It's a question that was bothering me for some time. In special relativity one can consider the "twin paradox", where one twins leaves earth and returns, staying young while the twin left behind grows older. This is not a paradox because the twin in the spaceship accelerates to change his direction when turning back- therefore he is not in an inertial frame.  However, in a universe with the geometry of a flat torus, one can go "around" the universe and return to earth without accelerating or experiencing any intrinsic curvature of the universe. In this situation the paradox rises again: which twin is older and which one is younger, considering that both twins' frames of reference are valid?
Take the following gedankenexperiment in which two astronauts meet each other again and again in a perfectly symmetrical setting - a hyperspherical (3-manifold) universe in which the 3 dimensions are curved into the 4. dimension so that they can travel without acceleration in straight opposite directions and yet meet each other time after time.  On the one hand this situation is perfectly symmetrical - even in terms of homotopy and winding number. On the other hand the Lorentz invariance should break down according to , so that one frame is preferred - but which one?  So the question is: Who will be older? And why?  And even if there is one prefered inertial frame - the frame of the other astronaut should be identical with respect to all relevant parameters so that both get older at the same rate. Which again seems to be a violation of SRT in which the other twin seems to be getting older faster/slower...  How should one find out what the preferred frame is when everything is symmetrical - even in terms of GRT?  And if we are back to a situation with a preferred frame: what is the difference to the classical Galilean transform? Don't we get all the problems back that seemed to be solved by RT - e.g. the speed limit of light, because if there was a preferred frame you should be allowed to classically add velocities and therefore also get speeds bigger than c ?!? (I know  is only a local theory but I don't understand why the global preferred frame should not 'override' the local one).  Could anyone please enlighten me (please in a not too technical way because otherwise I wouldn't understand!)  EDIT Because there are still things that are unclear to me I posted a follow-up question:
Looking to identify a short story I read in a sci-fi anthology as a child, about an angry, bullied somewhat misanthropic boy who is given immense near-omnipotent powers by an alien intelligence. I read this in English, sometime in the late 1990s/early 2000s. He goes on to use this power to destroy the human race, which causes him to age terribly in a very short amount of time. At the end of the story he is a lonely old man stripped of the powers he held briefly, and is alone on an isolated Earth. The aliens then proceed to invade Earth. How did Anakin Skywalker survive Order Sixty Six? . When activated, troopers would immediately perceive Jedi as a threat and kill them.  Since Anakin was a Jedi until around the time of the Order, how did the troopers know not to kill Anakin? Do multiple instances of the Extra Attack feature stack? It is true that $P(X&gt;t+s|X&gt;t)=P(X&gt;s)$ for certain values $t$ and $s$.  However, how can I show that this still holds if:   $T$ is a continuous random variable. That is $P(X&gt;T+s|X&gt;T)=P(X&gt;s)$ Both $T$ and $S$ are continuous RVs: $P(X&gt;T+S|X&gt;T)=P(X&gt;S)$   All the random variables are independent. What is the difference between an "API," a "framework," an "IDE," and an "engine?" I am just starting out trying to learn to develop games. At first I thought C++ and OpenGL were the tools a beginner would start out to make a game, but quickly found out the OpenGL was just a API for graphics. Then I thought C++ and SFML were what I needed to learn to make a game, but found out the SFML was a media framework that could be used to make games but wasn't a game engine.  I don't want to use a software that does all the work for me, I want to code my own game, the way a game programmer would. I don't understand the difference between a IDE, framework, API, and a engine; What separates them from each other? Do some game programmers use the SFML framework, while other programmers use a game engine? Type safety: Unchecked cast In my spring application context file, I have something like:  &lt;util:map id="someMap" map-class="java.util.HashMap" key-type="java.lang.String" value-type="java.lang.String"&gt;     &lt;entry key="some_key" value="some value" /&gt;     &lt;entry key="some_key_2" value="some value" /&gt;    &lt;/util:map&gt;   In java class, the implementation looks like:  private Map&lt;String, String&gt; someMap = new HashMap&lt;String, String&gt;(); someMap = (HashMap&lt;String, String&gt;)getApplicationContext().getBean("someMap");   In Eclipse, I see a warning that says:  Type safety: Unchecked cast from Object to HashMap  What did I do wrong? How do I resolve the issue? Atmospheric lighting in cycles How would I go a about making volume light beams?  I have have tried  various tutorials, but they must be for a older version. as applying them does seemingly nothing.  A link to a functioning tutorial would be greatly appreciated as well.  I don't have a picture or .blend to show, as my problem with atmospheric lighting does not pertain to any specific scene. "At least one" - singular or plural subject? Differential equation for Harmonic Motion Particle undergoes simple harmonic motion.  Initially Its displacement is $1$, velocity $1$ and acceleration is $-12$   Compute displacement and acceleration when the velocity is square root of $8$.  I am not familiar with this kind of application question.  So, its displacement is $1$, does that mean that $y=1$ when $x=0$ ? I am using MikTex 2.9 and TeXstudio for my TeX preparation. I would like to set the path to my figures folder, which contains all the figures, plots etc. required for my work.   I used \graphicspath to set the directory, but it gives me an error message:  \documentclass[12pt]{article}  \usepackage{graphicx}  \graphicspath{D:/LATEX/Reports@IIT/figures}  \begin{document} \includegraphics[width=4.5cm,height=4cm,angle=0]{logo.jpg}  \end{document}      Error: Package pdftex.def Error: File `logo.jpg' not found   However, when I prepend the filenames of the images with the same path in the \includegraphics command, it works just fine:  \documentclass[12pt]{article}  \usepackage{graphicx}  \begin{document}  \includegraphics[width=4.5cm,height=4cm,angle=0]{D:/LATEX/Reports@IIT/figures/logo.jpg}  \end{document}   Why is \graphicspath not setting the path? According to my logic,  $$\large\sqrt{x^2} = x^{2\times \frac{1}{2}} = x = x^{\frac{1}{2}\times 2}={(\sqrt{x})}^2$$  But when I look at the graphs of these guys, they're totally different.  Edit: Complex answers are okay, if you know what I mean. Why does the variance of the Random walk increase? Why does my ArrayList contain N copies of the last item added to the list? Help with Axis Placement when extruding I am just learning Blender. I am going through the .  The first part was good. I was able to follow it. Coming from SketchUp, it is taking some adjustment. But I am getting it.  I am just starting the second video. At the beginning, the instructor selects the top of the mug to extrude it. When he presses the E key and Enter, the red, green blue axis moves to the center of those vertices. That does not happen when I press the E key and Enter.   This is only 1:30 into the video.  How can I get the axis in the middle of this circle of vertices?  The next step is to scale. I need the reference point in the middle of this circle of vertices. I cannot left-click accurately enough to position the axis/reference point.  I am using 2.69. The video uses 2.64. Filling a polygon with a pattern of insets Custom cross-reference formats (e.g. "Equation 1" instead of just "1") I'm trying to make taint spread to a certain area in my world, as in, to a small NPC structure. Is there any way I can do this? I even tried using MCEdit to set the biome, but there's no "enter biome ID" thing, and Tainted Land isn't listed. How to style text in hyperref \url? I want to set color, font and similar style attributes for the text in a \url from the hyperref package. How do I go about doing it? All I have found are . I want to because I am using ttf fonts for my texts and I want my urls to be in that ttf and not the default (typewriter font?) that the hyperref package gives me. I started with just trying to get it in italics but didn't even succeed with that:  \documentclass{memoir} \usepackage{hyperref} \begin{document} Lala \url{\itshape www.example.com}. \end{document}     I am guessing there is some hook or something but I just can't find it. Maybe I have gone blind? How to save the render result within Blender Is it possible to render an image and have blender remember it, so that next time I open the file the render result will not be blank?  I am not talking about saving to an external file.  Specifically I would like to render an image, close blender, and come back later and be able to do some compositing on the render without having to waste a half an hour rendering it again (my GPU drivers are messed up so I have to use the CPU).  Currently the render result is blank when ever I open a file. How to use different domains for multilingual sites and keep users logged in? I have a site that is in French and English. I am able to successfully setup so that users can switch back and forth between the two domains to view either language, however I cannot seem to keep users logged in. When they are logged into the english domain and then switch to the french domain they are required to login again. However if they switch back to the english domain they are still logged in.  As an example we'll use www.english-domain.com as the base (default) domain and www.french-domain.com as the secondary domain  In settings.php I have modified the following, however it doesn't seem to be keeping them logged in. Am I missing something?  /**  * Drupal automatically generates a unique session cookie name for each site  * based on its full domain name. If you have multiple domains pointing at the  * same Drupal site, you can either redirect them all to a single domain (see  * comment in .htaccess), or uncomment the line below and specify their shared  * base domain. Doing so assures that users remain logged in as they cross  * between your various domains.  */ $cookie_domain = 'english-domain.com';   Note: That I have also flushed both site and browser cache as well as deleted any cookies left on my machine.  Update:  Here are my language settings: