I was curious if the curving of spacetime by mass/energy was actually a real thing or is it just a mathematical construct, a way of visualizing the force of gravity and explaining it and that there is not truly a "fabric of spacetime" and this fabric doesn't actually curve. The math simply explains our observations, but isn't literally what is happening.  I'm looking for a pure GR/SR answer to this, not asking about newer theories that build on GR. Did Einstein himself believe spacetime and its curvature was a real physical thing or did he know he was just using math and geometry to explain a phenomenon everyone can see/feel?
I get the curvature tensor in General Relativity, it is “just” math. Does space-time REALLY curves as a tangible thing, or is Einstein proposing a mathematical abstraction?  More naively, please allow, is space-time a real physical “something” like a “new ether”? If yes, does anyone have any idea what is it made off?
What does an icon on the tag mean, for example the "rg" before the sql-server tag as pictured below? Is a bivariate function that is a polynomial function with respect to each variable necessarily a bivariate polynomial? Understanding Group Permissions I'd appreciate help with a couple questions regarding standard file permissions. Assume that I'm not using access control lists.     If a file is associated with multiple groups, must all groups have the same permissions?   If the group permission includes write access, can any member of any group the file is associated with change the file permissions? Can such a person change the file owner?   Say now the file permission is rwxr----- (owner can do everything, but group members can only read). I am in the group associated with the file. I am also the file owner. Which permission am I granted? Read-only, or rwx? Which is correct: "on foot" or "by foot"? He went there on foot.   Or     He went there by foot.   Which one is correct, and on what basis? Is editing sources.list a good idea? This doubt is hitting me since 2-3 days, so I decided to test it myself. What I did is I installed Ubuntu 16.04.4 in VMWare and edited /etc/apt/sources.list and replaced xenial with bionic using:  sudo sed -i 's/xenial/bionic/g' /etc/apt/sources.list   Then I ran  sudo apt-get update &amp;&amp; sudo apt-get upgrade    I was able to upgrade to 18.04, confirmed by using lsb_release -a. However, I encountered some problems like:   apt does not have a stable CLI interface. Use with caution in scripts. unmet dependency of  python3-aptdaemon.pkcompat Release 'bionic-backports' for 'appstream' not found Broken Unity   I solved all the errors and installed gnome and gdm3.   So, I just want to know is it a good idea to edit sources.list? I know that my system might broke but is there any other specific reason not to use this.  Note: I tested this on Ubuntu 16.04 in VMWare to clear my doubts only. 19MB/s Net Speed but Only 2MB/s Usable Hi so i just got internet a about a month ago and my net speed is 19MB/s, I have tested it on loads of sites they all say the same thing(19MB/s), but it wont use more than around 2MB/s i have tried downloading different ways, and with more than 1 download at a time and I still get capped at about 2MB/s. It affects everything like loading websites and playing games as well. So if I'm downloading (max 2MB/s) and I load up a web page, every thing slows down, the download speed slows and web sites take for ever to load up, what can it be that makes it thus, and speed test say's 19MB/s I am unable to print documents using my HP printer. However, I am able to print test pages.  What I tried:  I have tried removing and re-installing the printer from the system settings but I can only print once (the first task). Then all the following tasks get stuck at the "Rendering Completed" status.  I additionally tried restarting the computer which leads in the same task mentioned above.  Moreover, I logged into the CUPS system ( 1.6.2) from http://localhost:631  From there, I removed the printer, restarted it and re-installed with no sucess.  System Specifications:  System : Ubuntu GNOME 13.04  Memory: 3.9 GB  CPU : AMD Athlon(tm) 64 X2 Dual Core Processor 4200+ × 2   OS-type : 32 bit  Graphics: Gallium 0.4 on NVA3  Printer: HP Deskjet 2050 j510 Series  IP Adress: localhost  I had no problems with my previous OS which was Pinguy OS 11.04 ( a distro based on Ubuntu 11.04) and it's the first time I use my printer at this OS.  Screenshots:   My printer stuck at "Rendering Completed" status.     From the CUPS web interface; you can see that "Rendering Completed" is stuck at the state of the print process.  **EDIT:**I tried printing from the pdf.js pdf viewer of firefox: I wait for about 5 minutes for a 10 pages document to print, the document is printed like faded and sort of in bold font (I accuse rendering for that). Then the status of the printer gets stuck at 'Rendering Completed' but I can print later on. Again, for the next prints I wait for 5 minutes minimum ( I wait for the 'Rendering....' bar to fill ) and the printing is weird. Can Reduce *really* not solve for x here? Proof that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then $p$ can't be written as a sum of two squares I'd appreciate your help showing that if $p\equiv3\,\left(\mbox{mod 4}\right)$   then p can't be written as a sum of two squares.  Thanks! ...on Finnis's view all distinct instances of basic goods are incommensurable — none is of more, less, or equal value with any other. &mdash;    I expect to see more or less than any other, but then equal shows up, and is followed by with. But since with any other cannot replace than any other, do you think this sentence illegitimately omits something? Search for singleton tags on questions Is there any way to search for questions that have a specific tag and only that lonely tag?  In other words, I'd like to be able to do something like the following to find questions that have a single tag and no other:  [mytag] tagcount:1   This would be very useful for tag cleanup situations to be able to do house-cleaning before nuking a tag. Most accurate Map of Arda before Beleriand was lost I am making my own Lotr themed Risk game and I am having trouble finding a very accurate map of Arda before beleriand was lost. This map should include Valinor and Numenor as well.  The best I can find so far appears to be a map from David Day's "Tolkien Bestiary"  I hope this is not a bad question, I am new. dpkg error: "trying to overwrite file, which is also in..." How to Store Fresh Herbs Evaluation of $\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7}$ Evaluation of $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} = $$   $\bf{My\; Try::}$ I have solved Using Direct formula::  $$\sin \frac{\pi}{n}\cdot \sin \frac{2\pi}{n}\cdot......\sin \frac{(n-1)\pi}{n} = \frac{n}{2^{n-1}}$$  Now Put $n=7\;,$ We get   $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7}\cdot \sin \frac{4\pi}{7}\cdot \sin \frac{5\pi}{7}\cdot \sin \frac{6\pi}{7}=\frac{7}{2^{7-1}}$$  So $$\sin \frac{\pi}{7}\cdot \sin \frac{2\pi}{7}\cdot \sin \frac{3\pi}{7} =\frac{\sqrt{7}}{8}$$  Now my question is how can we solve it without using Direct Formula, Help me  Thanks Search favorite questions by tag Favorite questions on Stack Overflow I feel is highly unused feature of all the features provided on the site. (This again is my personal opinion)  I have favorited around 90 question till date and it gets very very very difficult if I want to search for a question which I remember I have favorited, which defeats the entire purpose of one favouriting a question, so here is my small suggestion.  Favorites by tags  Screen 1        Screen 2        Please share your opinions on this. Testing whether two regression coefficients are significantly different (in R ideally) If this is a duplicate question, please point to the right way, but the similar questions I've found here haven't been sufficiently similar. Suppose I estimate the model $$Y=\alpha + \beta X + u$$  and find that $\beta&gt;0$. However, it turns out that $X=X_1+X_2$, and I suspect $\partial Y/\partial X_1 \ne \partial Y / \partial X_2$, and in particular, that $\partial Y/\partial X_1 &gt; \partial Y / \partial X_2$. So I estimate the model $$Y=\alpha + \beta_1 X_1 + \beta_2 X_2 +u$$and find significant evidence for $\beta_1,\beta_2&gt;0$. How can I then test whether $\beta_1&gt; \beta_2$? I considered running another regression $$Y=\alpha +\gamma(X_1 - X_2) + u$$ And testing whether $\gamma&gt;0$. Is this the best way?  Also, I need to generalize the answer to many variables, i.e. suppose we have $$Y=\alpha + \beta^1 X^1 + \beta ^2 X^2 + \dots + \beta^nX^n + u$$ where for each $j=1,\dots,n$, $X^j=X_1^j+X_2^j$, and I would like to test for each $j$ whether $\partial Y/\partial X_1^j \ne \partial Y / \partial X_2^j$.  By the way, I am primarily working in R. How can I find the points at which two circles intersect? Given the radius and $x,y$ coordinates of the center point of two circles how can I calculate their points of intersection if they have any? Vertex cover in a graph with maximum degree of 3 and average degree of 2. "invalid use of incomplete type" error with partial template specialization The following code:  template &lt;typename S, typename T&gt; struct foo {    void bar(); };  template &lt;typename T&gt; void foo &lt;int, T&gt;::bar() { }   gives me the error  invalid use of incomplete type 'struct foo&lt;int, T&gt;' declaration of 'struct foo&lt;int, T&gt;'   (I'm using gcc.) Is my syntax for partial specialization wrong? Note that if I remove the second argument:  template &lt;typename S&gt; struct foo {    void bar(); };  template &lt;&gt; void foo &lt;int&gt;::bar() { }   then it compiles correctly.