Let  $$f(x)= \begin{cases} e^{-1/x}\quad &amp;\text{for } x&gt;0 \\ 0\quad &amp;\text{for } x\leq 0. \end{cases} $$  Show that $f$ is smooth and $f^{(n)}(0)$ = $0$ for all $n\in \mathbb{N}.$  My professor gave us a hint, which was to show that $f^{(n)}(x)$ = $P_n(1/x)e^{-1/x}$ for $x&gt;0$, and some polynomial $P_n$. However, I've been trying to find the actual formula for the polynomial and can see the form it should take, but cannot figure out the polynomial itself. Is it necessary to actually define the polynomial, or just to prove that such a polynomial exists?  If the latter is the case, how would I go about proving this?
This is Spivak's Calculus on Manifolds ex. 2-25, he says     Define $f:\mathbb{R}\to \mathbb{R}$ by $f(x) =  \left\lbrace   \begin{array}{l}      e^{-x^{-2}} &amp;\text{ if } x \neq 0\\      0 &amp;\text{ otherwise } \\   \end{array}   \right.$. Show that $f$ is $C^{\infty}$ anf $f^{(i)}(0)=0\; \forall i$.   I haven't found a way to get a general solution. I only managed to prove:  $(1)$ $f$ is continuous at $0$.  This follows from $f(0)=0$ and $\lim_{x\to0} f(x)=\lim_{x\to 0}e^{-x^{-2}}=\lim_{x\to 0}\displaystyle\frac{1}{e^{1/x^2}}=0$ since $\displaystyle\frac{1}{x^2}\to+\infty$ if $x\to 0$ then $e^{1/x^2}\to+\infty$ if $x\to 0$, which means that $\displaystyle\frac{1}{e^{1/x^2}}\to 0$ if $x\to 0$ .  $(2)$$f'(0)=0$.  I got this by calculating the limit $f'(0)=\lim_{h\to 0}\displaystyle\frac{f(0+h)-f(0)}{h}=\lim_{h\to 0}\displaystyle\frac{f(h)}{h}=\lim_{h\to 0}\displaystyle\frac{e^{-h^{-2}}}{h}=\lim_{h\to 0}\displaystyle\frac{1/h}{e^{h^{-2}}}$.  The last one follows from L'Hôpital: $\lim_{h\to 0}\displaystyle\frac{\frac{d}{dh}(1/h)}{\frac{d}{dh}(e^{h^{-2}})}=\lim_{h\to 0}\displaystyle\frac{(-1/h^2)}{(e^{h^{-2}})(-2h^{-3})}=\lim_{h\to 0}\displaystyle\frac{h}{2e^{h^{-2}}}=\displaystyle\frac{1}{2}\left(\lim_{h\to 0}\displaystyle\frac{1}{e^{h^{-2}}}\right)\left(\lim_{h\to 0} h\right) = 0$.  $(3)$ $f''(0)=0$  I found that $\displaystyle\frac{df}{dx} =  \left\lbrace   \begin{array}{l}      2x^{-3}e^{-x^{-2}} &amp;\text{ if } x \neq 0\\      0 &amp;\text{ otherwise } \\   \end{array}   \right.$, then  I can get $f''(0)$ computing the limit $\lim_{h\to 0}\displaystyle\frac{f'(0+h)-f'(0)}{h} = \lim_{h\to 0}\displaystyle\frac{f'(h)}{h} = \lim_{h\to 0}\displaystyle\frac{2e^{-h^{-2}}}{h^4}= 2\lim_{h\to 0}\displaystyle\frac{1/h^4}{e^{h^{-2}}}=-2\lim_{h\to 0}\displaystyle\frac{-4h^{-5}}{-2h^{-3}e^{h^{-2}}}\\=4\lim_{h\to 0}\displaystyle\frac{1}{h^2e^{h^{-2}}}=4\lim_{h\to 0}\displaystyle\frac{h^{-2}}{e^{h^{-2}}})=-4\lim_{h\to 0}\displaystyle\frac{-2h^{-3}}{e^{h^{-2}}(-2h^{-3})}=-4\lim_{h\to 0}\displaystyle\frac{1}{e^{h^{-2}}}=0$, after using L'Hôpital twice.  But still I don't have a clue to prove that $f^{(i)}(0)=0$ for every $i$. One of the ideas I managed was to find a general form for $f^{(i)}$ but after I got $f'$ doesn't seem possible to do so considering that I would have to apply the product rule each time.
Duplicate auxiliary verbs in a sentence (may we / should we delete the second?) Height and minimal number of generators of an ideal How can I determine the height and the least number of generators of the ideal $ I=(xz-y^2,x^3-yz,z^2-x^2y) \subset K[x,y,z] $?      I tried to calculate the dimension of the vector space $I/I\mathfrak m$ with $\mathfrak m=(x,y,z)$, but I'm not able to find it.  I think the height is $2$, because the affine variety should be a curve, and also I know that the ideal is prime, so I need a chain $0\subset \mathfrak p \subset I$. Maybe $(xz-y^2)?$ Is it prime? Standard normal distribution is truncated to the left by 2 Would you please show me how to write the pdf distributions of the following   Standard normal distribution is truncated to the left by 2  Exponential distribution is truncated to the left by 2  Cauchy distribution is truncated to the left by 2  i do not know what does mean by truncated to the left by 2 Time Dilation inside a hollow shell Assuming I have a hollow shell with total mass $M$ and radius $r$. On the surface, the gravitational time dilation would be  $$\tau=t \cdot \sqrt{1-\frac{v_{esc}^2}{c^2}}$$  where  $$v_{esc} = \sqrt{\frac{2 \cdot G \cdot M}{r}}$$  but inside the shell there would be no gravitational field (Newton's shell theorem and Birkhoff's theorem).  But still, the escape velocity required to escape to infinity would be the same as on the surface, since inside the shell you could move without any accelerating or decelerating forces acting on you until you reach the surface, from where you would get pulled backwards.  So is the time dilation inside the hollow shell relative to a field free observer at infinity    zero (I assume it's not) or  the same as on the surface (my best guess), or something completely different?   I found some not really duplicate but related threads on the interior of black holes which did not really focus on the math, but I am more behind the calculations in terms of $M$ and $r$. dpkg error: "trying to overwrite file, which is also in..." For example:  $ sudo apt-get install curl Reading package lists... Done Building dependency tree Reading state information... Done You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies: bsh : Depends: libjline-java but it is not going to be installed groovy : Depends: libjline-java but it is not going to be installed rhino : Depends: libjline-java but it is not going to be installed E: Unmet dependencies.   Try 'apt-get -f install' with no packages (or specify a solution).   I get the same or similar errors when I attempt to install clojure1.3, leiningen, and several other packages.   When I try the suggestion made in the error message, this is what happens:  $ sudo apt-get -f install  Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following packages were automatically installed and are no longer required:   diffstat linux-headers-3.2.0-26-generic linux-headers-3.2.0-26 dh-apparmor dkms html2text libmail-sendmail-perl libsys-hostname-long-perl Use 'apt-get autoremove' to remove them. The following extra packages will be installed:   libjline-java Suggested packages:   libjline-java-doc The following NEW packages will be installed:   libjline-java 0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. 23 not fully installed or removed. Need to get 0 B/72.0 kB of archives. After this operation, 129 kB of additional disk space will be used. Do you want to continue [Y/n]? Y (Reading database ... 226243 files and directories currently installed.) Unpacking libjline-java (from .../libjline-java_1.0-1_all.deb) ... dpkg: error processing /var/cache/apt/archives/libjline-java_1.0-1_all.deb (--unpack):  trying to overwrite '/usr/share/java/jline.jar', which is also in package scala 2.9.2-400 Errors were encountered while processing:  /var/cache/apt/archives/libjline-java_1.0-1_all.deb E: Sub-process /usr/bin/dpkg returned an error code (1)  $ sudo apt-get upgrade Reading package lists... Done Building dependency tree        Reading state information... Done You might want to run 'apt-get -f install' to correct these. The following packages have unmet dependencies:  bsh : Depends: libjline-java but it is not installed  groovy : Depends: libjline-java but it is not installed  rhino : Depends: libjline-java but it is not installed E: Unmet dependencies. Try using -f. Why is it considered a bad practice to omit curly braces? Why does everyone tell me writing code like this is a bad practice?  if (foo)     Bar();  //or  for(int i = 0 i &lt; count; i++)     Bar(i);   My biggest argument for omitting the curly braces is that it can sometimes be twice as many lines with them. For example, here is some code to paint a glow effect for a label in C#.  using (Brush br = new SolidBrush(Color.FromArgb(15, GlowColor))) {     for (int x = 0; x &lt;= GlowAmount; x++)     {         for (int y = 0; y &lt;= GlowAmount; y++)         {             g.DrawString(Text, this.Font, br, new Point(IconOffset + x, y));         }      }  }  //versus using (Brush br = new SolidBrush(Color.FromArgb(15, GlowColor)))     for (int x = 0; x &lt;= GlowAmount; x++)         for (int y = 0; y &lt;= GlowAmount; y++)             g.DrawString(Text, this.Font, br, new Point(IconOffset + x, y));   You can also get the added benefit of chaining usings together without having to indent a million times.  using (Graphics g = Graphics.FromImage(bmp)) {     using (Brush brush = new SolidBrush(backgroundColor))     {         using (Pen pen = new Pen(Color.FromArgb(penColor)))         {             //do lots of work         }     }  } //versus using (Graphics g = Graphics.FromImage(bmp)) using (Brush brush = new SolidBrush(backgroundColor)) using (Pen pen = new Pen(Color.FromArgb(penColor))) {     //do lots of work }     The most common argument for curly braces revolves around maintance programming, and the problems that would ensue by inserting code between the original if statement and its intended result:  if (foo)     Bar();     Biz();     Questions:   Is it wrong to want to use the more compact syntax which the language offers? The people that design these languages are smart, I can't imagine they would put a feature which is always bad to use. Should we or Shouldn't we write code so the lowest common denominator can understand and have no problems working with it?  Is there another argument that I'm missing? I have a Nexus 7 (2013) and a Moto X (2014) running Lollipop. On both of these, if I get a heads-up notification (i.e., those that show up on top of the screen when the device is unlocked) and swipe down on it with 2 fingers, it 'breaks' further heads-up notifications:   I can't tap on them to open the responsible app For, e.g., an incoming phone call with 'answer' and 'dismiss' buttons, tapping the buttons does nothing. I can't answer phone calls anymore!   I can still swipe to the side to dismiss the notifications, and they still disappear on their own after a few seconds. I haven't been able to find a way to get the correct behavior back aside from a restart.  Yeah, the obvious workaround is to not two-finger swipe on them, but it's second nature as that same gesture expands notifications in other contexts.  I'm guessing the problem is generic, but I've triggered it on Hangouts and Phone notifications.  Do others experience the same thing? Any known 'fixes' aside from restarting? This might be a stupid question, but why exactly is work a path function?  In school, my teacher said it’s because it depends on the path. She gave an example of walking up a mountain, and said the work you expend depends on the path you take (straight path up or circling up around the mountain). But isn’t the work expended just equal to $E= mgh?$ I have a huge amount of 2D-coordinates, associated with a value, e.g.:    x   |   y   | value 27.50   52.15   12.51 61.83   13.32   57.56 36.23   21.83   41.73 40.46   85.67   25.20 ...   The data is not tabular and I Want the points between two data-points to be interpolated in some way (which way is not really clear, yet)  I want to preset the data as heatmap like this:   Is there any ready-to-use package for PSTricks or TikZ to do it? Infinite Series Convergence Verification I'm currently tackling the following infinite series in order to determine if it converges or not:  $$\sum_{n=1}^\infty \frac{\sqrt{n+1}-\sqrt{n}}{n}$$  What I've done is rearranged this to make use of the comparison test, shown below:  $$= \frac{\sqrt{n+1}-\sqrt{n}}{n} * (\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n+1}+\sqrt{n}})$$  $$= \frac{(n+1)-(n)}{n(\sqrt{n+1}+\sqrt{n})}$$  $$= \frac{1}{n(\sqrt{n+1}+\sqrt{n})}$$  $$= \frac{1}{n^{3/2}+\sqrt{n+1}} &lt; \frac{1}{\sqrt{n+1}} &lt; \frac{1}{\sqrt{n}} $$  However, $\frac{1}{n}$ is smaller than $\frac{1}{\sqrt{n}}$, therefore:  $$  \frac{1}{n}   &lt; \frac{1}{\sqrt{n}}$$   This implies divergence by the comparison test due to the fact $\frac{1}{n}$ diverges, and if a smaller series diverges any larger series must as well.  Is this right? If I've made any mistakes in any part, please tell me! How do I get 0-padded numbers in {} (brace expansion)? I'm trying to write a simple script to retrieve mem and swap usage from a list of hosts. Currently the only way I've been able to achieve this is to write 3 seperate scripts:  for a in {1..9}; do echo "bvrprdsve00$a; $(ssh -q bvrprdsve00$a "echo \$(free -m|grep Mem|/bin/awk '{print \$4}';free -m|grep Swap|/bin/awk '{print \$4}')")"; done &gt; /tmp/svemem.txt; for a in {10..99}; do echo "bvrprdsve0$a; $(ssh -q bvrprdsve0$a "echo \$(free -m|grep Mem|/bin/awk '{print \$4}';free -m|grep Swap|/bin/awk '{print \$4}')")"; done &gt;&gt; /tmp/svemem.txt; for a in {100..218}; do echo "bvrprdsve$a; $(ssh -q bvrprdsve$a "echo \$(free -m|grep Mem|/bin/awk '{print \$4}';free -m|grep Swap|/bin/awk '{print \$4}')")"; done &gt;&gt; /tmp/svemem.txt   The reason for this is that the hostname always ends in a 3 digit number and these hosts go from 001-218 so I've needed to do a different for loop for each set (001-009, 010-099, 100-218). Is there a way in which I can do this in one script instead of joining 3 together? What do you call somebody who asks a question and somebody who answers a question? The staircase paradox, or why $\pi\ne4$ How can I install the Google Apps Package (Play Store, ...) on my Android device? Can I perform a DNS lookup (hostname to IP address) using client-side Javascript? Is it possible to do a cross page POST to an ASP.NET, from a static HTML form? I've seen a lot of games in the LCS and All-Star, and they sometimes changed bot and top lanes.   When is this a good practice? What do you gain from doing this? Should I do this to avoid counters or if my solo laner has more sustain than theirs? How does the ternary operator work? Please demonstrate how the ternary operator works with a regular if/else block. Example:  Boolean isValueBig = value &gt; 100 ? true : false;     Exact Duplicate: Is infinite scrolling justifiable? Recently I've noticed a new trend in user interfaces. Facebook and Twitter do this, and I've come across it on other random, less famous sites. However, to keep things simple, I'll just use Facebook as an example.  When you first log into Facebook, you are presented with a "News Feed" that has a certain amount of posts from your friends. The feed extends down past the bottom of the viewport.  Then, when you scroll down, to see the feeds "below the fold", at some point as you move downward, it loads in more items. Apparently, this is called "infinite scrolling".  On my computer, this often causes my position in the feed to jump while I'm in the action of scrolling, which is really annoying.  What's particularly egregious about this UI concept is that there is footer information at the bottom of the page. I advertise on Facebook, and use the link at the bottom of the page to get to the advertisement management interface.  Because of this incremental load feature I'm describing, I often scroll down to the bottom to get to the link I want, when I reach the bottom of the page, it will suddenly load in new data, pushing the footer down again, and I have to scroll down again.  This whole thing seems completely pointless to me. What's the upside? A Facebook page is near a megabyte of data, so saving the few extra kilobytes of feed data for when the user scrolls down is just not worth it.  What's wrong with a link that just says "show more"?  Since this additional data retrieval thing seems to happen as soon as I start scrolling even a little anyway, why not just load that data in the first place?  To me this minor annoyance reeks of some engineers concept of a "neat trick" that does not have any actual appreciable upside.  Or am I wrong about that? Is there an advantage that I'm not seeing? Am I the only one who thinks it's stupid? Do some people like it? Does Google use the Meta Description Tag for Description of Page?