This will be an odd question, but... In the name , what is the meaning of word integral? Does it refer to integral such as adjective for noun integer? Or does it instead refer to noun integrity?  The reason I am asking... In Czech language this is translated (when I would translate it back literally) as a "Domain of integrity" ("" for czechs here). But I suspect the original meaning was perhaps intended to refer to integers, instead of integrity? Just wondering if the translation is perhaps inaccurate...  As pointed out in comments, this question might have answer to this , I just don't see what the answer is (it seems both the terms are referred there...)
Self-explanatory title really! A student today asked me why they were called integral domains -- and I realised that the word "integral" seems to be being used in a way totally unlike any other way I hear it used in mathematics. The student suggested that "integral" was used because the integers were an example -- but I didn't buy this because by that logic they could have been called "rational domains" or "real domains".  Any ideas anyone?
JQuery to load Javascript file dynamically What is a predicate in c#? Is it possible to change your single-player skin? In Minecraft, is it possible to customize your skin in single-player mode (not just in multiplayer)? How? how to stop firing oncomplete if save action has errors My requirement is to stop oncomplete call if save action has errors. I know I am doing some silly mistake.   Here I am unable to stop calling oncomplete function.  Requirement is, from VFP save method, data will get saved and then perform callout from oncomplete function. If there are any errors during save then stop making callouts.  Visualforce  &lt;apex:page id="hasErrorTestPage" controller="ErrorMessageFinder"  showHeader="false"&gt;     &lt;apex:pageMessages id="msgId"/&gt;     &lt;script&gt;         function performCallOutMethod()         {             var msg = document.getElementById('{!$Component.MSEForm.errorMsgInd}').value;             alert(msg);              if(msg == 'false' || typeof(msg) == 'undefined')             {                 alert('entering callout')                 performCallOut();                 return true;             }             else             {                 return false;             }         }     &lt;/script&gt;     &lt;apex:form id="MSEForm"&gt;         &lt;apex:actionFunction name="performCallOut" action="{!submit}" reRender="panelId,msgId"/&gt;          &lt;apex:outputPanel id="panelId"&gt;             &lt;apex:outputText value="{!hasError}" id="errorMsgInd"/&gt;         &lt;/apex:outputPanel&gt;         &lt;apex:inputText id="customerNm" value="{!CustomerName}"/&gt;&lt;br/&gt;         &lt;apex:commandButton value="Save" action="{!save}"                       oncomplete="return performCallOutMethod();" reRender="panelId,msgId"/&gt;     &lt;/apex:form&gt; &lt;/apex:page&gt;   Controller  public class ErrorMessageFinder {     public String CustomerName {get;set;}      public Boolean hasError {          get {              return ApexPages.hasMessages();          }            }      public PageReference save()     {            if(String.isBlank(CustomerName))         {             ApexPages.Message msg = new ApexPages.Message(ApexPages.Severity.Error, 'Name cannot be null');             ApexPages.addMessage(msg);             return null;         }         return null;      }       public PageReference submit()     {         try         {   //navigate to view page             PageReference pg = new PageReference('/'+ '0019000001UwnVS');             pg.setRedirect(true);             return pg;         }         catch (Exception ex)         {             System.debug('Error in submit ' + ex);             ApexPages.Message msg = new ApexPages.Message(ApexPages.Severity.Error, ex.getMessage());             ApexPages.addMessage(msg);              return null;         }     } }   Based on this code, if I hit save button without entering CustomerName then it is not showing error after save. It is showing error during oncomplete call and then navigating to view page, which I do not want.  Any help is appreciated. I have put a compiled code so that anyone can try with this code. Duplicates in sendable DE for MobileConnect Get string inside parentheses, removing parentheses, with regex What's an intuitive way to think about the determinant? In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a $2\times 2$ matrix by the formula. Our teacher showed us how to compute the determinant of an $n \times n$ matrix by breaking it up into the determinants of smaller matrices. Apparently there is a way by summing over a bunch of permutations. But the notation is really hard for me and I don’t really know what’s going on with them anymore. Can someone help me figure out what a determinant is, intuitively, and how all those definitions of it are related? Cardinal number subtraction We all know that $|\mathbb{N}| = \aleph_0$. Since $|\{-1\} \cup \mathbb{N}| = \aleph_0$ as well, I guess you could say that $\aleph_0 + 1 = \aleph_0$.  You can go on to derive that $\aleph_0 + \aleph_0 = \aleph_0$, and by induction $n \times \aleph_0 = \aleph_0$ (assuming that $n$ is finite, anyway).  Now, here's a thing: What is $\aleph_0 - \aleph_0$?  Well, $|\mathbb{P}| = \aleph_0$, and $|\mathbb{P} \backslash \mathbb{N}| = \aleph_0$, so perhaps $\aleph_0 - \aleph_0 = \aleph_0$?  No, wait. Consider the set $S = \{n \in \mathbb{N}: n &gt; 5\}$. Now we have $|S| = \aleph_0$, and yet $|\mathbb{N} \backslash S| = 5$. So maybe $\aleph_0 - \aleph_0 = 5$?  But that is absurd, since we can redefine $S$ to make the result any finite number we wish. Or we can define $S$ such that the result is countably infinite.  Does this mean that the notion of $\aleph_0 - \aleph_0$ simply has no definite answer? Or am I just being too simplistic here? If there are injective homomorphisms between two groups in both directions, are they isomorphic? If I have two groups, $G$ and $H$ and two injective homomorphisms $\phi:G \to H$ and $\psi: H \to G$, then by the first isomorphism theorem applied to $\phi$, we have that $G \cong \mathrm{Im} (\phi)$, a subgroup of $H$. Similarly, $H$ is isomorphic to a subgroup of $G$. For finite groups, this guarantees that $G \cong H$ but does this hold for infinite groups?  Weird things can happen for infinite groups, e.g $n\mathbb Z \subsetneq \mathbb Z$ but $n \mathbb Z \cong \mathbb Z$. I'm wondering if this kind of thing stops an isomorphism from occurring.  I think this question generalises to rings, modules, fields and so on. Can this be answered for all of these structures? Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$. Let $B$ be a compact subset of a metric space $X$. Let $x \in X \setminus B$. Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$.   Is this an existence proof? If yes, do I get to define $U,V$? So I have been extruding a cube to make a low poly banana and when I go from edit to object mode half the edges don't appear as well as half is dark.      I tried looking for an solution couldn't find one and its making me feel like I am doing something wrong so I decided to come here. Energy conservation in General Relativity $X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed What is time, does it flow, and if so what defines its direction? This is an attempt to gather together the various questions about time that have been asked on this site and provide a single set of hopefully authoritative answers. Specifically we attempt to address issues such as:   What do physicists mean by time? How does time flow? Why is there an arrow of time? Listing, zebra effect and broken lines In , I've found one solution to produce a zebra effect for listing. Unfortunately, this only works if the lines are not broken. Is there a way to fix this?  Here is a code with an example showing the problem:  \documentclass[10pt,a4paper]{article}     \usepackage[utf8x]{inputenc}     \usepackage{ucs}     \usepackage[T1]{fontenc}     \usepackage{lmodern}     \usepackage[frenchb]{babel}     \usepackage[x11names, svgnames]{xcolor}     \usepackage{graphicx}     \usepackage{listings}     \usepackage{amsfonts}     \usepackage{amssymb}      \makeatletter         % Patch line number key to call line background macro         \lst@Key{numbers}{none}{%             \def\lst@PlaceNumber{\lst@linebgrd}%             \lstKV@SwitchCases{#1}%             {none&amp;\\%              left&amp;\def\lst@PlaceNumber{\llap{\normalfont                         \lst@numberstyle{\thelstnumber}\kern\lst@numbersep}\lst@linebgrd}\\%              right&amp;\def\lst@PlaceNumber{\rlap{\normalfont                         \kern\linewidth \kern\lst@numbersep                         \lst@numberstyle{\thelstnumber}}\lst@linebgrd}%             }{\PackageError{Listings}{Numbers #1 unknown}\@ehc}}          % New keys         \lst@Key{linebackgroundcolor}{}{%             \def\lst@linebgrdcolor{#1}%         }         \lst@Key{linebackgroundsep}{0pt}{%             \def\lst@linebgrdsep{#1}%         }         \lst@Key{linebackgroundwidth}{\linewidth}{%             \def\lst@linebgrdwidth{#1}%         }         \lst@Key{linebackgroundheight}{\ht\strutbox}{%             \def\lst@linebgrdheight{#1}%         }         \lst@Key{linebackgrounddepth}{\dp\strutbox}{%             \def\lst@linebgrddepth{#1}%         }         \lst@Key{linebackgroundcmd}{\color@block}{%             \def\lst@linebgrdcmd{#1}%         }           % Line Background macro         \newcommand{\lst@linebgrd}{%             \ifx\lst@linebgrdcolor\empty\else             \rlap{%                 \lst@basicstyle                 \color{-.}% By default use the opposite (`-`) of the current color (`.`) as background                 \lst@linebgrdcolor{%                 \kern-\dimexpr\lst@linebgrdsep\relax%                 \lst@linebgrdcmd{\lst@linebgrdwidth}{\lst@linebgrdheight}{\lst@linebgrddepth}%                 }%             }%             \fi         }     \makeatother      \lstset{ %    * Number for lines         numbers=left,                   % Where to put the line-numbers         numberstyle=\footnotesize,      % Size of the fonts used for the line-numbers         stepnumber=1,                   % Step between two numbers         numbersep=5pt,                  % How far the line-numbers are from the code % %    * Back returns for long lines         breaklines=true,                % Sets automatic line breaking         breakatwhitespace=false,        % Automatic breaks only happen at whitespace ?         breakindent=0pt,                % Space before the text of a break         postbreak=\mbox{                % Character used at the begining of a break             \rotatebox[y=0.9ex]{180}{\color{Red} $\Lsh$}%         },     }  \begin{document}  \begin{lstlisting}[language=Python,linebackgroundcolor={\ifodd\value{lstnumber}\color{gray}\fi}] for i in range(5): for i in range(5): for i in range(5): for i in range(5): for i in range(5): for i in range(5):     sum += i**3 print sum  print('Fin de la boucle...') \end{lstlisting}  \end{document} Find the exact value of the infinite sum $\sum_{n=1}^\infty \big\{\mathrm{e}-\big(1+\frac1n\big)^{n}\big\}$ I recently watched  by Veritasium where he talks about the One Way Speed of Light and talks about the limiting case where in one direction the speed of light is $c/2$ while it's instantaneous in the other. He also says this is perfectly fine according to our Physics theories. He also points at Einstein's assumption in his famous 1905 paper where he assumes that the speed of light is same in all directions. This made me ask this question is taking the speed of light same in all directions an axiom of some sort? As I've often read no information can be sent at more than the speed of light but here one-way taking the speed to infinite makes no difference. So are all of our physics theories based on the assumption and what would happen if light turns out to be moving at different speeds in different direction? Will that enable transfer of information faster than the speed of light and is there any way for us knowing that the transfer happens faster than the speed of light? The video takes a Earth Mars case where he says it isn't possible for us to every realize this discrepancy but is there a more general proof which says it isn't possible Conventions for function notation I'm in year 11 right now and I just had a brief discussion with my maths teacher about function notation in trigonometry.  For a test, I wrote this,   sin(50)^2   I assumed that would be interpreted as sin(50)*sin(50)  But I was told the correct notation for this is is  sin^2 (50)   or optionally  (sin (50))^2   I'm curious if that is the 'proper' mathematical convention, or just how things are taught in high schools?  I'm Australian, in case there are some regional differences.  Thanks for the help! Copy a file back to local system with ssh If I'm logged in to a system via SSH, is there a way to copy a file back to my local system without firing up another terminal or screen session and doing scp or something similar or without doing SSH from the remote system back to the local system? URL to load resources from the classpath in Java