Show that $A=\{(x,|x|):x\in\mathbb{R}\}$ is not the image of an immersion $f:\mathbb{R}\to\mathbb{R}^2$.   Here are the definitions I know:   Let $M$ be a differentiable manifold. A tangent vector at $p\in M$ is a linear function $v:C^{\infty}(M)\to\mathbb{R}$ that is Leibnizian at $p$. The collection of these is the tanget space $T_pM$. The differential map of a differentiable map $f:M\to N$ at a point $p\in M$ is the function $df_p:T_pM\to T_{f(p)}N$ given by $df_p(v)(g)=v(g\circ f)$ where $v\in T_pM$ and $g\in C^{\infty}(N)$. An immersion is a differentiable map $f:M\to N$ such that $df_p$ is injective for every $p\in M$.   Thus, if we suppose that $f:\mathbb{R}\to\mathbb{R}^2$ is a differentiable map with $f(\mathbb{R})=A$, we should show that $df_p$ is not injective for some $p\in\mathbb{R}$. I really don't know what to do, but maybe we should use that $x\to |x|$ is not differentiable in the usual sense at $x=0$, but what tangent vectors should we take in order to show $df_p$ is not injective?  Thank you.
I have no idea how to prove that the set $\{(x, |x|): x\in\mathbb{R}\}$ is not the image of an immersion of $\mathbb{R}$ into $\mathbb{R}^2$  For example   If $f(t)=(t^3, |t|^3)$ then $f'(0)=(0, 0)$.
Why doesn't this C++0x code call the move constructor? How can I rename many files at once? Given a set $\{1,2,3,4\}$, how is the following relation $R$ antisymmetric?  $$R = \{(1, 2), (2, 3), (3, 4)\}$$  Note: Antisymmetric is the idea that if $(a,b)$ is in $R$ and $(b,a)$ is in $R$, then $a = b$. In my textbook  it says the above is antisymmetric which isn't the case as whenever $(a,b)$ is in $R$, $(b,a)$ is not. Cannot install R-forge package using install.packages If you activated a free upgrade to Windows 10 or bought and activated   Windows 10 from the Microsoft Store, you have a digital license for   your device. This means you can reinstall the same edition of Windows   10 that your device has a digital license for without entering a   product key. During reinstallation, if you’re asked to enter a product   key, you can skip this step. Windows 10 will automatically activate   online after the installation is complete.   source:   Does MS digital license 'fingerprint' hardware information  during the free upgrade / activation and store it (at Microsoft) , so that the next re-install queries MS for the fingerprint and activates Windows 10?  Thought the disk drive may change, I expect that none of the other fingerprinted hardware will change. I know that the rotation period of the moon equals its revolution period. It's just so astonishing that these 2 values have such a small difference. I mean, what is the probability of these 2 values to be practically the same? I don't believe this to be a coincidence. It's just too much for a coincidence. What could have caused this? Prove that subsequence converges to limsup Given a sequence of real numbers, $\{ x_n \}_{n=1}^{\infty}$, let $\alpha =$ limsup$x_n$ and $\beta = $ liminf$x_n$.   Prove that there exists a subsequence $\{ x_{n_k}\}$ that converges to $\alpha$ as $k \rightarrow \infty$.   Not sure how to start this without since I'm not given that the subsequence is bounded.. Is LAN with StarCraft 2 definitely not possible? How to make password-less login work I generated a key and copied it to my target:  ssh-keygen ssh-copy-id username@hostname   I left the the location of the key as default and did not set any passphrase.  For a small while after that, I was able to ssh username@hostname without being prompted for a password, and then it stopped working for some reason. (Before anyone jumps to the conclusion that this is the same question as , please read it through. Also I am looking for more comprehensive and detailed data than the poster in .)  Many years ago I used to use a tool called Line Speed Meter (or Monitor) by TCPIQ. This was a brilliant piece of software that would sit quietly in the background and several times a day (according to a selectable frequency) would reach out and test internet line performance against servers all over the world. In the course of a week (or more), thousands of servers would be tested and enough data would accumulate that one could analyze performance according to time of day, day of week, location of server, etc. Drill-down was possible to the point where you could see every server that was tested.  This data was invaluable in determining if one really was getting the performance being paid for. It was also infinitely more valuable than spot checks you perform at sites like  or even the better-regarded ones like  or  (both of which, incidentally, are biased toward USA users - which I am not one of anymore).  Unfortunately Line Speed Meter is no longer supported and the company that made it now has instead a tool called isposure that is nowhere near as flexible or transparent - it is geared to supplying data to ISPs rather than to the consumers. For example, according to their FAQ, it usually runs only once a day automatically - I'd like to start with running it every couple of hours for a week or two - and scale back once I have a good basis so that I can notice when performance changes. Also, the details of the tests are no longer visible to the consumer-user.  My question: is there a (preferably free) tool/service available today to automatically collect comprehensive line speed metrics so I can map out and analyze the performance I am getting on my Broadband internet connection?  EDIT 2012-02-13: I left something out of the above.  In my country, each ISP sells a package that implies performance beyond the borders of my small country. They own (or lease) the lines going across the ocean. They have full control of the bandwidth leading from their servers to the backbones in the USA and EU. Peformance through the cloud is very much relevant and although not entirely, certainly largely under the local ISP's control, since traditionally, the bottleneck has been traffic out of the country.  As an example, at the moment, my ISP is advertising widely their new transatlantic cable and how their performance now outstrips everyone else's. I haven't "felt" it - so I want a tool to quantify it.  Performance within my country is usually good. The problem is that almost all my work involves access to the Internet in North America, Europe, Pacific Rim and others. So I care about and want to document the end-to-end performance I am getting from my ISP - and if I switch, compare to what I get at a new ISP.  EDIT 2013-03-07: As @harrymc pointed out I neglected to mention OS. All the machines on my home LAN are (sadly) Windows (XP or 7). If A tool requires *nix, I have a couple ancient machines in storage I could probably re-purpose - but my preference is for a Windows-based tool. Thanks.  EDIT 2013-03-10: As a further clarification - a method to monitor speeds achieved during regular work provides half of what I'm looking for. What was nice about TCPIQ is that it would ferret out thousands of sites around the world and do brief d/l &amp; u/l tests against them to check speed and then log the results. This gave a valuable statistical database for analysis of speed overall, as well as during times of day, days of week and according to geographical targets. That's the sort of functionality I'd like to achieve (again). Stack Exchange & Area 51 sign up/ log in update Templated check for the existence of a class member function? Directing graph such that any outdegree would be at most 2 Let $G=(V,E)$ be a graph. Suppose that for every subgraph $G'=(V',E') , G' \subset G$, we have $|E'| \le 2|V'|$. Show it's possible to direct $G$ such that any outdegree would be at most $2$.   I tried proving it by induction. I've realized that the minimum degree at any subgraph of $G$ is at most 4. I tried removing the vertex $v_0$ with the minimal degree at a certain subgraph from the original graph, claiming that the graph from the remaining vertexes can be directed as required. How can I include the vertex $v_0$ in the new directed graph such that any outdegree would still be at most 2? Thanks When a new tag is created, should there be some kind of notification to the user?  This could serve a dual purpose:   To make the user check that this tag is really required.  Is it spelt right? Might there be a more appropriate tag to use instead? - See  To promote the creation of tag wikis.   These are both beneficial to the site.  Making users think twice about their tags reduces the amount of retagging edits that are needed and reduces noise.  Promoting tag wikis means that more new tags would come with explainations, making them more user-friendly for others.  Creating a new tag is a privilidge earned at 1500 reputation.  Hopefully the user has gained a feel for the ethos of SO by this point, and is either enjoying being a contributer or is mainly asking good, well recieved questions.  The user is probably also wanting to contribute to the site further (even if just to gain reputation), and so it makes sense to me that there should be a gentle push towards taking a little bit of time to write a short wiki for the tag you just created.  By no means should this be mandatory, and the precise format of this gentle push is more in the scope of UX guys to think up, but I wanted to see how people felt about some kind of discreet notification.  I thought of two forms this notification could take:  Option 1 - at the time of tagging      Option 2 - after posting      Update: Raphael has suggested in the comments that this could be implemented as a 10K user tool.  While I think that's a good idea on its own, I don't think it should be used instead.  I think that encouraging users with less than 10K to complete wikis for their own tags is likely to get some wikis written because there is a sense of ownership over your own tag, and by giving it a wiki you are giving it a better start in life. As with a child, users will want to equip their tag with the tools it will need to survive in the big wide world. Restoring MySQL database from physical files Is it possible to restore a MySQL database from the physical database files. I have a directory that has the following file types:  client.frm client.MYD client.MYI    but for about 20 more tables.  I usually use mysqldump or a similar tool to get everything in 1 SQL file so what is the way to deal with these types of files? How do the Azkaban escapees get their wands back? Sirius Black and a few Death Eaters escape Azkaban and seem to operate after that with their wands intact.  How come?  If a mere expulsion from Hogwarts entails wand destruction (as happened with Hagrid), shouldn't an Azkaban sentence entail the same?  And even if we imagine that the perceived impossibility of escape from Azkaban made it seem unnecessary to destroy the wand, this would apply to Sirius but not to Death Eaters.  PS: Two comments suggested that the escapees acquired new wands. It would be nice to have a more definitive answer than just a suggestion or a list of possibilities. Thanks.  PPS: Returning the personal property to a released prisoner is one thing, but the wand is a different matter: this is usually the tool of the crime, and I don't see how it can be argued that it should be returned and not destroyed. Again, Hagrid had his wand destroyed and he was not even sentenced to Azkaban.  PPPS: Maybe wand destruction is only done for non-fully-qualified wizards? At the start of A New Hope, when the corvette is being attacked, C3PO says something to the effect of:     There'll be no escape for the princess this time.   But when Luke stumbles onto Leia's message in R2D2, C3PO seems to not know who the figure in the message is:     I'm afraid I'm not quite sure, sir. I think she was a passenger on our last voyage. A person of some importance, sir - I believe.   Is C3PO feigning ignorance to protect her, or does he really not recognize her? In an integral domain every non-zero prime ideal is invertible implies every non-zero proper ideal is invertible. Is it common to review papers assigned to your supervisor? Is it common to supervisors to email their students papers to review (which were apparently assigned to the supervisor, and not to the student). The field is Computer Science, but I  would also like to hear from the other fields. How to configure idle period for auto locking screen I just installed a fresh Ubuntu 12.04 (Precise Pangolin).   I noticed that, there is no screensaver settings, which is fine. I like to control the period of time after which the blank screen is shown,which automatically locks the screen.  I do not want to enable screensaver. I only wish to be able to change the idle time period required to lock the screen automatically. It seems to me, it is by default around 10-15 minutes.