Proof that $d^2 \phi = 2\pi \delta^{(2)}(r)dx\wedge dy$ in polar coordinates around a puncture?
How to use Stokes theorem in the presence of holes?
So here is the problem      Solve $$4^x + 6^x = 9^x$$ for $x$.   I am trying to find its real solution.  I was trying in this way!! $$6^x\left( \frac {4^x}{6^x} + 1\right) = 9^x$$ $$\left({\frac 23}\right)^x +1=\left({\frac 32}\right)^x$$ but I'm stuck . . . Will anyone help ? Making an MFC application international I've got several large MFC applications here, and converting them into any other format is out of the question.  We're expanding into other markets, and would like to make the apps work in other languages and cultures.  So far, I've found occasional references as to what to do with Visual C++ version 6, with one mention that later versions of MFC have additional relevant features.  Searching MSDN gives me instructions on how to convert the apps to Unicode, which we already did.  I found nothing on MSDN on how to make languages multilingual once they're in Unicode, only a few older things using Google, and one book about internationalization using VC++6.  (We're using Visual Studio 2008 now, on XP and Vista.)    I make no claims for the strength of my Google-fu, and would be happy to be directed to things I've missed.  Is it reasonable to use the methods of VC++6, or should I use later features?  If I should use features later than that, where can I find some reference to them?  Is there something I should worry about other than setting the locale, converting all strings to resources, and duplicating resources in different languages?  I can find more things about .NET internationalization, but I'm inexperienced in .NET.  What will I find there that's applicable to VC++ and MFC?  Edit:  I just ran into difficulties trying to put Unicode strings into the String Table resource.  (As far as I can tell, Unicode strings need to be entered into the .rc file with a text editor, in L"0x0034" form.  I haven't found a resource editor way to do that.)  Any tips on that?  Any other resource I can use in VS2008 using VC++ and MFC?  Edit:  Somebody on a Microsoft forum suggested that I open the offending .rc file in Notepad, and save it in Unicode.  Unfortunately, this doesn't seem to fix the dialogs, although the stringtable seems to be working in Japanese. I know if $\ i\ $  of the following form $\ 3x^2 + (6y-3)x - y\ $ or $\ 3x^2 + (6y-3)x + y - 1, \ \ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$,  then $\ 12i + 5\ $ must be composite number, because:  $12(3x^2 + (6y-3)x + y - 1) + 5 = 36x^2 + (72y - 36)x + (12y - 7) = (6x + 12y - 7)(6x  + 1)$  How come if $\ i\ $ not of the following form $\ 3x^2 + (6y-3)x - y\ $ and $\ 3x^2 + (6y-3)x + y - 1\ $,  then $12i + 5$ must be prime? $\ x,y \in \mathbb{Z}^{+},i \in \mathbb Z_{\ge 0}$.  For example: $\ 5 = \ 3*1^2 + (6*1-3)*1 - 1\ $ ,when $\ x = y = 1\ $, as proved before,$\ 5*12+5\ $ must be composite number;  $\ 0, \ 1. \ 2.\ 3.\ 4\ $ can't of the following form $\ 3x^2 + (6y-3)x - y\ $ and $\ 3x^2 + (6y-3)x + y - 1\ $,  $\ 12*0 + 5 = 5\ $, $\ 5 $ is prime,$\ 12*1 + 5 = 17\ $, $\ 17 $ is prime, and so on, From command line open package installer How to Implement Page Object and Page Factory Pattern in Selenium Webdriver? I am a beginner in learning selenium Webdriver. I am not able to understand how page object and factory models works and how it can be beneficial?    Kindly help me explain it with detailed explanation. I am quite proficient with the keyboard and try to only use the mouse when necessary. Given that context I was very familiar with the shortcut keys in MS Outlook 2003 and never needed to use the mouse to use that product. Now my company has recently upgraded to MS Outlook 2010 and I am finding it impossible to operate it without having to use the mouse. In particular replying to meeting requests with the old shortcut keys no longer seems possible.   I have held down the ALT key and sometimes it shows some shortcuts on the ribbon but this seems to be limited to ALT+H which tends to alternate between designating "Accept" and "Tentative" without any keys for the other options. Also as this appears to switch between "Accept" and "Tentative" one can't even use this consistently without first having to check what it means for each particular instance.  Any help with this would be greatly appreciated! Subject/verb agreement when a title ends in a plural 1) "The book 'The Three Musketeers' is a wonderful example of..." Here we have a proper noun, a title that happens to end in a plural, and I have no sense that the verb should be plural. "Musketeers"/is seems proper/natural/correct, and 'are' not at all.   However, such certainty begins to wane in other examples: 2) "'The Heidi Chronicles' is a wonderful example of..." doesn't completely shake my confidence, but it moves the needle, at least a little bit. But, practically speaking, I wouldn't have a second thought about correcting a student who used 'are' instead of 'is'.  Here's the one that really bothers me: 3a) " 'The Basement Tapes' are a wonderful example of..." I don't think this sounds any less proper/natural/correct than 3b) " 'The Basement Tapes' is..." In fact, 3a seems less awkward than 3b.  Why? Why must we "avoid discussions in comments"? I've just started getting these messages:     Please avoid extended discussions in comments. Would you like to automatically move this discussion to chat?   Why should one avoid doing this? I think it adds to the site...  (Tagged  in more ways than one!) I need to concatenate two String arrays in Java.  void f(String[] first, String[] second) {     String[] both = ??? }   What is the easiest way to do this? The HDD I have my Ubuntu installed is about to fail. I would rather not lose 3 years worth of data, customisation and apps. I am looking for a way to move the complete system (SWAP included, because I'm not sure if I can relink the system to a new SWAP partition) to another HDD. But not the complete HDD&lt; only the partition containing Ubuntu, to a partition on a different HDD. Basically I'd like to do what I've been able to do with Norton Ghost for my Windows install. I thought about using Clonezilla but I think I would have issues with GRUB (Especially trying to boot from a different UUID than what is in the conf file). do you know of any way this could be done?  PS, my home directory is encrypted but that's not really an issue, because I can work around that.  EDIT: changed the explanation to make it clearer Where should I copy-paste-install software to? I'm installing IntelliJ, and the readme tells me to choose an installation folder. Where is a good, standard place to put installed programs? How to update Ubuntu terminal? How to update Ubuntu terminal? How to parse a date? I would like to know where gravity gets its energy to attract physical bodies?  I know that the law of conservation states that total energy of an isolated system cannot change. So gravity has to be getting its energy from somewhere, or else things like hydropower plants wouldn't be able to turn the power of the falling water into a spinning rotor.  Just to be clear, Lets create an example:  Lets say we have two objects with equal mass close to each other. So gravity does its job and it pulls each other closer, this gets turned into kinetic energy. This is where I'm lost. According to the law of conservation energy can't be created or destroyed and the kinetic energy comes from the gravitational pull so where does the gravitational pull gets its energy.  If that energy isn't being recycled from some where else then that means you have just created energy, therefore breaking the law of conservation. I am using XeTeX and have the following short document:  \documentclass[a4paper,12pt]{article} \usepackage{amsmath,amsthm} \usepackage[MnSymbol]{mathspec} \usepackage[no-sscript]{xltxtra} \defaultfontfeatures{Mapping=tex-text} \setallmainfonts{Minion Pro}  \begin{document}  $\phi, \varphi, \delta \ldots A \vee B$  \end{document}    Now no matter what font I am using the punctuation marks in math mode (commas, dots, \ldots etc.) come out in Computer Modern. Is there a way to change this? I am very confused about the probability formula $E[\min(X,k)]=\int_0^kxf(x)dx+k(1-F(k))$ I understand the proof of this first formula.  I am confused about where he got the first E[min(X, k)] formula. It seems to have just come out of nowhere. And also, can someone explain why the two formulas are correct please? I'm looking for an elegant way to show that, among non-negative numbers, $$ \max \{a_1 + b_1, \dots, a_n + b_n\} \leq \max \{a_1, \dots, a_n\} + \max \{b_1, \dots, b_n\} $$  I can show that $\max \{a+b, c+d\} \leq \max \{a,c\} + \max \{b,d\}$ by exhaustively checking all possibilities of orderings among $a,c$ and $b,d$.  But, I feel like there should be a more intuitive/efficient way to show this property for arbitrary sums like the one above. My lecturer has used some notation that I've never seen before: it is a (matrix) product symbol with a left-to-right arrow over the top. Does anybody know what this means?  Thanks in advance.  Edit: It looks like this: Value of the Hubble parameter over time There is something I don't understand about the  $H$, as it seems to clump two concepts together that I can't quite unify in my head. On the one side, we have  $$V = D H$$  which means that for a given distance $D$, there is a certain amount of new space created over time - and $H$ is simply the factor that makes this relationship work. So, for example, say we have two points 1 Mpc apart, this would mean they recede at about 70 km/s away from each other  (given our current approximation of $H$).  Now the thing I can't wrap my head around is that   $$T =\frac{1}{ H}$$  is also the age of the universe. Contrary to claims made on, say, Wikipedia this means that $H$ cannot possibly have been a constant throughout the past 13 billion years, because mathematically $1/H$ means that $H$ must be continually shrinking as the universe ages.  So if $H$ did start out as some huge value and is now shrinking over time, doesn't this mean the expansion of the universe is slowing down? Because if $H$ is shrinking, I'll get a lower value of $V$ today than I'll get tomorrow. Shouldn't the notation then be more like  $$V = DH(t)$$  So which one is it? If $1/H$ is simply the solution for $D=0$, how can we use it as the expansion-velocity-per-unit-of-distance at the same time? What's worse, how can literature say $H$ has probably been more or less constant forever and simultaneously assert that $1/H$ is the current age of the universe? What am I missing? UTF-8 -> ASCII in C language