Are the experiments that prove that time dilation exists, actually proving that an atomic clocks ability to keep time is affected by gravity/acceleration? How do we know 100% that that is not the case?  I am having trouble wrapping my head around how and why time dilation happens. I am not looking for a math explanation in any sense at all but a practical one. You can explain to me mathematically how a spinning spaceship creates a sense of gravity for people inside but I can take a bucket of water and spin it around and see and experience why that happens quite easily.  Is there any clear, practical explanation like this for time dilation?  Secondly how is this actually proven? How do we know for sure that atomic clock's frequencies don't fluctuate as gravity/acceleration changes? How do we know we are actually measuring "time"
Atomic clocks were used in Hafele and Keatings experiment which supposedly helped to prove time dilation.    How can we be sure other forces didn't act upon the clocks as a result of their being at different speeds and elevations? Perhaps the elapsed time may have been the same for all and only appeared different as a result of the clock's atoms behaving differently in different environments.  How might the presence of dark matter (presumably all over the place) impacted the clocks if say the dark matter is stationary relative to the earth's rotation?  What experiments have been done to test the constancy of clocks at different speeds and elevations?
Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous? Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$ be a function and suppose for any Cauchy sequence $(a_n)$ in $X$, $(f(a_n))$ is a Cauchy sequence in $Y$.  Is $f$ continuous?  Let $f$ be continuous, is it uniformly continuous? Is it possible to read a .PST file using C#? I would like to do this as a standalone application, not as an Outlook addin (if that is possible).   If have seen     to this mention  but I am looking to do this programmatically in C#.  I have looked at the  namespace but that appears to be just for Outlook addins.  appears to be able to read PST files, but this is in C (sorry Joel, I didn't ).  Any help would be greatly appreciated, thanks!  EDIT:   Thank you all for the responses! I accepted Matthew Ruston's response as the answer because it ultimately led me to the code I was looking for. Here is a simple example of what I got to work (You will need to add a reference to Microsoft.Office.Interop.Outlook):  using System; using System.Collections.Generic; using Microsoft.Office.Interop.Outlook;  namespace PSTReader {     class Program {         static void Main () {             try {                 IEnumerable&lt;MailItem&gt; mailItems = readPst(@"C:\temp\PST\Test.pst", "Test PST");                 foreach (MailItem mailItem in mailItems) {                     Console.WriteLine(mailItem.SenderName + " - " + mailItem.Subject);                 }             } catch (System.Exception ex) {                 Console.WriteLine(ex.Message);             }             Console.ReadLine();         }          private static IEnumerable&lt;MailItem&gt; readPst(string pstFilePath, string pstName) {             List&lt;MailItem&gt; mailItems = new List&lt;MailItem&gt;();             Application app = new Application();             NameSpace outlookNs = app.GetNamespace("MAPI");             // Add PST file (Outlook Data File) to Default Profile             outlookNs.AddStore(pstFilePath);             MAPIFolder rootFolder = outlookNs.Stores[pstName].GetRootFolder();             // Traverse through all folders in the PST file             // TODO: This is not recursive, refactor             Folders subFolders = rootFolder.Folders;             foreach (Folder folder in subFolders) {                 Items items = folder.Items;                 foreach (object item in items) {                     if (item is MailItem) {                         MailItem mailItem = item as MailItem;                         mailItems.Add(mailItem);                     }                 }             }             // Remove PST file from Default Profile             outlookNs.RemoveStore(rootFolder);             return mailItems;         }     } }   Note: This code assumes that Outlook is installed and already configured for the current user. It uses the Default Profile (you can edit the default profile by going to Mail in the Control Panel). One major improvement on this code would be to create a temporary profile to use instead of the Default, then destroy it once completed. If an edit was suggested to an answer to my question and I go to this edit, I see the checkmark How many connected components does $\mathrm{GL}_n(\mathbb R)$ have? I've noticed that $\mathrm{GL}_n(\mathbb R)$ is not a connected space, because if it were $\det(\mathrm{GL}_n(\mathbb R))$ (where $\det$ is the function ascribing to each $n\times n$ matrix its determinant) would be a connected space too, since $\det$ is a continuous function. But $\det(\mathrm{GL}_n(\mathbb R))=\mathbb R\setminus\{0\},$ so not connected.  I started thinking if I could prove that $\det^{-1}((-\infty,0))$ and $\det^{-1}((0,\infty))$ are connected. But I don't know how to prove that. I'm reading my notes from the topology course I took last year and I see nothing about proving connectedness... Say a creature owns multiple items with wording similar to this:  As an Action, you may speak the command word to [...]  Say the creature has deliberately obtained such items, perhaps by creating them itself, to have the same command word. Can this creature use one action to speak the shared command word and activate multiple items at once? The purpose of this question is not to cheese my DM. I am writing a short story and like to keep my D&amp;D fiction as close to rules-accurate as possible. Black is Back! New update with Achievement shows black backgrounds with black text Update: This is fixed on version 1.0.43 as per the accepted answer below.  Already saw ,  and . They are old and seem to be about previous versions.  I just updated to the version 1.0.41 of the Android app that brings the new feature - Achievements. Sadly, I think the stress of World Cup has made you guys not do a good job with it. It's buggy.  When I open my Achievements panel and tap on any post, I get this:    After I get this, it persists throughout the rest of the app. For example, if I go to the feed and open another question, I get this too.  Detailed reproduction steps are available in AD7six's answer.  My Device: Samsung Galaxy SIII LTE GT-I9305 Android: Stock ROM 4.3 Jelly Bean  Other Devices: - Device model, Android version, Reported by - HTC Desire, 4.2.2, xd6_ - Huawei Ascend, 4.2, Manu343726 - LG Optimus F3, 4.1.2, Rocket Hazmat - LG Optimus G, 4.3, Canadian Luke - Samsung Galaxy Nexus (Sprint), 4.3, Chrono Kitsune - Samsung Galaxy Note II, 4.3, Katja Christiansen - Samsung Galaxy SII GT-I9100, 4.1.2, AD7six - Sharp SH837W, 4.1.2, timrau - Sony Xperia SP, 4.3, pepoluan - Sony Xperia Z1, 4.3, user2313067 - Unknown, 4.2.1, Deer Hunter Why do you need 2 resistors when hooking up a transistor as a switch What is the point of R2 in the following diagram:   I get that R1 controls the current to the Base, but what does R2 do? I attempted to log out with the following actions:    This accomplished absolutely nothing since I remained logged in after clicking "log out" on the confirmation page. I then tried logging out of Stack Exchange from one of the network sites and repeated the steps above. The results were the same. I remained logged in no matter which combination of steps I performed.  In fact, the only way I could log out at all was to clear all of the cookies that were set. Date manipulation Function SSJS I want to delete the records from Data Extension based on Date Field in my DE i.e. I want to delete record which were created date before 30 days.But I didn't find date manipulation function in SSJS like dateadd() in ampscript Combo box in Java - nullpointer The banner ads are in the wrong format for the right-hand ad slot:    These kinds of ads are not useful (because most of it is cropped) and should be excluded from being shown in that spot. Apache error "Could not reliably determine the server's fully qualified domain name" NSXMLParser chokes on ampersand & Recently I saw a widget on the Stack Overflow website which says Stack&nbsp;Overflow now has its own currency called "unicoins". According to this widget, a user can earn reputation, customize Stack Overflow by changing its colors and the user can also remove the vote down button from their questions and answers by spending "unicoins".  This widget does not look real, because this is out of standard of the Stack Overflow website.  Also today is 1st April. Is Stack Overflow fooling us?  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Let $x,y,z\ge 0$, show that $$\dfrac{y}{xy+2y+1}+\dfrac{z}{yz+2z+1}+\dfrac{x}{zx+2x+1}\le\dfrac{3}{4}$$  I had solve $$\sum_{cyc}\dfrac{y}{xy+y+1}\le 1$$ becasuse After some simple computations, it is equivalent to  $$(1-xyz)^2\ge 0$$ Out-of-control starship results in some crew members landing in a rebirthed universe? How to find a good topic for a PhD research proposal? In many countries an application for a PhD position includes a written research proposal, so my questions is what are some advises/strategies to come up with a good topic/idea for a PhD research proposal and how can one assess the quality/fruitfulness of an idea? As an undergraduate student one just doesn't have the experience to foresee which ideas might have promising research results and which probably won't have. (And I doubt that potential supervisors have the time to comment on every idea of every potential applicant in cases where it is possible to establish some kind of contact before the actual application.) Do a "git export" (like "svn export")? I've been wondering whether there is a good "git export" solution that creates a copy of a tree without the .git repository directory. There are at least three methods I know of:   git clone followed by removing the .git repository directory.  alludes to this functionality but starts with "Just read the desired tree into the index..." which I'm not entirely sure how to do.  is a third party script that essentially does a git clone into a temporary location followed by rsync --exclude='.git' into the final destination.   None of these solutions really strike me as being satisfactory. The closest one to svn export might be option 1, because both those require the target directory to be empty first. But option 2 seems even better, assuming I can figure out what it means to read a tree into the index. How to prove positive semidefiniteness of two matrices through Schur Complement? Let matrices $X$ and $Y$ be positive semideﬁnite. Show that $X \succeq Y \succ 0$ is equivalent to $Y^{-1} \succeq X^{-1}$.   The teacher tells me that it is easy to prove via the Schur complement. But I could not get his idea. Could anyone help me and give me some intructions? Thanks in advance! In the Iron Man movie Tony flies the Iron Man armour from Miami to Afghanistan, roughly 7800 miles.    What is the maximum range of the Iron Man armour?   What sort of speed does the armour cruise at?  How long would the flight from Miami to Afghanistan have taken?   Could Iron Man circumnavigate the globe without stopping, and how long might this journey take him?   EDIT: Just found this (funny) image, New York to Washington in under an hour?!? :)