I'd like to know what is thought of when saying the universe is infinite.  I've read about what is meant when astronomers are speaking of a "flat" universe and I have read that its size is most likely infinite. Yet Google couldn't forward me to a site where it's actually explained what that should mean.    These are some sites I looked but which did not explain it:   , another Phys.SE question   , on esa.int , on www.space.com
This question is about cosmology and general relativity. I understand the difference between the universe and the observable universe. What I am not really clear about is what is meant when I read that the universe is infinite.   Does it have infinite mass or is it dishomogeneous?  How can the universe transition from being finite near the big bang and infinite 14 billion years later? Or would an infinite universe not necessarily have a big bang at all?
Why $\sum_{k=0}^m x^k $ is equal to $\dfrac{x^{m+1}-1}{x-1} $? Who can help me proving this (but you can no longer use $\sum$ in the proof):  $$\sum_{k=0}^m x^k=\dfrac{x^{m+1}-1}{x-1}$$ for any ring $A$ the matrix ring $M_n(A)$ is simple if and only if $A$ is simple Let integer $n\geq 1$. I have obtained that for any field $k$, the matrix ring $M_n(k)$ is simple, i.e., $M_n(k)$ contains no nonzero proper two sided ideals. Now I want to prove that: for any ring $A$ (not necessarily commutative), the matrix ring $M_n(A)$ is simple if and only if $A$ is simple.   How to prove? How a module can override theme suggestion? Basically it was  but I'll reword it, as I don't want to define a new theme (reason was already given).  What I want to do: in my module I have a file node--attach.tpl.php which is a modifed version of node.tpl.php. Now I want that, if view_mode = 'attach' then this file is used. Also, theme should be allowed to override it with its own node--attach.tpl.php in theme folder. C non-blocking keyboard input I'm trying to write a program in C (on Linux) that loops until the user presses a key, but shouldn't require a keypress to continue each loop.  Is there a simple way to do this? I figure I could possibly do it with select() but that seems like a lot of work.  Alternatively, is there a way to catch a ctrl-c keypress to do cleanup before the program closes instead of non-blocking io? USB-C Charging Speeds / Charger Outputs I own a 2016 Macbook (with the USB-C charging) - at the time and in response to all of the reports of poor cables and poor chargers, I bought a meter so I could see what the throughput of my cables/chargers was. However, having run some tests I'm now not sure what to consider normal / what is harmless to the battery.   For all tests I used an original Apple C-C cable - though was able to replicate with two third party cables I own.  I have the original Apple block charging at 14.3 V / 1.99 A and a third party block that charges the same.  Another third party block which advertises USB PD and 3 A is currently charging at 20 V / 1.43 A  I've done some research online but I can't find any articles that indicate what a normal USB PD output is relative to the devices that connect to it and would appreciate any guidance / assistance.   TL;DR: Have Apple charger which charges at one voltage/amperage. Bought several others advertised at PD which are higher voltage, lower amperage. Not sure what is 'safe', or 'fast', or does least overall damage to battery. Assume that $X$ and $Y$ are independent random variables with $X \sim \exp(\lambda)$ and $Y \sim \exp(\mu)$. It is impossible to obtain direct observations of $X$ and $Y$. Instead, we observe the random variables $Z$ and $W$, where $Z = \min\{X,Y\}$ and $W =1$ if $Z=X$ and $W=0$ if $Z=Y$.  How to find the joint distribution of $Z$ and $W$? What is the name of this theorem, and are there any caveats? For random variable $X$ that follows some distribution, $f(x)$ is the probability density function of that distribution if and only if $$\mathbb{E}[\phi(X)] = \int_{-\infty}^\infty \phi(x) f(x)dx$$ for all functions $\phi$.  Context: My professor used this in lecture to demonstrate a way to find the distribution of $cX$ given random variable $X$ that follows a specific distribution.   What is the name of this theorem? Have I missed any qualifications/caveats? I am in particular curious about whether "for all functions" is correct. My professor mentioned "positive, bounded, deterministic functions," but I am not sure what he meant by that. Where can I find a proof of this theorem? reason for exec in wrapper scripts I have seen wrapper script examples which in a nutshell are following:  #!/bin/bash  myprog=sleep echo "This is the wrapper script, it will exec "$myprog""  exec "$myprog" "$@"   As seen above, they use exec to replace the newly created shell almost immediately with the $myprog. One could achieve the same without exec:  #!/bin/bash  myprog=sleep echo "This is the wrapper script, it will exec "$myprog""  "$myprog" "$@"   In this last example, a new bash instance is started and then $myprog is started as a child process of the bash instance.  What are the benefits of the first approach? I am a first year Ph. D student, and I have been working on a new topic of research.  I worked day and night to prove something new and verified it. I began my Ph. D while my supervisor was on travel, but he returned before I had been here for long.  When he returned, he told me we have to stop and not continue without any reasonable reasons.  After a few months he began treating me poorly---always belittling and underestimating---although I got two grants from a top conference, made videos and podcasts, and constantly received praise for my good work and personality.    It seems that I have been treated as a competitor, not a collaborator.  Although agreeing with my approaches in private, my supervisor said false words to the committee to take the decision and stop my Ph. D work!  It is not surprising, as there was another former senior Ph. D student who I met on campus, who was not allowed to defend his Ph. D: this same supervisor claimed he was not able to finish his dissertation although he had published in a peer reviewed journal and having citation as well, it was scary and not familiar.  In front of committee, he lied, and said something without proving. I asked if he can make a detailed constructive criticism, but he could not.  All I can say that they are a gang literally, and he said words that show that he is jealous and doesn't want to see me having self confidence in my self. I know no one will believe me, but this is the truth.  Honestly, I am distressed and I'm losing sleep.  I had this dream where there were many people who saw me fail.  All things considered, I don't know whether what happened is good or bad, and I definitely don't know what I should do next!  I would be grateful for your advice and whether there is someone has encountered any similar situation. How to know the motherboard BIOS version using the command line Random point uniform on a sphere If $X=(x,y,z)$ is a random point uniform on the unit sphere in $\mathbb{R}^3$,  Are the  coordinates $x$, $y$, $z$ uniform in interval $(-1,1)$? Redirect STDERR and STDOUT to different variables without temporary files func() {     echo 'hello'     echo 'This is an error' &gt;&amp;2 }  a=$(func) b=???   I'd like to redirect the stderr to b variable without creating a temporary file.   echo $b  # output should be: "This is an error"   The solution that works but with a temporary file:  touch temp.txt exec 3&lt; temp.txt a=$(func 2&gt; temp.txt); cat &lt;&amp;3 rm temp.txt   So the question is, how do I redirect the stderr of the  function func to the variable b without the need of a temporary file? There was the following paragraph in the article that came under the title, “GOP and the rise of Anti-knowledge” written by  (October 29, 2015):     “English unfortunately doesn’t have a precise word for the German   “Fachidiot,” a narrowly specialized person accomplished in his own   field but a blithering idiot outside it. In any case, a surgeon is   basically a skilled auto mechanic who is not bothered by the sight of   blood and palpitating organs. We need the surgeon’s skills on pain of   agonizing death, and reward him commensurately but that does not make   him a Voltaire.”    In Japanese we have a word, 専門バカ, of which literal translation is “specialist idiot” meaning exactly same as defined the above - a narrowly specialized person accomplished in his own field but a blithering idiot outside it.  I’m really curious about the absence of short descriptions of the specialist who knows nothing more than his specialty area in English like German “Fachidiot,” and Japanese "専門バカ."  Isn’t there really any short word counterpart to “Fachidiot,” specialist’s ignorance, for instance, in English?  P.S.  I found "narrow-minded specialisit" as a German-to-English translation of "fachidiot" in the web site, Interglot translation dictionary. Non-Root Package Managers From my research, I seem to notice that all package managers insist on being used as a privileged user and must be installed into /.  Typically, what I like to do is create a throwaway account, compile some software, and install to $HOME for that account. I can try a variety of setups and then when I'm done, just destroy the account.  However, compiling software becomes tedious.  My experience is really just limited to yum, but I don't understand why I wouldn't be able to drop a repo file into ~/etc/yum.repos.d and have yum install everything into a home account.  Is there any reason why package managers must be used as a privleged user to install software? Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$? Is there anything special with the form:  $$\left|\frac{z-a}{1-\bar{a}{z}}\right|$$ ? With $a$ and $z$ are complex numbers.  In fact, I saw it in a problem:   If $|z| = 1$, prove that $|\frac{z-a}{1-\bar{a}{z}}| = 1$ If $|z| &lt; 1$ and $|a| &lt; 1$, prove that $|\frac{z-a}{1-\bar{a}{z}}| &lt; 1$   I can easily prove the first one with expansion:  $$z=\cos\theta + i\sin \theta \\ a = m +in$$  But it will be terrible to use in the second one. What's more, I found this form a little special so maybe there is some clever trick without using expansion?  Thank you a lot! What tools and technologies are used to build the Stack Exchange Network?  See also: One of the mysteries that my numerics class left me with are "stiff systems". Our prof didn't really manage to explain what they are. Now that the class (and the test) are over, I'm still curious.   The most concise definition about "stiff systems" I've found so far is: "There is no universally accepted definition of stiffness". So, if there is none, I don't expect one.   I'd be thankful about information on both of the following points:   From what kind of (real-world) problems do stiff systems arise? What's so "stiff" about them? What's the good thing - or the bad thing - that I'd worry about when I have to numerically solve such a real-world problem? How do I create a permanent Bash alias? Neutrino project in Tamil-Nadu India In , the Indian government is going to start a neutrino project.  Questions:   Is it harmful for people who stay there?   Will this project affect any living beings? Does it cause any pollution? What is the recommended way to encrypt user passwords in a database?