Could you tell me which is correct to say,  "Today is one of my friends' birthday." or "Today is my friend's birthday."  Thank you for your advice!
When we refer to a house that belongs to a family, we say "family's house". Pluralizing family gives us "families". Referring to the houses of several families, we say "families' houses". Forming the plural possessive in such a case is rather simple.  I encountered a more complex use of this recently - referring to a single house owned by a single family in a set of houses owned by a set of families.  We can say "the families' houses" to refer to all of the houses owned by all of the families.  We can say "one of the families' houses" to refer to one of the houses owned by one or more of the families. In this context, "one of" applies to "the families' houses".  If we were instead to apply "one of" to "the families", and want to refer to the one house owned by "one of the families", how would this be written?  My first though was "one of the families's house", taking "one of the families" as a single noun and appending 's to it. This looks (and sounds) a bit strange, though. "One of the families' house" and "one of the family's house"/"one of the family's houses" seem wrong to me, and I can't really determine the correct way to say/write this.  Is there a definitively correct way that this should be written?  EDIT: To clarify, although I think I made it clear, in this context, there are several families, and each owns one house. The goal of the sentence is to refer to one of those houses without stating which specific house. Example: "The players and their families want to celebrate their team's victory; this will likely entail a party at one of the families's house".
Strange grey bordering/outline in Illustrator I made a couple of vector shapes in AI and I get this strange thin border appearing around them. I can't get rid of it. Any suggestions?    Zoomed in: Deferring admission to a computer science graduate program I have terrible screen tearing, and I don't know how to go about trouble shooting it. I would LOVE to switch to Ubuntu completely, but with this tearing, it's just too irritating. It's only present on Ubuntu based distros, (including element OS). On Manjaro (Arch fork), it does not suffer from tearing.  PC Configuration:  Motherboard: Aorus Pro Wifi z390 CPU: Intel i7 9700k RAM: 32GB  GPU: Nvidia 1060 6GB DisplayPort output connects to monitor's HDMI  Monitor reported refresh rate 75hz, Ubuntu monitor settings reported refresh rate 74.97 (can change to 60hz, doesn't help). Try to do something vs. Try doing something Is there any differences between      Try to do   And     Try doing   Thank you in advence. Real UID, effective UID, and some systems even have a "saved UID". What's the purpose of all these, especially the last one? Can I use my single entry Schengen visa to go to a different Schengen country than originally planned? My friends and I planned to visit Germany, so I got a single entry tourist Schengen visa to Germany. However, my travel plans have changed since then, because my friends are going to Italy instead. I don't have time to apply for a visa to Italy, but I have been told that I can use my Schengen visa to enter Italy. Is that true? Please help me with a tabular problem Logistic regression with binary dependent and independent variables How should I use "eponym", "eponymous" and "namesake"? I was in London and visited the Tower Bridge. Afterwards, I went to a pub called The Tower Bridge. Just to be clear, the pub is named after the bridge.   Which of the following are correct?   The bridge is the eponym of the pub. The pub is the eponym of the bridge. From the pub, I could see the eponymous bridge. After I saw the bridge, I went to the eponymous pub. The bridge is the namesake of the pub. The pub is the namesake of the bridge. Why does rubbing feet against sheets create a spark? Sometimes when I'm in bed at night, I rub my feet against the sheets and light is created. What exactly is going on?     I'm familiar with static electricity, and the triboelectric effect (creating a charge by rubbing 2 insulators together), and electrostatic discharge (releasing charge by touching a grounded conductor).     I believe that all of that is what is at work here, but I'm slightly confused since in other cases, the triboelectric effect and electrostatic discharge seem to occur as two separate steps, with different substances involved, and here, the two effects seem to be occurring simultaneously, with the same substances.     Can someone help me with a step-by-step explanation of the dynamics of what is going on? Can the sum of the first $p$ factorials ever be a perfect power for $\ p>3\ $? Has $$\sum_{j=1}^p j!=q^r$$ , where q,p,r are positive integers, and r > 1 , a solution ?   I solved partially, if r is even, then RHS is a perfect square, and there is no doubt in that. Therefore, the left side must be a perfect square as well. But for p>3, the last digit of LHS is 3 which is not the last digit of any square. Hence, p&lt;4, and hence manually checking, only solutions are p =1,3. But i cannot generalize when r is odd.  Any solution to the next part would be helpful! Is it "legal" to CREATE and DROP #SomeTable more than once? Will something accelerate forever if a constant force is applied to it on a frictionless surface? If so how? I don't understand the fact that let's say hypothetically I was pushing a block on a frictionless surface, and I kept applying the same force on the object how it would keep accelerating! Wouldn't there have to be a certain velocity where it would stop getting faster? If not then what is actually acting on the object to cause it to accelerate infinitely? Fastest way to determine if an integer's square root is an integer I'm looking for the fastest way to determine if a long value is a perfect square (i.e. its square root is another integer):    I've done it the easy way, by using the built-in  function, but I'm wondering if there is a way to do it faster by restricting yourself to integer-only domain. Maintaining a lookup table is impractical (since there are about 231.5 integers whose square is less than 263).   Here is the very simple and straightforward way I'm doing it now:  public final static boolean isPerfectSquare(long n) {   if (n &lt; 0)     return false;    long tst = (long)(Math.sqrt(n) + 0.5);   return tst*tst == n; }   Note: I'm using this function in many  problems.  So no one else will ever have to maintain this code.  And this kind of micro-optimization could actually make a difference, since part of the challenge is to do every algorithm in less than a minute, and this function will need to be called millions of times in some problems.    I've tried the different solutions to the problem:   After exhaustive testing, I found that adding 0.5 to the result of Math.sqrt() is not necessary, at least not on my machine. The  was faster, but it gave incorrect results for n >= 410881.  However, as suggested by , we can use the FISR hack for n &lt; 410881. Newton's method was a good bit slower than Math.sqrt().  This is probably because Math.sqrt() uses something similar to Newton's Method, but implemented in the hardware so it's much faster than in Java.  Also, Newton's Method still required use of doubles. A modified Newton's method, which used a few tricks so that only integer math was involved, required some hacks to avoid overflow (I want this function to work with all positive 64-bit signed integers), and it was still slower than Math.sqrt(). Binary chop was even slower.  This makes sense because the binary chop will on average require 16 passes to find the square root of a 64-bit number. According to John's tests, using or statements is faster in C++ than using a switch, but in Java and C# there appears to be no difference between or and switch. I also tried making a lookup table (as a private static array of 64 boolean values).  Then instead of either switch or or statement, I would just say if(lookup[(int)(n&amp;0x3F)]) { test } else return false;.  To my surprise, this was (just slightly) slower. This is because . Getting key with maximum value in dictionary? Deleted comments are persisting in the global inbox I realise that there is other questions out there about the  (First open one I found is ), but this is marked .  I often get the below issue where it will notify me about comments on my  (in this case what appears to be the user re-writing after deleting their comment) that no longer exist.  Also, occasionally, I see comment snippets with text that doesn't match what is on the page when I view it.  Let me know if I should be trying to edit that one to remove the tag indicating that it has been fixed, and add the above as another answer. Customizing tray/taskbar date display in Ubuntu with GNOME 3 I have to give a presentation on the theorem in Real Analysis with a fellow student. While I've looked over the  and verified that, yes, step B does indeed follow logically from step A, etc. and have internalized the proof to the extent that I can replicate it myself on paper, I still feel that I have made little progress as to why this theorem works the way it does, i.e. what is the intuition behind the theorem, such that it should make sense that it follows the way it does. Therefore I ask: what kind of intuition is there about this theorem? It would also be helpful to understand how the theorem can be used effectively in analysis.  I have seen it referred to as a sort of L'Hopital's rule for sequences, which certainly seems to make sense, but I similarly have little intuitive understanding of how that rule works, either.  I am asking this question not only for my personal understanding, but also for the sake of being able to present it in an illuminating manner, such that the rest of the class can also come away with the same sort of intuition of how the theorem works and how it is useful. Any help would be greatly appreciated.  EDIT: I should point out that the formulation of the theorem we are being tasked to prove is the following:  Let $\lbrace a_n \rbrace$, $\lbrace b_n \rbrace$ be sequences, ${b_n}$ strictly increasing and unbounded. Then if $\lim\limits_{n \to \infty} \frac {a_{n+1} - a_n}{b_{n+1} - b_n} = l$ for some $l \in \mathbb R$, then $\lim\limits_{n \to \infty} \frac{a_n}{b_n} = l$ also. The Most Efficient Way To Find Top K Frequent Words In A Big Word Sequence Input: A positive integer K and a big text. The text can actually be viewed as word sequence. So we don't have to worry about how to break down it into word sequence. Output: The most frequent K words in the text.  My thinking is like this.    use a Hash table to record all words' frequency while traverse the whole word sequence. In this phase, the key is "word" and the value is "word-frequency". This takes O(n) time.  sort the (word, word-frequency) pair; and the key is "word-frequency". This takes O(n*lg(n)) time with normal sorting algorithm.  After sorting, we just take the first K words. This takes O(K) time.    To summarize, the total time is O(n+nlg(n)+K)， Since K is surely smaller than N, so it is actually O(nlg(n)).  We can improve this. Actually, we just want top K words. Other words' frequency is not concern for us. So, we can use "partial Heap sorting". For step 2) and 3), we don't just do sorting. Instead, we change it to be  2') build a heap of (word, word-frequency) pair with "word-frequency" as key. It takes O(n) time to build a heap;  3') extract top K words from the heap. Each extraction is O(lg(n)). So, total time is O(k*lg(n)).  To summarize, this solution cost time O(n+k*lg(n)).  This is just my thought. I haven't find out way to improve step 1). I Hope some Information Retrieval experts can shed more light on this question. Minecraft 1.7.4 server issues with loading world