$(a+b)^\beta \leq a^\beta +b^\beta$ for $a,b\geq0$ and $0\leq\beta\leq1$ It seems that $(a+b)^\beta \leq a^\beta +b^\beta$ for $a,b\geq0$ and $0\leq\beta\leq1$. However, I could not prove this nor the same result for a general concave and increasing function (for which it might not hold). If the inequality is true, does it follow from some general inequality or is there some other simple proof?
Prove that $(p+q)^m \leq p^m+q^m$ If $p,q$ are positive quantities and $0 \leq m\leq 1$ then Prove that $$(p+q)^m \leq p^m+q^m$$    Trial: For $m=0$, $(p+q)^0=1 &lt; 2= p^0+q^0$  and for $m=1$, $(p+q)^1=p+q =p^1+q^1$.   So, For $m=0,1$ the inequality is true.How I show that the inequality is also true for $0 &lt; m &lt; 1$.   Please help.
Can a person negotiate with banks on CD interest rates? I use a brick and mortar bank for some of my savings. I’d love to get a CD interest rate closer to what Ally or other online banks offer.   Are brick and mortar banks will to negotiate with customers on interest rates? For instance, if I went into a branch and said I’m moving my money to Ally unless the bank matches Ally’s rates, do I have any chance of getting them to? Which Ubuntu to use alongside Windows 7? What is the difference between the search() and match() functions in the ?  I've read the  (), but I never seem to remember it.  I keep having to look it up and re-learn it.  I'm hoping that someone will answer it clearly with examples so that (perhaps) it will stick in my head.  Or at least I'll have a better place to return with my question and it will take less time to re-learn it. Why "was" not "were" in "Nearly £20 was taken from my bank account" I've always said "$100 were taken" not "$100 was taken" because I thought $100 is plural. Could you explain why "was" not "were"? Any other helpful notes about the issue would be appreciated.     Nearly £20 was taken from my bank account Generated $\sigma$-algebras with cylinder set doesn't contain the space of continuous functions What are reaching fouls in the NBA? How to reinstall default packages In an attempt to uninstall wine I ran the command sudo apt-get purge wine* which as you may have guessed removed many of the default packages. I am wondering if there is a way to use the installation disk to reinstall the default packages without overwriting the file system. Not sure if it makes a difference or not, but I am duel booting Windows 8 and Ubuntu 14.04 which was updated from Ubuntu 12.04.  Thanks in advance. Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z}[X])$ is irreducible Let's say I have a (comparatively) lovely-looking document in LaTeX, full of lovingly typeset, (relatively) complex equations. Now, let's say some barbarians come along one day and decide that the document has to be put into Microsoft Word (2007). ...after the usual mourning period associated with such events, let's say I value my job (more specifically, the bread it provides) enough to get all the text and tables formatted and references organised into the Word document. –...related questions  and – Now I'm looking at the equations with fear and dread. One option of course is to just lift screenshots from the original document, but this is painstaking if I need to refer to parts of the equation in the text. Also, I might need to edit equations on the fly. Anyone know of a free application which allows embedding LaTeX math into MS Word? I've looked at  and  which do roughly what I want... they build LaTeX images from source and embed them into the Word document, allowing to edit the source later... but both are commerical. ...any help in these troubling times will be greatly appreciated. EDIT: Just a note that  offers a 30-day free trial and is working out really nicely... but still, it's not free. Might be a good solution for those with short-term needs, or money. Q&A blank in Stack Exchange Android app v1.0.95 - S10e/Android 10 To ensure that a new chapter starts on a right-hand page, the book class (and the report class with the options twoside and openright) will insert blank pages between chapters if necessary. However, these pages still contain the headers and footers as defined by the class. How can I remove those?  Note: I know that I can manually add \clearpage\thispagestyle{empty} at the appropriate points of my document. However, this seems awkward and is prone to fail if I change my document later. I'm looking for automatic solutions. We are given two independent standard normal random variables $X$ and $Y$. We need to find out the M.G.F of $XY$.  I tried as follows : \begin{align} M_{XY}(t)&amp;=E\left(e^{(XY)t}\right)\\&amp;=\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{(XY)t}f_X(x)f_y(y)dxdy \\ &amp;=\dfrac{1}{2\pi}\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{(xy)t}e^{\frac{-x^2}{2}}e^{\frac{-y^2}{2}}dxdy \\ &amp;= \dfrac{1}{2\pi}\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{-\dfrac{(x-ty)^2}{2}}e^{\dfrac{-t^2y^2}{2}}e^{\frac{-y^2}{2}}dxdy \\ &amp;= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{\dfrac{-t^2y^2}{2}}e^{\frac{-y^2}{2}}dy \\ &amp;= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{-y^{2}(\frac{1}{2}+t^2)}dy \\ &amp;= \dfrac{1}{\sqrt{1+2t^2}} \end{align} Is this correct ? How can you strip non-ASCII characters from a string? (in C#) How can you strip non-ASCII characters from a string? (in C#) What does \x mean in C/C++? Using ReplaceAll on SparseArray fatal: bad ownership or modes for chroot directory component "/" in SFTP I've been trying to configure sftp in a debian server.sshd_config:  Subsystem sftp internal-sftp  UsePAM no  Match User sftpUser ChrootDirectory /users/sftp/sftpUser ForceCommand internal-sftp AllowTcpForwarding no X11Forwarding no   Directories:   drwxr-xr-x   3 root      root      4096 Oct 20 10:59 users   drwxr-xr-x   3 root      root      4096 Oct 20 11:00 sftp   drwxr-xr-x   2 root      root      4096 Oct 20 11:00 sftpUser   cat /var/log/auth.log  Oct 20 10:58:22 w1 sshd[24634]: Accepted password for sftpUser from 201.156.103.213 port 34106 ssh2 Oct 20 10:58:22 w1 sshd[24636]: fatal: bad ownership or modes for chroot directory component "/"   sftpUser no home, bin/falseand member of user group.  In the client side I'm getting the famous Write failed: Broken pipe and then the connection is dropped.  Commenting the ChrootDirectorycommand in sshd_config makes the connection, but get's the user loose.  What am I doing wrong ? I'm stuck with the proof of the following:  Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$.   I've tried starting with $s_n \leq t_n \: \forall : n$ and the definitions of each limit (i.e. $|s_n - s| \leq \epsilon \: \forall \: n &gt; N_1$), but I'm not really getting very far. Any help is appreciated! why $1+\frac{1}{1+\frac{1}{\frac{1}{1+\dots}}}$ cannot be equal to $\frac{1-\sqrt{5}}{2}$  I know that some of you will answer because it is positive.I know it but I don't know how to prove it.Maybe the second part is positive how do you know that?Thanks in advance. Argue by contradiction to show that there is no surjection... from $A$ to $\{0,1\}^A$. When $A$ is any (possibly infinite) set  How does one go about this? Prove that, for any $n+1$ integers $a_{1},a_{2},....,a_{n+1}$, there exist two of the integers $a_{i}$ and $a_{j}$ with $i \neq j$, such that $a_{i} - a_{j}$ is divisible by $n$.  Please help me about this problem,I know there is lots of answers can find from Google, but I am not really understand them. I hope someone can give me more details explanation. I would really appreciate your help.