I have this question:  Considering $$(1+x)^n(1+x)^n=(1+x)^{2n}$$ and show that: $${n \choose 0}^2+{n \choose 1}^2+{n\choose 2}^2+....+{n\choose n}^2={2n\choose n}$$  I have attempted so far: $$(1+1)^n={n \choose 0}+{n \choose 1}+{n\choose 2}+....+{n\choose n}$$ $$(1+1)^{2n}={2n \choose 0}+{2n \choose 1}+{2n\choose 2}+....+{2n\choose 2n}$$ $$\left({n \choose 0}+{n \choose 1}+{n\choose 2}+....+{n\choose n}\right)^2={2n \choose 0}+{2n \choose 1}+{2n\choose 2}+....+{2n\choose 2n}$$ and this is the part where I am confused about. Please help in where I got wrong and what I should do to show it in the correct method. The question here is to prove that $${n \choose 0}^2+{n \choose 1}^2+{n\choose 2}^2+....+{n\choose n}^2={2n\choose n}$$
I was hoping to find a more "mathematical" proof, instead of proving logically $\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$.  I already know the logical Proof:  $${n \choose k}^2 = {n \choose k}{ n \choose n-k}$$  Hence summation can be expressed as:  $$\binom{n}{0}\binom{n}{n} + \binom{n}{1}\binom{n}{n-1} + \cdots + \binom{n}{n}\binom{n}{0}$$  One can think of it as choosing $n$ people from a group of $2n$  (imagine dividing a group of $2n$ into $2$ groups of $n$ people each. I can get $k$ people from group $1$ and another $n-k$ people from group $2$. We do this from $k = 0$ to $n$.
Which is better: ... or ... Is smell a bad way of determining whether meat is still good? I have always relied on my sense of smell to determine whether meat (most often chicken) is bad and I thought it was reliable. However, I have just heard that [1] smell is not always reliable, since some of the toxins produced by bacteria won't cause any smell whatsoever and you won't be able to notice that the meat is off until after you've eaten it.   Is this true? If yes, is this true for all meat or only for some? Is there a more reliable way of determining whether meat is still good.  [1] "Heard that" isn't very reliable; so that's why I'm asking the question here. :) Aligning equations within multline Does anyone know how prove the following?   Suppose that $f,g$ are holomorphic functions on a non-empty open connected set $\Omega \subset \mathbb{C}$ and that $|f|^2+ |g|^2$ is constant on $\Omega$. Show that $f$ and $g$ are constant on $\Omega$. MSTest Code Coverage How can I stop sending my hostname via DHCP? Ubuntu sends my computer's configured hostname to any DHCP server it tries to use. How do I stop it from sending a hostname? Provide a means to reward users and their answers (decoupled from bounties) If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module. If $m$ divides $n$, find a free resolution of $\mathbb{Z}/m$ as a $\mathbb{Z}/n$-module.   I have tried this one and got $0 \leftarrow \mathbb{Z}/m \leftarrow \mathbb{Z}/n \leftarrow \mathbb{Z}/n$.  The map $\mathbb{Z}/n \to \mathbb{Z}/n$ is multiplication by $m$, but I couldn't go further. Please help me. [[ 0*10%300 ]] works on AIX 6.1 but not on AIX 7.1 (ksh) I am looking to create some handwriting drill sheets.  Similar to what can be found here    I also found this question:  But I'd like to be able to simply write a paragraph of text with a known font and have the lines placed behind, perhaps like a watermark.  I have found a git repo with some nice lines in both portrait and landscape here   From the kindergarten example, I would like to overlay "phantom" text (probably a dotted cursive font) over the lines that are generated.  Any thoughts?  \documentclass[10pt,letterpaper]{article}  \usepackage[latin9]{inputenc} \usepackage{geometry} \geometry{verbose,tmargin=1cm,bmargin=1cm,lmargin=1cm,rmargin=1cm} \pagestyle{empty} \usepackage{dashrule} \usepackage{forloop} \makeatother  \setlength{\parindent}{0pt}  \begin{document}  \newcounter{ct} \forloop{ct}{1}{\value{ct} &lt; 13} { \vskip 10pt \rule[1.75ex]{7.5in}{1pt} \\ \hdashrule{7.75in}{1pt}{4mm} \\ \rule[-1.75ex]{7.5in}{1pt} \\ }  \vskip 1cm  \end{document}   *Edit: I also just found this source:    So some combination of these makes sense to me, I'm just not clear on how...  *Edit: Okay, So I've gotten pretty far here.  But it seems like several parameters need to be functions of each other some how.  The line spacing, the font size, etc.  Here is where I am:  \documentclass[10pt,letterpaper,english]{article}  \usepackage{babel}  \usepackage{blindtext}   \usepackage{setspace}  \usepackage[latin9]{inputenc} \usepackage{geometry} \geometry{verbose,tmargin=1cm,bmargin=1cm,lmargin=1cm,rmargin=1cm} \pagestyle{empty} \usepackage{dashrule} \usepackage{forloop} \makeatother  \usepackage{xcolor}  \newcommand{\baselinegrid}{%   \raisebox{0pt}[\height][0pt]{\makebox[0pt][l]{%   \begin{minipage}[t]{\textwidth}%     \begin{color}{red}%     \newcounter{ct}     \forloop{ct}{1}{\value{ct} &lt; 11}     {     %\vskip 15pt     \rule[0.5ex]{7.5in}{1pt}     \\     \hdashrule[0.75ex]{7.75in}{1pt}{4mm}     \\     \rule[1.2ex]{7.5in}{1pt}     \\     }     \end{color}   \end{minipage}% }}}  \begin{document}    \baselinegrid%  \begin{spacing}{2.5}    \usefont{T1}{wela}{m}{n}   \fontsize{30}{15}\selectfont   \setlength{\parskip}{0pt}   \setlength{\parindent}{0pt}    DURING the whole of a dull, dark, and soundless day in the autumn of the year, when the clouds hung oppressively low in the heavens, I had been passing alone, on horseback, through a singularly dreary tract of country; and at length found myself, as the shades of the evening drew on, within view of the melancholy House of Usher.  \end{spacing}  \end{document} How to send AT commands to modem once the connection is established? I have a zte 3g modem. I use carrier provided dialer for connection establishment. Once the ppp connection is active, i would like to send some AT commands(for ex. Query signal strength, AT+CSQ). But the dialer i use locks the /dev/ttyUSB0 port, which is the command port to send AT commands for my modem. So is there any way, to send the commands, once the connection is active?    Edit: I also tried the additional port /dev/ttyUSB1. But the port is flowing with random data from the modem. A sample is given below.  T^PREFMODE??                                                                     ^PREFMODE:8                                                                       OK                                                                               TC                                                                               ^DSDORMANT:1                                                                      +CSQ:19, 99                                                                       OK                                                                               T^SYSINFO                                                                        ^SYSINFO:2,3,0,4,255                                                              OK                                                                               TT^SYSINFO?                                                                      ^SYSINFO:2,3,0,4,255     I tried adding my commands, i even got output. But the response is very poor. Most of the times, my AT commands went unnoticed. Can I pass arguments to an alias command? I want to know if I can pass an argument with an alias command.  for example:  alias d="dmesg|grep -iw usb|tail -5"    Now d will print the last 5 lines. If I want to use d to print a different number of lines, I have to make change in the alias command declaration of d again.  Is there any way I can modify the declaration of an alias so that I don't have to retype the declaration to change the number of lines. Like incorporating passing the number of lines as an argument while declaring alias for d? Or is there some other method to solve this? How to effectively use the man pages? I am looking forward to mastering linux. I have been repeatedly told to refer to man pages when you are in doubt.   Though I am very comfortable with the commandline, I find it very difficult to quickly and efficiently use man to find help with commands. Hence I end up googling up how to get things done. It is even more painful when the system I am using does not have internet access and I have to walk back to my office to get help from the internet.  How does one master man pages? What are the most common tricks that can be used to quickly get whatever you want from the man page? What are the most commonly used keyboard shortcuts? I want to change my thesis topic (to the extent that the ethos is still in tact) but my advisor does not agree. What can I do? I am doing a phd where my aim is to solve a problem A which is very interesting. My supervisor has got a lot of connections in a technology B and he claims to be an expert in B. I came to this group because of these accomplishments. But soon I realized that B is not really a great technology for A.   So in my phd proposal I proposed a method which encompasses B but is more general (say R). Now I realize that I do not want to use B at all because my preliminary results are not encouraging. In fact, I am inclined to use E as a base technique which does not have any impact on R.  But my supervisor, who is interested in only publishing in B  and frowns upon everything else (it was a nightmare to convince him to let me work on R) is dead against it. He says that he is an expert in B and his vision of the group is not inclined with me trying other things.  There are other professors in my university who have worked in E and could collaborate but my supervisor won't collaborate with people from outside his kingdom. I also know professors from outside my university who could help me if needed. I could knock the doors of school committee in order to  help me but that would create a lot of problems as my supervisor wields a lot of power and influence everywhere.    Another query is, is it ethical to collaborate with people outside the university (of course after letting my supervisor know) for my PhD work. My Masters supervisor would be happy to help me out.  Edit: After looking at comments, I am thinking that after all, it is MY PhD. Yes, I have spent 1.5 years, but 1.5 years is left. Why should I spend any moment of my life on something I find to be not exciting, particularly less exciting because it is forced on me. And in those 1.5 years I learnt to write well and also learnt 'A' which is important. Its not total waste. My supervisor has his agenda, and why should I give up mine to make him happy? His vision is fine, but he does not have the right to force his vision on me. In-universe, a lightsaber is basically a controlled, fixed-length plasma stream that can be used as a sword. As such, the blade wouldn't have any mass (or infinitesimal mass if you want to get technical about E=mc2).  Anyone familiar with fencing or swordplay knows that the weight of the handle and blade, and thus the overall balance of the weapon, is critically important while fighting. A sword that is too blade-heavy is unwieldy; it is difficult to start, stop, and change the path of the blade as it's swung, which is vital to outmaneuvering your opponent in a duel. Conversely, a sword with too light a blade (equivalently, too heavy a hilt) is too easy to swing around. it's hard to "feel" where the blade is, and thus to make the minute changes needed to block a blow or land one. It can also be just as exhausting, in a different way, to wield such a sword; because most of the mass is behind your hand, it takes muscle power to point the blade forward instead of upward, while a balanced weapon would point directly forward from the user's hands. You also get no mechanical advantage while swinging it.  Obviously, out-of-universe the props were full "swords" that were then rotoscoped to paint the beam blade over the white wooden (later plastic) dowel that protruded from the saber. This gives the actual prop used by the actors a more realistic weight and balance, which can be seen in the duels (Darth Maul has to heave a bit to get that double-saber staff moving, and you can see the effort behind a full baseball swing of the sword by Luke or Vader in the original trilogy).  Was there any attempt in-universe to explain the weight and balance that an active lightsaber seemed to possess? How can I reset my Ability Points? I'm also curious about being able to reset my points, should I wish to change my ability configuration later.  So far (10h-ish) I've only seen one way, which is a potion sold by Kiera Metz for 1000g. However she's no longer available as a store, due to a story choice, so I'm not sure where else I could buy this potion.  Where might I buy this potion, and is it the only way to reset your ability points? What is the physical explanation for energy transport in simple electrical circuits? Keeping it simple, what is the physics approach to a current in a wire connected to a battery? How does the electric field behave?  I've read somewhere that, contrary to the common conception, electrons don't really flow as water in a hose so, what is really going on there? How is energy transported at the speed of light if electrons don't flow like very fast water? Is it possible to adjust the frame rate of viewport playback? Is it possible to play an animation in the viewport at a slow framerate?  Just because of the rather heavy scene it's already going at about 8 fps, but I would like to be able to play it even slower (an in a controlled manner) for the purpose of recording an animation with automatic keyframes. In what situation do you put a comma before "because"?   And, when do you not put a comma before "because"?   I've always assumed that you put a comma, because the clause introduced by "because" will always be independent, because it cannot stand on its own, because it wouldn't make sense on its own.   Is this the case? Could you think of an example where it doesn't take a comma? Why does James Bond use his real name? In the James Bond universe most characters go by a code name M, Q, the various 00s, In Skyfall Silva is a made up name.  Others such as Tanner, Robinson and Moneypenny, Trevelyan we're not so sure but we've got no reason to believe they're real.  However Bond uses his own name - we've even seen the grave of his parents to confirm this.  Why does Bond use his real name and (as far as we know) no one else does?