Adding strings in superscript 3
Convert a string/integer to superscript in C#
Show that the locus formed by z in the equation $z^3 + iz = 1$  never crosses the co-ordinate axes in the Argand plane. Further show that $$|z| = \sqrt{\dfrac{-\Im(z)}{2\Re(z)\Im(z) +1}}$$  Concept: $z=x+iy$ $z^3=(x+iy)^3=x^3-3xy^2+i(3x^2y-y^3)$ $iz=-y+ix$ Combined equation is $x^3-3xy^2-y-1+i(3x^2y-y^3+x)=0$ How do we proceed from here onward? How can I tell if sheet a exists in a workbook? I have the following code that's not working as expected:  If Sheets("a") &lt;&gt; "" Then MsgBox ("sheet a exists")   How can I tell if sheet a exists in the workbook? How can I create a Windows bootable USB stick using Ubuntu? I'd like to create a Windows bootable USB stick, but I don't have a Windows machine with me at the moment to do so. How could I achieve the same using Ubuntu? Function with limits only at irrational points This is the starred example 5.1.8 in Krantz' Real Analysis and Foundations.     Give an example of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ so that $\lim_{x\rightarrow c} f(x)$ exists when $c$ is irrational but does not exist when $c$ is rational.   I'm aware of the Dirichlet and Thomae functions but this seems slightly different. I'm not even sure a solution exists. Thanks for any help! Error Message:      "A reboot is pending. Please restart before starting Visual Studio".    I installed Visual Studio (Visual Basic 2015 and SQL 2015) and after what looked like a successful install I then got the "Reboot" window.   Not expecting it I had a bunch of things open. So I closed a bunch a bunch of things, and did the reboot myself.   Now I cannot run Visual Basic because it ONLY gives me the above error message in the pop-up window. It won't even let me do a re-install.   All it does is keep giving me the above error message.   If you have any advice that's better than a complete uninstall/reinstall I'm sure everyone in my situation would benefit from it.  How can I resolve this issue? How do you beta test an iphone app? How can you beta test an iPhone app?  I can get it on my own device, and anyone that gives me a device, I can run it on theirs, but is there a way to do a limited release via the app store for beta testing?  Related: Also, see  on getting your app onto phones without using the App Store. Does tourist visa (90 days) apply after a long-term visa ends in Schengen countries? How to find range $a_{75}$ of the term of the series $a_n=a_{n-1}+ {1 \over {a_{n-1}}} $ If $a_1=1$ and for n&gt;1$$a_n=a_{n-1}+ {1 \over {a_{n-1}}} $$ $a_{75}$ lies between (a) (12,15) (b) (11,12) (c) (15,18)  Now , in this question, I rewrote, $a_n-a_{n-1} = {1 \over {a_{n-1}}}$, to make a telescopic type and summed the terms to get $$a_n-a_{1}= {1 \over {a_{n-1}}}+ {1 \over {a_{n-2}}}...{1 \over {a_{1}}}$$ And as ${1 \over {a_{r}}} &lt;1$ for all r , I got $a_{75} &lt;76$ , but it's of no help as , the upper bound is much higher. I got no further ideas , on how to solve it, please help. Why is vote time missing in the SE data dump and SEDE? Alternative version of the Final Value theorem for Laplace Transform What are logic gates? So from what I understand transistors are one of the most important electronic elements for computers because of TTL (Transistors-Transistors Logic).  Is this true? If so, how exactly does this work? I have heard the terms "logic gate" and "logic block" tossed around. Why do objects in front and back view look reversed? Whenever I model objects that has different look at front and back. Viewing it from front perspective or orthographic the model displays back of the object. I had internet connection, but when I upgraded my VGA and restarted my PC then everything for eth0 has gone.     When I used ifdown:     ifdown: interface eth0 not configured   Is there something I could do?    EDIT: Content of /etc/network/interfaces    auto lo iface lo inet loopback   When I ifconfig I don't have these lines:    inet addr:192.168.1.5  Bcast:192.168.1.255  Mask:255.255.255.0 inet6 addr: fe80::219:5bff:fe5e:a5e/64 Scope:Link   In the Terminal typed:  sudo gedit /etc/network/interfaces   and then added these lines in the bottom:  auto eth0 iface eth0 inet dhcp   then I typed:    sudo ifdown eth0 sudo ifup eth0   when I send the second command I received the below message:    No DHCPOFFERS received No working leases in persistent database - sleeping   Any help? Difference between $:=$ and $=$ I am sorry if this is quite elementary question. But I always think, that why we use $:=$ at some places, instead of $=$. Is there any fundamental difference between these two? Before reading Terry Tao's blog (4 months ago), I had never seen a symbol like this (:=). Redirect parent window from an iframe action I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at  and I understand the proof, but I don't "get it". Can someone help me out with this ?   I find it hard to wrap my head around the idea of how the column space and the row space is related at a fundamental level. I have a char in c#:  char foo = '2';   Now I want to get the 2 into an int.  I find that Convert.ToInt32 returns the actual decimal value of the char and not the number 2.  The following will work:  int bar = Convert.ToInt32(new string(foo, 1));   int.parse only works on strings as well.    Is there no native function in C# to go from a char to int without making it a string?  I know this is trivial but it just seems odd that there's nothing native to directly make the conversion. Expectation of the Maximum of iid Gumbel Variables I keep reading in economics journals about a particular result used in random utility models. One version of the result is: if $\epsilon_i \sim_{iid}, $ Gumbel($\mu, 1), \forall i$, then:   $$E[\max_i(\delta_i + \epsilon_i)] = \mu + \gamma + \ln\left(\sum_i \exp\left\{\delta_i \right\} \right),  $$  where $\gamma \approx 0.52277$ is the Euler-Mascheroni constant.  I've checked that this makes sense using R, and it does.  The CDF for the Gumbel$(\mu, 1)$ distribution is:  $$G(\epsilon_i) = \exp(-\exp(-(\epsilon_i - \mu)))$$  I'm trying to find a proof of this and I've had no success.  I've tried to prove it myself but I can't get past a particular step.   Can anyone point me to a proof of this?  If not, maybe I can post my attempted proof up to where I get stuck. Shapefile layer no longer editable (toggle edit option greyed out) in QGIS? How can I reset $PATH to its default value in Ubuntu? I accidentally unset all the directories of $PATH while trying to add a new one in ~/.bashrc. I opened a new terminal window as I was editing and now $PATH is empty. I'm worried if I boot from another drive to find the $PATH I won't be able to boot into this drive again.  Basically, what is the default result of echo $PATH?