$[a,b)$ and $(a,b)$ are not homeomorphic
An Open Interval and a Half-open Interval are not Homeomorphic
possessive apostrophe for two things or persons What is the difference between the two sentences below? Are they the same or different in meaning?     Jack and Laura's new house is almost ready.      Jack's and Laura's new house is almost ready.   To my perception, the first says two of them have one house and the second says each one has a (different) house (which is almost ready). Is my perception correct?  EDIT: If I don't use "houses" instead of "house" (for my second interpretation), does it mean the sentence is wrong? A file manager with free icon placing option I would like to pause a render that I have started, with the intention of resuming it at a later time (but during the same Blender session).  How can I do this? What ways could you create a data extension that contains all the subscribers (SubscriberKeys) that are currently “in” a Journey (Journey Builder)?  The Journey I would like to do this for is a complex activity with multiple decision branches resulting in multiple possible journey durations. For example; if the customer didn’t open Email1 after 2 days, then send Email2; else Exit Journey. If the customer didn’t open Email2 after 2 days, then send Email3; else Exit Journey. Wait 2 more days, then End Journey. The minimum journey duration for the above example is 2 days, while the maximum is 6 days.  Note that I don’t want a list of subscribers that “entered” a Journey, but a list of the ones that are “currently in” a Journey and have not yet exited. Answer must account for non-standard Journey exits including Unsubscribes/Contact Deletes, etc. Firefox - Websites with custom fonts not displaying On Cesàro convergence: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $ I have this problem I'm working on. Hints are much appreciated (I don't want complete proof):  In a normed vector space, if $ x_n \longrightarrow x $ then  $ z_n = \frac{x_1 + \dots +x_n}{n} \longrightarrow x $   I've been trying adding and subtracting inside the norm... but I don't seem to get anywhere.  Thanks! Define ≡ in this situation? PHP function substr() error Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field I'm reading Galois Theory by Steven H. Weintraub (second edition), and finding that I'm at least somewhat short on the prerequisites. However the following proof looks wrong to me - am I misunderstanding something, or is it actually an incorrect proof?  Lemma 2.2.3. Let $F$ be a field and $R$ an integral domain that is a finite-dimensional $F$-vector space. Then $R$ is a field. Proof. We need to show that any nonzero $r \in R$ has an inverse. Consider $\{1, r, r^2, \cdots\}$. This is an infinite set of elements of $R$, and by hypothesis $R$ is finite dimensional as an $F$-vector space, so this set is linearly dependent. Hence $\sum_{i=0}^n{c_i r^i} = 0$ for some $n$ and some $c_i \in F$ not all zero.  It then goes on to show, given the above, that we can derive an inverse for $r$. However, if I consider examples like $r = 2 \in Q[\sqrt{2}]$, $r = \sqrt{2} \in Q[\sqrt{2}]$ or $r = 2 \in Q[X]/{&lt;X^2&gt;}$, the set $\{1, r, r^2, ...\}$ doesn't look linearly dependent to me. I do believe the lemma is true (and might even be able to prove it), but this does not look like a correct proof to me. Am I missing something? [Edit] Well yes, I am. Somehow I had managed to discount the possibility of any $c_i$ being negative, despite repeatedly looking at each fragment of the quoted text in an attempt to find what I might be misunderstanding. How can the following inequation be proven?  $$a^2 + b^2 + c^2 \ge ab + bc + ca$$ I have 4 copies of the Hot Questions widgets.  Sometimes, possibly after they reload on their own, I can only tap to open the last widget I added. If I tap on the question of any others nothing happens.  If I refresh each one then tapping works on the one I refreshed. WCF - Faults / Exceptions versus Messages We're currently having a debate whether it's better to throw faults over a WCF channel, versus passing a message indicating the status or the response from a service.  Faults come with built-in support from WCF where by you can use the built-in error handlers and react accordingly. This, however, carries overhead as throwing exceptions in .NET can be quite costly.  Messages can contain the necessary information to determine what happened with your service call without the overhead of throwing an exception. It does however need several lines of repetitive code to analyze the message and determine actions following its contents.  We took a stab at creating a generic message object we could utilize in our services, and this is what we came up with:  public class ReturnItemDTO&lt;T&gt; {     [DataMember]     public bool Success { get; set; }      [DataMember]     public string ErrorMessage { get; set; }      [DataMember]     public T Item { get; set; } }   If all my service calls return this item, I can consistently check the "Success" property to determine if all went well. I then have an error message string in the event indicating something went wrong, and a generic item containing a Dto if needed.  The exception information will have to be logged away to a central logging service and not passed back from the service.  Thoughts? Comments? Ideas? Suggestions?  Some further clarification on my question  An issue I'm having with fault contracts is communicating business rules.  Like, if someone logs in, and their account is locked, how do I communicate that? Their login obviously fails, but it fails due to the reason "Account Locked".  So do I:  A) use a boolean, throw Fault with message account locked  B) return AuthenticatedDTO with relevant information Book wherein humans were most dangerous/aggressive of numerous species I recall there were two warring alien empires, one of which was led by a race that could telepathically inflict pain on others, but when they tried to do it to humans, it would backfire. something about human's animal instincts being much closer to the surface than the other species.  Humans made the most aggressive and dangerous fighters, so both empires tried to reruit humanity, since the other races involved in the war were fairly monolithic and sided all with one empire or the other. Humanity was all splintered though.  The empire led by the telepathic race claimed that everyone in it was equal but in reality the telepathic race was in charge. How to resolve error "Handle bad layers" in QGIS? I'm new to QGIS and have been seeing what it can do for me – seems good. I imported shape file data from a UK Ordnance Survey data file and saved the file in a new folder. It was very good, doing all I wanted. I exported and got exactly what I wanted ( a tiff image).   I closed the file in order to try a new experiment. When I went to re-open it I got a dialogue box saying 'Handle Bad Layers. I don't understand what is at issue and what to do. Information on month old flights Where can I get flight info for a flight a month ago for an insurance claim? Flight stat sites will only show a weeks worth of data. "There are no comments" vs. "There is no comment" Today I found my first ever (to the best of memory) emoji in a post.        Having no idea this was even possible I came here looking for some history. I found there is an  tag. There seems to be consensus that they add little to no value on SE per  &amp;  we also know they don't translate well on all devices    I don't find any history other then this one post  where there is any focused attempt to support them. I don't find any history of discussion on supporting them.    Two questions   Is it possible to disable emojis across Stack Exchange?   If yes   Should we disable emojis across Stack Exchange? Restoring backups failing Ubuntu 16.04 In RSA, why is it important to choose e so that it is coprime to φ(n)? When choosing the public exponent e, it is stressed that $e$ must be coprime to $\phi(n)$, i.e. $\gcd(\phi(n), e) = 1$.   I know that a common choice is to have $e = 3$ (which requires a good padding scheme) or $e=65537$, which is slower but safer.  I also know that for two primes $p,q$, we have $\phi(pq) = (p - 1) (q - 1)$  Now, let me give a (simple) example:  Say I choose $e = 3$, and two random primes $p = 5$ and $q = 13$.  I can now compute $\gcd(3, \phi(5 \cdot 13)) = 3$.  This reveals that $3$ and $\phi(n)$ are not coprime. I assume this could also happen for large values of $p$ and $q$, and likewise for another $e$. I therefore assume that the RSA algorithm must check that $\gcd(e, \phi(pq)) = 1$. But let's assume it doesn't.  How does RSA become vulnerable if $\gcd(e, \phi(pq)) \neq 1$? What do the plus and minus signs mean in Objective-C next to a method?