Every emission object is darker than it should be
Why are lamps and mesh lights not illuminating anything?
What's the difference between stochastic and random? Chromatic Number The smallest number of colors needed to color a graph $G$ is called its chromatic number, and is often denoted $\chi(G)$.  Show that if graph $G$ is simple, then $$ \chi \ge \frac{V^2}{V^2 - 2E}. $$ Is "each" singular or plural? You goes with are. Does the presence of each change this?  How satisfied are each of you with the time you've spent in your respective  universities. How satisfied is each of you with the time you've spent in your respective  universities How to keep an SSD in shape on Yosemite? As of OS X 10.10, Apple uses kext signing. Ever since it has become  to manually activate TRIM for third-party SSDs on OS X. That means users with non-Apple SSDs will sooner or later see degrading write speeds.   So I am wondering what are my options to avoid this situation. Some approaches I could think of imply additional problems:   Are you aware of any alternative approach, that does not require disabling kext signing?  Are there tools or tricks to manually clean up a SSD (i.e. issue a TRIM command)? If I decided to temporarily use a tool like TRIM Enabler how long would it need to be activated, in order to clean the disk?   To summarize: No TRIM on Yosemite. What can I do? How to import a module given its name as string? I'm writing a Python application that takes as a command as an argument, for example:  $ python myapp.py command1   I want the application to be extensible, that is, to be able to add new modules that implement new commands without having to change the main application source. The tree looks something like:  myapp/     __init__.py     commands/         __init__.py         command1.py         command2.py     foo.py     bar.py   So I want the application to find the available command modules at runtime and execute the appropriate one.  Python defines an __import__ function, which takes a string for a module name:     __import__(name, globals=None, locals=None, fromlist=(), level=0)      The function imports the module name, potentially using the given globals and locals to determine how to interpret the name in a package context. The fromlist gives the names of objects or submodules that should be imported from the module given by name.       Source:    So currently I have something like:  command = sys.argv[1] try:     command_module = __import__("myapp.commands.%s" % command, fromlist=["myapp.commands"]) except ImportError:     # Display error message  command_module.run()   This works just fine, I'm just wondering if there is possibly a more idiomatic way to accomplish what we are doing with this code.  Note that I specifically don't want to get in to using eggs or extension points. This is not an open-source project and I don't expect there to be "plugins". The point is to simplify the main application code and remove the need to modify it each time a new command module is added. How to find out the terminal command of an application? I want to use some applications shown on the Unity Launcher in a terminal, but I need to know the appropriate command to run them. How can I achieve that? How can I attach a file to a post? I want to attach some files to a post on one of Stack Exchange sites. What is the best way to do this? How to evaluate $\int_{0}^{\pi }\theta \ln\tan\frac{\theta }{2} \, \mathrm{d}\theta$ How to test for equality of means for proportions? And two-sided or one-sided? I have the following problem: two groups A and B.  The proportions of married and single are ($p_1$) and ($p_2$) respectively, and the standard deviations ($s_1$) and ($s_2$). The sample consists of  ($n_1$) A and  ($n_2$) B observations.  Now I want to test whether the proportions are equal. I use the following formula to compute the z-statistic:  z = $abs(p_1 - p_2) / \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} }$  and test the hypothesis: h0: $p_1 \neq p_2$  An alternative approach I tried is to estimate the t-value based on the standard deviations (hence just comparing means), that is by the formula (assuming unequal variances):  t = $abs(p_1-p_2)/\sqrt{ s_1^2/n_1 + s_2^2/n_2 }$  My question are:    do you agree with my approach or; should it be two-sided or one-sided?   Thank you. Is there a shortcode for linking to a user profile in an answer? Sometimes I want to refer to a specific user in an answer.  Something along the lines of      "From the answers of UserA and UserB I have crafted this super useful solution..."   Is there a shortcode creating a link to a user profile using the user name? Similar to the @ we have in the comments, but in an answer. What is the simplest way of proving (to a non-mathematician) that the power set of the set of natural numbers has the same cardinality as the set of the real numbers, i.e. how to construct a bijection from $\mathcal{P}(\mathbb{N})$ to $\mathbb{R}$? I'm using Ubuntu 18.04.  I want to remove the digital clock from the lock screen because I just want to see a good wallpaper there, unobstructed.   How do I go about this? I know how to , but I'd like to add a bunch of different extensions and avoid the tedious job of adding every single one. Is there a command line tool or registry key I can use to accomplish this? I want to run a command when there are only 10 entities left. I know you can use /execute unless entity to detect when all matching mobs have died, but it doesn't work if you want to run a command when there are for example 10 entities left.  A command like /execute if entity @e[limit=10] does not work, it also triggers if there is only one entity.  Such a counting system could also be used to determine once there is only one player left in an arena, meaning they have won a battle or similar. I have a large set of pictures, and I want to apply predefined settings to each using the optional argument of the \includegraphics macro from the graphicx package.  I would like to define the settings once at the top of the file (like \def\settings{width=\textwidth, angle=90}), and then apply them to all of my images so that I can modify all of the settings from one place.  If I run the following code block, it gives me the versions of the error "Package xkeyval Error: 'width=\textwidth, angle=90' undefined in families 'Gin'", which seems to be because it's interpreting my entire group of settings as a single key input.  \documentclass{article} \usepackage{graphix} \begin{document}      \def\settingsA{width=\textwidth, angle=90}     \def\settingsB{page=1, scale=.1}     \def\settingsC{angle=90, scale=.5}      \begin{figure}         \includegraphics[\settingsA]{filename1.png}         \label{fig:oneofonehundred}     \end{figure}      \begin{figure}         \includegraphics[\settingsB]{filename2.pdf}         \label{fig:twoofonehundred}     \end{figure}      \begin{figure}         \includegraphics[\settingsC]{filename3.jpg}         \label{fig:threeofonehundred}     \end{figure}   \end{document}   What is the technique to modify this settings string so that it is interpreted correctly? How do I pass an optional argument that is interpreted as a set of arguments instead of one string?  Thanks in advance! I will be transferring between two international flights at Tokyo Narita airport. I'm scheduled to depart the same day I arrive. I am planning to stay within the airport during the layover. Does anyone know if I will require a transit visa at Narita? What's wrong with this equal probability solution for Monty Hall Problem? I'm confused about why we should change door in the Monty Hall Problem, when thinking from a different perspective  gives me equal probability.   Think about this first: if we have two doors, and one car behind one of them, then we have a 50/50 chance of choosing the right door.   Back to Monty Hall: after we pick a door, one door is opened and shows a goat, and the other door remains closed. Let's call the door we picked A and the  other closed door B. Now since 1 door has already been opened, our knowledge has changed such that the car can only be behind A or B. Therefore, the problem is equivalent to: given two closed doors (A and B) and one car, which door should be chosen (we know it's a 50/50 thing)? Then, not switching door = choosing A, and switching door = choosing B. Therefore, it seems that switching should be equally likely, instead of more likely.   Another way to think: no matter which door we choose from the three, we know BEFOREHAND that we can definitely open a door with a goat in the remaining two. Therefore, showing an open door with a goat reveals nothing new about which door has the car.   What's wrong with this thinking process? (Note that I know the argument why switching gives advantage, and I know experiments have been done to prove that. My question is why the above thinking, which seems legit, is actually wrong.) Thanks. finding examples for a non negative and continuous function for which the infinite integral is finite but the limit at infinity doesn't exist Question:  a. Find an example for a non-negative and continuous function s.t. $\int _0^\infty f(x)dx$ is finite but the following limit doesn't exist: $\lim_{x\to \infty} f(x)$.  b. Is it possible that $\int _0^\infty f(x)dx$ is bounded but $f(x)$ is not bounded?  What we did with A: We suggested the function $|sin(x^2)|$ which has periods that get smaller and smaller until they no longer imply on the sum. But since it's a trig function it doesn't have a determinate limit. Wolfram didn't have the integral value for that func, and we were wondering if it really converges and if it is really a good example.  What we did with B: We thought about the function: $f(x)=  { x\in \Bbb N },{e^{-x} \notin \Bbb N }$ and we had a disagreement whether f(x) is integrable. I said no because similarly to Dirichlet function, one sum's limit will be 0 while the other one's will be infinite. My partner said that if I find a $\delta&lt;1$ then the definition of Riemann's integral does hold and so this integral is equal to that of $f(x)=e^{-x}$ She was trying to say this functions integral will be 0, but still it won't be bounded. I disagreed. Who's right? Due to the implementation of Java generics, you can't have code like this:   public class GenSet&lt;E&gt; {     private E a[];      public GenSet() {         a = new E[INITIAL_ARRAY_LENGTH]; // error: generic array creation     } }   How can I implement this while maintaining type safety?  I saw a solution on the Java forums that goes like this:  import java.lang.reflect.Array;  class Stack&lt;T&gt; {     public Stack(Class&lt;T&gt; clazz, int capacity) {         array = (T[])Array.newInstance(clazz, capacity);     }      private final T[] array; }   But I really don't get what's going on. How to deal with downvotes without any explanation