Let $x,y,z\geq0$. Show that $x^4(y+z)+y^4(z+x)+z^4(x+y)\leq\frac{1}{12}(x+y+z)^5$
The inequality is homogeneous, but I am not sure if that helps. Also, I was thinking about considering a concave function and apply Jensen or Karamata on LHS.
Is there any anti-gravity material? I want to know if there is any  material. I am thinking of making flying vehicles which are made up of anti-gravity material so that they will not experience any gravity on them and can easily take off and be more fuel efficient. Is there any such thing? Or any workaround? Sum : $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3}$ LWJGL Eclipse Resource Deployment I'm still experimenting with some OpenGL in LWJGL, and as I get to know more and more about OpenGL, I start to do more complicated things like multi texturing, shadowmapping and more. Now, right now I'm using the src folder (src/res) for my resources, and use src/resourcename in my code. When I export it, this won't work on another location as the relative path is ofcourse incorrect. What would be the best way of deploying your game with all the resources in one file? Maybe put the files in the jar?  I would like to hear from a few people how they store and deploy their project with resources, thanks! Can a new save withdraw from the pokemon bank? I've been considering restarting my save and replaying the story with access to the entire National Pokedex (Minus a few Event-only Pokemon I'm missing) from the start. But I would first like to confirm: Can I withdraw my stored Pokemon from the bank if my game is reset? It seems that the only contributing factor is that I end up signed in with the same Nintendo Network ID, so as long as my network ID remains the same, I can withdraw stored Pokemon regardless of what save-game or cartridge is in my system, correct?  While I'm reasonably confident it works the way I think it does, past experience (My Pokemon Ranch being tied to specific trainer IDs, and something about the dream world being tied to the game's wireless settings) has made me just cautious enough to ask before attempting. How do I retrieve only the needed line from terminal output? Removal of statistically significant intercept term increases $R^2$ in linear model Should seemingly arbitrary things like "play piano for Church" or "intramural badminton" go on M.S. or PhD applications in sciences/engineering? And if they should, I assume that these things should go on a resume under something like "Activities and Community Service". I mean I consider being able to sight read music to play at my Church as a significant feat (how many people can do that?) and might sway one person on an admissions committee if he or she enjoys music and understands how hard one must have worked to accomplish this.   Badminton?.... perhaps not so much ALTER TABLE [dbo].[REMINDER_EMAIL]  WITH CHECK ADD  CONSTRAINT    [FK__REMINDER_EMAIL__FACILITY_ID] FOREIGN KEY ( [FACILITY_ID] )   REFERENCES [dbo].[FACILITY] ( [FACILITY_ID] )  ALTER TABLE [dbo].[REMINDER_EMAIL] CHECK CONSTRAINT [FK__REMINDER_EMAIL__FACILITY_ID] For an isolated system, can the entropy decrease or increase? In any sizable system, the number of equilibrium states are much, much greater then the number of non-equilibrium states. Since each accessible micro state is equally probably, it is overwhelmingly likely that we will find the system in an equilibrium state.   However, for a closed system, one that does not interact with any external system, the number of micro states is fixed. Therefore the entropy is fixed.  At any given time, the system will be in one of its micro states. Even if the system is in a micro state that does not look like equilibrium from a macroscopic point of view, its entropy will remain the same, since entropy is a property of all the micro states, not just a given micro state.   So, are the number of accessible micro states (and hence entropy) of an isolated system constant? Is there a RSS to the 'Stack Exchange Inbox'? How about to Activity? Just wondering if I'm missing something here. Also, I couldn't find any previous question related to this. Implement ```-style (fenced) Markdown code blocks Currently, Stack Exchange’s Markdown parser only allows four-space indents to represent code blocks:      // some code     // another line of code    and other Markdown implementations allow for an alternative syntax, that doesn’t require indenting each code line:  ``` // some code // another line of code ```   This is much more convenient to type out. It would be super useful if Stack Exchange could support this syntax.    By extension, this syntax also allow you to specify the source language right after the opening ```:  ```js // some code // another line of code ```   …which would then enable syntax highlighting for that specific language. Although it’s interesting metadata, I don’t think this feature is needed on Stack Overflow, as the syntax highlighting library works pretty well for various languages.  So, even if you would allow this syntax but ignore the ```language, this would greatly improve my productivity on Stack Exchange.  What do you think?  The moderators are currently collecting feedback regarding this feature, so please post a comment with your thoughts. Good idea? Bad idea? Don’t really care? What do you think are the benefits/drawbacks? Experiences? Let them know! Thanks! Why does python 2.5.2 have the following behavior  &gt;&gt;&gt;[2].extend([]) == [2] False  &gt;&gt;&gt; [2].extend([]) == None True  $ python --version Python 2.5.2   I assume I'm not understanding something here, but intuitively I'd think that [2].extend([]) should yield [2] Number of Homomorphisms from $\Bbb{Z}_m$ to $\Bbb{Z}_n$ This question coutesy of Allan Clark's "Elements of Abstract Algebra" (60$\zeta$).   Find the number of homomorphisms from $\Bbb{Z}_m\to \Bbb{Z}_n$ as a function of $m$ and $n$.  This is stumping me, can anyone help? Dr. StrangeNumbers or: How I Learned to Stop Worrying and Love Floating Point Arithmetic Word for "blissfully oblivious"? What is the name for the grammatical figure, where the present tense is substituted for a past event? I am preparing a book for publication which contains the following passage:     Stromberg in Religious Liberalism in Eighteenth-Century England   (1954) noted that 'the critical moment in the emergence of a positive   spirit of reform...consisted in the awareness of evil as being social   and remediable. It begins about 1750.'   My proof reader (my wife) has picked this out suggesting it should be the past tense. However if I say It began about 1750 it seems to convey too much certainty about the exact date. I just want to convey an impression of it being something which emerges in the mid-eighteenth century. I believe use of the present tense is the way to go.   But what is this artifice called, this use of the present to convey mild uncertainty? If you provide 0 as the dayValue in Date.setFullYear you get the last day of the previous month:  d = new Date(); d.setFullYear(2008, 11, 0); //  Sun Nov 30 2008   There is reference to this behaviour at . Is this a reliable cross-browser feature or should I look at alternative methods? Can you combine multiple lists with LINQ? Say I have two lists:  var list1 = new int[] {1, 2, 3}; var list2 = new string[] {"a", "b", "c"};   Is it possible to write a LINQ statement that will generate the following list:  var result = new []{      new {i = 1, s = "a"},     new {i = 1, s = "b"},     new {i = 1, s = "c"},     new {i = 2, s = "a"},     new {i = 2, s = "b"},     new {i = 2, s = "c"},     new {i = 3, s = "a"},     new {i = 3, s = "b"},     new {i = 3, s = "c"} };   ?  Edit: I forgot to mention I didn't want it in query syntax. Anyway, based on preetsangha's answer I've got the following:  var result = list1.SelectMany(i =&gt;  list2.Select(s =&gt; new {i = i, s = s})); What is wrong with this fake proof $e^i = 1$? Cloning a Non-Standard Svn Repository with Git-Svn