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To prove $\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$ if $a,b,c$ are non zero positive reals prove $$\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$$ I have used A.M  G.M  inequality as follows:  $$a^3+b^3+c^3 \ge 3abc$$ adding $abc$ both sides we get  $$a^3+b^3+abc \ge 4abc-c^3=c(4ab-c^2)$$ so  $$\frac{1}{a^3+b^3+abc} \le \frac{1}{c(4ab-c^2)}$$ But since $a,b,c$ are positive reals  $$4abc-c^3 &lt; 4abc$$ so  $$\frac{1}{4abc-c^3} \gt \frac{1}{4abc}$$  but i am unable to proceed here I have been trying to solve this problem for quite a while. I am still unsure of whether any of the avenues I have pursued have been of any use. Any advice will be much appreciated. Question: Let $V$ be a finite-dimensional inner product space, and let $E$ be an idempotent linear operator on $V$. Prove that if $EE^* = E^*E$, then $E$ is self-adjoint. (This is essentially exercise 5(a) in sec. 80 on p.162 of Paul R. Halmos, Finite-Dimensional Vector Spaces, but Halmos didn't assume that the dimension of $V$ is finite.) How to completely disable auto-save and versions in Mac OS X Lion? I hate the automatic saving and versioning in Mac OS X Lion.  Is there a way to completely turn it off?  (This has been asked before , but the answer and the linked article are completely wrong, so I thought I'd try again.) How to use (home) brew install python 3.6.5_1 I am trying to install python 3.6.5_1 using Homebrew by running the following:  $ brew install https://raw.githubusercontent.com/Homebrew/homebrew-core/f2a764ef944b1080be64bd88dca9a1d80130c558/Formula/python.rb   I am getting the following error:  Error: python contains a recursive dependency on itself: python depends on sphinx-doc sphinx-doc depends on python   How do I deal with it? The parmesan cheese in pasta recipe is quite expensive.....do we have cheaper alternative??  Are there are cheddar cheese which can replace parmesan cheese? Why I cannot ping 192.168.0.108 (my computer IP?) Finding $\lim_{n\to\infty}\binom{2n}{n}$ I found a limit problem which is $\lim_{n\to\infty}{2n \choose n}$ as we know that $\sum_{i=0}^{n}{n \choose i}^2={2n \choose n}$ now we could transform the given problem as $\lim_{n\to\infty} \sum_{i=0}^{n}{n \choose i}^2$ and as $\sum_{i=0}^{n}{n \choose i}^2$ is a polynomial in $n$, therefore, $\lim_{n\to\infty}\sum_{i=0}^{n}{n \choose i}^2=\infty$ Please tell me if my solution is correct or not,and if it is incorrect then please provide me a solution.  Thanks Number of labeled trees which have one vertex of degree 4 How many labeled trees which have one vertex of degree 4 can we have? A tree with $10$ vertices has $9$ edges, so the sum of the degrees of the vertices is $2\cdot 9=18$. If there is one vertex of degree $4$, the sum of the degrees of the other $9$ vertices must be $14$. I don't know what to do next. Should I count all possible vertex degrees? Word for "reading carefully" Sometimes we read books rather quickly and don't give them much (or any) thought, so the action 'reading' does not necessarily imply that we have given enough thoughts to any book we read.   Is there a word that describes reading a book with great care; usually but necessarily slowly over a long period of time. Django: manage.py does not print stack trace for errors CSS3 selector to find the 2nd div of the same class I need a CSS selector that can find the 2nd div of 2 that has the same class. I've looked at nth-child() but it's not what I want since I can't see a way to further clarify what class I want. These 2 divs will be siblings in the document if that helps.  My HTML looks something like this:  &lt;div class="foo"&gt;...&lt;/div&gt; &lt;div class="bar"&gt;...&lt;/div&gt; &lt;div class="baz"&gt;...&lt;/div&gt; &lt;div class="bar"&gt;...&lt;/div&gt;   And I want the 2nd div.bar (or the last div.bar would work too). I have the following problem.      Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ denote the Lebesgue measure of a set $A\subseteq \mathbb C$ and $G=f(D).$ Prove that $$l_2(G)=\pi\sum_{n=1}^\infty n|c_n|^2.$$   After a long struggle I managed to come up with the following formula $$l_2(G)=\iint_D |f\,&#39;(z)|^2dxdy.$$ It looks like the right thing to use because $$f\,&#39;(z)=\sum_{n=1}^\infty nc_nz^{n-1},$$ which is similar to what I have to prove. I understand that the $\pi$ will appear when I integrate something over the angle $\phi$ in polar coordinates. But I don't know how to find the square of the absolute value of the right-hand side to even start integrating...  EDIT: The function is supposed to be  (one-to-one). (I'm not perfectly sure I'm translating the term correctly. The (Polish) word in the statement of the problem was "jednolistna". I have not met it before.) Let  $f:X\times Y\to \mathbb{R}$. I could not come up with an example that satisfies $$\sup_{y\in Y}\inf_{x\in X}f(x,y)&lt;\inf_{x\in X}\sup_{y\in Y}f(x,y).$$ Any help would be appreciated. Thanks! Helix on a cylinder Let's say I write a mine.service file. Then I use systemctl enable mine.service.  If I later decide to edit mine.service, do I have to tell systemd that mine.service was changed? If so, how do I do that? The first given name of an author or editor in my bibliography should be written out, but all further first names should be abbreviated (initials with a period -- and space if there are multiple first names/initials, see below). I am using style=authoryear with BibLaTeX (v. 2.7) and Biber (v. 1.7).  My bibliography looks like this:     Doe, John Arthur Kyle (2013). The secret life of John Doe.   It should look like this:     Doe, John A. K. (2013). The secret life of John Doe.   Minimal example:  example.tex  \documentclass{scrartcl}  \usepackage[ngerman]{babel} \usepackage[babel,german=quotes]{csquotes} \usepackage[backend=biber,style=authoryear]{biblatex} \bibliography{bibliography}  \begin{document}  \cite[123]{Doe} \printbibliography  \end{document}   bibliography.bib  @book{Doe,     title = {The secret life of John Doe},     author = {Doe, John Arthur Kyle},     year = {2013} }   (Worked with said versions of BibLaTeX and Biber, Texmaker (v. 4.0.3) and MiKTeX (v. 2.9.4813).)  I could not find a solution in biblatex.def and firstinits=true abbreviates all first names, including the first one. After updating to 2.81, my previous works (done in 2.80) doesn't show colors Math symbol question: Vertical bar for ''evaluated at ...'' Reconciling different definitions of left-exact functor Product of path connected spaces is path connected