"Have you gone" vs. "have you ever gone"
What is the difference between "I've been to China" and "I've gone to China"?
Why did Slytherin House not get disbanded? Suspected adware infection of Mac How to manipulate 2D plots? Greatest comeback in 2nd leg match Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$ I'd like to show that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$ for all $x,y \ge 0$ and $p \in [0,1]$. I started out by defining a function $f(x,y)=(1+x)^p(1+y)^{1-p}$ and checking for critical points, but there are none in the domain I'm interested in. I think I'm probably missing a slick application of Jensen's or Holder's inequalities, but so far a proof eludes me. Any suggestions would be appreciated. Which is correct “Is that yours?” or “Is that your’s?”?  I ask because it is possessive, so I would think it would be the latter, but I typically use and see the former usage.  Are there particular cases in which one should be used instead of the other? Or is one simply correct and the other not?  This is one of the few things that still confuse me, so help is greatly appreciated. Which tool for fixing wobbly crank? Any idea what kind of tool should I but to fix the loose left crank on my bike? I found this  but not sure if it's gonna suit. Or is it enough to buy only nut driver? Find the volume of the tetrahedron with the vertices $P(1,1,1)$, $Q(1, 2, 3)$, $R(3, 1, 2)$, and $S(2, 3, 1)$. Tangent bundle : is a manifold I have studied what a differentiable manifold is, and what a tangent space at a given point is, and read the proof that its dimension is equal to the dimension of the manifold. Here a tangent vector is a derivation on the germs of differentiable functions at a point.  Now I am trying to see that the tangent bundle is indeed a differentiable manifold. I get that for an open set $U$ in Eucliden space, it takes the form $U \times \mathbb R^n$ with the vector $\sum a_i x_i$ giving the tangent vector $\sum a_i \partial /\partial x_i$. Now the book says that it is "obvious" that these glue up and give a differentiable manifold.  I may be stupid; but I will be grateful if somebody can hold my hand through this "obvious" statement. 14.04 --> 16.04 failed; apt (>= 1.0.1ubuntu2.13)' is not installed (This question was not resolved. Due to time constraints Computer was restored to factory settings.)  I was attempting to upgrade from 14.04 to 16.04, but apt threw this error:  The required dependency 'apt (&gt;= 1.0.1ubuntu2.13)' is not installed.   What does this mean, and how can I fix it?  When I type in the command apt-cache policy apt  Shageenth@shageenth-Inspiron-3451:~$ apt-cache policy apt apt: Installed: 1.0.1ubuntu2.11 Candidate: 1.0.1ubuntu2.11 Version table: *** 1.0.1ubuntu2.11 0 100 /var/lib/dpkg/status 1.0.1ubuntu2 0 500 http://us.archive.ubuntu.com/ubuntu/ trusty/main amd64  Packages   The error happens when I run the command:   sudo do-release-upgrade -d   and it produces the following:  Checking for a new Ubuntu release Get:1 Upgrade tool signature [198 B] Get:2 Upgrade tool [1,265 kB] Fetched 1,265 kB in 0s (0 B/s) authenticate 'xenial.tar.gz' against 'xenial.tar.gz.gpg' extracting 'xenial.tar.gz'  Reading cache  Checking package manager Reading package lists... Done Building dependency tree Reading state information... Done Building data structures... Done  Required depends is not installed  The required dependency 'apt (&gt;= 1.0.1ubuntu2.13)' is not installed. What is the logarithm of a kilometer? Is it a dimensionless number? In log-plots a quantity is plotted on a logarithmic scale. This got me thinking about what the logarithm of a unit actually is.  Suppose I have something with length $L = 1 \:\mathrm{km}$.  $\log L = \log \mathrm{km}$  It seems that the unit of $\log L$ is $\log \mathrm{km}$, but I can also say $L = 1000 \mathrm{\:m}$ and now:  $\log L = 3 + \log \mathrm{m}$  This doesn't appear to have any units at all.  This suggests that $\log \mathrm{km}$ and $\log \mathrm{m}$ are actually dimensionless numbers. But wait, I can do this with any unit! Does it actually make sense to talk about the logarithm of a unit, or some other function for that matter? I don't believe it's true that all submodules of a finitely generated free module are free just based on results from google. Is there a canonical simple example of a finitely generated free module with a submodule that is not free? I don't know anything about Star Trek except for the fact that it's about spaceships.  In  the person only wanted to watch the TV series. I want to watch every movie and TV series there is about Star Trek but I don't know where to start. I found this page on  which is about a time line. Do I need to watch it in this order? Are the movies and TV series related? From , I have tried the using \hspace{0pt} and   \makeatletter \renewcommand{\dddot}[1]{%   {\mathop{\kern\z@#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@    \hbox{\normalfont ...}\vss}}}} \renewcommand{\ddddot}[1]{%   {\mathop{\kern\z@#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@    \hbox{\normalfont....}\vss}}}} \makeatother   after loading mathtools but the problem still persists.    \documentclass{article} \usepackage{mathtools} \begin{document} \[ \dot{\mathbf{r}}\times\ddot{\mathbf{r}}\times\dddot{\mathbf{r}} \] \end{document}   In the case the font scheme I am using has any affect as well, the fonts are:  \usepackage[utf8]{inputenx} \renewcommand{\rmdefault}{ppl}                  \linespread{1.05}                   \usepackage[scaled]{helvet} \usepackage{courier}               \usepackage{eulervm} \normalfont       \usepackage[T1]{fontenc}            \usepackage{textcomp} How to add ground to DC circuit? I'm just a dabbler in electronic circuits and I'm really still kind of a noob. I am working on a project called the "Atari Punk Console" (Originally known as the "Stepped Tone Generator") but there are some parts of the wiring schematic that I don't understand, specifically the parts with the ground symbol.  Here is the schematic:   See those two ground symbols? I don't understand how to properly connect the circuit to ground at these points. How can I do this and be sure that I did it properly. I don't understand grounding at all so any pointers or clarification is also welcome. Thanks in advance. Has 'Ubuntu Software' been renamed to 'Snap Store'? My printer does not print light magenta. (Canon MP220 - CL-38) I have a Canon MP 220, with new cartridges. Prints fine everything, but it won't print light magenta. I cleaned it and deep cleaned it many times. I have no idea what could be the problem. Cartridge models: CL-38 and PG-37.  Any ideas ?  LE: Now it won't print magenta at all! What is the speed of time? command to display time (duration) in human readable form I spent 3 hours trying to write a command to format a time (duration) expressed as a floating number (in seconds) in a nice form as follows:  if the time is &lt; 60 write as a float with 3 decimals (i.e. milliseconds)  else let h,m,s be the hours, minutes and seconds corresponding to the time      if h=0 then write as m:ss (ss always with 2 digits, m can have only 1)      else write h:mm:ss (mm and ss with 2 digits, h can have only 1)   So 1.25678 gives 1.256, 125.35 gives 2:05, 18249 (which equals to 5*3600+4*60+9) gives 5:04:09.  I tried several packages for the comparison/formatting. No success. Can someone help me ? Thanks ! Is it supposed to be a HTML or an HTML