How do gravitons impact on general relativity?
How can gravitions exist without violating GR?
How to show this equality of $\rho(X,Y)$ Let I have $(S,\Sigma,\mu)$ be a probability space, then $X,Y \in \Sigma$. Define $\rho (X,Y)$ by $\rho (X,Y)$ = correlation between random variable $I_X$ and $I_Y$, where $I_X$ and $I_Y$ are the indicator function of $X$ and $Y$. Express $\rho (X,Y)$ in term of $\mu (X)$, $\mu (Y)$, $\mu(XY)$. Conclude that $\rho(X,Y)=0$, if and only if $X$ and $Y$ are independent.      How to show that: $$\rho(X,Y)= \frac{\mu(XY)\,\mu(X^cY^c)-\mu(XY^c)\,\mu(X^cY)}{(\mu(X) \,\mu(X^c)-\mu(Y) \,\mu(Y^c))^{1/2}};0&lt;\mu(X)&lt;1;\,0&lt;\mu(Y)&lt;1$$ Negative energy levels in the diagram for a hydrogen atom Is there any way to erase personal information from my phone that has already been stolen? My phone has been stolen. I do not want the thief to have access to any of my personal information such as calls, texts, and information saved in the phones contents. Is there any possible way I can get this information erased from my phone? Remember last selected items per page in global search Question about how to interpret arbitrary elements I have a question about how to interpret arbitrary elements. When we say:  &quot;let $x$ be an arbitrary integer&quot;  &quot;let $x$ be any integer&quot;  &quot;let $x$ be an integer&quot;   at the beginning of proofs, are we saying:  &quot;let the symbol $x$ represent one of the integers, although it is not clear which specific integer the symbol $x$ represents&quot;.  If not, please let me know how I should think about the above statements. Thank you for your time.  Edit: I had a follow up question to the one I asked earlier. What do mathematicians mean by “arbitrary element” or “any element”. I can understand when mathematicians say “let $x$ be $4$” as this means “the symbol $x$ represents $4$”. Could someone please elaborate on why the specific terms “arbitrary” and “any” are used? How to change color for a block of texts? Installing titling.sty onto an offline Windows machine I'm trying to compile a file for which I need titling.sty:  ! LaTeX Error: File `titling.sty` not found.  The two biggest issues are 1) it's on a Windows machine (I'm really only &quot;comfortable&quot; with the Unix command line) and 2) the machine cannot be connected to the internet (so automatic approaches are impossible). Assuming the error means I'm missing a package and following some related advice (e.g., ), I downloaded the file , transferred it to the offline machine, and unzipped it, then tried to run: latex titling.ins  But this gave me an error (given I can't copy-paste, I'm including what I think may be relevant):  ******************************************************** * No Configuration file found, using default settings. * ******************************************************** ! ************************************************** * Could not find your \batchfile=titling.ins. **************************************************. \processbatchFile ...****************************}                                                  \fi \else \ifDefault \Msg ... 1.1092 \process@first@batchfile  I have no idea how to proceed from this. How can I install titling.sty on this machine? I want to do:  cat update_via_sed.sh | sed 's/old_name/new_name/' &gt; new_update_via_sed.sh   in my program.  But I want to use variables, e.g.  old_run='old_name_952' new_run='old_name_953'   I have tried using them but the substitution doesn't happen (no error). I have tried:  cat update_via_sed.sh | sed 's/old_run/new_run/' cat update_via_sed.sh | sed 's/$old_run/$new_run/' cat update_via_sed.sh | sed 's/${old_run}/${new_run}/' Fire Adaptive Domain rendering like a solid cube When I render I see a solid cube which is Adaptive domain. But it shouldn't show up. I feel like I missed something. It is my first time simulating fire.  Candle with a solid domain blocking the fire    The material seems to be plugged in to the volume already :/  Node setup How do I drag and drop files into an application? I've seen this done in Borland's  environment, but I'm not sure how to go about it for a C# application I'm working on. Are there best practices or gotchas to look out for? At xmas, I had a cup of tea with some debris at the bottom from the leaves.  With less than an inch of tea left, I'd shake the cup to get a little vortex going, then stop shaking and watch it spin.  At first, the particles were dispersed fairly evenly throughout the liquid, but as time went on (and the vortex slowed, although I don't know if it's relevant) the particles would collect in the middle, until, by the time the liquid appeared to almost no longer be turning, all the little bits were collected in this nice neat pile in the center.  What's the physical explanation of the accumulation of particles in the middle?  My guess is that it's something to do with a larger radius costing the particles more work through friction... Why didn't Doc remember Marty's outfit? In Back to the Future 3, Doc (from 1955) sends Marty back in time wearing a cowboy outfit.  But then, Doc (from 1885) doesn't remember that it was him who offered this outfit to Marty.   Does this mean there are two different Docs (from differing timelines) or is it the same Doc but he somehow forgot the outfit. permission reporting on sudoers I'm looking for a good, consistent and easy way to do a report of sudoers permissions for every user and group on the system. For users, I want the report to include permissions granted via group membership. What makes a good jump scare? I am working on a game and would like to include a few jump scares. So here is a rather simple question — What makes a good jump scare and when and how should they happen? Insert into ... values ( SELECT ... FROM ... ) I am trying to INSERT INTO a table using the input from another table. Although this is entirely feasible for many database engines, I always seem to struggle to remember the correct syntax for the SQL engine of the day (, , , , and ).  Is there a silver-bullet syntax coming from an SQL standard (for example, ) that would allow me to insert the values without worrying about the underlying database? Revision history & community wiki links show the same page My pixel perfect picture gets blurry when compiled in LaTeX I have this picture:  (I have the original in Pixen and could export it to several different formats. I went with PNG though since it's lossless.)  When I open my picture in Preview (OS X) and zoom in, it looks fine. The characters have extremely sharp boarders between them and the background. However, when I put it into my LaTeX document, compile it to a PDF (using TeXShop) and then zoom in, it suddenly have gotten blurry (see screenshot: ).  Minimal working example:  \documentclass[12pt]{article} \usepackage{graphicx} \begin{document} \includegraphics[scale=1]{fig1.png} \end{document}   In the end, I would like to be able to scale the picture freely without there being any blurring at all. Show that $\mathbb{Q}\times \mathbb{Q}$ is denumerable I'm brand new to jQuery and have some experience using Prototype. In Prototype, there is a method to "flash" an element &mdash; ie. briefly highlight it in another color and have it fade back to normal so that the user's eye is drawn to it. Is there such a method in jQuery? I see fadeIn, fadeOut, and animate, but I don't see anything like "flash". Perhaps one of these three can be used with appropriate inputs? How can I report bugs with or request features for the Stack Exchange mobile apps? I'm using the official Stack Exchange mobile app for iOS or Android, and  My app crashed! It doesn't look good on my new smartphone. It won't run after I updated my device. I noticed another bug. I have an idea for a new feature which is currently not in the app.  How do I report bugs with or request features for the iOS and Android apps? Also:  I've used the built-in functionality of the app to report a bug or submit a feature request. Did I do the right thing? The app prompted me to post on Meta after an error occurred. Should I post about it here? I'm wondering why the latest version is from 2017. I've heard rumors that the apps are no longer supported; is that true? I can't find the app on my device's app store anymore! Where did it go?     We get a lot of bug reports from users about the mobile apps, which are unsupported, and at the moment, there is no centralized Q&amp;A pair for those to be closed as duplicates of (the relevant answers are spread across multiple unrelated questions). This question is an FAQ proposal, to serve as a directly-related duplicate target for those questions.