Proving we can partition $\Bbb R^+$ into two sets which are closed under addition and Zorn's lemma
Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition.
Why do some materials reflect (metals) and other materials reflect and refract (glass) from the quantum perspective? Recently I was asked to explain the difference between reflection and total internal reflection from a purely conceptual standpoint (no math).    Let me explain what I already know.  Reflection and refraction at the quantum level are the same thing.  Light is a photon.  A photon is a discrete particle that has wave characteristics (a finite wave traveling like a bullet).  As the photon travels it collides with electrons in the matter of the medium it is traveling in.  Depending on the energy of the photon and the allowed energy bands of the medium the photon cause the electron to jump up a level.  If the photon is absorbed then the medium will increase its motion (at the macro scale increasing its temperature).  If the photon is not absorbed it will be re-emitted (really as a new photon). I have read and watched Feynman's QED lectures and book and have a pretty good understanding of his process for determining how all these paths come together to give the net path of the photon.  The general rule of thumb is that the photon wants to take the path which requires the least amount of time.  I understand how this principle goes to explain refraction and reflection.    What I don't seem to understand is why does one material seem to cause a higher percentage of refraction compared to another (metal vs. glass).  What about the electron configuration of a the medium changes the net effect of the absorption and re-emissions of the photons?  Is there a change in the probability of the photon being re-emitted in a reverse direction?  Is there a farther distance the photon can travel before being incident onto an electron?  This is the part where my understand breaks down.    When you have hit the critical angle in a medium that refracts and the light completely reflects, are the photons moving is the same manner as they would be in a material that always reflects? How does this connect to the question in the previous paragraph?  I know I have a bunch of mini questions embedded in answering this one larger question.  Any help on any of the parts would be greatly appreciated? How do you make samba follow symlink outside the shared path This is Ubuntu server 10.04 64 and samba 3.4.7.  I have a shared directory /home/mit/share and another one /home/temp that I link into the shared one:  ln -s /home/temp /home/mit/share/temp   But on windows, after using internet, I cannot open S:/temp, but on Linux it is possible to access /home/mit/share/temp like expected.   This works if I link directories inside /home/mit/share/temp, so I guess samba is restricting to jump with a link outside/above the shared directory.   EDIT:  See also this question titled .  It seems best to put unix extensions = no into the global section and follow symlinks = yes and wide links = yes only into the shares section, where you really need it.  The unix extension flag has to live in the global section and not in the individual shares sections. But for security reasons it is better to use the other options only where you need it, and not globally. What is the best way to remove and prevent leaf stains on concrete pavers? Make wifi preferable over wired connection in Ubuntu 14.04? How do I make Ubuntu choose wifi connection over wired connection when both are connected to internet?  I edited the Ethernet connection to route using the preference "Use this connection only for resources on this network", which other people have reported having success with as a means of getting WiFi preferred, but it doesn't work for me. The issue persists. Ethernet is still being preferred for internet, and WiFi gets ignored.   another thing, is it possible to use wired connection as fail over connection to wireless? Is there a more accurate way to create a Javascript timer than setTimeout? Something that has always bugged me is how unpredictable the setTimeout() method in Javascript is.  In my experience, the timer is horribly inaccurate in a lot of situations. By inaccurate, I mean the actual delay time seems to vary by 250-500ms more or less. Although this isn't a huge amount of time, when using it to hide/show UI elements the time can be visibly noticeable.  Are there any tricks that can be done to ensure that setTimeout() performs accurately (without resorting to an external API) or is this a lost cause? WARNING DO NOT ATTEMPT TO RUN THIS ON A PRODUCTION MACHINE   In reading the  I generally follow what's going on with the following code:  :(){ :|:&amp; };:   excerpt of description     The following fork bomb was presented as art in 2002;   its exact origin is unknown, but it existed on Usenet prior to 2002.   The bomb is executed by pasting the following 13 characters into a    shell such as  or . It operates by defining   a function called ':', which calls itself twice, once in the   foreground and once in the background.   However the last bit isn't entirely clear to me. I see the function definition:  :(){ ... }   But what else is going on?  Also do other shells such as ksh, csh, and tcsh also suffer the same fate of being able to construct something similar? This is a question from the free Harvard online abstract algebra .  I'm posting my solutions here to get some feedback on them.  For a fuller explanation, see  post.  This problem is from assignment $5$.     Prove directly that distinct cosets do not overlap.   Let $H$ be a subgroup of $G$ and let $a$, $b$, and $c$ be elements of $G$ such that $b\not\in aH$ and $c$ is in both $aH$ and $bH$.  Then there are elements $h$ and $h^\prime$ in $H$ such that $c=ah$ and $c=bh^\prime$.  So $ah=bh^\prime$ and $b=ahh^{\prime -1}$.  But $hh^{\prime -1}\in H$ so $b\in aH$.  This contradicts our original assumption. Therefore, there can be no element in more than one distinct coset.  Again, I welcome any critique of my reasoning and/or my style as well as alternative solutions to the problem.  Thanks. Making an elastic waistband on clothing? How to find and replace using sed on records starting with Specific number l, ls & la -- what are the differences, and are there more of these commands? If $n\geq m$ then $(x^m+y^m)^{1/m} \ge (x^n+y^n)^{1/n}$ How to move Applications from phone to SD Card? What is the white smoke-like fog called? I have just migrated from Windows environment. I have installed Python 3.2 in a separate directory. How can I get the python installation path in Ubuntu shell?  Is there any way I can let the shell know/choose at runtime which python version is to be used for further code execution?  Are there any environment variables and search path kind of things in Ubuntu Linux as well? How do I remove Windows but keep Ubuntu? I want to keep Ubuntu but remove Windows. Is a strong background in maths a total requisite for ML? I'm starting to want to advance my own skillset and I've always been fascinated by machine learning. However, six years ago instead of pursuing this I decided to take a completely unrelated degree to computer science.   I have been developing software and applications for about 8-10 years now, so I have a good handle but I just can't seem to penetrate the maths side of machine learning/probabilities/statistics.   I start looking at learning material and on the first page it might include something which confuses me and immediately sets up a barrier in my learning.   Is a strong background in maths a total requisite for ML? Should I try and fill in the blanks of my maths before continuing with ML? Can self learning really work for just a developer without any hard computer science background?    Related question: How do you breed animals? How to install Eclipse C/C++ in Ubuntu 18.04 To clean install Eclipse for C/C++ on Ubuntu 18.04 I tried sudo apt install eclipse eclipse-cdt. It's showing an error, see the log file....  log file..... !SESSION Fri Aug 16 14:39:20 IST 2019 ------------------------------------------ !ENTRY org.eclipse.equinox.launcher 4 0 2019-08-16 14:39:20.065 !MESSAGE Exception launching the Eclipse Platform: !STACK java.lang.ClassNotFoundException: org.eclipse.core.runtime.adaptor.EclipseStarter     at java.base/java.net.URLClassLoader.findClass(URLClassLoader.java:471)     at java.base/java.lang.ClassLoader.loadClass(ClassLoader.java:588)     at java.base/java.lang.ClassLoader.loadClass(ClassLoader.java:521)     at org.eclipse.equinox.launcher.Main.invokeFramework(Main.java:626)     at org.eclipse.equinox.launcher.Main.basicRun(Main.java:584)     at org.eclipse.equinox.launcher.Main.run(Main.java:1438)     at org.eclipse.equinox.launcher.Main.main(Main.java:1414) How to Create Bootable Windows 10 image in Debian? Pseudocode but originally developed for Windows 7 iso file but applied for Windows 8 in the thread  but it does not work with Windows 10 iso  # https://rwmj.wordpress.com/2010/11/04/customizing-a-windows-7-install-iso/ # https://unix.stackexchange.com/a/312477/16920 $ dd if=../en_windows_10_x64_dvd.iso \     of=boot.img bs=2048 count=8 skip=734  $ mkisofs -o ../new-win.iso -b boot.img -no-emul-boot -c BOOT.CAT \     -iso-level 2 -udf \     -J -l -D -N -joliet-long -relaxed-filenames .   Unsuccessful output when run on Windows 10 image  dd if=/home/masi/Downloads/en_windows_10_multiple_editions_version_1511_x64_dvd.iso of=/home/masi/Downloads/boot.img bs=2048 count=8 skip=734 8+0 records in 8+0 records out 16384 bytes (16 kB) copied, 0.000392973 s, 41.7 MB/s    Some of the following fields have changed for the iso file used in dd   bs=2048 count=8  skip=734   How can you study which field values you can use for Windows 10 iso?  OS: Debian 8.5 64 bit Hardware: Asus Zenbook UX303UA Linux kernel: 4.6 of backports Related threads: ,  Motivation: I need Windows 10 to use Canon P-150 duplex scanner, but when I started my Windows, I got Error 0xC0000428 because Windows update has again broken things there and I use Windows otherwise so rarely; and I have no spare Windows left to make bootable media How to recursively download a folder via FTP on Linux I'm trying to ftp a folder using the command line ftp client, but so far I've only been able to use 'get' to get individual files.