$\text{Aut}(\Bbb Z_n) \cong U(n)$ I would like to show that if $G = \langle a \rangle$ with order $n$, then $\text{Aut}(G) \cong U(n)$.    I know that $\text{Aut}(G)$ is the group of bijections $\phi:G \rightarrow G$ under composition, but I don't know where to start.   Any help would be much appreciated.
$\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$. I've tried, but I can't solve the question. Please help me prove that:  $\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to  $U_n$.
I had the Facebook and Instagram apps installed on my iPhone (latest iOS). I deleted both apps and checked that there was no data from either app stored on iCloud (I went to Settings - iCloud - Manage storage and there was no mention of them storing anything on iCloud). Then I reinstalled Instagram, opened it, clicked on Login and the page still showed “Login as MYFACEBOOKNAME”.   How is this possible? How can I delete my FB login info once and for all on my iPhone? (mód 1) is not copied correctly Please consider the following MWE:  \documentclass{article} \usepackage[spanish]{babel} \selectlanguage{spanish} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc}  \begin{document} Works: áéà  Does not work: \(\pmod1\) \end{document}   The áéà text is copied correctly, but if I try to copy (mód 1) it says: (m´od 1).  Does anyone know why this happens? How is it solved?  Thanks! Using equations to draw out complex objects How do people come up with equations of curves to draw out complex objects?  Some popular examples would include:  &amp;  .  This  explains the rationale for the batman curve nicely.  But other than trial and error, I can't see a reasonable way of drawing the much more complicated PSY curve. I'm trying to make a logo that will be on the header of a website. What I'd like to achieve, and the reason I'm using Blender, is the projected shadows on the background looking perfectly real (as opposed to doing a drop-shadow in Photoshop).  The only thing is, the background has to be transparent because the website might use different background colors / gradients that make it impossible to match with the exported 3D background.  I made this dummy file (not the real logo... duh!):    If I use the file as is, the grey borders of the image will be impossible to match with the website's background color.  Here what I want:    So obviously, I tried masking the plane, which made the shadows disappear. I tried having a plane set to glass, which didn't work either.  And I looked around for a similar question, but didn't find one.   I am not trying to render , or a . I've seen  which has a very difficult solution to this problem, and  that is very similar but in Blender Internal.  is similar too but has never been solved. I'm only a beginner at Compositor, and I'm trying to learn more any chance I get, so any solution in Compositor is welcome if detailed enough. Although if there's a simpler or faster way, it's welcome too!  .blend file: Recurrence relation for number of ternary strings that contain two consecutive zeros The question is: Find a recurrence relation for number of ternary strings of length n that contain two consecutive zeros.    I know for ternary strings with length one, there are 0. For a length of 2, there is just 1 (00), and for a length of 3, there are 5 (000,001,002,100,200).    I did a similar problem, finding a relation for the number of bit strings of length n with two consecutive zeros: $$a_n = a_{n-1} + a_{n-2} + 2^{n-2}$$    Since you can add "1" to the end of all the $a_{n-1}$ strings, "10" to all the $a_{n-2}$ strings, and "00" any string of size $n-2$.    For the ternary string problem, I'm pretty sure you would replace the $2^{n-2}$ with $3^{n-2}$, but confused about the other terms of the relation. My guess is that it would have the coefficient $2$ in front of the other terms, since you can add either $1$ or $2$ to the end of $a_{n-1}$ and either $01$ or $02$ at the end of $a_{n-1}$.    So I believe the answer for the relation is: $$a_n = 2a_{n-1} + 2a_{n-2} + 3^{n-2}$$    How does that look? Record a macro of windows commands Using Windows 7, is there an in-built or third-party method of recording windows commands and being able to play them back?  As a very basic example of what I would like to be able to do, imagine having two desktop themes that you like to alternate between. The Windows commands would be something like   Open personalisation applet select second theme ok   (I'm calling this a macro because of the equivalent capability in Word.) I flagged a post today, typed out a fairly thorough explanation, clicked the button... and got told that I'd already flagged this post (for something else, which I'd forgotten about) so this flag wasn't allowed.  (This happens if the first flag is still active.)  The dialogue with my explanation was gone at this point, so if I wanted to communicate with the moderators about that, I was going to have to start over.  (In case you're wondering how this could happen, one flag was related to the post itself and one was related to the user -- there being no way to flag a user directly, you have to hang it off of a post.  But I've also seen this with cases where the first flag hangs around unhandled for a week or two and I've forgotten about it.)  Since the system is going to detect that I've already flagged a post and reject a second flag, can it do that after I click the "flag" link instead of waiting until the end of the process?  (Or gray out the link so I can't click it in the first place?)  2015-12-18: someone suggested to me in chat today that this had been implemented, so I tested again.  The behavior has not changed: I flagged a post using "other", then flagged it again (also "other") before the first flag was handled, and the flag was rejected only after I'd typed the message.  (I did this on a site where I'm a moderator so I could then clean up after myself.) Performance of Non Clustered Indexes on Heaps vs Clustered Indexes compares the performance for individual select/insert/delete/update and range select statements on a table organized as a clustered index vs that on a table organized as a heap with a non clustered index on the same key columns as the CI table.  Generally the clustered index option performed better in the tests as there is only one structure to maintain and because there is no need for bookmark lookups.  One potentially interesting case not covered by the paper would have been a comparison between a non clustered index on a heap vs a non clustered index on a clustered index. In that instance I would have expected the heap might even perform better as once at the NCI leaf level SQL Server has a RID to follow directly rather than needing to traverse the clustered index.   Is anyone aware of similar formal testing that has been carried out in this area and if so what were the results? Find a recurrence relation for the number of $n$-digit binary sequences with no pair of consecutive $1$s Find a recurrence relation for the number of $n$-digit binary sequences with no pair of consecutive $1$s.  I know my base case:$$a_1 = 2$$ for either $1$ or $0$. Normally to construct the recurrent relation I would say fill in $n-1$ spaces then add a $1$ or $0$, which is simply:$$a_n = 2\cdot a_{n-1}$$However, I don't believe this is right, because this doesn't take into account that there can't be any consecutive $1$s.  Even when I use the base case of $2$ digits:$$a_2 = 3$$for either $01$, $10$, or $00$, I still get the recurrence relation of:$$a_n = 3\cdot a_{n-2}$$which also does not account for $01$ and $10$ occurring consecutively.   So how would I express this? Can markdown mini replace an en dash with a normal hyphen in URLs? How to append something to an array? Is there a tool that combines zcat and cat transparently? When handling log files, some end up as gzipped files thanks to logrotate and others not. So when you try something like this:  $ zcat *   you end up with a command line like zcat xyz.log xyz.log.1 xyz.log.2.gz xyz.log.3.gz and then with:  gzip: xyz.log: not in gzip format   Is there a tool that will take the magic bytes, similar to how file works, and use zcat or cat depending on the outcome so that I can pipe the output to grep for example?  NB: I know I can script it, but I am asking whether there is a tool out there already. Execute vs Read bit. How do directory permissions in Linux work? Selenium + FF : How to fix error, Unable to connect to host 127.0.0.1 on port 7055 after 45000 ms? I want to start an Xorg server (which should only feature a minimal xterm terminal emulator from a newly created &quot;pristine&quot; tty). I have thought this to be as simple as issuing a xinit or maybe a startx however both have failed me. Therefore the question: How to manually run/init/start a Xorg server on a different VT/TTY? here the Longer background story to the question at hand: This is the backgroud and what I have tried already: Looking at man openvt I was able to run a shell on a newly created tty (or vt, cannot tell precisly the different vt&lt;&gt;tty) like this:  $&gt; sudo openvt -s /bin/bash  the -s of causes the direct switch to that new tty/vt on that new tty I wanted to manually start a Xorg server which according to man xinit should start with a minimal xterm application. On the new shell on the switched to tty I (created via openvt) I hence issues this command:  $&gt; su -c 'xinit xterm -display :4 -- :4' - mahr &amp;&gt;/tmp/output  and I receive this output $&gt; cat /tmp/output X.Org X Server 1.19.6 Release Date: 2017-12-20 X Protocol Version 11, Revision 0 Build Operating System: Linux 4.4.0-148-generic x86_64 Ubuntu Current Operating System: Linux scitech 4.15.0-70-generic #79-Ubuntu SMP Tue Nov 12 10:36:11 UTC 2019 x86_64 Kernel command line: BOOT_IMAGE=/vmlinuz-4.15.0-70-generic root=/dev/mapper/ubuntu--vg-root ro Build Date: 03 June 2019  08:10:35AM xorg-server 2:1.19.6-1ubuntu4.3 (For technical support please see http://www.ubuntu.com/support)  Current version of pixman: 0.34.0     Before reporting problems, check http://wiki.x.org     to make sure that you have the latest version. Markers: (--) probed, (**) from config file, (==) default setting,     (++) from command line, (!!) notice, (II) informational,     (WW) warning, (EE) error, (NI) not implemented, (??) unknown. (==) Log file: &quot;/home/mahr/.local/share/xorg/Xorg.4.log&quot;, Time: Thu Nov 28 10:57:53 2019 (==) Using system config directory &quot;/usr/share/X11/xorg.conf.d&quot; (EE)  Fatal server error: (EE) parse_vt_settings: Cannot open /dev/tty0 (Permission denied) (EE)  (EE)  Please consult the The X.Org Foundation support       at http://wiki.x.org  for help.  (EE) Please also check the log file at &quot;/home/mahr/.local/share/xorg/Xorg.4.log&quot; for additional information. (EE)  (EE) Server terminated with error (1). Closing log file.  Session terminated, terminating shell...xinit: giving up xinit: unable to connect to X server: Connection refused xinit: unexpected signal 2  ...terminated. Is there a difference between “?!” and “!?”? I have a custom search page for use in Customer Portal.  VF Page:    &lt;form name="frmSearch"&gt;      &lt;input type="text" name="keyword" id="keyword" value="{!portalSearchModel.searchTerm}" onkeypress="return noenter(event);" style="width:25%; height:25px;margin:0; padding: 0px 6px 0px;" placeholder="What are you looking for?" /&gt;      &lt;input type="button" id="btnSearch" name="btnSearch" value="Search" onclick="searchTerm()" class="go-we3" /&gt;   &lt;/form&gt;   &lt;script type='text/javascript'&gt;       function searchTerm(){         var searchTerm = document.getElementById("keyword").value;          var url="/apex/PortalSearch?s="+searchTerm;          window.location = url;          return false;     }          &lt;/script&gt;   Is there a way to call this apex method in Controller on button click along with the JavaScript?      public void logSearchTerm(){         portalSearchModel.searches();         Search_Log__c sl = new Search_Log__c();          sl.search_term__c = portalSearchModel.searchTerm;         sl.user__c = UserInfo.getUserId();              insert sl;  } Is there a way to configure your Stack Overflow account so that you receive notification emails when posts are made to specific tags?    Or do you simply have to log on to see new posts? What is the meaning of the beta-coefficient for an interaction term in a crossover study? I have asked a similar question here:  I am puzzled by the interpretation for an interaction term. In my data my Y is an interval variable with the health outcome of an experiment. I have used an interaction term in which I have interacted the condition with the predisposition of the subject considering health status at base level. They are both categorical variables (factor variables in R).  Now it gets complicated because the Condition was two treatments: in treatment A subjects got the placebo first and then the real medicine whereas in treatment B they go the real medicine first and the placebo second. All it changes is the order.   Health outcome = a + Condition * Health.Base   I have the worst state of health at the base level as my reference category I find that interaction with the Condition is statistically significant but I am not sure how to interpret this.  I use the lm() function of R (although my design looks more like an ANOVA) and in the output I get the b coefficient in an output that looks like this:  ConditionB:Health.Base.So.and.So         (Beta and p-value) ConditionB:Health.Base.Excellent         (Beta and p-value)   A statistically significant interaction term for those in Excellent health at baseline would mean that they are affected by the Condition B more than Condition A compared to the reference category people (Poor health at baseline). Is this right? What does the beta-coefficient represent?   If I would like to examine for each Health category at the base line separately without comparing to a reference category I would have to code each category as a dummy variable. However, in this case I would compare whether membership to a specific health status at the base line significantly changes between conditions compared to those who belong to the other health statuses at the base line. Is this right? Again, what does the beta-coefficient represent?  Would it be right to assume that the choice of the interaction between the Condition and the dummy variables is easier to interpret?  --- EDIT --- R output:  Call: lm(formula = HealthOutcome ~ Condition * HealthStatus,      data = datA)  Residuals:     Min      1Q  Median      3Q     Max  -1.5957 -0.5942 -0.2640  0.4423  2.4423   Coefficients:                                        Estimate Std. Error t value     Pr(&gt;|t|)     (Intercept)                            2.595652   0.053044  48.934  &lt; 2e-16 *** ConditionCondB                        -0.001449   0.077071  -0.019    0.985     HealthStatusSo.and.So                 -0.331693   0.078094  -4.247  2.35e-05 *** HealthStatusExcellent                 -0.836724   0.092692  -9.027  &lt; 2e-16 *** ConditionCondB:HealthStatusSo.and.So   0.137490   0.110612   1.243    0.214     ConditionCondB:HealthStatusExcellent  -0.199787   0.133943  -1.492    0.136     --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  Residual standard error: 0.8045 on 1059 degrees of freedom   (68 observations deleted due to missingness) Multiple R-squared:   0.16, Adjusted R-squared:  0.156  F-statistic: 40.33 on 5 and 1059 DF,  p-value: &lt; 2.2e-16 How to install "vanilla" TeXLive on Debian or Ubuntu?