In Two and Half Men, Charlie kept Alan around since they were brothers from the same mother. But why does Walden continue to keep him around? He's certainly not paying rent, helping and certainly not in love! I can understand why he would wanna keep Jenny and her girlfriend around (I mean, who wouldn't?), but why Alan?
I have watched the series of Two and a Half Men two times now till the latest episode, but can't understand why either Charlie or Walden just don't kick Alan out? I mean it is a bit cruel but they are shown to put up with him even though he gets up on their nerves (even got on my nerves).    Some examples off the top of my head:   He became the reason of his first engagement with Mia to be broken up.   He once set up his curtains on fire after ruining his day.   He invited his girlfriend/assistant to move in without his permission and started acting like he owned part of the house when he didn't even pay rent. He acted sickly and dying (Walden's time) and tricked him out of money for his care and comfort.   He deliberately tried to break his relations (Walden's) just so he can keep sponging off.   I know not the best examples, but if not in every episode then in every alternate episode he is shown to getting up on their nerves and they just keep on telling him to leave. Why don't they just kick him out? Why put up with such a leach?
A continuous, injective function $f: \mathbb{R} \to \mathbb{R}$ is either strictly increasing or strictly decreasing. I would like to prove the statement in the title.  Proof: We prove that if $f$ is not strictly decreasing, then it must be strictly increasing. So suppose $x &lt; y$.  And that's pretty much how far I got.  Help will be appreciated. If $$S=\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\cdots+\frac{n}{1+n^2+n^4}\,$$ find the value of $14S$.  The question can be simplified to:  Find $S=\sum\limits_{k=1}^n\,t_k$ if $t_n=\dfrac{n}{1+n^2+n^4}$.  As $1+n^2+n^4$ forms a GP, $$t_n=\frac{n(n^2-1)}{n^6-1}\,.$$ But I can't figure out how to solve further. It would be great if someone could help. Topology on cartesian product and product topology In an exam paper, there was a picture of an air stewardess in the aeroplane serving passengers. One of my pupils wrote the following:      The air stewardess works on an aeroplane.    Shouldn't it be the air stewardess works in an aeroplane? 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While searching on the internet, I found that the boxplot can provide an idea about how much they are associated; however, I was looking for a quantified value such as Pearson's product moment coefficient or Spearman's $\rho$. Can you please help me on how to do this? Or, inform on which method would be appropriate?  Would Point Biserial Coefficient be the right option? I'm automating a web application (the Mantis bug tracker) and I'm getting an interesting response header from it, called Refresh:  HTTP/1.x 200 OK ... Refresh: 0;url=my_view_page.php   It seems to be acting the same way that  does, and the meta refresh technique implies that it is an equivalent of a header in HTTP.  Problem is, I can't find any mention of the Refresh header in the  or any other definitive documentation on how it should be parsed and what the browser should do when it encounters it.  What's going on here? 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Is there a setting that i've overlooked that tells Visual Studio to treat Solution Folders as "real" folders, that is to create them in the file system and move files into it when I move them inside the solution into one of those folders?  Edit: Thanks. Going to make a suggestion for VS2010 then :) I can't not be pregnant This is a quote from a  where she talks about her miscarriage:     "It was awful," she told the Daily Mail's You magazine. "It happened really late, and I wasn't expecting it at all. "For a long time I just couldn't understand that it had happened and why. I was like 'But I can't not be pregnant. I'm already past the first trimester, and I have three car seats in my head.' To this day I feel like I'm missing that kid. But it wasn't meant to be and you have to trust the universe. "   What does she mean by saying "I can't not be pregnant?" Is this "can't not" construction a grammatical and common usage? Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics