What is the difference between these two phrases:  something he never did before  something he has never done before
I'm very confused about tenses.  I have examples in two different situations here. Situation 1: I went to New York two months ago and was talking to someone about it.  A. This is the first time I've been to New York. B. That was the first time I'd been to New York.  Which is correct in a daily conversation?  Situation 2: I'm going to New York next month and talking to someone.  A. This will be the first time I've been to New York. B. This is the first time I've been to New York. C. This will be the first time I go to New York.  Which sentence is appropriate?  According to a grammar article I've read before, I think it says that I can use this sentence in both cases:  This is the first time I've been to New York.  But I'm not sure if this is really so. So I'd like to know which tense I should use in each case. Could you explain it?
Why does water take a "wet" path? Allow me to explain what I mean... ((I am asking this question in the physics section in hopes of finding a "physics" argument))  I shall make it clear what I mean by demonstrating an experiment...  Hold your hand upright with your index finger pointing up towards the sky. Now drop about 2-3 drops of water right on top of your finger. Water drop should (theoretically) stay there. Now tilt your hand ever so slightly towards one side. The water drop rolls down your finger down your wrist leaving a trail towards the tilted side right?  Now's here comes my question. Now if you bring your finger and the hand to the initial upright state and place another water drop on the same finger. Now we see that the water drop takes the same trail! Why does it happen?  My argument: The water trail which was previously left off reduced the friction in that path by filling the gaps between the molecules. Hence the new drop tends to slide more on that path and hence ends up travelling in it.  ((This "effect" is also observed when tear drops keep rolling down in the same path. But that could just be because that path is the steepest.)) Why is English written and read from left-to-right as opposed to right-to-left, top-to-bottom, or (not even sure any language does this) bottom-to-top? Why is it OK to jump into the scope of an object of scalar type w/o an initializer? fake proof of $\forall a. \forall b. a = b \to 1 = 0$ I saw a less formal version of this fake proof that claimed to prove $2=1$ but because it assumed $a=b$ from the start I knew why it was wrong. It does seem however that the proof can be used to prove an certain implication I find very odd however. Clearly $\forall a. \forall b. a = b \to 0 = 1$ is false because I could pick $a$ and $b$ to be $1$ and then I would have derived 0 = 1 with no other conditions. So there must be something wrong with this proof that I am not seeing.  proof of $\forall a. \forall b. a = b \to 0 = 1$:  $a = b \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (assume)\\ a+a=a+b  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (x=y \to f(x) = f(y), f(x)=a+x)\\ 2a=a+b \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  (a+a=2a)\\ 2a-2b=a+b-2b \,\,\, (x=y \to f(x) = f(y), f(x)=x-2b)\\ 2(a-b)=a+b-2b \,\, (distributive)\\ 2(a-b)=a-b \,\,\,\,\,\,\,\,\,\,\,\,\,\, (-b=b-2b)\\ 2=1 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (x=y \to f(x) = f(y), f(x) =x/(a-b))\\ 1=0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (x=y \to f(x) = f(y), f(x) =x-1) $  now that I have "proven" $\forall a. \forall b. a = b \to 0 = 1$ I can now pick a and b to be 1.  $ 1 = 1 \to 1=0 \,\,\,\,\,\, (a=1, b=1, above)\\ 1 = 1 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\text{reflexivity})\\ 1=0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\text{modus ponens}) \\ $  So now it seems that I have managed to derive $1=0$. Clearly I messed up given that no one has been able to show $ZFC$ inconsistent and ostensibly this fits well within the most studied aspects of $ZFC$. Either that or a random guy that makes meme's online found the most profound result in mathematics ever.  So where have I missed the mistake? How does one use circular polarizers? What does it mean to "program to an interface"? Can you raise $\pi$ to a real power to make it rational? Why is the Cantor function not absolutely continuous? Is there an easy way to see that the Cantor function is not absolutely continuous that directly uses the definition of absolutely continuous? How can I read a file even when getting an "in use by another process" exception? bash: ./program: cannot execute binary file: Exec format error I'm trying to run a program, but an error happens like this:  bash: ./program: cannot execute binary file: Exec format error   The result of file program was:  program: ELF-32-bit LSB executable, ARM, EABI4 version 1 (SYSV), dynamically linked(uses share libs), for GNU/LINUX 2.6.16, not stripped   How can I fix this error?  I'm using Ubuntu 14.04.2 (amd64) with VMware. I also tried with Ubuntu i386, but result was same. Limit of $\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} }$ Direct way leads to nowhere.  $$\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} } = \lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{n \sqrt {1 +\frac{i}{n^2}} } = $$  $$ = \lim_{n\rightarrow \infty} \frac{1}{n} \cdot \lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {1 +\frac{i}{n^2}} } = 0 \cdot \sum^{n}_{i=1} 1 = 0 \cdot n $$  So, I tried to apply squeeze theorem:  $$\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} } \leq \lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{n}  = 1$$  Could you help me to figure out the lower bound? $f$ bounded but $f'$ isn't Did Luke attempt to retake the Jedi Temple on Coruscant? After the New Republic gains control of Coruscant through the Galactic Concordance, did Luke Skywalker attempt to reclaim (or even visit) the former and still very much intact Jedi Temple, now converted to be the deceased Emperor's palace?   If Luke is so insistent on saving a tree just because it belongs to the Jedi and the Empire stole it, then shouldn't reclaiming the Jedi Temple be even higher priority?  If he did, is he aware of the Sith shrine beneath it? Which is something even Yoda does not know. And if he didn't, why didn't he? Is there an intuitive reason the brachistochrone and the tautochrone are the same curve? The  asks what shape a hill should be so a ball slides down in the least time.  The  asks what shape yields an oscillation frequency that is independent of amplitude.  The answer to both problems is a cycloid.  Is there an intuitive reason why these problems have the same answer?  Proposed operational definition of "intuitive":  Imagine modifying the problem slightly, either to the  (tunnel through Earth), or by taking account of the finite radius of the ball.  If the answer to the original question is intuitive, we should easily be able  to tell whether these modified situations continue to have the same brachistochrone and tautochrone curves. I just proved that any finite group of order $p^2$ for $p$ a prime is abelian. The author now asks to show that there are only two such groups up to isomorphism. The first group I can think of is $G=\Bbb Z/p\Bbb Z\oplus \Bbb Z/p\Bbb Z$. This is abelian and has order $p^2$. I think the other is $\Bbb Z/p^2 \Bbb Z$.   Now, it should follow from the fact that there is only one cyclic group of order $n$ up to isomorphism that these two are unique up to isomorphism. All I need to show is these two are in fact not isomorphic. It suffices to show that $G$ as before is not cyclic. But this is easy to see, since we cannot generate any $(x,y)$ with $x\neq y$ by repeated addition of some $(z,z)$.  Now, it suffices to show that any other group of order $p^2$ is isomorphic to either one of these two groups. If the group is cyclic, we're done, so assume it is not cyclic. One can see that $G=\langle (1,0) ,(0,1)\rangle$. How can I move on? What are the differences between Generics in C# and Java... and Templates in C++? Spotlight with Halo applied is creating rings around light Here's a portion of a render I'm working on, which is backlit from behind by a spotlight with a "Halo" effect applied:    If you look carefully in the background, you'll notice the halo effect is creating "rings" of shadow as it fades.  How can I remove these rings and make the halo effect fade out smoothly?  I've played around with all the settings I can find without any success.  Here's my current settings for the spotlight: If I am thinking about using one of the 'fancy' top level domains such as .me or .tv should I be worried about how Google treats such domains?  Or to put it another way, if all other things are equal (good content, links etc.), will Google regard a .tv or .me as equal to a traditional .com or co.uk site? Using some matlab (I know it's cheating) I found that: $$k=0 =&gt; Result=1$$ $$k=1 =&gt; Result=1$$ $$k=2 =&gt; Result=2$$ $$k=3 =&gt; Result=5$$ $$k=4 =&gt; Result=15$$ $$k=5 =&gt; Result=52$$ $$k=6 =&gt; Result=203$$  This sequence is puzzling... Intuitive explanation for dividing by $n-1$ when calculating standard deviation? I was asked today in class why you divide the sum of square error by $n-1$ instead of with $n$, when calculating the standard deviation.  I said I am not going to answer it in class (since I didn't wanna go into unbiased estimators), but later I wondered - is there an intuitive explanation for this?!