center of free product of groups
Center of a free product
PayPal API Listener Website Payments Standard URI What's the weight of the earth? What's the weight of the earth?  That is, what's the weight of the earth in the gravitational field of the sun? I imagine it's either   $0$, because there's no net fall toward the sun, so the downward component of acceleration is $0$ and $F=ma=0$, or $Gm_1m_2/r^2\approx7\cdot10^{-11}\cdot6\cdot10^{24}\cdot2\cdot10^{30}/(1.5\cdot10^{11})^2\approx4\cdot10^{22}$ Newtons,   but which, and why? Can a level 1 Warlock Cast a 4th level spell from 8th level Sorcerer with their warlock pact magic spell slots? Can my player’s multiclass Warlock 1/Sorcerer 8 cast the higher level sorcerer spells from with their warlock pact magic spell slots?  My player argued that he could now cast his 4th level sorcerer spells with his level 1 warlock  spells slots. I think he interpreted pact magic as "highest spell slot possible",  so he figures that since he has 4th level spell slots, he interprets that the warlock spell slots are 4th level slots now too. What TV movie had lost space cadets who piloted yellow tripod robots, and a squid bad guy? I'm trying to find a movie I watched on TV, probably in the 90s. I think it was a made-for-TV movie.  It was about a ship with an AI. The ship was crewed by young adults. I think they were cadets?   There was an accident (like a warp drive malfunction) and they end up in another part of space where they meet a female alien whose ship had been shot down.  I remember they had manned robots they would pilot and they took them to a planet to get upgraded and they are attacked by more manned robots. They had three legs and they were yellow, I think.  I also remember the baddie had a holo-projector or something like that which allowed him to change his appearance. He commanded a massive fleet and he worked for someone he spoke to in a video chat that looked like a squid. Bevel is not symmetrical How to float until on left (or right) page? I have two sideways tables that I want on two facing pages so the reader can turn the entire book and read them both in one turn. For the moment I keep moving them around to keep them on facing pages. Is there a way I can make the first one float until it's on a left page and then place the other one right after? Or perhaps some similar way to automatically get them on the same spread? The dock doesn't work correctly after I enter my password, either. English questions and negation with *do* in syntax I want to prove that $\lim_{(x,y)\to (0,0)} \frac{\sin(x+y)}{x+y} = 1$  Is it sufficient to say that if $u = x+y$ then as $(x,y) \rightarrow (0,0)$ then $u \rightarrow 0$ and then evaluate $\lim_{u\to 0} \frac{\sin u}{u}$ which is 1 by L'Hôpital's rule?  I feel that this is somehow "cheating" and I would really like to prove it in a more rigorous way, i.e., by using an $\epsilon$ - $\delta$ argument.  I have tried for a long time to use the one variable version and prove that if $0 \lt \sqrt{x^2 + y^2} \lt \delta$ then somehow $0 \lt |x+y| \lt \delta$ and it would follow from the one variable case that $| \frac{\sin(x+y)}{x+y} - 1| \lt \epsilon$. However, because $|x+y| = 0$ when $y = -x$ I am not getting anywhere with this.  Any input on this would be helpful :) How do I restrict scaling to a specific axis? Im new to Blender but have some experience with 3dsmax. In blender, when i select some polys and try to scale them, i have no control on the amount and especially relative axis on which the scale would happen, where as with 3ds max i could scale restricted to x, y, z or any combination of using the handy widget. Is there a way to replicate this, or to get any control over the scale ?  thanks, jQuery: live() vs delegate() java.net.SocketException: Connection reset I am getting the following error trying to read from a socket. I'm doing a readInt() on that InputStream, and I am getting this error. Perusing the documentation this suggests that the client part of the connection closed the connection. In this scenario, I am the server.  I have access to the client log files and it is not closing the connection, and in fact its log files suggest I am closing the connection. So does anybody have an idea why this is happening? What else to check for? Does this arise when there are local resources that are perhaps reaching thresholds?    I do note that I have the following line:  socket.setSoTimeout(10000);   just prior to the readInt(). There is a reason for this (long story), but just curious, are there circumstances under which this might lead to the indicated error? I have the server running in my IDE, and I happened to leave my IDE stuck on a breakpoint, and I then noticed the exact same errors begin appearing in my own logs in my IDE.  Anyway, just mentioning it, hopefully not a red herring. :-( Limit of $x\left(\left(1 + \frac{1}{x}\right)^x - e\right)$ when $x\to\infty$ I am stuck on how to calculate the following limit: $$\lim_{x\to\infty}x\left(\left(1 + \frac{1}{x}\right)^x - e\right).$$  Definitely, it has to be through l'Hôpital's rule. We know that $\lim_{x\to\infty} (1+(1/x))^x = e$. So, I wrote the above expression as $$\lim_{x\to\infty}\frac{\left(1 + \frac{1}{x}\right)^x - e}{1/x}.$$Both numerator and denominator tend to zero as x tends to infinity. I applied l'Hôpital's rule twice, but I got the limit equal to infinity which is clearly wrong. In the book, it says that the limit should be $-e/2$.  Any help please? Also, in case someone managed to solve it, please tell me which numerator and denominator did you apply your l'Hôpital's rule to in both cases? Thanks a lot  Also, guys can you tell me how to write equations in this forum? Thanks. Why is my fire simulation so light? I have been struggling to get a match tip to actually look like it's on fire and no matter what material shaders I try, I can't seem to achieve what everyone else does, I can in other scenes, just not this one. :( Estimating the most important features in a k-means cluster partition FastBoot recovery failed: partition 'recovery' not support flash Will ArcGIS for Desktop run on Windows 10? Will ArcGIS 10 run on the new windows 10 technical preview?  I am setting up an interface on my Mac and there is the free option of downloading the Windows 10 technical preview, some sort of beta version, I'm assuming.   Just wanted to see if 10.1 will run ok on Windows 10 (ignoring the Mac aspect), as this is all I am using my windows interface for? HUD is not working after upgrade My HUD is not working after I upgraded from 11.04 (and or 11.10).  It prompts for command and it tries loading for 4 seconds then gives up. why do academics write peer reviews? Realtek RTL8723be doesn't have good reception