Let $a,b,c$ be positive reals, such that $\frac1a+\frac1b+\frac1c=a+b+c\ (\star)$, find the maximum of $\frac{1}{(2a+b+c)^2}+\frac{1}{(2b+c+a)^2}+\frac{1}{(2c+a+b)^2}$   This should be an application of Jensen's Inequality, so I have to find a concave function to maximize the sum, I thought to define;  $f(\frac1x):=\frac{1}{x+a+b+c},\quad\frac{\partial^2f(x)}{\partial x^2}&lt;0$  so $f$ is concave and then the sum above is;  $f(\frac1a)+f(\frac1b)+f(\frac1c)$   using Jensen it should be less or equal to   $3\cdot f(\frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}3)=3\frac{1}{\left(\frac3{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}+a+b+c\right)^2}\overset{(\star)}=3\frac{1}{\left(\frac3{a+b+c}+a+b+c\right)^2}$  and the denominator can be minimized with AM-GM;  $\left(\frac3{a+b+c}+a+b+c\right)^2\ge\left(2\sqrt{3\frac{a+b+c}{a+b+c}}\right)^2=12$  so the result is $\frac14$, but it is wrong, where does it fail, maybe AM-GM value cannot be attained ?
Let a,b,c be positive real number, such that  $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=a+b+c$. Prove that :  $\frac{1}{(2a+b+c)^2}+\frac{1}{(2b+c+a)^2}+\frac{1}{(2c+a+b)^2} \leq \frac{3}{16}$  Can anyone help me how to deal with it?
Singular values of the data matrix and eigenvalues of the covariance matrix $\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$. I have found that many USB wall chargers use a resistive voltage divider to set the D+ and D- pins to a specific voltage, usually between 2 and 3 volts.  Other USB wall chargers short the D+ and D- pins together with no connection to anything else.  From my experience some devices will not accept a charge rate above 500mA on the chargers using the voltage dividers, but will charge up to their max input on a charger with the data pins shorted. I have read things that suggest the opposite may be true as well, but have been unable to verify this. I am hoping to figure out which method provides the best compatibility with all USB devices. I often see output from TeX with the warning overfull hbox, badness 10000. What does this message mean? SQL Server Escape an Underscore How do I escape the underscore character?  I am writing something like the following where clause and want to be able to find actual entries with _d at the end.  Where Username Like '%_d' Sometimes it is convenient to refer to a group of equations by a letter and a number.  For example:   2+1   (A1) 2+4   (A2) x^4   (B3) x^e   (B4)   Placing  \tag{A\theequation}\label{eqn:thisone}\stepcounter{equation}} instead of just \label{eqn:thisone} works, but seems hacky (especially manually stepping the counter).   Is there a better way? I'd like to map an int value pbs_errno declared as a global in the library libtorque.so using ctypes.  Currently I can load the library like so:  from ctypes import * libtorque = CDLL("libtorque.so")   and have successfully mapped a bunch of the functions. However, for error checking purposes many of them set the pbs_errno variable so I need access to that as well. However if I try to access it I get:  &gt;&gt;&gt; pytorque.libtorque.pbs_errno &lt;_FuncPtr object at 0x9fc690&gt;   Of course, it's not a function pointer and attempting to call it results in a seg fault.  It's declared as int pbs_errno; in the main header and extern int pbs_errno; in the API header files.  Objdump shows the symbol as:  00000000001294f8 g    DO .bss   0000000000000004  Base        pbs_errno How does vendor restocking work in Fallout 4? I've been looking for 2mm EC ammo and according to the  it is supposed to restock every 48 hours at Arturo. However he has not gotten any new 2mm EC ammo for at least 10 restocks.   Do vendors restock on a random selection of items with some being more likely than others? Is there a max limit to how many different items a vendor will sell so that no new items will be added to stock after this limit is reached? Is there a level limit on what items can be restocked? I bought 2mm EC ammunition once from Arturo so I assume I'm high enough level if that's the case? I am very new to LaTeX and I am writing my thesis. In a chapter, I want to show my codes, which are in Mathematica.   I use the listings package as follows:  \documentclass{article}   \usepackage{listings}   \begin{document}   \lstset{language=Mathematica}    \begin{lstlisting}   Do[i^j,{i,1,10},{j,1,10}]   \end{lstlisting}    \end{document}   However, the output is much different to what the code looks like in the Mathematica IDE (even if I use other languages in \lstset{language=&lt;name&gt;}):    I read some questions on the site but, in some of them, the font or color of the codes were not the same. Could you please help me to display something like the picture below in LaTeX?  If it is important, I am using the BiDi package. What are the possible values of the Hibernate hbm2ddl.auto configuration and what do they do I really want to know more about the update, export and the values that could be given to hibernate.hbm2ddl.auto I need to know when to use the update and when not? And what is the alternative?  These are changes that could happen over DB:   new tables new columns in old tables columns deleted data type of a column changed a type of a column changed its attributes tables dropped values of a column changed    In each case what is the best solution? I got quite some trouble trying to prove this.  $$\sum_{k=m}^{n}(-1)^k\binom{n}{k}\binom{k}{m}=0, n&gt;m\geq 0$$  I tried using $$\binom{n}{m}\binom{m}{k}=\binom{n}{k}\binom{n-k}{m-k}$$ and then changing the summation boundaries as $k=0, n=n-m$, but that took me nowhere... I would really appreciate help. Thank you. Links being changed to cppreference.com Someone just edited a link in one of my answers to point to a 'better reference'.    I've no problem with links being changed in general, but this smells like part of a concerted attempt to redirect traffic.   Is this kind of activity acceptable on SO? Let $q \in \mathbb C$, $|q|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n:  n \in \mathbb N\}$ is dense in $S^1$.  My attempt: As $(q^n)$ is limited, there is subsequence $(q^{n_j})$ convergent. So given $\epsilon &gt; 0, \exists j_0 \in \mathbb N, \quad |q^{n_j} - q^{n_k}| &lt; \epsilon, \quad \forall j,k \geq j_0$. Supose that $n_j &gt; n_k$, then  $|q^{n_j} - q^{n_k}| = |q^{n_k}||q^{n_j - n_k} - 1| = |q^{n_j - n_k} - 1| &lt; \epsilon, \quad \forall j, k \geq j_0$. So, there is a subsequence of $(q^n)$ that converges to $1$.  Using this fact, how can I ensure that for all $w \in S^1$ there is a subsequence of $(q^n)$ that converges to  $w$? Technically, NULL = NULL is False, by that logic no NULL is equal to any NULL and all NULLs are distinct. Shouldnt this imply that all NULLs are unique and a unique index should allow any number of NULLs? Function that returns the second argument of Reap I'm planning a trip to recover a motorcycle from a distant city. The ideal trip would be flying to that location and coming back riding the bike, but that would require me to transport my riding gear with me in the plane: Leathers, boots, gloves, back protector and crash helmet.  All this is bulky and expensive stuff, and a crash helmet may be damaged if dropped or hit hard.   I don't expect any airline to allow me a crash helmet as hand luggage; and wearing it all the time is out of the question :-)   To complicate things, I would be coming back riding the bike, so any hard suitcase would have to be left behind.   How do people normally transport that kind of stuff? Prove by mathematical induction that $2^{3^n}+1$ is divisible by $3^{n+1}$ This is what I see:  GNU GRUB version 2.02~beta2-9ubuntu1  Minimal BASH-like line editing is supported. For the first word, TAB lists possible command completions. Anywhere else TAB lists possible device or file completions.  grub&gt; _   What should I do now? How to create a .deb file with dependency? I have a game made with the LÖVE framework and I want to package it into a .deb file so it can be easily installed by others and be easily findable through the Unity dash for example.   I also have the LÖVE framework love_0.7.2.deb which I want to be in the deb file also somehow.  Files that I have:  game_icon.png game .desktop the_game.love love_0.7.2.deb   Could someone explain to me how to do this? Is there a quick way to get the control that's under the mouse? I need to find the control under the mouse, within an event of another control.  I could start with GetTopLevel and iterate down using GetChildAtPoint, but is there a quicker way?