Is the Big Bang notion compatible with the flat space? (A problem with the **moment** of Big Bang, not with the place of it)
Did the Big Bang happen at a point?
How many ways can we let people into a movie theater if they only have half-dollars and dollars? My interest in combinatorics was recently sparked when I read about the many things that the Catalan numbers count, as found by Richard Stanley. I picked up a copy of Brualdi's Combinatorics, and while browsing the section on counting sequences I found a nice little puzzle that has definitely puzzled me.     Let $m$ and $n$ be nonnegative integers with $n\geq m$. There are $m+n$ people in line to get into a theater for which admission if $50$ cents. Of the $m+n$ people, $n$ have a $50$-cent piece and $m$ have a $\$ 1$ dollar bill. The box offices opens with an empty cash register. Show that the number of ways the people can line up so that change is available when needed is   $$ \frac{n-m+1}{n+1}\binom{m+n}{m}. $$     I first noted that the first person to enter must be one of the $n$ with a half-dollar. Now the register has a half-dollar change. The second person can be either a person with a half-dollar or a dollar. In the first case, the register will now have two half-dollars, in the second case, the register will now have one dollar bill. So it seems to me that when one of the $n$ people with a half-dollar enters, the number of half-dollars in the register increases by $1$, and when one of the $m$ people with a bill enters, the number of half-dollars decreases by $1$ but the number of bills increases by $1$.   I tried to model this by looking at paths in $\mathbb{Z}^2$. The $x$-axis is like the number of half-dollars, and the $y$-axis is the number of bills. You start at $(0,0)$, and you can take steps forward $(1,0)$ or backwards diagonally $(-1,1)$ corresponding to who enters, but you must always stay in the first quadrant of the plane without crossing over the axes. The goal is to make $m+n$ moves, and I figured maybe the number of such paths is counted by $\frac{n-m+1}{n+1}\binom{m+n}{m}$, but I'm not sure how to show this. I don't know if this observation simplifies the problem at all, as I don't know how to finish up. I'd be happy to see how this problem is done, thank you. Best WPF open source projects I have programmed with WPF for more than a year. When I started there weren't so many examples and guidelines to follow, now the situation is much more better.  I would like to ask you some help in listing the best WPF (open) source code examples or projects that you suggest to study in order to learn more about project structure, best practices, etc.  Thank you, Marco Is the neutral wire considered safe? Let $R$ be the subring of $M_2(\mathbb{R})$ defined by $$R=\left\{\begin{pmatrix}a&amp;b\\0&amp;d\end{pmatrix}\mid a\in\mathbb{Q},b,d\in\mathbb{R}\right\}$$  Show that $R$ is right-Artinian but not left-Artinian.  The "obvious" approach to show it is not left-Artinian is to find an $A\in R$ such that $$RA\supset R(A^2)\supset R(A^3)\supset\dots$$  is an infinite decreasing chain of proper left submodules of $R$ (as a module over itself). However, after spending a lot of time in this endeavor I've come to the conclusion that no such element $A$ exists (I can post a proof if anybody is interested but it'd be long and not worth the time for me otherwise). Does anybody have an idea for another approach I can take to this? I have no idea what else submodules of $R$ would look like. Does $\sqrt{a+b} \le \sqrt a + \sqrt b$ hold for all positive real numbers a and b? I thought of this a while ago, but can't make up a proof or a counterexample. Does anyone know more about this?  $$\sqrt{a+b}  \le \sqrt a + \sqrt b , \forall a,b \in \mathbb R_+$$  Moreover, what happens with more variables? Say:  $$\sqrt{x_1+x_2+...+x_n} \le \sqrt x_1 +\sqrt x_2 + ... + \sqrt x_n $$  with $ x_i \in\mathbb R_+ \forall i \in \{1,2,...,n\} $  Or when $i = \mathbb N$  PS: I tried looking on the internet for this but I don't know how I would call this. The following methods can be used to add CSS for specific versions of IE:   Paste the code into html.tpl.php:  &lt;!--[if lt IE 9]&gt;&lt;link type="text/css" rel="stylesheet" src="/sites/all/themes/mytheme/less/ie.css"&gt;&lt;![endif]--&gt; Make use of the plugin  Use drupal_add_css() function   I have always used option #1 above, and adding yet another plugin for something so simple seemed crazy. BUT working with LESS and calling in statically means the file does not get processed by the LESS pre-processor, so the question:  How do you include a stylesheet using drupal_add_css() for specific versions of IE? Why do backgrounded processes sometimes stop spontaneously? Sometimes, some time after I've backgrounded a process with bg in bash, when I press Enter in the same shell to redisplay the prompt (just to check that I'm still in bash when some output from the background process has been displayed), the background process seems to stop spontaneously.  If I do bg again the same problem recurs.  The only way to fix it seems to be fg.  Why does this happen? Conky does not work properly on Ubuntu 16.04 GNOME I have this strange problem with conky. It doesn't show properly on the desktop:    I've tried everything from editing conky widget file to reinstalling conky and conky manager (v.2.4). Who knows how to fix it? SSH: How to disable weak ciphers? Simple georeferenced 3D data format for building models Detect browser language in Rails Following on from this question:   It seems all feats carry over to your new form if your new form is capable of using them (so Mounted Combatant and Inspiring leader would usually be ruled as unusable in most circumstances). But what about Tough?     Your hit point maximum increases by an amount equal to twice your   level when you gain this feat. Whenever you gain a level thereafter,   your hit point maximum increases by an additional 2 hit points.   Would this increase your maximum HP by the same amount when in the form of a rat, or brown bear? Set the location in iPhone Simulator How can I set the location (as it's picked up in CoreLocation services) in the iPhone Simulator? Recommendations on a free library to be used for zipping files I need to zip and password-protect a file. Is there a good (free) library for this?  This needs to be opened by a third party, so the password protection needs to work with standard tools. How does Delta baryon decay conserve angular momentum? How do I make animations a steady speed? When I make an animation, it always starts out slow, then speeds up, then slows down at the end.  I cannot figure out how to fix this. Shapefile layer no longer editable (toggle edit option greyed out) in QGIS Cannot downgrade wpa_supplicant to fix Wi-Fi connection to a corporate network (WPA2/PEAP with no CA certificate) in Ubuntu 18.04 I recently upgraded my Ubuntu from 16.04 to 18.04 (clean install, full wipe). Now I can no longer connect to the corporate network at my office (perhaps related to these bugs: , ). What proved to fix this issue in Ubuntu 16.04 is the answer to ; Downgrade wpa_supplicant to 2.1.  However, even with the linode repository, this fix can no longer be applied. There is no version for wpa_supplicant below 2.6 to the best of my knowledge for Ubuntu 18.04.  My question is this: How can I downgrade wpa_supplicant to 2.1 for Ubuntu 18.04?  Configuration needed for corporate the wifi connection:   Security: WPA &amp; WPA2 Enterprise Authentication: Protected EAP (PEAP) Checkbox No CA certificate required. Inner authentication: MSCHAPv2 Custom Edit control inside a ExtJS Editor grid Big O, how do you calculate/approximate it?