Creating div's and textbox using Javascript
Creating a new DOM element from an HTML string using built-in DOM methods or Prototype
Let $A,B$ be real symmetric matrices, and for any $n\in \Bbb N^{+}$, and for all $x,y\in \Bbb R$, we have $$tr(xA+yB)^n=x^ntr(A^n)+y^ntr(B^n).$$  Show that $AB=0$.  My try:since $$(xA+yB)^n=x^nA^n+\binom{n}{1}x^{n-1}A^{n-1}yB+\cdots+\binom{n}{n-1}xA(yB)^{n-1}+y^nB^n$$ so $$tr\left(x^nA^n+\binom{n}{1}x^{n-1}A^{n-1}yB+\cdots+\binom{n}{n-1}xA(yB)^{n-1}+y^nB^n\right)=x^ntr(A^n)+y^ntr(B^n)$$  then  $$tr\left(\binom{n}{1}x^{n-1}A^{n-1}yB+\cdots+\binom{n}{n-1}xA(yB)^{n-1}\right)=0$$ so $$\binom{n}{1}x^{n-1}ytr(A^{n-1}B)+\cdots+\binom{n}{n-1}xy^{n-1}tr(AB^{n-1})=0$$ then I can't  I found this is a nice result, maybe this is an odd problem, and I can't solve it. Thank you. Can anyone explain how Windows effectively limits the CPU?  Last time I tried some tricks to let my laptop not overheat, so I tried limiting the CPU to 70%.   As for my suprise it stopped my laptop from overheating - even when launching CPU intensive applications which caused my laptop to overheat and shut down, didn't overheat my laptop anymore!  I looked at task manager but the "CPU intensive" application still uses ~95% of CPU and the windows task manager shows a total of 100% CPU usage.  I have limited the CPU usage here ("plugged in" option was on 70% [both min and max] when testing the CPU intensive application):    How does Windows do this magic?   I see 100% CPU usage yet my CPU does not overheat. And I do not see any performance differences.  Does Windows limit the CPU speed? Like, if I have a 2.4 GHz processor, on 70% it will only have 1.68 GHz when limited? How is this possible (changing the CPU speed)? I thought you can only underclock when in BIOS?  If not, how does Windows accomplish this if not by changing the speed?    Side question: If it's about the speed, is there a way to set the CPU max state to 200% ? Suppose that a box contains 10 red balls, 20 green balls, and 30 blue balls.   Suppose also that balls are drawn from the box one at a time at random.  What is the probability that all the red balls are drawn before the blue or green balls are themselves exhausted. What is the probability that as the last red ball is drawn, there remains at least one blue and one green left in the box.  The answer I was given is $\dfrac{7}{12}$ and a general equation is: $$ \dfrac{b g}{1-b}+\dfrac{b g}{1-g} $$ where $$ g=\dfrac{20}{60},b=\dfrac{30}{60} $$ but why? I'm about to use this diagram to make computing 3x3 determinants clear and easy for my students. I tried to add this picture to my lecture but it's blurred due to the bad quality of the picture. So i want to create exactly the same figure using latex.  I've searched and read a few topics about highlighting elements and diagonals in matrix but it does not work in this case. I want to add plus signs and minus signs as in the picture, and highlight elements in the matrice with elements outside the matrice.  I appreciate all your help and your advice. Thank you in advance ! Torsion on $\pi_1(X)$, $X$ connected and open in $\mathbb{R}^n$ Can the fundamental group of an open connected subset $X$ of $\mathbb{R}^n$ have a torsion element? The Follow Questions and Answers feature is now live across the Network How to format hard drive How do you make USB bootable on chromebook I have a chromebook series 5. I want to install Ubuntu onto a USB stick. When I downloaded Ubuntu onto my USB stick, it won't start up.  Do I first have to download Ubuntu onto my chromebook hard drive, open Ubuntu and then make USB stick bootable? And then return back to chrome?  Or is there a way to download directly onto USB and make the stick bootable?  I appreciate any help. Deductible expenses paid with credit card: In which tax year would they fall? If I paid my business expenses with a credit card during the month of December, which year can I deduct them from my taxes? 2011 (when the purchase was made) or 2012 (when I paid off the credit card)?  IRS pub reference greatly appreciated The Medibuntu Project has come to an end - what do I do now? Good color palette, usable for the color blind. I'm designing a chart right now, that will slowly have more items added to it. I'd like to create an list/order of colors for the items, that will be most usable for the color blind to differentiate between each item. Any ideas how/where to start? Countability of disjoint intervals According this problem/solution set from an MIT class (http://ocw.mit.edu/courses/mathematics/18-100c-analysis-i-spring-2006/exams/exam1_sol.pdf), the assertion:  "Every collection of disjoint intervals in R is countable."  is True, because "every interval contains a rational number", and the rationals are countable.  It seems to me this should be False, with possible counterexample:  { [x,x] | x is an element of R}  ie the set of all singelton intervals on R. Why isn't this a valid counterexample? Boot failure : No DEFAULT or UI configuration directive found! Inner product of two continuous maps is continuous Let f,g be continuous maps of S in $\mathbb{R}^n$, to $\mathbb{R}^m$. Show the inner product h(x) = $\langle f(x), g(x)\rangle$ is continuous.  My attempt: Since f,g are continuous, their product is a continuous mapping, so for $\epsilon \gt 0\: t.e\:  \delta \gt 0$ such that $\parallel x - a\parallel \lt \delta $ implies $\parallel \langle f(x),g(x)\rangle - \langle f(a)g(a)\rangle \parallel \le\parallel f(x)g(x) - f(a)g(a)\parallel \lt \epsilon$ by schwartz inequality. What is the " := " operator in (VB).NET or what is it good for? What is the limit of Lovers Leap? I know that SZ0 makes vertices straightened on the z-axis, but I'm wondering if there's a way to make vertices into a complete circle on the z-axis with a hotkey? What do two question marks together mean in C#? Ran across this line of code:  FormsAuth = formsAuth ?? new FormsAuthenticationWrapper();   What do the two question marks mean, is it some kind of ternary operator? It's hard to look up in Google. Cookie blocked/not saved in IFRAME in Internet Explorer I have two websites, let's say they're example.com and anotherexample.net. On anotherexample.net/page.html, I have an IFRAME SRC="http://example.com/someform.asp". That IFRAME displays a form for the user to fill out and submit to http://example.com/process.asp. When I open the form ("someform.asp") in its own browser window, all works well. However, when I load someform.asp as an IFRAME in IE 6 or IE 7, the cookies for example.com are not saved. In Firefox this problem doesn't appear.  For testing purposes, I've created a similar setup on  .  example.com uses cookie-based sessions (and there's nothing I can do about that), so without cookies, process.asp won't execute. How do I force IE to save those cookies?  Results of sniffing the HTTP traffic: on GET /someform.asp response, there's a valid per-session Set-Cookie header (e.g. Set-Cookie: ASPKSJIUIUGF=JKHJUHVGFYTTYFY), but on POST /process.asp request, there is no Cookie header at all.  Edit3: some AJAX+serverside scripting is apparently capable to sidestep the problem, but that looks very much like a bug, plus it opens a whole new set of . I don't want my applications to use a combination of bug+security hole just because it's easy.  Edit: the P3P policy was the root cause, full explanation below. Let $p$ be a prime number. If -1 is a square in $\mathbf F_p$, it means that there exists an element of order 4. So $p-1$ is divisible by 4 and therefore $p \equiv 1$ (mod 4). I wonder the converse is also true. Does $p \equiv 1$ (mod 4) mean the existence of an element whose square is $-1$ in $\mathbf F_p$?