Having trouble with a statement in sakurai modern qm I am facing trouble with a statement made after eq 5.2.4 in sakurai modern qm and the statement is $P_1(E-H_0-\lambda P_1VP_1)$ is singular in $P_1$ subspace. How one can we show the validity of the statement?  This question is similar to problem asked previously in this forum but haven't answered appropriately. So I am asking again here. The link for the  qn is
Can't Prove formula from Sakurai's Modern QM @ Perturbation Theory I am studying Perturbation Theory from J.J. Sakurai's textbook Modern Quantum Mechanics. I am having trouble proving formulas on page 299 (5.2.5) and (5.2.6) from the previous ones [mainly (5.2.4)]. Can anyone help?  This is a problem of working with perturbation theory when the non-perturbed state has degeneracy. To elaborate I do not get why $(E-H_0-\lambda P_1VP_1)$ is not singular.  More importantly is the equation below correct and if so why:  $$(E-\lambda P_1VP_1)|l \rangle = E_D^{(0)}|l \rangle $$  That is the core of my problem. If the above equation - in the framework of Sakurai's statement of the problem - is correct I can work out all the rest on my own. However I do not get why this equation would hold true.
How well will WCF scale to a large number of client users? Does anyone have any experience with how well web services build with Microsoft's WCF will scale to a large number of users?  The level I'm thinking of is in the region of 1000+ client users connecting to a collection of WCF services providing the business logic for our application, and these talking to a database - similar to a traditional 3-tier architecture.   Are there any particular gotchas that have slowed down performance, or any design lessons learnt that have enabled this level of scalability? How to insert a figure between two texts? I'd like to create something similar to what is give below: text on either side of a vertically centred image: The Download CSV link seems broken in SEDE query result window I've created   which works nice. I would really like to download those results as an CSV file. Therefor I click on the provided link on the right hand side of the screen called Download CSV. The result is:  The webpage cannot be found  https://data.stackexchange.com/stackoverflow/csv/202355?UserId=187606 I expected: A file in CSV format. It looks like a regression of  Queries without a parameter work as advertised. Can this be fixed? I'm using a curve modifier and an array modifier. My object is at one end of the curve but as soon as I choose the curve for the curve modifier it moves to the center. How can I fix the problem so that I can have the array going from one end of the curve to the other? You'd think that as soon as Harry met someone who was friends with his dad, he'd have asked about his family.  Certainly Lupin or Sirius could have told him something of his family, given that they were James' closest friends, and Sirius was "always welcome at the Potters".  Obviously there's nothing in the books, but did J.K. Rowling ever explain why there was no mention of any living relative other than Aunt Petunia, or even what happened to Harry's grandparents? I have a dataset in csv file format that I'm loading into pgfplotstable. I can't change the format of the original csv, and being financial data it has a column of item names and two or three explanatory data points at the top of the table. e.g.  \documentclass{article} \usepackage{array,datatool,pgfplotstable}   \begin{filecontents}{sample.csv}     item,2008,2009,2010,2011      ,Actual,Actual,Forecast,Forecast      ,GBP,GBP,GBP,GBP      ,('000),('000),('000),('000) rounded     Area 1 OP, 1000000,1500000,1750000,250000     Area 2 OP, 400000,500000, 450000,-50000     Area 51 OP, 300000,375000,390000,15000     No P, 1250000, 1000000, 950000, 50000     Residuals, 800000, 80000, 90000, 10000   \end{filecontents}  \pgfplotstableread[col sep=comma]{sample.csv}\mystuff   I'm stumped at the next stage which is to typeset \mystuff so I have comma separators and can add a row that sums the 'Area' columns. At the moment to display the table I have to do something like  \begin{document}  \pgfplotstableread[col sep=comma,header=false]{sample.csv}\mystuff  \pgfplotstabletypeset[   columns/0/.style={string type},   columns/1/.style={string type},   columns/2/.style={string type},   columns/3/.style={string type},   columns/4/.style={string type}]\mystuff  \end{document}   How do I tag the top three rows as string and the rest of the last four columns as integer type so I can format and transform them? Consecutive days missed I visited Stack Overflow for about 82 days in a row (starting January 21st), including yesterday (April 13th) and today (April 14th).  All times UTC of course (I thought that was clear but I've now added it to my question as a relevant info).  All visits are in a logged-in session of course.  I've just discovered that SO didn't "see" the April 13th visit, so my consecutive days visits were zeroed !!  It's a bug for sure. I have open tabs which my browser's history shows that are from April 13th. More than 20 of them. I saw this one:  , and it's clearly what happened to me. Please help! I'm so close to getting the Fanatic badge... How to show loading spinner in jQuery? In Prototype I can show a "loading..." image with this code:  var myAjax = new Ajax.Request( url, {method: 'get', parameters: pars,  onLoading: showLoad, onComplete: showResponse} );  function showLoad () {     ... }   In jQuery, I can load a server page into an element with this:  $('#message').load('index.php?pg=ajaxFlashcard');   but how do I attach a loading spinner to this command as I did in Prototype? Improving how suggested edits are displayed in your activity history Issue 1: I find it a bit irritating that when I click the Improve button and further edit it, even if I mark it as helpful, only Community takes the credit for the approval. I know the topic itself will contain the link to the suggested edit, but my activity history only says that I made an edit, and not approved a suggested edit. The same holds true for marking one as unhelpful. The activity history should say I rejected an edit, and then made an additional edit. I know it's not often, but sometimes I do try to go back to a suggested edit I've rejected, but when I improved and rejected, it's not traceable for me as it doesn't show in my history or in the question's revision history. So if I don't remember what user suggested it, it's lost to me.  This issue also plagues the  which are awarded for approving and rejecting suggested edits, which essentially you have done, you're just not receiving credit for it. I suggest that when I mark an edit as helpful or not helpful, both me and Community be listed as the approvers or rejecters, unless another user is already listed in the appropriate place.  Update: Here is  of the confusion this causes.  Update 2: Also, when improving a post and marking it as "not helpful", it would be nice to be able to  along with it.    Issue 2: Expanding off the idea that users need more notice for suggested edits that get rejected, the activity history page is much too useless when it comes to listing the suggested edits a user has made. Literally all it does is list them. A user still has to click through to each one in order to find out if it was approved or rejected, or even if it's still pending. Take, for example, this screen shot of the suggested edit history:    Is it really necessary to say I suggested a suggested edit? Really? Of course I suggested it, it's a suggested edit. This isn't !  A good way to fix this would be to change the second "suggested" word into one of: pending, approved, or rejected. This would be a quick way to tell which edits have been approved or rejected and which ones are still pending. Users would be able to target out rejected edits and review them for the reasons. Something like this: I have made a command  \newcommand{\mysec}[2]{%  \textbf{#1} \\ #2 }   I use it with \mysec{A}{B}.  But sometimes I also want a third parameter. I could make a whole new command for those situations where I need 3 parameters. But I thought if I could handle it with an optional parameter in my original command?  So I have to check if the third parameter is set and how to print if it is set. Why are these estimates to the German tank problem different? Direction of angular velocity Angular velocity is the rate of angular displacement about an axis. Its direction is determined by right hand rule.  According to right hand rule, if you hold the axis with your right hand and rotate the fingers in the direction of motion of the rotating body then thumb will point the direction of the angular velocity.  The direction of angular velocity is above or below the plane. But what does it mean? I mean in linear velocity the direction of velocity is in the direction of motion of body but what does it mean that body is moving in one direction while the direction of its angular velocity is in another direction? Why don't the Google Docs CTRL + ALT shortcuts work? I'm currently using Google Docs, but when I try to use the keyboard shortcuts like Ctrl+Alt+M it gives µ instead.    Is Chrome (9) overruling Google Docs in some way?  How can I get these shortcuts working?  I'm using Chrome dev 9 on a Dutch Windows 7 with a US-layout keyboard As the question says, I need to edit my /proc/acpi/wakeup file, but when I open it with any text editor, it shows blank file. cat command, on the other hand, displays correct file contents in terminal. When I navigate to that file in file manager, it says it's size is 0 bytes. I'm using Xubuntu 15.04. I'm thoroughly confused. If each suit is represented then we will have two cards from one suit and one card each from the remaining suits. So I am counting the ways this can happen like so -  ${52 \choose 1}$ ways of selecting the first card.  ${12 \choose 1}$ ways of selecting a card from the same suit as the first.  ${13 \choose 1}$ ways of selecting a card for each of the 3 remaining suits.  So I have -  $${52 \choose 1}{12 \choose 1}{13 \choose 1}{13 \choose 1}{13 \choose 1}$$  Then I have the sample space as $52 P 5$ as we are counting with regard to order on the numerator so we need to do the same for the denominator.  So I have $$\frac{{52 \choose 1}{12 \choose 1}{13 \choose 1}{13 \choose 1}{13 \choose 1}}{52P5}$$  Does that look correct? Did Snape ever commit murder? It's well known that Severus Snape was, in one way or another, a Death Eater since he was at Hogwarts. He hung out with the likes of Avery and Mulciber who became well known Death Eaters, he joined Voldemort right after he left Hogwarts, and he basically became Voldemort's right hand man until the very end.   However, in his time as a Death Eater did he ever kill anyone? We know that he was a Death Eater for around 3 years before Voldemort fell; Lily and James died at 21 and they were the same age as Snape, so it would have been 3 years since they left Hogwarts. Then after Voldemort came back in book 4, Snape would have been a Death Eater again for around 3 years. That's a total of around 6 years of Death Eaterhood with Voldemort.  So, in either of these periods, did Snape commit murder as part of his job as Death Eater? I'm not counting a certain death at the end of book 6 for reasons which are obvious if you've read book 7.   Clearly it's much more likely he killed before Voldemort fell since that roughly correlated with the period when he pledged allegiance only to Voldemort, before he became a teacher at Hogwarts. But did he ever do it? What is EFI in partition scheme? Recently I changed my partition scheme from MBR to GPT, and during after conversion I saw a another partition named EFI ... I don't have a good idea what does it do or its purpose ... but I think it's use for backup's.  So my question here is can you explain/tell me about EFI and it purpose? Can I charge my laptop with a charger that has different specifications? I'm using a common charger for both of my laptops.  The charger is specified at 20 Volts and 3.25 Amperes. My other laptop has a specification of 19 V and 3.42 A.  Is it safe to charge my laptop with chargers whose specifications exceed or are not equal to the mentioned specifications?  I have read somewhere that it may cause some damage if the voltage or amperage is not matching the specs. Is there a way to run Python on Android? Converting numpy dtypes to native python types