Recurrence problem with a game of probability
Gambler's ruin and coin toss
A dollar amount $\${-}67.9{-}$ is divisible by 72. What is this number? A client buys 72 turkeys. On the receipt, 2 digits are missing, the first and the last. So, the total price is $\${-}67.9{-}$. Find a way to make the price of each turkey round at the second digit without trial-and-error method.  So, the answer is obviously $\$5.11$/turkey, so $\$367.92$ total. I found that with with 2-3 trial-and-error.  I bypassed his problem by using python and made a script that tries every combination and assures it is divisible by 0.01.   Please help me find a mathematical way of solving this problem. Calling C/C++ from Python? What would be the quickest way to construct a Python binding to a C or C++ library?  (I am using Windows if this matters.) Retrieving Maxwell's equations from the minimum action principle I'm currently working at the start of Alexei Tsvelik's book Quantum Field Theory in Condensed Matter Physics. I'm kinda stumped on a few essential steps.  Starting with the action:  $$S = \int dt \int d^3 x \mathcal{L}_{\mathrm{EM}} = \int dt \int d^3 x \left( -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + j_\mu A^{\mu}\right).$$  In the absence of any sources $j_\mu = 0$ and $c=1$, which after pulling in the negative sign implies  $$S = \frac{1}{2}\int dt \int d^3 x  \,\,\left(E^2 - B^2\right)$$ The next step is to take $\vec{E} = -\vec{\nabla} \phi + \partial_t \vec{A}$ and $\vec{B} = \vec{\nabla} \times \vec{A}$. Letting $\phi = 0$ and directly inserting these into the above yields:  $$S = \frac{1}{2} \int dt \int d^3 x  \,\left( (\partial_t \vec{A})^2 - (\vec{\nabla} \times \vec{A})^2 \right) $$  Now considering $\vec{A} = \vec{A}_0 + \delta \vec{A}$ as some minimal change in our vector potential, with $\vec{A}_0$ gives a minimum for the action and putting this into the above expression we have: $$\delta S = \int dt \int d^3 x \left(\partial_0 \vec{A}_0 \partial_0 \delta \vec{A} + \left(\vec{\nabla} \times \vec{A}_0\right)\cdot\left(\vec{\nabla} \times \delta \vec{A} \right) \right)+ \mathcal{O}(A^2)\,\,\,\,\,\,\, (1)$$ where I have the term  $$\mathcal{O}(A^2) = \int dt \int d^3 x \left( (\partial_0 \vec{A}_0)^2 + (\partial_0 \delta \vec{A})^2 - (\vec{\nabla}\times\vec{A}_0)^2-(\vec{\nabla}\times \delta\vec{A})^2\right)$$  The next step is where I lose our author and go from "Oh ok I get this" to "completely clueless". He rewrites $\delta S$ as:  $$\delta S =  \int dt \int d^3 (\delta x) \vec{A} \mathcal{F(\vec{A}_0)}+ \mathcal{O}(A^2)$$  where $\mathcal{F} = \frac{\delta S}{\delta A}$. What is the purpose of $x\to \delta x$ in the above expression? Is the functional derivative $\mathcal{F}$ a vector quantity? He has it bolded with a bolded $\vec{A}$ in the above expression. One last question takes the above expression further. We assume $\delta \vec{A}$ vanishes at infinity, as it should, and integration by parts yields:  $$\delta S =  -\int dt \int d^3 x \left( \partial^2_0 \vec{A}_0 - (\vec{\nabla} \times \vec{\nabla})\times \vec{A}_0  \right)\delta \vec{A}\,\,\,\,\,\,\,\, (2)$$ How exactly did we go from (1) to (2)? Thanks I appreciate your time! I am curious if it is possible to have a VPS that has data on it that is not readable by the hosting provider, but is still usable on the VPS.  Obviously there are some things that you could do to prevent them from reading anything...   You could change all the passwords, including root. But then, they could still use some alternate boot to reset the password, or they could just mount the disk another way. So, you could encrypt the disk or at least some of the contents on the disk. But then it seems that if you decrypted the content, they could still "peer in" to see what you were doing at the console, because after all, the virtualization platform should allow this. And even if you could stop that, it seems they could just read the RAM of the VPS directly.   Of course, the VPS can store data on it and as long as the key is not on the VPS and the data is never decrypted there, then the host cannot get the data.  But it seems to me that if any point the data on the VPS is decrypted...for use on the VPS...then the hosting provider can get the data.  So, my two questions are:   Is this correct? Is it true that there is no way to 100% secure data on a VPS from a host from seeing it, while keeping it accessable by the VPS? If it is possible to make it 100% secure, then how? If it is not possible, then what is the closest you can get to hiding data from the web host? In my team, we use Selenium for automation of functional testing. We also do a lot of manual regression testing to make sure the appearance of the rendered web pages is ok. I would like to develop a framework, that would allow comparing screenshots of the "base" live web page, and the web page on the test server after applying the software update. I believe this will drastically reduce the amount of manual regression testing we need to do.   I would like the framework to be able to detect portions of web pages that were rendered differently from each other. An example of such functionality can be  (first image in the article)  I have never worked with image processing and would need ideas on what algorithms are already publicly available that would help me with the task. From what I can see, the simple pixel-by-pixel comparison would not work well, because it does not account for the concept of elements.  Or am I approaching this problem from a completely wrong direction? What EXACTLY happens when I ask to delete my account? I have meticulously read through similar questions, answers and their comments but I still do not understand how the deletion is done. From what I understand I email the customer service and then they do what exactly?     1) They disable my account so that I can no longer use it?      2) They reassign all of my questions and answers to some other   account? To which one?      3) They delete all my questions and answers so that they go away from   public view?   What is the flexibility (if it exists) of the deletion procedure? Is it possible to erase the traces of my former association with the site. Can I request any of the following be done?     4) All of my questions and answers be deleted from public view?      5) Some of my questions and answers be deleted? If yes what criteria?      6) Can I have all of my contributions be assigned to some other   account which runs as sort of a "collective account" and already has   tons of content so that it doesn't happen like my contributions just   move to some other url but still stay grouped under that location and   therefore that location can be discovered as representing my account?      7) Can I have my contributions dispersed through several "collective   accounts?      8) Can my current public name be removed from any comments to my posts   so that my posts cannot be identified later as once belonging to me?   Please be polite with me. I wouldn't have asked if I have found the answer in the other questions. The other questions are also old and it is not clear what the situation is as per 05/2012.  UPDATE (for those interested):  I asked the Stackoverflow team about the procedure. Apparently none of the points 2-8 I asked above are possible, only 1 will take place. The link to my user page from my contributions is going to be removed and replaced with non-clickable text of an anonymized user name like "user1234". Whether this number is the same as my current user_id is not clear. That's all the information I got.  I'm going now to rename myself to "Anonymous" then request the account deletion. Those interested in the procedure will witness the exact result on this page.  Farewell. Ok, so PHP isn't the best language to be dealing with arbitrarily large integers in, considering that it only natively supports 32-bit signed integers. What I'm trying to do though is create a class that could represent an arbitrarily large binary number and be able to perform simple arithmetic operations on two of them (add/subtract/multiply/divide).  My target is dealing with 128-bit integers.  There's a couple of approaches I'm looking at, and problems I see with them. Any input or commentary on what you would choose and how you might go about it would be greatly appreciated.  Approach #1: Create a 128-bit integer class that stores its integer internally as four 32-bit integers. The only problem with this approach is that I'm not sure how to go about handling overflow/underflow issues when manipulating individual chunks of the two operands.  Approach #2: Use the bcmath extension, as this looks like something it was designed to tackle. My only worry in taking this approach is the scale setting of the bcmath extension, because there can't be any rounding errors in my 128-bit integers; they must be precise. I'm also worried about being able to eventually convert the result of the bcmath functions into a binary string (which I'll later need to shove into some mcrypt encryption functions).  Approach #3: Store the numbers as binary strings (probably LSB first). Theoretically I should be able to store integers of any arbitrary size this way. All I would have to do is write the four basic arithmetic functions to perform add/sub/mult/div on two binary strings and produce a binary string result. This is exactly the format I need to hand over to mcrypt as well, so that's an added plus. This is the approach I think has the most promise at the moment, but the one sticking point I've got is that PHP doesn't offer me any way to manipulate the individual bits (that I know of). I believe I'd have to break it up into byte-sized chunks (no pun intended), at which point my questions about handling overflow/underflow from Approach #1 apply. The thing with these tags is, you can always (or almost always) use a specific product/technology/language tag which would render the company-name tag completely redundant (see Appendix C for sloppy non-proof). For example, what relevant information would  tag add to  already tagged with ? What would  tag add to  already tagged with ? Another way of looking at this: how much useful information do you lose if you don't tag a question with a company name? Note: I'm not talking about brand/company/organizations like , , ,  because those names also refer to websites/web services with which you can interact via API or from which you can extract data. Appendix C: The &quot;about&quot; test The relationship between a tag and a question can be expressed as such  This question is about [tag]  In case of ,  This question is about [microsoft]  Fair enough. But in reality people are using these company-name tags like this  This question is about something developed by or related to [microsoft]  But if that's the case, you might as well just go with  This question is about [something developed by or related to microsoft]  where  can be , , ... you get the idea. When I run LaTeX file its gives error as  "! Package minted Error: You must have pygmentize installed to use this package", you must have pygments installed. when I go through minted documentation minted itself required pygments, so Tex friend How do I install this pygment, I don't know the procedure to install. Please guid me or any link pygement.exe file iPhone Apps section not showing up in iTunes There are a number of services that will scan a website for vulnerabilities and allow the display of a trustmark image if the site passes their tests.  This allegedly increases the level of trust a potential client feel toward the website and therefore makes them more likely to complete a purchase.    McAfee claims it has studies which show significant increases in sales. Should I take their word for it?  What sort of increase should I really expect, if any?  Are these types of services worth the prices they charge? Does anyone know how to remove this dead bolt that has no screw heads to use a screw driver? Here's a picture below. I want to show that any two disjoint compact sets in a Hausdorff space $X$ can be separated by disjoint open sets. Can you please let me know if the following is correct? Not for homework, just studying for a midterm. I'm trying to improve my writing too.  My work:  Let $C$,$D$ be disjoint compact sets in a Hausdorff space $X$. Now fix $y \in D$ and for each $x \in C$ we can find (using Hausdorffness) disjoint open sets $U_{x}(y)$ and $V_{x}(y)$ such that $x \in U_{x}(y)$ and $y \in V_{x}(y)$. Now the collection $\{U_{x}: x \in C\}$ covers $C$ so by compactness we can find some natural k such that  $C \subseteq \bigcup_{i=1}^{k} U_{x_{i}}(y)$   Now for simplicity let $U = \bigcup_{i=1}^{k} U_{x_{i}}(y)$, then $C \subseteq U$ and let $W(y) = \bigcap_{i=1}^{k} V_{x_{i}}(y)$. Then $W(y)$ is a neighborhood of $y$ and disjoint from $U$.  Now consider the collection $\{W(y): y \in D\}$, this covers D so by compactness we can find some natural q such that $D \subseteq \bigcup_{j=1}^{q} W_{y_{j}}$.  Finally set $V = \bigcup_{j=1}^{q} W_{y_{j}}$, then $U$ and $V$ are disjoint open sets containing $C$ and $D$ respectively.   What do you think? When you hit up a site, such as cnn.com, does the TCP/IP packet contain the MAC address of your network card?  Could cnn.com theoretically record my MAC address? I add video files to the video editor,  Both audio and video channels appear but only the audio plays.  The video is just a white box.  Anyone know what is wrong.  Added video in WMV.  Also tried MP4 same thing. Can I check if the C# compiler inlined a method call? I'm writing an XNA game where I do per-pixel collision checks. The loop which checks this does so by shifting an int and bitwise ORing and is generally difficult to read and understand.  I would like to add private methods such as private bool IsTransparent(int pixelColorValue) to make the loop more readable, but I don't want the overhead of method calls since this is very performance sensitive code.  Is there a way to force the compiler to inline this call or will I do I just hope that the compiler will do this optimization?   If there isn't a way to force this, is there a way to check if the method was inlined, short of reading the disassembly? Will the method show up in reflection if it was inlined and no other callers exist?  Edit: I can't force it, so can I detect it? Zero-pad digits in string I need to cast single figures (1 to 9) to (01 to 09). I can think of a way but its big and ugly and cumbersome. I'm sure there must be some concise way. Any Suggestions When is it appropriate to use SQL Server Developer Edition? Do you use SQL Server Developer Edition on server-class machines in DEV and STAGING environments?  I am working on a large project where (if it passes the proof of concept stage) we will have several large geographically distributed enterprise class database servers running SQL Server 2012 Enterprise Edition.  We will have a production environment will initially have 3 servers, our Staging environment will have a minimum of 3 servers, and our Development environment will have 1 server (hosting three instances).  I was under the impression that we would only need to acquire Enterprise licences for the actual production servers, and we could get by with developer edition in our developer and staging environments because they are not "production".  Other sources have told me that we need to have an enterprise licence on all of the above mentioned machines, and that the developer edition is only meant for a single developer on their workstation.  Since developer edition has all the sexy features of Enterprise, I can't really see the value of it on a workstation class machine, especially for developing and testing the type of high availability system we are building.  If we have to fork out Enterprise licences for a dev server, that will just about kill our proof of concept stage, thus killing the project.  Forcing an enterprise licence on a staging environment will make management just want to skip staging altogether. Program does not wait before output Tony's Private life in Spider-Man: Homecoming I really don't know how to do this in terms of spoilers, but here goes...  After watching the end of Homecoming, it appears that       Tony and Pepper are very much back together.   Is there any in-universe explanation for this change from the way the universe seemed to be in Civil War?