Is the group $U(\mathbb{Z}_{54}) = \{1,5,7,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53\}$ a cyclic group? If so, how do I show this? This group was found as a group of units of the monoid $\mathbb{Z}_{54}$.
I'm trying to show that the group $U(Z_{54})$ is cyclic.  To start, I found the divisors of 54 = {1, 2, 3, 6, 9, 18, 27, 54}  Then I started to find the elements using the powers of a.  Where $U(Z_{54})= &lt;a&gt;$  Therefore, $&lt;a^{54/1}&gt;$={ e }  $ &lt;a^{54/2}&gt;$= { e, $ a^{36}$ }  and such, where the last would be &lt; $a^{54/54}$> = { e }  The things I would like to know is, how does this process show that it's cyclic? Is it because the first divisor (1) and the last divisor (54) is equal to e? Therefore creating a cycle?  Also, what would the &lt; a > to the powers finding the elements process be called?  Thank you!
Hierarchical clustering with mixed type data - what distance/similarity to use? In my dataset we have both continuous and naturally discrete variables. I want to know whether we can do hierarchical clustering using both type of variables. And if yes, what distance measure is appropriate? How can I determine the list of files in a directory from inside my C or C++ code?  I'm not allowed to execute the ls command and parse the results from within my program. Clarifications about Gstreamer I'm a bit confused about what Gstreamer does.   Does it come installed by default with Ubuntu ?  If so which version  is the default for 14.04?  Does it provide hardware accelerated video playback (decoding) ?   If so does it work with VA-API and VDPAU or is it an alternative to those?    Thanks for your help Does constant charging harm my Android cellphone? 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How can I go about recruiting people for play-by-chat games instead of in-person? Are there any online sites in which you can look for people that are looking to play in an online play-by-chat RPG Campaign? I'd like to determine the function corresponding to the following power series: $$x + \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n} \frac{x^{2n+1}}{2n+1}, $$ where $|x|&lt;1$. Let $a$ and $b$ be two integers such that $\left(a,b\right) = 1$. Prove that $\left(a+b, ab\right) = 1$.   $(a,b)=1$ means $a$ and $b$ have no prime factors in common  $ab$ is simply the product of factors of $a$ and factors of $b$.  Let's say $k\mid a+b$ where $k$ is some factor of $a$.  Then $ka=a+b$ and $ka-a=b$ and $a(k-l)=b$.  So $a(k-l)=b, \  a\mid a(k-1)$ [$a$ divides the left hand side] therefore $a\mid b$ [the right hand side].  But $(a,b)=1$ so $a$ cannot divide $b$.  We have a similar argument for $b$.  So $a+b$ is not divisible by any factors of $ab$.  Therefore, $(a+b, ab)=1$.  Would this be correct? Am I missing anything? I'd like to put some shared Javascript to Resource file to reuse it between several components. This Javascript is general purpose set of utility functions. However, functions from my resource file are not available from the Components. Static resource (named TestScript):  function helloWorld() {      return 42; }   Component:  &lt;ltng:require scripts="{!$Resource.TestScript}" &lt;ui:button label="Do job" press="{!c.doJob}"/&gt;   Controller:  doJob: function(cmp, event, helper) {      var i = helloWorld();      // failed with Uncaught ReferenceError: helloWorld is not defined }   I don't like to use  &lt;ltng:require scripts="{!$Resource.***resourceName***}" afterScriptsLoaded="{!c.afterScriptsLoaded}" /&gt;   because my static resource is a set of functions and I'd like just use them as functions later in my controllers. How can I achieve that (if possible)? If not, what's the suggested solution? 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I cannot use the unit step or other similar functions since they are just one step. I have been experimenting with various sigmoid curves and while I can get a single smooth step I cannot get to realize the staircase shape. The closest staircase function I have found is given in  in equation (18) and depicted in Fig. 4 and it is a close example of what I want (i.e generate a staircase in the range $x$ for arbitrary step heights and widths) but it is not smooth at all.  Regarding smooth steps, a likely starting point I found is  but it gives a smooth function of just a single step. I have been unable to modify the equation to make it into a staircase. I would like to specify arbitrary step heights and widths and generate a smooth staircase in the range $x$ specified.    Edit (Extra info): The smooth function I mention above has the problem that the upper, horizontal line is not equal in length to the lower, horizontal line which is why I have been unable to adapt it into a staircase function   Edit 2 Including some pictures     Edit 2 Plot of $s$ with a steep slope showing a different width on the first horizontal line Way to play shared game that is in use in steam? My brother just got me a new computer but he is always on so I can't play his steam shared games. Is there a way to play his games while he is playing a game too? Why didn't Dumbledore show his memory of Barty Crouch Jr's confession to Cornelius Fudge to confirm his point that Voldemort returned? Wouldn't that just prevent all the mumbo-jumbo happening in the 5th book? Is an Unarmed Strike considered a Weapon attack? For the purposes of the fighter Battle Master archetype, whenever a maneuver calls for a "weapon attack" does that include an unarmed strike? 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