What is the difference between "I haven't gone" and "I haven't been" in the below context?
Can I say: I have not gone to the cinema for a long time
Does the English language have an official Academy? For some languages, there are academies that decide topics such as grammar and spelling of things, for example, for the Spanish language, there are 22 academies in 22 different countries, all making decisions on spanish grammar in their country. I would like to know if there is something that serves the same purpose for the English language. Difference between $\mathbb C$ and $\mathbb R^2$ What are the basic differences between $\mathbb C$ and $\mathbb R^2$?  The points in these two sets are written as ordered pairs, I mean the structure looks similar to me. So what is the reason to denote these two sets differently? Markdown help page is broken (badly presented/formatted) on every site In the  I see:    Is that what the help is supposed to say and look like? Because it doesn't make sense to me.  As animuson points out in a comment, all of the highlighted example text is being shifted to the very left of each line. The entire page is broken. Extraneous solution from substituting in equations I came across this example of how you could end up with an extraneous solution but I was wondering how it arose. We have the equation: $$x^2+x+1=0 $$ Since x=0 does not satisfy the equation, you can divide by x on both sides which yields: $$x+1+\frac{1}{x}=0$$ which is equivalent to our first equation. From our first eqution we can conclude that: $$-x^2=x+1$$ We now substitute that into the second equation to get:$$x^2=\frac{1}{x}$$ which results in $$x^3=1$$ which is equivalent to our previous equation since x cannot be 0. However, one solution from our last equation is x=1, which is not a solution to our original equation. I have a vague idea that it may have to do with the fact that you get a cubic equation and you began with a quadratic, and that steps imply the following and not vice versa, but can you provide a very detailed answer as to why it arises? Can you please provide more examples? What is the correlation if the standard deviation of one variable is 0? $\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$ Finding Zero Divisors and Units of Rings $\mathbb{Z}_n \times \mathbb{Z}_m$ How do I exactly find the zero divisors and units of a ring in the form of $\mathbb{Z}_n \times \mathbb{Z}_m$? For instance, how does one calculate zero divisors and units of $\mathbb{Z}_6 \times \mathbb{Z}_2$? I understand that a zero divisor is a non-zero element of $R$ such that $rs = 0$ for some other non-zero element $s$ in $R$. But I don't really quite understand how to apply that to a ring of the form $\mathbb{Z}_n \times \mathbb{Z}_m$. Also, the units of $\mathbb{Z}_6 \times \mathbb{Z}_2$ would be elements that send some element $s$ in $\mathbb{Z}_n \times \mathbb{Z}_m$ to the multiplicative identity, right? (which would be 1)? How do I apply these concepts to "product" rings? Thank you. Too much sugar in soup -- how to make it less cloying? I was making this recipe: , which was from the old Lincoln Del, one of my mom's favourite places. I mistakenly doubled the sugar, putting in 1/2 a cup instead of a 1/4 of a cup. Given the 2/3 of a cup ketchup, which was Heinz, the soup turned out way, way too sweet.   Is there any way I can rescue this? Why is a person's location exclusively tied to their communicator? How should I install HPLIP binary/proprietary plugin/driver for my HP printer or printer+scanner? Let's assume that one has some modern HP printer and/or scanner which is  supported by HPLIP package from official repository. Or supported by newer version of HPLIP package (see  about installation).  From official documentation () it is known that device needs proprietary/binary driver or plugin.  How should one install such plugin? Why is the application of probability in QM fundamentally different from application of probability in other areas? I set up a new VPS instance of Ubuntu and am wondering what the easiest way is to get up and running with a basic LAMP stack (i.e. which packages are required, which configuration options need to be tweaked, if any, etc.). I've found  many queries to check edit counts, and I get different results for each one. Which one is the most accurate and up-to-date query? Or is there any way to see the edit count in stats? EM algorithm gaussian mixtures- derivation I'm trying to make sense of a derivation I'm following from the lecture notes of Stanford's ML course.  Specifically the notes are here:   I'm particular I'm interested in the step where they are optimizing the lower bound of the log-likelihood by finding the derivative of that function w.r.t the mean (top of page 7). I'm a bit stuck on going from line 1 to line 2 in that sequence.  To copy here (using abbreviated notation for space):  $$ \nabla \mu_s \sum_{i=1}^{m}\sum_{j=1}^{k} w_j^{(i)}\log\frac{\alpha_j \exp\psi_{j}^{(i)}\phi_j}{w_j^{(i)}}  \qquad(1) $$ $$ \nabla \mu_s \sum_{i=1}^{m}\sum_{j=1}^{k} w_j^{(i)}\psi_{j}^{(i)}\qquad(2) $$ where $\psi_{j}^{(i)} =-\frac{1}{2}(x^{(i)}-\mu_j)^T\Sigma^{-1}_{j}(x^{(i)}-\mu_j)$ and $\alpha_j=\frac{1}{(2\pi)^{n/2}|\Sigma_j|^{1/2}}$.  The quantity $\mu_s$ is the mean (of a multivariate gaussian distribution) for class $s$.    In short, I'm not sure how they get from (1) to (2).  From just changing the log of (1) I see:  $$ \nabla \mu_s \sum_{i=1}^{m}\sum_{j=1}^{k} w_j^{(i)}\log \left(\alpha_j \exp\psi_{j}^{(i)}\phi_j\right) - w_j^{(i)}\log w_j^{(i)} \qquad(1) $$  To get to their form (2) my impression is that they take pre-emptively discard terms that would contain derivatives of $w_j^{(i)}$ (w.r.t. $\mu_s$).  At least if you assume that much, you get to (2).  However, as they show earlier in that PDF,  $$ w_j^{(i)} = p(z^{(i)}=j | x^{(i)}) =\frac{p(x^{(i)}|z^{(i)}=j)p(z^{(i)}=j) }{\sum_j p(x^{(i)},z^{(i)}=j) } $$ i.e. this quantity is the posterior distribution of the class labels.  Intuitively, it seems like $\partial w_j^{(i)}/ \partial \mu_s$ should be non-zero.  After all, changing the location of the mean $\mu_s$ would change your posterior probability for a particular class given the same data.  (Right?).  Just from a mechanical point of view, $\partial w_j^{(i)}/\partial \mu_j$ would seemingly have some non-zero terms since $p(x^{(i)}|z^{(i)}=j) \sim \mathcal{N}(\mu_j, \Sigma_j)$.  If I work through the algebra, I get  $$ \frac{\partial w_j^{(i)}}{\partial \mu_j} = w_j^{(i)}(1-w_j^{(i)})\frac{\partial \psi^{(i)}_{j}}{\partial \mu_j} $$  What am I missing here?  If I take the log-likelihood (without even thinking about EM algorithm),    $$ \ell = \sum_{i=1}^{m}\log \sum_{j=1}^{k}p(x^{(i)}|z^{(i)}=j)p(z^{(i)}=j) $$ and differentiate w.r.t $\mu_s$, I correctly get  $$ \mu_s^* = \frac{\sum_{i=1}^{m}w_s^{(i)}x^{(i)}}{\sum_{i=1}^{m}w_s^{(i)}} $$  so I'm comfortable with the eventual result, just not the intermediate steps. How do filament LED bulbs work, looking very similar to incandescent bulbs? Filament LED bulbs use very narrow strips containing the LED, which look similar to filaments of an old incandescent bulb:  They look like a normal bulb with large "old-style" coal filaments, and are of about the same size and shape as the common standart E27 socket bulb.    There are two basic variants of these bulbs:     One "plain" as described, looking impressively similar to a classic bulb The other has a white coating on the glass, about 15mm wide next to the metal socket.   How are these filaments build? I could imagine the it's "pretty normal" LEDs, but lots of them, directly bonded to something? (There are references to COB)  How does the heat management work? Ii have seen speculations about filling the bulb with either helium or "some patented organic gas" - could that be enough for cooling?  How does the driver - which seems to be very small - work?  In the variant with white coating - hiding some components, I assume - is already not that much space, and in the plain variant, there seems to be almost no space for something nontrivial.  Below you see some bulbs of the white coating variant, switched on:   (Images with the permission of this )    Note how the LED filament bulbs are not so much similar to the tungsten filament bulbs recently being phased out - they are more resembling the long outdated carbon filament bulbs: How to prove that 2017 divides $1^{2017}+2^{2017}+\dots+2017^{2017}$? How can I prove that $2017 \mid 1^{2017}+2^{2017}+\dots+2017^{2017}$? I tried proving it with the use of modulo, however it was to no avail. Afterwards, I  tried researching, for possible theorems, that may help, but that bore no fruit, either. Can you guys please help me proving it, as I am stuck? Thanking you in advance. Why don't we pluralize "foot" in measurements? My son is going on 5 and starting to learn a lot about how we're created. I've been with my husband for 10 years, but nearly 6 years ago I had no other choice but to move 1000 miles away and after trying the long distance thing it didn't work so we split up.   I ended up going through a rough patch and started dating an abusive jerk. After a year I was pregnant. He threatened me trying to force me to have an abortion and I refused. His grandmother was chief of police and he scared me into moving back to Florida.  Eventually, when I was 4 months along in my pregnancy my husband and I picked up right where we left off. He loved me so much and accepted our son as if he was his own and has done an amazing job at raising him.   He enrolled in college while I was pregnant, got us a house and went above and beyond to care for us both (as after giving birth I became very ill due to rare brain and spine conditions). I've always tried to do right by my son. Since day one, even while with my husband, I tried to keep the sperm donor involved. But he wanted no part of his son.  My husband stepped up when the biological father refused. The biological father would pretend that he cared, only to tell me he never cared about him anyway and was just trying to manipulate me into stopping child support (which I did anyway after the first year: I felt it wasn't right to take his money when my son had a dad who loved, supported and wanted him from the beginning).  This is a very long story so I'll try and make it not so much of a novel. The sperm donor has made it very clear he wants nothing ever to do with my son. My husband has been Dad since the moment my son entered this world. I've always planned to tell my son as early as possible. I think now, or in a few years, it won't be as hard on him compared to us telling him when he's a teen and who knows how he'd handle it.   I just don't know what to do. We have an extremely happy loving family. My son adores his dad. We never argue, we have a very happy non-dysfunctional marriage. I just feel we have a very special family. But I refuse to lie to my son. What if we wait and someone else maliciously tells him? Or if, when he's older and finds out, he spirals out of control?  I'm so confused, I just want to do the right thing and desperately need advice. Especially from others who have been through similar circumstances. We're planning on meeting with therapists and I guess child development specialists, if someone could also point me in the right direction for what type of doctor is best suited in this situation.   Also my son is very happy. He has a wonderful life and I feel as if he would do fine, knowing how much he loves his dad and just beginning to really grasp the concept of life. I don't think it would change anything for him, if that makes sense. I've been worrying over this for 5 years and really need some answers! List AppDomains in Process How long will it take for Google's disavow tool to work? I had updated the disavow list on March 18th  with spammy domains which were linked to my site. Those links still show up in Google Search Console.  I Wanna know how long does it need to take effect?