I've always heard that gravity warpes spacetime, but I've never understood why and/or how. I'm only 12 so please keep it simple if possible.
Most of us have heard of Einstein's amazing equations which describe the universe around us, yet only some of us understand what the equations are actually saying.      What are these equations actually saying, and is there a simple (relatively) way to derive them?   Here they are, from :  $$R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda=\dfrac{8\pi G}{c^4}T_{\mu\nu}$$  I have a vague notion of what a tensor is (it describes things as an array and higher orders define more complex transformations), but I don't understand what all of these tensors are doing. And why is there a $c^{4}$  in the equation!?
Prove that $||x|-|y||\le |x-y|$ How to not have an equation in the first line using the align environment? Understanding linear regression In class we've seen that $a$ (the weights) must satisfy $$X^T (y-Xa) =0$$  Here $X$ is a $(n\times d)$ matrix (so we have $n$ samples in $\mathbb R^d$)  let's denote the residuals $r = y-Xa$. In our class notes, it is written that     The normal equations require the residuals to be orthogonal to each of the columns of $X$.   Why?     Therefore, the solution of the linear regression is a projection of $y$ onto the subspace spanned by $v_1 , \ldots , v_d$ (the columns of $X$)   Can you please explain this? Query Feature and Usage based entitlements license I need to query programmatically the Feature license and usage-based license which is available in UI. Could someone please highlight if we can query the same or is it available in UI only, Thanks. My keyboard doesn't have an Fn key so I can't get the coordinates to show up. I don't know how to get into the multiplayer settings and Shift + F3 doesn't work either.  How can I fix this? I'm trying to execute a string into my bash script. String contains variables (so I could substitute some CLI arguments depending on circumstances).  The problem is when I use variable containing --argument=value, everything breaks up, giving error:  Missing trailing-' in remote-shell command.   Example code:  #!/bin/bash  rsync_path="/usr/bin/rsync -azhrP" rsync_ssh_cmd="--rsh='ssh -o Compression=no'"  `$rsync_path /path/to/file $rsync_ssh_cmd username@localhost:/tmp`   If I just echo the last string from this code, it looks fine. But if i try to execute this string from script, it doesn't executes as expected.  I found a workaround - everything became OK if I try  eval "my-command-string"   instead of  `my-command-string`   or  (my-command-string)   Why does it so? How could I (should I really?) avoid to use eval?  I use bash 4.3.11(1)-release (x86_64-pc-linux-gnu) (Kubuntu 14.04.3). Python super() raises TypeError In Python 2.5, the following code raises a TypeError:  &gt;&gt;&gt; class X:       def a(self):         print "a"  &gt;&gt;&gt; class Y(X):       def a(self):         super(Y,self).a()         print "b"  &gt;&gt;&gt; c = Y() &gt;&gt;&gt; c.a() Traceback (most recent call last):   File "&lt;stdin&gt;", line 1, in &lt;module&gt;   File "&lt;stdin&gt;", line 3, in a TypeError: super() argument 1 must be type, not classobj   If I replace the class X with class X(object), it will work. What's the explanation for this? Joining lots of small polygons to form larger polygon using PostGIS? How to copy file path to clipboard? Is there a way to copy the currently selected filename, including the path, to the clipboard? Ideally this would be through the right-click menu.  Some degree of flexibility would be useful, like the ability to copy just the filename, path+filename, filename minus extension, etc. This isn't essential, though, as long as it can copy the path+filename.  The solution must be compatible with Windows XP, Windows Vista and Windows 7. Are we slightly lighter during the day and slightly heavier at night, owing to the force of the Sun's gravity? Using $g = \frac{Gm}{r^2}$, the force on a point mass located at 1 AU from the Sun ($m = 2 \cdot 10^{30} \text{ kg}$) is about ~0.006 N/kg.  Does that mean that, e.g., a 70 kg person is ~42g lighter during the day, and ~42g heavier at night? That seems like it could make a big difference for things like measuring gold bars or other weight-sensitive items. (Gold arbitrage: buy your gold during the day and sell it during the night! Risk-free profit!)  This makes me suspect that I'm overlooking something obvious, because a ~0.05% weight difference seems like something everyone would have noticed long ago. So what am I missing?    Edit: A few answers below indicate that there shouldn't be any weight difference, because the Earth orbits the Sun in free-fall. But if that's the reason, does that mean that a 1:1 tidally-locked Earth-Sun system wouldn't experience differential gravity from the Sun on opposite sides? That doesn't seem right. Ever since I upgraded to 15.10, fdisk -l reports 16 ram disks (/dev/ram0 ... /dev/ram15). I'm a bit unsure what those are needed for. Is it safe to delete them? If not, how can I get rid of that fdisk output? What is $A^-$ in topology? Hartshorne Algebraic Geometry Proposition 1.5 (page5) In a notherian topological space $X$, every nomempty closed subset $Y$ can be expressed as a finite union $Y=Y_1 \cup \cdots \cup Y_r$ of irreducible closed subsets $Y_i$. If we require that $Y_i \nsupseteq Y_j$ for $i \neq j$, then the $Y_i$ are uniquely determined. They are called the irreducible components of $Y$.    I understood the existence of such a representation. But I can't understand a sentence in  the uniqueness part, which I highlighted: Let $Y=Y_1 \cup \cdots \cup Y_r$ and $Y=Y&#39;_1 \cup \cdots \cup Y&#39;_s$ be two such representation. Then we can deduce that $Y_1=Y&#39;_1$ (the details are in the textbook). Now let $Z=(Y-Y_1)^-$, then $Z=Y_2 \cup \cdots \cup Y_r$ and also $Z=Y&#39;_2 \cup \cdots \cup Y&#39;_s$.  What is the operator $^-$? I understood it as a closure, is it right? But then how can I deduce that $Z$ is represented as $Z=Y_2 \cup \cdots \cup Y_r=Y&#39;_2 \cup \cdots \cup Y&#39;_s$? Why isn't temperature frame dependent? Sound of a next scene starts one moment earlier than video I saw this several times in different movies, but I don't understand what is this supposed to mean. The main idea is when the scene is about to change, for one second or so we still see the picture from the old scene, but the sound is already coming from from the upcoming scene. Why do they do this? What's the purpose of this technique? Interfaces with static fields in java for sharing 'constants' I'm looking at some open source Java projects to get into Java and notice a lot of them have some sort of 'constants' interface.  For instance,  has an interface called , and most other core classes implement this interface. The interface is riddled with static members. Is there a reason for this approach, or is this considered bad practice? Why not use enums where it makes sense, or a static class?  I find it strange to use an interface to allow for some sort of pseudo 'global variables'.  public interface PConstants {    // LOTS OF static fields...    static public final int SHINE = 31;    // emissive (by default kept black)   static public final int ER = 32;   static public final int EG = 33;   static public final int EB = 34;    // has this vertex been lit yet   static public final int BEEN_LIT = 35;    static public final int VERTEX_FIELD_COUNT = 36;     // renderers known to processing.core    static final String P2D    = "processing.core.PGraphics2D";   static final String P3D    = "processing.core.PGraphics3D";   static final String JAVA2D = "processing.core.PGraphicsJava2D";   static final String OPENGL = "processing.opengl.PGraphicsOpenGL";   static final String PDF    = "processing.pdf.PGraphicsPDF";   static final String DXF    = "processing.dxf.RawDXF";     // platform IDs for PApplet.platform    static final int OTHER   = 0;   static final int WINDOWS = 1;   static final int MACOSX  = 2;   static final int LINUX   = 3;    static final String[] platformNames = {     "other", "windows", "macosx", "linux"   };    // and on and on  } Been trying to find graphic novel for decades - at least I think it's a graphic novel I have seen lots of UI designer nowadays including this kind of illustration i want to know the name of this style so that i can also learn and try this in my designs. E: Could not get lock /var/lib/dpkg/lock-frontend - open (11: Resource temporarily unavailable) I'm trying to run this command in the terminal:  sudo apt install software-properties-common   This is the error message I get:  E: Could not get lock /var/lib/dpkg/lock-frontend - open (11: Resource temporarily unavailable)   E: Unable to acquire the dpkg frontend lock (/var/lib/dpkg/lock-frontend),     is another process using it? My HP Pavilion laptop has Ubuntu 18 version and I once tried to disable InBuilt webcam just for security reasons. But now it wasn't working and I reinstalled newer version of Ubuntu 20. But even after installing newer version the camera is still not working. When I open cheese it shows this screen. I assumed I might have deleted the drivers as I was doing package installation and removel that time. Please suggest me what should I do to get this working. Why hash the message before signing it with RSA?