Proof of a common expression in Combinatorics.
Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$
Let $U\subset\mathbb R^3$ be an open set, and $f:U\to \mathbb R$ be a smooth function. Suppose that the level set $S=f^{-1}(\{0\})$ is non-empty, and that at each $p\in S,$ the gradient $\overrightarrow \nabla f(p)$ is not the zero vector. Then $S$ is a smooth two-dimensional surface in $U$, and $p\mapsto \overrightarrow \eta(p)=\frac{1}{||\overrightarrow \nabla f(p)||}\overrightarrow \nabla f(p)$ defines a smooth unit-length normal vector field along $S$. At each $x\in U,$ write $H(f)_{(x)}$ for the $3\times 3$ Hessian matrix specified by $$(H(f)_{(x)})_{ij}=\frac{\partial^2f}{\partial x_i\partial x_j}(x).$$   Show that, at each $p\in S$, the second fundamental form $II_p: T_p(s)\times T_p(s)\to \mathbb R$ is the symmetric bilinear map    $$II_p(\overrightarrow v,\overrightarrow w)=\frac{-1}{||\overrightarrow \nabla f(p)||}\overrightarrow v\cdot H(f)_{(p)}\overrightarrow w,$$for all $\overrightarrow v ,\overrightarrow w \in T_p(s)$.       (Here, we view the tagent space $T_p(S)$ as the two-dimensional subspace $(span\{ {\overrightarrow \eta(p)}\})^{\bot}$ of $\mathbb R^3$.      Edit: Actually my question is why the second fundamental form under the usual definition can be written in this way.      Definition: The quadratic form $II_p$, defined in $T_p(S)$ by $II_p(v)=-&lt;d  N_p(v),v&gt;$ is called the second fundamental form of $S$ at $p$, where $dN_p:T_p(S)\to T_p(S)$ is the differential of the Gauss map.   Hopefully, I express this problem explicitly. I was just wondering how to prove this statement.   I took a diffrential geometry class last semester, and when I organized my notes this morning, I found this statement, but there was no proof...  Looking forward to an understandable explaination. Thanks in advance.     Edit 2:Furthermore, show that, at each point $p\in S$, the expression   $$\phi_p(z)=det\pmatrix{-H(f)_{(p)}-zI_{3\times 3} &amp; \overrightarrow \nabla f(p)\\\ \pm \overrightarrow \nabla f(p)&amp; 0}$$   (the underlying matrix here is $4\times 4$) defines a second-degree polynomial whose roots $\lambda_1$ and $\lambda_2$ are $||\overrightarrow \nabla f(p)||k_1$ and $||\overrightarrow \nabla f(p)||k_2$, where $k_1$ and $k_2$ are the principal curvatures of $S$ at $p$.      Also, if a non-zero vector $\pmatrix {\overrightarrow v \\c}$ lies in the kernel of the $4\times 4$ matrix $$\pmatrix{-H(f)_{(p)}-\lambda_jI_{3\times 3} &amp; \overrightarrow \nabla f(p)\\\ \pm \overrightarrow \nabla f(p)&amp; 0},$$   then $\vec v$ is a non-zero element of $T_p(S)$ and lies in the "principal direction" corresponding to $K_j$. Xcode - How to fix 'NSUnknownKeyException', reason: … this class is not key value coding-compliant for the key X" error? Caption on tabular environment I have got codes like the following (generated by Kile). How can I add a caption. I guess I should have used the table environment. Is there any solution now?  \newcommand{\mc}[3]{\multicolumn{#1}{#2}{#3}} \begin{center} \begin{tabular}{rccll}      &amp;   &amp; \mc{3}{c}{Colin}\\      &amp;   &amp; a &amp; \mc{1}{c}{b} &amp; \mc{1}{c}{c}\\\cline{3-5} Rose &amp; \mc{1}{c|}{A} &amp; \mc{1}{c|}{(1,2)} &amp; \mc{1}{c|}{(2,5)} &amp; \mc{1}{c|}{(4,4)}\\\cline{3-5}      &amp; \mc{1}{c|}{B} &amp; \mc{1}{c|}{(7,4)} &amp; \mc{1}{c|}{(3,5)} &amp; \mc{1}{c|}{(0,6)}\\\cline{3-5} \end{tabular} \end{center} Why is $5^{n+1}+2\cdot 3^n+1$ a multiple of $8$ for every natural number $n$? I have to show by induction that this function is a multiple of 8. I have tried everything but I can only show that is multiple of 4, some hints? The function is $$5^{n+1}+2\cdot 3^n+1 \hspace{1cm}\forall n\ge 0$$, because it is a multiple of 8, you can say that$$5^{n+1}+2\cdot 3^n+1=8\cdot m \hspace{1cm}\forall m\in\mathbb{N}$$. When using vagrant Stderr: VBoxManage: error: VT-x is not available (VERR_VMX_NO_VMX) I am trying to install open edx with  ubunty 16.04 and I get this error  There was an error while executing `VBoxManage`, a CLI used by Vagrant for controlling VirtualBox. The command and stderr is shown below.  Command: ["startvm", "c60f0594-d584-4e0b-b863-2801168e641e", "--type", "headless"]  Stderr: VBoxManage: error: VT-x is not available (VERR_VMX_NO_VMX) VBoxManage: error: Details: code NS_ERROR_FAILURE (0x80004005), component ConsoleWrap, interface IConsole   this occurs when using vagrant up "System" storage on macOS Sierra is 470GB! I was given the files for a mini linux, that boots directly into firefox. It works for all it should be doing, only that I do not get an internet connection.  We have 3 DNS servers in the network, which all work. I can ping them, too. But when trying to ping google.de or wget google.de I get a bad address error.  nslookup google.de works for some reason.  I tracked the issue down to my resolv.conf on the booted system not having the same contents as the resolv.conf that I put into the .iso file.  I tried understanding all the factors that go into creating and modifying resolv.conf. I'm not quite sure I got it all, but I definitely didn't find my solution there.  So as a last ditch effort, I tried making the resolv.conf file immutable using   :~# chattr +i /etc/resolv.conf   When rebuilding and booting again to my surprise my file was renamed to resolv.conf~ and in its place was the same standard file that has been haunting me.   The file contents make me believe it gets the information from the network itself. When starting the .iso in Virtualbox without internet access, my file is being kept as it is.  I tried changing /etc/dhcp/dhclient.conf to not get the information from the net, by deleting domain-name-server and domain-name-search from the request part of the file.  Didn't work unfortunately.  I don't have the NetworkManager installed. The iso is based on Ubuntu 14.04.  There is probably vital information missing. I'm happy to provide it.  UPDATE:  I think I found the file that clears resolv.conf.  It seems to be /usr/share/udhcpc/default.script  #!/bin/sh  # udhcpc script edited by Tim Riker &lt;Tim@Rikers.org&gt;  [ -z "$1" ] &amp;&amp; echo "Error: should be called from udhcpc" &amp;&amp; exit 1  RESOLV_CONF="/etc/resolv.conf" [ - n "$broadcast" ] &amp;&amp; BROADCAST="broadcast $broadcast" [ -n "$subnet" ] &amp;&amp; NETMASK="netmask $subnet"  case "$1" in     deconfig)         /bin/ifconfig $interface 0.0.0.0         for i in /etc/ipdown.d/*; do             [ -e $i ] &amp;&amp; . $i $interface         done         ;;      renew|bound)         /bin/ifconfig $interface $ip $BROADCAST $NETMASK          if [ -n "$router" ] ; then             echo "deleting routers"             while route del default gw 0.0.0.0 dev $interface ; do                 :             done              metric=0             for i in $router ; do                 route add default gw $i dev $interface metric $((metric++))             done         fi          echo -n &gt; $RESOLV_CONF # Start ----------------           [ -n "$domain" ] &amp;&amp; echo search $domain &gt;&gt; $RESOLV_CONF         for i in $dns ; do             echo adding dns $i             echo nameserver $i &gt;&gt; $RESOLV_CONF         done         for i in /etc/ipup.d/*; do             [ -e $i ] &amp;&amp; . $i $interface $ip $dns          done # End ------------------         ;; esac  exit 0   It's part of the udhcpc program. A tiny dhcp client, that is part of busybox  Will investigate further.    UPDATE2 AND SOLUTION:  I commented the part out (#Start to #End), that seemingly overwrites the /etc/resolv.conf file and sure enough. That was the culprit. So an obscure script caused all this trouble.  I changed the question to reflect, what actually needed to be known to solve my problem, so it would be easier to find for people with the same problem and so I could accept an answer.  Thanks for the help here in figuring things out. $127$ has an interesting property: It is the $31$st prime number and its rank ($31$) is also a prime. $31$ is the $11$th prime so its rank is also a prime. $11$ is also a prime number with a rank ($5$) that is also a prime. $5$ is the 3rd prime number and and so its rank ($3$) is also a prime. And finally $3$ is the $2$nd prime so its rank is also a prime... Is there a name for primes whose rank (index in the prime series) is also a prime? That is:     If $a_i$ is the $i$th prime number, then $i$ is a also a prime.   How about numbers like $127$ where going down ranks of ranks always produce prime numbers (down to rank $2$ of course)? The first $11$ :-) primes in this (infinite) series would be:  $3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, \dotsc$  Many thanks! Nested regression model I know how to , but I'd like to add a bunch of different extensions and avoid the tedious job of adding every single one. Is there a command line tool or registry key I can use to accomplish this? We can say ‘fried fish’, ‘baked potato’ or ‘minced pork’ using past participles for modifiers.  However, ‘roast’ is different - either ‘roast duck’ or ‘roasted duck’ works, it seems to me.  How should we analyze this?  Is ‘roast’ a noun modifying a noun here, or is it a different form of the past participle?  (According to freedictionary.com, in Middle English the past participle was ‘roste’ – does that mean that this was once a strong verb?) So I wish to create and object and set it's shading to 'smooth' using python. It seems simple enough but after trying every method I could find on the internet none of them worked for me. Here's a short example script where the object needs to have smooth shading:  import bpy from mathutils import Vector  verts = [(0,0,0),(0,5,0),(0.25,5,0),(0.25,0,0),(0,0,5),(0,5,5),(0.25,5,5),(0.25,0,5)] faces = [(0,1,2,3), (4,5,6,7), (0,4,5,1), (1,5,6,2), (2,6,7,3), (3,7,4,0)]  grat1mesh = bpy.data.meshes.new("grat1") grat1obj = bpy.data.objects.new("grat1", grat1mesh) grat1obj.location = (5, -5, 0) bpy.context.scene.objects.link(grat1obj) grat1mesh.from_pydata(verts,[],faces) grat1mesh.update(calc_edges=True)  grat2obj = grat1obj.copy() grat2obj.data = grat1mesh.copy() bpy.context.scene.objects.link(grat2obj) grat2obj.location += Vector((-0.125, 1.25, 1.25)) grat2obj.scale = (2, 0.5, 0.5)  bm = grat1obj.modifiers.new("Chop", 'BOOLEAN') bm.object = grat2obj bm.operation = 'DIFFERENCE' bpy.context.scene.objects.active = grat1obj bpy.ops.object.modifier_apply(modifier=bm.name) bpy.context.scene.objects.unlink(grat2obj)   I found this online and tried to place it at the end of the script:  smoothings = [] smoothings.append(face.use_smooth) me_ob.faces.foreach_set("use_smooth", smoothings)   However this just returned an error on the second line, perhaps as it was from an older website. Obviously using a simple bpy.ops.object.shade_smooth() didn't work and I'm kind of stumped as of what to do here! Also how do LEFT JOIN, RIGHT JOIN and FULL JOIN fit in? Crash or "segmentation fault" when data is copied/scanned/read to an uninitialized pointer MySQL Hashing Function Implementation I know that php has md5(), sha1(), and the hash() functions, but I want to create a hash using the MySQL PASSWORD() function.  So far, the only way I can think of is to just query the server, but I want a function (preferably in php or Perl) that will do the same thing without querying MySQL at all.  For example:  MySQL hash -> 464bb2cb3cf18b66  MySQL5 hash -> *01D01F5CA7CA8BA771E03F4AC55EC73C11EFA229  Thanks! Hydraulic Disk brake piston not fully retracting My front disk brake's piston doesn't fully retract.  While I can ride the bike, the brake is slightly rubbing.  (IE, if I just spin the wheel with my hand, it'll go around about one and a half times and then stop.) It's definitely slowing me down.   If this fixable?  The advice I got was that it's not worth the trouble and to buy a replacement set.  The current brakes are Avid Juicy 7s that are about 7 years old What are irreducibles in the ring $R$ of entire functions? I have had the last Ubuntu distro (10.something) installed for a year and love it.  But I have run in to an issue when I tried to upgrade it to the latest one it did not work and there has come some serious issues now with almost everything and I want to remove Ubuntu completely and install the new one.  How can I remove the operating system (I don’t mind formatting the whole thing if that’s what’s needed). Prove $\int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x = \frac{1}{2}\sqrt{\frac{\pi}{a}}\mathrm{e}^{-2\sqrt{ab}}$ This is Example 8.4.1 from Chapter 8 (The Normal Integral) of  by Boros and Moll. The authors outlined a solution method, which I will provide in full here. My question: is there another way to obtain this result?  Here is the solution of Boros and Moll with their notation.  Let \begin{equation} \mathrm{L}(a,b) := \int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x \label{eq:iic8-1} \tag{1} \end{equation} Making the substitution $t=x\sqrt{a}$ yields \begin{equation} \int\limits_{0}^{\infty} \mathrm{exp}(-ax^{2}-\frac{b}{x^{2}}) \mathrm{d} x = \frac{1}{\sqrt{a}} \int\limits_{0}^{\infty}  \mathrm{exp}(-t^{2}-\frac{ab}{t^{2}}) \mathrm{d} t \label{eq:iic8-2} \tag{2} \end{equation} Letting $ab=c$ we call the integral in equation \eqref{eq:iic8-2} $f(c)$, \begin{equation} f(c) = \int\limits_{0}^{\infty} \mathrm{exp}(-t^{2}-\frac{c}{t^{2}}) \mathrm{d} t \label{eq:iic8-3} \tag{3} \end{equation} so that  \begin{equation} \mathrm{L}(a,b) = \frac{f(ab)}{\sqrt{a}} \label{eq:iic8-4} \tag{4} \end{equation} In equation \eqref{eq:iic8-3} we let $y=\sqrt{c}/t$ \begin{equation} f(c) = \sqrt{c} \int\limits_{0}^{\infty} \mathrm{exp}(-y^{2}-\frac{c}{y^{2}}) y^{-2} \mathrm{d} y \label{eq:iic8-5} \tag{5} \end{equation}  Combining equations \eqref{eq:iic8-3} and \eqref{eq:iic8-5}, we have \begin{equation} f(c) = \frac{1}{2} \int\limits_{0}^{\infty} \mathrm{exp}(-t^{2}-\frac{c}{t^{2}}) \left(1+\frac{\sqrt{c}}{t^{2}} \right) \mathrm{d} t \label{eq:iic8-6} \tag{6} \end{equation} Now we let $s = t - \sqrt{c}/t$ \begin{equation} f(c) = \frac{\mathrm{e}^{- 2\sqrt{c}}}{2} \int\limits_{-\infty}^{\infty} \mathrm{exp}(-s^{2}) \mathrm{d} s = \frac{\sqrt{\pi}}{2} \mathrm{e}^{- 2\sqrt{c}} \lim_{z \to \infty} \mathrm{erf}(z) = \frac{\sqrt{\pi}}{2} \mathrm{e}^{- 2\sqrt{c}} \label{eq:iic8-7} \tag{7} \end{equation} Combining equations \eqref{eq:iic8-1}, \eqref{eq:iic8-4}, and \eqref{eq:iic8-7} yields our result. Why is the picture not inserted? When I pdflatex the text below it doesn't insert the picture, but writes the name of the file instead (pic).  And the geometry line gives the error  ! LaTeX Error: Missing \begin{document}.   The picture I am using is at  and the floatrow package .  Question  Can anyone see what is wrong?  Text  \documentclass[a4paper]{article} \pdfpagewidth=\paperwidth \pdfpageheight=\paperheight \usepackage{aeguill} \pdfcompresslevel=9 \usepackage[pdftex,colorlinks]{hyperref}  \usepackage{a4} \usepackage{fixltx2e} \usepackage[danish]{babel} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc}  \usepackage[draft]{graphicx} \usepackage{floatrow}  \usepackage{geometry}  \geometry{margin=10pt,paperwidth=\textwidth+20pt,paperheight=\textheight+20pt}  \begin{document}  The first paragraph.  \begin{figure}   \includegraphics{pic} \end{figure}  And the second.  \end{document}