Inspired by the construction of the non-mesurable Vitali set I thought about the group $\mathbb{R}/\mathbb{Q}$ (the additive group of the real numbers modulo the rationals). There must be some literature for this, but I haven't found any and it is hard to google. Does anyone know where to find results about this group?
Let $G = \mathbb{R} / \mathbb{Q}$. Is this an interesting group to study?  Is it isomorphic to any more natural mathematical objects?
Where do executables look for shared objects at runtime? I understand how to define include shared objects at linking/compile time. However, I still wonder how do executables look for the shared object (*.so libraries) at execution time.  For instance, my app a.out calls functions defined in the lib.so library. After compiling, I move lib.so to a new directory in my $HOME.  How can I tell a.out to go look for it there? I am writing a report in a 2 column page format. Now I want to insert a rather large picture that requires to use full width of the page and cut the columns horizontally. How can I achiveve this? Please see the images below for what I mean : I want to update Acrobat Reader for Firefox, but the download has the .bin extension. How can I install it? Ok. I was playing mcpe and I noticed that mossy cobblestone isn't in the crafting table. And I wanted to make a house that looks old. So if its possible to get mossy cobblestone without mods, how can I get it on survival? If it is possible can somebody please leave a step by step guide. Thank you. Last Logon Time of All Users - RDP server not on a domain I am attempting to get a list of "last logon time" of "all users" on Windows Server 2012, but currently only know of how to list a single user login, which is:  net user username | findstr /B /C:"Last logon"  Any ideas?  *P.S. This servers only purpose is to host the RDP connections; not tied to a domain/AD. Getting "Not found" message when running a 32-bit binary on a 64-bit system I have currently a strange problem on debian (wheezy/amd64).  I have created a chroot to install a server (i can't give any more detail about it, sorry). Let's call its path /chr_path/. To make things easy, I have initialized this chroot with a debootstrap (also wheezy/amd64).  All seemed to work well inside the chroot but when I started the installer script of my server I got : zsh: Not found /some_path/perl (the installer includes a perl binary for some reasons)  Naturally, I checked the /some_path/ location and I found the "perl" binary. file in chroot environment returns :  /some_path/perl ELF 32-bit LSB executable, Intel 80386, version 1 (SYSV), dynamically linked (uses shared libs), for GNU/Linux 2.2.5, not stripped   The file exists, seems ok, has correct rights. I can use file, ls, vim on it but as soon as I try to execute it - ./perl for example - I get : zsh: Not found ./perl.  This situation is quite understandable for me. Moreover :   I can execute other basic binaries (/bin/ls,...) in the chroot without getting errors I have the same problems for other binaries that came with the project When I try to execute the binary from the main root (/chr_path/some_path/perl), it works. I have tried to put one of the binaries with a copy of my ls. I checked that the access rights were the same but this didn't change anything (one was working, and the other wasn't) "Friendly" terminal color names in shell scripts? I'm aware of libraries in languages such as Ruby and Javascript to make colorizing your terminal scripts easier by using color names like "red".  But is there something like this for shell scripts in Bash, or Ksh, or whatever? The best stack-up possible with a four-layer PCB? I'm designing a 4 layer PCB and I know that the standard stack-up is  Signals GND VCC Singals  (GND and VCC can be switched depending on the layer with more signals) The problem is, I don't really want to connect all ground pins through vias, there are just too many of them ! maybe because I'm not used to 4 layer PCBs, anyway, I've read a  by Henry W. Ott about a different stack-up  GND Signals Signals GND  (Where the power is being routed with wide traces on the signal planes) According to him, this is the best stack-up possible with a four-layer PCB, for the following reasons: 1.Signal layers are adjacent to ground planes. 2.Signal layers are tightly coupled (close) to their adjacent planes. 3.The ground planes can act as shields for the inner signal layers. (I think this requires stitching ??) 4.Multiple ground planes lower the ground (reference plane) impedance of the board and reduce the common-mode radiation. (don't really understand this one) One problem is cross-talk, but I really don't have any signals in the third layer, so I don't think that corss-talk will be an issue with this stack-up,am I correct in my assumption ? Note: The highest frequency is 48MHz, there's a wifi module on the board too. iPhone's Bluetooth - What? Why? How? I googled a lot and read many documents. But I can't understand the concept behind iPhone's Bluetooth technology.  In my knowledge, Bluetooth was used for:   Transferring files between devices Wireless communication for things like video gaming and earphones   Recently I tried to copy some of the photos from my old Android mobile to my iPhone 4S via Bluetooth. It's quite simple in my older phones to connect with other devices by Bluetooth for file sharing. But I can't pair other devices (even another iPhone :( ) with my iPhone 4S.   Copying songs/videos may affect their iTunes as well as their business. But what about images? Why is the usage of Bluetooth restricted and what's the use of it being able to transfer images? Is there any other way to use iPhone's Bluetooth to transfer files? Maybe using an app? But I don't want to jailbreak my device.   I'm confused about the restrictions and Apple's thinking. CSS3's border-radius property and border-collapse:collapse don't mix. How can I use border-radius to create a collapsed table with rounded corners? Vim-latex adds extra brackets when typing empty brackets Is my US visa still valid after change in name? I'm a Malawian national and I got US Visa years back.  Recently, due to some reasons, I had to get my first name partially changed on my passport. I have got an affidavit certificate stating the reason and proof.  Since the visa on my old passport has still not expired, but there is a difference in first name, will I face any problem? A physical explanation for negative kelvin temperatures How do I load an mp4 video with multiple audio tracks into blender? Cam I have two GFCIs connected to one double gang box in my laundry room and if so can I do a single line connection so that they both can work independently?  I have a 12/2 going from the panel through conduit (copper wire) to the double gang box. I want to have each GFCI to run independently without using the load. Why, in case one shuts off the other isn't affected. the double gang box is standard. the device is a Leviton 20 amps. I was told that you can run 12/2 from the panel to the first GFCI black and white. Then connect the second GFCI using a jumper and attaching it to he second GFCI. Is this acceptable by code. we use a washer,dryer in the laundry and the garage heavy machinery. so why should this be a problem running two independent? Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets? $$\overline{z}=z^n$$ where $n\in \mathbb{N}$   So I have started with moving the polar representation as the expression is in the n-th power  $$rcis(-\theta)=r^ncis(n\theta)$$   I can not multiply both sides in $r^{(-n)}$ as $n\in \mathbb{N}$ how should I continue? Option clash for package xcolor Assume you have a flat table that stores an ordered tree hierarchy:  Id   Name         ParentId   Order  1   'Node 1'            0      10  2   'Node 1.1'          1      10  3   'Node 2'            0      20  4   'Node 1.1.1'        2      10  5   'Node 2.1'          3      10  6   'Node 1.2'          1      20   Here's a diagram, where we have [id] Name.  Root node 0 is fictional.                          [0] ROOT                           /    \                [1] Node 1          [3] Node 2               /       \                   \     [2] Node 1.1     [6] Node 1.2      [5] Node 2.1           /            [4] Node 1.1.1   What minimalistic approach would you use to output that to HTML (or text, for that matter) as a correctly ordered, correctly indented tree?   Assume further you only have basic data structures (arrays and hashmaps), no fancy objects with parent/children references, no ORM, no framework, just your two hands. The table is represented as a result set, which can be accessed randomly.   Pseudo code or plain English is okay, this is purely a conceptional question.  Bonus question: Is there a fundamentally better way to store a tree structure like this in a RDBMS?    EDITS AND ADDITIONS  To answer one commenter's ('s) question: A root node is not necessary, because it is never going to be displayed anyway. ParentId = 0 is the convention to express "these are top level". The Order column defines how nodes with the same parent are going to be sorted.  The "result set" I spoke of can be pictured as an array of hashmaps (to stay in that terminology). For my example was meant to be already there. Some answers go the extra mile and construct it first, but thats okay.  The tree can be arbitrarily deep. Each node can have N children. I did not exactly have a "millions of entries" tree in mind, though.  Don't mistake my choice of node naming ('Node 1.1.1') for something to rely on. The nodes could equally well be called 'Frank' or 'Bob', no naming structure is implied, this was merely to make it readable.  I have posted my own solution so you guys can pull it to pieces. nth fibonacci number in sublinear time