It has been said that when you try to jump you are exerting your force on the ground and the ground pushes you up.  I was wondering if this was an over-simplification or not.  If so, what's the full explanation?  Otherwise, I would like to know the derivation of this proof.  For clarification: If I jump and the ground is "pushing me up" (from a normal force), then assuming that this is not a simplification and is truly what happens, why is this true?  My current hypothesis is that it has something to deal with electromagnetism (pushing two repelling magnets together; how atoms swerve to avoid collisions, for example).
There is this famous example about the order difference between gravitational force and EM force. All the gravitational force of Earth is just countered by the electromagnetic force between the electrons on the surface of my feet and the ground.  But also wave-function of electrons(fermions) do not overlap each other due to Pauli exclusion principle.  So which one is the true reason of me not flopping inside the earth forever? Is it only one of them(my guess is Pauli exclusion) or is it both?
Would I be scrutinized at the border for traveling to the US twice in a quick succession? I am a British citizen visiting the USA in early August for a two week vacation with friends. After the holiday I'm returning to my home in the UK for three weeks. After that I will return to the USA and will be working for my British employer under the Visa Waiver Program for a period of 80-85 days.  I have read already that there is technically no problems with this under ESTA rules, but should I limit this working period to a shorter time frame (e.g. 70 or so days) so that it does not negatively affect the decision made by the POE officer? Ideally I would love to be over in the states for the near-full 85 day period.  Any advice or previous experiences anyone can offer?  Some clarifications: As confirmed by the immigration attorney, as I am employed by my UK employer, it will be regarded as a business trip, the important part being that I will be working officially as a consultant for my UK employer just overseas for the allotted 85 or so days Why did Dumbledore drink the potion in the attempt to get a Horcrux? When Albus Dumbledore  finally reaches the island with Harry Potter and finds out that you can't get near the potion, he made a cup and drank it. Why couldn't he just have poured the potion on the "floor"? Why couldn't he have made a big cup to hold almost everything inside the basin? If the dual spaces are isometrically isomorphic are the spaces isomorphic? I installed Windows 7, which ate Ubuntu's boot file. When starting up the computer, it now goes straight to Windows, without giving me the option of booting Ubuntu.   How can I get Ubuntu back? Where are the developer options in Android 4.2? Fixed background image with ios7 How to read such figures with % as "up (by) 120%" and "increased (by) 120%". Do they mean the same? Prove $kf(x)+f'(x)=0 $ when conditions of Rolle's theorem are satisfied . How to run two Thread Group in a loop - JMeter I'm facing problem here, There are 2 Thread Groups  and both are correlated ( run thread group consecutively is ticked in test plan ).  1st is user thread group and 2nd is Driver thread group. So thread group running concurrently but it is not happening in twice. I've put loop count for 2 they will run each independently after completion of first cycle. I want run this flow such a way that after completion of driver group request next do second iteration (start from user) thread group. So how to do this stuff ?  Loop will be :   User thread group Driver Thread group   After completing this once again start with :   User thread group Driver thread group   like this it should(loop) happen for multiple times What is the best way to return an error from a function when I'm already returning a value? Having different horizontally  distributed equations inside an align environment, only lines are numbered, but not individual equations.    How can the left equations be numbered using subequations? See also the question   \documentclass{article} \usepackage{amsmath} \begin{document}   \begin{subequations}     \begin{align}        %\label{a}         a &amp;= a ,&amp;       \label{b}         b &amp;= b ,\\       %\label{c}         c &amp;= c ,&amp;       \label{d}       d &amp;= d .     \end{align}   \end{subequations} \end{document}     The solution should be able to either:    attach a label to the left equations, or modify the labels on the right to read (1a,b) and (1c,d)    I couldn't find anything in the documentation of amsmath or other packages. Killing thread after some specified time limit in Java Matrix with labels nested in braces Was the singularity at Big Bang perfectly uniform and if so, why did the universe lose its uniformity? Let $U$ be an open set in $\mathbb R$. Then $U$ is a countable union of disjoint intervals.    This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as possible. A professor told me that there are many. So, I invite everyone who has seen proofs of this fact to share them with the community. I think it is a result worth knowing how to prove in many different ways and having a post that combines as many of them as possible will, no doubt, be quite useful. After two days, I will place a bounty on this question to attract as many people as possible. Of course, any comments, corrections, suggestions, links to papers/notes etc. are more than welcome. From this  by orlp:     If I strap a rocket booster to a rocket in space and fire it for one second, then the work provided is much higher when the rocket is flying fast compared to when the rocket was stationary. In both cases the rocket fires for the same duration but in the former case the rocket travels a much greater distance during this period. What gives? How do I remove hidden backup files? I have files ending in ~ in my home directory, and I want to delete them. How do I do that? How do I prevent this backup functionality? Defining coordinate reference system with rotation in GeoServer? I am using GeoServer and have a layer in EPSG:900913 ("Google Mercator").  I need to "rotate" the map around certain point (say, 1500000, 7000000) by certain degree (say, 30 degrees clockwise). How could I define such a coordinate system based on EPSG:900913?  GeoServer's  does not work for my purposes as I need to tile the map later on.  As far as I understand this, my only option is to define an own coordinate system. For GeoServer I'd need to . The configuration seems to be straightforward, but I have a difficulty defining my rotated CRS in .  I am wondering how to apply a rotation around certain point onto a CRS like Google Mercator:  PROJCS["WGS84 / Google Mercator",    GEOGCS["WGS 84",      DATUM["World Geodetic System 1984",        SPHEROID["WGS 84", 6378137.0, 298.257223563, AUTHORITY["EPSG","7030"]],        AUTHORITY["EPSG","6326"]],      PRIMEM["Greenwich", 0.0, AUTHORITY["EPSG","8901"]],      UNIT["degree", 0.017453292519943295],      AXIS["Longitude", EAST],      AXIS["Latitude", NORTH],      AUTHORITY["EPSG","4326"]],    PROJECTION["Mercator_1SP"],    PARAMETER["semi_minor", 6378137.0],    PARAMETER["latitude_of_origin", 0.0],    PARAMETER["central_meridian", 0.0],    PARAMETER["scale_factor", 1.0],    PARAMETER["false_easting", 0.0],    PARAMETER["false_northing", 0.0],    UNIT["m", 1.0],    AXIS["x", EAST],    AXIS["y", NORTH],    AUTHORITY["EPSG","900913"]]   My questions, specifically:   How to write a WKT which transform an existing CRS? My guess would be that I need a new PROJCS wrapping an existing one and adding a PROJECTION clause. How would I found out the projection id (like Mercator_1SP above) and the required parameters (the PARAMETER clauses)? Can I "reference" EPSG:900913 in CRS WKT instead of copy-pasting the whole PROJCS clause? I have a custom search page for use in Customer Portal.  VF Page:    &lt;form name="frmSearch"&gt;      &lt;input type="text" name="keyword" id="keyword" value="{!portalSearchModel.searchTerm}" onkeypress="return noenter(event);" style="width:25%; height:25px;margin:0; padding: 0px 6px 0px;" placeholder="What are you looking for?" /&gt;      &lt;input type="button" id="btnSearch" name="btnSearch" value="Search" onclick="searchTerm()" class="go-we3" /&gt;   &lt;/form&gt;   &lt;script type='text/javascript'&gt;       function searchTerm(){         var searchTerm = document.getElementById("keyword").value;          var url="/apex/PortalSearch?s="+searchTerm;          window.location = url;          return false;     }          &lt;/script&gt;   Is there a way to call this apex method in Controller on button click along with the JavaScript?      public void logSearchTerm(){         portalSearchModel.searches();         Search_Log__c sl = new Search_Log__c();          sl.search_term__c = portalSearchModel.searchTerm;         sl.user__c = UserInfo.getUserId();              insert sl;  } Is the set of real matrices diagonalizable in $\mathcal M_n(C)$ dense in the set of all companion matrices?