I have a problem where I am supposed to solve a PDE and then tell where in the plane that it has a unique solution, I found an almost identical question but didn't understand the answer that was given. I will post both below and I ask for  an explanation of the answer.  It seems like the basic premise of the answer is that if the characteristic curves do not intersect the intial curve then we cannot guarantee a unique solution. Why is this? Why would there necessarily be a unique solution if our characteristic curves did intersect the initial curve?    Question:  I am given a 1st order partial differential equation $y\frac{\partial \psi}{\partial x}+x\frac{\partial \psi}{\partial y}=0$ subjected to boundary condition $\psi(x,0)=\exp(−x^2)$. I have found that a solution is $ψ(x,y)=\exp(y^2−x^2)$. But I am asked when the solution is unique. Could someone please explain how to answer this? Thanks.  Answer:  Consider the parametric curves $x=Ae^{t}+Be^{−t}$, $y=Ae^{t}−Be^{−t}$, which satisfy $x′=y$, $y′=x$. Along such a curve any solution $\psi$ must be constant, according to the chain rule $$\frac{d}{dt} \psi (x(t),y(t))=\psi_x\frac{dx}{dt}+\psi_y\frac{dy}{dt}=0$$ Now the curve intersects $y=0$ if and only if $A$ and $B$ are either both positive (i.e. $x \gt \lvert y \rvert$), both negative ($x \lt \lvert y \rvert$), or both $0$ ($x=y=0$). So a boundary condition on $y=0$ produces uniqueness only in the regions $\lvert x \rvert \ge \lvert y \rvert$. In the region $\lvert y \rvert \gt \lvert x \rvert$ the solution is not unique. For example, you could add $f(y^2−x^2)$ to $\psi(x,y)$ where $f$ is differentiable with $f(s)=0$ for $s \le 0$.    EDIT  Maybe it is worth mentioning that I commented on the answer in the original question, but I did not get a reply. So asking again is necessary....
(a) Solve the equation $yu_x+xu_y=0$  with the condition $u(0,y) = e^{-y^2}$. (b) In which region of the xy plane is the solution uniquely determined?     I did the first part but I don't understand how to do part b.  I read   But I don't understand how the argument is based. A more clear explanation is appreciated
What is this element on the far right of schematic described as "270 OHM"? Is this some kind of inductor? What is its purpose in this device (this is a reference design of dc-dc power supply of sim900 taken from its datasheet) and what would be an exemplary part number? I'm given a matrix A, its dimensions are n x n.  I am required to prove that an invertible matrix B exists, such that the product of the matrices BA is triangular.  Any help? Is QImage able to open and render pure 16-bit images? How can I compute $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})$? How can I compute $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})$?  If $\varphi$ is one of this automorphism then $\varphi((1,0))=(1,0),(1,3),(3,3),(3,0)$ and $\varphi((0,1))=(0,1),(0,5),(2,1),(2,5)$.  So it seems to me that this groups contains 16 elements, am i right? And if I'm right there is a way to understand to which group of cardinality 16 this is isomorphic to?  EDIT: $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}\cong\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$. And $\mathbb{Z}/8\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$ are characteristic (am I right?), so $\mathrm{Aut}(\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z})\cong\mathrm{Aut}(\mathbb{Z}/8\mathbb{Z})\times\mathrm{Aut}(\mathbb{Z}/3\mathbb{Z})\cong\mathbb{Z}/8\mathbb{Z}^*\times\mathbb{Z}/3\mathbb{Z}^*$ and this group has cardinality 8, right? Where is my mistake? Are submodules of finitely generated free modules always free? I don't believe it's true that all submodules of a finitely generated free module are free just based on results from google. Is there a canonical simple example of a finitely generated free module with a submodule that is not free? How much creative independence are the Show runners accorded in Game of Thrones? Since Season 3, we have been observing that the show was taking a different direction than the books. Many important characters never made it into the show, others who are still alive were killed off.   As we know the show is supposed to be based on the books, not to be direct adaption of the books from print to television media, there's bound to be a certain degree of creative independence accorded to them by the author of the series, George R.R. Martin.   The question is, how much independence do they have? Do they still have to take GRRM on board with whatever changes they might intend to make? By "they" I mean D&amp;D and the whole writers crew for the show. Given the number of significant changes they have made, I reckon the answer would be a lot. Has GRRM or HBO ever commented on that? Cursor is set to move by increments. how to reset to normal? My cursor is moving by every selection by large increments!  Normally when I use Blender, I am able to select and move objects to any coordinate. Now, all of the sudden, when I select an object or mesh the selection moves by an inch or so in each direction.  How do I reset it to normal? Where are my installed applications? What is the best way to dump entire objects to a log in C#? What diceless tabletop RPG techniques are there? I am interested in using a diceless resolution system for my roleplaying, but am not sure what my options are in that vein. I can think of basically two categories of diceless tabletop role-playing systems - systems which eschew any randomisation, and those which randomise by some method other than dice.  Examples  Non-random tabletop systems  One option is of direct comparison of stats or other game statistics without a randomization factor. The obvious example here is Amber DRPG, but I suspect that there are other narrative driven RPGs out there that I don't know about.   Even in Amber, situation resolution it is rarely by direct comparison between stats. Narrative techniques can be used to manoeuvre two characters into a situation which narrows the gap between their ranking, widens it, or brings other stats into the picture - the classic example of course being Corwin relying on his stamina to stretch out conflicts with his siblings such that it becomes a dominant enough factor that he can escape.  Other randomisation techniques  I have seen card-based randomization system. Over the years, I've played quite a few home brew games based on the original SAGA Dragonlance system, which was similar to the SAGA Marvel superhero system.  The basic mechanics were that you had a hand size according to your experience, and for any skill check you would play a card from your hand and replace your card from the draw pile. If you had a card of the right suit as the skill check (a strength card on a strength check) you would also draw a card from the draw pile and add that to your skill check, so a small trump was often worth more than a medium value card not of the correct suit. When you were wounded, you would discard cards to the value of the damage, but those cards wouldn't be replenished - your hand size would be reduced until you were healed. One nice aspect of the system was that it simulated fatigue beautifully, you might start a scene with a spread of high, medium and low cards, but as your high cards were played they would more often than not get replaced by lower value cards, until you were left with a bunch of useless cards and feeling very vulnerable.    The only other system I've played without dice was Everway which, if I remember correctly, used a combination of these techniques, stats based resolution, card resolution and GM fiat.  What I would be interested in is what other diceless resolution techniques exist, ideally with examples of tabletop diceless resolution methods people have tried and how they found them to play. Is the Hogwarts Express in the magical or non-magical realm? This question is related to  and explores an apparent inconsistency, given only the Harry Potter movies as source material.  The Hogwarts Express line seems to have an inconsistent existence between magical (invisible to ordinary humans aka muggles) and non-magical (muggle) realms; platform 9-3/4 is clearly hidden from muggles, but in Chamber of Secrets, when Harry and Ron take the car (in the muggle world) to catch up to the train, they make no explicit crossing into a magical realm, suggesting that the train is traveling the muggle world despite its departure from a magical platform.  In which realm (magical or muggle) does the Hogwarts Express exist? How is this apparent inconsistency resolved? Why does Oracle 9i treat an empty string as NULL? Will factory resetting my device remove the Android upgrade? I have upgraded my phone to Android 2.1. I need to reset my phone but I don't want to lose the upgrade. Is it possible to do this? How? Execute an installed Python package as a script? If $\sum_{n=1}^{\infty} a_n$ converges, then so does $\sum_{n=1}^{\infty} a_n^{\frac{n}{n+1}}$ Given a positive sequence $(a_n)$. Prove that if $\displaystyle \sum_{n=1}^{\infty} a_n$ converges, then so does $\displaystyle \sum_{n=1}^{\infty} a_n^{\frac{n}{n+1}}$.   I tried to use the root test, and I was stuck with the case $\lim \sup \sqrt[n]{a_n}=1$. Maybe we need some trick here.  Thanks so much. How can I find help for new Ubuntu users? I've just started using Lubuntu 13.10. It's the first time I've ever used Linux.  Can any of you send me a tutorial-video for new users or explain the basics to me? Intuition for groups This is quite a non-standard question, certainly for mathematics, though I believe it is no less important (for me and my peers, i.e. grads).  The course I am reading so far introduced us to Groups, rings, fields, etc. in the first year, progressing to characters, reps etc in the latter years.  However, even with representation theory I still don't feel like I have a good intuition.  How do you gain intuition in such an area. I know there will be a response of doing examples, but this doesn't really help - I've done lots already.  Am I doomed. I want to run an app in 'kiosk' mode for public unattended access via touch screen (no mouse or keyboard attached). I can already automate a network user logon and then the app launches (but only in last state - not necessarily full screen - which I want) I know with Workgroup Manager you can hide the dock, use simple Finder etc...but need something more powerful. Ideally it needs to monitor if the app crashes out and restart it / log limited user off and back on again.  Thanks  Phil Tor browser not opening Attack moving or Attack Canceling I watch streams sometimes and i see the high elo Ranged AD move between each shot. Do they gain anything with this? And if so whats a good way to practice it.