Dual nature of Matter at gross level
What is the world's biggest Schrodinger cat?
Are there Ubuntu security notices feeds for specific releases Prove that $∂A$ is closed given $∂A = \text{Cl}(A) − \text{Int}(A)$ Similar questions have been asked, but none with the given information. My textbook doesn't give me the fact that  $∂A = \text{Cl}(A) − \text{Int}(A)$. If that were the case, I could just state that definition and note that it's the intersection of closed sets.   I have very little knowledge of set theory and proofs, so I'm not sure how else to prove this. As always, I appreciate any help.  The question comes from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa.   "Let $A$ be a subset of a topological space $X$. Prove that $∂A$ is closed given  $∂A = \text{Cl}(A) − \text{Int}(A)$" Define a mathematical set I'm thinking that one way Jeff and the Stack Overflow team could squeeze some extra money out of this site would be to automatically convert all Amazon links posted here into affiliate links, e.g. Stick "tag=codinghorror-20" (or more likely a new site-specific tag) onto every Amazon link. This would bring in some additional revenue every time someone purchased a book via a link on this site.   They could do similar things with other links as well. Amazon's simply the most obvious choice.  So my question is, would anyone have a problem with this?  I know I wouldn't mind, but I don't know how other people would react.  What does everyone think? Is this a horrible idea, a great idea, a waste of time? How can I bypass Activation Lock? Proof that every finite dimensional normed vector space is complete Can you read my proof and tell me if it's correct? Thanks.  Let $V$ be a vector space over a complete topological field say $\mathbb R$ (or $\mathbb C$) with $\dim(V) = n$, base $e_i$ and norm $\|\cdot\|$. Let $v_k$ be a Cauchy sequence w.r.t. $\|\cdot\|$. Since any two norms on a finite dimensional space are equivalent, $\|\cdot\|$ is equivalent to the $l^1$-norm $\|\cdot\|_1$ which means that for some constant $C$, $\varepsilon &gt; 0$, $k,j$ large enough, $$ \varepsilon &gt; \|v_j - v_k\| \geq C \|v_j - v_k\|_1 e_i= C \sum_{i=1}^n |v_{ji} - v_{ki}| \geq |v_{ji} - v_{ki}|$$ for each $1 \leq i \leq n$. Hence $v_{ki}$ is a Cauchy sequence in $\mathbb R$ (or $\mathbb C$) for each $i$. $\mathbb R$ (or $\mathbb C$) is complete hence $v_i = \lim_{k \to \infty} v_{ki} $ is in $\mathbb R$ (or $\Bbb C$) for each $i$. Let $v = (v_1, \dots , v_n) = \sum_i v_i e_i$. Then $v$ is in $V$ and $\|v_k - v\| \to 0$:  Let $\varepsilon &gt; 0$. Then $$ \|v_k - v\| \leq C \|v_k - v\|_1 = C \sum_{i=1}^n |v_{ki} - v_i| \leq C^{'}n \varepsilon$$  for $k$ large enough. How can I iterate over an enum? Is JavaScript's "new" keyword considered harmful? In another , a user pointed out that the new keyword was dangerous to use and proposed a solution to object creation that did not use new. I didn't believe that was true, mostly because I've used Prototype, Scriptaculous and other excellent JavaScript libraries, and everyone of them used the new keyword.  In spite of that, yesterday I was watching Douglas Crockford's talk at YUI theater and he said the exactly same thing, that he didn't use the new keyword anymore in his code ().  Is it 'bad' to use the new keyword? What are the advantages and disadvantages of using it? How do I run 32-bit programs on a 64-bit Debian/Ubuntu? I have a 64-bit (amd64 a.k.a. x86_64) Debian or Ubuntu installation. I need to run 32-bit (i386/i686) programs occasionally, or to compile programs for a 32-bit system. How can I do this with a minimum of fuss?  Bonus: what if I want to run or test with an older or newer release of the distribution? This is a  about DNS (Domain Name Service).   If my understanding of the DNS system is correct, the .com registry holds a table that maps domains (www.example.com) to DNS servers.   What is the advantage? Why not map directly to an IP address? If the only record that needs to change when I am configuring a DNS server to point to a different IP address, is located at the DNS server, why isn't the process instant? If the only reason for the delay are DNS caches, is it possible to bypass them, so I can see what is happening in real time? An example where the output of the k-medoid algorithm is different than the output of the k-means algorithm Can I disable GNOME legacy tray in 17.04? Linear Matrix Inequality of a matrix $T$ with $\| T \|_2 < \rho$ Assume $T \in \mathbb{R}^{d\times d}$. Let $\rho(T)$ be the spectral radius of $T$ and $\rho \ge 0$. Prove that  $\rho(T) &lt; \rho$ if and only if there exists a $P \succ 0$ satisfying $T^TPT - \rho^2 P \prec 0$.  I can prove the Necessity: If $P = I$, for any $x \in \mathbb{R}^d$,   \begin{equation} x^T T^T P T x - \rho^2 x^T P x = \| Tx \|^2 - \rho^2 \| x\|^2 &lt; 0 \end{equation} so $T^T PT - \rho^2 P \prec 0$  I get stuck in proving the Sufficiency. When do I clear US customs and immigration if connecting at a US airport to my final destination elsewhere in the US? I'm a British citizen travelling from London Heathrow (LHR) to the Hartford, Connecticut airport, and have a two hour stopover in Newark, New Jersey (EWR) on my itinerary. At what stage would I clear immigration and customs on arriving in the USA? LaTeX/BibTex not arranging citations by order of appearance I am having an issue with LaTex/Bibtex presenting my citations in order of appearance. I know there a plenty of questions on this and I have read some of them, but the solutions posted don't seem to do me any good. I am using MacTeX and TeXmaker.  I am writing a large document, like a Thesis. Chapters are split between individual Tex files and included using the \include command.  I have a large .bib file that is maintained by Mendeley (program I use for storing my references and producing a BibTex file).  I am using the package, cite (mainly for doing references like [1-3,6,9])  I have a pretty standard looking document.  \documentclass[11pt,a4paper]{report} \setcounter{secnumdepth}{3} \setcounter{tocdepth}{3} \newcommand{\doubleSignature}[3][Alex Mason]{% \parbox{\textwidth}{ \centering #3 \today\\ \vspace{2cm}  \parbox{7cm}{   \centering   \rule{6cm}{1pt}\\    #1  } \hfill \parbox{7cm}{   \centering   \rule{6cm}{1pt}\\   #2 } } } \usepackage[utf8]{inputenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} %\usepackage[sorting=none]{biblatex} %\bibliography{Alex} % Where journals.bib and phd-references.bi \usepackage[pdftex]{graphicx} \usepackage{enumerate} \usepackage{nomencl} \usepackage{wasysym} \usepackage{booktabs} \usepackage{cite} \usepackage[left=1in,right=1in,top=1in,bottom=1in]{geometry} \graphicspath{{./Diagrams/}} \usepackage[labelfont={sf,bf}, margin=1cm]{caption} \usepackage{subcaption} \renewcommand{\figurename}{Fig.} \renewcommand{\tablename}{Tab.} \renewcommand{\bibname}{References} \author{NAME (MEng)} \title{TITLE} \date{\today} \makenomenclature %------------------------------------------------ \begin{document} \maketitle \pagenumbering{roman} %-----------------------------------------------certificate of originality----------------      -------------------------------------------- \input{CertofOrigin} \begin{abstract} ABSTRACT \end{abstract} %-----------------------------------------------contents------------------------------------------------------------ \tableofcontents \listoffigures \addcontentsline{toc}{chapter}{List of Figures} \listoftables \addcontentsline{toc}{chapter}{List of Tables} \chapter*{Acknowledgements} \addcontentsline{toc}{chapter}{Acknowledgements} ACKNOWLEDGEMENTS \printnomenclature[1.5in] \addcontentsline{toc}{chapter}{Nomenclature} \pagebreak \pagenumbering{arabic} \include{2ndYearReport_CHAP1} \include{2ndYearReport_CHAP2} \include{2ndYearReport_CHAP3} \bibliographystyle{unsrt} \bibliography{Alex} %\printbibliography \end{document}   The goofy looking part at the top (\newcommand{doubleSig.....) is just for when i input a special page I made containing a certificate of originality spiel with two boxes for signatures and my university logo.  As you can see I am using the standard bibtex way of doing bibliographies by stating the style then the stating the bib file to use right at the end of the document. I have tried biblatex but can't get it to work. It says it hasn't produced the bib file but doesn't really state why.  As far as I am aware everything was working A-OK until I started on the second chapter and began adding citations in there. Now my citations are all messed up and won't correct. My first citation starts with [9]!!!  [1] appears in a figure caption later down the page, after 10, 11 and 12 have been called!  I have tried deleting all the working files (bbl, aux, log etc) but nothing happens. I get the same result. IIRC I used to have this problem with MS Word and is part of the reason why I switched to LaTex...yet it seems to have followed me!   bibtex does through a few warnings when it runs stating that some references are missing journal names or volume numbers. Would that be a reason?  What have I done wrong with my code? I have just downloaded "ubuntu-14.04.1-desktop-amd64.iso" and "ubuntu-14.04.1-server-amd64.iso" through "".   Please can someone confirm the maximum RAM that is supported.  According to many sites, Maximum RAM supported by the OS  ubuntu-14.04.1-desktop : " 2^64 bytes " -- Kindly confirm  ubuntu-14.04.1-server : " 2^64 bytes " -- Kindly confirm What happens if I unsubscribe from Apple Music, then resubscribe at a later date? I want to cancel my apple music subscription for now, but may want to bring it back this summer. Will all the music I added be permanently removed when I unsubscribe, or will re-subscribing retrieve all my previous music? I have already dual systems : Windows 7 and Ubuntu installed on my laptop. The Windows 7 had been installed when I bought this laptop, and I don't have a CD.  I just installed VitrualBox under Ubuntu, hoping that I could install another Windows in this virtual machine, so that I could run some applications of Windows even under Ubuntu.  But it seems that I need a CD or ISO of Windows to boot inside this virtual machine. Otherwise, there is an error: FATAL: No bootable medium found! System halted. My question is where I could find a CD or ISO?  At the moment, I am downloading ISO from this , but do not know if it will work. By the way, I am not sure if I should pick the 32-bit or 64-bit? As my CPU is 64-bit but my Ubuntu is i386.  Could anyone help? Thank you very much. What focal length lens do I need for photographing the moon? I would like to know whether it is possible to photograph moon without any flaws with a canon EOS 550d..   Also to get an image of the moon as close as the following image, what kind of equipment(lens,etc) is need to be used with the 550d Body Alternative for Notepad++?