Reading a book, I saw that this function (where $H$ is the standard Heaviside function)  $$ u(t,x)= H(x-0.5t) $$  could be the solution, in the sense of distributions, of this equation  $$ u_t + (u^{2})_{x} = 0 $$  i.e  $$ \int_{0}^{\infty} \left(\int^\ u f_{t} + u^{2}f_{x} dx \right) dt = -\int^\ u_0(x)f(0,x) dx $$ for any test function $f = f(t,x)$.  Can anyone help me do this verification by computing the above integral?  Regards
Consider an inviscid Burgers equation:  $u_t + u u_x = 0.$  With the initial data:  $ u(x,0) = \left\{ \begin{array}{ll}              0  \quad  &amp; \text{if} \quad x &lt; 0, \\[0.5em]              1  &amp; \text{if} \quad x &gt; 0.\end{array} \right. $  Possible weak solutions to this problem include, for example:  $ u_1(x,t) = \left\{ \begin{array}{ll}              0  \quad  &amp; \text{if} \quad x &lt; t/2, \\[0.5em]              1  &amp; \text{if} \quad x &gt; t/2.\end{array} \right. \hspace{10mm} $   $ u_2(x,t) = \left\{ \begin{array}{lll}              0  \quad  &amp; \text{if} \quad x &lt; 0, \\[0.5em]                      x/t &amp; 0 &lt; x &lt; t,\\[0.5em]              1  &amp; \text{if} \quad x &gt; t.\end{array} \right. $  Every material that provides such examples states that it is easy to check that these are both solutions of our problem. I do not see how that is done.  I know how to obtain the equation:  $ \int_{0}^\infty \int_{-\infty}^\infty [ q \phi_t + f(q) \phi_x ] dx dt + \int_{-\infty}^\infty q(x,0) \phi(x,0) dx = 0,  $  But I do not see how to "test" if the weak solutions provided above satisfy this. I tried writing the solutions as Heavside functions and integrating them, but I am not sure how to handle an arbitrary test function.  I assume there are some solution I could construct that would not be the weak solution to our initial data, how to check that?
Drop rates for set / legendary items Just wondering what the drop rates were for Set / Legendary items, or is it totally random? I imagine they would increase for later acts in Inferno. My monk is comfortable in Act 3 in Inferno, but I've only found one set item, so MF will increase chances for them? theorem numbering: section counter but remove the section number The MySQL manual at  covers this.  Usually I just dump the database and reimport it with a new name. This is not an option for very big databases. Apparently RENAME {DATABASE | SCHEMA} db_name TO new_db_name; .  This needs to work with , which stores things very differently than . iMessage doesn't say "delivered" or "not delivered" I sent an iMessage from my iPod touch to my friends iphone on Monday. It didn't say delivered or not delivered. It was just nothing. His phone is on (I know because my other friends are iMessaging him) and my friend said her messages don't say "delivered" at the bottom but they still talk. Mine always say delivered at the bottom. So is this an error, like it didn't go through, or is he just ignoring me? What do I need to know before upgrading the CPU of a laptop? I have an AMD laptop, which has AMD APU (e1-2500). I don't really like the speed of it and would love to upgrade to faster APU. There are some AMD APUs available in ebay. So the question is:   What do I need to check to be sure that a new APU will work without problems? How do you debug through a compressed JavaScript file? block certain dangerous commands? In my  High school there are few computer having Ubuntu 14.04 installed and many students know the Root  password, how  can I block/disable  certain dangerous  commands like :- rm -rf / rm -rf * :(){:|:&amp;};: to avoid Data loss ? I understand there is a method for solving simultaneous modular equations. For example; $$x = 2 \mod{3}$$ $$x = 3 \mod{5}$$ $$x = 2 \mod{7}$$ We find numbers equal to the product of every given modulo except one of them - giving $5 \cdot 7$, $3 \cdot 7$ and $3 \cdot 5$. We then find the multiplicative inverses of these numbers with modulo equal to the number missing from the product. The numbers found are then 2, 1 and 1 in this case. The value of x is then given by: $$x = 2 \cdot (5 \cdot 7) \cdot 2 + 3 \cdot (3 \cdot 7) \cdot 1 + 2 \cdot (3 \cdot 5) \cdot 1 = 233 = 23 \mod{3\cdot5\cdot7}$$  But I do not understand how this method correctly gives the value of $x$. I understand that the Chinese remainder theorem proves that there is a unique value of $0\le x \lt 3\cdot5\cdot7 \mod{3\cdot5\cdot7}$ but can someone please explain why this method finds this value of x? I am looking to set up Ubuntu Server as a headless NAS for my home. I would like to have file storage there, as well as a central hub for my MP3s and pictures.  What are the best packages out there to handle this? Can someone post a link to a good tutorial or post some tips?  One constraint I have is that it has to be Windows 7 friendly. By that I mean the shares and streaming should work for a Windows machine. This question led to a new package:      How can I display the full path of the tex file that I compile? It is a simple file, no include or input present.  Edit: I use MiKTeX 2.9 on a Windows 7 64-bit.  Less demanding: If not possible, then I will be satisfied with a partial path (at least one or two folders up + the filename). Can you explain more?  I do not know when we can use them in the sentence.  Can we use them instead of each other?     They have a beautiful home.   They have a beautiful house. I want to buy some stock photos like the ones below. What is this design called?  I don't even know how to start searching... Is there a better way than the following to check to see if a string is nil OR has a length of 0 in Ruby?  if !my_string || my_string.length == 0   return true else   return false end   In C# there's the handy  string.IsNullOrEmpty(myString)   Anything similar to that in Ruby? How to peel hard boiled eggs easily? This is one of the most annoying duties in the kitchen, for me.  What are yours tips and tricks to peel hard boiled eggs easily? Given a continuous random variable $X$ and the pdf find.... There is a pattern excercise:  First pattern: We have $X$ and the probability density function is usually:  $$ f(x) = \begin{cases}   {ax+bx^2} &amp;  {0\lt x \lt 1} \\   0 &amp; \text{else}   \end{cases} $$  Then we are given $E|X| = 0.6$   And usually asks to find:  a) $P\{X&gt;0.5\}$  b) $Var(X)$    Second pattern:  $$ f(x) = \begin{cases}   {cx^4} &amp;  {0\lt x \lt 2} \\   0 &amp; \text{else}   \end{cases} $$  Where c is a constant. Find:  a) Mean value of $X$  b) Variance of $X$     Third pattern (it's more like a reverse problem):  We are given $Y= \sqrt{X}$, find it's cumulative distribution function (CDF) and  it's probability density function (pdf).    Update:  The third one is transferred to a .  This is not homework. I'm self-studying and in need of some help. Thank you. So, I was given the following problem as part of a homework assignment.     Suppose $f'(x) &gt; 0$ in $(a,b)$. Prove that $f$ is strictly increasing in $(a,b)$, and let $g$ be its inverse function. Prove that $g$ is differentiable, and that   $$g'(f(x)) = \frac{1}{f'(x)}$$   I have proven that $f$ is strictly increasing in $(a,b)$, and I could prove that $g'(f(x)) = 1/f'(x)$ if I could prove that $g$ is differentiable. The problem is that I am having trouble with a proof of that. Any advice?   Also, as a reference, this is exercise 5.2 from Baby Rudin. Boolean Modifiers not working normally When using a Boolean modifier on an object, the "Operation" choices (Difference, Union, Intersect) are all not working normally. Choosing "Union" makes it do the effect of "Intersect" and "Difference" makes it do what "Intersect" is supposed to do. What's the deal here?   This is happening with following a simple tutorial. It's a simple primitive cube and primitive cylinder. No normals or doubles problems.   My specs are:  Macbook Pro 13" Retina early 2015 512GB i7 with 16gb RAM upgrade OS-X version 10.11.4 I am trying to make a simple pie chart with pgf-pie, but for a reason unknown to me the labels I use don't display correctly.   Here is the chunk of code I've copied from the manual into my .tex file :  \begin{tikzpicture}     \pie[color ={ cyan!10 , cyan!40, cyan!70,  cyan!100}, explode=0.1] {10/A, 20/B, 30/C, 40/D} \end{tikzpicture}   Here is a screenshot of what I get :    And here is a picture of what I'd like to obtain :   Does anybody knows why I get these numbers and how to get rid of them ?  edit : Here is a minimal working example   \documentclass{article} \usepackage[frenchb]{babel} \usepackage{pgf-pie}  \begin{document} \begin{tikzpicture}     \pie[color ={ cyan!10 , cyan!40, cyan!70,  cyan!100}, explode=0.1]{10/A, 20/B, 30/C, 40/D} \end{tikzpicture} \end{document} What is the shape of a black hole? I was thinking; what shape does a black hole have?. By 'Shape', I mean its form (e.g, circle , cylinder, sphere, torus, etc..).  We usually think of black holes as if they're plugholes (e.g, a flat circular object), but what if they're spherical? A spherical black hole would make much more sense. I would imagine than a black hole shaped like a basketball would be capable of pulling more mass towards it than a flat one, as it has a higher surface-to-volume ratio to do so.  Edit I know that it's probably a sphere, but when you think about it, a cylinder could also be a potential shape. Before today, I would visit Stack Overflow every day and click somewhere in the upper-left corner (I've forgotten what it said, and it's gone now!) and if I didn't have any notifications I would see a list of interesting questions from various sites in the Stack Exchange network which - I believe - were classified as "hot".  Now this new top bar has appeared, and I have no idea how to get this list any more. I'm not really sure what it was called so my searches have been useless as well.