For example when integrating by substitution one would write dx=2u.du. In this what does the dx mean and why do we get it in equations like the above? What i am basically asking is why does x=u^2 mean that dx=2u.du where do the du and dx come from??
$dx$ appears in differential equations, such us derivatives and integrals.  For example, a function $f(x)$ its first derivative is $\dfrac{d}{dx}f(x)$ and its integral $\displaystyle\int f(x)dx$. But I don't really understand what $dx$ is.
I am attempting to write a filter using something like sed or awk to do the following:   If a given pattern does not exist in the input, copy the entire input to the output If the pattern exists in the input, copy only the lines after the first occurrence to the output   This happens to be for a "git clean" filter, but that's probably not important. The important aspect is this needs to be implemented as a filter, because the input is provided on stdin.  I know how to use sed to delete lines up to a pattern, eg. 1,/pattern/d but that deletes the entire input if /pattern/ is not matched anywhere.  I can imagine writing a whole shell script that creates a temporary file, does a grep -q or something, and then decides how to process the input. I'd prefer to do this without messing around creating a temporary file, if possible. This needs to be efficient because git might call it frequently. If $\gcd(a,b)=1$, then there exists integers $x$ and $y$ such that $xa + yb = 1$ Did not find this from this website...  If $$ \gcd(a,b)=1,$$ then there exists integers $x$ and $y$ such that $$xa+yb=1.$$  Now, the tip is to use particular corollary, that states:  The class $[m]_{n}$ generates $\mathbb{Z}/n\mathbb{Z}\Leftrightarrow \gcd(m,n)=1.$  I am totally lost with the corollary.  Let's assume that $\gcd(m,n)=1$. Then, $[m]_{n}$ generates $\mathbb{Z}/n\mathbb{Z}$.  OK! Then what?  There is also follow up, where I have to prove the converse. I am familiar with Bezout's lemma. How does Cooper get out of the black hole? In the movie Interstellar,  Cooper (played by ) enters a black hole and sends a message to Murph (his daughter). Later, he is found by rangers, but I don't understand how he gets out of the black hole?  As far as I know, nothing can get out of a black hole, as it has extremely high gravity. $ d = \gcd(a,b)\Rightarrow\ \gcd(a/d,b/d) = 1$ Could someone please help me with this proof?     Suppose that $a, b \in N$, and $d = \gcd(a, b)$. Since $d$ divides $a$, we have $a = de$ for some integer $e,$ and similarly $b = df$ for some integer $f$. Prove that $\gcd(e, f) = 1$.   I understand why it works. Since d is all the common factors of $a$ and $b, e$ and $f$ had no common factors, therefore the $\gcd(e,f) = 1$. But how do I prove this?  Thanks in advance. Alignment of the titles in a ToC and \Roman{section} There is a problem with the alignment of the numbering of the ToC: When we use   \renewcommand{\thesection}{\Roman{section}}   the numbers encroach on the titles. When the command is commented out, the numbers are all aligned [from the first number on the left] but the titles do not begin at the same position.  What I want is to use \Roman{section} but with the numbers being displaced after V (or 9 in arabic) so that the first letters of the  titles are all aligned !  An illustration:  %\RequirePackage{fix-cm} \documentclass[a4paper, 11pt, twoside]{article}  \renewcommand{\thesection}{\Roman{section}}  \begin{document}   \tableofcontents \normalfont %\cleardoublepage  \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY} \section{QWERTY}  \end{document} Which folder is recycle bin in on Win7? How to check files there? I was using another account before. Let me call it account A. And I deleted a lot of files to the Recycle Bin at that time. But before I removed account A I forgot to empty the Recycle Bin.   My question is, are my files still there? I've searched and found that the Recycle Bin may live in $Recycle.Bin, but I can't open files there so I can't know what they are. Any help how to open deleted files or how to delete them permanently? What do I call someone who marries or has a sexual relationship with someone much younger than themselves? Their partner is someone who is at least 18 years old.    The term paedophile is not the correct word, because it refers to minors. Do badges get calculated in real time, or do they get calculated by an external service? How long does it take for a badge to be awarded, once its requirements are met?  Does Stack Overflow use any external services to do things, like maintenance, e-mailing or badge calculation? In "Knots and Primes: An Introduction to Arithmetic Topology", the author uses the following proposition   Let $h: Y \to X$ be a covering. For any path $\gamma : [0,1] \to X$ and any $y \in h^{-1}(x) (x = \gamma(0))$, there exists a unique lift $\hat{\gamma} : [0,1] \to Y$ of $\gamma$ with $\hat{\gamma}(0) = y$. Furthermore, for any homotopy $\gamma_t (t \in [0,1])$ of $\gamma$ with $\gamma_t = \gamma(0)$ and $\gamma_t(1) = \gamma(1)$, there exists a unique lift of $\hat{\gamma_t}$ such that $\hat{\gamma_t}$ is the homotopy of $\hat{\gamma}$ with $\hat{\gamma_t}(0) = \hat{\gamma}(0)$ and $\hat{\gamma_t}(1) = \hat{\gamma}(1)$.  The author then follows with; "In the following, we assume that any covering space is connected. By the preceding proposition, the cardinality of the fiber $h^{-1}(x)$ is independent of $x \in X$."   I am not sure why this results is true. This is my attempt at explaining it to myself. Take two different $x_1, x_2$ and their fibers $h^{-1}(x_1), h^{-1}(x_2)$ such that $y_1 \in h^{-1}(x_1)$ and $y_2 \in h^{-1}(x_2)$. Take two paths $\gamma_1, \gamma_2$ such that $\gamma_1(0) = x_1$ and $\gamma_2(0) = x_2$. Then we get two lifts $\hat{\gamma_1}, \hat{\gamma_2}$ with $\hat{\gamma_1}(0) = y_1$ and $\hat{\gamma_2}(0) = y_2$. Now, since our covering space is connected we can continuously deform $\hat{\gamma_1}$ into $\hat{\gamma_2}$ and conclude that every elements in $h^{-1}(x_1)$ is also in $h^{-1}(x_2)$ and vice versa. I feel that this is wrong but cannot figure out the right way to see this. Any help would be appreciated. Repeated error  I have been repeatedly encountering the following error when my laptop freezes. It may occur multiple times during the day, and should take several minutes to recover.  Source Windows  Summary Hardware error  Date ‎11/‎1/‎2018 10:11 AM  Status Not reported  Description A problem with your hardware caused Windows to stop working correctly.  Problem signature Problem Event Name: LiveKernelEvent Code:   144 Parameter 1:    3003 Parameter 2:    ffffad8c0b1a75c8 Parameter 3:    40010002 Parameter 4:    0 OS version: 10_0_17134 Service Pack:   0_0 Product:    256_1 OS Version: 10.0.17134.2.0.0.256.48 Locale ID:  1033   More details  Per my best reflection, the hardware error occurs when I am using the laptop on its docking station. Here is what I observe:   While using the laptop as normal, the "freeze" will occur and I can hardly operate the laptop: mouse cursor won't move continuously and Alt+Tab won't be able to switch across two windows. Shortly after (usually several minutes), things will go back to normal. Upon inspecting through the Performance tab in Task Manager, there had been no spike in anything, be it CPU, Memory or Disk. When I open the View reliability history tool, I will find another entry of Windows Hardware Error   Machine specification:   Thinkpad T480, i7-8650U (newly replaced module thanks to the manufacture)   With the same pair of docking stations, an older unit of T480 has had exactly the same problem.  Docking station: . Same issue has occurred on two identical docks with  slightly different monitors and accessories. OS specification: OS build 17134.376. Most recent collection of "reliability report":    Possible solutions   First, I would like to learn what exactly went wrong, hardware-wise: the manufacturer's support team did not bother too much with the precise error messages, and have ordered me to reinstall the Windows OS up to 5 times in total. [This is what they called "debugging".]   Please advise if you can decipher the recurrent "Hardware error".   Secondly, I will try to push the manufacturer to replace the docking stations, and see if the same "freezing problem' persists. What is a valid use case for an "execute only" file permission? I was reading up on chmod and its .  I saw that 1 is execute only.  What is a valid use case for an execute only permission?  To execute a file, one typically would want read and execute permission.  $ echo 'echo foo' &gt; say_foo $ chmod 100 ./say_foo $ ./say_foo bash: ./say_foo: Permission denied $ chmod 500 ./say_foo $ ./say_foo foo calculate slope for a road network I am working on a road network for routing operations using postgis and QGIS.    I am new to both of these tools and need to calculate the slope for each edge (by retrieving the Z value for the start and end node for the edge from a raster). Any ideas of the best approach to this?   I know postgis can dump points and get the value from a raster, but can I get the slope attribute to the edge without breaking the network topology (with edges that have corresponding start/end nodes) using this method?   thanks Why bother with the dual problem when fitting SVM? I'd say Microsoft have a way of bending the rules and I know that McLaren have won the championship. While this sounds strange, I believe it is correct English (sorry, I'm not native).   But when it's a small company, would you still use it this way? Is a company always plural, or are small companies singular? I.e., would you say Bakery Johnson makes fine bread or Bakery Johnson make fine bread? Is it My book seller, Woody's, have moved or is it has moved? Give $a, b. c$ be the lengths of a triangle and $a+ b+ c= 3$. Prove that: $$\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\geq  \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$$ My try  We have England MO inequality: $$\left ( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a} \right )^{2}\geq  \left ( a+ b+ c \right )\left ( \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c} \right )$$  and $$\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a} \geq  3=  a+ b+ c$$  But it cannot be used to prove the above inequality and how to use triangle inequality in this inequality. I need the help. Thanks! Can inner classes access private variables? class Outer {      class Inner {     public:         Inner() {}         void func() ;     };  private:     static const char* const MYCONST;     int var; };  void Outer::Inner::func() {     var = 1; }  const char* const Outer::MYCONST = "myconst";   This errors out when I compile with class Outer::Inner' has no member named `var' How to test for an hold item name in Minecraft? I'm trying to make a command block that spawns an item in front of you if you are holding an item with a specific name. I've got these two working scripts;  /testfor @p[r=10] {Inventory:[{tag:{display:{Name:"Item Name"}}}]}  /testfor @p[r=10] {SelectedItemSlot:0,Inventory:[{Slot:0b,­id:"minecraft:diamond_sword"}]­}   but I can't get them together. The block should only look for an item in the active slot. Is $\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dx$? C++ Struct Pointer Segfault How to view the system console message after Blender crashes? I am testing a script in Blender game engine. I can view the helpful message on the external system console when the script throws exception. Sometimes the bugs of the script cause the Blender to crash and close the console immediately.    Is there any way to view the message after the Blender crashes? How to write the message to a text file for later viewing? Any suggestion?