Can $\sin(xy)$ be written in terms of trigonometric functions of only $x$ or $y$? I am tempted to say yes, because the double- and half-angle formulae exist, and these would be special cases of $\sin(xy)$.  I first looked at the Taylor series,  $$\sin(xy) = \sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}y^{2n+1}}{(2n+1)!} = xy-\frac{x^3y^3}{6}+\frac{x^5y^5}{120}-\cdots$$  but as far as I know, neither $x$ nor $y$ can come out of the sum since they are not constants.
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 $y$.
Ubuntu 18.10 Cosmic not starting GNOME session on bootup how to bring up the menu at the bottom left hello everyone I have a little problem, how do I bring up the menu in the circle? when I select another object the menu becomes non-existent Do strong and weak interactions have classical force fields as their limits? Electromagnetic interaction has classical electromagnetism as its classical limit. Is it possible to similarly describe strong and weak interactions classically? Why doesn't the Java SDK installer set JAVA_HOME? I always wondered why I have to manually set the JAVA_HOME environment variable after installing the Java SDK.     JAVA_HOME=c:\Program Files\Java\jdk1.6.0_12   Visual Studio at least provides a batch file to set these kind of environment variables:     call "c:\Program Files\Microsoft Visual   Studio 9.0\VC\vcvarsall.bat"   Does Java have something similar? I'm trying to make a build script that should simply work after installing the Java SDK. I don't want people to mess with environment variables on their PC. Does the existence (now proved) of gravitational waves imply the existence of Gravitons? I studied the theoretical part about the Gravitational waves in General Relativity (linearization of gravity and small perturbations of the metric and so on).  But I was wondering about: since electromagnetic radiation is composed/carried by Photons (or better:the EM force), shall gravitational waves be composed/carried by Gravitons?   In the end:do gravitational waves imply the existence of gravitons? Or it's something unrelated and off topic? Tried to install both Windows and Ubuntu, now neither will boot Hardware:  I've got an SSD (c),an internal HDD (d), an external HDD (e) and a USB flash drive (f).   How this all went down:   I downloaded Ubuntu and used a program to install it on my (F) drive. I reset my computer, ran the (F) drive for boot, and installed Ubuntu. I reset my computer, and it forcibly automatically booted into Windows 10 instead of Ubuntu. I went into Windows 10, and used a program called Easy BCD, by NeoSmart. I followed their guide, but after step 6, it wasn't the same for me. Their guide said that upon rebooting, I should see a grub screen that lets me choose between Windows/Linux and a few other options, however my computer would only boot into Windows 10.  I decided to follow the rest of the guide regardless, however by the end it would let me boot 'windows' or 'linux', but the linux option would always break. It went to a screen with lots of commands, that I thought -may- be a barebones Linux or something, but it definitely wasn't Ubuntu.  I decided to reinstall Ubuntu using the USB flash drive. Upon resetting, no matter what hard drive or format I select for booting (besides the F drive), it says An operating system wasn't found. Try disconnecting any drives that don't contain an operating system. I can boot the F drive and run Ubuntu, but it's a one time version that doesn't save any of my data. When I went into the installation, it still says that I already have Ubuntu installed on the 4TB HDD and that I have Windows 10 installed on the SSD, however I cannot seem to boot into either of them.   TLDR: Windows 10/Ubuntu are both installed, Windows on the internal SSD and Ubuntu on the external HDD. Ubuntu's installation program on the (F) drive recognizes them, however, when I try to boot anything it says An operating system wasn't found.   When I press F11 and force it to boot from the SSD it says:  error: no such device: 83b6c64d-9a2c0467b-a02c-6374ee6ac3dc.  error: unknown filesystem. Entering rescue mode. . .  grub rescue&gt; _   and it will allow me to type some commands. Difference between "does have" and "has" Possible Duplicate:                 What is the difference between does have and has? For example, compare she does have a car and she has a car. Applying graduated renderer in PyQGIS? Just trying to run the following code already given in the pyqgis cookbook to create graduated colors symbology (as in example with 2 classes)  I define the source info for the shapefile but when I run execute the code within my test-plugin, it just adds the layer without classifiying. Just adds as usual, all polylines appear same.  Any solution or working code/method?  link to code in cookbook   from qgis.core import  (QgsVectorLayer, QgsMapLayerRegistry, QgsGraduatedSymbolRendererV2, QgsSymbolV2,QgsRendererRangeV2)  myVectorLayer = QgsVectorLayer('C:/my_test_shape_file.shp', 'test_shp_file', 'ogr')  myTargetField = 'target_field' myRangeList = [] myOpacity = 1  # Make our first symbol and range... myMin = 0.0 myMax = 50.0 myLabel = 'Group 1' myColour = QtGui.QColor('#ffee00') mySymbol1 = QgsSymbolV2.defaultSymbol(myVectorLayer.geometryType()) mySymbol1.setColor(myColour) mySymbol1.setAlpha(myOpacity) myRange1 = QgsRendererRangeV2(myMin, myMax, mySymbol1, myLabel) myRangeList.append(myRange1)  #now make another symbol and range... myMin = 50.1 myMax = 100 myLabel = 'Group 2' myColour = QtGui.QColor('#00eeff') mySymbol2 = QgsSymbolV2.defaultSymbol(myVectorLayer.geometryType()) mySymbol2.setColor(myColour) mySymbol2.setAlpha(myOpacity) myRange2 = QgsRendererRangeV2( myMin, myMax, mySymbol2, myLabel) myRangeList.append(myRange2)  myRenderer = QgsGraduatedSymbolRendererV2('', myRangeList) myRenderer.setMode(QgsGraduatedSymbolRendererV2.EqualInterval) myRenderer.setClassAttribute(myTargetField) myVectorLayer.setRendererV2(myRenderer) QgsMapLayerRegistry.instance().addMapLayer(myVectorLayer)      @Kelly's code works perfectly.   However I noticed that both the code (one I have typed in my first message and yours in the 2nd message) DO NOT work in QGIS v1.7.3 but WORKS in QGIS v1.8.0. I think this was a bug(?) that already resolved in v1.8.0  And one more question;  Do you have any sample code for (natural breaks, Jenks) classification of the "numeric_attribute_field" based on given number of classes (i.e. no of classes will be a variable in the code, say "n", and I will pass that from the plugin GUI i.e. n = spinBox.value() ) Could it be a structure of "How many -singular noun- is there?" We had an argument in our collage if it's possible to have such structure:     "How many -singular noun- is there?"   For example:      How many chair is there?   The claim of the supporters: This structure should be used in case that t is not known whether there is one one or more chairs there. While just in case that it is known certainty that there are more than one chair, it should be "how many chairs are there".    The claim of the opponents: This structure cannot be since using the phrase "how many" always should be with a plural noun.  Both sides are not native English speakers, then I would like to know what is the truth. Is there a word for someone who won't think critically? Is there a word for someone who won't think through issues and instead comes to others hoping that they will think for them and solve problems for them?  A word that is more specific to a lack of critical thinking skills than "lazy"? What makes SHA-256 secure? For example, RSA relies on a mathematically hard problem, factoring, while ECDSA or similar rely on discrete logarithm problem.  What makes SHA-256 and similar hash functions, of the same family, secure against pre-image and collision attacks? What's the math behind it? Remove vote UI elements beside own question and answer posts Currently, the system makes you think you can upvote/downvote your own question/answer but then you get notified that you can't. Why have the up/down vote buttons there then?   In general, if some operation is not allowed, then don't put UI for that up or make it disabled so it is apparently disabled - no post operation error panels please?  ps: hope this is not a dupe, but if so, just comment it and I will remove  (+a joke: a user notification panel message: I deleted this file. Is this a problem? Yes/No.) Retrieve Voice Memos from device? iOS seems to refuse to allow me to email "long" voice memos, and I'm unaware of any way to extract them from iTunes backups of my phone. How can I retrieve entire voice memos that aren't really short? Let me use the following example to explain my question: public string ExampleFunction(string Variable) {     return something; }  string WhatIsMyName = &quot;Hello World&quot;; string Hello = ExampleFunction(WhatIsMyName);  When I pass the variable WhatIsMyName to the ExampleFunction, I want to be able to get a string of the original variable's name. Perhaps something like: Variable.OriginalName.ToString()  Is there any way to do this? ［I'm Done］ button is available after ［Save Edits］ even when nothing was edited in First Posts queue Subscript labels in QGIS legends Is it possible to include subscript and superscript text in the title of a legend item in QGIS ?  I have tried using &lt;sub&gt;some text&lt;/sub&gt; but this just appears in the title with no change to the contents.  I am looking to have layer titles such as   Annual NO2 concentration (μg/m3)  The superscript is not a big issue, as I can use the character map to get a 3, but I don't know how to get the subscript. Where to declare environment variables? What are correct places for:   Global environment variables meant to affect all users? User-specific environment variables? Based on the following interchange between Harry and Dumbledore in the pivotal &quot;King's Cross&quot; chapter of Deathly Hallows, I was wondering when exactly Dumbledore put together that Harry would have to die in order to bring about Voldemort's ultimate demise:  &quot;And you knew this? You knew--all along?&quot; &quot;I guessed. But my guesses have usually been good,&quot; said Dumbledore (Deathly Hallows, 599, hardcover).  The conversation that spurred Harry's question includes material dating back to Halloween '81, but also specifically mentions the events of the graveyard from GoF. So, &quot;knowing all along&quot; could refer to everything (i.e., Harry must die, Voldemort ignorantly tethered him to life, etc.) or just that Voldemort tethered him to life. Other important information pertaining to this question can be found in OOTP, when Dumbledore confirms (if he didn't already know) that Harry contains part of Voldemort's soul. He uses the silver instrument to find out that Harry is just like Nagini, having the ability to share mind and soul with Voldemort. He also says the following:  &quot;I guessed, nearly fifteen years ago,&quot; said Dumbledore, &quot;when I saw the scar upon your forehead, what it might mean. I guessed that it might be the sign of a connection forged between you and Voldemort&quot; (Order of the Phoenix, 826-827, hardcover).  While confirming that he guessed that Harry might share a connection with Voldemort from the very beginning, Dumbledore does not confirm with this statement that he knew Harry would ultimately have to give up his life. In Philosopher's Stone, when Dumbledore drops off Harry on the Dursley's doorstep, we know he is not entirely truthful with McGonagall.  McGonagall asks: &quot;How in the name of heaven did Harry survive?&quot; &quot;We can only guess,&quot; said Dumbledore (Philosopher's Stone, 12, hardcover).  Of course, this was a lie. Dumbledore already knew how Harry survived, otherwise he wouldn't be delivering Harry to Lily's sister to extend the blood sacrifice. By proxy, he also knows that Voldemort will return b/c of Horcruxes, otherwise he wouldn't be so worried about Harry. Finally, when asked in the same scene if he could remove the scar, Dumbledore says:  &quot;Even if I could, I wouldn't. Scars can come in handy&quot; (Philosopher's Stone, 15, hardcover).  So, did Dumbledore know at this very moment that Harry would have to die? Did he at least guess it to be true during this scene? We know he gains perfect confirmation in OOTP, but did he know it &quot;all along,&quot; as a particular reading of the King's Cross interchange in DH would have us believe? Eliminating non-integer solutions to $ab / (2\sqrt{ab} + a + b)$ Shuffle an array with python, randomize array item order with python What's the easiest way to shuffle an array with python?