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The Android app crashes when trying to upload a picture from my phone Precise connection between complexification of $\mathfrak{su}(2)$, $\mathfrak{so}(1,3)$ and $\mathfrak{sl}(2, \mathbb{C})$ I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the proper orthochronous Lorentz group $SO(1,3)_+^{\uparrow}$ is given by \begin{equation} [J_i,J_j]=i \epsilon_{ijk} J_k \end{equation} \begin{equation} [J_i,K_j]=i \epsilon_{ijk} K_k \end{equation} \begin{equation} [K_i,K_j]=- i \epsilon_{ijk} J_k \end{equation} We can now define new generators with the old ones $N^{\pm}_i= \frac{1}{2}(J_i \pm i K_i)$ which satisfy \begin{equation} [N^{+}_i,N^{+}_j] = i \epsilon_{ijk} N^{+}_k ,\end{equation} \begin{equation} [N^{-}_i,N^{-}_j] = i \epsilon_{ijk} N^{-}_k ,\end{equation} \begin{equation} [N^{+}_i,N^{-}_j] = 0. \end{equation} where we can see that $N^{+}_i$ and $N^{-}_i$ make up a copy of the Lie algebra $\mathfrak{su}(2)$ each. My problem is to get what is going one here mathematically precise. Are the following statements correct and if not why:  When we build the new operators from the old generators we complexified $\mathfrak{so}(1,3)_+^{\uparrow}$ \begin{equation}(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} = \mathfrak{so}(1,3)_+^{\uparrow}\otimes \mathbb{C} \end{equation} We saw that $\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C}$ is isomorph to two copies of the complexified Lie algebra of $\mathfrak{su(2)}$: $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq \mathfrak{su(2)}_{\mathbb{C}} \oplus \mathfrak{su(2)}_{\mathbb{C}} $. Where exactly did we need that $\mathfrak{su(2)}$ is complexified here? The Lie algebras defined by $N^{\pm}_i$ are exactly those of $\mathfrak{su(2)}$ and we never use complex linear combination of $N^{\pm}_i$ or am I wrong here? $\mathfrak{su(2)}_{\mathbb{C}}$ is isomorph to $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$: \begin{equation}\mathfrak{su(2)}_{\mathbb{C}} \simeq (\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C} \end{equation} Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ denotes the complexified Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ correct?  Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ denotes the real Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C} \oplus (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ correct?  I looked this topic up in different books and each seemed to state something different. One book even used three differrent versions of $\mathfrak{sl}(2,\mathbb{C}) $ namely: $\mathfrak{sl}(2,\mathbb{C}) $, $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$ and $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{R}$.  states simply that $\mathfrak{sl}(2,\mathbb{C}) $ is the complexification of $\mathfrak{su(2)}$ without making any reference to $SL(2,\mathbb{C})$ which does not help me either. Any help would be great. Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by$ Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by.$  The required condition is $a^2&gt;8b^2$.I dont know how to prove it.I tried.  Let $(h,k)$ be the point from where tangents are drawn to the parabola $y^2=4ax$.Let $m_1,m_2$ be the slopes of the tangents.Equations of tangents are $y=m_1x+\frac{a}{m_1}$ and $y=m_2x+\frac{a}{m_2}$ and equations of normal to the parabola $x^2=4by$ is $y=m_1x-2bm_1-bm^3_1$ and $y=m_2x-2bm_2-bm^3_2$.  As $y=m_1x+\frac{a}{m_1}$ and $y=m_1x-2bm_1-bm^3_1$  are the same lines.Therefore,$\frac{a}{m_1}=-2bm_1-bm^3_1$  $bm^4+2bm^2+a=0$  And I am stuck.I cannot get the desired condition.Please help me. Installing Office 2010 on Ubuntu 15.04 using Wine I had installed Office 2010 32bit on Ubuntu 12.04 before using guides such as . However, due to package changes, I was not able to replicate the same on Ubuntu 15.04.  How does one install Office 10 using Wine? How to make a good glow effect around a sun? Flex-box: Align last row to grid How to define a Bijection between $(A^B)^C$ and $A^{B \times C}$ A question that may sound very trivial, apologies beforehand. I am wondering why $( \mathbb{Q}_{&gt;0} , \times )$ is not isomorphic to $( \mathbb{Q} , + )$. I can see for the case when $( \mathbb{Q} , \times )$, not required to be positive, one can argue the group contains elements with order 2 (namely all negatives). In the case of the requirement for all rationals to be positive this argument does not fly. What trivial fact am I missing here? What are all of the connected subsets of $\mathbb{Q}$? The answer are the singletons of $\mathbb{Q}$. I can show that the open intervals of $\mathbb{Q}$ are disconnected by choosing some irrational in the open set and using it to form a separation. But strangely enough, I am having a hard time seeing why the subset $\{p,q \}$ of $\mathbb{Q}$ is disconnected. If say $p &lt; q$ and we choose some irrational $p &lt; x &lt;q$, the separation of $\{p,q \}$ would, I think, be given by $ \{p\}$ and $\{q\}$. But these aren't open in $\mathbb{Q}$ ...? I'm trying to include a link to some code on OneDrive; the link includes an underscore, which is causing a lot of trouble.  I tried adding the entry like this:  @misc{code,     title = {Code},     url = {https://1drv.ms/link_link} }   I use \usepackage{url} and \bibliographystyle{IEEEtran}. Everything compiles fine and the bibliography shows the correct entry. However, if you click the link, it opens a page at https://1drv.ms/link - so ignoring the underscore and everything after it, which leads to a wrong page. Even copy and paste from the PDF doesn't work since the underscore is copied as space, which again is a broken link.  How can I add the link such that it can be clicked and copied properly? I tried \_, \usepackage{underscore}, {{link}}, different bibtex entries but nothing worked. Also all the related questions/answers don't answer mine. \foreach not behaving in axis environment Am hesitant to ask yet one more embarrassing question, but I can't seem to see the problem with the \foreach in the MWE. It should produce several horizontal lines but only produces the one with the yValue as specified in the \newcommand{\yValue}{0.3}:  Code: \documentclass{article} \usepackage{pgfplots}  \newcommand{\yValue}{0.3}  \begin{document} \begin{tikzpicture} \begin{axis} [  ymin=0,  ymax=1,  xmin=0,  xmax=1, ]  \foreach \yValue in {0.00,0.1,...,1.00} {     \draw [red] (axis cs:0,\yValue) -- (axis cs:1,\yValue); }  \end{axis} \end{tikzpicture} \end{document} I need to evaluate the limit without using l'Hopital's rule.  $$\lim_{x \to 0}{\frac{\sqrt{1+x\sin{x}}-\sqrt{\cos{2x}}}{\tan^{2}{\frac{x}{2}}}}$$ unable to reset root password of mysql I am trying to find a story where a children's nursery turns into a jungle I read this short story(I believe it was under novella length, but I could be mistaken) a while back, and I am trying to identify the title for a friend.   Plot:      Parents(male and female) get a computer aided/holographic nursery for their children. The mother becomes increasingly agitated as the children withdraw more. The children are barely even featured in the story, and the ending concludes with lions possibly eating the parents.   Publishing date: Years ago.   Tech level: Seemingly functional AI, not much more. When you suggest an edit to a post and view the post while it's pending, . But as noted in the , this linking hasn't been implemented for tag wikis/excerpts yet, partly because there are potentially two links to include and the wording is not entirely obvious, and partly because they're orders of magnitude less common.  Here's my suggested wording for the three cases.   Tag wiki excerpt only     Thanks for your edit!   This edit will be visible only to you until it is .  Tag wiki only     Thanks for your edit!   This edit will be visible only to you until it is .  Both     Thanks for your edit!   These edits will be visible only to you until they are peer reviewed (, ).        I can't directly address the prioritization of tag wiki edits, but I'd appreciate the corresponding  or  mark, at least. Regression for an outcome (ratio or fraction) between 0 and 1 I am thinking of building a model predicting a ratio $a/b$, where $a \le b$ and $a &gt; 0$ and $b &gt; 0$. So, the ratio would be between $0$ and $1$.  I could use linear regression, although it doesn't naturally limit to 0..1.  I have no reason to believe the relationship is linear, but of course it is often used anyway, as a simple first model.  I could use a logistic regression, although it is normally used to predict the probability of a two-state outcome, not to predict a continuous value from the range 0..1.  Knowing nothing more, would you use linear regression, logistic regression, or hidden option c? Expected Value Of Number of Removals From Urn Here's the question: "An urn contains b blue balls and r red balls. You repeatedly and independently remove balls from urn (without returning them) until the first blue ball is drawn. All balls currently in the urn have an equal probability of being selected each time a ball is removed. Define the random variable X as the number of balls that are drawn (number of red balls that are removed plus the first blue ball that is removed). Find E[X], the expected value of random variable X."  I'm having trouble finding a closed form expression for the expected value. Here's the work I've done: Last names that are English words with an extra 'e' I noticed that there are a lot of last names that have an 'e' at the end.  The pronunciation usually isn't changed from that of the base word.   Poole Steele Browne Clarke   Why do English words not have the e?  Maybe the answer to this question depends on which came first, pool or poole. How do I determine the size of my array in C? How do I determine the size of my array in C?   That is, the number of elements the array can hold? I've noticed this on occasion with a variety of applications. I've often thought it was because the output was cancelled early (ctrl+c, for example) or something similar, and zsh is filling in a new line character. But now curiosity has gotten the best of me, since it doesn't seem to do this in bash.   zsh    bash    The Sequence program is something I pulled from a book while reading on Java certifications and just wanted to see if it would compile and run. I did notice that it does not use the println() method from the System.out package/class. Instead it uses plain old print().  Is the lack of a new line character the reason I get this symbol?