Where/how to learn about dark matter? I am a physics student and as far as I know dark matter is beyond the topics included in my grade career. Are there any serious book on the subject? Any serious set of lectures somewhere in Internet?
Dark matter and dark energy references I've been looking for questions about dark matter, and I've read some very interesting answers. However, I desire too look into it deeply.   This is not actually a question. I'm asking the community to recommend interesting references to understanding dark matter and dark energy.   I accept all sort of references: notes, books, scientific papers etc.  Let us assume some background on classical physics, thermodynamics and basics about quantum theory.
Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$ Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$.  I started off by putting $\gcd(a+b, a-b) = d$. This implies that there are two relatively prime integers $x_1, x_2$, such that   $dx_1 = a+b$   $dx_2 = a -b$  Adding the first equation to the second gives us: $d(x_1 + x_2) = 2a$, and subtracting the second from the first gives us $d(x_1 - x_2) = 2b$. This implies that $d\mid2a, d\mid2b \implies \gcd(2a, 2b) = d \implies d = 2\cdot \gcd(a,b) \implies d = 2$.   This is based on the fact that $\gcd(x_1 + x_2, x_1 - x_2) = 1$, which is in fact the statement of the problem. How do I prove this? Can I somehow use the Euclidean Algorithm for this. Non-algebraically closed field in which every polynomial of degree $<n$ has a root My problem is to build, for every prime $p$, a field of characteristic $p$ in which every polynomial of degree $\leq n$ ($n$ a fixed natural number) has a root, but such that the field is not algebraically closed.  If I'm not wrong (please correct me if I am) such a field cannot be finite, by counting arguments. But on the other hand, the union of all finite fields (or of any ascending chain of finite fields) of characteristic $p$, which is what I get if I start with $F_p$ and add a root to each polynomial of degree $\leq n$ in each step, is the algebraic closure of $F_p$, hence algebraically closed. I don't see how I can control this process so that in the end I get a field that is not algebraically closed.  Any hint will be welcome. Thanks in advance. Name "Chapter" is not showed in the table of the content Why is $\omega$ the smallest $\infty$? In Episode 1: The Phantom Menace, Anakin's mother, Shmi, said that he does not have a father, although it seemed to me that she doubted that.  Is there a canonical answer as to who Anakin's father is?  If so, who? Including sections of source code as a listing, while being robust to source code edits Oneboxing a onebox from a different chat server generates a weird URL So if you try to onebox a onebox from a different chat server, . Basically, it generates this URL:    Now, obviously that isn't right. I've also tested this , and I'm assuming it's broken on chat.SO too. Can this be fixed? How do you interpret a significant but weak correlation? Set a default parameter value for a JavaScript function Classes and lessons at Hogwarts are presumably scheduled at fixed times of day. The first class might start, for example at nine in the morning.  To be on time for one's lessons, it is therefore crucial to know the current time but we know that electric watches of course do not work at Hogwarts because of all the magical interference. Are there magic watches? Is there maybe a spell to tell the time?  How do people at Hogwarts usually tell the time? Declare a global variable in a Blender session What's a command line way to find large files/directories to remove and free up space? Looking for a series of commands that will show me the largest files on a drive. Google Chrome (Windows) not displaying text/encoding correctly on certain websites I experience a problem similar to    Here's a screenshot on the website     I've tried:   Removing fonts I have installed Setting the encoding to Western, UTF-8, Greek, Automatic Changing the fonts used in Google Chrome Removing Google Chrome, including all application data/user profiles Running CCleaner Reinstalling Google Chrome, twice Installing Chromium (fonts display fine in Chromium) Turning off Windows ClearType Changing Windows/Chrome languages from English (Australia) to US, UK Restarting Windows several times after making changes   All of which did not work. I do not connect over VPN/proxy.   Can anyone give me any tips?  Windows 7 x64 Ultimate Is there a rule in British English about how to pronounce "either"? There are two common pronunciations of "either": British /ˈaɪðər/ and American /ˈiːðər/. If Americans are more or less consistent in this regard, then the Brits seem to be freely using both. In fact, from what I can tell, "either this or that" is more often in the first form, whereas "me either" is in the second. But I may be wrong. Is there any kind of an informal rule in the modern British English with regard to this? Should we have a tag for significant points in time? There are many discussions about the current incidence. For example:          Should we have a whole tag for this incidence? In general, if in the future there are more crises and incidences like this, should they have their own tags too? I feel it's fine, as we have specific tags for individual events like . And, should we have a  tag?    Related:        Note that these questions are specifically about the conflict 2019, while this question is simply about creating meta tags for significant events. why does loop-cut-and-slide ignore some faces but not others? Calibre No Longer Working Calibre stopped working recently. Tried uninstalling and reinstalling. When I open a terminal and call $calibre I get the following error message: Traceback (most recent call last):   File &quot;/usr/bin/calibre&quot;, line 20, in &lt;module&gt;     sys.exit(calibre())   File &quot;/usr/lib/calibre/calibre/gui_launch.py&quot;, line 73, in calibre     main(args)   File &quot;/usr/lib/calibre/calibre/gui2/main.py&quot;, line 543, in main     listener = create_listener()   File &quot;/usr/lib/calibre/calibre/gui2/main.py&quot;, line 514, in create_listener     return Listener(address=gui_socket_address())   File &quot;/usr/lib/calibre/calibre/utils/ipc/server.py&quot;, line 110, in __init__     self._listener._unlink.cancel() AttributeError: 'NoneType' object has no attribute 'cancel'  Do I need to install some dependency that the sudo apt install didn't get? I tried installing via command line and the software installer/updater GUI. I'm on version Ubuntu 20.04.1 LTS. I made sure that my software in general is up to date also. LaTeX macro expander I'm developing some scripts. One for the  that popped in Ubuntu 16.04 even though pcspkr is blacklisted. Another to fix a  where sound switches off from TV to the laptop.  What is the best directory to put my bash scripts in?  I've searched the Q&amp;A database but I get a bunch of different answers for a bunch of different scenarios.  /usr/local has been recommended by one person because it's in the PATH$. Is that the right thing to do?  Thank you for your advice.    Even though I accepted below answer over a year ago, I ended up using:   /usr/local/bin for scripts needing sudo powers /home/me/bin for scripts running at user level   After creating directory /home/me/bin you need to log out and in again for it to get into your PATH. 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.