My book, in mechanics, claimed that the normal forces were actually a result of electromagnetic force and that the "hand of the strongest weightlifter will crumble under the weight of a feather" if electromagnetic forces did not exist.  I imagine that it was due to some kind of attraction or repulsion between charges in the bodies. But I still cannot understand what exactly the statement tries to say. It also considers friction as a kind of normal reaction, so, I don't understand how the force actually sets up
It is commonly stated that there are four fundamental forces, or interactions, in nature. It is natural to consider which of those is responsible for the normal force we meet in elementary physics. What is the nature of the force exerted on a body by the floor on which it rests?  Of the four forces, the only one that seems relevant is the electromagnetic force, and I always assumed the repelling force in question is ultimately the EM one. The Wikipedia article on  seems to agree:     Molecular and quantum physics show that the electromagnetic force is the fundamental interaction responsible for contact forces.    An alternative explanation is that the force results from Pauli's Exclusion Principle; however, reasonable as it may seem, it doesn't seem to answer the question of what force is actually at play here. The exclusion principle isn't a force, or at least it's not normally described as such.  I've recently come across a rather forcefully presented argument that states that the normal force is a macroscopic force that is not directly reducible to the EM force, or to any of the other forces; rather, it does indeed result from Pauli's Exclusion Principle by way of interaction of electron wave functions. The argument further claims that attempts to correct the Wikipedia article have met with the usual "source needed" objections, and that this topic is either misunderstood by or poorly covered in the standard texts.  (The argument can be found ).   Which way is it?
Algebraic extension of perfect field in which every polynomial has a root is algebraically closed Let $F$ be a perfect field, i.e. every irreducible polynomial over $F$ has distinct roots in the algebraic closure of $F$. Suppose that $K$ is an algebraic extension of $F$ with the property that every non-constant $p(X) \in F[X]$ has a root in $K$.   I want to show that $K$ is algebraically closed, i.e. that $K = \overline{F}$, the algebraic closure of $F$.  If $K$ were a normal extension of $F$, this would follow immediately: Let $\alpha \in \overline{F}$ and $m(X)\in F[X]$ its minimal polynomial. By our assumption, $m(X)$ has at least one zero in $K$. Since $K$ is a normal extension of $F$, it thus contains all other roots of $m(X)$ including $\alpha$.  I don't have an idea as to how to show that $K$ is a normal extension. Let's say I have a document that has many pages and some pages have different formats. For example, most of the pages are fine with \documentclass{article} but there may be pages in the middle that I want to be \documentclass{letter}. I want my letter pages to be numbered inline with the other pages and have them show up in the Table of Contents.   Is it possible to change the class used for an individual or range of pages while still keeping things like page numbering consistent across them?   EDIT I'm trying to create a "welcome to the company" sort of document that has a cover page, table of contents, welcome letter, then various sections of information. I've never tried this in anything other than a word processor where it's easy to set "master pages" or "template pages" for individual pages in the document. So I could use a letterhead template page for the welcome letter part and a standard article-type template for the rest and still have the letter numbered/referenced in the ToC.   So the idea was to take advantage of the classes written, if possible, rather than have to format a single page by hand differently from the rest. When using an Int- or FloatProperty, is there any way of changing their min and max values dynamically?  I'm currently loading images on the fly and would like to limit a frame property to run from 0 to len(loaded_images) every time I call the loading operator. However, the property seems to be read-only and re-assigning gives the following error:  bpy.context.scene.project_props.bl_rna.properties['myIntProp'] = bpy.props.IntProperty(...) TypeError: bpy_prop_collection[key] = value: invalid, expected a StructRNA type or None, not a tuple   I understand the bpy.props constructor uses an intermediary tuple, but I'm not sure how to proceed. Is there any way around this? Export multiple objects to .obj It is possible to export all object from a scene into different obj files (each object in a obj file separately)? How does a Support Vector Machine (SVM) work? Sufficient Identification for Intra Schengen flights If one takes a direct flight from one Schengen member state to another, what's the minimum sufficient ID that needs to be provided to board the flight? How to use Mathematica functions in Python programs? After reading  It seems that there is no perfect solution for getting rid of the stretch at the poles of UV spheres.  According to , icospheres can be used for stretch-less UV mapping:     All faces have the same area, which may be useful for certain types of UV mapping containing non-organic textures. An example that comes to mind is an isocahedral die or billiard balls where stretch must be minimized near the point where the number is printed onto the ball.   Incidentally, a lowish-poly billiard ball is exactly what I'm attempting to create.  However, my attempts at neatly unwrapping an icosphere have mostly failed (smart UV unwrap works perfectly for using baked textures, however I was hoping for something more like the example in the 6th post of , with a nice clean rectangular map).  How can an icosphere (I assumed the term "geosphere" in the thread was referring to a (geodesic) sphere made of triangles, like an icosphere) be cleanly unwrapped? Or if it's not possible, is there another way to do this which I'm missing?    I'm planning on using the resulting sphere in the BGE. Alternate position of x tick labels Detecting unused packages in LaTeX document I have quite a bad habit of copy-and-pasting my packages from one LaTeX document to another. Is there any way we can detect unused packages (i.e. \usepackage commands) so that they can be removed from the LaTeX document? Subgroups of finitely generated groups are not necessarily finitely generated I was wondering this today, and my algebra professor didn't know the answer.      Are subgroups of finitely generated groups also finitely generated?    I suppose it is necessarily true for finitely generated abelian groups, but is it true in general?  And if not, is there a simple example of a finitely generated group with a non-finitely generated subgroup?  NOTE:  This question has been merged with another question, asked by an undergraduate.  For an example not involving free groups, please see Andreas Caranti's answer, which was the accepted answer on the merged question. MacOS dmg file question Not entirely sure if this is where to ask this, but I can't find another place to...  I've recently switched from Windows to Mac and I am trying to understand more in depth what I'm working with.  My question is: are .dmg files the equivalent of .zip files in Windows? I came to this theory when downloading a .dmg and when I opened it, it contained a .app and and text file read me. To me, this seems like a compressed file.   Is this correct? I am running Ubuntu 14.04 and Notepad++ 6.6.4 (through WINE)  Under the 'Run' tab, I clicked on "Launch in Firefox" , but nothing happened   I uninstalled Notepad++ and WINE, then shutdown. I re-installed Notepad++ and WINE and still have the same problem.   Does anyone know what the problem is? This is a  about securing public DNS resolvers   Open DNS servers seem pretty neat and convenient, as they provide IP addresses that we can use consistently across our company regardless of where they are located. Google and OpenDNS provide this functionality, but I'm not sure that I want these companies to have access to our DNS queries.  I want to set up something like this for use by our company, but I hear a lot about this being dangerous practice (particularly in regards to ) and I want to make sure that we do this right. What things do I need to keep in mind when building this type of environment? Where can I find a list of computers guaranteed to work with Ubuntu? Before I buy my next laptop, I'd like to make sure that it will work perfectly with Ubuntu. Is there a list of completely supported computers anywhere? I mean are we certain that they will be able to provide us a huge improvements (in some tasks) compared to clasical computers? Converting Degree Minutes to decimal degrees using Excel formula? I have location points collected from a garmin device stored in an excel sheet in Degree Minutes format --- W00208.172,N1046.977.  I am looking for an Excel formula to convert either to Decimal Degrees or Degrees Minutes seconds Format ? Why do rainbows have distinct colors? How do I force TeX to completely use the 1st column before writing to the second column?  Right now I dont have enough content to fill one whole page. So the left and right columns are equally filled, leaving 25% symmetrical empty page at bottom. How do I fill the left column first, and then the right?  \documentclass{article}  \usepackage{multicol} \usepackage[scale=0.9]{geometry} \usepackage{amssymb} \renewcommand{\labelitemi}{\tiny$\blacksquare$} \begin{document} \begin{multicols}{2}      %lots of text goes here  \end{multicols} \end{document} Suppose that a box contains 10 red balls, 20 green balls, and 30 blue balls.   Suppose also that balls are drawn from the box one at a time at random.  What is the probability that all the red balls are drawn before the blue or green balls are themselves exhausted. What is the probability that as the last red ball is drawn, there remains at least one blue and one green left in the box.  The answer I was given is $\dfrac{7}{12}$ and a general equation is: $$ \dfrac{b g}{1-b}+\dfrac{b g}{1-g} $$ where $$ g=\dfrac{20}{60},b=\dfrac{30}{60} $$ but why?