"Inverse Resolution" of a model of a countably categorical theory
Indiscernible to create descending chain of elementary models
Inequality involving rearrangement: $ \sum_{i=1}^n |x_i - y_{\sigma(i)}| \ge \sum_{i=1}^n |x_i - y_i|. $ Is there a hook that runs after a user logs in? I am writing a plugin that fetches some extended user info from a remote service and I need it to execute its function each time a user logs in.  Is there a hook that gets fired after login that I can add an action to? I was asked to prove the following:  If $\{ E_1,E_2,\ldots,E_n\}$ is a fully independent set of events, prove that:  $$ P\left(\bigcup_{i = 1}^n E_i \right) = 1 - \prod_{i = 1}^n \left[ 1 - P(E_i) \right]  $$  (Hint: use complements)  I got the following:  Let $\alpha = E_1\cup E_2 \cup \cdots \cup E_n = \{E^C_1 \cap E^C_2 \cap \cdots\cap E_n^C\}^C$. Then, $\alpha^C = \{ E^C_1 \cap E^C_2 \cap \cdots \cap E^C\} $.  Assuming that the set of complements of each ${E_i}$ is also fully independent, we have:  \begin{align} &amp; P(\alpha^C) = \prod_{i = 1}^n P(E_i^C)  = \prod_{i = 1}^n \left[ 1 - P(E_i) \right] \\ &amp; P(\alpha) = 1 - P(\alpha^C) = 1 - \prod_{i = 1}^n \left[ 1 - P(E_i) \right] \\ &amp; P(\bigcup_{i = 1}^n E_i) = 1 - \prod_{i = 1}^n \left[ 1 - P(E_i) \right] \end{align}  However, even though I know (by Googling it) that my assumption is true, I'm unable to prove it so I could validly use it as a lemma. Is there another way to prove this equality? How to disable windows snap target In a fresh install of Windows 10 Pro. When I drag a window a snap target position image is displayed. See image.    This is very frustrating as the image is often displayed under the place where I want to place the window. The affect being that the window is therefore snapped and not placed where I intended it to be.  How do I disable this?  To be clear I don't want to disable snapping altogether just the annoying destination image underlay  UPDATE  I do not have the option to "Allow the system to suggest companion windows when using Snap" Is it possible to make macros give different results at the beginning of a new sentence? Suppose that I want the macro "\secname" to write "Section" at the beginning of a new sentence, and "section" anywhere else. How can I do so? Any good resources for Lagrangian and Hamiltonian Dynamics? I'm taking a course on  and  Dynamics, and I would like to find a good book/resource with lots of practice questions and answers on either or both topics.  So far at my university library, I have found many books on both subjects, but not ones with good practice questions and answers. I have Schuam's outline of Lagrangian Dynamics, but didn't really find a lot of practice questions.  Any suggestions would be greatly appreciated! I'd like to update the label of an operator when it's called. But this :  def execute(self,context):     self.bl_label = "good night"   ... apparently doesn't work if the bl_label property has already been defined. Am I wrong ? Explicit example of a compact manifold of dimension $>2$ with strictly negative sectional curvature I am looking for examples of compact manifold of dimension $&gt;2$ with strictly negative sectional curvature (for dimension 2 it is well-known). Can anybody please help? Fixed points under action of conjugate group elements Lecture notes:  If $g$ and $h$ are conjugate then $fix(g)=fix(h)$. To see this, say $g=k^{-1}hk$  If $g.s=s \iff k^{-1}hk.s=s \iff h.(k.s)=(k.s)$  Me: I understand the last line- I just fail to see how it implies $fix(g)=fix(h)$  EDIT: As Ben S says I think there was a typo in the notes and they forgot modulus signs so- New Aim: show $|fix(g)|=|fix(h)|$ Differences between derivatives and strong derivatives Definition: Let $f$ be a real valued function. We say $f$ is $\mathbf{strongly}$ $\mathbf{differentiable}$ at $x = a$ if the following limits exists and is finite:  $$ \lim_{x \to a, y \to a, x \neq y} \frac{ f(x)-f(y)}{x-y} = f^*(a) $$  and we can $f^*(a)$ the strong derivative of $f$ at $a$. Why is this definition of derivative different than the usual one? What is the main crucial point to understand here that makes it different? How can you check for a #hash in a URL using JavaScript? Get index of parent div from child divs Find two elements that don't have a gcd in a subring of Gaussian integers I am using  to show the branch name in gnome terminal (Ubuntu 15.10) when working in a git repository. Based on the above I now have the below in my ~/.bashrc file:  # uncomment for a colored prompt, if the terminal has the capability; turned # off by default to not distract the user: the focus in a terminal window # should be on the output of commands, not on the prompt #force_color_prompt=yes    ...  # Add git branch if its present to PS1 parse_git_branch() {  git branch 2&gt; /dev/null | sed -e '/^[^*]/d' -e 's/* \(.*\)/(\1)/' } if [ "$color_prompt" = yes ]; then  PS1='${debian_chroot:+($debian_chroot)}\[\033[01;32m\]\u@\h\[\033[00m\]:\[\033[01;34m\]\w\[\033[01;31m\]$(parse_git_branch)\[\033[00m\]\$ ' else  PS1='${debian_chroot:+($debian_chroot)}\u@\h:\w$(parse_git_branch)\$ ' fi unset color_prompt force_color_prompt   As a result I now get:    so it works. But why has the coloring of my user@host been removed? And I would also expect that the branch name should be colored. Before it looked like this:    UPDATE: I have now tried this guide instead:    adding this to .bashrc:  parse_git_branch() {      git branch 2&gt; /dev/null | sed -e '/^[^*]/d' -e 's/* \(.*\)/ (\1)/' } export PS1="\u@\h \[\033[32m\]\w\[\033[33m\]\$(parse_git_branch)\[\033[00m\] $ "   and that works:    Notice in .bashrc I also have this (default):  # uncomment for a colored prompt, if the terminal has the capability; turned # off by default to not distract the user: the focus in a terminal window # should be on the output of commands, not on the prompt #force_color_prompt=yes   I have yet to find the reason why that snippet gives the correct result and the other version does not. Any input on this?  Here is the version of my .bashrc that has the old snippet enabled that does not work: Let us consider the following code and the algorithm corresponding to it    \documentclass{article}   \usepackage[utf8]{inputenc}    \usepackage[]{algorithm2e}     \begin{document}     \begin{center}    \begin{algorithm}[H]     \KwData{this text}     \KwResult{how to write algorithm with \LaTeX2e }     initialization\;     \While{not at end of this document}{      read current\;      \eIf{understand}{              go to next section\;       current section becomes this one\;       }{       go back to the beginning of current section\;      }     }     \caption{How to write algorithms}    \end{algorithm}    \end{center}    \end{document}   Which gives the following     I want to add running times for each step to my algorithm as shown below (for example)    So, I need to make my algorithm as a column in the table and one more column for running times of each step, how can I make my algorithm as a column inside a table? How to automate SSH login with password? How to automate SSH login with password? I'm configuring my test VM, so heavy security is not considered. SSH chosen for acceptable security with minimal configuration.  ex)  echo password | ssh id@server   This doesn't work.  I remember I did this with some tricks somebody guided me, but I can't remember now the trick I used... Recover accidentally removed files on NTFS partition I accidentally removed some files on a NTFS partition  How can I recover them staying inside Linux without rebooting ? Finding square root without using sqrt function? Let $a_n=\cos(a_{n-1}), L=[a_1,a_2,...,a_n,...].$ Is there an $a_0$ such that $L$ is dense in$[-1,1]?$ Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$