do I burn it to a disk to use to install Ubuntu? I am interested in using Ubuntu for the fist time, so I went to the Ubuntu download page (), to get the download file. When the file downloaded he file, it appeared in WinRAR, so I cannot burn the file to a DVD. Can anybody point me in the right direction as to how to burn this file to a DVD when the file appears as a WinRAR file? Any help would be great appreciated. Thanks :) I want OSM administrative boundaries from . I can get the geodata in different formats there, but the attribute data always is a separate JSON file. I want them both in just one file, or to be able to merge the information.   I haven't found a solution to merge the GeoJSON.   What's the workflow to get the two files joined and saved as a shapefile, for example? Which metric to use to evaluate highly imbalance classification model performance I have to do classification model to predict the possibilities of person getting cancer based on certain attributes. The data is highly imbalanced. As per client requirement I have to report model performance based on any one of these 4 metrics AUC_Weighted normalized root mean square log loss Accuracy  Can you suggest me which one should I use? Can I let my fake flash USB drive get its real capacity through low level format? Recently, I bought an USB flash drive which has a cool design.  The seller said it is 256GB however, I've already known that capacity would be fake. Anyway, I bought it cause of its cool design and want to use this USB flash drive anyway.  What should I do?  I've tried to find a proper tool(vid 1908 / pid 1320) but couldn't. Can I repair this through Low level format? Or partitioning this through Linux? Why are there scenes that will not render on GPU but will render fine on CPU? I've already tried GPU Rendering and it seem to be twice as fast as the CPU. However I have some situation where tried to render on the GPU, but it doesn't do anything. When I switch to CPU it renders.  I have some fur texture/materials using particle systems. I wonder if that causes the GPU not to render, I'm not really sure.