What happened before the Planck time in the big bang? I have seen a timeline for after the big bang but there is no description of what happened before the Planck time.
Does time stretch all the way back for infinity or was there a point when time appears to start in the universe?  I remember reading long ago somewhere that according to one theory time began shortly before the creation of the universe.  Does time have a starting point of note?
We are given with  $$S_n=\{(x,y)|x^2+y^2=n, \text{where }x,y,n \in \mathbb Z\text{ and }n\geq0  \}$$ Now observe that $-\sqrt n\leq x\leq\sqrt n $ and similarily, $-\sqrt n\leq y\leq\sqrt n $. This means that $-\left \lfloor{\sqrt n}\right \rfloor\leq x\leq \left \lfloor{\sqrt n}\right \rfloor$ and  $-\left \lfloor{\sqrt n}\right \rfloor\leq y\leq \left \lfloor{\sqrt n}\right \rfloor$. Through this result one can easily write code and figure out the cardinality of set $S_i$ but I was wondering whether an expression can be found that would allow one to directly compute $|S_i|$. Why does chmod +w not give write permission to other(o) When I run chmod +w filename it doesn't give write permission to other, it just gives write permission to user and group.  After executing this command  chmod +w testfile.txt   running ls -l testfile.txt prints  -rw-rw-r-- 1 ravi ravi 20 Mar 10 18:09 testfile.txt   but in case of +r and +x it works properly.  I don't want to use chmod ugo+w filename. Sharing a Bluetooth Keyboard and Mouse sequentially between two Macs I work out of my house where I have two computers: a mid-2010 Macbook Pro (for my day job) and a mid-2010 iMac (personal). During the workday, my iMac acts as a second display for my Macbook Pro via Mini Displayport.  Currently, I use a USB keyboard and mouse for my work computer and a Bluetooth keyboard and Magic Mouse + Magic Trackpad for use with my personal computer. I'd like to get rid of the wired keyboard and mouse and just use the Bluetooth stuff for both machines (but never simultaneously).   Is it possible to easily share the single set of input devices without having to repeatedly break and reestablish the pairing of the input device to the correct computer? I'm not opposed to buying some type of additional hardware to accomplish this, provided it's not prohibitively costly or cumbersome to use or configure.  It's not clear to me how I would tell the MacBook Pro to leave the devices alone so the iMac can have them for the evening. I want to avoid de-pairing and re-pairing the devices each time I want to shift them back or forth.  Basically is there software or hardware that works as a Bluetooth KVM switch? Cannot dismiss scrollbar in code box, blocks view This question is a duplicate of  (but this question can't be marked as a duplicate because that question is on meta.stackoverflow.com, and this question is on meta.stackexchange.com).  I'm seeing a bug on stackoverflow.com. I'm not sure if the culprit is Stack Overflow, Google Chrome, or macOS.  (Below I refer to a "code box". By that I mean some text that begins with 4 spaces when editing a question or answer.)  When I hover over a code box that contains one long line of code, a horizontal scrollbar appears. When I stop hovering over the code box, very frequently (maybe 3 out of 4 times) the scrollbar does not disappear. Clicking elsewhere on the page also does not dismiss the scrollbar. The only way to dismiss it is to refresh the page.  ...After experimenting more, it appears to always happen if I enter the box from below, but not always when I enter from above. It seems to be triggered by hovering over the lower half of the box.  Also if I enter from the top and scroll to the right, then until I refresh the page again that scrollbox will behave normally (only show a scrollbar when scrolling, and disappear after). In that case the scrollbar also has a different appearance, being much less tall (maybe a quarter of the height of the code box rather than half as tall).  Further, if I get a box and scrollbar into that functional state and then hover over the lower part of the box for a bit, then sometimes eventually it will break and stay broken until refresh again.  And in any case hovering over the lower part of the box causes the scrollbar to get taller.  Here is a screenshot of the issue:        Google Chrome 71.0.3578.98 (Official Build) (64-bit)  macOS High Sierra 10.13.6  Here is a code box that repros the issue for me:  fooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo   Here is another:  foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo foo For an ecology project, my lab group added vinegar to 4 tanks containing equal volumes of pond water, 1 control with no elodea (an aquatic plant) and 3 treatments with the same amount of elodea in each. The purpose of adding the vinegar was to reduce the pH. The hypothesis was that the tanks with elodea would go back to their normal pH quicker. This was indeed the case. We measured the pH of each tank daily for about two weeks. All the tanks eventually returned to their natural pH, but the length of time that this took was much shorter for the tanks with elodea.  When we told our professor about our experimental design, he said that there exists no statistical test that can be performed on the data to compare the control to the treatment. That because there was no replicate for the control (we only used one control tank) we cannot calculate variance and so we can't compare the sample means of the control and the treatment. So my question is, is this true? I definitely understand what he means. For example, if you took the height of one man and one woman, you can't draw conclusions about their respective populations. But we did 3 treatments, and the variance was small. It seems reasonable to assume that the variance would be similar in the control?  Update:  Thank you for the excellent answer. We got more water and elodea from the wetland and decided we would run the experiment again with smaller tanks but this time with 5 controls and 5 treatments. We were going to combine this with our original data but the starting pH of the tanks was different enough that it doesn't seem valid to consider the new experiment to be sampled from the same population as the original experiment.  We considered adding different amounts of elodea and trying to correlate speed of pH remediation (measured as time elapsed until pH returned to its original value) with amount of elodea, but we decided that wasn't necessary. Our objective is only to show that the elodea makes a positive difference, not to construct some kind of predictive model for exactly how the pH responds to differing amounts of elodea. It would be interesting to determine the optimal amount of elodea, but that's probably just the maximum amount that can survive. Trying to fit a regression curve to the data wouldn't be especially illuminating because of the various complicated changes that occur to the community when adding a large amount. The elodea dies, decomposes, new organisms start to dominate, and so on. if $x_i$ are IID  continuous random variables, with $E[x_i]=μ$, is the following correct?   $E\left[\prod_{i=1}^n(1+⍺\space x_i)^i\right] = \prod_{i=1}^nE\left[(1+⍺\space x_i)^i\right] =  \prod_{i=1}^n(1+⍺\space E\left[x_i\right])^i = (1+⍺\space μ)^n$  thanks! A No-Nonsense Introduction to Quantum Field Theory I've just been downvoted. How should I react? How do I escape a single quote ( ' ) in JavaScript? What is the sign of the  done on the system and by the system?  My chemistry book says when work is done on the system, it is positive. When work is done by the system, it is negative.  My physics book says the opposite. It says that when work is done on the system, it is negative. When work is done by the system, it is positive.  Why do they differ? How can I obtain total number of points? Editing a starred post in chat breaks display for user who starred it Steps to reproduce:   User A posts a message User B stars it, the star next to the post appears yellow User A edits the post The display of the star in User B's view becomes black, as if they haven't starred it   This is the worst bug ever in the history of StackExchange and makes the entire network completely unusable for me. I'm going to Yahoo! Answers until this is fixed. Open local folder from link Keyboard Layout indicator doesn't display current layout in Ubuntu 18.04 (empty drop down box) I see an empty drop down box where the language code of the  should be.  To reproduce the bug: Lock screen or log out and log in again.    I also get the same behavior on the lock screen at the time when I try to log in with my password:    After I tried the workaround (below) the language code on the login screen disappeared as soon as I typed the first character of my password.  Workaround (not permanent): Switching with keyboard (Super+space) or mouse.  System info: Ubuntu 18.04.1 LTS, GNOME 3.28.2.    Please  if also affects you. Why does sudo change the PATH? In Avengers: Endgame,       Thanos was able to stop and catch Mjolnir and Stormbreaker mid-air as both Captain America and Thor used/called them BEFORE he got the Infinity Stones.   Since this character is clearly not worthy and had no super powers at the time, how were they able to wield Thor's Mjolnir and Stormbreaker, when even the Hulk couldn't do it in The Avengers? That's basically the question, is there a "right" way to implement operator&lt;&lt; ? Reading  I can see that something like:  friend bool operator&lt;&lt;(obj const&amp; lhs, obj const&amp; rhs);   is preferred to something like   ostream&amp; operator&lt;&lt;(obj const&amp; rhs);   But I can't quite see why should I use one or the other.  My personal case is:  friend ostream &amp; operator&lt;&lt;(ostream &amp;os, const Paragraph&amp; p) {     return os &lt;&lt; p.to_str(); }   But I could probably do:  ostream &amp; operator&lt;&lt;(ostream &amp;os) {     return os &lt;&lt; paragraph; }   What rationale should I base this decision on?  Note:   Paragraph::to_str = (return paragraph)    where paragraph's a string. Quotient group $G/H$ is abelian iff $[G,G] \subseteq H$ On the , there is this line:      $p(\mathrm{height}|\mathrm{male}) = 1.5789$ (A probability distribution over 1 is OK. It is the area under the bell curve that is equal to 1.)    How can a value $&gt;1$ be OK? I thought all probability values were expressed in the range $0 \leq p \leq 1$. Furthermore, given that it is possible to have such a value, how is that value obtained in the example shown on the page? A familiar limit, but in general form .