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Manifold that is NOT smooth Could someone provide an example of a manifold that is not smooth? All manifolds that come to mind are smooth! By a manifold, I mean a hausdorff, second countable locally euclidean space. Allow questions to be saved as drafts prior to posting Visiting two Schengen countries, coming back home in between, do I apply for single-entry or multiple-entry visa? How to get a list of appendices? How to get a list of appendices like the list of figures and list of tables? How to set it on a separate page, like the list of figures and list of tables? When is $2^n \pm 1$ a perfect power Is there an easy way of showing that $2^n \pm 1$ is never a perfect power, except for $2^3 + 1 = 3^2 $?  I know that   gives the result directly, but I'm hopefully for a more elementary method.    I can show that it is never a square, except for $2^3 + 1$.  Proof: Cases $n=1, 2, 3$ are easily dealt with. Henceforth, let $n\geq4$.  $2^n -1 \equiv 3 \pmod{4}$ hence is never a square.     If $2^n +1 =x^2$, then $2^n = (x-1)(x+1)$ and both of these are powers of 2. Thus we must have $(x-1) = 2, (x+1) = 4$. This gives the solution of $2^3 + 1 = 3^2$.    Let's now do an odd prime.  Say $2^n +1 = x^p$. Then $2^n = x^p - 1 = (x-1)(x^{p-1}+x^{p-2} + \ldots +1)$, so both terms are powers of 2. We have $ x = 2^m+1$ is odd. But the other term is the sum of $p$ odd numbers, hence is odd. Since this is clearly greater than 1, we get a contradiction.  Say $2^n -1 = x^p$. Then $2^n = x^p + 1 = (x+1)(x^{p-1} - x^{p-2} + \ldots -x +1 )$, so both terms are powers of 2. We have $x = 2^m -1$ is odd. But the other term is the sum of $p$ odd numbers, hence is odd. Since $x^p + 1 \neq x+1$ except for $p=1$, this means that the term is greater than 1. Hence we get a contradiction. ZFC + $\exists$ Standard model $\rightarrow$ Con(ZFC + $\exists \omega$-model) $ZFC + \exists V_\alpha$ model of $ZFC \vdash Con(ZFC + \exists$ transitive standard model of $ZFC)$  and then  $ZFC + \exists$ transitive standard model of $ZFC \vdash Con(ZFC + \exists \omega-model$ of $ZFC)$  For the first one :   We can always find a countable extentional $M \subset V_\alpha$ elementary equivalent to $V_\alpha$. Let $M&#39;$ be the mostowski collapse of $M$. $M&#39; \approx M$ so $M&#39;$ is model of ZFC. And because $M&#39;$ is countable and transitive then $M&#39; \in V_\alpha$ (since $H_{\omega_1} \subset V_{\omega_1}$ and $\alpha$ is surely far larger than $\omega_1$).   So $V_\alpha$ is the model of '$\exists$ a standard transitive model of ZFC'.  For the second one :   I don't really know how to do it... Does anyone have an idea ? Why are effective pixels greater than the actual resolution? compares Canon EOS 550D and Canon EOS 500D cameras and mentions     18.7 million effective pixels   for 550D. However the best resolution possible using this camera is  5184 * 3456 = 17915904 ~ 17.9 million pixels   What are effective pixels, and why is that number greater than 17.9 million in this case? What are good ways to make pdflatex output copy-and-pasteable? One frequent aspect of pdflatex output that could be improved is to what extent one can select text output from the generated pdf-file and copy-and-paste it. This is essentially an issue of having the right output encoding. There are at least two important aspects to this:   In the ideal case, Unicode would be the output encoding. Ligature glyphs (such as ﬁ) are ideally de-ligated in the output (here: "fi").   It seems that output encodings are specified through the fontenc package. What is the range of possible output encodings, and most importantly: Is there a way to specify a Unicode output encoding and one that deals with ligatures in the intended way?  (Note: It is important to distinguish output from input encodings. How to use different input encodings in LaTeX has been documented widely. My understanding is that this is best handled by the inputenx package, as it supersedes the inputenc package.)    Quick guide to the solutions: The information in the answers and answer threads is a bit distributed, so here the short summary: one approach is to use \input glyphtounicode together with \pdfgentounicode=1; the other approach is to use cmap/mmap.  Addendum: Sometimes one will need to load the package accsupp and enclose one's macro definition in \BeginAccSupp{method=hex,unicode,ActualText=&lt;codepoint&gt;}[...]\EndAccSupp{} to generate a specific code point. See the caveat about non-BMP code points  and the fix (starting from accsupp version v0.4, 2012/11/18) , which provides the new unichar package option. My friend recently received a new digital drawing tablet (Wacom Intuos Pro). Among other features, there is a hole that resembles a charm hole found on mobile phones:    Other parts of the package (e.g. pen) do not have such holes, so it isn't possible to attach them using a band. What is the purpose of this hole? Is it just for attaching various charms and such? $S_3$ permutes the elements of $\mathbb{R}^3$, show that $\sigma (x_1, x_2, x_3) = ( x_{\sigma^{-1}(1)}, x_{\sigma^{-1}(2)}, x_{\sigma^{-1}(3)})$. Trying to find some simple SQL Server PIVOT examples.  Most of the examples that I have found involve counting or summing up numbers.  I just want to pivot some string data.  For example, I have a query returning the following.  Action1 VIEW   Action1 EDIT   Action2 VIEW   Action3 VIEW   Action3 EDIT     I would like to use PIVOT (if even possible) to make the results like so:  Action1 VIEW EDIT   Action2 VIEW NULL   Action3 VIEW EDIT     Is this even possible with the PIVOT functionality? A novel where a woman from "future" of 1999 speaks leet-speak When you reach 10k rep, you get to see other users deleted answers. I just think it's dishonest as a system to call a feature "delete," but show that to totally random strangers in the backroom. Due to the nature of the rep, given enough time, all of the regular users would eventually reach 10k rep, no matter how insensitive or jerk they are.  There's usually a good reason why people hit "delete" to their own work. They regret writing something totally bullshit or maybe they spoke too much about their own work that they shouldn't have.  First of all, is there really a practical reason why 10k rep user should view self-deleted answers? Does the benefit really outweigh the creepiness? (Imagine if total strangers could see your deleted messages in Gmail?)  What suggestions could we make to change it? Borrowing from the email metaphor, I think it makes sense for users to "empty trash" so the items gets seriously deleted. Imho, however the deletion is implemented is not important. It could be a soft delete, SQL Delete, or physical destruction of metal plates as long as the user or the 10k rep user no longer sees the deleted item.  What are the workarounds, if we can't change the behavior?  The least we could do is stop calling it "delete" if it's actually not a "delete." Call it "hide from public" or "actually not delete" or something.  Related user voice:        Edit: Recovery of a deleted-but-good answer is a practical benefit for the community, but I see that as creepy dangerous feature. IMHO, each user should have the rights to take back his or her perfectly good answer, and that the protecting privacy outweighs the benefit or creepiness.   I understand that it's all possible to find things out from Google cache or wherever, but why promote this as default behavior? As 10k rep user, I see others deleted answers by default. @codinghorror wrote:     But like any version control system, deletions are illusory.   Anything and everything that's implemented by human is more or less illusory including phone system and the medical records. Should the government start tapping into phone lines and scanning for words? Remember, this is a question of should had we given the opportunity to design a system. I know and understand the mean value theorem. But at the moment I don't have the intuition to understand the generalized mean value theorem     If $f$ and $g$ are continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$, then there exists a point $c\in(a,b)$ where$$[f(b)-f(a)]g'(c)=[g(b)-g(a)]f'(c).$$If $g'$ is never zero on $(a,b)$, then the conclusion can be stated as$$\frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}.$$   What is the intuition? And how can you prove this (generally) ? Alert on edit after a downvote Does construction of a Lightsaber require use of the Force? It's true that Lightsaber construction is a mandatory skill for a trained Force user, but I am not sure about it in reverse.  @DVK has commented on  that:     Grievous wasn't a Jedi (or a Force Sensitive) at all. He couldn't construct one.   I always thought that a Lightsaber was a normal piece of technology which even non-Force users could construct, and you only need Force sensitivity to master  Lightsaber skills. But, I could be wrong.  Does construction of a Lightsaber require use of the Force? If yes, what role does the Force play in the construction of a Lightsaber? What is a 'multi-part identifier' and why can't it be bound? I continually get these errors when I try to update tables based on another table. I end up rewriting the query, change the order of joins, change some groupings and then it eventually works, but I just don't quite get it.  What is a 'multi-part identifier'? When is a 'multi-part identifier' not able to be bound? What is it being bound to anyway? In what cases will this error occur? What are the best ways to prevent it?    The specific error from SQL Server 2005 is:     The multi-part identifier "..." could not be bound.   Here is an example:  UPDATE  [test].[dbo].[CompanyDetail]  SET Mnemonic = [dbBWKMigration].[dbo].[Company].[MNEMONIC],                 [Company Code] = [dbBWKMigration].[dbo].[Company].[COMPANYCODE]  WHERE [Company Name] = **[dbBWKMigration].[dbo].[Company].[COMPANYNAME]**   The actual error:     Msg 4104, Level 16, State 1, Line 3 The multi-part identifier   "dbBWKMigration.dbo.Company.COMPANYNAME" could not be bound. Add scratches to a material in Cycles? What are Covering Indexes and Covered Queries in SQL Server? Can you explain the concepts of, and relationship between, Covering Indexes and Covered Queries in Microsoft's SQL Server? Does a linear operator obeying Leibniz rule imply it is a differential operator?