A $d$-dimensional topological manifold $M$ is a topological space for which there exists continuous maps (charts) to $R^d$, such that every point $p \in M$ is mapped by at least one of those charts.  A differentiable manifold $M$ is a topological manifold with the additional requirement that its Atlas contains only charts whose transition maps are all differentiable.  I find this second definition (of differentiable manifolds) very strange, because it only makes reference to the charts in the Atlas of manifold $M$, but not to the manifold itself.  If I'm correct, it means we can construct a differentiable manifold out of a non-differentiable continuous function:     `Take the graph $M$ of a non-differentiable function (e.g. $f(x)=|x|$). This clearly is a $1$-dimensional topological manifold. Now we can simply take the Atlas that consists of the single chart map that maps every point on the graph to the $x \in \mathbb R$ that produced it. This clearly is a chart map, and it clearly has a chart transition map to itself that is differentiable.   So this means that manifolds that have "kinks" in them, like the graphs of non-differentiable functions, can still be differentiable manifolds.  Could even a function like the Weierstrass function be a differentiable manifold?  What does "differentiability" of a manifold intuitively mean, since it is not intuitively the same thing as the differentiability of a real function?
Take the manifold: The graph of $|x|$ on $(-1,1)$, with the induced topology from $\mathbb{R}^2$. This is a topological manifold, which is homeomorphic to $(-1,1)$ by projection. Is it a differentiable manifold? I believe it is, because we can take an atlas with only the projection, and then we will have only one transition map which is the identity, and therefore differentiable. But I'm not sure I get the definition of a differentiable manifold right...  Thanks!
I was asked to evaluate the following sum: $$\sum_{k=1}^{\infty}\frac{1}{k}\int_{\pi k}^{\infty}\frac{\sin(x)}{x}dx$$  I'm trying to use $$\int_{0}^{\infty} \frac{\sin(x)}{x}dx=\frac{\pi}{2}$$ However, it doesn't seem to work. Any help is greatly appreciated. I am trying to figure out this:  c = 1 def f(n):     print c + n  def g(n):     c = c + n  f(1) # =&gt; 2 g(1) # =&gt; UnboundLocalError: local variable 'c' referenced before assignment   Thanks! Deriving the Coulomb force equation from the idea of photon exchange? Since Newton's law of gravitation can be gotten out of Einstein's field equatons as an approximation, I was wondering whether the same applies for the electromagnetic force being the exchange of photons. Is there an equation governing the force from the exchange of photons? Are there any links which would show how the Coulomb force comes out of the equations for photon exchange? I know that my question is somewhat similar to the one posted here , but it doesn't really have an answer to my question specifically. Customize feed on stack exchange android app Why didn't all the Jedi at the Jedi Temple sense Anakin and his legion of Clones’ thoughts? If you go back to Attack of the Clones, Anakin and Obi-Wan sensed that poisonous alien worms were about to kill Padme. Back in Revenge of the Sith, the Jedi could take on the legion or if there were too many, they could at least escape. Center of $D_6$ is $\mathbb{Z}_2$ The center of $D_6$ is isomorphic to $\mathbb{Z}_2$.   I have that $$D_6=\left&lt; a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right&gt;$$ $$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$ My method for trying to do this has been just checking elements that could be candidates. I've widdled it down to that the only elements that commute with all of $D_6$ must be $\{e,a^3\}$ but I got there by finding a pair of elements that didn't commute for all other elements and I still haven't even shown that $a^3$ commutes with everything. For example, I have been trying to show now that $$a^3b=ba^3$$ and haven't gotten too far yet but if I had to answer a question like this on the exam, I feel it would be difficult, is there any kind of trick or hints other than brute force using the relations to get that $a^3$ commutes with everything?  For the solution once I have that the center of $D_6$ is what I think then as there is only one group of order $2$ up to isomorphism, it must be isomorphic to $\mathbb{Z}_2$.  Ideally a way that doesn't appeal to $D_6$ as symmetries of the hexagon if that seems possible. Output of Logistic Regression Prediction Dealing with "Find out who's going to buy the croissants" On Stack Overflow at the moment, there is a popular question which is currently high in the Hot Questions feed:   So far it has been closed twice (once by the community and a moderator and once by a moderator alone) but re-opened twice by the community. It is clearly not a [so] Stack Overflow question. Reading between the lines, there could be a valid Software Engineering question there, but it would be a very basic one (at the fizz-buzz competency level) and looks more like a populist attempt to farm reputation than a serious question.  Worst of all, even though there are some interesting answers on this question, they are all going to be completely useless to future visitors to the site, since they will be completely undiscoverable. If the question were at least "What is the best fair distribution algorithm?" it would still be off-topic (primarily opinion based), but at least it might help someone once the topic falls off the Hot Questions feed.  Although this has turned into just a bit of fun, and I don't want to kill that, enough is enough. While this question is open, unlocked, and still seen as a valid question, it is setting a bad example for the whole Stack Exchange network.   questions like this, especially on the grandfather Stack Exchange site, devalue the work we are putting in on every other site by making it appear acceptable to re-open populist questions like this after two moderators have classified it as off-topic.  What should we do about this question?   Should it be left open? Power to the people? Closed and locked by a moderator? Migrated to another site? &hellip;or something else? I've read  about how some publishers (e.g., Springer) allows authors to submit to arXiv before journal submission, but they make authors wait for a year after acceptance to submit to arXiv.   My question is: Is it okay to submit to arXiv during the time when the paper is under consideration by the journal, for example after it has been "revised and resubmitted" but before final decision? SQLiteException while inserting values in the database in Android Why does "kill -9 0" end my console session? I just wonder why kill -9 0 ends my tty console session?   What exactly is PID 0, it is not listed on a ps aux? What is the net charge of the Earth? I am trying to find out how to approach a relief type project. I have a background image and a flat plane mesh to sculpt on, but I actually don´t see any changes looking straight on top. When I tilt the perspective a bit the image disappears.  Is there a way to make the image appear at all perspectives? $\LaTeX$ and Mathematica Folding a cube (animation) I'm trying to fold a cube with an animation. I've tried inverse kinetics with no luck. Here are some images that explain what I'm trying to do. I'm trying to make this so that    when the upper bone is pushed down, the box is folded to the ground. when the upper bone is pulled up, it unfolds the box. How Does Kylo Ren not stop Chewbacca's blaster? How does Kylo Ren stop the blaster shot in the beginning when being shot at by Poe, only to be unaware of the coming blaster shot from Chewbacca, which seems much more predictable.   I am sure he knew Chewbacca, Rey and Finn where there as Chewbacca turns with the opening of the doors when Rey and Finn come in.  EDIT in regards to Richards Answer: I just want to expand a little more on question. What I don't understand is that Kylo Ren uses the force, so it is just not a matter him not hearing. When Poe shoots him from behind, he makes no sound, but yet he turns and stops the blaster and Poe. That has to be due to the power he commands in the Force.  And this is from Obi-Wan Kenobi:     "Well, the Force is what gives a Jedi his power. It's an energy field   created by all living things. It surrounds us and penetrates us; it   binds the galaxy together."―Obi-Wan Kenobi —   So it seems to me that it would be very hard for him even in a weakened state to not even notice that he was being shot at by virtue of the force being a part of Sith and Jedi's alike. Even if he was too weak to actually stop the blaster, the Force should have been able to warn him. For a moderately complex query I am trying to optimize, I noticed that removing the TOP n clause changes the execution plan.  I would have guessed that when a query includes TOP n the database engine would run the query ignoring the the TOP clause, and then at the end just shrink that result set down to the n number of rows that was requested.  The graphical execution plan seems to indicate this is the case -- TOP is the "last" step.  But it appears there is more going on.  My question is, how (and why) does a TOP n clause impact the execution plan of a query?  Here is a simplified version of what is going on in my case:  The query is matching rows from two tables, A and B.    Without the TOP clause, the optimizer estimates there will be 19k rows from table A and 46k rows from table B.  The actual number of rows returned is 16k for A and 13k for B.  A hash match is used to join these two results sets for a total of 69 rows (then a sort is applied).  This query happens very quickly.  When I add TOP 1001 the optimizer does not use a hash match; instead it first sorts the results from table A (same estimate/actual of 19k/16k) and does a nested loop against table B.  The estimated number of rows for table B is now 1, and the strange thing is that the TOP n directly affects the estimated number of executions (index seek) against B -- it appears to always be 2n+1, or in my case 2003.  This estimate changes accordingly if I change TOP n.  Of course, since this is a nested join the actual number of executions is 16k (the number of rows from table A) and this slows down the query.  The actual scenario is a bit more complex but this captures the basic idea/behavior.  Both tables are searched using index seeks.  This is SQL Server 2008 R2 Enterprise edition. The basic concept of Quotient Group is often a confusing thing for me,I mean can any one tell the intuitive concept and the necessity of the Quotient group, I thought that it would be nice to ask as any basic undergraduate can learn the intuition seeing the question. My Question is :   Why is the name Quotient Group kept,normally in the case of division, let us take the example of $\large \frac{16}{4}$ the Quotient of the Division is '$4$' which means that there are four '$4$'s in $16$, I mean we can find only $4$ elements with value $4$     So how can we apply the same logic in the case of Quotient Groups,like consider the Group $A$ and normal subgroup $B$ of $A$, So if $A/B$ refers to "Quotient group", then does it mean:           Are we finding how many copies of $B$ are present in $A$??, like in the case of normal division, or is it something different ??       I understood the Notion of Cosets and Quotient Groups,b ut I want a different Perspective to add Color to the concept. Can anyone tell me the necessity and background for the invention of Quotient Groups?  Note: I tried my level best in formatting and typing with proper protocol,if in case, any errors still persist, I beg everyone to explain the reason of their downvote (if any), so that I can rectify myself, Thank you. The code below compiles, but has different behavior for the char type than for the int types.  In particular     cout &lt;&lt; getIsTrue&lt; isX&lt;int8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;char&gt;::ikIsX  &gt;() &lt;&lt; endl;   result in 3 instantiations of templates for three types: int8, uint8 and char. What gives?  The same is not true for ints: int and uint32 which result in the same template instantiation, and signed int another.  The reason seems to be that C++ sees char, signed char and unsigned char as three different types. Whereas int is the same as a signed int. Is this right or am I missing something?  #include &lt;iostream&gt;  using namespace std;  typedef   signed char       int8; typedef unsigned char      uint8; typedef   signed short      int16; typedef unsigned short     uint16; typedef   signed int        int32; typedef unsigned int       uint32; typedef   signed long long  int64; typedef unsigned long long uint64;  struct TrueType {}; struct FalseType {};  template &lt;typename T&gt; struct isX {    typedef typename T::ikIsX ikIsX; };   // This  int==int32 is ambiguous //template &lt;&gt;            struct isX&lt;int  &gt;    { typedef FalseType ikIsX; };  // Fails template &lt;&gt;            struct isX&lt;int32  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;uint32 &gt;  { typedef FalseType ikIsX; };   // Whay isn't this ambiguous? char==int8 template &lt;&gt;            struct isX&lt;char  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;int8  &gt;  { typedef FalseType ikIsX; }; template &lt;&gt;            struct isX&lt;uint8 &gt;  { typedef FalseType ikIsX; };   template &lt;typename T&gt; bool getIsTrue(); template &lt;&gt;           bool getIsTrue&lt;TrueType&gt;() { return true; } template &lt;&gt;           bool getIsTrue&lt;FalseType&gt;() { return false; }  int main(int, char **t ) {    cout &lt;&lt; sizeof(int8) &lt;&lt; endl;  // 1    cout &lt;&lt; sizeof(uint8) &lt;&lt; endl; // 1    cout &lt;&lt; sizeof(char) &lt;&lt; endl;  // 1     cout &lt;&lt; getIsTrue&lt; isX&lt;int8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint8&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;char&gt;::ikIsX  &gt;() &lt;&lt; endl;     cout &lt;&lt; getIsTrue&lt; isX&lt;int32&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;uint32&gt;::ikIsX  &gt;() &lt;&lt; endl;    cout &lt;&lt; getIsTrue&lt; isX&lt;int&gt;::ikIsX  &gt;() &lt;&lt; endl;  }   I'm using g++ 4.something How to pronounce "Stannis's"? I know that the &quot;s&quot; becomes &quot;z&quot; when it comes after a voiced sound. And it is pronounced as &quot;s&quot; when it comes after a voiceless sound. I also know rules about possessives but I don't know how natives pronounce a word when it ends in S and &quot;apostrophe S&quot; is added to it. For example: &quot;Stannis's army&quot;. Do you add another syllable to it? Like Stannisiz? or just Stannis and ignore the possessive S? I really have difficulty in pronouncing words that end in S and possessive S is attached to it.