How can I approach this inequality?
Prove inequality $\sqrt{\frac{2a}{b+a}} + \sqrt{\frac{2b}{c+b}} + \sqrt{\frac{2c}{a+c}} \leq 3$
Using the uncertainty principle to estimate the ground state energy of hydrogen I have been reading through this  of the ground state energy of hydrogen  and others like it. In this one it says it is using the uncertainty principal but then proceeded to use the following: $$pr=\hbar$$ But why is it not using: $$pr=\frac{\hbar}{2}$$ which is more in line with the uncertainty principle and what comes earlier in the derivation. Removing empty lines in Notepad++ Kontact in Kubuntu 16.10 will not start because the Akonadi server will not start How do I preload a program for faster start? Is it possible to "preload"(actually not preload, but kinda emulate the startup) e.g. nautilus (or firefox) startup at login, so the first time I launch it myself it launches faster? Probably to start it on another desktop and close after some seconds? What can I use instead of the arrow operator, `->`? Output URL of open firefox tabs in terminal Let $A$ be a commutative ring, $I$ an ideal of $A$ and $M$ an module over $A$. Is it true that $\operatorname{Ann}(M/IM) = \operatorname{Ann}(M) + I$?    One inclusion is certainly true, but I don't know about the other one. If the above statement does not hold in general, does it perhaps for finitely generated modules? Let $G$ be a group and $H$ a subgroup with $[G:H]=n$. Is it true that $x^n\in H$ for all $x\in G$?   Remarks. The answer is positive whenever $H$ is normal, e.g., for $n=2$. In general, by using the normal core of $H$, one can find an $m\ge 1$ such that $x^m\in H$ for all $x\in G$. Lattices are congruence-distributive Let $p$ be a prime number, and let $\mathbb{F}_p$ be the field with $p$ elements. How many elements of $\mathbb{F}_p$ have cube roots in $\mathbb{F}_p$?   I had this question on an exam and after reviewing I am still not sure. Any help would be appreciated. Bump right-aligned text to next line iff no room Wrapping a mesh around point cloud with cavities I'm writing a script which shows the git log for a directory when I cd into it. Such a log can be overwhelming, containing hundreds of lines. So far I have been limiting that to a hard-coded 20 lines (... | head -n 20), which is fine on the screen at work, but too much on the smaller MacBook screen at home.  I would prefer the log to take up about half the (vertical) screen on either terminal. And "terminal" also changes: it's Gnome terminal at work, but iTerm2 at home. And I do not use screen or tmux.  How do I find the number of vertical lines available in a terminal from command line? 1.In classical mechanics, using Newton's laws, the ellipticity of orbits is derived. It is also said that the center of mass is at one of the foci.   2.Each body will orbit the center of the mass of the system.  My question is : Are the assumptions in 1 and 2 correct?   Follow up question : Assuming the distance from the centre of the mass to each body remains the same, do we have two bodies orbiting the centre of the mass of the system in an elliptical or circular orbit?   Finally : With elliptical orbits, if the heavier mass is supposed to be in one of the foci, if there is any significance to second focus, what is it? Is it a Lagrange point by any chance or does it have some other mathematical property? Explicitly accept change for PPA 'Label' When I run apt-get update on my ubuntu 17.10 I receive the following error:  E: Repository 'http://ppa.launchpad.net/ondrej/php/ubuntu artful InRelease' changed its 'Label' value from '***** The main PPA for PHP (5.6, 7.0, 7.1) with many PECL extensions *****' to '***** The main PPA for supported PHP versions with many PECL extensions *****' N: This must be accepted explicitly before updates for this repository can be applied. See apt-secure(8) manpage for details."   The man page doesn't really point to a solution.  My question is, how do I explicitly accept the change? Are comments safely hidden once the document is compiled? How to delete all lines that do NOT contain a certain word in Vim? In vim I can delete all lines that contain the word "price" with this  :g /price/d   How can I delete all lines that do NOT contain the word "price"? Where has all the RAM gone? Unable to cover get set in apex test class How to integrate $\int_0^{\infty} \frac{e^{ax} - e^{bx}}{(1 + e^{ax})(1+ e^{bx})}dx$ where $a,b > 0$. This  $$\ \int_0^{\infty} \frac{e^{ax} - e^{bx}}{(1 + e^{ax})(1+ e^{bx})}dx \text{ where } a,b &gt; 0. $$ is a problem that showed up on a GRE practice test. I believe you're supposed to use complex contour integration, but I'm not sure which contour to use. I think it's the keyhole contour, but I wasn't able to get anything useful.