Rank of product of a matrix and its transpose
Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$
ffmpeg synchonization of libfdk_aac (or libfacc) and libx264 How to synchronize libfdk_aac and libx264 streams?  ffmpeg -i pipe_libfdk_aac.fifo -i pipe_libx264.fifo -c copy -f flv flv_stream_pipe_or_rtmp.output   I'm always receiving the same asych in all cases: saving file on file system, rtmp-using or raw pipe out. Is it possible to syncronize them?   ffmpeg version N-66924-gaba61b2 Copyright (c) 2000-2014 the FFmpeg developers built on Jan 15 2015 12:50:29 with gcc 4.6 (GCC) 20120106 (prerelease) configuration: --arch=arm --cpu=cortex-a8 --target-os=linux --enable-runtime-cpudetect --prefix=/data/data/info.guardianproject.ffmpeg/app_opt --enable-pic --disable-shared --enable-static --cross-prefix=/home/qwerty/work/android-ndk-r10b86/android-ndk-r10b/toolchains/arm-linux-androideabi-4.6/prebuilt/linux-x86/bin/arm-linux-androideabi- --sysroot=/home/qwerty/work/android-ndk-r10b86/android-ndk-r10b/platforms/android-3/arch-arm --extra-cflags='-I/home/qwerty/x264 -I/home/qwerty/ffmpeg_build/mp3/build/include -I/home/qwerty/faac/output/include -I/home/qwerty/ffmpeg_sources/fdk-aac/build/include -I/usr/local/include/ -mfloat-abi=softfp -mfpu=neon' --extra-ldflags='-L/home/qwerty/x264 -L/home/qwerty/ffmpeg_build/mp3/build/lib -L/home/qwerty/ffmpeg_sources/fdk-aac/build/lib -L/home/qwerty/faac/output/lib -L/usr/local/lib/ -lrtmp -lpolarssl' --enable-version3 --enable-gpl --enable-nonfree --disable-doc --enable-yasm --enable-decoders --enable-encoders --enable-muxers --enable-demuxers --enable-parsers --enable-protocols --enable-filters --enable-avresample --enable-librtmp --enable-libfaac --enable-libfdk_aac --enable-libmp3lame --disable-indevs --enable-indev=lavfi --disable-outdevs --enable-hwaccels --enable-ffmpeg --disable-ffplay --disable-ffprobe --disable-ffserver --disable-network --enable-libx264 --enable-zlib --enable-muxer=md5 libavutil      54. 10.100 / 54. 10.100 libavcodec     56.  8.100 / 56.  8.100 libavformat    56.  9.100 / 56.  9.100 libavdevice    56.  1.100 / 56.  1.100 libavfilter     5.  1.106 /  5.  1.106 libavresample   2.  1.  0 /  2.  1.  0 libswscale      3.  1.101 /  3.  1.101 libswresample   1.  1.100 /  1.  1.100 libpostproc    53.  3.100 / 53.  3.100 Is reusing old code for a new assignment considered self plagiarism? How to protect yourself if you consider it to be, and a group partner does not? Blowing your own sail? How it this possible? Even if the gif is fake, the Mythbusters did it and with a large sail it really moves forward. What is the explanation? Motivation behind standard deviation? Is the "the ... the ..." construction governed by some grammatical rule? From this  by orlp:     If I strap a rocket booster to a rocket in space and fire it for one second, then the work provided is much higher when the rocket is flying fast compared to when the rocket was stationary. In both cases the rocket fires for the same duration but in the former case the rocket travels a much greater distance during this period. What gives? How do I show that $X = \left\{ (x,y) \in \mathbb{R}^2 \mid x \in \mathbb{Q}\text{ or }y \in \mathbb{Q}\right\}$ is path connected?    Note that $X$ is a topological space with subspace topology $\tau = \left\{ U \cap X \mid U\text{ is open in }\mathbb{R}^2\right\}$.  I only know two things (with respect to path connectedness):   A topological space $X$ is path connected iff for every $p,q \in X$, there is a continuous function $f: [a,b] \to X$ with $f(a)=p$ and $f(b) = q$.    Continuity $f$ is defined as for every open set in $X$, its preimage is open in $[a,b]$ (with subspace topology induced by $\mathbb{R}$ (with standard topology) )   Suppose $X$ and $Y$ are two topological spaces and $f: X \to Y$ is a continuous surjection. If $X$ is path connected, then $Y$ is path connected. I am completely new in Ubuntu. Just installed Ubuntu 14.04. On my other computer running Windows, I am using Avast, free edition.  Do I need anti-virus protection at all? Should I sign, watermark or otherwise label a framed print? And if so how? Prove that $ n^3 + 5n$ is divisible by 6 for all $n\in \textbf{N}$ Prove that $ n^3 + 5n  $ is divisible by 6 for all $ n \in \textbf{N} $.  I provide my proof below. Evaluating $\int \sqrt{1 + t^2} dt$? How do we solve (i.e. get the closed form of) $\int \sqrt{1 + t^2} dt$ ?   shows the closed form of it but not the steps in solving it.  I think I need some algebraic trick..   I thought of some hyperbolic functions and their derivatives and equalities:  If I make $ t = \sinh x$, then I can reduce the integral to  $$ \int \cosh ^2 x dx $$ though I'm not sure if this is valid. But from here, I cannot go on evaluating the integral... Detecting TCP Client Disconnect What issues should be considered when overriding equals and hashCode in Java? Does existence of magnetic monopole break covariant form of Maxwell’s equations for potentials? Absence of magnetic charges is reflected in one of Maxwell's fundamental equations: $$\operatorname{div} \vec B = 0. \tag1$$ This equation allows us to introducte concept of vector potential: $$\vec B = \operatorname{rot} \vec A.$$ Using this concept, it is possible to express Maxwell's equations in a graceful symmetric form: $$\nabla^2 \vec A - \frac{1}{c^2}\frac{\partial^2 \vec A}{\partial t^2} = - \frac{\vec j}{\epsilon_0 c^2}, \tag2$$ $$ \nabla^2 \phi -\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = - \frac{\rho}{\epsilon_0}. \tag3 $$  Noticing, that vector $\vec A$ and scalar $\phi$ potentials, as well as electric current density $\vec j$ and charge density $\rho$, form a 4-vector in Minkovsky space-time. Therefore, Maxwell's equations can be expressed in a covariant form, using d'Alembertian: $$\nabla_{\mu}\nabla^{\mu} A_{\nu} = \frac{j_{\nu}}{\epsilon_0}. \tag4$$  If magnetic monopols exist, Maxwell's equation $(1)$ will look as: $$\operatorname{div} \vec B = \mu_0 c \rho_{\mathrm{magnet}}.$$  As the divergence of $\vec{B}$ isn't equal to zero, it impossible to introduce concept of vector potential. Thus, the equation in the form of $(4)$ will not be possible to express. Yesterday I was copying a single, 8 GB file to a USB with a slow, 7 MB/s write speed, while my RAM is 3 GB. While copying the system froze, to the point where I couldn't even move the cursor.  I managed to log into the text console, and ran iotop, it showed that a process named kswapd0 was taking 99.99% of IO.  Are there workarounds so copying a large file doesn't make my system unusable? Eigenvalues of Householder matrix What would be the eigenvalues for a Householder matrix defined as: $H = I - 2 u u^T$? Can someone explain it  to me intuitively or with a simple proof? $\sum_{n=1}^\infty \log \frac{n+1}{n}$  $\sum_{n=1}^\infty \log \frac{n+1}{n}$ =  $\displaystyle\lim_{n \to{+}\infty}(\log2 - \log1)+(\log3-\log2)+...+(\log(n+1)-\log n)$=$\displaystyle\lim_{n \to{+}\infty}\log(n+1)\to \infty$. So, the series diverges.  Is my procedure correct? I would like to have a paper's bibliography entries inside its .tex file instead of outside in a .bib file. Is this possible with biblatex?  My reason for asking this is that I want to find the most convenient way to produce documents that uses biblatex from just one file to make collaboration on papers easier and to be able to handle submissions of papers to journals. Is it possible to plot a graph of any shape? In school, I have learnt to plot simple graphs such as $y=x^2$ followed by $y=x^3$.  A grade or two later, I learnt to plot other interesting graphs such as $y=1/x$, $y=\ln x$, $y=e^x$. I have also recently learnt about trigonometric graphs and circle equations. In the internet, I have seen users posting graphs of different shapes like a heart-shaped graph and a Batman logo-shaped graph. I am sure there are numerous more graphs that I have yet to see. Seeing that the graphs can be shaped into shapes like the Batman logo and a heart brings me to my question: Is it possible to plot a graph of any shape regardless of its complexity? Perhaps, shaped into an outline of a person or a landmark? Why or why not? Finding the contact point with SAT