Minimizing Height of a Table
Optimize table layout
A square with side $a$ is given. What is the average distance between two uniformly-distributed random points inside the square?  For more general "rectangle" case, see . The proof found there is fairly complex, and I am looking for a simpler proof for this special case. I expect it could be significantly simpler.  See also . Computation of likelihood when $n$ is very large, so likelihood gets very small? I think it would make sense to split up the flag "This is not an answer" into three flag choices:   This should be a comment This should be an edit This should be a new question   If we had that, it would make it easier (and therefore more efficient) for mods to figure out what to do with answers flagged thusly. Difference between Simple Past and Past Perfect What is the difference in meaning between the two examples below?   When she returned home from office, she was shocked to see that her husband and daughter were missing. (Simple Past)  When she returned home from office, she was shocked to see that her husband and daughter had been missing. (Past Perfect)    FIRST QUESTION: For me example #1 gives only idea of Status of her missing husband and daughter and example #2 gives idea of completed action/situation of missing husband and daughter before she reached home. Is this right? SECOND QUESTION: Is example #2 past perfect or past perfect continuous? Beamer: \nonumber equivalent for slides? Is there an equivalent command to \nonumber when one doesn't want to include certain slides in the numbering? I would like to leave outline and title slides out of my numbering, and instead just number slides with actual content. Other workarounds would be nice to know about as well.  Edit: Since a MWE example was requested, here is one. Suppose I have a title page and two other slides. The code below gives a slide counter that goes up to 3, and the title slide is slide 1. I would prefer to have the title slide not be numbered, the counter go up to 2, and the the numbering start at the slide after the title. Here is some code.  \documentclass{beamer}  \mode&lt;presentation&gt;  {    \usetheme{Dresden}  }   \setbeamertemplate  {footline}{\quad\hfill\insertframenumber/\inserttotalframenumber\strut\quad}    \begin{document}  \begin{frame}  \titlepage \end{frame}  \begin{frame} Some actual content here \end{frame}  \begin{frame} Some more actual content here \end{frame}  \end{document} Singular or plural verb after a series connected by "or" The following sentence refers to an apocalyptic story where money no longer has any value:     A can of sardines, radio batteries, or a bicycle is/are more precious than money.   Should I use is or are after the series? Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following:   $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over n!}\Big)=\sum_{n=0}^{\infty}\sum_{k=0}^n{x^ky^n \over {k!n!}}$$  But, where should I go next to get $e^{x+y}=\sum_{n=0}^{\infty}{(x+y)^n \over n!}$.  Thanks in advance. Calculus Question: $\int_0^1\frac{(x-x^2)^4}{1+x^2}dx$ Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$ I'd like your help with the following claim to prove:  $$\lim_{n \to \infty} \int_{0}^{\sqrt n}\left(1-\frac{x^2}{n}\right)^ndx=\int_{0}^{\infty} e^{-x^2}dx.$$  I think I should use the claim:     Let $a,b$ two real numbers and $\{f_n\}$ a sequence of continuous functions on $\left[a,b\right]$  which converges uniformly to $f$ on $[a,b]$. Then    $$\lim_{n\to\infty}\int_a^bf_n(t)dt=\int_a^bf(t)dt.$$   But for this, I must prove that $(1-\frac{x^2}{n})^n$ uniformly converges to $e^{-x^2}$.  How do I do that? One of the hardest and trickiest things to do is to prove this uniformly converges.. every function has its own way. I believe that the more examples I'll see the easiest it will be. Thanks again! Ted Shifrin on this site posed an interesting problem to me: show that   $$ \lim_{(x,y) \to (0,0)} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \frac\alpha\gamma + \frac\beta\delta &gt; 1, \,\,\,\,\text{where } \alpha, \beta, \gamma, \delta &gt;0$$  I think I've got the $(\Leftarrow)$ direction as follows: assume WLOG that $\gamma \leq \delta$. Then switch to polar coordinates and get   $$\frac {r^ {\alpha + \beta} (*) } {r^\gamma (1(*) + r^{\delta - \gamma}(*))},$$  where $(*)$ represents some trig stuff that is bounded near zero.   Edit: I need to make sure that $(1(*) + r^{\delta - \gamma}(*))$ is bounded below here.  The other direction is giving me some trouble. It seems we need a lower bound for the fraction (something to force to zero), and I'm not sure where to find one. In particular, I'm not sure what to do with the denominator. Any ideas? The differential equation $$ {d^2\alpha \over dt^2} + {g \over L} \cdot \alpha = 0 $$  describes a 1-dimensional mathematical pendulum, where $\alpha $ is the angle, $ g = 9.82 $, and $ L = 0.2 $ is the lentgh of the string. What is the position of the pendulum after 1 second if the velocity at time equals zero is $ 0 $ m/s and the angle at time equals zero is $ {\pi \over 60} $ radians.  To my understanding this is a homogenous equation, ergo  $$ y'' + Cy = 0$$  And with the help of the characteristic equation I get the complex roots $ Ci \, $ and $ -Ci $.  This is a farily simple differential equation, but I have a feeling my calculations so far are wrong (mainly because I didn't expect to see complex roots for this equation). Prove that the sum of digits of $(999...9)^{3}$ (cube of integer with $n$ digits $9$) is $18n$ Someone had posted a question on this site as to what would be sum of digits of $999999999999^3$ (twelve $9s$ ) equal to?   I did some computation and found the pattern that sum of digits of  $9^3 = 18$,  $99^3 = 36$,  $999^3 = 54$ and so on. So, I had replied that sum of digits of  $999999999999^3 = 12 \cdot 18 = 216$.  Can anybody help me prove this, that sum of digits of  $(\underbrace{999\dots9}_{n\text{ times}})^3 = 18n.$ Please provide me with the guidance to install oracle 11gr2 on Ubuntu 14.04. How do I create a WiFi hotspot sharing wireless internet connection (single adapter)? I know how to create a hotspot when connected to a wired network, but when I am using internet from a WiFi connection, this disconnects the wireless connection the moment I activate the hotspot.  In Windows I can use , which enables me to share the internet connection from the same wireless adapter as I am creating an access point on. As you can read on the :     Access Point mode allows you to create a hotspot using the same Wi-Fi card that you are using to access the Internet.   How do I do this on Ubuntu? Trying to find the original story (I read it in Russian translation, original was most likely English unless it was Lem).  The story plot had artificial birds designed to serve as cops. The birds started attacking humans who were percieved as being aggressive towards anything alive. So they created more advanced birds to protect against the first ones. This worked for a while until the second kind started attacking humans as well. It ended with the idea that they would need another generation to attack the second-generation birds.  The story wasn't very long. I read it a while back (mid-1980s, which means the original was written even earlier, probably before 1980).  Despite a somewhat Asimov-like "misbehaving robots" topic, I'm pretty sure it wasn't Asimov. I am using NCR18650 batteries with the tp4056 charging module.  When I connect the charger at 2.8&nbsp;V (fully discharged), the tp4056 IC is getting hot at a 1&nbsp;A charge current. Is it OK to reduce this charge current by changing the charge current selecting resistor? So the tp4056 IC heat would be reduced, wouldn't it?  For my application I don't need a quick charge. With a low current, charging time will be increased, but it does not matter for my application.  Will reducing the charge current affect the battery's lifetime? Or will it damage the battery?  I am trying to reduce the heat of the tp4056 IC by reducing the charging current, maybe from 1&nbsp;A to 500&nbsp;mA or less. The 18650 battery never heats for a 1&nbsp;A charge current. Definite Integral $\int_{-\infty}^{\infty}\frac{x\sin x}{(x^2+a^2)(x^2+b^2)}\,\mathrm{d}x$ $$\int_{-\infty}^{\infty}\frac{x\sin x}{(x^2+a^2)(x^2+b^2)}\,\mathrm{d}x$$  This is easy to evaluate with complex analysis but is there an elementary way (substitution, partial fractions, integration by parts)? How can I keep a folder synchronized to an external USB hard drive? I have a growing music collection which I manually keep in sync with an external USB drive. Sometimes I edit their ID3 tags, add or delete a file in either the hard drive or the USB drive, and I would like to keep those changes synchronized between both.  Does Ubuntu has something available that would help me with this scenario? Preferably something easy to use with a UI.  Update: To clarify my question, changes may happen on both the local hard drive or the USB drive, so the sync process must be on both directions. How can I enable multiple users login at a time in remote system? Recently we bought dedicated server with Windows Server 2012 Standard.It's allowing 2 people at a time.So I want to allow more people at a time.  How can I do this? "Track changes" in LaTeX I am writing a scientific paper with my collaborator. We have submitted a version to a journal and now we got a review from them, so we need to fix a few things. In Microsoft Word there is the option to "track changes", which displays the deleted text as crossed out and the new text bolded (if I remember the details correctly). Is there a way to achieve a similar result in LaTeX?  I was thinking to create a new file for the revised version and then write a script which would basically compare the two files and make the differences either bold or crossed out before sending it to latex. Is there a tool that already does it?