Check or falsify: There is a $C^\infty$-map  $S^1 \to S^1 \times S^1$ that is surjective.  Comments: I think it's not true, because $\dim S^1 = 1$ and $\dim S^1 \times S^1 = 2$ but I do not know how to justify.
My professor was lecturing today and he made this statement which I was unable to verify.  (I worded it nicer)  There is no map which is both smooth and onto from $S^1$ to $S^1$$\times$ $S^1$.  When he said this he included the original "one can clearly see" statement which is usually when the most people seem to be confused in his class.  Thus, I was hesitant to ask why.  So, please no one lecture me on the fact that I should have asked anyway.  If it is really that obvious, I would rather look stupid here than in class :)
Does anyone know if there are any video formats (besides the HUGE uncompressed avis) that you can render in that maintain the alpha channel? (i.e. for after FX etc.)   I used to use quicktime, but that's no longer an option.  I know you can render in png. or other image formats, but video is significantly smaller, and I often work on machines with less space &amp; resources. cardinality of the set of $ \varphi: \mathbb N \to \mathbb N$ such that $\varphi$ is an increasing sequence I know that the set of functions $ f:\mathbb N \to \mathbb N$ is uncountable, but what if we consider only $f$ such that $f$ is increasing? I want to know if this set is countable D: and also the case of bijective and increasing (clearly if the firstone holds then this too). I think that it could be possible that it's countable Cross Reference problem In particular, when $x=y=z$ isn't the above expression equal to 1 and not -1? Why does list(my_list) modify the object? How to prove that $A_5$ has no subgroup of order 30? How to make many copies of similar graphs systematically? I want to make various copies of a graph in tikz. The graph I want has 10 vertices. But only has the edges between the vertices a_i,a_(i+1),t,b.  I would like to automatically create 8 copies of this graph. and draw them. I just don't know how to code it. Here is my current code, but it adds all of the edges to the same copy of the graph:  \begin{tikzpicture} [main_node/.style={circle,fill=blue!20,draw,minimum size=2em,inner sep=3pt]}] \node[main_node] (8) at (0:0.5) {b}; \node[main_node] (9) at (180:0.5) {t}; \foreach \x in {0,1,...,7}     \node[main_node] (\x) at (360/8*\x:2) {$a_{\x}$}; \foreach \x in {0,1,...,7}{     \pgfmathsetmacro{\y}{int(mod(\x+1,8))}     \path[draw,thick] (\x) edge node {} (8);     \path[draw,thick] (\x) edge node {} (9);     \path[draw,thick] (\x) edge node {} (\y);     \path[draw,thick] (8) edge node {} (9);     \path[draw,thick] (8) edge node {} (\y);     \path[draw,thick] (9) edge node {} (\y);    }   I want each of the edge sets to be drawn seperately. I have used so many things like url removal request and deleted the pages etc., but still the errors are shown in my webmaster tools.  How can I remove the urls completely from Google? Today I found something strange on my newly installed Windows 7 OS - my "users" folder C:\Users was shared and "Everyone" had full access... I disabled that and now it seems fine, however I'm a bit scared actually, I'd like to know what caused that and if my system is compromised. closed form for $\binom{n}{0}+\binom{n}{3}+\binom{n}{6}+...+\binom{n}{n}$ I am trying to pickle some mangoes the Indian way. I boiled the jar in hot water, dried it and then added the mixture of spices, salt and mangoes to the jars after the mixture had cooled. I made the mixture in a frying pan to try and remove as much water as I could. After it was cooled, I put it in the jar and heated some mustard oil till it just barely started smoking. I let that cool and poured it over the mango spice mixture.  For the first three days it was fine. On the fourth day, I opened it to test the taste and ended up adding more salt. Now, there seems to be gas forming in the jar. I've noticed it a few times now. When I open the jar, I hear a whooshing noise as air escapes. It's quite a lot, since I opened the container about 3 times in a 10 hour period and each time gas escaped. Has this pickle gone bad? Or is this something that's normal? A Regular Tetrahedron is a cool Polyhedron. A regular tetrahedron has this property:  For any two of its vertices exists a third vertex, which forms a regular triangle with these 2 vertices.(It doesn't necessarily have to be a face of it).  Are there any other polyhedrons that have the same property?  I think there isn't such a polyhedron. But not sure how to prove it. I've tried proving that there isn't an irregular tetrahedron with this property, assuming that there is a pair of unequal sides.   Current answers are a little confused with the conditions of the problem. Undergraduate research field different from PhD field Upgrade SQL Server database Using Compatibility Level I've made a backup of a SQL Server database running on SQL Server 2008. I've restored that backup into a SQL Server 2012 instance. I've set the compatibility level on the restored instance to SQL Server 2012 (110).   What does this do to the database?   Does it, for example,    cause stored procedures to be examined for incompatibilities cause any updates to all of the data pages in the database Cause updates to to datapages as the are accessed in the future Cause errors in the future if stored procedures or views are executed that use incompatible syntax   Where in books on line, or else where, can I find more detail? How to use "tens of" and "hundreds of"? If I'm not mistaken, tens of means 10 to 99 and hundreds of means 100 to 999. Is this correct? I found in some dictionaries that tens of is actually not correct.  I also found that hundreds of could also mean any arbitrary large number. So how would people usually interpret hundreds of? Based on context?  In my case, I want to describe numbers of some items that are usually 50-90 but sometimes could be around 100-200, but definitely not as many as 300 or so. I want to really emphasize that there are a lot of these items. In this case, can I say "... have tens or even hundreds of ..."? Would people misinterpret what I actually want to say? The order of words in a clause: "tell me who is the real man" vs. "tell me who the real man is" Why do you need to know who I know on Google? I just created a SE account using my Google account. It said that it needs to "Know who you are on Google" (I guess my connections). As far as I know, SE is not a social network. I can't even follow other people (seeing when they post questions and answers).  What is the reason you need to know my Google connections?  This was on code review. Actually I have an account there, but for some reason it asked me to reconnect. This is the same on MSO; one sees this explained as "This app is requesting permission to associate you with your public Google profile": Prove that the derivative of an even differentiable function is odd, and the derivative of an odd is even. How would one go about proving that the rationals are not the countable intersection of open sets? I'm trying to prove that the rationals are not the countable intersection of open sets, but I still can't understand why   $$\bigcap_{n \in \mathbf{N}} \left\{\left(q - \frac 1n, q + \frac 1n\right) : q \in \mathbf{Q}\right\}$$   isn't a counter-example. Any ideas? Thanks! Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$?