I am learning about Lie algebra and have come to the point where my book defines the classical Lie algebras. I am investigating $so(3)$ as I know it to the be set of $3 \times 3$ skew-symm matrices. That is, $3 \times 3$ matrices $A$ which satisfy $A^T = -A$.  My book begins by letting $S \in gl(n, \mathbb{C})$  and defining the Lie subalgebra of $gl(n, \mathbb{C})$ by $$gl_S(n,\mathbb{C}) := \{A \in gl(n,\mathbb{C}) \mid A^{t}S = -SA\}$$ Then, when $n = 2l+1$ they take $S = \begin{pmatrix}1 &amp;  0 &amp;0\\ 0 &amp; 0 &amp; I_l \\ 0 &amp; I_l &amp; 0  \end{pmatrix}$ and define $\mathfrak{so}(2l+1, \mathbb{C}) = gl_S(2l+1, \mathbb{C})$ and call this the orthogonal Lie algebras.  Fine. Now, I want to examine $\mathfrak{so}(3)$. This is by the above defintion $\mathfrak{so}(3) := gl_S(3, \mathbb{C})$ where $S = \begin{pmatrix} 1 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 1 \\ 0 &amp; 1&amp; 0 \end{pmatrix}$. Let's see what elements of this matrix look like!  Let $A = \begin{pmatrix} a_1 &amp; a_2 &amp;a_3 \\ a_4 &amp; a_5 &amp; a_6 \\ a_7  &amp; a_8 &amp; a_9 \end{pmatrix}$ and $A^t = \begin{pmatrix} a_1 &amp; a_4 &amp;a_7 \\ a_2 &amp; a_5 &amp; a_8 \\ a_3 &amp; a_6 &amp; a_9 \end{pmatrix}$.  Now I compute $A^t S$ and $-SA$. They are $\begin{pmatrix} a_1 &amp; a_7 &amp;a_4 \\ a_2 &amp; a_8 &amp; a_5 \\ a_3 &amp; a_9 &amp; a_6 \end{pmatrix}$ and $\begin{pmatrix} a_1 &amp; a_2 &amp;a_3 \\ a_7 &amp; a_8 &amp; a_9 \\ a_4 &amp; a_5 &amp; a_6 \end{pmatrix}$ respectively.   I have noticed that this is not exactly the original definition I've seen before. Namely, it is not $A^t = -A$. Instead, the second matrix has its second and third row swapped and the first matrix has its second and third columns swapped.  So, are these definitions equivalent? Am I allowed to swap the row/column so that is matches the $A^t = -A$ defintion?
I am encountering two definitions of the special orthogonal lie algebra, and I would like to know if they are equivalent, and if there are advantages to working with one over the other.  If we begin with an $n$-dimensional vector space $V$ over a field $k$ and a chosen basis, we can define a bilinear form on $V$ by a matrix $S\in M_n(k)$, ie, let $\langle v,w\rangle=v^tSw$ for all $v,w\in V$.  Now $g\in GL_n(k)$ preserves the form ($\langle g(v),g(w)\rangle=\langle v,w\rangle$) if and only if $g^tSg=S$, so all such $g$ form a linear algebraic group $G$.  The tangent space at the identity of $G$ will be contained in that of $GL_n(k)$, so $T_eG\subset T_eGL_n(k)=M_n(k)$, and in fact, $T_eG=\{B\in M_n(k)\mid B^tS+SB=0\}$.  $T_eG$ becomes a lie algebra, $Lie(G)$, if we define the bracket to be the commutator of two matrices.  Now, if $S=I_n$, it follows that $G=O_n(k)$ is the orthogonal group of matrices satisfying $g^tg=I_n$, and $Lie(G)=\mathfrak{so}_n$ is the lie algebra of antisymmetric matrices.  In Humphrey's Introduction to Lie Algebras and Representation Theory, he defines $\mathfrak{so}_n$ to be all matrices $B$ satisyfing $B^tS+SB=0$, where $$ S=\begin{pmatrix} 1&amp;0&amp;0\\ 0&amp;O&amp;I_l\\ 0&amp;I_l&amp;O \end{pmatrix} \hspace{.5in}\text{or}\hspace{.5in} S=\begin{pmatrix} O&amp;I_l\\ I_l&amp;O \end{pmatrix} $$ depending on the parity of $n$.  The matrices obtained in this way are not antisymmetric, nor is the group $G$ preserving the form defined by $S$ the orthogonal group $O_n(k)$.  Are the two groups obtained from considering different $S$ isomorphic?  Are the two lie algebras isomorphic?  If so, why would one prefer one form to the other?
Confidence interval for the p-value estimate when doing Monte Carlo testing The second paragraph of the  section of the Wikipedia article on resampling statistics, the values of a confidence interval for a p-value from a MC sampling is given:     After $N$ random permutations, it is possible to obtain a confidence interval for the p-value based on the Binomial distribution. For example, if after $N = 10000$ random permutations the p-value is estimated to be $\hat{p}=0.05$, then a 99% confidence interval for the true $p$ (the one that would result from trying all possible permutations) is $[0.044, 0.056]$.   What's the formula to calculate it? Radius of convergence of $\sum_{n=0}^\infty n!x^{n^2}$ The question is to calculate the radius of convergence of $\sum\limits_{n=0}^\infty n!x^{n^2}$.  I dont know how to calculate R.O.C for such type of questions involving $n^2$ as power of x. How do we do this? Any help appreciated. For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable? I recently read the well known theorem that for a group $G$ and $H$ a normal subgroup of $G$, then $G$ is solvable if and only if $H$ and $G/H$ are solvable. In my book, only the fact that $G$ is solvable implies $H$ is solvable was proven. I was able to show that if $H$ and $G/H$ are solvable, then so is $G$, but I can't quite show that $G$ is solvable implies $G/H$ is solvable.   My idea was this. Since $G$ is solvable, there exists a normal abelian tower $$ G=G_0\supset G_1\supset\cdots\supset G_r=\{e\}. $$ I let $K_i=G_i/(H\cap G_i)$, in hopes of getting a sequence $$ G/H\supset K_1\supset\cdots\supset K_r=\{e\}. $$ My hunch is that the above is also a normal abelian tower. However, I'm having trouble verifying that $K_{i+1}\unlhd K_i$ and that $K_i/K_{i+1}$ is abelian.  Writing $H_i=H\cap G_i$, I take $gH_{i+1}\in K_{i+1}$ for some $g\in G_{i+1}$. If $g&#39;H_i\in K_i$, then I want to show $g&#39;H_igH_{i+1}g&#39;^{-1}H_i$ is still in $K_{i+1}$, but manipulating the cosets threw me off. I also tried to use either the second or third isomorphism theorems to show that $K_i/K_{i+1}$ is abelian, but I'm not clear on how to apply it exactly. I'd be grateful to see how this result comes through. Thank you. Presentation $\langle x,y,z\mid xyx^{-1}y^{-2},yzy^{-1}z^{-2},zxz^{-1}x^{-2}\rangle$ of group equal to trivial group Problem: Show that the group given by the presentation $$\langle x,y,z  \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2} \rangle $$ is equivalent to the trivial group.   I have tried all sorts of manners to try to show that the relations given by the presentation above imply that $x=y=z=e$. However, I am stuck and would appreciate any hints as to how I should move forward. cron ignores variables defined in ".bashrc" and ".bash_profile" I have defined "SHELL" variable in /etc/crontab file:  [martin@martin ~]$ grep SHELL /etc/crontab  SHELL=/usr/local/bin/bash [martin@martin ~]$ file /usr/local/bin/bash /usr/local/bin/bash: ELF 32-bit LSB executable, Intel 80386, version 1 (FreeBSD), dynamically linked (uses shared libs), for FreeBSD 8.0 (800107), stripped [martin@martin ~]$    In addition, all my scripts in /etc/crontab file are started under user "martin". However /home/martin/.bash_profile(for login shell) and /home/martin/.bashrc(for non-logging shell) contain some variables which are ignored in case of cron job, but are used in case I log into machine over SSH or open new bash session. Why cron ignores those variables? Isn't cron simply executing "/usr/local/bin/bash my-script.sh" with permissions for user "martin"? Transformation of Random Variable $Y = X^2$ I'm learning probability, specifically transformations of random variables, and need help to understand the solution to the following exercise:      Consider the continuous random variable $X$ with probability density function $$f(x) = \begin{cases} \frac{1}{3}x^2 \quad  -1 \leq x \leq 2, \\ 0 \quad \quad \text{elsewhere}. \end{cases}$$ Find the cumulative distribution function of the random variable $Y = X^2$.   The author gives the following solution:   For $0 \leq y \leq 1: F_Y(y) = P(Y \leq y) = P(X^2 \leq y) \stackrel{?}{=} P(-\sqrt y \leq X \leq \sqrt y) = \int_{-\sqrt y}^{\sqrt y}\frac{1}{3}x^2\, dx = \frac{2}{9}y\sqrt y.$  For $1 \leq y \leq 4: F_Y(y) = P(Y \leq y) = P(X^2 \leq y) \stackrel{?}{=} P(-1 \leq X \leq \sqrt y) = \int_{-1}^{\sqrt y}\frac{1}{3}x^2\, dx = \frac{1}{9} + \frac{1}{9}y\sqrt y.$  For $y &gt; 4: F_{Y}(y) = 1.$    Previous to this exercise, I've managed to follow the solutions of two similar (obviously simpler) problems for a strictly increasing and strictly decreasing function of $X$, respectively. However in this problem, I don't understand the computations being done, specifically:    How does the three intervals $0 \leq y \leq 1$, $1 \leq y \leq 4$ and $y &gt; 4$ are determined? In the two previous problems I've encountered, we only considered one interval which was identical to the interval where $f(x)$ was non-zero. In the case where $0 \leq y \leq 1$, why does $P(X^2 \leq y) = P(-\sqrt y \leq X \leq \sqrt y)$ and not $P(X \leq \sqrt y)$? I have put question marks above the equalities that I don't understand.   I think I have not understand the theory well enough. I'm looking for an answer that will make me understand the solution to this problem and possibly make the theory clearer. Page Object Model or External property file for Selenium Object Repository I am a beginner in Selenium WebDriver (Java with TestNg). I'm trying to create a framework of my own.  My question here is what would be the better option to store my objects (basically all the webelements of my website under test), Page Object Model or External Property file? Why? Is there a better method than those? Please help. Skip certain tables with mysqldump How to make `sudo` preserve $PATH? I am trying to make a histogram with the fitted curve plotted on it. Here's the mwe:   \documentclass[a4paper]{scrartcl}     \usepackage{tikz}     \usepackage{pgfplots,pgfplotstable}     \usepackage{filecontents}     \pgfplotsset{compat=1.10}     \usepackage[active,tightpage]{preview}     \setlength\PreviewBorder{2pt}     \newcommand{\plots}{0.611201}     \newcommand{\plotm}{2.19882}     \begin{filecontents}{data.dat}     2 0.0629921259843     3 0.0236220472441     4 0.0314960629921     5 0.125984251969     6 0.0629921259843     7 0.102362204724     8 0.110236220472     9 0.0551181102362     10 0.0629921259843     11 0.0314960629921     12 0.0236220472441     13 0.0314960629921     14 0.0629921259843     15 0.0551181102362     16 0.0393700787402     17 0.0472440944882     18 0.00787401574803     19 0.0393700787402     24 0.00787401574803     27 0.00787401574803     33 0.00787401574803     \end{filecontents}     \begin{document}     \begin{preview}      \begin{tikzpicture}     \begin{axis}[     yticklabel style={/pgf/number format/fixed},     scaled y ticks = false,     minor y tick num={1},     xtick pos=left,     legend cell align = left,     legend style={draw=none},     xlabel = {group size},     ylabel = {ratio}     ]     \addplot[blue,ybar,fill, fill opacity=0.3, bar width = 0.8,] table {data.dat};     \addplot[red, line width = 1,domain=1:40,samples=100] {1/(x*sqrt(2*pi)*\plots)*exp(-(ln(x)-\plotm)^2/(2*\plots^2))};     \legend{empirical,lognormal fit}     \end{axis}     \end{tikzpicture}     \end{preview}     \end{document}     As you can see, the legend is not really good. How can I change the legend so it looks more like this? extended stars-and-bars problem(where the upper limit of the variable is bounded) The problem of counting the solutions $(a_1,a_2,\ldots,a_n)$ with integer $a_i\geq0$ for $i\in\{1,2,\ldots,n\}$ such that $$a_1+a_2+a_3+....a_n=N$$ can be solved with a stars-and-bars argument. What is the solution if one adds the constraint that $a_i\leq r_i$ for certain integers $r_1,\ldots,r_n$?  e.g. for $n=3$, $N=6$ and $(r_1,r_2,r_3)=(3,3,2)$, the tuple $(a_1,a_2,a_3)=(2,3,1)$ is a solution, but $(2,1,3)$ is not a solution because $a_3=3&gt;2=r_3$. Creating and configuring .bash_profile I was setting path for anaconda. an answer to a question  has the following solution. I searched for .bash_profile, but I can't seem to find it. How do I setup this file and get it to run on startup? Determining the characteristic of a field Difference between a += 10 and a = a + 10 in java? I'm reading  and I'm trying to understand the proof under "Degeneracy in One Dimension".  Here's what it says:     Considering a one-dimensional quantum system in a potential $V(x)$ with degenerate states $|\psi_1\rangle$ and $|\psi_2\rangle$ corresponding to the same energy eigenvalue $E$, writing the time-independent Schrödinger equation for the system:       $$-\frac{\hbar^2}{2m}\frac{ \partial^2\psi_1}{ \partial x^2} + V\psi_1 =E\psi_1$$       $$-\frac{\hbar^2}{2m}\frac{ \partial^2\psi_2}{ \partial x^2} + V\psi_2 =E\psi_2$$   Multiplying the first equation by $\psi_2$ and the second by $\psi_1$ and subtracting one from the other, we get:       $$\psi_1\frac{d^2}{d x^2}\psi_2-\psi_2\frac{d^2}{d x^2}\psi_1=0$$   Integrating both sides       $$\psi_1\frac{\partial \psi_2}{\partial x}-\psi_2\frac{\partial \psi_1}{\partial x}=constant$$   In case of well-defined and normalizable wave functions, the above constant vanishes, provided both the wave functions vanish at at least one point, and we find: $\psi_1(x)=c\psi_2(x)$ where $c$ is, in general, a complex constant. So, the two degenerate wave functions are not linearly independent. For bound state eigenfunctions, the wave function approaches zero in the limit $x\to\infty$ or $x\to-\infty$, so that the above constant is zero and we have no degeneracy.   Here are the parts I don't understand:   How does integrating $\psi_1\frac{d^2}{d x^2}\psi_2-\psi_2\frac{d^2}{d x^2}\psi_1=0$ yield $\psi_1\frac{\partial \psi_2}{\partial x}-\psi_2\frac{\partial \psi_1}{\partial x}=constant$.  Without having explicit formulas for $\psi_1(x)$ and $\psi_2(x)$, how do we just apparently pull them both out of the integral? I don't understand how the constant in the last equation vanishes just because $\psi_1$ and $\psi_2$ vanish at infinity. Even if it does vanish, how does that imply $\psi_1=c\psi_2$? Input impedance of Arduino Uno analog pins? I'm using the  on some small robotic arm motors, and reading in the analog voltage with an Arduino Uno.  I've had reasonably good results, but only after putting in an RC filter on the output.  However, in the application note on the datasheet, it says not to do that:  "the addition of an RC filter to the output of a sensor IC can result in undesirable device output attenuation — even for DC signals."  Then it gives a formula to calculate attenuation, but it depends on knowing the input impedance of whatever is reading the signal, so that's what I'm after here. Enable fractional scaling for Ubuntu 18.04 Is it possible to group several objects and then manipulate them (scale, rotate, translate) as one? How to run BitTorrent Sync? I am in Ubuntu 12.04 (precise) 64bit and downloaded the btsync executable from  but running it doesn't seem to have any effect. Double-clicking the executable does nothing, and running ./btsync in the terminal also exits immediately without any output.  Has anyone successfully ran this under Ubuntu, and if so, is there any special configuration changes that need to be made to make it work? How does HOTP keep in sync? My understanding of  is that each password is unique and based on a counter.    $$PASSWORD = HOTP_1(K,C)$$  Where $C$ is an incremental counter.  What I wish to know, is how you keep the client and server in sync?  Looking at current systems (google authenticator), there appears to be no indication if the actual 'counter' number, but you can generate successive passwords at will.  What happens if, for example, the password you have generated uses $C = 5$, but the server is expecting $C = 3$?  Also, what is the 'integrity check' for?  Google cant seem to come up with a solution.