If $G$ is a group with $xy=yx$ for $x,y\in G$, and  $x^m=y^n=e$, then: (1) $o(xy)|mn$, where $o(xy)$ is the order of $xy$, and (2) if $(m,n)=1$ then $o(xy)=mn$.   Well, I've already proved (1), but I'm having trouble with (2), I was thinking that maybe, if we already know (1), $o(xy)|mn$ then we just have to show that $mn|o(xy)$, that way we get equality. So we get: $mn|o(xy)$ iff $o(xy)=mn*l$ for some $l\in\Bbb Z$. On the other hand, we know that $(xy)^{mn}=e$ then it's obvious that for any $l\in\Bbb Z$, $(xy)^{mn*l}=e^l=e$, we get what we wanted, but if there's an infinite number of $l's$, what does this means? I feel that what I'm doing is wrong since I didn't used the principal hypothesis: $(m,n)=1$, but this is the only way so far I've been able to do something.
Another homework problem from my Group Theory class.  Let $a,b$ be elements of a group, $G$.  Let $\operatorname{ord}(a)=m$ and $\operatorname{ord}(b)=n$. Let $a$ and $b$ commute. Prove: If $m$ and $n$ are relatively prime, then the $\operatorname{ord}(ab)=mn$.  So I am having trouble starting off on this problem.  Here is what I believe I know:  $a^m=e$ and $b^n=e$ [as defined by the order of the elements] which leads $a^m=b^n$  $\operatorname{ord}(ab)=mn=\operatorname{ord}(ba)=nm$ as defined by commuting.  $\gcd(m,n)=1$ as defined by $m,n$ being relatively prime.    I don't know how to start this proof. Do I want to to go the route of $a^mb^n=e$?  Then $(ab)^{mn}=e$?   Any tips to start this would be greatly appreciated!!!
Correct usage of will and would How can I repair grub? (How to get Ubuntu back after installing Windows?) For which $n$ is the sum: $$\sum_{k=0}^{n}10^k$$ a prime number? Are they finite? I have a small issue with configuring the lidR  for parametrizing the extracted raster files with grid_metrics() function.  My question is how to set in the catalog options that:   be able to overwrite the results with a second run if I want to run my script several times and I do not want to name each output differently. instead of .tif I would like to get .grd files as output.   My current code (which is working) is the following:  normalizedctg = catalog(workingdirectory) opt_output_files(normalizedctg) &lt;- paste(workingdirectory,"/clipped/normalized/{XLEFT}_{YBOTTOM}_height",sep="") height = grid_metrics(normalizedctg, max(Z),res = resolution)   But with a second run on the same code I get the following error:  Error in .getGDALtransient(x, filename = filename, options = options,  :    filename exists; use overwrite=TRUE Javascript to sort contents of select element is there a quick way to sort the items of a select element? Or I have to resort to writing javascript?  Please any ideas.  &lt;select size="4" name="lstALL" multiple="multiple" id="lstALL" tabindex="12" style="font-size:XX-Small;height:95%;width:100%;"&gt; &lt;option value="0"&gt; XXX&lt;/option&gt; &lt;option value="1203"&gt;ABC&lt;/option&gt; &lt;option value="1013"&gt;MMM&lt;/option&gt; &lt;/select&gt; Digitally selectable filter I have asked a  but I did not get an answer, as it was too specific. So I will build on it but make it a bit more broad spectrum.  I would like to make a digitally selectable BPF using transistors to select a component in the filter; resulting in the center frequency being changed.  I have used FilterPro to generate quick second order filters and get a rough filter response.  For all the designs, a Multiple-feedback was selected:  In the first picture, a center frequency at 108KHz with a bandwidth @ -3dB of 103K to 113KHz is given.    For the second, a center frequency at 117.5KHz with a bandwidth @ -3dB of 112.8K to 122.2KHz is given.    The only difference between these two circuits is R3.  If I were to implement a circuit as shown below, would it work and what transistor would be recommended? I have selected the  as the Opamp in the circuit design as that is what was used for the .    For the transistors, I was thinking of using the MOSFETs (2N7002) used in the ST7590 design as well. In single-blind peer-review, can you reveal your identity without the editor's consent? In my field (chemistry), review is always done in a single blind process, i.e. the author does not know who the referees were, but the referees do know who the authors are. In ten years, I have never seen a referee breaching this anonymity, e.g. by signing their review, unless they got prior approval by the editor (in order to continue discuss things further with the authors, once the manuscript was accepted for publication).  So, it seemed logical to assume that in single blind peer-review, the reviewer should not disclose his identity without the editor's consent. Yet, some people have told me ( and ; also IRL a friend from human sciences) that they have seen people sign their reviews, or write emails to them after the review but before the paper is published.  So, in single-blind peer-review, can you reveal your identity without the editor's consent? Does it depends on the customs of each field, or is there a hard rule? (in which case, the few anecdotes I heard were outliers) Proving the general limit $\lim_{n\to\infty} \frac{an+c}{bn+d} = \frac{a}{b}$ Cannot get 2 raster files of same cell size to align ArcMap 10.2 I've recently installed Fedora 19.  is installed. But its own examples don't work. When I enter gedti the output is:  bash: gedti: command not found... Similar command is: 'gedit'   And it works correctly but when I enter powertop, the output is:  bash: powertop: command not found...   With no suggestion about the package. Why? Should I edit some configuration files or update some cache or ... ? How to remove app which is greyed out, can't be deleted, won't open and doesn't appear in apps list? I have this app with a greyed out icon which I'm trying to delete.    I tap and hold, and it asks me if I want to delete it and its data. I confirm and nothing happens.  I try to open it and nothing happens.  I go to Settings>General>Usage and it's not in the list.  I reboot the device and it's still here.  Is there another way to force deletion of this (ghost) app? Escaping WP_Query tax_query when term has special character(s) Here's my issue. I have a custom post type that I'm attaching a custom taxonomy to. This taxonomy will allow the admin to provide search suggestions that will ensure the post is found when a user searches. This taxonomy is also used on the front-end to autocomplete the search field with these suggestions.  The system works fine, except when the term contains a forward slash, a space or other "special" characters.  &lt;?php $keyword = 'Ski-in%2FSki-out';//Submitted via $_GET $keyword = urldecode($keyword); // (string)'Ski-in/Ski-out'  $taxQuery = new WP_Query(array(     'post_type' =&gt; 'ml_properties',     'tax_query' =&gt; array(         'relation' =&gt; 'OR',         array(             'taxonomy' =&gt; 'mc_tax_lifestyle',             'field' =&gt; 'name',             'terms' =&gt; array($keyword),             'operator' =&gt; 'IN'         )     ) )); ?&gt;   I've looked into using the escaping features provided by WordPress, but haven't had any luck there. I thought this would be a simple problem, but either my Google-fu is not strong, or I'm just not having any luck.  I've also verified that that Ski-in/Ski-out is the exact string stored in MySQL. I've looked into using esc_attr(), $wpdb-&gt;esc_like() and esc_sql() but none have an effect.  Perhaps I'm missing something obvious? what should I do to stop the cases environment compressing the first line of the following? \begin{equation} \alpha_{mn} = \begin{cases} \frac{16\mu^2}{mn\pi^2\left[\pi^2(m^2/a^2 + n^2/b^2) - \mu^2\right]} &amp;\mbox{for } m \mbox{ and } n \mbox{ odd;}  \\ 0 &amp;\mbox{for } m \mbox{ or } n \mbox{ even;} \end{cases} \\[1ex] \end{equation}  I'm not sure to do tex on this site, but if anyone tries it they see the first case is compressed because it's a fraction. Kinematic Viscosity How would you define ? What does it physically represent? Around the Internet I've found it defined as just a ratio, and that's it.  I saw in an answer that I can think of it as "diffusivity of momentum". What does this mean? What attracted Padmé to Anakin? Search for singleton tags on questions Is there any way to search for questions that have a specific tag and only that lonely tag?  In other words, I'd like to be able to do something like the following to find questions that have a single tag and no other:  [mytag] tagcount:1   This would be very useful for tag cleanup situations to be able to do house-cleaning before nuking a tag. What's the difference between centre of mass & centre of gravity for massive bodies? My book says:     For most of the small objects, both are same. But for mammoth ones, they are really different ones. And in a gravity-less environment, COG is absent; COM still exists.   Ok, what's the big deal when things are small and big? How these two :  &amp; ? I'm looking for a term for audio in form of the word visual.  Visual is defined as      of or relating to the sense of sight   What could you call the sense of hearing? Also, what do you call this form of a word? (i.e verb, noun...)  It should fit the x in the following sentence:     Is it possible to fall in love with someone without visual or x contact?   I'm not a native English speaker, so I couldn't find the terms for searching this. Scorched3d will not launch Scorched3d worked then stopped. Unistalled and reinstalled did not help. Running Ubuntu 16.04.04LTS. I think it was after installing new release.  I think the problem is the Ubuntu 16.04.4LTS. The newer version has no Scorched3d in the Ubuntu Software Center. I have 3 systems. After updating to new version. Scorched3D will not install and work after updating.  ron@RonsComputer:~$ sudo objdump -p $(type -p scorched3d) | grep NEEDED  NEEDED libwx_gtk2u_adv-3.0.so.0  NEEDED libwx_gtk2u_core-3.0.so.0 NEEDED libwx_baseu-3.0.so.0 NEEDED libfreetype.so.6 NEEDED libz.so.1 NEEDED libSDL-1.2.so.0 NEEDED libSDL_net-1.2.so.0 NEEDED libexpat.so.1  NEEDED libstdc++.so.6 NEEDED libm.so.6 NEEDED libgcc_s.so.1 NEEDED libpthread.so.0 NEEDED libc.so.6   ron@ron-desktop ~ $ objdump -p $(type -p scorched3d) | grep NEEDED | awk '{print $2}' | xargs -n 1 dpkg -S  libwxgtk3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_gtk2u_adv-3.0.so.0.2.0 libwxgtk3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_gtk2u_adv-3.0.so.0 libwxgtk3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_gtk2u_core-3.0.so.0.2.0 libwxgtk3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_gtk2u_core-3.0.so.0 libwxbase3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_baseu-3.0.so.0.2.0 libwxbase3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_baseu-3.0.so.0 libfreetype6:amd64: /usr/lib/x86_64-linux-gnu/libfreetype.so.6.12.1 libfreetype6:i386: /usr/lib/i386-linux-gnu/libfreetype.so.6 libfreetype6:amd64: /usr/lib/x86_64-linux-gnu/libfreetype.so.6 libfreetype6:i386: /usr/lib/i386-linux-gnu/libfreetype.so.6.12.1 zlib1g:i386: /lib/i386-linux-gnu/libz.so.1.2.8 zlib1g:amd64: /lib/x86_64-linux-gnu/libz.so.1 zlib1g:i386: /lib/i386-linux-gnu/libz.so.1 zlib1g:amd64: /lib/x86_64-linux-gnu/libz.so.1.2.8 libsdl1.2debian:amd64: /usr/lib/x86_64-linux-gnu/libSDL-1.2.so.0.11.4 libsdl1.2debian:i386: /usr/lib/i386-linux-gnu/libSDL-1.2.so.0 libsdl1.2debian:i386: /usr/lib/i386-linux-gnu/libSDL-1.2.so.0.11.4 libsdl1.2debian:amd64: /usr/lib/x86_64-linux-gnu/libSDL-1.2.so.0 libsdl-net1.2:amd64: /usr/lib/x86_64-linux-gnu/libSDL_net-1.2.so.0 libsdl-net1.2:i386: /usr/lib/i386-linux-gnu/libSDL_net-1.2.so.0 libsdl-net1.2:amd64: /usr/lib/x86_64-linux-gnu/libSDL_net-1.2.so.0.8.0 libsdl-net1.2:i386: /usr/lib/i386-linux-gnu/libSDL_net-1.2.so.0.8.0 libexpat1:i386: /lib/i386-linux-gnu/libexpat.so.1.6.0 libexpat1:amd64: /lib/x86_64-linux-gnu/libexpat.so.1 libexpat1:amd64: /lib/x86_64-linux-gnu/libexpat.so.1.6.0 libexpat1:i386: /lib/i386-linux-gnu/libexpat.so.1 libstdc++6:amd64: /usr/share/gdb/auto-load/usr/lib/x86_64-linux-gnu/libstdc++.so.6.0.21-gdb.py libstdc++6:i386: /usr/lib/i386-linux-gnu/libstdc++.so.6 libstdc++6:amd64: /usr/lib/x86_64-linux-gnu/libstdc++.so.6 libstdc++6:amd64: /usr/lib/x86_64-linux-gnu/libstdc++.so.6.0.21 libstdc++6:i386: /usr/lib/i386-linux-gnu/libstdc++.so.6.0.21 libstdc++6:i386: /usr/share/gdb/auto-load/usr/lib/i386-linux-gnu/libstdc++.so.6.0.21-gdb.py libc6:amd64: /lib/x86_64-linux-gnu/libm.so.6 libc6:i386: /lib/i386-linux-gnu/libm.so.6 libgcc1:amd64: /lib/x86_64-linux-gnu/libgcc_s.so.1 libgcc1:i386: /lib/i386-linux-gnu/libgcc_s.so.1 libc6:i386: /lib/i386-linux-gnu/libpthread.so.0 libc6:amd64: /lib/x86_64-linux-gnu/libpthread.so.0 libc6:i386: /lib/i386-linux-gnu/libc.so.6 libc6:amd64: /lib/x86_64-linux-gnu/libc.so.6 ron@ron-desktop ~ $ How to identify if a webpage is being loaded inside an iframe or directly into the browser window? I am writing an iframe based facebook app. Now I want to use the same html page to render the normal website as well as the canvas page within facebook. I want to know if I can determine whether the page has been loaded inside the iframe or directly in the browser?