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I am currently in the process of learning how to prove statements by induction. For the most part, I understand but I am at most clueless when it comes to prove some inequalities.  For example, I am having a hard time proving that: $n &lt; 2^n$ is true. (Even if it's clear that it is indeed true!)  In the induction step, I am stuck with: $n + 1 &lt; 2^{n + 1}$  I don't know what would be the assumption I should make from this statement.  Anyone could point me in the right direction? Thanks! How to find the mime type of a file in python? Did the date of Frodo's rescue out of the fire of Mordor have any particular meaning in The Lord of the Rings? Prove that the difference between two rational numbers is rational This is a terribly simple question I'm sure, but I can't find a work-around in my proof. I must prove that the difference between two rational numbers is thus rational. Here is my attempt:     Let $a$ and $b$ be rational numbers. Therefore, \begin{align}  a=\frac{\lambda}{\beta},\:b=\frac{\xi}{\zeta},\:\ni\:\lambda,\beta,\xi,\zeta\in\mathbb{Z},\tag{1} \end{align} which gives us \begin{align}  a-b=\frac{\lambda}{\beta}-\frac{\xi}{\zeta}=\frac{\lambda\zeta-\xi\beta}{\beta\xi}.\tag{2} \end{align}   So I have shown that $a$ and $b$ are rational numbers which, by definition, can be represented by the quotient of two integers. But now how do I tackle the problem of the difference? By definition the difference between two integers is an integer. Does that require that this difference is thus rational?  Thank you for your time, Possible Duplicate:          As answered in  I want to use git diff --color=always filename | less -r to get a colored output of my git diffs. Since I use this a lot it would be nice to set up an alias for this. But how to include the filename in this command?  I tried the following without success:  alias gdiff='git diff --color=always $1 | less -r'   Typing gdiff example.php displays the content of example.php with less but ignores the diff part.  Any ideas? How does one change to engineering for graduate school after undergrad in a science field? I've always had an interest in electrical engineering (actually all engineering in general) and I've decided that I'd like to pursue a graduate degree.  I guess I'm wondering if I should really try to do this or just keep it as a fun hobby I spend all my free time doing.  So, the question is,  how does one change fields between undergrad and grad school?  A specific example follows, but please provide answers that address a more general case, so that your answers will be helpful to other people with the same general question.  I already have a BS and MS in chemistry with a minor in applied math.I published three papers during my MS, one first author in the top journal in my specialty and two as coauthor in an average and a so-so journal.  I hope those will count for something.  I finished my BS with around 3.1 and my MS with a 3.9 gpa.  I'm sure that my MS advisor would write me a great letter.  My GRE scores are too old to count so I'd have to retake it or try and get a pass on it since I already have a graduate degree.  Since finishing school I've been working for 4 years as a chemist.  I can get great recommendations from my supervisor who is a PhD chemist.      I've been working for a while to cover some of the things I would have learned in a BSEE to try and make up for my lack of one.  I worked through some texts on analog/digital design, lots and lots of programming, built a balancing robot that got a bit of internet attention, and I'm currently working on an autonomous robotics project that's looking really good so far.  After that I have more project ideas than I have time or money for.    I want to apply for PhD programs (specifically in controls/robotics) but I'm worried that my application will get tossed right away because I don't have a BSEE.  Maybe I should just apply to MS programs? This should be a fairly trivial problem. I'm trying to open an ofstream using a std::string (or std::wstring) and having problems getting this to work without a messy conversion.  std::string path = ".../file.txt";  ofstream output;  output.open(path);   Ideally I don't want to have to convert this by hand or involve c-style char pointers if there's a nicer way of doing this? Prove that for for all $p,q\in \mathbb{Z}$, $q&gt;0$ we have: $$ \left| \sqrt{2}-\frac{p}{q} \right| &gt; \frac{1}{3q^2}. $$  To be honest, I do not know where to start - any help would be appreciated. Why aliases in a non-interactive Bash shell do not work I have this alias set in my system /etc/bashrc file:  alias root="sudo !!"   The intention of this being to run the last used command using sudo of course. When used, it of course appears to substitute the last command in history to the bashrc file upon shell initialization, and not the actual command that you would get if you were to run sudo !! in an interactive shell. I've also tried alias root="sudo fc -s" to no avail.  I realize this is probably something to do with how the BASH does command substitutions, but can someone explain why this is, and provide a usable substitute?  I'm running BASH version 3.2.51(1)-release (x86_64-apple-darwin13). How is light emitted by an incandescent lamp Isomorphism between $\mathbb{C}^* $ under multiplication with $\mathbb{C}$ under addition. Does there exists an Isomorphism between $\mathbb{C}^* $ under multiplication with $\mathbb{C}$ under addition.  Attempt In my opinion It does not exists.  If there exists such an isomorphism say $\phi :\mathbb{C}^* \to \mathbb{C}$ then, from the fact that Isomorphism takes identity to identity we have $$\phi \left( 1 \right)=0$$  From the fact      Given $\phi:G \to H$ be an isomorphism, then For a fixed integer $k$ and a fixed group element $b$ in $G$, the   equation $x^k=b$ has the same number of solutions in $G$ as does   the equation $x^k=\phi\left(b\right)$ in $H$.   We have $x^4=1$ and $x^4=0$ must have same number of solutions, but the solutions to these equations are 4 and 1 only. Who is this running kid wearing a shirt with Uchiha symbol? There's a kid wearing a shirt with Uchiha symbol, but I have no idea who s/he is.    Who is this kid? Is it possible to see a list of all my edits using a query? vim-clang-format: clang-format is not found Are there irreducible polynomials that are partially solvable by radicals? In general, polynomials of degree 5 or higher, with rational coefficients, are not solvable by radicals. There are some exceptions and trivial cases, for example $x^5-5x+12$ and $x^{100}+x^{25}+3$, respectively.  All irreducible polynomials with rational coefficients, that I am aware of, have either none or all roots that can be expressed by radicals.  Do partially-solvable irreducible polynomials exist? If they do, are there any examples known?  I mean, for example, an irreducible polynomial of $n$th degree that can be split over $Q(\sqrt{2})$ into two polynomials of 3rd and $(n-3)$th degrees, where the polynomial of $(n-3)$th is not solvable by radicals.    At  Alexander Gruber wrote:     Polynomials which are not solvable by radicals have (at least one) root that cannot be written by any combination of the operations of addition, multiplication, and the taking of nth roots.   I am not sure if he meant only reducible polynomials, and/or with non-rational coefficients, as polynomials that have "at least one" root that cannot be expressed by radicals. How can I add a PPA from a previous release? Is it bad to defrag a ssd? Or just pointless? I've read on some websites that it's bad to defrag an ssd. But I'm reading on this website now and one person was like "You didnt do any harm its just not necessary to defrag it." I feel like that's wrong since every other website I've read says it will damage the ssd. Redirect, Change URLs or Redirect HTTP to HTTPS in Apache - Everything You Ever Wanted to Know About mod_rewrite Rules but Were Afraid to Ask Enabling tftpd  sudo launchctl load -F /System/Library/LaunchDaemons/tftp.plist   Start:  sudo launchctl start com.apple.tftpd   See logs syslog -w  com.apple.xpc.launchd[1] (com.apple.tftpd[30687]) &lt;Warning&gt;: Service exited with abnormal code: 1   Or with log stream --process tftp --level debug  Filtering the log data using "process BEGINSWITH[cd] "tftp"" Timestamp                       Thread     Type        Activity             PID    TTL 2020-01-21 12:35:44.903240+0200 0x5292a    Activity    0x6afa0              31479  0    tftpd: (libsystem_info.dylib) Retrieve User by Name 2020-01-21 12:35:44.903949+0200 0x5292a    Default     0x0                  31479  0    tftpd: recvfrom: Resource temporarily unavailable   I also tried to run tftpd directly without launchctl sudo /usr/libexec/tftpd -d -i /private/tftpboot/ with the same results.  Earlier I used this configuration and it worked fine. But after some changes (may be OS update) it became broken. Current MacOS version is 10.14.6 (18G2022)    How to fix tftpd?    Appendix. Configuration file /System/Library/LaunchDaemons/tftp.plist has default content:  &lt;?xml version="1.0" encoding="UTF-8"?&gt; &lt;!DOCTYPE plist PUBLIC "-//Apple Computer//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd"&gt; &lt;plist version="1.0"&gt; &lt;dict&gt;     &lt;key&gt;Disabled&lt;/key&gt;     &lt;true/&gt;     &lt;key&gt;Label&lt;/key&gt;     &lt;string&gt;com.apple.tftpd&lt;/string&gt;     &lt;key&gt;ProgramArguments&lt;/key&gt;     &lt;array&gt;         &lt;string&gt;/usr/libexec/tftpd&lt;/string&gt;         &lt;string&gt;-i&lt;/string&gt;         &lt;string&gt;/private/tftpboot&lt;/string&gt;     &lt;/array&gt;     &lt;key&gt;inetdCompatibility&lt;/key&gt;     &lt;dict&gt;         &lt;key&gt;Wait&lt;/key&gt;         &lt;true/&gt;     &lt;/dict&gt;     &lt;key&gt;InitGroups&lt;/key&gt;     &lt;true/&gt;     &lt;key&gt;Sockets&lt;/key&gt;     &lt;dict&gt;         &lt;key&gt;Listeners&lt;/key&gt;         &lt;dict&gt;             &lt;key&gt;SockServiceName&lt;/key&gt;             &lt;string&gt;tftp&lt;/string&gt;             &lt;key&gt;SockType&lt;/key&gt;             &lt;string&gt;dgram&lt;/string&gt;         &lt;/dict&gt;     &lt;/dict&gt; &lt;/dict&gt; &lt;/plist&gt;   Update1 It seems port is in use. Why? Who?  $ sudo lsof -i:69 COMMAND PID USER   FD   TYPE             DEVICE SIZE/OFF NODE NAME launchd   1 root   23u  IPv6 0x55f5dbd8d897c847      0t0  UDP *:tftp launchd   1 root   57u  IPv4 0x55f5dbd8d897e687      0t0  UDP *:tftp launchd   1 root   59u  IPv6 0x55f5dbd8d897c847      0t0  UDP *:tftp launchd   1 root   60u  IPv4 0x55f5dbd8d897e687      0t0  UDP *:tftp   launchd uses UDP port 69. Why?  $ ps -ef | grep tftp 501 94727 48073   0  4:16PM ttys007    0:00.00 grep --color=auto tftp