Let the dihedral group $D_{12}=\{xy\mid x^6=y^2=1,xy=yx^{-1}=yx^5\}$  Order 2 subgroups are: $\{1, x^3\}$, $\{1, y\}$, $\{1, xy\}$, $\{1, x^2 y\}$,$\{1, x^3 y\}$, $\{1, x^4 y\}$,$\{1, x^5y\}$.  Order 3 subgroups are: $\{1, x, x^5\}$, $\{1, x^2, x^4\}$.  Order 6 subgroups are: $\{1, x, x^2, x^3, x^4, x^5\}$.  Does $D_{12}$ have a subgroup of order 4?
It seems that $\{e\}, \{e,s\}, \{e,rs\}, \{e,r^2s\},...,\{e,r^{n-1}s\}, \{e,r,r^2,...,r^{n-1}\}, D_n$ are always subgroups of $D_n$. Especially when $n$ is odd, these seem to be the only subgroups. But when n is even, say $n=4$, then there are also $\{e,s,r^2,r^2s\}$ and $\{e,rs,r^2,r^3s\}$. It makes me wonder, is there a general formula/algorithm for finding all subgroups of $D_n$ when $n$ is even?
C++11 does not deduce type when std::function or lambda functions are involved What kind of distribution is this "almost" uniformly distributed data for calls/week? My supervisor asked me to find out which distribution represents a particular situation.  I have a VoIP generator that generates calls "uniformly" distributed between callers. This means that the volume per caller distribution is "almost" uniformly distributed between a minimum and maximum. So by running a test with 10000 users and a min value equal to 30 calls per week and a max value equal to 90 calls/week i obtain that not all the users respect this limits: we have some users that generate &lt;30 calls and some other users that generate >90 calls. It is clear that the obtained distribution is not uniform.  The situation is this:    He said that i have to perform a sort of numerical process in order to find some formulas that could define this distribution. Initially, as wrote before,we wanted to obtain a uniform (min,max) distribution (the green area in the figure) but this is not the case as proved with chi-square test.  Moreover the curve in figure is not symmetric, the probability of call generation below 30 call/week and greater that 90 call/week are not identical (it is high for 90calls/week). The variability of the number of generated call increases with the increasing call generation rates.   "Actually implementation of this distribution is nothing but assigning different call rates in a range for users in domain which indicates implementation of several delta functions. As the call rates increases the variability of the generated calls also increases with the average call rate and this leads to the asymmetric behavior of the curve." [cit. from the Voipgenerator documentation]  Someone can help me?I think that now i cannot use Q-Q plot because i don't know which theoretical distribution i have to use in order to compare it with my empirical data.  Sorry if I have stressed with a similar problem a few weeks ago, but initially we thought we could change the implementation, but now we cannot. Hence i have to discover the type of the distribution i obtained and i don't know how can i do this. What is the closure of an open ball $B_X(\mathbf{a},r)$ in $X=\mathbb{R}^n$? Suppose we have the open ball $B_{X}(\mathbf{a},r)$ and the closed ball $\bar{B}_{X}(\mathbf{a},r)$ of radius $r$ about $\mathbf{a}\in\mathbb{R}^n=X$ with the Euclidean metric $d_2$.  What is the closure of an open ball $B_{X}(\mathbf{a},r)$ in $(\mathbb{R}^n,d_2)$?  First, I have defined the sets like this:  $$B_X(\mathbf{a},r)=\{\mathbf{y}\in\mathbb{R}^n:d_2(\mathbf{a},\mathbf{y})&lt;r\}$$ $$\bar{B}_X(\mathbf{a},r)=\{\mathbf{y}\in\mathbb{R}^n:d_2(\mathbf{a},\mathbf{y})\le r\}$$  Then, let $U=B_X(\mathbf{a},r)$ and claim the closure is $$\bar{U}=\{\mathbf{x}\in\mathbb{R}^n:\exists (\mathbf{x}_n)\in U\text{s.t}~\mathbf{x}_n\to \mathbf{x}\}=\bar{B}_X(\mathbf{a},r).$$  Since closed balls are closed then we have $$\bar{U}\subset\bar{B}_X(\mathbf{a},r).$$  But, how can I prove the other direction $\bar{B}_X(\mathbf{a},r)\subset\bar{U}$?  Since it's a containment argument I know that I have to take any element from $\bar{B}_X(\mathbf{a},r)$ and show it is in $\bar{U}=\{\mathbf{x}\in\mathbb{R}^n:\exists (\mathbf{x}_n)\in U\text{s.t}~\mathbf{x}_n\to \mathbf{x}\}$.  But I do not know how to get an expression for $\mathbf{x}_n$.  I would appreciate any help. How to cd into a directory with this name "-2" (starting with the hyphen)? I have a directory named:  -2   I want to cd into it but the cd complains:  bash: cd: -2: invalid option   With no success, I've tried:  cd "-2" cd '-2' cd \-2   Any solution?  Edit: no file browsers like mc, etc. available on the server. How to break out of multiple loops? Given the following code (that doesn't work):  while True:     #snip: print out current state     while True:         ok = get_input("Is this ok? (y/n)")         if ok.lower() == "y": break 2 #this doesn't work :(         if ok.lower() == "n": break     #do more processing with menus and stuff   Is there a way to make this work? Or do I have do one check to break out of the input loop, then another, more limited, check in the outside loop to break out all together if the user is satisfied? I refer to , which I understand says that the branch points of a nonconstant holomorphic map $F: X \to Y$ between Riemann surfaces $X$ and $Y$, both of which are not necessarily compact, form a discrete subset of range $Y$ of the map $F$. (Not sure if 'connected' is part of the definition of Riemann surfaces in other textbooks, but it is here.) Question 1: Do I understand correctly, and is this indeed true? What else I understand:  $A$ is a discrete subspace/subset (I just assume everything here is the usual subspace topology) of $B$ if and only if every singleton subset of $A$ is open in $A$ if and only if every point of $A$ is isolated.  This overflow question () seems to suggest this is not true but is true if $F$ is proper. However, I'm not sure the definitions of Riemann surface are the same (maybe some answers use definitions where Riemann surfaces are not necessarily connected or something).  2.1. I know $F$ is 'discrete' (meaning that its fibres are discrete; here, i refer to a ): see definition before Lemma 3.1, on p. 7), by Proposition II.3.12, and open, by open mapping theorem, which is Proposition II.3.8, but I'm not sure these help show $F$ is proper or are otherwise helpful. Ostensibly 'discrete' maps don't map discrete subsets to discrete subsets.   This stackexchange question () seems to suggest this is true.  I think this is true if $F$ is injective or at least something like locally injective. (Please don't make me type up all the stuff I did for this part.) However based on the question in (3) and based on the , which I think uses the term 'branch points' to mean the same thing as what Miranda means by 'ramification points' (and thus is different from Miranda's 'branch points'), it seems that no neighbourhood $U$ of a ramification point $p \in U \subseteq X$ of $F$ is such that the restriction $F|_U$ is injective.  Kinda related:    This page () says    Now, the ramification and branch points must form a discrete set  I don't think the page assumes anything like $F$ is proper.  The  text in (2.1) actually seems to assume proper based on the '3.7. Proper holomorphic maps' part, but said part refers to 'Lemma 3.17', so I think proper can be replaced with closed. Maybe $F$ is closed or something.  Kinda related (see Question 2 below):     says    If $F$ is proper then the image $\Delta = F(R)$ is discrete in $Y$.   This seems to have all the same assumptions as with Miranda. Ostensibly, either Miranda is wrong or there's some higher level machinery that allows us to omit $F$ proper.  Question 2: To possibly generalise this, what are the minimum requirements on $F, X$ and $Y$ to make $F$ map discrete subsets to discrete subsets? I mean, I'm not sure we really need 'holomorphic' here. I have a feeling this applies perhaps to maps that are just open/closed, continuous and discrete and spaces that are just Hausdorff and locally compact or something.  (again) Kinda related: Creating A Bootable USB Of Windows 8.1 On OS X? Reasons for wanting create such a usb include:   PC is unbootable and you want to prepare a bootable USB using the ISO file that you have on your external hard drive. to install Windows 8.1 on Mac without Boot Camp.   What are the necessary steps to create such a Windows 8.1 bootable usb on a Mac ? Drop rates for set / legendary items Just wondering what the drop rates were for Set / Legendary items, or is it totally random? I imagine they would increase for later acts in Inferno. My monk is comfortable in Act 3 in Inferno, but I've only found one set item, so MF will increase chances for them? On the recent changes to flagging and limits Looks like flagging has been affected by releases today. Are these intended effects?   Higher daily limits (appear doubled for me, flagging of posts limit also doubled).  Explained by Shog9 below. Soft comment deletion  - e.g. for "accept rate" obsolete comments - it does in fact work, but the immediate feedback is gone.  This is fixed now. Has the throttling rate changed?  Seems possible to repeated flag without waiting 5 seconds.  If it was broken, it isn't anymore. Old soft deletions that were neither categorised as "helpful" or "declined" have all become "helpful".   If limits have been updated, can we get the new formula published? Not changed, as confirmed by Shog9. How can I use quotas in samba? I have a simple set-up with just one public share on my configuration file. How can I restrict its size? Would it be possible to create a wand more powerful than the Elder Wand? I'm looking to find records in a table that match a specific number that the user enters.  So, the user may enter 12345, but this could be 123zz4-5 in the database. I imagine something like this would work, if PHP functions worked in MySQL. SELECT * FROM foo WHERE preg_replace(&quot;/[^0-9]/&quot;,&quot;&quot;,bar) = '12345'  What's the equivalent function or way to do this with just MySQL? Speed is not important. $A=\begin{pmatrix}0&amp;0&amp;1\\ 0&amp;1&amp;0\\ 1&amp;0&amp;0\end{pmatrix}$  How do I show that there are no solutions for $X^2=A$ in $M_3(R)$?  I tried $X^2=\begin{pmatrix}a_{11}&amp;a_{12}&amp;a_{13}\\ a_{21}&amp;a_{22}&amp;a_{23}\\ a_{31}&amp;a_{32}&amp;a_{33}\end{pmatrix}^2=A$, but I couldn't seem to find anything that shows there are no solutions. How should I approach this? I have  a  requirement where a future method need to be called from batch apex ,I came across various articles stating that a future method could not be called from the batch. Is there any alternate solutions for that? Some icons are correct, and some are whack. Turns out it's not just the hot questions list, but the whole icon set. I've already read The Hobbit and The Lord of the Rings, which I assume would be the best start in any case. I also got about halfway through The Silmarillion several years ago and am quite familiar with the legendarium as a whole, but for the sake of argument let's assume I'm starting from scratch.  What is the best order for a budding scholar of Middle-earth to read Tolkien's books? Or does it even matter once you've covered the core novels? Chronological order clearly isn't the most logical approach here, and even publication date isn't necessarily best.   The Hobbit The Lord of the Rings The Silmarillion The Children of Hurin The History of Middle-earth Unfinished Tales The Letters of J.R.R. Tolkien Anything else   Note: After asking this question I found  on Reddit. It covers even more material and in more detail than my original question or any answers here. Why have a letter in a word when it’s silent in pronunciation, like the b in debt?  Can anyone please clarify my uncertainty here? I have an enum construct like this:  public enum EnumDisplayStatus {     None    = 1,     Visible = 2,     Hidden  = 3,     MarkedForDeletion = 4 }   In my database, the enumerations are referenced by value. My question is, how can I turn the number representation of the enum back to the string name.   For example, given 2 the result should be Visible. Navigating lookup relationships with dot notations, to retrieve field values I need to access the value from a roll-up field on my Opportunities, from it's related Opportunity Product.  When I use oli.Opportunity.Software_Line_Item__c the value retrieved is always null.  So I've also tried oli.OpportunityId.Software_Line_Item__c which causes an error in the compiler     Invalid foreign key relationship: OpportunityLineItem.OpportunityId   &amp; just to make sure that it's not the roll-up field calculation that's not working as expected, I've tested oli.Opportunity.Name which is also returning null.  I know this is basic but I don't have Eclipse installed so I can't check the correct reference to use. Could someone please let me know how I can navigate this relationship, to retrieve the value from the Software_Line_Item__c field? How to get a toddler to start chewing and eating solids? My son is 18 months old and still doesn't want to chew and eat solid foods. We have to blend soups and other food into purée for him to accept it. If there is even a slight solid, he will automatically push it out of his mouth with his tongue.  I have had a breakthrough with breakfast where I don't blend as much and leave it a little chunky. He ate it but still pushed chunks of soft banana that were bigger out.  Some of my friends have kids that are 4-5 months younger, who are eating chicken cutlets.  Any ideas or proven methods for getting a child to start eating solids?