I am attempting to write out the basic process for finding the absolute value of some number.  In ASCII, this is what I am attempting to write.  |X| = {         x  if x &gt; 0         0  if x = 0         -x if x &lt; 0       }   This is what I have so far.  \begin{equation} \left|x\right| = \biggl\{ x\ if\ x &gt; 0 \biggr\} \end{equation}   I cannot figure out how to add the vertical spacing to allow the other two lines.
I want to write a conditional expression such as the following.    What is the best way to express such conditional expressions in Latex?
How is the Schroedinger equation a wave equation? Wave equations take the form:  $$\frac{ \partial^2 f}  {\partial t^2} = c^2 \nabla ^2f$$  But the Schroedinger equation takes the form:  $$i \hbar \frac{ \partial f}  {\partial t} = - \frac{\hbar ^2}{2m}\nabla ^2f + U(x) f$$  The partials with respect to time are not the same order. How can Schroedinger's equation be regarded as a wave equation?  And why are interference patterns (e.g in the double-slit experiment) so similar for water waves and quantum wavefunctions? Limits, find a limit that exist in absolute value but not outside the absolute value. Let I be an open interval that contains the point c and let f be a function that is defined on I except possibly at the point c. Suppose that lim |f(x)| as x->c exists. Give an example to show that lim f(x) as x->c may not exist.   Not really sure of an example for this? How to disable auto brightening LCD How do I time a method's execution in Java? If $r,s,t$ are prime numbers and $p,q$ positive integers s.t. $\text{lcm}(p,q)$ is $r^2s^4t^2$, then the number of ordered pairs $(p,q)$ is? If $r,s,t$ are prime numbers and $p,q$ are the positive integers such that LCM of $p,q$ is $r^2s^4t^2$, then the number of ordered pairs $(p,q)$ is?  My attempt to the solution:  Let $r^a s^b t^c$ be the LCM where $a=2,b=4,c=2$  Then the cases that arise are  I case   1) $p$ has $r^as^bt^c$ then there are a total of $(2+1)(4+1)(2+1)$ options  II case   1) $p$ has $r^as^b$ or $s^bt^c$,$r^at^c$ then there are total of $(2+1)(4+1)(2+1)(3)$ options.  The answers comes out to be $45*4$ but the answer is $45*5$. Which case am I missing? Why does Google not consider dot in usernames of Gmail addresses? E.g.: If your Gmail ID is abc.xyz@gmail.com, it considers this the same as abcxyz@gmail.com.  Why is this so? I've been battling the following two questions for more than a day today.  Write a regular expression (comprised of {a, b}) that contain at least two b and do not contain abb. Write a regular expression (comprised of {a, b}) that contain abba and do not contain baa.  For the first part I wrote a basic one to start off with that doesn't handle the must two-b: b* OR a* OR (a+ AND b)* OR a*  I ran a few &quot;truth tables&quot; inputs over it and it seems ok. But the moment I add the requirement, things go haywire. Mostly because when pushing in those &quot;b&quot;s, you can suddenly use the OR to break things up. Same thing on the second one.... Any help would be greatly appreciated! Norah What species of insect is this? I saw this insect near a beach in Corsica France: Should I use the datetime or timestamp data type in MySQL? Would you recommend using a  or a  field, and why (using MySQL)?   I'm working with PHP on the server side. I tried the first experiment in chapter 5 in the Metafontbook: ran mf, typed \relax and drawdot (35,70); showit;.  This opened a window entitled Metafont, but containing nothing.  What's missing?  Using TeXlive 2020 on Linux. PS. After adding drawdot (65,70); showit; I see both dots. PS II. Which have now disappeared, after I changed window-manager workspace and back. PS III.  Typing showit; before the first drawdot opens a window and allows the first dot to be seen, but after having switched workspaces and back, I haven't found any way of bringing the picture back, short of drawing something more and showing it. Is it legal to use car manufacturers' names in a game? Is it legal to use car manufacturers' names in a game?   For example:   to describe Audi's quattro drive system? to display a rendering of an Audi? to include a photo (to which I have rights) of an Audi? Smoothing/generalizing polygon in QGIS? Is it possible to smooth or generalize a polygon in QGIS, perhaps by using a plugin like the generalizer-plugin for lines?  I could convert my polygons to lines, smooth them and convert them back to polygons but this is not working for my polygons because of several reasons. How many different/unique $4$-letter arrangements are there of the letters in the word Mississauga?  I'm thinking we need permutations? I already tried finding each letter and still don't get it... any help?   So far I've broken it off to:-  $M=1$  $I=2$  $S=4$  $A=2$  $U=1$  $G=1$  However, I don't really know what to do next. What is the relationship between a chi squared test and test of equal proportions? Suppose that I have three populations with four, mutually exclusive characteristics.  I take random samples from each population and construct a crosstab or frequency table for the characteristics that I am measuring.  Am I correct in saying that:   If I wanted to test whether there is any relationship between the populations and the characteristics (e.g. whether one population has a higher frequency of one of the characteristics), I should run a chi-squared test and see whether the result is significant. If the chi-squared test is significant, it only shows me that there is some relationship between the populations and characteristics, but not how they are related. Furthermore, not all of the characteristics need to be related to the population.  For example, if the different populations have significantly different distributions of characteristics A and B, but not of C and D, then the chi-squared test may still come back as significant. If I wanted to measure whether or not a specific characteristic is affected by the population, then I can run a test for equal proportions (I have seen this called a z-test, or as prop.test() in R) on just that characteristic.   In other words, is it appropriate to use the prop.test() to more accurately determine the nature of a relationship between two sets of categories when the chi-squared test says that there is a significant relationship? Can I run two 4k (3840x2160) displays at 60hz on a Macbook Pro 13" 2015 I think the title says it all. Can I run two 4k (3840x2160) displays, both at 60hz on a Macbook Pro 13' 2015? I understand they might need to be SST and each connected to a Thunderbolt 2 port. But I cannot get a confirmation whether that setup would work.    says:     Simultaneously supports full native resolution on the built-in display and up to 3840 by 2160 pixels on up to two external displays, both at millions of colors.   …   Support for up to 5120-by-2880 resolution at 60Hz on a single external display (model with AMD Radeon R9 M370X only)   Unfortunately, it does not state at what frequency the two displays can be run.    says:     With OS X Yosemite v10.10.3 and later, most single-stream 4K (3840x2160) > displays are supported at 60Hz operation on the following Mac computers: … MacBook Pro (Retina, 13-inch, Early 2015) and later. MacBook Pro (Retina, 15-inch, Mid 2014) and later   Unfortunately it does not state whether two displays can be run at 3840 by 2160 at 60hz How to understand Cauchy's proof of AM-GM inequality(the last step) Howto wildcard SELECT * to query all fields of objects in APEX? I have created sObject and I need all the fields and I have many. Is there a way I can directly get all of them rather then typing each one of them. Scale a div to fit in window but preserve aspect ratio What is the most efficient way to write the old-school:  StringBuilder sb = new StringBuilder(); if (strings.Count &gt; 0) {     foreach (string s in strings)     {         sb.Append(s + ", ");     }     sb.Remove(sb.Length - 2, 2); } return sb.ToString();   ...in LINQ? LM317 Heat sink, how big?