the continuity of total variation function of a continuous function of bounded variation
Is the total variation function uniform continuous or continuous?
Does C# have extension properties? I lock my computer a lot, and was wondering if there was a way to get the lock screen when I hit Ctrl + Alt + L. Right now when I use that key combination it turns the screen black, and I have to hit a key or click to bring up the screen where it shows the option to enter your password or switch users, as well as displaying the desktop background behind it.   Is there anyway to avoid having to hit that extra key, to bring up the screen I want? And is there anyway to keep it that way? It's really a matter of personal preference, and I don't see any reason why not to have it stay that way as my monitor eventually is turned off and/or my computer suspended. Difference between 2 numbers I need the perfect algorithm or C# function to calculate the difference (distance) between 2 decimal numbers.  For example the difference between: 100 and 25 is 75 100 and -25 is 125 -100 and -115 is 15 -500 and 100 is 600  Is there a C# function or a very elegant algorithm to calculate this or I have to go and handle every case separately with ifs.  If there is such a function or algorithm, which one is it? For some reason, the profile-hover view thingy is not loading at all(on Stackoverflow). I initially thought that it might be because of , but its been closed with the status, no-repro. Also, this is now persisting for the past 1 hour for me.  I tried to clear the cache and stuffs but nothing seems to be working, as the hover view is still not loading. Is this some sort of  or an issue with the browser? I'm using Firefox 25.0.1 on Win7 Pro(x86) machine. Here is a snapshot of it. Here's an example of a Google Docs Spreadsheet script that sends emails using Google spreadsheets. I want the email Sent in HTML not plain text. Can someone help me out?  function sendEmails() {   var sheet = SpreadsheetApp.getActiveSheet();   var startRow = 2;  // First row of data to process   var numRows = 2;   // Number of rows to process   // Fetch the range of cells A2:B3   var dataRange = sheet.getRange(startRow, 1, numRows, 2)   // Fetch values for each row in the Range.   var data = dataRange.getValues();   for (i in data) {     var row = data[i];     var emailAddress = row[0];  // First column     var message = row[1];       // Second column     var subject = "Sending emails from a Spreadsheet";     MailApp.sendEmail(emailAddress, subject, message);   } } How to create a new object instance from a Type One may not always know the Type of an object at compile-time, but may need to create an instance of the Type.    How do you get a new object instance from a Type? Proving inequality $(a+\frac{1}{a})^2 + (b+\frac{1}{b})^2 \geq \frac{25}{2}$ for $a+b=1$ $a _{1} = \frac{1}{2}, a _{2} = 1, a _{n}= \frac{1}{2} a_{n-1} + \sqrt{a _{n-2} }$  Show by the induction that it's $\le 4$ and also by the induction that $a_{n+1}-a_n \ge 0$  If it comes to first I don't know what to do and if it comes to second I tried many different ways but I couldn't prove it. How should an educational institute woo good faculty? What steps must an educational institution wanting to recruit good faculty take? There is a cycle in the academia:    Good doctorates from top universities invariably join other top universities as faculty members.  Good students aspire to study in top universities under excellent faculty members.   Suppose a new educational institution or department springs up and desires to set up a top-quality group, how does it break into this cycle? I suppose this could take some years, but is money-power the only thing that could be wielded by the institute to achieve its end? Is a diode needed for this simple mini solenoid circuit? Bibliography error: "Use of \blx@bbl@verbadd@i doesn't match its definition. \verb" Currently having problems with compiling my Master's thesis after downloading the newest version of TeXStudio and MiKTeX. The error message is as follows:  Use of \blx@bbl@verbadd@i doesn't match its definition. \verb   The only other place I found a similar question regarding this error message was found here:   I am not sure how this applies to my problem, but from what I understand this is due to some problem in an external file. If you need the bibliography-entry, I can provide you with that on short notice, though it is rather long (which is why I didn't include it here in the first place).  I am using Mendeley Desktop v1.16.1 to automatically generate my BibTeX database.   I am using biblatex as follows:  \usepackage[backend = bibtex,         style = numeric-comp,         sorting = none,         natbib = true,         doi = false,         isbn = true,         hyperref = true]{biblatex}   I also have some code to change how the bibliography output, but I doubt it is causing the problem, as commenting it out did not help.  Hope anyone knows what causes my problem and how to fix it.  Thanks. Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics How do I definitively show that there are only two propagating degrees of freedom in the Lorenz Gauge $\partial_\mu A^\mu=0$ in classical electrodynamics.  I need an clear argument that   involves the equations of motion for just the potentials $A^0$ and $\mathbf{A}$, and not the electric and magnetic fields.   includes sources $\rho$ and $\mathbf{J}$ in the equations of motion.  This is to justify the assertion that there are degrees of freedom that decouple from the rest of system. does not critically rely on quantum field theoretic arguments (although any supplementary remarks are welcome).     To illustrate the level of clarity I expect, I provide an argument in the Coulomb gauge $\nabla\cdot \mathbf{A} = 0$:  Of the four field degrees of freedom, the gauge condition $\nabla\cdot \mathbf{A} = 0$ removes one degree of freedom (the longitudinally polarized EM waves).  To show that among the remaining three degrees of freedom only two are propagating, consider the field equations of motion in the Coulomb gauge:  \begin{align} \nabla^2 A^0 &amp;= -\rho/\epsilon_0,\\ \big[\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2\big]\mathbf{A}&amp;= \mu_0 \mathbf{J}-\nabla \frac{1}{c^2}\frac{\partial}{\partial t}A^0. \end{align}  The first equation is NOT a wave equation for $A^0$, and thus does not propagate.  The final equation IS a wave equation, and describes the propagation of two degrees of freedom (Gauss' law in first eq. can be solved, and then inserted into the second equation to show that $\mathbf{A}$ only couples to the Solenoidal part of the current). Where does pp (PAR) unpack add (-a) files? GUI for watching logs (tail and grep) Could you recommend a GUI application with powerful log watching capabilities?  Generally it would work as tail -f in GUI, but on top of that following features would be very useful:   filtering out some lines based on (regular) expressions  coloring some lines based on (regular) expressions interactive search saveable configuration easily applicable to different files notifications based on (regular) expressions   A similar tool on Windows is  and its paid version - Sturm-Liouville Questions In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on.  The first issue I have is that my book began with the statement that given  $$L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  the problem $L[y] \ = \ f$ can be re-cast in the form $L[y] \ = \ \lambda y$.  Now it could be a typo on their part but I see no justification for the way you can just do that!  More importantly though is the motivation for Sturm-Liouville theory in the first place. The story as I know it is as follows:  Given a linear second order ode   $$F(x,y,y',y'') = L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  it is an exact equation if it is derivable from a differential equation of one order lower, i.e.  $$F(x,y,y',y'') = \frac{d}{dx}g(x,y,y').$$  The equation is exact iff  $$a''(x) - b'(x) + c(x) = 0. $$  If $F$ is not exact it can be made exact on multiplication by a suitable integrating factor $\alpha(x)$.  This equation is exact iff  $$(\alpha(x)a(x))'' - (\alpha(x)b(x))' + \alpha(x)c(x) = 0 $$  If you expand this out you get the Adjoint operator  $$L^*[\alpha(x)] \ = \ (\alpha(x)a(x))'' \ - \ (\alpha(x)b(x))' \ + \ \alpha(x)c(x) \ = 0 $$  If you expand $L^*$ you see that we can satisfy $L \ = \ L^*$ if $a'(x) \ = \ b(x)$ &amp; $a''(x) \ = \ b'(x)$ which then turns $L[y]$ into something of the form  $$L[y] \ = \ \frac{d}{dx}[a(x)y'] \ + \ c(x)y \ = \ f(x).$$  Thus we seek an integratiing factor $\alpha(x)$ so that we can satisfy this &amp; the condition this will hold is that $\alpha(x) \ = \ \frac{1}{a(x)}e^{\int\frac{b(x)}{a(x)}dx}$  Then we're dealing with:  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \alpha(x)f(x)$$  But again, by what my book said they magically re-cast this problem as  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \lambda \alpha(x) y(x)$$  Then calling  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ ( \alpha(x)c(x)y \ - \ \lambda \alpha(x) )y(x) \ = \ 0$$  a Sturm-Liouville problem.  My question is, how can I make sense of everything I wrote above? How can I clean it up &amp; interpret it, like at one stage I thought we were turning our 2nd order ode into something so that it reduces to the derivative of a first order ode so we can easily find first integrals then the next moment we're pulling out eigenvalues &amp; finding full solutions - what's going on? I want to be able to look at $a(x)y'' \ + \ b(x)y' \ + \ c(x)y \ = \ f(x)$ &amp; know how &amp; why we're turning this into a Sturm-Liouville problem in a way that makes sense of exactness &amp; integrating factors, thanks for reading! find the value of limit Find the value of $$ \lim_{n\to\infty} \sum_{k=n+1}^{2n} \frac{1}{k} $$ I think it become a series with each denominator approach infinity. so the value of the limit is 0. am i true ? At some point I accidentally ran off a cliff with my War Horse which resulted in a death. After respawning, I noticed my War Horse was gone and the default Tennessee Walker was my main horse.   How did the War Horse disappear and is it possible to retrieve it? "Naturally occurring" non-Hausdorff spaces? It is not difficult for a beginning point-set topology student to cook up an example of a non-Hausdorff space; perhaps the simplest example is the line with two origins. It is impossible to separate the two origins with disjoint open sets. It is also easy for a beginning algebraic geometry student to give a less artificial example of a non-Hausdorff space: the Zariski topology on affine $n$-space over an infinite field $k$, $\mathbf{A}_{k}^{n}$, is not Hausdorff, due to the fact that polynomials are determined by their local behavior. Open sets here are in fact dense. I am interested in examples of the latter form. The Zariski topology on $\mathbf{A}_{k}^{n}$ exists as a tool in its own right, and happens to be non-Hausdorff. As far as I'm aware, the line with two origins doesn't serve this purpose. What are some non-Hausdorff topological spaces that aren't merely pathological curiosities? I've always wondered why the number of protons in the Universe exactly matches the number of electrons. They are such different particles with totally different cross sections. So, first of all, is this true? And, secondly, how could it be that they have been produced in the same amount after the Big-Bang? I want something similar to "preview" in macs.  For example: I want an image editor that ONLY does simple adjustments like increase/decrease contrast, saturation, exposure, color tinting.... rotate, flip vertically, flip horizontally, make black and white, change size or format, crop.  THATS IT.  I know gimp can do all those things but its a bit overkill.  I just want to right click an image, open it with this magical program i just described, do a few quick adjustments, and then save and exit.  Nothing really fancy.  Anyone know of anything like this?  Btw I am using ubuntu 12.04 :) It rocks and I am glad I switched from mac, i just need to replace this one piece of software.