Use of Present Perfect
How do the tens­es and as­pects in English cor­re­spond tem­po­ral­ly to one an­oth­er?
Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$ Add a "voted to close" activity event somewhere on the user profile page Often when I vote to close, I want to go back and check the comments on the question, to see if it's been improved, to comment further, or to answer the question.  It would be helpful to add a bit of a trail so I can find those questions again. I'm suggesting adding an event to the Activity page on the user profile, because this is the logical spot for the given action.  I realize the Activity page is public, but I don't see why this content couldn't be added only for the currently logged-in user.  An alternative would be to add a new tab on the user profile page that only lists current-user content, and is not shown publicly. I'm sure there are other events, data, or statistics that could be shown here as well, so I'm certain a new tab like this wouldn't be underutilized.    One thing that could be done instead of this is to always add a comment stating why I voted to close. This works well enough, but only some of the time -- for example, if there is already a vote to close as a duplicate and a comment with a link to that duplicate, I'm not going to add a comment myself if I have nothing to add aside from a close vote.  Another way to do it would be to favourite the question. This only works well to a point since questions that get voted to close don't necessarily end up being downvoted (i.e., for my "closed favourites," I couldn't just look at the last page of my favourites).    Related: Jon Skeet  that votes to close on your own questions be shown in the Recent Activity area, which is not what I'm requesting here. Floating sidebar can get too long and obscure footer Sometimes, when you create a question which returns a lot of similar questions, it makes the height of the floating "Similiar Questions" box too big to fit on the page. As a result, it blocks some of the footer on the right hand side. It's actually doing it as I type this question.  I think a max height based on the view port - footerHeight should fix it. I got a stomach flu or I got the stomach flu or I got stomach flu? Schengen Visa four years- 90 days stay I have been granted multiple entry Schengen Visa valid for 4 years with the duration of stay of 90 days. Does it mean only 90 days of stay in total during a 4 years period?  I'll be travelling to Schengen Area in September for 25 days, then after 2 month or so I will travel again for 30 days. Similarly, I'll be traveling to the Schengen Area in the future depending on the business meetings.  How will I know, that how many days are left? On what basis the calculations are done?  Also, please explain the "90/180" rule. How to get a drive formatted with exfat working? In Ubuntu 13.04 there was an option to use the PPA by Relan in order to get ExFat functionality. Seeing that in Ubuntu this functionality is now available from the repositories without the PPA I tried this but failed to make it work.  So my question is how do I enable exfat support? On found exact duplicates, should these be voted to close directly against a preferred, or ultimate, canonical question?   Or is a chain of possible duplicates (where one question is closed as duplicate of another, which itself is a duplicate of another, and so on and so on) allowable or OK?  One example of this situation can be found . Let $f_1 : [-1, 1] \rightarrow \mathbb{R};\: f_1(0) = 0 $ be a continuously differentiable function and $\lambda &gt; 1$. Consider the sequence of functions defined inductively by $f_k(x) := \lambda f_{k-1}\left(\dfrac{x}{\lambda}\right);\: k \geq 2;\: x\in [-1, 1]$. Find the pointwise limit of the sequence of functions $(f_n)$.   I figured out that $f_k(0)=0 \:\forall\: k\implies f_k\rightarrow 0 $ when $ x=0$. Also $f_k$ is continuously differentiable $\forall \:k$.   $\Bigg($Since $\quad\displaystyle\lim_{h\rightarrow0}\dfrac{f_2(x+h)-f_2(x)}{h}=\lambda\lim_{h\rightarrow0}\dfrac{f_1\left(\frac{x+h}{\lambda}\right)-f_1\left(\frac{x}{\lambda}\right)}{h}.\: $Since$ f_1 $is continuously differentiable we see $ f_2 $ is differentiable and its derivative is continuous and hence $f_k$. $\Bigg)$.  Further since $f_1(0)=0$. So I assumed $f_1(x)=x\cdot g(x)$, where $g(x)$ is continuously differentiable and $g(0)\neq0$. Assume $x\neq0,\:f_2(x)=\lambda\cdot\dfrac{x}{\lambda}g\left(\dfrac{x}{\lambda}\right)=x\cdot g\left(\dfrac{x}{\lambda}\right)\implies f_3(x)=x\cdot g\left(\dfrac{x}{\lambda^2}\right)$ $\implies f_k(x)=x\cdot g\left(\dfrac{x}{\lambda^{k-1}}\right)$.   So taking limit as $k\rightarrow \infty$, we get $\displaystyle \lim_{k\rightarrow\infty}f_k(x)=x\cdot g(0)$. Now I am not sure how to proceed or whether there is something wrong here. Please provide hints or suggestions.  $\rule{17cm}{1pt}$  To sum up (using the arguments made by $\textbf{zhw}$ and $\textbf{Paul Frost}$)  When $\displaystyle x\neq0, \lim_{n\rightarrow\infty}f_{n+1}(x) = \lim_{n\rightarrow\infty}x\,\frac{f_1(x/\lambda^n)}{x/\lambda^n} = \lim_{n\rightarrow\infty}x\,\frac{f_1(x/\lambda^n)-f_1(0)}{x/\lambda^n}=x\:f'(0).$ Thus  $$f_n\rightarrow\begin{cases}0&amp;x=0\\x\:f'_1(0) &amp;x\neq0\end{cases}=f $$ Background  I am a first year calculus student so I would prefer if answers remained in Layman's terms.  It is common knowledge and seems to me a mantra that I keep hearing over and over again to "not treat differentials/derivatives as fractions".  I am of course, in particular, referring to Leibniz notation.  However, aside from a quick response such as "oh, it's because its not a fraction but rather a type of operator", I never really got a full answer as to why we can't treat it as such. It just kind of sits at the edge of taboo in my mind where it sometimes gets used and sometimes doesn't.  Confusion is further compounded when a lot of things seem to just work out if we treat them just as fractions (e.g. u-substitution / related-rates)    Example     Air is being pumped into a balloon at a rate of $100cm^3/s$. We want the rate of change of radius when the radius is at $25cm$.   $$\text{we are given}\ \frac{dv}{dt}=100cm^3/s$$ $$\text{we want}\ \frac{dr}{dt}\ \text{when}\ r=25cm$$ Thus we will solve this by using the relation $v=\frac{4}{3}\pi r^3$ $$\frac{dv}{dt}=\frac{dv}{dr}\frac{dr}{dt}$$ $$\frac{dv}{dt}\frac{dr}{dv}=\frac{dr}{dt}$$ $$100\frac{1}{4\pi r^2}=\frac{1}{25\pi}$$ So the answer is $\frac{dr}{dt}=\frac{1}{25\pi}$ when $r=25cm$  *Note the manipulation of derivatives just as if they were common fractions using algebra.    Question  When exactly can I treat differentials/derivatives as fractions and when can I not?   Please keep in mind that at the end of the day, I am a first year college student. An answer that is easy to understand is preferred over one that is more mathematically rigorous but less friendly to a beginner such as me. How do I spell the possessive form of children? In the sentence below, are the usages of apostrophes correct?     In research about parent stereotypes, the correlation between parents’ beliefs about their early elementary age children’s’ mathematics abilities and the children’s actual abilities increased as children aged. How can I add custom '.desktop' application launchers to Ubuntu Dock? I have made .desktop files for Ubuntu 16.04 to access my Google Chrome profiles, but once I upgraded to Ubuntu 17.10 (it worked up to Ubuntu 17.04) they got removed from the dock and I can't pin them back. Sample File: #!/usr/bin/env xdg-open  [Desktop Entry] Version=1.0 Type=Application Terminal=false Icon[en_US]=google-chrome Name[en_US]=Google Chrome Exec=google-chrome --profile-directory=Default Comment[en_US]=Chrome Name=Google Chrome Comment=Chrome Browser Icon=google-chrome Categories=Browser;Internet; I'm a C++ guy learning Java. I'm reading Effective Java and something confused me. It says never to write code like this:  String s = new String("silly");   Because it creates unnecessary String objects. But instead it should be written like this:  String s = "No longer silly";   Ok fine so far...However, given this class:  public final class CaseInsensitiveString {     private String s;     public CaseInsensitiveString(String s) {         if (s == null) {             throw new NullPointerException();         }         this.s = s;     }     :     : }  CaseInsensitiveString cis = new CaseInsensitiveString("Polish"); String s = "polish";    Why is the first statement ok? Shouldn't it be   CaseInsensitiveString cis = "Polish"; How do I make CaseInsensitiveString behave like String so the above statement is OK (with and without extending String)? What is it about String that makes it OK to just be able to pass it a literal like that? From my understanding there is no "copy constructor" concept in Java? If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact. How do I recover my accidentally lost Windows partitions after installing Ubuntu? I have a Toshiba satellite A-200 laptop with a Vista OS on it with 4 NTFS partitions  (C:) Vista  (D:) Entertainment  (E:) Work  (F:) Sources   and I wanted to start using Ubuntu instead. So I tried it first from the live CD and everything was OK and all the partitions were shown and working and so I decided to install Ubuntu to replace Vista on the (C:) drive. After I did that I can no longer find my folders and files on the (D:), (E:), (F:) partitions and the only file system that is shown is one 198 GB although my HDD is 320 GB.  I can't access the lost data on the remaining 120 GB which I hope is still there and not totally lost  I am now working from the live CD but I am unable to install testdisk. Can I recover the Vista partitions by the product recovery CD to get my laptop back to the factory settings? Can I recover the NTFS partitions using a recovery program for Windows or will that make the problem worse?   I need these data badly as I don't have a backup for them. Prove $\chi(G)\chi(\bar{G}) \geq n$ for chromatic number of graph and its complement Is it ever revealed what Crookshanks is? When Hermione first got her pet thing Crookshanks, there was a mention that it looked kinda like a cat, but wasn't.  Was it ever revealed what Crookshanks really was? Is there a limit of $\cos(n!)$? I encountered a problem today to prove that $(X_n)$ with $X_n = \cos(n!)$ does not have a limit (when $n$ approaches infinity). I have no idea how to do it formally. Could someone help? The simpler the proof (by that I mean less complex theorems are used) the better. Thanks Random errors on various sites when viewing main page Occasionally (very occasionally) when returning to the main page of a site by clicking on the logo I get the following message:     It was not possible to perform this tag search at this time due to an unexpected error.   This is odd because I'm not performing a tag search.  So far this has happened on both Movies &amp; TV and Cooking.  I have heard that others are getting this same message. Cleaning unwanted overlapping duplicate meshes I have a model with hundreds of mesh objects. Unfortunately I clicked shift+D to duplicate many and did not notice. Later after working a while, I was rendering, and noticed there are double meshes overlapping each other.   How can I find all duplicate meshes and delete them. I could do it manually,  just clicking,  but it would take a long time and if I miss something i will not notice.   Is there any option,  or can anybody suggest to me,  how can I find all duplicate mesh and delete them.  Model with all double (duplicate) mesh  Final Result Find the value of : $\cos x \cos 2x...\cos 999x$ given that $x=\frac {2\pi}{1999}$ Given $x=\frac {2\pi}{1999}$  Find the value of $$\cos x \cos 2x \cos 3x ...\cos 999x$$  So I tried expanding $\sin {2000x}=2\sin 1000x \cos 1000x$  Then rewriting  $\cos 1000x= \cos {(999x+x)}$   No luck so far.