$$a\cdot x_1+b\cdot x_2+c\cdot x_3+...+qx_n=\text{constant}$$ is called a linear equation because it represents the equation of a line in an $n$ dimensional space. So "linear" comes from the word "line". Basically there should not be any higher power of $x$ failing which the graph of the function will not be a straight line.  similarly  $$a(x)y+b(x)y'+c(x)y''+d(x)y'''+...+q(x)=0$$ is also called linear differential equation because all the derivatives have power=1 which is similar to the above definition of a linear equation.  A function f is called linear if: $$f(x+y)=f(x)+f(y)$$ and $$f(c\cdot x)=c\cdot f(x).$$ Here c is a constant. In this definition of linearity of function $f$ what does the word linear means? How does it relate to a straight line?  Finally what does the term linear means in case of linear vector spaces? Where is the reference to a straight line?  So, whether linear is just a word used in different contexts? Does it have different meaning in different situation? Or linearity refers to some relation to a straight line?
$a_1x_1+a_2x_2+a_3x_3+...+a_nx_n=$ is called a linear equation because it represents the equation of a line in an n dimensional space. So "linear" comes from the word "line".Basically there should not be any higher power of x failing which the graph of the function will not be a straight line.  simillarly  $a(x)y+b(x)y'+c(x)y"+d(x)y'''+...+q(x)=0$ is also called linear differential equation because all the derivatives have power equal to 1 which is similar to the above definition of a linear equation.  A function f is called linear if: $f(x+y)=f(x)+f(y)$ and $f(cx)=cf(x)$. Here c is a constant. In this definition of linearity of function "$f$" what does the word linear means? How does it relate to a straight line?  Finally what does the term linear means in case of linear vector spaces? Where is the reference to a straight line?  So, whether linear is just a word used in different contexts? Does it have different meaning in different situation? Or linearity refers to some relation to a straight line? At Least please explain how the linearity of function f and linear vector space relate to the equation of a line.
How do I reset an old iPod? Disclaimer Yes, I am fully aware that what I am asking about is totally stupid and that anyone who would wish to try such a thing in production code should be fired and/or shot.  I'm mainly looking to see if can be done. Now that that's out of the way, is there any way to access private class members in C++ from outside the class?  For example, is there any way to do this with pointer offsets? (Naive and otherwise non-production-ready techniques welcome) Update As noted in the comments, I asked this question because I wanted to write a blog post on over-encapsulation (and how it affects TDD).  I wanted to see if there was a way to say &quot;using private variables isn't a 100% reliable way to enforce encapsulation, even in C++.&quot;  At the end, I decided to focus more on how to solve the problem rather than why it's a problem, so I didn't feature some of the stuff brought up here as prominently as I had planned, but I still left a link. At any rate, if anyone's interested in how it came out, here it is:   (I suggest reading it before you decide that I'm crazy). How to change/rewrite the lost password url? I used iThemes Security (formerly Better WP Security) hide back-end functionality to change the login url to /signin however clicking the lost password link /wp-login.php?action=lostpassword now generate a 404  Manually going to /signin?action=lostpassword works.  I came across the following code below which is suppose to make the change to however after some testing it doesn't seem to work  //******************************************************* //Function to change to lost password url //******************************************************** add_filter('site_url',  'wplogin_filter', 10, 3); function wplogin_filter( $url, $path, $orig_scheme ){     $old  = array( "/(wp-login\.php)/");     $new  = array( "signin/"); //this can be change to login or whatever or may remain there     return preg_replace( $old, $new, $url, 1); }   add to .htaccess  RewriteRule ^signin$ wp-login.php   How can i modify the above code in order to change the url for the lost password? Intuitive explanation for formula of maximum length of a pipe moving around a corner? For one of my homework problems, we had to try and find the maximum possible length $L$ of a pipe (indicated in red) such that it can be moved around a corner with corridor lengths $A$ and $B$ (assuming everything is 2d, not 3d):    My professor walked us through how to derive a formula for the maximum possible length of the pipe, ultimately arriving at the equation $L = (A^{2/3} + B^{2/3})^{3/2}$.  The issue I have is understanding intuitively why this formula works, and exactly what it's doing. I understand the steps taken to get to this point, but there's an odd symmetry to the end result -- for example, is the fact that $\frac{2}{3}$ and its inverse are the only constants used just a coincidence, or indicative of some deeper relationship?  I also don't quite understand how the formula relates, geometrically, to the diagram. If I hadn't traced the steps myself, I would have never guessed that the formula was in any way related to the original problem.  If possible, can somebody give an intuitive explanation as to why this formula works, and how to interpret it geometrically?    Here's how he found the formula, if it's useful:  The formula is found by finding the maximum possible length of the pipe by expressing the length in terms of the angle $\theta$ formed between the pipe and the wall, and by taking the derivative to find when $\frac{dL}{d\theta} = 0$, which is the minimum of $\frac{dL}{d\theta}$ and is therefore when $L$ is the smallest:  $$ L = \min_{0 \leq \theta \leq \frac{\pi}{2}} \frac{A}{\cos{\theta}} + \frac{B}{\sin{\theta}} \\ 0 = \frac{dL}{d\theta} = \frac{A\sin{\theta}}{\cos^2{\theta}} - \frac{B\cos{\theta}}{\sin^2{\theta}} \\ 0 = \frac{A\sin^3{\theta} - B\cos^3{\theta}}{\sin^2{\theta}\cos^2{\theta}} \\ 0 = A\sin^3{\theta} - B\cos^3{\theta} \\ \frac{B}{A} = \tan^3{\theta} \\ \theta = \arctan{\left( \frac{B}{A} \right)^{\frac{1}{3}}} \\ $$  At this point, we can substitute $\theta$ back into the original equation for $L$ by interpreting $A^{1/3}$ and $B^{1/3}$ as sides of a triangle with angle $\theta$ and hypotenuse $\sqrt{A^{2/3} + B^{2/3} }$:  $$ \cos{\theta} = \frac{A^{1/3}}{ \sqrt{A^{2/3} + B^{2/3} }} \\ \sin{\theta} = \frac{B^{1/3}}{ \sqrt{A^{2/3} + B^{2/3} }} \\ \therefore L = A^{2/3} \sqrt{A^{2/3} + B^{2/3} } + B^{2/3} \sqrt{A^{2/3} + B^{2/3} } \\ L = (A^{2/3} + B^{2/3}) \sqrt{A^{2/3} + B^{2/3} } \\ L = (A^{2/3} + B^{2/3})^{3/2} \\ $$  The equation for the formula for the maximum length of the pipe is therefore $L = (A^{2/3} + B^{2/3})^{3/2}$. How to share files through the local network? I am trying to draw the below given figure. Please help me to plot it in Tikz. I am only able to plot concentric circles. . I would like to have different colors in between different circles. Also, the text should be along the curve. The arrow in between the two concentric circles does not have to overlap with the circle on the right hand side.  I would be grateful if you could help me in plotting this in Tikz. Please let me know if you need more information. Thanks!  \documentclass[conference]{IEEEtran}  \ifCLASSINFOpdf  \else  \fi  \usepackage[usenames,dvipsnames]{color} \usepackage[T1]{fontenc} \usepackage{tikz} \usetikzlibrary{positioning} \usetikzlibrary{arrows} \usetikzlibrary{shapes.multipart} \usepackage[caption=false]{subfig} \usepackage{tabularx,tikz} \usepackage{graphicx} \usepackage{color} \usepackage{pdfpages} \begin{document}  \title{Practice Tex}   \maketitle  \section{Introduction}  \begin{figure} \begin{center} \begin{tikzpicture}     \draw(0,0) circle (0.4cm)   node[draw=none,sloped,above]{cxvzxcv};         \draw(0,0) circle (0.75cm) node[draw=none,sloped,above]{ratio};    \draw(0,0) circle (1.25cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,2.2) {22};    \draw(0,0) circle (2cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,3.8) {33333};    \draw(0,-9) circle (0.4cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,-9) {fsgsdgf};      \draw(0,-9) circle (0.75cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,-7.8) {fgdsfg};    \draw(0,-9) circle (1.25cm) node[midway,sloped] at (0,-9) {dfsdfad};    \draw(0,-9) circle (2.8cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,-6.4) {dfsdfsd};    \draw(0,-9) circle (3.2cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,-6) {dfsdfsd};    \draw(0,-9) circle (3.6cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,-5.6) {sfdsfd};    \draw(0,-9) circle (4.4cm);   \node[draw=none,align=center, font=\scriptsize,text width = 1.25cm] at (0,-4.8) {sdfsf};     \pgfsetlinewidth{3pt}    \draw [-implies][-&gt;] (0,-4) -- (0,-4.6);   \end{tikzpicture}  \end{center} \end{figure} \end{document} Why do some materials reflect (metals) and other materials reflect and refract (glass) from the quantum perspective? Recently I was asked to explain the difference between reflection and total internal reflection from a purely conceptual standpoint (no math).    Let me explain what I already know.  Reflection and refraction at the quantum level are the same thing.  Light is a photon.  A photon is a discrete particle that has wave characteristics (a finite wave traveling like a bullet).  As the photon travels it collides with electrons in the matter of the medium it is traveling in.  Depending on the energy of the photon and the allowed energy bands of the medium the photon cause the electron to jump up a level.  If the photon is absorbed then the medium will increase its motion (at the macro scale increasing its temperature).  If the photon is not absorbed it will be re-emitted (really as a new photon). I have read and watched Feynman's QED lectures and book and have a pretty good understanding of his process for determining how all these paths come together to give the net path of the photon.  The general rule of thumb is that the photon wants to take the path which requires the least amount of time.  I understand how this principle goes to explain refraction and reflection.    What I don't seem to understand is why does one material seem to cause a higher percentage of refraction compared to another (metal vs. glass).  What about the electron configuration of a the medium changes the net effect of the absorption and re-emissions of the photons?  Is there a change in the probability of the photon being re-emitted in a reverse direction?  Is there a farther distance the photon can travel before being incident onto an electron?  This is the part where my understand breaks down.    When you have hit the critical angle in a medium that refracts and the light completely reflects, are the photons moving is the same manner as they would be in a material that always reflects? How does this connect to the question in the previous paragraph?  I know I have a bunch of mini questions embedded in answering this one larger question.  Any help on any of the parts would be greatly appreciated? Why would someone use WHERE 1=1 AND &lt;conditions&gt; in a SQL clause (Either SQL obtained through concatenated strings, either view definition)  I've seen somewhere that this would be used to protect against SQL Injection, but it seems very weird.  If there is injection WHERE 1 = 1 AND injected OR 1=1 would have the same result as injected OR 1=1.  Later edit: What about the usage in a view definition?    Thank you for your answers.  Still, I don't understand why would someone use this construction for defining a view, or use it inside a stored procedure.  Take this for example:  CREATE VIEW vTest AS SELECT FROM Table WHERE 1=1 AND table.Field=Value Find the limit of $\sum\limits_{k=1}^n\left(\sqrt{1+\frac{k}{n^2}}-1\right)$ I stumbled upon a  with the following paragraph:     MBIE has launched an investigation into claims Chinese galvanised   steel coil imports have been dumped on the local market, causing   "material injury" to New Zealand industry, and the case will end up on   the table of newly appointed Commerce Minister Jacqui Dean.   I thought that sentence was grammatically wrong because the author should put the word "that" (as a conjunction) right after the word "claim". I don't see why the author could drop the conjunction. Is this a prevailing phenomenon in American or Australian English? Why does ( exit 1 ) not exit the script? I have a script, that does not exit when I want it to.  An example script with the same error is:  #!/bin/bash  function bla() {     return 1 }  bla || ( echo '1' ; exit 1 )  echo '2'   I would assume to see the output:  :~$ ./test.sh 1 :~$   But I actually see:  :~$ ./test.sh 1 2 :~$   Does the () command chaining somehow create a scope? What is exit exiting out of, if not the script? I am trying to prove $(a)$ is a maximal ideal in $\mathbb{Z}$,  if and only if $a$ is prime number.  Now I wrote: assume $a \in\mathbb{Z}$, while it's not prime number we can write as $a=xy$, for some integers $x$ and $y$. then $(a)\subset (x)$ and $(a)\subset (y)$  while if $(a)$ is maximal ideal, it's not exist $(k)$ such that $(a)\subset (k)\subset\mathbb{Z}$. so $(a)$ cannot be maximal ideal. we can say by contradictory if $a$ is prime, $(a)$ is maximal ideal.  is this correct?  and how to prove the other way. If monotone decreasing and $\int_0^\infty f(x)dx <\infty$ then $\lim\limits_{x\to\infty} xf(x)=0.$ I noticed that I can only perform certain actions such as commenting a finite number of times in a given period of time. Obviously, rate limiting is in place to prevent accidental misuse or intentional abuse of certain features.  Where else is rate limiting applied on Stack Exchange sites, and what are the limits? jQuery Galleria 'fatal error: no theme found.' in IE Why is observing 100 heads for a fair coin flips surprising? Assume that we have a fair coin. We flip it 100 times. The outcome is all heads. Why is it surprising? Doesn't all outcomes have the same probability? Any particular outcome, including irregular ones, would have the same very small probability. Why is it that observing an irregular outcome is less surprising to us than a regular one? Only squares instead of letters and numbers are displayed in my calculator and other apps in Ubuntu 18.04 How do I install Netbeans? I used NetBeans for Java development on Windows and remember finding it in the Ubuntu Software Center on Natty, but now it's gone. Is there a PPA I can use or will it be re-added to the official repos? Prove that a polynomial is irreducible or the field contains a $p$th root An exercise from Lang:  Let $k$ be a field. Let $n$ be a nonnegative integer. Consider the polynomial $x^{p^n}-a \in k[x]$. Assume that $\text{char}(k) = p &gt; 0$. Prove that either the polynomial $x^{p^n}-a$ is irreducible over the field $k$ or there exists a $b\in k$ such that $a = b^p$.  How to prove this statement? I am a little bit confused with this statement. I need to enter the command line alone with no GUI to install the official NVIDIA drivers. I'm trying to get my computer to work over HDMI and someone on Youtube shows how but you have to be in the Nvidia settings program.