Find $\sqrt{(-1)^2}$
Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$?
$d = \gcd(c,m),\ ac≡bc\pmod{\! m}\Rightarrow a≡b\pmod {\!m/d}\ $ [Congruence Cancellation & Division Rule] How would you show that if $ac≡bc$ $\mod m$ and $\gcd(c,m)=d$, then  $a≡b$ $\mod \frac{m}{d}$?  Any help would be much appreciated! Sheafs and closed immersion Where did they go?   Where do they went?   Can you please advise which of above forms is correct?  I'm pretty sure that first one is fine but wondering about second one.   Can I use a second form as well or it's incorrect? Do I need an airport transit visa for Dublin, Ireland en route to the UK? I am Haitian, and traveling with a Haitian passport. I have to go to London with a work permit (tier 2 visa) on the 29th of July but I have 2 hour transit in Dublin.  Do I need a transit visa? Can I get it at the airport?  I also have a valid US visa. How to implement glob in C# "As a beginner, I am happy that I learned a new thing."  You will recognize this as your DoReMi-s.( at 4:52)  What is the meaning of the "AS" there and when do we use that?  In depth answers would be appreciated.  Similarly:     Do I have your word as a lawman not to shoot me down like a dog in the street?    Can you explain this sentence?   Can I understand it as a lawman wouldn't shoot down people on the street or wouldn't have his word or what? I want to know how to get the X and Y position of HTML elements such as img and div in JavaScript relative to the browser window. I have a file with the lines like below.  title1:A1 title2:A2 title3:A3 title4:A4 title5:A5  title1:B1 title2:B2 title3:B3 title4:B4 title5:B5  title1:C1 title2:C2 title3:C3 title4:C4 title5:C5  title1:D1 title2:D2 title3:D3 title4:D4 title5:D5  How can I achieve this?   title1    title2     title3    title4 A1         A2         A3         A4 B1         B2         B3         B4 C1         C2         C3         C4 D1         D2         D3         D4 How do I gunzip all files recursively in a target directory? I'd like to know what is the command to use to gunzip all files in a target directory recursively? I tried to use the unzip command but it didn't work.  I tried the command from Why is super.super.method(); not allowed in Java? I read  and thought that would easily be solved (not that it isn't solvable without) if one could write:   @Override public String toString() {     return super.super.toString(); }   I'm not sure if it is useful in many cases, but I wonder why it isn't and if something like this exists in other languages.  What do you guys think?  EDIT: To clarify: yes I know, that's impossible in Java and I don't really miss it. This is nothing I expected to work and was surprised getting a compiler error. I just had the idea and like to discuss it. Looking for Space Exploration Book which is collection of different sub-stories The Book is about an earthship exploring the frontiers that I read as a kid (80's). All the stories are told by the same character but different encounters and events.    The first story is about the crew exploring the planet in a number of separate ships when one of them is taken over by a telepathic alien. As the ship returns however the surprise is sprung on the alien that the Mother ship is Computer controlled. Another is the ship shrinking and then finding after its shrunk beyond the atom that it comes back in as gigantic compared to the Universe in a closed loop.   Hopefully that's enough to trigger a memory. How can I determine which process owns a hotkey in Windows? Given a hotkey, how can I find which program owns it? When can you build up all representations from the fundamental and antifundamental ones? Under what conditions can you determine all representations of a Lie algebra from the fundamental and antifundamental ones using just the tensor product, direct sum and Clebsch-Gordan decomposition? I think this is true for $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{C})$, or at least that's what physics books lead me to believe!   Is it true in general for semisimple Lie algebras? Why can’t the Doctor’s companions understand babies and horses? The Doctor claimed to speak both horse and baby. The reason human companions can understand every language is because the TARDIS translates. So why can't anyone else understand babies and horses? "They're using a cell phone" vs. "They're using cell phones" The usage of singular and plural has always been confusing for me.   I often see sentences like these         People are using cell phones.   People are using a cell phone.      Does the first sentence mean everyone has a phone and they are all using their own?  Does the second sentence mean they are sharing one cell phone?   If I see a group of people holding a cell phone in their hand(s), &lt;- even this is confusing for me, should I use the first sentence then?   Another example: you see two men, and both of them are carrying a bag. Which sentence should I say/use?        They are carrying backpacks          They are carrying a backpack      Could you please make it clear for me? I have some thoughts on this. First, I want to say I am no expert on cryptography, I just know some stuff, and I took a cryptography class in University. I am very interested in this topic.  I understand that many security experts out there might get angry just by the discussion of these things, but I hope you can spare me :)  My Questions:  Would it be possible to break an RSA key, in for example 1 week of time, if the cracker have already spent X number of years building an index of primes by performing every permutation of existing prime keys up to 2^2048 ?  I understand this would take an immense amount of time to do, but it would only be done once.  Given the public key, you would be able to look up what the two primes are, and hence retrieve the private key instantly.  When I consider who might break such a code, I am primarily thinking about NSA and other Governments that might have spent years in research and have the resources and interest in doing so.  Breaking an 256 AES would then be easier, as AES keys are often generated using the RSA private key.  Would it be possible to build such an index ? How long would that take in permutations, and perhaps with the fastest computer in the world ?  I hope we can have a nice chat about this. We are just discussion theory here. I hope you can offer me some insight on this topic.  Thanks! Where is the pet store? I was playing sims freeplay on my iPod yesterday but could not find the pet store. Where is the pet store in the preteens are here update? Execute bash script from URL Is it possible to keep publishing under my professional (maiden) name, different from my married legal name? Proving Cauchy condensation test