How to split a beveled surface to two parts to use different materials
Add different materials to different parts of a mesh?
Open existing file, append a single line Prove that if $A$ is transitive, then $\mathcal{P}(A)$ is transitive too. My book on set theory says     Prove that is a set $A$ is transitive, then $\mathcal{P}(A)$ is also transitive, where $\mathcal{P}(A)$ is the power set of $A$.   I don't know understand how. Let $A=\{a,b\}$. Clearly it is transitive. Now $\mathcal{P}(A)=\{\{a\},\{b\},\{a,b\},\emptyset\}$. This is clearly not transitive. Referencing the reference? Oddly wide spaces in subsection title I have a problem, this is what it looks like:     And this is the code:  \subsection{Bakterienst\"amme und Anzuchtbedingungen}   Where does the huge whitespace come from? If the universe is expanding, what is it expanding into? When the big bang happened where did it occur?  When the big bang happened how did it occur?  Where did the energy come from? Energy can not be created or destroyed does that mean, energy has existed before the universe was here? Center figure that is wider than \textwidth I have a figure that is wider than the \textwidth of my document. (I don't want to change the \textwidth.) As a result, the leftmost part of the figure is flush with the left margin, while the rightmost part spills into the right margin too much. I'd like to center the figure, so that it will exceed the \textwidth equally on both sides. I have the following code  &lt;?php if ($handle = opendir('C:/xampp/htdocs/movies')) {     while (false !== ($file = readdir($handle))) {         if ($file != "." &amp;&amp; $file != "..") {             echo $file."&lt;br /&gt;\n";         }     }     closedir($handle); } ?&gt;   When it does have mb language such as japanese, it doesn't display properly instead it display like kyuukyoku Choujin R ?????~? rather then kyuukyoku Choujin R 究極超人あ～る  Anyway to make it display the correct name or make it still download-able by others?  Thanks for helping me :) Explain why $\mathbb{Z \times Z}$ and $\mathbb{R \times R}$ is not a field Wi-Fi not working on Lenovo ThinkPad E570 (Realtek RTL8821CE) Are there ways to determine if a wall is load bearing? Other than looking up blue prints, which many homeowners may not have, are there ways to determine if a wall is load bearing?  Methods I can think of might include:   Going up in the attic to check if ceiling trusses run perpendicular to the wall If the wall is an exterior wall   Any others? Let's assume a very simplified situation. There is only one seller asking for 10$ and there is only one buyer bidding 20$. At what price would the transaction go?   Some answers say it won't, because the ask price has to match the bid price. () Other answers say, that the time ordering is important. () Which answer is correct? What is the final price in the presented situation? Why is my find not recursive? I am running the following command, but it is not performed recursively:  find . -name *.java   I know there are java files further down in the current directory but it is performing the find on the current directory only. I am using OS X, 10.9. Calculate the last digit of $3^{347}$ I am pretty sure it shouldn't, but want to make sure that I am doing this correctly in a proposal that I am working on.  Should job titles like "Project Manager" ever be capitalized, and if so when?  In particular I am concerned about capitalization in sentences like this where the title refers to a specific person, not the role in general.     "The Project Manager will   evaluate the alternatives and offer a   solution."   I'm a lot more certain that it should NOT be capitalized in sentences like this.     "A project manager is the person who   manages the budget and deliverable for   a project." Lazy loading properties after an insert What happened to Rhaegal? In Game of Thrones S08E03 ("The Long Night"), the Night King's dragon tore at the one Jon Snow rides (Rhaegal). He falls off of it and we don't see the dragon anymore, is it dead? List of generally believed conjectures which cannot all be true There are some conjectures which most leading experts believe in albeit no one can prove it yet. For example: $\mathcal{P} \neq \mathcal{NP}$, the Riemann hypothesis or the Collatz conjecture.  My question is whether there is a set of such conjectures which cannot all be true, eg. is a statement such as "Either $\mathcal{P} = \mathcal{NP}$ or the Riemann hypothesis is wrong" true? Placing math environments inside math environments I have a macro which appears throughout my document in various places:  \newcommand{\placefraction}[2]{     $\frac{#1}{#2}$ }   Sometimes it is placed with regular text. Sometimes it is placed within other math modes. However, the document won't compile if it is placed inside another math mode.  None of my math ever appears in displayed formulas, it is always part of the paragraph.   How can I have a macro which sometimes appears inside math mode, sometimes outside, but itself contains math mode? Missing abstract with amsart class I'm not very good at LaTeX but I've been using it to write my dissertation. Have just come to the end and trying to format everything correctly. Whenever I try to put my Abstract paragraph into the form \begin{abstract} (writing here) \end{abstract} and click play the whole abstract section just disappears. What am I doing wrong?  \documentclass[12pt]{amsart}  \begin{document}  \begin{abstract} (my abstract here) \end{abstract} \end{document} How do I get the Glare node to output transparent instead of Black background