I have heared that there is connection between number theory and physics. I specially ask that:          Can anyone give a concrete example which shows this connection?   I want to see an example which is related to (integral) quadratic forms or theta series.          Let $f$ to be a quadratic form; i.e a homogeneous polynomial of degree two (in any number of variables), for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$. Also let's assume that all of the co-efficeints of $f$ are integral.    Now let's fix a  (integral) quadratic form $f$ in $m$ variables, and let $n$ varies in $\mathbb{N}$; let's define:   $$  N(n)  :=  N(f,n)  =  N\big(f(x_1, ..., x_m),n\big)  =  \#  \{(x_1, ..., x_m) \in \mathbb{Z}^m : f(x_1, ..., x_m)=n \}  ;  $$ note that $N(n)$ is the number of integral solutions to the equation $f=n$; now by a  associated to the quadratic form $f$ we mean the following:   $$  \Theta(q)  :=  \Theta_f(q)  =  \sum_{n\in \mathbb{N}_0}N(n)q^n  =\sum_{(x_1, ..., x_m) \in \mathbb{Z}^m}q^{f(x_1, ..., x_m)}  ;  $$   also if we assume that $q=e^{2\pi i z}$;  then there is theorem which states that $\Theta_f(q)$ is a modular form in $z$.      I have asked my question  again.
As a Graduate Mathematics student, my interest lies in Number theory. I am curious to know if Number theory has any connections or applications to physics. I have never even heard of any applications of Number theory to physics. I have heard Applications of linear algebra and analysis to many branches of physics, but not number theory.  Waiting forward in receiving interesting answers!
I recently made my pen-drive bootable with Ubuntu 14.04. But I need to unboot my pen-drive now. I formatted my pen-drive once,But the size of the pen-drive still shows 2.25 MB. Whereas,this is an 8GB pen-drive. How can I get rid of this problem. I'm still using Windows 7. How do I get my players to form a PC party without just forcing them to? I believe that it is generally a good thing that all the PCs travel together, as a "party". This way the GM has to manage one environment – not, say, five. The problem is how the GM can ensure such a relationship between the characters.  One simple and really great solution is to require the players to design their characters so that they know each other. This has the added advantage of cross-backstories – as the players try to adjust their stories to fit together, they get and create hooks for more narrative!  However, I am now experimenting with giving the players complete freedom at character creation and later meeting in-game. No luck so far.  It is not easy, as trusting someone with your life (what happens during combat) is not a trivial thing. Furthermore, people generally like others with the same behaviour/interests/skin colour. Fantasy is all about diversity – you do not character-generate a strong man with a sword, no, you generate a half-ogre with an axe never before seen!  How can the GM get players, with diverse characters who are strangers to each other, decide to have their PCs act as a coherent group. I absolutely want to avoid asking my players to just solve this through metagaming ("Hey Joe, think of a way to get together with those, I can't story-tell two separate groups at the same time!").  I'm not having the stereotypical problem where a bunch of PCs meet for the first time and are automatically so paranoid of strangers that they start trying to kill each other. My players have just created a bunch of characters who have their own legitimate interests, and are faithfully following those in completely different directions that don't result in a classic "party". I put the PCs in great peril together and everyone escaped by their own devices, but now I have to figure out get the guy who ran for the hills back together with the couple that entered the city. And, I want to do it by resorting only to reasonable in-game events, not out-of-game metagame suggestions. Java Generics: Generic type defined as return type only I'm looking at some GXT code for GWT and I ran across this use of Generics that I can't find another example of in the Java tutorials. The class name is  if you want to look at all of the code.  Here are the important parts:  private RpcMap map;  public &lt;X&gt; X get(String property) {   if (allowNestedValues &amp;&amp; NestedModelUtil.isNestedProperty(property)) {     return (X)NestedModelUtil.getNestedValue(this, property);   }   return map == null ? null : (X) map.get(property); }   X is defined nowhere else in the class or anywhere in the hierarchy, and when I hit "go to declaration" in eclipse it just goes to the &lt;X&gt; in the public method signature.  I've tried to call this method with the following two examples to see what happens:  public Date getExpiredate() {     return  get("expiredate"); }  public String getSubject() {     return  get("subject"); }   They compile and show no errors or warnings.  I would think at the very least I would have to do a cast to get this to work.  Does this mean that Generics allow a magic return value that can be anything and will just blow up at runtime?  This seems counter to what generics are supposed to do.  Can anyone explain this to me and possibly give me a link to some documentation that explains this a little better?  I've went through Sun's 23 page pdf on generics and every example of a return value is defined either at the class level or is in one of the parameters passed in. QGtkStyle was unable to detect the current GTK+ theme When I start any Qt app, I get this message in the terminal, and the Qt app doesn't use the same theme as my Gtk apps:  QGtkStyle was unable to detect the current GTK+ theme.   How do I make QGtkStyle detect my current GTK+ theme?  It seems that LXDE is using a settings daemon, and there is no gtkrc-2.0 file in my home folder. I'm trying to educate myself on Granger Causality. I've read the posts on this site and several good articles online.  I also came across a very helpful tool, the , that allows you to enter your time series and calculate the Granger Stats.  Below, is the output from the sample data included on the site. I have also taken a crack at interpreting the results.   My Questions:     Is my interpretation directionally correct?   What key insights have I overlooked?   Also what is the meaning and interpretation of the CCF charts?  (I'm assuming CCF is cross correlation.)     Here are the results and plots that I have interpreted:  Summary of computational transaction Raw Input   view raw input (R code) Raw Output  view raw output of R engine Computing time  2 seconds R Server    'Herman Ole Andreas Wold' @ wold.wessa.net  Granger Causality Test: Y = f(X) Model   Res.DF  Diff. DF    F   p-value Complete model  356          Reduced model   357 -1  17.9144959720894    2.94360540545316e-05  Granger Causality Test: X = f(Y) Model   Res.DF  Diff. DF    F   p-value Complete model  356          Reduced model   357 -1  0.0929541667364279  0.760632773377753       My interpretation:    Test was based upon 357 data points and was performed with a lag value of 1 The p-value of 0.0000294 means I can reject the null hypothesis that x does not cause y for the Y = f(x). The p-value of .76 allows me to accept the null for X = f(Y) The fact that first hypothesis was rejected and second accepted is a good thing  I'm a little rusty on my F-test so I don't really have anything to say on this for now. I'm also not sure how to interpret the CCF graph.   I really appreciate it if any of you who are well versed with Granger-causality could let me know if I'm interpeting this correctly and also fill in some of the blanks.   Thanks for your help. Calculate $\tan9^{\circ}-\tan27^{\circ}-\tan63^{\circ}+\tan81^{\circ}$ Calculate $\tan9^{\circ}-\tan27^{\circ}-\tan63^{\circ}+\tan81^{\circ}$?  The correct answer should be 4. Why is glass transparent? Frequently when I ask a question I have to include quite a bit of code for reference purposes.    It would be nice to be able to bold the important lines in the code.  For example this code:  &lt;DataTrigger Value="True"&gt;     &lt;DataTrigger.Binding&gt;         &lt;MultiBinding Converter="{StaticResource DisableWorkItemConverter}"&gt;             &lt;Binding ElementName="MainForm" Path="PickedWorkItemID"/&gt;             &lt;Binding Path="Id"/&gt;         &lt;/MultiBinding&gt;     &lt;/DataTrigger.Binding&gt;     &lt;Setter Property="IsEnabled" Value="False"/&gt;     &lt;Setter Property="IsSelected" Value="False"/&gt; &lt;/DataTrigger&gt;   Is not way long, but the important part (as seen in ) is the second to last line.  However, if I try to bold it, it ends up looking like this:  &lt;DataTrigger Value="True"&gt;     &lt;DataTrigger.Binding&gt;         &lt;MultiBinding Converter="{StaticResource DisableWorkItemConverter}"&gt;             &lt;Binding ElementName="MainForm" Path="PickedWorkItemID"/&gt;             &lt;Binding Path="Id"/&gt;         &lt;/MultiBinding&gt;     &lt;/DataTrigger.Binding&gt;     &lt;Setter Property="IsEnabled" Value="False"/&gt;     **&lt;Setter Property="IsSelected" Value="False"/&gt;** &lt;/DataTrigger&gt;   Not what I was going for.  It would be nice to be able to draw attention to specific lines of code when needed.    (One suggestion of a different color background sounds nice too (ie a yellow background to draw the eye to the relevant code.)  If there is some other way to bold in code please let me know. Pressing "W" turned my cursor into a circle How to run a specific program as root without a password prompt? BAD signature when trying to verify SHA256SUMS for Bionic Beaver ISO I am attempting to verify my download of Bionic Beaver on Windows 7.  I feel like I am missing something very obvious.  I went to  and clicked on both SHA256SUMS and SHA256SUMS.gpg  It seems like this does not download the files, but opens the text in the files in new tabs.  I used Ctrl-A and then copied the text into Notepad, naming the files SHA256SUMS and SHA256SUMS.gpg.  I made sure that windows didn't add any weird extensions when saving.  I manually downloaded both keys I used from https://keyserver.ubuntu.com, one with fingerprint  C598 6B4F 1257 FFA8 6632 CBA7 4618 1433 FBB7 5451   and the other with fingerprint  8439 38DF 228D 22F7 B374 2BC0 D94A A3F0 EFE2 1092   After I downloaded both keys I certified them with my personal key using Kleopatra, after verifying the key fingerprint with those at .  When I ran   gpg --verify SHA256SUMS.gpg SHA256SUMS   I got   gpg: Signature made 11/29/18 16:27:43 US Mountain Standard Time gpg:                using DSA key 46181433FBB75451 gpg: BAD signature from "Ubuntu CD Image Automatic Signing Key &lt;cdimage@ubuntu.c om&gt;" [full] gpg: Signature made 11/29/18 16:27:43 US Mountain Standard Time gpg:                using RSA key D94AA3F0EFE21092 gpg: BAD signature from "Ubuntu CD Image Automatic Signing Key (2012) &lt;cdimage@u buntu.com&gt;" [full]   When I ran   gpg --verbose --verify SHA256SUMS.gpg SHA256SUMS   I got  gpg: armor header: Version: GnuPG v1 gpg: Signature made 11/29/18 16:27:43 US Mountain Standard Time gpg:                using DSA key 46181433FBB75451 gpg: using pgp trust model gpg: BAD signature from "Ubuntu CD Image Automatic Signing Key &lt;cdimage@ubuntu.c om&gt;" [full] gpg: binary signature, digest algorithm SHA512, key algorithm dsa1024 gpg: Signature made 11/29/18 16:27:43 US Mountain Standard Time gpg:                using RSA key D94AA3F0EFE21092 gpg: BAD signature from "Ubuntu CD Image Automatic Signing Key (2012) &lt;cdimage@u buntu.com&gt;" [full] gpg: binary signature, digest algorithm SHA512, key algorithm rsa4096   I tried copying the data from SHA256SUMS and SHA256SUMS.gpg tabs over TOR instead, in case it was a local network problem, but no dice either.  What am I missing here? Is there some way to download the SHA256SUMS and SHA256SUMS.gpg files directly, as opposed to copying the data into a text editor and saving it?  Any help would be very appreciated. I am very confused and frustrated at this point. How can I output a list of all installed packages with installation date and additional information? When closing a question as a duplicate there is a text box which says: "search, or enter a question link or numeric id". You can copy and paste a question link into here, but this fails if the URL starts with https:// instead of http://, with the message "Your search returned no matches; please try a different search". What is the rule to convert the 'mm' notation to the 'optical zoom' notation? I searched a bit and found this one:  optical zoom = maximum focal length / minimum focal length   For example a 18-55mm lens would have a 3x optical zoom, and a 18-200mm lens would have a 11x optical zoom. Is that right? Diamond moderator privileges vs. 10k rep privileges? As we are in election period, I don't get the difference between getting to 10k reputation point and the diamond. What additional controls do you earn with the diamond ? Why do Tcler suggest to brace your `expr`essions? How to find answer to the sum of series $\sum_{n=1}^{\infty}\frac{n}{2^n} $ I have put his on wolfram and obtained answer as follows:  $\sum_{n=1}^{\infty}\frac{n}{2^n} = 2$  And the series is convergent too because $\lim_{n\to\infty} \frac {n}{2^n} = 0$  However I am wondering if there is a convenient way to solve this; I don't think you can represent it by a geometric progression either. So how do we have to do it on paper? How to fix "W: Duplicate sources.list entry"? I keep getting this warning whenever I try to run sudo apt-get update.  W: Duplicate sources.list entry http://archive.ubuntu.com/ubuntu/ precise-updates/main i386 Packages (/var/lib/apt/lists/archive.ubuntu.com_ubuntu_dists_precise-updates_main_binary-i386_Packages) W: You may want to run apt-get update to correct these problems   Below is the output from /etc/apt/sources.list file:  deb http://archive.ubuntu.com/ubuntu precise main restricted deb-src http://archive.ubuntu.com/ubuntu precise main restricted  deb http://archive.ubuntu.com/ubuntu precise-updates main restricted deb-src http://archive.ubuntu.com/ubuntu precise-updates main restricted  deb http://archive.ubuntu.com/ubuntu precise universe deb-src http://archive.ubuntu.com/ubuntu precise universe deb http://archive.ubuntu.com/ubuntu precise-updates universe deb-src http://archive.ubuntu.com/ubuntu precise-updates universe  deb http://archive.ubuntu.com/ubuntu precise multiverse deb-src http://archive.ubuntu.com/ubuntu precise multiverse deb http://archive.ubuntu.com/ubuntu precise-updates multiverse deb-src http://archive.ubuntu.com/ubuntu precise-updates multiverse  deb http://archive.ubuntu.com/ubuntu precise-security main restricted deb-src http://archive.ubuntu.com/ubuntu precise-security main restricted deb http://archive.ubuntu.com/ubuntu precise-security universe deb-src http://archive.ubuntu.com/ubuntu precise-security universe deb http://archive.ubuntu.com/ubuntu precise-security multiverse deb-src http://archive.ubuntu.com/ubuntu precise-security multiverse   How do I fix it? Credit score just tanked ~50 points and I'm not sure why I check my credit score monthly using a credit tracking app provided by my credit card company. I use a reminder to check on the same day each month, and have been tracking my credit score for almost two years.  In the last year, the app switched from a TransUnion model to VantageScore 3.0. I have TransUnion data for the entire duration, but only about a year's worth of VantageScore data.  I have good credit, and my TransUnion score has increased steadily from ~740 to ~780 in the last two years.  My VantageScore started around ~760 and increased to ~800 last month. This morning when I checked my credit score, it had dropped just over 50 points to ~750.  My TransUnion score has stayed solid at ~780.  My questions are:  Should I be concerned about this sudden change in score?  Is there any way I can determine what changed to account for the sudden change in credit score? Time-Stream Crossing and Forgetfulness In The Day of the Doctor, it's explained that the Doctor doesn't remember having interacted with himself, and if he interacts with a future version of himself, the younger version forgets.  But in Time Crash, 10th solves the problem of the TARDISes occupying the same space because he remembers himself doing it from when he was 5th.  If this is true, 5th wouldn't have wondered whether or not he was going to regenerate when he did.  Does the TARDIS occupying the same space with itself re-instate those memories? Or is it just possible that I'm asking a ridiculous question that will never have an answer because Doctor Who.  :D