Consider the set $S= \{\frac{p}{2^n} : p,n \in \mathbb{Z}\}$. Is it dense in $\mathbb{R}$ ?? To me intuitively it seems to be dense but I cannot come up with any analytic proof or disprove .
Let $S=\{\frac{m}{2^n}| n\in\mathbb{N}, m\in\mathbb{Z}\}$, is  $S$ a dense set on $\mathbb{R}$?
How to include graphics with spaces in their path? I want to import graphics into my main input file using the macro \includegraphics. It does not work if the filename contains spaces.   also discusses this subject, but there is no solution there.  My compilation routine is latex->dvips->ps2pdf (because of PSTricks). Proving Binomial Identity without calculus How to establish the following identities without the help of calculus:  For positive integer $n, $ $$\sum_{1\le r\le n}\frac{(-1)^{r-1}\binom nr}r=\sum_{1\le r\le n}\frac1r $$  and  $$\sum_{0\le r\le n}\frac{(-1)^r\binom nr}{4r+1}=\frac{4^n\cdot  n!}{1\cdot5\cdot9\cdots(4n+1)}$$ Medium with refractive index less than unity? How to Map a subset of list elements to a function? What really allows airplanes to fly? One single command to update everything in Ubuntu? Relation bewteen Hessian Matrix and Curvature According to ,  It describes the local curvature of a function.   AFAIK, for one-variable function $f(x)$, its local curvature is  $$\kappa = \frac{|f''|}{(1 + f'^2)^{3/2}},$$ and its Hessian matrix is  $$\mathcal{Hess}(f) = [f''],$$ right? And here is my problem, I think the local curvature is not just described by its Hessian matrix, because $f'$ also has its role in it, doesn't it?  And furthermore, for 2-variable function $f(x,y)$, its Hessian matrix is  $$\mathcal{Hess}(f) = \left[ \begin{array}{cc} f_{xx}'' &amp; f_{xy}'' \\ f_{xy}'' &amp; f_{yy}'' \end{array} \right].$$ How does it relate to the local curvature of $f(x,y)$? What would be an appropriate opposite of "exceed"? Multiple Inheritance #include&lt;iostream&gt; using namespace std;  class A  {    int a;    int b;    public:    void eat()    {       cout&lt;&lt;"A::eat()"&lt;&lt;endl;    } };  class B: public A {    public:    void eat()    {        cout&lt;&lt;"B::eat()"&lt;&lt;endl;     }  };  class C: public A {     public:    void eat()     {        cout&lt;&lt;"C::eat()"&lt;&lt;endl;     }  };  class D: public B, C {  };  int foo(A *ptr) {  ptr-&gt;eat();  } main() {  D obj; foo(&amp;(obj.B)); //error. How do i call with D's B part.  }   The above foo call is a compile time error.  I want to call foo with obj's B part without using virtual inheritance. How do i do that.   Also, in case of virtual inheritance, why the offset information need to be stored in the vtable. This can be determined at the compile time itself. In the above case, if we pass foo with D's object, at compile time only we can calculate the offset of D's A part. If I've got a file (or files) selected in Windows Explorer, the only way I've found to "unselect" them is click with the mouse on some empty space.  I thought Esc would do this, but it doesn't. Also changing focus with Tab or Shift + Tab doesn't work; the file(s) are still selected even if focus goes to the address bar for instance.  Does anyone know of a keyboard shortcut to do this? Wubi with 14.10 doesn't work - failed to partition the selected disk I tried to install Ubuntu 14.10 on my computer with Wubi. The installation appeared to work in Windows, and told me to reboot.   After rebooting the Ubuntu installer continued the installation, until it stopped with an error message: Failed to partition the selected disk with the detail This probably happened because are too many primary partitions in the partition table. From where do the permanent magnets get energy from? I have a doubt about permanent magnets. If a magnet is permanent it can attract some materials permanently.  Attracting something involves energy. If a permanent magnet can do this forever, from where does this energy come from? How can it not run out of energy? Doesn't it contradict with the laws of energy? You "show" someone a picture. You "---" someone a song? In , we have a verb meaning &quot;to show&quot; corresponding to &quot;to see/to look&quot;, and we have a different verb corresponding to &quot;to hear/to listen&quot;:  inti tara stampa (you look at a picture.) ---- jiena nurilek stampa (I show you a picture) inti tisma' diska (you listen to a song.) ----- jiena nsemmagħlek diska (I 'show' you a song)  In English, do you &quot;show someone a song&quot; (sounds weird to me) or is there a verb that corresponds more directly to the Maltese insemmgħek? So I tried a few different answers that I found here, and I feel that this is possibly a duplicate question, but everything I have tried from the other solutions hasn't worked.  Basically I am trying to bake a material to a texture that will be used in Unity, but every test gives me a black background image.  In the image below, the top image texture is the mask (black and white, white being the leaves), and the bottom image texture is the colour. Simple excel math wrong Firstly I do not want to round my numbers, I am using base 10 figures so excel should have no bother doing simple add and subtract.  A|  665.110000000000000000000000000000 B|  4.990000000000000000000000000000 C|  =a1+b1 D|  670.790000000000000000000000000000 E|  1,335.890000000000000000000000000000 F|  =e1-(c1+d1)   I should get -5 however F1  returns   -4.999999999999770000000000000000   It is a financial document therefore I want to avoid rounding, more to the point. Why should I need to? I've put cell formatting to Currency with 2 decimal placed and I will get the correct display value but for all calculations I will still get -4.9999...7 which leads to further errors with conditional formatting etc. How to get Lookup ID from Search API with REST I'm trying to get the ID of a lookup field when using search api. My query looks something like this:  /_api/Search/query?querytext='ListId:12345678-1234-1234-1234-123456789012'&amp;selectproperties='Title,Company'   Where Company is the lookup field, in a specific list.  The result looks like this:  &lt;d:element m:type="SP.KeyValue"&gt;   &lt;d:Key&gt;Company&lt;/d:Key&gt;   &lt;d:Value&gt;Some company&lt;/d:Value&gt;   &lt;d:ValueType&gt;Edm.String&lt;/d:ValueType&gt; &lt;/d:element&gt;   I have created a managed property which looks like this: "I", "me" and "myself" Possible Duplicate:           What is correct? We are a family of four: my father, my mother, my brother and me. or We are a family of four: my father, my mother, my brother and I. or We are a family of four: my father, my mother, my brother and myself. How to make partitions mount at startup? How do you make HDDs and partitions mount at startup on Ubuntu?  I always keep my media and data files on separate partitions - one 2tb HDD and a 400 odd gig partition.  I have been trying to swap the default folder location for the home directory to my other drives.  It works when I change the /home/user_name/.config/user-dirs.dirs file to the directories I want, but only until I reboot the system. I am only assuming that the drives not mounting is the problem with it, but it would be helpful either way.  If any one knows another reason for why the directories change back to default each time that would be good also. Program does not wait before output Is the wave-particle duality a real duality? I often hear about the wave-particle duality, and how particles exhibit properties of both particles and waves. However, I wonder, is this actually a duality? At the most fundamental level, we 'know' that everything is made up out of particles, whether those are photons, electrons, or maybe even strings. That light for example, also shows wave-like properties, why does that even matter? Don't we know that everything is made up of particles? In other words, wasn't Young wrong and Newton right, instead of them both being right?