Which is the most popular Chiclet keyboard in the world same as laptop?        What proxy to use for online  flash games?
Stuff like polls, recommendations based on subjective constraints, puzzles, webcomics etc. do not belong on the serious main SE sites, where professionals should be considered at work and having just a few spare minutes ("code's compiling") to answer questions, so they should not be distracted by such things. However, I'd also like to have a home for these things still using the SE engine. For the reason I mentioned before, this needs to be a separate place though. Let's call it four.[sitename-here].com.
Properties of p-adic valuations Let $n$ be a integer not equal to zero, $p$ a prime and $$|n|_p = p^{-\operatorname{ord}_p(n)}; |0|_p=0$$  Proposition: $|xy|_p=|x|_p|y|_p$  We want to show: $\operatorname{ord}_p(ab)=\operatorname{ord}_{p}(a)+\operatorname{ord}_p(b)$ So if we suppose there is a largest integral $v$ such that $p^{v}|(ab)$, then we can split up v such that $p^{v_1}|a,p^{v_2}|b$. If we suppose that there exist $v_1,v_2$ with them both being the largest integers such that $p^{v_1}|a, p^{v_2}|b$,  with $v_1+v_2= : v $ then $p^v|ab$. (I don't believe this to be a formal proof but rather my best own "idea" I can come up with...)  The script also mentions that the padic valuations' sums are bounded: $|x+y|_p\le \sup\{|x|_p,|y|_p\}$, how to show this?   Edit:   According to Robert Israel and Bill Dubuques answers, the first proposition is proven, if I understood correctly:   Given $ord(a)= v$ and $ord(b) = u$ which is the same as $a:= p^{v}m, b:= p^{u}n$ with $gcd(m,p)=gcd(n,p)=1$, because $ab=p^{u+v}mn$ with $gcd(mn,p)=1$ it follows that $ord(ab)=ord(a)+ord(b)$ and so $|xy|_{p}=p^{-ord(xy)}=p^{-(ord(x)+ord(y))}=|x|_{p}|y|_{p}$  Now to show: $|x+y|_{p}\le sup\{|x|_{p},|y|_{p} \}$  and the second can be tackled like this: set $a:=p^{v}m. b:=p^{u}n$, then checking two cases $1:(u=v)$ and $2:(v&lt;u)$: $(a+b)=p^{v}m+p^{u}n= p^{v}(m+p^{u-v}n)$ so in case of (u=v) it follows that  $ord(a+b)= u = sup\{ord(a),ord(b)\}$  and in case of $(v&lt;u)$ because the biggest v is $v:= u-1$: $ord(a+b)\le p^{u-1}(m+pn) \le p^{u}(m+n)= sup\{ord(a),ord(b)\}$  It seems that I did something wrong in both cases... How to avoid WSP deployment globally using Visual Studio 2010? How can we avoid the Globally Deployment of a WSP using Visual Studio 2010? multi-layer board copper pour I have a 6-layer board where the internal 4 planes are +15,GND,VCC,-15.  I was wondering if there was any advantage to doing a copper pour on the top and bottom layers?  I would probably leave them floating as I don't want to use micro-vias to say tie it to GND?    Is this actually a bad idea?  I.e. floating copper=antenna.  Would it be just as acceptable to have 4 layer board with the top layer having copper pour to VCC and the bottom having a pour to GND and keep the two internals as +-15?  Note that this is for a fairly low speed circuit which has some analog and digital parts. Fantasy piece involving different worlds and a crippled protagonist This is a novel that I think I remember reading somewhere in the 1990s courtesy of my brother (who does not remember the book). Only a few details are coming to mind at the moment. It's a male protagonist. Something was not quite right with his legs, which led to him being nicknamed Frog or Leapfrog due to his usual method of locomotion, which consisted of pushing off from his legs to his hands and vice versa. Something happens and he winds up moving between a number of worlds (I want to say that he fell into a well or a hole in a field). There was a part which stuck with me where he's in a bottomless hole and finds he keeps falling faster and faster without end with the rushing wind becoming actively painful (I don't remember if he actually reaches a terminal velocity or if some aspect of the world prevented it while still allowing the rushing air to affect him physically).  Unfortunately, that's all I can remember. I don't remember whether he eventually gains companions on the voyage or how it ended. The most vivid memory is that of the nonstop falling. Wi-Fi only works after waking MacBookPro7,1 from sleep Been having some issues with the STA drivers from bcrmwl-kernel-source. While the drivers do function, they only seem to function after putting the laptop to sleep, by either closing the lid, or manually putting it to sleep, and waking it up. The laptop here is a MacBookPro7,1 (Mid 2010 13-inch MacBook Pro), running Ubuntu 17.04.  While the issue isn't bad enough to deter me from using the laptop, it is not an elegant solution, and I believe that there can be a solution found somewhere.  For reference, here are the outputs from the Wireless Info script for both before and after: , . Hopefully these are of some use. If you need additional information about the computer, I will be happy to answer. Object-oriented shell for *nix I'm trying to load a field value (machine name: field_theme_image) from a content of node value 3, in "page--front.tpl.php".  &lt;?php   $node=node_load(3);   print $node-&gt;field_theme_image[$node-&gt;language][0]['uri']; ?&gt;   Using the above code I'm not getting full URL value. Please see the image below. I can see the path value in Devel themer. Modular arithmetic: division, fractions, solving linear congruences I want to ask a question about modular arithmetic. I know, that modular multiplicative inverse exists only if modulo and integer are relatively prime. I want to know, are there any ways of division in modular arithmetic, if modulo and integer aren't relatively prime? I tried to find info about that, but failed. How to access object attribute given string corresponding to name of that attribute What does var prime = num != 1; mean? Spin or rotate an image on hover Install on, install in, install to When I say "programs to install on a new PC" it sounds alright to me, but I'm not sure if it's the correct usage. Which one of the following should I use?   Programs to install on a new PC Programs to install in a new PC Programs to install to a new PC Let $|A|=\alpha, |B|=\beta, |C|= \gamma$ be cardinals and $\beta \leq \gamma$. Prove $\alpha ^{\beta}\le \alpha ^{\gamma}$.  So from the given we know that there's an injection $f:B\to C$ and some functions $h:B\to A, g: C\to A$. We want to prove there's an injection $l_1:A\to C$. It appears that $f$ doesn't help here.  Trying to take representatives from $A$ and show they're in $C$ and there's an injection doesn't work so maybe the function should be $l_2: h \to g$ but I don't know how to work with it. Why does centering not work with calligra? Is it possible to query the content of the Time-based Workflow Queue from an Apex unit test to assess that a specific Workflow Rule has been executed and added a new item to the queue? I didn't see anything alike in the documentation, but wish it could be possible to improve testing and stability overtime.  Thanks List user's folder access permissions Finder: keep sort-by per-directory Some directories make sense to be sorted by date modified, while others make more sense to be kept sorted alphabetically.   How can I make it so that finder will apply a specified sort by selection on a per-directory basis?   If no such feature exists (In which case I hope somebody at APPL is paying attention), I suppose using/remapping the keyboard shortcuts (Defaults being Cmd+Ctrl+5 for Modified and Cmd+Ctrl+1 for Alphabetical by Name) is the next best thing. How do I explain current and voltage to a lay man? A man asked me this question in suburban train when I told him that I am an electrical engineer. Of course I replied him that current is the flow of electrons and in order for the current to flow there should be a potential difference between them. But the man didn't know what electron or potential difference was either. How to turn off Wireless power management permanently Is it possible to search for punctuation/special characters? I tried to find a related question here, but was unsuccessful.  Someone on SO asked ... I tried searching SO and was presented with the search options page. Even Google fails at this search.  So my question is, is there some way to search for an exact word (in quotes) that won't cause it to fail?