I'm trying to simplify an expression of the form $\sin(f(t)\cos(\omega t + \phi))$.  The nonlinear ordinary differential equation I'm working on involves terms like this (where the equation is to be solved for the unknown function $f(t)$).  Of course, one way would be to use the first several terms of the Taylor series for the sine.  But I'm curious if there is any analytical expression for an angle product like $\sin(\alpha \beta)$.
Can we express a trigonometric function for the product of two angles as a function of trigonometric functions of its factors?  For example: Is there a formula for $\sin(xy)$ as a function of $\sin x$ and $\sin y$ or other trigonometric functions of $x$ and $y$.
Flex HTTPservice and POST, Sending Files? I use a basic Post to send data to a Django server.  The data consists of a base64 encoded 640*380 PNG image dynamically created by the flex component.  &lt;mx:HTTPService id="formSend" showBusyCursor="true"      useProxy="false" url="http://127.0.0.1/form/"      method="POST" result="formSentConfirmation(event)"    fault="formSendingFailed(event)"/&gt;    private function sendForm(url:String, message:String, meteo:Number):void {     formSend.url = url;     var params:Object = { message: message, image_data: getEncodedImage() };      snapButton.label = "sending ...";     formSend.send(params); }   On the server side i can see that the data is in the request.POST not in request.FILES. That means the image is not send as a File with multiencode HTTP.   Will i get into trouble on a real server ? since the limit is 200k for urlencoded POST var. How to make HTTPservice send the data as a file? Any other solutions?   Thanks Show function is continuous at origin $$ \begin{cases}xy^2\sin\left(\frac 1 y\right), \text {if $y$ $\neq$ 0} \\ 0, \text{if $y$ = 0} \end{cases}$$  I'm not sure how to work out this limit as $x$ goes to infinity. If it were just the y part, I know that $y^2$ goes to 0 and $\sin$ is a limited function, so it has to go to 0, but I don't know what to do with the $x$ term. How many bandits must agree? Problem  A group of $n~(1 \leq n \leq 30)$ bandits hid their stolen treasure in a room.  The treasure needs to be locked away.  The bandits want to ensure that at least $k~(1 \leq k \leq n)$ of the bandits must agree in order to retrieve the treasure.  They have decided to place multiple locks on the door such that the door can be opened if and only if all the locks are opened.  Each lock may have up to n keys, distributed to a subset of the bandits.  A group of bandits can open a particular lock if and only if someone in the group has a key to that lock.  Given $n$ and $k$, how many locks are needed such that if the keys to the locks are distributed to the bandits properly, then every group of bandits of at least size $n$ can open all the locks, and no smaller group of bandits can open all the locks?  For example, if $n=3$ and $k=2$, only 3 locks are needed - keys to lock 1 can be given to bandits 1 and 2, keys to lock 2 can be given to bandits 1 and 3, and keys to lock 3 can be given to bandits 2 and 3.  No single bandit can open all the locks, but any group of 2 bandits can open all the locks.  My thoughts  At first glance, I thought it was just a simple combinations problem ($nCr$), and the provided test cases confirmed my suspicions: when $n=3$ and $k=2$, $\binom{3}{2}=3$.  Likewise, when $n=5$ and $k=3$, $\binom{5}{3}=10$.  However, two cases where this doesn't work are where $n=2$ and $k=1$ and $n=5$, $k=4$ (it gives $2$ and $5$, respectively, when the correct answer according to the judging data should be $1$ and $10$).  Next, I bothered a colleague who's a math major, and he briefly glanced at it and suggested it might be a hypergeometric distribution problem.  However, if this is correct, I'm having trouble wrapping my head around it - all the examples I can find have 2 distinct categories, e.g. red and green marbles being drawn from an urn, and I don't see that here.  An example in the terms of this problem would be extremely helpful.  If this isn't correct, where do I go from here?  I've been spinning my wheels on it for quite a while.  Background  This is for homework for a CS class (I'm a CS major with 2 semesters of calc plus discrete).  I'd like to be able to understand the math behind it, because once I can understand it, I can program it.  This problem originally came from an ACM ICPC regional contest this year, but my professor has made it into a challenge homework problem. pylab.ion() in python 2, matplotlib 1.1.1 and updating of the plot while the program runs I'm trying to display models that have shading drawn or baked directly on the textures, but I can't figure out how to disable Blender's default scene lighting. If there are no lights in the scene, everything is black, but I want to see the textured objects fully bright. How do I disable the lighting?  NOTE: Just before posting this, I found the "Shadeless" material property, which does more or less what I want. However, is there a way to globally toggle the default lighting? Hook into the loop via a plugin, and output something after every X post? I have a plugin where I want to output ads after X number of posts on the home page. The home page is step 1, but things like archives should be possible too once I get the code for the home page.  How do I hook in to post loops and say something like "after every loop, increment a counter, and then if the counter = my number, output an ad". I can write all the logic for this code myself, but where to hook/implement my code is confusing. What are these fuse-looking cartridge case components on small, battery operated RF tranceivers? I have recently opened up 3 dead, battery-operated RF transceivers lying around, both wearable and key-chains. I've noticed that all three had components that looked like fuses (cartridge case), soldered down, that are quite large compared to the size of the circuit. See the pictures below. I'm not posting full circuit pictures because I don't know if I should, I can update if they are necessary.  Device 1:    Device 2:    Device 3:    I can't see why such a device will require a fuse, and I couldn't find any other common components manufactured in cartridge cases, maybe I'm not using the correct name. I don't see them on any applications notes or reference designs for the ICs used either.  I've tried to figure out how they are connected to the circuit, I don't have schematics and PCBs are more than 2 layers so they may not be complete, but see the schematics I've figured below.  Device 1:    Device 2:    Device 3:    Also, when I do a continuity test between the terminals of the components with a multimeter, resistance starts in ~50 Ohms range and goes down to short as I keep the probes on its terminals so my best guess is something for ESD protection, maybe Varistors or Thermistors?   Does anyone know what these components are and why they are there? Minecraft Empty Chests Command Block I am using the following Command to spawn a new chest at the given coordinates with stuff in it at each new round:  /setblock 264 4 997 minecraft:chest 0 replace {Items:[{id:274,Count:1,Slot:0},{id:275,Count:1,Slot:1}]}   The problem with that is that when replacing the old chest all the stuff in the old chest is thrown out. And that's pretty much not what I want. I need to find a way to replace a chest which might have stuff in it without throwing those things out.  Someone else said I should replace the given position for one tick with lava, which I tried. Works in 90% of the tries (in the other 10% the items from the chest are thrown to far away - out of the spawned lava block), but the problem with that is that I can't use it in wooden areas because it might burn everything down.  So what can I do to fix that problem, because otherwise my custom made Pv.P map is not gonna work?  Thanks very much! Disable drag and drop option for multiple file uploader These are the screenshots of multiple file uploader from the same SharePoint 2010 site. Some users are seeing the first and some are seeing the second one. Is it possible to disable the second view(drag and drop) by editing upload.aspx in the layouts folder?      Thank you Proving an entire, complex function with below bounded modulus is constant. How would I show that an entire function $f$ with the property $|f(z)| \geq 1$ must be constant?    I'm aware of Liouville's theorem, just not sure how to apply it here. No human players, just AI and have the ability to scroll around and see what the AIs are doing with the turn management being also automated, so the AIs just end turn when they are done. Oh, and all of this hopefully without any sort of fog of war.  This way, I can see how the AI manages resources and units and maybe learn a thing or two as a civ newb. Let $A$ be a $3\times 4$ and $b$ be a $3\times 1$ matrix with integer entries.Suppose that the system $Ax=b$ has a complex solution. Then which of the following are true? (CSIR December 2014)         $Ax=b$ has an integer solution.    $Ax=b$ has a rational solution.   The set of real solutions to $Ax=0$ has a basis consisting of rational solutions.   If $b$ is not equal to zero then $A$ has positive rank.      Is it possible to say that 1 and 2 are true as $Ax=b$ has a complex solution? If not  what we have to understand from the statement "the system $Ax=b$ has a complex solution " ? What about 3 and 4  ? Integrating Factor Method. Why Can We Set the Constant of Integration to Zero? My textbook gives a demonstration of the use of the integrating factor method for solving first order ordinary differential equations:  $\dfrac{dx}{dt} + p(t)x = r(t)$  $\implies g(t)\dfrac{dx}{dt} + g(t)p(t)x = g(t)r(t)$  $\dfrac{d}{dt}[g(t)x]$  We use the product rule and compare the expression to the left-hand side of the above equation:  $g(t)\dfrac{dx}{dt} + \dfrac{dg}{dt}x = g(t)\dfrac{dx}{dt}+g(t)p(t)x$  $\therefore \dfrac{dg}{dt} = g(t)p(t)$  The above equation is separable.  $\therefore \int\dfrac{dg}{g(t)} = \int p(t) dt$  $\implies ln[g(t)] = \int p(t)dt$  My textbook then says that, "We can set the constant of integration to zero -- any function that satisfies $\dfrac{dg}{dt} = g(t)p(t)$ will do for our purposes". However, it gives no explanation as to why this is the case. Why can we set the constant of integration to zero? Why is any function that satisfied $\dfrac{dg}{dt} = g(t)p(t)$ appropriate for our purposes?  I would greatly appreciate it if people could please take the time to elaborate on this. Apparently I inadvertently irked a moderator by creating a tag.  The moderator told me that my tag was "meta" and therefore not appropriate.  Having been a SO user for a long time (and a lurker on MetaSO), it was not clear to me what makes my tag "meta" and why meta tags would be inappropriate.  Many of the questions I see on SO aren't really about the topics in the title or the existing tag list.  For example, , isn't really about C++ or SetWindowText (or the Windows API as tagged).  It's simply somebody who had a bug, couldn't find it, and tried to outsource the debugging.  This seems to be a common category of question on SO.  So I started tagged questions like this with the label "findmybug".  Lots of frequent SO contributors regularly answer questions in this category rather than voting to close them as "Too Localized".  I took that as a cue that I should also try to answer rather than voting to close.  But leaving these questions open and tagged with "c++" and "winapi" even though the question isn't really about C++ or the Windows API seemed misleading.  Someone searching for answers on using SetWindowText from C++ probably isn't making exactly the same mistake and won't find that entry useful.  (In the end, the sample question was closed as "Too Localized", but this doesn't seem common for these types of questions.)  The Tags page says:     A tag is a keyword or label that categorizes your question with other, similar questions. Using the right tags makes it easier for others to find and answer your question.   It seems to me that the "findmybug" tag does group these similar questions together.  So from that perspective, it seems to fit.  Lots of users seem to enjoy answering these types of questions, so it makes sense they'd want a way to find them.  If you want to suggest that all the questions I would tag as "findmybug" should be closed as "Too Localized", that's fine.  But I'm trying to ask the more general question:  What makes a tag meta, and are meta tags inappropriate? What visa does my wife need to travel to the UK with me, if I'm going there on a business visa? I am resident of India and just got UK Business visa ( 6 months, multiple entry) .. I am going for a week on business trip. I want to take my wife along with me and at the end of trip.. I want to extend my trip for 3 days and we want to spend some time on leisure.   What visa I should apply for my wife.. visitor visa? should she mention in the visa form that she is going along with me and staying with me while I am on work and later we are spending time on leisure.   While filling my business visa form, there was a field asking if anyone else is going with me.. I selected no.. as there were no plans.  Also is it ok to spend time on leisure on UK Business visa Is a passport needed when flying within Mexico as a U.S permanent resident? Proof of Frullani's theorem How can I prove the Theorem of Frullani? I did not even know all the hypothesis that $f$ must satisfy, but I think that this are  Let $\,f:\left[ {0,\infty } \right) \to \mathbb R$ be a a continuously differentiable function such that $$ \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = 0, $$ and let $ a,b \in \left( {0,\infty } \right)$. Prove that $$ \int\limits_0^{\infty}  {\frac{{f\left( {ax} \right) - f\left( {bx} \right)}} {x}}dx  = f\left( 0 \right)\left[ {\ln \frac{b} {a}} \right] $$  If you know a more general version please give it to me )= I can´t prove it. Stone-Weierstrass theorem with $p(1/x)$ I am trying to prove that for a continuous $f\colon[1,\infty)\rightarrow\mathbb R$ and $f(x)\to a$ as $x\to\infty$ it could be approximated by $g(x)=p(1/x)$ where $p$ is a polynomial. Is there any evolutionary/physiological advantage to multi-lateral symmetry? Just wondering ; I get the impression a large number of multicellular organisms are multi-lateral in their physical structure. Why would evolution/mutation have retained multi-lateral symmetry in an organism? What advantage/s does symmetry provide an organism?  EDIT:  By multi-lateral symmetry I mean at-least bi-lateral symmetry (as in the case of many beings such as elephants, crabs and fishes), but perhaps more than bi-lateral too (starfish come to mind)  END EDIT: Output the common lines (similarities) of two text files (the opposite of diff)? Diff is a great tool to display the changes between two files. But how to display the similarities of two text files (while ignoring the differences)?  I.e. sample input:  a: Foo Bar X Hello World 42  b: Foo Baz Hello World 23   Pseudo output (something like this):  @@ 2,3 =Hello World   Just sorting both files and using comm is not enough, because in that case the line information is lost.