I am a Masters' student applying from India to US graduate schools for a Ph.D. in Mathematics(some of them are top departments like Stanford, MIT, UCLA, Berkeley, Illinois Urbana-Champaign). I wish to work at the interface of Analysis and Differential Geometry. Hence, I wanted to explore Geometric Analysis and took extensive differential geometry courses, a winter course on Geometric Measure Theory and did self-study on Geometric Analysis. However, as we did not have anyone in our department working in these areas, I did my Masters' thesis (ongoing) on noncommutative geometry. For this, I studied operator algebras, followed by compact quantum groups and am currently working with quantum isometries of noncommutative manifolds.  I am a little confused about what I should include or exclude from my Statement of Purpose(SOP). I have written that I am interested in both Analysis and Geometry and have mentioned: (1) Geometric Analysis, and (2) Operator Algebras and Noncommutative Geometry, as the fields that interest me. Then I explain my background in both. Is this a valid approach? Or will it give the impression that I don't have a well formed research interest and work against my selection at the graduate school?     While I was towards the end of my school life, I had a publication modelling a small physical problem. I am currently developing it into pedagogic material with a retired professor in the USA. Also, during my undergraduate studies I had two more publications in applied mathematics (control theory). I am including the first one in my SOP to illustrate my motivation in research from younger days. But should I include the other publications? I am afraid that I might run the risk of giving the committee the false impression that I am too inclined in applied mathematics.
I want to change my field from molecular biophysics to theoretical ecology. While they are basically required some same basic background (physics, math, programming), the rest is quite irrelevant. In my current draft of this part, about a half is about this the problem and the result of the work, another half is for my contribution and what I've learnt. Having said that, should I focus less on describing the work and leave space for the thing that will be useful for the next research?  Below is copy from .     What have you already done? What problems have you solved, or at least worked on? What independent projects have you been part of? What were your key contributions? What did you learn? What did you teach the world? How do your results compare to what was already known? What original ideas are you most proud of? Be specific, technical, credible, and confident (but not arrogant). Refer the reader to your web page for more details. Have a web page with more details: preprints, project reports, source code, videos, etc.
To consume REST services there is a lot of boilerplate code needed:   Build Apex Objects to model the responses Code to deserialize REST responses to those classes Handling of HTTP requests, responses and errors ...   Did anyone come up with patterns or best practices codified into an open source library or framework that one could just reuse in a similar project without reinventing the callout-wheel over an over again?  Like the zillions of trigger-frameworks that exist as repos on Github... I sometimes hear stories where people compare their feelings in winter in different places in the world.  It goes like     in city X the temperature was the same as in city Y, but the humidity made me feel much colder...   or      oh well, -20°C would be cold, but the humidity was low, so it felt OK   so it implies that humidity somehow makes it feel colder. I am talking about temperatures below freezing (-30...0°C).  Does this have any physical explanation, or is it some sort of ? What focal length or range of focal lengths are usually used for portraits?   What is used by professionals? Why would one select a certain focal length? What are the advantages and disadvantages of different choices, and are there certain situations where an unusual choice might be better? List all the directories under the current one cat files with directory Is there a command to show the directory/file name when cat files?  For example: assume two files f1.txt and f2.txt are under ./tmp  ./tmp/f1.txt ./tmp/f2.txt    Then when I do cat ./tmp/*.txt, only the content of files will be shown. But how to firstly show the file name, then the content?, e.g.:   (The needed command):  ./tmp/f1.txt:    This is from f1.txt  and so on  ./tmp/f1.txt:  This is from f2.txt ...   Is there a command to do it? (There seems to be no option for 'cat' to show the file names) Example of modules that are projective but not free; torsion-free but not free How do I make a bootable Ubuntu USB? How do I erase an iPhone? I had an iPhone on loan from a friend to try it out and now I need to give it back. How do I make sure that everything is erased from it? SMS, Calendar, Phone Logs, contacts? Show that every nonzero integer has balanced ternary expansion? Install software from packages, or build it from source? Browser can't access/find relative resources like CSS, images and links when calling a Servlet which forwards to a JSP Python re.sub with a flag does not replace all occurrences How do I recover my Google account (or Gmail) password or username? I have forgotten the password or username for my Google account, or I'm unable to access my account for other reasons.  How can I recover my account?  (This Q&amp;A is meant as a general description for all Google and Gmail username/password questions. See: ) Proof that all abelian simple groups are cyclic groups of prime order A deck of 52 cards is shuffled, and we deal a bridge of 13 cards. Let $X$ be the number of aces and $Y$ be the number of spades. Show that $X$, $Y$ are uncorrelated.  Here is what I did:  $Cov(X,Y) = E[XY]-E[X]E[Y]$   uncorrelated means $Cov(X,Y) = 0$, hence  $E[XY]=E[X]E[Y]$   $E[X] =  \sum_{k=0}^{k=4} k \frac{\dbinom{4}{k} \dbinom{48}{13-k}}{\dbinom{52}{13}} $   $E[Y] =  \sum_{k=0}^{k=13} k \frac{\dbinom{13}{k} \dbinom{39}{13-k}}{\dbinom{52}{13}} $   Are the summations above correct? and how do I calculate $E[XY]$? bash: warning: shell level (1000) too high, resetting to 1 Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles. (using disk/washer)  I saw no example of this problem anywhere.. I saw an example how to solve it without calculus but I'm afraid that'll be deductions on my test.. I have the volume of a cylinder and the method of area between two curves but this is out of my reach.... I don't know what I can try... How can I remove a key from a Python dictionary? I'd like to create a Yahoo! account. The signup page won't let me proceed without entering a cell phone number, and it won't recognize my Google Voice number as a valid cell phone number. Is there a workaround to be able to create a Yahoo! account without having a cell phone that can receive SMS messages? Is there a function to round a float in C or do I need to write my own?       float conver = 45.592346543;   I would like to round the actual value to one decimal place, conver = 45.6.