I have to write \Leftrightarrow quite often, and i am currently using a macro for it, but it isn't working too well. Is it possible to make it so that when i type LRA in a math field, that it then outputs it as a \Leftrightarrow? Thanks a lot for any and all help!
I am very new to being able to use LaTeX to type up mathematical papers, and I just got my "system" set up. Currently I have installed TexWorks, and MikTex, and I have put together a 'hello world' article which seems to run well.   I have also used the syntax "\usepackage{amsmath}" and "\usepackage{amssymb}" in the beginning of my .tex file to write math symbols, only because some online tutorial said so.   My question is, firstly, are those the only packages I need to use for writing math, and secondly, I remember reading about some shortcuts(?) that can be used for writing math symbols, and/or user defined shortcuts that can be 'set' in the beginning so as to reduce the amount of LaTeX code needed to write mathematical symbols. However, I do not know if this is true, and if it is, how to go about 'installing' such a thing.   Does such a thing exist? Am I able to make my own shortcuts?  Thanks!
"It looks straightforward, but actually it isn't" In , I asked about proof of statements which are simple but incorrect.   Here, I ask about statements which seems, at a first glance, straightforward, but if we try to write a proof, we can see it's much harder than it looked. So I expect the answers to contain:   the statement; why it looks easy to prove; why actually it isn't. My dad's old TV! When I switch it off after watching and get my hands closer, my hands' hair gets attached to the screen. And they stick for sometimes! Now, I know how a cathode ray oscilloscope works. It is just the clever use of Hall - Effect . But why does my hair get stuck to the screen? What is the cause? Note, I checked this for my LED TV but it didn't work (I couldn't perceive anything) unknowingly. What's the magic in my old TV?? What is the Taylor series of $$\ln \frac{1-x^2}{1+x^2}$$   ?   I started by evaluating the first derivatives but the more I go, the more complicated they are and I can't identify a pattern. How do I bake a texture using Cycles bake I created a interior scene and  i don't know how to bake a texture using cycles bake. Can I mention professors' names in the statement of purpose? For PhD applications, I have mentioned that I am interested in some research areas. Besides, I am thinking about mentioning some professors' names, so that it shows I have read their websites and I am not just sending out generic statements applicable to many schools.  But is it good to mention some professors' names in the statement of purpose, if I don't know them personally? In what cases is it good, and in what cases it isn't?  For example, is it good to only mention    those whose research directions are aligned with my research interests, or/and those whose books or notes I have heard of, or better, might have read some parts of, or/and those who are already established and famous (such as having certain titles), or/and those that are directing the PhD graduate programs, or/and those that are heads of the departments?    What will other professors who are not mentioned think? How will you do, if it were you? Is it possible to have site-specific badges? Does Earth's Rotation Affect Its Shape? How to ensure that all traffic goes through VPN connection once it has been started? I connect to the internet through a VPN provider along with other precautions in order to ensure privacy (against traffic analysis by agencies of a repressive government). I currently use  to configure and establish the VPN connection. Tunnelblick does not allow automatically connecting to the provider at or before login so I need to connect manually each time, and sometimes the connection is lost.  I need any and all traffic to go through the VPN connection at all times; whenever the VPN isn't connectable for whatever reason, I want the fallback to be no connection. I need to ensure   that OS X does not connect to the internet on my regular internet connection, or at all, during boot (until the login screen, for time synchronization and other possible "call home" stuff etc.). What services, if any, try to do this, and how can I practically analyze and change what's going on with networking during boot? Is there a way to connect to VPN earlier than right after login? that if my VPN connection is temporarily lost, applications do not continue to communicate over my regular unencrypted connection.    I basically want the system to act as if the network connection is lost altogether if the VPN connection fails, but currently they just continue to work as if nothing happened. What can I do to my system so that once a VPN is established, all traffic can only flow through that connection and no traffic can flow if the connection drops? submitting a GET form with query string params and hidden params disappear How to make a press proof with crop marks and color bars? A standard proof for press needs crop marks and color bars as below    With the package crop, the crop marks can be easily added.  But how about the color bar ?  Is there a way or a package to make it ?  An example is   \documentclass[b5paper,10pt]{article} \usepackage[utf8]{inputenc} \usepackage[cross,a4,center,info,axes]{crop}  \title{Test} \author{User} \begin{document} \maketitle \section{Fox} The quick brown fox jumps over a lazy dog. \end{document} This post is in reference to the following blog post:   To start off, a quick introduction of myself; my name is Jascha Drel, Product Manager for Advertising at Stack Overflow. In this role, I am part of ’s organization.   Our team is continuously making improvements to the ads experience on our sites. One of the changes we’re excited about sharing is the new Report this Ad feature now available on some of our sites. We want to continue to give our communities more native control over their ad experience. We needed a quantifiable way to receive feedback and this feature will allow that.  We enforce strict  on the types of advertisers we work with but cannot fully control what is shown due to how advertising delivery works. However, being able to hear from our users about problematic ads will be a huge benefit as we can address those problems faster and with more accuracy. The Report this Ad feature will allow us to quickly see and investigate ads reported by you which matters to us because we want to hear your thoughts.  Furthermore, this new feature ensures that the members of our communities have a built-in way to report ads they consider problematic when we test any new ad formats/solutions.  We are launching this feature on 5 sites initially (, , , , and ) to get a good understanding of what we can expect in terms of numbers of reports whenever launching this to the entire network. Prior to launching it network-wide, we’ll share general results we’ve seen here.  What happens once an ad is reported?  All reports will be consolidated into a dashboard that the team will look at several times per week to spot irregularities in terms of the total number of reports, report categories, and on an advertising campaign basis. We will not be replying to individual reports, but we will reach out to our advertisers, to share feedback or notify them if the ad being reported does not fit our guidelines. We may also evaluate our policies in response to large numbers of reports not covered by them. Since we don’t have a baseline for the number of ads that are going to be reported using this new feature, we will consider reevaluating the process if the current process doesn’t scale.    We’d like to share an update on the progress of this feature:   We've launched this feature on 5 SE sites (Server Fault, Database Administrators, Arqade, Game Development, and Unix &amp; Linux) initially for logged in users only, and for anonymous users afterward. The report this ad button was shown several million times (of which roughly 65% paid ads and 35% house ads). We showed several dozens of different ad campaigns. So far we have received 0 reports. We've tested the feature several ways, found some minor bugs (which have been/are being fixed), however, in the vast majority of the scenarios the feature works well (we’re happy to be ).   We’re currently graduating this feature and are releasing it network-wide this week. Feel free to comment below if you have any questions/comments. Play audio with Python I am running Ubuntu Gnome 16.10 (64bit). Gnome-shell version 3.20.4.   When I start my computer, I eventually see GDM3 offering a login prompt, as expected.   After I login, my monitors flash a little (like they're unplugged and then replugged in), and then I am brought to the gnome 3 desktop.   However, if I hit  CTRL + ALT + F1, I am brought back to GDM3, offering a login prompt.   If I then hit CTRL + ALT + F2,  I am brought back to my Gnome 3 desktop.  So it seems like GDM3 continues to run in TTY1 while the desktop runs in TTY2.   Questions   Is this how things are supposed to work? I thought the desktop was supposed to use the same TTY (or whatever you call it) as the display manager.  Am I using more graphical resources by having things setup this way? Is there a way to have GDM3 pass off the display to Gnome Desktop after I login? Is there any advantage do doing #3? The weighted sum of two independent Poisson random variables Using wikipedia I found a way to calculate the probability mass function resulting from the sum of two Poisson random variables. However, I think that the approach I have is wrong.  Let $X_1, X_2$ be two independent Poisson random variables with mean $\lambda_1, \lambda_2$, and $S_2 =  a_1 X_1+a_2 X_2$, where the $a_1$ and $a_2$ are constants, then the probability-generating function of $S_2$ is given by $$ G_{S_2}(z) = \operatorname{E}(z^{S_2})= \operatorname{E}(z^{a_1 X_1+a_2 X_2}) G_{X_1}(z^{a_1})G_{X_2}(z^{a_2}). $$ Now, using the fact that the probability-generating function for a Poisson random variable is $G_{X_i}(z) = \textrm{e}^{\lambda_i(z - 1)}$, we can write the probability-generating function of the sum of the two independent Poisson random variables as $$ \begin{aligned} G_{S_2}(z) &amp;= \textrm{e}^{\lambda_1(z^{a_1} - 1)}\textrm{e}^{\lambda_2(z^{a_2} - 1)}  \\ &amp;= \textrm{e}^{\lambda_1(z^{a_1} - 1)+\lambda_2(z^{a_2} - 1)}. \end{aligned} $$ It seems that the probability mass function of $S_2$ is recovered by taking derivatives of $G_{S_2}(z)$ $\operatorname{Pr}(S_2 = k) = \frac{G_{S_2}^{(k)}(0)}{k!}$, where $G_{S_2}^{(k)} = \frac{d^k G_{S_2}(z)}{ d z^k}$.  Is this is correct? I have the feeling I cannot just take the derivative to obtain the probability mass function, because of the constants $a_1$ and $a_2$. Is this right? Is there an alternative approach?  If this is correct can I now obtain an approximation of the cumulative distribution by truncating the infinite sum over all k? Show that G is a group, if G is finite, the operation is associative, and cancellation law holds Let $G$ be a non-empty finite set with an associative binary operation so that cancellation law holds, i.e. $ab=ac$ or $ba=ca$ implies $b=c$, for any choices of $a,b,c$ in $G$. Assume that there is an identity element $e$ in $G$. Show that $G$ is a group.  Proof: To show $G$ is a group, conditions must hold. Suppose that $ab=ac$, then $b=eb=(a^{-1}a)b =a^{-1}(ab)=a^{-1}(ac) =(a^{-1}a)c=ec=c$. So left cancellation holds in $G$.  2) Suppose $ba=ca$, then $b=be=b(aa^{-1}) =(ba)a^{-1}=(ca)a^{-1} =c(aa^{-1})=ce=c$. So right cancellation holds in $G$.  3) If $a$ exists in $G$ then $aa^{-1}=a^{-1}a=e$, where $a$ is an inverse of $a^{-1}$. Since inverses are unique, $(a^{-1})^{-1}=a$.  4) Let $x$ be the inverse of $ab$. Then $(ab)x=e$. By associativity we have $a(bx)=aa^{-1}$. Through left cancellation we have  $bx=a^{-1}bx=ea^{-1}=b(b^{-1}a^{-1})$ and $x=b^{-1}a^{-1}$.  Thus $(ab)^{-1}=b^{-1}a^{-1}$. So all conditions hold, $G$ is a group.  Is this proof correct? I know to show $G$ is a group these conditions have to be met. This is all I have to show right? Footnote to a footnote How can I make a footnote to a footnote? E.g.,  Playfair's axiom\footnote{so called in honor of mathematician John Playfair, etc, etc., but was actually discovered by Proclus\footnote{Greek mathematician, etc, etc.}} How to place a float at the top of a floats-only page? I just got a short algorithm on my page, and it keeps getting aligned at the center of the page. But I want to have it aligned at the top, what do I have to do? Is it possible to get the data off of the hard drive of a PlayStation 3 that has crashed? I have already voided the warranty so I am not worried about that and I have access to quite a few tools.   I have a new PlayStation and a few saves that were not backed up. I am not worried about them but if it is possible I would love to get it moved over.   The old PlayStation will not turn on at all and I think I am going to have to pull the drive. My main concern is if I am able to transfer the files without running into any issues. I have heard that some of the stuff is specific to each PlayStation, is that true? How is light affected by gravity? After upgrading from Ubuntu 13.04 server, the www pages won't display themselves.     Error 403 Forbidden You don't have permission to access / on this server.   I've changed access control saves from apache2.2 to apache2.4, but that didn't work. What else could I do?