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Let $q \in \mathbb C$, $|q|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n:  n \in \mathbb N\}$ is dense in $S^1$.  My attempt: As $(q^n)$ is limited, there is subsequence $(q^{n_j})$ convergent. So given $\epsilon &gt; 0, \exists j_0 \in \mathbb N, \quad |q^{n_j} - q^{n_k}| &lt; \epsilon, \quad \forall j,k \geq j_0$. Supose that $n_j &gt; n_k$, then  $|q^{n_j} - q^{n_k}| = |q^{n_k}||q^{n_j - n_k} - 1| = |q^{n_j - n_k} - 1| &lt; \epsilon, \quad \forall j, k \geq j_0$. So, there is a subsequence of $(q^n)$ that converges to $1$.  Using this fact, how can I ensure that for all $w \in S^1$ there is a subsequence of $(q^n)$ that converges to  $w$? Checking if a double (or float) is NaN in C++ How to delete all lines that do NOT contain a certain word in Vim? In vim I can delete all lines that contain the word "price" with this  :g /price/d   How can I delete all lines that do NOT contain the word "price"? Does Hard reboot damage my Ubuntu Questions about right ideals How Do I Allow A Standard User to Install Programs? Let $L$ denote the set of all lines in the Euclidean plane $\Bbb R^2$. We say that two lines $L$ and $M$ are parallel if either $L = M$ or $L$ and $M$ have no points in common.  Using Euclid’s Parallel Postulate prove that parallelism is an equivalence relation on the set $L$.  I need some help determining if I have the right idea about this.  $L=L$ which means that $L$ is parallel to itself. So this is reflexive.  Suppose we have $L || M$. Then Either $L=M$ or $L$ and $M$ share no points. So Then $M=L$ or $M$ and $L$ share no points. Then $M||L$. So this is symmetric.  Suppose we have $L || M$ and $M || N$. So then either $L=M$ or $L$ and $M$ share no points and either $M=N$ or $M$ and $N$ share no points. So we have that either $L=N$ or $L$ and $N$ share no points. So $L || N$. So then this is transitive.  So from all of this, parallelism is an equivalence relation. I'm not sure if this is correct or not. Any help would be appreciated. What is the bug about: Keyword gets duplicated even when the query is not modified.  Impact: As far as i can tell - cosmetic.  How to replicate: Go to:  and keep pressing Search button. With each search there will be a duplicate python keyword appended to search box.  Best guess for what is causing it: I think you have somewhere a piece of code that appends tag as a keyword without checking if it is already present. My reasoning comes from this url:  not having the same issue - the tag is not present in it. Suppose you have a group isomorphism given by the first isomorphism theorem: $$G/\ker(\phi) \simeq \operatorname{im}(\phi)$$ What can we say about the group $\ker(\phi)\times \operatorname{im}(\phi)$? In particular, when does the following hold: $$G\simeq \ker(\phi)\times \operatorname{im}(\phi)?$$ I ask this question because i want to prove that $GL_n^+(\mathbb{R}) \simeq SL_n(\mathbb{R}) \times \mathbb{R}^*_{&gt;0}$, with $GL_n^+(\mathbb{R})$ the group of matrices with positive determinant. I proved that $SL_n(\mathbb{R})$ is a normal subgroup and that $GL_n^+(\mathbb{R})/  SL_n(\mathbb{R}) \simeq \mathbb{R}^*_{&gt;0}$, using the surjective homomorphism $\det(M)$. I tried something with semidirect products but I got stuck. How do I appropriately examine the dimensionality of binary data? I have 72 binary variables and, at a theoretical level, I am trying to identify groups of variables that seem to vary together. In practice, I am struggling with how to analyze this data properly. I am using R and the psych package. Here is what I have tried and what the outcomes were:  Estimated a tetrachoric correlation matrix (ran without problem) Estimated MSA (relatively strong values, ran without problem) Ran parallel analysis to identify how many factors to extract (ran without problem) EFA using fa() to estimate factors (non-positive definite matrix and strangely high RMSEA) Tried different factor counts, smoothing/unsmoothing, estimation methods, rotations methods, and removing variables (problems persisted) Tried to focus instead on eBIC (always returned as NULL in output) EFA using (inappropriately) Pearson correlations (RMSEA behaved)  From looking at other questions and answers on here, I am now aware that tetrachoric matrices can be problematic like this. However, I don't know what to do with that information. Accept the problems as a limitation and move forward? Conduct a different type of analysis? I considered IRT as is recommended, but am unaware of an exploratory way to conduct IRT analyses. I understand that most of my options are imperfect. My goal is to find an analysis that is at least arguably appropriate for binary variables and that will inform the ways in which I group these variables in my theorizing. Given this, my questions are:  Can I trust EFA on a tetrachoric matrix despite problematic output? Can I somehow recover the lost eBIC value and use that? Is exploratory IRT possible/desirable? Are there other analyses, in R, that can be used to examine the dimensionality of binary data?  I appreciate all feedback! I love learning about data analysis and hearing about how I can improve. How do I Uninstall Windows 10 or Windows 10 preview? How to evaluate a math expression given in string form? Empty intersection and empty union If $n\ne 4$ is composite, then $n$ divides $(n-1)!$. I have a proof and need some feedback. It seems really obvious that the statement is true but it is always the obvious ones that are a little trickier to prove. So I would appreciate any feedback. Thank you!  Here is what I am asked to prove:     If $n$ is composite then $(n-1)! \equiv 0 \pmod n$.   Proof:  $n$ is composite $\implies n=ab$ where $a,b \in \mathbb{Z}$ and $0&lt;a,b&lt;n$.  Case 1: If $a=b$ then $n=a^{2}$. Now $n \mid (n-1)! \implies a \mid (n-1)!$, so $$\begin{aligned} (n-1)! &amp;\equiv 1\times 2\times \dotsb \times a \times\dotsb\times (n-a)\times\dotsb\times (n-1) \\ &amp;\equiv 1\times 2\times \dotsb\times a \times\dotsb\times -a\times\dotsb\times -1 \\ &amp;\equiv 0 \pmod n \end{aligned}$$  Case 2: $0&lt;a&lt;b&lt;n$.  Then, since $a \mid n$, $b \mid n$ and $n \mid (n-1)!$ we have that $a \mid (n-1)!$ and $b \mid (n-1)!$.  So this implies $(n-1)! \equiv 1\times 2\times \dotsb\times a \times\dotsb\times b\times\dotsb\times (n-1) \equiv 0 \pmod n$, Q.E.D. Prove: $\gcd(a,b) = \gcd(a, b + at)$. Why does $A\sin{k(x+c)}=a\sin{kx}+b\cos{kx}$ imply that $A=\sqrt{a^2+b^2}$ and $\tan{c}=-b/a$? I don't understand this. These identities are given in the online notes for MIT's 18.01 calculus class. It's related to taking the sum of two trig functions and transforming them into a single trig function. I can use the formulas to do this, but I am having trouble finding anything on the internet to use a proof for my understanding.  Thanks for any responses!  EDITED (added after the first answer):  Sorry, the identity goes like this:  $A\sin{k(x-c)}=a\sin{kx}+b\cos{kx}$ ,  with the relationships  $a=A\cos{kc}$ , $b=-A\sin{kc}$ , $A=\sqrt{a^2+b^2}$ , and $\tan{kc}=-b/a$ HTML form readonly SELECT tag/input If low-cost index funds are considered the best investment, why are there so many high-cost, managed funds? The general consensus is that the single best indicator of a fund's future performance is the its expense ratio . If you look for more than 5 minutes online you discover the  approach to investing; the , etc. It looks like the less you pay for a fund (and therefore the less it's managed), the better it does in the long run.  So why is it that in peoples' 401(k)s there are so many high-cost actively managed funds? Furthermore, why are there so many high-cost actively managed funds in the first place? Shouldn't the market reflect the fact that actively-managed funds typically don't beat the index they benchmark to and and react by purging itself of these crappy funds? In the 3d viewport I am using a border region to only render things inside the selected region. I did this with CtrlB. I cannot get rid of the region. How can I make the border disappear? Square roots not on same height This part C\frac{1}{\sqrt{G}}\sqrt{\rho} doesn't seem to be on the same height as this part \sqrt{b}\sqrt{l} when compiled. What am I doing wrong?  \begin{flalign*} T=C\frac{1}{\sqrt{G}}\sqrt{\rho}\sqrt{b}\sqrt{l} &amp;&amp; \end{flalign*}