Let $a$, $b$, $c$, $d$ and $e$ be positive numbers. Prove that: $$\frac{a-b}{b+c}+\frac{b-c}{c+d}+\frac{c-d}{d+e}+\frac{d-e}{e+a}+\frac{e-a}{a+b}\geq0$$ Let $e=\min\{a,b,c,d,e\}$. So, we can prove this inequality in any cases except the following two cases.  $d\geq b\geq c\geq a\geq e$ and  $a\geq b\geq d\geq c\geq e$.   I suggest that in this topic we'll say about these two cases only. We have here the following things (thanx to Mida, which showed me this possibility). Let $c\geq d$. Thus, by C-S $$\sum_{cyc}\frac{a-b}{b+c}=\frac{a-b}{b+c}+\frac{b-c}{c+d}+\frac{c-d}{d+a}+\frac{d-a}{a+b}+$$ $$=\frac{d-c}{d+a}+\frac{a-d}{a+b}+\frac{c-d}{d+e}+\frac{d-e}{e+a}+\frac{e-a}{a+b}=$$ $$=-4+\frac{a+c}{b+c}+\frac{b+d}{c+d}+\frac{a+c}{a+d}+\frac{b+d}{a+b}+$$ $$+(c-d)\left(\frac{1}{d+e}-\frac{1}{d+a}\right)+\frac{e-d}{a+b}+\frac{d-e}{e+a}=$$ $$=-4+(a+c)\left(\frac{1}{b+c}+\frac{1}{a+d}\right)+(b+d)\left(\frac{1}{c+d}+\frac{1}{a+b}\right)+$$ $$+\frac{(c-d)(a-e)}{(d+e)(d+a)}+\frac{(d-e)(b-e)}{(e+a)(a+b)}\geq$$ $$\geq-4+\frac{4(a+c)}{b+c+a+d}+\frac{4(b+d)}{c+d+a+b}=0.$$ Id est, it's enough to prove our inequality for $d\geq c$. Now, by using C-S for the expressions $$\frac{b-c}{c+d}+\frac{c-d}{d+e}+\frac{d-e}{e+b}+\frac{e-b}{b+c},$$ $$\frac{c-d}{d+e}+\frac{d-e}{e+a}+\frac{e-a}{a+c}+\frac{a-c}{c+d},$$ $$\frac{d-e}{e+a}+\frac{e-a}{a+b}+\frac{a-b}{b+d}+\frac{b-d}{d+e}$$ and $$\frac{e-a}{a+b}+\frac{a-b}{b+c}+\frac{b-c}{c+e}+\frac{c-e}{e+a}$$ respectively, we obtain: $(1)$ $$\sum_{cyc}\frac{a-b}{b+c}\geq\frac{(d-e)(b-a)}{(e+a)(e+b)}+\frac{(a-e)(a-c)}{(b+c)(a+b)},$$ $(2)$ $$\sum_{cyc}\frac{a-b}{b+c}\geq\frac{(a-e)(b-c)}{(a+c)(a+b)}+\frac{(a-b)(d-b)}{(b+c)(c+d)},$$ $(3)$ $$\sum_{cyc}\frac{a-b}{b+c}\geq\frac{(a-b)(d-c)}{(b+c)(b+d)}+\frac{(c-b)(c-e)}{(c+d)(d+e)}$$ and $(4)$ $$\sum_{cyc}\frac{a-b}{b+c}\geq\frac{(c-b)(d-e)}{(c+e)(c+d)}+\frac{(d-c)(d-a)}{(e+a)(d+e)}.$$ Since $d\geq c$ and $e=\min\{a,b,c,d,e\}$, it's enough to check $12$ cases and we can see that in ten of them our inequality is true. For example, the case $a\geq d\geq b\geq c\geq e$ follows from $(2).$
let $a,b,c,d,e$ are positive real numbers,show that $$\dfrac{a-b}{b+c}+\dfrac{b-c}{c+d}+\dfrac{c-d}{d+e}+\dfrac{d-e}{e+a}+\dfrac{e-a}{a+b}\ge 0$$  My try:  I have solved follow Four-variable inequality:  let $a,b,c,d$ are positive real numbers,show that $$\dfrac{a-b}{b+c}+\dfrac{b-c}{c+d}+\dfrac{c-d}{d+a}+\dfrac{d-a}{a+b}\ge 0$$ poof:since \begin{align*} &amp;\dfrac{a-b}{b+c}+\dfrac{b-c}{c+d}+\dfrac{c-d}{d+a}+\dfrac{d-a}{a+b}=\dfrac{a+c}{b+c}+\dfrac{b+d}{c+d}+\dfrac{c+a}{d+a}+\dfrac{d+b}{a+b}-4\\ &amp;=(a+c)\left(\dfrac{1}{b+c}+\dfrac{1}{d+a}\right)+(b+d)\left(\dfrac{1}{c+d}+\dfrac{1}{a+b}\right)-4 \end{align*} By Cauchy-Schwarz inequality we have $$\dfrac{1}{b+c}+\dfrac{1}{d+a}\ge\dfrac{4}{(b+c)+(d+a)},\dfrac{1}{c+d}+\dfrac{1}{a+b}\ge\dfrac{4}{(c+d)+(a+b)}$$ so we get $$\dfrac{a-b}{b+c}+\dfrac{b-c}{c+d}+\dfrac{c-d}{d+a}+\dfrac{d-a}{a+b}\ge\dfrac{4(a+c)}{(b+c)+(d+a)}+\dfrac{4(b+d)}{(c+d)+(a+b)}-4=0$$ Equality holds for $a=c$ and $b=d$  By done!  But for  Five-variable inequality,I can't prove it.Thank you
Can we have a way to access Area 51 on the iOS and android app?  More sites would flourish due to more traffic. Proving that $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ if $n$ is greater than 2 Often in my papers I want to mention theorems in the introduction that are introduced in more detail and proved in a later section.  I'd like to include the statement of these theorems twice over, each time numbered as the theorem would naturally appear in its later section.  My current solution (using amsthm) is, e.g.  %(in the introduction) \newtheorem*{thm:associativity}{Theorem \ref{thm:associativity}} \begin{thm:associativity} Lorem ipsum ... \end{thm:associativity}  %(in a later section) \begin{thm} \label{thm:associativity} Lorem ipsum ... \end{thm}      Can anyone think of a cleaner solution? Getting Started with Unit Testing Unit testing is, roughly speaking, testing bits of your code in isolation with test code. The immediate advantages that come to mind are:  Running the tests becomes automate-able and repeatable You can test at a much more granular level than point-and-click testing via a GUI    My question is, what are the current &quot;best practices&quot; in terms of tools as well as when and where to use unit testing as part of your daily coding? Lets try to be somewhat language agnostic and cover all the bases. $\operatorname{Supp}(M)=V(\operatorname{Ann}M)$ if $M$ is finitely generated $f$ is irreducible $\iff$ $G$ act transitively on the roots Let $F$ be a field and $f\in F[X]$ a separable polynomial. Let $K_f$ be the splitting field of $f$ and $G_f=\text{Gal}(K_f/F)$ its Galois group. Show that $f$ is irreducible $\iff$ $G_f$ acts transitively on the roots on $f$. My work $\implies :$ Let $f\in F[x]$ irreducible, $\alpha \in K_f$ a root and $\alpha _1,...,\alpha _n$ the other roots of $f$ (so $K_f=F(\alpha ,\alpha _1,...,\alpha _n)$). What I have to show (I think) is that for all $i$ there is a $\sigma \in G_f$ s.t. $\sigma \alpha =\alpha _i.$ Q1) Is it right ? Q2) Is $f$ the minimal polynomial of $\alpha $ ? (To me it is, but my teacher said that it can be not the case, but I don't understand why). Since $\alpha $ and $\alpha _1$ have the same minimal polynomial, there is an isomorphism $\sigma :\alpha \longmapsto \alpha _1$. We can observe that $\sigma : F(\alpha )\longrightarrow K_f$. Now $F(\alpha ,\alpha _1)/F(\alpha )$ is algebraic, then we can extend $\sigma $ to a morphism $F(\alpha ,\alpha _1)\longrightarrow K_f$. Now, we can continue like that and extend $\sigma $ as $\psi: K_f\longrightarrow K_f$ s.t. $\psi(\alpha )=\alpha _1$. Q3) The fact that $\psi(\alpha )=\alpha _1$ comes from the fact that $\psi|_{F(\alpha )}=\sigma $, right ? The bijectivity is obvious since it's a field homomorphism and that $\sigma (K_f)=K_f$. Q4) How to conclude that $G_f$ act transitively on the roots ? I don't have particular problem for the converse. If $l$ and $n$ are any positive integers, is there a  proof of the identity     $$\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}(l-k)^n=n!\;$$   which uses the Inclusion-Exclusion Principle?  (If necessary, restrict to the case where $l\ge n$.)    This question is closely related to   and also Is there any research or data on horizontal vs. vertical navigational menus? I've seen many sites using both horizontal (top) navigation, vertical (left) navigation, and both across web sites for a long time. My understanding is that they're both conventions, but I would assume there must be some data out there supporting one route over the other.   My question is: are any studies or data for or against horizontal or vertical navigation (as suggested above) with regards to primary navigation (specifically on a smaller site or web app).  I've read some interesting things that are flipping me back and forth between liking horizontal more than vertical and vice versa. For example if you look at Jacob Nielsen's post on  you get the feeling that more important content should be on the left -- but you could take that as it being more important for the navigation (the user's escape routes) to be in that valuable space. If you're interested in grammar, as I am, I am sure you have delved into a thought process about infinitive to, and like me, you have probably questioned what it is, or what it could be defined as. My mind always wants to place it in the Eight Parts of Speech, and I know this is foolish of me to some regard, so I stray from it, but I still wish to designate it under something that can be explain it. So, this is the motivation for this post, this question: What is the infinitive to, and why?  Here are my thoughts: There are three real possibilities, going back through the times. First, we can say that it is part of the verb that comes after it (to go, to run, etc.). This could be supported by the claim that you are not supposed to "split" infinitives ("To boldly go"), but only that.   Secondly, we could consider it a subordinator. I came across this designation most recently in my studies of modern and transformational grammar. As of right now, it is still the one I accept, especially as a result of the trend of designating more phrases as clauses. It works, except it's weird. Do we call this a unique subordinator as a result of how it operates? For example, to-infinitive clauses call for non-finite verb forms, which typically have an implied subject as a part of their clause. For example, "I want to go to the store." In the infinitive clause, "I" is the implied subject, as a understand. So, again, this works. However, things get a little weird when other subordinators come into play. For example, "I'm excited for you to win the competition." Now, if I'm right, those of you will say that "for" is a preposition here, as a result of the new-grammar stuff, but I'm going to count it as a subordinator, as defined by my own beliefs and that of a lot of other grammars. So, how do the clauses work in such a sentence? According to transformational grammar, "for you to win the competition" is defined as an ordinary clause. So, does that mean that it also contains a secondary clause within itself (the infinitive clause part)? If so, how do those two clauses work in conjunction to each other when one is so integrated? "For you to win the competition" looks to just be one clause, but if that's the case, doesn't "to" have to represent something else?  This moves us on to the third way to look at things, which is to consider "to" as auxiliary verb, a defective one at that. There are a lot of things to support this, and it even allows for the split infinitive. Furthermore, it calls for a non-finite verb form to follow, like an auxiliary. But, as I said before, it's defective in the way that is all it can function as.  As I said, I still accept to as a subordinator, but I want to be proven wrong or right. Wrong by showing evidence of it being something else, or right by showing how clauses operate when it seems to be layered up. How to add startup applications in lubuntu? Is there a 'Startup applications' app for lubuntu? Or how do I add apps to startup? I attempted a tutorial to add entries to ~/.config/autostart, but it did not work. Isn't there an easy way? Any ideas please? Follow up: instance control in Java without enum Sort thebibliography by citation order How can I tell what version of Linux I'm using? Often times I will ssh into a new client's box to make changes to their website configuration without knowing much about the server configuration. I have seen a few ways to get information about the system you're using, but are there some standard commands to tell me what version of Unix/Linux I'm on and basic system information (like if it is a 64-bit system or not), and that sort of thing?  Basically, if you just logged into a box and didn't know anything about it, what things would you check out and what commands would you use to do it? I'm curious to see some real examples of where simply capturing the same photo in RAW (and being processed by someone who can do it justice) has significantly improved the photo at the end of the process.  I understand what RAW is and why you might want to use it over JPEG, however, I'd like to actually see some examples where it has allowed for a better result. More control over tone, conversion from the more detailed data to 8-bit RGB etc.  Does anyone have or know of some processed RAW+JPEG shots for exact comparison? Is MacBook Pro warming during gaming normal? I have a 2010 MacBook Pro MC371 (Core i5, GeForce 320M, 4Gb RAM).   When I launch a game either new one (i.e. Starcraft 2) or rather old (Unreal Tournament 2004, bootcamped) my MacBook gets rather warm, even hot in the top left corner, around the place where magsafe charger is plugged in. The temperature is more than 40C. The fact that confuses me is that this warming occures even when I play old games.  Is that OK? Pull-up and Pull-down Resistor Usage on Input or Output MCU Pins Are pull-up/down resistors (whether internal or external) only needed for MCU INPUT pins? In contrast, an MCU pin configured as an OUTPUT "knows what level it's at" because it does the driving - a "floating" MCU OUTPUT pin tied to some input of another circuit doesn't make sense, because the state of the MCU pin can only be high or low... do I have this right? Now, upon MCU bootup or failure, it may be beneficial to have a pull-up/down tied to this "MCU output to IC input" line to ensure that the input to some IC is never floating.  Maybe I just answered my own question here... pull-up/down resistors can be used on both input and output pins, depending on application? In this video, a college basketball player steals the ball, races down court, and avoids the defender by throwing a pass to himself off the backboard.    In all the (disorganized) basketball games I've played in, a shot had to hit the rim before the shooter was allowed to rebound. And that looks like a shot to me.  Obviously the officials allowed it, but I couldn't find an explicit mention in the . Is this ruled as a shot, and if so, why is it legal? I'd be interested in the answer for any level of basketball, though NCAA is the most relevant. Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$ How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$? I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me. (This series arises in studying the first passage time of a simple random walk.) Total number paths between two nodes in a complete graph How to determine the value of a domain name? I'd like to register a second-level domain with the .com extension and I noticed that it has already been registered and the owner put it for sale and let the possibility to make an offer in their site or by contacting them.  I saw that domains are generally sold for extremely different prices, so I'm confused.  Are there any general criteria to decide how much I should offer to purchase a domain?