The space-time invariance principle states that:  $(c^2)(\Delta t^2) - l^2 = \rm invariant$ for any inertial frame.  Consider two events that occur at the same time in inertial frame $S$, and frame $S_1$ moves with translational speed $v$ relative to $S$. It seems that by the length contraction, the distance between these events should be shorter in $S_1$ because the events are moving, yet since these events do not happen simultaneously in frame $S_1$, then $(c^2)(\Delta t^2) - l^2$ for frame $S_1$ is larger than for $S$, which is a contradiction. Can someone please explain the issue in this contradiction (without using Minkowski diagrams)?
I am having a serious trouble finding an agreement between the two facts in Special Relativity:  (1) given two events $A$ and $B$ in spacetime separeted by a spacelike interval $s$, the distance in space between the two events will be the shortest in the reference frame which the two events occur simultaneously:   The invariance of the interval says $s^2=(\Delta x)^2-(c\Delta t)^2$ and $\Delta x$ is minimum when $\Delta t=0$  (2) the length contraction says $L=\dfrac{L_0}{\gamma}$ where $\gamma=(1-(v/c)^2)^{-1/2}$ , $L_0$ is the proper length of an object in a reference frame which the object is at rest and $L$ is the length of the object in a reference frame which the object moves with speed $v$ in the $x$ direction.  This says that the lenght of the object in the $x$ direction is the largest in the comoving reference frame of this object.  The question is: According to (1), the reference frame in which two events $A$ and $B$ occur simultaneously is the reference frame that gives the shortest distance in space between the two events. Let $A$ and $B$ be the two end points of a meter stick in a reference frame in which the stick is at rest and at time $t=0$. This two events occur simultaneously, so the distance in space between the two is the shortest acording to (1). But according to (2) shouldn't this distance be the largest?  What is wrong?
Is quantum entanglement affected by time dilation? Is quantum entanglement affected by time dilation? Let's say one of the entangled pair is accelerated to very high speed. When both the entangled particles are observed at the same time, will they have the opposite spin? onKeyUp not firing for ESC in IE Why an abstract class implementing an interface can miss the declaration/implementation of one of the interface's methods? A curious thing happens in Java when you use an abstract class to implement an interface: some of the interface's methods can be completely missing (i.e. neither an abstract declaration or an actual implementation is present), but the compiler does not complain.  For example, given the interface:  public interface IAnything {   void m1();   void m2();   void m3(); }   the following abstract class gets merrily compiled without a warning or an error:  public abstract class AbstractThing implements IAnything {   public void m1() {}   public void m3() {} }   Can you explain why? fleqn document class option, long text lines, and hyperref package When the fleqn option is used, the vertical spacing before an equation placed right after a "long" text line (exactly \linewidth I would say) is improperly set up. See below:    From the mathmode user guide, this could probably be fixed through the \topsep length. Is it the right way to proceed? The problem is reproduced by the code below:  \documentclass[12pt,fleqn]{book} \usepackage{amsmath} \usepackage{hyperref} \begin{document} \chapter{plo} \section{plo} eoiu oieu oeiuoe ueoi ueoi ueoiu eoiu oeu oe iupaêio umùaeoiuaem oeiu eoiu eoiu oeee. eoiu oieu oeiuoe ueoi ueoi ueoiu eoiu oeu oe iupaêio umùaeoiuaemm \begin{equation} \cos\pi=-1 \end{equation} eoiu oieu oeiuoe ueoi ueoi ueoiu eoiu oeu oe iupaêio umùaeoiuaem oeiu eoiu eoiu oeee. eoiu oieu oeiuoe ueoi ueoi ueoiu eoiu oeu oe iupaêio umùae \begin{equation} \cos\pi=-1 \end{equation} \end{document}   There is something wrong involving hyperref, amsmath, and fleqn. Removing one of those removes the problem. I want to use some constants, height and width, when creating an ellipse but it won't work with \newcommand. Here is a minimal example:  \documentclass[12pt, a4paper]{article}  \usepackage{amsthm, tikz} \usetikzlibrary{arrows, shapes, trees, positioning} \begin{document}  \begin{figure} \centering \begin{tikzpicture}       \newcommand*{\width}{2}%     \newcommand*{\height}{0.5*\width}%      \draw[fill=gray!40] (0, 0) ellipse (\width and \height); \node (Y) at (0,0) {\(Y\)};      \end{tikzpicture} \end{figure}  \end{document}    I get the following error:  error: line 14:  Package PGF Math Error: Unknown operator `a' or `an' (in '2and 0.5*2'). ...ay!40] (0,0) ellipse (\width and \height) Let $f$ be an analytic function such that $\operatorname{Im}(f(z))=\operatorname{Re}(f(z))^2$. Prove that $f$ is constant. How to solve this question? Also it would be great if someone could tell me from which book this question is from? Gmail filters/labels: Force email to stay in Inbox? How can I change the default location for bluetooth file transfers? I have an Android 2.3 device (Samsung Galaxy Y Duos, S6102) and want all files that I receive via bluetooth to save to my SD card.  How can I do that? How to install hp scanjet 200 in Ubuntu 12.04? Unable to scan with hp scan-jet 200. When I type lsusb command in terminal it shows the scanner details.  I opened the simple scan which displays no scanners found, and x sane image scanners also displays the same error. Word for in-house jargon I am trying to use following apex class with invocable method but its not showing in process builder.  global with sharing class SlackOpportunityPublisher {      private static final String slackURL = 'https://testsalesforceapps.slack.com/';      global class Oppty {         @InvocableVariable(label='Opportunity Name')         global String opptyName;         @InvocableVariable(label='Stage')         global String stage;     }      @InvocableMethod(label='Post to Slack')     global static void postToSlack(List&lt;Oppty&gt; oppties) {         Oppty o = oppties[0]; // If bulk, only post first to avoid overloading Slack channel         Map&lt;String,Object&gt; msg = new Map&lt;String,Object&gt;();         msg.put('text', 'The following opportunity has changed:\n' + o.opptyName + '\nNew Stage: *' + o.stage + '*');         msg.put('mrkdwn', true);         String body = JSON.serialize(msg);             System.enqueueJob(new QueueableSlackCall(slackURL, 'POST', body));     }      global class QueueableSlackCall implements System.Queueable, Database.AllowsCallouts {          private final String url;         private final String method;         private final String body;          global QueueableSlackCall(String url, String method, String body) {             this.url = url;             this.method = method;             this.body = body;         }          global void execute(System.QueueableContext ctx) {             HttpRequest req = new HttpRequest();             req.setEndpoint(url);             req.setMethod(method);             req.setBody(body);             Http http = new Http();             HttpResponse res = http.send(req);         }      }  } Assumptions of linear models and what to do if the residuals are not normally distributed I am a little bit confused on what the assumptions of linear regression are. So far I checked whether:  all of the explanatory variables correlated linearly with the response variable. (This was the case) there was any collinearity among the explanatory variables. (there was little collinearity). the Cook's distances of the datapoints of my model are below 1 (this is the case, all distances are below 0.4, so no influence points). the residuals are normally distributed. (this may not be the case)  But I then read the following:  violations of normality often arise either because (a) the distributions of the dependent and/or independent variables are themselves significantly non-normal, and/or (b) the linearity assumption is violated.  Question 1 This makes it sound as if the independent and depend variables need to be normally distributed, but as far as I know this is not the case. My dependent variable as well as one of my independent variables are not normally distributed. Should they be? Question 2 My QQnormal plot of the residuals look like this:   That slightly differs from a normal distribution and the shapiro.test also rejects the null hypothesis that the residuals are from a normal distribution: &gt; shapiro.test(residuals(lmresult)) W = 0.9171, p-value = 3.618e-06  The residuals vs fitted values look like:   What can I do if my residuals are not normally distributed? Does it mean the linear model is entirely useless? Give permission to debian-transmission to write into a www-data directory Do neutrinos not couple to the Higgs field? Why does Calculate button remain gray in Group Stats Plugin for QGIS? In my case I failed to use  plugin (2.0.30) with QGIS 2.14.11 and 2.18.3 on windows64.  Tested on several datasets: points, polygons, and csv.- But I am not sure what I have missed.    I want to click on this Calculate button. If there are 5 points on the surface of a sphere, then there is a closed half sphere, containing at least 4 of them.  It's in a pigeonhole list of problems. But, I think I have to use rotations in more than 1 dimension.  Regards If I want to shoot darker shots, is it better to increase the lighting and then darken in post to retain clarity? I'm just getting started - in my last shoot, I was using a 6D with a 50mm/1.4.  In low light, the pictures looked fantastic on the viewfinder but on the computer screen, appeared extremely grainy.  For future shoots, if I want to achieve something like this:     Should I shoot with way more light / lower ISO to retain clarity, and then just darken in post?  What kind of setup would be needed to achieve the above?    With my setup -   Camera Canon EOS 6D  Lens EF50mm f/1.4 USM  Focal Length 50mm  Shutter Speed 1/40 s  Aperture f/1.8  ISO/Film 6400  I was getting very grainy results that I had to soften in post (in case you want to see my attempts):      Any tips would be much appreciated. When I run the following query in query editor I get this error.  Error     Unknown error parsing query   Query  SELECT Id, OwnerId,Account.OwnerId FROM Contact where OwnerId = Account.OwnerId limit 100   Does anybody know what the problem is? This question is related to  and explores an apparent inconsistency, given only the Harry Potter movies as source material.  The Hogwarts Express line seems to have an inconsistent existence between magical (invisible to ordinary humans aka muggles) and non-magical (muggle) realms; platform 9-3/4 is clearly hidden from muggles, but in Chamber of Secrets, when Harry and Ron take the car (in the muggle world) to catch up to the train, they make no explicit crossing into a magical realm, suggesting that the train is traveling the muggle world despite its departure from a magical platform.  In which realm (magical or muggle) does the Hogwarts Express exist? How is this apparent inconsistency resolved? Natural Sort in MySQL