I just finish reading a pdf about group cohomology that explain a lot about it. It goes from the definition to Galois's cohomology. But there is one think I would like to know and I think I haven't enough read about cohomology to know : What is the intuition behind all of this theory ? If someone could explain me or just give me some references I would appreciate. Thank you very much !
Group cohomology is one thing I don't seem to get my head around.  I understand $H^0$ as being the "fixed points" but when it comes to anything higher I have no idea what the notions are meant to be capturing...if anything.  For example I can see that 1-cocycles are homomorphisms with a specific property but that property doesn't seem to mean anything to me...and I think it should!  Is there an intuition or example that explains the meaning or is it just something that "stuck" and seems to work in lots of places?
Effect of space time relativity on the age of the universe? So we all heard about the twins paradox to explain einstein's time space relativity.  Wikipedia Quote :" In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin sees the other twin as traveling, and so, according to a naive application of time dilation, each should paradoxically find the other to have aged more slowly. "  So what if the universe has been travelling at varying speeds (increasing or descreasing), wouldn't this effect our measurements on age of the universe? Internal or external oscillator I always use the internal oscillator that pics have as I have never found the need to run anything at higher frequency than 8 MHz (which is the fastest the pics I use tend to be able to go). Are there any reasons, beyond going above 8 MHz, that mean I should use an external oscillator? It just seems like one more thing to go wrong to me but I'd be interested to hear what others do. Uniform convergence for sequences of functions What are recursive triggers? How can recursive triggers be controlled? What are recursive triggers? How can we control them, or prevent infinite recursion?  Are these any risks inherent in them, and any limits that we should be aware of?  Please explain with an example. Fastest way to try all passwords Suppose you have a computer with a password of length $k$ in an alphabet of $n$ letters.  You can write an arbitrarly long word and the computer will try all the subwords of $k$ consecutive letters. What is the smallest word that contains all combinations of $k$ letters as subword? (i.e. the fastest way to hack the computer :) )  The smallest word that contains $n^k$ subwords of size $k$ has length $k-1+n^k$ and based on some easy cases, we would like to prove that it is in fact possible to find a word of such length that contains all possible passwords. The problem can be translated into a problem in graph theory, by taking as vertices all words of length $k$.  We tried $k=2$, where you can prove the conjecture by induction. For $n=2$ and small $k$ it also works. Skyrim crashes as soon as I boot up into the menu. How do I fix this? Whenever I boot up Skyrim, it crashes as soon as it reaches the main menu with no dialog or anything. I thought this was related to my using dual monitors somehow, but I disconnected the other monitor and the game still crashed. I tried running the game in different resolutions, both full-screen and windowed, at various quality settings; no dice. My computer does meet the minimum requirements to play Skyrim.  How do I fix this problem? double perpendicular symbol (for independence) I found from  that I could use the independence symbol:     \newcommand\independent{\protect\mathpalette{\protect\independenT}{\perp}} \def\independenT#1#2{\mathrel{\rlap{$#1#2$}\mkern2mu{#1#2}}}    I was wondering how this works. Can anybody explain? I don't know the commands mathpalette, rlap, and mkern2mu. Hide trivial comments I noticed that there is a definition of the pvalue in my textbook is defined says the following:    I have no idea why it is written with a supremum. I've spent hours pondering this, does anyone have enough of a background to help me with this? Thank you! Vertex of a pentagon-Does the algorithm always stop? To each vertex of a pentagon,we assign an integer $x_{i}$ with sum $s=\sum x_{i}&gt;0$. If x,y,z are the numbers assigned to 3 successive vertices and if $y&lt;0$,then we replace $(x,y,z)$ by $(x+y,-y,y+z)$.This step is repeated as long as there is a y&lt;0.Decide if the algorithm always stops. Running macOS Sierra(Version 10.12.2 16C67).  I installed  version wine-1.8.6 (through a .pkg file). I opened up a game and it worked fine.  The next day when I downloaded Origin installer, it wouldn't open. When I double clicked on it, the Wine icon showed up in the dock and said it was open, but no windows showed up. When I clicked on another program(or on the desktop), Wine closed. Now when I try to start that same game I opened when I first installed Wine, the same thing happens that happened when I started Origin installer.  When I go to the terminal and type wine {path of .exe} I got this:   I installed wine tricks using brew install winetricks and it installed fine, yet it didn't change my situation.  winecfg works properly, and so does sh winetricks. What characters/symbols are allowed within the CSS class selectors? I know that the following characters are invalid, but what characters are valid?  ~ ! @ $ % ^ &amp; * ( ) + = , . / ' ; : " ? &gt; &lt; [ ] \ { } | ` # Change cursor color after log out on ubuntu 18.04.2 LTS I've set my cursor color on gnome-tweaks to be DMZ-Black, which works fine when I log in and lock the computer. But, after I log out, the cursor color changes to white.  Is there a way to change the color of the cursor after log out?  My gnome-tweaks setting: Why do polarized sunglasses remove (most) reflections on car windows?  I know that both lenses of polarized sunglasses are polarized in the same direction and I can see that reflections in my own windshield and the windows of other cars appear, disappear, or change depending on how I tilt my head, but why?  Why is the light in those reflections polarized?  edit: And why is most of the light in the various reflections on the interior of my own windshield and various windows/windshields at what seem like a variety of angles all polarized in a sufficiently narrow set of ways as to make polarized sunglasses work? I know that I can make certain blocks only placeable on a specified block, but what is the command to give an item that can only BREAK certain blocks? A post can be locked.   What does it mean to lock a post? Who can lock a post? When should a post be locked? Extra semicolon near answers header On all the SE sites that I checked, the text that normally says something like “1 Answer” now says “1 Answer;” (with a semicolon seemingly in superscript):    The HTML for that part of the page looks like this (the semicolon is on the fourth line from bottom):              &lt;div id="answers"&gt;                  &lt;a name="tab-top"&gt;&lt;/a&gt;                 &lt;div id="answers-header"&gt;                     &lt;div class="subheader answers-subheader"&gt;                         &lt;h2&gt;                                 1 Answer                         &lt;/h2&gt;                         &lt;div&gt;                             &lt;div id="tabs"&gt;         &lt;a href="/questions/18587301/which-is-good-il-editor?answertab=active#tab-top" title="Answers with the latest activity first"&gt;             active         &lt;/a&gt;         &lt;a href="/questions/18587301/which-is-good-il-editor?answertab=oldest#tab-top" title="Answers in the order they were provided"&gt;             oldest         &lt;/a&gt;         &lt;a class="youarehere" href="/questions/18587301/which-is-good-il-editor?answertab=votes#tab-top" title="Answers with the highest score first"&gt;             votes         &lt;/a&gt; &lt;/div&gt; ;                         &lt;/div&gt;                     &lt;/div&gt;                 &lt;/div&gt; $n$th derivative of $e^x \sin x$ Are Destiny characters backwards and/or cross-compatible? From what I heard and remember hearing correctly, characters that are leveled up through the PS3 can be transferred to the PS4 version (assuming that both consoles have the game).  So what if I wanted to go back to the PS3 version with some friends? I only have the PS4 version at the moment--would my PS4 characters transfer over to the PS3 side? What if I wanted to go to the 360 version too to play with some other people too? What is the logical form of the definition of validity? My text book gives the definition of validity as "An argument is valid IFF if the premises are true, then the conclusion must be true". Using a conditional elimination on the RHS of the IFF yields "An argument is valid IFF the premises are false or the conclusion is true".   Yet, this seems strange to me. Am I doing something incorrect with my manipulation or is this definition actually true? Surely the criteria for a valid argument goes beyond a false set of premises or a true conclusion.