Need to split right bracket at side of matrix into three smaller brackets
Matrix with labels nested in braces
Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$ I have no idea where to even start. WolframAlpha cant compute it either.  $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$  I think it can be done with series, but I am not sure, can someone help a little so I can get a start??  Thanks! some \newcommand instructions not working [Firmware Bug]: TSC_DEADLINE disabled due to Errata - what should I do about this? I saw several bugs and error messages when I was looking by chance into the tool called "logs".  Being a newbie in Ubuntu I have no idea how to deal with these bugs, where to report them or how to fix them.  Any help to guide me here?  The bug I have (which is listed under important) reads:  [Firmware Bug]: TSC_DEADLINE disabled due to Errata; please update microcode to version: 0x52 (or later) Three side-by-side tables I am using this LaTeX code to generate three tables side by side. However, it showing errors and not giving satisfactory result. Can anyone please help me to short out the problem.  \begin{table} \centering \makebox[0pt][c]{\parbox{1.2\textwidth}{% \begin{minipage}[b]{0.265\hsize}\centering \begin{tabular}{ | c | c |c | } \multicolumn{3}{c} \hline {$q=1/10$} \\ \hline T &amp; $r_h$ &amp; $\omega$ \\  \hline 0.233 &amp; 0.135 &amp; 2.38486-0.34667I  \\  0.256 &amp; 0.140 &amp; 2.38092-0.35837I \\ 0.273 &amp; 0.145 &amp; 2.38049-0.35982I \\  0.278 &amp; 0.146 &amp; 2.38007-0.36130I \\  0.281 &amp; 0.147 &amp; 2.37964-0.36282I \\  \hline 0.284 &amp; 0.761 &amp; 2.54150-2.03209I \\ 0.285 &amp; 0.765 &amp; 2.54515-2.04292I \\ 0.289 &amp; 0.800 &amp; 2.57799-2.13761I \\ 0.302 &amp; 0.900 &amp; 2.67987-2.40712I \\ 0.312 &amp; 1.000 &amp; 2.79214-2.67552I \\ \hline \end{tabular} \end{minipage} \hfill \begin{minipage}[b]{0.265\hsize}\centering \begin{tabular}{ | c | c | c | } \multicolumn{3}{c} \hline {$q=1/9$} \\ \hline T &amp; $r_h$ &amp; $\omega$ \\ \hline 0.227 &amp; 0.150 &amp; 2.34385-0.41094I  \\  0.245 &amp; 0.155 &amp; 2.34215-0.41723I \\ 0.260 &amp; 0.160 &amp; 2.34056-0.42416I \\  0.271 &amp; 0.165 &amp; 2.33908-0.43175I \\  0.276 &amp; 0.167 &amp; 2.33852-0.43497I \\  \hline 0.280 &amp; 0.720 &amp; 2.50248-1.92299I \\ 0.283 &amp; 0.750 &amp; 2.52893-2.00424I \\ 0.286 &amp; 0.780 &amp; 2.55660-2.08533I \\ 0.289 &amp; 0.800 &amp; 2.57568-2.13931I \\ 0.302 &amp; 0.900 &amp; 2.67807-2.40842I \\ \hline \end{tabular} \end{minipage} \hfill \begin{minipage}[b]{0.265\hsize}\centering \begin{tabular}{ | c | c | c |} \multicolumn{3}{c} \hline {$q=1/8$} \\ \hline T &amp; $r_h$ &amp; $\omega$ \\  \hline 0.246 &amp; 0.180 &amp; 2.29733-0.50611I  \\  0.255 &amp; 0.185 &amp; 2.29654-0.51364I \\ 0.263 &amp; 0.190 &amp; 2.29589-0.52172I \\  0.269 &amp; 0.195 &amp; 2.29534-0.53033I \\  0.273 &amp; 0.199 &amp; 2.29496-0.53758I \\  \hline 0.275 &amp; 0.675 &amp; 2.46067-1.80426I \\ 0.277 &amp; 0.700 &amp; 2.48131-1.87194I \\ 0.288 &amp; 0.800 &amp; 2.57245-2.14170I \\ 0.302 &amp; 0.900 &amp; 2.67555-2.41027I \\ 0.317 &amp; 1.000 &amp; 2.78870-2.67796I \\ \hline \end{tabular} \end{minipage}% }} \end{table} How does the SQL injection from the "Bobby Tables" XKCD comic work? Just looking at:   (Source: )  What does this SQL do:  Robert'); DROP TABLE STUDENTS; --   I know both ' and -- are for comments, but doesn't the word DROP get commented as well since it is part of the same line? Understanding divison by monic polynomial in $R[x]$ where $R$ is an arbitrary ring I read "Algebra: Chapter 0" by P.Aluffi. I encountered a topic where it says you can divide any polynomial in $R[x]$($R$ is any ring) by a monic polynomial(that is, a polynomial of the form $x^d + \sum\limits_{i=0}^{d-1} a_i x^{i}$).   It says if $g(x)$ is any polynomial and $f(x)$ is a monic polynomial. Then $\exists \ \ \ q(x), r(x) \in R[x],\: \deg r(x) &lt; \deg f(x): g(x) = f(x)q(x) + r(x).$  I have two questions regarding this:  1) Why $f(x)$ needs to be monic?   2) How can we prove it? The book talk about induction and that if $\deg  g(x) = n &gt; d = deg \ f(x)$, then $\forall a \in R$ we have $ax^n = ax^{n-d}f(x) + h(x)$ for some $h(x): \deg \ h(x) &lt; n$. It's true, but how do I take it from here? What is the difference between object keys with quotes and without quotes? I just downloaded a DVD which is for a movie. After uncompressing the DVD file I see .VOB, .IFO and .BUP files. If I open the first .VOB file it shows me the DVD menu but I can't interact with it. So, my question is... is Totem able to play those kind of files? If so, how can I achieve that? What other app could I use in order to reproduce those files?  Edit  Using VLC didn't work neither. I forgot to mention I have already tried that. Let me rephrase: if I open the first video file it shows the DVD menu, BUT I can't interact with it. Taylor expansion for $\arcsin^2{x}$ I stumbled upon this particular expansion that was included in  post.    $$ \displaystyle \arcsin^{2}(x) = \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n^{2} \binom{2n}{n}} (2x)^{2n}$$    This caught my eye because I remember trying to derive a Taylor series for $\arcsin^{2}(x)$ a while ago without much success.  Can anyone prove this or point me to a material that would show a proof of this identity?  EDIT :  Feel free to use any mathematical apparatus at hand. I'm not interested in a proof fit for a certain level, nor am I looking for utmost elegance (though that would be lovely). Many other solutions of the Cauchy's Functional Equation By reading the  on the Wiki,  it is said that      On the other hand, if no further conditions are imposed on f, then (assuming the axiom of choice) there are infinitely many other functions that satisfy the equation. This was proved in 1905 by Georg Hamel using Hamel bases. Such functions are sometimes called Hamel functions.   Could anyone give a more explicit explanation of these many other solutions?   Besides the trivial solution of the form $f(x)=C x$, where $C$ is a constant, and the solution above constructed by the Hamel Basis, are there any more solutions existing? Recently I booked a flight using my points. I am  planning to fly to Seattle WA from Indianapolis Indiana.  Stop over is Chicago.  On the day of my flight I am supposed to be in Chicago.   So I was wondering if I can directly board my Seattle flight from Chicago thereby not showing up in Indianapolis.  Is this okay?  Will United airlines cancel my entire journey if I do not show up in Indianapolis? Simplify the expression $\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\cdots +\binom{n+k}{k}$ Starting with the episode "Crossroads Part II" the      final five Cylons begin to hear Bob Dylan's "All Along the Watchtower" in their heads.   Have the creators/writers ever stated why they picked that song in particular and what it might mean to the story? Individual ColorFunction with RGBColor Why is Array.Length an int, and not a uint. This bothers me (just a bit) because a length value can never be negative.   This also forced me to use an int for a length-property on my own class, because when you  specify an int-value, this needs to be cast explicitly...  So the ultimate question is: is there any use for an unsigned int (uint)? Even Microsoft seems not to use them. Java equivalent of unsigned long long? In C++, I enjoyed having access to a 64 bit unsigned integer, via unsigned long long int, or via uint64_t. Now, in Java longs are 64 bits, I know. However, they are signed.  Is there an unsigned long (long) available as a Java primitive? How do I use it? Removed Python 3 and now Ubuntu Software Center, terminal and other applications don't work I am a new Ubuntu user running Ubuntu 13.04.  I've been trying to install and use a DeDRM tool, but I've been having problems doing so. I was following some directions that said I needed a version of Python 2.7, but not anything Python 3.0 or later (because 3.0 versions are missing some necessary libraries?). I tried to install Python 2.7.5.6, but it didn't work. I thought it was probably because I had the later version of Python, so I went through terminal and removed Python 3.3, so I could install the earlier version of Python.  Now that I've uninstalled Python 3.3, a lot of applications no longer work, including the terminal and the Ubuntu Software Center. I have no idea how to fix this problem now. I understand what a decibel is, the spirit of the question is different: why do we still use decibels (and with at least two different, context-specific definitions), if we have the feat of scientific notation? I dare to claim that the change of 5e-1 (which is a half) is much more readable than 3dB (voltage? power?). 1e6 is more clear than 120dB, etc. Just for the joy of adding? 3dB + 7dB is 10dB, but 2e5+3e6 is errrm... I see many new datasheets from the top vendors (LTC, for example) use the V/V convention, which is closer to the scientific notation than to the decibels. An upcoming paradigm change? Solving 4 Quadratic Simultaneous Equations Consider the equations:   $$\begin{cases} ab+c+d=3\\ bc+d+a=5\\ cd+a+b=2\\ da+c+b=6 \end{cases}$$   Where $a,b,c,d \in \mathbb{R}$. How can we find $a,b,c,d$?   The furthest I've got is by adding the first two equations and the last two equations together to get: $ab+bc+c+d+d+a=(b+1)(a+c)+2d=8$ and $(d+1)(a+c)+2b=8$ We can rearrange these (assuming $b,d \neq -1$) to get $$\frac{8-2d}{b+1}=\frac{8-2b}{d+1} \implies -d^{2}+3d+4=-b^{2}+3b+4$$ Which leads us further to $$b^{2}-d^{2}-3(b-d)=0 \implies (b-d)(b+d-3)=0$$ Now since $b=d$ gives us an absurdity (equations 1 and 4 reduce to 3=6), we must have $b+d=3$. This is as far as I can get unfortunately, since trying the same approach with the other pairs doesn't work in the same way.  Help would be much appreciated; the problem is from the BMO1 2004 paper. What temperature is achieved in focus point by 5000 flat 1x1cm mirrors onto a satellite dish? There is this  ("R5800 Solar Death Ray") where teenager built a 5000 mirror device which concetrates the solar rays which is showing the potency of the mosaic method of concentrating sunlight using elliptical dish. It can fire the wood, boil the water in seconds, melt an aluminum, glass or even burn the whole in concrere or rock.  What is the approximate maximum temperature (could be a range) is achieved by this mirror device in its focus point on sunny day?       The R5800 solar concentrator, replaced by the new R23k, is made from   an ordinary fiberglass satellite dish. It is covered in about 5800   3/8" (~1cm) mirror tiles. When properly aligned, it can generate a   spot the size of a dime with an intensity of 5000 times normal   daylight. This intensity of light is more than enough to melt steel,   vaporize aluminum, boil concrete, turn dirt into lava, and obliterate   any organic material in an instant. It stands at 5'9" and is 42"   across.   Technical details &amp; calculations:     Calculations: Area of dish aperture (elliptical)= (pi x 102cm x   73cm)/4 = 5848 sq cm       Area of focal point (circular)= pi(0.6cm)^2 = 1.14 sq cm      Dish Type= 0.7m x 1m offset focus elliptical.      Concentration Power= 5000x      Output Power Estimate = 560 watts