HTML 5 validation CSS Pseudoclass invalid use with required attribute
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How to open recovered files in word? My computer restarted while I was asleep because of system updates (need to turn that off), and when I opened up Word the first time after it restarted, it gave me a pane on the left hand side that showed me all the documents I had opened.  I chose one of the documents but I want to open them all, how can I bring that pane back?    Update:  I forgot to mention I'm on Word 2007.  I've found that if you go to      C:\Users*YOURUSER*\AppData\Roaming\Microsoft\Word   you can see the autorecovery files.  Mine are still in there, but I don't want to touch them, I'd rather bring back that pane and do it right just in case they disappear as soon as I double click haha... Explanation behind Second Derivative of a Parametric Equation Formula Is there a good word for a square-rectangle relationship? Any given square is always a rectangle, but a rectangle isn't necessarily a square, so squares and rectangles have a _ relationship. I've been noticing this sort of thing everywhere ever since I noticed that I didn't know a good word for it. I've been calling it a container relationship because one class of things is contained within a larger class of things. However, a "container" relationship doesn't sound very good and doesn't really convey the meaning on its own. A more elegant word would be helpful.  Edit: A specific definition would be something like: a is always b. b is sometimes a. Divisibility Proof with Induction - Stuck on Induction Step I'm working on a problem that's given me the run around for about a weekend. The statement:  For all $m$ greater than or equal to $2$ and for all $n$ greater than or equal to $0$, $m - 1$ divides $m^n - 1$.  Using induction over $n$, the base step comes easy since $m^n - 1$ is $0$ when $n = 0$.  My induction hypothesis is letting $k \geq 0$ and assuming that $m - 1$ divides $m^k - 1$. In order to show that $m - 1$ divides $m^{k+1} - 1$, I obviously need to use the induction hypothesis. However, no matter where I try to use the fact that $m^k - 1 = (m-1)a$ for some $a$ in the integers, the expression $m^{k+1} - 1$ always becomes more difficult to get to $m^{k+1} - 1$ being equal to $(m-1)b$ for some $b$ in the integers.   In other words, I can't figure out any actually helpful way to apply the induction hypothesis with the goal of proving the next step.  Any help would be appreciated! How to compute the integral $\int_{0}^{\infty} \frac{x^{1/3}}{1+x^{2}} \ dx$ I want to compute this integral $$\displaystyle\int_{0}^{\infty} \frac{x^{1/3}}{1+x^{2}} \ dx$$  What I did was the following. I substituted $x=t^{6}$, so that my $dx= 6t^{5} \ dt$ and so the integral changes to $$\int_{0}^{\infty} \frac{t^2}{1+t^{12}} \cdot 6 t^{5} \ dt =6 \cdot \int_{0}^{\infty} \frac{t^7}{1+t^{12}} \: dt$$  Now If I substitute $t^{4}=v$ then what I will be having is the following integral $$\frac{6}{4}\cdot \int_{0}^{\infty} \frac{v}{1+v^{3}} \ dv$$  Now I can write $1+v^{3} = (1+v) \cdot (1-v+v^{2})$ and so I have   \begin{align*} \int_{0}^{\infty} \frac{v}{1+v^{3}}\: dv &amp;= \int_{0}^{\infty}\biggl[\frac{1}{3}\cdot \frac{v+1}{1-v+v^{2}} - \frac{1}{3} \cdot \frac{1}{1+v}\biggr]\: dv \end{align*}  Now the point is that the integral of $1/(1+v) \to \infty$, so I am not sure if this is the right way to do. Can anyone suggest anything? How can I make an object follow a path exactly I want to make this sphere follow a path, but everything I try creates a giant distance from the actual path. I guess what I'm asking is how do I get the blue line extending from the path to the sphere to be basically 0 at all times? Where is the preferred place to set the PATH envvar?  ~/.profile or /etc/environment?  What is the case when PATH is set in both places? Is the end result a concatenation of both values set in those two places? Canonical URL for a home page and trailing slashes Eclipse returns error message "Java was started but returned exit code = 1" Play a Sound with Python What's the maximum value for an int in PHP? Ignoring the special libraries that allow you to work with very big numbers, what's the largest int value you can store in PHP? Infinite Cartesian product of countable sets is uncountable Let $\{E_n\}_{n\in\mathbb{N}}$ be a sequence of countable sets and let $S=E_1\times\cdots\times E_n\times\cdots $. Show $S$ is uncountable. Prove that the same statement holds if each $E_n=\{0,1\}$.  By the definition of Cartesian product of sets,   $$\displaystyle S=\Pi_{n\in\mathbb{N}} \{f\colon\mathbb{N}\rightarrow\cup_{n\in\mathbb{N}}E_n\mid\forall n, f(n)\in E_n\}$$  If $E_n=\{ 0,1\}$, then  $$\displaystyle S_{01}=\Pi_{n\in\mathbb{N}}\{0,1\}=E^{\mathbb{N}}$$, where $E=\{0,1\}$.  By a theorem, $\cup_{n\in\mathbb{N}} E_n$ is countable since the sequence is countable.  I'm not sure how to go on from here to show $S$ is uncountable. Can we say anything about the function $f$ that maps a countable $\mathbb{N}$ to another countable union of sequence of countable sets? Trying to format USB (using Linux/fdisk) so it shows up on my Mac I have a USB drive that I can see/access/modify/etc in Linux/Ubuntu but it doesn't show up on my Mac OSX.  I did sudo fdisk /dev/sdb (where my USB is) and then g to create a new GPT partition table, n to make a new partition, used all default settings, then w to save and exit.  Then I did sudo mkfs.fat -F32 /dev/sdb1 to format the partition with a FAT32 filesystem.  I ejected the USB and put it in my Mac and it doesn't show up / get detected.  What did I miss? When should we use proximity rule in "either/or", and "neither/nor"? According to this , if at least one of the nouns involved is plural then it should take the plural form of the verb. Otherwise, it should take the singular form of the verb. But in the last part it says not all grammars agree to those rules thus proximity rule applies. However, it doesn't explain when to apply the proximity rule and when not to use the former rules. I am not able to install LanguageTool extension for LibreOffice. Getting exception when I try to install it.   p.s. Somebody says that the solution is installing openoffice.org-java-common package but Ubuntu 14.04 does not have this package in its repository.  System details:  OS: Ubuntu 14.04 LTS  LibreOffice: 4.2 (4.2.3.3)  LanguageTool: 2.5  Java: Oracle JDK 1.8.0_05     Exception message: Sophomore's dream: $\int_0^{1} x^{-x} \; dx = \sum_{n=1}^\infty n^{-n}$ In the solution of the so-called , one of the key steps is to compute $$\int_0^1 x^n (\log x)^n~\mathrm dx$$ using the change of variables $x = \exp\left(-\frac{u}{n+1}\right)$ to obtain the Gamma function.  This substitution to me, looks like it was pulled out of thin air. Can someone help me motivate it? How would I have thought of this substitution? Is "a total of 10 payments" singular or plural? A total of 10 payments were made.   OR     A total of 10 payments was made.   Which is correct? Or can both be correct? How to prove $A+ A^T$ symmetric, $A-A^T$ skew-symmetric. I have often heard people say "x of his" or "x of mine". But since his and mine point out ownership, isn't using "of" here doubles that ownership? Wouldn't it be more appropriate to say "x of him" or "x of them" etc.?  Example:     I was a good friend of your dad's. He told me all about you` show that  $$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$$  using different ways  thanks for all