Proof: $A=B^2$ for some positive definite symmetric $B$
Square root of Positive Definite Matrix
How do I find a recursion formula to solve the following question:  In how many sequences of length $n$, with elements from $\{{0,1,2,3,4}\}$, the difference between every 2 adjacent elements is $1$ or $-1$?  I've been trying to solve the problem for the last 5 hours maybe, I'm sure that I'm missing something..   This is a problem in combinatorics, and the solution must be by finding a recursion formula for the problem.   best solution I've come so far is that if we say that $a_n$ is the solution to the question, so let's count how many sequences of size $n-1$ there are that meet the requirement, and we complete the first element to make it a sequence of $n$ elements. so for $n-1$ there $a_{n-1}$ sequences, and for each sequence we have $2$ options to choose the first element (one option is to take a plus $1$ number from the number that start the $n-1$ sequence, and the other option to to take a minus $1$ number from the number that start the $n-1$ sequence), only problem as you figured out is that if the $n-1$ sequnce starts with $0$ or $4$ there is only $1$ option left for the first element in the sequnce.. so $a_n=2a_{n-1}$ won't work.  I think that the solution may involve using $2$ or maybe $3$ recursion formulas.  Hope I was clear enough. XIV pages in page counts? If $ \operatorname{Tr}(M^k) = \operatorname{Tr}(N^k)\;\forall 1\leq k \leq n$, then how do we show the $M$ and $N$ have the same eigenvalues? How are PCA and classical MDS different? How about MDS versus nonmetric MDS? Is there a time when you would prefer one over the other? How do the interpretations differ? The top bar is showing (again) the suggested edits count and 10K flags count, like it was before.  On Stack Overflow it looks like this: (that's the original color, same as in old top bar)        And on this very Meta site:        Same top bar. But different colors! Consistency broken.  I've just noticed the suggested edits count on Meta:        Black on black, I can see just a floating number - really confusing and out of any context.  In the name of Consistency, can we please have the same color on all sites? How to get file creation & modification date/times in Python? I have a script that needs to do some stuff based on file creation &amp; modification dates but has to run on Linux &amp; Windows.  What's the best cross-platform way to get file creation &amp; modification date/times in Python? The third formula on the  states that $$\varphi (mn) = \varphi (m) \varphi (n) \cdot \dfrac{d}{\varphi (d)} $$ where $d = \gcd(m,n)$.  How is this claim justified?  Would we have to use the Chinese Remainder Theorem, as they suggest for proving that $\varphi$ is multiplicative? Well today as I was going home I got a strange email, in my private inbox (email associated with S.O.) asking me if I had solved a question on S.O. that I didn't ask. To be more precise I had given an answer.   After reading:       and several other email related issues on S.O. I decided to let you all know about this because probably I am not the only one.   So the mail subject was:     Re: Deleted facebook account - Cannot access my apps?   which refers to   And the body (masked last name of sender):      Hi Jimmy,      Just wondering if you ever got this question answered.  If not, maybe   a brief chat with a html wizpert could help: link   .      Good luck getting this resolved!      Cheers,      Michael      ------------------------- Michael W%^%^GF Founder, CEO @ Wizpert   Now I know that my email address is not visible and I read about the MD5 regarding gravatar but I don't think this is the case. I thought about some methods and sure it's possible via metadata, github, etc for someone to get my mail, but I do not dare to think that this has reached an automated level, so that some service could easily get the email of S.O. members.   I didn't get annoyed by the message, since the CEO took the time to contact me :-p but someone else could get annoyed by these kinds of messages, since they come in the important inbox of gmail.   What is more strange is that Wizpert seems to be a nice site and also is active in the Careers 2.0 if I am not mistaken.   Another thing that crossed my mind is that since Wizpert needs unanswered questions to promote their services, possibly this is an action of Wizpert trying to attract people into their services.   Well that's it, I hope I did good to let you know (maybe I did't include the correct tags). Wrong posts number on Review section How to compute the marginal likelihood given two multivariate Normal distribution? Given that $a$~$\mathcal{N}(\mu, \Sigma)$ and $\mu$~$\mathcal{N}(0, \sigma^2 I)$, Is $p(a|\Sigma, \sigma)=\int p(a|\mu, \Sigma)p(\mu|\sigma)d\mu$ again a Normal distribution? And how to estimate the mean and variance respectively? Thanks a lot! How to install with Wubi when it says "Could not retrieve the required disk image files"? How do you keep user.config settings across different assembly versions in .net? How to prove the property of the Lemoine point of a triangle? Overlapping text in Database Schema SEDE Over at , there is overlapping text in the Database Schema under Close-As-Off-Topic-Reason-Types:  that makes it really hard to read.  This is also happening underneath &quot;Pending Flags&quot;, and seemingly with every long line in the schema.  This would be fine if you could increase the width of the list as you can with the height, which would in theory separate the lines, but you can't.  Can we fix this, please?  UPDATE: I have found that  post is almost exactly the same as mine, but steady on with duplicate votes.   comment says that it will be fixed, but that question was asked more than five years ago, and in that time the problem has not been resolved.  UPDATE 2: It seems that  would help.  I changed a little CSS/HTML, and came up with this:  Although mine is a little cruder than it could be.  This is just an example. What is the difference between a relative phase and a global phase? In particular, what is a phase? I know that $re^{i\theta} = x + iy$ for any complex number $x + iy$ by Euler's formula. How do you calculate relative and global phase? How to prove that $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$ How would one prove that $$\lim_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}=\varphi$$  where $F_n$ is the nth Fibonacci number and $\varphi$ is the Golden Ratio? nmake fatal error U1034: syntax error : separator missing 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. How to get the insert ID in JDBC? Ubuntu 12.04 comes with Gimp 2.6, so I installed Gimp 2.8 using the instructions below:   sudo apt-get purge gimp*   sudo add-apt-repository ppa:otto-kesselgulasch/gimp   sudo apt-get update   sudo apt-get install gimp    When try to open help, I get the message below. How do I install and configure the help for 2.8 on Ubuntu?