math induction help?
Solving binomial theorem via induction
Is it possible to follow a user automatically? I've been looking around and I can't seem to find an automatic (i.e. buzz- or Twitter-like) way to follow users inside Stack Exchange. Why haven't the developers added such a feature? It could help beginner users follow other beginners with the same interests. We would be able to learn from each others' questions and pitfalls. Also, it would reduce double posting. Reopened questions get a "modified" time stamp in the questions list just showed up on 's front page like this:    ... which makes no sense, because the question hasn't been modified since May 25th, and the person who edited it wasn't CatLord anyway:    Turns out what actually happened was that CatLord cast a fifth reopen vote to reopen the question.    So that stamp in the questions list should probably say this instead:     reopened 4m ago CatLord 10.7k   Or just:     reopened 4m ago When to use "click" and "click on" How to render a background image in 2.8? How do I use variables in a sed command? I tried the following code to replace QQ with ZZ, but it doesn't do what I want:  var1=QQ sed -i 's/$var1/ZZ/g' $file   However, this code does what I want:  sed -i 's/QQ/ZZ/g' $file   How do I use variables in sed? Redirect www to non-www using Apache htaccess does not work I want to count the total number of files in particular directory that ends with ".mp4" extension.  I am getting following command:  ls -F |grep -v / | wc -l   It count all the files in particular directory, but I want the count of files that end with .mp4 extension.  Is there any Ubuntu command for that? Which version of Ubuntu will work on my PowerPC? I'm using iMac PowerPC G5 1.8 GHz, 1GB RAM, 160GB HDD, and 10.5.4 Leopard OS.  I want to run Ubuntu on my iMac. Please can anyone tell me which version will run in my system? Prove that if $B$ is the set of rationals in $[0,1]$ with a finite subcover, then: $1 \leq \sum_{k=1}^n m^*(I_k)$ I have the following homework problem:  Let $B$ be the set of rational numbers in the interval $[0,1]$, and let $\{I_k\}_{k=1}^n$ be a finite collection of open intervals that covers $B$. Prove that $\sum_{k=1}^n m^*(I_k) \geq 1$.    This is my approach, First we have that $B = \mathbb{Q}\cap [0,1]$. Now, $B \subset \bigcup_{k=1}^nI_k$, where $I_k$ are open intervals.In particular $0 \in B \wedge 1 \in B$.Now, since $B \subset \bigcup_{k=1}^nI_k$ then there exist positive numbers, $\epsilon_0, \epsilon_1$ such that: $$(0-\epsilon_0,0+\epsilon_0)\subset \bigcup_{k=1}^nI_k$$ $$(1-\epsilon_0,1+\epsilon_0)\subset \bigcup_{k=1}^nI_k.$$   Clearly, $-e_0 \notin [0,1]$ and $e_1 \notin [0,1]$.I am not sure, if so far I am in the right track. At 9th level, the Armorer Artificer (from Tasha's Cauldron of Everything) gains the &quot;Armor Modifications&quot; feature :  You learn how to use your artificer infusions to specially modify your Arcane Armor. That armor now counts as separate items for the purposes of your Infuse Items feature: armor (the chest piece), boots, helmet, and the armor’s special weapon. Each of those items can bear one of your infusions, and the infusions transfer over if you change your armor’s model with the Armor Model feature. In addition, the maximum number of items you can infuse at once increases by 2, but those extra items must be part of your Arcane Armor.  This special weapon is called 'Thunder Gauntlets' (a simple melee weapon) for the 'Guardian' Armor Model, and 'Lightning Launcher' (a simple ranged weapon) for the 'Infiltrator' Model. But I am wondering which Artificer Infusions are appropriate for this special weapon ? Is it only the 'Enhanced Weapon' and 'Radiant Weapon' infusions, or are there others ? printing a block of text that matches a pattern Using just shell script, how to search a text file and list all whole blocks of lines that has inside some text (simple grep criteria).   The text file has blocks of lines separated by "-----------------" (precisely, each block start with "\n\n\n--------------------"... about 50 chars "-").  A sample could be:  ------------------------------- Abracadabra, blablablalbalba blablablabla, banana    ------------------------------- Text, sample text, sample text, sample text Text, sample text, sample text, sample text Text, sample text, sample text, sample text Text, sample text, sample text, sample text   -------------------------------   Text, sample text, sample text, sample text banana. Sample text, sample text, sample text, sample text Text, sample text, sample text, sample text   Lets consider the word "banana" the search criteria. So, the blocks listed would be:  ------------------------------- Abracadabra, blablablalbalba blablablabla, banana   ------------------------------- Text, sample text, sample text, sample text banana. Sample text, sample text, sample text, sample text Text, sample text, sample text, sample text   EDIT:  Testing answers to try awk, like: awk 'BEGIN{RS="\n------------"}/INFO/{print}' where INFO is what was searched for. I cannot get the whole block. So, follows a real sample and the result:  A REAL SAMPLE (including the first 3 new lines):      ------------------------------------------------- Diretório separado do nome o arquivo: adis, IWZLM (/home/interx/adis/src/IWZLM.SRC) Gerando rotina em linguagem C: (yla5 adis IWZLM -if) .INFO =>Rotina BLOQUEADA (status 'M'): Geracao ignorada (use -is para ignorar checagem do status)  [  OK-I ] IWZLM (adis) - Lista lay: Geracao ignorada do codigo em C.    ------------------------------------------------- Diretório separado do nome d arquivo: adis, ADISA (/home/interx/adis/src/ADISA.SRC) Gerando rotina em linguagem C: (yla5 adis ADISA -if) .ERRO: Falha inesperada  Compilando o programa: (ycomp adis ADISA -exe adis/exe/ADISA.temp.exe ) adis/exe/ADISA.temp.exe => adis/exe/ADISA  [  OK   ] ADISA (adis) - Menu A : Gerada e compilada com sucesso.    ------------------------------------------------- Diretório separado do nome o arquivo: adis, ADISD1 (/home/interx/adis/src/ADISD1.SRC) Gerando rotina em linguagem C: (yla5 adis ADISD1 -if) .ATENCAO: Definicao nao localizada  Compilando o programa: (ycomp adis ADISD1 -exe adis/exe/ADISD1.temp.exe ) adis/exe/ADISD1.temp.exe => adis/exe/ADISD1  [  OK   ] ADISD1 (adis) - Menu : Gerada e compilada com sucesso.   I cannot get the whole block, just the line containing "INFO", like a ordinary grep, either setting or not ORS:  $ cat file  | awk 'BEGIN{RS="\n------------"}/INFO/{print}'  .INFO =&gt;Rotina BLOQUEADA (status 'M'): Geracao ignorada (use -is para ignorar checagem do status)   NOTES: It is the awk from AIX 7.1, not gawk. My friend and I have been wracking our heads with this one for the past 3 hours... We have 2 point masses, $m$ and $M$ in a perfect world separated by radius r.  Starting from rest, they both begin to accelerate towards each other. So we have the gravitational force between them as:  $$F_g ~=~ G\frac{Mm}{r^2}$$  How do we find out at what time they will collide?  What we're having trouble with is this function being a function of $r$, but I have suspected it as actually a function of $t$ due to the units of $G$ being $N(m/kg)^2$.  I've tried taking a number of integrals, which haven't really yielded anything useful.  Any guidance?  No, this is not an actual homework problem, we're just 2 math/physics/computer people who are very bored at work :) If votes works differently on meta sites how does that of accepting answers work? In the faq of meta sites you can read something like the following:     Voting here works a bit differently   from other Stack Exchange sites. On   Meta Stack Overflow, voting is used to   express agreement or disagreement, not   to point out a lack of quality or   helpfulness. Please don't be concerned   if you receive downvotes – members of   the community may simply disagree with   your bug, feature request, support   issue, or the nature of the   discussion.   My question is: If votes works differently on meta sites how does that of accepting answers work?  Take for instance a question on a meta site that is meant for the community to discuss some issue. What does it mean if the question is accepted? Accepting an answer is up to the author of the question so it cannot be a sign of what answer the community concludes from the discussion. It could mean that the discussion is over but that should be up to the community and not the author of the question. Is it so that accepting an answer for a discussion question on a meta site does not have a particular function? How to add a browser tab icon (favicon) for a website? Some of code I read in trigger is like this    for(account a: [select id  from account where id in: mapAccount.keySet()]) {     //code      }    some are like   List&lt;Account&gt; listAccount = [select id  from account where id in: mapAccount.keySet()];   for(Account a: listAccount ){   //coode   }   Are those same from performance view?which one is better? Permitting scp but not ssh - without scponly Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$ What will be the one's digit of the remainder in: $\left|5555^{2222} + 2222^{5555}\right|\div 7=?$ What will be the ones digit of the remainder in: $$\frac{\left|5555^{2222} + 2222^{5555}\right|} {7}$$ How to measure time taken by a function to execute I need to get execution time in milliseconds.  I originally asked this question back in 2008. The accepted answer then was to use  However, we can all agree now  that using the standard  API is more appropriate. I am therefore changing the accepted answer to this one. How can I backup my programs/applications, so that after I reinstall a new one, I can still use the backup-ed ones? Is there a way, that I could backup all my programs, all apps, put them in a directory or something, so that I could install them quickly next time, without having to manually going over the app center, downloading them from terminal, etc?  Note that I might import them from a distro other than the one I exported them on. So, say I'm in Ubuntu, exported some stuff, and then installed BT and wanted to import them from there.