Consider a surface diffeomorphic to a cylinder (the boundary is 2 curves diffeomorphic to circles). Suppose the boundary curves are geodesic, can the surface have non-vanishing gaussian curvature (i.e. $K \neq 0$ everywhere)?  I tried to use the Gauss-Bonnet theorem, but I'm not sure how.
The question is as follows.  Imagine we have two closed geodesics on a surface, where these two geodesics bound a cylinder together and the geodesics are diffeomorphic to circles.  Can the cylinder have $K&gt;0$ everywhere or $K&lt;0$ everywhere?  Here $K$ is Gaussian Curvature.   What's confusing me about this question is that a cylinder has Gaussian Curvature of $K=0$ everywhere, so wouldn't the answer to both questions be no?  But then I don't get the point of the question with the geodesics.  Thank you in advance
How to prove sum of powers property of roots of unity? Product of path connected spaces is path connected Will product of path connected topological spaces be necessarily path connected? Why or why not?  Give me some hints. Thank you. Is it fair to buy a return ticket I know I will never use just because it's cheaper? I am planning a trip to Asia and then back to Rio. The thing is that I am traveling on flexible dates, so I am thinking of buying a one way ticket to Hong Kong, and I was planning to do the same when I want to come back.  But what I noticed is that, for example, a one way ticket from Melbourne to Rio costs no less than $2700 dollars for the month of May.  When I do a search for a round-trip from Melbourne to Rio and back, I can find flights for $2200.  I know I will not be using that return to Melbourne, but I'd still prefer to save those $500.   Can there be any issue with buying a round-trip with a return that I will never use? Editing/Creating files in C:\Program Files\WindowsApps When deriving the quadratic equation as shown in the  () the main proof contains the step: $$ \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}} $$ the square root is taken from both sides, so why is  $$\sqrt{4a^2} = 2a$$  in the denominator and not $$ \sqrt{4a^2} = 2\left |a  \right | $$ Could somebody explain this to me? Thank you very much Local server updates for the network I have setup one computer on our network as the file server. Because Internet here in Tanzania is both slow and expensive I would like that one system to download all the updates and then the other 10 computers on the network to get those update files from the server. I'm a bit of a noobie to Ubuntu, but really want to learn how to get this working smoothly so as to help other NGOs and schools here in Tanzania. Brendon Proving the two given vectors are orthogonal I am given the vectors $\mathbf{w}, \mathbf{v}, \mathbf{u}$ in $\mathbb{R}^n$ such that $\mathbf{u} \neq 0$ and $\mathbf{w} = \mathbf{v} - \frac{\mathbf{u}\,\bullet\,\mathbf{v}}{\|\mathbf{u}\|^2}\bullet\mathbf{u}$. I am asked to show that the vector $\mathbf{w}$ is orthogonal to $\mathbf{u}$.     So far, I have written out the definition of orthogonal: two vectors are orthogonal if and only if their dot product is zero.   So what we need to prove is $\mathbf{w}\bullet\mathbf{u} = 0$ where $\mathbf{w}\bullet\mathbf{u}$ is defined as $\mathbf{w}^T\bullet\mathbf{u}$.   However, I have been stuck on this problem for about an hour and haven't made any significant progress from here. How do we go about proving that the vectors are orthogonal? What's wrong with these two cron jobs? I have the following cron jobs defined.  55  8   *   *   3   /usr/bin/php /home/mark/dev/processes/customClient/events.php &gt; /home/mark/dev/processes/customClient/events-`date +%Y-%m-%d --date='last Wednesday'`-`date +%Y-%m-%d`.csv 0   9   *   *   3   /usr/bin/echo 'The csv for last week, trying my hand at automatiging this' | /usr/bin/mutt &lt;emailaddress&gt; -s 'Events from `date +%Y-%m-%d --date='last Wednesday'`-`date +%Y-%m-%d`' -a '/home/mark/dev/processes/customClient/events-`date +%Y-%m-%d --date='last Wednesday'`-`date +%Y-%m-%d`.csv'   It seems to work properly if I run it the above command directly from the command line. But when I checked the running of the script this morning I got an e-mail stating (I'm paraphrasing because I accidentally deleted them) that the back ticks weren't closed properly. Given function $g$ continuous in line $[0,\infty)$, positive and strongly increasing that upholds $\lim_{t\to\infty}  g(t) = L$. Calculate the limit:  $$ \lim_{n\to\infty} \int\limits_1^2 g(nx) \, dx. $$  At the moment what I think that I should do is swap the $g(nx)$ with $L$ as it approaches infinity but from there on I am just not sure whats my next step is. How widely-accepted is "What do you got?" to Americans? Watching , I noticed that Melanie Griffith twice asked    I recognise this as an American construction which sounds strange to me — Brits invariably say either "What have you got?" or "What do you have?". But I'd be interested to know if it's considered "normal" by most/all Americans, or if it's regionally or otherwise restricted. Calculate the factorial of an arbitrarily large number, showing all the digits C# has the keyword called . VB.NET lacks this keyword. How have the Visual Basic programmers gotten around the lack of this keyword? Do they implement they own iterator class?  Or do they try and code to avoid the need of an iterator?  The  keyword does force the compiler to do some coding behind the scenes.  has a good example of that. A charged particle undergoing an acceleration radiates photons.  Let's consider a charge in a freely falling frame of reference.  In such a frame, the local gravitational field is necessarily zero, and the particle does not accelerate or experience any force. Thus, this charge is free in such a frame. But, a free charge does not emit any photons. There seems to be a paradox. Does a freely falling charge in a gravitational field radiate? When to pronounce ‹s› as /z/ in the middle of words? Recovering added/staged file after doing git reset --hard HEAD^? Why can't variable names start with numbers? Where can I ask questions regarding Electronics? How can I rewire half switched receptacles? In my bedroom the bottom half of all receptacles are controlled by one switch. Is it possible to have just one receptacle with top and bottom hot all the time?  The top half is hot all the time the bottom half is controlled by one switch.  Keep in mind there are six other receptacles in this room that are the same way controlled by the same switch. When I took the receptacle out the right side tab had been removed.  The top right screw had a black wire, the bottom right had a red wire, the top and bottom left screws had separate white wires.  I changed the receptacle out as suggested, I capped off the red wire and I capped off the white wire that was attached to the bottom left screw. I attached the other white wire to the top left screw and attached the black wire to the top right screw. I now have none of the receptacles working at all. What did I do wrong? Is cancer caused by vitamin B17 deficiency? crossfade between 2 videos using ffmpeg I've been trying to achieve a crossfade transition between 2 video clips using ffmpeg but have failed so far. I'm new to ffmpeg and am mostly relying on tweaking what I can find in the documentation and existing examples online. From what I read so far, using either the blend or overlay filter should help in achieving what I'm after but I can't figure out the command line details to get it to work.  The fade and concat filters are great for fade-out of video 1, fade-in to video 2 and concat the 2 into 1 clip type transitions but I'd appreciate help in getting a command to transition from video 1 to video 2 without any going to black in between. I couldn't find any examples for exactly this problem anywhere, maybe I'm looking for the wrong keywords...?  More speficially, my videos are mp4s (h264 video, no sound, in case that matters), each is 5 seconds long and I'm after a transition from approx. 4.5s of video 1 to 0.5s of video 2.  Similar to what this tutorial does using MLT and frames (see 2:25 for an example fade), though I'm looking for a way to do this just in ffmpeg without calling any other progs.   Any pointers or maybe a command line to get a fade like this would be much appreciated, thanks very much!