I'm trying to find explicit form of volume of "deformed torus":  Suppose we have a curve $\gamma(t)$ in $\mathbb{R}^n$, $t\in[0,1]$. The curve closed and smooth : $\gamma(0)=\gamma(1)$,$\gamma'(0)=\gamma'(1)$. I'm defining "deformed torus" as $T_r=\{x\in\mathbb{R}^n \mid |\gamma(t)-x|\leq r\}$ for some $t$.  I guess that volume must be something like $\approx A r^{n-1}$.   So I understand that I need to integrate volume form. I always do it by pullback volume form in parameter space and integrate then. But here I have difficulties with entering any parametrization. Could you help me here?
Let $\gamma:[0,1] \to \mathbb R^n$ be a smooth closed curve, such that $\gamma(0)=\gamma(1), \gamma'(0)=\gamma'(1), |\gamma'(a)|=1$ and let $B_r=\{x| \exists t, |\gamma(t)-x|&lt;r\}$. How can i show that the volume of $B_r$ is equal to $br^{(n-1)}$ for $r &lt;\varepsilon$ for some $\varepsilon&gt;0$. And how can I find $b$? It is intuitively obvious. But how can one prove it? I have no idea.
What does it mean for two objects to "touch"? If you've ever been annoyingly poked by a geek, you might be familiar with the semi-nerdy obnoxious response of      "I'm not actually touching you! The electrons in the atoms of my   skin are just getting really close to yours!"   Expanding on this a little bit, it seems the obnoxious geek is right. After all, consider Zeno's paradox. Every time you try to touch two objects together, you have to get them halfway there, then quarter-way, etc. In other words, there's always a infinitesimal distance in between the two objects.   Atoms don't "touch" each other; even the protons and neutrons in the nucleus of an atom aren't "touching" each other.  So what does it mean for two objects to touch each other?   Are atoms that join to form a molecule "touching"? I suppose the atoms are touching, because their is some overlap, but the subatomic particles are just whizzing around avoiding each other. If this is the case, should "touching" just be defined relative to some context? I.e, if I touch your hand, our hands are touching, but unless you pick up some of my DNA, the molecules in our hands aren't touching? And since the molecules aren't changing, the atoms aren't touching either?  Is there really no such thing as "touching"? Can HTML checkboxes be set to readonly? I thought they could be, but as I'm not putting my money where my mouth was (so to speak) setting the readonly attribute doesn't actually seem to do anything.  I'd rather not use Disabled, since I want the checked check boxes to be submitted with the rest of the form, I just don't want the client to be able to change them under certain circumstances. How to get PHP 7.0 running with Apache2 I am totally new to Linux, servers and websites. I recently set up Apache2, Python, PHP, and MySQL for setting up a LAMP server. I am using Xubuntu 16.04 and PHP 7.0.11. I followed a guide and placed a PHP file named testing.php in it with:  &lt;?php phpinfo(); ?&gt;    It doesn't show anything on localhost/testing.php All it shows is the code itself. What are good methodologies for creating sprocs that reduce the pain of debugging? And what tools are out there for debugging stored procedures?  Perhaps most importantly, what are indications to look out for that errors are happening in a sproc and not in the code? I hope I'm not all over the board too terribly bad here. Votes for answers to any of the above. Thanks.  For what it's worth, I work in a .NET environment, SQL servers. Simplify $\prod_{k=1}^5\tan\frac{k\pi}{11}$ and $\sum_{k=1}^5\tan^2\frac{k\pi}{11}$ My question is:  If $\tan\frac{\pi}{11}\cdot \tan\frac{2\pi}{11}\cdot \tan\frac{3\pi}{11}\cdot \tan\frac{4\pi}{11}\cdot \tan\frac{5\pi}{11} = X$ and $\tan^2\frac{\pi}{11}+\tan^2\frac{2\pi}{11}+\tan^2\frac{3\pi}{11}+\tan^2\frac{4\pi}{11}+\tan^2\frac{5\pi}{11}=Y$ then find $5X^2-Y$.  I couldn't find any way to simplify it. Please help. Thanks. Is "Is it a girl or a boy?" really calling the infant an "it"? So, my boss comes in, railing that "English is a stupid language!" Since this is pretty much a thrice-weekly occurrence 'round these parts, I barely raised an eyebrow, and waited for him to continue.   "Mary just wrote to tell us that she's back from maternity leave, and I want to congratulate her and ask whether she had a girl or a boy, but I can't do it without calling the child an 'it'!" I blinked, then confirmed that yes, he believes the it in "Is it a boy or a girl" is the impersonal pronoun, the same word you'd apply to an apple or a house.  Is it the 'thing' pronoun, really? Or is it just a placeholder of some sort? I used to believe the latter: I gave the boss a mini-lecture about  and the . He wasn't convinced, however, and now he's got me doubting too. (Harumph. I really should know better than to listen to the boss.)  I'm not asking about politeness, here; the former title was to be taken somewhat facetiously. I'm wondering about the grammar: what role is that "it" playing in that sentence? Is it a personal pronoun (and thus the infant has grounds for feeling offended) or a dummy pronoun (and thus those who perceive a politeness issue are just misunderstanding the grammar)? Render looks darker than Look Dev mode I have been texture painting in look dev mode. But when i try to render it, everything looks very dark. I have 3 sun lights in my scene, ambient turned on in Eevee settings. What am i missing? I really like the look of my scene in Look Dev mode, how can i achieve the same look when my scene is rendered too? The choice between the gerund and the infinitive in a certain construction I am pretty much sure that for native speakers the issue I am going to bring up might look as an uncalled question as they can easily figure out which form of a verbal part of speech should be used, be it a gerund, an infinitive, or a bare infinitive. However, it can be pretty much misleading for a foreigner because the same construction works differently with different words and I don’t see a logical explanation. For example:  The purpose of this article is to analyze the issue…  My hobby is reading. One may assume, and many foreigners really come to this conclusion, that it’s OK to switch the “gerund” and “infinitive” as the meaning will stay the same, that is:  The purpose of this article is analyzing the issue… (same as 1 above)  My hobby is to read. (same as 2 above)   However, as I was told, this assumption is erroneous. Another example: His task was watching after them. (not good) His task was to watch after them. Is there any principle which governs the choice of the right form in such cases? Do I have to re-enter the Schengen area to activate my second visa when traveling on back-to-back Schengen visas? I currently have a valid multiple entry Schengen visa that expires on Feb 10th (issued by the Spanish consulate). I need to visit the Czech Republic from Feb 9th to Feb 15th. For this, I am getting a visa starting Feb 11th from the Czech consulate as I was advised that I cannot have two simultaneous visas.   Is it possible for me to enter Czech republic on the current visa on Feb 9th, and stay past its expiry up to Feb 15th using the new visa without leaving the Czech Republic? Will there be any likely problem either while entering or exiting Schengen zone (I am entering Schengen in Frankfurt)? How can I set the value of auto property backing fields in a struct constructor? Given a struct like this:  public struct SomeStruct {     public SomeStruct(String stringProperty, Int32 intProperty)     {         this.StringProperty = stringProperty;         this.IntProperty = intProperty;     }      public String StringProperty { get; set; }     public Int32 IntProperty { get; set; } }   Of course, a compiler error is generated that reads The 'this' object cannot be used before all of its fields are assigned to.  Is there a way to assign values to the backing fields or the properties themselves, or do I have to implement properties the old-fashioned way with my own explicit backing fields? I have an older Nvidia video card and my PC crashes when installing the driver for the video card in Ubuntu 16.04. When I get to the login screen but no further: after entering my password, the system went into a loop and returned always back to the login screen [with funny stripes present].   How do I fix this problem? A group such that $a^m b^m = b^m a^m$ and $a^n b^n = b^n a^n$ ($m$, $n$ coprime) is abelian? Let $(G,.)$ be a group and $m,n \in\mathbb Z$ such that $\gcd(m,n)=1$. Assume that $$ \forall a,b \in G, \,a^mb^m=b^ma^m,$$ $$\forall a,b \in G, \, a^nb^n=b^na^n.$$  Then how prove $G$ is an abelian group?  Some context: Some of these commutation relations often imply that $G$ is abelian, for example , or . This looks like another example of this phenomenon, but the same techniques do not apply. How do you make HDDs and partitions mount at startup on Ubuntu?  I always keep my media and data files on separate partitions - one 2tb HDD and a 400 odd gig partition.  I have been trying to swap the default folder location for the home directory to my other drives.  It works when I change the /home/user_name/.config/user-dirs.dirs file to the directories I want, but only until I reboot the system. I am only assuming that the drives not mounting is the problem with it, but it would be helpful either way.  If any one knows another reason for why the directories change back to default each time that would be good also. Is there a way to disable access to certain Settings components? How can I programmatically pause an NSTimer? I'm using an NSTimer to do some rendering in an OpenGL based iPhone app.  I have a modal dialog box that pops up and requests user input.  While the user is providing input I'd like to "pause" i.e. something like this:  [myNSTimer pause];   I'm using this syntax because I've been doing things like:  [myNSTimer invalidate];   when I want it to stop.  How can I programmatically pause the NSTimer? Correct the "about" page to reference the new "on hold" policy which of the following is NOT a possible value of $(e^{f})''(0)$?? Let $f$ be an analytic function on $\bar{𝐷} = \{z \in \mathbb{C}: |z| \le 1\}$. Assume that $|𝑓(𝑧)| ≤ 1$ for each $z\in \bar{D}$. Then, which of the following is NOT a possible value of $(e^{f})''(0)$??  $(A) 2$  $(B) 6$  $(C) \frac{7}{9}e^{\frac{1}{9}}$  $(D) √2 + 𝑖√2$  So I take a look at Cauchy Integral Formula and then $$(e^{f})''(0)=\frac{1}{\pi}\int_{|z|=1}\frac{e^{f(z)}}{z^3}$$  Taking modulus on both sides we have $$|(e^{f})''(0)| \le \frac{e}{\pi}2\pi=2e$$  Hence (b) cannot be the choice.   Is this alright!!  Thanks for the help! I tried to export my Geogebra graphs into tikz-code. In General it works fine, but plot x^1/3 doesn't work:  \documentclass[10pt]{article} \usepackage{pgf,tikz} \usetikzlibrary{arrows} \pagestyle{empty} \begin{document} \definecolor{ccqqqq}{rgb}{0.8,0,0} \definecolor{qqttcc}{rgb}{0,0.2,0.8} \definecolor{cqcqcq}{rgb}{0.75,0.75,0.75} \begin{tikzpicture}[line cap=round,line join=round,&gt;=triangle 45,x=1.0cm,y=1.0cm] \draw [color=cqcqcq,dash pattern=on 1pt off 1pt, xstep=1.0cm,ystep=1.0cm] (-4,-4) grid (4,4); \draw[-&gt;,color=black] (-4,0) -- (4,0); \foreach \x in {-4,-3,-2,-1,1,2,3} \draw[shift={(\x,0)},color=black] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $\x$}; \draw[-&gt;,color=black] (0,-4) -- (0,4); \foreach \y in {-4,-3,-2,-1,1,2,3} \draw[shift={(0,\y)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize $\y$}; \draw[color=black] (0pt,-10pt) node[right] {\footnotesize $0$}; \clip(-4,-4) rectangle (4,4); \draw[color=qqttcc, smooth,samples=100,domain=-4.0:4.0] plot(\x,{(\x)*(\x)*(\x)}); \draw[color=ccqqqq, smooth,samples=100,domain=-4.0:4.0] plot(\x,{((\x))^(1/(3))}); \draw [color=qqttcc](2.06,4.18) node[anchor=north west] {$f(x)=x^3$}; \draw [color=ccqqqq](2.06,1.18) node[anchor=north west] {$f(x)=\sqrt[3]{x}$}; \begin{scriptsize} \draw[color=qqttcc] (-1.74,-7.16) node {$f$}; \end{scriptsize} \end{tikzpicture} \end{document}      Error line 21: !Missing number, treated as zero.   So plot has problems with the negative x I suppose, but I don't know. Any help would be great! I googled it for nearly 1 hour - but no real answers.  As suggestet it rotated and mirrored the graph - works great!:  \documentclass[10pt]{article} \usepackage[utf8]{inputenc} \usepackage{pgf,tikz} \usetikzlibrary{arrows} \pagestyle{empty} \begin{document} \definecolor{ccqqqq}{rgb}{0.8,0,0} \definecolor{qqttcc}{rgb}{0,0.2,0.8} \definecolor{cqcqcq}{rgb}{0.75,0.75,0.75} \begin{tikzpicture}[line cap=round,line join=round,&gt;=triangle 45,x=1.0cm,y=1.0cm] \draw [color=cqcqcq,dash pattern=on 1pt off 1pt, xstep=1.0cm,ystep=1.0cm] (-4,-4) grid (4,4); \draw[-&gt;,color=black] (-4,0) -- (4,0); \foreach \x in {-4,-3,-2,-1,1,2,3} \draw[shift={(\x,0)},color=black] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize \x}; \draw[-&gt;,color=black] (0,-4) -- (0,4); \foreach \y in {-4,-3,-2,-1,1,2,3} \draw[shift={(0,\y)},color=black] (2pt,0pt) -- (-2pt,0pt) node[left] {\footnotesize \y}; \draw[color=black] (0pt,-10pt) node[right] {\footnotesize 0}; \clip(-4,-4) rectangle (4,4); \draw[color=qqttcc, smooth,samples=100,domain=-4.0:4.0] plot(\x,{(\x)*(\x)*(\x)}); \begin{scope}[yscale=-1,xscale=1] \draw[rotate=90, color=ccqqqq, smooth,samples=100,domain=-4.0:4.0] plot(\x,{(\x)*(\x)*(\x)}); \end{scope} \draw [color=qqttcc](2.06,4.00) node[anchor=north west] {$f(x)=x^3$}; \draw [color=ccqqqq](2.06,1.00) node[anchor=north west] {$f(x)=\sqrt[3]{x}$}; \begin{scriptsize} \draw[color=qqttcc] (-1.74,-7.16) node {f}; \end{scriptsize} \end{tikzpicture} \end{document} I've been told that when "the" is proceeded by a vowel sound, like "apple" or "hour", it's pronounced as "thee" and not as "thu".  But after listening to a couple of songs, I noticed that sometimes this "rule" is not followed. Take for example the two Katy Perry's songs, "Roar" and "The one that got away". In the first she sings "I got thee eye of thu tiger", but in the second she sings "Thu one that got away".  I don't know if it was sang this way to better suit the song melody (I understand nothing about those techniques), but I got confused. What's the correct pronunciation?  Thanks in advance. how to use these hash functions in python? In this article , specifically in the section 4.1 of the initialization phase,the following cryptographic hash functions have been chosen: \begin{align} h_1&amp;:   \mathbb G\to\mathbb Z_q^*\\ h_2&amp;:  \{0,1\}^*\times\{0,1\}^*\times\mathbb G\to\mathbb Z_q^*\\ h_3&amp;:  \{0,1\}^*\to\mathbb Z_q^* \end{align} where G is an additive group with order q. Can anyone explain to me what's the meaning of their notations and how to use them for example in SHA algortihm? Or maybe give me a simple example with real inputs?