Assume that $M$ is connected. Ideally, I would like $U$ to be connected, though that's not strictly necessary.  If this were true, it would instantly make integration on compact oriented manifolds much easier, since we can simply pass to Euclidean space $\mathbb{R}^n$ in order to evaluate the integral. Intuitively, it seems like it must be true. Consider the following examples:   $\mathbb{S}^n$ – we use the stereographic projection $\mathbb{T}^n \cong (\mathbb{R}/\mathbb{Z})^n$ – we consider $U=(0,1)^n$ $\mathbb{RP}^n$ – we consider $U_0=\{[1:x^1:\cdots:x^n]\, \big\vert\, x^1,\ldots,x^n\in\mathbb{R}^n\}$ If $M$ and $N$ have smooth charts $(U,(x^i)$ and $(V,(y^j))$ defined almost everywhere, respectively, then $M\times N$ has smooth chart $(U\times V, (x^1,\ldots,x^n,y^1,\ldots,y^m))$ defined almost everywhere.   However, I have no clue about how to prove a statement like this, and I suspect that any counterexample would be very difficult to find.
I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded into the Euclidean space of the same dimension. So I wonder if all manifolds can be embedded into a same dimensional Euclidean space when we cut out a measure zero set? Can anyone prove it or disprove it by giving me some counterexamples?
I've always wondered this - why can't you declare variables after a case label in a switch statement?  In C++ you can declare variables pretty much anywhere (and declaring them close to first use is obviously a good thing) but the following still won't work:  switch (val)   {   case VAL:     // This won't work   int newVal = 42;     break; case ANOTHER_VAL:     ...   break; }     The above gives me the following error (MSC):     initialization of 'newVal' is skipped by 'case' label   This seems to be a limitation in other languages too.  Why is this such a problem? Why my differential equations become True? This week we rolled out a series of changes to  of the question close workflow.  You can read more about the details of these changes .    For context, the overall effort to improve the question close experience will be optimized for Stack Overflow. Given Stack Overflow’s scale, many of the pain points we hear about with question closing and reopening are unique to Stack Overflow and won’t make sense for other network sites. However some  changes are general improvements that should improve usability and guidance overall. Wherever possible, we plan to roll these changes out network wide as well. Any time we roll out a change network wide, we’ll make sure to cross-post here. This first phase is being rolled out to the network and is why I’m posting today.  Let us know if you run into any issues or have questions regarding the network here. Closure of $\{\frac{1}{n}\}:=M$ In $\mathbb{R}$ with standard topology the closure of the above set $M$ is $M \cup 0$. I fail showing this by purely topological means. I.e. not using the fact that this holds in the reals as a metric space and deduce it for the induced topology.  I assumed that there exist a closed subset $A$ containing $M$ but not 0. Hence, the complement of $A$ is an open subset containing 0. Does this already yield a contradiction or do I need a different approach?  Could you help me on this? Dynamic Keyword Arguments in Python? Does python have the ability to create dynamic keywords?  For example:  qset.filter(min_price__usd__range=(min_price, max_price))   I want to be able to change the usd part based on a selected currency. How do I remove/delete a folder that is not empty? I don't know what exactly it's called, by cache refresh I mean, refresh the page after clearing its cache. I don't want to clear the entire browser cache.  I can't seem to cache refresh my pages. In Firefox, I know it to be Shift+Refresh.  In Chrome, I've tried Ctrl+R, Ctrl+Refresh, Alt+Refresh, Shift+Refresh but none of them work. New phone: How to transfer game progress? I'm planning to buy a new Android phone and I'd like to be able to transfer the progress I have made at several games to the new device, so I won't have to play all the levels again from scratch.  I currently have an HTC Desire and will buy an HTC Evo 3D.  There's probably no simple/general solution to this.  But I was wondering if anyone did something similar and could provide some advice (e.g. where to look for files that might hold game progress information). What is the difference between private and protected members of C++ classes? What is the difference between private and protected members in C++ classes? I understand from best practice conventions that variables and functions which are not called outside the class should be made private—but looking at my MFC project, MFC seems to favor protected. What's the difference and which should I use? How can I get a views exposed filter to persist across multiple views? I have 2 separate views that return a set of results for each of my content types. The fields are different, but the filters are exactly the same - Sort by Location / Distance. This Distance / Proximity filter is exposed, and it is set to 'remember' the last entry. I followed the tutorial here  to be able to filter results by distance. It works great.  In views, to remember the last entry, Views uses sessions. But, each view has its own individual $_SESSION variable. That means what I set for one view does not carry over to the next, and even each view can have different values, which is not what I want. Here is a dump of the session after they have been set:  [views] =&gt; Array     (         [search_pages] =&gt; Array             (                 [default] =&gt; Array                     (                     )              )          [baby_clothes_pages] =&gt; Array             (                 [default] =&gt; Array                     (                         [distance] =&gt; Array                             (                                 [postal_code] =&gt; 11211                                 [search_distance] =&gt; 100                                 [search_units] =&gt; mile                             )                      )              )          [baby_gear_pages] =&gt; Array             (                 [default] =&gt; Array                     (                         [distance] =&gt; Array                             (                                 [postal_code] =&gt; 22343                                 [search_distance] =&gt; 100                                 [search_units] =&gt; mile                             )                      )              )      )   What I would like to do is create a small module or snippet of code that will simply allow someone to type in their 5 digit zip code and then it update all the values across all my pages with this filter. If it is changed on any page, then it changes it for the rest of the pages as well. It also needs to work so that it affects the results on the next page load so that it correctly shows them filtered.  Here is a snippet of code I was experimenting with to try to set this value, but it's the wrong direction.   $all_views = views_get_all_views();            // get an array of all views in use on the site  foreach ($all_views['category_pages']-&gt;display as $key =&gt; $value) {  //check to see if the structure exists I am looking to modify if(array_key_exists('postal_code', $all_views['category_pages']-&gt;display[$key]-&gt;display_options['filters']['distance']['value'])) {  //if so, set its value $all_views['category_pages']-&gt;display[$key]-&gt;display_options['filters']['distance']['value']['postal_code'] = '28152';  //try setting the session instead $_SESSION['views']['category_pages'][$key]['distance']['postal_code'] = '28152';     }   What's the correct direction? How do I get this value changed? Could it be done in the template.php file with preprocess function? Should I do it in a module, and if so, what do I need to alter?  Very lost at this point. Thanks for the direction! Access Denied using TakeOwn.exe I have got this file that I can't delete. It happened after a system crash, so the CHKDSK kicked in upon next reboot. After that, I can't delete the file.  THis is on Windows Home Server, and the file is one of those hidden Thumbs.db, and my WHS reports a "File conflict" on the file, the reason: Access Denied"  What I have tried so far, running as an Administrator: Delete: Access Denied TakeOwn.exe : Access Denied Attrib.exe -s -h : Access Denied cacls.exe : Access Denied   (even whe trying to display the current owner) Rename: File not found (its hidden) Re-boot in to safe mode and tried the above: Access Denied Running the above commands as Local System: Access Denied   Tizer Unlocker: Fail (Doesn't find anything locking the file) File Assasin: Crashes ("FindRemoteFileHandles returned NULL value.")  I have used the CHKDSK /f again, rebooted since some suggestions is that the file has been corrupted, but that didn't change anything. Any suggestions ?  Update: In a total separate scenario, I experienced a similar situation; Access Denied when trying to take ownership of a file. It turned out that the file was owned by TrustedInstaler, which is not possible to override just by being an Administrator. In this case, one can use psexec /ids cmd to elevate yourself to System and then TakeOwn.exe /a For example:  $ sudo apt-get install curl Reading package lists... Done Building dependency tree Reading state information... Done You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies: bsh : Depends: libjline-java but it is not going to be installed groovy : Depends: libjline-java but it is not going to be installed rhino : Depends: libjline-java but it is not going to be installed E: Unmet dependencies.   Try 'apt-get -f install' with no packages (or specify a solution).   I get the same or similar errors when I attempt to install clojure1.3, leiningen, and several other packages.   When I try the suggestion made in the error message, this is what happens:  $ sudo apt-get -f install  Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following packages were automatically installed and are no longer required:   diffstat linux-headers-3.2.0-26-generic linux-headers-3.2.0-26 dh-apparmor dkms html2text libmail-sendmail-perl libsys-hostname-long-perl Use 'apt-get autoremove' to remove them. The following extra packages will be installed:   libjline-java Suggested packages:   libjline-java-doc The following NEW packages will be installed:   libjline-java 0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. 23 not fully installed or removed. Need to get 0 B/72.0 kB of archives. After this operation, 129 kB of additional disk space will be used. Do you want to continue [Y/n]? Y (Reading database ... 226243 files and directories currently installed.) Unpacking libjline-java (from .../libjline-java_1.0-1_all.deb) ... dpkg: error processing /var/cache/apt/archives/libjline-java_1.0-1_all.deb (--unpack):  trying to overwrite '/usr/share/java/jline.jar', which is also in package scala 2.9.2-400 Errors were encountered while processing:  /var/cache/apt/archives/libjline-java_1.0-1_all.deb E: Sub-process /usr/bin/dpkg returned an error code (1)  $ sudo apt-get upgrade Reading package lists... Done Building dependency tree        Reading state information... Done You might want to run 'apt-get -f install' to correct these. The following packages have unmet dependencies:  bsh : Depends: libjline-java but it is not installed  groovy : Depends: libjline-java but it is not installed  rhino : Depends: libjline-java but it is not installed E: Unmet dependencies. Try using -f. I just received a PhD invitation letter from a German University which mentions that my remuneration will be 2/3 of full TV-L 13 position. I do not know what a TV-L 13 position is (Google is of no help here), and didn't want to be rude, so I am asking here first. Is this a common position or specific to Germany? If anyone knows, currently how much remuneration corresponds to a TV-L position? Using conditional statements in ArcGIS Field Calculator? I have two fields, "P_Area" and "PC_Area", whose sizes I'd like to compare in a third field, "Areas_Equal". The "Areas_Equal" field should contain a "Yes" if the other two are equal and a "No" if they aren't.  How can I do this with Field Calculator and its Python Parser? I've watched the Vampire Knight anime. I really like the story so far but find it frustrating that the anime doesn't finish off the story. I'm trying to figure out if it is worth reading the manga.  Does the manga contain more story than the anime? Does it finish the story off? I want to create buffer polygon(with curved edges) from  points like this using ArcGIS for Desktop.   distance toward North from a point 5, toward South 1.5, toward west 1, toward East 2.   all this distance is attached as attribute in point in different fields. Do I need to upgrade RAM from 4GB to 8GB for Android development? I'm an android developer and have laptop with 4GB RAM. When I compile my project the RAM is used at 100%.    If I add more RAM will it solve the problem? How to make a font with Adobe Illustrator? I have recently created my own font/typeface with Adobe Illustrator, and I would like to use it as a real font (e.g I would like to use it as I type something).   Is this possible in Adobe Illustrator or would I need a different software?   If so what program would be good for creating a font? I'm going to reset my Android tab because its internal memory is low.  Is it safe to erase the system data of my android tablet (not rooted)? If I restore the factory setting of my Android, will the uninstallable apps be also removed or only those that can be uninstalled? Why does the speed of light in vacuum have no uncertainty? I could understand that the definition of a second wouldn't have an uncertainty when related to the transition of the Cs atom, so it doesn't have an error because it's an absolute reference and we measure other stuff using the physical definition of a second, like atomic clocks do.  But why doesn't the speed of light have uncertainty? Isn't the speed of light something that's measured physically?  Check out that .