I'm taking an intro to topology course, and am having trouble with this question. What is the fundamental group of $\mathbb{C^*}/\{e,a\}$, where $e$ is the identity homomorphism and $az=\overline{z}$.   My thoughts are that $\mathbb{C^*}$ is a $G$-space with $G=\{e,a\}$, so if I know what $\mathbb{C^*}/\{e,a\}$ is isomorphic to then I'll be able to figure out the fundamental group for this quotient space. But so far, we only know the fundamental group of simply-connected spaces, the circle, and products of spaces, so I guess it has to be isomorphic to one of these spaces.   Can someone please help out?
What is the fundamental group of $(\mathbb{C} \setminus {\{0\}})~/~\{e,a\}$, where $e$ is the identity homeomorphism and $az = -\bar{z}$?  Clearly this is homeomorphic to the half cylinder , which is homotopy equivalent to the half circle.  But what is the fundamental group of the half circle?
Can you link external USB hard drives in a RAID-style system? When should I use a fixed or variable time step? URW-Garamond with mathdesign is typesetting vertical double-quote as vertical double-quote I just started using URW-Garamond with mathdesign, and it's typesetting straight double-quote character as a straight double-quote rather than a right double-quote.  How can I fix this?  That is, when I put this in my header:  \usepackage[T1]{fontenc} \usepackage[urw-garamond]{mathdesign}   I get a nice Garamond in my output, but " in my input file produces a Garamond version of the same character in the output.  If I use two single-quotes instead, it comes out properly as a double right-quote.  I could switch to typing '' or set up a macro, but this is undesirable for more than one reason.  Is there any way to use urw-garamond without this non-traditional behavior for "? Fancyhdr does not apply same header/footer on chapter and non-chapter pages I accidentally set an object as camera, how do I undo this? I have a problem with rendering in cycles. In blender render my object is rendering fine, but in cycles when I pushed F12 I got only this:    One more thing: when I click 0 on keyboard I'm not in camera persp, but in object as camera, it looks like invisible layer, I don't know how to undo this, and be in camera persp again.  Does anybody have any idea how to fix this? Tikz: shading region bounded by several curves I know that my effort below can be improved upon considerably.  In particular, there must be a way of shading the region in a single scope, as opposed to subdividing as I have done (thus necessitating a hidden gray boundary).  Neither am I fond of sample points, but this is where my research has led me.  Your constructive criticism is appreciated.    \documentclass{article} \usepackage{tikz} \begin{document}  \begin{tikzpicture} %graph \draw[draw=gray!50!white,fill=gray!50!white]      plot[smooth,samples=100,domain=0:1] (\x,{0}) --      plot[smooth,samples=100,domain=1:0] (\x,{1}); \draw[fill=gray!50!white] plot[smooth,samples=100,domain=1:2.71828] (\x,{ln(\x)}) --      plot[smooth,samples=100,domain=2.71828:1] (\x,{1}); \draw[domain=0:4] plot (\x,{1}); \draw[samples=100,domain=.25:4] plot (\x,{ln(\x)}) node[above left]{$y=\ln x$}; %coordinate grid \draw (-.5,0)--(4.5,0) node[right]{$x$}; \draw (0,-1.5)--(0,2.5) node[above]{$y$}; \foreach \x in {1,2,3,4}     \draw (\x,2pt)--(\x,-2pt) node[below] {$\x$}; \foreach \y/\ytext in {1,2}     \draw (2pt,\y)--(-2pt,\y) node[left] {$\y$};     %labels  \node at (.75,.5) {$\mathcal{D}$}; \end{tikzpicture}  \end{document} barchart with pgfplot - missing bar I am using PGFplot to make a barchart. There are 4 entries in my csv file and unfortunately I have only 3 bars in my figure.   my LaTeX code:  \begin{figure}     \centering     \begin{tikzpicture}          \begin{axis}[             x tick label style={/pgf/number format/1000 sep=},                 xlabel=Sequence,             ylabel=Heigth,                 ymin=0.7,             enlargelimits=0.05,             legend style={at={(0.5,-0.15)},                 anchor=north,legend columns=-1},             ybar interval=0.7,             width=\textwidth,             ]             \addplot table [col sep=comma,trim cells=true,y=height] {numericaltests/data/test.csv};              \addplot[red,sharp plot,update limits=false]                 coordinates {(-1,1.00) (34,1.00)}                 node[above] at (axis cs:15,1.00) {};             \legend{Ca2Fe2As2}         \end{axis}     \end{tikzpicture}     \caption{Sequence}     \label{fig:sequenceHeight} \end{figure}   my test.csv file:  sequence,height 1,2.0 2,1.5 3,1.8 4,2.4   Barchart: If $a\neq 1$, find $(a+1)(a^2+1)(a^4+1)\ldots(a^{2^n}+1)$. If $a\neq 1$, find $$(a+1)(a^2+1)(a^4+1)\ldots(a^{2^n}+1)$$   Or i.e.     If $a\neq 1$, find $\prod_{i=0}^n(a^{2^i}+1)$.   It really does seem like $$(a+1)(a^2+1)(a^4+1)\ldots(a^{2^n}+1)=a^{2^n+2^{n-1}+\ldots+2+1}+a^{2^n+2^{n-1}+\ldots+2+1-1}+a^{2^n+2^{n-1}+\ldots+2+1-2}+\ldots+a^2+a+1$$  Or i.e. $$\prod_{i=0}^n(a^{2^i}+1)=\sum_{i=0}^{2^n+2^{n-1}+\ldots+2+1}a^i$$  Is there an interesting proof of this? I'd appreciate any help. Low budget alien movie about the Earth being cooked I’m looking for an alien movie I watched years ago, mid to late 2000’s about the Earth being cooked or heated up by aliens. The poster is a reflection of the Earth heating up with an orange glow on an aliens eye. It was a low budget movie and I think the one I watched was pirated from China. Again low budget, bad quality and the name could’ve been something worthy he word scorch or something like that. The alien in the movie had legs with knees that went inwards and it had red eyes. Evaluate $\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^{1/n} \ \mathrm{d}x$ Why is the last digit of $n^5$ equal to the last digit of $n$? I tried to prove this question but without a success: Let $K_1$ and $K_2$ be definable sets, prove that $K_1\cup K_2$ is definable.  What I tried to do is to assume: $K_1=Ass(X)=\{ v|v \vDash X \}$ and $K_2=Ass(Y)=\{ v|v \vDash Y \}$ and to say that $K_1\cup K_2=Ass(X \cap Y)$. but I couldn't prove that: $K_1\cup K_2 \supseteq Ass(X \cap Y)$ .   I would like to get help with this proof Unsealed meat in freezer Below is my situation:  Is completely (vacuum) sealing leftover meat in the freezer absolutely necessary? I would always just use a plastic bag to cover the meat and fold it around couple of times so it's almost as good as sealed (usually grocery plastic bag is much bigger than the piece of meat) and have never had a problem with it last a decade or so I have been cooking.  But recently I got this new roommate who insists on me using special small plastic bags with sealers to use and he is being very paranoid, obsessive about it.  So my question is: Is he right? Why is this sequence of equicontinuous functions uniformly bounded? Let $\left\{f_{n}\right\}$ be a sequence of equicontinuous functions where $f_n: [0,1] \rightarrow \mathbf{R}$.  If $\{f_n(0)\}$ is bounded, why is $\left\{f_{n}\right\}$ uniformly bounded? For each positive even integer $n$, set   $$P_n = \displaystyle {{n} \choose {\frac{n}{2}}}\frac{1}{2^n}.$$  Show that $\displaystyle \lim_{n \to \infty} P_n$ exists and determine its value.   Here's what I have so far: each $P_n$ can be thought of as the probability of tossing a fair coin $n$ times, and obtaining exactly $\frac{n}{2}$ heads. My expectation is that $P_n \to 1$, since in any large trial, one would expect to record just about as many heads as tails. But I'm not sure how to mathematically justify this hunch.  Hints or solutions are greatly appreciated. Loopy integral $\int xe^x\sin(x)$ How do I diagnose not being able to reach a specific website as an end user? Placement of the asterisk in pointer declarations I've recently decided that I just have to finally learn C/C++, and there is one thing I do not really understand about pointers or more precisely, their definition.  How about these examples:   int* test; int *test; int * test; int* test,test2; int *test,test2; int * test,test2;   Now, to my understanding, the first three cases are all doing the same: Test is not an int, but a pointer to one.  The second set of examples is a bit more tricky. In case 4, both test and test2 will be pointers to an int, whereas in case 5, only test is a pointer, whereas test2 is a "real" int. What about case 6? Same as case 5? Pre-1980s Sci-fi movie about killer aliens Microsoft Edge keeps taking over PDF file open/association