How can I solve the ODE $2y'' + ty' -2y=10, y(0) =y’(0)=0 $ by Laplace transform?
Laplace transform of integral constant
Calculate the limit including binomial coefficients Biblatex citation order I'm facing a problem with biblatex citation order.  Using this configuration:  \documentclass {article} \usepackage[brazil]{babel} \usepackage[utf8]{inputenc} \hyphenation{es-ta-bi-li-da-de} \usepackage{setspace} \singlespacing \usepackage[natbib=true,style=numeric]{biblatex} \addbibresource{frameworks.bib}   Being frameworks.bib this:  @Article{JONHSON88, author = {Ralph Johnson, Brian Foote}, title = {Designin Reusable Classes}, journal = {Journal of Object-Oriented Programming SIGS}, year = {1988}, key = {Johnson:88}, volume = {1}, number = {5}, pages = {22-35}, month = jun }  @book{UML-F:00, author = {Fontoura M., Pree W., Rumpe B.}, editor = {Addison-Wesley}, title = {The UML Profile for Framework Architectures}, year = {2000}, }   The PDF is generated with a text like:  Something about something[4], and moo foo other children[1].   This citation [4] is the first in all text. How can I order this correctly to [1]...[2] ...? Prove $-(-a)=a$ using only ordered field axioms I need to prove for all real numbers $a$, $-(-a) = a$ using only the following properties:     I tried by starting with P12 written using symbolic logic:  $\begin{align} \forall a,b \in P \to &amp; a \cdot b \in P \Leftrightarrow True \\ \forall a \in P \land \forall b \in P \to &amp; a \cdot b \in P \Leftrightarrow True \\ \lnot \left[ \left( \forall a \in P \right) \land \left( \forall b \in P \right) \right] \lor &amp; a \cdot b \in P \Leftrightarrow True \\ \exists a \notin P \lor \exists b \notin P \lor &amp; a \cdot b \in P \Leftrightarrow True \\ \end{align} \\ $  and played with evaluating it with $b = -1$, $-a$ and so on. But, felt this wasn't really a solution because the last symbolic logic statement above is ambiguous.  Not to mention--other than P12--I haven't used any of the axioms listed above whatsoever. ( And I've not bothered to prove any of the theorems of symbolic logic I have used. )  Can anyone give me a start on this problem?  Many thanks in advance.  Edit  Thank you, everyone! .  Best regards! Adding coordinate system to layout in QGIS composer? In 5e's version of the flesh to stone spell, if the spell is maintained for the full minute duration, the effect (petrification) lasts until "removed".  How can this petrification be removed? When is it best to write "can't" versus writing "cannot"? Are they interchangeable in every situation? Device : Moto X Play ( Rooted, Stock, Android 6.0.1, msm8916 Board, XT1562)  Why do I want to limit charging ?  ( Let's leave out whether my reasons are valid or not and accept that I want to, if you disagree with my reasons)  To improve the longevity of battery , I wish to limit charging when it reaches 90%, as explained here    Is it possible?  Yes, at least on my previous device Honor 6, as brought out here , with two different ways of doing it  I have tried  unsuccessfully to identify the file responsible in which changing values would limit charging  I am looking for the system file responsible for stopping charging, with it's true path (without symlinks)  (as I saw on on my previous device, locating the path of file was a big challenge after identifying it) If $|G| = pq$,  $p,q$ primes, $p \gt q, q \nmid p-1 $, then how do I prove $G$ is cyclic without using Sylow's theorems? Hide details of a chapter from TOC and list of tables, figures and algorithms On  we try really hard to fix spelling, grammar, and capitalization mistakes in question titles in order to avoid the . But occasionally someone enters a question with an all-lowercase title ().  It would be useful if the system could nudge the user toward capitalizing titles property by showing a hint when the user appears to be capitalizing incorrectly. For instance:    As  and  note in comments, the feature should also try to detect and advise against Title Case and ALL CAPS. Device that makes metal malleable I’m a big fan of ampersands (possibly too much …) and I’d like to replace all mentions of “et al.” (included in citations) by \&amp; al. – both in the main text (I use natbib citations of the form [Author1 et al., 1999]), and in the actual bibliography?  How can I do that, considering that BibTeX is inserting the words for me, and I have no control over the text? Preferably, I’d like to do this without creating my own bibliography style sheet (since I want to make this work regardless of the actual BibTeX style that is used), but if anyone can tell me how I need to edit the style sheet file, that would be fine as well (makebst doesn’t seem to offer an option for this …). I've been using SSH Tunnel manager to easily set up and stop port tunneling over SSH to various servers at my company. That saved me the pain of opening up a new terminal window which will hang out there just for the sake of having a tunnel open.  This worked great (well , sort of) but SSH Tunnel Manager is a PowerPC application. With Lion, it is not supported any more.  What is a good replacement? I was working on a Beamer presentation, and noticed that, if I open a group scope and use a \tiny, the font size will bet set, but there will be a large space between lines. However, if I issue a \par at the end of the paragraph, it works as intended:  \documentclass[t]{beamer} \newcommand{\shortlipsum} {Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis. Curabitur dictum gravida mauris. Nam arcu libero, nonummy eget, consectetuer id, vulputate a, magna. Donec vehicula augue euneque. Pellentesque habitant morbi tristique senectus et netus et malesuada fames ac turpis egestas. Mauris ut leo. Cras viverra}  \begin{document} \begin{frame} \shortlipsum  {\tiny \shortlipsum\par   % &lt;--- removing this \par will break line spacing } \end{frame} \end{document}   Here's a comparison between not using and using \par:    Am I doing it the right way? Why is \par the right way? I've always had a LOST.DIR folder on my external SD card, which until now has been empty.  Today my phone complained that my SD card was damaged and wouldn't mount it, so I pulled it out and looked at it on my computer.  There's now a 70237 file inside LOST.DIR which is 128 MB.  It starts with something like "&euml;&lt;MSWIN4.1", seemingly to indicate that it's a dump of a  (which makes no sense).  What did Android dump here and and why?  (I think I may have disconnected my phone from my PC yesterday without ejecting the drives first, so that might be the "why", though it was from Windows XP and not Windows 98.)  Is there a simple way to recover the information in this file? How to delete OS from boot menu After installing fedora and Centos and other OS now I have them all showing in my Boot menu [F2] on start up not the menu form the OS even after deleting them and reinstaling other OS - Ubuntu -so It just shows one when the OS starts but when I go to UEFI setting I still find all the old ones they instruct you to use DEL to take them out but that does not work persist port routing from 80 to 8080 I use amazon EC2 instance which works via ubuntu. By default according security restrictions I can't bin my application to port 80, so I just bind it port 8080 and then set routing redirect from port 80 to 8080 via the following command:  iptables -t nat -A PREROUTING -p tcp --dport 80 -j REDIRECT --to 8080   But I found that when I reboot the server this settings no longer active untill I invoke this command again.  So my question is how to enable port's redirect work even if system was rebooted? PHP replacing special characters like à->a, è->e Converse of Schur's Lemma in finite dimensional vector spaces I am trying to prove (or disprove) the converse of Schur's Lemma in finite dimensional vector spaces. I am not sure if it holds in this case, but I have tried to apply the idea that proves it in representation theory (see for example Theorem 4.3  that uses Maschke's theorem or in questions  and ).  The converse of Schur's Lemma in finite dimensional vector spaces is:     Let $V$ be a finite dimensional vector space over the complex numbers. Let $S$ be a set of endomorphisms of $V$ and assume that every endomorphism $A$ of $V$ such that $$AB=BA\text{ for all }B\in S$$ is of the form $\lambda I, \ \lambda\in\mathbb{C}$. Then $V$ is a simple $S$-space.   (*an endomorphism of $V$ is a linear operator from $V$ to $V$  *$V$ is a simple $S$-space if the only $S$-invariant subspaces of $V$ are $V$ itself and the zero subspace)  What I've tried so far:  Assume to the contrary that $V$ is not a simple $S-$space. Then, there exists a subspace $W$ of $V$ such that $W\not =\{0\}$, $W\not= V$ and $W$ is $S$-invariant. Let $W'=V\setminus W$. Then, since $V$ is finite, we can easily prove that $V=W\oplus W'$ (this is my attempt to translate Maschke's theorem in vector spaces). Consequently, for every $v\in V$, there exists unique $w\in W$ and $w'\in W'$, such that $v=w+w'$. Define the projection $P:V\rightarrow V$ by $Pv=w$ for every $v\in V$.   What is left to prove is that $PB=BP$ for all $B\in S$. Then $P$ is clearly not a scalar and thus we have the contradiction we are looking for.  There is a difficulty in showing that $PB=BP$ for all $B\in S$, because $W'$ might be $S$-invariant or not. More precisely:   Let $v\in V=W\oplus W'$ and $v\not =0_V$.  If $v\in W$ then $Bv\in W$ because $W$ is $S$-invariant, therefore: $PBv=Bv$ and $BPv=Bv$.  If $v\in W'$ then $Pv=0$, hence $BPv=B\cdot 0=0$, and  (i) if $Bv\in W'$ then $PBv=0$.  (ii) if $Bv\in W$ then $PBv=Bv$ and this is where the problem occurs.  Any hints, ideas or counterexamples would be very helpful. I have 16GB RAM. Do I need 32GB swap? I read many places that the rule of thumb for swap space is to double the amount of physical RAM. However, 32 GB does seem a LOT. Do I need that much? Do I need it at all with this high amount of physical RAM?