Does $\left(-3\right)^\frac{2}{2}$ not equal $\sqrt{\left(-3\right)^2}$?
I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &amp;= i \\\\\ \frac1{\sqrt{-1}} &amp;= \frac1i \\\\  \frac{\sqrt1}{\sqrt{-1}} &amp;= \frac1i \\\\  \sqrt{\frac1{-1}} &amp;= \frac1i \\\\  \sqrt{\frac{-1}1} &amp;= \frac1i \\\\  \sqrt{-1} &amp;= \frac1i \\\\  i &amp;= \frac1i \\\\  i^2 &amp;= 1 \\\\  -1 &amp;= 1 \quad !!? \end{align}
Adding dynamic graphics to an animation He called her, emailed her, texted her, tweeted her—all to no use.   Strictly speaking, I would need to write texted her and tweeted her, but I'm omitting and to convey a rhythm and sense of urgency. What do you call this kind of construction, and is it accepted grammar (at least for creative writing, if not in an academic context)? 500 Error on index.php file Windows wildcards with files having more than 3 characters extensions On Windows 7, I have a directory containing the following four files:   xxx.txt xxx.txt2 xxx2.txt xxx2.txt2   With the file have more than 3 character as a file extension, the behavior of the wildcard * seems odd:  dir *.txt  10/13/2014  04:14 PM                 6 xxx.txt 10/13/2014  04:17 PM                 6 xxx2.txt 10/13/2014  04:17 PM                 6 xxx2.txt2 10/13/2014  04:14 PM                 6 xxx.txt2            4 File(s)             24 bytes            0 Dir(s)   6,660,980,736 bytes free   I would expect to see only the two files with txt extension and not the ones with txt2 (like it would on a Linux machine). Does MS-DOS ignore the truncate the file extension to 3 characters or does it add automatically another wildcard at the end? If I want to delete only the files with txt extension and not the one with txt2, which command should I use? Thanks Why too many capacitors in parallel for Vdd supply net? Can't we just add all to replace with one big capacitor? Analyze a circuit with two diodes and two voltage rails I want to find \$I\$, \$I_{D2}\$ and \$V\$ in this circuit. The diodes are made of Silicium. I'm a beginner and normally I would assume the Diode bias and then apply KVL to see if it is true. However, I've only encountered circuits where the wire forms a loop.    How am I supposed to apply KVL in this case? If voltage is the difference in potential between two points, how can I find the voltage of V if one point is outside of the circuit, therefore, outside of KVL application? How to assign a SSL Certificate to IIS7 Site from Command Prompt Can you advise me whether it is possible or not to assign a SSL Certificate to a website in IIS7 using the APPCMD application?  I am familiar with the command to set the HTTPS Binding  appcmd set site /site.name:"A Site" /+bindings.[protocol='https',bindingInformation='*:443:www.mysite.com']   and how to obtain current mappings   %windir%\system32\inetsrv\Appcmd   but can not seem to find any way to map a site to a certificate (say the certificates hash for example) Simpler proof - Non atomic measures Suppose that $(X,\mathcal{E},\mu)$ is a non-atomic finite measure space (i.e. for every $E \in \mathcal{E}$ with $\mu(E)&gt;0$ there exists $F \subset E$ measurable such that $0&lt;\mu(F) &lt;\mu(E)$.)  a) Prove that for every $ \varepsilon &gt;0$ there is a finite partition of $X$ in measurable subsets $X_1,..,X_n$ such that $\mu(X_i)\leq \varepsilon$.  b) Prove that for every $\alpha \in [0,\mu(X)]$ there exists $E \in \mathcal{E}$ with $\mu(E)=\alpha$.    I guess that a) is given to prove b) more easily. I have the following idea of solution for a) (inspired from , which proves b)  )  Denote  $$ \Gamma = \{ (X_1,..,X_n) : n &gt;&gt; \mu(X)/\varepsilon, X_i \text{ are disjoint }, \mu(X_i)\leq \varepsilon \}$$ ordered by componentwise inclusion. Totally ordered parts $(Y_\alpha)$ of $\Gamma$ have an upper bound element $(\bigcup Y_\alpha^i)_{i=1}^n$ which is still in $\Gamma$.   By Zorn's lemma $\Gamma$ has maximal elements. If a maximal $(X_1..X_n)$ element is not a partition, then we can replace it with something better, of the form $(X_1,..,X_n\cup A)$ where $A \subset X \setminus (X_1 \cup..\cup X_n)$ and $\mu(A)&gt;0$.     My questions are:  1) Is my solution of a) correct? I feel that the part with the upper bound element may not work, since there might be noncountable unions.  2) What is a simpler solution of a)? I was trying to solve internet problems, and seem to have made things worse: Now when I start the computer it gets stuck on "Booting system without full network configuration".   Does anyone know how to make it unstuck?   Alternatively: It got messed up when I changed the contents of /etc/network/interfaces, so presumably would work again if I changed it back. How can I edit this file during boot?   I'm using Ubuntu 12.10. How does a magnet work? Install packages through yum from file Is there any way to install packages through yum from a file?  Something like:   yum -f packages.txt  I couldn't find anything in the man pages.  packages.txt would contain something like:  bash bc binutils bzip2 bzip2-libs ca-certificates cairo How to prove polynomials with degree $n$ does not form a vector space? This is one of my linear algebra problems:     Prove that polynomials of degree $n$ does not (The professor made these words bold intentionally) form a vector space.   From what I read, the set of polynomials of degree $n$ should be a vector space, because:   There is an "One" and a "Zero" in this set; We can find inverse for addition and multiplication from this set; It follows all the axioms of addition. It follows all the axioms of scalar multiplication.   Then can someone give me some hints to prove it does not form a vector space? I have a form with a lot of controls on it.  How can I detect when the mouse leaves the form?  I've tried wiring up a MouseLeave event for every single control and the form, but that does not work because those events fire all the time as the mouse passes over controls.  Is there a way that actually works.? Similar to , what's the best way to stay up to date with the latest stable releases of Thunderbird? Problems with # inside a verbatim that's inside a note I'm trying to use a # character inside a verbatim that's inside a note :   \begin{frame}[fragile]  \note{\begin{verbatim}  # \end{verbatim} }  \end{frame}   This produces the following errror :   >   ! Illegal parameter number in definition of \beamer@todo. >    to be read again>  >                     >   l.5 }   How can I write the # character in a way that doesn't break the latex compilation ?  Thanks in advanced,  PS : This is related to the following problem in pandoc : Can you allocate an array with something equivalent to make_shared? Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1 How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the origin with probability $1$?  Edit: note that while returning to the origin is guaranteed $(p = 1)$ in $1$ and $2$ dimensions, it is not guaranteed in higher dimensions; this means that something in a correct justification for the $1$- or $2$-d case must fail to extend to $3$-d (or fail when the probability for each direction drops from $\frac14$ to $\frac16$). Proving that for a smooth function if $f(\frac 1 k)=0 :\forall k\in \mathbb N$ then $f(x)=0 :\forall x\in[-1,1]$ Show that $2^n > n^2$ whenever n>4. Let $S(n)$ be the proposition "$2^n &gt; n^2$"  Basis step: $S(5)$ is true, since $2^5 = 32$, $ 32 &gt; 25 = 5^ 2$  Inductive step:  Induction hypothesis:   Assume $S(K)$ is true, $2^K &gt; K^2$.  Then  $2^{K+1} = 2.2^K &gt; K^2 + K^2$   I don't know how to proceed after this. How to complete this proof? I would like to have my whole document showing a blank line (or space) between paragraphs and no indentation. I know how to do it line by line, but is there a way to change that with just one or a few lines at the beginning of the document? Like a package you can install, or a new command?