Problem Statement:     Let $G$ be a finite group, say a group with $n$ elements, and let $S$ be a nonempty   subset of $G$. Suppose $e \in S$, and that $S$ is closed with respect to multiplication. Prove   that $S$ is a subgroup of $G$. (HINT: It remains to prove that $G$ is closed with respect to   inverses. Let $G = \{a_1, \ldots ,a_n\}$; one of these elements is $e$. If $a_i \in G$, consider the distinct elements $\{a_ia_1, a_ia_2, \ldots a_ia_n\}$.   Incomplete Solution:  Let $G = \{a_1, \ldots, a_n\}.$ Let $a_i \in G.$ Then any number of $a_ia_1, a_ia_2, \ldots, a_ia_n \in S,$ since $S$ is closed under multiplication. One of those $a_ia_j$ is $e$ since $e \in S.$ Then either $(a_i)^{-1}$ or $(a_j)^{-1} \in S.$  To complete the proof I was told:     You need to argue why one of them must be $e$. Show $f(x)=a_i \cdot x$ is an injection, then since it's a map from a finite set to itself that implies it's also a surjection.   Let $a_i \cdot x = a_i \cdot y$. Then $x = y$. So, $f$ is injective. Not too sure if that's correct, but I have a follow up question:  For $x = 1, 2, 3, \ldots, n$; $f(x) \in \{a_i, a_i \cdot 2, a_i \cdot 3, \ldots  a_i \cdot n\}$. So, $f: \{1, 2, 3, ..., n\} \to \{a_i, a_i \cdot 2, a_i \cdot 3, ..., a_i \cdot n\}$. Is that correct? If so, how do we know $f$ maps from a finite set to itself?
Let $G$ be a finite group, say a group with $n$ elements, and let $\emptyset \neq S \subseteq G$. Suppose $e \in S$, and that $S$ is closed with respect to multiplication. Prove that $S$ is a subgroup of $G$.    Let $G = \{a_1, \ldots, a_n\}.$ Let $a_i \in G.$ Then any number of $a_ia_1, a_ia_2, \ldots, a_ia_n \in S,$ since $S$ is closed under multiplication. One of those $a_ia_j$ is $e$ since $e \in S.$ Then either $a_i$ or $a_j$ is $a^{-1}.$ Would this argument work?
Parse JSON with R What is Hawking radiation and how does it cause a black hole to evaporate? .NET DateTime.Now returns incorrect time when time zone is changed How to create a Geometric Polygonal Art portrait I've been interested in creating Geometric Polygonal art illustrations, i.e. making polygons and shapes with different color shading represent a face, or an object, see  set of .  How are they achieved? Is there some script for Illustrator which alters the image adding the shapes according to the colors or shades? How would I transform a normal image of Marilyn Monroe for example, to a polygonal illustration of her? How to self study Linear Algebra I have no idea if this question is appropriate for this forum, but I hope you guys can overlook the fact that I asked it on a wrong forum (if I did) and still help me answer it (of course, if this is indeed the wrong forum for the type of question I'm about to ask, do please say so).  I am a 16 year old guy who is passionate about physics and as a result wants to increase his knowledge in mathematics, the language of physics. I've read and heard a lot about Linear Algebra and how crucial it is to physics and I am deeply motivated to self study this intriguing part of mathematics. However, I have limited knowledge of maths. I (for example) know basic algebra, trig, diff/int calc and some analytical geometry, but I wouldn't say I master these subjects past the high school curriculum. Now my question is: Would you kind people say I am able to self study Linear algebra or is it too tough and/or does it require too much of a math background? And are there any good books out there for BEGINNERS in LA? I found this (free) e-book called: Elementary Linear Algebra by Kenneth Kuttler and another one called 'Linear Algebra: Theory and its applications' also by Kuttler? Are these any good? Or would you recommend other books? If you guys have any tips regarding books for LA but also tips in general, please, I'd appreciate them! Why does a bezier curve twist when rotated? If I create a bezier curve and rotate it in edit mode it does not accordingly rotate the tilt of the handles which creates an undesirable twisting along the curve.  Here is an example using a bezier circle with a square used for a bevel object:   Here it is after being rotated 90° about the y-axis:   How do I prevent/fix this? Plotting a 3d surface in tikz, with a limit to the infinity Does JavaScript have the interface type (such as Java's 'interface')? Singleton: How should it be used I think the following is true:  If $q$ is coprime to $a$ then $a\mid (nq-1)$ for some $n\in\mathbb Z$.  I have started to first reducing $n\in\mathbb Z$ to $n\in\{1,\ldots,a-1\}$ and then considering all the cases $q = ka + j$ for $j=1,2,3,\ldots,a-1$. I stopped at $j=3$ and found that this is kinda messy. Does anyone have an elegant proof for this (maybe using some results from number theory)?  [Note: question has been generalized from prime $q\nmid a$ to $\,q$ coprime to $a$] Which parts are from Eminems real life? In "" we saw how Bunny Rabbit got into the rapping gang and won the heart of millions of rap lovers. Now in Eminem's original life too similar thing happened. We know how Dr. Dre helped him getting into an industry which is dominated by mainly black people in USA. In fact,  called the film semi-autobiographical. So I want to know which parts are direct picturisations of his real life? SSH access to office host behind NAT router I would like to access the ssh port of my office linux host from home. Unfortunately the host is located behind a NAT router. So, the IP address is not publicly available. There is however access to another internet host (Server) which is unfortunately only non-root access. After a while of searching I do not find a suitable solution.  Following setup:   Office PC (linux, root access) behind NAT (IP not public) but full Internet access. Server PC (linux, no root access) static and public IP and full Internet access. Home PC (linux, root access) behind NAT (IP not public) but full Internet access.   Possible connections: Office PC --> Server &lt;-- Home PC  Not possible: Office PC &lt;-X- Server -X-> Home PC  Neither the Home PC, nor the Server can initiate access to the Office PC. But both the Office PC and the Home PC can initiate connections to the Server.  Reverse SSH tunnel not possible: I tried a method called reverse ssh-tunnel. Unfortunately this requires GatewayPorts on Server set to "yes" in /etc/ssh/sshd_config, where I have no root access.  In principle it should be possible:   0) On the Server I start a userspace program which listens on 2 ports (1 incoming, 1 outgoing)  1) On my office PC I run a another program which keeps a TCP connection open to the outgoing port on the server.  2) From home I connect to Server's incoming port.  There should be a standard solution for this out there.  What is the quickest and cleanest solution to solve this?  Frank Why is uniform prior on log(x) equal to 1/x prior on x? I'm trying to understand Jeffreys prior. One application is for 'scale' variables like the standard deviation $\sigma$ (or its square, the variance $\sigma^2$) of Gaussian distributions. It is often said that using a uniform prior over $\sigma$ is not really non-informative and instead one should either:   Use instead $\ln \sigma$ as the free parameter, with a uniform prior (this is often called a log-uniform prior) Or keep using $\sigma$ as the free parameter but use $1/\sigma$ as the prior (which is not uniform).   Why are the above two methods/priors equivalent? I feel it has something to do with the fact that the derivative of ln $\sigma$ is $1/\sigma$ but I can't take the next step.   Also, why does this even matter, in simple language with minimal jargon? I see all these complicated explanations online involving the Fisher information matrix but in the end all I see is that the above log-uniform or $1/\sigma$ priors preferentially weight lower values of $\sigma$ more highly. Why? If possible, a simple analytic example or python snippet would be very helpful. Why were the people of Rhun and Harad allied with Sauron?  Did they join him out of fear, or did Sauron say he would give them something if they joined him?  I know the men of Dunland were angry at Rohan; did Rhun and Harad have something against Gondor? When using an Int- or FloatProperty, is there any way of changing their min and max values dynamically?  I'm currently loading images on the fly and would like to limit a frame property to run from 0 to len(loaded_images) every time I call the loading operator. However, the property seems to be read-only and re-assigning gives the following error:  bpy.context.scene.project_props.bl_rna.properties['myIntProp'] = bpy.props.IntProperty(...) TypeError: bpy_prop_collection[key] = value: invalid, expected a StructRNA type or None, not a tuple   I understand the bpy.props constructor uses an intermediary tuple, but I'm not sure how to proceed. Is there any way around this? The following MWE compiles (for me)  \documentclass{article}  \usepackage{pgfplots} \pgfplotsset{compat=newest} \usetikzlibrary{external} \tikzexternalize[prefix=figures/externalized/]   \begin{document} \begin{figure}     \centering     \begin{tikzpicture}          \begin{axis}  %           \addplot3[surf,mesh/rows=129,mesh/cols=129,mesh/ordering=y varies] table[x=X,y=Y,z=psi] {figures/data/LidDrivenCavity/Lid_SPIKE_128x128_Re1000_psi.dat};              \addplot3[contour gnuplot,mesh/rows=129,mesh/cols=129,mesh/ordering=y varies] table[x=X,y=Y,z=psi] {figures/data/LidDrivenCavity/Lid_SPIKE_128x128_Re1000_psi.dat};          \end{axis}      \end{tikzpicture}     \caption{ciao} \end{figure} \end{document}   where the data file is . If I de-comment the first \addplot3 and comment the second one, then I get an error. The final part of the log file (the interesting one, I think) is  ===== 'mode=convert with system call': Invoking 'pdflatex -shell-escape -halt-o n-error -interaction=batchmode -jobname "figures/externalized/prova-figure0" "\ def\tikzexternalrealjob{prova}\input{prova}"' ======== \openout3 = `prova.auxlock'.  runsystem(pdflatex -shell-escape -halt-on-error -interaction=batchmode -jobname  "figures/externalized/prova-figure0" "\def\tikzexternalrealjob{prova}\input{pr ova}")...executed.  \openout3 = `prova.auxlock'.   ./prova.tex:22: Package tikz Error: Sorry, the system call 'pdflatex -shell-esc ape -halt-on-error -interaction=batchmode -jobname "figures/externalized/prova- figure0" "\def\tikzexternalrealjob{prova}\input{prova}"' did NOT result in a us able output file 'figures/externalized/prova-figure0' (expected one of .pdf:.jp g:.jpeg:.png:). Please verify that you have enabled system calls. For pdflatex,  this is 'pdflatex -shell-escape'. Sometimes it is also named 'write 18' or som ething like that. Or maybe the command simply failed? Error messages can be fou nd in 'figures/externalized/prova-figure0.log'. If you continue now, I'll try t o typeset the picture.  See the tikz package documentation for explanation. Type  H &lt;return&gt;  for immediate help.  ...                                                l.22    \end{tikzpicture}  ?  PGFPlots: reading {figures/data/LidDrivenCavity/Lid_SPIKE_128x128_Re1000_psi.da t} Runaway argument? {\pgfkeyssetvalue {/pgfplots/execute at begin plot visualization}{}\pgfkeyssetv alue \ETC. ./prova.tex:22: TeX capacity exceeded, sorry [main memory size=5000000]. &lt;argument&gt; ...5e-1],2Y7.753628e-3]}{2Y8.20894\ETC.  l.22    \end{tikzpicture}  If you really absolutely need more capacity, you can ask a wizard to enlarge me.   Here is how much of TeX's memory you used:  20761 strings out of 493054  547383 string characters out of 6134761  5000001 words of memory out of 5000000  23958 multiletter control sequences out of 15000+600000  3640 words of font info for 14 fonts, out of 8000000 for 9000  1141 hyphenation exceptions out of 8191  62i,4n,84p,728b,1519s stack positions out of 5000i,500n,10000p,200000b,80000s ./prova.tex:22:  ==&gt; Fatal error occurred, no output PDF file produced!   similar (or the same) error appears if externalization is not enabled/used.  My TeX distribution is TeX Live on Mac OS X 10.9.5 and I use TeXShop as editor.  Once I (partially) solved a  on Windows+MikTeX+TeXniccenter by modifying/adding options to the command executed when compile button is pressed. I say partially since I actually solved the problem for some data file. Then I experienced the same problem with larger datafile and/or different plots (I don't remember anymore) and didn't find a way out, no matter how much I extended the memory.  What should I do now? Bridged adapter in Virtual Box - cannot obtain IP I have virtual box with RHEL installed. My virtual box is running on windows 7 64 -bit. I configured bridged connection, but when I am restarting network interfaces, I receive following error:  Determining IP information for eth1... failed   On Windows XP it works without problems. I disabled Windows Firewall and installed latest Virtual Box, still without success.   NAT is working without any problems. What did I missed?  UPDATE  Ouput of ifconfig -a is:  eth0      Link encap:Ethernet  HWaddr 08:00:27:B6:FC:69           inet addr:10.0.2.15  Bcast:10.0.2.255  Mask:255.255.255.0           inet6 addr: fe80::a00:27ff:feb6:fc69/64 Scope:Link           UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1           RX packets:430 errors:0 dropped:0 overruns:0 frame:0           TX packets:346 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:1000           RX bytes:36520 (35.6 KiB)  TX bytes:31455 (30.7 KiB)  eth1      Link encap:Ethernet  HWaddr 08:00:27:C3:84:4A           inet6 addr: fe80::a00:27ff:fec3:844a/64 Scope:Link           UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1           RX packets:1464 errors:0 dropped:0 overruns:0 frame:0           TX packets:17 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:1000           RX bytes:135419 (132.2 KiB)  TX bytes:4230 (4.1 KiB)  lo        Link encap:Local Loopback           inet addr:127.0.0.1  Mask:255.0.0.0           inet6 addr: ::1/128 Scope:Host           UP LOOPBACK RUNNING  MTU:16436  Metric:1           RX packets:0 errors:0 dropped:0 overruns:0 frame:0           TX packets:0 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:0           RX bytes:0 (0.0 b)  TX bytes:0 (0.0 b)   Output of route -n  Kernel IP routing table Destination     Gateway         Genmask         Flags Metric Ref    Use Iface 10.0.2.0        0.0.0.0         255.255.255.0   U     0      0        0 eth0 169.254.0.0     0.0.0.0         255.255.0.0     U     1002   0        0 eth0 0.0.0.0         10.0.2.2        0.0.0.0         UG    0      0        0 eth0   Output of dhclient -v eth1 is:  Listening on LPF/eth1/08:00:27:c3:84:4a Sending on   LPF/eth1/08:00:27:c3:84:4a Sending on   Socket/fallback DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 6 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 9 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 15 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 18 (xid=0x6afb6c4c) DHCPDISCOVER on eth1 to 255.255.255.255 port 67 interval 13 (xid=0x6afb6c4c) No DHCPOFFERS received. No working leases in persistent database - sleeping.   Moreover the same machine with the same configuration works well on Windows XP. The problem is only on Windows 7. In 5e's version of the flesh to stone spell, if the spell is maintained for the full minute duration, the effect (petrification) lasts until "removed".  How can this petrification be removed? Why is polynomial regression considered a special case of multiple linear regression? If polynomial regression models nonlinear relationships, how can it be considered a special case of multiple linear regression?  Wikipedia notes that "Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function $\mathbb{E}(y | x)$ is linear in the unknown parameters that are estimated from the data."  How is polynomial regression linear in the unknown parameters if the parameters are coefficients for terms with order $\ge$ 2? proving $\left(1+\frac 1n\right)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ using the binomial theorem