I'm trying to type the following sequence recursively on Overleaf but I'm not exactly sure how to format it exactly as it is shown below. More specifically, I need help on how to type the info past the left brace.
Is there an intelligent way to left-align the parts right of &amp; of the two cases environments shown below (if x >= 0, otherwise, if y &lt; 0, otherwise). I wonder if there is a better way than using manual spaces:  \begin{align*}  u(x) &amp;=   \begin{cases}    0        &amp; \text{if } x \geq 0 \\    1        &amp; \text{otherwise}   \end{cases}   \\  \tau &amp;=   \begin{cases}    a+b+c+d  &amp; \text{if } y &lt; 0 \\    1        &amp; \text{otherwise}   \end{cases} \end{align*}
Are there any examples of subspaces  of $\ell^{2}$ and $\ell^{\infty}$ which are not closed? Control cannot fall through from one case label Is there a typo in the name of the "Joga Bounito" hat? We have currently a hat called "Joga Bounito":    Bounito? Which language is that?  It is clearly a latin-derived language, so not English, German, Arabic, Chinese...   In Portuguese it should be "Joga Bonito" (which means "Plays beautifully"). I'm not sure about Spanish, but I guess that it should be then "Juega Bonito". I'm not sure about Italian either, but I guess that it should be "Gioca Bel". In French (not sure) it should be "Joue Beau". In Romen (really unsure) it should be "Joacă Frumos".   So, the best guess is that it is in Portuguese. Further this hat is basically a soccer ball, and in this year we had the FIFA World Cup in Portuguese-speaking Brazil, which is known for being a country where soccer is the main sport. So "Joga Bounito" would really be Portuguese.  But there is a small problem. It has a typo: It should not be "Bounito", it should be "Bonito". Detecting the system DPI/PPI from JS/CSS? I'm working on a kind of unique app which needs to generate images at specific resolutions according to the device they are displayed on. So the output is different on a regular Windows browser (96ppi), iPhone (163ppi), Android G1 (180ppi), and other devices. I'm wondering if there's a way to detect this automatically.  My initial research seems to say no. The only suggestion I've seen is to make an element whose width is specified as "1in" in CSS, then check its offsetWidth (see also ). Makes sense, but iPhone is lying to me with that technique, saying it's 96ppi.  Another approach might be to get the dimensions of the display in inches and then divide by the width in pixels, but I'm not sure how to do that either. Do we know how Senju Hashirama died? This is a comic book I read no later than 1993 (so is pre-1993).  And probably after 1988 though I can't be certain.  The protagonist is a huge barbarian with an axe, who seems like an old man, who is defending a series of gigantic walls against an invading army, apparently single-handedly. They keep having to fall back to inner walls. He has to get a massage at one point (not that kind of comic btw).  It's not a super-hero or funny comic, it's more of a serious graphic novel.  That's about it, sorry it's so vague. How to Setup GIMP 2.10 on Ubuntu 16.04 (Xenial Xerus) Prime factorizations of consecutive numbers, can they both contains powers of greater than or equal to $1000$? .NET Casting Generic List Accented letters with IM FELL fonts For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$ Meta-analysis of means and medians in R? I've been asked to review the literature for all studies that estimate the mean and median Blood Lead Level (BLL) in a certain country, and then perform a meta-analysis to come up with one overall value for the mean and median level.   Is such a thing typically done? I'm not interested in any "effect size," just a univariate measure.   I'd like to create a forest plot and do some of the standard tests of heterogeneity across studies. Is there a package and function in R that will allow me to do a meta-analysis of mean and median BLLs?  Just to clarify, this would be a separate meta-analysis for means, and a separate meta-analysis for medians. Similar matrices and field extensions What are the minimum contributions required for co-authorship For the command   sudo apt-get update   I get following error:  W: Failed to fetch http://ch.archive.ubuntu.com/ubuntu/dists/maverick-updates/multiverse/binary-i386/Packages.gz   407  Proxy Authentication Required (The ISA Server requires authorization to fulfill the request. Access to the Web Proxy filter is denied. )   I am running Ubuntu 10.10 installed on Windows XP using VirtualBox. For Internet connections I am using proxy server with an authentication. I tried to use gnome-network-proxy tool to set proxy settings system-wide.   After that /etc/environment has been updated by http_proxy variable with the format http://my_proxy:port/, there were no authentication data.  I checked this with Firefox. Browser asked my for login and password and everything was working fine. It was unfortunately not the case for apt-get. I have also tried to do as described . Unfortunately it does not work.  May it be somehow related to the fact that a proxy is in a Windows domain, any ideas?  My proxy name is http-proxy. Is '-' a special character here? Is accessing a variable in C# an atomic operation? I've been raised to believe that if multiple threads can access a variable, then all reads from and writes to that variable must be protected by synchronization code, such as a "lock" statement, because the processor might switch to another thread halfway through a write.  However, I was looking through System.Web.Security.Membership using Reflector and found code like this:  public static class Membership {     private static bool s_Initialized = false;     private static object s_lock = new object();     private static MembershipProvider s_Provider;      public static MembershipProvider Provider     {         get         {             Initialize();             return s_Provider;         }     }      private static void Initialize()     {         if (s_Initialized)             return;          lock(s_lock)         {             if (s_Initialized)                 return;              // Perform initialization...             s_Initialized = true;         }     } }   Why is the s_Initialized field read outside of the lock? Couldn't another thread be trying to write to it at the same time? Are reads and writes of variables atomic? Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar with all major results and counterexamples.  If one replaces the underlying field by a ring, and therefore looks at modules, things become more tricky. Many pathologies occur that one maybe would not expect coming from linear algebra.  I am looking for a list of such pathologies where modules behave differently than vector spaces. This list should not only be a list of statements but all phenomena should be illustrated by an example.  To start the list I will post an answer below with all pathologies that I know from the top of my head. This should also explain better what kind of list I have in mind. Why are the even and odd Regge trajectories degenerate? Durrett Example 1.9 - Pairwise independence does not imply mutual independence? Correct position of "only" Which is grammatically correct?  I can only do so much in this time.  or  I can do only so much in this time.