Let $X=S^1\times I$ where $S^1\subset \mathbb{C}$ and $I=[0,1]\subset \mathbb{R}$ also define the equivalence relation ~ by $(x,t)$~$(y,s)$ if and only if $xt=ys$.   Prove that $X$/~ is homeomorphic to the unit disc $D^2=\{x\in \mathbb{R^2}: |x|\le1\}=\{x\in \mathbb{C}: |x|\le1\}$ with the induced topology.  Any hints would be appreciated.
Let $\sim$ be the equivalence relation on $S_1 \times I$ given by $(x,t)\sim(y,s)$ if and only if $xt=ys$ where $I=[0,1]$  How do I prove that $(S_1 \times I)/\sim$ is homeomorphic to the unit disk $D^2$ with the induced topology?
Today I was going through my text book exercise and got hold of the following question.     Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ is a non empty convex subset of $X$ then prove that $\operatorname{Closure}(A)$ is also a convex subset of $X$.   I gave it a try but could not succeed. Roman numerals with over- and underline This question led to a new package:      (using a variation of egreg's code, and stating this in the manual; the package isn't really new now, sorry for the delayed announcement)  I have read about "How to get Roman numerals" but asked myself how to get Roman numerals formatted as in     i.e. with fitted over- and underline (without manually using boxes and such)? I'm curious for your answer(s)! Find the general solution of $4u_{xx}+5u_{xy}+u_{yy}+u_x+u_y=2$ How do I solve the following second order partial differential equation?  $4u_{xx}+5u_{xy}+u_{yy}+u_x+u_y=2$  I have classified the equation to be hyperbolic and changed variables to obtain the canonical form as $u_{\epsilon\nu}={1\over{3u_\nu}}-{8\over9}$ which I believe is correct but I am struggling to find the general solution? A step by step solution would be appreciated. AttributeError: MapDocObject: Unable to save on mxd.save/saveACopy? I'm trying to use arcpy to save an .mxd and am having an issue.  It doesn't matter if I use mxd.save or mxd.saveACopy I get the same error.  I know it doesn't look like I am doing anything w/ it here, but it's just a portion of my code:  import arcpy import arcpy.mapping as map arcpy.env.overwriteOutput = True  env.workspace = r"C:\MGIS\FinalProject2.gdb     mxd = map.MapDocument(r"C:\MGIS\Finalproject2.mxd") df = map.ListDataFrames(mxd)[0]  for df in map.ListDataFrames(mxd): df.scale = 2000000  #Add layers into map document addLayer = map.Layer(OKXcounties) addLayer1 = map.Layer(contourLabelText) addLayer2 = map.Layer(contoursClipLyr) addLayer3 = map.Layer(clipOKXLyr) map.AddLayer(df, addLayer, "AUTO_ARRANGE") map.AddLayer(df, addLayer1, "AUTO_ARRANGE") map.AddLayer(df, addLayer2, "AUTO_ARRANGE") map.AddLayer(df, addLayer3, "AUTO_ARRANGE")  # Add contour Labels for lyr in map.ListLayers(mxd):     if lyr.name == "contourLabelText":         lyr.showLabels = True     if lyr.name == "Contour Features":         lyr.transparency = 100  mxd.saveACopy(mxd)   I get the following error:   Traceback (most recent call last):   File "C:\Python26\ArcGIS10.0\Lib\site-packages\pythonwin\pywin\framework\scriptutils.py", line 312, in RunScript     exec codeObject in __main__.__dict__   File "C:\MGIS\geog485\FinalProject2\mxdtest.py", line 7, in &lt;module&gt;     mxd.saveACopy(mxd)   File "C:\Program Files\ArcGIS\Desktop10.0\arcpy\arcpy\utils.py", line 181, in fn_     return fn(*args, **kw)   File "C:\Program Files\ArcGIS\Desktop10.0\arcpy\arcpy\_mapping.py", line 668, in saveACopy     self._arc_object.saveACopy(file_name) AttributeError: MapDocObject: Unable to save.  Check to make sure you have write access to the specified file and that there is enough space on the storage device to hold your document.   Space is not the issue, I have 60 GB of free space and I've checked the windows properties of the .mxd file.  Read-only box is unchecked and I've "allowed" all permissions for all groups:    Does anyone have any idea how if this is a bug or how I can fix this?  I'm using ArcInfo 10.0 SP4. How can I insert a newline in a framebox? I have several frameboxes in my document, using the following command:  \framebox(115,115){ R-Sq: \newline For example}    I want to force a new line, but neither \newline or \\ appear to work. Is there any other way of doing this? Different colors for different types of blocks in beamer Dirac delta function as a limit of sinc function Does Ubuntu follow the Linux Standard Base (LSB)? Auto comment "possible duplicate of ..." missing on voting to close as duplicate Please refer to the following question:    The  had posted an  a while back.  I posted a comment referring to the same and voted to close the question as a duplicate of the referenced question.  Surprisingly, an auto comment saying "possible duplicate of ..." was not generated.  I agree that the referenced question shows up in the linked questions list but an auto comment ought to be generated.  Seems like a bug.    EDIT: I understand that the above behavior is by-design.  However, I'd say that the behavior in the following case should also be the same:  It's interesting to see that the auto comment would be generated if you were to add a comment linking to an answer and then voting to close the question as duplicate.  For example, observe the following question: "Scaled $L^p$ norm" and geometric mean Does the penalty to movement speed while grappling stack if you grapple two opponents? In 5e, a player who is grappling has their movement speed halved. However, a player with two hands can grapple two opponents.  As the wording seems to indicate that the movement speed penalty applies due to the grappling status, does that mean that a player grappling two opponents is doubly penalized?  If so, how would it apply: is your speed reduced to 0 ([1 - 0.5] - 0.5), or a quarter of your regular speed ([1 / 2] / 2)? Why do elementary matrix operations not affect the row space of a given matrix? Ideal in a ring of continuous functions I need to prove that the language $\mathscr L=\{0^n \big|\text{n is the square of a natural number}\}$ is non regular using the pumping lemma   My try:  $\mathscr L=\{\overbrace{\epsilon}^{0^2},\overbrace{0}^{1^2},\overbrace{0000}^{2^2},\overbrace{000000000}^{3^2},\overbrace{0000000000000000}^{4^2},\dots\}$  Suppose that $\mathscr L$ is regular so $\exists$ a word '$x$' with length of at least $n$   $|x|\geq n $ such that  $(1)\,\,\,|uv|\leq n$  $(2)\,\,\,|v|\geq 1$  $(3)\,\,\,uv^iw \in \mathscr L$  Now, let as choose the word $\color{blue}{x=0^{n^2}}$  $|x|\geq n$ so we can use $(1)-(3)$  $x=\underbrace{000000\dots}_{n^2\text{ times}}$  $x=uv0^kw=\underbrace{uv0^k}_{\text{n zeros   }}\overbrace{w}^{n^2-n\text{ zeros }}$  Let as choose $\color{blue}{i=??}$      I'm difficult choosing the word '$\color{blue}x$' and choosing  $\color{blue}i$ to come to contradiction with the lemma Can we know the fundamental nature of space and time? How do I add arrowheads to circular arcs? If $x_1 = 3$, $x_{n+1} = \frac{1}{4-x_n}$ for $n \geq 1$, prove the sequence is bounded below by $0$, above by $4$. If $x_1 = 3$, $x_{n+1} = \dfrac{1}{4-x_n}$ for $n \geq 1$, prove the sequence is decreasing and bounded below by $0$, above by $4$.  Want to show: $0 &lt; x_{n+1} &lt; x_n &lt; 4$  I decided to do the bounded part first.    Base Case, $n=1$  $0 &lt; 3 &lt; 4$ [works]  IH  Suppose $0 &lt; x_k &lt; 4$, for some $k\in\mathbb{N}$  IS  Show $0 &lt; x_{k+1} &lt; 4$  We know $x_{k+1} = \dfrac{1}{4-x_k}$  We also know that $0 &lt; x_k &lt; 4$, by IH.  So we have:  $$\dfrac{1}{4} &lt; x_{k+1} &lt; \infty$$  But this doesn't show the boundness. Where have I went wrong? It seems like I'm doing something wrong when using foreach variables to indicate node numbers. The two foreach loops are anchoring the paths to the east of each node. Why does this happen?    \documentclass{article}  \usepackage{tikz} \usetikzlibrary{backgrounds,fit,patterns,decorations.pathreplacing,calc}  \makeatletter \tikzset{   labelnode/.style={draw=none, fill=none, black},   latentnode/.style={draw, minimum width=5mm, shape=circle, ultra thick, black},   dagconn/.style={arrows=-&gt;, black, thick}, } \makeatother  \begin{document} \begin{tikzpicture}[scale=1]  \pgfmathsetmacro{\colgap}{1.6}  \pgfmathsetmacro{\rowgap}{1.3}   \node[labelnode] (x1) at (\colgap*0,\rowgap*1) {$X_1$};  \node[labelnode] (x2) at (\colgap*1,\rowgap*1) {$X_2$};  \node[labelnode] (x3) at (\colgap*2,\rowgap*1) {$X_3$};   \node[latentnode,blue] (x4) at (\colgap*0,\rowgap*0) {$X_4$};  \node[latentnode,red] (x5) at (\colgap*1,\rowgap*0) {$X_5$};  \node[labelnode] (x6) at (\colgap*2,\rowgap*0) {$X_6$};   \foreach \c in {1,...,2} {    \foreach \r in {1,...,2} {      \pgfmathsetmacro{\st}{(((\r-1)*3)+\c)}      \pgfmathsetmacro{\en}{\st+1}      \draw[dagconn] ([xshift=0.2cm]x\st) to ([xshift=-0.9cm]x\en);    }  }   \foreach \c in {1,...,3} {    \foreach \r in {1,...,1} {      \pgfmathsetmacro{\st}{((\r-1)*3)+\c}      \pgfmathsetmacro{\en}{\st+3}      \draw[dagconn] (x\st) to (x\en);    }  } \end{tikzpicture} \end{document} "Naturally occurring" non-Hausdorff spaces? It is not difficult for a beginning point-set topology student to cook up an example of a non-Hausdorff space; perhaps the simplest example is the line with two origins. It is impossible to separate the two origins with disjoint open sets. It is also easy for a beginning algebraic geometry student to give a less artificial example of a non-Hausdorff space: the Zariski topology on affine $n$-space over an infinite field $k$, $\mathbf{A}_{k}^{n}$, is not Hausdorff, due to the fact that polynomials are determined by their local behavior. Open sets here are in fact dense. I am interested in examples of the latter form. The Zariski topology on $\mathbf{A}_{k}^{n}$ exists as a tool in its own right, and happens to be non-Hausdorff. As far as I'm aware, the line with two origins doesn't serve this purpose. What are some non-Hausdorff topological spaces that aren't merely pathological curiosities? I need rules to drop some malicious Apache connections I drop all traffic on ports except on 80 for my web server.  I have some rules like this on iptables:  iptables -A INPUT -p tcp -m tcp --dport 80 -m string --string "cgi" --algo bm --to 1000 -j DROP   Someone who have more can share? I know always the bad hackers still updating, but some of them always begin with the same code. I need to Drop the connection based on some criteria. Here are some Apache log (I remove ips but each atack come from the same):  Attack 1: This I don't know what are trying to do, but make it 50 times from the same ip  GET / HTTP/1.1  301 224 -   Mozilla/5.0 (X11; Linux i686) AppleWebKit/537.22 (KHTML, like Gecko) Chrome/25.0.1364.152 Safari/537.22 GET / HTTP/1.1  302 3387    -   Mozilla/5.0 (X11; Linux i686) AppleWebKit/537.22 (KHTML, like Gecko) Chrome/25.0.1364.152 Safari/537.22   Attack 2: this try to get information about server only.  GET / HTTP/1.1  301 224 http://myip:80/ Go-http-client/1.1 GET / HTTP/1.1  302 3228    http mywebsite  Go-http-client/1.1 GET /es/ HTTP/1.1   200 40947   https mywebsite Go-http-client/1.1   Attack 3: they try get access to a login page vulnerability  GET /userlogin/login.aspx HTTP/1.1  302 186 -   -   Attack 4: this try to access a cgi at first request, (see my first iptables rule to drop this)  GET /hndUnblock.cgi HTTP/1.0    302 186 -   Wget(linux) GET /tmUnblock.cgi HTTP/1.0 302 186 -   Wget(linux)   I'm very new with server this 4 attacks are from only last 12 hours... Have thousands per week.