Show that the group is trivial. Show that the following group is identity:   $$G=\langle x,y,z  \mid xyx^{-1}=y^{2}\, , \, yzy^{-1}=z^{2}\, , \, zxz^{-1}=x^{2} \rangle.$$  This group is its own derived group. So all I get is group is perfect. Am I correct? What else I missed? Should I use tietze transformation, I am not getting anywhere with them.
Presentation $\langle x,y,z\mid xyx^{-1}y^{-2},yzy^{-1}z^{-2},zxz^{-1}x^{-2}\rangle$ of group equal to trivial group Problem: Show that the group given by the presentation $$\langle x,y,z  \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2} \rangle $$ is equivalent to the trivial group.   I have tried all sorts of manners to try to show that the relations given by the presentation above imply that $x=y=z=e$. However, I am stuck and would appreciate any hints as to how I should move forward.
Turn a string into a valid filename? I have a string that I want to use as a filename, so I want to remove all characters that wouldn't be allowed in filenames, using Python.  I'd rather be strict than otherwise, so let's say I want to retain only letters, digits, and a small set of other characters like "_-.() ". What's the most elegant solution?  The filename needs to be valid on multiple operating systems (Windows, Linux and Mac OS) - it's an MP3 file in my library with the song title as the filename, and is shared and backed up between 3 machines. How does gravity escape a black hole? My understanding is that light can not escape from within a black hole (within the event horizon).  I've also heard that information cannot propagate faster than the speed of light.  It would seem to me that the gravitational attraction caused by a black hole carries information about the amount of mass within the black hole.  So, how does this information escape?  Looking at it from a particle point of view: do the gravitons (should they exist) travel faster than the photons? How to determine the size of a package while using apt prior to downloading? When using apt-get install &lt;package_name&gt;, and there are dependencies that need to be downloaded, the terminal outputs names of additional packages and total size, and asks for confirmation before downloading.   But, when dependencies are satisfied and nothing but the named package needs to be downloaded there is no size output and no confirmation.  When using Synaptic, I can see the total size that new packages that will use after installation but no way to see the size that needs to be downloaded, except to go from package to package and use properties to see the compressed size.  I would like to know if there is a way to see the size of a package(s) in terminal and Synaptic prior to downloading and installing it/them? BufferedWriter not writing everything to its output file I already know how to do it on Mac OS X (plenty of instructions online).  But is it possible to do it on Ubuntu? If yes, how? Comparing SVM and logistic regression Can someone please give me some intuition as to when to choose either SVM or LR? I want to understand the intuition behind what is the difference between the optimization criteria of learning the hyperplane of the two, where the respective aims are as follows:     SVM: Try to maximize the margin between the closest support vectors LR: Maximize the posterior class probability    Let's consider the linear feature space for both SVM and LR.  Some differences I know of already:   SVM is deterministic (but we can use Platts model for probability score) while LR is probabilistic. For the kernel space, SVM is faster (stores just support vectors) diophantine equation: $x^2 +y^2 = z^n$ Prove that $x^2+y^2=z^n$ has a solution $(x, y, z)$ in $\mathbb{N}$ for all $n\in\mathbb{N}$  I tried to prove this by induction, but couldn't. ( This was probably because the solution for some $n$ isn't necessarily related to the solution for $n+1$)  I can't seem to see any other way other than induction for proving the statement. Any help/hints on solving this problem, or any alternative approach will be highly appreciated :) Is there a place where I can look what updates I've installed? 10 hour layover in Moscow Russia, do I need a transit visa? On June 29th 2015, I will be traveling for the first time to Italy from USA. I will have a 10 hour layover in Moscow Russia. My question is, do I need a visa to visit the city? If $E$ has measure zero, then does $E^2$ have measure zero? Currently I'm just working through measure theory just to see if its something I would like to take.  Unfortunately I am stuck on this problem from Carothers.     If $m^*(E)=0$, then $m^*(E^2)=0$.   Where $m^*$ denotes outer measure and $$E^2=\{x^2:x\in E\}.$$  I toyed with the idea that $I_k &lt; 1 \Rightarrow I^2_k &lt; I_k$.  However I am at a loss as to how to set up a chain of inequalities (which is what I am assuming I need). In my cssh config file, I have:  terminal_font=6x13   this default font is too small for me.   man cssh says, I can specify other font size with the -f argument, while using "standard X font notation".  is a wikipedia page containing some other fonts types. Unfortunately, none of these work for me:  $ cssh host1 host2 -f "7x14" Fatal: Unrecognised font used (7x14). Please amend $HOME/.clusterssh/config with a valid font (see man page).   Why does "7x14" not work?  How can I use bigger fonts with cssh ? Unable to catch JsonMappingException How many permutations of $\{1, \ldots, n\}$ exist such that none of them contain $(i, i+1)$ (as a sequence) for $i \in {1,...,(n-1)}$? How can I select first occurrence between two patterns including them. Preferably using sed or awk.  I have:  text something P1 something content1 content2 something P2 something text something P1 something content3 content4 something P2 something text   I want the first occurrence of the lines between P1 and P2 (including P1 line and P2 line):  something P1 something content1 content2 something P2 something Finding the limit of $(1-\cos x)/x^2$ I was reading Brian Greene's "Hidden Reality" and came to the part about Hawking Radiation. Quantum jitters that occur near the event horizon of a black hole, which create both positive-energy particles and negative-energy particles, produce the so-called Hawking radiation. However, I do not understand why only the negative-energy particles are being absorbed into the black hole, while the positive-energy particles shoot outward. Shouldn't there be 50/50 chance that each type of particle is being absorbed by the black hole? Also, the book mentions that a negative-energy particle would appear to an observer inside the black hole as positive. Why? The integral $\int_0^\infty \dfrac{x \sin(x)}{x^2+1} dx$ How to calculate: $$\int_0^\infty \frac{x \sin(x)}{x^2+1} dx$$ I thought I should find the integral on the path  $[-R,R] \cup \{Re^{i \phi} : 0 \leq \phi \leq \pi\}$.  I can easily take the residue in $i$ $$  Res_{z=i} \frac{x \sin(x)}{x^2+1}  \quad = \quad \frac{i (e^{ii}-e^{-ii})}{2i}  \quad = \quad \frac{i(\frac1e-e)}{2i} \quad = \quad \frac1{2e}-\frac e2 $$ I know that the integral on $[-R,R]$ will get nearby the real value of $ \frac12\int_0^\infty \frac{x \sin(x)}{x^2+1} dx$. That means that I'm done if I show that the integral on the other part of the path will be small, here is an attempt to do so:  $$ \left|\frac{x \sin(x)}{x^2+1} \right|  \quad \leq \quad  R \cdot \frac{|\sin x|}{R^2-1} $$ I could rewrite $\sin x$ but it becomes nasty. Can you help me with this? Is anti-gravity (i.e. repulsive gravity) possible in theoretical physics? Is  (i.e. repulsive gravity) possible in string theory?  I have read some articles about scientists making assumptions about the existence of anti-gravity, but is it possible in string theory? How can I keep low resolution textures pixelated? RAM not clearing, alway in standby I'm having a problem with my ram on windows 7. The ram is clearing or programs are using and dumping it faster than it can clear. I have 8gb of ram I'm only using 4gb and the rest is in Standby. At best time I can get 100mb of free but its mostly at 15mb or less. I suspect a memory leak but I have no way to figure out what program it is. I have not installed any new hardware recently and newly installed programs have bin closed. Does any one have an idea of what is going on and how to fix it?   I'm sorry this is so little information but that is all I have as well.  Edit: I tried RamMap and found out the my "Mapped File" has 3,291,784k ram in standby how do I figure out what program and how do I limit it's ability to set ram in standby?  Edit: I keep getting a problem when I will have 0 free ram and my computer will stop working properly. It will become outrageously slow where it will take 3-4 minutes to start task manager so I can close some programmes. When I free up some ram by closing some programmes every thing runs normally. It does not matter which programs I closes only that I free up some ram. I know a lot of you say it's supposed to be like that and that's how windows works. But its obviously a problem and my computer has never worked that way neither has any of my other computers. I would like to fix this please help me.