definite integral $\int_{0}^{1}\sqrt[3]{x\log\frac 1x} $
Computing: $\int_0^1 \sqrt[3]{x\log{\frac{1}x}} \, \mathrm dx$
Prove that a closed ball in a metric space is a closed set   My attempt: Suppose $D(x_0, r)$ is a closed ball. We show that $X \setminus D $ is open. In other words, we need to find an open ball contained in $X \setminus D$.   Pick $$t \in X-D \implies d(t,x_0) &gt; r \implies d(t,x_0) - r &gt; 0 $$ Let $B(y, r_1)$ be an open ball, and pick $z \in B(y,r_1)$. Then, we must have $d(y,z) &lt; r_1 $. We need to choose $r_1$ so that $d(z,x_0) &gt; r$. Notice by the triangle inequality  $$ d(x_0,t) \leq d(x_0,z) + d(z,t) \implies d(z,x_0) \geq d(x_0,t) - d(z,t) &gt; d(x_0,t) - r_1.$$  Notice, if we pick $r_1 = d(t,x_0)-r$ then we are done.  Is this correct? Do I have to pay taxes in the US if my online store sells to US customers even though I don't live in the US? I  own a website and I sell digital content. I'm a nonresident alien with the US. Most of my customers are US based so I was wondering will I have to pay any taxes to the US?  I have read that      For nonresidents, taxable income is separated into two categories:         Income that is effectively connected with a U.S. trade or business, after the deductions that are allowed, is subject to U.S. income tax   at graduated rates - the same rates that apply to U.S. citizens and   residents - or lower rates if the provisions of a tax treaty apply.   Income that is subject to U.S. income tax, for being U.S. source income, but that is not effectively connected with a U.S. trade or   business, is subject to a flat 30% tax      Is this true and do I belong to any of these categories?  If I form an LLC or C corp am I liable for this withholding tax?  My customers are based in the US and I believe all income is considered FDAP income according to the IRS:     Fixed, Determinable, Annual, or Periodical (FDAP) income is all   income, except:         Gains derived from the sale of real or personal property (including market discount and option premiums, but not including original issue   discount)   Items of income excluded from gross income, without regard to the U.S. or foreign status of the owner of the income, such as tax-exempt   municipal bond interest and qualified scholarship income      The thing is the IRS also says:     Effectively Connected Income, after allowable deductions, is taxed at   graduated rates. These are the same rates that apply to U.S. citizens   and residents. FDAP income generally consists of passive investment   income; however, in theory, it could consist of almost any sort of   income. FDAP income is taxed at a flat 30 percent (or lower treaty   rate) and no deductions are allowed against such income. Effectively   Connected Income should be reported on page one of Form 1040NR. FDAP   income should be reported on page four of Form 1040NR.   In the last quote, it’s like they say it only applies to passive income but if they feel like it, they can apply this to all income?? I want to close one tab of chrome. I tried killing the pid of the tab, but the tab doesn't close instead says "Aw, Snap! Something went wrong while displaying this webpage. To continue, reload or go to another page". How can I get rid of this. Ubuntu 19.10 change desktop environment If a subsequence of a Cauchy sequence converges, then the whole sequence converges. Let $(X,d)$ be a metric space, and say $(x_n)$ is a Cauchy sequence such that it has a convergent subsequence $(x_{n_k})$ that converges to $x$. We show $x_n \to x$. Let $\epsilon &gt; 0$. Take $N &gt;0$ such that for all $n,m &gt; N$, we have   $$d(x_n,x_m) &lt; \frac{\epsilon}{2}.$$  By hypothesis, we can take also $K &gt;0$ such that for all $n_k &gt; K$, we have  $$ d(x_{n_k},x) &lt; \frac{\epsilon}{2}.$$  Put $M = \max \{N,K\}$. Therefore, for all $n,m,n_k &gt; M$, we have  $$ d(x_n,x) \leq d(x_n, x_{n_k}) + d( x_{n_k},x) &lt; \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.$$  Hence, $x_n \to x$ as desired.  Is this a correct approach? Thank you very much in advance. I need to find a way of proving that the square roots of a finite set    of different primes are linearly independent over the field of    rationals.    I've tried to solve the problem using elementary algebra  and also using the theory of field extensions, without success. To  prove linear independence of two primes is easy but then my problems  arise. I would be very thankful for an answer to this question. Column validation for US phone number format Which one is correct to say: "It's me" or "It's I"? I was taught at school that the following expression is not grammatically correct:  Who is there? It's me.  The correct one is:  Who is there? It's I.  Can you let me know which one is accurate? I'm writing a mildly dystopian scenario where competition is taken to vicious extremes, and I was wondering if there was a word that I could use (specifically a noun that was a synonym of "competition") to describe the sort of vicious, cutthroat ... _________ (the word that I want) that was going on.  I'm specifically looking for a word that doesn't refer to a one-off event (like "contest" or "tug-of-war"), but rather is more about the state of things. No reputation for editing questions? I want to use some constants, height and width, when creating an ellipse but it won't work with \newcommand. Here is a minimal example:  \documentclass[12pt, a4paper]{article}  \usepackage{amsthm, tikz} \usetikzlibrary{arrows, shapes, trees, positioning} \begin{document}  \begin{figure} \centering \begin{tikzpicture}       \newcommand*{\width}{2}%     \newcommand*{\height}{0.5*\width}%      \draw[fill=gray!40] (0, 0) ellipse (\width and \height); \node (Y) at (0,0) {\(Y\)};      \end{tikzpicture} \end{figure}  \end{document}    I get the following error:  error: line 14:  Package PGF Math Error: Unknown operator `a' or `an' (in '2and 0.5*2'). ...ay!40] (0,0) ellipse (\width and \height) Computing sums of divisors in $O(\sqrt n)$ time? I have a sequence: $1,3,5,8,10,14,16,20,23,27,\dots$  I know that the recursive relation is:  $$p[i] := p[i-1] + \text{number of factors of $i$}, \quad \text{with $p[1]=1$.}$$  How do I solve this recursive relation in $O(n^{0.5})$ time? I have to write a program for it, and $n$ can be as large as $10^9$. So, some appropriate precomputation and prestorage is allowed, provided that they comply (memory and time wise) with the range of $n$. Server 2012 deduplication: run on Hyper-V host or guest VM? I have a Windows Server 2012 Hyper-V host with VMs on a dedicated RAID-5 volume. One VM is also running Server 2012 and is being used to serve files. The other VMs are smaller and mostly used for temporary testing scenarios.   The main file server VM has two dynamically expanding VHDX files - one for the OS and one for storage. The VHDX being used for storage has quite a few duplicated files on it.  Should I run Server 2012 deduplication on the volume containing the storage VHDX file on the host, or within the file server VM itself? I understand that running it within the VM will not decrease the size of the VHDX file on the host, but my goal is to enable as much storage capacity within the VHDX file itself. I also understand that if I run it on the host, I will have to take the VM offline to do it. Downtime isn't a concern in this instance, just the best way to do this. Feed can show entries for deleted posts, which viewing them fails Just noticed this:  Someone @replied to me in a question which appropriately showed up in my feed.  I go to click on that notice in the feed, and it presents me with "Loading the question failed."  I'd show a picture, but I don't think that it's entirely relevant - I believe it's reproducible if an answer is posted, someone makes a reply, the answer is removed, then that someone else goes back in the feed to try and revisit the post.  Alternatively, why not synchronize the interface between what a 10K user sees on the site, and what they see in the app?  As a 10K user, I would expect to see deleted questions and answers. I want to prove that if we have a commutative integral domain $D$ with field of fractions $F\neq D$ then $F$ is not finitely generated as a $D$-module. (In this question it may be the case that $1\not\in D$.)    My original plan was to localize at a maximal ideal and then use Nakayama's lemma but as we don't have a $1$ our localization may not be local (in fact we need not have maximal ideals).  So I'm a bit lost as to where to go?  Thanks for any help. Named parameters in JDBC TikZ: Bend text so that it follows a line I have two arrows going from a node on the left to a node on the right. Both arrows are slightly bent. I want to label the arrows with text that is bent just like the arrows. This is as far as I've got:  \begin{tikzpicture} \node (One) at (-3,0) [shape=circle,draw] {One};  \node (Two) at (3,0) [shape=circle,draw] {Two}; \draw [-&gt;, thick] (One) to [bend right=45]  (Two); \draw [-&gt;, thick] (One) to  [bend left=45] (Two); \node (mental) at (0,1.75) {Some bent text}; \node (non-mental) at (0,-1.75) {Some more bent text}; \end{tikzpicture} Why does my apple not weigh 500 tons? Related to this question:   $E=mc^2$ - energy equals mass. So, if an object has gravitational potential energy relative to another object, it should have additional mass to represent that extra energy.  So, if I have an apple in my hand, it should have quite a large amount of gravitational potential energy relative to a black hole at the edge of the observable universe. I did a quick calculation of how much this gravitational potential energy would be (using Newton's Law of Gravitation) and it came out to be the equivalent of around 500 tons (and that's just for one black hole!).  So, why does my apple not weigh a lot more than it appears to?  EDIT: Here's my calculation of the GPE:  Assumptions:   The apple has mass $m_a=0.3 kg$ The black hole has a mass equal to 1000 solar masses, $m_b=2\times10^{33} kg$ Distance between the apple and black hole is 60 billion light years   Obviously $GPE=mgh$ can't be used, because the gravitational acceleration is not constant and varies with distance, so it has to be integrated.  I used Newton's Law of Gravitation to provide the acceleration:  $$ F=\frac{G{m_1}{m_2}}{h^2} $$  A small change in height produces a small change in energy, which can be used to integrate:  $$ \begin{eqnarray*} \delta E&amp;=&amp;F \delta h\\ \int_0^EdE&amp;=&amp;G{m_a}{m_b}\int_1^R\frac{1}{h^2}dh \end{eqnarray*} $$  I'm integrating the force starting at 1m from the center of the black hole, as the force goes to infinity at $h=0$. It's a little crude, but it should provide a conservative estimate.  $$ \begin{eqnarray*} E&amp;=&amp;G{m_a}{m_b}(1-\frac{1}{R})\\ E&amp;\approx&amp; G{m_a}{m_b}\;\;\;\;\;\;(R&gt;&gt;1)\\ G&amp;=&amp;6.673\times 10^{-11}\;\;N(\frac{m}{kg})^2\\ G{m_a}{m_b}&amp;=&amp;4\times 10^{22}\;\;Nm\\ E&amp;=&amp;mc^2\\ \Delta m&amp;=&amp;\frac{4\times 10^{22}}{(3\times 10^8)^2}\\ &amp;\approx&amp;444,000\;\;kg \end{eqnarray*} $$  So, a little short of 500 tons, but still pretty big! How to prove $n^5 - n$ is divisible by $30$ without reduction How can I prove that prove $n^5 - n$ is divisible by $30$? I took $n^5 - n$ and got $n(n-1)(n+1)(n^2+1)$ Now, $n(n-1)(n+1)$ is divisible by $6$. Next I need to show that $n(n-1)(n+1)(n^2+1)$ is divisible by $5$. My guess is using Fermat's little theorem but I don't know how. Graphics file extensions and their order of inclusion when not specified When using 's \includegraphics you can specify the file extension or not. For generic use and perhaps portability, extensions are commonly dropped. The types of "approved" inclusions and their order of precedence depends on the driver/compiler being used. What are these driver-defaults?  Related: