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Describing groups with given presentation? $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I'm trying to describe the groups with presentations $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I have some problems getting a good picture of what they look like...   For the first one, $xy=yx$ allows us to say that any element in the group can be written as $x^iy^j$ for some $i,j$. More precisely, using $x^5=y^3$, we can always reduce $j$ to $0,1$ or $2$, whereas $i$ could have any value. So I was thinking about $\mathbb{Z} \times C_3$. But the other way around (reducing $i$ this time) it's also $C_5 \times \mathbb{Z}$. But then I should be able to prove that both descriptions are equivalent, which I can't do...  For the second, I proceed similarly and get to the same kind of problem...  Could you tell me where I'm going wrong?  Thank you very much in advance! Showing that the set of fixed points is a smooth manifold Let $M$ be a smooth manifold and Let $G$ be a finite group acting on $M$ by diffeomorphisms. Show that the set of fixed point $F=\{m \in M: g.m=m \}$ is a smooth manifold. I am unable to deal with the set of fixed points. It doesn't seem to be an open set. It would have been trivial if it was open. Any hints? Compile TeX (directly) into SVG using the command line Is there a way to directly compile TeX code into the SVG image format (rather than going the detour over TeX => PDF => SVG)?  I found this question:   But I don't want to do it online - it should be a command-line call. Centering \vdots in a system of many equations I have a system of equations with an arbitrary number of equations (k). I'd like to use \vdots to compactly describe the system, like so:  \begin{align*}   R(-1) &amp;= \sum_{i=1}^m A(i)R(i-1) \\   R(-2) &amp;= \sum_{i=1}^m A(i)R(i-2) \\         &amp;\vdots                    \\   R(-k) &amp;= \sum_{i=1}^m A(i)R(i-k) \end{align*}     I'd like the dots, however, to be centered with = or the entire equation. Is there an elegant way of centering a column or row with AMSMath? I'm currently using an unholy combination of whitespace operators (\; \, etc.) to get the job done. Get the generated SQL statement from a SqlCommand object? I have the following code:  Using cmd As SqlCommand = Connection.CreateCommand     cmd.CommandText = "UPDATE someTable SET Value = @Value"     cmd.CommandText &amp;= " WHERE Id = @Id"     cmd.Parameters.AddWithValue("@Id", 1234)     cmd.Parameters.AddWithValue("@Value", "myValue")     cmd.ExecuteNonQuery End Using   I wonder if there is any way to get the final SQL statment as a String, which should look like this:  UPDATE someTable SET Value = "myValue" WHERE Id = 1234   If anyone wonders why I would do this:   for logging (failed) statements for having the possibility to copy &amp; paste it to the Enterprise Manager for testing purposes The  of March 01, 2013 put it's distance at 14 hrs 04 mins 23 secs of light-travel time from Earth. A  (earlier today) tweet says it is 14 hrs 04 mins 22 secs of light-travel time from Earth. An  tweet put it still further out at 14 hrs 04 mins 25 secs of light-travel time.  The Voyager 2 tweets on Voyager 1's proximity to Earth too has reduced similarly to  earlier today from  reported on Feb 28 2013  It could quite simply be a typo ... If not a typo then my curiosity is piqued; I was under the impression the Voyagers are outbound from the Solar System now without any major in-system activities now.   How is it the Voyagers are closer to Earth than before? How do I determine how many spell slots I have if I multiclass into more than one spellcasting class? For example, I have a character that is a level 3 Wizard, a level 5 Eldritch Knight Fighter, a level 4 Arcane Trickster Rogue, a level 4 Paladin, and a 4 Warlock. How many spell slots do I have and how do I determine this? Where can game assets for Blender be downloaded? Is there any place where I can download surroundings of a game as a 3d model or maybe there re some inspired artists where they put them for download. I'm trying to find rooms from Wall-e. Some rooms, or outside surroundings or some interior surroundings. Where can I find things like that? Inserting an image in plain TeX I used the OzTeX implementation of TeX, also I used to typeset my docs in plain TeX, not LaTeX for me, so I had no problem to insert an image using the OzTeX, now I'm back to TeX docs, am using the TeXWorks frontend, but I can't insert an .eps image file, it seems the \includegraphics macro its only for LaTeX, so I can't use it, any idea on how to insert an image when using plain TeX? On every bootup it's the same:  /dev/sda1: clean, 908443/38690816 files, 44176803/154733312 blocks   Is it some kind of option Ubuntu uses to ensure filesystem consistency or is there something wrong with my HDD? fsck takes up to 30s while booting and so about triples the time needed otherwise.  Full output (partly in German):  Begin: Loading essential drivers ... done. Begin: Running /scripts/init-premount ... done. Begin: Mounting root file system ... Begin: Running /scripts/local-top ... done. Begin: Running /scripts/local-premount ... done. Begin: Running /scripts/local-bottom ... done. done. Begin: Running /scripts/init-bottom ... done. fsck von util-linux 2.20.1 /dev/sda1: sauber, 908443/38690816 Dateien, 44176803/154733312 Blöcke udevd[623]: unknown key 'SYSFS{idVendor}' in /lib/udev/rules.d/45-libticables.rules:6  udevd[623]: invalid rule '/lib/udev/rules.d/45-libticables.rules:6'   * Starting mDNS/DNS-SD daemon                                                 [ OK ]  * Starting Reload cups, upon starting avahi-daemon to make sure remote queues are populated                                                                   [ OK ]  * Starting configure network device security                                  [ OK ]  * Starting bluetooth daemon                                                   [ OK ]  ####* Starting all other stuff Can I deduct a contribution to a traditional IRA if I am also contributing to a 401(k) in the same tax year? Is my baby's gender an example of Schrodinger's cat? apt-get build-dep is unable to find a source package I need to run a program as a different user + make it run with admin privileges (Run as administrator).  I use the '' for running the program as a different user  How can I run the program as admin with that command? (or a different one so will give me my requirements)  EDIT :  I need 2 things   Run as a different user Run as admin - not just as the admin user but also with full permissions   As this image How can I write a unit test to determine whether an object can be garbage collected? CPU clock speed versus CPU core count - higher GHz, or more cores for SQL Server? We are beginning to provision a set of physical servers for a virtual cluster of SQL Server 2016 nodes within VMware. We will be utilizing Enterprise Edition licenses.  We plan on setting up 6 nodes, but there is a bit of a debate on what the ideal way to provision the physical servers with regards to CPU clock speed versus CPU core count.  I know this is largely dependent on transaction volume and number of databases stored among other software-specific factors, but is there a general rule of thumb that is advised?  For instance, is a dual 8-core, 3.2 GHz physical server (16 cores) more preferential to a dual 16-core, 2.6 GHz server (32 cores)?  Has anyone come across a white paper that further delves into this type of topic? What equipment and settings do I use for panning shots of moving objects? How do you go about getting great panning shots of things like cars and bicycles so that the background is nice and blurred but the target is sharp as can be. I have tried on many occasion to do panning shots and I just can't get the setup right.  What lens setup, shutter speed would be a good place to start? linux find command to move files Does the point at which salt is added to a dish affect how the flavor is changed? In the example  $\cup^{i=1}_{\infty}F_i$   How do you get the i=1 directly below the union symbol and the infinity directly above?