so the exercise goes:  Let $E, F$ and $G$ be given sets and let the functions $f: E \to F$ and $g: F \to  G$ be given. Consider the claim:  if $f$ is surjective and $g$ is surjective  $\implies g\circ f$ is surjective.  The claim is either true or false. If it is true, prove it. If it is false, give example of 3 sets of $E,F,G$ and two functions $f,g$ where the fuctions $f:E\to F$ and $g: F\to G$ are surjective but $g\circ f$ isn't surjective.  How should I think about all of this? We've had so few tasks in our course on this subject
Suppose we have two functions, $f:X\rightarrow Y$ and $g:Y\rightarrow Z$. If both of these functions are onto, how can we show that $g\circ f:X\rightarrow Z$ is also onto?
Why are year, pound and mile in the singular form in the phrases below?   five-year-old children  20 pound note 10 mile run   Is that because they're acting as adjectives, which are always invariable in English?  Is it incorrect to say...   five-years-old children? 20 pounds note? 10 miles run? Rigify generated rig, hand IK controller has unexpected rotation $(K[x,y,z]/(x^2+y^3+z^7))_{(x,y,z)}$ is a UFD. How to figure if it is safe to travel to the area I am interested in? When planning a trip, I like to make sure I will feel safe enough (I am talking about war/terrorism risk, not pickpockets), or at least I like to have an idea about the risks. I know the risk could be subject to political views (for example the political stability of a region might not be acknowledged by every country) and therefore I am mostly interested in a Western country view.  In order to figure that, I thought about checking travel operators offers. I suppose if they do not organise trips to a specific region, it might be due to political instability risks. But it might be due to a lack of tourist interest too. As I tend to travel without their help, I am not really considering this kind of source as very helpful.  So I am interested in one (or two, for cross-checking) source(s) of information with a western view (I mean I understand the view could be biased) regarding the security of ideally any area in the world. My expectation is that, the same way I can browse a tour operator's catalog of visited areas, I could browse through the areas and see if it is recommended or not to travel there. General Information about Eigenvalues for an 3x3 symmetric matrix How do we find general information about the eigenvalues of an arbitrary 3x3 symmetric matrix without resorting to explicitly computing the solutions to the cubic characteristic equation?:  \begin{bmatrix}     a   &amp; b &amp; c  \\     b    &amp; d &amp; e \\     c &amp; e &amp;f \end{bmatrix}  (In this case the sum and product of the eigenvalues, and the sum of the reciprocals of the eigenvalues.) Since diagonalization preserves the trace and determinant of matrices, would it be best to just find the corresponding diagonal matrix and read off the sum and product of the eigenvalues from there? The problem is I can't think of a way to diagonalize a matrix without explicitly computing the eigenvalues. I am a Canadian citizen, and I intend to travel to the US for a student clinical elective at a university in Texas for the months of September and October. The university will help me obtain an F-1 status. However, I want to attend a few conferences and educational seminars which are being held in November/December for which I will be using a B1/B2 visa.  Will I have to leave the US and come back in November using a B1/B2 visa, or is there another way to change my visa status, or can I apply for both visas upon entering the US? I've made a simple snowfall particle simulation (using gandalf3's solution: ). Is there a way to loop the animation, so the transition between last frame and first frame is not visible? Is there a setting to do it? Dismiss Windows 8 Volume Box (top left corner) When adjusting the volume in Windows 8.1, a box appears at the top left corner of the screen. This box sticks around for ~13 seconds before it fades away. If you attempt to dismiss it by click on it, it switches to the music app.  How do you disable or dismiss this annoying box? At the very least, can the time visible be adjusted? More times than not, it's in the way. Why is potential energy negative when orbiting in a gravitational field? How to fix "W: Duplicate sources.list entry"? I keep getting this warning whenever I try to run sudo apt-get update.  W: Duplicate sources.list entry http://archive.ubuntu.com/ubuntu/ precise-updates/main i386 Packages (/var/lib/apt/lists/archive.ubuntu.com_ubuntu_dists_precise-updates_main_binary-i386_Packages) W: You may want to run apt-get update to correct these problems   Below is the output from /etc/apt/sources.list file:  deb http://archive.ubuntu.com/ubuntu precise main restricted deb-src http://archive.ubuntu.com/ubuntu precise main restricted  deb http://archive.ubuntu.com/ubuntu precise-updates main restricted deb-src http://archive.ubuntu.com/ubuntu precise-updates main restricted  deb http://archive.ubuntu.com/ubuntu precise universe deb-src http://archive.ubuntu.com/ubuntu precise universe deb http://archive.ubuntu.com/ubuntu precise-updates universe deb-src http://archive.ubuntu.com/ubuntu precise-updates universe  deb http://archive.ubuntu.com/ubuntu precise multiverse deb-src http://archive.ubuntu.com/ubuntu precise multiverse deb http://archive.ubuntu.com/ubuntu precise-updates multiverse deb-src http://archive.ubuntu.com/ubuntu precise-updates multiverse  deb http://archive.ubuntu.com/ubuntu precise-security main restricted deb-src http://archive.ubuntu.com/ubuntu precise-security main restricted deb http://archive.ubuntu.com/ubuntu precise-security universe deb-src http://archive.ubuntu.com/ubuntu precise-security universe deb http://archive.ubuntu.com/ubuntu precise-security multiverse deb-src http://archive.ubuntu.com/ubuntu precise-security multiverse   How do I fix it? Laptop w/o battery US bound flight? Bisecting K-means using Dynamic Time Warping I'm trying to cluster time series of different length and I came up to an idea to use DTW as a similarity measure, which seems to be adequate, but the thing is, I cannot use it with K-means, since it's hard to define centroids based on time series which can have different length/phase. So I was thinking about Hierarchical clustering, since it seems appropriate to combine with DTW, but it's not scalable. So my next thought is to try with bisecting k-means that seems scalable, since it is based on K-means step repetitions. My idea is next, by steps:   Take two signals as initial centroids (maybe two signals that have smallest similarity, calculated using DTW) Assign all signals to two initial centroids Repeat the procedure on the biggest cluster   In this way I could use DTW as distance measure, that could be useful since my data may be shifted, skewed, and avoid calculating centroids. At the end I could take one signal from each cluster that is the most similar with others in cluster (some kind of centroid/medioid).  What do you think about this approach and about the scalability? Operator Theoretic Perspectives on Bayesian Inference Prove that $ \int _a^b\:f\left(x\right)\:\:dx\:=\:\int _a^b\:f\left(a+b-x\right)\:\:dx $ This is the prove- $y=a+b-x$. Then $dy/dx=-1$ and thus $$\int_a^bf(a+b-x)dx= \int_b^a-f(y)dy = \int_a^bf(x)dx$$  But I dont understand the last part of this - Why ? $$\int_b^a-f(y)dy = \int_a^bf(x)dx$$  I know that $\int_b^a-f(y)dy = \int_a^bf(y)dy $. But something is not clear to me. Printing a file to a printer in Ruby When to use LinkedList over ArrayList in Java? Multiple Apple Developer Accounts Why is printf better than echo? How to replace ${} placeholders in a text file? I opened Cairo-dock in my terminal with: open cairo-dock and that worked but if I close the terminal it goes with it. How can I keep it open regardless of the terminal?