How can I handle two separate file part as one (mount, virtual join, etc)?
A virtual file containing the concatenation of other files
How do I show that if there are functions $f,g$ such that$$ f(g(x)+g(y))=bx+cy $$holds for all $b,c\in\mathbb{R}$, then we necessarily have $b=c$? Is this even true? It seems so, but I'm just not sure how to explain it in detail and using theory from functional equations. Any ideas?  Extra: In addition, I'd like to prove that there are no pair of functions $f,g$ such that$$ y^2+z^2=f(x,g(y-x)+g(z-x)). $$ Any ideas on how to prove this? Plus, does anyone know some literature regarding this type of problems? Question: Best way to derive an ellipse formula when given bunch of points (say 60 points which when connected draws an ellipse)?  Background: Using R Momocs library function  which returns confidence ellipse coordinates. Would be nice to abstract those points into a formula.  You can see my ellipse points in the image below;    Any suggestions much appreciated, Thanks How to get a list of applications installed in Ubuntu dash? In Gnome, we have applications grouped up based on their type (like - games, accessories, et al). But in Unity, we need to manually type in the Unity to get the App.  Is there any keyword/utility to display the applications in GUI?  Ubuntu Version : 14.04 Hardware : Dell Inspiron 3437 Well, I have come to understand persons with the following traits will be usually be Sorted to Slytherin   Resourcefulness  Cunning Ambition Leadership Qualities Self-Preservation Determination  Cleverness Fraternity  Power   On which trait(s) was Snape Sorted into Slytherin?  One more point: Lily Evans was Sorted before Snape, and when she was placed in Gryffindor, obviously Snape wanted to be in Gryffindor too, because Lily is his only friend.   So why is he Sorted into Slytherin? Is there a stronger reason for the Sorting Hat to override his preference and Sort him into Slytherin? 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? Why does 1366x768 resolution exist? I know that there's  but it doesn't have any real answers despite having been viewed 12,400 times, and the fact that it's been closed. With that in mind...  Why in the world is 1366x768 resolution a real thing? It has an aspect ratio of 683:384, which is the weirdest thing I've ever heard of while living in a 16:9 world.   All screens and resolutions I've been familiar with have been 16:9 aspect ratio. My screen, 1920x1080, is 16:9. The 720p that I'm familiar with is 1280x720, also 16:9. 4K that I'm familiar with, 3840x2160, is also 16:9. Yet 1366x768 is 683:384, a seemingly wild break from the standard.  I know there are plenty of other resolutions all over the place, but 1366x768 seems to dominate most of the mid priced laptop world and also seems unique to the laptop world. Why don't laptops use 1280x720 or something else as a standard? I'm cooking two different dishes in the oven at the same time that each require different temperatures. One is a roasted vegetable dish that says to cook at 475 for 35-40 minutes and the other is parmesan crusted pork chops at 350 for 40-45 minutes. What do you suggest I do? Prettiest way to typeset "C++" (cplusplus)? Writing "C++" in plain text results in an ugly setting, as the '+' signs are too big and too spaced:    I've seen around the web several marcos for typesetting the "C++" symbol, ranging from mild kerning to shrinking, raising and lowering the '+' signs.  What is your version of a \cpp{} macro? Prove that $$\nabla_{\mathrm X} \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) = - \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}$$   My proof is below. I am interested in other proofs.    My proof  Let   $$f (\mathrm X) := \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B)$$  Hence,  $$\begin{array}{rl} f (\mathrm X + h \mathrm V) &amp;= \mbox{tr} (\mathrm A (\mathrm X + h \mathrm V)^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A (\mathrm X ( \mathrm I + h \mathrm X^{-1} \mathrm V))^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A ( \mathrm I + h \mathrm X^{-1} \mathrm V)^{-1} \mathrm X^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A ( \mathrm I - h \mathrm X^{-1} \mathrm V + O (h^2)) \mathrm X^{-1} \mathrm B)\\ &amp;= \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) - h \, \mbox{tr} ( \mathrm A \mathrm X^{-1} \mathrm V \mathrm X^{-1} \mathrm B) + O (h^2)\\ &amp;= f (\mathrm X) - h \, \mbox{tr} ( \mathrm X^{-1} \mathrm B \mathrm A \mathrm X^{-1} \mathrm V ) + O (h^2)\end{array}$$  Thus, the directional derivative of $f$ in the direction of $\mathrm V$ at $\mathrm X$ is  $$D_{\mathrm V} f (\mathrm X) = - \mbox{tr} ( \mathrm X^{-1} \mathrm B \mathrm A \mathrm X^{-1} \mathrm V ) = - \mbox{tr} ( (\mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top})^\top \mathrm V ) = - \langle \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}, \mathrm V \rangle$$  and, lastly,  $$\nabla_{\mathrm X} \mbox{tr} (\mathrm A \mathrm X^{-1} \mathrm B) = - \mathrm X^{-\top} \mathrm A^\top \mathrm B^\top \mathrm X^{-\top}$$ How to know when to change a chain? A bike shop employee used a tool he said would check the wear on the chain, and said that it should be changed.   He said it took a measurement that meant the chain had expanded, but it is still hard for me understand why I should change the chain.   He said that the reason was that the expanded chain links will cause the chain rings to wear down.   That is one argument and I have no experience with it, to know if it is true.   But are there more reasons? How do I know when the chain should be changed?   My first thought is to use it until it breaks, but it probably is not the wisest option. What is the limit of:  $$\lim_{n\to \infty} \frac{n^1+\dots+n^n}{1^n+\dots+n^n} = ?$$  I did some computation with big numbers, I guess it is in the interval $\left(\frac{1}{2},1\right)$. Java map with values limited by key's type parameter Is there a way in Java to have a map where the type parameter of a value is tied to the type parameter of a key? What I want to write is something like the following:  public class Foo {     // This declaration won't compile - what should it be?     private static Map&lt;Class&lt;T&gt;, T&gt; defaultValues;      // These two methods are just fine     public static &lt;T&gt; void setDefaultValue(Class&lt;T&gt; clazz, T value) {         defaultValues.put(clazz, value);     }      public static &lt;T&gt; T getDefaultValue(Class&lt;T&gt; clazz) {         return defaultValues.get(clazz);     } }   That is, I can store any default value against a Class object, provided the value's type matches that of the Class object. I don't see why this shouldn't be allowed since I can ensure when setting/getting values that the types are correct.  EDIT: Thanks to cletus for his answer. I don't actually need the type parameters on the map itself since I can ensure consistency in the methods which get/set values, even if it means using some slightly ugly casts. "Contact us"... and then what? Compactness of $\operatorname{Spec}(A)$ In an exercise in Atiyah-Macdonald it asks to prove that the prime spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ as a topological space $X$ (with the Zariski Topology) is compact.   Now because the basic open sets $X_f = \{\mathfrak{p} \in \operatorname{Spec} (A) : \{f\} \not\subseteq \mathfrak{p} \}$ form a basis for the Zariski Topology it suffices to consider the case when   $$X = \bigcup_{i \in I} X_{f_i}$$   where $I$ is some index set. Then taking the complement on both sides we get that   $$\emptyset = \bigcap_{i \in I} X_{f_i}^c$$  so there is no prime ideal $\mathfrak{p}$ of $A$ such that all the $f_i$'s are in $\mathfrak{p}$. Now from here I am able to show that the ideal generated by the $f_i$'s is the whole ring as follows. Since there is no prime ideal $\mathfrak{p}$ such that all the $f_i \in \mathfrak{p}$, it is clear that there is no $\mathfrak{p}$ such that $(f_i) \subseteq \mathfrak{p}$ for all $i \in I.$ Taking a sum over all the $i$ then gives $$\sum_{i \in I} (f_i) = (1).$$  Now here's the problem:          How do I show from here that there is an equation of the form $1 = \sum_{i \in J} f_ig_i,$ where $g_i \in A$ and $J$ some finite subset of $I$?       This part has been giving me a headache. I am not sure if the finiteness bit has to do with algebra, topology or the fact that we are dealing with prime ideals.  This is not a homework problem but rather for self-study.  $\textbf{Edit:}$ I have posted my answer below after the discussion with Dylan and Pierre. Are all stormtroopers as poor soldiers as the ones in the movies? In the Star Wars films we consistently witness just how poor soldiers stormtroopers are. They show very sub-par marksmanship, crappy house searching tactics, and are easily fooled and defeated by Ewoks and Rebels.   That being said, I remember reading a character guide that defined stormtroopers as the very best of the best (best marksman, above average intelligence, etc.). So, in the realm of the Expanded Universe, does the mold that the movies set continue to hold true or are they just the exceptions to the rule? How to relate combinatorial proof to bipartite graphs? The question is - Let $k \geq 1$ be an integer and consider a sequence $n_1,n_2,\cdots,n_k$ of positive integers. Use a combinatorial proof to show that  $\binom{n_1}{2} + \binom{n_2}{2} +\cdots+ \binom{n_k}{2} \leq \binom{n_1+n_2+\cdots+n_k}{2}$  How can i prove this without using induction?  Additionally, For each $i$ with $1 \leq i \leq k$,consider the complete graph on $n_i$ vertices. How many edges does this graph have?  I thought of making graphs and figuring out a relation but still wasnt able to. How can I set an external display (monitor) as the primary one? Currently the MacBook's display works as primary, so every application I open gets open on the 13" and I have to drag it to the 24". I also don't have access to the dock on my 24".  Is is possible to turn off the MacBook screen and use only the one attached via DVI?  (I've just connected my 24" HP LP2475w via Mini DisplayPort to DVI to my MacBook Pro 13", which works great except for this one thing.) I enjoy going fast and my current bicycle tops too early, what kind of upgrades would I need to make to achieve a taller final gear? Is there a pronoun I can use as a gender-neutral pronoun when referring back to a singular noun phrase?  Each student should save his questions until the end. Each student should save her questions until the end. I'm writing a report, in LaTeX. And I need to include, as appendices, other latex documents I have written. I am currently doing this using pdfpages and it works. However, the inserted PDF has no page numbers and it overlays my base documents page numbering, resulting in no page numbers on the PDF pages. My questions:   How to solve the above problem. Is there a better way to do this.