If every Jordan curve $J \subset D$ is null-homotopic in a domain $D \subset \mathbb{R}^2$, then is it clear that every closed curve $C \subset D$ is also null-homotopic in $D$?  Apparently the set $\mathbb{R}^2 \setminus C$ is an union of countably many Jordan domains $D_i$, but it's not clear to me that the result for every $\partial D_i$ is enough. What is the right approach?  EDIT: Before you mofos shut down my question with silly down votes I ask to check out the following link   There this question is answered with seemingly simple compactness argument. It would be nice if someone of you can explain why those argument are enough?
Problem. Assume that $U$ is an open and connected subset of $\mathbb R^2$, and $\gamma :[0,1]\to U$ is a closed curve, which is not null-homotopic in $U$ and not necessarily simple closed. Show that there exists a simple closed curve, which is not null-homotopic in $U$.  I am sure that this is a standard problem, and I am looking for an elegant, if possible, solution or a reference. I have in mind the following sketch of proof:  a. Construct a polygonal closed curve $\tilde\gamma$ which is homotopic with $\gamma$ in $U$.  b. Construct $\tilde\gamma$ so that it intersects itself in finitely many points, and not whole segments.  c. Use induction on the number of self–intersections of $\tilde\gamma$, in order to show that $\tilde\gamma$ is written as a product (in the fundamental group's sense) of simple closed polygonal curves.
Getting the expectation symbol to behave like $\sum$ instead of $\Sigma$ How can I get an expectation symbol which behaves like $\sum$?  Specifically, I want the underscore to be like that of a summation.  At the moment I use   $$\mathbb{E}_{x\in A}$$    However, I do not like this.  It feels like I am writing a sum as   $$\Sigma_{x\in A}$$    Whereas the alternative  $$\sum_{x\in A}$$   is much nicer when in a centered equation.  (The underscore appears directly below the sigma, instead of slight infront and below.)  Is there a way to make an expectation symbol which behaves like $$\sum$$ rather then $$\Sigma$$? How do I change the "Selected Items" color? I was trying to change the selected items color from the default orange to something else, however, the option isn't there, as it used to be in the Appearance properties in previous versions of ubuntu. Any idea where it could be in the beta? Why do we need a Stack Overflow in Portuguese? I have noticed there is a Stack Overflow in language differs from English which is global and mostly used by programmers.  I think there is no need for a site in other languages. Otherwise we will end up with 127 StackOverflow sites. I am afraid that we will lose the amount of people sharing data. How to export Polygons to CSV with coordinates? I am very new to QGIS, so please bear with me.  I have a ".shp" file, which is contains a set of boundaries for suburbs of Australia. I opened it to QGIS and I would like to export it into a CSV file so that the final file would be like following.  POSTCODE, LATITUDE, LONGITUDE  When I select "Layer > Open attribute table" I can see the "postcode" column.  I had a look at a similar question here     Which told me to simply go to "Layer > Save as" and then type "GEOMETRY=AS_XY" or "GEOMETRY=AS_XYZ" in the "Layer" section of the "OGR creation options".  I followed exactly, but the latitude and longitude values are not being returned. How to switch language keyboard combination? In the Peter Jackson movie, once the One Ring is thrown into the volcano in Mount Doom, why does the mountain and tower collapse, and the ground disappears everywhere where the orcs are standing but not where the humans are? Where are all the orcs running to, what do they do next? How to use inotifywait to watch a directory for creation of files of a specific extension I have seen .      You should consider using inotifywait, as an example:  inotifywait -m /path -e create -e moved_to |     while read path action file; do         echo "The file '$file' appeared in directory '$path' via '$action'"         # do something with the file     done    My question is that, the above script watches a directory for creation of files of any type, but how do I modify the inotifywait command to report only when a file of certain type/extension is created (or moved into the directory) - e.g. it should report when any .xml file is created.  WHAT I TRIED:  I have run the inotifywait --help command, and have read the command line options. It has --exclude &lt;pattern&gt; and --excludei &lt;pattern&gt; commands to EXCLUDE files of certain types (by using regEx), but I need a way to INCLUDE just the files of a certain type/extension. Is it possible to choose which OS will be loaded using a hardware switch? Is there a good casual online place to practice/learn bridge? I know that Pogo.com (which powers Yahoo Games) has a long running online bridge game and that there are a handful of places for serious tournament/MasterPoint level bridge players online (though most seem based on software written a decade or more ago) but are there any places to play, practice and learn bridge in a more casual manner? Ideally via a well designed modern piece of software? How to sail downwind faster than the wind? Recently a  set a record for sailing a wind-powered land vehicle directly down wind, and a speed faster than wind speed. Wikipedia has a  talking about it, but it doesn't explain exactly how it works.  I can imagine some configurations, but I don't get how a system could be set up to push forward with a tailwind (when the cart is moving slower than the wind) and push forward with a head wind (when the cart is moving faster than the wind).  Could someone help explain how this is possible? Installing IE8 in wine - error rename/sha1sum mismatch winetest.cat Please note, this is an informative QA post rather than a question specific, due to this issue popping up many a time.  QUESTION / ISSUE  How to solve the issue of installing IE8 (Internet Explorer) in wine (any version - to my knowledge).  Related Wine Errors:  Terminal install method  Installing IE8:  winetricks -q ie8   should give you a similar error:  Executing w_do_call ie8 Executing load_ie8 Executing wine iexplore -unregserver Executing w_do_call msls31 msls31 already installed, skipping Using native,builtin override for following DLLs: iexplore.exe itircl itss jscript msctf mshtml shdoclc shdocvw shlwapi urlmon wininet xmllite Executing winetricks_early_wine regedit /S C:\windows\Temp\_ie8\override-dll.reg Using builtin override for following DLLs: updspapi Executing winetricks_early_wine regedit /S C:\windows\Temp\_ie8\override-dll.reg Downloading http://winezeug.googlecode.com/svn/trunk/winetricks_files/winetest.cat to /home/user/.cache/winetricks/ie8 --2016-09-08 23:03:12--  http://winezeug.googlecode.com/svn/trunk/winetricks_files/winetest.cat Resolving winezeug.googlecode.com (winezeug.googlecode.com)... 66.102.1.82, 2a00:1450:400c:c06::52 Connecting to winezeug.googlecode.com (winezeug.googlecode.com)|66.102.1.82|:80... connected. HTTP request sent, awaiting response... 404 Not Found 2016-09-08 23:03:13 ERROR 404: Not Found.  ------------------------------------------------------ Downloading http://winezeug.googlecode.com/svn/trunk/winetricks_files/winetest.cat failed ------------------------------------------------------   GUI Install via Winetricks GUI method  or using the GUI method, you should see a error dialog similar to this: Distinguishing whether "up" is a preposition or a part of a phrasal verb in this case In the sentences below, “bill” and “hill” are both nouns.    He ran up a big hill. He ran up a big bill.   Can someone explain how one distinguishes whether “up” is a preposition or a part of a phrasal verb? Is there any way, shape, or form that ultraviolet light is recorded by a digital sensor?  Ideally, this question is related strictly to the sensor, without even a glass lens in between which could cut or block UV light.  But of course I don't take pictures with a bare sensor, I also have a lens, so I'd also like to know how much UV gets to the sensor to begin with. Google Maps API v3, jQuery UI Tabs, map not resizing What is the Necessity of $...$ Around Numbers? I typed up an assignment with a lot of numbers in LaTeX and turned it in to my adviser.  When he returned it to me with some corrections, one of which was a lot of $ around my numbers.  I understand that $ are necessary around some mathematical formulas, but I was just using numbers like -2 and 4.  I did not see any change in the font of the numbers when I added the $ either.  Since my adviser did not want to give me an explanation, what is the rationale? Why do quantum effects dominates when the thermal de Broglie wavelength is on the order of or larger than the interparticle distance? We say that when the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. On the other hand, when the thermal de Broglie wavelength is on the order of or larger than the interparticle distance, quantum effects will dominate and the gas must be treated as a Fermi gas or a Bose gas, depending on the nature of the gas particles.  $(V/N)^{1/3} &lt; \lambda_{th} $. My question is why do quantum effects dominate? I understand how to define include shared objects at linking/compile time. However, I still wonder how do executables look for the shared object (*.so libraries) at execution time.  For instance, my app a.out calls functions defined in the lib.so library. After compiling, I move lib.so to a new directory in my $HOME.  How can I tell a.out to go look for it there? I have recently installed ubuntu 11.10. But when i try to install a software it tries to 'updating cache' but after a while, it stops, showing:                            Failed to download repository information                           Check your internet connection   so, now i am totally unable to install any software in my ubuntu.  Please,let me know how to solve the problem. Libvirt: how to pass qemu command line args? I have the current libvirt 1.2.20 i want to pass to my machine create with virt-manager some qemu flags. I edited the file and add    &lt;/devices&gt;   &lt;qemu:commandline&gt;           &lt;qemu:arg value='-L pc-bios'/&gt;   &lt;/qemu:commandline&gt; &lt;/domain&gt;   But doesn't work i also tried to edit from virsh  virsh edit domain   and put   &lt;qemu:commandline&gt;           &lt;qemu:arg value='-L pc-bios'/&gt;   &lt;/qemu:commandline&gt;   But give me this error  errore: XML document failed to validate against schema: Unable to validate doc against /usr/share/libvirt/schemas/domain.rng Element domain has extra content: qemu:commandline  Failed. Try again? [y,n,i,f,?]:    How to pass correctly qemu command line? Thanks What is the most efficient way to deep clone an object in JavaScript? What is the most efficient way to clone a JavaScript object? I've seen obj = eval(uneval(o)); being used, but . I've done things like obj = JSON.parse(JSON.stringify(o)); but question the efficiency.  I've also seen recursive copying functions with various flaws.  I'm surprised no canonical solution exists.