Say we have the earth and a person on it and an astronaut travelling away from the earth in a spaceship in a straight line and then coming back in a straight line at the same speed.   The person on the earth measures the time between when the astronaut leaves and when he comes back as 1 year. This is a proper time since the observer has not moved in his frame of reference. Thus according to the relation $t = \gamma t_o$ the astronaut must have experienced a longer time.   The person on the spaceship measures the time from takeoff to landing as 1 year. Since he's standing at the same point in his frame of reference this is a proper time. Thus according to the relation $t = \gamma t_o$ the person on earth must have experienced a longer time.   The above is contradictory. Either the astronaut experiences a longer time or the person on earth does. The issue is that either the astronaut is thought to be moving relative to the earth or the earth is thought to be moving relative to the astronaut. What's the explanation? Can anyone help me understand?  Okay. I’ve read through the below answers, as well as watched a few videos and the Wikipedia page on it, however, i don’t see any of them explaining it. They claim that the acceleration solves the issue, however, don’t we arrive at the same problem? In the earth point of view the astronaut is rotating, but you might as well see the earth as rotating in the astronauts view. In other words, same problem.
I read a lot about the classical twin paradox recently. What confuses me is that some authors claim that it can be resolved within , others say that you need . Now, what is true (and why)?
Repeat the same reference in footnote on different pages Scenario: SQL Server 2014 CU6. 1 AG with 2 databases (compatibility level 100). Failover Cluster with 3 nodes on Windows Server 2012 R2. Primary Availability Replica on node 1 and node 2. Secondary Availability Replica on node 3 (no votes). Async commit, manual failover, readable secondary. Node 3 is used to run reports on these 2 databases (out of 8 databases running outside AG on node 1 and 2).   Problem: Looking to patch these from SQL 2014 to SP1 and latest CU. Not my build and no experience with AG or FC so far. My "test" environment is also used for development, so no room for mistakes.   Questions:  What's the best order to patch the 3 nodes ?   Do I first have to remove the databases from AG on the node I'm patching ?  Do I need to take the Secondary out when patching it ?  No experience on this, and I can't find clear answers for my scenario (maybe not supported?) in the docs: How do I tell if ACR will work with my camera and how do I get the correct version for Photoshop? I am a beginner photographer with a Canon 7d and MacBook Pro. I'm not sure if Adobe camera Raw is compatible with either and really need advice on how to go about downloading the correct version. Is Con modifier × level added to HP every level-up? I see the  on the user page.  It seems to edit posts sometimes.  What is it? And why has it downvoted over 1800 times? See also: Let $n$ is a natural number. Prove that $$\left\lfloor\frac{n}{1}\right\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+....+\left\lfloor\frac{n}{n}\right\rfloor+\left\lfloor{\sqrt{n}}\right\rfloor$$   is even.   I used induction and the inequality $$x-1&lt;\lfloor{x}\rfloor\le{x}$$ to prove it.  Is there any other, nicer way to do it? Prove $\lim_{x\to0}\sqrt{4-x}=2$ using the precise definition of limits. (Epsilon-Delta) Prove $$\lim_{x\to0}\sqrt{4-x}=2$$ using the precise definition of limits. (Epsilon-Delta)  I am not sure how to link $0&lt;\left |x  \right |&lt;\delta$ with $\left |\sqrt{4-x}-2\right |&lt;\epsilon$ .  EDIT (Trying it out now)  I worked till here, then I basically got stuck. Prove that: area of triangle $S_{ABC}=p \cdot R$, where p is half perimeter of triangle $DEF$ Prove that: area of triangle $S_{ABC}=p \cdot R$, where p is half perimeter of triangle $DEF$. $AD,CF, BE$ are altitudes. Is there a simple way? Windows 10 100% Disk Usage after Startup After startup for 5 minutes my system is very slow. The Task Manager shows that the disk usage is at 100%. (Later this number is around 2-4 %, but sometimes for 1-2 minutes go up again at 100%.)  I searched a lot on Google. For example I read , and also other pages. I opened two topic on Microsoft answer.  and one in Hungarian.  But I didn't found the solution yet. So I would like to ask Community's help. :)  Here the steps I already done:   I disabled everything at startups (including services) and the issue is the same.  I disabled Prefetch and Superfetch, the Baground Intellingent Transmission, Windows Search and Windows Defender but the issue is still persists. I made a health check on HDD, I made defrag and file system error check, but there are no errors. Also I tried to reinstall the HDD's driver. I made a full hardware check up, with no errors. The Windows and all the drivers are up-to-date. I tried a clean start. The issue is the same. I deleted Dropbox and all antivirus. I turned of the MSI mode.      I using also Linux, where this error is not occurs. Good to know, that Windows had 4 partitions, but I deleted 2 of them for the Linux. I don't know what partitions was these but both was only 10 - 20 MB.  I using a Dell Inspiron 15R SE 7520 with Windows 10 Home.  Personally I think it can be the Intel Rapid Store Technology "driver", because the problem started around when I installed it. (However I'm not sure.) And my hardware not supported yet by Intel, so I have no the newest driver. However I deleted Intel's software and driver and the issue is the same.    Thank you in advance for your assistance! Density of ring of algebraic integers in $\mathbb C$ Clearly, $\mathbb Z$ is not dense in $\mathbb Q$ (and not dense in $\mathbb C$).  But why is the ring of algebraic integers $\overline{\mathbb Z}$ dense in $\overline{\mathbb Q}$?  In particular, if this is so, it is dense in $\mathbb C$. I want to show that the two annuli $$A=\{r&lt;|z-z_0|&lt;R\} $$ $$A'=\{r'&lt;|z-z_0'|&lt;R'\} $$ are conformally equivalent (i.e. there exists a biholomorphic map between the two) iff $$\frac{R}{r}=\frac{R'}{r'}. $$  Sufficiency:  Suppose the ratios of the radii are the same. A suitable scaling (by $\frac{r'}{r}$) will make $A$ congruent to $A'$, and a translation will map the scaled $A$ onto $A'$. This is clearly a conformal homeomorphism.  Necessity:  The annulus $A$ has the following set of periods $$\left\{ \pm \frac{2 \pi}{\log (R/r)} \right\} $$ (this step required some computations which I will omit)  Similarly $A'$ has   $$\left\{ \pm \frac{2 \pi}{ \log(R'/r')} \right\}$$ as its set of periods. Since the set of periods is a conformal invariant. The ratios of the radii must indeed by the same.  Is this proof correct?   Thanks! "Fatal error: Class name must be a valid object or a string in C:\xampp\htdocs\uc\includes\common.inc on line 7506"  The error above showed up unexpectedly. I was adding a "Long text" field to a content-type, but when I expected to be brought to a saved field configuration page, I instead got a Fatal error on every page.  I had a look at includes\common.inc:  /**  * Get the entity controller class for an entity type.  */ function entity_get_controller($entity_type) {   $controllers = &amp;drupal_static(__FUNCTION__, array());   if (!isset($controllers[$entity_type])) {     $type_info = entity_get_info($entity_type);     $class = $type_info['controller class'];     $controllers[$entity_type] = new $class($entity_type); // Line 7506   }   return $controllers[$entity_type]; }   However, I just don't understand Drupal at a low enough level to understand the relationships between these variables and functions.  I also searched Drupal for people with a similar problem:  (1)  who claims that I have to install the submodule "Relation Endpoints Field", but searching Drupal, .  (2) :  DELETE FROM cache; DELETE FROM cache_filter; DELETE FROM cache_menu; DELETE FROM cache_page; DELETE FROM watchdog;   (via , misleading name, I know.)  Clearing those tables from the cache did not help.  (3) And another user . Because I can't find a module specifically pertaining to old search facets, I went into MySQL system table, and marked the Search module's line with a 0 status, indicating to stop loading the module altogether.  On that same link as (3), one user suggested enabling the , so I dropped all the tables, imported the latest stable .sql, and installed the module, but after recreating the problem (trying to add a field to a content-type), that as well concluded fruitless.  Still, the problem remains  The Fatal error shows up on every page, making the site completely non-functional. Any help or direction would be immensely appreciated. Thank you for looking. How to check if element is visible after scrolling? I'm loading elements via AJAX. Some of them are only visible if you scroll down the page. Is there any way I can know if an element is now in the visible part of the page? How do you minimise hand washing between every step to keep tools clean? Say you cooking beef steak:  You have to unwrap the beef from its packaging, lay it down, get blood on your hands, wash hands, sprinkle salt, wash hands, grind pepper, rub the seasoning into the beef, wash hands, grab the tongs etc..  Is it normal to wash hands after every step or is there a trick to avoid getting all the tools dirty?  If you watch TV, they don't ever wash as much as I do. In reading about the  protocol I noticed how much its proofs-of-work apparently depend on  of the SHA-256 hash function. And so  presumably do many other applications.  How do researchers go about proving uniformity of proposed hash functions? Is it just an empirical matter of running many inputs through the function and observing the distribution of outputs, or can one formally prove the property? Where could one (non-crypto-expert) learn about how such proofs are constructed and look one up e.g. for the case of SHA-256? Sharing a Bluetooth Keyboard and Mouse sequentially between two Macs Let $V,W$ be two vector spaces over a field $F$ and let $f:V \to W$ be a linear mapping. I have going over some facts about vector spaces (not necessarily finite dimensional) and I have been trying to apply the rank nullity theorem to derive a simple lemma.  I have not been able to complete the proof and was wondering if I need to use something other then $\ker f + \mathrm{Im}\; f = \dim V$.  How do we show that for ever vector subspace $K \subset W$, $ \dim(f^{-1}(K)) = \dim(K \cap \mathrm{Im}\; f) + \dim( \mathrm{Ker}\; f)$? Where is the rest of Yoda's species? We never seem to see any beings of Yoda's species - except, of course, Yoda himself.  This leads us to the question of where the rest of the li'l critt'rs are. Yoda had to have parents, correct? So what happened to them? Was Yoda, put simply, just one of the last of a dying species? I'm trying to install bang and olufsen drivers, or HP hd audio driver, as only 2 of the 4 speakers work and the 2 amps don't work in ubuntu  lspci -v | grep -i audio 00:03.0 Audio device: Intel Corporation Broadwell-U Audio Controller (rev 09)     Subsystem: Hewlett-Packard Company Broadwell-U Audio Controller 00:1b.0 Audio device: Intel Corporation Wildcat Point-LP High Definition Audio Controller (rev 03)     Subsystem: Hewlett-Packard Company Wildcat Point-LP High Definition Audio Controller   The volume is terrible. Is there any way to fix this?  The speakers work perfectly on the pre-installed Windows 8. The title word filter is one of the worst ideas ever implemented on SO