Finding the volume of a torus
Volume of the torus with an integral
Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$ Let $a_1, a_2, a_3, \dots , a_n$ be positive real numbers whose sum is $1$. Prove that $$\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \ldots +\frac{a_n^2}{a_n+a_1} \geq \frac12\,.$$  I thought maybe the Cauchy and QM inequalities would be helpful. But I can't see how to apply it. Another thought (might be unhelpful) is that the sum of the denominators on the left hand side is $2$ (the denominator on the right hand side). I would really appreciate any hints. Applications of ultrafilters I'm looking for some interesting applications of ultrafilters and also everything of interest involving ultrafilters. Do you know some applications or interesting things involving ultrafilters?  I'm at the beginning, so I'd prefer some applications that also a beginner could read. Ant-Man, shrinking, and weight I am trying to make a top down shooter that could potentially have hundreds of objects at once at one point. I need to check collision with projectiles, walls, and enemies. I am currently using ArrayLists which probably won't suffice later on and was wondering if someone had a good method or knows a good method for checking all of those objects at once. This is going to be a game for smartphones and is built in java. Any ideas would be great or if you could point me in some direction that would be fantastic.  Thanks in advance! How to determine the cause of MacBook battery not charging? I see that the question &quot;why is my MacBook not charging?&quot; gets asked a lot. The problem is, there are multiple reasons. Possible points of failure are: charger, cable, magsafe connector, DC input board, logic board, battery. (Have I missed any?). What is the best sequence for a user to follow, to diagnose the problem? By way of example, here are the first steps that I usually try:  Make sure the charger is working. Either use your charger on a different Mac, or test a spare charger with your Mac.  Check the connection. There might be gunk in your port that's stopping the connector from lying flat.  Try resetting the SMC and PRAM.  If the computer boots, run Coconut Battery to determine battery status.   This is where I get a bit stuck. For example, I have a machine here that boots fine when plugged in, but not from battery. Disconnecting the MagSafe causes the machine to switch off. Sounds like it's a dead battery, except that Coconut reports the battery as detected, in good condition, but not charging. What steps should I take next? Specifically, I'm asking for a generic algorithm to help determine the source of the fault (preferably without swapping out lots of components one by one)? Network simplex method, leaving and entering variables I have read so many things already that I'm more confused than before. So decided to ask this by myself.  My paladin just got his Find Steed spell, and after some tweaking and talking, he decided to get a Worg and I was okay with it. This last session was his first time using it and I have few doubts when it comes to combat.  I know that while mounted, he uses the mounts movement and actions. But what about when unmounted? What if the combat began with him walking by his mount's side?   Does the mount share his initiative? Are the actions that the mount do, part of the Paladin's actions? Or are they both separately, having his Action, Movement and Bonus Action and then the mounts Action and movement?   We've stated that it can understand him but not talk back, he just telepathically receives feelings as a feedback to his commands, knowing that the mount understood what he needed. Last session he just said things to the mount like "I tell the mount to protect the ranger and the druid" and in case they got hit, it would attack them, but it didn't really do anything at all, as I didn't know when it would act and I had too many things already to remember about the mount. Plotting two variables as lines using ggplot2 on the same graph Integral equation problem An operator has closed range if and only if the image of some closed subspace of finite codimension is closed. Is there a fairly simple way to divide a circle into equal parts using Adobe Illustrator CC? The value of $\lim_{n \to \infty}\int_{0}^{1}nx^ne^{x^2}dx$ The value of $\lim_{n \to \infty}\int_{0}^{1}nx^ne^{x^2}dx$ is _____ I tried by taking the odd $n$ values as in that case, the integral I suppose was easier to calculate. So, denote $I_n=n\int_{0}^{1}e^{x^2}x^ndx$, then we have : $I_1=\frac{e-1}{2},\\I_3=\frac{3}{2},\\I_5=5(\frac{e}{2}-1),\\I_7=7(3-e)$ Then I tried calculating (random) values using calculator as integration was getting cumbersome. $I_{31}=2.488$, $I_{51}=2.57$. I do not see any kind of recurrence so that I can find a general term for odd $n$. I also tried the method given . The idea was that since integral doesn't depend on $n$ considering $I(n)=\int_{0}^{1}e^{x^2}x^ndx$, then $I'(n)=\int_{0}^{1}x^n\ln(x)e^{x^2}dx$, but this also doesn't lead me to any conclusion as well. I believe I couldn't get to a proper approach to tackle this question. Can someone please help with the idea that involves in solving these type of questions? How precisely do I need to post close to 7:00 (somewhere) to get hero of time? Timezones... are complicated. We have full hour offsets, half hour offsets...  And time keeping isn't always accurate. So does the Hero Of Time hat assume full or half hour offsets (or literally any timezone) and how close to 7:00 at somewhere do I have to post for my hat? Hi Blender coders and artists.  I am updating one of my add-ons for Blender 2.8 and have run into some  issues finding the appropriate way to hide a collection that is added to my scene.  I want to deselect the eyeball on the toolbar with a python command in a script but I don't want to disable the collection in the view-port entirely as it is used as a collection in a particle system.   I just want to deselect the eyeball circled in green below.  Anyone know the python API for this? What is the likelihood of ever discovering the graviton? How would one look for and confirm existence of a graviton?  Someone was speaking to me about perhaps one day discovering the graviton, but to me it seems unlikely, although I'm young and essentially quite naive, so am coming to you physicists to ask    What actually is the likelihood of finding it?  How would we find it? Chat Feature-Request: Thumbs-Up Add an additional form of "I agree" besides starring or pinning (since pinning is restricted to a smaller subset to begin with) or something that we can use to not clog up the right hand star bar.  This is because sometimes you want to say "I agree" but you don't want to star it because ... let's face it, that's not star-worthy, and you don't want to clutter the stream with fifty thousand "like totes for sure, hella right!" because then it becomes one big back-patting party and you lose all semblance of conversation.  So I would propose a fourth "flag" for chat, that allows us to say "I {agree with|like|am of the same opinion as} this comment" and that it would only be useful in the live-chat (and maybe but not really in the transcript) and that it have no other useful semantics (no badges, no list of's [unless the list is of comments you've upthumbed]). How can I copy the contents of a folder to another folder in a different directory using terminal? I am trying to copy the contents of a folder to another folder in a different directory using terminal.  Would somebody be able to provide me an example of the command line syntax required to achieve this? Can you recommend a Java library for reading (and possibly writing) CSV files? Silencing "Your disk is almost full" notification After upgrading to macOS Sierra, I'm getting the notification that "Your disk is almost full. Save space by optimizing storage.":    The options appear to be to store my files in iCloud, automatically delete files, or manually delete files:    My problem/irritation is that I have 80GB free of my 440GB volume.  The question: is there a way to silence this notification in a (semi-)permanent fashion, or to change the threshold it uses for the notification? Deleting missing data with CountryData