Weird artifacts when rendering on Nvidia GPU
Cause of black artifacts in render?
Counterexample: $G \times K \cong H \times K \implies G \cong H$ I am having a hard time finding a counterexample for the statement: $G \times K \cong H \times K \implies G \cong H$  I think this should be true for abelian, finite groups. But is this true in general? What would be a counterexample?  Any hints appreciated!  Thanks Using have gone to / been to / been in I understand the difference between these constructions: "has gone to", "has been to", "has been in":   "has gone to" &mdash; there or on his way to; "has been to" &mdash; someone has been there but he is on his way back; "has been in" &mdash; someone is still there;   But now I am puzzled by next statements:   Have you ever been to / in Greece? &mdash; Yes, it's a very beautiful country; When did you last go to Canada? &mdash; I've never been to Canada.   If I use "has you ever gone to Greece" construction in the first case or "I have never gone to Canada" in the second statement will it be uncorrect? Or just will sound odd? May I say "I have never been in Canada" in the second sentence. Or in these case all of the variants are possible, the problem is in how people usually say it (in what how it perfectly sounds)? In which cases is a comma/period placed inside or outside of parentheses? Are there cases where a period is placed inside parentheses? In which cases the period (or a comma) should be placed inside the parentheses? What's the simplest way to list conflicted files in Git? Monty hall problem extended. I just learned about the Monty Hall problem and found it quite amazing. So I thought about extending the problem a bit to understand more about it. In this modification of the Monty Hall Problem, instead of three doors, we have four (or maybe $n$) doors, one with a car and the other three (or $n-1$) with a goat each (I want the car).  We need to choose any one of the doors. After we have chosen the door, Monty deliberately reveals one of the doors that has a goat and asks us if we wish to change our choice.  So should we switch the door we have chosen, or does it not matter if we switch or stay with our choice?  It would be even better if we knew the probability of winning upon switching given that Monty opens $k$ doors. When I try to create a reaction from chemmacros/reactions I got the following error. I dunno what I am missing. Please help. Thank you.     line 11: Undefined control sequence. \end   line 11: Missing number, treated as zero. \end   line 11: Missing number, treated as zero. \end     Minimum Working Example    \documentclass[12pt,a4paper]{article} \usepackage{chemmacros} \chemsetup{modules={reactions}} \begin{document} \begin{reaction}     A + B -&gt; C \end{reaction} \end{document} How can I convert the binary string $x_bin="0001001100101" to its numeric value $x_num=613 in Perl? What 70s (60s?) TV show / TV movie had a generation ship whose inhabitants did not know they were on a spaceship? I have the vaguest memory of a TV show (maybe a BBC import?) in which the protagonists were moving through the sections of a massive generation ship whose inhabitants had forgotten that they were on a spaceship... I know there was an RPG with the same plot (which, after all, is a pretty great blank slate) but I definitely feel I saw this on a static-filled 13" B&amp;W TV back in the day... What kind of ship was flying away from Rey in her flashback? The ship flying away from Rey in her flashback of being left on Jakku looks like one of the ships in the group that takes her away when she is captured.  If so, and it is a First Order ship that dropped her off, this is an interesting Easter egg. Determine resultant velocity of an elastic particle-particle collision in 3d space So I have two particles that have collided in 3 dimensional space. I want the particles to rebound off of each other in an elastic manner. How do I determine the resultant velocity vector if I know: Both particles' initial velocity vector, their masses, and their initial position vectors relative to the origin.  I tried to follow the instructions on  but I am not sure how to make their equations work without polar coordinates / how to translate my data to polar coordinates.  Also, if you answer my question in terms of polar coordinates, can you explain which angle is theta and which is phi? Since there are 2 different conventions, I am not really sure which angle is being referred to in the explanation on the aforementioned website.  EDIT: This isn't for a class. I'm writing a physics engine for a gravity simulation I am making using Unity. I haven't taken kinematics since I was in high school, which is why I am so rusty. Why is $\omega$ the smallest $\infty$? I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. So if we have a set $\Omega$ and $|\Omega| = \aleph_i$, then (assuming continuum hypothesis) the cardinality of $2^{\Omega}$ is $|2^{\Omega}| = \aleph_{i+1} &gt; \aleph_i$ and we have $\aleph_{i+1} = 2^{\aleph_i}$.  I am fine with these argument.  What I don't understand is why should there be a smallest $\infty$? Is there a proof (or) an axiom that there exists a smallest $\infty$ and that this is what we address as "countably infinite"? or to rephrase the question "why can't I find a set whose power set gives us $\mathbb{N}$"?  The reason why I am asking this question is in "some sense" $\aleph_i = \log_2(\aleph_{i+1})$. I do not completely understand why this process should stop when $\aleph_i = \aleph_0$.  (Though coming to think about it I can feel that if I take an infinite set with $\log_2 \aleph_0$ elements I can still put it in one-to-one correspondence with the Natural number set. So Is $\log_2 \aleph_0 = \aleph_0$ (or) am I just confused? If $n \rightarrow \infty$, then $2^n \rightarrow \infty$ faster while $\log (n) \rightarrow \infty$ slower and $\log (\log (n)) \rightarrow \infty$ even slower and $\log(\log(\log (n))) \rightarrow \infty$ even "more" slower and so on).  Is there a clean (and relatively elementary) way to explain this to me?  (I am totally fine if you direct me to some paper (or) webpage. I tried googling but could not find an answer to my question) Center Gravity in Unity Search in favorited questions? What's the derivative of: $ \sqrt{x+\sqrt{{x}+\sqrt{x+\cdots}}}$? let $ y=\displaystyle \sqrt{x+\sqrt{{x}+\sqrt{x+\cdots}}}$, i'm really interesting to know how do I find :  $\displaystyle \frac{dy}{dx}$ ?.  Note: I have used the definition of derivative of the square root function but i don't succed .  Thank you for any help Extraneous solution from substituting in equations I came across this example of how you could end up with an extraneous solution but I was wondering how it arose. We have the equation: $$x^2+x+1=0 $$ Since x=0 does not satisfy the equation, you can divide by x on both sides which yields: $$x+1+\frac{1}{x}=0$$ which is equivalent to our first equation. From our first eqution we can conclude that: $$-x^2=x+1$$ We now substitute that into the second equation to get:$$x^2=\frac{1}{x}$$ which results in $$x^3=1$$ which is equivalent to our previous equation since x cannot be 0. However, one solution from our last equation is x=1, which is not a solution to our original equation. I have a vague idea that it may have to do with the fact that you get a cubic equation and you began with a quadratic, and that steps imply the following and not vice versa, but can you provide a very detailed answer as to why it arises? Can you please provide more examples? Ping faster than light I just discovered very strange thing while testing my internet connection. My ping is smaller than it should be. For example ping time to Arizona State University is about 14ms.  eryk@eryk-pc:~$ ping www.asu.edu PING www.asu.edu.cdn.cloudflare.net (104.16.51.14) 56(84) bytes of data. 64 bytes from 104.16.51.14: icmp_seq=1 ttl=60 time=13.8 ms   I'm living in Poznań in Poland so my distance to Arizona State University in a straight line (very optimistic assumption) is about 10000 km. Considering the fact that ping time is a time for two directions (to the target and back to home). So my ping packet have to cross 20000 km distance. Speed of light is 300 000 km per second that is 300km per millisecond. So the smallest possible time to ping Arizona State University with my packet travelling at a speed of light is  20000/300=67ms   I have noticed similar results for servers located in Australia and some other States of America. Is my internet connection five times faster than light? uname -r and available linux-headers- packages do not match I was asked today in class why you divide the sum of square error by $n-1$ instead of with $n$, when calculating the standard deviation.  I said I am not going to answer it in class (since I didn't wanna go into unbiased estimators), but later I wondered - is there an intuitive explanation for this?! How to type latin small letter c with cedilla I run Ubuntu on a DELL XPS 13 notebook that has an US keyboard configured as US international. After upgrading to version 13.10 I can no loger get a cedilla by typing '+C. Instead, I get a "ć".  What should I do to get a "ç"?  I am using Ubuntu Gnome 13.10 64 bit. In Random forest algorithm, Breiman (author) constructs similarity matrix as follows:   Send all learning examples down each tree in the forest If two examples land in the same leaf increment corresponding element in similarity matrix by 1 Normalize the matrix with number of trees   He says:     The proximities between cases n and k form a matrix {prox(n,k)}. From their definition, it is easy to show that this matrix is symmetric, positive definite and bounded above by 1, with the diagonal elements equal to 1. It follows that the values 1-prox(n,k) are squared distances in a Euclidean space of dimension not greater than the number of cases.    In his implementation, he uses sqrt(1-prox), where prox is a similarity matrix, to convert it to distance matrix. I guess it has something to do with the "sqaured distances in a Euclidean space"-quoted above.  Can somebody shine a little light on why it follows that 1-prox are squared distances in a Euclidean space and why he uses squared root to get distance matrix?