What is the Euclidean inner product intuition? The Euclidean inner product $&lt;x,y&gt;$ of the vectors $x , y \in \mathbb{R}^n$ is defined by:  $\langle x,y\rangle = x_1y_1 + x_2y_2 + x_3y_3 + ... + x_ny_n$  I am unable to find the intuition behind this? Why do we need this inner product? What happens to the vectors under equation and what does the scalar answer tell us?  P.S. I have already seareched it, most the articles take it into geometric terms and introduce $\cos\theta$. Can anybody help me visualize thing without angular stuff?  Edit: Question addressed  uses the angle to explain geometric dot product. I am wondering if there exist some explanation for dot product in Vector Space.
What does the dot product of two vectors represent? I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent.  The product of two numbers, $2$ and $3$, we say that it is $2$ added to itself $3$ times or something like that.  But when it comes to vectors $\vec{a} \cdot \vec{b}$, I'm not sure what to say. "It is $\vec{a}$ added to itself $\vec{b}$ times" which doesn't make much sense to me.
I am trying to print a List of Appendices (LoA) for my thesis, separate from the Table of Contents (ToC).   I wish for my ToC to look like this, with depth up to subsections (2?):  CONTENTS  CHAPTER   I. Introduction......1    1.1 Section title...1     1.1.1 Subsection...1  Appendices ..... 2   and for an LoA on a new page, with depth 0, thus:  APPENDICES  APPENDIX A. Proof of Theorem 1...1 B. Proof of Theorem 2...1   I am trying to achieve this using the report class, in a way that allows me to type appendices using the hierarchy that I obtain from the chapter, section commands. This is as opposed to defining a custom myappendix command into which I insert my appendix content. (I realize that I only need the titles from the appendix chapters for the LoA, but I'd like for the appendix content to have such structure.)   For instance, this approach would solve my problem:   \chapter{Introduction} \section{Section title} \subsection{Subsection} % \begin{appendices} is also acceptable \appendix   % Some settings are changed now \chapter{Proof of Theorem 1} \chapter{Proof of Theorem 2} % \end{appendices} if environment is begun   whereas the following approach would not:  \chapter{Introduction} \section{Section title} \subsection{Subsection} \myappendix{Proof of Theorem 1} \myappendix{Proof of Theorem 2}   I realize several questions have been asked regarding this before, but none of them quite solved my problem. Notably,  defines a custom appendix command;  defines two custom commands to maintain some hierarchy, but I will have to do more work to handle the inclusion of another layer (subsection) into the appendix.    is the closest to what I want: it does give me a separate LoA, but it also adds appendices to ToC and I can't quite figure out how to suppress the entires in ToC. The moment I use   \addcontentsline{toc}{\setcounter{tocdepth}{-1}}   the line   \xpatchcmd{\@chapter}{\addcontentsline{toc}}{\addcontentsline{toa}}{}{}%   fails because I am not adding anything to the ToC anymore.   My approach: Since I was using titlesec and tocloft packages anyway, I tried to do the following:   Stop ToC once appendix command is issued Start a new counter for appendices using tocloft Using the &lt;after-code&gt; option of titlesec, add contents line after every chapter.    My MWE   \documentclass[a4paper]{report} \usepackage{lipsum} % for arbitrary text  \usepackage{tocloft,etoolbox,titlesec} \newcommand{\listappendixname}{List of Appendices} \newlistof{appendix}{apc}{\listappendixname}  \makeatletter \g@addto@macro{\appendix}{     \clearpage     \addcontentsline{toc}{chapter}{Appendices}     \addtocontents{toc}{\protect\setcounter{tocdepth}{-1}}     \titleformat      {\chapter}      [display]     {\huge\bfseries\filcenter}      {\Large \MakeUppercase{\@chapapp} \thechapter}      {1.5ex}{}     {\refstepcounter{appendix} \addcontentsline{apc}{appendix}{\protect{\@chapapp \thechapter}}} } \makeatother  \title{Title} \author{Author}  \begin{document} \maketitle \tableofcontents \clearpage \listofappendix  \chapter{Introduction} \lipsum[1] \section{Section} \lipsum[2] \subsection{Subsection} \lipsum[3]  \appendix  \chapter{Proof of Theorem One} \lipsum[4] \chapter{Proof of Theorem Two} \lipsum[5]  \end{document}   produces the following LoA:  List of Appendices  Appendix...... 5   It doesn't get the Appendix title or number, and fails after the first one.  Could you please help me obtain a separate LoA in a way that I can still use LaTeX's chapter, section, ... hierarchy? (I am okay with using different packages, the book class, or any approach other than mine.) How to get video duration, dimension and size in PHP? I am looking for a multivariable calculus book that is really physics oriented.  Anyone know of any?  EDIT: My wife is looking to brush up on multivariable at the same time she needs to brush up on the related physics. What does ** (double star/asterisk) and * (star/asterisk) do for parameters? After turning census block data into a raster, how do I redistribute the density to reflect land cover types I plan on rasterizing (terminology?) census block data relating to density to a regular 30 m by 30 m grid for use with NLCD data.  I then hope to distribute the density of the original census block among each result set of grid cells within the original polygon according to the land cover type dictated by the NLCD dataset.  For example, let's imagine a string of city blocks with a park, all found within one census block.  If within this census block the average density is 1000 people/km^2, we ultimately would want the resulting grid cell containing the park to have a lower population density than the grid cells containing the surrounding apartments.  This information may be captured by the NLCD dataset, with the park being classified as '21' (open but developed land) versus 24 for apartment buildings which could be classified as '24' (high density development).  Is there a method, or set of methods, within ArcMap that would allow me to:   Redistribute the density based on the NLCD value Perform this redistribution only within the original census block polygon   I don't know if this is possible, but I figured I would ask.  Edit  Here's what I have been trying:  I converted the census block data to a raster, snapping it to the NLCD land cover data and ensuring the output grid cells were the same size.  Each cell contains the same info as the original census block (housing umbers, population, etc). I have been trying to figure out a way to edit the new grid so that if the NLCD grid cell contains a land cover type that doesn't support a business or housing (park, forest, farm, etc.) that it's housing and population values area set to 0.  My next move was going to be figuring out how to subtract the non-residential/non-business grid cells area from the area of the original census block, and force a recalculation of the density.  Converted the altered census block layer to grid cells again, with each having the new density value, and once again set the cells' housing, population, and density values that don't support a business or housing (park, forest, farm, etc.) to 0.  Kind of circular, and I'm not sure of all the steps (raster calculator for editing individual grid cells based on another raster's grid cell value?), but it seems to logically work out if I can get the parts to work.  On the other hand, @dof1985's answer seems to simplify the first part of my workflow (by directing me to the correct toolset), but I will still have to nullify the grid cells where parks, etc., do exits after the density has been recalculated for the census data. Is my US visa still valid after change in name? I'm a Malawian national and I got US Visa years back.  Recently, due to some reasons, I had to get my first name partially changed on my passport. I have got an affidavit certificate stating the reason and proof.  Since the visa on my old passport has still not expired, but there is a difference in first name, will I face any problem? What punctuation should follow after a list introduced by a colon? The whole department: John, Jeff, Jean, and the accountant [?] were thrown into jail. ORA-00979 not a group by expression I sometimes hear stories where people compare their feelings in winter in different places in the world.  It goes like     in city X the temperature was the same as in city Y, but the humidity made me feel much colder...   or      oh well, -20°C would be cold, but the humidity was low, so it felt OK   so it implies that humidity somehow makes it feel colder. I am talking about temperatures below freezing (-30...0°C).  Does this have any physical explanation, or is it some sort of ? Is there a LaTeX trick to prevent a PDF viewer from copying the line number? A PDF output is obtained by compiling the following code. \documentclass{article} \usepackage{xcolor} \usepackage{listings}  \lstset {     language={[LaTeX]TeX},         numbers=left,         numbersep=1em,         numberstyle=\tiny,     frame=single,     framesep=\fboxsep,     framerule=\fboxrule,     rulecolor=\color{red},     xleftmargin=\dimexpr\fboxsep+\fboxrule\relax,     xrightmargin=\dimexpr\fboxsep+\fboxrule\relax,     breaklines=true,     basicstyle=\small\tt,     keywordstyle=\color{blue},     commentstyle=\color[rgb]{0.13,0.54,0.13},     backgroundcolor=\color{yellow!10},     tabsize=2,     columns=flexible,     morekeywords={maketitle}, }  \begin{document}  \begin{lstlisting} \documentclass{article} \usepackage{listings} \title{Sample Document} \author{John Smith} \date{\today} \begin{document} \maketitle Hello World! % This is a comment. \end{document} \end{lstlisting}  \end{document}  I attempted to copy the code only inside Acrobat Reader. Unfortunately, the line numbers also got copied as shown on the following screen shot.  The line numbers are useful, but readers want not to copy them. Is there a LaTeX trick to prevent a PDF viewer from copying the line number? If $X$ is compact and $T:X \to Y$ continuous and bijective, show that $T$ is homeomorphism. Let $X$ and $Y$ be metric spaces, $X$ compact, and $T:X \to Y$ bijective and continuous. Show that $T$ is a homeomorphism.   My attempt:  We need only show that $T^{-1}$ is continuous.  Let $M \subset Y$ be closed. Then, from the surjectiveness of $T$, there exists a $A \subset X$ such that $T(A)=M$. Since $M$ is closed, by the continuity of $T$, we must have that $A$ is closed.  Now $$T^{-1}(T(A))=T^{-1}(M)$$ i.e, $A=T^{-1}(M)$. Since $A$ and $M$ are closed, we must have that $T^{-1}$ is continuous.  My problem with this, however, is that I am not using the fact that $X$ is compact anywhere in the proof. Is there perhaps another way that I can show this, using the compactness of $X$?   Any hints will be appreciated. How do I upgrade from 14.04 LTS to 16.04.1 LTS? So, today is July 21st and . gives me an option to download 16.04.1 LTS. However, attempting to upgrade gets me this message -  $ sudo do-release-upgrade Checking for a new Ubuntu release No new release found   What is wrong? Does it matter that I am running apt-get update behind apt-cacher-ng?  Note: Commenting out the files in apt.conf.d that used apt-cacher-ng did not make a difference.  Edit: Explanation for possible duplicates:  deals with upgrade from 12.04LTS to 14.04LTS. My question is specific to 14.04 LTS to 16.04.1 LTS upgrade. Old answers, while good for archival purposes are less relevant in a fast changing package like ubuntu. Moreover none of the answers answer what is wrong. The link  pointed out is mentioned in one of the comments. This Q &amp; A is more helpful for people upgrading today than the work around mentioned in "". How to find orphaned files and images which are not linked from any css stylesheet or from any node? Is there a way to list all unused files which reside inside public file system and inside a theme directory and list them, or maybe even have an option to delete them automatically?  I mean files which are currently not linked from any css stylesheet or from any node. On company 401k matching. Clarification needed I realized I don't fully understand something. It almost embarrassing to admit.  In 2016 we have 18k max contribution limit for 401k. Does this mean that employee can contribute at most 18k or that total contributions made on behalf of employee should not exceed 18k (company match)  To make is simple, if you make 100k, by contributing 18% you got your limit. But if your company matches 5%, actual contributions will be 18.9k. In the above example, if one wants to maximize benefit should one contribute 18% or less? Are there any other methods to apply to solving simultaneous equations? We are asked to solve for $x$ and $y$ in the following pair of simultaneous equations:  $$\begin{align}3x+2y&amp;=36 \tag1\\ 5x+4y&amp;=64\tag2\end{align}$$  I can multiply $(1)$ by $2$, yielding $6x + 4y = 72$, and subtracting $(2)$ from this new equation eliminates $4y$ to solve strictly for $x$; i.e. $6x - 5x = 72 - 64 \Rightarrow x = 8$. Substituting $x=8$ into $(2)$ reveals that $y=6$. I could also subtract $(1)$ from $(2)$ and divide by $2$, yielding $x+y=14$. Let $$\begin{align}3x+3y - y &amp;= 36 \tag{1a}\\ 5x + 5y - y &amp;= 64\tag{1b}\end{align}$$ then expand brackets, and it follows that $42 - y = 36$ and $70 - y = 64$, thus revealing $y=6$ and so $x = 14 - 6 = 8$. I can even use matrices! $(1)$ and $(2)$ could be written in matrix form: $$\begin{align}\begin{bmatrix} 3 &amp;2 \\ 5 &amp;4\end{bmatrix}\begin{bmatrix} x \\ y\end{bmatrix}&amp;=\begin{bmatrix}36 \\ 64\end{bmatrix}\tag3 \\ \begin{bmatrix} x \\ y\end{bmatrix} &amp;= {\begin{bmatrix} 3 &amp;2 \\ 5 &amp;4\end{bmatrix}}^{-1}\begin{bmatrix}36 \\ 64\end{bmatrix} \\ &amp;= \frac{1}{2}\begin{bmatrix}4 &amp;-2 \\ -5 &amp;3\end{bmatrix}\begin{bmatrix}36 \\ 64\end{bmatrix} \\ &amp;=\frac12\begin{bmatrix} 16 \\ 12\end{bmatrix} \\ &amp;= \begin{bmatrix} 8 \\ 6\end{bmatrix} \\ \\ \therefore x&amp;=8 \\ \therefore y&amp;= 6\end{align}$$  Question Are there any other methods to solve for both $x$ and $y$? If we had an airtight container that was large enough, say extended out to the Karman line, would there still be a pressure gradient? If we assume that the pressure at the bottom was $1\ \mathrm{atm}$, would it have the same exact pressure gradient at 6 miles up on the outside? This is assuming that the gravity is still $9.8\ \mathrm{m/s^2}$. Why is polyploidy lethal for some organisms while for others is not? is the multiplication of number of chromosomal sets from 2n to 3n (triploidy), 4n (tetraploidy) and so on. It is quite common in plants, for example many crops like wheat or Brassica forms. It seems to be rarer in animals but still it is present among some amphibian species like Xenopus.  As I know in mammals polyploidy is lethal (I don't mean tissue - limited polyploidy). I understand that triploidy is harmful due to stronger influence of maternal or paternal epigenetic traits that cause abnormal development of placenta, but why there is no tetraploid mammals? What is the closed form for $\sum_{n=1}^\infty \frac1n - \frac1{n+1/p}$? A while ago, I started to look at expressions of the following form: $$ S_p:=\sum_{n=1}^\infty \frac1n - \frac1{n+1/p}, $$  where $p$ is prime, because otherwise things get too complicated for me at the moment.  What I found so far is the following: $$ S_p= p -\log(2p) - \frac12 \pi \cot\left(\frac\pi p\right) + T_p, $$ where $T_p$ contains $\frac{p-1}2$ terms $t_{p,k}$ of the form:  \begin{cases} \pm 2 \sin\left(\frac{k\pi}{2p}\right) \log\left(\sin\left(\frac{k\pi}{2p}\right)\right) \\ \pm 2 \cos\left(\frac{k\pi}{2p}\right) \log\left(\sin\left(\frac{k\pi}{2p}\right)\right)\\ \pm 2 \sin\left(\frac{k\pi}{2p}\right) \log\left(\cos\left(\frac{k\pi}{2p}\right)\right)  \\ \pm 2 \cos\left(\frac{k\pi}{2p}\right) \log\left(\cos\left(\frac{k\pi}{2p}\right)\right)  \end{cases}   An example can be found .     How does the closed form for $\sum_{n=1}^\infty \frac1n - \frac1{a+n}$ look like? (EDIT: where $a=1/p$) Compute Engine - 1000 GB disk cant decrease size i have a huuuuge problem.  Created a snapshot and image from my instance...instance has 1000GB of disk.  Now i need to reduce it but everytime i try to create an image or a snapshot from that, i get the same error "Disk need to be at least 1000GB".  There is a way to reduce disk size directly on instance? Some workaround? I use to work at night and it is eye-hurting to watch at lightbulb-bright screens (I've got two - laptop's built-in panel and an external 18-inch CRT) in a dark environment. So I adjust my screens to be darker.  Can I do this a software way instead of pressing monitors' buttons that many times every time?