Show that if $m &lt;n$ then $\mathbb{R}^m$ seen as subset of $\mathbb{R}^n$ has zero measure.  A set $X \subset \mathbb{R}^m$ have zero measure if for all $\epsilon &gt; 0$ it is possible to obtain a sequence of open cubes $m$-dimensional $C_1 , C_2 , ..., C_i , ...$ such that $X \subset \bigcup_{i=1}^{\infty} C_i$ and $\sum_{i=1}^{\infty} vol(C_i) &lt; \epsilon$  I have tried to use the following theorem:     If $m&lt;n$ and  $f:U \rightarrow \mathbb{R}^n$ of class $C^1$ in a open set $U \subset \mathbb{R}^m$ then $f(U)$ has null measure in $\mathbb{R}^n$.   I use the function $f: \mathbb{R}^m \rightarrow \mathbb{R}^n$, $f(x_1,...,x_m)=(x_1,...,x_m,0,...0)$.  My doubt is if this function satisfies the hypotheses of the theorem and the question.
I would appreciate it if someone could refer me to a proof (or simply give one here) for the statement in the title. That is:  If $k&lt;n$, then every $k-$dimensional subspace of $\mathbb{R}^{n}$ has $n-$dimensional Lebesgue measure zero.  I've seen some proofs that use Sard's lemma but I'm not really familiar with that subject and I've never seen a proof of said lemma so I'd appreciate a proof that doesn't use it if possible.  Thanks in advance!
Test the convergence of the series For $a&gt;0$, test the convergence of the infinite series: $$\sum a^{\ln(n)}$$ Which test should be better to use here? I tried the D'Alembert's ratio test. $$\lim_{n\to \infty} {\frac{a^{\ln(n)}}{a^{\ln(n+1)}}}= \lim_{n\to \infty}a^{ln{\frac{n}{n+1}}}=a^0=1$$ So the test failed.  UPDATE: I observed that $${a^{\ln(n)}}=e^{{\ln(a)}{\ln(n)}}={n^{\ln(a)}}={{1\over n^{ln({1\over a})}} }$$Further by test of auxiliary series $$\sum \frac{1}{n^p}$$ we know that criterion for convergence is $p&gt;1$. Here $p=ln({1\over a})$.Hence the series will be convergent when $ln(a)&lt;-1$ i.e. when $0&lt;a&lt;1/e$, since $a&gt;0$ is already given.Finally this solved my question. Why do we multiply the magnetic flux by the winding number? Difference between deportation and removal As I understand it, there are two mechanisms for throwing somebody out of the UK: deportation and removal. Informally, people seem to call both of these "deportation".  What are the differences between them? Who do they apply to? Do they have different effects on the person's ability to return to the UK later? How does Cypher exit/enter the Matrix without an operator? How can I override the OnBeforeUnload dialog and replace it with my own? What is the difference between a predicate and function? I need to to understand the difference between predicates and functions in the context of Clasual Form Logic in order to define the Herbrand universe.  If I have p(x) :- q(f(x)) would I be right in saying that p and q are predicates while f is a function because it is "nested"?  By this thinking then if I have p(x) :- q(x) both p and q are predicates and I have no functions?  If this is incorrect then how can I tell the difference between a predicate and function? How do you tell if caps lock is on using JavaScript? How do you tell if caps lock is on using JavaScript?  One caveat though: I did google it and the best solution I could find was to attach an onkeypress event to every input, then check each time if the letter pressed was uppercase, and if it was, then check if shift was also held down. If it wasn't, therefore caps lock must be on. This feels really dirty and just... wasteful - surely there's a better way than this? I'm trying to include a file with a specific shape (cup) into another TikZ image, in order to use the cup as a "stamp" to draw more complex figure. The file is located in /usr/share/tikzit/shapes/cap.tikz  \documentclass{article} \usepackage[svgnames]{xcolor} \usepackage{tikz} \usetikzlibrary{decorations.markings} \usetikzlibrary{shapes.geometric} \pagestyle{empty}  \pgfdeclarelayer{edgelayer} \pgfdeclarelayer{nodelayer} \pgfsetlayers{edgelayer,nodelayer,main}  \tikzstyle{none}=[inner sep=0pt] \tikzstyle{node1}=[cap,fill=White,draw=Black]  \tikzstyle{simple}=[-,draw=Black,line width=2.000] \tikzstyle{arrow}=[-,draw=Black,postaction={decorate},decoration={markings,mark=at position .5 with {\arrow{&gt;}}},line width=2.000] \tikzstyle{tick}=[-,draw=Black,postaction={decorate},decoration={markings,mark=at position .5 with {\draw (0,-0.1) -- (0,0.1);}},line width=2.000]  \usepackage[graphics,tightpage,active]{preview} \PreviewEnvironment{tikzpicture} \newlength{\imagewidth} \newlength{\imagescale}  \begin{document} \begin{tikzpicture}     \begin{pgfonlayer}{nodelayer}         \node [style=node1] (0) at (-2.5, 1.5) {};     \end{pgfonlayer} \end{tikzpicture}  \end{document}   The code i want to include is the following:  \begin{tikzpicture}         \path [use as bounding box] (-2,-2) rectangle (2,2);         \begin{pgfonlayer}{nodelayer}                 \node [style=none] (0) at (0, 0.75) {};                 \node [style=none] (1) at (-0.5, 0) {};                 \node [style=none] (2) at (0.5, 0) {};         \end{pgfonlayer}         \begin{pgfonlayer}{edgelayer}                 \draw [in=0, out=90] (2.center) to (0.center);                 \draw [in=270, out=-90, looseness=0.75] (1.center) to (2.center);                 \draw [in=90, out=180] (0.center) to (1.center);         \end{pgfonlayer} \end{tikzpicture}   But it seems I miss how to specify the directory of cup. I get this error:     Package pgfkeys Error: The key '/tikz/cap' requires a value. I am   going to ignore this key.   It seems like I have to deal with tikzstyle or pgfkey, but I don't know how to specify an external file. How long does it take for badges to be awarded? How are they generated? How to combine multiple conditions to subset a data-frame using "OR"? Login screen resolution So I just installed Ubuntu 16.04 on my Dell XPS 13. When I get to the login screen it starts at 3200x1800 for a quick second, then I get a way lower resolution like 800x600 but as soon as I login it adjusts to 3200x1800. How would I get 3200x1800 permanently on my login screen? I didn't have this issue on 14.04. Elmah: How to get JSON HTTP request body from error report As it was suggested  I am using cat command to concatenate several .mp3 files into one .mp3 file. Imagine, I have following .mp3 files in the current folder: 001001.mp3 001002.mp3 001003.mp3 001004.mp3 001005.mp3  or, like this: 096001.mp3 096002.mp3 096003.mp3 096004.mp3  I need to concatenate these .mp3 files in there ascending sequence, i.e.  001001.mp3+001002.mp3+001003.mp3+etc. In order to join these .mp3 files into one I am executing following command in the current folder: cat *.mp3 &gt; final.mp3  I tested the final .mp3 file and it is what I am expected, but I need to be sure that above command picks files in there ascending sequence. Can I be sure that above command always concatenates files in the ascending sequence? Thank you Sir! Use the view from other cameras in the scene as VideoTexture in Cycles I have a scene with monitors that display video from cameras in the same scene (or other scenes, but all of them are linked i.e., same objects in the scene). The cameras shown on the monitors do not view the monitors, so no looping should occur.  I found  that shows how to get the effect in the Game Engine.  I would like to do this in cycles.  Essentially I would like a material input node that corresponds to the output of the composition of a particular scene. The incoming image should honor the render settings of the particular scene and any compositing applied to it.  The Render Layer node does not appear to honor the resolution or compositing settings of the scene it comes from, and is not available to apply as a texture in the material node editor (that I can find).  I understand that this could be done by rendering the animation from the various cameras, then applying that video as textures to the monitors, then rendering the camera of the monitors. That's super tedious if your making changes to the whole scene, and I don't want to do that. What can I use to typeset MATLAB code in my document? This question led to a new package:      Does anybody have a quick and easy way to typeset some relatively long MATLAB code in the appendix? I looked up a few methods, and the mcode package emerged as a possibility, but it seems to be unmaintained...  I have a ton of equations in my code, so I'd prefer a solution that avoids having to use $ to denote equations.  Also, if possible, if there's a way to keep all of MATLAB's "natural" coding colors (e.g. green for comments), that would be great...  Any idea? Removing and adding permission using numerical notation on the same line What class of Devil Fruit does the Gomu Gomu no Mi come under? Considering the three types of Devil Fruits that has been explained are 1. Paramecia Type - where the user can get super human abilities and change their body parts into weapons, etc. 2. Zoan type - where the use can turn into a particular animal and inherit its abilities 3. Logia Type - where the user's body inherits a particular element and the body becomes like air which cannot be touched or damaged by non Haki users.  So what exactly is the type of Gomu Gomu no Mi? It cannot be Zoan or Logia. It's very close to a Paramecia but still doesn't have a logical explanation to be of Paramecia type! So is it some special type of Devil Fruit? Leave Convert to Edit button enabled all the time in the moderator flag list In the moderator flags list, there is a &quot;Convert to Edit&quot; button on answers which are posted by the person who posted the question. This button is disabled at times, and I am not sure that the algorithm used is useful. My proposal is that the Convert to Edit button in the flag list should always be enabled, as it seems fairly frequent that it's disabled when it would be useful. Justification When the button is disabled, it is required that we pop open the answer view in the flag list, and use the regular Mod menu and choose convert to edit there. When we do that, however, we experience this issue:  (that should also be fixed, by the way!) Here are two items currently in the queue:    Note that one has the button enabled, while the other does not. The only differences I can see in those two answers is the age and the number of answers. Neither question has an accepted answer, but the one with the button disabled only has that single answer. I've only just now noticed that difference with these two posts, so I'm not 100% sure if that's the determining factor. 1 It is perhaps coincidental, but these buttons being (dis)abled are reverse of the action needing to be taken. The one with the disabled button needed to be converted to an edit; the one with the enabled button was an attempt to answer their own question. If I'm right about the determining factor here (that the answer is the only one), I'm not sure that's useful. A good bit of the time, an asker will post an answer due to people requesting additional information. In such cases, it seems like it will be likely there are no other answers. 1  Note 1: I was looking to create a userscript to enable these buttons, and I found something that I think is related as to the reason. It looks like the ones which are disabled are on answers which have comments on them. I noticed the title attribute of those buttons says, &quot;There are comments on the post. Please review&quot;. That does seem to make some sense in theory. However, it still seems like I am often converting them, anyway. With the above-linked, separate issue regarding how the 'mod' link on the expanded view works, this is still a bit of an impediment to our workflow. It occurred to me to check if the button was reenabled once you expanded the view, but it does not. Maybe that's an alternate solution. My plan for my userscript is to enable the button, but change the text color (and leave the title attribute), to point out that there are comments. When rolling percentages, do 1d100 and two d10s (percentiles) share the same probabilities? I know d100s are shunned because they take too much time to roll, and 1d10+1percentile is really fun, but do they share the same probabilities? Is it better to use a (digital) d100?  Also, do d100s have 0? You can't roll a 0 with percentiles, I think. How can I check if a program exists from a Bash script?