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Rotating a gyroscope's output values into an earth-relative reference frame For whatever reason, I seem to have gotten myself twisted into quite a confusion on how to process data from my 3 axis gyro and 3 axis accelerometer to get gyro rotation values related to the earth's reference frame. i.e. Given my accelerometer measures gravity (we assume it's stable) along the sensors (x,y,z) reference frame, how do I adjust the values of my gyro, measured along the sensors same (x,y,z) reference frame such that I am actually measuring the rotating about the Earth's frame x (North), y (West), and z (downwards) direction?  I just can't seem to wrap my head around the use of the rotation matrix. Do I simply employ a 3d rotation matrix, and multiply the vector, \$[gyro_x, gyro_y, gyro_z]^T\$, by a rotation matrix, R? I then assume the rotation angles, are somehow derived from the filtered accelerometer outputs, but I'm having trouble deciding on what those should be.  I have yet to find a concise reference on how to move the rotation and acceleration values from the sensor reference frame into the earth global reference frame. Perhaps someone can help? Thanks Determine the number of lines within a text file Is there an easy way to programmatically determine the number of lines within a text file? ''More than one row returned by a subquery used as an expression'' postgres error I am trying to realize a postgres request :   select distinct nom_reg_12  from region_15,repartition  where st_intersects(region_15.geom,                    (select geom                     from repartition                     where id_espece='Tetrarti'))=true;    but i get the following error :      More than one row returned by a subquery used as an expression   in the first time I tried this and it work fine :   select distinct nom_reg_12  from region_15,repartition  where st_intersects(region_15.geom,repartition.geom)=true;   and then I tried this a part to get the geometry I want :   select geom from repartition where id_espece='Tetrarti' Prove that $X^\ast$ separable implies $X$ separable cardinality of the Borel $\sigma$-algebra of a second countable space Second countability by itself doesn't restrict the cardinality of a topological space, since every set with the trivial topology is a second countable space, but it seems natural to ask whether second countability restricts the cardinality of the Borel $\sigma$-algebra of the space.  Can the cardinality of the Borel $\sigma$-algebra of a second countable space be arbitrarily big? If this is the case is there a simple construction for a second countable space with an arbitrarily big Borel $\sigma$-algebra? What is difference between have/get/make someone do something? What is the relationship between the focal length and field of view for an equal-area ("equisolid") projection fisheye lens?  Alternatively, what is the relationship between the 180-degree image circle size and the focal length for such a lens?  Ultimately I want to find the field of view of a Tokina 10-17 mm fisheye zoom at focal lengths other than 10 mm on APS-C (). Should inner tube have excess length? If $\left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\;,$ Then $\left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right)$ If $\displaystyle \left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\;,$ Then $\left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right) = $  $\bf{My\; Try::}$ Given $\displaystyle \left(1+\sin \phi\right)\cdot \left(1+\cos \phi\right) = \frac{5}{4}\Rightarrow 1+\sin \phi\cdot \cos \phi+\sin \phi+\cos \phi = \frac{5}{4}.$  So $\displaystyle \sin \phi+\cos \phi+\sin \phi \cdot \cos \phi = \frac{1}{4}\Rightarrow \left(\sin \phi+\cos \phi\right) = \frac{1}{4}-\sin \phi \cdot \cos \phi.$  Now $\displaystyle \left(1-\sin \phi\right)\cdot \left(1-\cos \phi\right) = 1-\left(\sin \phi+\cos \phi\right)+\sin \phi \cdot \cos \phi =\frac{3}{4}+\sin 2\phi$  Now How can I calculate $\sin 2\phi.$  Help me, Thanks I'm studying measure theory and two question came to my mind:        If $f:\mathbb{R}\to\mathbb{R}$ is monotone and $B\subseteq\mathbb{R}$ is borel, is the image $f(B)$ borel?   If $f:\mathbb{R}\to\mathbb{R}$ is a monotone function (say, non-decreasing), does there exist a sequence of continuous functions $f_n:\mathbb{R}\to\mathbb{R}$ converging pointwise to $f$?      Here's the motivation for those questions: Let $(X,M)$ and $(Y,N)$ be measure spaces.   It's known that if $\mu$ is a measure on $(X,M)$ and $f:X\to Y$ is measure, then we have the pushforward measure $f_*\mu(A)=\mu(f^{-1}(A))$ on $(Y,N)$. What if we were to define a "pullback measure"? Given a function $f:X\to Y$ such that $f(M)\subseteq N$ (i.e. $f$ maps measurable sets to measurable sets) and a measure $\nu$ on $N$, the natural formula would be $f^*\nu(A)=\nu(f(A))$. So we ask if is there a good amount of functions which map measurable sets to measurable sets in $\mathbb{R}$, and monotone functions seem like good candidates (for strictly monotone, continuous functions, the result is valid and there are several answers on the web). If this were true, maybe we could use some convergence argument to solve the problem above. Can a warlock use one of their Thirsting Blade attacks to grapple or shove? Object selection box with eyedropper I know that editing the boot options line and adding 'nomodeset' solves the problem of my laptop during LiveCD mode, what I don't know is how to set it at boot up through Grub2 after I've installed Ubuntu.  So, my question is; how do I set nomodeset before I boot into Ubuntu? Over time, I've noticed an increasing practice among moderators of the smaller sites (i.e. not in the Trilogy) to move extended comment chains to chat, just because they're too long. Is this a good idea? In the past, I've taken the position on many occasions that I believe that comments serve two purposes only: clarification, and asking for clarification.  I take this position because as a mod I want the freedom to purge a comment trail that is clearly running off the rails without having to argue with anyone about politeness definitions. But that's not the same thing as two pedantic C++ language lawyers debating the finer points of &quot;undefined behavior.&quot;  I give wide latitude to such discussions, as long as the participants don't start throwing spears, because I believe they are instructive. The SE software even has accommodations for this.  Long comment chains are automatically collapsed, so as not to disrupt the Q&amp;A format of the post.  And moving a comment chain to chat is not even a move, really; to do an actual move, you have to perform the conversion, and then delete all of the copied comments from the post. Here is a recent example:  and it's accompanying . What I find amusing is that the OP's detailed POV about the term, where he floats a theory about why everyone thinks that &quot;bugged&quot; is not a thing (that conversation took place in the comments, and is now safely sequestered in the chat room) gets to stay prominently featured in his question, at the very top of the post. Now you could argue that the chat room is a better environment to have these discussions anyway, and you'd be right.  I could argue that if we're going to do that, then why not just automatically convert long discussions to chat rooms? The problem is not so much where the comments reside.  I don't understand why mods are taking it on themselves to fix something that isn't hurting the site (at least in my opinion).  In addition, every time I see this happen, the participants in the conversation immediately lose interest in it.  The only comment that was posted in that room after all the comments were moved is most likely this one:  It wouldn't surprise me if the folks using &quot;bugged&quot; to refer to a certain software quality are the same folks who refer to their programs as &quot;codes&quot; and who believe textspeak is the pinnacle of modern English.  Which nobody will ever see. :) Some mods are really aggressive about this.  I discussed this with a mod on Physics, and he told me that they routinely move discussions when they think they might run off the rails. How do volume control headphones work? A recent question over on  got me to wondering how volume control headphones work.  I don't mean volume control which works by , but the kind of volume control that signals to the device to increase or decrease the signal output.  For instance, according to the , the volume controls are only for apple devices:     Universal Compatibility and Microphone Communication   LP comes with 2 cables for universal compatibility with all modern mobile and audio devices. The 3-Button remote microphone cable is designed for the latest Apple devices, including iPhone®,iPad®, iPod®, and Macbook series. The long audio-only cable and 1/4" adapter is universally compatible with all audio devices and professional equipment.   However this sort of device apparently works with some android phones and the question implies that it works with some computer sound cards on Windows machines too, but not being an Apple buyer (and only recently getting my first smartphone) I haven't seen this sort of thing before. I've done a quick google search, but can't find anything on standards for this technology.   How does this work?   Is it something as simple as shorting left/right/mic channels? If so, could using headphones of this type with equipment that doesn't support this technology, damage that equipment? Is it a serial signal, for instance sent over the mic channel? Does it just require a headphone  connector or a headphone  +microphone TRRS connector?   Does this technique have a name?  Incidentally, is someone could answer this, I can probably look up the rest of the information myself. *8') Is it as standard as it appears, or is everyone just following Apple? Is this something that is patented?   If so, who holds the patent and do they charge a license fee for people to use this technology? If not, is there an open standard for it?    There is some excellent information in  to the question  but it doesn't answer the specifics of how this type of volume (etc.) control works, since it now seems to be rather more common and standard. The series $\sum_{n=2}^{\infty }\frac{1}{\log (n!)}$ I'm trying to find out whether this Series $\sum_{n=2}^{\infty } a_{n}$ converges or not when $$a_{n}=\frac{1}{\log (n!)}$$  I tried couple of methods, among them: d'Alembert $\frac{a_{n+1}}{a_{n}}$,   Cauchy condensation test $\sum_{n=2}^{\infty } 2^{n}a_{2^n}$, and they both didn't work for me.  Edit: I can't use stirling, and integral.  Thank you "Contact us"... and then what? I am able to parse strings containing date/time with time.strptime  &gt;&gt;&gt; import time &gt;&gt;&gt; time.strptime('30/03/09 16:31:32', '%d/%m/%y %H:%M:%S') (2009, 3, 30, 16, 31, 32, 0, 89, -1)   How can I parse a time string that contains milliseconds?  &gt;&gt;&gt; time.strptime('30/03/09 16:31:32.123', '%d/%m/%y %H:%M:%S') Traceback (most recent call last):   File "&lt;stdin&gt;", line 1, in &lt;module&gt;   File "/usr/lib/python2.5/_strptime.py", line 333, in strptime     data_string[found.end():]) ValueError: unconverted data remains: .123 When used like this:  import static com.showboy.Myclass;  public class Anotherclass{}   what's the difference between import static com.showboy.Myclass and import com.showboy.Myclass? Figure refs stop working after I added a new counter I was using the amspset document class for generating exam practice problems. I had wanted to add some referencing to the problems so I started a new counter like this (adapted from some Tex Exchange post)  \newcounter{P} \newcommand{\plb}[1]{\refstepcounter{P}\label{#1} \textbf{P-\ref{#1}.}}  \begin{problem} \plb{p1} A parallel-wire transmission line is constructed of \#6 AWG copper wire (diameter $d=0.162$ in., conductivity $\sigma_c=5.8\times 10^7$ S/m) with a 12-inch separation in air. Assuming no leakage between the two wires, find $R'$, $L'$, $G'$, and $C'$. Assume a working frequency of 1 MHz. \end{problem}  \begin{solution} Using Table.~\ref{tab:tlparam} and Fig.~\ref{fig:tlempl},  \[ Rs = \sqrt{\pi f \mu_c/\sigma_c} = 2.61\times10^{-4}~ \Omega\] \[ R' = \frac{2R_s}{\pi d} = 4.04\times10^{-2} ~\Omega\text{/m} \] \[ L' = \frac{\mu_0}{\pi} \ln{\frac{2D}{d}}=2.0~\mu \text{H/m} \] \[ C' = \frac{\pi \epsilon}{\displaystyle\ln{\frac{2D}{d}}}=5.56 ~\text{pF/m} \] $G'=0$ because the problem states that there is no leakage between the two wires. \end{solution}  \begin{figure}[ht] \begin{centering}     \includegraphics{fig24}     \caption{(Fig.2-4 from FAE) A few examples of transmission lines.} \end{centering} \label{fig:tlempl} \end{figure}  \begin{table}[ht] \caption{(Table 2-1 from FAE)Transmission-line parameters $R'$, $L'$, $G'$, and $C'$ for three types of lines.}     \begin{tabular}{c}         \hspace{3cm}\includegraphics{tb21}     \end{tabular} \label{tab:tlparam} \end{table} ... and more   Now all the figure references stop working, I get "Fig. " instead of "Fig. 1". The table references are alright though. Any idea what the problem could be?