Is it possible to find a subset $A$ of the real line $\mathbb R$ such that the Lebesgue measure of  $A$ minus its interior is positive ?
When studying weak border conditions (in Sobolev Spaces), the usual motivation for the weak meaning of inequalities is that the boundary of most open sets in $\mathbb{R}^n$ has zero Lebesgue measure. But is there any open set $U\subseteq\mathbb{R}^n$ such that it's boundary has positive Lebesgue measure? I really can't think of anything like that.
I was trying to give the wording "isomorphic groups have the same structure" a precise interpretation, at least for the finite order case. A positive answer to my final question would reach the goal, as it would translate a "$\cong$" between groups into a "$=$" between sets.    For $n$ positive integer, let be:   $I_n:=\{1,\dots,n\}$; $G$, $\overline G$ groups of order $n$; $\psi\colon G \rightarrow \overline G$ isomorphism; $f$, $\bar f$ bijections; $\theta$, $\bar \theta$ embeddings; in general, $\varphi^{(\alpha)}$ the isomorphism between symmetric groups on sets of the same cardinality, defined by $\sigma \mapsto (g \mapsto (\alpha\sigma\alpha^{-1})(g))$, where $\alpha$ is a bijection between the sets; $S_n$ the symmetric group of degree $n$.   Visually:  $$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c}    I_n     &amp;   &amp;    I_n      \\     \da{f}         &amp;           &amp;     \da{\bar f}          \\    G &amp; \ras{\psi} &amp;    \overline G  \\     \da{\theta}         &amp;           &amp;     \da{\bar \theta}          \\    \operatorname{Sym}(G) &amp; \ras{\varphi^{(\psi)}} &amp;    \operatorname{Sym}(\overline G)  \\     \da{\varphi^{(f)}}         &amp;           &amp;     \da{\varphi^{(\bar f)}}          \\    S_n &amp;  &amp;    S_n  \\ \end{array} $$     Is it $(\varphi^{(f)}\circ\theta)(G)=(\varphi^{(\bar f)}\circ\bar \theta)(\overline G)$? Do particles in a sound wave ever move transversally? I'm trying to visualize how sound waves work and I was curious about something.   So sound moves in longitudinal waves, which I think I understand.  There is a really good  explaining this.  In the video the air particles are depicted as moving from left to right (longitudinally). But do they ever move up and down even a little bit? Are they on some kind of fixed axis where they can only move longitudinally and never move even the slightest bit in a vertical direction? Is $\mathbb{Z}[x]/(x^2+1)$ isomorphic to $\mathbb{Z}[i]$? I am making a skydome and I have a texture hooked up with an emission shader for it, however, since the dome is blue it only emits blue light. Is there a way that I can have the whole sky illuminated and have the light reaching the rest of my scene be a different colour? I'm trying to install the drivers to my wireless adapter ():   I have already tried the following:  sudo add-apt-repository ppa:hanipouspilot/rtlwifi sudo apt update sudo apt install rtl8192eu-dkms   But I'm getting the following error:  E: Unable to locate package rtl8192eu-dkms   I have also tried manually downloading and installing it, but I'm getting this:  make make ARCH=x86_64 CROSS_COMPILE= -C /lib/modules/4.6.0-kali1-amd64/build M=/root/Desktop/20140812_rtl8192EU_linux_v4.3.1.1_11320  modules make[1]: *** /lib/modules/4.6.0-kali1-amd64/build: No such file or directory.  Stop. Makefile:1323: recipe for target 'modules' failed make: *** [modules] Error 2   edit: lsusb output  root@kali:~# lsusb Bus 002 Device 002: ID 8087:0024 Intel Corp. Integrated Rate Matching Hub Bus 002 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 001 Device 004: ID 8087:07da Intel Corp.  Bus 001 Device 003: ID 1bcf:2c0e Sunplus Innovation Technology Inc.  Bus 001 Device 002: ID 8087:0024 Intel Corp. Integrated Rate Matching Hub Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 004 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub **Bus 003 Device 002: ID 2357:0107**   Bus 003 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$ Are the foods in the Star Wars Universe the same as ours'? Do they have strawberries or potatoes or something? I know that they might have some food other than the food that we also have, but I'm wondering if they have any food like ours'. What is the difference between local and .local in ubuntu? Echoing path in ubuntu returns following colon separated paths,  /home/stack/bin:/home/stack/.local/bin:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/usr/local/games:/snap/bin   In these paths at some places there is local while at others .local is used what is the difference between both? Projection is an open map Let $X$ and $Y$ be (any) topological spaces. Show that the projection  $\pi_1$ : $X\times Y\to X$   is an open map. It seems like I'm doing something wrong when using foreach variables to indicate node numbers. The two foreach loops are anchoring the paths to the east of each node. Why does this happen?    \documentclass{article}  \usepackage{tikz} \usetikzlibrary{backgrounds,fit,patterns,decorations.pathreplacing,calc}  \makeatletter \tikzset{   labelnode/.style={draw=none, fill=none, black},   latentnode/.style={draw, minimum width=5mm, shape=circle, ultra thick, black},   dagconn/.style={arrows=-&gt;, black, thick}, } \makeatother  \begin{document} \begin{tikzpicture}[scale=1]  \pgfmathsetmacro{\colgap}{1.6}  \pgfmathsetmacro{\rowgap}{1.3}   \node[labelnode] (x1) at (\colgap*0,\rowgap*1) {$X_1$};  \node[labelnode] (x2) at (\colgap*1,\rowgap*1) {$X_2$};  \node[labelnode] (x3) at (\colgap*2,\rowgap*1) {$X_3$};   \node[latentnode,blue] (x4) at (\colgap*0,\rowgap*0) {$X_4$};  \node[latentnode,red] (x5) at (\colgap*1,\rowgap*0) {$X_5$};  \node[labelnode] (x6) at (\colgap*2,\rowgap*0) {$X_6$};   \foreach \c in {1,...,2} {    \foreach \r in {1,...,2} {      \pgfmathsetmacro{\st}{(((\r-1)*3)+\c)}      \pgfmathsetmacro{\en}{\st+1}      \draw[dagconn] ([xshift=0.2cm]x\st) to ([xshift=-0.9cm]x\en);    }  }   \foreach \c in {1,...,3} {    \foreach \r in {1,...,1} {      \pgfmathsetmacro{\st}{((\r-1)*3)+\c}      \pgfmathsetmacro{\en}{\st+3}      \draw[dagconn] (x\st) to (x\en);    }  } \end{tikzpicture} \end{document} understanding TTNavigator There are two checkboxes for Run As Administrator in the Properties dialog for a shortcut in Windows 7. One is under the Shortcut tab in the Advanced Properties dialog, and the other is under the Compatibility tab.  I've seen this under Windows 7 and Windows Server 2008. It might be in Vista and other versions, but I'm not sure.  What is the difference between these two settings? Direction of $d\mathbf{l}$ A solid sphere has charge $q$ and radius $R$. Find the potential at a point a distance $r$ from the center of the sphere where $r&gt;R$, using infinity as the reference point.  My attempt:  From Gauss' theorem we may deduce that $\displaystyle\mathbf{E}=\frac{1}{4\pi\epsilon_0}\frac{q}{r'^2}\hat{\mathbf{r}}$ where $r'$ is the distance of an arbitrary point from the center of the sphere provided $r'&gt;R$.  $V=-\int_\infty^\mathbf{r}\mathbf{E}\cdot d\mathbf{l}$  My question: What is $d\mathbf{l}$? Since we are traversing in the direction opposite to $\hat{\mathbf{r}}$, I think it should be $-dr'\hat{\mathbf{r}}$. But when I use it to find $V$, I get a sign error. Please help! Why is propagation of uncertainties quadratic rather than linear? How do I distinguish between a command, utility, or builtin for getting documentation? How to capture all std outputs How to prevent accidental clicks in the Save button when saving a node? I need to prevent accidental clicks in the Save button when a node is being saved.  I found this   which does exactly this. But the problem is that it uses javascript (jquery). Then if the user disable javascript this module is useless.  My problem is that in my website the user get points when creating nodes, so I have to prevent this in a more secure way.  How can I do this?   What do you think of these ideas?:  1) Maybe I would forbid the node being saved more than one time    programmatically instead of disabling the Save button.   2) Or better: I would allow the user to just save a new node every 5 minutes. Smart Phone's camera socket I just want ask how the socket in the image is called?   I couldn't really find and the closer I could get is only Raspberry's module. Introductory Calculus textbook for physics? I have the summer to prepare myself for a high school calculus-based physics course; however, I will have to study calculus by myself. I've completed a course in precalculus and I'm beginning to read .   I am looking for a textbook that has mathematical rigor (i.e. not memorizing formulas), yet also provides adequate foundations for a physics course. From similar stack exchange questions, my top candidates are  and . Which one of these textbooks, if any, would be the best choice for my situation? Help with table multicolumn