So I know that subgroups of $(\mathbb{R},+)$ are either isomorphic to $x \mathbb{Z}$ for $x\in \mathbb{R}$, or dense in $\mathbb{R}$.  I don't see how (if?) this descends to the quotient, though. Denote $G = \mathbb{R}/\mathbb{Z}$. If $H \leq G$, then by the third iso theorem there is a group $\mathbb{Z} \leq \tilde{H} \leq \mathbb{R}$ such that $H \cong \tilde{H}/\mathbb{Z}$. How to proceed? I'm very lost.    (Also, from this it supposedly follows that for $x\in\mathbb{R}$ the image of  $$ \phi_x:\mathbb{Z} \to G, \quad n \mapsto nx + \mathbb{Z} $$ is dense in $G$ if $x$ is irrational. I also don't see this: For any $x$, the image of $\phi_x$ is $x\mathbb{Z} + \mathbb{Z}$, is this not cyclic? The above result would then imply that the image is not dense. Somewhere is an error in my thought process.)
Let $T=\{z:|z|=1 \mathbin{\text{and}} z \in \mathbb C\}$ be the unit circle in the complex plane, considered as a topological group under complex multiplication and the usual topology. Show that every subgroup of $T$ is either dense in $T$ or finite.  How to prove the denseness?
Insert HTML markup into Gmail signature Is there any way to add HTML markup signature in Gmail settings? I mean, insert some HTML markup without using rich editor, just inserting HTML? How to find relevant and new open research topics? Is it true that an ideal is primary iff its radical is prime? I am travelling from Panama to Los Angeles and around the States for approx 80 days then flying back to Panama and making my way up Central America for approx 3 months before flying back to LA to fly home to Australia.    Will this be OK with an ESTA?    Does it count as spending too long in the States?    I have been told that Panama is not far enough away. Is the function differentiable at $0$? Let $$f(x) = \begin{cases}\begin{align*}&amp;\cos{\dfrac{1}{x}}, &amp;x \neq0 \\ &amp;0, &amp;x=0. \end{align*}\end{cases}$$ Is the function $F(x) = \displaystyle \int_{0}^x f dx$ differentiable at $0$?  We can see that the function $f(x)$ is continuous everywhere except $x=0$. In order to show differentiability we will need to show the derivatives from the left and right are equal. So we need to show that $$\lim_{h \to 0^+} \dfrac{F(x+h)-F(x)}{h} = \lim_{h \to 0^-} \dfrac{F(x+h)-F(x)}{h}.$$ The derivative from the right is $$\lim_{h \to 0^+} \dfrac{F(x+h)-F(x)}{h} = \lim_{h \to 0^+} \dfrac{\displaystyle \int_{0}^x\cos{\dfrac{1}{x+h}}-\int_{0}^x\cos{\dfrac{1}{x}}}{h}$$ and the derivative from the left is $$\lim_{h \to 0^-} \dfrac{F(x+h)-F(x)}{h} = \lim_{h \to 0^-} \dfrac{\displaystyle \int_{0}^x\cos{\dfrac{1}{x+h}}-\int_{0}^x\cos{\dfrac{1}{x}}}{h}.$$ I am not sure how to proceed. When GMing a game, how much should you prepare in advance? Obviously you need to have some high level idea of what's going on in the world around the players, and where you are heading with the story, but in an ongoing game at some point players will be players and will head off in a different direction to what you planned.  How far ahead should you be planning to minimise or eliminate being caught off guard, but also try to minimise the amount of 'wasted content' you come up with but never gets used? Compiler warning: null reference exception I have the following code in Visual Studio 2005.      Dim OutFile As System.IO.StreamWriter     Try         OutFile = New System.IO.StreamWriter(Filename)        // Do stuff with OutFile     Catch Ex As Exception        // Handle Exception     Finally        If OutFile IsNot Nothing Then OutFile.Close()     End Try   But VS2005 brings up the warning for the line "If OutFile IsNot.." that      Variable 'OutFile' is used before it has been assigned a value. A null reference exception could result at runtime.   Is there some way of removing this warning by subtly altering the code or is there just a better way of doing what I'm trying to do?  Thanks  Rob I mean to ask  that $7$ is an arithmetic  expression  but it can also  be  written  as $7x^0$.  which is a constant polynomial  expression.  Every polynomial  expression  is an algebraic  expression  so with  this logic is $7$ an algebraic  expression  or an arithmetic  expression.  Further $7=5+2$, is it an arithmetic  equation  or algebraic  equation  because  it can also be written  in  $7x^0 = 5x^0 + 2x^0$ form. please help.  Thank you. change environment of a running process How might it be possible to alter some variable in the env of an already running process, for example through /proc/PID/environ? That "file" is read-only.  Need to change or unset the DISPLAY variable of a long-running batch job without killing it. Detecting an undefined object property What's the best way of checking if an object property in JavaScript is undefined? PHP Session data not being saved I have one of those "I swear I didn't touch the server" situations. I honestly didn't touch any of the php scripts. The problem I am having is that php data is not being saved across different pages or page refreshes. I know a new session is being created correctly because I can set a session variable (e.g. $_SESSION['foo'] = "foo" and print it back out on the same page just fine. But when I try to use that same variable on another page it is not set! Is there any php functions or information I can use on my hosts server to see what is going on?  Here is an example script that does not work on my hosts' server as of right now:  &lt;?php session_start(); if(isset($_SESSION['views']))     $_SESSION['views'] = $_SESSION['views']+ 1; else     $_SESSION['views'] = 1;  echo "views = ". $_SESSION['views']; echo '&lt;p&gt;&lt;a href="page1.php"&gt;Refresh&lt;/a&gt;&lt;/p&gt;'; ?&gt;   The 'views' variable never gets incremented after doing a page refresh. I'm thinking this is a  problem on their side, but I wanted to make sure I'm not a complete idiot first.  Here is the phpinfo() for my hosts' server (PHP Version 4.4.7): Why must a field whose a group of units is cyclic be finite? Let $F$ be a field and $F^\times$ be its group of units. If $F^\times$ is cyclic, then show that $F$ is finite.  I'm a bit stuck. I know that I can represent $F^\times = \langle u \rangle$ for some $u \in F^\times$ and that we must have that $|F^\times| = o(u)$, where $o(u)$ denotes the order of $u$ in $F^\times$.  I tried assuming $o(u) = \infty$, but I'm not sure exactly where to go from there. I was wondering if I could get a hint. WIth Vimeo Plus account, how to hide/remove the "play" button in the middle of an embedded video? Tree level QFT and classical fields/particles I had to write the proof to show that an Ideal $P$ of a commutative ring $R$ is prime Ideal if $R/P$ is an integral domain.       let $a,b\in R$ s.t. $ab\in P$ ,        $$ab+P=P\implies(a+P)(b+P)=P\implies\overline a \overline b=0$$  and as $R/P$ is an integral domain either $a=0$ or $b=0$ $\implies a+P=P$ or  $b+P=P$.        $\therefore$ either $a\in P$ or $b\in P$.   Hence, $P$ is a prime Ideal.    Am I correct in writing this or am I making some error... At some point I accidentally ran off a cliff with my War Horse which resulted in a death. After respawning, I noticed my War Horse was gone and the default Tennessee Walker was my main horse.   How did the War Horse disappear and is it possible to retrieve it? POSTFIX main.cf permission denied How to resolve ambiguous column names when retrieving results? I have two tables in my database:  NEWS table with columns:   id - the news id  user - the user id of the author)   USERS table with columns:   id - the user id   I want to execute this SQL:   SELECT * FROM news JOIN users ON news.user = user.id    When I get the results in PHP I would like to get associative array and get column names by $row['column-name']. How do I get the news ID and the user ID, having the same column name? Let $R$ be a commutative ring with finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_n$. Let $M$ be a finitely generated module. Then there exists an element $x\in M$ such that $\frac{x}{1}\not\in\mathfrak m_iM_{{\mathfrak m}_i}$ for every $i=1,\dots,n$.    I cannot prove that such an element there exists. I was trying to prove it by induction on $n$. If $n=1$ it is true by Nakayama, so suppose it is true for $n-1$. Then for every $i$ I can find an $x_i\in M$ such that $\frac{x_i}{1}\not\in \mathfrak m_jM_{\mathfrak m_j}$ for every $j\neq i$. If $\frac{x_i}{1}\not\in \mathfrak m_iM_{\mathfrak m_i}$ for some $i$ we are done, so suppose $\frac{x_i}{1}\in\ m_iM_{m_i}$ for every $i$. I don't know how to go on, could you help me? I came across the :    Which states:     $\pi$ Pi      Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that   every possible number combination exists somewhere in pi.  Converted   into ASCII text, somewhere in that infinite string of digits is the   name of every person you will ever love, the date, time and manner    of your death, and the answers to all the great questions of    the universe.   Is this true? Does it make any sense ?