Retagging to include tags for possible solutions
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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 want to find an example of a homomorphism $f:G\to H$ such that $A$ is a normal subgroup of $G,$ but $f(A)$ is not so in $H.$  I know that if $f$ was onto $f(A )$ must be normal , but otherwise i want to find an example! To my understanding, "Control" can have two meanings in statistics.   Control group: In an experiment, no treatment is given to the member of the control group.  Ex: Placebo vs Drug: You give drugs to one group and not to the other (control),  which is also referred as "controlled experiment".  Control for a variable: Technique of separating out the effect of a particular independent variable. Some of the other names given to this techniques are, " accounting for", "holding constant", "controlling for", some variable.  For example: In a Football viewing study (like or not like),  you may want to take out the effect of gender as we think gender causes bias, that is, male may like it more than female.     So, my question is for point (2). Two questions:  How do you "control"/ "account for" variables, in general. What techniques are used? (In terms of regression, ANOVA framework).   In above example, does choosing male and female randomly constitutes control? That is, is "randomness" one of the techniques for controlling other effects? A get() like method for checking for Python attributes If I had a dictionary dict and I wanted to check for dict['key'] I could either do so in a try block (bleh!) or use the get() method, with False as a default value.  I'd like to do the same thing for object.attribute. That is, I already have object to return False if it hasn't been set, but then that gives me errors like     AttributeError: 'bool' object has no attribute 'attribute' How to avoid the pop up message while downloading an application (.exe) using selenium I am trying to download an application and check if it's downloaded into my local system or not. But when I click the download link I have a popup with Save or Cancel options, which is stopping me from downloading the file. I googled and found some solutions and tried to change the profile of Firefox by adding the below code, but it is not working.  firefoxProfile.setPreference("browser.download.folderList", 2); firefoxProfile.setPreference("browser.download.alertOnEXEOpen", false); firefoxProfile.setPreference("browser.helperApps.neverAsksaveToDisk", "application/x-msexcel,application/excel,application/x-excel,application/excel,application/x-excel,application/excel,application/vnd.ms-excel,application/x-excel,application/x-msexcel"); firefoxProfile.setPreference("browser.download.manager.showWhenStarting", false); firefoxProfile.setPreference("browser.download.manager.focusWhenStarting", false); firefoxProfile.setPreference("browser.helperApps.alwaysAsk.force", false); firefoxProfile.setPreference("browser.download.manager.alertOnEXEOpen", false); firefoxProfile.setPreference("browser.download.manager.closeWhenDone", false); firefoxProfile.setPreference("browser.download.manager.showAlertOnComplete", false); firefoxProfile.setPreference("browser.download.manager.useWindow", false); firefoxProfile.setPreference("browser.download.manager.showWhenStarting", false); firefoxProfile.setPreference("services.sync.prefs.sync.browser.download.manager.showWhenStarting", false);   This is the code I added for setting up Firefox preferences, but I still get the popup. There is a common story that airline ticket prices rise significantly if you buy them less than two weeks before departure.  And that any time before that will yield standard prices.  Is that advice valid for flying during holiday seasons?  As an example: if I want to fly coast to coast USA on December 7th, must I buy my ticket ASAP, or would I get a similar price buying on November 21st? How to check if jQuery.ajax() request header Status is "304 Not Modified"? $A$ is an $n\times n$ matrix of complex numbers. Prove that if $\lambda$ is an eigenvalue of $A^2,$ then $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A.$    If $\lambda$ is an eigenvalue of $A^2,$ we have $\lambda\alpha=A^2\alpha$ for some $\alpha.$ Then how can we find a $\beta$ s.t. $\sqrt{\lambda}\beta=A\beta?$ of Stack Overflow, all content posted on Stack Exchange sites by their users (i.e. you wonderful people) has been provided to the whole universe under . For my fellow non-lawyers, that license basically means:   Anyone can use any Stack Exchange posts at any time without having to ask for permission Making money off of the copied content is permitted You don't even have to copy stuff from here verbatim; you can just use it as a starting point and make whatever edits you want There are just two rules you have to follow:   You have to provide . Simple links to the original post and author info are just fine. You have to  and allow other people to use your content, as long as they follow these very same rules. How meta!    (If you ever forget any of that, and want to refresh your memory, the license info is linked to in the footer of every page.)  There are, in fact, a lot of people who republish varying amounts of our content for assorted reasons. Unfortunately, there are some Stack Content Republishers Attributing Poorly and/or Excelling at Ranking (SCRAPERs, for short).  In this context, "attributing poorly" means any use that doesn't follow our attribution rules or make any other reasonable attempt at give credit. This can get pretty egregious; I've seen SCRAPERs that not only don't link back to SE originals, but also use fake author info and post dates to make it harder to find originals. By "excelling at ranking," I'm referring to copycat sites that end up higher in Google results than the original SE sites do for the same content. That's not necessarily wrong, but in some cases, it indicates inappropriate SEO hackery.  So, the question is: what can you do if you spot a SCRAPER? What prevents investors from buying high yield stocks and selling them as soon as their dividend is paid out? What prevents investors from buying high yield stocks and selling them as soon as their dividend is paid out?   I guess I am confused: If you know the yield and when the dividend will be paid out, why not buy a share an instant before the dividend is paid out and then sell it?  Do you have to have held the share for some amount of time to get the dividend? Is there too much volatility in the instant the dividend is paid out? Mysql: Select rows from a table that are not in another If $S_n$ has cyclic subgroups of these orders or not? This question was asked in my quiz and I am having difficulties in solving it .  Question : For a positive integer $n$ , let $S_n$ denote permutation group on $n$ symbols. Choose correct statements from below. (a) For a positive integer $n$ and for every $m$ with $1\leq m \leq n , S_n $ has a cyclic subgroup of order $m$; (b) For every positive integer $n$ and for every $m$ with $n&lt;m &lt;n!$, $S_n$ has a cyclic subgroup of order $m$; (c) There exists positive integers $n$ and $m$ with $n&lt;m&lt;n!$ such that $S_n$ has a cyclic subgroup of order $m$; (d) For every positive integer $n$ and for every group $G$ of order $n$, $G$ is isomorphic to a subgroup of order $S_n$ .  Attempt: (a) is true as I can choose an element of length $m$ which will generate cyclic subgroup of order $m$. (b) Choose $n=4$ and $m=19$ to contradict (b). (c) I dont know if there can be found any such case and if yes how to do so . (d) I dont know about it . kindly tell. Answer:   A,C,D are true What would you call a person who doesn't want to learn anything new? I encountered the following induction proof on a practice exam for calculus:  $$\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$$  I have to prove this statement with induction.  Can anyone please help me with this proof? Understanding QGIS buffer tool units Today my lecturer put up on the board that:  If $\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}$ exists and $a_n&gt;0$ then   $\displaystyle \limsup\limits_{n\to\infty}\left(a_n^{\frac{1}{n}}\right)=\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$  however I am not sure why this is true, can somebody give me a hint or something as to how to go about proving this.  thanks for any help I'm not that into dynamic programming languages but I've written my fair share of JavaScript code. I never really got my head around this prototype-based programming, does any one know how this works? var obj = new Object(); obj.prototype.test = function() { alert('Hello?'); }; var obj2 = new obj(); obj2.test();  I remember a lot discussion I had with people a while back (I'm not exactly sure what I'm doing) but as I understand it, there's no concept of a class. It's just an object, and instances of those objects are clones of the original, right? But what is the exact purpose of this &quot;.prototype&quot; property in JavaScript? How does it relate to instantiating objects? Update: correct way var obj = new Object(); // not a functional object obj.prototype.test = function() { alert('Hello?'); }; // this is wrong!  function MyObject() {} // a first class functional object MyObject.prototype.test = function() { alert('OK'); } // OK  Also these  really helped a lot. Find the default application for a certain extension Given xdg-open and an extension, is there a way to get the application which xdg-open is set to for that particular extension?  For example given xdg-open and .jpg the result is eog. There is a type and search drodown which will appear after user type in the value. I need to click on the dropdown which contains the name of the drop down field.   The problem here is that there is no unique attribute present (No unique class, no unique name) &amp; the id keeps on changing after every login so cannot take id also. Ubuntu server: Cannot setup dummy driver alsa (LXC container) // modprobe issue