Exercise :     Show that a smooth solution of the initial value problem   $$\begin{cases} u_t + uu_x = 0 \\ u(x,0) = \cos \pi x \end{cases}$$   must satisfy $u = \cos[\pi(x-ut)]$. Also, show that when $t=1/u$, the function $u$ stops to exist (as a single-value function).   Attempt :  $$\frac{\mathrm{d}x}{u} = \frac{\mathrm{d}t}{1} = \frac{\mathrm{d}u}{0} $$  $$\implies u_1 = u, \; \; u_2 = x - ut$$  Which means that the solutions are given by $u_1 = F(u_2) \Rightarrow u = F(x-ut)$. Thus, taking into account the initial value : $F(x) = \cos \pi x.$ Now, for $x:= x-ut$ we get $F(x-ut) = \cos[\pi(x-ut)]$ which $\in C^\infty$ and thus a smooth solution.  Question : How would one proceed to show the second part of the exercise ? It gives as a hint to check the graphs of $\arccos(u)$ and $\pi(x-ut)$ as functions of $u$, which makes sense, since :  $$\arccos(u)=\pi(x-ut)$$  But I can't seem how to grasp out a graph or figure out a solution to it.
I have shown that a smooth solution of the problem $u_t+uu_x=0$ with $u(x,0)=\cos{(\pi x)}$ must satisfy the equation $u=\cos{[\pi (x-ut)]}$. Now I want to show that $u$ ceases to exist (as a single-valued continuous function) when $t=\frac{1}{\pi}$.  When $t=\frac{1}{\pi}$, then we get that $u=\cos{(\pi x-u)}$.  With single-valued function is it meant that the function is 1-1 ?  If so, then we have that $\cos{(2 \pi-u)}=\cos{(4 \pi -u)}$, i.e. for two different values of $x$, we get the same $u$, and so for $t=\frac{1}{\pi}$, $u$ is not 1-1.  But if this is meant, how are we sure that for $t \neq \frac{1}{\pi}$ the function is single-valued?
What Thunderbolt to USB 3.0 cables/adapters exist for OS X and Mac hardware? I have the Late 2013 Macbook Pro, which has two USB 3.0 ports and two Thunderbolt 2 ports. I currently use both USB 3.0 ports and one TB2 port. I want to connect a third USB 3.0 device to the machine, but I don't want to use a USB splitter on one USB port as I'm concerned the devices would draw too much power from the bus. I have no current use for the second TB2 port and I would love to be able to use it for this purpose.   Browsing extensively online, I've found numerous docks, such as , that connect via a Thunderbolt cable. This would work, but it's an expensive, 'bulkier' solution (and not the one I'm looking for).   I'm aware that a USB 3.0 device wouldn't be able to take advantage of Thunderbolt's I/O speeds, but it's certainly compatible, which is all I'd need. After all, Apple even sells a . I also know that Thunderbolt is not only used for digital video purposes (along with the TB > Firewire option, there exists ).   There are almost no Thunderbolt to USB 3.0 cables/adapters on the market. I'd think this hardware would be more prevalent given the prominence of USB devices out there. What adapters are available for OS X? I want to completely replace Ubuntu with Lubuntu I accidentally issued some commands I thought were safe in the terminal, and I ended up with no System Settings and lots of other missing functionality, but the OS still boots normally. I've decided I want to install Lubuntu instead of fixing Ubuntu for a fresh start and since my laptop is running on a 1GHz dual core AMD CPU.  I created a bootable USB as I usually do and placed Lubuntu 17.10 amd64 using Rufus 2.14.1086. I have GRUB that was installed with Ubuntu upon first install I still have Win8.1 which I still want to keep.  PS: I also tried creating a startup disk from Ubuntu  The question is: How can I boot from USB with GRUB2 so I can overwrite Ubuntu? My server is still vulnerable to heartbleed even after I update OpenSSL I have an Ubuntu 12.04 server. I have updated the OpenSSL package in order to fix the heartbleed vulnerability. But I am still vulnerable even, even though I have restarted the web server, and even the whole server.  To check my vulnerability I used:       dpkg gives:  dpkg -l |grep openssl ii  openssl  1.0.1-4ubuntu5.12   Secure Socket Layer (SSL) binary and related cryptographic tools   (launchpad.net/ubuntu/+source/openssl/1.0.1-4ubuntu5.12) Show all inbox items (notifications) in app The SE app currently only shows some inbox items:   Answers Comments Chats   When I see those I assume I'm all caught up, but it turns out there are more on the web which aren't displayed in the app:   Bounty expiration Elections Edits on your own posts Maybe others?   I was about to tag this  but realized it's probably intentional.  I imagine the reason for not showing them is because they're not something you can act on from within the app. Bounties can't be offered or awarded1 and elections can't be voted on.  But then again chats can't be replied to either. Badge descriptions can't be viewed (on the Achievements side). Selecting those launches the browser. Why then shouldn't bounties or elections launch the browser too? Hiding them just means I'll miss it if I haven't visited in browser recently.  1 Though accepting answers does award the bounty How to fix very slow Ubuntu booting? When I boot up my my newly installed copy of Ubuntu on my custom rig w/4GB DDR3 RAM and a 3 Ghz Intel Core i7, it takes about a minute and a half at the purple loading screen, and then after logging in it takes a while for the icons to show up on the screen. This computer could run and boot Windows 7 faster, so  i have a problem here. I have read this topic:   There, the issue is handling two different generations of PHP (PHP 4 vs PHP 5), but what I need is to make all but one of my server accounts work under 5.3 and just a single domain under 5.2.  I cannot make this through Apache models, and the answer from  does not apply since in my case I can't use a different port - this is a regular website  on the same server. How to add citation for a whole paragraph? If I write a report and a whole paragraph is based on one source but not cited verbally it seems unnecessary to include the reference after every sentence:  Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua (source 1). Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat (source 1).  Now I found a Phd thesis that puts the reference after the last dot of the paragraph to indicate that it is based on this very source:  Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. (source 1)  Is this a common practice? Is it generally understood? What are other ways to add the reference for a whole paragraph? Suppose I have a line segment of length $L$.  I now select two points at random along the segment.  What is the expected value of the distance between the two points, and why? XPath - Get node with no child of specific type How to draw normal map? How can I draw (not bake) normal map over my model, like a texture in texture paint mode? Render cycles.  Thank you in advance! Problem with Refsection I can't seem to get the reference within my {refsection}.  \documentclass[a4paper]{article}  \newcommand{\myparagraph}[1]{\paragraph{#1}\mbox{}\\}  \usepackage[     backend=bibtex,     isbn=false,     url=false,     doi=false,     eprint=false, ]{biblatex}  %Point to the bibliography db \addbibresource{mybib.bib}  \begin{document}  \include{myparagraph}  \end{document}   My paragraph file  \myparagraph{\fullcite{dante1320}} \begin{refsection}  \cite{manzoni1827}  \printbibliography[heading=subbibliography] \end{refsection}   I get the first citation (dante1320) right, but the second prints as [manzoni1827]. I don't get any error when I run Bibtex concerning the missing reference. But I got it when I run Latex: LaTeX Warning: Citation 'manzoni1827' on page 2 undefined on input line 5. The result of a Google search for &quot;error&quot; and &quot;PREP line&quot; is:  &quot;error&quot; &quot;on line&quot;: 439,000,000 results &quot;error&quot; &quot;in line&quot;: 60.600.000 results &quot;error&quot; &quot;at line&quot;: 35,800,000 results  While on is the indisputable number one, in and at have also very many hits. In the following context, which one should I use.  __ the first line you create an instance of XXX and XXX leads to the runtime error __ the last line.  I think, both prepositions are different. Although the word create refers to the action which is executed while runtime, the former part refers to the actual code that is written. In contrast, the latter part is figurative. I don't refer to the written code. The error does not really occur in the code but the code is the representation of the assembly which fails. Thus, I have two bests guess:  On the first line you create an instance of XXX and XXX leads to the runtime error at the last line. In the first line you create an instance of XXX and XXX leads to the runtime error on the last line. I have a WordPress site which is 90% based on images.  I've installed Wp-Super-Cache, now it seems to load a little bit faster.   I've made a speed test on Pingdom Tools:   Page size 4.7MB Load time 5.09s Requests 153 Perf. grade 74/100   And also the Time Spent per Content Type:   Image    88.23%     So the problem is obviously the images.   My plan is to try as much as I can to optimize the images in terms of size, and then to add CDN support.  Is this enough in terms of improving the loading of images? If a product of relatively prime integers is an $n$th power, then each is an $n$th power Show that if $n$, $a$, $b$, and $c$ are positive integers with $\gcd(a, b) = 1$ and $ab = c^n$, then there are positive integers $d$, and $e$ such that $a = d^n$ and $b = e^n$.   I know that (by Bezout) $\gcd\left(a,b\right) = 1$ implies $ax + by = 1$ for some integers $x$ and $y$, and also that $\gcd\left(a^n,b^n\right) = 1$, but this does not help me. In ArcGIS there is a Erase Point tool how can this be done in R Discourage questions being marked as answered within an hour or so of being posted Isomorphism between $\mathfrak o(4,\mathbb R)$ and $\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $ I just fit a model in lme4, and I'm wondering what the heck I fit...  I have individuals id, and each is measured pass/fail on items that can be described using two factors, f1 and f2. My theory says that id and f2 can interact. So I want to compare these models:  # f2 doesn't matter m1 &lt;- lmer(pass ~ f1 + (1|id), family="binomial")  # f2 is a fixed effect, doesn't interact with individuals m2 &lt;- lmer(pass ~ f1 + f2 + (1|id), family="binomial")  # f2 is a nested random effect, interacting with individuals m3 &lt;- lmer(pass ~ f1 + (1|f2/id), family="binomial")   First, am I right that there's a random effect of each level of f2, and each individual's level of f2 is partially pooled, depending on the number of observations for that individual?  Second, does it make sense to fit f2 as a fixed effect with random individuals nested within each level of f2? How does one write that?  Third, what's this, with f2 and id reversed in the formula? I typed it by accident, but now I want to understand it.  m3weird &lt;- lmer(pass ~ f1 + (1|id/f2), family="binomial") According to , the only restriction on self-deleting an answer is that you can't delete an answer that's been accepted.  But I've also heard of users getting in trouble for deleting (non-accepted) answers, and there's the CC-SA license to consider, so -- quite aside from what is technically possible -- my question is when it's permitted to delete your own answer.  Some cases to consider:   You now believe the answer is incorrect. Your answer is correct, but another answer covers the same territory better. You aren't satisfied with the quality/rigor of your answer, e.g. it's speculative and you didn't back it up (on a site that expects that). Something in the answer embarrasses you and you'd rather just make it go away. Your answer wasn't well-received by the community (whether that's "meh" or "make the rep bleeding stop please"). (I'm sure there are others; this isn't meant to be an exhaustive list.)   Does voting matter, either in absolute terms or in comparison to other answers?  For example, maybe it's ok to delete a lower-voted one but not the top-voted answer, or one with score > N??  There is a badge, Disciplined, for deleting your own post of score 3 or higher.  This tells me that it's sometimes ok to delete upvoted answers.  But I'm not sure how much policy we can infer from badges.  Could we have some clarification please? How does matchmaking work in Play mode? How does Hearthstone match you with opponents in Play mode, for both Ranked and Unranked? Does your record in Ranked affect matchmaking in Unranked (and vice-versa)?