I need proof this property of operators, where should I start?, A and B are operators, and $$\hat{e}Be^{-A} = B+[A,B]+ \frac{1}{2!}[A,[A,B]]+\frac{1}{3!}[A,[A,[A,B]]]+...  $$ Thx
I'd like to show that, given to hermitian operators $A,G$ on a Hilbert space $\mathscr{H}$, the following identity holds: $$ e^{iG\lambda}A e^{-iG\lambda} = A + i\lambda [G,A] + \frac{\left(i\lambda\right)^2}{2!}[G,[G,A]]+\ldots+\frac{(i\lambda)^n}{n!}\underbrace{[G,[G,[G,\ldots[G}_{n\ times},A]]]\ldots]+\ldots $$ where $\lambda$ denotes a real parameter and $[\ \!,]$ indicates the commutator.  This is a proof left to the reader by Sakurai in his books on Modern Quantum Mechanics.
How do you choose Spells Known when creating a new character above 1st level? It's not clear to me how to choose Spells Known when creating a new character at a higher level.  Let's take the Bard as a working example first:  Bard level 2:   Spells Known: 5 Slots per Spell level: 1st &times; 3   Bard level 3:   Spells Known: 6 Slots per Spell level: 1st &times; 4, 2nd &times; 2   I'm interpreting this to mean that, if an existing 2nd-level Bard gains a new level (level 3), he's gaining 1 (one) new spell known. He can then choose the new spell known to be either 1st or 2nd, correct?  What I don't understand then, is the case of creating a new character starting at a higher level; say, creating a new Bard at level 3.  In that case you're forced to choose 6 spells, which must be from the list of 1st- and 2nd-level spells, correct?  When making this 3rd-level Bard from scratch, though, can I select 2nd level spells for all 6 spells known?  As another example, say you're creating a very high level Bard from scratch, starting at level 15. The book says your Spells Known is 19, and your spell slots are 1st × 4, 2nd × 3, 3rd × 3, 4th × 3, 5th × 2, 6th × 1, 7th × 1, and 8th × 1.  So, for those 19 Spells Known, you can choose any spells from 1st through 8th spell level. Am I free then to select as many spells as I can at 8th spell level (5 in number), working backwards, and in the end ignoring the 1st, 2nd, 3rd level spells entirely?  As another example, the Ranger doesn't know any spells at 1st level, nor have any spell slots, but the book says “You know two 1st-level spells of your choice from the ranger spell list.” Does it mean at 2nd level and above?  Long story short: When choosing Spells Known for a spellcasting character created above 1st level, am I allowed to choose my Spells Known at whatever spell level I like, just so long as I have at least one spell slot of that level?  I'm working on a D&amp;D 5e character generator and need to clarify this. I've seen lately that new question editing is stored somewhere for later use if we wander off the new question without posting it and then return back to the page. Which is a good functionality addition in it's own right.  What I'd like to also have is the Discard this draft functionality that would clear form data and forget it. Somehow we find an answer in the meantime and the question isn't valid any more.  So having a Discard saved draft or something similar would be a nice contra-functionality to auto-save to remove stored data. The way it's implemented now seems too hidden hidden and unpredictable to users.  Additional edit  When adding hidden auto functionality usually means creating the opposite as well by either giving the possibility to preference of not having automatism at all or giving the possibility to clear what's been auto saved.  Let's see how GMail doesn't work (and I think we all agree that from UX perspective it does the job as expected:   User clicks Compose message User starts writing a new email message User decides to not send the message and checks their inbox User clicks Compose message again A clean form is presented to user (old one being part of drafts)   I'm not suggesting we should have something like Saved drafts but we surely should have the possibility to clear previous question draft.  Since this new functionality was introduced I should point out that I would use the discard functionality more frequently than the saved one. I do welcome saving messages, but discarding of saved drafts is anything but a rare situation one might want to do. It wouldn't be a problem if this was a 1% against 99%. To me it's more like 50/50.  Best possible solution  When providing saved drafts there should (from the UX perspective) be the opposite functionality as well. Namely discarding of it. But first thing we should be asking ourselves is:    do users more frequently need pre-filled saved question draft or do users more frequently need empty form with the possibility to populate it with a saved draft   User testing should provide the usage frequency between the two and the one that's more likely to happen (my guess is that there would be a big difference between the two) should be implemented. So one of these:   Pre-populate question fields with saved draft on question form load, while also providing a button to clear the draft and start writing anew without the need to manually clear all the fields (which would be the optimal solution in case of a single field form) Question form should always load empty while also having a button that would pre-populate it with a saved draft.   I think this would be the best consensus related to saved question drafts and their usage. But as we've seen this issue has been swept under the rug for an indefinite time. Identification of fuzzy insect in Pennsylvania I live in south eastern Pennsylvania, USA. About a month ago I came across this very peculiar insect. Its body was about an inch long, it was fuzzy, and it had a long proboscis(?) and long orange legs.  I've been wondering about it ever since. Can anyone identify it? How to understand Cauchy's proof of AM-GM inequality(the last step) The AM-GM inequality: $$a_1a_2\cdots a_n\leq\left(\frac{a_1+\cdots + a_n}{n}\right)^n$$  the trivial case: $a_1a_2 \leq \left(\frac{a_1+a_2}{2}\right)^2 $ is self-evident.  then cauchy use this fact repeatedly. He get: $$a_1a_2\cdots a_{2^m} \leq \left(\frac{a_1+a_2+\cdots +a_{2^m}}{2^m}\right)^{2^m}.\tag{$\ast$}$$  This is easy to understand, and natural, but next step:  he let  $$b_1 = a_1,\ b_2=a_2,\ \ldots,\  b_n=a_n,\  b_{n+1}=\cdots=b_{2^m}=\frac{a_1+a_2+\cdots a_n}{n}$$  replace $a_1, a_2,\ldots ,a_{2^m}$ with $b_1, b_2, \ldots b_{2^m}$ in $(\ast)$ and simplified both left and right. He get the answer. I can repeat his proof, but I don't know why Cauchy can got this idea. How does he got it? Losing permissions by adding capability? I observed the following phenomenon that I can not explain. After adding the CAP_SYS_ADMIN capability, unshare is no longer able to write to /proc/self/setgroups.   In fact, writing to this file requires the capability, but that is achieved by changing the user namespace. So why does adding the capability to the parent process prevents writing to this file?  me@myhost:~$ unshare -r root@myhost:~# exit logout  me@myhost:~$ sudo setcap cap_sys_admin=ep /usr/bin/unshare me@myhost:~$ unshare -r unshare: cannot open /proc/self/setgroups: Permission denied  me@myhost:~$ sudo setcap cap_sys_admin= /usr/bin/unshare me@myhost:~$ unshare -r root@myhost:~#   By the way: I am running Ubuntu 16.04.4 LTS with kernel version 4.4 and the version of util-linux (including unshare) is 2.27.1. How to update Wine to the latest version? I've just happened to read this  on MO (that of course has been closed) and some of the answers to a similar  on MSE.   I know almost nothing of nonstandard analysis and was asking myself if something like the sentence « $1- 0.999 \dots$ is a nonzero positive infinitesimal» could be easily expressed and proved in nonstandard analysis.  First of all, what is 0.999... ? If we take the usual definition as a series or as a limit of a sequence of rationals, then it will still be a real number and equal to $1$ (I guess by "transfer principle", but please correct me if I'm wrong).  Instead, let's define  $$0.9_N:=\sum_{i=1}^N 9\cdot 10^{-i} $$  where $N\in{}^*\mathbb{N}\setminus\mathbb{N}$ is an infinite nonstandard natural number. This $0.9_N$ is a legitimate element of ${}^*\mathbb{R}$, expressed as $0.$ followed by an infinite number of "$9$" digits.  What can be said about $\epsilon_N:=1-0.9_N$ ? Is there an elementary proof that $\epsilon_N$ is a positive infinitesimal of ${}^*\mathbb{R}$ ? (by "elementary" I mean just order and field axioms and the intuitive facts about infinitesimals, like that for $x$ infinite $1/x$ is infinitesimal etc.; no nonprincipal ultrafilters &amp; C). Is there a way to head/tail a document and get the reverse output; because you don't know how many lines there are in a document?  I.e. I just want to get everything but the first 2 lines of foo.txt to append to another document. How to panelize gerber files? I am planning on ordering a panel with four different designs on it. Two were made in Altium, while the other two were made in Eagle.  Altium has a great tool for panelizing boards designed in it, but I don't believe it can put designs from other programs on the panel.  How can I panelize these PCBs? How to create a custom UI? I was watching this  on making a custom UI tab using Python, and in the line from bpy.types import Menu, Panel, UIList, I see the names of objects that look like they can all be used to make different UI elements. Does bpy.types contain all the objects that people use when they want to make custom UI elements? Show $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A+B)$ Fire (at, on, in, to) target How to correctly choose the preposition in "fire (at, on, in, to) target"? I'm trying to condense the space around examples in gb4e, with the goal of removing unnecessary whitespace around examples in an abstract. Here's what gb4e gives me now:    What I'd like is for the examples to essentially look like part of the running text, as follows:    In sum, I'd like to remove the left indent before the top level item numbers, remove the vertical space between examples, and remove the vertical space above and below the exe environment.  I think \vspace commands might work here, but it would get very tedious to do this every time, and I'd also like to do it the 'right' way.  (I also think linguex might be able to do this without too much fuss, but I'm used to using gb4e so I'd prefer to stick with it if there's a way to do this.)  MWE:  \documentclass{article}  \usepackage{gb4e} \usepackage{lipsum}  \begin{document}     \lipsum[1]     \begin{exe}         \ex Blah \begin{xlist}             \ex Blah             \ex Blah         \end{xlist}         \ex Blah         \ex Blah     \end{exe}     \lipsum[1] \end{document} Ubuntu 18.04 bond with netplan duplicate MAC address after cloning I have installed Google Drive on Windows Server 2008 R2. When I login to the server, Google Drive starts automatically and synchrnoizes files, pulling changes down from the cloud. All good so far.  However, I would like the sync to run even if I am not logged in. Is there any way to start the sync process automatically, maybe as a scheduled task?  I have other jobs on the server which distribute the files by FTP so I would like the server copies of the files to be up to date. Incompatibility between etextools and etoolbox command \dolistloop (\forlistloop)? I am wondering whether there is any difference between a gerund acting as subject and an infinitive acting as a subject. Which packages should be loaded after hyperref instead of before? Give a combinatorial proof to show the following for all integers $n \geq 2$. $$2^{n-2} n (n -1) = \sum\limits_{k=2}^n k (k - 1) \binom{n}{k}.$$  I'm completely stumped. I just have no idea how to do this. What I've tried so far has been simplifying the right hand side slightly to $\sum\limits_{k=2}^n \binom{n}{k - 2}$, and then proving that this somehow is equal to some manipulation of the left hand side. I can't seem to find any breakthroughs that way though.  If anyone could help out or point me in the right direction that would be great.  EDIT: I've actually just realized my simplification of the right hand side is wrong. Now I'm even more stuck. Split List into Sublists with LINQ Is there any way I can separate a List&lt;SomeObject&gt; into several separate lists of SomeObject, using the item index as the delimiter of each split?  Let me exemplify:  I have a List&lt;SomeObject&gt; and I need a List&lt;List&lt;SomeObject&gt;&gt; or List&lt;SomeObject&gt;[], so that each of these resulting lists will contain a group of 3 items of the original list (sequentially).  eg.:   Original List: [a, g, e, w, p, s, q, f, x, y, i, m, c] Resulting lists: [a, g, e], [w, p, s], [q, f, x], [y, i, m], [c]   I'd also need the resulting lists size to be a parameter of this function.