If a department places no limit to the number of recommendation letters, how many are appropriate?  Are 5 OK? Are 7 too many?  Of course this depends on how strong the letters are - but let's assume that all of them are equally strong.
I know the minimum number of reference letters required at many U.S. PhD programs in the STEM fields is three -- and that it is also not uncommon for someone to submit a fourth letter.  What about a fifth letter?  Would a fifth letter of recommendation be perceived as "too many"?
Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle Prove that $\sin(2A)+\sin(2B)+\sin(2C)=4\sin(A)\sin(B)\sin(C)$ when $A,B,C$ are angles of a triangle   This question came up in a miscellaneous problem set I have been working on to refresh my memory on several topics I did earlier this year. I have tried changing $4\sin(A)\sin(B)\sin(C)$ to  $$4\sin(B+C)\sin(A+C)\sin(A+B)$$ by making substitutions by reorganizing $A+B+C=\pi$. I then did the same thing to the other side to get $$-2(\sin(B+C)\cos(B+C)+\sin(A+C)\cos(A+C)+\sin(A+B)\cos(A+B))$$ and then tried using the compound angle formula to see if i got an equality. However the whole thing became one huge mess and I didn't seem to get any closer to the solution. I am pretty sure there is some simpler way of proving the equality, but I can't seem to figure it out. Maybe there is a geometric interpretation or maybe it can be done using just algebra and trig. Any hint's would be appreciated (I would prefer an algebraic approach, but it would be nice to see some geometric proofs as well) I have a string that contains both double-quotes and backslashes that I want to set to a variable in Python. However, whenever I try to set it, the quotes or slashes are either removed or escaped. Here's an example:  &gt;&gt;&gt; foo = 'baz "\"' &gt;&gt;&gt; foo 'baz ""'   So instead of baz "\" like I want I'm getting baz "". If I then try to escape the backslash, it doesn't help either:  &gt;&gt;&gt; foo = 'baz "\\"' &gt;&gt;&gt; foo 'baz "\\"'   Which now matches what I put in but wasn't what I originally wanted. How do you get around this problem? How do I detect what .NET Framework versions and service packs are installed? A similar question was asked , but it was specific to .NET 3.5. Specifically, I'm looking for the following:   What is the correct way to determine which .NET Framework versions and service packs are installed? Is there a list of registry keys that can be used? Are there any dependencies between Framework versions? java.lang.OutOfMemoryError: PermGen space Exception Java: Sorting user-input array upon return "Forgot" vs "Forget" Is the following correct, or is there more to it?  "I forgot his name" — I knew his name, but I forgot it.  "I forget his name" — I keep forgetting his name. Where using "forget" basically means that you tried to remember this information before as well, but you couldn't then either. QGIS 2.99 "qgis._core import error: DLL load failed" I am trying to run a standalone application. However, it does not run the program because of mentioned error in title. Here is my code:  from qgis.core import * from qgis.gui import * from qgis.utils import * from PyQt5.QtCore import * from PyQt5.QtGui import *  app = QgsApplication([], True) path = "C:/OSGeo4W64/apps/qgis-dev" app.setPrefixPath(path, True) app.initQgis()  canvas = QgsMapCanvas() title = "PyQGIS Standalone Application Example" canvas.setWindowTitle(title) canvas.setCanvasColor(Qt.white)  layer_info = 'LineString?crs=epsg:4326' layer = QgsVectorLayer(layer_info, 'MyLine', "memory") pr = layer.dataProvider() linstr = QgsFeature() wkt = "LINESTRING (1 1, 10 15, 40 35)" geom = QgsGeometry.fromWkt(wkt) linstr.setGeometry(geom) pr.addFeatures([linstr]) layer.updateExtents() QgsProject.instance().addMapLayer(layer)  canvas.setExtent(layer.extent()) canvas.setLayers([layer]) canvas.zoomToFullExtent() exitcode = app.exec_() QgsApplication.exitQgis() sys.exit(exitcode)   When I run it, I get this error:  Traceback (most recent call last):   File "C:/Users/DELL/Desktop/MyMap/myMap.py", line 1, in &lt;module&gt;     from qgis.core import *   File "C:\OSGeo4W64\apps\qgis-dev\python\qgis\__init__.py", line 72, in &lt;module&gt;     from qgis.core import QgsFeature, QgsGeometry   File "C:\OSGeo4W64\apps\qgis-dev\python\qgis\core\__init__.py", line 34, in &lt;module&gt;     from qgis._core import *   ImportError: DLL load failed: The specified module could not be found.  Python version is: 3.6 PyQt   version is: PyQt5.9 Windows 10 - 64bit  I also add many path:  %SystemRoot%\system32 %SystemRoot% %SystemRoot%\System32\Wbem %SYSTEMROOT%\System32\WindowsPowerShell\v1.0\ C:\OSGeo4W64 C:\OSGeo4W64\apps C:\OSGeo4W64\apps\gdal-dev C:\OSGeo4W64\apps\gdal-dev\bin C:\OSGeo4W64\apps\gdal-dev\pymod3\Lib\site-packages C:\OSGeo4W64\apps\grass\grass-7.2.2\bin C:\OSGeo4W64\apps\Python36 C:\OSGeo4W64\apps\Python36\Scripts C:\OSGeo4W64\apps\Python36\DLLs C:\OSGeo4W64\apps\Python36\lib\site-packages C:\OSGeo4W64\apps\qgis-dev C:\OSGeo4W64\apps\qgis-dev\bin C:\OSGeo4W64\apps\qgis-dev\lib C:\OSGeo4W64\apps\qgis-dev\python C:\OSGeo4W64\apps\qgis-dev\python\qgis C:\OSGeo4W64\apps\Qt5 C:\OSGeo4W64\apps\Qt5\lib C:\OSGeo4W64\bin C:\OSGeo4W64\apps\qgis-dev\python C:\OSGeo4W64\apps\Python36\lib C:\OSGeo4W64\include C:\Users\Mustafa Uçar\AppData\Roaming\QGIS\QGIS3\profiles\default C:\Users\Mustafa Uçar\AppData\Roaming\QGIS\QGIS3\profiles\default\QGIS   How can I fix this error? A property of every real polynomial. Question: For every polynomial $P(x)$ of degree at n we have $$\sum_{i=0}^{n+1}(-1)^i\binom{n+1}{i}P(i)=0$$  Well I was inducting on degree of the polynomial, for $n=0$ its true and I got a bit confused in the inductive step, actually I cant figure the inductive step.   Is this property (if true) a famous one?  Thank you for your help. Show that a matrix $A$ is singular if and only if $0$ is an eigenvalue. I can't find the missing link between singularity and zero eigenvalues as is stated in the following proposition:  A matrix $A$ is singular if and only if $0$ is an eigenvalue.  Could anyone shed some light? Assign VNC password using script I am using a script to automatically set up a computer. I need to assign a password to the vnc server for the user, which is normally done using the vncserver command. However, it prompts for the user to enter and re-enter their password, neither of which the script is capable of doing. So, how can I set up the VNC password without an interactive prompt?   Thanks. Why was my "shopping list" question closed? My question was closed for being a "shopping list question", even though it was very much related to the topic of the site. Why? Merging bib files from within LaTeX I am an Indian citizen residing in the United States. My wife and I are planning to travel to the UK for a week. We plan to apply for the UK standard visitor visa from the US. I was filling out the application and I have a few questions.   Travel History - The application asks for travel history to the United States, Canada, Australia and New Zealand. Considering that we're resident aliens in the United States, should we consider all the times we've traveled back to the US from India? Travel History - Should we also list out the times we've traveled to India from the US? Cost of the trip - Hypothetically if we plan to spend a total of ~4,000 USD on this trip, should I split up the cost and write down 2,000 USD on my application and 2,000 USD on my wife's application? Income - From what I've read on the forum, my wife must say that "someone else" i.e. me - is paying for her costs and mention my GWF number in the remarks on her application. Would that be sufficient considering that I'll be sending all income documents with my application? Savings - We have joint savings accounts. Should I mention the same amount on both our applications? Posting the documents - Is the up-to-date advice to send all documents in one packet? Or would sending them in separate packets at the same time be better? My thought was to annotate my wife's packet with my GWF number if we were to send them separately. Bank statements - We get e-statements. Would printing those out be enough? Restricting a user to a certain directory on SSH login I am having trouble restricting a user to a fixed directory when they connect with SSH  I want the user restricted to /var/www/home  however I connected myself and they are free to move about the system  I set the home directory to /var/www/home so they go there upon login however this doesnt restrict them.  I have tried a few things from suggestions in Google but nothing so far seems to work / do what I want  For example I have changed the chmod of www to 755 and root owned but still no effect How to properly typeset multiletter quantities, e.g. Re or Nu (Reynolds, Nusselt numbers) postfix discard or change Message-ID header based on another header line I want to change the Message-ID header or better yet, discard it. I know everyone is going to ask "Why would you want to do that?" because I know it sounds weird to even write it.  Well I am developing a system that saves an incoming e-mail in EML format, changes it and sends it to the user, at some point the user can request to have the original EML sent to him/her. The issue is that when I re-send the EML it has the same Message-ID header which causes exchange servers to drop the re-send.  Of course I am hoping there is some option through postfix to do this, otherwise I will need to parse and change the EML but I rather not do that.  I tried to do this with "header_checks" and it works BUT I need to be able to do this conditionally, meaning only on a specific occasion, which I can identify by looking at the other headers, I want to remove the Message-ID. How to write a survey paper? Multiple "order by" in LINQ I have two tables, movies and categories, and I get an ordered list by categoryID first and then by Name.  The movie table has three columns ID, Name and CategoryID. The category table has two columns ID and Name.  I tried something like the following, but it didn't work.  var movies = _db.Movies.OrderBy( m =&gt; { m.CategoryID, m.Name }) How did the Death Star move? In the Star Wars saga, one of the most iconic weapons of the Empire was the Death Star: a self sustained weapons platform the size of a moon, with the capacity to destroy a planet. in A New Hope, we see the first Death Star in close proximity to the planet Alderaan, holding the planet hostage as the Empire tries to deduce the location of the Rebel Alliance's headquarters. In Return of the Jedi, we see the second Death Star in "dry dock" over the forest moon of Endor as it is being constructed.  My question is: How did the Death Star move? I imagine a spherical space station the size of a moon would be utterly massive, requiring a massive amount of thrust to move it. If it was constructed in low orbit of an inhabited planet/moon, the gravitational field would be immense, consisting of the planet/moon's gravity, and the Death Star's own considerable gravity. Breaking that gravitational field would be a massive task, yet we see no thrusters or engines on either Death Star. Were they hidden, stored under the station's surface while not in use? Was the station towed by another vessel? Is speed of light and sound rational or irrational in nature? Just as circumference of circle will remain $\pi$ for unit diameter, no matter what standard unit we take, are the speeds of light and sound irrational or rational in nature ?   I'm talking about theoretical speeds and not empirical, which of course are rational numbers.