Could someone give me a suggestion to solve this problem?  Let $\frak{g}$ a nilpotent Lie álgebra. Prove that there is an ideal  $\frak{h}\subseteq\frak{g}$ such that $\dim(\frak{g}) = \dim(\frak{h}) + 1$
There is an exercise in Humphreys's An Introduction to Lie Algebras and Representation Theory: "Any nilpotent Lie algebra contains a codimension 1 ideal".  The proof I am thinking of is the following.  Suppose the Lie algebra $L$ satisfies $L\neq[L,L]$. Then $\pi\colon L\to L/[L,L]$ is a projection onto an abelian algebra. Choose a codimension 1 subspace $I\subset L/[L,L]$. Then $I$ is an ideal and its preimage $\pi^{-1}(I)$ is necessarily an ideal in $L$, because any subspace containing the derived algebra is an ideal. Moreover, $\pi^{-1}(I)$ has codimension 1. $\square$  The only condition on $L$ is that $L\neq[L,L]$, which holds for solvable Lie algebras as well. Is this proof correct? How come Humphreys didn't set the exercise with "nilpotent" replaced by "solvable"?
How to transform negative values to logarithms? I would like to know how to transform negative values to Log(), since I have heteroskedastic data. I read that it works with the formula Log(x+1) but this doesn't work with my database and I continue getting NaNs as result. E.g. I get this Warning message (I didn't put my complete database because I think with one of my negative values is enough to show an example):   &gt; log(-1.27+1) [1] NaN Warning message: In log(-1.27 + 1) : NaNs produced &gt;    Thanks in advance  UPDATE:  Here is an histogram of my data. I'm working with palaeontological time series of chemical measurements, E.g the difference between variables like Ca and Zn is too big, then I need some type of data standardization, that is why I'm testing the log() function.   This is my Where to find my Favorites in Google Chrome? General trigger bulkification - best practices A general question asked from time to time on stack exchange is question of trigger bulkification.  Therefore I have 2 questions:   Which best practices are you using? What is the best way to handle updates of more than 200 records? (In case of more than 200 records the trigger is executed more than once.) Infinite metric space has open set $U$ which is infinite and its complement is infinite Let $(X,d)$ be a metric space where $X$ is an infinite set. Prove that the space has an open set $U$ such that both $U$ and its complement are infinite sets.   I have considered if $d$ is the discrete metric, then one such $U$ exists. But if $d$ is not discrete, how do we find such $U$? iPad won't turn on after a complete battery drain My iPad mini retina is out of warranty,  I forgot to plug in my iPad after the critical battery warning and it went some minutes following the warning. I was not worried because same thing had happened a few times before and iPad was perfectly okay after charging it. However, this time I remember seeing a flash of blue screen immediately after it went black. Then, I plugged it in an waited for a while and tried starting it. Ever since, it fails to start hanging on the apple logo with white background screen.   I tried to connect to my Mac but nothing showed up in iTunes. I am worried now. Can I troubleshoot this or is it broken and needing repair? Should the weight of question upvotes be reduced? After thinking about    which I declined for the reasons , and    I have a proposal.  How about if question upvotes were only worth +5 rep instead of +10? And this was applied retroactively? (answer upvotes would be unchanged from the current behavior, so this would be question specific.)  This seems to be a better solution to the particular  problem  without all the rather serious negative repercussions of increasing the punitive value and/or cost of downvotes across the board. I would like to search my table having a column of first names and a column of last names.  I currently accept a search term from a field and compare it against both columns, one at a time with       select * from table where first_name like '%$search_term%' or      last_name like '%$search_term%';   This works fine with single word search terms but the result set includes everyone with the name "Larry". But if someone enters a first name then a space, then a last name, I want a narrower search result. I've tried the following without success.         select * from table where first_name like '%$search_term%' or last_name      like '%$search_term%' or concat_ws(' ',first_name,last_name)      like '%$search_term%';   Any advice?  EDIT:  The name I'm testing with is "Larry Smith".  The db stores "Larry" in the "first_name" column, and "Smith" in the "last_name" column.  The data is clean, no extra spaces and the search term is trimmed left and right.  EDIT 2:  I tried Robert Gamble's answer out this morning.  His is very similar to what I was running last night.  I can't explain it, but this morning it works.  The only difference I can think of is that last night I ran the concat function as the third "or" segment of my search query (after looking through first_name and last_name). This morning I ran it as the last segment after looking through the above as well as addresses and business names.    Does running a mysql function at the end of a query work better than in the middle? How to set systemd for automatically force fsck disks after crash (hard poweroff)?  When I used sysvinit (in Arch Linux) as /sbin/init I used the hack: in the rc.local I create /forcefsck file; in the rc.local.shutdown I remove it. At boot-up rc.sysinit enable force fsck if /forcefsck exists.  How to do same in systemd? Maybe it has built-in facilities for automatic fsck after crush? Will C++ exceptions safely propagate through C code? How can we prevent "assembly" tag being tagged to .NET assembly question I am seeing this happen again and again. Sometimes people tag the .NET and C# questions with "assembly" tag while in reality it belongs for Assembly programming questions. What does ` (backquote/backtick) mean in commands? I came across the following command:  sudo chown `id -u` /somedir   and I wonder: what is the meaning of the ` symbol. I noticed for instance that while the command above works well, the one below does not:  sudo chown 'id -u' /somedir Why didn't (or did) I or some other user get a hat although the requirements were (or weren't) fulfilled? During Winter Bash, users can earn hats. But sometimes it seems that a hat was awarded even though its requirements weren't (yet) met, or vice versa.  What could be the reason for this? How did the writers not notice that the mailbox of Leela's parents says "Turanga," and not "Leela"?  Turanga Leela is Leela's proper name, with Leela being her surname, so shouldn't it follow that her folks would be the same? Deferring admission to a computer science graduate program deals with various ways of rewarding the finding of duplicates, a tough and increasingly necessary janitorial task.  Here's a concrete suggestion:   Reward the finder of a duplicate with +10 points if the question gets closed as a dupe. If multiple duplicates are found, reward all finders whose suggestion was voted for at least once by another user. (The limitation is to avoid gaming by making frivolous suggestions when it's obvious that a question is going to get closed as a duplicate.) Reward each following closevoter who picks one of the previously suggested dupes with +2 points if the question gets closed. Punish everyone involved (dupe-finders and dupe-voters) with -5 points, and revoke all rep gained from the dupe-voting, if a question closed as duplicate gets reopened.   A punishment for mis-voting is going to be controversial but in my eyes, it is absolutely essential. Finding a good dupe is a work of love: You need to make sure you understand the OP's question, and you need to scan every potential duplicate for whether it really contains an answer that will help the OP. Everyone dupe-votes carelessly from time to time, so the threat of punishment needs to be present to keep everyone on their toes, just as the threat of downvoting does when you post an answer.  If you think +10 is too high a reward: Consider that finding a duplicate can often be more work than writing an answer, and a correct answer usually nets you at least 10 points, if not much more.   If you think -5 is too much of a punishment: -2 might work as well, although I really think there should be a harsh punishment on an unfair dupe-closing. Ubuntu 20.04 Failed to Set MokListRT: Invalid Parameter I have a MacBook Pro and only use Ubuntu on it. I just upgraded from 18.04 to 20.04. Now when I start the computer it says: Failed to Set MokListRT: Invalid Parameter Could not create mokListRT: Invalid Parameter Importing MOK states has failed: import_mok_state() failed: Invalid Parameter Continuing boot since secure mode is disabled.  Does anyone know EXACTLY how I can fix this?? Step-by-step process I don't want to mess anything up trying to fix this. What is the difference between g++ and gcc? What is the difference between g++ and gcc?  Which one of them should be used for general c++ development? Proving an infinite number of primes of the form 6n+1 I'm looking to schedule 16 players to play a round robin tournament with each other such that there are 4 players at each table.  I'd like for each player to play with each other player exactly once over the course of 5 rounds.  I just took a look at this post:  , and I was wondering if anyone knows if the case n=m=4 has been solved, and if so what that solution is. Calculate posterior distribution (gamma-prior, poisson-likelihood) I want to calculate the posterior distribution given a gamma-prior and a poisson likelihood. The task is from a textbook and I just have the solutions (without a walkthrough). Please find all given information in the following:  Given information:  $m$ countries each with a population of $n_i$; $y_i$ is the number of people having a disease; $y_i$ is a poisson random variable with mean $\frac{n_i*\lambda}{100}$  The prior distribution for $\lambda$: $\lambda \sim Gamma(a_0,b_0)$  Task:  Show that the posterior distribution is also a gamma distribution with $a_m = a_0 + \sum_{i = 1}^{m}y_i$ and $b_m = b_0 + \sum_{i = 1}^{m} \frac{n_i}{100}$  My solution:  $Posterior \propto Likelihood \times Prior$  $p(\theta|y) \propto \prod_{i=1}^{m}(\theta^{y_i}e^{-\theta})\times \theta^{a_0-1}e^{-b_0\theta}$  $p(\theta|y) \propto \theta^{\sum_{i = 1}^{m}y_i}e^{-m\theta}\times \theta^{a_0-1}e^{-b_0\theta}$  $p(\theta|y) \propto \theta^{\sum_{i = 1}^{m}y_i+a_0-1}e^{-\theta(m+b_0)}$  $a_m = a_0 + \sum_{i = 1}^{m}y_i$  $b_m = b_0 + m \neq b_0 + \sum_{i = 1}^{m} \frac{n_i}{100}$  I think I am doing a mistake, because I don't know how to incorporate the information that the mean of $y_i$ is given as $\frac{n_i*\lambda}{100}$. Or do I just have to transform $b_m$ somehow? Can someone may give me a hint?   Thank you already in advance!  Edit 1: Considering first advices  $Likelihood: p(y | \theta) \propto \prod_{i=1}^{m}(\frac{n_i\lambda}{100})^{y_i}e^{-\frac{n_i\lambda}{100}}$  $\propto \prod_{i = 1}^{m} ((\frac{n_i}{100})^{y_i})\lambda^{\sum_{i=1}^{m}y_i}e^{-\lambda\sum_{i=1}^{m}\frac{n_i}{100}}$  $\propto \lambda^{\sum_{i=1}^{m}y_i}e^{-\lambda\sum_{i=1}^{m}\frac{n_i}{100}}$  Is the transformation of $\prod_{i=1}^{m}$ I did from the first to the second step correct?