If $x,y,z>0$ and $x+y+z=1$ then $\frac{xyz}{(1-x)(1-y)(1-z)}\le\frac{1}{8}$ If $x,y,z&gt;0$ and $x+y+z=1$, then: $$\frac{xyz}{(1-x)(1-y)(1-z)}\le\frac{1}{8}$$    $$\frac{xyz}{(1-x)(1-y)(1-z)}=\frac{x}{(1-x)}\frac{y}{(1-y)}\frac{z}{(1-z)}$$ Let $\frac{x}{(1-x)}=a$, $\frac{y}{(1-y)}=b$, $\frac{z}{(1-z)}=c$ I am stuck here.
How to establish this inequality: $(1-a)(1-b)(1-c) \geq 8abc$ for $a+b+c=1$? Let $a$, $b$, and $c$ be positive real numbers such that $a+b+c = 1$. Then how to establish the following inequality? $$ (1-a)(1-b)(1-c) \geq 8abc.$$  My effort:   Since $a+b+c =1$, we can write $$  (1-a)(1-b)(1-c) = 1 - (a+b+c) +(ab+bc+ca) - abc = 1 - 1  +(ab+bc+ca) - abc =  (ab+bc+ca) - abc = abc(a^{-1} + b^{-1} + c^{-1} ) - abc = 3abc (\frac{a^{-1} + b^{-1} + c^{-1}}{3}) - abc.$$ What next?
biblatex: per-entry language switching According to biblatex manual (p.24), the hyphenation field in a bib-entry switches language on per-entry basis:     hyphenation field    The language of the bibliography entry. The identifier must be a language name   known to the babel package. This information may be used to switch hyphenation   patterns and localize strings in the bibliography.   This functionality seemed to not work. I just used an example of bib base that comes with biblatex documentation to print the entire bibliography.   \documentclass[a4paper]{article} \usepackage[T1]{fontenc} \usepackage[german,american]{babel} \usepackage{csquotes} \usepackage[style=numeric,backend=biber]{biblatex} \addbibresource{biblatex-examples.bib} \begin{document} \nocite{*} \printbibliography \end{document}   The bibliography contains few entries in German. However in the printed list of references they are formatted with english strings.     [8] Averroes. Drei Abhandlungen über die Conjunction des separaten Intellects   mit dem Menschen. Von Averroes (Vater und Sohn), aus dem   Arabischen übersetzt von Samuel Ibn Tibbon. Ed. and trans. by J. Hercz.   Berlin: S. Hermann, 1869.   Per-entry language switching is provided by the  bundle of bst style for BibTeX. Looking through biblatex internal code, I've got an impression that biblatex needs major re-implementation in order to provide similar functionality. Or am I wrong? Nvidia graphics card no longer works in Windows 10 (error: Code 43) but works in Ubuntu Currently, when I boot into Windows, one monitor stays black (undetected), and the other can only display 800x600 resolution.  When I look at Device Manager &gt; Display Adapters &gt; NVIDIA GeForce GTX 970 &gt; General &gt; Device Status, it says "Windows has stopped this device because it has reported problems. (Code 43)".  Details &gt; Problem Code says "00002b". Details &gt; Status says:  01802400 DN_HAS_PROBLEM DN_DISABLEABLE DN_NT_ENUMERATOR DN_NT_DRIVER   When I then click into the Events tab, it says "Device PCI\VEN_10DE&amp;DEV_13C2&amp;SUBSYS_29763842&amp;REV_A1\4&amp;25438c51&amp;0&amp;0008 requires further installation."  I have a EVGA Nvidia GeForce GTX 970 graphics card with the latest drivers (376.19 from 12/1/2016).  My Windows 10 Pro has the latest updates and is at Version 1607, OS Build 14393.576.  My ASRock Z170 Pro4 motherboard BIOS is also updated to the latest version (7.00 from 10/4/2016).  I have dual Acer monitors with 1920x1080 resolution.  I downloaded Ubuntu 16.04.1 LTS onto a bootable USB drive and booted into the Ubuntu operating system instead of Windows. Both monitors immediately worked at full 1920x1080 resolution. So the hardware seems functional! My remaining challenge is to figure out why my video card has stopped working within Windows.  What I Have Tried   I've tried using Display Driver Uninstaller from Guru3d in Safe Mode to delete all drivers and try to install fresh.   I've tried many different versions of the driver dating back more than 12 months. I've tried reseating the card. I've tried moving the card to a different slot. EVGA's phone support told me to reinstall windows (Windows &gt; Reset this PC &gt; Keep my files). Unfortunately, I followed their advice (and now don't have any of my programs), and I still get Code 43 with the EVGA Nvidia card.   What I Have NOT Tried   I would love to try installing this card onto a different Windows 10 computer to see what happens, but I don't have access to any other computers. If you live northeast of Atlanta, let me know. ;-)   I'd appreciate any other ideas you have!. How can a list be created, whose items are both    enumerated (a,b,c,d,...)  and allow for setting a description via \item[texthere] ?   Im am using MikTex on Windows. Two definitions of $\limsup$ Proof a $2^n$ by $2^n$ board can be filled using L shaped trominoes and 1 monomino Suppose we have an $2^n\times 2^n$ board. Prove you can use any rotation of L shaped trominoes and a monomino to fill the board completely.  You can mix different rotations in the same tililng. How to set up passwordless SSH access for root user I need to configure a machine so software installation can be automated remotely via SSH. Following , I was able to setup SSH keys so my user can access the machine without a password, but I still need to manually enter my password when I use sudo, which obviously an automated process shouldn't have to do.  Although my /etc/ssh/sshd_config has PermitRootLogin yes, I can't seem to be able to log in as root, presumably because it's not a "real" account with a separate password.  How do I configure SSH keys, so a process can remotely log in as root on Ubuntu? Prove the following in $S_n$:  Let $\alpha = (a_1,a_2,\ldots,a_s)$ be a cycle and let $\pi$ be a permutation in $S_n$. Then $\pi\alpha\pi^{-1}$ is the cycle $(\pi(a_1),\ldots,\pi(a_s)).$     I'm not sure where to even start. ARMA lag selection for ARMA-GARCH models Sometimes I wonder if he is actually/really  as lavish as he   pretends to be.   I know that this sentence can work without either actually or really, but if we were to use one of them to make an impact on the listener, which one should it be? I hope I'm right in thinking this is the site where you post general computer-related problems...  A problem I have when playing games (since you won't be pressing multiple keys simultaneously anytime else), is that when I press up/left and then another key, it fails to 'register' as if that last key is never pressed. This doesn't happen when I'm pressing any other two arrow keys, e.g. down/left or up/right. The weird thing is that when I'm playing with WASD instead of the arrow keys, i still have this problem! Meaning W+A+another button doesn't work.  If anyone knows what I should, I'd appreciate the help. (My reflexes are fast and I can always circumvent this problem, but it's quite annoying and shouldn't exist to begin with!) Does personified Death really exist in Harry Potter? I went through the last book, Harry Potter and the Deathly Hallows, yesterday and a question popped in my mind: Does Death, the supposed creator of the Deathly Hallows, really exist in the Harry Potter universe? Dumbledore thought otherwise and I am with the same school of thought.  By the way who created the deathly Hallows actually?  Update:  1) Why were the brothers so sure that the hooded figure was death(couldn't it be a more powerful wizard playing with them)  2) Couldn't, Beedle and the brothers create this story? Can wizards really cast spells in armor? Where can I have a genuine Ainu cultural experience in Hokkaido? During my next trip to Japan I may end up in Hokkaido. I've read something about this mysterious indigenous group called Ainu/Aynu/Aino/Ezo, but I must admit my understanding of the whole thing is rather limited right now.   What I wish to know is: is there such a thing like an authentic Ainu village/town/whole-region where Ainu people live according to their tradition, as opposed to something staged for tourists?  Edit: given the comments below I wish to specify better what I mean with the word 'authentic'. Of course, I don't expect pristine landscapes and Ainu people living in blissful harmony with fluffy penguins (semi-quote) but I won't be surprised to find out there still are authentic villages in remote areas of Hokkaido where mass tourism is not an issue at all.  People are subject to tourist's attention (tourist coming from different countries, at least) everywhere, not only when they "made a conscious decision to live a traditional life". This doesn't actually mean anything. Indian people are surprisingly differents (in local customs) from Italians and Italians take a lot of pictures during their travels there, still the tourism isn't banned in India. Graph for relationship between two ordinal variables What is an appropriate graph to illustrate the relationship between two ordinal variables?  A few options I can think of:   Scatter plot with added random jitter to stop points hiding each other. Apparently a standard graphic - Minitab calls this an "individual values plot". In my opinion it may be misleading as it visually encourages a kind of linear interpolation between ordinal levels, as if the data were from an interval scale. Scatter plot adapted so that size (area) of point represents frequency of that combination of levels, rather than drawing one point for each sampling unit. I have occasionally seen such plots in practice. They can be hard to read, but the points lie on a regularly-spaced lattice which somewhat overcomes the criticism of the jittered scatter plot that it visually "intervalises" the data. Particularly if one of the variables is treated as dependent, a box plot grouped by the levels of the independent variable. Likely to look terrible if the number of levels of the dependent variable is not sufficiently high (very "flat" with missing whiskers or even worse collapsed quartiles which makes visual identification of median impossible), but at least draws attention to median and quartiles which are relevant descriptive statistics for an ordinal variable. Table of values or blank grid of cells with heat map to indicate frequency. Visually different but conceptually similar to the scatter plot with point area showing frequency.   Are there other ideas, or thoughts on which plots are preferable? Are there any fields of research in which certain ordinal-vs-ordinal plots are regarded as standard? (I seem to recall frequency heatmap being widespread in genomics but suspect that is more often for nominal-vs-nominal.) Suggestions for a good standard reference would also be very welcome, I am guessing something from Agresti.  If anyone wants to illustrate with a plot, R code for bogus sample data follows.  "How important is exercise to you?" 1 = not at all important, 2 = somewhat unimportant, 3 = neither important nor unimportant, 4 = somewhat important, 5 = very important.  "How regularly do you take a run of 10 minutes or longer?" 1 = never, 2 = less than once per fortnight, 3 = once every one or two weeks, 4 = two or three times per week, 5 = four or more times per week.  If it  would be natural to treat "often" as a dependent variable and "importance" as an independent variable, if a plot distinguishes between the two.  importance &lt;- rep(1:5, times = c(30, 42, 75, 93, 60)) often &lt;- c(rep(1:5, times = c(15, 07, 04, 03, 01)), #n=30, importance 1            rep(1:5, times = c(10, 14, 12, 03, 03)), #n=42, importance 2            rep(1:5, times = c(12, 23, 20, 13, 07)), #n=75, importance 3            rep(1:5, times = c(16, 14, 20, 30, 13)), #n=93, importance 4            rep(1:5, times = c(12, 06, 11, 17, 14))) #n=60, importance 5 running.df &lt;- data.frame(importance, often) cor.test(often, importance, method = "kendall") #positive concordance plot(running.df) #currently useless   A related question for continuous variables I found helpful, maybe a useful starting point: I was loading an image resource with the following code and it works fine from my dev environment (netbeans) but threw an exception as soon as it ran from the desktop.  new ImageIcon(Images.class.getResource("images/highlighter24.gif"));   I traced it down to my file having different case than in the code.  Fine, my bad, but why the inconsistent behavior?  Gremlins?  Note: I'm running on windows before you ask. Named parameters in JDBC What hardware does Ubuntu Touch support? With the announcement of the Ubuntu Phone OS I'd like to know what phone (and tablet) models are supported at this time.  Note from foss &amp; Oli: We are making this the master question for all future "Will this work on &lt;insert random tablet/phone/device here&gt;?!" questions Finding my house in minecraft I wandered too far to go look for sheep and now I'm on a completely different island.  How do I find my way back to my house? Install windows 7 through virtual box Python and regular expression with Unicode