I'm really embarrassed to ask but what is the nilradical of the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$, i.e. the set of ad-nilpotent elements of $\mathfrak{gl}_n(\mathbb{C}) = \mathrm{Mat}_n(\mathbb{C})$$? This must be standard knowledge but I couldn't find a reference.  Clearly, all nilpotent matrices and all diagonal matrices are in the nilradical. What else?
I find it intuitive enough that the radical of $\mathfrak{gl}_n\mathbb F$ is the scalar matrices, but I have trouble finding an easy, but complete proof:  Proof. Let $\mathfrak s$ denote the scalar matrices. Clearly $\mathfrak s\subset\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$. Suppose that $\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$ is generated by more than one element, so that $X\in\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$, but $X\notin\mathfrak s$. We can change basis such that $\mathrm{rad}(\mathfrak{gl}_n\mathbb F)$ is upper-triangular. (The change of basis leaves the scalar matrices invariant.) Then $X$ is upper-triangular.  I want to conclude that there exists a $Y\in\mathfrak{gl}_n\mathbb F$ s.t. $[X,Y]$ is not upper-triangular. How can I see that $Y$ always exists, except by waving hands?
What are the rules for calling the base class constructor? What are the C++ rules for calling the base class constructor from a derived class? For example, I know in Java, you must do it as the first line of the subclass constructor (and if you don't, an implicit call to a no-arg super constructor is assumed - giving you a compile error if that's missing). What happens to a lens when you turn the focus ring during AF mode? What happens to a lens when you turn the focus ring during AF mode?  I have read that this causes damage, or can cause damage. But what is actually happening? I've done this recently with my Canon EF 50mm 1.8 lens.  Is a lens damaged after doing this once or does it have to happen multiple times? Why are there $1 / 1.602176634 \times 10^{-19}$ electrons in a coulomb? Why that exact number of electrons in one coulomb? who decided it? there is nothing wrong with the number, it just seems slightly messy. Why didn't the scientific community just settle on an easier number, such as $1\times10^{-19}$ for example? Homemade coconut milk separates into different layers after a day in the fridge I made some coconut milk. After a day in the fridge there was two layers of liquid in the jar, coconut water and solid coconut which I found out afterwards that it is called coconut butter! When I stir them together to  have my coconut milk back, it didn't work. Is there a way to prevent this? If not, is there a way to have coconut milk again? What is the US Visa Waiver Program and what is an ESTA? How to split bibliography into "works cited" and "works not cited"? I need two bib-like sections: a "References" giving works explicitly cited in my text, and a "Bibliography" (or "Further reading" if you prefer) giving everything from my bibfile that didn't already appear in "References". The first is of course completely standard, but the second... I'm stumped.  Further loose constraints: I'm somewhat committed to biblatex and the biblatex-apa style. But I'll happily accept an answer not related to them if there's any chance I can adapt it. Extended Euclidean Algorithm: backward vs. forward How do I increase the size of swapfile without removing it in the terminal? Is there a way to increase my existing "swapfile" without having to destroy and re-create it? I would like to up my swap space from 1GB to 2GB.  Currently it is set up as such:  $ sudo swapon -s Filename                Type        Size    Used    Priority /swapfile               file        1048572 736640  -1 $ ls -lh /swapfile -rw------- 1 root root 1.0G Nov  9  2016 /swapfile   I'm using Ubuntu 14.04. Best word for "unable to change" How to search for email of a particular date from years ago in Gmail There are similar questions and several answers. I tried to follow the methods provided in the answers.  They did not work. So I am going to ask it again with more specifics.  I only knew the date of the email in the in-box in 2012 and the content of it but did not know the exact wording of the subject. Is it possible to find this email? How to evaluate $\int_0^{2\pi} \frac{d\theta}{A+B\cos\theta}$? I'm having a trouble with this integral expression:  $$\int_0^{2\pi} \frac{d\theta}{A+B \cos\theta}$$  I've done this substitution: $t= \tan(\theta/2)$  and get: $\displaystyle \cos\theta= \frac{1-t^2}{1+t^2}$ and $\displaystyle d\theta=\frac{2}{1+t^2}dt$ where $\displaystyle \cos^2\theta/2=\frac{1}{1+t^2}$  then the integral becomes:  $$\int\frac{2 \, dt}{(A-B)t^2+(A+B)}= \sqrt\frac{A+B}{A-B} \arctan \left(\left(\sqrt\frac{A+B}{A-B} \right) t\right)$$   However, I'm not sure about the new limits since $\tan$ has period $\pi$ so what I have to do at this point to decide the new limits? And of course if you find some mistake in what I've done before, please let me know! How do I run a program with commandline arguments using GDB within a Bash script? How do I set exposure with a manual flash? I recently got an old Canon AE-1 film camera back in working order. I'd like to use my modern 430EXII Speedlight flash with it, however since there is no TTL metering, I have no idea how much the flash will affect the exposure of my shot (note that I have confirmed that the AE-1 will trigger the flash). I'd rather not discover that my shots were either blown out or way to dark after going through several rolls of film.  Are there any basic rules-of-thumb or basic calculations for anticipating how much a burst from the 430EXII will affect the exposure of a subject at various distances? Why isn't Higgs coupling considered a fifth fundamental force? How to find the number of anti-symmetric relations? At this time, Wubi does not work with Windows 8 default boot-loader. Thus at this point Wubi would not work on a new Windows 8 machine.      Citation fom    Question  Recently I've learned of , a way to install Ubuntu right from Windows.  Besides installing from Windows, I would like to know key differences between a regular "alongside Windows" or Dual-boot Ubuntu installation and one done with Wubi. Are there any disadvantages (for example a performance penalty) or incompatibilities I should look out for when using Wubi? Do I need a visa for a connecting flight to Japan? I'm going to the Philippines this October and would like to layover in Japan. I found a reasonably priced ticket with a 15-hour layover to Haneda, however, I have to transfer to Narita. Do I need a visa since I'm going to have to stay at a hotel and ride a bus to travel to Narita airport? Thank you! Say, I have foo-1.2.3.deb which depends on perl and python, however, running command:  dpkg -i ./foo-1.2.3.deb   won't install these dependencies. So I must apt-get install perl python by hand.  How to make dpkg -i install these dependencies for me automatically? As a physics student, I have occasionally run across the gamma function  $$\Gamma(n) \equiv \int_0^{\infty}t^{n-1}e^{-t} \textrm{d}t = (n-1)!$$  when we want to generalize the concept of a factorial.  Why not define the gamma function so that   $$\Gamma(n) = n!$$  instead?  I realize either definition is equally good, but if someone were going to ask me to choose one, I would choose the second option.  Are there some areas of mathematics where the accepted definition looks more natural?  Are there some formulas that work out more cleanly with the accepted definition? Is adjusting p-values in a multiple regression for multiple comparisons a good idea? Lets assume you are a social science researcher/econometrician trying to find relevant predictors of demand for a service. You have 2 outcome/dependent variables describing the demand (using the service yes/no, and the number of occasions). You have 10 predictor/independent variables that could theoretically explain the demand (e.g., age, sex, income, price, race, etc). Running two separate  multiple regressions will yield 20 coefficients estimations and their p-values. With enough independent variables in your regressions you would sooner or later find at least one variable with a statistically significant correlation between the dependent and independent variables.   My question: is it a good idea to correct the p-values for multiple tests if I want to include all independent variables in the regression? Any references to prior work are much appreciated.