Quoted from .     We called an integral domain $R$ a Euclidean domain if there exists a function $f$ from $R$ to strictly positive integers such that:   1) For $a,b$ non zero in $R$, $f(ab)\ge f(a)$.   2) If $a,b\in R$, $b\neq 0$, then we can write $a=bq+r$ with $q,r\in R$ such that either $r=0$ or $f(r)&lt;f(b)$.     One of the answers below say that the definition is equivalent if we remove 1) above. To prove this, we take  $$ g(a)=\min_{x\in R^* a} f(x), R^*=R\backslash \{0\}. $$  Obviously $g$ satisfies 1). Now I am struggling to show that the new function $g$ satisfy 2), given that $f$ satisfies 2). Any hints?
Can anyone make me understand in simple language why the second condition for being an Euclidean domain is superfluous ?  Why $v(a)  \leq v(ab)$ is not needed?  How we can deduce from the first one?
Showing that $|f(z)| \leq \prod \limits_{k=1}^n \left|\frac{z-z_k}{1-\overline{z_k}z} \right|$ How can I delete an account on a specific site? My account has 20 (!) sites associated to it.  I only use it for 3-5 sites.  To reduce clutter, I would like to delete my profiles that I do not use much.  I have went through , but when I enter the profile URL as:  http://russian.stackexchange.com/users/5772/alex   it gives the error message:     oops! this isn't a link to a valid profile   What procedure do I go through to delete my profile only on certain sites? Is the Schrödinger equation derived or postulated? I'm an undergraduate mathematics student trying to understand some quantum mechanics, but I'm having a hard time understanding what is the status of the Schrödinger equation.  In some places I've read that it's just a postulate. At least, that's how I interpret e.g. the following quote:     Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger.    -- Richard Feynman   (from the Wikipedia entry on the Schrödinger equation)  However, some places seem to derive the Schrödinger equation: just search for "derivation of Schrödinger equation" in google.  This motivates the question in the title: Is the Schrödinger equation derived or postulated? If it is derived, then just how is it derived, and from what principles? If it is postulated, then it surely came out of somewhere. Something like "in these special cases it can be derived, and then we postulate it works in general". Or maybe not?  Thanks in advance, and please bear with my physical ignorance. View my questions with unaccepted answers There is this new acceptance rate feature that appears beneath my signature card when I ask a question.  How do I display all my questions with unaccepted answers? How can I easily see the list of questions that have not been checked as accepted? Switching motherboard without new install? Could the granting lag for each badge be added to the FAQ? What is the complexity of finding SHA-1 collisions? Given the , what is the complexity of finding a SHA-1 collision?  Marc Steven's  is still detailing about 260. Does the previous complexity still hold? How do I set a user environment variable? (permanently, not session) This is irritating me. I seen several suggestions (all using different files and syntax) and none of them worked.  How do I set an environment variable for a specific user? I am on debian squeeze. What is the exact syntax I should put in the file to make ABC = "123"? Design a custom Beamer theme from scratch How to run .sh script with keybind I am asked to solve $$u_x+u_y=1$$ If is was homogeneous i.e., $u_x+u_y$ the answer would be $u(x,y)=f(y-x)$ where $f$ is an arbitrary function. I have found the following set of solutions: $$u(x,y)=\lambda x +(1-\lambda)y$$ where $\lambda$ is an arbitrary constant(real or imaginary). I just have no idea what method other then trial and error would have lead me here. Any ideas? Thanks! Client-Side Rendering/JS Link documentation? Has there been any sort of documentation or anything like that released for SharePoint 2013 Client-Side Rendering/JS Link?  Anthing unofficial? Any dates for when documentation might be released? If I take the $k \in \mathbb{N}$ power of 10 and mod it by a large prime, I notice that the remainders repeat at some point.  For instance $10^k mod~7$ seems to repeat every $8$th value of $k$.  Given $k$ (the power of ten), and a large prime number $p$, how can I compute the size of the cycle?  I tried writing a python script that computes the first instance of a repeat. However, I don't know if this works since a single instance of a repeat may not necessarily mean a full cycle. Bash: command not found How can gravity affect light? I understand that a black hole bends the fabric of space time to a point that no object can escape.   I understand that light travels in a straight line along spacetime unless distorted by gravity. If spacetime is being curved by gravity then light should follow that bend in spacetime.  In Newton's Law of Universal Gravitation, the mass of both objects must be entered, but photon has no mass, why should a massless photon  be affected by gravity in by Newton's equations?  What am I missing? The subscript size for formulas in beamer presentation First of all sorry because this question may be duplicated. I don't know why the question was uploaded with another user's name and I can't edit it (may be you can delete the previous question with the same title). The question is the following:  I am creating a beamer presentation where I have a lot of formulas. Between the packages used I have \usepackage{amsmath} and the theme for the presentation is \usetheme{Warsaw}.  As it can be seen in the attached picture, the size of the sub(super)script is bigger than the normal writting.     Anyone knows a set of basic packages in order to get the classical formulas, without these size changes?  The code is the following  \documentclass[]{beamer}  \setbeamercovered{transparent}   %--------------------------% % To add enumeration index %--------------------------% \addtobeamertemplate{navigation symbols}{}{%     \usebeamerfont{footline}%     \usebeamercolor[fg]{footline}%     \hspace{1em}%     \insertframenumber / \inserttotalframenumber  } \setbeamercolor{footline}{fg=black} \setbeamerfont{footline}{series=\bfseries}   %---------------------------------------------------------------------------------%  \usepackage{amsmath,amssymb}    % Theme for the presentation \usetheme{Warsaw} % Change the colour \usecolortheme{spruce}  \setbeamertemplate{footline}{}  % Delete the footline with the names    \begin{document}    \begin{frame} \frametitle{\textbf{Algorithm}: step \texttt{s}}   \scriptsize  \textbf{STEP \texttt{s}}  \vspace{0.25cm}  Let $B^{(s-1)}$ be the variable  for the $\mathit{(s-1)}$-th step, let's take the sets $I_+^{(s-1)}$ and $I_-^{(s-1)}$ of length $r_+^{(s-1)}$ and $r_-^{(s-1)}$ respectively,and define:\\  \vspace{0.25cm}  \begin{center}  $B^{(s+)}= \sqrt{\tfrac{(r_+^{(s-1)} + 1)*r_-^{(s-1)}}{r_+^{(s-1)} + r_-^{(s-1)} + 1}} $  \end{center}   \end{frame}  \end{document} How to suppress scientific notation when printing float values? Here's my code:  x = 1.0 y = 100000.0     print x/y   My quotient displays as 1.00000e-05.  Is there any way to suppress scientific notation and make it display as 0.00001? I'm going to use the result as a string. How can I get out of the debt-trap as quickly as possible? I have a credit card with 30% interest and a real estate loan with 11% interest. The amount of debt on my card is lower than the real estate loan.   I want to redeem the installments of the real estate loan, but I can't do it because the credit card debt takes all my money.    What can I do in this situation? How can I get out of the debt-trap as quickly as possible?   I have a 4-room house and 2 cars, and 2 children who have to go to school, that's why there are a lot of fix expenses. Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is... Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n\mathbb Z$. Prove that either every element of $H$ is even or exactly half of its elements are even. (Gallian, Contemporary Abstract Algebra, Exercise 24, Chapter 3.) Perfect set without rationals