I have to show that $\mathbb{C}/\mathbb{Z}$ is isomorphic to the multiplicative group $\mathbb{C} \setminus \{0\}$.   Proof. Let $f:\mathbb{C} \setminus \{0\} \rightarrow \mathbb{C}/\mathbb{Z}$ be the map $$ f(\alpha) = \alpha \mathbb{Z}.$$ This map has inverse $f^{-1}(\alpha \mathbb{Z})  = \alpha$ so it is bijective. Furthermore, $$f(\alpha  \beta) = (\alpha \beta)\mathbb{Z} = \alpha \mathbb{Z} \beta \mathbb{Z} =f(\alpha) f(\beta)$$ and $f(1) = \mathbb{Z}$, so the map $f$ is also a homomorphism.  Question.  Is this correct? It feels a bit like I am cheating.
Is there a "nice" way to think about the quotient group $\mathbb{C} / \mathbb{Z}$?  Bonus points for $\mathbb{C}/2\mathbb{Z}$ (or even $\mathbb{C}/n\mathbb{Z}$ for $n$ an integer) and how it relates to $\mathbb{C} / \mathbb{Z}$.  By "nice" I mean something like:   $\mathbb{R}/\mathbb{Z}$ is isomorphic to the circle group via the exponential map $\theta \mapsto e^{i\theta}$, and $\mathbb{C}/\Lambda$ is a complex torus for $\Lambda$ an integer lattice (an integer lattice is a discrete subgroup of the form $\alpha\mathbb{Z} + \beta\mathbb{Z}$ where $\alpha,\beta$ are linearly independent over $\mathbb{R}$.)   Intuitively, it seems like it should be something like a circle or elliptic curve.
Are there any places to leave luggage in Brussels, for short and/or long periods of time? The runtime arguments are as follows: $1 is the path to the file containing the list of files $2 is the path to the directory containing the files What I want to do is check that each file listed in $1 exists in the $2 directory  I'm thinking something like:  for f in 'cat $1' do if (FILEEXISTSIN$2DIRECTORY) then echo '$f exists in $2' else echo '$f is missing in $2' sleep 5 exit fi done   As you can see, I want it so that if any of the files listed in $1 don't exist in the directory $2, the script states this then closes. The only part I can't get my head around is the (FILEEXISTSIN$2DIRECTORY) part. I know that you can do [ -e $f ] but I don't know how you can make sure its checking that it exists in the $2 directory. Groups of prime squared order Is the humanoid appearance of most alien species in the Star Trek universe ever canonically addressed? The vast majority of intelligent species in the Star Trek universe are bipedal, humanoid creatures.  From the lot I've seen of TOS, TNG, DS9, Voyager, Enterprise, and the movies, it seems that this coincidence goes largely unquestioned.  Metaknowledge grants that it is much easier to have human actors portray human-like beings, and so this was probably done due to budgetary/technological limitations.  However, I'm wondering if an in-universe reason for this was ever given or explored? Test coverage for setCookies method I am adding cookie to my page in the controller using  ApexPages.currentPage().setCookies(new Cookie[]{CookieName});  When am trying to get the code coverage, my test class is not covering the above line.  How to get the code coverage for the above setcookie method in the test class. Convert a string representation of a hex dump to a byte array using Java? I am looking for a way to convert a long string (from a dump), that represents hex values into a byte array.  I couldn't have phrased it better than the person that posted .  But to keep it original, I'll phrase it my own way: suppose I have a string "00A0BF" that I would like interpreted as the  byte[] {0x00,0xA0,0xBf}   what should I do?  I am a Java novice and ended up using BigInteger and watching out for leading hex zeros. But I think it is ugly and I am sure I am missing something simple. Determining a matrix from its characteristic polynomial I'd like a couple of Atmega328s to run the same program, but they're going to interface with each other and need unique IDs. Is there any such ID or serial number hard-coded on the chip? Given any uncountable subset $S$ of the unit interval. Then $S$ clearly has an accumulation point and indeed uncountably many (which might also be a nice exercise). So my question is: Is there an accumulation point, that again lies in $S$? Why doesn't my Bash script recognize aliases? Does $\lim_{z\rightarrow0}\frac{\bar{z}^2}{z^2}$ exist? I have detached a process from my terminal, like this:  $ process &amp;   That terminal is now long closed, but process is still running, and I want to send some commands to that process's stdin. Is that possible? What techniques have been used to repeat geometry or other detail to the infinite horizon? How to give name to topics created using LDA? Best practices for API versioning? Are there any known how-tos or best practices for web service REST API versioning?  I have noticed that . Is this the only way or are there other ways to accomplish the same goal? If there are multiple ways, what are the merits of each way? Why is compactness so important? I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel theorem, and a proof continuous functions on R from closed subintervals of R are bounded. It seems like such a strange thing to define; why would the fact every open cover admits a finite refinement be so useful? Especially as stating "for every" open cover makes compactness a concept that must be very difficult thing to prove in general - what makes it worth the effort?  If it helps answering, I am about to enter my third year of my undergraduate degree, and came to wonder this upon preliminary reading of introductory topology, where I first found the definition of compactness. Spaces and em-dash In using the em-dash, where do you put the spaces? Which of the following is correct?     School is based on the three R’s —reading, writing, and ’rithemtic.   or     School is based on the three R’s— reading, writing, and ’rithemtic.   or     School is based on the three R’s — reading, writing, and ’rithemtic. How is the notation $X\sim N(\mu,\sigma^2)$ read? Is it $X$ follows a normal distribution? Or $X$ is a normal distribution? Or perhaps $X$ is approximately normal..  What if there are several variables that follow (or whatever the words is) the same distribution? How is it written? "the 90s were the best decade ever" or "the 90s was the best decade ever"  "the 90s" and "decade" are both referring to the same singular period of 10 years, so "was" seems to make more sense. "90s were" rolls off the tongue better though. According with Mumford the answer is yes, but there are some obscure points in the proof.  We know that there exists a very ample line bundle on $A$ since every abelian variety is projective. Hence, via some corollaries of the Theorem of the Cube, we get an ample line bundle whose restriction to the kernel of $n_A$ is both the trivial bundle and very ample. This should imply that the dimension of $\ker n_A$ is $0$.  Furthermore, Mumford says that the fact that the dimension of the kernel is $0$ implies the surjectivity of $n_A$.  Is anybody able to clarify these two points to me?  Thanks in advance!