Ring Theory - Isomorphism
showing $\psi: R\to \mathbb C$ is ring isomorphism.
Suppose $f$ is an isometric (i.e., distance preserving) function on $\mathbb{E}^2$ such that $f(0,0) = (0,0)$.  Then I want to show that $f$ is necessarily linear.  Now $f$ is linear iff $f$ is both additive and homogenous.  The following is an attempted proof for the homogeneity of f (missing the last step); still more, I have no idea how to argue for the additivity of $f$.  Any ideas?  Let $x \in \mathbb{E}^2$ and $\alpha \in \mathbb{R}$   We know that $\forall x \in \mathbb{E}^2$, $\Vert x - 0 \Vert = \Vert f(x) - f(0)\Vert = \Vert f(x) - 0 \Vert$ so that $\Vert x\Vert = \Vert f(x)\Vert$.  From this we immediately have the following facts:  $\Vert x \Vert = \Vert f(x) \Vert$  $\Vert \alpha x \Vert = \Vert f(\alpha x) \Vert$  We can then argue that since $\Vert \alpha x\Vert = |\alpha| \Vert x \Vert = |\alpha| \Vert f(x) \Vert = \Vert \alpha f(x) \Vert$, we also have that $\Vert f(\alpha x) \Vert = \Vert \alpha f(x)\Vert$.  Finally, we have that  $\Vert \alpha x - x \Vert = \Vert \alpha f(x) - f(x) \Vert$   iff $ \Vert (\alpha - 1)x \Vert = \Vert (\alpha - 1) f(x) \Vert$   iff $|\alpha - 1| \Vert x \Vert = |\alpha - 1| \Vert f(x) \Vert$  Since the last of these statements is in fact true, we now have $\Vert \alpha x - x \Vert = \Vert \alpha f(x) - f(x) \Vert$ as desired.  Now at this point it seems like I have all of the facts required to assert that $f(\alpha x) = \alpha f(x)$, but I can't figure out how to formally state why without illegally appealing to visual intuition.  Any ideas? Primary decomposition of a monomial ideal 0 ram free on ICS Python scope: "UnboundLocalError: local variable 'c' referenced before assignment" How to run untrusted Ruby code inside a safe sandbox? If the lines intersect the vertices of the square. The area of the square is $1$ and the lines also intersect the midpoints of the square lines. How to find the area of shaded region? How do I attach rope to a mesh I am trying to make a pirate ship and am trying to attach rope to the sails. my rope is made using a path, and soft body physics. I can not find a way to attach my rope to the sail. and I can not use pinned vertices because this is not a cloth modifier. How can I use Gmail (in Chrome) as the default email handler in Windows 8?  I used to use GmailNotifier in Windows 7, but that doesn't seem to work anymore. Clicking on a mailto: link simply opens Chrome with a blank page.  I am not interested in using the Mail app built into Windows 8. Mcrypt extension is missing in 14.04 server for mysql The php mcrypt extension on ubuntu 14.04 shows that it is installed properly, but phpmyadmin says:   The mcrypt extension is missing. Please check your PHP configuration.   My current php config is . If you need anymore information, please contact me through the comments.  $ sudo apt-get install php5-mcrypt     Reading package lists... Done     Building dependency tree            Reading state information... Done     php5-mcrypt is already the newest version.     0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. I recently saw a question titled "" (10k+ users only). I thought, "surely someone has asked this question before", and began looking for a duplicate.  is a question with the exact same title, just with a little different punctuation and casing. It was "Jquery vs Javascript" and "jQuery vs. Javascript?". The only difference being the ., ?, and the casing.  Since we now , I propose that when checking for questions with duplicate names, certain punctuation be striped out to prevent the duplicate from occurring in the first place.  As pointed out in the comments, if isn't already, white-space and non-printable characters may want to be removed for duplicate detection as well. What is Chromatic Aberration? Install pandas to python3.5 instead of 2.7 How can I control a servo motor with the PWM output of a microcontroller? I wanted to control the direction of servo motor with a switch. I am using an Atmega32.  Outputs are PIND4 and PIND5 and the input is PINA3, but the two servo motors turn only in 1 direction.  #ifndef F_CPU  #define F_CPU 1000000UL   #endif  #include &lt;avr/io.h&gt; #include &lt;util/delay.h&gt; #include &lt;avr/interrupt.h&gt;   int main(void) { DDRD= 0xFF ; DDRA=0x00; PORTA =0x00; // FAST PWM TCCR1A |= 1&lt;&lt;COM1A0 | 1&lt;&lt;COM1A1 | 1&lt;&lt;WGM11 |1&lt;&lt;COM1B0|1&lt;&lt;COM1B1 ; TCCR1B= 1&lt;&lt;WGM12|1&lt;&lt;WGM13 ; ICR1 = 19999 ;  TCCR1B |=1&lt;&lt;CS10 ; while (1) {    if(PORTA==0x08) //if pina3 is 1    {     OCR1A = ICR1 -2000 ;   //turn servo 1     _delay_ms(1000);          OCR1B=ICR1-2000;    //turn servo 2     _delay_ms(1000);     }     if(PORTA==0x00)// if pina3 is 0     {       OCR1A = ICR1 -900 ; //turn servo 1 in the other direction       _delay_ms(1000);               OCR1B=ICR1-900;     //turn servo 2 in the other direction       _delay_ms(1000);      } }       return 0 ;  } What is a single word for a person who, as a habit, exaggerates a lot when relating information?   So for example, they relate something that happened but exaggerate so much that they are including clear falsehood and they are well known for doing that. I'm running Bash scripting, but sometimes I get confused which of these commands I use belong to who. Sometimes man xxx works, sometimes doesn't, so I use --help or info, mostly one of these works to show description of the command. Can anyone tell me how I would know what command belongs to what? Bash builtin, GNU utility, etc. How to make program autostart only in GNOME I use both GNOME and KDE SC. In GNOME, I use Docky but I do not use it with KDE. So, I would like Docky to start up only when I login into GNOME not KDE. In KDE, there is an option to make a program autostart only in KDE. But I cannot find any such option in GNOME.    How can I do this?    Thanks in advance. Work done in adiabatic process How to limit zoom level in OpenLayers 2.x I would like to have only 3 levels of zoom available (from level 6 to level 8).  demolayer2 = new OpenLayers.Layer.WMS( "abc...","http://localhost:8080/geoserver/gwc/service/wms", {layers: 'def...', transparent:"true", format: 'image/png', numZoomLevels: 3, minZoomLevel: 6 },{isBaseLayer:false},  { tileSize: new OpenLayers.Size(256,256)});  map.addLayer(demolayer2);   map.addControl(new OpenLayers.Control.LayerSwitcher({'div':OpenLayers.Util.getElement('layerswitcher')}));  map.zoomToExtent(new OpenLayers.Bounds(-4.601615515076983,39.8769407866263,-3.0527184873294764,41.16710732525929)); Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$ Prove that for any natural number $n&gt;1$, we have $(2n)!&lt;(n(n+1))^n$    I tried using induction, but I failed in that approach, I rather found it was untrue, but on several case testing, I found it to be true. I am out of ideas here, please help. Thank you. Electrical Machines based on Electric Fields Our teacher asked us this question during the lecture for Power Electromagnetics. We know that Electric Fields and Magnetic Fields are easily transformable into one another. And they are used greatly in various systems.   He asked us why is it that almost all the electric machines are still based on exchange of magnetic energy? Why isn't electric field used for power exchange and interaction through air gaps for a machine?