Why are there only two groups of order $21$?
Number of non-isomorphic groups of order 21
I'm currently working on a scene containing 1000+ objects which all need to be set to a different draw type to save on computationally expenses in the viewport. But doing this for every object separately is really tedious. Therefor I'm wondering if there's a way to change the maximum draw type of multiple objects simultaneously. Preferably only the selected objects. Is it possible for a physical object to have an irrational length? For a Google account, you are allowed to sync Contacts and People details. What is the difference between these two? Is there a simple way to explain how Kepler's third law follows from the inverse square law that of gravity  (and laws of motion)  For example for Kepler's second law we can say it's because Gravity is a central force  What is it about the law of gravity which ultimately gives rise to Kepler's third law? What happens to water level when ice XII melts? There exists this famous idea that if all floating icebergs melt, water level will stay the same (because the water replaced by ice is the volume of the melted ice). Now,   Is this always so, if you allow more exotic forms of ice (take e.g. ice XII with density 1.29) ? Is this always so on/in earth ?   EDIT:  I was looking a possibility that you would have water as a liquid that contains pieces of water as a solid. No land is assumed to exist here. "Ice" is always floating or sunken(if possible).  To reformulate, can Ice be sunken in water in any circumstance (pressure, temperature)?  EDIT 2:  Wikipedia: "Ice VII has a density of about 1.65 g cm-3 (at 2.5 GPa and 25°C)".   And water as an incompressible liquid has a density of 1 g cm-3. This means that ice VII is at the bottom of the "lake" and will expand when it melts. Is this true? And are there environments in earth where this (or similar) can happen? Do file descriptors optimise writing to files? Auto Package download for TeXLive I use MiKTeX on Windows and quite satisfied with it. Recently I started switching all my tasks toward open-source alternatives, and in the course I would love to use Linux. In Linux TeXLive is available as alternative to MiKTeX. The thing I really like about MiKTeX, is its ability to install packages automatically. Can I do the same in TeXLive too? Is there a way I can enable such or install some plugins for it?  I am using Fedora 18, if that's needed. Sending and Parsing JSON Objects in Android The Half-Elven and their choices Earendil and Elwing were given the choice of kindreds after their journey to Valinor, This choice was extended to their sons (specifically sons, not daughters or grandchildren)  Elrond chose the race of the Eldar and his brother Elros chose to be counted amongst the Edain. Thus far Manwë's proclamation has been fulfilled.  There are some inconsistencies here:   Arwen was given the choice, but Elladan and Elrohir remained as Eldar in Imladris Tar-Vardamir was not given the choice The offspring of Imrazor and Mithrelas in Dol Amroth dont seem to have been given a choice   I can think of only one explanation, that the Mannish blood is dominant over the blood of the Eldar, thus, As Vardamir was the offspring of two mortals he did not fall under Manwë's proclamation despite being the son of a half elven.  However this does not seem to explain the other inconsistencies, Is it possible that Elladan and Elrohir and also the princes of Dol Amroth weren't considered important enough by Manwë or Eru to give them the choice.  Also as a side note Tolkien states there were only three unions between man and elf, but this seems to ignore Turin/Finduilas and Imrazor/Mithrelas  My question is, Why was Arwen given the choice and not the sons of Elros? Date api for managing off days in an year? In this  to a question about a series, a theorem was stated:     If $A= \{a_i \}$ is a set such that $\sum_{i = 1}^{\infty} \frac{1}{a_i}$ converges, then $d(A) = 0$, where $d(A)$ is the  of the set.   My background in number theory is basically zero and all my attempts to prove this have been utterly unseccessful; would anyone please help me, or at least provide me with a hint to prove it? How can I use an align environment flush left? Is there a way possible to imitate the align environment for math mode commands, without having all writing being placed toward the center on the output page?  MWE:  \documentclass{article} \usepackage{mathtools}  \begin{document} \begin{align*} \cos\theta_1 \cos\theta_2-\sin\theta_1\sin\theta_2 &amp;= \cos(\theta_1 +\theta_2) \\ \sin\theta_1 \cos\theta_2 + \cos\theta_1 \sin\theta_2 &amp;= \sin(\theta_1+\theta_2) \end{align*}  \end{document} if $(x_0,y_0)$ is local extrema in $ax^2 + by^2 + cxy + dx + ey + l$ then its global too. An exercise on a book asks me to prove that if a point $(x_0,y_0)$ is a local extrema for the function $$f(x,y) = ax^2 + by^2 + cxy + dx + ey + l$$ then it's also a global extrema.  The exercise asks me to consider that $g(t) = f(x_0+ht,y_0+kt)$ is a parabola, but I couldn't understand how it helps. Maybe since it's a parabola, I know that this derivative has to be $0$ in some point, and also that this function $g(t)$ has a global extrema point, but I couldn't derive anything from this. I also tried to find the derivatives, equal them to $0$ and solve the system. I got:  $$x_0 = -\frac{2 b d - c e}{4 a b - c^2}, y_0 = -\frac{   c d - 2 a e}{-4 a b + c^2}$$  But when I try to find   $$f\left(-\frac{2 b d - c e}{4 a b - c^2},-\frac{   c d - 2 a e}{-4 a b + c^2}\right)$$ so I can try to prove:  $$f(x,y)-f(x_0,y_0)$$  is either greater or less $0$ depending on the case (so I can do $f(x,y) \le f(x_0,y_0)$ and $f(x,y) \ge f(x_0,y_0)$)  things get hairy. I have the string   a.b.c.d   I want to count the occurrences of '.' in an idiomatic way, preferably a one-liner.  (Previously I had expressed this constraint as "without a loop", in case you're wondering why everyone's trying to answer without using a loop). What's a big-vocabulary word for someone with a big vocabulary? What's a big-vocabulary word for someone with a big vocabulary?  I'd like to say I'm looking for a "_____". How would I use Takeown to take ownership of all folders on one drive? I have a hard drive which I need to recover specific files from but all the folders in the K:\ drive are denying me access even though I took owner ship of the full drive I.E      C:\Windows\System32>takeown /F K:\   This granted me access to the drive and see the folders but I can not access any of the folders after that.   Is there a way in which I can use Takeown to apply ownership to every directory on the drive> Why do mathematicians use single-letter variables? Series rearrangement and Riemann's theorem I think understand that when I have a conditionally convergent series, it consists of series of positive and negative values which are divergent and thus one can find such permutation of indices $\phi : \mathbb{N}\rightarrow\mathbb{N}$ so the rearranged series sums up to an arbitrary value, diverges or oscillates.   This is how I understand what Riemann's rearrangement theorem says, but how do I use it practically? For example, when I have the conditionally convergent series:  $$\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n}}$$  and I want to rearrange it to be divergent or to sum up to certain value M? How do I define such bijections $\phi$ ?  I appreciate all help. I am trying to execute a web service call to our backend system when the opportunity is closed/won. I am attempting to do a  POST to our endpoint. All the api call is really carrying the account name and a few structural detail back (All listed below). My problem is that when the trigger executes I get the following error in the debug logs "System.CalloutException: You have uncommitted work pending. Please commit or rollback before calling out".  The Process Builder Passing in the company name:   APEX Class  public class Play2 { @TestVisible static HttpResponse res;  @InvocableMethod public static void sendNotification(List&lt;String&gt; name) {        HttpRequest req = new HttpRequest();     Http http = new Http();     String payload = JSON.serialize(         new Map&lt;String, String&gt; {             'companyName' =&gt; name.get(0),             'salesforceAccountType' =&gt; 'Company',             'salesforceSynchStatus' =&gt; 'Enabled'         }     );      req.setEndpoint('https://TheEndPoint');     req.setMethod('POST');     req.setHeader('Authorization', 'The Mighty API KEY');     req.setHeader('Content-Type', 'application/json');      req.setBody( payload );     req.setCompressed(true);       try {         System.debug('Point 1');         res = http.send(req);         System.debug('Point 2');     } catch(System.CalloutException e) {         System.debug('Callout error: '+ e);         System.debug(res.toString());     } } } Unable to mount Windows (NTFS) filesystem due to hibernation