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At the moment I'm working with the . For this I read the following paper: .  But I have some mathematical problems with this formalism. I don't really see why this is a variation principle since it is opartor-valued.   The action principle is defined by: $\delta S  = G_1 -G_2$, where $S$ is the action operator $S = \int_{t_1}^{t_2} dt ~L$ and $G$ the generator operators of the transformations.  What does it mean that the action principle is operator-valued? For the classical action where $S$ is real we search for a stationary point but for operators? I also have a problem that $\delta S \neq 0$. What interpretation has this additional term $G_1 - G_2$? Proof of the square root inequality $2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}$ Proof of "Dual normed vector space is complete" Book recommendation on the history of PDE/ODE? Fast way of copying IMAP to IMAP on CentOS I'm trying to copy of IMAP emails from one server to my new server but after Googling for hours and trying several tools, nothing is working. Is there a tool that actually works?  I am copying hundreds of thousands of emails so using a email client is out of the question, I tried it, and at the rate of a email per 2 seconds, it would take a few weeks or months!  www.linux-france.org/prj/imapsync/ is commonly mentioned, however, the tool is no longer available.  I tried www.athensfbc.com/imap_tools/ which just hangs and does nothing.  Any migration tools that work? Any thoughts on a semidecadian badge for people with five years service and an average of N points per year? Given that Stack Overflow is close to its fifth birthday as a public site, has any thought been given to a 'semidecadian' badge for people who have:   been registered with the site five years (a semi-decade, hence the name), scored more than N points per year on average (where N might be 1000).   The idea behind 'average of N points per year' is to allow slow starters to gain the badge eventually, rather than ruling them out in year 1.  The details are obviously subject to tweaking.  This might be a gold badge (the platinum version would be the decadian badge, with a larger value of N required — and no-one eligible for another 5 years or so).  These would complement the current Yearling badge (which is silver). How to find correct size inner tube Why does the weighing balance restore when tilted and released I'm talking about a Weighing Balance shown in the figure:    Press &amp; Hold on onside of the horizontal beam and then release it. It makes some oscillations and comes back to equilibrium like shown in the figure.  Both the pans are of equal equal masses. When the horizontal beam is tilted by an angle using external force, the torque due to these pan weights are equal in magnitude &amp; opposite in direction. Then why does it come back to it position? What's making it to come back? Are there examples of third-(or higher)-order linear differential equations in physics or applied mathematics? The classical second-order linear ordinary differential equation is that named after Sturm and Liouville: formally, \begin{equation} (pu')'=ru. \end{equation} It arises naturally in many physical situations, for example through radial considerations of Schrödinger's equation. A higher-order "analogue" of it is the Orr$-$Sommerfeld equation, which, after relabelling the coefficients, may be written as \begin{equation} (\phi u'')''=\psi u. \end{equation} It also arises naturally, namely from certain simplifications applied to Navier$-$Stokes' (in)famous equation, and describes to great accuracy the cross-stream behaviour of channel fluid flow.  I have been thinking long and hard about whether there are additional higher-order linear differential equations that emerge naturally from our mathematical models of the world. It even seems pretty much all of the linear partial differential equations (heat, wave, Schrödinger, etc.) are of second order. Does anybody know of higher-order examples? I'd like to do something like this:  &gt;&gt; foo = @() functionCall1() functionCall2()   So that when I said:  &gt;&gt; foo()   It would execute functionCall1() and then execute functionCall2().  (I feel that I need something like the )  EDIT:  functionCall1 and functionCall2 are not necessarily functions that return values. I know than an ellipsis can be used to show a broken sentence, the same way an em dash can be used. What I don't know is that if the two are interchangeable and it is writer preference or if different meaning is conveyed by their use.  Are ellipses and em dashes commutable? How to prove that $4^m + 5^m$ is divisible by 9 when m is an odd number I am trying to use congruence theorems, specifically Euler's Theorem for a proof. Is doing two PhDs a good path? How should we cope with 12 month old waking in the night? We are trying to get our 12 month olds sleep routine improved now that both of us will be returning to work.  We've been working on getting the settling routine sorted out each night, and haven't quite yet got to him falling asleep on his own, but he does now settle quickly at the end of the routine in our arms.  We will be moving to trying to get him to settle in his cot soon.  Once settled he will tend to do about 3-4 hours in his cot before waking screaming.  When cuddled he will settle down in about 10 minutes and can generally be put back in his cot.  However often he will wake several times.  Last night he went down at 8, woke at 10, woke at 12, woke at 1am.  Each time putting him back in his cot.   At that point we gave up and brought him into our bed where he slept through to 6am.  When waking up he is often on either all fours or standing in the cot whilst screaming.  The main problem is that he's no longer sleeping "solidly" and will often poke and prod his mum in his sleep, and she therefore doesn't get a great nights sleep.  And with us both back to work we both need our sleep as much as possible. Is propagation of uncertainties linear? I'm in doubt with one thing: let's imagine that we have $n+1$ quantities, $n$ of them being directly measured, and the other one being related to the first $n$ by a function $f : \mathbb{R}^n \to \mathbb{R}$. If the $x_i$ is one of the quantities being directly measured and if $\Delta x_i$ is it's uncertainty, then I've learned that the uncertainty of the quantity $y = f(x_1, \dots x_n)$ is given by:  $$\Delta y=\sum_{i=1}^{n}{\frac{\partial f}{\partial x_i}\Delta x_i}$$  My problem in understanding this is: well, if $f$ is such a function, it's derivative is the linear transformation $df : \mathbb{R}^n \to \mathbb{R}$ given by:  $$df=\sum_{i=1}^{n}\frac{\partial f}{\partial x_i}dx_i$$  Where $dx_i : \mathbb{R}^n \to \mathbb{R}$ is given by $dx_i(a_1 \dots a_n) = a_i$. Hence, what we are saying, is that $\Delta y = df ( \Delta x_1, \dots \Delta x_n)$, in other words, we are saying that the uncertainty of $y$ is given by a linear function of the vector of uncertainties of the quantities $x_i$. Why should that be true? I mean, what's the reasoning behind this? I really didn't get the point, my teacher just gave the equation without justifying or anything like that.  Any help and reference is appreciated. how to change the distance between tick mark and tick label So from what I understand transistors are one of the most important electronic elements for computers because of TTL (Transistors-Transistors Logic).  Is this true? If so, how exactly does this work? I have heard the terms "logic gate" and "logic block" tossed around. I am trying to find a book (young adult) where the devil is the protaganist, and other gods appear as characters I believe it was written by a teenager, and might have a sequel.   It has been many years since I read it, but I believe it had the devil acting like a regular human. He wasn't evil. I think it incorporated a lot of (gods/characters) norse, roman, Christian, and pagan. I would like to protect the formatting of a cell (or a range of cells) in a Google Spreadsheet, but still allow changing the data in that (or those) cell(s).  For example, I want to add a validation rule on a cell that says the cell has to contain a number between 1 and 10, and I would like others to be able to change the value in that cell, but not the validation rule itself.  How do I do that?  So far I was only able to completely protect the cell (content + formatting) or not at all (the user can change the content as well as the formatting).  To me this seems to be an important functionality when adding access rights to spreadsheets. Is it possible that it is completely missing from Google Spreadsheets? How do I limit the number of rows returned by an Oracle query after ordering?